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    import cirq
except ImportError:
    print("installing cirq...")
    !pip install --quiet cirq
    print("installed cirq.")

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 refer 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.

Some external high-performance simulators also provide an interface to Cirq. These can often provide results faster than Cirq's built-in simulators, especially when working with larger circuits. For details on these tools, see the external simulators section.

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()

(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 <a href=""><code>cirq.Simulator</code></a> and passing the circuit into its run method:

from cirq import Simulator
simulator = Simulator()
result =


The method run() returns a Result. As you can see, the object result contains the result of any measurements for the simulation run.

The actual measurement results depend on the seeding from random seed generator (numpy). You can set this using numpy.random_seed.

Another run can result in a different set of measurement results:

result =


The simulator is designed to mimic what running a program on a quantum computer is actually like.

Accessing the state vector

In particular, the run() methods (run() and run_sweep()) on the simulator do not give access to the state vector 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 state vector:

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

print(np.around(result.final_state_vector, 3))
[0.5+0.j  0. +0.5j 0. +0.5j 0.5+0.j ]

simulate() returns a SimulationTrialResult containing the final state, as seen above. The built-in Cirq simulator returns a StateVectorTrialResult , which includes a number of utilities for analyzing the final state vector.

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 dtype of np.complex128 if higher precision is needed.

Expectation values

For applications that measure expectation values of observables, the simulate_expectation_values() method provides a simple interface for returning just the desired expectation values. This can be more efficient than returning the entire state vector, particularly when handling multiple results at once, or when using an external simulator.

XX_obs = cirq.X(q0) * cirq.X(q1)
ZZ_obs = cirq.Z(q0) * cirq.Z(q1)
ev_list = simulator.simulate_expectation_values(
    observables=[XX_obs, ZZ_obs],
[(1+0j), 0j]

simulate_expectation_values() returns a list of expectation values, one for each observable provided.

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))
[ 1  2 10 20]

More concretely, 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
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])
[0. 0. 1. 0.]
# second qubit in order flipped
result = simulator.simulate(c, qubit_order=[q_stay, q_flip])
[0. 1. 0. 0.]

Stepping through a circuit

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 method is named simulate_moment_steps:

circuit = cirq.Circuit()
for i, step in enumerate(simulator.simulate_moment_steps(circuit)):
    print('state at step %d: %s' % (i, np.around(step.state_vector(), 3)))
state at step 0: [0. +0.5j 0.5+0.j  0.5+0.j  0. -0.5j]
state at step 1: [0. +0.5j 0.5+0.j  0.5+0.j  0. +0.5j]
state at step 2: [0.5+0.j  0. +0.5j 0. +0.5j 0.5+0.j ]
state at step 3: [1.+0.j 0.+0.j 0.+0.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_vector(), 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 to lie entirely in the computation basis state for the binary expansion of the passed integer.

Alternate stepping behavior

For simulators that do not support simulate_moment_steps, it is possible to replicate this behavior by splitting the circuit into "chunks" and passing the results of each chunk as the initial state for the next chunk:

chunks = [cirq.Circuit(moment) for moment in basic_circuit()]
next_state = 0  # represents the all-zero state
for i, chunk in enumerate(chunks):
    result = simulator.simulate(chunk, initial_state=next_state)
    next_state = result.final_state_vector
    print(f'state at step {i}: {np.around(next_state, 3)}')
state at step 0: [0. +0.5j 0.5+0.j  0.5+0.j  0. -0.5j]
state at step 1: [0. +0.5j 0.5+0.j  0.5+0.j  0. +0.5j]
state at step 2: [0.5+0.j  0. +0.5j 0. +0.5j 0.5+0.j ]
state at step 3: [1.+0.j 0.+0.j 0.+0.j 0.+0.j]

The added cost of passing state vectors around like this is nontrivial; for this reason, this workaround should only be used with simulators that do not support simulate_moment_steps.

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 cirq.ParamResolver. A cirq.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_vector, 2))
[1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.6 +0.6j   0.25-0.25j  0.25-0.25j -0.1 -0.1j ]
[0. +0.5j 0.5+0.j  0.5+0.j  0. -0.5j]
[-0.1 +0.1j   0.25+0.25j  0.25+0.25j  0.6 -0.6j ]
[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 provides this value at run time.

Parameterized values are most useful in defining what we call a sweep. A sweep is a sequence of trials, where each trial is a run with a particular set of parameter values.

Running a sweep returns a Result for each set of fixed parameter values and repetitions.

For instance:

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,
for result in results:

Above we see that assigning different values to gate parameters yields different results for each trial in the sweep, and that each trial is repeated repetitions times.

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 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 =, repetitions=100)
Counter({0: 59, 1: 41})

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))
[[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.

External simulators

There are a few high-performance circuit simulators which provide an interface for simulating Cirq Circuits. These projects are listed below, along with their PyPI package name and a description of simulator methods that they support.

Project name PyPI package Description
qsim qsimcirq Implements cirq.SimulatesAmplitudes and cirq.SimulatesFinalState. Recommended for deep circuits with up to 30 qubits (consumes 8GB RAM). Larger circuits are possible, but RAM usage doubles with each additional qubit.
qsimh qsimcirq Implements cirq.SimulatesAmplitudes. Intended for heavy parallelization across several computers; Cirq users should generally prefer qsim.
qFlex qflexcirq Implements cirq.SimulatesAmplitudes. Recommended for shallow / low-entanglement circuits with more than 30 qubits. RAM usage is highly dependent on the number of two-qubit gates in the circuit.
quimb quimb Cirq-to-quimb translation layer provided in contrib/quimb. In addition to circuit simulation, this allows the use of quimb circuit-analysis tools on Cirq circuits.