Standard Gates

GateU(θ, ϕ, λ[, γ = 0])

Single qubit generic unitary gate $U(\theta, \phi, \lambda, \gamma = 0)$, where $\theta$, $\phi$, and $\lambda$ are the Euler angles specified in radians, and $\gamma$ is a global phase.

See also GateU3, GateP, GateU2, GateU1

Matrix representation

\[\operatorname{U}(\theta, \phi, \lambda, \gamma = 0) = \mathrm{e}^{i\gamma} \begin{pmatrix} \cos\left(\frac{\theta}{2}\right) & -\mathrm{e}^{i\lambda}\sin\left(\frac{\theta}{2}\right) \\ \mathrm{e}^{i\phi}\sin\left(\frac{\theta}{2}\right) & \mathrm{e}^{i(\phi+\lambda)}\cos\left(\frac{\theta}{2}\right) \end{pmatrix}\]

Examples

julia> @variables θ ϕ λ
3-element Vector{Symbolics.Num}:
 θ
 ϕ
 λ

julia> GateU(θ, ϕ, λ)
U(θ, ϕ, λ)

julia> matrix(GateU(2.023, 0.5, 0.1))
2×2 Matrix{ComplexF64}:
  0.53059+0.0im       -0.843394-0.0846217im
 0.743864+0.406375im   0.437915+0.299594im

julia> c = push!(Circuit(), GateU(θ, ϕ, λ), 1)
1-qubit circuit with 1 instructions:
└── U(θ,ϕ,λ) @ q[1]

julia> push!(c, GateU(π/8, π/2, π/4), 2)
2-qubit circuit with 2 instructions:
├── U(θ,ϕ,λ) @ q[1]
└── U(π/8,π/2,π/4) @ q[2]

julia> power(GateU(θ, ϕ, λ), 2), inverse(GateU(θ, ϕ, λ))
(GateU(θ, ϕ, λ, 0π)^2, GateU(-θ, -λ, -ϕ, 0π))

Decomposition

Since $U$ gate, is the most general single qubit unitary matrix, all other matrices are defined from it.

julia> decompose(GateU(θ, λ, ϕ))
1-qubit circuit with 1 instructions:
└── U(θ,λ,ϕ) @ q[1]

GateP(λ)

Single qubit Phase gate.

λ is the phase angle in radians.

Matrix representation

\[\operatorname{P}(\lambda) = \operatorname{U}(0, 0, \lambda) = \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i\lambda} \end{pmatrix}\]

Examples

julia> @variables λ
1-element Vector{Symbolics.Num}:
 λ

julia> GateP(λ)
P(λ)

julia> matrix(GateP(1.989))
2×2 Matrix{ComplexF64}:
 1.0+0.0im       0.0+0.0im
 0.0+0.0im  -0.40612+0.91382im

julia> c = push!(Circuit(), GateP(λ), 1)
1-qubit circuit with 1 instructions:
└── P(λ) @ q[1]

julia> push!(c, GateP(π/2), 2)
2-qubit circuit with 2 instructions:
├── P(λ) @ q[1]
└── P(π/2) @ q[2]

Decomposition

julia> decompose(GateP(λ))
1-qubit circuit with 1 instructions:
└── U(0,0,λ) @ q[1]

GateU1(λ)

Single qubit rotation \operatorname{U1}(\lambda) about the Z axis.

Equivalent to GateP.

Matrix representation

\[\operatorname{U1}(\lambda) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\lambda} \end{pmatrix}\]

Examples

julia> @variables λ
1-element Vector{Symbolics.Num}:
 λ

julia> GateU1(λ)
U1(λ)

julia> matrix(GateU1(0.519))
2×2 Matrix{ComplexF64}:
 1.0+0.0im       0.0+0.0im
 0.0+0.0im  0.868316+0.496012im

julia> c = push!(Circuit(), GateU1(λ), 1)
1-qubit circuit with 1 instructions:
└── U1(λ) @ q[1]

julia> push!(c, GateU1(π/2), 2)
2-qubit circuit with 2 instructions:
├── U1(λ) @ q[1]
└── U1(π/2) @ q[2]

julia> power(GateU1(λ), 2), inverse(GateU1(λ))
(GateU1(λ)^2, GateU1(-λ))

Decomposition

julia> decompose(GateU1(λ))
1-qubit circuit with 1 instructions:
└── U(0,0,λ) @ q[1]

GateU2(ϕ, λ)

Single qubit rotation \operatorname{U2}(\phi, \lambda) about the X+Z axis.

Matrix representation

\[\operatorname{U2}(\phi,\lambda) = \frac{1}{\sqrt{2}}e^{-(\phi+\lambda)/2}\begin{pmatrix} 1 & -e^{i\lambda} \\ e^{i\phi} & e^{i(\phi+\lambda)} \end{pmatrix}\]

Examples

julia> @variables ϕ λ
2-element Vector{Symbolics.Num}:
 ϕ
 λ

julia> GateU2(ϕ, λ)
U2(ϕ, λ)

julia> matrix(GateU2(2.023, 0.5))
2×2 Matrix{ComplexF64}:
  0.707107+0.0im       -0.620545-0.339005im
 -0.308969+0.636033im  -0.576077+0.410044im

julia> c = push!(Circuit(), GateU2(ϕ, λ), 1)
1-qubit circuit with 1 instructions:
└── U2(ϕ,λ) @ q[1]

julia> push!(c, GateU2(π/2, π/4), 2)
2-qubit circuit with 2 instructions:
├── U2(ϕ,λ) @ q[1]
└── U2(π/2,π/4) @ q[2]

julia> power(GateU2(ϕ, λ), 2), inverse(GateU2(ϕ, λ))
(GateU2(ϕ, λ)^2, GateU2(-3.141592653589793 - λ, π - ϕ))

Decomposition

julia> decompose(GateU2(ϕ, λ))
1-qubit circuit with 1 instructions:
└── U3(π/2,ϕ,λ) @ q[1]

GateU3(θ, ϕ, λ)

Single qubit generic unitary gate U3(\theta, \phi, \lambda).

This gate is equivalent to the generic unitary gate GateU, differing from it only by a global phase of $\frac{\phi + \lambda + \theta}{2}$.

Matrix representation

\[\operatorname{U3}(\theta,\phi,\lambda) = \begin{pmatrix} \cos\left(\frac{\theta}{2}\right) & -e^{i\lambda}\sin\left(\frac{\theta}{2}\right) \\ \mathrm{e}^{i\phi}\sin\left(\frac{\theta}{2}\right) & \mathrm{e}^{i(\phi+\lambda)}\cos\left(\frac{\theta}{2}\right) \end{pmatrix}\]

Examples

julia> @variables θ ϕ λ
3-element Vector{Symbolics.Num}:
 θ
 ϕ
 λ

julia> GateU3(θ, ϕ, λ)
U3(θ, ϕ, λ)

julia> matrix(GateU3(2.023, 0.5, 0.1))
2×2 Matrix{ComplexF64}:
  0.53059+0.0im       -0.843394-0.0846217im
 0.743864+0.406375im   0.437915+0.299594im

julia> c = push!(Circuit(), GateU3(θ, ϕ, λ), 1)
1-qubit circuit with 1 instructions:
└── U3(θ,ϕ,λ) @ q[1]

julia> push!(c, GateU3(π/8, π/2, π/4), 2)
2-qubit circuit with 2 instructions:
├── U3(θ,ϕ,λ) @ q[1]
└── U3(π/8,π/2,π/4) @ q[2]

julia> power(GateU3(θ, ϕ, λ), 2), inverse(GateU3(θ, ϕ, λ))
(GateU3(θ, ϕ, λ)^2, GateU3(-θ, -λ, -ϕ))

Decomposition

julia> decompose(GateU3(θ, ϕ, λ))
1-qubit circuit with 1 instructions:
└── U(θ,ϕ,λ) @ q[1]

GateX()

Single qubit Pauli-X gate.

