/**
 * IonQ QIS gate matrices.
 *
 * All gates are defined as 2×2 unitaries on the computational basis {|0⟩, |1⟩}.
 * Names match IonQ's native gate set (h, x, y, z, s, si, t, ti, v, vi, rx, ry, rz).
 */
import { Gate2x2, Gate4x4 } from './statevector.js';
export declare const Id: Gate2x2;
export declare const H: Gate2x2;
export declare const X: Gate2x2;
export declare const Y: Gate2x2;
export declare const Z: Gate2x2;
export declare const S: Gate2x2;
export declare const Si: Gate2x2;
export declare const T: Gate2x2;
export declare const Ti: Gate2x2;
export declare const V: Gate2x2;
export declare const Vi: Gate2x2;
/** Rotation around X axis by angle θ (radians). */
export declare const Rx: (theta: number) => Gate2x2;
/** Rotation around Y axis by angle θ (radians). */
export declare const Ry: (theta: number) => Gate2x2;
/** Rotation around Z axis by angle θ (radians). */
export declare const Rz: (theta: number) => Gate2x2;
/** Rz(π/2) — phase rotation by a half-turn. Equal to S up to global phase. */
export declare const R2: Gate2x2;
/** Rz(π/4) — phase rotation by a quarter-turn. Equal to T up to global phase. */
export declare const R4: Gate2x2;
/** Rz(π/8) — phase rotation by an eighth-turn. */
export declare const R8: Gate2x2;
/**
 * U3(θ, φ, λ) — general single-qubit unitary; the basis gate of OpenQASM 2.0.
 *
 *   ┌  cos(θ/2)           −e^(iλ)·sin(θ/2)      ┐
 *   └  e^(iφ)·sin(θ/2)    e^(i(φ+λ))·cos(θ/2)   ┘
 *
 * Key identities:
 *   U3(π, 0, π)   = X           U3(π/2, 0, π) = H
 *   U3(θ, 0, 0)   = Ry(θ)       U3(0, 0, λ)   = U1(λ)
 */
export declare const U3: (theta: number, phi: number, lambda: number) => Gate2x2;
/**
 * U2(φ, λ) = U3(π/2, φ, λ) — equatorial gate. U2(0, π) = H.
 */
export declare const U2: (phi: number, lambda: number) => Gate2x2;
/**
 * U1(λ) = diag(1, e^(iλ)) — phase gate; equal to Rz(λ) up to global phase.
 * U1(π/2) = S,  U1(π/4) = T,  U1(π) = Z.
 */
export declare const U1: (lambda: number) => Gate2x2;
/**
 * XX(θ) = exp(−iθ/2 · X⊗X) — Ising-XX interaction; IonQ native gate.
 * XX(π/2) is maximally entangling. XX(0) = I.
 */
export declare const Xx: (theta: number) => Gate4x4;
/**
 * YY(θ) = exp(−iθ/2 · Y⊗Y) — Ising-YY interaction; IonQ native gate.
 */
export declare const Yy: (theta: number) => Gate4x4;
/**
 * ZZ(θ) = exp(−iθ/2 · Z⊗Z) — Ising-ZZ interaction; IonQ native gate.
 * Diagonal: diag(e^(−iθ/2), e^(iθ/2), e^(iθ/2), e^(−iθ/2)).
 * Identity: ZZ(θ) = CNOT · (I⊗Rz(θ)) · CNOT.
 */
export declare const Zz: (theta: number) => Gate4x4;
/**
 * XY(θ) — XY interaction gate.
 * Acts as exp(iθ/2 · (X⊗X + Y⊗Y)/2) in the |01⟩/|10⟩ subspace.
 * XY(0) = I,  XY(π) = iSWAP,  XY(π/2) = √iSWAP.
 */
export declare const Xy: (theta: number) => Gate4x4;
/** iSWAP = XY(π): swaps qubits and multiplies by i. */
export declare const ISwap: Gate4x4;
/** √iSWAP = XY(π/2): square root of iSWAP. */
export declare const SrSwap: Gate4x4;
/**
 * GPI(φ) = [[0, e^{−iφ}], [e^{iφ}, 0]] — IonQ hardware-native single-qubit gate.
 *
 * Key identities:
 *   GPI(0)   = X        GPI(π/2) = Y
 *   GPI(φ)†  = GPI(φ)  (Hermitian: self-adjoint and self-inverse, GPI² = I)
 */
export declare const Gpi: (phi: number) => Gate2x2;
/**
 * GPI2(φ) = (1/√2)[[1, −ie^{−iφ}], [−ie^{iφ}, 1]] — IonQ hardware-native half-rotation.
 *
 * Key identities:
 *   GPI2(0)     = Rx(π/2)    GPI2(π/2)  = Ry(π/2)
 *   GPI2(φ)⁻¹  = GPI2(φ+π)  GPI2(0)²   = X  (up to global phase)
 */
export declare const Gpi2: (phi: number) => Gate2x2;
/**
 * MS(φ₀, φ₁) — Mølmer-Sørensen entangling gate; IonQ's native two-qubit operation.
 *
 * MS(0, 0)       = XX(π/2)   (maximally entangling)
 * MS(π/2, π/2)   = YY(π/2)
 * MS(φ₀, φ₁)|00⟩ always produces a superposition of |00⟩ and |11⟩ with equal probability.
 */
export declare const Ms: (phi0: number, phi1: number) => Gate4x4;
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