Matrix modeling of quantum computation

Property of quantum operations in matrix

Unitarity

U†U=UU†=IU^\dag U = U U^\dag = I

where U†U^\dag is the Hermitian (Hermitean) adjoint of UU

Gates: 1-qubit example

QNOT

QNOT≡[0110]QNOT \equiv \begin{bmatrix}
   0 & 1 \\
   1 & 0
\end{bmatrix}

Same as Pauli-X (Pauli-1)

Rotation

Uθ≡[cos⁡(θ)sin⁡(θ)−sin⁡(θ)cos⁡(θ)]U_\theta \equiv \begin{bmatrix}
\cos(\theta) & \sin(\theta) \\
-\sin(\theta) & \cos(\theta)
\end{bmatrix}

Square root of NOT

SRN≡12[1−111]SRN \equiv \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & -1 \\
1 & 1
\end{bmatrix}

Hadamard

H≡12[111−1]H \equiv \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}

Gates: 2-qubit example

Controlled phase

CPHASE≡[100001000010000eiϕ]\text{CPHASE} \equiv \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{i\phi}
\end{bmatrix}

Swap

SWAP≡[1000001001000001]\text{SWAP} \equiv \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}

Reference(s)

  • L. Spector, Automatic Quantum Computer Programming: A Genetic Programming Approach. NY: Springer, 2004.

  • M. Soeken, D. M. Miller, R. Drechsler. On quantum circuits employing roots of the Pauli matrices. arXiv:1308.2493, 2013.