Benchmarking quantum devices beyond classical capabilities

Karol \.Zyczkowski; Marcin Rudzi\'nski; Rafa{\l} Bistro\'n; Ryszard Kukulski

arxiv: 2502.02575 · v4 · submitted 2025-02-04 · 🪐 quant-ph

Benchmarking quantum devices beyond classical capabilities

Rafa{\l} Bistro\'n , Marcin Rudzi\'nski , Ryszard Kukulski , Karol \.Zyczkowski This is my paper

Pith reviewed 2026-05-23 03:32 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum volumequantum benchmarkingheavy outputquantum computinguniversal gate setcircuit ensemblescalability

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The pith

Modified Quantum Volume test uses restricted universal circuits to identify heavy outputs directly without classical simulation.

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

The paper modifies the Quantum Volume benchmark so that larger quantum devices can be tested even when classical computers cannot simulate the circuits. The original test needs classical computation to locate the most probable outcomes, which grows exponentially costly with qubit number. The authors generate circuits from a deliberately limited ensemble drawn from a gate set that is still universal, making the heavy-output subspace identifiable by direct inspection. This change removes the classical bottleneck while the circuits retain the ability to perform general quantum computation.

Core claim

The modified Quantum Volume test adopts a carefully restricted circuit ensemble generated from a gate set that remains universal for quantum computation, allowing direct determination of the heavy-output subspace. This overcomes the scalability barrier of the Quantum Volume test beyond classical computational limits while still probing the key features of universal quantum computing.

What carries the argument

The restricted circuit ensemble generated from a universal gate set, which enables direct identification of the heavy-output subspace.

If this is right

Quantum Volume benchmarking becomes feasible for devices whose circuits exceed classical simulation capacity.
The test remains architecture-independent and applicable across different quantum hardware platforms.
The benchmark continues to assess capability for general quantum computation rather than a narrow task.
The exponential classical cost that previously limited Quantum Volume is eliminated for the modified ensemble.

Where Pith is reading between the lines

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

The same restriction technique might be applied to other benchmarks that rely on identifying probable outcomes to extend their reach beyond classical simulation.
If the approach works, it opens the possibility of designing additional classically tractable yet universal circuit families for testing specific quantum resources.
One could test whether performance on the modified test predicts success on concrete algorithms such as variational quantum eigensolvers on the same device.

Load-bearing premise

The restricted ensemble still captures the essential features of universal quantum computation that the original Quantum Volume test was designed to measure.

What would settle it

Running both the original and modified tests on devices with few enough qubits for full classical simulation and finding that the two tests give uncorrelated pass/fail results on the same hardware would falsify the claim that the modified version continues to probe the same key features.

Figures

Figures reproduced from arXiv: 2502.02575 by Karol \.Zyczkowski, Marcin Rudzi\'nski, Rafa{\l} Bistro\'n, Ryszard Kukulski.

**Figure 3.** Figure 3: FIG. 3: Diagram of the modified quantum volume [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Comparison of single parity ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: An example of parity preserving circuit for [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Exponent [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: The difference between exponents [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Exponent [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Exponent [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Comparison of heavy output probability [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Heavy output probability [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12: Ranges of the scaling factor [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗

read the original abstract

Rapid development of quantum computing technology has led to a wide variety of sophisticated quantum devices. Benchmarking these systems becomes crucial for understanding their capabilities and paving the way for future advancements. The Quantum Volume (QV) test is one of the most widely used benchmarks for evaluating quantum computer performance due to its architecture independence. However, as the number of qubits in a quantum device grows, the test faces a significant limitation. It requires determining the subspace of the most probable outcomes, a task that is typically performed via classical simulation of the quantum circuit and therefore incurs an exponential computational cost. In this work, we propose modifications to the QV test, by adopting a carefully restricted circuit ensemble generated from a gate set that remains universal for quantum computation, that allows for the direct determination of the heavy-output subspace. Crucially, the modified circuits remain capable of general quantum computation. This approach overcomes the scalability barrier of the Quantum Volume test beyond classical computational limits, while still probing the key features of universal quantum computing.

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.

