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REVIEW 2 major objections 7 minor 87 references

Verifiable blind probabilistic error cancellation

T0 review · 2 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that probabilistic error cancellation, a standard quantum error mitigation technique, can be delegated to a fully malicious server with perfect blindness and verifiable, composably secure outputs.

desk verdict First composable-security protocol for PEC; the main theorem survives my checks, but practicality claims are untested and the proof is dense. read the letter →

arxiv 2607.25704 v1 pith:J63KP3OL submitted 2026-07-28 quant-ph cs.CR

Bo Yang , Elham Kashefi , Harold Ollivier This is my paper · ORCID
classification quant-phcs.CR MSC 81P6894A60 PACS 03.67.-a03.67.Dd
keywords verifiableblindquantumcomputationprobabilisticerrorcancellationmitigationcomposablesecuritymeasurement-basedone-timepadWalsh-Hadamardtransformnoisebenchmarking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Probabilistic error cancellation (PEC) lets a client estimate ideal expectation values by sampling from a quasi-probability decomposition of the inverse noise channel, but it assumes the noise model is accurate and the server faithfully executes the sampled corrections. This paper claims that both assumptions can be dropped: in the VBPEC protocol the client interleaves computation rounds with trap test rounds, uses the trap outputs to estimate the server's effective Pauli deviation via a Walsh–Hadamard transform, and accepts the PEC-corrected estimate only when that estimated deviation is close to the client's reference noise model. The main theorem states that VBPEC constructs an ideal "Secure Delegated Probabilistic Error Cancellation" resource with perfect blindness and security error exponentially small in the number of computation and test rounds, within the abstract cryptography framework. A sympathetic reader would care because it would give the first end-to-end cryptographic reliability guarantee for a quantum error mitigation pipeline, at the cost of only additional repetitions rather than additional qubits.

What carries the argument

The central mechanism is the combination of the quantum one-time pad (QOTP) twirl, general trap patterns, and the inverse Walsh–Hadamard transform. The QOTP converts any server deviation into an effective stochastic Pauli channel whose $Z$ component alone flips measurement outcomes, so PEC's inverse channel can be realized entirely by classical bit flips on the client's decoded outcomes. Test rounds use random general trap patterns, and because $\mathbb{E}[(-1)^{\langle \mathrm{set}(P), S_i\rangle} \tilde{y}_i] = \tilde{p}_T(P)$, the signed average over $N_t$ traps gives an unbiased estimator of the server's Pauli-deviation probability. The client assembles these estimates into the statistic $\widetilde{\Delta} = D_\Lambda(\tilde{p}_T, \tilde{p}_{\mathrm{PEC}})$, a coarse-grained $\ell^1$ distance between the estimated and reference deviation distributions, and accepts iff $\widetilde{\Delta} \le \epsilon_t$. The proof then bounds three fluctuations---the PEC estimator, the trap-statistic estimator, and the computation/test round mismatch---using Hoeffding, McDiarmid, and Serfling inequalities.

What would settle it

Search numerically for an adversarial sequence of Pauli deviations across rounds that passes the test condition $\widetilde{\Delta} \le \epsilon_t$ while the PEC-corrected computation estimate satisfies $|\widetilde{o}_C - \mathrm{Tr}[\rho O]| > \epsilon$ with non-negligible probability. Theorem 2's security bound says this joint event has probability at most three exponentially small terms, so any concrete strategy beating that bound would refute the central claim.

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Extended reading notes

Core claim

The central discovery, stated as Theorem 2, is that VBPEC $\delta$-constructs the SDPEC resource: for any accepted non-abort output, the probability that it deviates from the ideal expectation value $\mathrm{Tr}[\rho O]$ by more than $\epsilon$ is negligible in both $N_c$ and $N_t$, and under honest noise matching the reference model $E_{\mathrm{PEC}}$, the probability of accepting a correct mitigated estimate converges to one exponentially. The protocol achieves this by turning trap-based verification from deterministic pass/fail checks into a statistical benchmark of the server's effective Pauli deviation distribution, and by proving that the residual bias of the PEC estimator is controlled by the same statistic used for acceptance.

Load-bearing premise

The protocol assumes the client already has a correct-enough description of the server's noise as a stochastic Pauli channel with only polynomially many relevant terms, and that the server's true deviation, after the one-time-pad randomization, really is of that form; the protocol verifies consistency with this description but does not learn it from scratch.

