Gate Conventions

The object of the simulator is to include gates which are useful in the simulation of near-term quantum devices.

One-Qubit Gates

The following single qubit quantum gates are implemented by the simulator

• X-rotation, Y-rotation and Z-rotation

  These gates, \(R_x(\theta)\), \(R_y(\theta)\) and \(R_z(\theta)\), implement rotations of the input state about an axis of the Bloch sphere. The member functions are called rotateX, rotateY and rotateZ. They are defined as follows:

  \[\begin{split}\begin{align*} R_x(\theta) &= \begin{bmatrix}\cos(\theta/2)&-i\sin(\theta/2)\\-i\sin(\theta/2)&\cos(\theta/2)\end{bmatrix},\\ R_y(\theta) &= \begin{bmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{bmatrix},\\ R_z(\theta) &= \begin{bmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{bmatrix}. \end{align*}\end{split}\]

  Sometimes an equivalent version of the Z rotation, the phase shift, is more useful. This is defined as followed:

  \[\begin{split}\text{phase}(\theta) = e^{i\theta/2}R_z(\theta) = \begin{bmatrix}1&0\\0&e^{i\theta}\end{bmatrix}.\end{split}\]

  It is the same as a Z-rotation, up to the global phase \(e^{i\theta/2}\), which means interchanging them in a circuit will have no effect on measured outcomes. (However, different amplitudes in the state vector will have different phases depending which is used.) It is accessible using the phase member function.

• Pauli X, Pauli Y and Pauli Z

  These gates, \(\sigma_x\), \(\sigma_y\) and \(\sigma_z\), are commonly used special cases of the general X-, Y- and Z-rotations, so they have their own member functions: pauliX, pauliY and pauliZ. They are defined as follows:

  \[\begin{split}\begin{align} \sigma_x &= iR_x(\pi) =\begin{bmatrix}0&1\\1&0\end{bmatrix},\\ \sigma_y &= iR_y(\pi)= \begin{bmatrix}0&-i\\i&0\end{bmatrix},\\ \sigma_z &= iR_z(\pi)= \begin{bmatrix}1&0\\0&-1\end{bmatrix}. \end{align}\end{split}\]

  As before, the global phases of \(i\) do not matter, so, for example, every time \(R_x(\pi)\) is used it a circuit, it is valid to replace it with \(\sigma_x\).

• Hadamard

  The Hadamard gate is defined as follows:

  \[\begin{split}H = \frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix},\\\end{split}\]

  and is accessible using the hadamard member function. It can be written in terms of X- and Y-rotations as follows: \(H = \sigma_xR_y(\pi/2)\). THe Hadamard gate is mainly used for creating an equal superposition of states.

• Arbitary unitary

  The user can specify an arbitrary one-qubit unitary gate with the function unitary.

Two-Qubit Gates

Controlled operations

Most two-qubit operations of interest for near-term devices are controlled operations, which means that the state of one of the qubits (the control) determines whether a single qubit unitary is applied to another qubit (the target). If that unitary that is applied to the target qubit is \(U\), then the general form of a controlled operation is as follows

\[\begin{split}\text{controlled-}U = \begin{bmatrix}I&0\\0&U\end{bmatrix}\end{split}\]

(This matrix assumes that the control is qubit 1 and the target is qubit 0. For more information about qubit indexing, see [link].) The simulator implements the following controlled operations:

• Controlled X-rotation, Y-rotation and Z-rotation

  These use \(U=R_x(\theta),R_y(\theta),R_z(\theta)\) as defined above. The corresponding function names are controlRotateX, controlRotateY, controlRotateZ.

• Controlled-Pauli gates

  When \(U=\sigma_x\), the gate is commonly called the Controlled-NOT (or CNOT) gate. The corresponding function name is controlNot.

  When \(U=\sigma_z\), the gate is commonly called Controlled-Z (or CZ). Here we give it the function name controlZ.

  Taking \(U=\sigma_y\) the gate is called controlY.

• Controlled-Phase

  This is given by \(U=\text{phase}(\theta)\). The corresponding function name is controlPhase.

• Controlled-H

  This is given by \(U=H\). The corresponding function name is controlHadamard.

• Controlled-unitary

  The user can specify an arbitrary controlled gate with a one-qubit unitary with controlUnitary.

Number preserving

The full Hilbert space \(\mathcal{H}\) of \(n\) qubits can be written as a direct sum of the Hilbert spaces \(\mathcal{H}_i\) with a fixed Hamming weight \(i\) for \(i=0,...,n\). Number preserving gates preserve these Hilbert spaces. For example, the gate \(\sigma_Z\) is number preserving but \(\sigma_X\) is not.

An arbitrary one-qubit number preserving gate is the phase gate. An arbitary two-qubit number preserving gate preserves the \(|00\rangle\), \(\{|01\rangle, |10\rangle \}\) and \(|11\rangle\) subspaces and takes the form

\[\begin{split}\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & u_{00} & u_{01} & 0 \\ 0 & u_{10} & u_{11} & 0 \\ 0 & 0 & 0 & e^{i\theta} \end{bmatrix}\end{split}\]

where the 2x2 sub-matrix of \(u_{ij}\) is a unitary matrix \(U\). Number preserving gates are useful when simulating quantum chemistry on a quantum computer and for this reason we also provide a number preserved simulator [link, will explain why useful with NP simulator].

• Number preserved X- and Y-rotations

  These gates use \(U = R_x(\theta)\) and \(U = R_y(\theta)\). The corresponding function names are npRotateX and npRotateY. In the first case, the gate is equivalent to \(e^{-i\theta/2(XX+YY)}\) and in the second \(e^{-i\theta/2(YX-XY)}\).

• Swap and fermionic swap

  Taking \(U=\sigma_x\) swaps the state of two qubits, the function name for this is swap. A fermionic swap gate acts as a swap gate but for fermions, this is useful when using fermion-to-qubit mappings when simulating quantum chemistry. It can be accessed with fswap and is equivalent to a swap gate followed by a controlled-Z gate.

• Number preserved Hadamard

  This is given by \(U=H\) and can be accessed with npHadamard.

• Arbitary number preserved gate

  The user can specify an arbitrary \(U\) with npUnitary.