# Simulation¶

Cirq comes with a built in Python simulator for testing out small circuits. This simulator can shard its simulation across different processes/threads and so take advantage of multiple cores/CPUs.

Currently the simulator is tailored to the gate set from Google’s Xmon architecture, but the simulator can be used to run general gate sets, assuming that you provide an implementation in terms of this basic gate set.

Here is a simple circuit

import cirq
from cirq import Circuit
from cirq.devices import GridQubit
from cirq.google import ExpWGate, Exp11Gate, XmonMeasurementGate

q0 = GridQubit(0, 0)
q1 = GridQubit(1, 0)

def basic_circuit(meas=True):
sqrt_x = ExpWGate(half_turns=0.5, axis_half_turns=0.0)
cz = Exp11Gate()
yield sqrt_x(q0), sqrt_x(q1)
yield cz(q0, q1)
yield sqrt_x(q0), sqrt_x(q1)
if meas:
yield XmonMeasurementGate(key='q0')(q0), XmonMeasurementGate(key='q1')(q1)

circuit = Circuit()
circuit.append(basic_circuit())

print(circuit)
# prints
# (0, 0): ───X^0.5───@───X^0.5───M('q0')───
#                    │
# (1, 0): ───X^0.5───@───X^0.5───M('q1')───


We can simulate this by creating a cirq.google.Simulator and passing the circuit into its run method:

from cirq.google import XmonSimulator
simulator = XmonSimulator()
result = simulator.run(circuit)

print(result)
# prints something like
# q0=1 q1=1


Run returns an TrialResult. As you can see the result contains the result of any measurements for the simulation run.

The actual measurement results here depend on the seeding numpys random seed generator. (You can set this using numpy.random_seed) Another run, can result in a different set of measurement results:

result = simulator.run(circuit)

print(result)
# prints something like
# q0=1 q1=0


The simulator is designed to mimic what running a program on a quantum computer is actually like. In particular the run methods (run and run_sweep) on the simulator do not give access to the wave function of the quantum computer (since one doesn’t have access to this on the actual quantum hardware). Instead the simulate methods (simulate, simulate_sweep, simulate_moment_steps) should be used if one wants to debug the circuit and get access to the full wave function:

import numpy as np
circuit = Circuit()
circuit.append(basic_circuit(False))
result = simulator.simulate(circuit, qubit_order=[q0, q1])

print(np.around(result.final_state, 3))
# prints
# [-0.5-0.j   0. -0.5j  0. -0.5j -0.5+0.j ]


Note that the simulator uses numpy’s float32 precision (which is complex64 for complex numbers).

## Qubit and Amplitude Ordering¶

The qubit_order argument to the simulator’s run method determines the ordering of some results, such as the amplitudes in the final wave function. The qubit_order argument is optional. When it is omitted, qubits are ordered ascending by their name (i.e. what their __str__ method returns).

The simplest qubit_order value you can provide is a list of the qubits in the desired ordered. Any qubits from the circuit that are not in the list will be ordered using the default __str__ ordering, but come after qubits that are in the list. Be aware that all qubits in the list are included in the simulation, even if they are not operated on by the circuit.

The mapping from the order of the qubits to the order of the amplitudes in the wave function can be tricky to understand. Basically, it is the same as the ordering used by numpy.kron:

outside = [1, 10]
inside = [1, 2]
print(np.kron(outside, inside))
# prints
# [ 1  2 10 20]


More concretely, the k’th amplitude in the wave function will correspond to the k’th case that would be encountered when nesting loops over the possible values of each qubit. The first qubit’s computational basis values are looped over in the outer-most loop, the last qubit’s computational basis values are looped over in the inner-most loop, etc:

i = 0
for first in [0, 1]:
for second in [0, 1]:
print('amps[{}] is for first={}, second={}'.format(i, first, second))
i += 1
# prints
# amps[0] is for first=0, second=0
# amps[1] is for first=0, second=1
# amps[2] is for first=1, second=0
# amps[3] is for first=1, second=1


We can check that this is in fact the ordering with a circuit that flips one qubit out of two:

q_stay = cirq.NamedQubit('q_stay')
q_flip = cirq.NamedQubit('q_flip')
c = cirq.Circuit.from_ops(cirq.X(q_flip))

# first qubit in order flipped
result = simulator.simulate(c, qubit_order=[q_flip, q_stay])
print(abs(result.final_state).round(3))
# prints
# [0. 0. 1. 0.]

# second qubit in order flipped
result = simulator.simulate(c, qubit_order=[q_stay, q_flip])
print(abs(result.final_state).round(3))
# prints
# [0. 1. 0. 0.]


