The Quantum Graph Recurrent Neural Network

Author: Jack Ceroni

This demonstration investigates quantum graph recurrent neural networks (QGRNN), which are the quantum analogue of a classical graph recurrent neural network, and a subclass of the more general quantum graph neural network ansatz. Both the QGNN and QGRNN were introduced in this paper (2019).

The Idea

A graph is defined as a set of nodes along with a set of edges, which represent connections between nodes. Information can be encoded into graphs by assigning numbers to nodes and edges, which we call weights. It is usually convenient to think of a graph visually:

../_images/graph.png

In recent years, the concept of a graph neural network (GNN) has been receiving a lot of attention from the machine learning community. A GNN seeks to learn a representation (a mapping of data into a low-dimensional vector space) of a given graph with feature vectors assigned to nodes and edges. Each of the vectors in the learned representation preserves not only the features, but also the overall topology of the graph, i.e., which nodes are connected by edges. The quantum graph neural network attempts to do something similar, but for features that are quantum-mechanical; for instance, a collection of quantum states.

Consider the class of qubit Hamiltonians that are quadratic, meaning that the terms of the Hamiltonian represent either interactions between two qubits, or the energy of individual qubits. This class of Hamiltonians is naturally described by graphs, with second-order terms between qubits corresponding to weighted edges between nodes, and first-order terms corresponding to node weights.

A well known example of a quadratic Hamiltonian is the transverse-field Ising model, which is defined as

\[\hat{H}_{\text{Ising}}(\boldsymbol\theta) \ = \ \displaystyle\sum_{(i, j) \in E} \theta_{ij}^{(1)} Z_{i} Z_{j} \ + \ \displaystyle\sum_{i} \theta_{i}^{(2)} Z_{i} \ + \ \displaystyle\sum_{i} X_{i},\]

where \(\boldsymbol\theta \ = \ \{\theta^{(1)}, \ \theta^{(2)}\}\). In this Hamiltonian, the set \(E\) that determines which pairs of qubits have \(ZZ\) interactions can be represented by the set of edges for some graph. With the qubits as nodes, this graph is called the interaction graph. The \(\theta^{(1)}\) parameters correspond to the edge weights and the \(\theta^{(2)}\) parameters correspond to weights on the nodes.

This result implies that we can think about quantum circuits with graph-theoretic properties. Recall that the time-evolution operator with respect to some Hamiltonian \(H\) is defined as:

\[U \ = \ e^{-it H}.\]

Thus, we have a clean way of taking quadratic Hamiltonians and turning them into unitaries (quantum circuits) that preserve the same correspondance to a graph. In the case of the Ising Hamiltonian, we have:

\[U_{\text{Ising}} \ = \ e^{-it \hat{H}_{\text{Ising}} (\boldsymbol\theta)} \ = \ \exp \Big[ -it \Big( \displaystyle\sum_{(i, j) \in E} \theta_{ij}^{(1)} Z_{i} Z_{j} \ + \ \displaystyle\sum_{i} \theta_{i}^{(2)} Z_{i} \ + \ \displaystyle\sum_{i} X_{i} \Big) \Big]\]

In general, this kind of unitary is very difficult to implement on a quantum computer. However, we can approximate it using the Trotter-Suzuki decomposition:

\[\exp \Big[ -it \Big( \displaystyle\sum_{(i, j) \in E} \theta_{ij}^{(1)} Z_{i} Z_{j} \ + \ \displaystyle\sum_{i} \theta_{i}^{(2)} Z_{i} \ + \ \displaystyle\sum_{i} X_{i} \Big) \Big] \ \approx \ \displaystyle\prod_{k \ = \ 1}^{t / \Delta} \Bigg[ \displaystyle\prod_{j \ = \ 1}^{Q} e^{-i \Delta \hat{H}_{\text{Ising}}^{j}(\boldsymbol\theta)} \Bigg]\]

where \(\hat{H}_{\text{Ising}}^{j}(\boldsymbol\theta)\) is the \(j\)-th term of the Ising Hamiltonian and \(\Delta\) is some small number.

This circuit is a specific instance of the Quantum Graph Recurrent Neural Network, which in general is defined as a variational ansatz of the form

\[U_{H}(\boldsymbol\mu, \ \boldsymbol\gamma) \ = \ \displaystyle\prod_{i \ = \ 1}^{P} \Bigg[ \displaystyle\prod_{j \ = \ 1}^{Q} e^{-i \gamma_j H^{j}(\boldsymbol\mu)} \Bigg],\]

for some parametrized quadratic Hamiltonian, \(H(\boldsymbol\mu)\).

Using the QGRNN

Since the QGRNN ansatz is equivalent to the approximate time evolution of some quadratic Hamiltonian, we can use it to learn the dynamics of a quantum system.

Continuing with the Ising model example, let’s imagine we have some system governed by \(\hat{H}_{\text{Ising}}(\boldsymbol\alpha)\) for an unknown set of target parameters, \(\boldsymbol\alpha\) and an unknown interaction graph \(G\). Let’s also suppose we have access to copies of some low-energy, non-ground state of the target Hamiltonian, \(|\psi_0\rangle\). In addition, we have access to a collection of time-evolved states, \(\{ |\psi(t_1)\rangle, \ |\psi(t_2)\rangle, \ ..., \ |\psi(t_N)\rangle \}\), defined by:

\[|\psi(t_k)\rangle \ = \ e^{-i t_k \hat{H}_{\text{Ising}}(\boldsymbol\alpha)} |\psi_0\rangle.\]

We call the low-energy states and the collection of time-evolved states quantum data. From here, we randomly pick a number of time-evolved states from our collection. For any state that we choose, which is evolved to some time \(t_k\), we compare it to

\[U_{\hat{H}_{\text{Ising}}}(\boldsymbol\mu, \ \Delta) |\psi_0\rangle \ \approx \ e^{-i t_k \hat{H}_{\text{Ising}}(\boldsymbol\mu)} |\psi_0\rangle.\]

This is done by feeding one of the copies of \(|\psi_0\rangle\) into a quantum circuit with the QGRNN ansatz, with some guessed set of parameters \(\boldsymbol\mu\) and a guessed interaction graph, \(G'\). We then use a classical optimizer to maximize the average “similarity” between the time-evolved states and the states prepared with the QGRNN.

As the QGRNN states becomes more similar to each time-evolved state for each sampled time, it follows that \(\boldsymbol\mu \ \rightarrow \ \boldsymbol\alpha\) and we are able to learn the unknown parameters of the Hamiltonian.

../_images/qgrnn3.png

A visual representation of one execution of the QGRNN for one piece of quantum data.

Learning an Ising Model with the QGRNN

We now attempt to use the QGRNN to learn the parameters corresponding to an arbitrary transverse-field Ising model Hamiltonian.

Getting Started

We begin by importing the necessary dependencies:

import pennylane as qml
from matplotlib import pyplot as plt
import numpy as np
import scipy
import networkx as nx
import copy

We also define some fixed values that are used throughout the simulation.

qubit_number = 4
qubits = range(qubit_number)

In this simulation, we don’t have quantum data readily available to pass into the QGRNN, so we have to generate it ourselves. To do this, we must have knowledge of the target interaction graph and the target Hamiltonian.

Let us use the following cyclic graph as the target interaction graph of the Ising Hamiltonian:

ising_graph = nx.cycle_graph(qubit_number)

print(f"Edges: {ising_graph.edges}")
nx.draw(ising_graph)

Out:

Edges: [(0, 1), (0, 3), (1, 2), (2, 3)]
../_images/graph1.png

We can then initialize the “unknown” target parameters that describe the target Hamiltonian, \(\boldsymbol\alpha \ = \ \{\alpha^{(1)}, \ \alpha^{(2)}\}\). Recall from the introduction that we have defined our parametrized Ising Hamiltonian to be of the form:

\[\hat{H}_{\text{Ising}}(\boldsymbol\theta) \ = \ \displaystyle\sum_{(i, j) \in E} \theta_{ij}^{(1)} Z_{i} Z_{j} \ + \ \displaystyle\sum_{i} \theta_{i}^{(2)} Z_{i} \ + \ \displaystyle\sum_{i} X_{i},\]

where \(E\) is the set of edges in the interaction graph, and \(X_i\) and \(Z_i\) are the Pauli-X and Pauli-Z on the \(i\)-th qubit.

For this tutorial, we choose the target parameters by sampling from a uniform probability distribution ranging from \(-2\) to \(2\), with two-decimal precision.

matrix_params = [[0.56, 1.24, 1.67, -0.79], [-1.44, -1.43, 1.18, -0.93]]

In theory, these parameters can be any value we want, provided they are reasonably small enough that the QGRNN can reach them in a tractable number of optimization steps. In matrix_params, the first list represents the \(ZZ\) interaction parameters and the second list represents the single-qubit Z parameters.

Finally, we use this information to generate the matrix form of the Ising model Hamiltonian in the computational basis:

def create_hamiltonian_matrix(n, graph, params):

    matrix = np.zeros((2 ** n, 2 ** n))

    # Creates the interaction component of the Hamiltonian
    for count, i in enumerate(graph.edges):
        m = 1
        for j in range(0, n):
            if i[0] == j or i[1] == j:
                m = np.kron(m, qml.PauliZ.matrix)
            else:
                m = np.kron(m, np.identity(2))
        matrix += params[0][count] * m

    # Creates the bias components of the matrix
    for i in range(0, n):
        m1 = m2 = 1
        for j in range(0, n):
            if j == i:
                m1 = np.kron(m1, qml.PauliZ.matrix)
                m2 = np.kron(m2, qml.PauliX.matrix)
            else:
                m1 = np.kron(m1, np.identity(2))
                m2 = np.kron(m2, np.identity(2))
        matrix += (params[1][i] * m1 + m2)

    return matrix

# Prints a visual representation of the Hamiltonian matrix
ham_matrix = create_hamiltonian_matrix(qubit_number, ising_graph, matrix_params)
plt.matshow(ham_matrix, cmap='hot')
plt.show()
../_images/ising_hamiltonian.png

Preparing Quantum Data

The collection of quantum data needed to run the QGRNN has two components: (i) copies of a low-energy state, and (ii) a collection of time-evolved states, each of which are simply the low-energy state evolved to different times. The following is a low-energy state of the target Hamiltonian:

low_energy_state = [
    (-0.054661080280306085+0.016713907320174026j),
    (0.12290003656489545-0.03758500591109822j),
    (0.3649337966440005-0.11158863596657455j),
    (-0.8205175732627094+0.25093231967092877j),
    (0.010369790825776609-0.0031706387262686003j),
    (-0.02331544978544721+0.007129899300113728j),
    (-0.06923183949694546+0.0211684344103713j),
    (0.15566094863283836-0.04760201916285508j),
    (0.014520590919500158-0.004441887836078486j),
    (-0.032648113364535575+0.009988590222879195j),
    (-0.09694382811137187+0.02965579457620536j),
    (0.21796861485652747-0.06668776658411019j),
    (-0.0027547112135013247+0.0008426289322652901j),
    (0.006193695872468649-0.0018948418969390599j),
    (0.018391279795405405-0.005625722994009138j),
    (-0.041350974715649635+0.012650711602265649j)
    ]

This state can be obtained by using a decoupled version of the Variational Quantum Eigensolver algorithm (VQE). Essentially, we choose a VQE ansatz such that the circuit cannot learn the exact ground state, but it can get fairly close. Another way to arrive at the same result is to perform VQE with a reasonable ansatz, but to terminate the algorithm before it converges to the ground state. If we used the exact ground state \(|\psi_0\rangle\), the time-dependence would be trivial and the data would not provide enough information about the Hamiltonian parameters.

We can verify that this is a low-energy state by numerically finding the lowest eigenvalue of the Hamiltonian and comparing it to the energy expectation of this low-energy state:

res = np.vdot(low_energy_state, (ham_matrix @ low_energy_state))
energy_exp = np.real_if_close(res)
print(f"Energy Expectation: {energy_exp}")


ground_state_energy = np.real_if_close(min(np.linalg.eig(ham_matrix)[0]))
print(f"Ground State Energy: {ground_state_energy}")

Out:

Energy Expectation: -7.244508985189101
Ground State Energy: -7.330689661291242

We have in fact found a low-energy, non-ground state, as the energy expectation is slightly greater than the energy of the true ground state. This, however, is only half of the information we need. We also require a collection of time-evolved, low-energy states. Evolving the low-energy state forward in time is fairly straightforward: all we have to do is multiply the initial state by a time-evolution unitary. This operation can be defined as a custom gate in PennyLane:

def state_evolve(hamiltonian, qubits, time):

    U = scipy.linalg.expm(-1j* hamiltonian * time)
    qml.QubitUnitary(U, wires=qubits)

We don’t actually generate time-evolved quantum data quite yet, but we now have all the pieces required for its preparation.

Learning the Hamiltonian

With the quantum data defined, we are able to construct the QGRNN and learn the target Hamiltonian. Each of the exponentiated Hamiltonians in the QGRNN ansatz, \(\hat{H}^{j}_{\text{Ising}}(\boldsymbol\mu)\), are the \(ZZ\), \(Z\), and \(X\) terms from the Ising Hamiltonian. This gives:

def qgrnn_layer(param1, param2, qubits, graph, trotter_step):

    # Applies a layer of RZZ gates (based on a graph)
    for count, i in enumerate(graph.edges):
        qml.MultiRZ(2 * param1[count] * trotter_step, wires=[i[0], i[1]])

    # Applies a layer of RZ gates
    for count, i in enumerate(qubits):
        qml.RZ(2 * param2[count] * trotter_step, wires=i)

    # Applies a layer of RX gates
    for i in qubits:
        qml.RX(2 * trotter_step, wires=i)

As was mentioned in the first section, the QGRNN has two registers. In one register, some piece of quantum data \(|\psi(t)\rangle\) is prepared and in the other we have \(U_{H}(\boldsymbol\mu, \ \Delta) |\psi_0\rangle\). We need a way to measure the similarity between these states. This can be done by using the fidelity, which is simply the modulus squared of the inner product between the states, \(| \langle \psi(t) | U_{H}(\Delta, \ \boldsymbol\mu) |\psi_0\rangle |^2\). To calculate this value, we use a SWAP test between the registers:

def swap_test(control, register1, register2):

    qml.Hadamard(wires=control)
    for i in range(0, len(register1)):
        qml.CSWAP(wires=[int(control), register1[i], register2[i]])
    qml.Hadamard(wires=control)

After performing this procedure, the value returned from a measurement of the circuit is \(\langle Z \rangle\), with respect to the control qubit. The probability of measuring the \(|0\rangle\) state in this control qubit is related to both the fidelity between registers and \(\langle Z \rangle\). Thus, with a bit of algebra, we find that \(\langle Z \rangle\) is equal to the fidelity.

Before creating the full QGRNN and the cost function, we define a few more fixed values. Among these is a “guessed” interaction graph, which we set to be a complete graph. This choice is motivated by the fact that any target interaction graph will be a subgraph of this initial guess. Part of the idea behind the QGRNN is that we don’t know the interaction graph, and it has to be learned. In this case, the graph is learned automatically as the target parameters are optimized. The \(\boldsymbol\mu\) parameters that correspond to edges that don’t exist in the target graph will simply approach \(0\).

