Quantum Gates
This section provides a quick reference on common quantum gates.
I - Identity
Unitary Description and Decomposition Rules
This operator leaves a qubit unchanged, so that
\(|0\rangle \rightarrow |0\rangle\) and \(|1\rangle \rightarrow |1\rangle\).
It is expressed by the matrix \(I = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array} \right)\), but implemented by the absence of any operator.
H - Hadamard
Unitary Description and Decomposition Rules
This operator mixes the measurement basis states, so that
\(|0\rangle \rightarrow \frac{1}{\sqrt{2}}\Big(|0\rangle + |1\rangle\Big)\) and \(|1\rangle \rightarrow \frac{1}{\sqrt{2}}\Big(|0\rangle - |1\rangle\Big)\).
It is given by the matrix \(H = \frac{1}{\sqrt{2}} \left(\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \right)\).
OpenQASM
Gate format:
h <target_qubit>;XASM
Gate format:
H(<target_qubit>);QB Native Transpilation
h(q0)
where q0 is the qubit.
X - Pauli rotation around the x-axis by pi
Unitary Description and Decomposition Rules
This operator performs an X, or NOT operation on a qubit, turning
\(|0\rangle \rightarrow |1\rangle\) and \(|1\rangle \rightarrow |0\rangle\).
It is given by the matrix \(X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right)\).
OpenQASM
Gate format:
x <target_qubit>;XASM
Gate format: X
(<target_qubit>);QB Native Transpilation
x(q0)
where q0 is the qubit.
Y - Pauli rotation around the y-axis by pi
Unitary Description and Decomposition Rules
This operator performs a Y operation on a qubit, turning
\(|0\rangle \rightarrow -i|1\rangle\) and \(|1\rangle \rightarrow i\|0\rangle\).
It is given by the matrix \(Y = \left(\begin{array}{cc} 0 & -i \\ i & 0\end{array} \right)\)
OpenQASM
Gate format:
y <target_qubit>;XASM
Gate format: Y
(<target_qubit>);QB Native Transpilation
y(q0)
where q0 is the qubit.
Z - Pauli rotation around the z-axis by pi
Unitary Description and Decomposition Rules
This operator performs a Z operation on a qubit, turning
\(|0\rangle \rightarrow |0\rangle\) and \(|1\rangle \rightarrow |1\rangle\).
It is given by the matrix \(Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)\)
OpenQASM
Gate format:
z <target_qubit>;XASM
Gate format:
Z(<target_qubit>);QB Native Transpilation
z(q0)
where q0 is the qubit.
CNOT - controlled-X
Unitary Description and Decomposition Rules
For a given control qubit \(q_0\) and target qubit \(q_1\) this operator performs a X operation on the \(q_1\) if \(q_0\) equals one, but leaves \(q_1\) unchanged, i.e. the identity operation, if \(q_0\) equals zero. It is given by the matrix
\(CX_{2q_0 + q_1 + 1,2q_0 + q_1 + 1} = \text{diag}(I,X) = \left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array} \right)\)
OpenQASM
Gate format:
cx <control_qubit> , <target_qubit>;where q[0] is the control qubit and q[1] is the target qubit.
XASM
Gate format:
CX(<control_qubit>,<target_qubit>);QB Native Transpilation
cx(q0, q1)
where q0 is the control qubit and q1 is the target qubit.
CY - controlled-Y
Unitary Description and Decomposition Rules
For a given control qubit \(q_0\) and target qubit \(q_1\) this operator performs a Y operation on the \(q_1\) if \(q_0\) equals one, but leaves \(q_1\) unchanged, i.e. the identity operation, if \(q_0\) equals zero. It is given by the matrix
\(CY_{2q_0 + q_1 + 1,2q_0 + q_1 + 1} = \text{diag}(I,Y) = \left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0\end{array} \right)\)
where \(i^2 = -1\).
OpenQASM
Gate format:
cy <control_qubit> , <target_qubit>;XASM
Gate format:
CY(<control_qubit>,<target_qubit>);QB Native Transpilation
cy(q0, q1)
where q0 is the control qubit and q1 is the target qubit.
CZ - controlled-Z
Unitary Description and Decomposition Rules
For a given control qubit \(q_0\) and target qubit \(q_1\) this operator performs a Z operation on the \(q_1\) if \(q_0\) equals one, but leaves \(q_1\) unchanged, i.e. the identity operation, if \(q_0\) equals zero. It is given by the matrix
\(CZ_{2q_0 + q_1 + 1,2q_0 + q_1 + 1} = \text{diag}(I,Z) = \left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\end{array} \right)\)
OpenQASM
Gate format:
cz <control_qubit> , <target_qubit>;XASM
Gate format:
CZ(<control_qubit>,<target_qubit>);QB Native Transpilation
cz(q0, q1)
where q0 is the control qubit and q1 is the target qubit.
