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arxiv: 2606.21283 · v1 · pith:RGDSJ2C3new · submitted 2026-06-19 · 🪐 quant-ph

Reducing measurement error with adaptivity

James Byrne , Noah Linden , Paul Skrzypczyk This is my paper

Pith reviewed 2026-06-26 14:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords adaptive measurementsmeasurement errornoisy qubit measurementsfeed-forwardquantum information processingtwo-outcome measurementserror mitigation
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0 comments X

The pith

Adaptive measurement circuits can reduce errors in noisy qubit measurements better than any non-adaptive circuit with the same number of measurements.

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

The paper shows that adaptivity in measurement circuits, where earlier results guide later choices, allows for greater reduction in overall error when using multiple noisy measurements on qubits compared to doing all measurements simultaneously. This holds for a large class of noisy two-outcome qubit measurements, with the advantage appearing at just three measurements. The benefit can become arbitrarily large as the number of measurements increases. Readers would care because this offers a way to improve the accuracy of quantum measurements using existing noisy devices more effectively through circuit design rather than hardware improvements.

Core claim

We show that adaptivity, also called feed-forward, is a powerful resource in reducing the error of measurement circuits that combine multiple uses of a noisy measuring device. Adaptive measurement circuits can in general reduce overall measurement errors further than any non-adaptive measurement circuit could with the same number of measurements. In particular, for a large class of noisy two-outcome qubit measurements, such an adaptive advantage can exist when as few as three measurements are used. We also show that the adaptive advantage is unbounded across the class of noisy two-outcome qubit measurements, as the number of uses of the device increases.

What carries the argument

Adaptive measurement circuits with feed-forward, in which the results of previous measurements influence the choice of further measurements and processing.

If this is right

  • Methods exist for finding optimal measurement circuits in both adaptive and non-adaptive cases.
  • An adaptive advantage exists for a large class of noisy two-outcome qubit measurements at three uses.
  • The adaptive advantage is unbounded as the number of uses of the device increases.
  • General results on the limits of such circuits apply to measuring a qubit and more generally a qudit.

Where Pith is reading between the lines

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

  • Similar adaptive advantages might exist for other types of quantum measurements or systems beyond the specified class.
  • The optimal circuit finding methods could be applied to design better protocols in quantum computing tasks.

Load-bearing premise

The existence of a large class of noisy two-outcome qubit measurements for which the adaptive advantage holds at three uses, making the class representative for the general statements.

What would settle it

A concrete noisy two-outcome qubit measurement where the lowest error with any three-measurement adaptive circuit is no better than with the best non-adaptive circuit.

Figures

Figures reproduced from arXiv: 2606.21283 by James Byrne, Noah Linden, Paul Skrzypczyk.

Figure 1
Figure 1. Figure 1: FIG. 1. An illustration of multiple measurement schemes with two uses of a measuring device [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The three candidate non-adaptive circuits to be opti [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The general form of a class of circuits where one such circuit will be the optimal adaptive circuit with three imperfect [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. A plot of the values of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Example [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Contour plots of the relative adaptive advantage for [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The general non-adaptive circuit we can use to [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The simplified optimal non-adaptive circuit for an [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The general adaptive circuit we can use to improve the measurement [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. A non-adaptive circuit using the semi-perfect [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. An example of the shape of the achievable sets [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The achievable error set [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. An optimal circuit for [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The achievable error set [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 19
Figure 19. Figure 19: , it takes on a very similar shape. Indeed, the lower left boundary of E¯(MSIC, 2) between the axes is exactly shrunk by a factor of 1 2 towards the origin com￾pared to E¯(MSIC, 1). Since these sets are defined recur￾sively, we deduce that this pattern must continue for all N, and hence that: v(MSIC, N) = n (0, 1), [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. An adaptive circuit using the SIC-POVM [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. For three uses of [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
read the original abstract

We show that adaptivity, also called feed-forward, is a powerful resource in reducing the error of measurement circuits that combine multiple uses of a noisy measuring device. In previous work, it has been shown that error in measurements can be mitigated by using the measuring device multiple times. So far, the most effective protocols have been parallel measurement schemes, where all measurements are simultaneous. We extend this idea to adaptive measurement schemes, where the results of previous measurements can influence our choice of processing further down the circuit. We show that adaptive measurement circuits can in general reduce overall measurement errors further than any non-adaptive measurement circuit could with the same number of measurements. In particular, we show that for a large class of noisy two-outcome qubit measurements, such an adaptive advantage can exist when as few as three measurements are used. We also show that the adaptive advantage is unbounded across the class of noisy two-outcome qubit measurements, as the number of uses of the device increases. As part of this work, we devise and use methods for finding optimal measurement circuits in both the non-adaptive and adaptive cases. In addition, we prove general results about the limits of such circuits, both in measuring a qubit, and more generally, a qudit.

