# Source code for cirq.experiments.cross_entropy_benchmarking

# Copyright 2019 The Cirq Developers
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and

from typing import (Any, Dict, Iterable, List, NamedTuple, Optional, Sequence,
Set, Tuple, Union)
import dataclasses
import numpy as np
import scipy
from matplotlib import pyplot as plt
from cirq import circuits, devices, ops, protocols, sim, work

CrossEntropyPair = NamedTuple('CrossEntropyPair', [('num_cycle', int),
('xeb_fidelity', float)])

@dataclasses.dataclass
class CrossEntropyDepolarizingModel:
"""A depolarizing noise model for cross entropy benchmarking.

The depolarizing channel maps a density matrix ρ as

ρ → p_eff ρ + (1 - p_eff) I / D

where I / D is the maximally mixed state and p_eff is between 0 and 1.
It is used to model the effect of noise in certain quantum processes.
This class models the noise that results from the execution of multiple
layers, or cycles, of a random quantum circuit. In this model, p_eff for
the whole process is separated into a part that is independent of the number
of cycles (representing depolarization from state preparation and
measurement errors), and a part that exhibits exponential decay with the
number of cycles (representing depolarization from circuit execution
errors). So p_eff is modeled as

p_eff = S * p^d

where d is the number of cycles, or depth, S is the part that is independent
of depth, and p describes the exponential decay with depth. This class
stores S and p, as well as possibly the covariance in their estimation from
experimental data.

Attributes:
spam_depolarization: The depolarization constant for state preparation
and measurement, i.e., S in p_eff = S * p^d.
cycle_depolarization: The depolarization constant for circuit execution,
i.e., p in p_eff = S * p^d.
covariance: The estimated covariance in the estimation of
spam_depolarization and cycle_depolarization, in that order.
"""
spam_depolarization: float
cycle_depolarization: float
covariance: Optional[np.ndarray] = None

[docs]@protocols.json_serializable_dataclass(frozen=True)
class CrossEntropyResult:
"""Results from a cross-entropy benchmarking (XEB) experiment.

Attributes:
data: A sequence of NamedTuples, each of which contains two fields:
num_cycle: the circuit depth as the number of cycles, where
a cycle consists of a layer of single-qubit gates followed
by a layer of two-qubit gates.
xeb_fidelity: the XEB fidelity after the given cycle number.
repetitions: The number of circuit repetitions used.
"""
data: List[CrossEntropyPair]
repetitions: int

[docs]    def plot(self, ax: Optional[plt.Axes] = None,
**plot_kwargs: Any) -> plt.Axes:
"""Plots the average XEB fidelity vs the number of cycles.

Args:
ax: the plt.Axes to plot on. If not given, a new figure is created,
plotted on, and shown.
**plot_kwargs: Arguments to be passed to 'plt.Axes.plot'.
Returns:
The plt.Axes containing the plot.
"""
show_plot = not ax
if not ax:
fig, ax = plt.subplots(1, 1, figsize=(8, 8))
num_cycles = [d.num_cycle for d in self.data]
fidelities = [d.xeb_fidelity for d in self.data]
ax.set_ylim([0, 1.1])
ax.plot(num_cycles, fidelities, 'ro-', **plot_kwargs)
ax.set_xlabel('Number of Cycles')
ax.set_ylabel('XEB Fidelity')
if show_plot:
fig.show()
return ax

[docs]    def depolarizing_model(self) -> CrossEntropyDepolarizingModel:
"""Fit a depolarizing error model for a cycle.

Fits an exponential model f = S * p^d, where d is the number of cycles
and f is the cross entropy fidelity for that number of cycles,
using nonlinear least squares.

Returns:
A DepolarizingModel object, which has attributes coefficient
representing the value S, decay_constant representing the value
p, and covariance representing the covariance in the estimation
of S and p in that order.
"""
# Get initial guess by linear least squares with logarithm of model
x = [depth for depth, fidelity in self.data if fidelity > 0]
y = [np.log(fidelity) for _, fidelity in self.data if fidelity > 0]
fit = np.polynomial.polynomial.Polynomial.fit(x, y, 1).convert()
p0 = np.exp(fit.coef)

# Perform nonlinear least squares
x = [depth for depth, _ in self.data]
y = [fidelity for _, fidelity in self.data]

def f(d, S, p):
return S * p**d

params, covariance = scipy.optimize.curve_fit(f, x, y, p0=p0)

return CrossEntropyDepolarizingModel(spam_depolarization=params[0],
cycle_depolarization=params[1],
covariance=covariance)

