Basic Gates

isQ supports the following quantum gates.

X

X(qbit q)

Pauli X gate. $$X=\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$$

Y

Y(qbit q)

Pauli Y gate. $$Y=\begin{bmatrix} 0 & -i\\ i & 0 \end{bmatrix}$$

Z

Z(qbit q)

Pauli Z gate. $$Z=\begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$$

H

H(qbit q)

Hardmard gate. $$H=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}$$

S

S(qbit q)

Phase gate. $$S=\begin{bmatrix} 1 & 0\\ 0 & i \end{bmatrix}$$ Note that $S^2=Z$.

T

T(qbit q)

(\pi/8) gate. $$T=\begin{bmatrix} 1 & 0\\ 0 & e^{i\pi/4} \end{bmatrix}$$ Note that $T^2=S$.

Rx

Rx(double theta, qbit q)

Rotation about the X axis. $$Rx(\theta)=e^{-i\theta X/2}=\begin{bmatrix} \textrm{cos}\frac{\theta}{2} & -i\textrm{sin}\frac{\theta}{2}\\ -i\textrm{sin}\frac{\theta}{2} & \textrm{cos}\frac{\theta}{2} \end{bmatrix}$$

Ry

Ry(double theta, qbit q)

Rotation about the Y axis. $$Ry(\theta)=e^{-i\theta Y/2}=\begin{bmatrix} \textrm{cos}\frac{\theta}{2} & -\textrm{sin}\frac{\theta}{2}\\ \textrm{sin}\frac{\theta}{2} & \textrm{cos}\frac{\theta}{2} \end{bmatrix}$$

Rz

Rz(double theta, qbit q)

Rotation about the Z axis. $$Rz(\theta)=e^{-i\theta Z/2}=\begin{bmatrix} e^{-i\theta/2} & 0\\ 0 & e^{i\theta/2} \end{bmatrix}$$

X2P

X2P(qbit q)

$\pi/2$ rotation about the X axis. $$X2P=Rx(\pi/2)=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -i\\ -i & 1 \end{bmatrix}$$

X2M

X2M(qbit q)

$-\pi/2$ rotation about the X axis. $$X2M=Rx(-\pi/2)=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & i\\ i & 1 \end{bmatrix}$$

Y2P

Y2P(qbit q)

$\pi/2$ rotation about the Y axis. $$Y2P=Rx(\pi/2)=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix}$$

Y2M

Y2M(qbit q)

$-\pi/2$ rotation about the Y axis. $$Y2M=Rx(-\pi/2)=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1\\ -1 & 1 \end{bmatrix}$$

U3

U3(double theta, double phi, double lambda, qbit q)

Generic single-qubit rotation gate with 3 Euler angles. $$U3(\theta,\phi,\lambda)=\begin{bmatrix} \textrm{cos}\frac{\theta}{2} & -e^{i\lambda}\textrm{sin}\frac{\theta}{2}\\ e^{i\phi}\textrm{sin}\frac{\theta}{2} & e^{i(\phi+\lambda)}\textrm{cos}\frac{\theta}{2} \end{bmatrix}$$ Note that $$U3\left(\theta,-\frac{\pi}{2},\frac{\pi}{2}\right)=Rx(\theta)$$ $$U3(\theta,0,0)=Ry(\theta)$$

CNOT

CNOT(qbit c, qbit t)

Controlled-NOT gate. $$CNOT=\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{bmatrix}$$

CZ

CZ(qbit c, qbit t)

Controlled-Z gate. $$CZ=\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix}$$

Toffoli

Toffoli(qbit c1, qbit c2, qbit t)

Toffoli gate. $$Toffoli=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$

GPhase

GPhase(double theta)

Global phase gate. $$GPhase(\theta)=e^{i\theta}$$ Note that this gate does not have any qbit-type parameter.