Desk Editor's Note private letter to a colleague

The paper sketches a restricted universal-gate-set ensemble for QV that would let heavy outputs be read off directly, but supplies no circuit construction or check that the restriction preserves the original test's meaning.

read the letter

The main claim is that a carefully chosen subset of circuits, still generated from a universal gate set, lets you identify the heavy-output subspace without exponential classical simulation. If the construction works, it would remove the main scaling limit on QV while keeping the benchmark relevant to universal quantum computation. That is the one concrete new element the abstract puts forward. The paper correctly flags the exponential cost in the standard protocol and points to a possible workaround. Beyond that, there is little to evaluate. No explicit ensemble is defined, no circuit family is written down, and no argument is given that the restriction still exercises the same computational features the original QV was designed to probe. The weakest assumption therefore sits right at the center: whether the restricted set remains diagnostic. Without the details, that cannot be checked. The stress-test note is accurate on this point; nothing load-bearing can be falsified because nothing technical is shown. This is a short note aimed at people who already work on quantum volume and related benchmarks. A reader who wants to see whether the idea can be made rigorous might find it worth a quick look, but the current version offers no evidence that the fix succeeds. I would send it to peer review so the authors can supply the missing construction and any supporting checks; the underlying scaling problem is real and worth addressing even if this particular route needs substantial work.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes modifications to the Quantum Volume (QV) test to overcome its scalability limitation. The original test requires classical simulation to identify the heavy-output subspace, incurring exponential cost as qubit number grows. The authors suggest using a carefully restricted circuit ensemble generated from a gate set that remains universal for quantum computation; this is claimed to permit direct determination of the heavy-output subspace. The modified circuits are asserted to remain capable of general quantum computation and to continue probing the key features of universal quantum computing.

Significance. If the central claim holds, the work would be significant because it would enable QV-style benchmarking of devices with qubit counts beyond the reach of classical simulation while retaining architecture independence and diagnostic power for universal QC. The absence of any explicit construction, however, prevents confirmation that the restriction achieves this without circularity or loss of generality.

major comments (2)

[Abstract] Abstract: the claim that the restricted ensemble 'still probes the key features of universal quantum computing' is presented without any derivation, explicit gate-set definition, or verification that the restriction preserves the original test's diagnostic power; this is the load-bearing assumption for the central claim.
[Abstract] Abstract: no circuit construction, ensemble definition, or argument is supplied showing how direct heavy-output identification is achieved while the gate set remains universal and the circuits retain general computational capability.

minor comments (1)

The abstract is concise but contains no equations, tables, or figures, which is atypical for a technical proposal of this type and makes independent assessment impossible from the given material.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and for identifying points where the presentation of our modified Quantum Volume test can be strengthened. We address the major comments below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the restricted ensemble 'still probes the key features of universal quantum computing' is presented without any derivation, explicit gate-set definition, or verification that the restriction preserves the original test's diagnostic power; this is the load-bearing assumption for the central claim.

Authors: We agree that the abstract, owing to length constraints, states the claim without accompanying derivation or definitions. The manuscript introduces the restricted ensemble at a conceptual level but does not supply the requested explicit gate-set definition or verification in the provided sections. We will revise by adding both an explicit gate-set definition and a derivation (including verification that diagnostic power is retained) to the main text and a concise reference in the abstract. revision: yes
Referee: [Abstract] Abstract: no circuit construction, ensemble definition, or argument is supplied showing how direct heavy-output identification is achieved while the gate set remains universal and the circuits retain general computational capability.

Authors: We acknowledge the absence of an explicit circuit construction, ensemble definition, or detailed argument in the abstract (and, per the referee's observation, in the manuscript as reviewed). The current text asserts the existence of such a construction but does not exhibit it. In the revision we will insert a dedicated subsection containing the circuit construction, ensemble definition, and the argument demonstrating direct heavy-output identification while preserving universality and general computational capability. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The provided abstract and description contain no equations, fitted parameters, self-citations, or derivation steps. The proposal modifies the QV test via a restricted universal gate-set ensemble to enable direct heavy-output identification. This is a methodological suggestion whose validity rests on external verification of the ensemble's properties rather than any internal reduction to inputs by construction. No load-bearing claim reduces to self-definition, renaming, or fitted prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on the unverified premise that the restricted ensemble preserves universality and benchmarking relevance.

pith-pipeline@v0.9.0 · 5713 in / 1012 out tokens · 24489 ms · 2026-05-23T03:32:07.672462+00:00 · methodology

discussion (0)

Reference graph

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We note that some parts of the below calculations were inspired by [39]

Solvable counterpart of parity quantum volume circuit Now we are prepared to present a modification of the parity-preserving quantum volume circuit which does not substantially affect its behavior but enables us to obtain a close analytical formula for an average heavy output probability in the presence of introduced noise model. We note that some parts o...

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Its general formula for the heavy output frequency looks exactly like in the previous case (A3)

Solvable counterpart of double-parity quantum volume circuit Now we consider the circuit with double-parity preservation - parity on two, complementary, randomly selected subsets of qubits. Its general formula for the heavy output frequency looks exactly like in the previous case (A3). The only difference is the diagonal gatesu diag (6) appearing each tim...

