# Gates¶

In OpenQASM we refer to unitary quantum instructions as gates.

## Built-in gates¶

We define a mechanism for parameterizing unitary matrices to define new quantum gates. The parameterization uses a built-in universal gate set of single-qubit gates and a two-qubit entangling gate (CNOT) [BBC+95]. This basis is not an enforced compilation target but a mechanism to define other gates. For many gates of practical interest, there is a circuit representation with a polynomial number of one- and two-qubit gates, giving a more compact representation than requiring the programmer to express the full \(2^n \times 2^n\) matrix. However, a general \(n\)-qubit gate can be defined using an exponential number of these gates.

We now describe this built-in gate set. All of the single-qubit unitary gates are built-in and parameterized as

When `a`

is a quantum
register, the statement `U(θ, ϕ, λ) a;`

means apply `U(θ, ϕ, λ) a[j];`

for each index `j`

into register `a`

. The
values \(\theta\in [0,2\pi)\), \(\phi\in [0,2\pi)\), and
\(\lambda\in
[0,2\pi)\) in this base gate are angles whose precision is implementation
dependent [1]. This specifies any element of \(U(2)\) up to a
global phase. For example `U(π/2, 0, π) q[0];`

, applies a Hadamard gate to qubit `q[0]`

.

New gates are associated to a unitary transformation by defining them as a sequence of built-in or
previously defined gates. For example the `gate`

block

```
gate h q {
U(π/2, 0, π) q;
}
```

defines a new gate called `h`

and associates it to the unitary matrix of the Hadamard gate. Once we have
defined `h`

, we can use it in later `gate`

blocks. The definition does not imply that `h`

is
implemented by an instruction `U(π/2, 0, π)`

on the quantum computer. The implementation is up to
the user and/or compiler, given information about the instructions supported by a particular target.

Controlled gates can be constructed by adding a control modifier to an existing gate. For example,
the NOT gate is given by `X = U(π, 0, π)`

and the block

```
gate CX a, b {
ctrl @ U(π, 0, π) a, b;
}
CX q[1], q[0];
```

defines the gate

that applies a bit-flip `X`

to `q[0]`

if `q[1]`

is one and otherwise applies the identity gate.
The control modifier is described in more detail later.

Throughout the document we use a tensor order with higher index qubits on the left. In this tensor order,
`CX q[0], q[1];`

is represented by the matrix

Given the gate definition we have already given, the statement `CX a, b;`

describes a CNOT gate that
flips the target qubit `b`

if and only if the control qubit `a`

is one. The
arguments cannot refer to the same qubit. For convenience, gates automatically broadcast over registers. If `a`

and `b`

are quantum registers
*with the same size*, the statement `CX a, b;`

means apply `CX a[j], b[j];`

for each index `j`

into
register `a`

. If instead `a`

is a qubit and `b`

is a quantum register, the
statement means apply `CX a, b[j]`

for each index `j`

into register `b`

. Finally, if `a`

is a
quantum register and `b`

is a qubit, the statement means apply `CX a[j], b;`

for each
index `j`

into register `a`

.

From a physical perspective, the gates \(e^{i\gamma}U\) and \(U\) are equivalent although they differ by a global
phase \(e^{i\gamma}\). When we add a control to these gates, however, the global phase becomes a relative phase
that is applied when the control qubit is one. To capture the programmer’s intent, a built-in global phase gate
allows the inclusion of arbitrary global phases on circuits. The instruction `gphase(γ);`

adds a global phase
of \(e^{i\gamma}\) to the scope containing the instruction. For example

```
gate rz(tau) q {
gphase(-tau/2);
U(0, 0, tau) q;
}
ctrl @ rz(π/2) q[1], q[0];
```

constructs the gate

and applies the controlled gate

## Hierarchically defined unitary gates¶

For new gates, we associate them with a corresponding unitary transformation by a sequence of built-in gates. For example, a CPHASE operation is shown schematically in Fig. 3 corresponding OpenQASM code is

```
gate cphase(θ) a, b
{
U(0, 0, θ / 2) a;
CX a, b;
U(0, 0, -θ / 2) b;
CX a, b;
U(0, 0, θ / 2) b;
}
cphase(π / 2) q[0], q[1];
```

Note that this definition does not imply that `cphase`

must be implemented with
this series of gates. Rather, we have specified the unitary
transformation that corresponds to the symbol `cphase`

. The particular
implementation is up to the compiler, given information about the basis
gate set supported by a particular target.

In general, new gates are defined by statements of the form

```
// comment
gate name(params) qargs
{
body
}
```

where the optional parameter list `params`

is a comma-separated list of variable
parameters, and the argument list `qargs`

is a comma-separated list of qubit
arguments. The parameters are identifiers with arbitrary-precision numeric types.
The qubit arguments are identifiers. If there are no
variable parameters, the parentheses are optional. At least one qubit
argument is required. The arguments in `qargs`

cannot be indexed within the body
of the gate definition.

```
// this is ok:
gate g a
{
U(0, 0, 0) a;
}
// this is invalid:
gate g a
{
U(0, 0, 0) a[0];
}
```

Only built-in gate statements, calls to previously defined gates, and
timing directives can appear in `body`

. For example, it is not valid to
declare a classical register in a gate body. Looping constructs over these quantum
statements are valid.