Matrix representation

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

Examples

julia> GateX()
X

julia> matrix(GateX())
2×2 Matrix{Int64}:
 0  1
 1  0

julia> c = push!(Circuit(), GateX(), 1)
1-qubit circuit with 1 instructions:
└── X @ q[1]

julia> push!(c, GateX, 2)
2-qubit circuit with 2 instructions:
├── X @ q[1]
└── X @ q[2]

Decomposition

julia> decompose(GateX())
1-qubit circuit with 1 instructions:
└── U(π,0,π) @ q[1]

GateY()

Single qubit Pauli-Y gate.

Matrix representation

\[\operatorname{Y} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\]

Examples

julia> GateY()
Y

julia> matrix(GateY())
2×2 Matrix{ComplexF64}:
 0.0+0.0im  0.0-1.0im
 0.0+1.0im  0.0+0.0im

julia> c = push!(Circuit(), GateY(), 1)
1-qubit circuit with 1 instructions:
└── Y @ q[1]

julia> push!(c, GateY, 2)
2-qubit circuit with 2 instructions:
├── Y @ q[1]
└── Y @ q[2]

Decomposition

julia> decompose(GateY())
1-qubit circuit with 1 instructions:
└── U(π,π/2,π/2) @ q[1]

GateZ()

Single qubit Pauli-Z gate.

Matrix representation

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

Examples

julia> GateZ()
Z

julia> matrix(GateZ())
2×2 Matrix{Float64}:
 1.0   0.0
 0.0  -1.0

julia> c = push!(Circuit(), GateZ(), 1)
1-qubit circuit with 1 instructions:
└── Z @ q[1]

julia> push!(c, GateZ, 2)
2-qubit circuit with 2 instructions:
├── Z @ q[1]
└── Z @ q[2]

Decomposition

julia> decompose(GateZ())
1-qubit circuit with 1 instructions:
└── P(π) @ q[1]

GateH()

Single qubit Hadamard gate.

Matrix representation

\[\operatorname{H} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]

Examples

julia> GateH()
H

julia> matrix(GateH())
2×2 Matrix{Float64}:
 0.707107   0.707107
 0.707107  -0.707107

julia> c = push!(Circuit(), GateH(), 1)
1-qubit circuit with 1 instructions:
└── H @ q[1]

julia> push!(c, GateH, 1)
1-qubit circuit with 2 instructions:
├── H @ q[1]
└── H @ q[1]

julia> power(GateH(), 2), inverse(GateH())
(GateH()^2, GateH())

Decomposition

julia> decompose(GateH())
1-qubit circuit with 1 instructions:
└── U(π/2,0,π) @ q[1]

GateHXY()

Single qubit HXY gate.

Matrix representation

\[\operatorname{HXY} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 - i \\ 1 + i & 0 \end{pmatrix}\]

Examples

julia> GateHXY()
HXY

julia> matrix(GateHXY())
2×2 Matrix{ComplexF64}:
      0.0+0.0im       0.707107-0.707107im
 0.707107+0.707107im       0.0+0.0im

julia> c = push!(Circuit(), GateHXY(), 1)
1-qubit circuit with 1 instructions:
└── HXY @ q[1]

julia> power(GateHXY(), 2), inverse(GateHXY())
(GateHXY()^2, GateHXY())

Decomposition

julia> decompose(GateHXY())
1-qubit circuit with 5 instructions:
├── H @ q[1]
├── Z @ q[1]
├── H @ q[1]
├── S @ q[1]
└── U(0,0,0,-1π/4) @ q[1]

GateHXZ()

The HXZ gate is an alias for the Hadamard gate. It applies a transformation that puts the qubit in an equal superposition of |0⟩ and |1⟩.

Matrix representation

\[\operatorname{HXZ} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]

Examples

julia> GateHXZ()
H

julia> matrix(GateHXZ())
2×2 Matrix{Float64}:
 0.707107   0.707107
 0.707107  -0.707107

julia> c = push!(Circuit(), GateHXZ(), 1)
1-qubit circuit with 1 instructions:
└── H @ q[1]

julia> power(GateHXZ(), 2), inverse(GateHXZ())
(GateH()^2, GateH())

Decomposition

julia> decompose(GateHXZ())
1-qubit circuit with 1 instructions:
└── U(π/2,0,π) @ q[1]

GateHYZ()

Single qubit HYZ gate.

Matrix representation

\[\operatorname{HYZ} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ i & -1 \end{pmatrix}\]

Examples

julia> GateHYZ()
HYZ

julia> matrix(GateHYZ())
2×2 Matrix{ComplexF64}:
 0.707107+0.0im             0.0-0.707107im
      0.0+0.707107im  -0.707107+0.0im

julia> c = push!(Circuit(), GateHYZ(), 1)
1-qubit circuit with 1 instructions:
└── HYZ @ q[1]

julia> power(GateHYZ(), 2), inverse(GateHYZ())
(GateHYZ()^2, GateHYZ())

Decomposition

julia> decompose(GateHYZ())
1-qubit circuit with 5 instructions:
├── H @ q[1]
├── S @ q[1]
├── H @ q[1]
├── Z @ q[1]
└── U(0,0,0,-1π/4) @ q[1]

GateID()

Single qubit identity gate

Matrix representation

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

Examples

julia> GateID()
ID

julia> matrix(GateID())
2×2 Matrix{Float64}:
 1.0  -0.0
 0.0   1.0

julia> c = push!(Circuit(), GateID(), 1)
1-qubit circuit with 1 instructions:
└── ID @ q[1]

julia> power(GateID(), 2), inverse(GateID())
(GateID(), GateID())

Decomposition

julia> decompose(GateID())
1-qubit circuit with 1 instructions:
└── U(0,0,0) @ q[1]

GateS()

Single qubit $S$ gate (or phase gate). It is defined as the square root of the $Z$ gate.

See also GateSDG, GateZ, Power

Matrix representation

\[\operatorname{S} = \sqrt{\operatorname{Z}} = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}\]

Examples

julia> GateS()
S

julia> matrix(GateS())
2×2 Matrix{Complex{Int64}}:
 1+0im  0+0im
 0+0im  0+1im

julia> c = push!(Circuit(), GateS(), 1)
1-qubit circuit with 1 instructions:
└── S @ q[1]

julia> push!(c, GateS, 2)
2-qubit circuit with 2 instructions:
├── S @ q[1]
└── S @ q[2]

julia> power(GateS(), 2), inverse(GateS())
(GateZ(), Inverse(GateS()))

Decomposition

julia> decompose(GateS())
1-qubit circuit with 1 instructions:
└── U(0,0,π/2) @ q[1]

GateSDG()

Single qubit S-dagger gate (conjugate transpose of the S gate).

See also GateS, GateZ, Power, Inverse

Matrix representation

\[\operatorname{S}^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}\]

Examples

julia> GateSDG()
S†

julia> matrix(GateSDG())
2×2 Matrix{Complex{Int64}}:
 1+0im  0+0im
 0+0im  0-1im

julia> c = push!(Circuit(), GateSDG(), 1)
1-qubit circuit with 1 instructions:
└── S† @ q[1]

julia> push!(c, GateSDG, 2)
2-qubit circuit with 2 instructions:
├── S† @ q[1]
└── S† @ q[2]

julia> power(GateSDG(), 2), inverse(GateSDG())
((Inverse(GateS()))^2, GateS())

Decomposition

julia> decompose(GateSDG())
1-qubit circuit with 1 instructions:
└── U(0,0,-1π/2) @ q[1]

GateT()

Single qubit T gate. It is defined as the square root of the $S$ gate, $Z^{\frac{1}{4}}$.