# Defines some fixed values

reg1 = list(range(qubit_number))  # First qubit register
reg2 = list(range(qubit_number, 2 * qubit_number))  # Second qubit register

control = 2 * qubit_number  # Index of control qubit
trotter_step = 0.01  # Trotter step size

# Defines the interaction graph for the new qubit system

new_ising_graph = nx.Graph()
new_ising_graph.add_nodes_from(range(qubit_number, 2 * qubit_number))
new_ising_graph.add_edges_from([(4, 5), (5, 6), (6, 7), (4, 6), (7, 4), (5, 7)])

print(f"Edges: {new_ising_graph.edges}")
nx.draw(new_ising_graph)

Out:

Edges: [(4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)]
../_images/graph2.png

With this done, we implement the QGRNN circuit for some given time value:

def qgrnn(params1, params2, time=None):

    # Prepares the low energy state in the two registers
    qml.QubitStateVector(np.kron(low_energy_state, low_energy_state), wires=reg1+reg2)

    # Evolves the first qubit register with the time-evolution circuit to
    # prepare a piece of quantum data
    state_evolve(ham_matrix, reg1, time)

    # Applies the QGRNN layers to the second qubit register
    depth = time / trotter_step  # P = t/Delta
    for i in range(0, int(depth)):
        qgrnn_layer(params1, params2, reg2, new_ising_graph, trotter_step)

    # Applies the SWAP test between the registers
    swap_test(control, reg1, reg2)

    # Returns the results of the SWAP test
    return qml.expval(qml.PauliZ(control))

We have the full QGRNN circuit, but we still need to define a cost function. We know that \(| \langle \psi(t) | U_{H}(\boldsymbol\mu, \ \Delta) |\psi_0\rangle |^2\) approaches \(1\) as the states become more similar and approaches \(0\) as the states become orthogonal. Thus, we choose to minimize the quantity \(-| \langle \psi(t) | U_{H}(\boldsymbol\mu, \ \Delta) |\psi_0\rangle |^2\). Since we are interested in calculating this value for many different pieces of quantum data, the final cost function is the average negative fidelity* between registers:

\[\mathcal{L}(\boldsymbol\mu, \ \Delta) \ = \ - \frac{1}{N} \displaystyle\sum_{i \ = \ 1}^{N} | \langle \psi(t_i) | \ U_{H}(\boldsymbol\mu, \ \Delta) \ |\psi_0\rangle |^2,\]

where we use \(N\) pieces of quantum data.

Before creating the cost function, we must define a few more fixed variables:

N = 15  # The number of pieces of quantum data that are used for each step
max_time = 0.1  # The maximum value of time that can be used for quantum data

We then define the negative fidelity cost function:

def cost_function(params):

    global iterations

    # Separates the parameter list
    weight_params = params[0:6]
    bias_params = params[6:10]

    # Randomly samples times at which the QGRNN runs
    times_sampled = [np.random.uniform() * max_time for i in range(0, N)]

    # Cycles through each of the sampled times and calculates the cost
    total_cost = 0
    for i in times_sampled:
        result = qnode(weight_params, bias_params, time=i)
        total_cost += -1 * result

    # Prints the value of the cost function
    if iterations % 5 == 0:
        print(
            "Fidelity at Step " + str(iterations) + ": " + str((-1 * total_cost / N)._value)
            )
        print("Parameters at Step " + str(iterations) + ": " + str(params._value))
        print("---------------------------------------------")

    iterations += 1

    return total_cost / N

The last step is to define the new device and QNode, and execute the optimizer. We use Adam, with a step-size of \(0.5\):

# Defines the new device

qgrnn_dev = qml.device("default.qubit", wires=2 * qubit_number + 1)

# Defines the new QNode

qnode = qml.QNode(qgrnn, qgrnn_dev)

iterations = 0
optimizer = qml.AdamOptimizer(stepsize=0.5)
steps = 300
qgrnn_params = list([np.random.randint(-20, 20)/50 for i in range(0, 10)])
init = copy.copy(qgrnn_params)

# Executes the optimization method

for i in range(0, steps):
    qgrnn_params = optimizer.step(cost_function, qgrnn_params)

Out:

