mimiqcircuits.decomposition.rules.gridsynth

Ross–Selinger gridsynth: Clifford+T approximation of Z-rotations.

The synthesis itself is implemented in Rust in the mimiqcircuits._core extension.

References

• N. J. Ross and P. Selinger, “Optimal ancilla-free Clifford+T approximation of z-rotations”, Quantum Inf. Comput. 16, 901 (2016), arXiv:1403.2975.

Classes

GridSynthRewrite([epsilon, seed])

Rewrite rule that approximates GateRZ with a Clifford+T sequence, using the Ross–Selinger gridsynth algorithm.

class mimiqcircuits.decomposition.rules.gridsynth.GridSynthRewrite(epsilon=1e-10, seed=0)

Bases: RewriteRule

Rewrite rule that approximates GateRZ with a Clifford+T sequence, using the Ross–Selinger gridsynth algorithm.

For GateRZ(λ) the rule emits a sequence of GateH, GateS, GateSDG, GateZ, GateT and GateX whose unitary V satisfies min_φ ‖RZ(λ) − e^{iφ}V‖ ≤ epsilon in operator norm. The approximation holds up to a global phase, which is dropped. The T-count is about 3·log₂(1/epsilon) and, except for rare failures of the integer factoring step, optimal among ancilla-free Clifford+T circuits with this error. Angles that are multiples of π/4 give their exact short decomposition.

The factoring step is randomized. For a fixed seed, the same angle and tolerance always give the same sequence. Results are cached per angle, since circuits often repeat the same rotation.

The synthesis runs in the compiled mimiqcircuits._core extension without holding the GIL, so several threads can synthesize in parallel.

Parameters:

• epsilon (float) – Operator-norm tolerance, 1e-100 <= epsilon < 1 (default 1e-10). A rotation takes a few milliseconds at 1e-10 and up to about a second at 1e-100.

• seed (int) – Seed of the randomized factoring step, an integer in [0, 2**64) (default 0).

Raises:

• ValueError – If epsilon or seed is out of range.

• TypeError – If epsilon is not a real number or seed is not an integer.

Supported operations:

• GateRZ(λ) with a numeric angle λ.

Symbolic angles are not matched. GateRX, GateRY and GateU must first be reduced to Z-rotations, for example with ZYZRewrite and ToZRotationRewrite. CliffordTBasis does this and uses this rule by default.

Example

>>> from mimiqcircuits import GateRZ, GateT
>>> from mimiqcircuits.decomposition import GridSynthRewrite
>>> def tcount(c):
...     return sum(isinstance(i.operation, GateT) for i in c)
>>> # Approximate one rotation to the default tolerance 1e-10
>>> circ = GridSynthRewrite().decompose_step(GateRZ(0.123), [0], [], [])
>>> # A coarser tolerance gives fewer T gates
>>> coarse = GridSynthRewrite(epsilon=1e-4).decompose_step(GateRZ(0.123), [0], [], [])
>>> tcount(coarse) < tcount(circ)
True

References

N. J. Ross and P. Selinger, “Optimal ancilla-free Clifford+T approximation of z-rotations”, Quantum Inf. Comput. 16, 901 (2016), arXiv:1403.2975.

epsilon: float = 1e-10

seed: int = 0

matches(op)

Check if this rule can decompose the operation.

Parameters:

op (Operation) – The operation to check.

Returns:

True if the operation is a GateRZ with a numeric angle.

Return type:

bool

decompose_step(op, qubits, bits, zvars)

Decompose the Z-rotation into Clifford+T gates.

Parameters:

• op (Operation) – The GateRZ to decompose.

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

• bits (Sequence[int]) – Classical bit indices (unused).

• zvars (Sequence[int]) – Z-variable indices (unused).

Returns:

A Circuit containing the Clifford+T approximation.

Raises:

• DecompositionError – If the operation is not a numeric GateRZ.

• ValueError – If the angle is not finite.

Return type:

Circuit

__init__(epsilon=1e-10, seed=0)