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arxiv: 2512.13949 · v2 · pith:CCTHPMT7new · submitted 2025-12-15 · 🪐 quant-ph · math-ph· math.MP

Coherence Response in Noisy Quantum Measurements

Zachariah Malik , Quinn Langfitt , Zain Saleem This is my paper

Pith reviewed 2026-05-25 07:10 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum readoutmeasurement noisecoherence responsePOVMerror mitigationquantum computingCPTP mapsassignment matrix
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The pith

Quantum readout probabilities split into separate responses from populations and coherences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a general expression for observed measurement probabilities when arbitrary CPTP noise precedes a computational-basis measurement. It writes the ideal state in populations x and coherences y to obtain the linear relation z = A x + C y, with A the usual column-stochastic assignment matrix and C a new matrix built from the off-diagonal elements of the effective POVM. Standard classical models appear only when every POVM element is diagonal, so C directly measures the extra information carried by coherences. Readers would care because the extra term affects how accurately one can invert readout errors on hardware where coherences survive the noise channel. Experiments reported in the paper indicate that using the full expression raises recovery fidelity and cuts the overhead of certain mitigation routines.

Core claim

Writing the ideal post-circuit state ρ̃ in terms of its populations x and coherences y, the observed probability vector z satisfies z = A x + C y, where A is the familiar classical assignment matrix and C is a coherence-response matrix constructed from the off-diagonal matrix elements of the effective POVM in the computational basis. The classical model z = A x arises if and only if all POVM elements are diagonal; in this sense C quantifies accessible information about coherent readout distortions and interference between computational-basis states, all of which are invisible to models that retain only A.

What carries the argument

The coherence-response matrix C, assembled from the off-diagonal entries of the effective POVM in the computational basis, which supplies the linear map from coherences y to their contribution in the observed probability vector z.

If this is right

  • Readout recovery that uses only A leaves residual error traceable to C y.
  • Including C raises the fidelity of the recovered state compared with classical inversion.
  • Selective Pauli twirling can be performed with exponentially lower circuit overhead once C is known.
  • The separation z = A x + C y supplies a complete linear model for any CPTP noise followed by a fixed-basis measurement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Estimating C from calibration circuits that prepare known superpositions would let experimenters quantify coherence leakage without full POVM tomography.
  • The same decomposition could be applied after rotating the measurement basis, extending the framework to non-computational measurements.
  • Devices whose C is found to be small could safely continue using classical models, while those with large C would require the extended correction.

Load-bearing premise

The noise before the computational-basis measurement is an arbitrary CPTP map whose effective POVM can have nonzero off-diagonal elements in that basis.

What would settle it

Prepare a state whose populations x are known but whose coherences y are nonzero, apply the noisy measurement, and test whether the deviation of the observed z from A x exactly matches the prediction C y.

read the original abstract

Readout error models for noisy quantum devices almost universally assume that measurement noise is classical: the measurement statistics are obtained from the ideal computational-basis populations by a column-stochastic assignment matrix $A$. This description is equivalent to assuming that the effective positive-operator-valued measurement (POVM) is diagonal in the measurement basis, and therefore completely insensitive to quantum coherences. We relax this assumption and derive a fully general expression for the observed measurement probabilities under arbitrary completely positive trace-preserving (CPTP) noise preceding a computational-basis measurement. Writing the ideal post-circuit state $\tilde{\rho}$ in terms of its populations $x$ and coherences $y$, we show that the observed probability vector $z$ satisfies $z = A x + C y$, where $A$ is the familiar classical assignment matrix and $C$ is a coherence-response matrix constructed from the off-diagonal matrix elements of the effective POVM in the computational basis. The classical model $z = A x$ arises if and only if all POVM elements are diagonal; in this sense $C$ quantifies accessible information about coherent readout distortions and interference between computational-basis states, all of which are invisible to models that retain only $A$. Our numerical experiments show that incorporating $C$ into readout recovery can improve fidelity over classical inversion and enable selective Pauli twirling with exponentially reduced circuit overhead. This work therefore provides a natural, fully general framework for coherence-sensitive readout modeling on current and future quantum devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript derives a general expression for observed readout probabilities under arbitrary CPTP noise preceding a computational-basis measurement. Writing the ideal post-circuit state in terms of populations x and coherences y, it obtains the decomposition z = A x + C y, where A is the standard column-stochastic assignment matrix collecting diagonal POVM elements and C is the coherence-response matrix built from off-diagonal elements of the effective POVM. The classical model z = A x holds if and only if all POVM elements are diagonal in the computational basis; numerical experiments are reported to show fidelity gains from incorporating C and reduced overhead for selective Pauli twirling.

Significance. If the decomposition is valid, the work supplies a parameter-free, fully general extension of readout models that quantifies coherence-induced distortions invisible to classical assignment matrices. This framework directly connects to practical error mitigation by enabling coherence-sensitive recovery and lower-overhead twirling protocols. The explicit separation of A and C, together with the tautological characterization of the classical limit, provides a clean conceptual tool for analyzing measurement noise on current hardware.

minor comments (2)
  1. The abstract states that C is 'constructed from the off-diagonal matrix elements of the effective POVM,' but the main text should include an explicit component-wise formula for the entries of C (e.g., in terms of the matrix elements of M_k) to make the construction immediately reproducible without re-deriving the trace.
  2. The numerical experiments paragraph would be strengthened by reporting the precise circuit depths, qubit counts, and noise-model parameters used to generate the fidelity comparisons; without these, it is difficult to assess how representative the reported gains are.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing the conceptual utility of the A/C decomposition, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation of z = A x + C y follows directly from writing the post-circuit state in the computational basis, isolating populations x and coherences y, and applying the trace against the effective POVM elements M_k; this is an immediate algebraic consequence of linearity of the trace with no fitted parameters renamed as predictions, no self-citations invoked as load-bearing premises, and no ansatz or uniqueness theorem smuggled in. The observation that the classical model holds iff all M_k are diagonal is tautological under the same construction and does not constitute a circular reduction. The paper remains self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that readout noise is realized by an arbitrary CPTP channel followed by a fixed computational-basis measurement; C is constructed rather than postulated as an independent entity.

axioms (1)
  • domain assumption Readout noise is realized by an arbitrary CPTP map preceding a computational-basis measurement.
    Stated explicitly as the modeling choice that is relaxed from the classical case.
invented entities (1)
  • Coherence-response matrix C no independent evidence
    purpose: To encode the contribution of off-diagonal POVM elements to observed probabilities.
    C is defined by construction from the effective POVM; no independent experimental signature outside the model is supplied in the abstract.

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discussion (0)

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Reference graph

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