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

2 major / 2 minor

Summary. The manuscript claims that adaptive (feed-forward) measurement circuits outperform non-adaptive ones in reducing overall error for the same number of uses of a noisy device. For a large class of noisy two-outcome qubit measurements, an adaptive advantage exists already at three uses; the advantage is unbounded as the number of uses grows. The authors supply explicit constructions, methods for finding optimal adaptive and non-adaptive circuits, and general bounds that apply both to qubits and to qudits.

Significance. If the constructions and comparisons hold, the work establishes adaptivity as a concrete resource for measurement-error mitigation beyond what parallel schemes achieve. The n=3 explicit advantage and the unboundedness result across a broad noise class, together with the optimization methods, would be useful contributions to quantum information processing.

major comments (2)
  1. [Theorem/Construction for n=3 (likely §4 or §5)] The central existence claim for the adaptive advantage at n=3 rests on an unspecified 'large class' of noisy two-outcome qubit measurements. Without an explicit, checkable definition of this class (e.g., the precise noise model or parameter range in the relevant theorem), it is impossible to verify that the stated constructions indeed lie inside the class and that the comparison to the optimal non-adaptive circuit is correctly performed.
  2. [Unboundedness result (likely §6)] The unboundedness statement as n increases likewise depends on the same class definition. If the class is defined only implicitly via the existence of the constructions, the claim risks being circular; an independent characterization of the noise family is needed to confirm the result is not tautological.
minor comments (2)
  1. [Methods for optimal circuits] Notation for the adaptive circuit (feed-forward maps, decision trees) should be introduced with a small diagram or explicit pseudocode in the methods section to make the optimality-finding algorithm reproducible.
  2. [General results for qudits] The general qudit bounds are stated without an accompanying example or numerical check; adding one small qudit instance would clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comments point by point below. The concerns raised are valid regarding clarity, and we will revise the manuscript to provide an explicit, independent definition of the noise class.

read point-by-point responses

  1. Referee: [Theorem/Construction for n=3 (likely §4 or §5)] The central existence claim for the adaptive advantage at n=3 rests on an unspecified 'large class' of noisy two-outcome qubit measurements. Without an explicit, checkable definition of this class (e.g., the precise noise model or parameter range in the relevant theorem), it is impossible to verify that the stated constructions indeed lie inside the class and that the comparison to the optimal non-adaptive circuit is correctly performed.

    Authors: We agree that the definition requires greater explicitness for independent verification. The class is characterized by a specific noise model on two-outcome qubit measurements (POVMs with a tunable bias parameter in a defined interval), but the presentation in the theorems may not make the membership conditions immediately checkable. In the revised manuscript we will insert a dedicated paragraph or subsection that states the precise noise model, the parameter range, and the conditions under which the n=3 constructions belong to the class and outperform the optimal non-adaptive circuit. revision: yes

  2. Referee: [Unboundedness result (likely §6)] The unboundedness statement as n increases likewise depends on the same class definition. If the class is defined only implicitly via the existence of the constructions, the claim risks being circular; an independent characterization of the noise family is needed to confirm the result is not tautological.

    Authors: The class is defined by the noise model and parameter range independently of any particular construction; the constructions are then shown to lie inside that class while achieving growing advantage. Nevertheless, the referee's observation that the current wording risks appearing circular is fair. We will revise §6 (and the preceding sections) to first state the class definition in full, then present the family of constructions and the general bounds that establish unbounded advantage within that independently defined class. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes its central claims through explicit constructions of adaptive and non-adaptive measurement circuits, direct comparisons of their error performance on a specified class of noisy two-outcome qubit measurements, and methods for identifying optimal circuits in both settings. These elements are presented as independent contributions that enable the stated advantage at n=3 and its unbounded growth, without any reduction of predictions to fitted inputs, self-definitional equivalences, or load-bearing self-citations that collapse the argument to prior unverified results by the same authors. The derivation chain remains self-contained against external benchmarks via the explicit constructions and general limit proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are described in the abstract; the work appears to rely on standard quantum measurement theory.

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    Then, there exists a quantum channelAsuch thatA(|i⟩ ⟨i|) =ρ i for alli

    Lemma 1 - Construction of measure-and-prepare channel without measurements Letρ i ∈ D(H)for eachi∈[d]. Then, there exists a quantum channelAsuch thatA(|i⟩ ⟨i|) =ρ i for alli. Proof.For eachi, choose|Ψ i⟩ ∈ H⊗Hto be a purification ofρ i. Then, define a completely positive map Λ by Kraus operatorsK i =|Ψ i⟩ ⟨i|. We have: X i∈[d] K † i Ki = X i∈[d] |i⟩ ⟨Ψi|Ψ...

