# Programs and Gates¶

Note

If you’re running locally, remember set up the QVM and quilc in server mode before trying to use them: Setting Up Server Mode for PyQuil.

## Introduction¶

Quantum programs are written in Forest using the `Program`

object. This `Program`

abstraction will help us
compose Quil programs.

```
from pyquil import Program
```

Programs are constructed by adding quantum gates to it, which are defined in the `gates`

module. We can import all
standard gates with the following:

```
from pyquil.gates import *
```

Let’s instantiate a `Program`

and add an operation to it. We will act an `X`

gate on qubit 0.

```
p = Program()
p += X(0)
```

All qubits begin in the ground state. This means that if we measure a qubit without applying operations on it, we expect to receive
a 0 result. The `X`

gate will rotate qubit 0 from the ground state to the excited state, so a measurement immediately
after should return a 1 result. More details about gate operations are explained in Introduction to Quantum Computing.

We can print our pyQuil program (`print(p)`

) to see the equivalent Quil representation:

```
X 0
```

This isn’t going to be very useful to us without measurements. In pyQuil 2.0, we have to `DECLARE`

a memory space
to read measurement results, which we call “readout results” and abbreviate as `ro`

. With measurement, our whole program
looks like this:

```
from pyquil import Program
from pyquil.gates import *
p = Program()
ro = p.declare('ro', 'BIT', 1)
p += X(0)
p += MEASURE(0, ro[0])
print(p)
```

```
DECLARE ro BIT[1]
X 0
MEASURE 0 ro[0]
```

We’ve instantiated a program, declared a memory space named `ro`

with one single bit of memory, applied
an `X`

gate on qubit 0, and finally measured qubit 0 into the zeroth index of the memory space named `ro`

.

Awesome! That’s all we need to get results back. Now we can actually see what happens if we run this program on the Quantum Virtual Machine (QVM). We just have to add a few lines to do this.

```
from pyquil import get_qc
...
qc = get_qc('1q-qvm') # You can make any 'nq-qvm' this way for any reasonable 'n'
executable = qc.compile(p)
result = qc.run(executable)
print(result)
```

Congratulations! You just ran your program on the QVM. The returned value should be:

```
[[1]]
```

For more information on what the above result means, and on executing quantum programs on the QVM in general, see The Quantum Computer. The remainder of this section of the docs will be dedicated to constructing programs in detail, an essential part of becoming fluent in quantum programming.

## The Standard Gate Set¶

The following gates methods come standard with Quil and `gates.py`

:

- Pauli gates
`I`

,`X`

,`Y`

,`Z`

- Hadamard gate:
`H`

- Phase gates:
`PHASE(theta)`

,`S`

,`T`

- Controlled phase gates:
`CZ`

,`CPHASE00(alpha)`

,`CPHASE01(alpha)`

,`CPHASE10(alpha)`

,`CPHASE(alpha)`

- Cartesian rotation gates:
`RX(theta)`

,`RY(theta)`

,`RZ(theta)`

- Controlled \(X\) gates:
`CNOT`

,`CCNOT`

- Swap gates:
`SWAP`

,`CSWAP`

,`ISWAP`

,`PSWAP(alpha)`

The parameterized gates take a real or complex floating point number as an argument.

## Declaring Memory¶

Classical memory regions must be explicitly requested and named by a Quil program using the `DECLARE`

directive.
Details about can be found in the migration guide or in `pyquil.quil.Program.declare()`

.

In pyQuil, we declare memory with the `.declare`

method on a `Program`

. Let’s inspect the function signature

```
# pyquil.quil.Program
def declare(self, name, memory_type='BIT', memory_size=1, shared_region=None, offsets=None):
```

and break down each argument:

`name`

is any name you want to give this memory region.`memory_type`

is one of`'REAL'`

,`'BIT'`

,`'OCTET'`

, or`'INTEGER'`

(given as a string). Only`BIT`

and`OCTET`

always have a determined size, which is 1 bit and 8 bits respectively.`memory_size`

is the number of elements of that type to reserve.`shared_region`

and`offsets`

allow you to alias memory regions. For example, you might want to name the third bit in your readout array as`q3_ro`

.`SHARING`

is currently disallowed for our QPUs, so we won’t focus on this here.

Now we can get into an example.

```
from pyquil import Program
p = Program()
ro = p.declare('ro', 'BIT', 16)
theta = p.declare('theta', 'REAL')
```

Warning

`.declare`

cannot be chained, since it doesn’t return a modified `Program`

object.