Fidelity at Step 0: 0.9918346272467008
Parameters at Step 0: [-0.18, 0.3, 0.32, -0.08, 0.22, 0.28, 0.0, 0.04, -0.26, -0.34]
---------------------------------------------
Fidelity at Step 1: 0.9982958958675725
Parameters at Step 1: [-0.6799981741676951, 0.7999982222290656, 0.8199988800891312, 0.4199985008189975, 0.7199987506613634, -0.2199988652959371, -0.4999965632247722, -0.4599971332618679, 0.2399970101066845, 0.1599984088859147]
---------------------------------------------
Fidelity at Step 2: 0.990490615156777
Parameters at Step 2: [-1.0098221172022992, 1.1336805430938313, 1.1430272204750902, 0.7644358118180916, 1.0421203687079919, -0.5468227660845358, -0.8209215857161252, -0.8014895139877191, 0.5869815972734128, 0.47692466959999924]
---------------------------------------------
Fidelity at Step 3: 0.9762351051354551
Parameters at Step 3: [-1.2299363744628509, 1.3661666433666075, 1.3415230886864924, 1.030456044669323, 1.2378781697230516, -0.7575172669172114, -1.0130201555167162, -1.0581775588121731, 0.8610525437144596, 0.6563901663908569]
---------------------------------------------
Fidelity at Step 4: 0.9784260543002337
Parameters at Step 4: [-1.3467580918064672, 1.5070467394602267, 1.4191992723432134, 1.2363074986621367, 1.3105574652271725, -0.8578216277740998, -1.0787194096943145, -1.2453915798444228, 1.082957803098034, 0.7007286251069664]
---------------------------------------------
Fidelity at Step 5: 0.9760924766644138
Parameters at Step 5: [-1.3975050155126143, 1.5874154907714464, 1.4253477829297985, 1.3972088582056272, 1.310768053020758, -0.8907455190190542, -1.0697339005273567, -1.3825354197935542, 1.264473178391723, 0.6691484100078874]
---------------------------------------------
Fidelity at Step 6: 0.9791342696605839
Parameters at Step 6: [-1.3916054226388823, 1.6158640373620028, 1.3729978909141598, 1.5194931625164307, 1.2515225501166842, -0.8679964251616565, -0.9987099125166444, -1.4761909187915563, 1.4117656894362285, 0.5767242683721397]
---------------------------------------------
Fidelity at Step 7: 0.9787421089376458
Parameters at Step 7: [-1.3419795434786161, 1.6037009979053718, 1.2793404242059645, 1.6096566051847816, 1.1498502520134959, -0.8049890167819886, -0.8822831819835857, -1.5340642754199918, 1.530248443597091, 0.44241619983839797]
---------------------------------------------
Fidelity at Step 8: 0.9820315749997621
Parameters at Step 8: [-1.2488997936415513, 1.5514040173101722, 1.1474714836863231, 1.6683325278610117, 1.0085180485340484, -0.7045659446371773, -0.7220052474377402, -1.5561429872624357, 1.6212374403240934, 0.2699385653926958]
---------------------------------------------
Fidelity at Step 9: 0.9906449268083709
Parameters at Step 9: [-1.1168651248534744, 1.4629872578303456, 0.9855814851930917, 1.6969881487826513, 0.8353653676865135, -0.57428268457325, -0.524589866529066, -1.5440633182793588, 1.68617934363009, 0.06842824811303153]
---------------------------------------------
Fidelity at Step 10: 0.99582195327501
Parameters at Step 10: [-0.964031092107903, 1.354674684306868, 0.814393606152222, 1.7042986703019174, 0.650940374604191, -0.43328323068490904, -0.31101171334022504, -1.5090252857025748, 1.731326080636684, -0.1402782545374336]
---------------------------------------------
Fidelity at Step 11: 0.9961926541501808
Parameters at Step 11: [-0.8036684420752858, 1.238468601757792, 0.648795252860385, 1.6970845839900839, 0.46994924470041266, -0.29523086642904206, -0.0966467937581893, -1.4596190993696592, 1.7617532916742076, -0.34043090654836505]
---------------------------------------------
Fidelity at Step 12: 0.9983158950218793
Parameters at Step 12: [-0.6312660733884548, 1.1116800858652522, 0.4972845049612349, 1.6699268716653712, 0.2975512423191221, -0.16726719667887802, 0.11884361341846977, -1.3889433232896733, 1.773276350359844, -0.5223183946029325]
---------------------------------------------
Fidelity at Step 13: 0.9967000138255077
Parameters at Step 13: [-0.4676461753735439, 0.9917792527825551, 0.36890033998622973, 1.6368149409927473, 0.1458560667857181, -0.058079424182490846, 0.3155504611747129, -1.3145273809480482, 1.7766599241241958, -0.6769959196369855]
---------------------------------------------
Fidelity at Step 14: 0.9945001638616079
Parameters at Step 14: [-0.31794007513498357, 0.8838593182597574, 0.274152383590202, 1.5986807415684292, 0.023998424267510157, 0.02334325807987936, 0.484463374084153, -1.2376354781826886, 1.7724156182238344, -0.792919523723297]
---------------------------------------------
Fidelity at Step 15: 0.9913345835898904
Parameters at Step 15: [-0.18832820993560723, 0.7934498472478128, 0.2192041202380814, 1.5582912583375383, -0.06231146312915761, 0.07191800841202414, 0.6173058799196036, -1.1617573784335757, 1.7624859746860937, -0.8642178255228796]
---------------------------------------------
Fidelity at Step 16: 0.9889565034466684
Parameters at Step 16: [-0.08560961522854411, 0.7266587755855446, 0.21178056237429993, 1.5174343908488523, -0.10602854111569737, 0.08120012393307255, 0.7038563895141848, -1.089491099254555, 1.7477885167776666, -0.8832941887432142]
---------------------------------------------
Fidelity at Step 17: 0.9925729160049567
Parameters at Step 17: [-0.01530712818391683, 0.688186047241436, 0.2546346524352683, 1.478007577277352, -0.10461799053446165, 0.04908916100233172, 0.7372715330783467, -1.023558091459499, 1.72924671045939, -0.8487271410430316]
---------------------------------------------
Fidelity at Step 18: 0.9929504851720798
Parameters at Step 18: [0.030032491434869538, 0.6705379228419254, 0.32736077326483304, 1.4416928946464962, -0.07563674255845146, -0.007189452883322685, 0.7339963088570366, -0.965283352682299, 1.7091399150791209, -0.7831498975610103]
---------------------------------------------
Fidelity at Step 19: 0.9965405348346449
Parameters at Step 19: [0.049351359919739754, 0.6743590703990037, 0.4278756421344065, 1.4090284612866684, -0.02163430171942262, -0.08514460113615971, 0.693373791202934, -0.9156335387084984, 1.6874429857641309, -0.6900461171429496]
---------------------------------------------
Fidelity at Step 20: 0.9966665201201759
Parameters at Step 20: [0.05278414037474171, 0.690320672688792, 0.5389740191005465, 1.3798222192408045, 0.04246657361378194, -0.1704614872588347, 0.6327107569705728, -0.8732384948075647, 1.665370037015241, -0.5873476637991509]
---------------------------------------------
Fidelity at Step 21: 0.9985601378649517
Parameters at Step 21: [0.036744860702025185, 0.7211548377193117, 0.6624797948658447, 1.3541418526093136, 0.11662055186990422, -0.2630293857462455, 0.5463270732728669, -0.8391813354875706, 1.641691390380928, -0.47517726904059726]
---------------------------------------------
Fidelity at Step 22: 0.9989726690473872
Parameters at Step 22: [0.0099853113729, 0.7590819662221863, 0.786559829395425, 1.331306986027622, 0.19078421742491747, -0.3533747180286354, 0.44798226535748187, -0.811516385783778, 1.6171600388645548, -0.3653610979369781]
---------------------------------------------
Fidelity at Step 23: 0.999047032083147
Parameters at Step 23: [-0.026725503966848224, 0.8028384497425541, 0.9074123764373417, 1.3108986407404344, 0.2601589904988729, -0.43682334045593063, 0.3386032585919626, -0.7903088349898605, 1.5906634543837104, -0.2636408610946702]
---------------------------------------------
Fidelity at Step 24: 0.9987705154428619
Parameters at Step 24: [-0.07086221861622898, 0.8494889117930519, 1.0190044786807104, 1.292312641664198, 0.31796110307647885, -0.5068103469275773, 0.22176642815390943, -0.7752768448722682, 1.561023616126524, -0.1781655177903615]
---------------------------------------------
Fidelity at Step 25: 0.9988100329375428
Parameters at Step 25: [-0.11779944704122991, 0.8944605594186567, 1.1143459278348542, 1.2748222812428165, 0.35783783243731665, -0.5572620061886935, 0.10437661028468458, -0.765460064780762, 1.52771962327713, -0.11647520632647335]
---------------------------------------------
Fidelity at Step 26: 0.9985757201454402
Parameters at Step 26: [-0.1620749955989192, 0.9340651187861451, 1.1928230771213455, 1.2582183861869087, 0.3819403680851732, -0.5908063527356947, -0.0052458861753093555, -0.7587107231668222, 1.4928816938483498, -0.07549378112681823]
---------------------------------------------
Fidelity at Step 27: 0.9976302566535219
Parameters at Step 27: [-0.20132070244284592, 0.9664369998249582, 1.2536060950940575, 1.2422878678254046, 0.3904514403341251, -0.6078223076765568, -0.10356163561282461, -0.7542507212062988, 1.4569850261955124, -0.05477860988229539]
---------------------------------------------
Fidelity at Step 28: 0.9983882356575454
Parameters at Step 28: [-0.23255417027769784, 0.987815950434486, 1.2909889004408277, 1.2265674025512432, 0.37752274884372017, -0.6025799670865759, -0.1859803393905809, -0.752257031132375, 1.4186113777017493, -0.06138955414451268]
---------------------------------------------
Fidelity at Step 29: 0.9978671373289247
Parameters at Step 29: [-0.2566856271475478, 1.0006492046651103, 1.3125780415283235, 1.2115530344170848, 0.3520816340739027, -0.5841604105400039, -0.2540974192141527, -0.7518168751351452, 1.380097816028637, -0.0842201282781858]
---------------------------------------------
Fidelity at Step 30: 0.9974929437984331
Parameters at Step 30: [-0.27124830423752155, 1.001805037820717, 1.3148505271380608, 1.1972028099405516, 0.3108355440242823, -0.5493333047969007, -0.30386761874823015, -0.7534487198975702, 1.340572547803093, -0.12730767306292412]
---------------------------------------------
Fidelity at Step 31: 0.9987904515719516
Parameters at Step 31: [-0.27331253281188894, 0.9875366027856964, 1.2947325486214651, 1.183881268184288, 0.2512304445465742, -0.49559813556132276, -0.3302261854169147, -0.7582194250788556, 1.2993209544142166, -0.1938896756224774]
---------------------------------------------
Fidelity at Step 32: 0.9990327822738706
Parameters at Step 32: [-0.26795379123456736, 0.9646777852539683, 1.2642320526274793, 1.172242080195256, 0.18538030083281698, -0.4350974227693908, -0.3416288874251854, -0.7653120532473738, 1.2589997312286312, -0.26913506648646457]
---------------------------------------------
Fidelity at Step 33: 0.9996630784088545
Parameters at Step 33: [-0.25498616785829376, 0.9326639083646915, 1.2250032532283779, 1.1631299925830865, 0.11566158561604338, -0.370123583270212, -0.3372023926341056, -0.7759491224912733, 1.219973281394287, -0.35046354300650573]
---------------------------------------------
Fidelity at Step 34: 0.9995646137376791
Parameters at Step 34: [-0.23931125925929084, 0.8979390115443661, 1.18517083195646, 1.1562392191477264, 0.049407091226932126, -0.3080683093766753, -0.3256263558112948, -0.7885433711096579, 1.1837377130938038, -0.42860701069958995]
---------------------------------------------
Fidelity at Step 35: 0.9996465904308229
Parameters at Step 35: [-0.21955712185077875, 0.8579684028917567, 1.1461751770792745, 1.153596712136502, -0.010617573336647104, -0.25156592871335065, -0.3032907108221233, -0.806211033906433, 1.150742645372461, -0.5010224120719031]
---------------------------------------------
Fidelity at Step 36: 0.9994156035961047
Parameters at Step 36: [-0.19868629481528227, 0.8165858084509231, 1.1121003293501985, 1.1548201771770399, -0.06107266557805362, -0.20402041326536352, -0.2755913764320364, -0.827857105634376, 1.1216160185109565, -0.5633578965665793]
---------------------------------------------
Fidelity at Step 37: 0.9993165244122285
Parameters at Step 37: [-0.1779889897836405, 0.7749703004823995, 1.0870344529373086, 1.1611755563451476, -0.09753314112151815, -0.16984814116498834, -0.24407479905269586, -0.8549288607994256, 1.0972908304783051, -0.6106194470473059]
---------------------------------------------
Fidelity at Step 38: 0.9996276075039349
Parameters at Step 38: [-0.15915307597784445, 0.7354708885031871, 1.072332703887529, 1.1720124226103679, -0.11964806996345806, -0.14955951002483486, -0.2122772329988582, -0.8862271155143, 1.0778211101348336, -0.6420226972112081]
---------------------------------------------
Fidelity at Step 39: 0.9990796004168293
Parameters at Step 39: [-0.142525631080849, 0.6994887039221028, 1.0634937491532876, 1.1842620087197029, -0.13411912965193842, -0.13670846657035737, -0.1829062246986572, -0.9176867201800266, 1.0615334788002615, -0.6645525202471699]
---------------------------------------------
Fidelity at Step 40: 0.9991081504737204
Parameters at Step 40: [-0.12978142843589885, 0.6682722355308486, 1.067173079974738, 1.2008548828199337, -0.13307343789244414, -0.13917897231406814, -0.15733943829829156, -0.9527752044623491, 1.0504893600171643, -0.6695713382045293]
---------------------------------------------
Fidelity at Step 41: 0.9994006109153413
Parameters at Step 41: [-0.12115567316443504, 0.6424397976328438, 1.0824788700855221, 1.2211349307434558, -0.11825805101753337, -0.15543553765641152, -0.1367627010490742, -0.9905091080053223, 1.0443773039911841, -0.6588951948060155]
---------------------------------------------
Fidelity at Step 42: 0.9994774988967781
Parameters at Step 42: [-0.11571425014381993, 0.6214062734740367, 1.1051772812625795, 1.2432828086933019, -0.09506646644529715, -0.1802470006258153, -0.12088226665017486, -1.0287060346219037, 1.0418592834612284, -0.6384405334852695]
---------------------------------------------
Fidelity at Step 43: 0.9994472365267485
Parameters at Step 43: [-0.11343136573993991, 0.6053750054673392, 1.1345586980326912, 1.2670816384279806, -0.0645914205975793, -0.21272564092891239, -0.11021940656600411, -1.0669698557220983, 1.042885163548047, -0.6093818606417603]
---------------------------------------------
Fidelity at Step 44: 0.9997444727808593
Parameters at Step 44: [-0.11493137902455217, 0.5956040002841007, 1.1720296957398517, 1.2933199126033281, -0.025840937747204952, -0.25441678362223674, -0.10664260949985985, -1.1056396577415977, 1.0485202738504023, -0.5705427079314583]
---------------------------------------------
Fidelity at Step 45: 0.99988159413535
Parameters at Step 45: [-0.1181641198295314, 0.5899525287207167, 1.2117280803719108, 1.3197972842747712, 0.014517460349924967, -0.29859722278693634, -0.10769621831761414, -1.1424715828989647, 1.0567394027897858, -0.5293778103982675]
---------------------------------------------
Fidelity at Step 46: 0.9998573514345785
Parameters at Step 46: [-0.12175656133105636, 0.5867654275419255, 1.2499010203636594, 1.3451356909319643, 0.05248875951969164, -0.3410987405398154, -0.11141149487295963, -1.176254626328141, 1.0661176290234677, -0.4903933522447272]
---------------------------------------------
Fidelity at Step 47: 0.9998292007097693
Parameters at Step 47: [-0.12487131421130249, 0.5859454109604079, 1.284613548937081, 1.3689706154338066, 0.0852664799312279, -0.379774628371075, -0.11801000957728758, -1.2060029474462826, 1.0767548088559875, -0.45664961742963583]
---------------------------------------------
Fidelity at Step 48: 0.9997365697977294
Parameters at Step 48: [-0.12654857023039606, 0.586429393841211, 1.313585556214431, 1.3905244254034577, 0.11041058221068717, -0.4121675022983979, -0.12627431817292728, -1.230996950201297, 1.087855998062673, -0.43089443071099853]
---------------------------------------------
Fidelity at Step 49: 0.9997528113932673
Parameters at Step 49: [-0.12584831617269926, 0.5874242095681329, 1.3349964372480259, 1.4091922470968539, 0.12563466677970891, -0.43619264265287366, -0.13540569126088883, -1.2504754631024455, 1.0988857079981462, -0.41566649516961174]
---------------------------------------------
Fidelity at Step 50: 0.9995771492573322
Parameters at Step 50: [-0.12308220081986324, 0.5882956053892081, 1.3498112566839737, 1.4250593957050575, 0.1327111946665708, -0.45293666365919977, -0.14442696202432756, -1.2653186413203268, 1.109248000229557, -0.4091049260651626]
---------------------------------------------
Fidelity at Step 51: 0.9996582070356129
Parameters at Step 51: [-0.1167464756855651, 0.5885180846568769, 1.3553538821219644, 1.437311180522153, 0.12773708284845237, -0.45934041737317816, -0.153191885507882, -1.2738828621497664, 1.1187160226528092, -0.41542403253932575]
---------------------------------------------
Fidelity at Step 52: 0.9996955432159077
Parameters at Step 52: [-0.10771034556633162, 0.5880448359118233, 1.3537658157861867, 1.4464530329091365, 0.11384863623479319, -0.45786035704962497, -0.16146420142746504, -1.2774690460612694, 1.1271196329598958, -0.43122725720747185]
---------------------------------------------
Fidelity at Step 53: 0.9997991619034833
Parameters at Step 53: [-0.09627329672367939, 0.5868963942580755, 1.3462380053806309, 1.4527280567240575, 0.09247903186228626, -0.44976296196126986, -0.1693346453161187, -1.2765780952688226, 1.1344036427582234, -0.45493168233995446]
---------------------------------------------
Fidelity at Step 54: 0.9996928121478973
Parameters at Step 54: [-0.08345955687710035, 0.5853498150099569, 1.3350713003972525, 1.456813442186792, 0.06682602891064508, -0.43769031387004786, -0.1769697735889833, -1.2725521174090646, 1.1406837825893554, -0.48306702794086753]
---------------------------------------------
Fidelity at Step 55: 0.9998899365649595
Parameters at Step 55: [-0.06751148109701582, 0.583214665299736, 1.3183722482806561, 1.4575330258805324, 0.033384228036375245, -0.41943921471437035, -0.18540640162811964, -1.2629340228252863, 1.1456369135708773, -0.5191713211994088]
---------------------------------------------
Fidelity at Step 56: 0.9998624340125136
Parameters at Step 56: [-0.05131752850714959, 0.5811990289809753, 1.3013023835929207, 1.4569478769178683, -0.0004006580174740204, -0.4007625392880318, -0.19430310803813716, -1.251647479542463, 1.1498364761138966, -0.5553743416586863]
---------------------------------------------
Fidelity at Step 57: 0.9999132721039935
Parameters at Step 57: [-0.034655781202251414, 0.5797721835560955, 1.284992930419999, 1.4547640992807522, -0.03367428060470313, -0.3831140718715586, -0.20534639523532677, -1.2379072945156397, 1.1532505168291682, -0.5904570588092755]
---------------------------------------------
Fidelity at Step 58: 0.9998775148230059
Parameters at Step 58: [-0.018859726755114874, 0.579075691631346, 1.271532927647943, 1.4521537374570472, -0.06335771928892965, -0.36867465578447683, -0.21781255421386045, -1.2238816583685157, 1.1562215101775397, -0.6213031578896728]
---------------------------------------------
Fidelity at Step 59: 0.9998457347810404
Parameters at Step 59: [-0.0043922861721972534, 0.5793702002250531, 1.2627489076560674, 1.4496008479846716, -0.087487791151152, -0.3593321319413558, -0.2321843903928341, -1.2102636747738713, 1.1588836805134917, -0.6457585646371806]
---------------------------------------------
Fidelity at Step 60: 0.9997727108398051
Parameters at Step 60: [0.008381004961572968, 0.5806993140477965, 1.2595896924677998, 1.447526804103137, -0.10478843190020545, -0.35617567351770024, -0.2484759318134681, -1.1977686151876175, 1.1613353047857047, -0.6624979557651268]
