SpinHalfOperator

Base class: Operator

Base class for spin-1/2 symbolic operators.

Operators

`CNOT(integer site0, integer site1) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the CNOT (controlled-NOT) gate, taking two individual site indices.

Parameters

site0: The control qubit site index.
site1: The target qubit site index.

Example

var cnot1 = CNOT(0, 1)

`DualUnitary(real J, integer i, integer j) -> SpinHalfOperator`

Returns a spin-1/2 dual unitary operator: $\exp(i(\frac{\pi}{4} X(i) X(j) + \frac{\pi}{4} Y(i) Y(j) + J_z Z(i) Z(j)))$ where $i, j$ are the site indices.

Parameters

J: The Jz constant parameter.
i: Site index i.
j: Site index j.

Example

var dual_u = DualUnitary(0.25, 0, 1)  // Dual unitary with Jz=0.25

`Flip(integer site) -> SpinHalfOperator`

Creates a spin-1/2 OperatorPermute that flips the spin at a single site.

Parameters

site: The site index where the spin is flipped.

`Flip(const List<integer> sites) -> SpinHalfOperator`

Creates a spin-1/2 OperatorPermute that flips spins at specified sites.

Parameters

sites: Vector of site indices where spins are flipped.

`FlipAll() -> SpinHalfOperator`

Creates a spin-1/2 OperatorPermute that flips all spins.

`Floquet(const List<real> Js, integer i, integer j) -> SpinHalfOperator`

Returns a spin-1/2 Floquet $(J_x, J_y, J_z) = \exp(i(J_x X(i) + J_y Y(i) + J_z Z(i)) Z(j))$ where $i, j$ are the site indices.

Parameters

Js: Vector of exactly three parameters [Jx, Jy, Jz].
i: Site index i.
j: Site index j.

Example

var flqt = Floquet([0.2, 0.3, 0.8], 1, 2)  // Floquet operator

`Hgate(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the Hadamard gate.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var h0 = Hgate(0)

`Permute(const List<integer> permutation, const List<integer> sites) -> SpinHalfOperator`

Creates a spin-1/2 OperatorPermute that permutes spins according to a specified permutation.

Parameters

permutation: Vector defining the permutation of spins (destination indices).
sites: Vector of site indices where the permutation is applied.

`Phase(real phi) -> SpinHalfOperator`

Returns a global phase spin-1/2 operator $P(\phi) = e^{-i\phi}$ .

Parameters

phi: The phase angle.

Example

var ps = Phase(3.14159)  // π phase

`PhaseShift(real phi, integer site) -> SpinHalfOperator`

Returns a spin-1/2 PhaseShift $(\phi) = \exp(i\phi)$ on the $|1\rangle$ state.

Parameters

phi: The rotation angle.
site: The site index where the operator acts.

Example

var ps_half = PhaseShift(1.5708, 0)

`Proj0(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the projector $|0\rangle\langle 0|$ .

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var p0_op = Proj0(0)

`Proj1(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the projector $|1\rangle\langle 1|$ .

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var p1_op = Proj1(1)

`Reflect() -> SpinHalfOperator`

Creates a spin-1/2 OperatorPermute that reflects all spins about the center.

`RotEuler(real phi, real theta, real omega, integer site) -> SpinHalfOperator`

Returns a general spin-1/2 rotation operator RotEuler $(\phi,\theta,\omega) = RZ(\omega)RX(\theta)RZ(\phi) = \exp(-i \omega Z/2) \exp(-i \theta X/2) \exp(-i \phi Z/2)$ .

Parameters

phi: The first rotation angle.
theta: The second rotation angle.
omega: The third rotation angle.
site: The site index where the operator acts.

Example

var rot0 = RotEuler(1.57, 0.78, 3.14, 0)  // General rotation on site 0

`RotX(real phi, integer site) -> SpinHalfOperator`

Returns a spin-1/2 RotX $(\phi) = \exp(-i\phi X/2)$ rotation operator around the X-axis.

Parameters

phi: The rotation angle.
site: The site index where the operator acts.

Example

var rx_half = RotX(1.5708, 0)  // π/2 rotation on site 0

`RotY(real phi, integer site) -> SpinHalfOperator`

Returns a spin-1/2 RotY $(\phi) = \exp(-i\phi Y/2)$ rotation operator around the Y-axis.

Parameters

phi: The rotation angle.
site: The site index where the operator acts.

Example

var ry_quarter = RotY(0.7854, 1)  // π/4 rotation on site 1

`RotZ(real phi, integer site) -> SpinHalfOperator`

Returns a spin-1/2 RotZ $(\phi) = \exp(-i\phi Z/2)$ rotation operator around the Z-axis.

Parameters

phi: The rotation angle.
site: The site index where the operator acts.