## Stepping through a Circuit¶

Often when debugging it is useful to not just see the end result of a circuit, but to inspect, or even modify, the state of the system at different steps in the circuit. To support this Cirq provides a method to return an iterator over a Moment by Moment simulation. This is the method simulate_moment_steps:

circuit = Circuit()
circuit.append(basic_circuit())
for i, step in enumerate(simulator.simulate_moment_steps(circuit)):
print('state at step %d: %s' % (i, np.around(step.state(), 3)))
# prints something like
# state at step 0: [ 0.5+0.j   0.0+0.5j  0.0+0.5j -0.5+0.j ]
# state at step 1: [ 0.5+0.j   0.0+0.5j  0.0+0.5j  0.5+0.j ]
# state at step 2: [-0.5-0.j  -0.0+0.5j -0.0+0.5j -0.5+0.j ]
# state at step 3: [ 0.+0.j  0.+0.j -0.+1.j  0.+0.j]


The object returned by the moment_steps iterator is a XmonStepResult. This object has the state along with any measurements that occurred during that step (so does not include measurement results from previous Moments). In addition, theXmonStepResult contains set_state() which can be used to set the state. One can pass a valid full state to this method by passing a numpy array. Or alternatively one can pass an integer and then the state will be set lie entirely in the computation basis state for the binary expansion of the passed integer.

## Gate sets¶

The xmon simulator is designed to work with operations that are either a GateOperation applying an XmonGate, a CompositeOperation that decomposes (recursively) to XmonGates, or a 1-qubit or 2-qubit operation with a KnownMatrix. By default the xmon simulator uses an Extension defined in xgate_gate_extensions to try to resolve gates that are not XmonGates to XmonGates.

So if you are using a custom gate, there are multiple options for getting it to work with the simulator:

• Define it directly as an XmonGate.
• Provide a CompositeGate made up of XmonGates.
• Supply an Exentension to the simulator which converts the gate to an XmonGate or to a CompositeGate which itself can be decomposed in XmonGates.

## Parameterized Values and Studies¶

In addition to circuit gates with fixed values, Cirq also supports gates which can have Symbol value (see Gates). These are values that can be resolved at run-time. For simulators these values are resolved by providing a ParamResolver. A ParamResolver provides a map from the Symbol’s name to its assigned value.

from cirq import Symbol, ParamResolver
val = Symbol('x')
rot_w_gate = ExpWGate(half_turns=val)
circuit = Circuit()
circuit.append([rot_w_gate(q0), rot_w_gate(q1)])
for y in range(5):
resolver = ParamResolver({'x': y / 4.0})
result = simulator.simulate(circuit, resolver)
print(np.around(result.final_state, 2))
# prints
# [1.+0.j 0.+0.j 0.+0.j 0.+0.j]
# [ 0.85+0.j    0.  -0.35j  0.  -0.35j -0.15+0.j  ]
# [ 0.5+0.j   0. -0.5j  0. -0.5j -0.5+0.j ]
# [ 0.15+0.j    0.  -0.35j  0.  -0.35j -0.85+0.j  ]
# [ 0.+0.j  0.-0.j  0.-0.j -1.+0.j]


Here we see that the Symbol is used in two gates, and then the resolver provide this value at run time.

Parameterized values are most useful in defining what we call a Study. A Study is a collection of trials, where each trial is a run with a particular set of configurations and which may be run repeatedly. Running a study returns one TrialContext and TrialResult per set of fixed parameter values and repetitions (which are reported as the repetition_id in the TrialContext object). Example:

resolvers = [ParamResolver({'x': y / 2.0}) for y in range(3)]
circuit = Circuit()
circuit.append([rot_w_gate(q0), rot_w_gate(q1)])
circuit.append([XmonMeasurementGate(key='q0')(q0), XmonMeasurementGate(key='q1')(q1)])
results = simulator.run_sweep(program=circuit,
params=resolvers,
repetitions=2)
for result in results:
print(result)
# prints something like
# repetition_id=0 x=0.0 q0=0 q1=0
# repetition_id=1 x=0.0 q0=0 q1=0
# repetition_id=0 x=0.5 q0=0 q1=1
# repetition_id=1 x=0.5 q0=1 q1=1
# repetition_id=0 x=1.0 q0=1 q1=1
# repetition_id=1 x=1.0 q0=1 q1=1


where we see that different repetitions for the case that the qubit has been rotated into a superposition over computational basis states yield different measurement results per run. Also note that we now see the use of the TrialContext returned as the first tuple from run: it contains the param_dict describing what values were actually used in resolving the Symbols.

TODO(dabacon): Describe the iterable of parameterized resolvers supported by Google’s API.

## Simulation Configurations and Options¶

The xmon simulator also contain some extra configuration on the simulate commands. One of these is initial_state.This can be passed the full wave function as a numpy array, or the initial state as the binary expansion of a supplied integer (following the order supplied by the qubits list).

A simulator itself can also be passed Options in it’s constructor. These options define some configuration for how the simulator runs. For the xmon simulator, these include

num_shards: The simulator works by sharding the wave function over this many shards. If this is not a power of two, the smallest power of two less than or equal to this number will be used. The sharding shards on the first log base 2 of this number qubit’s state. When this is not set the simulator will use the number of cpus, which tends to max out the benefit of multi-processing.
min_qubits_before_shard: Sharding and multiprocessing does not really help for very few number of qubits, and in fact can hurt because processes have a fixed (large) cost in Python. This is the minimum number of qubits that are needed before the simulator starts to do sharding. By default this is 10.