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Quantum advantage for noisy channel discrimination

Zane M. Rossi, Jeffery Yu, Isaac L. Chuang, Sho Sugiura

arXiv:2105.08707v1quant-ph

TL;DR

It is unclear whether coherent-access advantages in channel discrimination persist when the process is noisy. This paper studies noisy single-qubit rotation discrimination with parameterized noise and coherent, incoherent, and hybrid protocols. It finds a finite-noise advantage region and a noise-dependent transition in optimal strategies.

  • Problem

    The paper investigates the gap between constructive studies of noise effects and prior access-model results that do not directly address noisy classification.

    Source: §I · p. 1

  • Method

    The authors study binary classification between two channel hypotheses with continuously parameterized noise and compare coherent, incoherent, and hybrid protocols.

    Source: §I · p. 1, §IV · p. 8

  • Results

    The coherent-access advantage persists at finite noise, but its region in (σ, δ) shrinks inverse-polynomially with query complexity, and hybrid protocols can improve performance.

    Source: §I · p. 2, §V · p. 9

  • Takeaways & Limitations

    The optimal coherence length depends on noise, following the standard quantum limit at small noise and becoming more classical in character at large noise.

    Source: §V · p. 9

    The study excludes adaptive hybrid and entangled protocols, though its limited protocols are asymptotically optimal in the small-noise and small-mean-separation limits.

    Source: §V · p. 9

Abstract

from arXiv · show

Many quantum mechanical experiments can be viewed as multi-round interactive protocols between known quantum circuits and an unknown quantum process. Fully quantum "coherent" access to the unknown process is known to provide an advantage in many discrimination tasks compared to when only incoherent access is permitted, but it is unclear if this advantage persists when the process is noisy. Here, we show that a quantum advantage can be maintained when distinguishing between two noisy single qubit rotation channels. Numerical and analytical calculations reveal a distinct transition between optimal performance by fully coherent and fully incoherent protocols as a function of noise strength. Moreover, the size of the region of coherent quantum advantage shrinks inverse polynomially in the number of channel uses, and in an intermediate regime an improved strategy is a hybrid of fully-coherent and fully-incoherent subroutines. The fully coherent protocol is based on quantum signal processing, suggesting a generalizable algorithmic framework for the study of quantum advantage in the presence of realistic noise.

I. INTRODUCTION

The paper addresses whether coherent-access advantages can persist under finite noise through a constructive study of noisy channel classification. It finds a noise-dependent transition between coherent and incoherent protocols and identifies improved hybrid strategies.

I. INTRODUCTION

  • The study maps where quantum advantage exists for a binary classification task with finite, continuously parameterized noise.The task compares discrimination protocols as a function of signal and noise parameters.Source: §I · p. 1
  • The work addresses a gap between noise-fragile classification results and quantum advantages shown under noise only information-theoretically.The authors seek a constructive investigation of noise effects for realistic inference tasks.Source: §I · p. 1
  • Below a noise threshold, coherent access can outperform incoherent access; above it, the reverse holds, and below it hybrid protocols can perform even better.The hybrid strategy alternates coherent computation and measurement, with an optimized coherence length.Source: §I · pp. 1–2
  • Quantum signal processing (QSP) provides the basis for the coherent protocol and is extended here to a noisy context.The analysis relies on known QSP properties and guarantees.Source: §I · p. 2
  • The paper first treats a noiseless instance, then noisy discrimination, and finally hybrid protocols.Closed-form limits are used to analyze behavior in the noisy setting.Source: §I · p. 2

A. Problem statement

The rotation discrimination game asks which of two angle distributions governs repeated single-qubit channel rotations. The paper compares serial, non-adaptive protocols that either measure after every use or retain coherence between channel applications.

A. Problem statement

  • The rotation discrimination game distinguishes two channels whose rotation angles are sampled from distributions fixed across queries.The queried party must identify the underlying distribution with high confidence and low query complexity.Source: A. Problem statement · p. 2
  • The normal angle distributions are parameterized by mean and standard deviation, with δ denoting the separation between their means.The noise strength is σ, and the mean separation is δ = |µ0 − µ1|.Source: A. Problem statement · p. 2
  • The protocols are serial and non-adaptive, using one qubit and making the strategy independent of intermediate measurement results.The channel is not applied jointly to multiple qubits.Source: A. Problem statement · p. 2
  • The problem exhibits a transition: coherent access is strictly better without noise, while sufficiently large noise favors fully incoherent access.The paper uses these limits to study the intermediate regime.Source: A. Problem statement · p. 2
  • Incoherent access requires a complete measurement between channel applications, while coherent access permits intermediate quantum gates instead.Incoherent measurement outcomes are processed by a classical thresholding procedure.Source: A. Problem statement · p. 2, Figure 1 · p. 3

B. Prior work

Prior work establishes access-model comparisons and noisy channel discrimination results, but leaves a gap around constructive comparisons for noisy classification. QSP and QSVT offer flexible tools for inference, although their noisy-setting applications have been limited.

