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Clifford testing: algorithms and lower bounds

Marcel Hinsche, Zongbo Bao, Philippe van Dordrecht, Jens Eisert, Jop Briët, Jonas Helsen

TL;DR

This work resolves the natural unitary analogue of stabilizer testing by introducing a four-query, inverse-free Clifford tester that distinguishes Clifford unitaries from those ε-far from Clifford, and proves that the tester is tolerant via a non-commutative Gowers U^3 framework. It establishes a tight fidelity relation between Clifford fidelity and stabilizer fidelity, enabling reductions to stabilizer testing and enabling efficient auxiliary-free single-copy Clifford testers with near-linear resource scaling; it also proves a surprising lower bound Ω(n^{1/4}) for auxiliary-free single-copy Clifford testing. The theoretical core rests on a detailed analysis of the Clifford commutant, including the t=4 case, and a novel connection to matroid theory and PPT-unitary design concepts, which may have independent interest for quantum information and additive combinatorics. Overall, the results advance both practical property-testing tools for quantum devices and the mathematical understanding of Clifford structures under restricted access models, with implications for memoryless and single-copy quantum testing paradigms.

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.

Clifford testing: algorithms and lower bounds

TL;DR

This work resolves the natural unitary analogue of stabilizer testing by introducing a four-query, inverse-free Clifford tester that distinguishes Clifford unitaries from those ε-far from Clifford, and proves that the tester is tolerant via a non-commutative Gowers U^3 framework. It establishes a tight fidelity relation between Clifford fidelity and stabilizer fidelity, enabling reductions to stabilizer testing and enabling efficient auxiliary-free single-copy Clifford testers with near-linear resource scaling; it also proves a surprising lower bound Ω(n^{1/4}) for auxiliary-free single-copy Clifford testing. The theoretical core rests on a detailed analysis of the Clifford commutant, including the t=4 case, and a novel connection to matroid theory and PPT-unitary design concepts, which may have independent interest for quantum information and additive combinatorics. Overall, the results advance both practical property-testing tools for quantum devices and the mathematical understanding of Clifford structures under restricted access models, with implications for memoryless and single-copy quantum testing paradigms.

Abstract

We consider the problem of Clifford testing, which asks whether a black-box -qubit unitary is a Clifford unitary or at least -far from every Clifford unitary. We give the first 4-query Clifford tester, which decides this problem with probability . 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 -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 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.

Paper Structure

This paper contains 44 sections, 46 theorems, 176 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Clifford testing.
  3. Inverse-free Clifford testing.
  4. Tolerant Clifford testing.
  5. Single-copy Clifford testing.
  6. Summary of results
  7. Technical overview
  8. From stabilizer testing to Clifford testing.
  9. Expected stabilizer fidelity and the auxiliary-free tester
  10. Clifford group forms an approximate unitary design for PPT operators
  11. Organization of this work
  12. Preliminaries
  13. Inner products over F2
  14. Weyl operators and stabilizer states
  15. The characteristic distribution
  16. ...and 29 more sections

Key Result

Theorem 1.1

There exists a quantum algorithm that, given an $n$-qubit unitary $U$, makes 4 queries to $U$ and for any $\varepsilon>0$, has the following completeness and soundness guarantees:

Theorems & Definitions (102)

  • Theorem 1.1: One-sided 4-query Clifford tester
  • Theorem 1.2: Two-sided 4-query Clifford tester
  • Theorem 1.3: Inverse theorem for the $Q^3$ norm
  • Theorem 1.4: Efficient auxiliary-free, single-copy Clifford tester
  • Theorem 1.5: Lower bound for auxiliary-free, single-copy Clifford testers
  • Theorem 1.6: Clifford group is an approximate $t=o(n^{1/4})$-design for PPT operators
  • Definition 2.1: Standard inner product
  • Definition 2.2: Dual of subspace
  • Definition 2.3: Self-orthogonal subspace, self-dual subspace
  • Definition 2.4
  • ...and 92 more