---------------------------------------------
Fidelity at Step 61: 0.999780827709714
Parameters at Step 61: [0.01909797144686237, 0.5832292737274423, 1.2640247275721554, 1.4463805516185642, -0.1129425547030803, -0.36150916550403545, -0.267404289385826, -1.187080581646848, 1.1635990684701452, -0.6689774147180123]
---------------------------------------------
Fidelity at Step 62: 0.999808168158736
Parameters at Step 62: [0.027905032228909082, 0.5864669710827904, 1.2750805933139595, 1.4463033399265706, -0.11250744283286447, -0.37457023406410855, -0.2881605783982173, -1.1786398687238406, 1.1656020826645257, -0.666008871067185]
---------------------------------------------
Fidelity at Step 63: 0.9998548091256995
Parameters at Step 63: [0.035299063031327474, 0.5897168777071272, 1.2912297644365056, 1.4472746060795163, -0.10510414420013081, -0.39368682521839066, -0.3096536254084067, -1.1725501854957112, 1.1671566218333023, -0.6555791753225414]
---------------------------------------------
Fidelity at Step 64: 0.9999037069964792
Parameters at Step 64: [0.0418815267714097, 0.5922150299101131, 1.3102058029478096, 1.449115390383419, -0.09285188170899843, -0.416680689996342, -0.33075519781109286, -1.168752259973914, 1.168045873571082, -0.640236123403879]
---------------------------------------------
Fidelity at Step 65: 0.9999246406433501
Parameters at Step 65: [0.04829108705674348, 0.5933961363308191, 1.3295024801701152, 1.451494818076525, -0.07868535063518896, -0.44066890713809964, -0.3503195623475613, -1.1667777631995717, 1.1681448292091399, -0.6233418461259689]
---------------------------------------------
Fidelity at Step 66: 0.9998866529728345
Parameters at Step 66: [0.054990646400591335, 0.5929714866395723, 1.3475420631521589, 1.4541780082209224, -0.06478406516638432, -0.4635982065859475, -0.367668914093558, -1.1661843607147455, 1.1674078817134135, -0.6073574766751283]
---------------------------------------------
Fidelity at Step 67: 0.9999098513016205
Parameters at Step 67: [0.06305970317875366, 0.5894425173204019, 1.362279806065894, 1.457201816175589, -0.05293376636334743, -0.4837715235887784, -0.38173243002587237, -1.1674220982195875, 1.1651062924666915, -0.594733597363039]
---------------------------------------------
Fidelity at Step 68: 0.9998660737935083
Parameters at Step 68: [0.07207131811893552, 0.5836763740576104, 1.3731394847850424, 1.4602536559593045, -0.04417940302564178, -0.500178959137041, -0.39288037868101516, -1.169729608023709, 1.161707190111347, -0.5863686970050932]
---------------------------------------------
Fidelity at Step 69: 0.9999129411159435
Parameters at Step 69: [0.08262588608079113, 0.574810479604329, 1.3780673500067586, 1.4633174442657115, -0.04043810439157602, -0.5109593138014746, -0.40059235668512994, -1.1733561658529568, 1.156749837842348, -0.5847647387158366]
---------------------------------------------
Fidelity at Step 70: 0.9999204175249918
Parameters at Step 70: [0.09353640799320935, 0.5647304323063623, 1.3794737376476547, 1.4663251015063614, -0.03994340760723002, -0.5179451077951711, -0.40634844089445255, -1.1775732993830843, 1.151165123449546, -0.5873654574983501]
---------------------------------------------
Fidelity at Step 71: 0.9999188294218071
Parameters at Step 71: [0.10439909428941072, 0.5539946810211575, 1.3778815045617863, 1.4692807280461864, -0.04216929238300462, -0.521672517131992, -0.4107648593458895, -1.1822576620042096, 1.1451991959499215, -0.5934612963667989]
---------------------------------------------
Fidelity at Step 72: 0.9998604104737938
Parameters at Step 72: [0.1150711419134814, 0.5428094649719107, 1.3736920817157159, 1.4723117761807147, -0.047000405497170064, -0.5223555994970089, -0.414287638179804, -1.187491318807621, 1.1388891215693562, -0.6029561557441782]
---------------------------------------------
Fidelity at Step 73: 0.9999430308973932
Parameters at Step 73: [0.1256137433970921, 0.5304311299324498, 1.364379429769975, 1.4758445198215686, -0.05652412917964197, -0.5178998350909616, -0.4174842021241018, -1.1944011422191803, 1.131549276053809, -0.618992065967143]
---------------------------------------------
Fidelity at Step 74: 0.9999164917980096
Parameters at Step 74: [0.1352240671048545, 0.5186721462999183, 1.3544791514439096, 1.4795190877049202, -0.06662586038443774, -0.5124582922772923, -0.42092281691525507, -1.2016505027993059, 1.124376089573319, -0.6358352457562463]
---------------------------------------------
Fidelity at Step 75: 0.9999447368621411
Parameters at Step 75: [0.1433517389997682, 0.5077377878377025, 1.3439833619291048, 1.4839036749660373, -0.07720888813729462, -0.5061759141332505, -0.4260363440256178, -1.2102020610623234, 1.1171661798181443, -0.6538662573020569]
---------------------------------------------
Fidelity at Step 76: 0.9999371167409344
Parameters at Step 76: [0.1500140190045107, 0.4980752316911107, 1.3345300738606132, 1.4887111518450673, -0.08677048568017348, -0.5005964210616465, -0.43256554322458635, -1.219315335276509, 1.1103510852672205, -0.6708789782047512]
---------------------------------------------
Fidelity at Step 77: 0.9999422443337805
Parameters at Step 77: [0.1548386249560046, 0.4900488617967078, 1.3268507273328314, 1.49410275596857, -0.09430363221716345, -0.4967256792177868, -0.4413051029923251, -1.2291959841357525, 1.1040339321643222, -0.6857726663067597]
---------------------------------------------
Fidelity at Step 78: 0.9999238559846356
Parameters at Step 78: [0.15816023191744086, 0.4835880087905452, 1.3215248542660816, 1.499899606624128, -0.09948259802800598, -0.4949866314201788, -0.45179307232821825, -1.239408282708757, 1.0983243101263098, -0.6979516019681357]
---------------------------------------------
Fidelity at Step 79: 0.9998947527867448
Parameters at Step 79: [0.1598049053610818, 0.47891485857613925, 1.3193948259446773, 1.5062519491417448, -0.10136287159746196, -0.49640463581192196, -0.4647400306369159, -1.2500852220050942, 1.0932619266576435, -0.7063956771055033]
---------------------------------------------
Fidelity at Step 80: 0.9999239374484553
Parameters at Step 80: [0.15968640176034107, 0.47623857742848363, 1.3215516035038206, 1.5133445108042078, -0.09869941981715959, -0.5023790625052149, -0.4810894704739998, -1.2613481265326896, 1.0888831966015242, -0.709782922818542]
---------------------------------------------
Fidelity at Step 81: 0.999948470725103
Parameters at Step 81: [0.15903121689591235, 0.47455604185793715, 1.3263353473424122, 1.5205295668382561, -0.09350921353968285, -0.5109460782478329, -0.4986121270465538, -1.2722717718240208, 1.085019857104728, -0.7100977773081497]
---------------------------------------------
Fidelity at Step 82: 0.9999366087435407
Parameters at Step 82: [0.15850219731430015, 0.4732269015028527, 1.3323677097014557, 1.5273815258143946, -0.08721313304564966, -0.5206595953771328, -0.5159691422708877, -1.2823461352567127, 1.08150103237917, -0.7087956210664609]
---------------------------------------------
Fidelity at Step 83: 0.9999578753805831
Parameters at Step 83: [0.15876122870928364, 0.4717539889208967, 1.3389858240280035, 1.5336558651468708, -0.07983759313470387, -0.531645417340478, -0.5331916992279223, -1.2912136020041343, 1.078157462787459, -0.7059802614905673]
---------------------------------------------
Fidelity at Step 84: 0.9999562859848723
Parameters at Step 84: [0.15998950538828352, 0.46987222594640976, 1.3453745056781694, 1.5392301575813734, -0.0727606140942332, -0.5424516612950055, -0.5491990471098366, -1.2987740326465576, 1.0749304304852945, -0.7031076480354954]
---------------------------------------------
Fidelity at Step 85: 0.9999228784481042
Parameters at Step 85: [0.16259932393819493, 0.46723784978087973, 1.3510020439148975, 1.5439781690404197, -0.06661231505507895, -0.5525379692017052, -0.5636186962362523, -1.3048278506884046, 1.0717015957679998, -0.7009024320958638]
---------------------------------------------
Fidelity at Step 86: 0.9999267299255452
Parameters at Step 86: [0.16836953998618337, 0.4625743331186897, 1.354370824497011, 1.547342987670039, -0.06270296629337258, -0.5610776835106112, -0.5757496874720254, -1.3083583567906822, 1.068061298085744, -0.7009301098350672]
---------------------------------------------
Fidelity at Step 87: 0.9999539062194203
Parameters at Step 87: [0.17706182486991237, 0.4559909932922185, 1.3550043012605686, 1.5493683852536209, -0.06147149350529153, -0.567534538603623, -0.5855838960013352, -1.3095163493047912, 1.0640384884320284, -0.7037221397941468]
---------------------------------------------
Fidelity at Step 88: 0.9999375340731136
Parameters at Step 88: [0.1868411682003187, 0.4487769452876402, 1.354198171678452, 1.5506991362680713, -0.06179202566244289, -0.572554209006431, -0.5939668641662594, -1.3094725665544014, 1.0600401284357561, -0.708003630693759]
---------------------------------------------
Fidelity at Step 89: 0.9999452209763122
Parameters at Step 89: [0.19823455064505652, 0.44058748194066133, 1.3509939304430245, 1.5511797992868592, -0.06428899198213606, -0.5757041510507358, -0.601117863559753, -1.3079073427777828, 1.0559608562613465, -0.7146251024356468]
---------------------------------------------
Fidelity at Step 90: 0.9999533531894214
Parameters at Step 90: [0.21043439829521546, 0.4320595524880997, 1.3463316808943995, 1.5512027849227317, -0.06823345854195782, -0.5776080355311207, -0.6077936351302407, -1.305379521016447, 1.0519900966765465, -0.7228294616693073]
---------------------------------------------
Fidelity at Step 91: 0.9999482377985015
Parameters at Step 91: [0.22258131884660334, 0.42383575872142953, 1.3410192677440258, 1.5511322451252496, -0.07274782428256824, -0.5789636125943975, -0.614715204002582, -1.302483776213029, 1.0483402298473565, -0.7316818371041457]
---------------------------------------------
Fidelity at Step 92: 0.9999576342273339
Parameters at Step 92: [0.23412628512023217, 0.41639544583068705, 1.3358218745171693, 1.5513108208713804, -0.07713588301552442, -0.5804647359056426, -0.6227699371318457, -1.29963212390933, 1.0451605613454913, -0.7405203785739566]
---------------------------------------------
Fidelity at Step 93: 0.9999727698188148
Parameters at Step 93: [0.24453212095330634, 0.4100441400134776, 1.3315228065798237, 1.5519269263778077, -0.08074493839523109, -0.5825878464036203, -0.6319817828330716, -1.2972190092792801, 1.0425036708669353, -0.7485460033992009]
---------------------------------------------
Fidelity at Step 94: 0.9999536250599137
Parameters at Step 94: [0.2536722703488573, 0.4047163590726245, 1.3280973887080771, 1.5528955787872132, -0.08347035077885467, -0.5852439420251464, -0.6417777542876416, -1.2953414143018134, 1.0403125102601947, -0.755512034118745]
---------------------------------------------
Fidelity at Step 95: 0.9999374137250309
Parameters at Step 95: [0.26103507537725984, 0.4009139089583318, 1.3261420351258584, 1.5546937868438435, -0.08446635447884701, -0.5893556788424157, -0.65363447984888, -1.2945073876821842, 1.0388087430656394, -0.7607797218924264]
---------------------------------------------
Fidelity at Step 96: 0.9999374430735666
Parameters at Step 96: [0.26627778353324005, 0.3989196213921769, 1.3262941049535335, 1.5577437040216018, -0.0830341263130707, -0.5956816237748449, -0.668542648219778, -1.2951902905794082, 1.0381178975629617, -0.7638221259352528]
---------------------------------------------
Fidelity at Step 97: 0.9999649962536792
Parameters at Step 97: [0.2697847785808088, 0.39826938809292944, 1.3281747322189905, 1.561947624046751, -0.07949687637201738, -0.6038947845882106, -0.6859370315702844, -1.2973315413297546, 1.038061526677841, -0.7650278039241306]
---------------------------------------------
Fidelity at Step 98: 0.9999680100655207
Parameters at Step 98: [0.27271796356922473, 0.39790234793181534, 1.330559862331582, 1.566421385013026, -0.07551557234134834, -0.6122839897830252, -0.7031685695024871, -1.2999467407306893, 1.0381884951626206, -0.7657447634063109]
---------------------------------------------
Fidelity at Step 99: 0.9999595967720151
Parameters at Step 99: [0.2754978092770754, 0.39740785939847073, 1.3329187656288328, 1.5709769654558308, -0.07156215758556388, -0.6203943327686694, -0.719667344436063, -1.3028301018713266, 1.038348575369489, -0.7664758729296259]
---------------------------------------------
Fidelity at Step 100: 0.9999691823205074
Parameters at Step 100: [0.2787195506309466, 0.39613177135110544, 1.3345447334346552, 1.5756188983879296, -0.06806192154371864, -0.627881145467017, -0.7351851429420309, -1.3059974283184748, 1.0383453956803301, -0.7679358743646278]
---------------------------------------------
Fidelity at Step 101: 0.9999619187430177
Parameters at Step 101: [0.28242894636354865, 0.3941413617317143, 1.335479267494555, 1.5800706763364583, -0.06526765767770006, -0.6344739933622472, -0.7492072726913137, -1.3091003914506294, 1.038150294608112, -0.7701324583381439]
---------------------------------------------
Fidelity at Step 102: 0.9999527247255329
Parameters at Step 102: [0.2871335910640331, 0.39093393639234225, 1.3353433022708152, 1.584278472971393, -0.06355716737124557, -0.6398872952154574, -0.7614264174470093, -1.3120450306286773, 1.0375848139552188, -0.7736042137182007]
---------------------------------------------
Fidelity at Step 103: 0.9999515087413056
Parameters at Step 103: [0.2933050376932996, 0.3860404487541233, 1.3337755584108504, 1.5881898954764302, -0.06324796246744374, -0.6438856506025266, -0.7716056939368332, -1.3147448766603498, 1.0364772154865878, -0.7788250794420518]
---------------------------------------------
Fidelity at Step 104: 0.9999675825710751
Parameters at Step 104: [0.3008025878139133, 0.37959472310447573, 1.3309036751336876, 1.5917705693945272, -0.06415186981343302, -0.6466373568076824, -0.7799700701153659, -1.317171006296594, 1.0348254855853776, -0.7855574595540972]
---------------------------------------------
Fidelity at Step 105: 0.999963746275596
Parameters at Step 105: [0.3088450633120314, 0.3724504767796725, 1.3275806019833747, 1.5951023660493973, -0.06558320660968625, -0.6487570170855949, -0.7872103416969299, -1.3194046097416412, 1.032895159344219, -0.7928421744279232]
---------------------------------------------
Fidelity at Step 106: 0.9999700820140445
Parameters at Step 106: [0.3171910993892471, 0.3648639835104649, 1.3243046328861554, 1.5982874179327842, -0.0671603503119517, -0.6506556187610064, -0.7938392326798476, -1.3215284432992322, 1.0307296748903971, -0.8002770965620059]
---------------------------------------------
Fidelity at Step 107: 0.9999719230711291
Parameters at Step 107: [0.32535216077061707, 0.3573429169358276, 1.3214957744448017, 1.6013913477063242, -0.06846711483943341, -0.6527000200317139, -0.8003590156529284, -1.3236232324916415, 1.028486597834076, -0.8073183277096091]
---------------------------------------------
Fidelity at Step 108: 0.9999704259176454
Parameters at Step 108: [0.3329986920706661, 0.3501863547749545, 1.3192680004739101, 1.6044694211790305, -0.06917601723569679, -0.6551678922869517, -0.8072809087079148, -1.3257835416202473, 1.0262475057816136, -0.813620320598846]
---------------------------------------------
Fidelity at Step 109: 0.9999707168742463
Parameters at Step 109: [0.33998041479735225, 0.34364365212175774, 1.3181363309815315, 1.6076105586239462, -0.06917365171601826, -0.658241855114755, -0.8147859085270887, -1.3280243362252648, 1.0241020915541021, -0.8190015935969763]
---------------------------------------------
Fidelity at Step 110: 0.9999620589656709
Parameters at Step 110: [0.3460423583913361, 0.3378693854411911, 1.317849107573066, 1.6108044671696549, -0.06823790133041444, -0.662061454074891, -0.8233949240935231, -1.3304100784557746, 1.0220571285441684, -0.8232926159057777]
---------------------------------------------
Fidelity at Step 111: 0.9999584134890723
Parameters at Step 111: [0.35107705015323165, 0.3329995731612195, 1.318762522003613, 1.6141968846358399, -0.06612681776667563, -0.6669555529780533, -0.8338466286064476, -1.3330303389384777, 1.020113758621977, -0.8263798910593158]
---------------------------------------------
Fidelity at Step 112: 0.9999738155717663
Parameters at Step 112: [0.3551015929094727, 0.3289772288315669, 1.320341831355963, 1.6177230807833847, -0.06299787239164033, -0.6727464025727425, -0.8464442930459265, -1.3358252556034642, 1.0182343928369553, -0.8285450055481617]
---------------------------------------------
Fidelity at Step 113: 0.9999756646761303
Parameters at Step 113: [0.35874167432563875, 0.3253652359278932, 1.3222239301261025, 1.6211422839553296, -0.05984117723587664, -0.6784929360424193, -0.8592066116829397, -1.338460774651548, 1.0164419406180722, -0.8305134681878302]
---------------------------------------------
Fidelity at Step 114: 0.9999632639053777
Parameters at Step 114: [0.36224136399520773, 0.3219381474441058, 1.3240473921763805, 1.6243697990828445, -0.05697943055729214, -0.6838898111277703, -0.8717711089033766, -1.3408279751056573, 1.0146925419723878, -0.8326343910256307]
---------------------------------------------
Fidelity at Step 115: 0.9999668306911149
Parameters at Step 115: [0.3661666312383166, 0.31816758562783154, 1.3251122730719682, 1.6273587352316352, -0.054927480071705025, -0.6885799157666435, -0.8845328813242739, -1.3427311562359716, 1.012805347737836, -0.8358068058694881]
---------------------------------------------
Fidelity at Step 116: 0.9999754262812004
Parameters at Step 116: [0.37067123661742024, 0.3139181266791284, 1.3252926209140343, 1.63006555267173, -0.05381331395093721, -0.6924085884467933, -0.8968698569941312, -1.3441423791478109, 1.0108210366991226, -0.8400656201548946]
---------------------------------------------
Fidelity at Step 117: 0.9999695765369236
Parameters at Step 117: [0.37551193859655563, 0.30942234244863465, 1.3250715263661343, 1.632541596793348, -0.053322135969472426, -0.6956156209933725, -0.9083586665283568, -1.3451943259101402, 1.008852882232944, -0.8448540224550405]
---------------------------------------------
Fidelity at Step 118: 0.9999704499616886
Parameters at Step 118: [0.3809867404554024, 0.30431566381658504, 1.3237609869734193, 1.6347345280858476, -0.05357966866203419, -0.6980657635836122, -0.9190687419076997, -1.3458179041827725, 1.0068498778842818, -0.8504909559059793]
---------------------------------------------
Fidelity at Step 119: 0.9999755406629498
Parameters at Step 119: [0.38704494386526944, 0.2987017809134776, 1.322100112216846, 1.6368192018235117, -0.054239309681938824, -0.7001794349389332, -0.9290719847797951, -1.3461376589728842, 1.0048849088674854, -0.8565799861007863]
---------------------------------------------
Fidelity at Step 120: 0.9999789011267138
Parameters at Step 120: [0.39331096352645406, 0.29294291177961673, 1.320681871596173, 1.6389160289170315, -0.054877407594790976, -0.7023326927126207, -0.938493381656362, -1.3463437842275554, 1.0030652486348512, -0.8625479083263631]
