Unifying and Extending Strong Simulation of Quantum Circuits

2026-09-30

AI Summary

This study addresses the absence of a unified framework integrating structural and algebraic tractability in the classical simulation of quantum circuits. We propose functional aggregate queries (FAQs) as a unifying formalism, introducing the novel criterion of symmetric width for tensor layouts. By combining the InsideOut evaluation scheme with semiring algebra and factorized representation techniques, our approach enables efficient and exact simulation. This work recovers known complexity bounds, establishes the polynomial-time simulability of Clifford circuits, and identifies new tractable regimes that surpass existing limitations, thereby laying a novel algorithmic foundation for quantum circuit simulation.

Abstract

We establish functional aggregate queries (FAQs) as a unifying language for exact classical simulation of quantum circuits. A circuit becomes a sum-product query: factors encode gates, internal wire variables are aggregated, and free boundary variables index transition amplitudes. The central insight is that distinct sources of simulation tractability can be exploited within the same InsideOut evaluation scheme. The query specifies what is computed; the evaluation plan, semiring, and representation of intermediate factors determine the cost. This view unifies structural and algebraic simulation guarantees. With explicit factor representations, FAQ evaluation recovers the treewidth bound for tensor-network contraction and yields finer sparsity-sensitive bounds via fractional covers. Over a formal phase semiring, compressed intermediate factors recover rank-width-based simulation for compatible quadratic phase representations. For Clifford circuits, affine-quadratic factors are closed under multiplication and marginalization and remain polynomial in size, yielding polynomial-time exact amplitude computation without any bounded-width assumption. Beyond these recoveries, the framework yields a new tractability criterion: tensor layout symmetry width. This parameter combines local cut-rank with separator symmetry through exact tree-tensor representations. We give a constructive evaluation bound and exhibit a circuit family with bounded tensor layout symmetry width but unbounded phase-graph rank-width and circuit line-graph treewidth. These results establish representation-aware FAQ evaluation as a common algorithmic foundation for classical simulation and a systematic route to new tractable regimes.

Problem

• quantum circuit simulation
• strong simulation
• functional aggregate queries
• tractability
• tensor network contraction

Innovation

• Functional Aggregate Queries
• Quantum Circuit Simulation
• Tensor Layout Symmetry Width
• InsideOut Evaluation
• Clifford Circuits