Notice that the `.declare`

method returns a reference to the memory we’ve just declared. We will need this reference
to make use of these memory spaces again. Let’s see how the Quil is looking so far:

```
DECLARE ro BIT[16]
DECLARE theta REAL[1]
```

That’s all we have to do to declare the memory. Continue to the next section on Measurement to learn more about
using `ro`

to store measured readout results. Check out Parametric Compilation to see how you might use
`theta`

to compile gate parameters dynamically.

## Measurement¶

There are several ways you can handle measurements in your program. We will start with the simplest method – letting
the `QuantumComputer`

abstraction do it for us.

```
from pyquil import Program, get_qc
from pyquil.gates import H, CNOT
# Get our QuantumComputer instance, with a Quantum Virutal Machine (QVM) backend
qc = get_qc("8q-qvm")
# Construct a simple Bell State
p = Program(H(0), CNOT(0, 1))
results = qc.run_and_measure(p, trials=10)
print(results)
```

```
{0: array([1, 1, 0, 1, 0, 0, 1, 1, 0, 1]),
1: array([1, 1, 0, 1, 0, 0, 1, 1, 0, 1]),
2: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
3: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
4: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
5: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
6: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
7: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])}
```

The method `.run_and_measure`

will handle declaring memory for readout results, adding `MEASURE`

instructions for each
qubit in the QVM, telling the QVM how many trials to run, running and returning the measurement results.

You might sometimes want finer grained control. In this case, we’re probably only interested in the results on
qubits 0 and 1, but `.run_and_measure`

returns the results for all eight qubits in the QVM. We can change our program
to be more particular about what we want.

```
from pyquil import Program
from pyquil.gates import *
p = Program()
ro = p.declare('ro', 'BIT', 2)
p += H(0)
p += CNOT(0, 1)
p += MEASURE(0, ro[0])
p += MEASURE(1, ro[1])
```

In the last two lines, we’ve added our `MEASURE`

instructions, saying that we want to store the result of qubit 0
into the 0th bit of `ro`

, and the result of qubit 1 into the 1st bit of `ro`

. The following snippet could be a
useful way to measure many qubits, in particular, on a lattice that doesn’t start at qubit 0 (although you can
use the compiler to re-index your qubits):

```
qubits = [5, 6, 7]
# ...
for i, q in enumerate(qubits):
p += MEASURE(q, ro[i])
```

Note

The QPU can only handle `MEASURE`

final programs. You can’t operate gates after measurements.

### Specifying the number of trials¶

Quantum computing is inherently probabilistic. We often have to repeat the same experiment many times to get the
results we need. Sometimes we expect the results to all be the same, such as when we apply no gates, or only an `X`

gate. When we prepare a superposition state, we expect probabilistic outcomes, such as a 50% probability measuring 0 or 1.

The number of shots (also called trials) is the number of times to execute a program at once. This determines the length of the results that are returned.

As we saw above, the `.run_and_measure`

method of the `QuantumComputer`

object can handle multiple executions of a program.
If you would like more explicit control for representing multi-shot execution, another way to do this is
with `.wrap_in_numshots_loop`

. This puts the number of shots to be run in the representation of the program itself,
as opposed to in the arguments list of the execution method itself. Below, we specify that our program should
be executed 1000 times.

```
p = Program()
... # build up your program here...
p.wrap_in_numshots_loop(1000)
```

Note

Did You Know?

The word “shot” comes from experimental physics where an experiment is performed many times, and each result is called a shot.

## Parametric Compilation¶

Modern quantum algorithms are often parametric, following a hybrid model. In this hybrid model, the program ansatz (template of gates) is fixed, and iteratively updated with new parameters. These new parameters are often determined by an update given by a classical optimizer. Depending on the complexity of the algorithm, problem of interest, and capabilities of the classical optimizer, this loop may need to run many times. In order to efficiently operate within this hybrid model, parametric compilation can be used.

Parametric compilation allows one to compile the program ansatz just once. Making use of declared memory regions, we can load values to the parametric gates at execution time, after compilation. Taking the compiler out of the execution loop for programs like this offers a huge performance improvement compared to compiling the program each time a parameter update is required. (Some more details about this and an example are found here.)

