All simulator gate methods accept multiple qubits: qsim.H(q0, q1, q2) applies H to each independently.
Identity — I
Leaves the qubit state unchanged.
Simulator
Gate package
gate.I() *matrix.Matrix
// Accepts an optional repetition count: gate.I(n) returns the n-fold tensor product.
Pauli-X — X
Bit-flip gate. Maps |0⟩ → |1⟩ and |1⟩ → |0⟩. Quantum analogue of the classical NOT gate.
Matrix
Simulator
Gate package
Pauli-Y — Y
Combined bit-flip and phase-flip. Maps |0⟩ → i|1⟩ and |1⟩ → -i|0⟩.
Simulator
Gate package
Pauli-Z — Z
Phase-flip gate. Leaves |0⟩ unchanged and maps |1⟩ → -|1⟩.
Simulator
Gate package
Hadamard — H
Creates an equal superposition from a basis state: |0⟩ → (|0⟩ + |1⟩)/√2, |1⟩ → (|0⟩ - |1⟩)/√2. Fundamental to most quantum algorithms.
Matrix
H = (1/√2) | 1 1 |
| 1 -1 |
S Gate — S
Applies a π/2 (90°) phase shift to |1⟩. Equivalent to T².
Simulator
Gate package
gate.S() *matrix.Matrix
// Matrix: [[1, 0], [0, i]]
T Gate — T
Applies a π/4 (45°) phase shift to |1⟩. Used in fault-tolerant gate sets.
Simulator
Gate package
gate.T() *matrix.Matrix
// Matrix: [[1, 0], [0, exp(iπ/4)]]
Phase Rotation — R(theta)
Applies a phase of exp(i·theta) to |1⟩, leaving |0⟩ unchanged.
R(θ) = | 1 0 |
| 0 exp(i·theta) |
q.Theta(k) to get the standard QFT angle 2π / 2^k.
Simulator
Gate package
gate.R(theta float64) *matrix.Matrix
// Standard QFT angle helper:
theta := gate.Theta(k) // 2 * math.Pi / math.Pow(2, float64(k))
gate.R(theta)
RX — Rotation Around X Axis
Rotates the Bloch sphere vector by theta radians around the X axis.
RX(θ) = | cos(θ/2) -i·sin(θ/2) |
| -i·sin(θ/2) cos(θ/2) |
gate.RX(theta float64) *matrix.Matrix
RY — Rotation Around Y Axis
Rotates the Bloch sphere vector by theta radians around the Y axis. Produces real-valued superpositions.
RY(θ) = | cos(θ/2) -sin(θ/2) |
| sin(θ/2) cos(θ/2) |
gate.RY(theta float64) *matrix.Matrix
RZ — Rotation Around Z Axis
Rotates the Bloch sphere vector by theta radians around the Z axis. Applies a relative phase between |0⟩ and |1⟩.
RZ(θ) = | exp(-i·θ/2) 0 |
| 0 exp(i·θ/2) |
gate.RZ(theta float64) *matrix.Matrix
Universal Gate — U(theta, phi, lambda)
The most general single-qubit unitary up to a global phase. Every single-qubit gate is a special case of U.
U(θ,φ,λ) = | cos(θ/2) -exp(i·λ)·sin(θ/2) |
| exp(i·φ)·sin(θ/2) exp(i·(φ+λ))·cos(θ/2) |
| Gate | U parameters |
|---|
| H | U(π/2, 0, π) |
| X | U(π, 0, π) |
| Z | U(0, 0, π) |
| S | U(0, 0, π/2) |
| T | U(0, 0, π/4) |
qsim.U(math.Pi/2, 0, math.Pi, q0) // equivalent to H
gate.U(theta, phi, lambda float64) *matrix.Matrix
Example: Bell state via U parameters
package main
import (
"fmt"
"math"
"github.com/itsubaki/q"
"github.com/itsubaki/q/quantum/gate"
)
func main() {
qsim := q.New()
q0 := qsim.Zero()
q1 := qsim.Zero()
// H via U, then CNOT
qsim.U(math.Pi/2, 0, math.Pi, q0)
qsim.CNOT(q0, q1)
for _, s := range qsim.State() {
fmt.Println(s)
}
// [00][ 0]( 0.7071 0.0000i): 0.5000
// [11][ 3]( 0.7071 0.0000i): 0.5000
_ = gate.U(math.Pi/2, 0, math.Pi) // gate matrix directly
}