---------------------------------------------
Fidelity at Step 121: 0.9999805974747978
Parameters at Step 121: [0.3994724184276103, 0.28723095564053364, 1.3192844618119195, 1.6410257256243213, -0.055214914824290494, -0.7046612022543123, -0.9475386902958111, -1.3465682713464002, 1.001459711273059, -0.868078671789324]
---------------------------------------------
Fidelity at Step 122: 0.9999776972201694
Parameters at Step 122: [0.4053697020523372, 0.2817518952184357, 1.3183268871909553, 1.643233147188186, -0.05510610404915501, -0.7073342903646016, -0.9563234811807184, -1.3468978140778394, 1.0001022397220145, -0.872971845704736]
---------------------------------------------
Fidelity at Step 123: 0.9999730869531723
Parameters at Step 123: [0.41099360416098013, 0.27661557993975416, 1.3183903011479048, 1.6456545435677146, -0.05453484459865441, -0.710475460425631, -0.9649009871180091, -1.3473801603951079, 0.9990075498368671, -0.8771861687839271]
---------------------------------------------
Fidelity at Step 124: 0.9999757043274963
Parameters at Step 124: [0.4162510926470969, 0.2717324710945155, 1.3191867431897601, 1.6484648379221867, -0.053046171510678926, -0.7144703154559339, -0.9739679574720589, -1.3482207595415412, 0.9982798148899359, -0.8805092467341217]
---------------------------------------------
Fidelity at Step 125: 0.9999831071267143
Parameters at Step 125: [0.4212157294059932, 0.2671278073829876, 1.3208485576256803, 1.6515742056714695, -0.051072873534062604, -0.7189571892077739, -0.9831871286985219, -1.3492992985000976, 0.9978150247357753, -0.8832925936026619]
---------------------------------------------
Fidelity at Step 126: 0.9999800502741776
Parameters at Step 126: [0.4258304803154631, 0.2628015348419753, 1.322576103509064, 1.6547282217328532, -0.04901727259077763, -0.7234002955708773, -0.9922258325738191, -1.3504893545077803, 0.9975241476585869, -0.8858653921420795]
---------------------------------------------
Fidelity at Step 127: 0.999984253640506
Parameters at Step 127: [0.43014442630061056, 0.25868690873059363, 1.3240528613827867, 1.657857631389491, -0.047087685568584815, -0.7275751147813105, -1.00102221378504, -1.3517377997881197, 0.9973592845901739, -0.8884676881460879]
---------------------------------------------
Fidelity at Step 128: 0.9999815621781017
Parameters at Step 128: [0.4342474947003735, 0.25468250766471245, 1.3250608082262136, 1.6609254323711848, -0.0454602413512257, -0.7313106186446107, -1.0095491550284497, -1.353004449699596, 0.9972786993423773, -0.8913329456660455]
---------------------------------------------
Fidelity at Step 129: 0.9999749203279624
Parameters at Step 129: [0.43828868033187524, 0.2507339797713747, 1.3259657360952608, 1.6639692251407665, -0.04423732639627555, -0.7346198312241684, -1.017741785522529, -1.3542354623451425, 0.997236371907943, -0.8945655022370658]
---------------------------------------------
Fidelity at Step 130: 0.9999834155837806
Parameters at Step 130: [0.44249513928258005, 0.24653040306162236, 1.3262773111368178, 1.6670270506989011, -0.04370634050950086, -0.7372511491001004, -1.0258454978354004, -1.3554121508254215, 0.9971815425965311, -0.8987263635407147]
---------------------------------------------
Fidelity at Step 131: 0.9999843047093948
Parameters at Step 131: [0.44662627031508617, 0.2423288408400945, 1.3261078794234142, 1.6699493390380813, -0.04356251616217691, -0.7393949056552869, -1.0336202982832152, -1.3565045133903642, 0.9971185609488726, -0.9032365078623498]
---------------------------------------------
Fidelity at Step 132: 0.9999805708115902
Parameters at Step 132: [0.4506341367367542, 0.23819936793492752, 1.3256677578018876, 1.6727233225737634, -0.043638344726336875, -0.7412146163040286, -1.0410454472963724, -1.3575100642862223, 0.9970466843308186, -0.9078502956878469]
---------------------------------------------
Fidelity at Step 133: 0.9999848139657832
Parameters at Step 133: [0.4545912429808756, 0.23404123760127696, 1.324996072175742, 1.6753963110904075, -0.043834865568968186, -0.7428366929831622, -1.048332836540861, -1.3584348303868834, 0.9969536660917491, -0.9125615675357505]
---------------------------------------------
Fidelity at Step 134: 0.9999825718980218
Parameters at Step 134: [0.4584279371592802, 0.2300414525837428, 1.3247252928222295, 1.677991163691852, -0.043977558780503384, -0.7444909542643103, -1.0553028345072406, -1.3592844893711566, 0.9968544812772046, -0.9170150826462645]
---------------------------------------------
Fidelity at Step 135: 0.9999838951828618
Parameters at Step 135: [0.4621382229469667, 0.22606720373351205, 1.3244691931108554, 1.6804903156912678, -0.04388265097535359, -0.7462825168234908, -1.062277085779582, -1.360073745565299, 0.9967386980878402, -0.9211551550856614]
---------------------------------------------
Fidelity at Step 136: 0.9999824784588653
Parameters at Step 136: [0.46573094404052606, 0.22218648245433192, 1.3246036871228646, 1.6829350170970572, -0.043495549937033674, -0.7483250061268732, -1.0691916011324232, -1.3608082940377562, 0.9966148553885258, -0.924864944920965]
---------------------------------------------
Fidelity at Step 137: 0.9999844204058552
Parameters at Step 137: [0.4691665762480434, 0.21829418758173436, 1.3246903022951173, 1.6852722136598322, -0.04263885474266841, -0.7506944700595976, -1.0763112061299038, -1.3614873941276366, 0.9964682866850386, -0.928060163579988]
---------------------------------------------
Fidelity at Step 138: 0.999981074378395
Parameters at Step 138: [0.4724835716645587, 0.2145049377725422, 1.3251259128289934, 1.6875346505803719, -0.041523943498307715, -0.7532784602034222, -1.0833319831342216, -1.3620989339182574, 0.9963078754896275, -0.9308430754925966]
---------------------------------------------
Fidelity at Step 139: 0.9999795341590785
Parameters at Step 139: [0.47570079563668827, 0.2106587381633647, 1.3254196683382355, 1.6896652488659856, -0.04007702689921555, -0.7560711026919471, -1.0905213768357342, -1.362611511128764, 0.9961036723792209, -0.9332696152915033]
---------------------------------------------
Fidelity at Step 140: 0.9999889992016171
Parameters at Step 140: [0.47898375000729126, 0.2067913373020297, 1.3262842079716255, 1.6917932283805504, -0.03851562036395674, -0.7590541683418879, -1.097685648625456, -1.3630140081676458, 0.9958597307579654, -0.935521013127338]
---------------------------------------------
Fidelity at Step 141: 0.999987152994515
Parameters at Step 141: [0.4820979864855765, 0.2030562123457122, 1.3268750998011958, 1.6937422986913524, -0.03705166232259487, -0.7618206345540147, -1.1045286788348445, -1.3633311333858664, 0.9956043895794627, -0.9377007330127332]
---------------------------------------------
Fidelity at Step 142: 0.9999837616045216
Parameters at Step 142: [0.4852152795745083, 0.19933276937130542, 1.3274710255573734, 1.6956023299086598, -0.03577461253637882, -0.7643978601280781, -1.1111527147563243, -1.363538821944383, 0.9953193029481325, -0.9399955381588703]
---------------------------------------------
Fidelity at Step 143: 0.9999814459257635
Parameters at Step 143: [0.48839916484759044, 0.19550491221578772, 1.3277946418636415, 1.6973704451352887, -0.034798868893791066, -0.7666648753845143, -1.1176717543209973, -1.3636173732484602, 0.9949934317787731, -0.9426268272160425]
---------------------------------------------
Fidelity at Step 144: 0.9999850374815137
Parameters at Step 144: [0.49165454993302277, 0.19157234440392285, 1.3278888659113894, 1.6990801687879322, -0.03415337895580718, -0.7686115048888433, -1.1241119264423507, -1.3635854280358073, 0.994637952532866, -0.9456455164014339]
---------------------------------------------
Fidelity at Step 145: 0.9999861831973936
Parameters at Step 145: [0.4948516637510453, 0.18771062041704265, 1.3279261867569543, 1.700750702088882, -0.03371833064936751, -0.7703351317991944, -1.130353693041737, -1.3635155517339068, 0.9943020382966017, -0.9488152011860097]
---------------------------------------------
Fidelity at Step 146: 0.9999849895773791
Parameters at Step 146: [0.4979179564776596, 0.1839432903772568, 1.3277729869065453, 1.7023684683632287, -0.033421590116644154, -0.7718574676594108, -1.1364400887029087, -1.3634364571146622, 0.9939995210390251, -0.9520625040412555]
---------------------------------------------
Fidelity at Step 147: 0.9999867654127862
Parameters at Step 147: [0.5008672133519527, 0.18029084997712289, 1.3277142937941975, 1.704005832696915, -0.03317473744913395, -0.7733145786739568, -1.1424194449958749, -1.3633827960967202, 0.9937519907295783, -0.955299499225928]
---------------------------------------------
Fidelity at Step 148: 0.9999887913045802
Parameters at Step 148: [0.503651682297394, 0.17679445321469484, 1.3277508023942615, 1.7056658715580102, -0.032895307435838114, -0.774767538847392, -1.1482963708561775, -1.3633806485383175, 0.9935767702098116, -0.958419192994982]
---------------------------------------------
Fidelity at Step 149: 0.999985035956237
Parameters at Step 149: [0.506274433952535, 0.1734509832221617, 1.327941771421969, 1.7073681258447921, -0.032527136133335424, -0.7762749830110723, -1.154080538510611, -1.3634484542330767, 0.9934845033090591, -0.9613654595093701]
---------------------------------------------
Fidelity at Step 150: 0.9999878877045175
Parameters at Step 150: [0.508749462189302, 0.17018243939953917, 1.328248119010844, 1.7091637132690014, -0.03196923377526107, -0.7779311273700311, -1.1599394135997274, -1.363615635769636, 0.9934987056822612, -0.9641197343895163]
---------------------------------------------
Fidelity at Step 151: 0.9999872199533634
Parameters at Step 151: [0.5110306887058557, 0.16704257626956867, 1.328412707125246, 1.7109354590868575, -0.03128792053631156, -0.7796047782485971, -1.1656706186958778, -1.3638443013173356, 0.9935892840802129, -0.9666568943461408]
---------------------------------------------
Fidelity at Step 152: 0.9999846672973843
Parameters at Step 152: [0.5132716374928659, 0.16400649350758884, 1.3290566977033718, 1.7127763720480835, -0.030506251567427905, -0.7814239682976281, -1.171216767607195, -1.3641084371106356, 0.9937345066138964, -0.9689872541512644]
---------------------------------------------
Fidelity at Step 153: 0.9999867351528788
Parameters at Step 153: [0.5154380642884647, 0.16101081197847836, 1.329681664784569, 1.7146132461957844, -0.029644046624950387, -0.7832743982470529, -1.1766086526335258, -1.3643966310971727, 0.9939355408878404, -0.9711722947871008]
---------------------------------------------
Fidelity at Step 154: 0.9999862241484375
Parameters at Step 154: [0.5174946573552897, 0.15803941295449758, 1.3298958677221986, 1.7163569021894192, -0.028766525059498282, -0.7850159870432324, -1.1817991369158976, -1.3646823960544083, 0.9941758238430728, -0.9732732065211951]
---------------------------------------------
Fidelity at Step 155: 0.9999830014109053
Parameters at Step 155: [0.5195040670339732, 0.15504745752530472, 1.329762082040617, 1.7180207634581337, -0.02790811151081738, -0.7866483216268434, -1.1867998664498791, -1.3649424237865677, 0.9944472518870449, -0.9753472278100357]
---------------------------------------------
Fidelity at Step 156: 0.9999864511000253
Parameters at Step 156: [0.521716676233046, 0.15176583730252977, 1.32957639767777, 1.719723619205503, -0.02705181275335927, -0.7883125895766997, -1.1918166235153695, -1.3651207890826964, 0.9947525014995762, -0.977549037634112]
---------------------------------------------
Fidelity at Step 157: 0.9999826513650695
Parameters at Step 157: [0.5239964399597866, 0.14844427349129943, 1.329535680851092, 1.7214165735621738, -0.026241741176397944, -0.7899683259992069, -1.1967061184083099, -1.365230019906992, 0.995068887255867, -0.979764123843502]
---------------------------------------------
Fidelity at Step 158: 0.9999882498026169
Parameters at Step 158: [0.5262861536543998, 0.14504362765356396, 1.3292303431819674, 1.7230464001713983, -0.025476035965192906, -0.7915423527347953, -1.2016220671627142, -1.365248690384345, 0.9954066878505755, -0.982048867852817]
---------------------------------------------
Fidelity at Step 159: 0.9999857804291704
Parameters at Step 159: [0.5285783161964774, 0.14174543081676313, 1.3293491757849916, 1.7246782438510426, -0.02475599728313822, -0.7931466879198291, -1.206408566689147, -1.3652104189458465, 0.995741455288656, -0.9842759850445346]
---------------------------------------------
Fidelity at Step 160: 0.9999873512494414
Parameters at Step 160: [0.5307594882904066, 0.13849537736632228, 1.3291680039841776, 1.7262214114795207, -0.024073128205469314, -0.7946463806378471, -1.2112733331631036, -1.3651025135799986, 0.9960957414433877, -0.9865214428462055]
---------------------------------------------
Fidelity at Step 161: 0.9999886461524623
Parameters at Step 161: [0.5328458040346352, 0.13532214379578952, 1.328966681660508, 1.7277096320780783, -0.023409888019261096, -0.7961063813366783, -1.2162034351667637, -1.364932886473924, 0.996457568976065, -0.9887460480457212]
---------------------------------------------
Fidelity at Step 162: 0.9999902032474028
Parameters at Step 162: [0.5348340653861403, 0.13225505344324726, 1.328847208373762, 1.7291647442219011, -0.02275233771033779, -0.7975569869396825, -1.2211816293722073, -1.3647211936191672, 0.9968308645104235, -0.9909191256609178]
---------------------------------------------
Fidelity at Step 163: 0.9999889132157431
Parameters at Step 163: [0.5366843777290922, 0.12933385256430174, 1.3286998070140525, 1.7305499865191734, -0.022101471015799637, -0.7989569513789231, -1.2260919913498218, -1.3644891300526791, 0.9972069997644573, -0.9929949238177518]
---------------------------------------------
Fidelity at Step 164: 0.999989502893319
Parameters at Step 164: [0.5385235144431655, 0.12657531222828117, 1.3291695036954272, 1.731969724717606, -0.021470826093723947, -0.8004348306344797, -1.2308069913603101, -1.3642503959184866, 0.9975698806550973, -0.9949470347889933]
---------------------------------------------
Fidelity at Step 165: 0.9999843597471785
Parameters at Step 165: [0.5402555117761144, 0.12386269780869166, 1.3294479006466213, 1.7333139145782528, -0.020846006937108762, -0.801838413403773, -1.2355098657138137, -1.3639852050723429, 0.9979320419695976, -0.9968610486132228]
---------------------------------------------
Fidelity at Step 166: 0.9999892815397706
Parameters at Step 166: [0.5419763943442601, 0.12107381862368885, 1.3296300899917965, 1.7346385301625344, -0.020234793790248003, -0.8032058316245257, -1.2402922602982387, -1.3636810725471509, 0.9982968131525182, -0.9988249045527734]
---------------------------------------------
Fidelity at Step 167: 0.9999868476099629
Parameters at Step 167: [0.5436811281664722, 0.118345847639199, 1.3299671848217822, 1.7359542632234002, -0.01966693203088115, -0.8045450432272869, -1.2448980877577691, -1.3633785403963068, 0.9986538360575402, -1.0007502576104403]
---------------------------------------------
Fidelity at Step 168: 0.999988211891881
Parameters at Step 168: [0.5453907419233148, 0.11562234491250503, 1.3303961942731146, 1.737252260673979, -0.019144809206808718, -0.8058398287593344, -1.249321291250448, -1.3630740644029284, 0.9989949914004753, -1.002662997387518]
---------------------------------------------
Fidelity at Step 169: 0.9999928232199977
Parameters at Step 169: [0.5470298089619259, 0.11294845433054324, 1.3306209946783156, 1.7384770651864752, -0.018675792225203727, -0.8070111888957845, -1.2535154412184717, -1.3627845973676904, 0.9993249606501416, -1.0045453852920951]
---------------------------------------------
Fidelity at Step 170: 0.9999887321883951
Parameters at Step 170: [0.5486670595875374, 0.11037292141096658, 1.3311378474407562, 1.739711282598617, -0.01825573027893023, -0.8081728492461591, -1.2574421460778868, -1.3625190854901312, 0.9996410514327894, -1.0063623030994178]
---------------------------------------------
Fidelity at Step 171: 0.9999920762371789
Parameters at Step 171: [0.5502349300327534, 0.10779993295101048, 1.3313410830723522, 1.7408785660065607, -0.01787842457962252, -0.8092012976753657, -1.26118384893814, -1.362280184910283, 0.9999541327857622, -1.008177363562292]
---------------------------------------------
Fidelity at Step 172: 0.9999868204329156
Parameters at Step 172: [0.5518084300106664, 0.10527005598833068, 1.3317547409661767, 1.7420676848269074, -0.017533696191392507, -0.8102232480918031, -1.2647257065399566, -1.3620720984869334, 1.0002613731708783, -1.0099566960612834]
---------------------------------------------
Fidelity at Step 173: 0.9999913559311914
Parameters at Step 173: [0.5533847676018934, 0.1026629991782988, 1.3321004972759545, 1.7432814900492706, -0.01719997151269994, -0.811207265200446, -1.2681875031077672, -1.3619056071299658, 1.0005797750452579, -1.011770234432686]
---------------------------------------------
Fidelity at Step 174: 0.9999913214040246
Parameters at Step 174: [0.5548821612889252, 0.10011060185804076, 1.3323360404668545, 1.7444747043954105, -0.01687189563828634, -0.8121357317212428, -1.2715295445300454, -1.3617802772055234, 1.000907299920358, -1.0135406958891202]
---------------------------------------------
Fidelity at Step 175: 0.9999860300741775
Parameters at Step 175: [0.5563210452484355, 0.0976058452896269, 1.3326247104410003, 1.7456829155834255, -0.01652412298005788, -0.8130683214357861, -1.2748043515034768, -1.3616955370247923, 1.0012482944684757, -1.0152569520851642]
---------------------------------------------
Fidelity at Step 176: 0.9999869130137046
Parameters at Step 176: [0.5576730167381697, 0.09506111198725749, 1.332764702975683, 1.7469248277458853, -0.016100468936757248, -0.8140193217285754, -1.2782028243333425, -1.361661260556583, 1.0016330592746054, -1.0169543314815557]
---------------------------------------------
Fidelity at Step 177: 0.9999877217147968
Parameters at Step 177: [0.5589274852929289, 0.09258159463488037, 1.3328613000011675, 1.7481529641823588, -0.015629042269218567, -0.8149771282707365, -1.2816037761035932, -1.361648318523111, 1.0020367177663734, -1.0185712748653197]
---------------------------------------------
Fidelity at Step 178: 0.9999872658409166
Parameters at Step 178: [0.5602276069540725, 0.09013573257411094, 1.3335546522621782, 1.7494895597083502, -0.015105424148538587, -0.8161053194032152, -1.2850558032987738, -1.3616335521110063, 1.0024591565155927, -1.0201225639762055]
---------------------------------------------
Fidelity at Step 179: 0.9999875682031998
Parameters at Step 179: [0.5614533265372388, 0.08765171564005021, 1.333926828438151, 1.750790180328323, -0.01456972519685228, -0.8171790361138784, -1.2886525881175384, -1.361586512781489, 1.0029150679738548, -1.0217108790652585]
---------------------------------------------
Fidelity at Step 180: 0.9999901610429807
Parameters at Step 180: [0.5626325362460528, 0.08529226687868985, 1.334338451916551, 1.75203049108332, -0.01408242264275763, -0.8181988672647009, -1.2920751971204063, -1.3615175757134808, 1.0033517777316945, -1.0232391229953068]
