mimiqcircuits.decomposition.basis.clifford_t

Clifford+T decomposition basis.

Classes

CliffordTBasis([sk_depth, rotations])

Decomposition basis targeting the Clifford+T gate set.
class mimiqcircuits.decomposition.basis.clifford_t.CliffordTBasis(sk_depth=None, rotations=None)[source]

Bases: DecompositionBasis

Decomposition basis targeting the Clifford+T gate set.

Clifford+T is the standard universal gate set of error-corrected quantum computing. The terminal operations of this basis are those of StimBasis plus GateT and GateTDG, so the two bases always keep the same Clifford and stabilizer operations:

  • single-qubit Cliffords: GateID, GateX, GateY, GateZ, GateH, GateHXY, GateHYZ, GateS, GateSDG, GateSX, GateSXDG, GateSY, GateSYDG;

  • two-qubit Cliffords: GateCX, GateCY, GateCZ, GateSWAP, GateISWAP, GateISWAPDG, GateDCX, GateECR;

  • Pauli rotations by +-pi/2: GateRXX, GateRYY, GateRZZ, and RPauli with a non-identity Pauli string;

  • the non-Clifford GateT and GateTDG;

  • measurements, resets and measure-and-resets in the X, Y and Z bases, MeasureXX, MeasureYY, MeasureZZ and MeasurePauli;

  • Pauli noise: PauliX, PauliY, PauliZ, Depolarizing1, Depolarizing2 and PauliNoise;

  • barriers and annotations;

  • classical operations, which StimBasis does not keep;

  • an IfStatement whose body is terminal in this basis, with any condition. StimBasis only keeps a conditional GateX, GateY or GateZ on a single bit.

Non-terminal operations are rewritten by the first rule that matches, in this order:

  1. SpecialAngleRewrite: rotations by multiples of pi/4 to exact Clifford+T.

  2. ToffoliToCliffordTRewrite: GateCCX to Clifford+T with 7 T gates.

  3. A GateU whose angles are all zero (a pure global phase) is dropped.

  4. ZYZRewrite: GateU to RZ*RY*RZ.

  5. ToZRotationRewrite: GateRX, GateRY to GateRZ and Cliffords.

  6. The rotations rule (default GridSynthRewrite): remaining GateRZ to an approximate Clifford+T sequence.

  7. CanonicalRewrite: any other operation, toward GateU and GateCX.

The rotations rule only sees angles that step 1 could not decompose exactly. The body of an IfStatement goes through the same pipeline and stays conditional. No T-count optimization is performed on the result.

Parameters:

  • sk_depth (int | None) – Use SolovayKitaevRewrite with this recursion depth as the rotations rule.

  • rotations (RewriteRule) – Rule that synthesizes the remaining Z-rotations. The default GridSynthRewrite() approximates each rotation to operator-norm error 1e-10. The rule must match GateRZ with a numeric angle and emit only gates of this basis.

Raises:

ValueError – If both sk_depth and rotations are given, or if rotations does not match GateRZ.

Decomposing an operation that no rule handles, such as a noise channel other than the Pauli ones above, raises a DecompositionError.

Example

>>> from mimiqcircuits import Circuit, GateH, GateT, GateRZ
>>> from mimiqcircuits.decomposition import decompose, CliffordTBasis
>>> from mimiqcircuits.decomposition import GridSynthRewrite
>>> from symengine import pi
>>> c = Circuit()
>>> c.push(GateRZ(pi/4), 0)  # special angle, decomposed exactly
1-qubit circuit with 1 instruction:
└── RZ((1/4)*pi) @ q[0]

>>> decomposed = decompose(c, CliffordTBasis())

>>> # An arbitrary angle is approximated to 1e-10 by gridsynth
>>> c2 = Circuit()
>>> c2.push(GateRZ(0.123), 0)
1-qubit circuit with 1 instruction:
└── RZ(0.123) @ q[0]

>>> decomposed2 = decompose(c2, CliffordTBasis())

>>> # A looser error bound per rotation gives fewer T gates
>>> basis = CliffordTBasis(rotations=GridSynthRewrite(epsilon=1e-6))
>>> decomposed3 = decompose(c2, basis)

>>> # Solovay-Kitaev instead of gridsynth
>>> decomposed4 = decompose(c2, CliffordTBasis(sk_depth=0))
rotations: RewriteRule
__init__(sk_depth=None, rotations=None)[source]
isterminal(op)[source]

Check if an operation is terminal in the Clifford+T basis.

Parameters:

op (Operation) – The operation to check.

Returns:

True if the operation is terminal in StimBasis, is GateT, GateTDG or a classical operation, or is an IfStatement whose body is terminal in this basis.

Return type:

bool

decompose(op, qubits, bits, zvars)[source]

Decompose a non-terminal operation toward Clifford+T.

Applies the rules listed in the class docstring, in that order.

Parameters:

  • op (Operation) – The operation to decompose.

  • qubits (Sequence[int]) – Qubit indices for the operation.

  • bits (Sequence[int]) – Classical bit indices for the operation.

  • zvars (Sequence[int]) – Z-variable indices for the operation.

Returns:

A Circuit containing the decomposed instructions.

Raises:

DecompositionError – If the operation cannot be decomposed.

Return type:

Circuit