Quantum Circuits Library: Three-qubit gates

ToffoliGate()

Three-qubit Toffoli gate, also known as the CCX (controlled-controlled-NOT) gate.

Circuit Representation

q_0: ──■──
       │
q_1: ──■──
     ┌─┴─┐
q_2: ┤ X ├
     └───┘

Matrix Representation

\[Toffoli = |0 \rangle \langle 0| \otimes I \otimes I + |1 \rangle \langle 1| \otimes CXGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}\]

QFT3Gate()

Three-qubit Quantum Fourier Transform (QFT) gate, where the QFT operation on n-qubits is given by:

\[|j\rangle \mapsto \frac{1}{2^{n/2}} \sum_{k=0}^{2^n - 1} e^{2\pi ijk / 2^n} |k\rangle\]

Circuit Representation

     ┌──────┐
q_0: ┤      ├
     │      │   
q_1: ┤ QFT3 ├ 
     │      │   
q_2: ┤      ├
     └──────┘ 

Matrix Representation

\[M = \frac{1}{2\sqrt{2}} \begin{pmatrix} 1 1 1 1 1 1 1 1 \\ 1 \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} i -\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} -1 -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} -i \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \\ 1 i -1 -i 1 i -1 -i \\ 1 -\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} -i \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} -1 \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} i -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \\ 1 -1 1 -1 1 -1 1 -1 \\ 1 -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} i \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} -1 \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} -i -\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \\ 1 -i -1 i 1 -i -1 i \\ 1 \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} -i -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} -1 -\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} i \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \\ \end{pmatrix}\]

CSwapGate()

Three-qubit, controlled SwapGate, also known as the Fredkin gate.

Circuit Representation

q_0: ─■─
      │
q_1: ─X─
      │
q_2: ─X─

Matrix Representation

\[CSwapGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix}\]

CCZGate()

Three-qubit controlled-controlled Z gate.

Circuit Representation

q_0: ─■─
      │
q_1: ─■─
      │
q_2: ─■─

Matrix Representation

\[CCZGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{pmatrix}\]

PeresGate()

Three-qubit Peres gate. This gate is equivalent to ToffoliGate followed by the CNotGate in 3 qubits. Reference: https://doi.org/10.1103/PhysRevA.32.3266

Circuit Representation

q_0: ──■─────■──          
       │   ┌─┴─┐
q_1: ──■───┤ X ├
     ┌─┴─┐ └───┘
q_2: ┤ X ├──────
     └───┘

Matrix Representation

\[PeresGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \end{pmatrix}\]

RCCXGate()

Three-qubit relative (or simplified) Toffoli gate, or the CCX gate. This gate is equivalent to ToffoliGate up to relative phases. The advantage of this gate is that its implementation requires only three CNot (or CX) gates. Reference: https://arxiv.org/pdf/1508.03273.pdf

Matrix Representation

\[RCCXGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\ \end{pmatrix}\]

MargolusGate()

Three-qubit Margolus gate, which is a simplified ToffoliGate and coincides with the Toffoli gate up to a single change of sign. The advantage of this gate is that its implementation requires only three CNot (or CX) gates. Reference: https://arxiv.org/pdf/quant-ph/0312225.pdf

Matrix Representation

\[MargolusGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix}\]

CiSwapGate()

Three-qubit controlled version of the iSwapGate. Reference: https://doi.org/10.1103/PhysRevResearch.2.033097

Circuit Representation

q_0: ─────■─────
          │
      ┌───────┐
q_1: ─┤       ├─
      │ iSwap │   
q_2: ─┤       ├─ 
      └───────┘ 

Matrix Representation

\[CiSwapGate = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\ 0 & 0 & 0 & 0 & 0 & i & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix}\]