---------------------------------------------
Fidelity at Step 181: 0.9999924934552326
Parameters at Step 181: [0.5637724360868879, 0.08301598861453113, 1.3346625456129624, 1.7531977142841724, -0.013653694695672566, -0.8191386884772068, -1.2953676293760934, -1.3614089750352865, 1.0037750754379051, -1.0247469653569756]
---------------------------------------------
Fidelity at Step 182: 0.9999896337416619
Parameters at Step 182: [0.5648823856161473, 0.08081632764422314, 1.3349008987446196, 1.7542874802301704, -0.013285444609570781, -0.8199977576502516, -1.298518194796666, -1.3612559517700367, 1.0041812470182039, -1.02623464086704]
---------------------------------------------
Fidelity at Step 183: 0.9999886552526498
Parameters at Step 183: [0.5659564124151729, 0.07863183162668098, 1.3348109422735874, 1.7552690439422955, -0.012978665788255614, -0.8207337329914272, -1.3015909474075096, -1.361033993599378, 1.004578488805329, -1.0277434686827096]
---------------------------------------------
Fidelity at Step 184: 0.9999896963669672
Parameters at Step 184: [0.5670407395555168, 0.0764062248474207, 1.3345306186077732, 1.7561739406827335, -0.012682726906812027, -0.8214300293705896, -1.304667668449513, -1.3607179965360254, 1.0049775004301271, -1.0292546807793863]
---------------------------------------------
Fidelity at Step 185: 0.9999892555185942
Parameters at Step 185: [0.5681657453963701, 0.0742498846666968, 1.3346425503959116, 1.7570930899011632, -0.012365431398008667, -0.822222072957483, -1.3076478743542113, -1.3603657170326973, 1.0053656310008658, -1.030665191966978]
---------------------------------------------
Fidelity at Step 186: 0.9999873008804145
Parameters at Step 186: [0.5692469331809209, 0.07214852738005786, 1.334726456522285, 1.7579694445496343, -0.01199912569255848, -0.8230416451427469, -1.310601063990293, -1.359981830405764, 1.005761025868443, -1.031971760971129]
---------------------------------------------
Fidelity at Step 187: 0.9999902918781893
Parameters at Step 187: [0.5703082261411033, 0.0700651796476419, 1.334896749157116, 1.7588420377632485, -0.011550108087819182, -0.8239479011347973, -1.3136022649324255, -1.3595645721844367, 1.006177612859388, -1.0331729024355931]
---------------------------------------------
Fidelity at Step 188: 0.9999885080739047
Parameters at Step 188: [0.5713983164659022, 0.06808727518476335, 1.3355403183830272, 1.7597628961459177, -0.011100614964728651, -0.8249430455093314, -1.3164774463852682, -1.35915795659551, 1.0065828112639668, -1.0342885329375568]
---------------------------------------------
Fidelity at Step 189: 0.9999862800292353
Parameters at Step 189: [0.572434804434997, 0.06612847682904653, 1.3359856559587966, 1.7606464812958231, -0.010677275161958846, -0.825864146318007, -1.3193148628423925, -1.3587522835441175, 1.0069937130870932, -1.0354150953922756]
---------------------------------------------
Fidelity at Step 190: 0.999986941039955
Parameters at Step 190: [0.5734157712423219, 0.0642115062042154, 1.3362412133150423, 1.7614910453000487, -0.01031387695515812, -0.8266793210934649, -1.3220600557304993, -1.358363678558335, 1.0074003729498613, -1.0365720553497828]
---------------------------------------------
Fidelity at Step 191: 0.9999890579779093
Parameters at Step 191: [0.5742763144740772, 0.062338167532080394, 1.3359942272632255, 1.762246736010568, -0.01000955999141036, -0.8273153806758138, -1.3247240659672528, -1.3580029613090359, 1.0078091831397777, -1.037765435378209]
---------------------------------------------
Fidelity at Step 192: 0.9999798614554044
Parameters at Step 192: [0.5750348275406495, 0.06050912454155909, 1.3354045794993685, 1.7629382965909832, -0.009732835043158979, -0.8278312897232609, -1.3273137762418075, -1.3576734333506415, 1.0082177525611982, -1.0389580051847715]
---------------------------------------------
Fidelity at Step 193: 0.9999867211457386
Parameters at Step 193: [0.575541937870243, 0.058638268388376595, 1.333829424344813, 1.7635015625270305, -0.00933054933450848, -0.8282103517661018, -1.330068048247978, -1.3573865132373175, 1.0086873691308977, -1.040078785952071]
---------------------------------------------
Fidelity at Step 194: 0.9999885173075377
Parameters at Step 194: [0.5759330636780624, 0.05686170692563526, 1.332260716345552, 1.7640508985116936, -0.008758432038375216, -0.8287101056883912, -1.3328695949297107, -1.3571284961271388, 1.0091873394591213, -1.0409664786968056]
---------------------------------------------
Fidelity at Step 195: 0.9999914414076407
Parameters at Step 195: [0.5761698087898832, 0.05520787437158809, 1.3306213537620428, 1.764541391429165, -0.007951988821247653, -0.8293609243539548, -1.3357305590793547, -1.3568807266091802, 1.0097224300509535, -1.041523503066072]
---------------------------------------------
Fidelity at Step 196: 0.9999909358323312
Parameters at Step 196: [0.5764460927363217, 0.05365713280830934, 1.3297462583876818, 1.7650963298974642, -0.006946505580886105, -0.8303293111980891, -1.3385944472376565, -1.356607494314803, 1.0102670663121784, -1.041766147095695]
---------------------------------------------
Fidelity at Step 197: 0.9999906039213291
Parameters at Step 197: [0.5767220976396684, 0.05221967959575094, 1.3291124851999803, 1.7656255362200648, -0.005933869704823914, -0.8313403443762636, -1.341316897342956, -1.3563194698910808, 1.0107906440580432, -1.0418902860638228]
---------------------------------------------
Fidelity at Step 198: 0.999989638431839
Parameters at Step 198: [0.5770661079103651, 0.05082746036319119, 1.3287969456999924, 1.7661289073141504, -0.004936116431480546, -0.8324038971128354, -1.343923477449557, -1.355964787344698, 1.0112947775188423, -1.0419333188145385]
---------------------------------------------
Fidelity at Step 199: 0.999991606617489
Parameters at Step 199: [0.5775094191382846, 0.04945618421796187, 1.328710317887879, 1.7666051097500668, -0.004059246822290077, -0.8334229380088882, -1.3463756706953927, -1.355537991382031, 1.0117698517998235, -1.042021085504792]
---------------------------------------------
Fidelity at Step 200: 0.9999887382906469
Parameters at Step 200: [0.578030459572647, 0.048039145898378995, 1.328377894191988, 1.7669658843418832, -0.0034115315624715353, -0.8342053910707248, -1.3486503101922291, -1.3550075140083484, 1.0122104791186208, -1.0422898758521553]
---------------------------------------------
Fidelity at Step 201: 0.9999925504526961
Parameters at Step 201: [0.5786045080319537, 0.046515376553358335, 1.3274020566780673, 1.7671434184496464, -0.0030863047854294773, -0.8345862731337464, -1.350754920349443, -1.3543502995139975, 1.012623699363371, -1.0428631724087638]
---------------------------------------------
Fidelity at Step 202: 0.999987255051033
Parameters at Step 202: [0.5792017647238181, 0.04507292120345454, 1.3266199087241262, 1.7673322669051341, -0.002871042638470988, -0.8349088800492177, -1.3527100967128156, -1.3537096189628486, 1.013010919594112, -1.043485932366876]
---------------------------------------------
Fidelity at Step 203: 0.9999888907573472
Parameters at Step 203: [0.5797387075603089, 0.04364080969232375, 1.325521237782578, 1.7674533054630355, -0.0027846252235358806, -0.8350404370721993, -1.3546345976760379, -1.3530614356430213, 1.0134144707666521, -1.044209093510324]
---------------------------------------------
Fidelity at Step 204: 0.9999881322007121
Parameters at Step 204: [0.5802396206810376, 0.0422928753246885, 1.324706835881735, 1.7676251500038638, -0.0026872600106955867, -0.8352189681322226, -1.3565439873059322, -1.3524655967666366, 1.0138380231743556, -1.0448691491124078]
---------------------------------------------
Fidelity at Step 205: 0.9999894671754493
Parameters at Step 205: [0.5806914874725001, 0.04104581140421217, 1.3244634196493874, 1.7679304385405523, -0.0024719827456277303, -0.8355911829340839, -1.358539534277575, -1.3519640678407858, 1.0143193814162696, -1.0453599992552314]
---------------------------------------------
Fidelity at Step 206: 0.9999887550530641
Parameters at Step 206: [0.58112054461415, 0.03988085878168208, 1.324829539004447, 1.768372668192599, -0.0021429946527134463, -0.8361566271138334, -1.3605549445614127, -1.3515688560968233, 1.0148434248146052, -1.0456880730062577]
---------------------------------------------
Fidelity at Step 207: 0.9999886259446119
Parameters at Step 207: [0.5814592119397126, 0.038760976413281766, 1.3251588665269407, 1.7688472494478302, -0.0017760094832740304, -0.8367119264841479, -1.3625314031351388, -1.3512812908547716, 1.0154026011795458, -1.0459490260414839]
---------------------------------------------
Fidelity at Step 208: 0.9999881139367572
Parameters at Step 208: [0.5818449253018668, 0.03765135420895377, 1.325802181057237, 1.7694035248856217, -0.0014506017523894395, -0.837284168467825, -1.36437529011688, -1.351070052402854, 1.0159605514890004, -1.0462281768586885]
---------------------------------------------
Fidelity at Step 209: 0.999987619645847
Parameters at Step 209: [0.5821125428709006, 0.036466933319086334, 1.325516216879589, 1.7698410345525648, -0.0012633434400915718, -0.8375101782469347, -1.3660476637189252, -1.350942916995247, 1.0165309547632595, -1.046658068686272]
---------------------------------------------
Fidelity at Step 210: 0.99999025342367
Parameters at Step 210: [0.582340225734264, 0.03517995166803719, 1.3246358185219782, 1.7702078978309685, -0.0011893135878432212, -0.8374859194741336, -1.3675400627278114, -1.3508856113579193, 1.0171132484957168, -1.0472076010069815]
---------------------------------------------
Fidelity at Step 211: 0.9999884765567478
Parameters at Step 211: [0.5827442110195769, 0.03389662460962347, 1.3246663160569445, 1.7707286210834383, -0.001116984343657353, -0.8376557949861562, -1.3689037468637544, -1.3508563250802832, 1.0176748261746098, -1.0477187392767007]
---------------------------------------------
Fidelity at Step 212: 0.9999904561856239
Parameters at Step 212: [0.5832025352877948, 0.032631330812688694, 1.3250559757662133, 1.771294272313985, -0.0010245966853708556, -0.8379075762354566, -1.3701751164810776, -1.3508489587041526, 1.0182246556232561, -1.0481672409301965]
---------------------------------------------
Fidelity at Step 213: 0.9999901858354218
Parameters at Step 213: [0.5836605704891774, 0.031376665348989655, 1.3255095674442012, 1.7718537512835941, -0.0009259331585511649, -0.8381631597635073, -1.3713892089714161, -1.3508567602125896, 1.0187724587122158, -1.0485744541707018]
---------------------------------------------
Fidelity at Step 214: 0.999988435759673
Parameters at Step 214: [0.5841644644043431, 0.0301670059306356, 1.326257830920721, 1.7724447901025937, -0.0008609609656053234, -0.8384547421229761, -1.3725448268533085, -1.3508618728232376, 1.0192939089925728, -1.0489831592110408]
---------------------------------------------
Fidelity at Step 215: 0.9999895762160621
Parameters at Step 215: [0.5846071363430934, 0.028952547931390647, 1.3266199248264239, 1.7729538086743766, -0.0008886357076636716, -0.8385738392256091, -1.3736919164157642, -1.3508571862340193, 1.019813802841107, -1.049476081419914]
---------------------------------------------
Fidelity at Step 216: 0.9999865439963951
Parameters at Step 216: [0.5850557559413209, 0.027766459985979788, 1.327061556266125, 1.7734498368139349, -0.0009720306347003305, -0.8386553607704775, -1.3748193624390446, -1.3508389982062428, 1.020319208258268, -1.0500056998757614]
---------------------------------------------
Fidelity at Step 217: 0.9999885827559089
Parameters at Step 217: [0.5854117281207973, 0.026620914277741707, 1.327155156248737, 1.7738595553203997, -0.0010939920041759354, -0.8386143736767395, -1.3759722960439218, -1.3508104401024605, 1.0208262718879615, -1.050558252967573]
---------------------------------------------
Fidelity at Step 218: 0.999988349649449
Parameters at Step 218: [0.5858236667875478, 0.025519249618372042, 1.3277427224692333, 1.7743297264414288, -0.0011998850761463797, -0.8386915295261786, -1.3771535054686013, -1.350765659221492, 1.0213253207632875, -1.051067274653289]
---------------------------------------------
Fidelity at Step 219: 0.9999927873362311
Parameters at Step 219: [0.5862885686180744, 0.02446858506824528, 1.328768317162256, 1.7748567588791937, -0.001287037408261083, -0.8388806807238663, -1.3783355403979574, -1.3507124604760232, 1.0218139081740487, -1.051532499293655]
---------------------------------------------
Fidelity at Step 220: 0.9999904472208753
Parameters at Step 220: [0.5868062589518784, 0.023450048412156033, 1.330064725519595, 1.7754166728690355, -0.001396332986240341, -0.8391133113320853, -1.3794656098399214, -1.3506545039230118, 1.0222811084089933, -1.0520065794444358]
---------------------------------------------
Fidelity at Step 221: 0.999989842546006
Parameters at Step 221: [0.5872844947166628, 0.022425904775236594, 1.3309321801533682, 1.7758978858197463, -0.0015924064284546574, -0.8391785376958127, -1.3805132302165977, -1.3505976173625995, 1.0227283344789568, -1.0525785601205644]
---------------------------------------------
Fidelity at Step 222: 0.9999870275056041
Parameters at Step 222: [0.5877537904766912, 0.021392833563420723, 1.3315298501957273, 1.7763328917515049, -0.0018538439441622395, -0.8391290190289682, -1.3814592840869555, -1.3505514496180628, 1.0231558695945016, -1.0532272656902077]
---------------------------------------------
Fidelity at Step 223: 0.9999884661453955
Parameters at Step 223: [0.5884048263614704, 0.020321388108284218, 1.332843939475398, 1.7769053981589935, -0.0021503852953904654, -0.8392136781452616, -1.3822951490721656, -1.3505225289686806, 1.0235581477181113, -1.0539266248019814]
---------------------------------------------
Fidelity at Step 224: 0.9999884532070813
Parameters at Step 224: [0.5889900674029886, 0.01924770428584472, 1.3336934487269372, 1.7774144968686065, -0.002489047082506203, -0.8391561943730307, -1.3830499513386436, -1.3505289113292414, 1.0239430903605118, -1.0546918015277298]
---------------------------------------------
Fidelity at Step 225: 0.9999898475733888
Parameters at Step 225: [0.5895506048504998, 0.018194378508048982, 1.3344672926785306, 1.7779352447027705, -0.0028199249136769576, -0.8390843005297155, -1.3837557963187326, -1.3505831368072339, 1.024316936801018, -1.0554693272819546]
---------------------------------------------
Fidelity at Step 226: 0.9999907428905219
Parameters at Step 226: [0.5901977370756432, 0.01720730735481824, 1.3358847432853092, 1.7785804675311812, -0.0031111741874706956, -0.839195479627143, -1.3844284679651042, -1.3506652385280777, 1.0246693620289369, -1.0561995176813221]
---------------------------------------------
Fidelity at Step 227: 0.9999906102497097
Parameters at Step 227: [0.590844867359659, 0.01627999061901328, 1.3374478668238698, 1.7792678723385187, -0.0033879803093705956, -0.8393568484949371, -1.385075003542551, -1.350777094926472, 1.0249992681104725, -1.0569197423061372]
---------------------------------------------
Fidelity at Step 228: 0.9999910334499799
Parameters at Step 228: [0.591543823070571, 0.015365295171680707, 1.339289895838081, 1.7800321846247285, -0.00368734918107617, -0.8395639503160266, -1.3856726658278233, -1.3509285028095877, 1.0252961632984325, -1.057697135594466]
---------------------------------------------
Fidelity at Step 229: 0.999988409038041
Parameters at Step 229: [0.5922323634726342, 0.01449344407434168, 1.341072365846745, 1.7807940042253934, -0.004020935168399662, -0.8397365382906985, -1.386217667855764, -1.351093930735195, 1.0255610966677864, -1.058518138916406]
---------------------------------------------
Fidelity at Step 230: 0.9999901039286563
Parameters at Step 230: [0.5928132620427989, 0.013598646052713477, 1.3420267384272355, 1.7814540200052944, -0.004481787594141734, -0.8396210033567889, -1.3866913758321575, -1.3513050735492125, 1.0257866556443531, -1.059548334044546]
---------------------------------------------
Fidelity at Step 231: 0.9999915196733226
Parameters at Step 231: [0.5932533362477989, 0.012721532992385855, 1.3421873232196984, 1.782005536688833, -0.004981545274549092, -0.8393000221813757, -1.3871439407418964, -1.3515519569762011, 1.0259807639210963, -1.0606729599407296]
---------------------------------------------
Fidelity at Step 232: 0.9999884483776887
Parameters at Step 232: [0.5936264913568032, 0.011918917506283057, 1.3422479640294946, 1.7825284524150424, -0.005397205500562751, -0.8390286721940902, -1.387612788995174, -1.3518038286523277, 1.02615340247575, -1.0617009680045686]
---------------------------------------------
Fidelity at Step 233: 0.9999893754195205
Parameters at Step 233: [0.5939147481832755, 0.011224914908710732, 1.342500451027886, 1.7830627156054486, -0.005544808905161805, -0.8390145839131689, -1.3881919560065117, -1.3520723348736075, 1.026317765175582, -1.0624413935125638]
---------------------------------------------
Fidelity at Step 234: 0.9999869842920284
Parameters at Step 234: [0.5941250004028764, 0.010646979257375278, 1.343025322096422, 1.783601059347741, -0.0053556782844705235, -0.8393379560831443, -1.3889179381326244, -1.352327019736342, 1.0264676445133947, -1.0628130998079022]
---------------------------------------------
Fidelity at Step 235: 0.9999893059352454
Parameters at Step 235: [0.5942117178606191, 0.010163933474947767, 1.3432180484802942, 1.784028868450274, -0.004968590479311849, -0.8397604447145343, -1.38973655486462, -1.3525316255797117, 1.026605482439269, -1.0629541374178728]
---------------------------------------------
Fidelity at Step 236: 0.9999896434642385
Parameters at Step 236: [0.5943484019952318, 0.009694666764780353, 1.3434811896287295, 1.78436741956631, -0.004450894883286583, -0.8403327344086806, -1.390590184555642, -1.352592763454403, 1.0266775758984168, -1.0629430110938438]
---------------------------------------------
Fidelity at Step 237: 0.9999901073736958
Parameters at Step 237: [0.5945756158662212, 0.009196184216358051, 1.3436384200091407, 1.7845877212992054, -0.003988396553925689, -0.8408677281742165, -1.3913982477537676, -1.3524894743047176, 1.0266746373408528, -1.0629792281037513]
---------------------------------------------
Fidelity at Step 238: 0.9999927691264081
Parameters at Step 238: [0.594703642656405, 0.008692772398369507, 1.3427089405740413, 1.784543553512466, -0.0036730206376294873, -0.8410659516918019, -1.3921709162345033, -1.3522501730040704, 1.0266266948791514, -1.0631680015636504]
---------------------------------------------
Fidelity at Step 239: 0.9999917315733722
Parameters at Step 239: [0.5948067580090348, 0.008218803138169877, 1.3414399845389282, 1.7843878034662675, -0.0034434592965103803, -0.8411340592696075, -1.3929314198469678, -1.3519230900200534, 1.0265522515142744, -1.0634263593280144]
---------------------------------------------
Fidelity at Step 240: 0.9999905424814088