Editorial extensions

If this is right

  • A client with only single-qubit prepare-and-send capabilities can obtain error-mitigated expectation values from an untrusted quantum server while keeping the computation perfectly blind.
  • PEC's cancellation operations reduce to classical bit flips in the client's post-processing, so no additional noisy quantum gates are inserted into the delegated circuit.
  • Under honest noise that matches the reference model, the probability of accepting a within-$\epsilon$ estimate converges to one exponentially in $N_c$ and $N_t$; under moderate model mismatch it remains high as long as $D_{\mathrm{mis}} < \min(\epsilon/\|\vec q\|_1, \epsilon_t)$.
  • Verification no longer rejects once noise exceeds a fixed threshold; noise consistent with the PEC model is actively cancelled, so the practical acceptance probability improves without changing the security guarantee.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same trap post-processing already reconstructs the server's effective Pauli distribution, so the test-round data could serve as a standalone noise-benchmarking primitive that certifies sparse Pauli noise models without a separate calibration step.
  • Beyond the paper: the proof mechanism of estimator-valued tests plus concentration bounds should transfer to other QEM schemes whose correction is a classical function of the decoded outcomes, such as probabilistic error amplification, though the paper only sketches this direction.
  • Beyond the paper: feeding the estimated $\tilde{p}_T$ back into an updated $E_{\mathrm{PEC}}$ over successive blocks could make the protocol noise-agnostic, replacing a priori model trust with statistical certification.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper introduces Verifiable Blind Probabilistic Error Cancellation (VBPEC), a delegated quantum computation protocol that integrates probabilistic error cancellation (PEC) into the universal blind quantum computing framework. The authors formalize the ideal functionality as the Secure Delegated Probabilistic Error Cancellation (SDPEC) resource in the abstract cryptography framework, and prove that VBPEC constructs it with perfect blindness and an exponentially small security error. The protocol uses test rounds based on general trap patterns to estimate the server's effective Pauli deviation via a Walsh-Hadamard post-processing, and compares this estimate to the reference PEC noise model through a statistical test statistic. The main result, Theorem 2, claims that VBPEC delta-constructs the SDPEC resource with negligible delta in both the number of computation rounds and the number of test rounds, while also providing active error cancellation that improves acceptance probability under matched and moderately mismatched honest noise.

Significance. If the proof can be repaired, this is a substantial contribution to both quantum error mitigation and verified delegated computation. It provides the first composable cryptographic treatment of a QEM technique, formalizing PEC as an ideal resource and showing that it can be securely delegated to a fully malicious server. The extension of trap-based verification from deterministic pass/fail outcomes to statistical benchmarking via the Walsh-Hadamard transform is conceptually novel and technically useful. The protocol inherits the absence of quantum-space overhead from PEC and from recent verification protocols, with overhead only in additional repetitions. The concentration proofs are detailed and largely sound, and the noise-robustness analysis in Section V is a significant improvement over threshold-based verification, since correctly mitigated estimates are accepted with high probability under matched honest noise. The construction is not circular: the reference noise model E_PEC is an input, protocol parameters are fixed, and the security proof does not fit the data.

