"""
Alleviating barren plateaus with local cost functions
=====================================================
.. meta::
:property="og:description": Local cost functions are cost formulations for variational quantum circuits that are more robust to barren plateaus.
:property="og:image": ../demonstrations/local_cost_functions/Cerezo_et_al_local_cost_functions.png
.. related::
tutorial_barren_plateaus Barren plateaus in QNNs
*Author: Thomas Storwick (tstorwick@gmail.com). Posted: 9 Sep 2020. Last updated: 28 Jan 2021.*
Barren Plateaus
---------------
:doc:`Barren plateaus ` are large regions of the cost function's parameter space
where the variance of the gradient is almost 0; or, put another way, the
cost function landscape is flat. This means that a variational circuit
initialized in one of these areas will be untrainable using any gradient-based
algorithm.
In `"Cost-Function-Dependent Barren Plateaus in Shallow Quantum Neural
Networks" `__ [#Cerezo2020]_, Cerezo et al. demonstrate the
idea that the barren plateau
phenomenon can, under some circumstances, be avoided by using cost functions that only have
information from part of the circuit. These *local* cost functions can be
more robust against noise, and may have better-behaved gradients with no
plateaus for shallow circuits.
.. figure:: ../demonstrations/local_cost_functions/Cerezo_et_al_local_cost_functions.png
:align: center
:width: 50%
Taken from Cerezo et al. [#Cerezo2020]_.
Many variational quantum algorithms are constructed to use global cost functions.
Information from the entire measurement is used to analyze the result of the
circuit, and a cost function is calculated from this to quantify the circuit's
performance. A local cost function only considers information from a few qubits,
and attempts to analyze the behavior of the entire circuit from this limited scope.
Cerezo et al. also handily prove that these local cost functions are bounded by the
global ones, i.e., if a global cost function is formulated in the manner described
by Cerezo et al., then the value of its corresponding local cost function will always be
less than or equal to the value of the global cost function.
In this notebook, we investigate the effect of barren plateaus in
variational quantum algorithms, and how they can be mitigated using
local cost functions.
We first need to import the following modules.
"""
import pennylane as qml
from pennylane import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import LinearLocator, FormatStrFormatter
np.random.seed(42)
######################################################################
# Visualizing the problem
# -----------------------
#
# To start, let's look at the task of learning the identity gate
# across multiple qubits. This will help us visualize the problem and get
# a sense of what is happening in the cost landscape.
#
# First we define a number of wires we want to train on. The work by
# Cerezo et al. shows that circuits are trainable under certain regimes, so
# how many qubits we train on will effect our results.
wires = 6
dev = qml.device("default.qubit", wires=wires, shots=10000)
######################################################################
# Next, we want to define our QNodes and our circuit ansatz. For this
# simple example, an ansatz that works well is simply a rotation along X,
# and a rotation along Y, repeated across all the qubits.
#
# We will also define our cost functions here. Since we are trying to
# learn the identity gate, a natural cost function is 1 minus the probability of measuring the
# zero state, denoted here as :math:`1 - p_{|0\rangle}`.
#
# .. math:: C = \langle \psi(\theta) | \left(I - |0\rangle \langle 0|\right) | \psi(\theta) \rangle =1-p_{|0\rangle}
#
# We will apply this across all qubits for our global cost function, i.e.,
#
# .. math:: C_{G} = \langle \psi(\theta) | \left(I - |00 \ldots 0\rangle \langle 00 \ldots 0|\right) | \psi(\theta) \rangle = 1-p_{|00 \ldots 0\rangle}
#
# and for the local cost function, we will sum the individual contributions from each qubit:
#
# .. math:: C_L = \langle \psi(\theta) | \left(I - \frac{1}{n} \sum_j |0\rangle \langle 0|_j\right)|\psi(\theta)\rangle = 1 - \sum_j p_{|0\rangle_j}.