@classmethod
def _from_json_dict_(cls, data, repetitions, **kwargs):
return cls(data=[CrossEntropyPair(d, f) for d, f in data],
repetitions=repetitions)

def __repr__(self) -> str:
return ('cirq.experiments.CrossEntropyResult('
f'data={[tuple(p) for p in self.data]!r}, '
f'repetitions={self.repetitions!r})')

[docs]def cross_entropy_benchmarking(
sampler: work.Sampler,
qubits: Sequence[ops.Qid],
*,
benchmark_ops: Sequence[ops.Moment] = None,
num_circuits: int = 20,
repetitions: int = 1000,
cycles: Union[int, Iterable[int]] = range(2, 103, 10),
scrambling_gates_per_cycle: List[List[ops.SingleQubitGate]] = None,
simulator: sim.Simulator = None,
) -> CrossEntropyResult:
r"""Cross-entropy benchmarking (XEB) of multiple qubits.

A total of M random circuits are generated, each of which comprises N
layers where N = max('cycles') or 'cycles' if a single value is specified
for the 'cycles' parameter. Every layer contains randomly generated
single-qubit gates applied to each qubit, followed by a set of
user-defined benchmarking operations (e.g. a set of two-qubit gates).

Each circuit (circuit_m) from the M random circuits is further used to
generate a set of circuits {circuit_mn}, where circuit_mn is built from the
first n cycles of circuit_m. n spans all the values in 'cycles'.

For each fixed value n, the experiment performs the following:

1) Experimentally collect a number of bit-strings for each circuit_mn via
projective measurements in the z-basis.

2) Theoretically compute the expected bit-string probabilities
$P^{th, mn}_|...00>$,  $P^{th, mn}_|...01>$, $P^{th, mn}_|...10>$,
$P^{th, mn}_|...11>$ ... at the end of circuit_mn for all m and for all
possible bit-strings in the Hilbert space.

3) Compute an experimental XEB function for each circuit_mn:

$f_{mn}^{meas} = \langle D * P^{th, mn}_q - 1 \rangle$

where D is the number of states in the Hilbert space, $P^{th, mn}_q$ is the
theoretical probability of a bit-string q at the end of circuit_mn, and
$\langle \rangle$ corresponds to the ensemble average over all measured
bit-strings.

Then, take the average of $f_{mn}^{meas}$ over all circuit_mn with fixed
n to obtain:

$f_{n} ^ {meas} = (\sum_m f_{mn}^{meas}) / M$

4) Compute a theoretical XEB function for each circuit_mn:

$f_{mn}^{th} = D \sum_q (P^{th, mn}_q) ** 2 - 1$

where the summation goes over all possible bit-strings q in the Hilbert
space.

Similarly, we then average $f_m^{th}$ over all circuit_mn with fixed n to
obtain:

$f_{n} ^ {th} = (\sum_m f_{mn}^{th}) / M$

5) Calculate the XEB fidelity $\alpha_n$ at fixed n:

$\alpha_n = f_{n} ^ {meas} / f_{n} ^ {th}$

Args:
sampler: The quantum engine or simulator to run the circuits.
qubits: The qubits included in the XEB experiment.
benchmark_ops: A sequence of ops.Moment containing gate operations
between specific qubits which are to be benchmarked for fidelity.
If more than one ops.Moment is specified, the random circuits
will rotate between the ops.Moment's. As an example,
if benchmark_ops = [Moment([ops.CZ(q0, q1), ops.CZ(q2, q3)]),
Moment([ops.CZ(q1, q2)]) where q0, q1, q2 and q3 are instances of
Qid (such as GridQubits), each random circuit will apply CZ gate
between q0 and q1 plus CZ between q2 and q3 for the first cycle,
CZ gate between q1 and q2 for the second cycle, CZ between q0 and
q1 and CZ between q2 and q3 for the third cycle and so on. If
None, the circuits will consist only of single-qubit gates.
num_circuits: The total number of random circuits to be used.
repetitions: The number of measurements for each circuit to estimate
the bit-string probabilities.
cycles: The different numbers of circuit layers in the XEB study.
Could be a single or a collection of values.
scrambling_gates_per_cycle: If None (by default), the single-qubit
gates are chosen from X/2 ($\pi/2$ rotation around the X axis),
Y/2 ($\pi/2$ rotation around the Y axis) and (X + Y)/2 ($\pi/2$
rotation around an axis $\pi/4$ away from the X on the equator of
the Bloch sphere). Otherwise the single-qubit gates for each layer
are chosen from a list of possible choices (each choice is a list
of one or more single-qubit gates).
simulator: A simulator that calculates the bit-string probabilities
of the ideal circuit. By default, this is set to sim.Simulator().

