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 sometimes 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.
In [2]:
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)
We can simulate this by creating a cirq.Simulator and
passing the circuit into its run method:
In [3]:
from cirq import Simulator
simulator = Simulator()
result = simulator.run(circuit)
print(result)
The method run() returns an TrialResult. 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:
In [4]:
result = simulator.run(circuit)
print(result)
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:
In [4]:
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))
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.
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:
In [5]:
outside = [1, 10]
inside = [1, 2]
print(np.kron(outside, inside))
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.:
In [6]:
i = 0
for first in [0, 1]:
for second in [0, 1]:
print('amps[{}] is for first={}, second={}'.format(i, first, second))
i += 1
We can check that this is in fact the ordering with a circuit that flips one qubit out of two:
In [5]:
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))
In [6]:
# second qubit in order flipped
result = simulator.simulate(c, qubit_order=[q_stay, q_flip])
print(abs(result.final_state).round(3))
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:
In [8]:
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)))
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.
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:
$\rho \rightarrow (1-p) \rho + p X \rho X$
Let us see an example in a simulator:
In [9]:
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'))
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.
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.
In [10]:
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))
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
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).
For instance:
In [11]:
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)
Above 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 Symbol objects.
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:
In [12]:
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'))
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:
In [13]:
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))
We see that we have access to the density matrix at the
end of the simulation via final_density_matrix.
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.
Note: In general, these simulators are optimized for specific use cases. Before choosing a simulator, make sure it supports the behavior that you need!
| Project name | PyPI package | Description |
|---|---|---|
| qsim | qsimcirq | Implements SimulatesAmplitudes and 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 SimulatesAmplitudes. Intended for heavy parallelization across several computers; Cirq users should generally prefer qsim. |
| qFlex | qflexcirq | Implements 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. |