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try:
import cirq
except ImportError:
print("installing cirq...")
!pip install --quiet cirq
print("installed cirq.")
import numpy as np
import cirq
from cirq.contrib.svg import SVGCircuit
Set up Random Circuits
We create a set of 10 random, two-qubit circuits which uses SINGLE_QUBIT_GATES to randomize the circuit and SQRT_ISWAP as the entangling gate. We will ultimately truncate each of these circuits according to cycle_depths. Please see the XEB Theory notebook for more details.
exponents = np.linspace(0, 7/4, 8)
exponents
array([0. , 0.25, 0.5 , 0.75, 1. , 1.25, 1.5 , 1.75])
import itertools
SINGLE_QUBIT_GATES = [
cirq.PhasedXZGate(x_exponent=0.5, z_exponent=z, axis_phase_exponent=a)
for a, z in itertools.product(exponents, repeat=2)
]
SINGLE_QUBIT_GATES[:10], '...'
([cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=0.0), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=0.25), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=0.5), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=0.75), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=1.0), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=1.25), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=1.5), cirq.PhasedXZGate(axis_phase_exponent=0.0, x_exponent=0.5, z_exponent=1.75), cirq.PhasedXZGate(axis_phase_exponent=0.25, x_exponent=0.5, z_exponent=0.0), cirq.PhasedXZGate(axis_phase_exponent=0.25, x_exponent=0.5, z_exponent=0.25)], '...')
import cirq_google as cg
from cirq.experiments import random_quantum_circuit_generation as rqcg
q0, q1 = cirq.LineQubit.range(2)
# Make long circuits (which we will truncate)
n_circuits = 10
circuits = [
rqcg.random_rotations_between_two_qubit_circuit(
q0, q1,
depth=100,
two_qubit_op_factory=lambda a, b, _: cirq.SQRT_ISWAP(a, b),
single_qubit_gates=SINGLE_QUBIT_GATES)
for _ in range(n_circuits)
]
# We will truncate to these lengths
max_depth = 100
cycle_depths = np.arange(3, max_depth, 9)
cycle_depths
array([ 3, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93])
Emulate coherent error
We request a \(\sqrt{i\mathrm{SWAP} }\) gate, but the quantum hardware may execute something subtly different. Therefore, we move to a more general 5-parameter two qubit gate, cirq.PhasedFSimGate.
This is the general excitation-preserving two-qubit gate, and the unitary matrix of PhasedFSimGate(θ, ζ, χ, γ, φ) is:
[[1, 0, 0, 0],
[0, exp(-iγ - iζ) cos(θ), -i exp(-iγ + iχ) sin(θ), 0],
[0, -i exp(-iγ - iχ) sin(θ), exp(-iγ + iζ) cos(θ), 0],
[0, 0, 0, exp(-2iγ-iφ)]].
This parametrization follows eq (18) in https://arxiv.org/abs/2010.07965. Please read the docstring for cirq.PhasedFSimGate for more information.
With the following code, we show how SQRT_ISWAP can be written as a specific cirq.PhasedFSimGate.
sqrt_iswap_as_phased_fsim = cirq.PhasedFSimGate.from_fsim_rz(
theta=-np.pi/4, phi=0,
rz_angles_before=(0,0), rz_angles_after=(0,0))
np.testing.assert_allclose(
cirq.unitary(sqrt_iswap_as_phased_fsim),
cirq.unitary(cirq.SQRT_ISWAP),
atol=1e-8
)
We'll also create a perturbed version. Note the \(\pi/16\) phi angle:
perturbed_sqrt_iswap = cirq.PhasedFSimGate.from_fsim_rz(theta=-np.pi/4, phi=np.pi/16,
rz_angles_before=(0,0), rz_angles_after=(0,0))
np.round(cirq.unitary(perturbed_sqrt_iswap), 3)
array([[1. +0.j , 0. +0.j , 0. +0.j , 0. +0.j ],
[0. +0.j , 0.707+0.j , 0. +0.707j, 0. +0.j ],
[0. +0.j , 0. +0.707j, 0.707+0.j , 0. +0.j ],
[0. +0.j , 0. +0.j , 0. +0.j , 0.981-0.195j]])
We'll use this perturbed gate along with the GateSubstitutionNoiseModel to create simulator which has a constant coherent error. Namely, each SQRT_ISWAP will be substituted for our perturbed version.
def _sub_iswap(op):
if op.gate == cirq.SQRT_ISWAP:
return perturbed_sqrt_iswap.on(*op.qubits)
return op
noise = cirq.devices.noise_model.GateSubstitutionNoiseModel(_sub_iswap)
noisy_sim = cirq.DensityMatrixSimulator(noise=noise)
Run the benchmark circuits
We use the function sample_2q_xeb_circuits to execute all of our circuits at the requested cycle_depths.
