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Gate pipelines

A gate pipeline is a list of gates that are traversable in both directions.

How to use them​

We can name them:​

name : H | Y

We can execute them directly with an action:​

1 -> H | Y

Once named they can be traversed in reverse:​

1 -> name | name <-

Here, the first name is traversed from left to right and the second time from right to left. It's equivalent to:

1 -> H | Y | Y | H

Angle convention​

Rotation angles are expressed in half-turns (multiples of π). So RX(0.5) means a rotation of 0.5π = 90°. Decimal literals are accepted: RZ(0.25) = π/4 rotation.

Available gates​

Pauli-X (NOT) gate​

Flips |0⟩ to |1⟩ and |1⟩ to |0⟩. Equivalent to a classical NOT.

N  or  X
[[0, 1],
 [1, 0]]

Pauli-Y gate​

Combines a bit-flip and a phase-flip. Maps |0⟩ → i|1⟩ and |1⟩ → -i|0⟩.

Y
[[0, -i],
 [i,  0]]

Pauli-Z gate​

Flips the phase of |1⟩.

Z
[[1,  0],
 [0, -1]]

Hadamard gate​

Creates superpositions.

H
[[1/√2,  1/√2],
 [1/√2, -1/√2]]

Phase gate (S)​

Applies a phase of i to |1⟩.

S
[[1, 0],
 [0, i]]

Conjugate phase (S†) gate​

Inverse of S. Applies -i to |1⟩.

ST
[[1,  0],
 [0, -i]]

π/8 gate (T)​

Applies a phase of e^(iπ/4) to |1⟩.

T
[[1, 0],
 [0, e^(iπ/4)]]

Conjugate T (T†) gate​

Inverse of T. Applies phase -π/4 to |1⟩.

TT
[[1, 0],
 [0, e^(-iπ/4)]]

√X gate (SX)​

Square root of the X gate.

SX
[[1+i, 1-i],
 [1-i, 1+i]] / 2

√X† gate (SXdg)​

Inverse of SX.

SXDG

V gate​

Alias for √X in the TKET convention.

V

V† gate​

Inverse of V.

VDG

Rotation-X gate​

Rotates around the X axis by angle θ (in half-turns).

RX(theta)
[[cos(θπ/2),   -i·sin(θπ/2)],
 [-i·sin(θπ/2), cos(θπ/2)]]

Rotation-Y gate​

Rotates around the Y axis by angle θ.

RY(theta)
[[cos(θπ/2), -sin(θπ/2)],
 [sin(θπ/2),  cos(θπ/2)]]

Rotation-Z gate​

Rotates around the Z axis by angle θ.

RZ(theta)
[[e^(-iθπ/2), 0         ],
 [0,           e^(iθπ/2)]]

IBM U1(λ) gate​

Diagonal single-qubit gate; 1 angle in half-turns.

U1(lambda)

IBM U2(φ, λ) gate​

Single-qubit gate; 2 angles in half-turns.

U2(phi, lambda)

IBM U3(θ, φ, λ) gate​

Full SU(2) single-qubit gate; 3 angles in half-turns.

U3(theta, phi, lambda)

TKET TK1(α, β, γ) gate​

Euler decomposition; 3 angles in half-turns.

TK1(alpha, beta, gamma)

PhasedX gate​

Rotation around a phase-shifted axis; 2 angles (exponent, phase) in half-turns.

PX(exponent, phase)   or   PHASEDX(exponent, phase)

Controlled-NOT (CX / CNOT)​

Flips the target qubit if the control is |1⟩.

CNOT(control)  or  CX(control)
[[1,0,0,0],
 [0,1,0,0],
 [0,0,0,1],
 [0,0,1,0]]

Controlled-Y (CY)​

Applies Y to the target if the control is |1⟩.

CY(control)

Controlled-Z (CZ)​

Applies Z to the target if the control is |1⟩.

CZ(control)

Controlled-Hadamard (CH)​

Applies H to the target if the control is |1⟩.