See also GateTDG, GateS, GateZ, Power

Matrix representation

\[\operatorname{Z} = \operatorname{Z}^{\frac{1}{4}} = \begin{pmatrix} 1 & 0 \\ 0 & \exp(\frac{i\pi}{4}) \end{pmatrix}\]

Examples

julia> GateT()
T

julia> matrix(GateT())
2×2 Matrix{ComplexF64}:
 1.0+0.0im       0.0+0.0im
 0.0+0.0im  0.707107+0.707107im

julia> c = push!(Circuit(), GateT(), 1)
1-qubit circuit with 1 instructions:
└── T @ q[1]

julia> push!(c, GateT, 2)
2-qubit circuit with 2 instructions:
├── T @ q[1]
└── T @ q[2]

julia> power(GateT(), 2), power(GateT(), 4), inverse(GateT())
(GateS(), GateZ(), Inverse(GateT()))

Decomposition

julia> decompose(GateT())
1-qubit circuit with 1 instructions:
└── U(0,0,π/4) @ q[1]

GateTDG()

Single qubit T-dagger gate (conjugate transpose of the T gate).

See also GateT

Matrix Representation

\[\operatorname{T}^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & \exp(\frac{-i\pi}{4}) \end{pmatrix}\]

Examples

julia> GateTDG()
T†

julia> matrix(GateTDG())
2×2 Matrix{ComplexF64}:
 1.0+0.0im       0.0+0.0im
 0.0+0.0im  0.707107-0.707107im

julia> c = push!(Circuit(), GateTDG(), 1)
1-qubit circuit with 1 instructions:
└── T† @ q[1]

julia> push!(c, GateTDG, 2)
2-qubit circuit with 2 instructions:
├── T† @ q[1]
└── T† @ q[2]

julia> power(GateTDG(), 2), power(GateTDG(), 4), inverse(GateTDG())
((Inverse(GateT()))^2, (Inverse(GateT()))^4, GateT())

Decomposition

julia> decompose(GateTDG())
1-qubit circuit with 1 instructions:
└── U(0,0,-1π/4) @ q[1]

GateSX()

Single qubit $\sqrt{X}$ gate.

See also GateSXDG, GateX, Power

Matrix representation

\[\operatorname{SX} = \sqrt{\operatorname{X}} = \frac{1}{2} \begin{pmatrix} 1+i & 1-i \\ 1-i & 1+i \end{pmatrix}\]

Examples

julia> GateSX()
SX

julia> matrix(GateSX())
2×2 Matrix{ComplexF64}:
 0.5+0.5im  0.5-0.5im
 0.5-0.5im  0.5+0.5im

julia> c = push!(Circuit(), GateSX(), 1)
1-qubit circuit with 1 instructions:
└── SX @ q[1]

julia> push!(c, GateSX, 2)
2-qubit circuit with 2 instructions:
├── SX @ q[1]
└── SX @ q[2]

julia> power(GateSX(), 2), inverse(GateSX())
(GateX(), Inverse(GateSX()))

Decomposition

julia> decompose(GateSX())
1-qubit circuit with 4 instructions:
├── S† @ q[1]
├── H @ q[1]
├── S† @ q[1]
└── U(0,0,0,π/4) @ q[1]

GateSXDG()

Single qubit $\sqrt{X}^\dagger$ gate (conjugate transpose of the $\sqrt{X}$ gate)

See also GateSX, GateX, Power, Inverse

Matrix representation

\[\operatorname{SXDG} = \sqrt{\operatorname{X}}^\dagger = \frac{1}{2} \begin{pmatrix} 1-i & 1+i \\ 1+i & 1-i \end{pmatrix}\]

Examples

julia> GateSXDG()
SX†

julia> matrix(GateSXDG())
2×2 Matrix{ComplexF64}:
 0.5-0.5im  0.5+0.5im
 0.5+0.5im  0.5-0.5im

julia> c = push!(Circuit(), GateSXDG(), 1)
1-qubit circuit with 1 instructions:
└── SX† @ q[1]

julia> push!(c, GateSXDG, 2)
2-qubit circuit with 2 instructions:
├── SX† @ q[1]
└── SX† @ q[2]

julia> power(GateSXDG(), 2), inverse(GateSXDG())
((Inverse(GateSX()))^2, GateSX())

GateR(θ, ϕ)

Single qubit parametric rotation $\operatorname{R}(\theta, \lambda)$ gate.

It performs a rotation of $\theta$ radians for the target qubit around an XY-plane axis of the Bloch sphere determined by $\cos(\phi)\mathbf{x} + \sin(\phi)\mathbf{y}$.

Matrix representation

\[\operatorname{R}(\theta,\phi) = \begin{pmatrix} \cos\frac{\theta}{2} & -ie^{-i\phi}\sin\frac{\theta}{2} \\ -ie^{i\phi}\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\]

Examples

julia> @variables θ ϕ
2-element Vector{Symbolics.Num}:
 θ
 ϕ

julia> GateR(θ, ϕ)
R(θ, ϕ)

julia> matrix(GateR(2.023, 1.989))
2×2 Matrix{ComplexF64}:
 0.53059+0.0im       -0.77458+0.344239im
 0.77458+0.344239im   0.53059+0.0im

julia> c = push!(Circuit(), GateR(θ, ϕ), 1)
1-qubit circuit with 1 instructions:
└── R(θ,ϕ) @ q[1]

julia> push!(c, GateR(π/2, π/4), 2)
2-qubit circuit with 2 instructions:
├── R(θ,ϕ) @ q[1]
└── R(π/2,π/4) @ q[2]

Decomposition

julia> decompose(GateR(θ, ϕ))
1-qubit circuit with 1 instructions:
└── U3(θ,-1.5707963267948966 + ϕ,1.5707963267948966 - ϕ) @ q[1]

GateRX(θ)

Single qubit parametric rotation $\operatorname{R}_X(\theta)$ gate.

It performs a rotation of $\theta$ radians around the X-axis of the Bloch sphere of the target qubit.

Matrix representation

\[\operatorname{R}_X(\theta) = \begin{pmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRX(θ)
RX(θ)

julia> matrix(GateRX(1.989))
2×2 Matrix{ComplexF64}:
 0.544922+0.0im            0.0-0.838487im
      0.0-0.838487im  0.544922+0.0im

julia> c = push!(Circuit(), GateRX(θ), 1)
1-qubit circuit with 1 instructions:
└── RX(θ) @ q[1]

julia> push!(c, GateRX(π/2), 2)
2-qubit circuit with 2 instructions:
├── RX(θ) @ q[1]
└── RX(π/2) @ q[2]

Decomposition

julia> decompose(GateRX(θ))
1-qubit circuit with 1 instructions:
└── U(θ,-1π/2,π/2) @ q[1]

GateRY(θ)

Single qubit parametric rotation $\operatorname{R}_Y(\theta)$ gate.

It performss a rotation of $\theta$ radians around the Y-axis of the Bloch sphere of the target qubit.

Matrix representation

\[\operatorname{R}_Y(\theta) = \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRY(θ)
RY(θ)

julia> matrix(GateRY(1.989))
2×2 Matrix{Float64}:
 0.544922  -0.838487
 0.838487   0.544922

julia> c = push!(Circuit(), GateRY(θ), 1)
1-qubit circuit with 1 instructions:
└── RY(θ) @ q[1]

julia> push!(c, GateRY(π/2), 2)
2-qubit circuit with 2 instructions:
├── RY(θ) @ q[1]
└── RY(π/2) @ q[2]

Decomposition

julia> decompose(GateRY(θ))
1-qubit circuit with 1 instructions:
└── U(θ,0,0) @ q[1]

GateRZ(λ)

Single qubit parametric rotation $\operatorname{R}_Z(\lambda)$ gate.

