- Compilation/
Pauli Product Rotations
Pauli Product Rotations
Parametrized gates
We list typical parametrized gates with their PPR decomposition. The cost of a gate is critically dependent on its angle, making it impractical to assign a fixed cost. To actually determine the cost, floating point angles need to be discretized and transformed to the (Clifford + T) gate set.
| Operator | PPR | Comment |
|---|---|---|
| PauliRot | e^{-i \frac{\phi}{2} P} | note the factor -\frac{1}{2} in the exponent |
| RZ | e^{-i \frac{\phi}{2} Z} | same logic for RX and RY |
| PhaseShift | e^{i \frac{\phi}{2}}e^{-i \frac{\phi}{2} Z} | |
| ControlledPhaseShift | e^{i \frac{\phi}{4}}e^{-i \frac{\phi}{4} Z_0} e^{-i \frac{\phi}{4} Z_1} e^{i \frac{\phi}{4} Z_0 Z_1} | |
| CRX | e^{i \frac{\phi}{4} Z_0 X_1} e^{-i \frac{\phi}{4} X_1} | |
| CRY | e^{i \frac{\phi}{4} Z_0 Y_1} e^{-i \frac{\phi}{4} Y_1} | |
| CRZ | e^{i \frac{\phi}{4} Z_0 Z_1}e^{-i \frac{\phi}{4} Z_1} | |
| PSWAP | e^{-i (\frac{\pi}{4} - \frac{\phi}{2})} e^{i (\frac{\pi}{4} - \frac{\phi}{2}) Z_0 Z_1} e^{-i \frac{\pi}{4} X_0 X_1} e^{-i \frac{\pi}{4} Y_0 Y_1} | Note that the XX and YY coefficients are static |
| SingleExcitation | e^{i \frac{\phi}{4}( XY - YX)} | Givens rotation, all generators commute |
| DoubleExcitation | e^{i \frac{\phi}{2^4}(XXXY + XXYX - XYXX + XYYY - YXXX + YXYY - YYXY - YYYX)} | all generators commute |
MultiControlledPhaseShift
Let us denote a controlled PhaseShift gate with multiple controls as a MultiControlledPhaseShift, or, C_{1, 2, .., n-1}(R_\phi). In code, this can be realized as
qp.ctrl(qp.PhaseShift(phi, wires=[n]), control=range(1, n))
For n=3 qubits, the PPR decomposition takes the following form:
Note that we collected generators with the same Pauli weight in a single exponent for the sake of brevity. Formally, this can be written as
where P_Z(\ell, n) is the sum of all possible pure Z Pauli words with Pauli weight \ell on n qubits. We define P_Z(0, n) as the identity, so this PPR term is just a global phase.
This can alternatively be understood in code:
import pennylane as qp
from itertools import combinations
def MultiControlledPhaseShift(phi, n):
qp.exp(1j * phi/(2**n) * I(range(n)))
for i in range(1, n+1):
for ops in combinations([Z(jj) for jj in range(n)], i):
qp.exp(1j * (-1)**i * phi/(2**n) * qp.prod(*ops))
And can be easily verified:
from functools import partial
import numpy as np
n = 4
phis = np.linspace(0, np.pi, 50)
U_decomp = qp.matrix(MultiControlledPhaseShift, wire_order=range(n))
@partial(qp.matrix, wire_order=range(n))
def U(phi, n):
qp.ctrl(qp.PhaseShift(phi, wires=[n-1]), control=range(n-1))
all(np.allclose(U_decomp(phi, n), U(phi, n)) for phi in phis)
# True
PCPhase
The PCPhase gate can be decomposed into individual MultiControlledPhaseShift operations, as detailed here. So together with the above decomposition of MultiControlledPhaseShift, we can decompose any PCPhase gate into PPRs.