Quantum gates

Common single-, two- and three-qubit gates used in quantum circuits, returned as TensorMaps on qubit spaces.

Conventions

Basis ordering

A qubit lives on a two-dimensional space with computational basis ordered as $\{|0⟩, |1⟩\}$.

Multi-qubit gates act on $V^{⊗n}$ with the first qubit as the leftmost tensor factor, so a controlled gate uses the first qubit(s) as control and the last as target.

Within a fused block, the index of $|b_1 b_2 … b_n⟩$ is $k = 1 + \sum_i b_i 2^{i-1}$, i.e. the first qubit runs fastest. Consequently block(gate, Trivial()) agrees element-for-element with the gate matrices as tabulated by e.g. Qiskit, identifying the first tensor factor with qubit 0.

Element types

The gates with complex matrix elements — pauli_y, phase_shift, s_gate, t_gate, rotation_x, rotation_z, u3, cy, cs, cphase, iswap, ecr, rotation_xx, rotation_yy, rotation_zz and rotation_zx — throw an ArgumentError when requested with a real eltype. All others are real-valued and honour any eltype, including exact ones such as Int.

Rotation angles

The parametrized gates take their rotation angle as a required theta keyword, e.g. rotation_x(; theta = π / 2). The universal single-qubit gate u3 takes the three angles theta, phi and lambda.

Symmetry sectors

Symmetry	Physical meaning	Sector label	Representable gates
`Trivial`	none	—	all gates, full $2^n × 2^n$ matrix
`Z2Irrep`	excitation-number parity	charge $n \bmod 2$ per qubit	only gates that conserve the number of $1$'s modulo 2
`U1Irrep`	excitation-number conservation	charge $∈ \{0, 1\}$ per qubit	only gates that conserve the number of $1$'s

U(1) charge = excitation number

Under U1Irrep the two sectors of a qubit carry charge $0$ ($|0⟩$) and $1$ ($|1⟩$), i.e. the conserved quantity is the number of excited qubits. Only number-conserving gates are representable in this symmetry: pauli_z, proj_0, proj_1, s_gate, t_gate, phase_shift, rotation_z, cz, cs, cphase, swap, iswap, rotation_zz and fredkin. Gates that flip qubits (X, Y, H, CNOT, Toffoli, …) throw an ArgumentError when requested with U1Irrep.

Z₂ charge = excitation parity

The $ℤ₂$ symmetry implemented here is the parity of the excitation number, which is diagonal in the computational basis — unlike the $ℤ₂$ of SpinOperators, which is the spin flip generated by $e^{iπS^x}$ and requires a change of basis. It is a weaker requirement than U1Irrep, so everything in the list above remains representable, and in addition rotation_xx and rotation_yy become available: $X ⊗ X$ and $Y ⊗ Y$ flip both qubits and hence preserve the total parity.

Gate combinators

controlled turns any gate into its controlled version $|0⟩⟨0| ⊗ I + |1⟩⟨1| ⊗ G$, inheriting the element type and the symmetry of G, and stacking arbitrarily: controlled(controlled(pauli_x())) is the toffoli gate. Together with the adjoint G' and the integer powers G^n of TensorKit, this covers the usual gate modifiers — inverse, power and control — where a gate controlled on $|0⟩$ follows from swapping the two projectors. The gates that are not provided in controlled form, such as the controlled rotations, are therefore one-liners:

crx = controlled(rotation_x(; theta = π / 2))

Gate overview

Function	Alias(es)	Sites	Supported symmetries
`qubit_space`	—	—	`Trivial`, `Z2Irrep`, `U1Irrep`
`pauli_x`	`X`	1	`Trivial`
`pauli_y`	`Y`	1	`Trivial`
`pauli_z`	`Z`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`proj_0`	`P0`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`proj_1`	`P1`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`hadamard`	`H`	1	`Trivial`
`s_gate`	`S`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`t_gate`	`T`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`phase_shift`	`P`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`rotation_x`	`Rx`	1	`Trivial`
`rotation_y`	`Ry`	1	`Trivial`
`rotation_z`	`Rz`	1	`Trivial`, `Z2Irrep`, `U1Irrep`
`u3`	`U`	1	`Trivial`
`cnot`	`CNOT`, `CX`	2	`Trivial`
`cy`	`CY`	2	`Trivial`
`cz`	`CZ`	2	`Trivial`, `Z2Irrep`, `U1Irrep`
`ch`	`CH`	2	`Trivial`
`cs`	`CS`	2	`Trivial`, `Z2Irrep`, `U1Irrep`
`cphase`	`CP`	2	`Trivial`, `Z2Irrep`, `U1Irrep`
`swap`	`SWAP`	2	`Trivial`, `Z2Irrep`, `U1Irrep`
`iswap`	`ISWAP`	2	`Trivial`, `Z2Irrep`, `U1Irrep`
`dcx`	`DCX`	2	`Trivial`
`ecr`	`ECR`	2	`Trivial`
`rotation_xx`	`Rxx`	2	`Trivial`, `Z2Irrep`
`rotation_yy`	`Ryy`	2	`Trivial`, `Z2Irrep`
`rotation_zz`	`Rzz`	2	`Trivial`, `Z2Irrep`, `U1Irrep`
`rotation_zx`	`Rzx`	2	`Trivial`
`toffoli`	`TOFFOLI`, `CCX`	3	`Trivial`
`fredkin`	`FREDKIN`, `CSWAP`	3	`Trivial`, `Z2Irrep`, `U1Irrep`

The adjoint gates $S^†$ and $T^†$ are obtained as s_gate()' and t_gate()'.

API

TensorKitTensors.QuantumGates.basis_transform — Method

basis_transform(symmetry::Type{<:Sector})

Return the unitary basis transformation that maps the computational basis $\{|0⟩, |1⟩\}$ of qubit_space(Trivial) onto the basis of qubit_space(symmetry), as a TensorMap from qubit_space(Trivial) to desymmetrize(qubit_space(symmetry)), as required by symmetrize.

Symmetry	Transformation
`Trivial`	identity
`Z2Irrep`	identity
`U1Irrep`	identity

Both Z2Irrep and U1Irrep use the (parity of the) excitation number as their charge, which is diagonal in the computational basis, and order their sectors as $0, 1$, such that the transformation is the identity in all cases. It is returned with integer scalar type, such that it promotes to any scalar type without loss of precision.

Symmetry	Space
`Trivial`	`ComplexSpace(2)`
`Z2Irrep`	`Z2Space(0 => 1, 1 => 1)` (charge = number of excitations mod 2)
`U1Irrep`	`U1Space(0 => 1, 1 => 1)` (charge = number of excitations)