Returns:
A CrossEntropyResult object that stores and plots the result.
"""
simulator = sim.Simulator() if simulator is None else simulator
num_qubits = len(qubits)

if isinstance(cycles, int):
cycle_range = [cycles]
else:
cycle_range = list(cycles)

# These store the measured and simulated bit-string probabilities from
# all trials in two dictionaries. The keys of the dictionaries are the
# numbers of cycles. The values are 2D arrays with each row being the
# probabilities obtained from a single trial.
probs_meas = {
n: np.zeros((num_circuits, 2**num_qubits)) for n in cycle_range
}
probs_exp = {
n: np.zeros((num_circuits, 2**num_qubits)) for n in cycle_range
}

for k in range(num_circuits):

# Generates one random XEB circuit with max(num_cycle_range) cycles.
# Then the first n cycles of the circuit are taken to generate
# shorter circuits with n cycles (n taken from cycles). All of these
# circuits are stored in circuits_k.
circuits_k = _build_xeb_circuits(qubits, cycle_range,
scrambling_gates_per_cycle,
benchmark_ops)

# Run each circuit with the sampler to obtain a collection of
# bit-strings, from which the bit-string probabilities are estimated.
probs_meas_k = _measure_prob_distribution(sampler, repetitions, qubits,
circuits_k)

# Simulate each circuit with the Cirq simulator to obtain the
# wavefunction at the end of each circuit, from which the
# theoretically expected bit-string probabilities are obtained.
probs_exp_k = []  # type: List[np.ndarray]
for circ_k in circuits_k:
res = simulator.simulate(circ_k, qubit_order=qubits)
state_probs = np.abs(np.asarray(res.final_state)  # type: ignore
)**2
probs_exp_k.append(state_probs)

for i, num_cycle in enumerate(cycle_range):
probs_exp[num_cycle][k, :] = probs_exp_k[i]
probs_meas[num_cycle][k, :] = probs_meas_k[i]

fidelity_vals = _xeb_fidelities(probs_exp, probs_meas)
xeb_data = [
CrossEntropyPair(c, k) for (c, k) in zip(cycle_range, fidelity_vals)
]
return CrossEntropyResult(  # type: ignore
data=xeb_data, repetitions=repetitions)

[docs]def build_entangling_layers(qubits: Sequence[devices.GridQubit],
two_qubit_gate: ops.TwoQubitGate
) -> List[ops.Moment]:
"""Builds a sequence of gates that entangle all pairs of qubits on a grid.

The qubits are restricted to be physically on a square grid with distinct
row and column indices (not every node of the grid needs to have a
qubit). To entangle all pairs of qubits, a user-specified two-qubit gate
is applied between each and every pair of qubit that are next to each
other. In general, a total of four sets of parallel operations are needed to
perform all possible two-qubit gates. We proceed as follows:

The first layer applies two-qubit gates to qubits (i, j) and (i, j + 1)
where i is any integer and j is an even integer. The second layer
applies two-qubit gates to qubits (i, j) and (i + 1, j) where i is an even
integer and j is any integer. The third layer applies two-qubit gates
to qubits (i, j) and (i, j + 1) where i is any integer and j is an odd
integer. The fourth layer applies two-qubit gates to qubits (i, j) and
(i + 1, j) where i is an odd integer and j is any integer.

After the layers are built as above, any empty layer is ejected.:

Cycle 1:                            Cycle 2:
q00 ── q01    q02 ── q03            q00    q01    q02    q03
|      |      |      |
q10 ── q11    q12 ── q13            q10    q11    q12    q13

q20 ── q21    q22 ── q23            q20    q21    q22    q23
|      |      |      |
q30 ── q31    q32 ── q33            q30    q31    q32    q33

Cycle 3:                           Cycle 4:
q00    q01 ── q02    q03            q00    q01    q02    q03

q10    q11 ── q12    q13            q10    q11    q12    q13
|      |      |      |
q20    q21 ── q22    q23            q20    q21    q22    q23

q30    q31 ── q32    q33            q30    q31    q32    q33

Args:
qubits: The grid qubits included in the entangling operations.
two_qubit_gate: The two-qubit gate to be applied between all
neighboring pairs of qubits.