The first step is to build our parametric program, which functions like a template for all the precise programs we will run. Below we create a simple example program to illustrate, which puts the qubit onto the equator of the Bloch Sphere and then rotates it around the Z axis for some variable angle theta before applying another X pulse and measuring.

```
import numpy as np
from pyquil import Program
from pyquil.gates import RX, RZ, MEASURE
qubit = 0
p = Program()
ro = p.declare("ro", "BIT", 1)
theta_ref = p.declare("theta", "REAL")
p += RX(np.pi / 2, qubit)
p += RZ(theta_ref, qubit)
p += RX(-np.pi / 2, qubit)
p += MEASURE(qubit, ro[0])
```

Note

The example program, although simple, is actually more than just a toy example. It is similar to an experiment which measures the qubit frequency.

Notice how `theta`

hasn’t been specified yet. The next steps will have to involve a `QuantumComputer`

or a compiler
implementation. For simplicity, we will demonstrate with a `QuantumComputer`

instance.

```
from pyquil import get_qc
# Get a Quantum Virtual Machine to simulate execution
qc = get_qc("1q-qvm")
executable = qc.compile(p)
```

We are able to compile our program, even with `theta`

still not specified. Now we want to run our program with `theta`

filled in for, say, 200 values between \(0\) and \(2\pi\). We demonstrate this below.

```
# Somewhere to store each list of results
parametric_measurements = []
for theta in np.linspace(0, 2 * np.pi, 200):
# Get the results of the run with the value we want to execute with
bitstrings = qc.run(executable, {'theta': [theta]})
# Store our results
parametric_measurements.append(bitstrings)
```

In the example here, if you called `qc.run(executable)`

and didn’t specify `'theta'`

, the program would apply
`RZ(0, qubit)`

for every execution.

Note

Classical memory defaults to zero. If you don’t specify a value for a declared memory region, it will be zero.

## Gate Modifiers¶

Gate applications in Quil can be preceded by a gate modifier. There are three supported modifiers:
`DAGGER`

, `CONTROLLED`

, and `FORKED`

. The `DAGGER`

modifier represents the dagger of the gate. For instance,

```
DAGGER RX(pi/3) 0
```

would have an equivalent effect to `RX(-pi/3) 0`

.

The `CONTROLLED`

modifier takes a gate and makes it a controlled gate. For instance, one could write the Toffoli gate in any of the three following ways:

```
CCNOT 0 1 2
CONTROLLED CNOT 0 1 2
CONTROLLED CONTROLLED X 0 1 2
```

Note

The letter `C`

in the gate name has no semantic significance in Quil. To make a controlled `Y`

gate, one cannot write `CY`

, but rather one has to write `CONTROLLED Y`

.

The `FORKED`

modifier allows for a parametric gate to be applied, with the specific choice of parameters conditional on a qubit value. For a parametric gate `G`

with k parameters,

```
FORKED G(u1, ..., uk, v1, ..., vk) c q1 ... qn
```

is equivalent to

```
if c == 0:
G(u1, ..., uk) q1 ... qn
else if c == 1:
G(v1, ..., vk) q1 ... qn
```

extended by linearity for general `c`

. Note that the total number of parameters in the forked gate has doubled.

All gates (objects deriving from the `Gate`

class) provide the
methods `Gate.dagger()`

, `Gate.controlled(control_qubit)`

, and `Gate.forked(fork_qubit, alt_params)`

that
can be used to programmatically apply the `DAGGER`

, `CONTROLLED`

, and `FORKED`

modifiers.

For example, to produce the controlled-NOT gate (`CNOT`

) with
control qubit `0`

and target qubit `1`

```
prog = Program(X(1).controlled(0))
```

To produce the doubly-controlled NOT gate (`CCNOT`

) with
control qubits `0`

and `1`

and target qubit `2`

you can stack
the `controlled`

modifier, or simply pass a list of control qubits

```
prog = Program(X(2).controlled(0).controlled(1))
prog = Program(X(2).controlled([0, 1]))
```