It performs a rotation of $\lambda$ radians around the Z-axis of the Bloch sphere for the target qubit.

Matrix representation

\[\operatorname{RZ}(\lambda) = \begin{pmatrix} e^{-i\frac{\lambda}{2}} & 0 \\ 0 & e^{i\frac{\lambda}{2}} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRZ(θ)
RZ(θ)

julia> matrix(GateRZ(1.989))
2×2 Matrix{ComplexF64}:
 0.544922-0.838487im       0.0+0.0im
      0.0+0.0im       0.544922+0.838487im

julia> c = push!(Circuit(), GateRZ(θ), 1)
1-qubit circuit with 1 instructions:
└── RZ(θ) @ q[1]

julia> push!(c, GateRZ(π/2), 2)
2-qubit circuit with 2 instructions:
├── RZ(θ) @ q[1]
└── RZ(π/2) @ q[2]

Decomposition

julia> decompose(GateRZ(θ))
1-qubit circuit with 1 instructions:
└── U(0,0,θ,(-1//2)*θ) @ q[1]

GateSWAP()

Two qubit SWAP gate.

See also GateISWAP, GateCSWAP

Matrix representation

\[\operatorname{SWAP} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\]

Examples

julia> GateSWAP()
SWAP

julia> matrix(GateSWAP())
4×4 Matrix{Float64}:
 1.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  0.0  1.0

julia> c = push!(Circuit(), GateSWAP(), 1, 2)
2-qubit circuit with 1 instructions:
└── SWAP @ q[1:2]

julia> push!(c, GateSWAP, 3, 4)
4-qubit circuit with 2 instructions:
├── SWAP @ q[1:2]
└── SWAP @ q[3:4]

julia> power(GateSWAP(), 2), inverse(GateSWAP())
(Parallel(2, GateID()), GateSWAP())

Decomposition

julia> decompose(GateSWAP())
2-qubit circuit with 3 instructions:
├── CX @ q[1], q[2]
├── CX @ q[2], q[1]
└── CX @ q[1], q[2]

GateISWAP()

Two qubit ISWAP gate.

See also GateSWAP.

Matrix representation

\[\operatorname{ISWAP} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\]

Examples

julia> GateISWAP()
ISWAP

julia> matrix(GateISWAP())
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+1.0im  0.0+0.0im
 0.0+0.0im  0.0+1.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  1.0+0.0im

julia> c = push!(Circuit(), GateISWAP(), 1, 2)
2-qubit circuit with 1 instructions:
└── ISWAP @ q[1:2]

julia> push!(c, GateISWAP, 3, 4)
4-qubit circuit with 2 instructions:
├── ISWAP @ q[1:2]
└── ISWAP @ q[3:4]

julia> power(GateISWAP(), 2), inverse(GateISWAP())
(GateISWAP()^2, Inverse(GateISWAP()))

Decomposition

julia> decompose(GateISWAP())
2-qubit circuit with 6 instructions:
├── S @ q[1]
├── S @ q[2]
├── H @ q[1]
├── CX @ q[1], q[2]
├── CX @ q[2], q[1]
└── H @ q[2]

GateCX()

Two qubit Controlled-$X$ gate (or CNOT).

Matrix representation

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

Examples

julia> GateCX(), numcontrols(GateCX()), numtargets(GateCX())
(GateCX(), 1, 1)

julia> matrix(GateCX())
4×4 Matrix{Float64}:
 1.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  0.0  1.0
 0.0  0.0  1.0  0.0

julia> c = push!(Circuit(), GateCX(), 1, 2)
2-qubit circuit with 1 instructions:
└── CX @ q[1], q[2]

julia> power(GateCX(), 2), inverse(GateCX())
(Control(GateID()), GateCX())

Decomposition

julia> decompose(GateCX())
2-qubit circuit with 1 instructions:
└── CX @ q[1], q[2]

GateCY()

Two qubit Controlled-$Y$ gate.

Matrix representation

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

Examples

julia> GateCY(), numcontrols(GateCY()), numtargets(GateCY())
(GateCY(), 1, 1)

julia> matrix(GateCY())
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0-1.0im
 0.0+0.0im  0.0+0.0im  0.0+1.0im  0.0+0.0im

julia> c = push!(Circuit(), GateCY(), 1, 2)
2-qubit circuit with 1 instructions:
└── CY @ q[1], q[2]

julia> power(GateCY(), 2), inverse(GateCY())
(Control(GateID()), GateCY())

Decomposition

julia> decompose(GateCY())
2-qubit circuit with 3 instructions:
├── S† @ q[2]
├── CX @ q[1], q[2]
└── S @ q[2]

GateCZ()

Two qubit Controlled-$Z$ gate.

Matrix representation

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

Examples

julia> GateCZ(), numcontrols(GateCZ()), numtargets(GateCZ())
(GateCZ(), 1, 1)

julia> matrix(GateCZ())
4×4 Matrix{Float64}:
 1.0  0.0  0.0   0.0
 0.0  1.0  0.0   0.0
 0.0  0.0  1.0   0.0
 0.0  0.0  0.0  -1.0

julia> c = push!(Circuit(), GateCZ(), 1, 2)
2-qubit circuit with 1 instructions:
└── CZ @ q[1], q[2]

julia> power(GateCZ(), 2), inverse(GateCZ())
(Control(GateID()), GateCZ())

Decomposition

julia> decompose(GateCZ())
2-qubit circuit with 3 instructions:
├── H @ q[2]
├── CX @ q[1], q[2]
└── H @ q[2]

GateCH()

Two qubit controlled-Hadamard gate.

Matrix representation

\[\operatorname{CH} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}\]

Examples

julia> GateCH(), numcontrols(GateCH), numtargets(GateCH)
(GateCH(), 1, 1)

julia> matrix(GateCH())
4×4 Matrix{Float64}:
 1.0  0.0  0.0        0.0
 0.0  1.0  0.0        0.0
 0.0  0.0  0.707107   0.707107
 0.0  0.0  0.707107  -0.707107

julia> c = push!(Circuit(), GateCH(), 1, 2)
2-qubit circuit with 1 instructions:
└── CH @ q[1], q[2]

julia> power(GateCH(), 2), inverse(GateCH())
(Control(GateH()^2), GateCH())

Decomposition

julia> decompose(GateCH())
2-qubit circuit with 7 instructions:
├── S @ q[2]
├── H @ q[2]
├── T @ q[2]
├── CX @ q[1], q[2]
├── T† @ q[2]
├── H @ q[2]
└── S† @ q[2]

GateCS()

Two qubit Controlled-S gate.

See also GateS, Control.

Matrix representation

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

Examples

julia> GateCS(), numcontrols(GateCS()), numtargets(GateCS())
(GateCS(), 1, 1)

julia> matrix(GateCS())
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  1.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+1.0im

julia> c = push!(Circuit(), GateCS(), 1, 2)
2-qubit circuit with 1 instructions:
└── CS @ q[1], q[2]

julia> power(GateCS(), 2), inverse(GateCS())
(GateCZ(), Control(Inverse(GateS())))

Decomposition

julia> decompose(GateCS())
2-qubit circuit with 1 instructions:
└── CP(π/2) @ q[1], q[2]

GateCSDG()

Adjoint of two qubit Controlled-S gate.

See also GateS, Control.