#
# It might be clear to some readers now why this function can perform better.
# By formatting our local cost function in this way, we have essentially divided
# the problem up into multiple single-qubit terms, and summed all
# the results up.
#
# To implement this, we will define a separate QNode for the local cost
# function and the global cost function.
#
#
def global_cost_simple(rotations):
for i in range(wires):
qml.RX(rotations[0][i], wires=i)
qml.RY(rotations[1][i], wires=i)
return qml.probs(wires=range(wires))
def local_cost_simple(rotations):
for i in range(wires):
qml.RX(rotations[0][i], wires=i)
qml.RY(rotations[1][i], wires=i)
return [qml.probs(wires=i) for i in range(wires)]
global_circuit = qml.QNode(global_cost_simple, dev)
local_circuit = qml.QNode(local_cost_simple, dev)
def cost_local(rotations):
return 1 - np.sum(local_circuit(rotations)[:,0])/wires
def cost_global(rotations):
return 1 - global_circuit(rotations)[0]
######################################################################
# To analyze each of the circuits, we provide some random initial
# parameters for each rotation.
#
RX = np.random.uniform(low=-np.pi, high=np.pi)
RY = np.random.uniform(low=-np.pi, high=np.pi)
rotations = [[RX for i in range(wires)], [RY for i in range(wires)]]
######################################################################
# Examining the results:
#
print("Global Cost: {: .7f}".format(cost_global(rotations)))
print("Local Cost: {: .7f}".format(cost_local(rotations)))
print("--- Global Circuit ---")
print(global_circuit.draw())
print("--- Local Circuit")
print(local_circuit.draw())
######################################################################
# With this simple example, we can visualize the cost function, and see
# the barren plateau effect graphically. Although there are :math:`2n` (where :math:`n` is the
# number of qubits) parameters, in order to plot the cost landscape
# we must constrain ourselves. We will consider the case where all X rotations
# have the same value, and all the Y rotations have the same value.
#
# Firstly, we look at the global cost function. When plotting the cost
# function across 6 qubits, much of the cost landscape is flat, and
# difficult to train (even with a circuit depth of only 2!). This effect
# will worsen as the number of qubits increases.
#
def generate_surface(cost_function):
Z = []
Z_assembler = []
X = np.arange(-np.pi, np.pi, 0.25)
Y = np.arange(-np.pi, np.pi, 0.25)
X, Y = np.meshgrid(X, Y)
for x in X[0, :]:
for y in Y[:, 0]:
rotations = [[x for i in range(wires)], [y for i in range(wires)]]
Z_assembler.append(cost_function(rotations))
Z.append(Z_assembler)
Z_assembler = []
Z = np.asarray(Z)
return Z
def plot_surface(surface):
X = np.arange(-np.pi, np.pi, 0.25)
Y = np.arange(-np.pi, np.pi, 0.25)
X, Y = np.meshgrid(X, Y)
fig = plt.figure()
ax = fig.add_subplot(111, projection="3d")
surf = ax.plot_surface(X, Y, surface, cmap="viridis", linewidth=0, antialiased=False)
ax.set_zlim(0, 1)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter("%.02f"))
plt.show()
global_surface = generate_surface(cost_global)
plot_surface(global_surface)
######################################################################
# However, when we change to the local cost function, the cost landscape
# becomes much more trainable as the size of the barren plateau decreases.
#
#
local_surface = generate_surface(cost_local)
plot_surface(local_surface)
######################################################################
# Those are some nice pictures, but how do they reflect actual
# trainability? Let us try training both the local and global cost
# functions.
# To simplify this model, let's modify our cost function from
#
# .. math:: C_{L} = 1-\sum p_{|0\rangle},
#
# where we sum the marginal probabilities of each qubit, to
#
# .. math:: C_{L} = 1-p_{|0\rangle},
#
# where we only consider the probability of a single qubit to be in the 0 state.
#
# While we're at it, let us make our ansatz a little more like one we would encounter while
# trying to solve a VQE problem, and add entanglement.
def global_cost_simple(rotations):
for i in range(wires):
qml.RX(rotations[0][i], wires=i)
qml.RY(rotations[1][i], wires=i)
qml.broadcast(qml.CNOT, wires=range(wires), pattern="chain")
return qml.probs(wires=range(wires))
def local_cost_simple(rotations):
for i in range(wires):
qml.RX(rotations[0][i], wires=i)
qml.RY(rotations[1][i], wires=i)
qml.broadcast(qml.CNOT, wires=range(wires), pattern="chain")
return qml.probs(wires=[0])
global_circuit = qml.QNode(global_cost_simple, dev)
local_circuit = qml.QNode(local_cost_simple, dev)
def cost_local(rotations):
return 1 - local_circuit(rotations)[0]
def cost_global(rotations):
return 1 - global_circuit(rotations)[0]