  42. [42]

    To begin with, suppose thatN= 0

    Proposition 2 - The achievable error set is the set of achievable errors A pair of errors(ε 0, ε1)can be achieved using an adaptive circuit withNmeasurements ofMif and only if(ε 0, ε1)∈ ¯E(M, N), where: ¯E(M, N) = conv(E(M, N)).(D3) Proof.We proceed by induction. To begin with, suppose thatN= 0. With no measurements available, the input state is irrelevan...

  43. [43]

    Proposition 3 - Lower bound onη (A) For anyMandN⩾0: η(A)(M, N)⩾η(M) N .(D9) Proof.Let (ε 0, ε1)∈E(M, N). By definition of the single circuit achievable set (55), we can find ε(x) 0 , ε(x) 1 ∈E(M, N−1) for allx∈[m], and q0,q 1 ∈ R(M) with: 38 ε0 +ε 1 = X x∈[m] ε(x) 0 q(x) 0 +ε (x) 1 q(x) 1 ⩾min x∈[m] n ε(x) 0 +ε (x) 1 o ×  X x∈[m] ε(x) 0 ε(x) 0 +ε(x) 1 q...

  44. [44]

    Then, the action of the dephasing channel followed byC N is the same as the action of measuring with the imperfectZ measurementZ ε0,ε1

    Lemma 4 - Reproducing an imperfectZ measurement LetC N be any circuit with error rates(ε 0, ε1). Then, the action of the dephasing channel followed byC N is the same as the action of measuring with the imperfectZ measurementZ ε0,ε1. Proof.Letρ=ρ 00 |0⟩ ⟨0|+ρ01 |0⟩ ⟨1|+ρ10 |1⟩ ⟨0|+ρ11 |1⟩ ⟨1|, a general qubit state. We will show that the two circuits: •Dep...

  45. [45]

    Using Proposition 3, we already have thatη(M) N ⩽η (A)(M, N)⩽η (NA)(M, N), so it suffices to prove thatη (NA)(M, N)⩽η(M) N

    Proposition 6 - Sufficient condition for there to be no adaptive advantage for anyN If(0, η(M))∈ ¯E(M,1), then for anyN⩾0: η(A)(M, N) =η (NA)(M, N) =η(M) N ,(D16) Proof.This result is trivially true forN= 0, so we as- sume thatN⩾1. Using Proposition 3, we already have thatη(M) N ⩽η (A)(M, N)⩽η (NA)(M, N), so it suffices to prove thatη (NA)(M, N)⩽η(M) N. (...

  46. [46]

    When|ψ⟩=|0⟩, this circuit never makes an error, al- ways outputtingj= 0 correctly

    The post-processing functionfoutputs 1 if any of the measurement results are 1, and 0 if all are 0. When|ψ⟩=|0⟩, this circuit never makes an error, al- ways outputtingj= 0 correctly. When|ψ⟩=|1⟩, the only way for an error to occur is if every measurement resulted in a 0, which happens with probabilityq N. Therefore, we have total error: η=ε 0 +ε 1 = 0 +q ...

  47. [47]

    Proof.IfN= 0, (D18) is trivially true, ase(M,0)∪ g(M,0) ={(0,1),(1,0)}=E(M,0), so their convex hulls are also the same

    Proposition 7 - Achievable error set is a polygon For anyMandN⩾0: conv(e(M, N)∪g(M, N)) = ¯E(M, N),(D18) whereg(M, N) ={(1−ε 1,1−ε 0) : (ε0, ε1)∈e(M, N)} is the reflection ofe(M, N)in the lineε 0 +ε 1 = 1. Proof.IfN= 0, (D18) is trivially true, ase(M,0)∪ g(M,0) ={(0,1),(1,0)}=E(M,0), so their convex hulls are also the same. Then, forN⩾1, we use induction,...

  48. [48]

    Lemma 9 - Large deviation bound on binomial distribution Letp⩽ 1

  49. [49]

    Then: P Bin(n, p)⩾ 1 2 n ⩽ 2 p p(1−p) n ,(D24) whereBin(n, p)is the binomial distribution withntrials and probability of successp. Proof.We derive this from the result of Hoeffding [26, Theorem 1] (see also [27, Theorem 1]), which states that if 0⩽p < a <1, then: P(Bin(n, p)⩾an)⩽e −nD(a∥p),(D25) where: D(a∥p) =alog a p + (1−a) log 1−a 1−p .(D26) Assuming ...