Parameters at Step 240: [0.5949800696134573, 0.007805234926347038, 1.3406572208595793, 1.784297829954337, -0.003224236022222644, -0.8413117231071137, -1.3936942408910464, -1.3515670053333104, 1.0264749143994978, -1.063658562179518]
---------------------------------------------
Fidelity at Step 241: 0.9999885826045259
Parameters at Step 241: [0.5949962815780454, 0.007512344113460753, 1.3394914506452924, 1.7841052004053317, -0.002924095968009788, -0.8414742961671783, -1.3945856033212118, -1.351167893492728, 1.0264217099709094, -1.063775584532843]
---------------------------------------------
Fidelity at Step 242: 0.9999883606888723
Parameters at Step 242: [0.5948245777087302, 0.0073659552955860266, 1.3380419916583655, 1.7838336882096821, -0.0024508502494072974, -0.8417114284919093, -1.3956439248697783, -1.3507514127294864, 1.026415173035054, -1.063672136033635]
---------------------------------------------
Fidelity at Step 243: 0.9999861511179472
Parameters at Step 243: [0.5945540945025368, 0.00734998275974998, 1.3368001062312966, 1.783572564000042, -0.0017556868828075458, -0.8421690874043849, -1.3968405142949893, -1.3503339085157124, 1.0264641452937509, -1.0632846088543018]
---------------------------------------------
Fidelity at Step 244: 0.9999911580594937
Parameters at Step 244: [0.5941391817848185, 0.0074079919451457265, 1.3352179770944113, 1.7832192624843257, -0.0008842166745967229, -0.8426835128816891, -1.3980958687205038, -1.3499101993315223, 1.0265699236460317, -1.0626600448263706]
---------------------------------------------
Fidelity at Step 245: 0.9999898212508886
Parameters at Step 245: [0.593766426167929, 0.007461086818384863, 1.3337732245594254, 1.782883782463429, -3.520519593994305e-05, -0.843206839122646, -1.399266288825347, -1.3494930655712347, 1.0266859103435684, -1.0620202439793718]
---------------------------------------------
Fidelity at Step 246: 0.9999856379788754
Parameters at Step 246: [0.5935274087548836, 0.007438726797372935, 1.3325679928963028, 1.782559960006288, 0.0007574560546551006, -0.8437377803837636, -1.4002772006975712, -1.3490425506377353, 1.0268113796108818, -1.061391656073096]
---------------------------------------------
Fidelity at Step 247: 0.9999883239594803
Parameters at Step 247: [0.5933535967481981, 0.007343159535556303, 1.3311877546850395, 1.7821986098049338, 0.001414719058850634, -0.8441183829929937, -1.4011219020749626, -1.3485820724101096, 1.0269495375873934, -1.0608717265285883]
---------------------------------------------
Fidelity at Step 248: 0.9999900698059424
Parameters at Step 248: [0.5934131491707247, 0.00715239199806403, 1.330460118591667, 1.7819551746946658, 0.0018896686076640776, -0.8444998490167378, -1.4017840398455415, -1.3481123911230533, 1.027081520629623, -1.0605114227579961]
---------------------------------------------
Fidelity at Step 249: 0.9999885539374775
Parameters at Step 249: [0.5935062758562192, 0.006879797546502684, 1.3294961145063628, 1.781681104239747, 0.002145344361266706, -0.8446309996057094, -1.4022846754026792, -1.3476699145790654, 1.027214005311259, -1.0603643602112434]
---------------------------------------------
Fidelity at Step 250: 0.9999895300027376
Parameters at Step 250: [0.5936777625599198, 0.006599806437940118, 1.3289779761618017, 1.7815249122821493, 0.002275726765265315, -0.8447460865153723, -1.402721063135856, -1.347291975458325, 1.0273548112675543, -1.060335920350216]
---------------------------------------------
Fidelity at Step 251: 0.9999897825367936
Parameters at Step 251: [0.593862963547294, 0.006332944207955403, 1.3287970691222821, 1.781478031764532, 0.0023094209749266086, -0.8448313009439179, -1.4031274797944795, -1.347011456821376, 1.0275121600687303, -1.0604008336432516]
---------------------------------------------
Fidelity at Step 252: 0.9999896387188143
Parameters at Step 252: [0.5939058088165182, 0.006097596632468126, 1.3282866949915018, 1.7814292255724553, 0.002266040513625809, -0.8447432600265276, -1.4035341399018701, -1.3468518581022193, 1.0276909463279056, -1.0605455629161815]
---------------------------------------------
Fidelity at Step 253: 0.9999887068802289
Parameters at Step 253: [0.59387067843434, 0.005906120856296461, 1.32777561050331, 1.781422298638998, 0.0022220968178467077, -0.8446309645301731, -1.4039513992891786, -1.3467865416862939, 1.0278852875637112, -1.060682381012993]
---------------------------------------------
Fidelity at Step 254: 0.9999912082647311
Parameters at Step 254: [0.5937380705822191, 0.005765497119177365, 1.3275153758191351, 1.7815164473039649, 0.0022604555415742266, -0.8445948794870505, -1.4044082361672299, -1.3468767773236676, 1.0281166962573705, -1.0607094184704775]
---------------------------------------------
Fidelity at Step 255: 0.9999913261847386
Parameters at Step 255: [0.5936578158955296, 0.005640217289234351, 1.3278276470580708, 1.7817332040436074, 0.002350305575703594, -0.8447120575987057, -1.4048494338234274, -1.3470353050263304, 1.0283499935773348, -1.0606584350011485]
---------------------------------------------
Fidelity at Step 256: 0.9999884429061434
Parameters at Step 256: [0.5937387657258045, 0.005466537421689071, 1.3289761481135578, 1.782106391856814, 0.0024479747093570704, -0.8450064258141284, -1.4051979002680197, -1.3472456516461127, 1.0285697676625802, -1.0605701923627995]
---------------------------------------------
Fidelity at Step 257: 0.9999891831757101
Parameters at Step 257: [0.5937851969295652, 0.005196756021029536, 1.3294245827802798, 1.7823535410337858, 0.0024088866250306104, -0.8450219960253053, -1.4053849083966263, -1.3474686685044446, 1.0287637259980984, -1.0606193114296032]
---------------------------------------------
Fidelity at Step 258: 0.9999883857699693
Parameters at Step 258: [0.5940953819353095, 0.004791092327584531, 1.3305076450871693, 1.7826956874110953, 0.0022010109558538555, -0.8450446134557336, -1.4053616165625418, -1.3476661487748742, 1.0289057839928066, -1.060837114189176]
---------------------------------------------
Fidelity at Step 259: 0.9999904796568865
Parameters at Step 259: [0.5944964245890103, 0.004190556706255601, 1.3309402564751298, 1.7829042001339033, 0.0016739683970465424, -0.8446549383648442, -1.4050618203636571, -1.347832416441113, 1.0289875015471048, -1.0613979396555118]
---------------------------------------------
Fidelity at Step 260: 0.9999895446462769
Parameters at Step 260: [0.5948290412691644, 0.003611728925779746, 1.3310376718637225, 1.7830523350642844, 0.0011174288273581834, -0.844163954881152, -1.4047616223944055, -1.348002212413368, 1.0290679993304543, -1.0619883701766766]
---------------------------------------------
Fidelity at Step 261: 0.9999875391094621
Parameters at Step 261: [0.5951669123789316, 0.0031310396097699476, 1.331789961880425, 1.7833168983733616, 0.000681157510524022, -0.8439073978707629, -1.4045768433706556, -1.3482025491651533, 1.0291692895659659, -1.0624320524464426]
---------------------------------------------
Fidelity at Step 262: 0.9999906306246412
Parameters at Step 262: [0.5952821276050037, 0.002775226252607156, 1.3320799556119916, 1.7834941541314508, 0.0003834410190620781, -0.8436445142949486, -1.404537104631228, -1.3484310545032105, 1.0292994237971727, -1.062717230258624]
---------------------------------------------
Fidelity at Step 263: 0.9999918475423802
Parameters at Step 263: [0.5955068278307881, 0.002537066041621898, 1.333574274178206, 1.783890447565905, 0.00023900989459528507, -0.8437857783409928, -1.4046369850364349, -1.3486683103850958, 1.0294464368228422, -1.0628231123748586]
---------------------------------------------
Fidelity at Step 264: 0.9999912431486708
Parameters at Step 264: [0.5955932148174262, 0.002386430386773306, 1.3347469769816265, 1.7842223336425767, 0.00016586603537737705, -0.8439075571128305, -1.4048322277067897, -1.3489091954001837, 1.0296144174254342, -1.0628521286989452]
---------------------------------------------
Fidelity at Step 265: 0.9999872555933262
Parameters at Step 265: [0.5956772571815188, 0.002257828793900233, 1.33578997508499, 1.7845332412487767, 5.7699104043846676e-05, -0.843988917135868, -1.4050290899242444, -1.3491239263836743, 1.029776966527158, -1.0629292663204126]
---------------------------------------------
Fidelity at Step 266: 0.9999902967524766
Parameters at Step 266: [0.5957507501519791, 0.002068783031136052, 1.3361309650648827, 1.7847050976583787, -0.00020519692950900033, -0.8438051992942112, -1.4051204013078076, -1.3492778991144794, 1.029910772991523, -1.063194799291699]
---------------------------------------------
Fidelity at Step 267: 0.9999900150551528
Parameters at Step 267: [0.5960916717608045, 0.001792095457352826, 1.3372109116920048, 1.7849799192210827, -0.0005592198163297445, -0.8437340950967995, -1.405069258993, -1.349362115887662, 1.0299998148528058, -1.0635698848235604]
---------------------------------------------
Fidelity at Step 268: 0.9999901632430063
Parameters at Step 268: [0.5964938257348139, 0.0014818649446677085, 1.3381974555803442, 1.7852420652228995, -0.0009791586947237767, -0.8436098082259162, -1.4049470696253095, -1.3494147180174358, 1.0300676204942645, -1.0640370491853084]
---------------------------------------------
Fidelity at Step 269: 0.9999903697305814
Parameters at Step 269: [0.5970118277779763, 0.0011219116396615045, 1.3393961902403144, 1.7855517098796023, -0.0014572778116721133, -0.8435008592080591, -1.4047356296715312, -1.34945268329791, 1.030103626036143, -1.064599942972532]
---------------------------------------------
Fidelity at Step 270: 0.9999899367559645
Parameters at Step 270: [0.5975015615937334, 0.0007479805980981278, 1.340286482112039, 1.785845478008108, -0.001971802490065613, -0.8433015328756424, -1.4044857384421614, -1.349519912456144, 1.030121254199464, -1.0652548452467254]
---------------------------------------------
Fidelity at Step 271: 0.9999900678666692
Parameters at Step 271: [0.5977691910068161, 0.0004258803750859548, 1.3403369377028158, 1.7860300864203622, -0.00241999652252484, -0.8429562359241124, -1.4042866622671257, -1.3496564051307558, 1.0301453455775575, -1.0658908588260874]
---------------------------------------------
Fidelity at Step 272: 0.9999874390289892
Parameters at Step 272: [0.5978043725676705, 0.00022655979324939457, 1.3399444553783821, 1.7861742417428905, -0.0026566307691544863, -0.8426747036087735, -1.4042431189145252, -1.3498703592182126, 1.030191903838473, -1.066332331123344]
---------------------------------------------
Fidelity at Step 273: 0.9999852404955787
Parameters at Step 273: [0.5976018832312114, 0.00019732576339516204, 1.3394361789334397, 1.7863019114132654, -0.002520678451761485, -0.84265540315487, -1.404425981261165, -1.3501473692207824, 1.0302708567239884, -1.0663787728334129]
---------------------------------------------
Fidelity at Step 274: 0.999991202216739
Parameters at Step 274: [0.5970193753976241, 0.00039122226824866557, 1.3383976223932375, 1.7862643127485072, -0.0018034291770251108, -0.8429631976270391, -1.404918488619218, -1.350438857017038, 1.0303917093774562, -1.0657749577970645]
---------------------------------------------
Fidelity at Step 275: 0.9999868730098144
Parameters at Step 275: [0.5964002312016697, 0.0006386686899270823, 1.3372686174563626, 1.7861038705095236, -0.0008960319344269613, -0.843421216990226, -1.4054938764832425, -1.3506018283136876, 1.0304883342850093, -1.0649185421353626]
---------------------------------------------
Fidelity at Step 276: 0.9999871021993375
Parameters at Step 276: [0.5959968024125873, 0.0008543240692973162, 1.3365676350952018, 1.78585275692967, 1.8806409563042127e-05, -0.8440244978214125, -1.406039485419777, -1.3505103596775951, 1.030531301231813, -1.0639819437643518]
---------------------------------------------
Fidelity at Step 277: 0.9999881185344416
Parameters at Step 277: [0.5957871447329421, 0.0009511001755770216, 1.3354265535238314, 1.785356367154387, 0.0006376402592192541, -0.8443413411173908, -1.4064349034510109, -1.350115097279099, 1.0304975653635302, -1.0633064271646853]
---------------------------------------------
Fidelity at Step 278: 0.9999906556766786
Parameters at Step 278: [0.5956367530266714, 0.0009847656361054722, 1.3336692706093733, 1.7846757019204984, 0.0009634945439259536, -0.8443164401766956, -1.4067434739871398, -1.3495484340103792, 1.0304263826065396, -1.062916102860791]
---------------------------------------------
Fidelity at Step 279: 0.9999894442481566
Parameters at Step 279: [0.5955525976517577, 0.001028907117684992, 1.3321084157952712, 1.78401261978164, 0.001164747539451552, -0.8442449016140025, -1.4070533814616768, -1.3489321291968648, 1.0303594700642262, -1.0626365307718746]
---------------------------------------------
Fidelity at Step 280: 0.999989762379587
Parameters at Step 280: [0.5955296466388412, 0.0011362950193639534, 1.331190926052615, 1.783475476175838, 0.001338438536987609, -0.8442954311854832, -1.407434080324662, -1.3483249178832246, 1.0303222541973678, -1.0623605557760991]
---------------------------------------------
Fidelity at Step 281: 0.9999865518287322
Parameters at Step 281: [0.5955545184217964, 0.0013231465148655075, 1.3311625356047956, 1.78314061313324, 0.0015410837043886285, -0.8445508065681544, -1.4078983402622387, -1.3477988954190034, 1.0303360583279675, -1.0620348186940016]
---------------------------------------------
Fidelity at Step 282: 0.9999885658815401
Parameters at Step 282: [0.59561242446879, 0.0015227890917823526, 1.3317033960996465, 1.7829985284014171, 0.0016868396520360817, -0.8448606401220743, -1.4083455925713972, -1.3474149162023767, 1.0303933476864877, -1.0617804808654006]
---------------------------------------------
Fidelity at Step 283: 0.9999886200387803
Parameters at Step 283: [0.5957255375318595, 0.0016497018583908577, 1.3324425509618274, 1.783020973690081, 0.001659143486992421, -0.8450490961123054, -1.40864684396356, -1.347207439471713, 1.030483142356627, -1.0617566525637303]
---------------------------------------------
Fidelity at Step 284: 0.999989658576399
Parameters at Step 284: [0.5959609043974802, 0.0015499456597282543, 1.3331254019793428, 1.7832049882435026, 0.0012922693222957348, -0.8449006011213077, -1.4085712839597364, -1.3472519572169899, 1.030580980547529, -1.0621844020557074]
---------------------------------------------
Fidelity at Step 285: 0.9999894649394983
Parameters at Step 285: [0.5960238507252131, 0.0013189631337330305, 1.332539860511835, 1.783301927654444, 0.0007433866489563661, -0.8442854509445332, -1.4082711162144352, -1.347499484338802, 1.0306917016644268, -1.0628767596250766]
---------------------------------------------
Fidelity at Step 286: 0.9999849933233228
Parameters at Step 286: [0.5960627205395083, 0.0010609043148659702, 1.3322430338627351, 1.7835243539681294, 0.00034286537921595847, -0.8438166939869268, -1.4079035520248064, -1.3479187962707286, 1.0308251683085796, -1.0634261212436664]
---------------------------------------------
Fidelity at Step 287: 0.9999879639253013
Parameters at Step 287: [0.5956899682731177, 0.0009154963726379061, 1.3311635607559102, 1.7836488028761535, 0.0004225855939777025, -0.8434963287575707, -1.407661671454534, -1.3485458555802754, 1.0310382483211102, -1.0634353812611523]
---------------------------------------------
Fidelity at Step 288: 0.9999861925231072
Parameters at Step 288: [0.5951550758551492, 0.0009283519295401207, 1.3304978498841862, 1.7837725159484372, 0.0010039543254848613, -0.8436443722100078, -1.4076374485189134, -1.3491935214571396, 1.0313008332555351, -1.0628211566014651]
---------------------------------------------
Fidelity at Step 289: 0.9999899160547191
Parameters at Step 289: [0.5943319215746347, 0.0011024983291010803, 1.3290715959896324, 1.7835802504401053, 0.001939345423603396, -0.843900829201413, -1.407855162572034, -1.3496824841474095, 1.0315948437820224, -1.0616977278993973]
---------------------------------------------
Fidelity at Step 290: 0.9999899406341382
Parameters at Step 290: [0.593879437917849, 0.0013024934116531413, 1.3292359330494528, 1.7834789968444222, 0.0028823150174223234, -0.8445658740971359, -1.408146953358147, -1.349884604304381, 1.0318369775129694, -1.060448351856046]
---------------------------------------------
Fidelity at Step 291: 0.999987701120682
Parameters at Step 291: [0.5938227296665606, 0.001395519180852045, 1.3297453114468043, 1.7831820333246586, 0.0033488197044209502, -0.8449975130724344, -1.408343231895112, -1.3496242833250376, 1.0319721784111542, -1.0595916756870527]
---------------------------------------------
Fidelity at Step 292: 0.9999895591448354
Parameters at Step 292: [0.594030472531836, 0.001275204988229672, 1.3289723092333423, 1.782426923070065, 0.002887692115341309, -0.8444637224643667, -1.4082817262878289, -1.3488816570547606, 1.031993478449861, -1.05965962408314]
---------------------------------------------
Fidelity at Step 293: 0.9999892316742227
Parameters at Step 293: [0.5944107428465928, 0.0011561190186811987, 1.3287859463486562, 1.7817881428685853, 0.002200021375265551, -0.8439060424703603, -1.408222823638352, -1.3480851698187917, 1.0320067372168447, -1.0599710816436352]
---------------------------------------------
Fidelity at Step 294: 0.9999899316350848
Parameters at Step 294: [0.5947992758125891, 0.0011229866430632221, 1.3293308466414726, 1.7813714966253529, 0.001521404070670901, -0.8435042668478548, -1.4082559739400768, -1.3474186267317105, 1.0320785139540976, -1.0602894796597422]
---------------------------------------------
Fidelity at Step 295: 0.9999907505093709
Parameters at Step 295: [0.5951524351616371, 0.0011663624769124455, 1.3305982924157018, 1.7812371317727713, 0.000988489193023588, -0.8433565124020631, -1.4083561337808042, -1.3469875194740446, 1.0322186618691107, -1.0605020314513662]
---------------------------------------------
Fidelity at Step 296: 0.9999881552104773
Parameters at Step 296: [0.5953267513068672, 0.0012511861842933924, 1.3319337927383572, 1.7813510571695326, 0.0006205158363415861, -0.8433069686202627, -1.4084495824717296, -1.3469315332264586, 1.032453125529733, -1.0606305908474725]
---------------------------------------------
Fidelity at Step 297: 0.9999886313248761
Parameters at Step 297: [0.5952913194747004, 0.001267158715709055, 1.332815657793257, 1.7816823291053976, 0.00036993673278882546, -0.8431800027361622, -1.4083585518645125, -1.3473569073207656, 1.0327821979296068, -1.0607704079399491]
---------------------------------------------
Fidelity at Step 298: 0.999989473771392
Parameters at Step 298: [0.595385647586571, 0.0011875072416154175, 1.3342622514353542, 1.7822505911816997, 0.00014488435321944392, -0.8431903850464094, -1.408104846362179, -1.34797102144586, 1.0331094484959207, -1.0609579558064077]
---------------------------------------------
Fidelity at Step 299: 0.9999884925835485
Parameters at Step 299: [0.595518478464883, 0.0009458696659425904, 1.3352816336854778, 1.7828634413147242, -0.0001669195299291341, -0.8430304233164108, -1.4076043914415428, -1.34873729802631, 1.0334059085675407, -1.0613246722884933]
---------------------------------------------

