Symbolic Synthesis of Clifford Circuits and Beyond

Abstract

Path sums are a convenient symbolic formalism for quantum operations with applications to the simulation, optimization, and verification of quantum protocols. Unlike quantum circuits, path sums are not limited to unitary operations, but can express arbitrary linear ones. Two problems, therefore, naturally arise in the study of path sums: the unitarity problem and the extraction problem. The former is the problem of deciding whether a given path sum represents a unitary operator. The latter is the problem of constructing a quantum circuit, given a path sum promised to represent a unitary operator. In this paper, we show that the unitarity problem is co-NP-hard in general, but that it is in P when restricted to Clifford path sums. We then provide an algorithm to synthesize a Clifford circuit from a unitary Clifford path sum. The circuits produced by our extraction algorithm are of the form C1-H-C2, where C1 and C2 are Hadamard-free circuits and H is a layer of Hadamard gates. We also provide a heuristic generalization of our extraction algorithm to arbitrary path sums. While this algorithm is not guaranteed to succeed, it often succeeds and typically produces natural looking circuits. Alongside applications to the optimization and decompilation of quantum circuits, we demonstrate the capability of our algorithm by synthesizing the standard quantum Fourier transform directly from a path sum.

arXiv commentIn Proceedings QPL 2022, arXiv:2311.08375

References in corpus (23)

• Polynomial-time T-depth Optimization of Clifford+T circuits via Matroid Partitioning
• Automated optimization of large quantum circuits with continuous parameters
• Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
• The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
• Reducing T-count with the ZX-calculus
• Towards Large-scale Functional Verification of Universal Quantum Circuits
• Hadamard-free circuits expose the structure of the Clifford group
• There and back again: A circuit extraction tale
• ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
• Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations
• T-count optimization and Reed-Muller codes
• Generators and relations for n-qubit Clifford operators
• Quantum circuits and low-degree polynomials over F_2
• Fast Stabiliser Simulation with Quadratic Form Expansions
• A simple encoding of a quantum circuit amplitude as a matrix permanent
• Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus
• A Finite Presentation of CNOT-Dihedral Operators
• Computing quopit Clifford circuit amplitudes by the sum-over-paths technique
• Circuit Extraction for ZX-diagrams can be #P-hard
• The phase/state duality in reversible circuit design
• Generators and Relations for Un(Z[1/2,i])
• Generators and Relations for Real Stabilizer Operators
• Generators and Relations for the Group On(Z[1/2])

Cited by in corpus (2)

• Fast algorithms for classical specifications of stabiliser states and Clifford gates
• Polynomial-Time Classical Simulation of Hidden Shift Circuits via Confluent Rewriting of Symbolic Sums