######################################################################
# Of course, now that we've changed both our cost function and our circuit,
# we will need to scan the cost landscape again.
global_surface = generate_surface(cost_global)
plot_surface(global_surface)
local_surface = generate_surface(cost_local)
plot_surface(local_surface)
######################################################################
# It seems our changes didn't significantly alter the overall cost landscape.
# This probably isn't a general trend, but it is a nice surprise.
# Now, let us get back to training the local and global cost functions.
# Because we have a visualization of the total cost landscape,
# let's pick a point to exaggerate the problem. One of the worst points in the
# landscape is :math:`(\pi,0)` as it is in the middle of the plateau, so let's use that.
rotations = np.array([[3.] * len(range(wires)), [0.] * len(range(wires))])
opt = qml.GradientDescentOptimizer(stepsize=0.2)
steps = 100
params_global = rotations
for i in range(steps):
# update the circuit parameters
params_global = opt.step(cost_global, params_global)
if (i + 1) % 5 == 0:
print("Cost after step {:5d}: {: .7f}".format(i + 1, cost_global(params_global)))
if cost_global(params_global) < 0.1:
break
print(global_circuit.draw())
######################################################################
# After 100 steps, the cost function is still exactly 1. Clearly we are in
# an "untrainable" area. Now, let us limit ourselves to the local cost
# function and see how it performs.
#
rotations = np.array([[3. for i in range(wires)], [0. for i in range(wires)]])
opt = qml.GradientDescentOptimizer(stepsize=0.2)
steps = 100
params_local = rotations
for i in range(steps):
# update the circuit parameters
params_local = opt.step(cost_local, params_local)
if (i + 1) % 5 == 0:
print("Cost after step {:5d}: {: .7f}".format(i + 1, cost_local(params_local)))
if cost_local(params_local) < 0.05:
break
print(local_circuit.draw())
######################################################################
# It trained! And much faster than the global case. However, we know our
# local cost function is bounded by the global one, but just how much
# have we trained it?
#
cost_global(params_local)
######################################################################
# Interestingly, the global cost function is still 1. If we trained the
# local cost function, why hasn't the global cost function changed?
#
# The answer is that we have trained the global cost a *little bit*, but
# not enough to see a change with only 10000 shots. To see the effect,
# we'll need to increase the number of shots to an unreasonable amount.
# Instead, making the backend analytic by setting shots to ``None``, gives
# us the exact representation.
#
dev.shots = None
global_circuit = qml.QNode(global_cost_simple, dev)
print(
"Current cost: "
+ str(cost_global(params_local))
+ ".\nInitial cost: "
+ str(cost_global([[3.0 for i in range(wires)], [0 for i in range(wires)]]))
+ ".\nDifference: "
+ str(
cost_global([[3.0 for i in range(wires)], [0 for i in range(wires)]])
- cost_global(params_local)
)
)
######################################################################
# Our circuit has definitely been trained, but not a useful amount. If we
# attempt to use this circuit, it would act the same as if we never trained at all.
# Furthermore, if we now attempt to train the global cost function, we are
# still firmly in the plateau region. In order to fully train the global
# circuit, we will need to increase the locality gradually as we train.
#
def tunable_cost_simple(rotations):
for i in range(wires):
qml.RX(rotations[0][i], wires=i)
qml.RY(rotations[1][i], wires=i)
qml.broadcast(qml.CNOT, wires=range(wires), pattern="chain")
return qml.probs(range(locality))
def cost_tunable(rotations):
return 1 - tunable_circuit(rotations)[0]
dev.shots = 10000
tunable_circuit = qml.QNode(tunable_cost_simple, dev)
locality = 2
params_tunable = params_local
print(cost_tunable(params_tunable))
print(tunable_circuit.draw())
locality = 2
opt = qml.GradientDescentOptimizer(stepsize=0.1)
steps = 600
for i in range(steps):
# update the circuit parameters
params_tunable = opt.step(cost_tunable, params_tunable)
runCost = cost_tunable(params_tunable)
if (i + 1) % 10 == 0:
print(
"Cost after step {:5d}: {: .7f}".format(i + 1, runCost)
+ ". Locality: "
+ str(locality)
)
if runCost < 0.1 and locality < wires:
print("---Switching Locality---")
locality += 1
continue
elif runCost < 0.1 and locality >= wires:
break
print(tunable_circuit.draw())
