Simulation

Cirq comes with built in Python simulators for testing small circuits. The two main types of simulations that Cirq supports are pure state and mixed state. The pure state simulations are supported by cirq.Simulator and the mixed state simulators are supported by cirq.DensityMatrixSimulator.

The names pure state simulator and mixed state simulators refers to the fact that these simulations are for quantum circuits; including unitary, measurements, and noise that keeps the evolution in a pure state or a mixed state. In other words, there are some noisy evolutions that are supported by the pure state simulator as long as they preserve the purity of the state.

Introduction to pure state simulation

Here is a simple circuit

import cirq

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

def basic_circuit(meas=True):
    sqrt_x = cirq.X**0.5
    yield sqrt_x(q0), sqrt_x(q1)
    yield cirq.CZ(q0, q1)
    yield sqrt_x(q0), sqrt_x(q1)
    if meas:
        yield cirq.measure(q0, key='q0'), cirq.measure(q1, key='q1')

circuit = cirq.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.Simulator and passing the circuit into its run method:

from cirq import Simulator
simulator = Simulator()
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 = cirq.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) by default, but that the simulator can take in a a dtype of np.complex128 if higher precision is needed.

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 outermost 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(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 = cirq.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_vector(), 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 StepResult. 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, the StepResult 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.

Monte Carlo simulations of noise

Some noise models can be thought of as randomly applying unitary evolutions with different probabilities. Such noise models are amenable to Monte Carlo simulation. An example of such a noise model is the bit flip channel. This channel randomly applied either does nothing (identity) or applies a Pauli cirq.X gate:

$\rho \rightarrow (1-p) \rho + p X \rho X$

Lets see a use of this in a simulator

q = cirq.NamedQubit('a')
circuit = cirq.Circuit(cirq.bit_flip(p=0.2)(q), cirq.measure(q))
simulator = cirq.Simulator()
result = simulator.run(circuit, repetitions=100)
print(result.histogram(key='a'))
# prints something like
# Counter({1: 17, 0: 83})

As expected, the bit is flipped about 20 percent of the time.

Channels that support this sort of evolution implement the SupportsMixture protocol. Also note that this functionality is currently only supported in the pure state simulator and not in the density state simulator. If the mixed state simulator encounters a mixture, it will treat it as a general channel.

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.

import sympy
rot_w_gate = cirq.X**sympy.Symbol('x')
circuit = cirq.Circuit()
circuit.append([rot_w_gate(q0), rot_w_gate(q1)])
for y in range(5):
    resolver = cirq.ParamResolver({'x': y / 4.0})
    result = simulator.simulate(circuit, resolver)
    print(np.round(result.final_state, 2))
# prints something like
# [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 = [cirq.ParamResolver({'x': y / 2.0}) for y in range(3)]
circuit = cirq.Circuit()
circuit.append([rot_w_gate(q0), rot_w_gate(q1)])
circuit.append([cirq.measure(q0, key='q0'), cirq.measure(q1, key='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.

Mixed state simulations

In addition to pure state simulation, Cirq also supports simulation of mixed states. Even though this simulator is not as efficient as the pure state simulators, they allow for a larger class of noisy circuits to be run as well as keeping track of the simulation’s density matrix. This later fact can allow for more exact simulations (for example the pure state simulator’s Monte Carlo simulation only allows sampling from the density matrix, not explicitly giving the entries of the density matrix like the mixed state simulator can do). Mixed state simulation is supported by the cirq.DensityMatrixSimulator class.

Here is a simple example of simulating a channel using the mixed state simulator

q = cirq.NamedQubit('a')
circuit = cirq.Circuit(cirq.H(q), cirq.amplitude_damp(0.2)(q), cirq.measure(q))
simulator = cirq.DensityMatrixSimulator()
result = simulator.run(circuit, repetitions=100)
print(result.histogram(key='a'))
# prints something like
# Counter({0: 61, 1: 39})

We create a state in an equal superposition of 0 and 1 then apply amplitude damping which takes 1 to 0 with something like a probability of 0.2. We see that instead of about 50 percent of the timing being in 0, about 20 percent of the 1 has been converted into 0, so we end up with total around 60 percent in the 0 state.

Like the pure state simulators, the mixed state simulator supports run and run_sweeps methods. The cirq.DensityMatrixSimulator also supports getting access to the density matrix of the circuit at the end of simulating the circuit, or when stepping through the circuit. These are done by the simulate and simulate_sweep methods, or, for stepping through the circuit, via the simulate_moment_steps method. For example, we can simulate creating an equal superposition followed by an amplitude damping channel with a gamma of 0.2 by

q = cirq.NamedQubit('a')
circuit = cirq.Circuit(cirq.H(q), cirq.amplitude_damp(0.2)(q))
simulator = cirq.DensityMatrixSimulator()
result = simulator.simulate(circuit)
print(np.around(result.final_density_matrix, 3))
# prints
# [[0.6  +0.j 0.447+0.j]
#  [0.447+0.j 0.4  +0.j]]

We see that we have access to the density matrix at the end of the simulation via final_density_matrix.