from cirq.experiments.xeb_sampling import sample_2q_xeb_circuits
sampled_df = sample_2q_xeb_circuits(sampler=noisy_sim, circuits=circuits,
cycle_depths=cycle_depths, repetitions=10_000)
sampled_df.head()
100%|██████████| 117/117 [00:12<00:00, 9.59it/s]
Compute fidelity assuming SQRT_ISWAP
In contrast to the XEB Theory notebook, here we only have added coherent error (not depolarizing). Nevertheless, the random, scrambling nature of the circuits shows circuit fidelity decaying with depth (at least when we assume that we were trying to use a pure SQRT_ISWAP gate)
from cirq.experiments.xeb_fitting import benchmark_2q_xeb_fidelities
fids = benchmark_2q_xeb_fidelities(sampled_df, circuits, cycle_depths)
fids.head()
%matplotlib inline
from matplotlib import pyplot as plt
xx = np.linspace(0, fids['cycle_depth'].max())
plt.plot(xx, (1-5e-3)**(4*xx), label=r'Exponential Reference')
plt.plot(fids['cycle_depth'], fids['fidelity'], 'o-', label='Perturbed fSim')
plt.ylabel('Circuit fidelity')
plt.xlabel('Cycle Depth $d$')
plt.legend(loc='best')
<matplotlib.legend.Legend at 0x7fe21babc040>
Optimize PhasedFSimGate parameters
We know what circuits we requested, and in this simulated example, we know what coherent error has happened. But in a real experiment, there is likely unknown coherent error that you would like to characterize. Therefore, we make the five angles in PhasedFSimGate free parameters and use a classical optimizer to find which set of parameters best describes the data we collected from the noisy simulator (or device, if this was a real experiment).
fids_opt = simulate_2q_xeb_fids(sampled_df, pcircuits, cycle_depths, param_resolver={'theta': -np.pi/4, 'phi': 0.1})
import multiprocessing
pool = multiprocessing.get_context('spawn').Pool()
from cirq.experiments.xeb_fitting import \
parameterize_circuit, characterize_phased_fsim_parameters_with_xeb, SqrtISwapXEBOptions
options = SqrtISwapXEBOptions(
characterize_theta=True,
characterize_phi=True,
characterize_chi=False,
characterize_gamma=False,
characterize_zeta=False
)
p_circuits = [parameterize_circuit(circuit, options) for circuit in circuits]
res = characterize_phased_fsim_parameters_with_xeb(
sampled_df,
p_circuits,
cycle_depths,
options,
pool=pool,
xatol=1e-3,
fatol=1e-3)
Simulating with theta = -0.785 phi = 0 Loss: 0.265 Simulating with theta = -0.685 phi = 0 Loss: 0.607 Simulating with theta = -0.785 phi = 0.1 Loss: 0.0794 Simulating with theta = -0.885 phi = 0.1 Loss: 0.251 Simulating with theta = -0.885 phi = 0.2 Loss: 0.277 Simulating with theta = -0.81 phi = 0.05 Loss: 0.133 Simulating with theta = -0.71 phi = 0.05 Loss: 0.4 Simulating with theta = -0.842 phi = 0.0875 Loss: 0.119 Simulating with theta = -0.817 phi = 0.138 Loss: 0.0326 Simulating with theta = -0.82 phi = 0.181 Loss: 0.0302 Simulating with theta = -0.764 phi = 0.194 Loss: 0.0166 Simulating with theta = -0.724 phi = 0.247 Loss: 0.0695 Simulating with theta = -0.798 phi = 0.275 Loss: 0.0589 Simulating with theta = -0.795 phi = 0.231 Loss: 0.0111 Simulating with theta = -0.739 phi = 0.244 Loss: 0.0359 Simulating with theta = -0.799 phi = 0.197 Loss: 0.00333 Simulating with theta = -0.831 phi = 0.234 Loss: 0.0834 Simulating with theta = -0.78 phi = 0.204 Loss: -0.000941 Simulating with theta = -0.785 phi = 0.17 Loss: 0.00865 Simulating with theta = -0.787 phi = 0.185 Loss: 0.00101 Simulating with theta = -0.768 phi = 0.192 Loss: 0.0115 Simulating with theta = -0.792 phi = 0.196 Loss: -0.000596 Simulating with theta = -0.785 phi = 0.215 Loss: -0.000775 Simulating with theta = -0.773 phi = 0.223 Loss: 0.000963 Simulating with theta = -0.787 phi = 0.202 Loss: -0.00146 Simulating with theta = -0.783 phi = 0.192 Loss: 0.000499 Simulating with theta = -0.784 phi = 0.209 Loss: -0.00146 Simulating with theta = -0.791 phi = 0.207 Loss: -0.000215 Simulating with theta = -0.783 phi = 0.205 Loss: -0.00147 Simulating with theta = -0.786 phi = 0.198 Loss: -0.00129 Simulating with theta = -0.785 phi = 0.206 Loss: -0.00155 Simulating with theta = -0.78 phi = 0.209 Loss: -0.00124 Simulating with theta = -0.785 phi = 0.204 Loss: -0.00155 Simulating with theta = -0.787 phi = 0.205 Loss: -0.00139 Simulating with theta = -0.784 phi = 0.205 Loss: -0.00155 Simulating with theta = -0.786 phi = 0.205 Loss: -0.00151 Simulating with theta = -0.784 phi = 0.205 Loss: -0.00156 Simulating with theta = -0.785 phi = 0.203 Loss: -0.00153 Simulating with theta = -0.785 phi = 0.205 Loss: -0.00156 Simulating with theta = -0.784 phi = 0.206 Loss: -0.00154 Simulating with theta = -0.785 phi = 0.205 Loss: -0.00156
xx = np.linspace(0, fids['cycle_depth'].max())
p_depol = 5e-3 # from above
plt.plot(xx, (1-p_depol)**(4*xx), label=r'Exponential Reference')
plt.axhline(1, color='grey', ls='--')
plt.plot(fids['cycle_depth'], fids['fidelity'], 'o-', label='Perturbed fSim')
plt.plot(res.fidelities_df['cycle_depth'], res.fidelities_df['fidelity'], 'o-', label='Refit fSim')
plt.ylabel('Circuit fidelity')
plt.xlabel('Cycle Depth')
plt.legend(loc='best')
plt.tight_layout()
