Gate¶

The below table shows built-in gates in Qgate.

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

Qgate also suports controlled gate and adjoint.

Controlled gate

All gates except for Swap gate works as controlled gate. Every contolled gate is able to have arbitary number of control bits.
Adjoint

All gates except for Swap gate have their adjoint.

1 qubit gate¶

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

Tokens surrounded by <> is optional, may appear 0- or 1-time according to a gate to be created.

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

Control bits

ctrl(qregs). specify control bits. It’s required to create a controlled gate. A comma-separated list of qregs, a list of qregs, or their mixture is accepted.
GateType<(parameters)>

GateType is a 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.

Examples:

# Hadamard gate
H(qreg0)

# Controlled X gate (CX gate)
ctrl(qreg0).X(qreg1)

# 2-control-bit X gate (Toffoli gate)
ctrl(qreg0, qreg1).X(qreg2)

# Rx gate (1 parameter)
Rx(0.)(qreg)

# Adjoint of Rx gate
Rx(0.).Adj(qreg)

# adjoint of 3-bit-controlled U3 gate
# control bits are given by a python list.
ctrlbits = [qreg0, qreg1, qreg2]  # creating a list of control bits
ctrl(ctrlbits).U3(theta, phi, _lambda).Adj(qreg3)

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

Gates in this section are single qubit 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

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

# 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¶

Gates in this section are single qubit gates with one parameter.

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 for 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 : Matrix exponential for I and Z matrices.

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

# 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 use U1 gate as Rz gate if you need quantum circuits compatible with OpenQASM.

U2 gate, single qubit gate with 2 parameters¶

Gates in this section are single qubit gates with two parameters.

U2(phi, lambda) : u2 gate defined in OpenQASM. Global phase differs from the original definition.

\[\begin{split}U_2(\phi, \lambda) = U_3(\frac{\pi}2, \phi, \lambda) = \frac{1}{\sqrt{2}} \begin{pmatrix} e^{-i \frac{\phi + \lambda}2} & - e^{-i \frac{\phi - \lambda}2} \\ e^{i \frac{\phi - \lambda}2} & e^{i \frac{\phi + \lambda}2} \end{pmatrix}\end{split}\]

# examples

u2gate = U2(phi, _lambda)  # U2 gate

cu2 = ctrl(qreg0).U2(phi, _lambda)(qreg1) # controlled U2 gate.
u2dg = U2(phi, _lambda).Adj(qreg)         # Adjoint of U2 gate

U3 gate, single qubit gate with 3 parameters¶

Gates in this section are single qubit gates with three parameters.

U3(theta, phi, lambda) : u3 gate defined in OpenQASM, global phase differs from the original definition.

\[\begin{split}U_3(\theta, \phi, \lambda) = \begin{pmatrix} e^{-i \frac{\phi + \lambda}2} cos(\frac{\theta}2) & - e^{-i \frac{\phi - \lambda}2} sin(\frac{\theta}2) \\ e^{i \frac{\phi - \lambda}2} sin(\frac{\theta}2) & e^{i \frac{\phi + \lambda}2} cos(\frac{\theta}2) \end{pmatrix}\end{split}\]

# examples

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

cu3 = ctrl(qreg0).U3(theta, phi, _lambda)(qreg1)  # Controlled U3 gate
u3dg = U3(theta, phi, _lambda).Adj(qreg)          # Adjoint of U3 gate

Macro gate¶

Expi is the macro gate that Qgate currently implements.

Expi(theta)(gatelist)

gatelist : list of pauli and/or identity gates.
theta : angle of rotation.

Expii gate is for rotation of a product of pauli and identity gates. This gate is able to have (multiple-)controll bits, and adjoint is also available.

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

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

Examples:

# 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 Expi gate
u3dg = U3(math.pi., math.pi / 4.., math.pi / 8.).Adj(qreg)   # Adjoint of Expi gate

# exp(i * math.pi * X), identical to Rx(math.pi).
Expi(math.pi)(X(qreg))

# can have a sequence of pauli operators
Expi(math.pi / 2.)(X(qreg0), Y(qreg1), Z(qreg2))

# Can be a controlled gate
ctrl(qreg0).expi(math.pi)(Y(qreg1))

# Supports adjoint
expi(math.pi).Adj(Y(qreg1))

2 qubit gate¶

Swap is the only 2 qubit gate currently qgate implements.

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

Swap does not have any control-bits nor adjoint.

# examples
swap = Swap(qreg0, qreg1)