# 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
`numpy`

s 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 `Symbol`

s.

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`

.