# 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

$\begin{split}U(\theta,\phi,\lambda) := \left(\begin{array}{cc} \cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i(\phi+\lambda)}\cos(\theta/2) \end{array}\right).\end{split}$

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 . This specifies any element of $$U(2)$$ up to a global phase. For example U(π/2, 0, π) q;, applies a Hadamard gate to qubit q.

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, q;


defines the gate

$\begin{split}\mathrm{CX} := I \oplus X = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)\end{split}$

that applies a bit-flip X to q if q 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, q; is represented by the matrix

$\begin{split}\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right)\end{split}$

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. Fig. 1 The two-qubit controlled-NOT gate is contructed from built-in single-qubit gates and the control modifier. If a and b are qubits, the statement CX a,b; applies a controlled-NOT (CNOT) gate that flips the target qubit b iff the control qubit a is one. If a and b are quantum registers, the statement applies CNOT gates between corresponding qubits of each register. There is a similar meaning when a is a qubit and b is a quantum register and vice versa.¶ Fig. 2 The single-qubit unitary gates are built-in. These gates are parameterized by three real parameters $$\theta$$, $$\phi$$, and $$\lambda$$. If the argument q is a quantum register, the statement applies size(q) gates in parallel to the qubits of the register.¶

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, q;


constructs the gate

$\begin{split}R_z(\tau) = \exp(-i\tau Z/2) = \left(\begin{array}{cc} e^{-i\tau/2} & 0 \\ 0 & e^{i\tau/2} \end{array}\right) = e^{-i\tau/2}\left(\begin{array}{cc} 1 & 0 \\ 0 & e^{i\tau} \end{array}\right)\end{split}$

and applies the controlled gate

$\begin{split}I\otimes R_z(\pi/2) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{-i\tau/2} & 0 \\ 0 & 0 & 0 & e^{i\tau/2} \end{array}\right).\end{split}$

## 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, q; Fig. 3 New gates are defined from previously defined gates. The gates are applied using the statement name(params) qargs; just like the built-in gates. The parentheses are optional if there are no parameters. The gate $${cphase}(\theta)$$ corresponds to the unitary matrix $${diag}(1,1,1,e^{i\theta})$$ up to a global phase.¶

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;
}


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;
qubit qr1;
qubit qr2;
qubit qr3;
g qr0, qr1, qr2, qr3; // ok
g qr0, qr2, qr1, 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,qr1[j],qr2,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

\begin{align}\begin{aligned}C^n_U = I_1 \otimes I_2 ... \otimes I_n \otimes U^{c_1*c_2*...*c_n}\\N^n_U = I_1 \otimes I_2 ... \otimes I_n \otimes U^{1 - c_1*c_2*...*c_n}\end{aligned}\end{align}

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 a;
qubit b;
qubit f;
reset f;
ctrl(3) @ x a, a, a, f;
negctrl(3) @ ctrl @ x a, b, a, b, f;
negctrl @ ctrl(2) @ negctrl @ x a, b, a, a, f;
negctrl(2) @ ctrl @ x b, a, b, 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;
}