Returns:
A list of ops.Moment, with a maximum length of 4. Each ops.Moment
includes two-qubit gates which can be performed at the same time.
"""
interaction_sequence = _default_interaction_sequence(qubits)
return [
ops.Moment([two_qubit_gate(q_a, q_b)
for (q_a, q_b) in pairs])
for pairs in interaction_sequence
]

def _build_xeb_circuits(
qubits: Sequence[ops.Qid],
cycles: Sequence[int],
single_qubit_gates: List[List[ops.SingleQubitGate]] = None,
benchmark_ops: Sequence[ops.Moment] = None,
) -> List[circuits.Circuit]:
if benchmark_ops is not None:
num_d = len(benchmark_ops)
else:
num_d = 0
max_cycles = max(cycles)

if single_qubit_gates is None:
single_rots = _random_half_rotations(qubits, max_cycles)
else:
single_rots = _random_any_gates(qubits, single_qubit_gates, max_cycles)
all_circuits = []  # type: List[circuits.Circuit]
for num_cycles in cycles:
circuit_exp = circuits.Circuit()
for i in range(num_cycles):
circuit_exp.append(single_rots[i])
if benchmark_ops is not None:
for op_set in benchmark_ops[i % num_d]:
circuit_exp.append(op_set)
all_circuits.append(circuit_exp)
return all_circuits

def _measure_prob_distribution(sampler: work.Sampler, repetitions: int,
qubits: Sequence[ops.Qid],
circuit_list: List[circuits.Circuit]
) -> List[np.ndarray]:
all_probs = []  # type: List[np.ndarray]
num_states = 2**len(qubits)
for circuit in circuit_list:
trial_circuit = circuit.copy()
trial_circuit.append(ops.measure(*qubits, key='z'))
res = sampler.run(trial_circuit, repetitions=repetitions)
res_hist = dict(res.histogram(key='z'))
probs = np.zeros(num_states, dtype=float)
for k, v in res_hist.items():
probs[k] = float(v) / float(repetitions)
all_probs.append(probs)
return all_probs

def _xeb_fidelities(ideal_probs: Dict[int, np.ndarray],
actual_probs: Dict[int, np.ndarray]) -> List[float]:
num_cycles = sorted(list(ideal_probs.keys()))
return [
_compute_fidelity(ideal_probs[n], actual_probs[n]) for n in num_cycles
]

def _compute_fidelity(probs_exp: np.ndarray, probs_meas: np.ndarray) -> float:
_, num_states = probs_exp.shape
pp_cross = probs_exp * probs_meas
pp_exp = probs_exp**2
f_meas = np.mean(num_states * np.sum(pp_cross, axis=1) - 1.0)
f_exp = np.mean(num_states * np.sum(pp_exp, axis=1) - 1.0)
return float(f_meas / f_exp)

def _random_half_rotations(qubits: Sequence[ops.Qid],
num_layers: int) -> List[List[ops.OP_TREE]]:
rot_ops = [
ops.X**0.5, ops.Y**0.5,
ops.PhasedXPowGate(phase_exponent=0.25, exponent=0.5)
]
num_qubits = len(qubits)
rand_nums = np.random.choice(3, (num_qubits, num_layers))
single_q_layers = []  # type: List[List[ops.OP_TREE]]
for i in range(num_layers):
single_q_layers.append(
[rot_ops[rand_nums[j, i]](qubits[j]) for j in range(num_qubits)])
return single_q_layers

def _random_any_gates(qubits: Sequence[ops.Qid],
op_list: List[List[ops.SingleQubitGate]],
num_layers: int) -> List[List[ops.OP_TREE]]:
num_ops = len(op_list)
num_qubits = len(qubits)
rand_nums = np.random.choice(num_ops, (num_qubits, num_layers))
single_q_layers = []  # type: List[List[ops.OP_TREE]]
for i in range(num_layers):
rots_i = []  # type: List[ops.OP_TREE]
for j in range(num_qubits):
rots_i.extend([rot(qubits[j]) for rot in op_list[rand_nums[j, i]]])
single_q_layers.append(rots_i)
return single_q_layers

def _default_interaction_sequence(
qubits: Sequence[devices.GridQubit]
) -> List[Set[Tuple[devices.GridQubit, devices.GridQubit]]]:
qubit_dict = {(qubit.row, qubit.col): qubit for qubit in qubits}
qubit_locs = set(qubit_dict)
num_rows = max([q.row for q in qubits]) + 1
num_cols = max([q.col for q in qubits]) + 1

l_s = [set() for _ in range(4)
]  # type: List[Set[Tuple[devices.GridQubit, devices.GridQubit]]]
for i in range(num_rows):
for j in range(num_cols - 1):
if (i, j) in qubit_locs and (i, j + 1) in qubit_locs:
l_s[j % 2 * 2].add((qubit_dict[(i, j)], qubit_dict[(i, j + 1)]))

for i in range(num_rows - 1):
for j in range(num_cols):
if (i, j) in qubit_locs and (i + 1, j) in qubit_locs:
l_s[i % 2 * 2 + 1].add(
(qubit_dict[(i, j)], qubit_dict[(i + 1, j)]))

l_final = []  # type: List[Set[Tuple[devices.GridQubit, devices.GridQubit]]]
for gate_set in l_s:
if len(gate_set) != 0:
l_final.append(gate_set)

return l_final