######################################################################
# A more thorough analysis
# ------------------------
#
# Now the circuit can be trained, even though we started from a place
# where the global function has a barren plateau. The significance of this
# is that we can now train from every starting location in this example.
#
# But, how often does this problem occur? If we wanted to train this
# circuit from a random starting point, how often would we be stuck in a
# plateau? To investigate this, let's attempt to train the global cost
# function using random starting positions and count how many times we run
# into a barren plateau.
#
# Let's use a number of qubits we are more likely to use in a real variational
# circuit: n=10. We will say that after
# 400 steps, any run with a cost function of less than 0.9 (chosen
# arbitrarily) will probably be trainable given more time. Any run with a
# greater cost function will probably be in a plateau.
#
# This may take up to 15 minutes.
#
samples = 10
plateau = 0
trained = 0
opt = qml.GradientDescentOptimizer(stepsize=0.2)
steps = 400
wires = 8
dev = qml.device("default.qubit", wires=wires, shots=10000)
global_circuit = qml.QNode(global_cost_simple, dev)
for runs in range(samples):
print("--- New run! ---")
has_been_trained = False
params_global = [
[np.random.uniform(-np.pi, np.pi) for i in range(wires)],
[np.random.uniform(-np.pi, np.pi) for i in range(wires)],
]
for i in range(steps):
# update the circuit parameters
params_global = opt.step(cost_global, params_global)
if (i + 1) % 20 == 0:
print("Cost after step {:5d}: {: .7f}".format(i + 1, cost_global(params_global)))
if cost_global(params_global) < 0.9:
has_been_trained = True
break
if has_been_trained:
trained = trained + 1
else:
plateau = plateau + 1
print("Trained: {:5d}".format(trained))
print("Plateau'd: {:5d}".format(plateau))
samples = 10
plateau = 0
trained = 0
opt = qml.GradientDescentOptimizer(stepsize=0.2)
steps = 400
wires = 8
dev = qml.device("default.qubit", wires=wires, shots=10000)
tunable_circuit = qml.QNode(tunable_cost_simple, dev)
for runs in range(samples):
locality = 1
print("--- New run! ---")
has_been_trained = False
params_tunable = [
[np.random.uniform(-np.pi, np.pi) for i in range(wires)],
[np.random.uniform(-np.pi, np.pi) for i in range(wires)],
]
for i in range(steps):
# update the circuit parameters
params_tunable = opt.step(cost_tunable, params_tunable)
runCost = cost_tunable(params_tunable)
if (i + 1) % 10 == 0:
print(
"Cost after step {:5d}: {: .7f}".format(i + 1, runCost)
+ ". Locality: "
+ str(locality)
)
if runCost < 0.5 and locality < wires:
print("---Switching Locality---")
locality += 1
continue
elif runCost < 0.1 and locality >= wires:
trained = trained + 1
has_been_trained = True
break
if not has_been_trained:
plateau = plateau + 1
print("Trained: {:5d}".format(trained))
print("Plateau'd: {:5d}".format(plateau))
######################################################################
# In the global case, anywhere between 70-80% of starting positions are
# untrainable, a significant number. It is likely that, as the complexity of our
# ansatzâ€”and the number of qubitsâ€”increases, this factor will increase.
#
# We can compare that to our local cost function, where every single area trained,
# and most even trained in less time. While these examples are simple,
# this local-vs-global cost behaviour has been shown to extend to more
# complex problems.
##############################################################################
# References
# ----------
#
# .. [#Cerezo2020]
#
# Cerezo, M., Sone, A., Volkoff, T., Cincio, L., and Coles, P. (2020).
# Cost-Function-Dependent Barren Plateaus in Shallow Quantum Neural Networks.
# `arXiv:2001.00550 `__
#