major comments (2)
  1. [Section IV.C, Eq. (38) and Eq. (47)] The McDiarmid bound in Eq. (47) is applied to the deviation |˜∆−Δ_T|, with Δ_T defined in Eq. (38) as E[˜∆|p_T]. However, the independence of the test-round variables required for the bounded-difference argument holds only conditionally on the actual test set S_T and the fixed Pauli sequence, so Eq. (47) controls |˜∆−E[˜∆|S_T, Pauli sequence]|, not |˜∆−E[˜∆|p_T]|. The gap between these two conditional means is not bounded, and E[˜∆|S_T, Pauli sequence] can depend on the detailed composition of S_T beyond the empirical distribution p_T, so this gap can be non-negligible. This breaks the chain leading from Eq. (42) to Eq. (43) and therefore the proof of Theorem 2 as written. The fix is local: redefine Δ_T := E[˜∆|S_T, Pauli sequence], use Jensen's inequality to obtain D_Λ(p_T, p_PEC) ≤ Δ_T in Lemma 6, and update Eq. (38) and the surrounding text accordingly.
  2. [Theorem 2 and Section IV.C, Eq. (53)] The security proof of Theorem 2 fixes the protocol parameter in Eq. (53) as ϵ_t = γ_t = γ_s = ϵ/(4∥q∥₁), whereas the theorem statement allows any 0 ≤ ϵ_t < ϵ/∥q∥₁. With the choices in Eq. (53), the proof only covers the single value ϵ_t = ϵ/(4∥q∥₁); for other admissible ϵ_t, the inequality γ_c + ∥q∥₁(ϵ_t + γ_t + γ_s) ≤ ϵ may fail. The statement and proof should be reconciled by either stating the theorem for this specific choice or by introducing the slack s = ϵ − ∥q∥₁ϵ_t > 0 and choosing γ_c = s/4, γ_t = γ_s = s/(4∥q∥₁), which preserves the exponential bounds. The lack of consistency is compounded by the correctness proof taking ϵ_t = ϵ/2 and the noise-robustness section taking ϵ_t = ϵ/4, without clarifying that these are illustrative choices for a protocol whose ϵ_t is a single fixed parameter.
minor comments (7)
  1. [Resource 1 and Resource 2] The description of the filtered server interface ('when e=0, the interface does not send any information nor take inputs') is easy to misread; please clarify that the server receives only the public information and otherwise has a null interface in the honest case.
  2. [Protocol 3] The set N_G^odd(S) and the variables d_v are used in Step 5 before being defined; add a pointer to Appendix D where the general trap construction is explained.
  3. [Eq. (12) and Protocol 4] The estimator ˜p_T(P) is defined twice, once in the text around Eq. (12) and again inside Protocol 4; unify the notation to avoid duplication and potential inconsistency.
  4. [Section IV.C, before Eq. (38)] There is a stray period after the definition of μ_C in the sentence introducing Eq. (38); this is a minor typographical error.
  5. [Appendix E, Lemma 6 proof] The equality D_Λ(p_T, p_PEC) = D_Λ(E[˜p_T|p_T], p_PEC) is used without comment; it follows from Lemma 5 and linearity of the conditional expectation, but this should be stated explicitly for clarity.
  6. [Section II.B] The crucial assumption that the client's preparation imperfections are independent of the secret QOTP parameters is stated in passing; it is load-bearing for the twirling argument and should be listed as an explicit assumption in Section II.B alongside the definition of E_PEC.
  7. [References] Several references are to unpublished or not-yet-indexed works (e.g., Refs. [24], [28], [59]); please provide arXiv identifiers or DOIs where available for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: VBPEC's security proof is a concentration-based derivation from a fixed input noise model, not a fit or a self-citation chain.

full rationale

The claimed derivation chain is self-contained and does not reduce to its inputs. The reference model E_PEC (equivalently p_PEC and q) is an explicit protocol input and is never fitted to the server's trap data; the trap statistic p~_T is used only to test consistency with p_PEC. The central bound Lemma 6 derives |mu_C - Tr[rho O]| <= ||q||_1 D_Lambda(p_C,p_PEC) using the inversion identity q * p_PEC = e_0 and Young's inequality; this is a mathematical consequence of PEC, not a restatement of the acceptance test. The acceptance test supplies D_Lambda(p~_T,p_PEC) <= eps_t, and the proof separately controls fluctuations of the mitigated estimator, the test statistic, and the computation/test split by Hoeffding, McDiarmid, and Serfling inequalities; none of these bounds is calibrated to the data. Consequently, 'every accepted output is within eps' is a derived statistical guarantee rather than a definitional equivalence. The main self-citations (VBOE [29], general traps [27], UBQC composability [36]) are either independently established, used for comparison, or derived in the appendices; none imports an unverified uniqueness theorem or ansatz. The reviewer-noted conditioning concern around Eq. (47) is, even if valid, a proof-completeness issue about the conditioning in McDiarmid's inequality, not a circular reduction of the claim to its inputs, so it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No data are fitted; protocol parameters are chosen by the proof, and E_PEC is an input from the user. The axioms list the standard UBQC/PEC assumptions and the prior results invoked. No new physical entities are introduced; the BDPEC and SDPEC resources are formal ideal functionalities, not physical postulates.