With the learned parameters, we construct a visual representation of the Hamiltonian to which they correspond and compare it to the target Hamiltonian, and the initial guessed Hamiltonian:

new_ham_matrix = create_hamiltonian_matrix(
    qubit_number, nx.complete_graph(qubit_number), [qgrnn_params[0:6], qgrnn_params[6:10]]
)

init_ham = create_hamiltonian_matrix(
    qubit_number, nx.complete_graph(qubit_number), [init[0:6], init[6:10]]
)

fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(6, 6))

axes[0].matshow(ham_matrix, vmin=-7, vmax=7, cmap='hot')
axes[0].set_title("Target Hamiltonian", y=1.13)

axes[1].matshow(init_ham, vmin=-7, vmax=7, cmap='hot')
axes[1].set_title("Initial Guessed Hamiltonian", y=1.13)

axes[2].matshow(new_ham_matrix, vmin=-7, vmax=7, cmap='hot')
axes[2].set_title("Learned Hamiltonian", y=1.13)

plt.subplots_adjust(wspace=0.3, hspace=0.3)
plt.show()
../_images/hamiltonian_comparison.png

These images look very similar, indicating that the QGRNN has done a good job learning the target Hamiltonian.

We can also look at the exact values of the target and learned parameters. Recall how the target interaction graph has \(4\) edges while the complete graph has \(6\). Thus, as the QGRNN converges to the optimal solution, the weights corresponding to edges \((1, 3)\) and \((2, 0)\) in the complete graph should go to \(0\), as this indicates that they have no effect, and effectively do not exist in the learned Hamiltonian.

# We first pick out the weights of edges (1, 3) and (2, 0)
# and then remove them from the list of target parameters

qgrnn_params = list(qgrnn_params)

zero_weights = [qgrnn_params[1], qgrnn_params[4]]

del qgrnn_params[1]
del qgrnn_params[3]

Then, we print all of the weights:

target_params = matrix_params[0] + matrix_params[1]

print(f"Target parameters: {target_params}")
print(f"Learned parameters: {qgrnn_params}")
print(f"Non-Existing Edge Parameters: {zero_weights}")

Out:

Target parameters: [0.56, 1.24, 1.67, -0.79, -1.44, -1.43, 1.18, -0.93]
Learned parameters: [0.5958460805811168, 1.33650946569539, 1.7835767870051, -0.8428389553658877, -1.4068803387384392, -1.3495612547209628, 1.03363436004853, -1.0618808945949776]
Non-Existing Edge Parameters: [0.000540998047389223, -0.0005928152653526596]

The weights of edges \((1, 3)\) and \((2, 0)\) are very close to \(0\), indicating we have learned the cycle graph from the complete graph. In addition, the remaining learned weights are fairly close to those of the target Hamiltonian. Thus, the QGRNN is functioning properly, and has learned the target Ising Hamiltonian to a high degree of accuracy!

References

  1. Verdon, G., McCourt, T., Luzhnica, E., Singh, V., Leichenauer, S., & Hidary, J. (2019). Quantum Graph Neural Networks. arXiv preprint arXiv:1909.12264.

Total running time of the script: ( 0 minutes 0.000 seconds)

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