B. Prior work

  • Access-model comparisons have shown coherent advantages for some inference problems, but prior results differ in task and treatment of noise.Work proves noisy advantage information-theoretically for regression, while classification results such as [1] can lose advantage under finite noise.Source: B. Prior work · p. 3
  • Constructive evidence for performance gaps under physically motivated access restrictions can inform future generalizations.The paper frames this motivation in the context of quantum mechanics as a resource convex theory.Source: B. Prior work · p. 3
  • Existing noisy-channel hypothesis-testing bounds generally do not compare coherent and incoherent access specifically for classification.The authors state that no existing work investigates this relative advantage for even simple classification games such as RDGs.Source: B. Prior work · p. 3
  • QSP and QSVT efficiently transform unitary eigenvalues and embedded-operator singular values, but their noisy-inference applications have been limited.These methods reproduce algorithms for tasks including Hamiltonian simulation, quantum linear systems, and factoring.Source: B. Prior work · pp. 3–4

II. NOISELESS DISCRIMINATION

The noiseless section establishes a simpler baseline for the later noisy analysis. It asks whether coherent, including QSP-based, discrimination can outperform incoherent protocols and what resource model suffices for optimal discrimination.

II. NOISELESS DISCRIMINATION

  • The noiseless case permits closed-form results and provides intuition for the later introduction of noise.The section considers its questions analytically and then with a concrete example.Source: §II · p. 4
  • The section frames its analysis around questions that become important and non-trivial in the general noisy setting.These include the relative speedup and the simplest resource model for optimal discrimination.Source: §II · p. 4
  • The section asks whether coherent, including QSP-based, protocols can provably outperform incoherent protocols and by how much.It also asks which resource model is sufficient for optimal discrimination across channel hypotheses.Source: §II · p. 4

A. Sufficiency of QSP protocols for optimal noiseless discrimination

The section studies noiseless discrimination between two fixed rotations and characterizes coherent and incoherent access protocols. It establishes that coherent access matches or exceeds incoherent performance, with QSP providing a sufficient protocol family for optimal noiseless discrimination.

A. Sufficiency of QSP protocols for optimal noiseless discrimination

  • A noiseless RDG asks the querent to distinguish the fixed rotations exp{iθ0σx} and exp{iθ1σx}.Source: A. Sufficiency of QSP protocols for… · p. 4
  • A QSP protocol uses phase-angle lists together with classical descriptions of preparations and projective measurements, and QSP protocols include incoherent protocols as a special case.When rj = 1, the protocol reduces to an incoherent access protocol.Source: A. Sufficiency of QSP protocols for… · p. 4
  • For noiseless RDGs, coherent protocols match or exceed incoherent performance, and an N-query coherent non-adaptive protocol perfectly decides b with optimal N=O(δ^-1).Source: A. Sufficiency of QSP protocols for… · p. 5
  • QSP protocols suffice for optimal noiseless discrimination because coherent access protocols can always outperform incoherent ones.The proof notes that coherent access strictly contains the protocols available under incoherent access.Source: A. Sufficiency of QSP protocols for… · p. 5

B. A simple concrete example

The example compares QSP with majority-vote incoherent protocols for fixed-axis rotation discrimination. QSP has strictly lower error for nonzero signal separation, while an adaptive incoherent strategy improves on plain majority vote but does not achieve perfect discrimination for nonorthogonal channels.

B. A simple concrete example

III. NOISY DISCRIMINATION

The section asks whether the performance gap between coherent and incoherent access survives finite noise. It reports a transition boundary for quantum advantage and examines it numerically and analytically.

III. NOISY DISCRIMINATION

  • Finite noise produces a transition boundary between regions of quantum advantage for different access models.Source: §III · p. 6
  • The section tests whether the coherent-versus-incoherent performance gap for RDGs remains robust under finite noise.Source: §III · p. 6
  • The transition boundary is investigated first numerically and then analytically.Source: §III · p. 6

A. Quantum advantage for noisy discrimination

The noisy-discrimination analysis compares QSP and majority-vote protocols at equal query complexity and finds a region of coherent advantage at low signal and noise. It also reports poor performance for both protocols at high noise and analytically characterizes the advantage boundary in two simplified limits.

A. Quantum advantage for noisy discrimination

B. Transition boundaries in the low-separation limit

In the low-separation limit, the transition between coherent and incoherent access can be analyzed by comparing their error probabilities. The transition boundary shrinks with increasing query complexity.

B. Transition boundaries in the low-separation limit

  • For noisy RDG with sufficiently small angular separation, optimal QSP phase angles tend to zero, yielding the simple QSP protocol.Simple QSP uses zero phase angles and a projective measurement optimized for the hypothesis channels.Source: B. Transition boundaries in the… · p. 7
  • The analysis compares simple-QSP and majority-vote error probabilities to identify the transition boundary for small angular separation.The analytic treatment extends the comparison into a low-separation region where simulations are numerically unstable.Source: B. Transition boundaries in the… · pp. 7–8
  • σ^2 ≈ log(M)/M = log(N)/N as N grows; for M = 1, the boundary occurs at σ = ln 2/2.The M = 1 boundary agrees well with numerical simulation.Source: B. Transition boundaries in the… · p. 8

IV. HYBRID PROTOCOLS FOR LARGE N AND OPTIMAL COHERENCE LENGTH

This section asks how to allocate a large query budget between coherent and incoherent access when channel noise is known. It derives an optimal coherence length for hybrid protocols and finds distinct noise-dependent regimes.

