Gate

Qgate implements various gates as shown Table 1. Multi-control-bit and adjoint are also supported.

Type Gate
1 qubit gate  
No parameter ID, H, X, Y, Z, S, T
1 parameter Rx(theta), Ry(theta), Rz(theta), U1(_lambda), Expii(theta), Expia(theta)
2 parameters U2(phi, _lambda)
3 parameters U3(theta, phi, lambda)
Composed gate Expi(theta)(gatelist)
2 qubit gate Swap(qreg0, qreg1)

1 qubit gate

To create a 1 qubit gate, the following syntax is used.

Tokens surrounded by <> may appear 0- or 1-time according to gates to be declared.

<cntr(qregs).>GateType<(paramters)><.Adj>(qreg)

  • Control bits

    cntr(qregs). specify control bits. It appears only when controlled gates are decalared. A comma-seperated list of qregs, a list of qregs, or their mixture is accepted.

  • GateType<(parameters)>

    GateType is the gate name, such as H, Rx and Expii. If a specified gate type does not have any parameter, (paramters) is omitted.

  • <.Adj>

    Specifying a gate is adjoint of GateType. All gates except for Swap gate support adjoint. Gates such as H and X are hermite, so their adjoint is identical. In these cases, .Adj is simply ignored.

  • (qreg)

    Qreg instance as a target qubit(qreg).

I, X, Y, Z, H, S, T gate, Single qubit gate without parameter

These gates are 1 qreg gates without parameters.

I, X, Y, Z : Identity and Pauli gates

\[\begin{split}I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\end{split}\]

H: Hadamard gate S, T: Phase shfit gates

\[\begin{split}H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, S = \begin{pmatrix} 1 & 0 \\ 0 & {e}^{\frac{i\pi}4} \end{pmatrix}, T = \begin{pmatrix} 1 & 0 \\ 0 & {e}^{\frac{i\pi}4} \end{pmatrix}\end{split}\]
# examples

igate = I(qreg)  # I gate
xgate = X(qreg)  # X gate
ygate = Y(qreg)  # Y gate
zgate = Z(qreg)  # Z gate

hgate = H(qreg)  # H gate
sgate = S(qreg)  # S gate
tgate = T(qreg)  # T gate

cx = ctrl(qreg0).X(qreg)         # CX gate
ccx = ctrl(qreg0, qreg1).X(qreg) # Toffoli gate

S.Adj(qreg)                      # Adjoint of S gate
ctrl(qreg0).S.Adj(qreg1)         # Adjoint of controlled S gate

Rx, Ry, Rz, U1, Expii, Expiz gate, single qubit gate with one parameter

These gates are 1 qreg gates without parameters.

Rx(theta), Ry(theta), Rz(theta) : Rotation around X, Y, Z axes

\[ \begin{align}\begin{aligned}\begin{split}Rx(\theta) = e^{-i{\theta}X / 2} = \begin{pmatrix} cos(\frac{\theta}2) & - i sin(\frac{\theta}2) \\ - i sin(\frac{\theta}2) & cos(\frac{\theta}2) \end{pmatrix}\end{split}\\\begin{split}Ry(\theta) = e^{-i{\theta}Y / 2} = \begin{pmatrix} cos(\frac{\theta}2) & - sin(\frac{\theta}2) \\ sin(\frac{\theta}2) & cos(\frac{\theta}2) \end{pmatrix}\end{split}\\\begin{split}Rz(\theta) = e^{-i{\theta}Z / 2} = \begin{pmatrix} e^{-i{\theta}/2} & 0 \\ 0 & e^{i{\theta}/2} \end{pmatrix}\end{split}\end{aligned}\end{align} \]

U1(theta) : Phase shift gate with a given angle. This gate comes from OpenQASM specification.

\[\begin{split}U_1(\lambda) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 \\ 0 & e^{i\lambda} \end{pmatrix}\end{split}\]

Expii, Expiz : Exponents of I and Z matrices.

\[ \begin{align}\begin{aligned}\begin{split}Expii(\theta) = e^{i{\theta}I} = \begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{i\theta} \end{pmatrix}\end{split}\\\begin{split}Expiz(\theta) = e^{i{\theta}Z} = \begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix}\end{split}\end{aligned}\end{align} \]
# examples

rxgate = Rx(theta)(qreg)  # Rx gate
rygate = Ry(theta)(qreg)  # Ry gate
rzgate = Rz(theta)(qreg)  # Rz gate

u1gate = U1(theta)(qreg)  # U1 gate
expiigate = Expii(theta)(qreg)  # exp(i * theta * I) gate
expizgate = Expiz(theta)(qreg)  # exp(i * theta * Z) gate

crz = ctrl(qreg0).Rz(theta)(qreg)  # controlled Rz gate
eizdg = Expiz(theta).Adj(qreg)     # Adjoint of Expiz gate

Note

Rz gate definition is different from that defined in OpenQASM. Please us U1 gate as Rz gate if you need quantum circuits compatible with OpenQASM.

U2 gate, single qubit gate with 2 parameters

U2(phi, lambda) : u2 gate defined in OpenQASM.

\[U_2(\phi, \lambda) = U_1(\phi + \frac{\pi}2)R_x(\frac{\pi}2)U_1(\lambda - \frac{\pi}2)\]
# examples

u2gate = U2(phi, _lambda)  # U2 gate

cu2 = ctrl(qreg0).u2(pha, _lambda)  # CX gate
cu2dg = .Adj(qreg)                 # Adjoint of S gate

U3 gate, single qubit gate with 3 parameters

U3(theta, phi, lambda) : u3 gate defined in OpenQASM.

\[U_3(\theta, \phi, \lambda) = U_1(\phi + 3\pi)R_x(\frac{\pi}2)U_1(\theta + \pi)R_x(\frac{\pi}2)U_1(\lambda - \frac{\pi}2)\]
# examples

u3gate = U3(theta, phi, _lambda)  # U3 gate

cu3 = ctrl(qreg0).u3(theta, pha, _lambda)  # Controlled U3 gate
cu2dg = .Adj(qreg)                         # Adjoint of U3 gate

Composed gate

Expi is the only composed gate currently qgate implements.

Expi(theta)(gatelist) : Exponent of tensor product of gates in gatelist.

Expi gate is allowed to have (multiple-)controll bits.

Gates in gatelist should be Pauli and identity operators. This gate applies exponent of tensor product of gates in gatelist to multiple qregs.

If there are sets of gates which have the same target qreg, these gates are fused to one gate before calculating tensor product.

\[Expi(\theta)(gatelist) = e^{i \theta [P_0 \otimes P_1 \otimes P_2 \otimes ... \otimes P_N]}\]

where \(P_i\) is a matrix product of operators that shares a target qreg.

# examples
gatelist = [Z(qreg0), Z(qreg1), X(qreg2), ... ]
expigate = Expii(theta)(gatelist)          # Expii gate with a given gatelist

cexpi = ctrl(qreg).expii(theta)(gatelist)  # Controlled U3 gate
cu2dg = .Adj(qreg)                         # Adjoint of U3 gate

2 qubit gate

Expi is the only composed gate currently qgate implements.

Swap(qreg0, qreg1) : Swapping qreg0 and qreg1.

Swap does not have neigher any control-bits nor adjoint.

# examples
swap = Swap(qreg0, qreg1)