Gates

Gates define single-qubit or multi-qubit unitary operations that change the state of a qubit register in a deterministic fashion. The general form of a gate is given by the gate name followed by the (comma-separated list of) qubit operand(s), e.g., X q[0]:

 gate qubit-arguments

Parameterized unitary operations are represented by parameterized gates. The general form of a parameterized gate is given by the gate name followed by its (comma-separated list of) parameter(s) that is enclosed in parentheses, which in turn is followed by the (comma-separated list of) qubit operand(s), e.g., CRk(2) q[0], q[1]:

 gate(parameters) qubit-arguments

Note that the parameters, either single or a list of multiple parameters, appear within parentheses directly following the gate name. We distinguish between the parameters of a parameterized gate, in terms of number literals, and the qubit arguments that a (parameterized) gate acts on.

Grammar for gates

gate:
  identifier parameters_opt qubit-arguments

parameters:
  ( parameter-list )

parameter-list:
  parameter
  parameter-list , parameter

parameter:
  integer-literal
  floating-literal

qubit-arguments:
  qubit-argument-list

qubit-argument-list:
  qubit-argument
  qubit-argument-list , qubit-argument

qubit-argument:
  qubit-variable
  qubit-index

qubit-variable:
  identifier

qubit-index:
  index

A few examples of gates are shown below.

// A single-qubit Hadamard gate
H q[0]

// A two-qubit controlled-Z gate (control: q[0], target: q[1])
CZ q[0], q[1]

// A parameterized single-qubit Rx gate (with π/2 rotation around the x-axis)
Rx(pi/2) q[0]

// A parametrized two-qubit controlled phase-shift gate (control: q[1], target: q[0])
CRk(2) q[1], q[0]

The gates that are supported by the cQASM language are listed in the standard gate set.

Single-gate-multi-qubit (SGMQ) notation

It is possible to pass multiple qubits as an argument to a single-qubit gate, by making use of the single-gate-multi-qubit (SGMQ) notation. The single-qubit gate will then be applied to each qubit, respectively.

Note

SGMQ notation does not imply that the gates are, necessarily, executed in parallel on the target device. SGMQ notation is nothing other than syntactic sugar, whereby a series of instruction statements can be written as one. Moreover, SGMQ notation should not be confused with multiple-qubit gates, e.g., X q[0,1] means X q[0]; X q[1], and does not represent the 2-qubit gate XX q[0], q[1]. Note that the latter 2-qubit gate XX is currently not supported by the cQASM language, see the standard gate set below.

If the name of the qubit register is q, then the following can be passed as an argument to the single-qubit gate:

• the whole qubit register q;

• a slice thereof q[i:j], where \(0 \leq i < j < N\);

• or a list of indices can be passed q[i,], where \(0 \leq i < N\),

with \(N\) the size of the qubit register. The following slicing convention is adopted: a slice q[i:j] includes qubits q[i], q[j], and all qubits in between. The code block below demonstrates some examples.

Example

The whole qubit registerA slice of the qubit registerA list of indices of the qubit register

qubit[5] q

X q  // is semantically equivalent to:
X q[0]; X q[1]; X q[2]; X q[3]; X q[4]

qubit[5] q

X q[1:3]  // is semantically equivalent to:
X q[1]; X q[2]; X q[3]

qubit[5] q

X q[0,2,4]  // is semantically equivalent to:
X q[0]; X q[2]; X q[4] 

In the above examples we have used the semicolon ; to separate statements occurring on the same line.

Standard gate set

Name	Description	Example statement
I	Identity gate	I q[0]
H	Hadamard gate	H q[0]
X	Pauli-X	X q[0]
X90	Rotation around the x-axis of \(\pi/2\)	X90 q[0]
mX90	Rotation around the x-axis of \(-\pi/2\)	mX90 q[0]
Y	Pauli-Y	Y q[0]
Y90	Rotation around the y-axis of \(\pi/2\)	Y90 q[0]
mY90	Rotation around the y-axis of \(-\pi/2\)	mY90 q[0]
Z	Pauli-Z	Z q[0]
S	Phase gate	S q[0]
Sdag	S dagger gate	Sdag q[0]
T	T	T q[0]
Tdag	T dagger gate	Tdag q[0]
Rx	Arbitrary rotation around x-axis	Rx(pi) q[0]
Ry	Arbitrary rotation around y-axis	Ry(pi) q[0]
Rz	Arbitrary rotation around z-axis	Rz(pi) q[0]
CNOT	Controlled-NOT gate	CNOT q[0], q[1]
CZ	Controlled-Z, Controlled-Phase	CZ q[0], q[1]
CR	Controlled phase shift (arbitrary angle)	CR(pi) q[0], q[1]
CRk	Controlled phase shift (\(\pi/2^k\))	CRk(2) q[0], q[1]