IV. HYBRID PROTOCOLS FOR LARGE N AND OPTIMAL COHERENCE LENGTH

  • The authors examine the transition region through a separate limit that makes it analytically tractable and instructive.They use the large-noise limit to expand the digamma functions and derive an approximate solution.Source: §IV · p. 8
  • A ξ-hybrid protocol runs ξ-length QSP sequences N/ξ times and takes a majority vote; ξ = N and ξ = 1 recover coherent and incoherent access.The choice of coherence length addresses how to split a large query budget when noise magnitude is known.Source: §IV · p. 8
  • In the small-separation, large-N limit, the hybrid error expression is minimized by a unique ξmin relation.The analysis assumes large N permits analytic continuation of the discrete expressions.Source: §IV · p. 8
  • When ξmin exceeds N, a fully coherent protocol is simply optimal.The analytic continuation permits ξmin to exceed N or fall below 1.Source: §IV · p. 8, Figure 4 · p. 9
  • In the large-noise regime, the coherence length is set to keep accumulated error constant, matching the classical inverse-variance behavior.The small-noise scaling instead decays more slowly than this classical limit.Source: §IV · pp. 8–9
  • At σ/π around 0.2, ξmin falls below 1 and incoherent access minimizes error; around 0.01, fully coherent access is optimal, with hybrids optimal between.The analytically continued ξmin is not bounded by 1 or N.Source: §IV · p. 9, Figure 4 · p. 9
  • Optimal coherence length follows ξ ∝ σ^-2 only at large noise, but approaches ξ ∝ σ^-1 at small noise.Figure 4 plots the N = 100 example and shows the transition between these scaling regimes.Source: §IV · pp. 8–9, Figure 4 · p. 9

V. CONCLUSION AND DISCUSSION

The paper finds that coherent-protocol advantage persists at finite noise, while its parameter-space region shrinks inverse-polynomially with query complexity. Hybrid protocols can improve performance, and the appendix describes an adaptive incoherent strategy based on Bayesian updating and measurement-basis selection.

V. CONCLUSION AND DISCUSSION

  • The coherent-access advantage region in noise and signal strength shrinks inverse-polynomially in area as query complexity increases.Numerical approximations agree with the analytic form in various limits.Source: §V · p. 9
  • For hybrid protocols, the optimal coherence length follows standard-quantum-limit dependence on noise at small noise and more classical dependence at large noise.The authors compute the optimal allocation between coherent and incoherent subroutines in suitable limits.Source: §V · p. 9
  • The work applies QSP methods to noisy settings and provides numerical and analytical methods for studying protocol robustness.The authors describe these channels as minimal examples where QSP-based methods can fully characterize quantum advantage.Source: §V · p. 9
  • The protocols studied are not exhaustive: adaptive hybrid and entangled protocols are excluded, though the analyzed limits are asymptotically optimal for these limited protocols.The stated limits are small noise and small mean separation.Source: §V · p. 9
  • Bounds on adaptive incoherent protocols are generally difficult to compute beyond specific noise models, including for more exotic models such as non-Markovian noise.The robustness of coherent-access advantage for general noise models remains open.Source: §V · pp. 9–10

Appendix A: Adaptive incoherent strategy

  • The adaptive incoherent strategy updates hypothesis probabilities by Bayes’ theorem after each measurement and thresholds the final probabilities to choose the more likely hypothesis.The measurement basis can be selected to maximize mutual information between the hypothesis and measurement outcome.Source: Appendix A · pp. 10–11

Appendix B: Numerical methods

The numerical methods combine analytic calculations for classical success probabilities with Monte Carlo estimation and repeated optimization for QSP protocols. The appendix also derives the majority-vote success probability by retaining first-order terms in the small-separation parameter.

Appendix B: Numerical methods

  • The plotted error probabilities are evaluated or simulated separately on a 0.001 × 0.001 grid.The authors identify the noisy-qsp-rdg repository as containing the relevant code.Source: Appendix B · p. 11
  • Classical success probability is computed by analytically integrating over the two angle distributions, with the normal-distribution case evaluated in closed form.The error probability is the complement of success, and majority votes are calculated using Eq. (6).Source: Appendix B · p. 11
  • QSP success probabilities are estimated by Monte Carlo sampling of channel angles, and the phase angles and prepared states are optimized to minimize error.Optimization is repeated because local minima may prevent a single run from finding the global optimum.Source: Appendix B · p. 11

Appendix C: Majority vote for Helstrom bound

  • For majority voting over 2M + 1 samples, the appendix derives the success probability by applying the majority-vote function and retaining first-order terms in δ.The collected terms include 1/2 by binomial symmetry and a partial-sum identity.Source: Appendix C · p. 11

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