assumptions (7)
  • domain assumption Reference E_PEC is a stochastic Pauli channel, invertible in Pauli transfer representation, with known support Lambda of polynomial size in |V|.
    Assumed in Section II.B and used in Protocol 4. VBPEC verifies consistency but does not learn E_PEC; if the true noise has large components outside Lambda, the test statistic detects them only as coarse-grained events.
  • domain assumption QOTP twirls any server-side deviation into an effective stochastic Pauli channel; X errors immediately before X-basis measurements are harmless, so PEC can be reduced to I/Z corrections.
    Used in Section III to justify blind PEC and the I/Z-only quasi-probability decomposition. Standard in UBQC blindness proofs; not machine-checked here.
  • domain assumption General trap patterns from Kapourniotis et al. satisfy Lemma 4 and the Walsh-Hadamard relation, yielding unbiased estimators of Pauli deviation probabilities.
    Used in Section IV.B and Appendix D. The construction is cited from prior work, with a derivation in Appendix D.
  • domain assumption UBQC is perfectly blind and composably secure, and UBOE therefore perfectly constructs the BDOE resource.
    Used in UBPEC proof and in VBPEC correctness and security hybrids. The composable security of UBQC is an external prior result.
  • domain assumption Client preparation imperfections are modeled as a single deviation independent of secret parameters, absorbed into the server-side deviation.
    Stated in Section III. Standard in UBQC-type protocols.
  • standard math Hoeffding, McDiarmid, and Serfling concentration inequalities.
    Used throughout Section IV.C and Appendix F.
  • standard math Observable estimation problems reduce without loss of generality to binary +/-1 observables via Naimark dilation.
    Section II.A.

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Cite this review

Pith. "Pith review of Verifiable blind probabilistic error cancellation." pith.science (2026). https://pith.science/paper/J63KP3OL

@misc{pith2026260725704,
  author       = {Pith},
  title        = {Pith review of: Verifiable blind probabilistic error cancellation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J63KP3OL}},
  note         = {Machine review of arXiv:2607.25704}
}
read the original abstract

Quantum error mitigation (QEM) is an essential tool for mitigating hardware noise without incurring space overhead. Yet, its reliability depends on modeling, calibration, and implementation, leaving end-to-end security on untrusted quantum hardware unresolved. We address this problem by introducing verifiable blind probabilistic error cancellation (VBPEC), the first secure verification protocol against a fully malicious adversary that integrates QEM. VBPEC brings probabilistic error cancellation (PEC), a widely studied QEM technique, within the scope of composable security by formalizing delegated mitigation as a cryptographic resource in the abstract cryptography framework. The protocol performs PEC with perfect blindness and an exponentially small security error. VBPEC retains the absence of quantum-space overhead from recent statistically-secure verified quantum computation protocols and from PEC. The only overhead takes the form of additional repetitions due to the QEM procedure. To achieve this, we extend trap-based verification from deterministic pass/fail checks to statistical tests that benefit from QEM and develop a new proof technique that integrates the corresponding additional deviation sources. Rather than merely tolerating honest noise below a fixed threshold, VBPEC actively cancels it, enabling correctly mitigated estimates to be accepted with high probability without compromising security. Our framework thus establishes an essential route towards secure, reliable, and practical delegated quantum computation on near-future quantum hardware: VBPEC fundamentally improves the practicality of verification.

Figures

Figures reproduced from arXiv: 2607.25704 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustration of the PEC process for a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic illustration of a single round of the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Schematic illustration of the SDPEC resource, where [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic illustration of the VBPEC protocol. Each small graph represents one delegated MBQC instance. Blue [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The schematic illustrations of correctness and security [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. A schematic illustration of a specific general trap [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Geometric action of PEC on two-qubit trap-response vectors in the Walsh–Hadamard representation. Since the [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

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Reference graph

Works this paper leans on

87 extracted references · 52 canonical work pages

  1. [1]

    The Simulator setse= 1

  2. [2]

    The Simulator prepares EPR pairs for each qubit that the Server is supposed to receive in Protocol 4

  3. [3]

    The Simulator sends all half-EPR pairs to the Server and instructs the Server to perform measure- ments at random angles, and retrieves the alleged measurement outcomes

  4. [4]

    The Simulator forwards the second half of the EPR pairs, the chosen angles, and the received bits to the SDPEC resource

  5. [5]

    The Simulator samples at random indices within [Nc +Nt] to define the sets S C, and S T

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    Client” and a potentially malicious “Server

    The Simulator decides the trap patterns{S i}i∈ST associated with each test round in S T and passes this to the SDPEC resource. Following the security proof of UBQC, the information passed per round, together with the type of each round, is sufficient for the SDPEC resource to generate per-round measurement results following the same probability dis- tribu...

