Block-local commutation analysis

[taalexander]

Based on prerequisite work in #4982.

This PR adds a read-only block-local analysis for determining whether two Quake operations are proven to commute. A later transformation will use it to find rewrites across intervening commuting operations.

The analysis assigns virtual-qubit identifiers to supported scalar Quake values, builds operation-local control and target views, and applies a centralized set of exact commutation rules. Identity tracking remains a separate internal analysis and translation unit for separation of concerns, but it is included in this PR as an implementation dependency of CommutationAnalysis. It is not proposed as the shared identity-analysis API. #4983 is superseded by this combined review unit.

The current rules cover disjoint support, the same operation or exact adjoint, computational-basis diagonal operations, common Pauli axes, Pauli-product parity, diagonal actions on controls, compatible controlled targets, and mutually exclusive controls. Unsupported or ambiguous cases remain indeterminate.

The following AI-generated candidates are recorded for later validation rather than included in this PR.

Future extensions

Structurally provable candidates

• Two controlled custom unitaries sharing a control but acting on disjoint targets.
• Controlled custom unitaries with opposite polarity on a shared control and no target-control crossover.
• An all-I/Z ExpPauli acting on the controls of another controlled operation.
• Controlled all-I/Z Pauli products that are diagonal but are not currently classified as computational-basis diagonal.
• A dynamic ExpPauli word produced by a compile-time constant expression.
• Symbolic Pauli words whose equality or anti-commutation parity is known by another analysis.

Parameter-dependent candidates

• An identity-valued rotation, such as Rx(0), against an otherwise noncommuting operation on the same target.
• Equivalent parameter expressions represented by different SSA values, such as %theta and %theta + 0.
• PhasedRx operations whose phase parameters are equivalent modulo the exact phase period.
• U2 or U3 parameter combinations that reduce to a known axis rotation.
• Cross-operation equivalences exposed by exact constant parameters but not already handled by the X-, Y-, and Z-axis families.
• Special angles at which generally noncommuting Pauli rotations become identity or another commuting operator.

These cases may be better handled by folding and canonicalization before commutation analysis.

Permutation candidates

• Swap(a,b) with CZ(a,b), which is invariant under exchanging its two qubits.
• Swap(a,b) with a symmetric Pauli product such as XX, YY, or ZZ on (a,b).
• Swap(a,b) with any multi-qubit operation whose target action is invariant under the same permutation.
• Equivalent larger qubit permutations represented by different sequences of Swap operations.

Custom-unitary candidates

• Different custom-unitary symbols containing identical constant matrices.
• Different diagonal custom unitaries acting on the same target.
• Different custom-unitary matrices that can be proven to commute exactly.
• Custom operators exposing a common generator, eigenspace, or diagonalizing basis.
• Controlled custom operators for which only target-action properties are needed by a generic controlled rule.

Basis-aware candidates

• Operations diagonal in the same noncomputational basis.
• Controls whose effective basis is established by surrounding conjugations, such as H-C(U)-H.
• Pauli or Clifford conjugation that transforms two operations into a recognized commuting family.
• Mutually exclusive controls established by classical or quantum dataflow rather than opposite static polarity on one control operand.

These require sequence-level basis, Clifford, or predicate tracking rather than only an operation-pair query.

Cases requiring a different semantic contract

• Reordering operations that commute only up to global phase.
• State-dependent commutation on a proven invariant subspace.
• Reordering or eliminating diagonal gates around computational-basis measurement.
• Relations involving measurement, reset, or quantum channels.
• Incremental preservation of cached results across block mutation.

[github-actions]

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