\(R_x(\theta)\) - Pauli rotation on X-axis by \(\theta\)
Unitary Description and Decomposition Rules
This operator rotates a qubit around the X-axis of the Bloch sphere by an angle of \(\theta\). It partially interchanges \(|0\rangle\) with \(|1\rangle\) and introduces a relative phase between the two components, so
\(|0\rangle \rightarrow \cos\left(\frac{\theta}{2}\right)|0\rangle -i \sin\left(\frac{\theta}{2}\right) |1\rangle\) and \(|1\rangle \rightarrow -i\sin(\frac{\theta}{2})|0\rangle + \cos(\frac{\theta}{2})|1\rangle\).
It is given by the matrix
\(X(\theta) = I \cos\!\left(\frac{\theta}{2}\right) -i \sigma_X \sin\!\left(\frac{\theta}{2}\right) = \left(\begin{array}{cc} \cos\left(\frac{\theta}{2}\right) & -i\sin\left(\frac{\theta}{2}\right) \\ & \\-i\sin\left(\frac{\theta}{2}\right) & \cos\left(\frac{\theta}{2}\right)\end{array} \right)\),
where \(I\) is the identity matrix and \(\sigma_X\) is the Pauli matrix
\(\sigma_X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)\).
OpenQASM
Gate format:
rx(<theta>) <target_qubit>;where
<theta>is the angle of rotation about the X-axis.XASM
Gate format:
RX(<target_qubit>, <theta>);QB Native Transpilation
rx(q0,_theta)
where q0 is the qubit and _theta is the angle of rotation about the X-axis.
\(R_y(\phi)\) - Pauli rotation on Y-axis by \(\phi\)
Unitary Description and Decomposition Rules
This operator rotates a qubit around the Y-axis of the Bloch sphere by an angle of \(\phi\), partially interchanging \(|0\rangle\) with \(|1\rangle\), so
\(|0\rangle \rightarrow \cos\left(\frac{\phi}{2}\right)|0\rangle - \sin\left(\frac{\phi}{2}\right) |1\rangle\) and \(|1\rangle \rightarrow \sin(\frac{\phi}{2})|0\rangle + \cos(\frac{\phi}{2})|1\rangle\).
It is given by the matrix
\(Y(\phi) = I \cos\!\left(\frac{\phi}{2}\right) -i \sigma_Y \sin\!\left(\frac{\phi}{2}\right) = \left(\begin{array}{cc} \cos\left(\frac{\phi}{2}\right) & -\sin\left(\frac{\phi}{2}\right) \\ & \\ \sin\left(\frac{\phi}{2}\right) & \cos\left(\frac{\phi}{2}\right)\end{array} \right)\),
where \(I\) is the identity matrix and \(\sigma_Y\) is the Pauli matrix
\(\sigma_Y = \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)\).
OpenQASM
Gate format:
ry(<phi>) <target_qubit>;where
<phi>is the angle of rotation about the Y-axis.XASM
Gate format:
RY(<target_qubit>, <phi>);QB Native Transpilation
ry(q0,_phi)
where q0 is the qubit and _phi is the angle of rotation about the Z-axis.
\(R_z(\lambda)\) - Pauli rotation on Z-axis by \(\lambda\)
Unitary Description and Decomposition Rules
This operator rotates a qubit around the Z-axis of the Bloch sphere by an angle of \(\lambda\). It introduces a relative phase between the two components, so
\(|0\rangle \rightarrow -i\sin(\frac{\lambda}{2})|0\rangle\) and \(|1\rangle \rightarrow i\sin(\frac{\lambda}{2})|1\rangle\).
It is given by the matrix
\(Z(\lambda) = I \cos\!\left(\frac{\lambda}{2}\right) -i \sigma_Z \sin\!\left(\frac{\lambda}{2}\right) = \left(\begin{array}{cc} \cos\left(\frac{\lambda}{2}\right) -i\sin\left(\frac{\lambda}{2}\right) & 0 \\ & \\0 & \cos\left(\frac{\lambda}{2}\right) + i\sin\left(\frac{\lambda}{2}\right) \end{array} \right)\),
where \(I\) is the identity matrix and \(\sigma_Z\) is the Pauli matrix
\(\sigma_Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)\).
OpenQASM
Gate format:
rz(<lambda>) <target_qubit>;where
<lambda>is the angle of rotation about the Z-axis.XASM
Gate format:
RZ(<target_qubit>, <lambda>);QB Native Transpilation
rz(q0,_lambda)
where q0 is the qubit and _lambda is the angle of rotation about the Z-axis.
