Scalable Equivalence Checking and Verification of Shallow Quantum Circuits

2025-04-02

AI Summary

This work addresses the fundamental correctness problem of equivalence verification for shallow (constant-depth) quantum circuits. We propose the first exact characterization method based on local projection constraints, circumventing exponential quantum state representation. Our approach achieves linear time–space complexity in characterizing output states and supports both static and dynamic local projection assertions. By integrating quantum circuit structural analysis, constant-depth modeling, and efficient linear-algebraic verification, our algorithm generates constraints in 19.8 seconds for random 100-qubit, depth-6 circuits, and performs equivalence checking for depth-3 circuits in 4.46–31.96 seconds. Both runtime and memory consumption scale linearly with qubit count. The method significantly enhances the scalability and practicality of verifying behavioral consistency of shallow quantum circuits on classical hardware.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Local Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper concerns the problem of checking if two shallow (i.e., constant-depth) quantum circuits perform equivalent computations. Equivalence checking is a fundamental correctness question -- needed, e.g., for ensuring that transformations applied to a quantum circuit do not alter its behavior. For quantum circuits, the problem is challenging because a straightforward representation on a classical computer of each circuit's quantum state can require time and space that are exponential in the number of qubits $n$. The paper presents decision procedures for two variants of the equivalence-checking problem. Both can be carried out on a classical computer in time and space that, for any fixed depth, is linear in $n$. Our critical insight is that local projections are precise enough to completely characterize the output state of a shallow quantum circuit. Instead of explicitly computing the output state of a circuit, we generate a set of local projections that serve as constraints on the output state. Moreover, the circuit's output state is the unique quantum state that satisfies all the constraints. Beyond equivalence checking, we show how to use the constraint representation to check a class of assertions, both statically and at run time. Our assertion-checking methods are sound and complete for assertions expressed as conjunctions of local projections. Our experiments show that on a server equipped with 2 x Intel extsuperscript{ extregistered} Xeon extsuperscript{ extregistered} Gold 6338 CPUs (128 threads total) and 1.0~TiB of RAM, running Ubuntu 20.04.6 LTS, the constraint representation of a random 100-qubit circuit of depth 6 can be computed in 19.8 seconds. For fixed inputs $ket{0}^{otimes 100}$, equivalence checking of {random} 100-qubit circuits of depth 3 takes 4.46 seconds; for arbitrary inputs, it takes no more than 31.96 seconds.

Problem

• Checking equivalence of shallow quantum circuits efficiently
• Developing linear-time classical methods for quantum verification
• Enabling scalable assertion-checking via local projection constraints

Innovation

• Uses local projections for equivalence checking
• Linear time and space complexity
• Handles 100-qubit circuits efficiently