You can achieve the oft-used control-off gate (flip the target qubit
`1`

if the control qubit `0`

is zero) with

```
prog = Program(X(0), X(1).controlled(0), X(0))
```

The gate `FORKED RX(pi/2, pi) 0 1`

may be produced by

```
prog = Program(RX(np.pi/2, 1).forked(0, [np.pi]))
```

## Defining New Gates¶

New gates can be easily added inline to Quil programs. All you need is a matrix representation of the gate. For example, below we define a \(\sqrt{X}\) gate.

```
import numpy as np
from pyquil import Program
from pyquil.quil import DefGate
# First we define the new gate from a matrix
sqrt_x = np.array([[ 0.5+0.5j, 0.5-0.5j],
[ 0.5-0.5j, 0.5+0.5j]])
# Get the Quil definition for the new gate
sqrt_x_definition = DefGate("SQRT-X", sqrt_x)
# Get the gate constructor
SQRT_X = sqrt_x_definition.get_constructor()
# Then we can use the new gate
p = Program()
p += sqrt_x_definition
p += SQRT_X(0)
print(p)
```

```
DEFGATE SQRT-X:
0.5+0.5i, 0.5-0.5i
0.5-0.5i, 0.5+0.5i
SQRT-X 0
```

Below we show how we can define \(X_0\otimes \sqrt{X_1}\) as a single gate.

```
# A multi-qubit defgate example
x_gate_matrix = np.array(([0.0, 1.0], [1.0, 0.0]))
sqrt_x = np.array([[ 0.5+0.5j, 0.5-0.5j],
[ 0.5-0.5j, 0.5+0.5j]])
x_sqrt_x = np.kron(x_gate_matrix, sqrt_x)
```

Now we can use this gate in the same way that we used `SQRT_X`

, but we will pass it two arguments
rather than one, since it operates on two qubits.

```
x_sqrt_x_definition = DefGate("X-SQRT-X", x_sqrt_x)
X_SQRT_X = x_sqrt_x_definition.get_constructor()
# Then we can use the new gate
p = Program(x_sqrt_x_definition, X_SQRT_X(0, 1))
```

Tip

To inspect the wavefunction that will result from applying your new gate, you can use
the Wavefunction Simulator
(e.g. `print(WavefunctionSimulator().wavefunction(p))`

).

## Defining Parametric Gates¶

Let’s say we want to have a controlled RX gate. Since RX is a parametric gate, we need a slightly different way of defining it than in the previous section.

```
from pyquil import Program, WavefunctionSimulator
from pyquil.quilatom import Parameter, quil_sin, quil_cos
from pyquil.quilbase import DefGate
import numpy as np
# Define the new gate from a matrix
theta = Parameter('theta')
crx = np.array([
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, quil_cos(theta / 2), -1j * quil_sin(theta / 2)],
[0, 0, -1j * quil_sin(theta / 2), quil_cos(theta / 2)]
])
gate_definition = DefGate('CRX', crx, [theta])
CRX = gate_definition.get_constructor()
# Create our program and use the new parametric gate
p = Program()
p += gate_definition
p += H(0)
p += CRX(np.pi/2)(0, 1)
```

`quil_sin`

and `quil_cos`

work as the regular sines and cosines, but they support the parametrization. Parametrized
functions you can use with pyQuil are: `quil_sin`

, `quil_cos`

, `quil_sqrt`

, `quil_exp`

, and `quil_cis`

.

Tip

To inspect the wavefunction that will result from applying your new gate, you can use
the Wavefunction Simulator
(e.g. `print(WavefunctionSimulator().wavefunction(p))`

).

## Defining Permutation Gates¶

Note

`quilc`

supports permutation gate syntax since version `1.8.0`

. pyQuil introduced support in version `2.8.0`

.

Some gates can be compactly represented as a permutation. For example, `CCNOT`

gate can be represented by the matrix

```
import numpy as np
from pyquil.quilbase import DefGate
ccnot_matrix = np.array([
[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1],
[1, 0, 0, 0, 0, 0, 1, 0]
])
ccnot_gate = DefGate("CCNOT", ccnot_matrix)
# etc
```

It can equivalently be defined by the permutation

```
import numpy as np
from pyquil.quilbase import DefPermutationGate
ccnot_gate = DefPermutationGate("CCNOT", [0, 1, 2, 3, 4, 5, 7, 6])
# etc
```

## Pragmas¶

`PRAGMA`

directives give users more control over how Quil programs are processed or simulated but generally do not
change the semantics of the Quil program itself. As a general rule of thumb, deleting all `PRAGMA`

directives in a Quil
program should leave a valid and semantically equivalent program.

