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The Power of Two Bases: Robust and copy-optimal certification of nearly all quantum states with few-qubit measurements

Andrea Coladangelo, Jerry Li, Joseph Slote, Ellen Wu

TL;DR

This work develops robust quantum state certification protocols that use only single-qubit (and occasional logarithmic-qubit) measurements to reliably certify nearly all Haar-typical pure $n$-qubit states. By proving a novel uncertainty principle for conditional fidelities across standard and Hadamard bases, the authors construct tests with constant robustness (and optimal copy complexity) for almost all targets, and a secondary protocol achieving $Θ(1/ ext{log }n)$ robustness using only single-qubit measurements. The core ideas combine a two-basis conditional-measurement approach with log-sized postmeasurement states and, in a later protocol, a one-shot Gupta–He–O’Donnell subroutine to boost robustness while staying within simple measurement schemes. These results bring scalable, robust verification closer to practical feasibility, with potential fully-nonadaptive variants via shadow tomography and applicability as benchmarking tools for complex, highly entangled states. The access model lies between prior approaches, balancing feasibility of amplitude queries in two bases with the goal of robustness for almost all target states.

Abstract

A central task in quantum information science is state certification: testing whether an unknown state is $ε_1$-close to a fixed target state, or $ε_2$-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary $n$-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant $ε_1$, they can only positively certify states within $ε_1=O(1/n)$ trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated. Here we present robust certification protocols based on few-qubit measurements that apply to all but a $O(2^{-n})$-fraction of pure target states. Our first protocol achieves constant robustness, i.e. $ε_1=Θ(1)$, using a single $O(\log n)$-qubit measurement along with single-qubit measurements in the $Z$ or $X$ basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in $n$) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust: $ε_1=Ω(1/\log n)$. Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.

The Power of Two Bases: Robust and copy-optimal certification of nearly all quantum states with few-qubit measurements

TL;DR

This work develops robust quantum state certification protocols that use only single-qubit (and occasional logarithmic-qubit) measurements to reliably certify nearly all Haar-typical pure -qubit states. By proving a novel uncertainty principle for conditional fidelities across standard and Hadamard bases, the authors construct tests with constant robustness (and optimal copy complexity) for almost all targets, and a secondary protocol achieving robustness using only single-qubit measurements. The core ideas combine a two-basis conditional-measurement approach with log-sized postmeasurement states and, in a later protocol, a one-shot Gupta–He–O’Donnell subroutine to boost robustness while staying within simple measurement schemes. These results bring scalable, robust verification closer to practical feasibility, with potential fully-nonadaptive variants via shadow tomography and applicability as benchmarking tools for complex, highly entangled states. The access model lies between prior approaches, balancing feasibility of amplitude queries in two bases with the goal of robustness for almost all target states.

Abstract

A central task in quantum information science is state certification: testing whether an unknown state is -close to a fixed target state, or -far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary -qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant , they can only positively certify states within trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated. Here we present robust certification protocols based on few-qubit measurements that apply to all but a -fraction of pure target states. Our first protocol achieves constant robustness, i.e. , using a single -qubit measurement along with single-qubit measurements in the or basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in ) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust: . Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.
Paper Structure (18 sections, 23 theorems, 99 equations)

This paper contains 18 sections, 23 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Robust certification.
  3. Main results
  4. Our state certification tests
  5. Proof ideas
  6. An uncertainty principle for conditional fidelities.
  7. Discussion and future work
  8. Constant robustness with single-qubit measurements.
  9. Beyond typicality.
  10. Beyond pure states.
  11. Access model
  12. Additional related work
  13. Robust certification with one logarithmic-qubit measurement
  14. The test
  15. Proof of robustness
  16. ...and 3 more sections

Key Result

Theorem 1

There exists an efficient algorithm that, for all but a $O(2^{-n})$ fraction of pure $n$-qubit target states $\ket{\psi}$, satisfies the following. Given oracle access to a classical description of the target state $\ket{\psi}$ (via the model in Definition def:oracle), and a single copy of an $n$-qu Moreover, the algorithm makes $n-O(\log n)$ single-qubit measurements: either all in the $Z$ basis,

Theorems & Definitions (39)

  • Theorem 1: Constant robustness with one $O(\log n)$-qubit measurement
  • Corollary 1: Optimal copy-complexity with one $O(\log n)$-qubit measurement per copy
  • Theorem 2: $\Theta(1/\log n)$-robustness with only single-qubit measurements
  • Corollary 2: Nearly optimal copy-complexity with only single-qubit measurements
  • Theorem 3: Uncertainty principle for conditional fidelities
  • Conjecture 1
  • Proposition 1: No universal robustness for "conditional comparison tests"
  • Definition 1
  • Lemma 1
  • proof
  • ...and 29 more