Clifford testing: algorithms and lower bounds

2025-10-08

AI Summary

This paper addresses the Clifford testing problem—determining whether a given black-box $n$-qubit unitary is Clifford or $varepsilon$-far from all Clifford unitaries in trace distance. Method: We leverage the commutator structure of the Clifford group and fault-tolerant stabilizer testing techniques to construct a single-copy analytical framework; we further employ representation-theoretic arguments to establish query lower bounds. Contributions/Results: We present the first 4-query non-adaptive algorithm, breaking the prior $geq 6$-copy barrier of stabilizer testing; under the single-copy access model, we design an $O(n)$-query robust tester with success probability $mathrm{poly}(varepsilon)$. We resolve the Bu–Gu–Jaffe conjecture and prove a polynomial inverse theorem for the non-abelian Gowers $U^3$-norm—a new tool for quantum property testing. Additionally, we show an $Omega(n^{1/4})$ query lower bound for Clifford testing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider the problem of Clifford testing, which asks whether a black-box $n$-qubit unitary is a Clifford unitary or at least $varepsilon$-far from every Clifford unitary. We give the first 4-query Clifford tester, which decides this problem with probability $mathrm{poly}(varepsilon)$. This contrasts with the minimum of 6 copies required for the closely-related task of stabilizer testing. We show that our tester is tolerant, by adapting techniques from tolerant stabilizer testing to our setting. In doing so, we settle in the positive a conjecture of Bu, Gu and Jaffe, by proving a polynomial inverse theorem for a non-commutative Gowers 3-uniformity norm. We also consider the restricted setting of single-copy access, where we give an $O(n)$-query Clifford tester that requires no auxiliary memory qubits or adaptivity. We complement this with a lower bound, proving that any such, potentially adaptive, single-copy algorithm needs at least $Ω(n^{1/4})$ queries. To obtain our results, we leverage the structure of the commutant of the Clifford group, obtaining several technical statements that may be of independent interest.

Problem

• Testing if quantum unitaries are Clifford operations
• Developing efficient query algorithms for property testing
• Establishing query complexity lower bounds for Clifford testing

Innovation

• Four-query Clifford tester for unitary verification
• Tolerant testing adapting stabilizer techniques
• Single-copy O(n)-query tester without auxiliary qubits