Matrix representation

\[\operatorname{CS}^{\dagger} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -i \end{pmatrix}\]

Examples

julia> GateCSDG(), numcontrols(GateCSDG()), numtargets(GateCSDG())
(Control(Inverse(GateS())), 1, 1)

julia> matrix(GateCSDG())
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  1.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0-1.0im

julia> c = push!(Circuit(), GateCSDG(), 1, 2)
2-qubit circuit with 1 instructions:
└── C(S†) @ q[1], q[2]

julia> power(GateCSDG(), 2), inverse(GateCSDG())
(Control((Inverse(GateS()))^2), GateCS())

Decomposition

julia> decompose(GateCSDG())
2-qubit circuit with 1 instructions:
└── CP(-1π/2) @ q[1], q[2]

GateCSX()

Two qubit Controled-SX gate.

See also GateSX, GateCSXDG, Control.

Matrix representation

\[\operatorname{CSX} =\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1+i}{2} & \frac{1-i}{2} \\ 0 & 0 & \frac{1-i}{2} & \frac{1+i}{2} \end{pmatrix}\]

Examples

julia> GateCSX(), numcontrols(GateCSX()), numtargets(GateCSX())
(GateCSX(), 1, 1)

julia> matrix(GateCSX())
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.5+0.5im  0.5-0.5im
 0.0+0.0im  0.0+0.0im  0.5-0.5im  0.5+0.5im

julia> c = push!(Circuit(), GateCSX(), 1, 2)
2-qubit circuit with 1 instructions:
└── CSX @ q[1], q[2]

julia> power(GateCSX(), 2), inverse(GateCSX())
(GateCX(), Control(Inverse(GateSX())))

Decomposition

julia> decompose(GateCSX())
2-qubit circuit with 3 instructions:
├── H @ q[2]
├── CU1(π/2) @ q[1], q[2]
└── H @ q[2]

GateCSXDG()

Two qubit CSX-dagger gate. (Control on second qubit)

See also GateSX, GateCSXDG, Control.

Matrix representation

\[\operatorname{CSX}^{\dagger} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 0 & 0 & \frac{1-i}{2} & \frac{1+i}{2} \\ 0 & 0 & \frac{1+i}{2} & \frac{1-i}{2} \end{pmatrix}\]

Examples

julia> GateCSXDG(), numcontrols(GateCSXDG()), numtargets(GateCSXDG())
(Control(Inverse(GateSX())), 1, 1)

julia> matrix(GateCSXDG())
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.5-0.5im  0.5+0.5im
 0.0+0.0im  0.0+0.0im  0.5+0.5im  0.5-0.5im

julia> c = push!(Circuit(), GateCSXDG(), 1, 2)
2-qubit circuit with 1 instructions:
└── C(SX†) @ q[1], q[2]

julia> power(GateCSXDG(), 2), inverse(GateCSXDG())
(Control((Inverse(GateSX()))^2), GateCSX())

Decomposition

julia> decompose(GateCSXDG())
2-qubit circuit with 3 instructions:
├── H @ q[2]
├── CU1(-1π/2) @ q[1], q[2]
└── H @ q[2]

GateCU(θ, ϕ, λ)

Controlled-$\operatorname{U}(\theta, \phi, \lambda)$ gate.

See also Control, GateU.

\[\operatorname{CU}(\theta, \phi, \lambda, \gamma) = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{i\gamma} \cos\left(\frac{\theta}{2}\right) & -e^{i\gamma} e^{i\lambda}\sin\left(\frac{\theta}{2}\right) \\ 0 & 0 & e^{i\gamma} \mathrm{e}^{i\phi}\sin\left(\frac{\theta}{2}\right) & e^{i\gamma} \mathrm{e}^{i(\phi+\lambda)}\cos\left(\frac{\theta}{2}\right) \end{pmatrix}\]

Examples

julia> @variables θ ϕ λ γ
4-element Vector{Symbolics.Num}:
 θ
 ϕ
 λ
 γ

julia> GateCU(θ, ϕ, λ, γ), numcontrols(GateCU(θ, ϕ, λ, γ)), numtargets(GateCU(θ, ϕ, λ, γ))
(GateCU(θ, ϕ, λ, γ), 1, 1)

julia> matrix(GateCU(2.023, 0.5, 0.1, 0.2))
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im       0.0+0.0im             0.0+0.0im
 0.0+0.0im  1.0+0.0im       0.0+0.0im             0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.520013+0.105412im  -0.809771-0.250491im
 0.0+0.0im  0.0+0.0im  0.648302+0.546057im   0.369666+0.380622im

julia> c = push!(Circuit(), GateCU(θ, ϕ, λ, γ), 1, 2)
2-qubit circuit with 1 instructions:
└── CU(θ,ϕ,λ,γ) @ q[1], q[2]

julia> push!(c, GateCU(π/8, π/2, π/4, π/7), 1, 2)
2-qubit circuit with 2 instructions:
├── CU(θ,ϕ,λ,γ) @ q[1], q[2]
└── CU(π/8,π/2,π/4,π/7) @ q[1], q[2]

julia> power(GateCU(θ, ϕ, λ, γ), 2), inverse(GateCU(θ, ϕ, λ, γ))
(Control(GateU(θ, ϕ, λ, γ)^2), GateCU(-θ, -λ, -ϕ, -γ))

Decomposition

julia> decompose(GateCU(θ, λ, ϕ, γ))
2-qubit circuit with 7 instructions:
├── P(γ) @ q[1]
├── P((1//2)*(λ + ϕ)) @ q[1]
├── P((1//2)*(-λ + ϕ)) @ q[2]
├── CX @ q[1], q[2]
├── U((-1//2)*θ,0,(1//2)*(-λ - ϕ)) @ q[2]
├── CX @ q[1], q[2]
└── U((1//2)*θ,λ,0) @ q[2]

GateCP(θ)

Controlled-$\operatorname{P}(\lambda)$ gate.

See also Control, GateRZ.

Examples

julia> @variables λ
1-element Vector{Symbolics.Num}:
 λ

julia> GateCP(λ), numcontrols(GateCP(λ)), numtargets(GateCP(λ))
(GateCP(λ), 1, 1)

julia> matrix(GateCP(1.989))
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im       0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  1.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  -0.40612+0.91382im

julia> c = push!(Circuit(), GateCP(λ), 1, 2)
2-qubit circuit with 1 instructions:
└── CP(λ) @ q[1], q[2]

julia> push!(c, GateCP(π/8), 1, 2)
2-qubit circuit with 2 instructions:
├── CP(λ) @ q[1], q[2]
└── CP(π/8) @ q[1], q[2]

julia> power(GateCP(λ), 2), inverse(GateCP(λ))
(GateCP(2λ), GateCP(-λ))

Decomposition

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> decompose(GateCP(θ))
2-qubit circuit with 5 instructions:
├── P((1//2)*θ) @ q[1]
├── CX @ q[1], q[2]
├── P((-1//2)*θ) @ q[2]
├── CX @ q[1], q[2]
└── P((1//2)*θ) @ q[2]

GateCRX(θ)

Controlled-$\operatorname{R}_X(\theta)$ gate.

See also Control, GateRX.