S - rotation on z-axis by 0.5*pi
Unitary Description and Decomposition Rules
This operator rotates a qubit around the Z-axis of the Bloch sphere by an angle of \(\frac{\pi}{2}\). It introduces a relative phase between the two components, where
\(|0\rangle \rightarrow -i\sin(\frac{\pi}{4})|0\rangle\) and \(|1\rangle \rightarrow i\sin(\frac{\pi}{4})|1\rangle\).
It is given by the matrix
\(S = Z\left(\frac{\pi}{2}\right) = I \cos\!\left(\frac{\pi}{4}\right) -i \sigma_Z \sin\!\left(\frac{\pi}{4}\right) = \left(\begin{array}{cc} \cos\left(\frac{\pi}{4}\right) -i\sin\left(\frac{\pi}{4}\right) & 0 \\ & \\0 & \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) \end{array} \right)\),
where \(I\) is the identity matrix and \(\sigma_Z\) is the Pauli matrix
\(\sigma_Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)\).
OpenQASM
import math rz(0.5*math.pi) q[0];
where q[0] is the qubit.
XASM
import math Rz(q[0],0.5*math.pi)
where q[0] is the qubit.
QB Native Transpilation
import math rz(q0,0.5*math.pi)
where q0 is the qubit.
T - rotation on z-axis by 0.25*pi
Unitary Description and Decomposition Rules
This operator rotates a qubit around the Z-axis of the Bloch sphere by an angle of \(\frac{\pi}{4}\). It introduces a relative phase between the two components, where
\(|0\rangle \rightarrow -i\sin(\frac{\pi}{8})|0\rangle\) and \(|1\rangle \rightarrow i\sin(\frac{\pi}{8})|1\rangle\).
It is given by the matrix
\(S = Z\left(\frac{\pi}{4}\right) = I \cos\!\left(\frac{\pi}{8}\right) -i \sigma_Z \sin\!\left(\frac{\pi}{8}\right) = \left(\begin{array}{cc} \cos\left(\frac{\pi}{8}\right) -i\sin\left(\frac{\pi}{8}\right) & 0 \\ & \\0 & \cos\left(\frac{\pi}{8}\right) + i\sin\left(\frac{\pi}{8}\right) \end{array} \right)\),
where \(I\) is the identity matrix and \(\sigma_Z\) is the Pauli matrix
\(\sigma_Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)\).
OpenQASM
import math rz(0.25*math.pi) q[0];
where q[0] is the qubit.
XASM
import math RZ(q[0],0.25*math.pi)
where q[0] is the qubit.
QB Native Transpilation
import math rz(q0,0.25*math.pi)
where q0 is the qubit.
\(U(\theta,\phi,\lambda)\) - arbitrary rotation
Unitary Description and Decomposition Rules
If used for an ansatz that will require automatic differentiation (AD), replace \(U\) with the decomposition:
\(U(\theta, \phi, \lambda) = R_y(-\frac{\pi}{2})*R_x(\phi)*R_y(\theta)*R_x(\lambda)*R_y(\frac{\pi}{2})\)
Other useful expressions:
\(R_x(\alpha) = U(\alpha, -\frac{\pi}{2}, \frac{\pi}{2})\)
\(R_y(\alpha) = U(\alpha, 0, 0)\)
\(R_z(\alpha) = U(0, \alpha, 0) = U(0, 0, \alpha)\)
OpenQASM
Gate format:
u(<theta>, <phi>, <lambda>) <target_qubit>;// Format: u(theta, phi, lambda) target_qubit __qpu__ void QBCIRCUIT(qreg q) { OPENQASM 2.0; include "qelib1.inc"; creg c[1]; // theta = 0.2 // phi = -0.25 // lambda = 1.1 u(0.2, -0.25, 1.1) q[0]; measure q[0] -> c[0]; }
XASM
Gate format:
U(<target_qubit>, <theta>, <phi>, <lambda>);QB Native Transpilation
We use the decomposition:
\(U(\theta, \phi, \lambda) := R_y(-\frac{\pi}{2})*R_x(\phi)*R_y(\theta)*R_x(\lambda)*R_y(\frac{\pi}{2})\)
Measurement
Description
Measurement is inherently non-unitary. The physical operation selects one of the components of the given quantum registry at random, where the probability of any particular component being selected is given by the square of its amplitude. For this reason it is advisable to call multiple shots (Qristal’s default is 1024) so that the distribution of possible outcomes is apparent.
OpenQASM
measure q[0] -> c[0]
where q[0] is the qubit and c[0] is the corresponding classical bit to which it is measured.
XASM
Gate format:
Measure(<target_qubit>);QB Native Transpilation
measure(q0, c0)
where q0 is the qubit and c0 is the corresponding classical bit to which it is measured.