  50. [50]

    (D29) LetX N = (X k)N k=1 be a vector of iid random variables with this same distribution

    Lemma 10 - Large deviation bound on multinomial distribution LetXbe a random variable with distribution: X= ( 0with probability1−t; xwith probability t m−1 for allx∈[m]\ {0}. (D29) LetX N = (X k)N k=1 be a vector of iid random variables with this same distribution. Then, for anyj̸= 0: (a) Ift⩽ m−1 m , (i.e.0more likely than the rest) then: P(#{X k =j}⩾#{X...

  51. [51]

    As before, we apply Lemma 9: P(#{X k =j}⩽#{X k = 0}) ⩽ NX n=0 N n (m−2)t m−1 N−n (m−1)−(m−2)t m−1 n × 2 q (m−1)(1−t) (m−1)−(m−2)t t (m−1)−(m−2)t n = NX n=0 N n (m−2)t m−1 N−n 2 q t(1−t) m−1 n = (m−2)t m−1 + 2 q t(1−t) m−1 N .(D37)

  52. [52]

    (D38) Then,E ∈ ¯Ed(M, N), and soNadaptive uses ofMcan reproduce one use of the generalised symmetric imperfect ZmeasurementS d, 1 d η(A) d (M,N)

    Lemma 11 - Reproducing a generalised symmetric imperfectZmeasurement For any POVMMandN⩾0, define the matrixE= (εj|i)j,i∈[d] by: εj|i = ( 1− 1 d η(A) d (M, N)ifj=i; 1 d(d−1) η(A) d (M, N)ifj̸=i. (D38) Then,E ∈ ¯Ed(M, N), and soNadaptive uses ofMcan reproduce one use of the generalised symmetric imperfect ZmeasurementS d, 1 d η(A) d (M,N). Proof.LetE ′ ∈ ¯E...

  53. [53]

    Lemma 12 - Lower bound onη d for generalised symmetric imperfectZ Letm⩾dandt∈[0,1]. Then: ηd(Sm,t)⩽(d−1) m m−1 t.(D45) Proof.Consider the following circuit, making use of one measurement ofS m,t: •Take input|i⟩ ∈ {|0⟩,|1⟩, ...,|d−1⟩}; •Apply channel mapping|i⟩inddimensions to|i⟩ inmdimensions; •Measure withS m,t; •Post-process the measurement result with ...

  54. [54]

    Then, for any POVMMandN⩾0, we have: 1 d′−1 η(A) d′ (M, N)⩽ 1 d−1 η(A) d (M, N).(D48) Proof.By Lemma 11, one use ofS d, 1 d η(A) d (M,N) can be reproduced byNadaptive uses ofM

    Proposition 13 - Lower bound onη (A) d Letd ′ ⩽d. Then, for any POVMMandN⩾0, we have: 1 d′−1 η(A) d′ (M, N)⩽ 1 d−1 η(A) d (M, N).(D48) Proof.By Lemma 11, one use ofS d, 1 d η(A) d (M,N) can be reproduced byNadaptive uses ofM. Then, we can apply Lemma 12, substitutingd7→d ′,m7→d, and t7→ 1 d η(A) d (M, N) to obtain: η(A) d′ (M, N)⩽η d′ Sd, 1 d η(A) d (M,N)...

  55. [55]

    Proposition 15 - Asymptotic upper bound on η(NA) d for an arbitrary POVM LetMbe any POVM. Then, for anyd⩾2: η(NA) d (M, N) =O (η2(M)(2−η 2(M))) 1 4 N .(D50) Proof.First, we note that ifη 2(M) = 1, then this bound is trivial, so we may assume that our measurementMis non-trivial, i.e.η 2(M)<1. By Corollary 5, one use ofMcan reproduce one use ofZ 1 2 η2(M), ...

  56. [56]

    Letd, ℓ⩾2, with integersq⩾0 andr∈[ℓ]defined by integer division such thatd=ℓq+r

    Proposition 16 - Tighter asymptotic upper bound onη (NA) d for an arbitrary POVM LetMbe any POVM. Letd, ℓ⩾2, with integersq⩾0 andr∈[ℓ]defined by integer division such thatd=ℓq+r. Then: η(NA) d (M, N) =O κN ,(D52) where: κ= (ℓ−2)ηℓ(M) ℓ(ℓ−1) + 2 q ηℓ(M)(ℓ−ηℓ(M)) ℓ2(ℓ−1) 1− q(d−ℓ+r) d(d−1) .(D53) Proof.We note first that we have replacedmin (157) by ℓ, to a...