Matrix representation

\[\operatorname{CRX}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ 0 & 0 & -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateCRX(θ), numcontrols(GateCRX(θ)), numtargets(GateCRX(θ))
(GateCRX(θ), 1, 1)

julia> matrix(GateCRX(1.989))
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im       0.0+0.0im            0.0+0.0im
 0.0+0.0im  1.0+0.0im       0.0+0.0im            0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.544922+0.0im            0.0-0.838487im
 0.0+0.0im  0.0+0.0im       0.0-0.838487im  0.544922+0.0im

julia> c = push!(Circuit(), GateCRX(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── CRX(θ) @ q[1], q[2]

julia> push!(c, GateCRX(π/8), 1, 2)
2-qubit circuit with 2 instructions:
├── CRX(θ) @ q[1], q[2]
└── CRX(π/8) @ q[1], q[2]

julia> power(GateCRX(θ), 2), inverse(GateCRX(θ))
(GateCRX(2θ), GateCRX(-θ))

Decomposition

julia> decompose(GateCRX(θ))
2-qubit circuit with 5 instructions:
├── P(π/2) @ q[2]
├── CX @ q[1], q[2]
├── U((-1//2)*θ,0,0) @ q[2]
├── CX @ q[1], q[2]
└── U((1//2)*θ,-1π/2,0) @ q[2]

GateCRY(θ)

Controlled-$\operatorname{R}_Y(\theta)$ gate.

See also Control, GateRY.

Matrix representation

\[\operatorname{CRY}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ 0 & 0 & \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateCRY(θ), numcontrols(GateCRY(θ)), numtargets(GateCRY(θ))
(Control(GateRY(θ)), 1, 1)

julia> matrix(GateCRY(1.989))
4×4 Matrix{Float64}:
 1.0  0.0  0.0        0.0
 0.0  1.0  0.0        0.0
 0.0  0.0  0.544922  -0.838487
 0.0  0.0  0.838487   0.544922

julia> c = push!(Circuit(), GateCRY(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── CRY(θ) @ q[1], q[2]

julia> push!(c, GateCRY(π/8), 1, 2)
2-qubit circuit with 2 instructions:
├── CRY(θ) @ q[1], q[2]
└── CRY(π/8) @ q[1], q[2]

julia> power(GateCRY(θ), 2), inverse(GateCRY(θ))
(Control(GateRY(2θ)), Control(GateRY(-θ)))

Decomposition

julia> decompose(GateCRY(θ))
2-qubit circuit with 4 instructions:
├── RY((1//2)*θ) @ q[2]
├── CX @ q[1], q[2]
├── RY((-1//2)*θ) @ q[2]
└── CX @ q[1], q[2]

GateCRZ(θ)

Controlled-$\operatorname{R}_Z(\theta)$ gate.

See also Control, GateRZ.

Matrix representation

\[ \operatorname{CRZ}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{-i\frac{\lambda}{2}} & 0 \\ 0 & 0 & 0 & e^{i\frac{\lambda}{2}} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateCRZ(θ), numcontrols(GateCRZ(θ)), numtargets(GateCRZ(θ))
(Control(GateRZ(θ)), 1, 1)

julia> matrix(GateCRZ(1.989))
4×4 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im       0.0+0.0im            0.0+0.0im
 0.0+0.0im  1.0+0.0im       0.0+0.0im            0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.544922-0.838487im       0.0+0.0im
 0.0+0.0im  0.0+0.0im       0.0+0.0im       0.544922+0.838487im

julia> c = push!(Circuit(), GateCRZ(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── CRZ(θ) @ q[1], q[2]

julia> push!(c, GateCRZ(π/8), 1, 2)
2-qubit circuit with 2 instructions:
├── CRZ(θ) @ q[1], q[2]
└── CRZ(π/8) @ q[1], q[2]

julia> power(GateCRZ(θ), 2), inverse(GateCRZ(θ))
(Control(GateRZ(2θ)), Control(GateRZ(-θ)))

Decomposition

julia> decompose(GateCRZ(θ))
2-qubit circuit with 4 instructions:
├── RZ((1//2)*θ) @ q[2]
├── CX @ q[1], q[2]
├── RZ((-1//2)*θ) @ q[2]
└── CX @ q[1], q[2]

GateECR()

Two qubit ECR echo gate.

Matrix representation

\[\operatorname{ECR} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 & i \\ 1 & 0 & -i & 0 \\ 0 & i & 0 & 1 \\ -i & 0 & 1 & 0 \end{pmatrix}\]

Examples

julia> GateECR()
ECR

julia> matrix(GateECR())
4×4 Matrix{ComplexF64}:
      0.0+0.0im            0.0+0.0im       …       0.0+0.707107im
      0.0+0.0im            0.0+0.0im          0.707107+0.0im
 0.707107+0.0im            0.0-0.707107im          0.0+0.0im
      0.0-0.707107im  0.707107+0.0im               0.0+0.0im

julia> c = push!(Circuit(), GateECR(), 1, 2)
2-qubit circuit with 1 instructions:
└── ECR @ q[1:2]

julia> power(GateECR(), 2), inverse(GateECR())
(Parallel(2, GateID()), GateECR())

Decomposition

julia> decompose(GateECR())
2-qubit circuit with 3 instructions:
├── RZX(π/4) @ q[1:2]
├── X @ q[1]
└── RZX(-1π/4) @ q[1:2]

GateDCX()

Two qubit double-CNOT (control on first qubit and then second) OR DCX gate.

Matrix representation

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

Examples

julia> GateDCX()
DCX

julia> matrix(GateDCX())
4×4 Matrix{Float64}:
 1.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0
 0.0  0.0  0.0  1.0
 0.0  1.0  0.0  0.0

julia> c = push!(Circuit(), GateDCX(), 1, 2)
2-qubit circuit with 1 instructions:
└── DCX @ q[1:2]

julia> power(GateDCX(), 2), inverse(GateDCX())
(Inverse(GateDCX()), Inverse(GateDCX()))

Decomposition

julia> decompose(GateDCX())
2-qubit circuit with 2 instructions:
├── CX @ q[1], q[2]
└── CX @ q[2], q[1]

GateRXX(θ)

Parametric two qubit $X \otimes X$ interaction \operatorname{R}_{XX}(\theta) gate.

It corresponds to a rotation of $\theta$ radians along the XX axis of the two-qubit Bloch sphere.

See also GateRYY, GateRZZ, GateRZX, GateXXplusYY, GateXXminusYY.

Matrix representation

\[\operatorname{R}_{XX}(\theta) = \begin{pmatrix} \cos\left(\frac{\theta}{2}\right) & 0 & 0 & -i\sin\left(\frac{\theta}{2}\right) \\ 0 & \cos\left(\frac{\theta}{2}\right) & -i\sin\left(\frac{\theta}{2}\right) & 0 \\ 0 & -i\sin\left(\frac{\theta}{2}\right) & \cos\left(\frac{\theta}{2}\right) & 0 \\ -i\sin\left(\frac{\theta}{2}\right) & 0 & 0 & \cos\left(\frac{\theta}{2}\right) \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRXX(θ)
RXX(θ)

julia> matrix(GateRXX(θ))
4×4 Matrix{Complex{Symbolics.Num}}:
     cos((1//2)*θ)                  0                  0  -im*sin((1//2)*θ)
                 0      cos((1//2)*θ)  -im*sin((1//2)*θ)                  0
                 0  -im*sin((1//2)*θ)      cos((1//2)*θ)                  0
 -im*sin((1//2)*θ)                  0                  0      cos((1//2)*θ)

julia> c = push!(Circuit(), GateRXX(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── RXX(θ) @ q[1:2]

julia> push!(c, GateRXX(π/2), 1, 2)
2-qubit circuit with 2 instructions:
├── RXX(θ) @ q[1:2]
└── RXX(π/2) @ q[1:2]

Decomposition

julia> decompose(GateRXX(θ))
2-qubit circuit with 7 instructions:
├── H @ q[1]
├── H @ q[2]
├── CX @ q[1], q[2]
├── RZ(θ) @ q[2]
├── CX @ q[1], q[2]
├── H @ q[2]
└── H @ q[1]

GateRYY(θ)

Parametric two qubit $Y \otimes Y$ interaction \operatorname{R}_{YY}(\theta) gate.