The statements in the body can only refer to the symbols given in the parameter or argument list, and these symbols are scoped only to the subroutine body.

An empty body corresponds to the identity gate.

Gates must be declared before use and
cannot call themselves. The statement `name(params) qargs;`

applies the gate,
and the variable parameters `params`

can have any numeric type.

The gate can be applied to any combination of qubit registers *of the same size*, as shown in the following example.

The quantum circuit given by

```
gate g qb0, qb1, qb2, qb3
{
// body
}
qubit qr0[1];
qubit qr1[2];
qubit qr2[3];
qubit qr3[2];
g qr0[0], qr1, qr2[0], qr3; // ok
g qr0[0], qr2, qr1[0], qr3; // error! qr2 and qr3 differ in size
```

has a second-to-last line that means

```
// FIXME: insert translation of algorithmic block from TeX source.
for j ← 0, 1 do
g qr0[0],qr1[j],qr2[0],qr3[j];
```

We provide this so that user-defined gates can be applied in parallel like the built-in gates.

## Quantum gate modifiers¶

A gate modifier is a keyword that applies to a gate. A modifier \(m\) transforms a gate \(U\) to a new gate \(m(U)\) acting on the same or larger Hilbert space. We include modifiers in OpenQASM both for programming convenience and compiler analysis.

The modifier `ctrl @`

replaces its gate argument \(U\) by a
controlled-\(U\) gate. If the control bit is 0, nothing happens to the target bit.
If the control bit is 1, \(U\) acts on the target bit. Mathematically, the controlled-\(U\)
gate is defined as \(C_U = I \otimes U^c\), where \(c\) is the integer value of the control
bit and \(C_U\) is the controlled-\(U\) gate. The new quantum argument is prepended to the
argument list for the controlled-\(U\) gate. The quantum argument can be a register, and in this
case controlled gate broadcast over it (as it shown in examples above for CNOT gate). The modified
gate does not use any additional scratch space and may require compilation to be executed.

We define a special case, the controlled *global* phase gate, as
\(ctrl @ gphase(a) = U(0, 0, a)\). This is a single qubit gate.

```
// Define a controlled Rz operation using the ctrl gate modifier.
// q1 is control, q2 is target
gate crz(θ) q1, q2 {
ctrl @ rz(θ) q1, q2;
}
```

The modifier `negctrl @`

generates controlled gates with negative polarity, ie conditioned on a
controlled value of 0 rather than 1. Mathematically, the negative controlled-\(U\) gate is
given by \(N_U = I \otimes U^{1-c}\), where \(c\) is the integer value of the control bit
and \(N_U\) is the negative controlled-\(U\) gate.

```
// Define a negative controlled X operation using the negctrl gate modifier.
// q1 is control, q2 is target
gate neg_cx(θ) q1, q2 {
negctrl @ x q1, q2;
}
```

`ctrl`

and `negctrl`

both accept an optional positive integer parameter `n`

, specifying the
number of control arguments (omission means `n=1`

). `n`

must be a compile-time constant. For an `N`

qubit operation,these operations are mathematically defined as

where \(c_1\), \(c_2\), …, \(c_n\) are the integer values of the control bits and \(C^n_U\) are the n-bit controlled-\(U\) and n-bit negative controlled-\(U\) gates, respectively.

```
// A reversible boolean function
// Demonstrates use of ``ctrl(n) @`` and ``negctrl(n) @``
qubit[3] a;
qubit[2] b;
qubit f;
reset f;
ctrl(3) @ x a[1], a[0], a[2], f;
negctrl(3) @ ctrl @ x a[0], b[1], a[2], b[0], f;
negctrl @ ctrl(2) @ negctrl @ x a[0], b[0], a[2], a[1], f;
negctrl(2) @ ctrl @ x b[1], a, b[0], f;
```

The modifier `inv @`

replaces its gate argument \(U\) with its inverse
\(U^\dagger\). This can be computed from gate \(U\) via the following rules

The inverse of any gate \(U=U_m U_{m-1} ... U_1\) can be defined recursively by reversing the order of the gates in its definition and replacing each of those with their inverse \(U^\dagger = U_1^\dagger U_2^\dagger ... U_m^\dagger\).

The inverse of a controlled operation is defined by inverting the control unitary. That is,

`inv @ ctrl @ U = ctrl @ inv @ U`

.The base case is given by replacing

`inv @ U(θ, ϕ, λ)`

by`U(-θ, -λ, -ϕ)`

and`inv @ gphase(a)`

by`gphase(-a)`

.

```
// Define a negative z rotation and the inverse of a positive z rotation
gate rzm(θ) q1 {
inv @ rzp(θ) q1;
}
// Equivalently, this can be written as
gate rzm(θ) q1 {
rzp(-θ) q1;
}
```

The modifier `pow(k) @`

replaces its gate argument \(U\) by its \(k\)th
power \(U^k\) for some positive integer or floating point number \(k\) (not necessarily
constant). In the case that \(k\) is an integer, the gate can be implemented (albeit
inefficiently) by \(k\) repetitions of \(U\) for \(k > 0\) and \(k\)
repetitions of `inv @ U`

for \(k < 0\).

```
// define x as the square of sqrt(x) ``sx`` gate
gate x q1 {
pow(2) @ sx q1;
}
```