In pyQuil, `PRAGMA`

directives play many roles, such as controlling the behavior of gates in noisy simulations,
or commanding the Quil compiler to perform actions in a certain way. Here, we will cover the basics of two very
common use cases for including a `PRAGMA`

in your program: qubit rewiring and delays. For a more comprehensive
review of what pragmas are and what the compiler supports, check out The Quil Compiler. For more information about
`PRAGMA`

in Quil, see
A Practical Quantum ISA, and
Simulating Quantum Processor Errors.

### Specifying A Qubit Rewiring Scheme¶

Qubit rewiring is one of the most powerful features of the Quil compiler. We are able to write Quil programs which are agnostic to the topology of the chip, and the compiler will intelligently relabel our qubits to give better performance.

When we intend to run a program on the QPU, sometimes we write programs which use specific qubits targeting a specific device topology, perhaps to achieve a high-performance program. Other times, we write programs that are agnostic to the underlying topology, thereby making the programs more portable. Qubit rewiring accommodates both use cases in an automatic way.

Consider the following program.

```
from pyquil import Program
from pyquil.gates import *
p = Program(X(3))
```

We’ve tested this on the QVM, and we’ve reserved a lattice on the QPU which has qubits 4, 5, and 6, but not qubit 3. Rather than rewrite our program for each reservation, we modify our program to tell the compiler to do this for us.

```
from pyquil.quil import Pragma
p = Program(Pragma('INITIAL_REWIRING', ['"GREEDY"']))
p += X(3)
```

Now, when we pass our program through the compiler (such as with `QuantumComputer.compile()`

) we will get native Quil
with the qubit reindexed to one of 4, 5, or 6. If qubit 3 is available, and we don’t want that pulse to be applied to
any other qubit, we would instead use `Pragma('INITIAL_REWIRING', ['"NAIVE"']]`

. Detailed information about the
available options is here.

Note

In general, we assume that the qubits you’re supplying as input are also the ones which you prefer to operate on, and so NAIVE rewiring is the default.

### Asking for a Delay¶

At times, we may want to add a delay in our program. Usually this is associated with qubit characterization. Delays
are not regular gate operations, and they do not affect the abstract semantics of the Quil program, so they’re implemented with a `PRAGMA`

directive.

```
# ...
# qubit index and time in seconds must be defined and provided
# the time argument accepts exponential notation e.g. 200e-9
p += Pragma('DELAY', [qubit], str(time))
```

Warning

These delays currently have effects on the real QPU. They have no effect on QVM’s even when those QVM’s have noise models applied.

Warning

Keep in mind, the program duration is currently capped at 15 seconds, and the length of the program is multiplied by the number of shots. If you have a 1000 shot program, where each shot contains a 100ms delay, you won’t be able to execute it.

## Ways to Construct Programs¶

PyQuil supports a variety of methods for constructing programs however you prefer.
Multiple instructions can be applied at once, and programs can be added together. PyQuil can also produce a
`Program`

by interpreting raw Quil text. You can still use the more pyQuil 1.X style of using
the `.inst`

method to add instruction gates. Thus, the following are all valid programs:

```
# Preferred method
p = Program()
p += X(0)
p += Y(1)
print(p)
# Multiple instructions in declaration
print(Program(X(0), Y(1)))
# A composition of two programs
print(Program(X(0)) + Program(Y(1)))
# Raw Quil with newlines
print(Program("X 0\nY 1"))
# Raw Quil comma separated
print(Program("X 0", "Y 1"))
# Chained inst; less preferred
print(Program().inst(X(0)).inst(Y(1)))
```

All of the above methods will produce the same output:

```
X 0
Y 1
```

The `pyquil.parser`

submodule provides a front-end to other similar parser
functionality.

### Fixing a Mistaken Instruction¶

If an instruction was appended to a program incorrectly, you can pop it off.

```
p = Program(X(0), Y(1))
print(p)
print("We can fix by popping:")
p.pop()
print(p)
```

```
X 0
Y 1
We can fix by popping:
X 0
```

## QPU-allowable Quil¶

Apart from `DECLARE`

and `PRAGMA`

directives, a program must break into the following three regions, each optional:

- A
`RESET`

command. - A sequence of quantum gate applications.
- A sequence of
`MEASURE`

commands.

The only memory that is writeable is the region named `ro`

, and only through `MEASURE`

instructions. All other
memory is read-only.

The keyword `SHARING`

is disallowed.

Compilation is unavailable for invocations of `DEFGATE`

s with parameters read from classical memory.