It corresponds to a rotation of $\theta$ radians along the YY axis of the two-qubit Bloch sphere.

See also GateRXX, GateRZZ, GateRZX, GateXXplusYY, GateXXminusYY.

Matrix representation

\[\operatorname{R}_{YY}(\theta) = \begin{pmatrix} \cos\left(\frac{\theta}{2}\right) & 0 & 0 & i\sin\left(\frac{\theta}{2}\right) \\ 0 & \cos\left(\frac{\theta}{2}\right) & -i\sin\left(\frac{\theta}{2}\right) & 0 \\ 0 & -i\sin\left(\frac{\theta}{2}\right) & \cos\left(\frac{\theta}{2}\right) & 0 \\ i\sin\left(\frac{\theta}{2}\right) & 0 & 0 & \cos\left(\frac{\theta}{2}\right) \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRYY(θ)
RYY(θ)

julia> matrix(GateRYY(θ))
4×4 Matrix{Complex{Symbolics.Num}}:
    cos((1//2)*θ)                  0                  0  im*sin((1//2)*θ)
                0      cos((1//2)*θ)  -im*sin((1//2)*θ)                 0
                0  -im*sin((1//2)*θ)      cos((1//2)*θ)                 0
 im*sin((1//2)*θ)                  0                  0     cos((1//2)*θ)

julia> c = push!(Circuit(), GateRYY(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── RYY(θ) @ q[1:2]

julia> push!(c, GateRYY(π/2), 1, 2)
2-qubit circuit with 2 instructions:
├── RYY(θ) @ q[1:2]
└── RYY(π/2) @ q[1:2]

Decomposition

julia> decompose(GateRYY(θ))
2-qubit circuit with 7 instructions:
├── RX(π/2) @ q[1]
├── RX(π/2) @ q[2]
├── CX @ q[1], q[2]
├── RZ(θ) @ q[2]
├── CX @ q[1], q[2]
├── RX(-1π/2) @ q[1]
└── RX(-1π/2) @ q[2]

GateRZX(θ)

Parametric two qubit $Z \otimes X$ interaction \operatorname{R}_{ZX}(\theta) gate.

It corresponds to a rotation of $\theta$ radians about ZX.

See also GateRXX, GateRYY, GateRZZ, GateXXplusYY, GateXXminusYY.

Matrix representation

\[\operatorname{RZX}(\theta) =\begin{pmatrix} \cos(\frac{\theta}{2}) & -i\sin(\frac{\theta}{2}) & 0 & 0 \\ -i\sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}) & 0 & 0 \\ 0 & 0 & \cos(\frac{\theta}{2}) & i\sin(\frac{\theta}{2}) \\ 0 & 0 & i\sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}) \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRZX(θ)
RZX(θ)

julia> matrix(GateRZX(θ))
4×4 Matrix{Complex{Symbolics.Num}}:
     cos((1//2)*θ)  -im*sin((1//2)*θ)                 0                 0
 -im*sin((1//2)*θ)      cos((1//2)*θ)                 0                 0
                 0                  0     cos((1//2)*θ)  im*sin((1//2)*θ)
                 0                  0  im*sin((1//2)*θ)     cos((1//2)*θ)

julia> c = push!(Circuit(), GateRZX(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── RZX(θ) @ q[1:2]

julia> push!(c, GateRZX(π/2), 1, 2)
2-qubit circuit with 2 instructions:
├── RZX(θ) @ q[1:2]
└── RZX(π/2) @ q[1:2]

Decomposition

julia> decompose(GateRZX(θ))
2-qubit circuit with 5 instructions:
├── H @ q[2]
├── CX @ q[1], q[2]
├── RZ(θ) @ q[2]
├── CX @ q[1], q[2]
└── H @ q[2]

GateRZZ(θ)

Parametric two qubit $Z \otimes Z$ interaction \operatorname{R}_{ZZ}(\theta) gate.

It corresponds to a rotation of $\theta$ radians along the ZZ axis of the two-qubit Bloch sphere.

See also GateRXX, GateRYY, GateRZX, GateXXplusYY, GateXXminusYY.

Matrix representation

\[\operatorname{R}_{ZZ}(\theta) = \begin{pmatrix} e^{-i\frac{\theta}{2}} & 0 & 0 & 0 \\ 0 & e^{i\frac{\theta}{2}} & 0 & 0 \\ 0 & 0 & e^{i\frac{\theta}{2}} & 0 \\ 0 & 0 & 0 & e^{-i\frac{\theta}{2}} \end{pmatrix}\]

Examples

julia> @variables θ
1-element Vector{Symbolics.Num}:
 θ

julia> GateRZZ(θ)
RZZ(θ)

julia> matrix(GateRZZ(θ))
4×4 Matrix{Complex{Symbolics.Num}}:
 cos((-1//2)*θ) + im*sin((-1//2)*θ)  …                           0
                          0                                      0
                          0                                      0
                          0             cos((-1//2)*θ) + im*sin((-1//2)*θ)

julia> c = push!(Circuit(), GateRZZ(θ), 1, 2)
2-qubit circuit with 1 instructions:
└── RZZ(θ) @ q[1:2]

julia> push!(c, GateRZZ(π/2), 1, 2)
2-qubit circuit with 2 instructions:
├── RZZ(θ) @ q[1:2]
└── RZZ(π/2) @ q[1:2]

Decomposition

julia> decompose(GateRZZ(θ))
2-qubit circuit with 3 instructions:
├── CX @ q[1], q[2]
├── RZ(θ) @ q[2]
└── CX @ q[1], q[2]

GateXXminusYY(θ, β)

Parametric two qubit $X \otimes X - Y \otimes Y$ interaction $\operatorname{(XX-YY)}(\theta, \beta)$ gate, where $\theta$ and $\beta$ are the rotation and phase angles.

See also GateRXX, GateRYY, GateRZZ, GateRZX, GateXXplusYY.

Matrix Representation

\[\operatorname{(XX-YY)}(\theta, \beta) = \begin{pmatrix} \cos(\frac{\theta}{2}) & 0 & 0 & -i\sin(\frac{\theta}{2})e^{-i\beta} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -i\sin(\frac{\theta}{2})e^{i\beta} & 0 & 0 & \cos(\frac{\theta}{2}) \end{pmatrix}\]

Examples

julia> @variables θ β
2-element Vector{Symbolics.Num}:
 θ
 β

julia> GateXXminusYY(θ, β)
XXminusYY(θ, β)

julia> matrix(GateXXminusYY(θ, β))
4×4 Matrix{Complex{Symbolics.Num}}:
          cos((1//2)*θ)                          …  sin(-β)*sin((1//2)*θ) - im*cos(-β)*sin((1//2)*θ)
                      0                                                            0
                      0                                                            0
 sin(β)*sin((1//2)*θ) - im*cos(β)*sin((1//2)*θ)                        cos((1//2)*θ)

julia> c = push!(Circuit(), GateXXminusYY(θ, β), 1, 2)
2-qubit circuit with 1 instructions:
└── XXminusYY(θ,β) @ q[1:2]

julia> push!(c, GateXXminusYY(π/2, 0.0), 1, 2)
2-qubit circuit with 2 instructions:
├── XXminusYY(θ,β) @ q[1:2]
└── XXminusYY(π/2,0π) @ q[1:2]

Decomposition

julia> decompose(GateXXminusYY(θ, β))
2-qubit circuit with 14 instructions:
├── RZ(-β) @ q[2]
├── RZ(-1π/2) @ q[1]
├── SX @ q[1]
├── RZ(π/2) @ q[1]
├── S @ q[2]
├── CX @ q[1], q[2]
├── RY((1//2)*θ) @ q[1]
├── RY((-1//2)*θ) @ q[2]
├── CX @ q[1], q[2]
├── S† @ q[2]
├── RZ(-1π/2) @ q[1]
├── SX† @ q[1]
├── RZ(π/2) @ q[1]
└── RZ(β) @ q[2]

GateXXplusYY(θ, β)

Parametric two qubit $X \otimes X + Y \otimes Y$ interaction $\operatorname{(XX+YY)}(\theta, \beta)$ gate, where $\theta$ and $\beta$ are the rotation and phase angles.