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The most insightful modification from our point of view is the uncontrolled interaction with the environment in the faulty realization of two-qubit gates

Dissipative noise model One may generalize the above calculations to encompass other sorts of errors in numerous ways. The most insightful modification from our point of view is the uncontrolled interaction with the environment in the faulty realization of two-qubit gates. The adjustments in calculations are quite minor. The new formula for heavy output f...

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To make the presentation more apparent we investigate simplified, exponential formulas (A10) for single parity circuit, and (A24) for double parity circuit

Comparison of analytical and numerical results We finish this chapter by illustrating how well the formulas for heavy output frequency derived using modified circuits suit the simulations of actual proposed circuits. To make the presentation more apparent we investigate simplified, exponential formulas (A10) for single parity circuit, and (A24) for double...

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For 6-qubit tests presented in the main text, 50 randomly generated circuits were executed 650 times

Device testing For performing the QV, single parity, and double parity tests we executed corresponding circuits onIBM Brisbane device. For 6-qubit tests presented in the main text, 50 randomly generated circuits were executed 650 times. Heavy output subspaces for all circuits were determined through noiseless classical computation. The heavy output probab...

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Simulation of the IBM device was performed withAerSimulator

Device simulation Simulations with rescaled errors on the IBM simulator were performed to test the performance of the proposed tests for different noise scales and to compare their results with those of the Quantum Volume test. Simulation of the IBM device was performed withAerSimulator. Using fake backend we modify error parameters i.e.readout errorfor a...

work page 2000
[69]

How- ever, as we discussed in the main body of the work and above Appendices, different tests have different sensitivities for various error models

Discussion The presented simulations show agreement between the proposed measures with the Quantum Volume test. How- ever, as we discussed in the main body of the work and above Appendices, different tests have different sensitivities for various error models. Therefore discrepancies are inevitable in some scenarios. We argue that the proposed tests exhib...

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[70]

Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1)

Computing estimator ̃hU for standard QV circuit The equation for the heavy output probability for fixedNandTwith the noise model(Ω z)M z=1 is expressed as ̃hU = ∫C=(U 1,...,UM) dC⟨ΩM(UM . . .Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1). . . U † M),Π C⟩,(D1) whereCis a random circuit of the size(N, T)that containsTlayers of random subsystem permutations and random two-qubit gates...

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2N P(M) 0 −1 2N −1 M−1 ∏ z=1 az

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[72]

Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1)

Computing the estimator ̃hm U for parity preserving circuits Using similar notation as in previous section, we can express heavy output probability ̃hm U for a quantum volume circuit preserving parity onmqubits as ̃hm U = ∫Cm=(U1,...,UM) dCm⟨ΩM(UM . . .Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1). . . U † M),Π Cm⟩.(D6) Based on the scheme description and exponential distribution...

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Estimatingh U for a given noise model type(Ω z)M z=1 Having established concise expressions for heavy output frequency in different circuits we may leverage them, in particular scenarios, to obtain an estimator of original heavy output frequency. a. No estimation for general noise channelsΩ z Let us fix Ω 1 =Z ⊗N λ and Ω2, . . . ,ΩM =1 l withZ λ =∣0⟩⟨0∣+e...

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[74]

Meanwhile, for the Eq

2N P(M) 0 −1 2N−1 . Meanwhile, for the Eq. (D10) we get ̃hm U = 1 2 ∑ p tr((Π m,p⊗1 l⊗N−m 2 )Ω(s) M (σm,p⊗ρ⊗N−m ∗ )) = 1 2( N m) ∑ p,{q1,...,qm}⊂{1,...,N} tr(πq1,...,qm(Πm,p⊗1 l⊗N−m 2 )π † q1,...,qmΩM(πq1,...,qm(σm,p⊗ρ⊗N−m ∗ )π † q1,...,qm)) = 1 2N( N m) ∑ p,q1,...,qm ∑ i,j∶⊕m k=1iqk=p,⊕m k=1jqk=p tr(∣i⟩⟨i∣ΩM(∣j⟩⟨j∣)) = 1 2N( N m) ∑ p,q1,...,qm ∑ i,j w(M)...