CH(control)

Flipped-Control variants (FCX, FCY, FCZ, FCH)​

These are the same as their C* counterparts but with control and target swapped. Useful when writing pipelines where the qubit flowing through the pipeline is the control rather than the target:

FCNOT(target)  or  FCX(target)
FCY(target)
FCZ(target)
FCH(target)

Controlled-U1 phase gate (CU1)​

Applies a phase shift of φ to the target if the control is |1⟩.

CU1(phi, control)

SWAP gate​

Exchanges the states of two qubits.

SWAP(target)

Controlled-Rx (CRX)​

Controlled rotation around X; 1 angle + control qubit.

CRX(angle, ctrl)

Controlled-Ry (CRY)​

Controlled rotation around Y; 1 angle + control qubit.

CRY(angle, ctrl)

Controlled-Rz (CRZ)​

Controlled rotation around Z; 1 angle + control qubit.

CRZ(angle, ctrl)

Echoed Cross-Resonance gate (ECR)​

Native 2-qubit gate for IBM superconducting hardware.

ECR(ctrl)

iSWAP gate​

Parameterised imaginary-SWAP; angle in half-turns.

ISWAP(angle, other)

Maximal iSWAP (ISWAPMAX)​

Equivalent to ISWAP(1).

ISWAPMAX(other)

ZZMax gate​

Maximum ZZ interaction (≡ ZZPhase(1)).

ZZMAX(other)

ZZPhase gate​

exp(-i·angle·π/2 · ZZ); angle in half-turns.

ZZPH(angle, other)

XXPhase gate​

exp(-i·angle·π/2 · XX); angle in half-turns.

XXPH(angle, other)

YYPhase gate​

exp(-i·angle·π/2 · YY); angle in half-turns.

YYPH(angle, other)

Fermionic Simulation gate (FSim)​

2 angles + partner qubit.

FSIM(theta, phi, other)

TKET TK2 gate​

Canonical 2-qubit interaction; 3 angles + partner qubit.

TK2(a, b, c, other)

PhasedISWAP gate​

2 angles + partner qubit.

PHISWAP(p, t, other)

Toffoli (CCX / TOFFOLI) gate​

Double-controlled NOT. Flips the target only if both controls are |1⟩.

TOFFOLI(ctrl1, ctrl2)  or  CCX(ctrl1, ctrl2)
8×8 identity with the bottom-right 2×2 swapped (|110⟩ ↔ |111⟩)

CSWAP / Fredkin gate​

Controlled-SWAP: swaps target ↔ other when ctrl=|1⟩.

CSWAP(ctrl, other)  or  FREDKIN(ctrl, other)

XXPhase3 gate​

3-qubit simultaneous XX interactions on all pairs; 1 angle + 2 partner qubits.

XXP3(angle, q1, q2)

Measurement​

Measures the qubit in the computational basis and stores the result in a classical bit.

M(bit)  or  MEASURE(bit)

If no bit argument is given, the result is stored in the default bit for that qubit index. When the target is * with no argument, measure_all() is used.

* -> M        # measure every qubit
q 0 -> M      # measure qubit 0 into default bit
q 0 -> M(b0)  # measure qubit 0 into named classical bit b0

Reset​

Resets a qubit to |0⟩.

R  or  RESET
q 0 -> RESET   # return qubit to ground state

BARRIER​

Prevents the compiler from reordering gates across this point. One joint barrier is placed across all targeted qubits:

q 0 -> BARRIER
[q 0, q 1] -> BARRIER
* -> BARRIER

PHASE (global phase)​

Adds a global phase of angle·π to the whole circuit. Global phase is not physically observable but is useful for circuit identities.

q 0 -> PHASE(0.5)    # adds a global phase of π/2

CIRCBOX (sub-circuit box)​

Wraps a named instruction pipeline into a pytket CircBox sub-circuit:

bell_prep : H  FCX(q 1)
[q 0, q 1] -> CIRCBOX(bell_prep)

The named pipeline is compiled into an isolated sub-circuit and embedded as an opaque box in the parent circuit. This is useful for hierarchical circuit design.