See also GateRXX, GateRYY, GateRZZ, GateRZX, GateXXminusYY.

Matrix representation

\[\operatorname{(XX+YY)}(\theta, \beta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\frac{\theta}{2}) & -i\sin(\frac{\theta}{2})e^{i\beta} & 0 \\ 0 & -i\sin(\frac{\theta}{2})e^{-i\beta} & \cos(\frac{\theta}{2}) & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\]

Examples

julia> @variables θ β
2-element Vector{Symbolics.Num}:
 θ
 β

julia> GateXXplusYY(θ, β)
XXplusYY(θ, β)

julia> matrix(GateXXplusYY(θ, β))
4×4 Matrix{Complex{Symbolics.Num}}:
 1                                 0                  …  0
 0                     cos((1//2)*θ)                     0
 0  sin(-β)*sin((1//2)*θ) - im*cos(-β)*sin((1//2)*θ)     0
 0                                 0                     1

julia> c = push!(Circuit(), GateXXplusYY(θ, β), 1, 2)
2-qubit circuit with 1 instructions:
└── XXplusYY(θ,β) @ q[1:2]

julia> push!(c, GateXXplusYY(π/2, 0), 1, 2)
2-qubit circuit with 2 instructions:
├── XXplusYY(θ,β) @ q[1:2]
└── XXplusYY(π/2,0) @ q[1:2]

Decomposition

julia> decompose(GateXXplusYY(θ, β))
2-qubit circuit with 14 instructions:
├── RZ(β) @ q[1]
├── RZ(-1π/2) @ q[2]
├── SX @ q[2]
├── RZ(π/2) @ q[2]
├── S @ q[1]
├── CX @ q[2], q[1]
├── RY((-1//2)*θ) @ q[2]
├── RY((-1//2)*θ) @ q[1]
├── CX @ q[2], q[1]
├── S† @ q[1]
├── RZ(-1π/2) @ q[2]
├── SX† @ q[2]
├── RZ(π/2) @ q[2]
└── RZ(-β) @ q[1]

GateCSWAP()

Three-qubit, controlled-$\operatorname{SWAP}$ gate.

See also Control, GateU.

Examples

julia> GateCSWAP(), numcontrols(GateCSWAP()), numtargets(GateCSWAP())
(GateCSWAP(), 1, 2)

julia> matrix(GateCSWAP())
8×8 Matrix{Float64}:
 1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0
 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0

julia> c = push!(Circuit(), GateCSWAP(), 1, 2, 3)
3-qubit circuit with 1 instructions:
└── CSWAP @ q[1], q[2:3]

julia> power(GateCSWAP(), 2), inverse(GateCSWAP())
(Control(Parallel(2, GateID())), GateCSWAP())

Decomposition

julia> decompose(GateCSWAP())
3-qubit circuit with 3 instructions:
├── CX @ q[3], q[2]
├── C₂X @ q[1:2], q[3]
└── CX @ q[3], q[2]

GateC3X()

Four qubit, triply-controlled $X$ gate.

Examples

julia> GateC3X(), numcontrols(GateC3X()), numtargets(GateC3X())
(Control(3, GateX()), 3, 1)

julia> matrix(GateC3X())
16×16 Matrix{Float64}:
 1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     1.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  1.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  1.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  1.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  1.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  1.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  1.0  0.0

julia> c = push!(Circuit(), GateC3X(), 1, 2, 3, 4)
4-qubit circuit with 1 instructions:
└── C₃X @ q[1:3], q[4]

julia> power(GateC3X(), 2), inverse(GateC3X())
(Control(3, GateID()), Control(3, GateX()))

Decomposition

julia> decompose(GateC3X())
4-qubit circuit with 31 instructions:
├── H @ q[4]
├── P(π/8) @ q[1]
├── P(π/8) @ q[2]
├── P(π/8) @ q[3]
├── P(π/8) @ q[4]
├── CX @ q[1], q[2]
├── P(-1π/8) @ q[2]
├── CX @ q[1], q[2]
├── CX @ q[2], q[3]
⋮   ⋮
├── CX @ q[1], q[4]
├── P(π/8) @ q[4]
├── CX @ q[3], q[4]
├── P(-1π/8) @ q[4]
├── CX @ q[2], q[4]
├── P(π/8) @ q[4]
├── CX @ q[3], q[4]
├── P(-1π/8) @ q[4]
├── CX @ q[1], q[4]
└── H @ q[4]

GateCCX()

Three-qubit, doubly-controlled $X$ gate.

Examples

julia> GateCCX(), numcontrols(GateCCX()), numtargets(GateCCX())
(Control(2, GateX()), 2, 1)

julia> matrix(GateCCX())
8×8 Matrix{Float64}:
 1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0
 0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0

julia> c = push!(Circuit(), GateCCX(), 1, 2, 3)
3-qubit circuit with 1 instructions:
└── C₂X @ q[1:2], q[3]

julia> power(GateCCX(), 2), inverse(GateCCX())
(Control(2, GateID()), Control(2, GateX()))

Decomposition

julia> decompose(GateCCX())
3-qubit circuit with 15 instructions:
├── H @ q[3]
├── CX @ q[2], q[3]
├── T† @ q[3]
├── CX @ q[1], q[3]
├── T @ q[3]
├── CX @ q[2], q[3]
├── T† @ q[3]
├── CX @ q[1], q[3]
├── T @ q[2]
├── T @ q[3]
├── H @ q[3]
├── CX @ q[1], q[2]
├── T @ q[1]
├── T† @ q[2]
└── CX @ q[1], q[2]

GateCCP(λ)

Three-qubit, doubly-controlled phase gate.

Examples

julia> @variables λ
1-element Vector{Symbolics.Num}:
 λ

julia> GateCCP(λ), numcontrols(GateCCP(λ)), numtargets(GateCCP(λ))
(GateCCP(λ), 2, 1)

julia> matrix(GateCCP(1.989))
8×8 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  …  0.0+0.0im       0.0+0.0im
 0.0+0.0im  1.0+0.0im  0.0+0.0im  0.0+0.0im     0.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  1.0+0.0im  0.0+0.0im     0.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  1.0+0.0im     0.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im     0.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  …  0.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im     1.0+0.0im       0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im     0.0+0.0im  -0.40612+0.91382im

julia> c = push!(Circuit(), GateCCP(λ), 1, 2, 3)
3-qubit circuit with 1 instructions:
└── CCP(λ) @ q[1:2], q[3]

julia> push!(c, GateCCP(π/8), 1, 2, 3)
3-qubit circuit with 2 instructions:
├── CCP(λ) @ q[1:2], q[3]
└── CCP(π/8) @ q[1:2], q[3]

julia> power(GateCCP(λ), 2), inverse(GateCCP(λ))
(GateCCP(2λ), GateCCP(-λ))

Decomposition

julia> @variables λ
1-element Vector{Symbolics.Num}:
 λ

julia> decompose(GateCCP(λ))
3-qubit circuit with 5 instructions:
├── CP((1//2)*λ) @ q[2], q[3]
├── CX @ q[1], q[2]
├── CP((-1//2)*λ) @ q[2], q[3]
├── CX @ q[1], q[2]
└── CP((1//2)*λ) @ q[1], q[3]