Example

var rz_full = RotZ(6.28318, 2)  // 2π rotation on site 2

`SWAP(integer site0, integer site1) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the SWAP gate, taking two individual site indices.

Parameters

site0: The first site index.
site1: The second site index.

Example

var swap1 = SWAP(0, 1)

`Sgate(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the S gate.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var s0 = Sgate(0)

`Sminus(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for Spin-minus operator.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var sm1 = Sminus(1)

`Splus(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for Spin-plus operator.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var sp0 = Splus(0)

`Tgate(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for the T gate.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var t0 = Tgate(0)

`Translate(integer shift) -> SpinHalfOperator`

Creates a spin-1/2 OperatorPermute that translates spins by a specified shift.

Parameters

shift: Number of sites by which the spins are shifted (default: 1).

`UnitaryXYZ(const List<real> Js, integer i, integer j) -> SpinHalfOperator`

Returns a spin-1/2 UnitaryXYZ $(J_x, J_y, J_z)$ operator: $\exp(i(J_x X(i) X(j) + J_y Y(i) Y(j) + J_z Z(i) Z(j)))$ where $i, j$ are the site indices.

Parameters

Js: Vector of exactly three parameters: [Jx, Jy, Jz].
i: Site index i.
j: Site index j.

Example

var xyz01 = UnitaryXYZ([0.5, 1.0, 1.5], 0, 1)  // coupling between sites 0 and 1

`X(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for Pauli-X operator.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var px0 = X(0)

`XX(integer site0, integer site1) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $X_i \cdot X_j$ operator, taking two individual site indices.

Parameters

site0: The first site index ( $i$ ).
site1: The second site index ( $j$ ).

Example

var pxx1 = XX(1, 2)

`XXPYY(integer site0, integer site1) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $X_i \cdot X_j + Y_i \cdot Y_j$ operator, taking two individual site indices.

Parameters

site0: The first site index.
site1: The second site index.

Example

var xxpyy1 = XXPYY(0, 1)

`Y(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for Pauli-Y operator.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var py1 = Y(1)

`YY(integer site0, integer site1) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $Y_i \cdot Y_j$ operator, taking two individual site indices.

Parameters

site0: The first site index.
site1: The second site index.

Example

var pyy1 = YY(0, 1)

`Z(integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for Pauli-Z operator.

Parameters

site: The site index where the operator acts (exactly 1 site required).

Example

var pz2 = Z(2)

`ZN(const List<integer> sites) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $\sum_{i \in \text{sites}} Z_i$ operator.

Parameters

sites: Vector of site indices where the operator acts (minimum 1 site).

Example

var zn_s = ZN([0])     // Single site: use brackets (vector format) 
var zn_m = ZN([0, 1, 2, 3])  
// ZN(0) would be INVALID - missing brackets

`ZZ(integer site0, integer site1) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $Z_i \cdot Z_j$ operator, taking two individual site indices.

Parameters

site0: The first site index.
site1: The second site index.

Example

var pzz1 = ZZ(0, 2)

`ZZNN(const List<integer> sites) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $\sum_{i \in \text{sites}} Z_i \otimes Z_{i+1}$ operator.

Parameters

sites: Vector of site indices where the operator acts (minimum 2 sites).

Example

var zznn_s = ZZNN([0, 1])     // Minimum sites: use brackets (vector format)  
var zznn_m = ZZNN([0, 1, 2, 3])  
// ZZNN(0, 1) would be INVALID - missing brackets

`ZZNNN(const List<integer> sites) -> SpinHalfOperator`

Returns a spin-1/2 OperatorNamed for $\sum_{i \in \text{sites}} Z_i \otimes Z_{i+2}$ operator.

Parameters

sites: Vector of site indices where the operator acts (minimum 3 sites).

Example

var zznnn_s = ZZNNN([0, 1, 2])     //  use brackets (vector format)  
var zznnn_m = ZZNNN([0, 1, 2, 3, 4])  
// ZZNNN(0, 1, 2) would be INVALID - missing brackets

`operator_dense(const RealMatrix matrix, integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorDense parametrized by the given matrix.

Parameters

matrix: The matrix parametrizing the operator.
site: The site index where the operator acts.

`operator_dense(const RealMatrix matrix, const List<integer> sites) -> SpinHalfOperator`

Returns a spin-1/2 OperatorDense parametrized by the given matrix.

Parameters

matrix: The matrix parametrizing the operator.
sites: Vector of site indices where the operator acts.

`operator_dense(const ComplexMatrix matrix, integer site) -> SpinHalfOperator`

Returns a spin-1/2 OperatorDense parametrized by the given matrix.

Parameters

matrix: The matrix parametrizing the operator.
site: The site index where the operator acts.

`operator_dense(const ComplexMatrix matrix, const List<integer> sites) -> SpinHalfOperator`

Returns a spin-1/2 OperatorDense parametrized by the given matrix.