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    The Client also computes a prop v = M j∈NG(v) ainit j ∈{0,1}for allv∈V

    The Client generates secret parameters: (a) (Xrandomization) The Client chooses a random bit a init v ∈{0,1}forv∈I and sets a init v = 0 forv∈V\I. The Client also computes a prop v = M j∈NG(v) ainit j ∈{0,1}for allv∈V. (b) (Zrandomization) The Client chooses a random bit r v∈{0,1}for allv∈V. (c) (randomization for blindness) The Client chooses a randomθ v...

  8. [8]

    Forv∈I, the Client sequentially sends each qubit in O v∈I Rzv(θv)X ainit v v ! [ρinit]

    The Client prepares and sends to the Server all single qubits corresponding tov∈V. Forv∈I, the Client sequentially sends each qubit in O v∈I Rzv(θv)X ainit v v ! [ρinit]. Forv∈V\I, the Client sends|+ θv⟩

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Show all 87 references
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    The Server applies the entangling operationG= Y (u,v)∈E CZu,v to prepare the resource state according to the graphG

  2. [10]

    Once the Client receives the mea- surement outcome bj∈{0,1}for allj∈S X,v∪SZ,v, whereS X,v =f −1 G (v),S Z,v ={j|v∈N G (fG (j))}, the Client computes the adaptive angle updateϕ ′ v

    For eachv∈V, the Client and Server interactively perform the MBQC process. Once the Client receives the mea- surement outcome bj∈{0,1}for allj∈S X,v∪SZ,v, whereS X,v =f −1 G (v),S Z,v ={j|v∈N G (fG (j))}, the Client computes the adaptive angle updateϕ ′ v. The Client then comp...

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    Protocol 6Universal Blind Observable Estimation (UBOE) Public Information:Public information (C,G,f G,Nc)

    The Client performs the remaining classical post-processing upon the measurement results of all qubitsv∈Vand returns ˜y∈{−1,1}. Protocol 6Universal Blind Observable Estimation (UBOE) Public Information:Public information (C,G,f G,Nc). Inputs from Client:The target computationC...

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    The Client chooses the total number of roundsN c

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    For every round indexed byi∈[N c], the Client delegates the target computation with the UBQC protocol (Protocol 5)

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    Blind Delegated Observable Es- timation (BDOE)

    Upon receiving and decoding the result of the computation roundi∈[N c], the Client assigns the result to ˜y i. The Client then sets ˜o= 1 Nc X i∈[Nc] ˜yi and returns ˜oas the final result. Based on MBQC, the UBQC protocol is given as Pro- tocol 5. Based on UBQC, the UBOE proto...

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    The interface is filtered so that whene= 0, the interface does not send any information nor take inputs

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    Processing by the Resource:

    Fore= 1, the Resource receives a quantum stateσandF, a list of instructions so that the resource producess∈R. Processing by the Resource:

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    Ife= 0, it setso= 1 Nc NcX i=1 yi withy i = 2Yi−1 whereY i∼B(p), sampled from a Bernoulli distributionB(p) with p= Tr [ρ|+O⟩⟨+O|]

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    It forwardsoto the Client. perfectly constructs the BDOE resource, leaking only pub- lic information(C,G,f G,Nc). Appendix D: General trap patterns In the test rounds of VBPEC, we use general trap pat- terns introduced in [27], which can be seen as stabilizer tests. We review ...

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    Since VBOE is a special case of VBPEC that omits PEC, its test statistic can also be described as the identity-reference special case of ˜∆ in Eq

    Noise-robustness of VBOE under honest noise We now analyze the noise robustness of VBOE in the honest-noise setting. Since VBOE is a special case of VBPEC that omits PEC, its test statistic can also be described as the identity-reference special case of ˜∆ in Eq. (12), by repl...

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    We assume that each round is independently affected by the same stochas- tic Pauli channel with deviation distribution⃗ p∈R|ΩV| ≥0

    Noise-robustness of VBPEC under honest noise mismatch We now analyze the noise-robustness of VBPEC when the Server is honest, but the actual noise does not exactly match the noise model used for PEC. We assume that each round is independently affected by the same stochas- tic ...

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    Applying PEC locally to the first qubit in a two-qubit system Let us first consider a two-qubit system and take the noise modelE PEC = (1−p 1)II+p 1ZIas an illustrative example. The probability distributions ofE PEC andE−1 PEC can be expressed in vector form, respectively, as ...

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    Letd:= 2 |V| , and identify each u∈{0,1} |V| with the Pauli operation Zu := |V|O i=1 Zui

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