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Precise estimation for the maximally depolarizing noises To calculate the estimator ̃hU for depolarizing noise we notice that⟨J Φϵ , J1 l⟩=4−3ϵwhich givesa z = (4−3ϵ)N−1 4N−1

2N⟨v ,̃h∗ U⟩−1 2N −1 .(D14) d. Precise estimation for the maximally depolarizing noises To calculate the estimator ̃hU for depolarizing noise we notice that⟨J Φϵ , J1 l⟩=4−3ϵwhich givesa z = (4−3ϵ)N−1 4N−1 . Moreover,P (M) 0 =(1−ϵ/2) N that gives ̃hU = 1 2 +(p ∗−1

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[76]

(D16) Using that to Eq

(2−ϵ)N −1 2N −1 ( (4−3ϵ)N −1 4N −1 ) M−1 .(D15) To calculate ̃hm U we notice that tr((Πm,pz ⊗1 l⊗N−m 2 )Ω(s) z (σm,pz−1⊗ρ⊗N−m ∗ )) =tr (Πm,pzΦ⊗m ϵ (σm,pz−1)) = 1 2m−1∑ i,j tr(∣i⟩⟨i∣Φ⊗m ϵ (∣j⟩⟨j∣))δ⊕j=pz−1δ⊕i=pz = 1 2m−1 m ∑ d=0 ∑ i,j∶ρH(i,j)=d (ϵ/2) d(1−ϵ/2)m−dδ⊕j=pz−1δ⊕i=pz = m ∑ d=0 ( m d)(ϵ/2) d(1−ϵ/2)m−dδpz⊕d=pz−1= 1+(−1)pz−pz−1(1−ϵ)m 2 . (D16) Using ...

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Thus the information obtained from multiple ̃hm U is in fact redundant, and any of them can be used to estimate the value of ̃hU

(2−ϵ)N −1 2N −1 ( (4−3ϵ)N −1 4N −1 ) M−1 , ̃hm U = 1+(1−ϵ) mM 2 (10) (see derivation in Appendix D 3 d). Thus the information obtained from multiple ̃hm U is in fact redundant, and any of them can be used to estimate the value of ̃hU. VI. BENCHMARKING FUTURE QUANTUM COMPUTERS Natural idea for characterizing quantum computers is to use some low level bench...

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We note that some parts of the below calculations were inspired by [39]

Solvable counterpart of parity quantum volume circuit Now we are prepared to present a modification of the parity-preserving quantum volume circuit which does not substantially affect its behavior but enables us to obtain a close analytical formula for an average heavy output probability in the presence of introduced noise model. We note that some parts o...

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Its general formula for the heavy output frequency looks exactly like in the previous case (A3)

Solvable counterpart of double-parity quantum volume circuit Now we consider the circuit with double-parity preservation - parity on two, complementary, randomly selected subsets of qubits. Its general formula for the heavy output frequency looks exactly like in the previous case (A3). The only difference is the diagonal gatesu diag (6) appearing each tim...

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The most insightful modification from our point of view is the uncontrolled interaction with the environment in the faulty realization of two-qubit gates

Dissipative noise model One may generalize the above calculations to encompass other sorts of errors in numerous ways. The most insightful modification from our point of view is the uncontrolled interaction with the environment in the faulty realization of two-qubit gates. The adjustments in calculations are quite minor. The new formula for heavy output f...

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To make the presentation more apparent we investigate simplified, exponential formulas (A10) for single parity circuit, and (A24) for double parity circuit

Comparison of analytical and numerical results We finish this chapter by illustrating how well the formulas for heavy output frequency derived using modified circuits suit the simulations of actual proposed circuits. To make the presentation more apparent we investigate simplified, exponential formulas (A10) for single parity circuit, and (A24) for double...

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For 6-qubit tests presented in the main text, 50 randomly generated circuits were executed 650 times

Device testing For performing the QV, single parity, and double parity tests we executed corresponding circuits onIBM Brisbane device. For 6-qubit tests presented in the main text, 50 randomly generated circuits were executed 650 times. Heavy output subspaces for all circuits were determined through noiseless classical computation. The heavy output probab...

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Simulation of the IBM device was performed withAerSimulator

Device simulation Simulations with rescaled errors on the IBM simulator were performed to test the performance of the proposed tests for different noise scales and to compare their results with those of the Quantum Volume test. Simulation of the IBM device was performed withAerSimulator. Using fake backend we modify error parameters i.e.readout errorfor a...

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How- ever, as we discussed in the main body of the work and above Appendices, different tests have different sensitivities for various error models

Discussion The presented simulations show agreement between the proposed measures with the Quantum Volume test. How- ever, as we discussed in the main body of the work and above Appendices, different tests have different sensitivities for various error models. Therefore discrepancies are inevitable in some scenarios. We argue that the proposed tests exhib...

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Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1)

Computing estimator ̃hU for standard QV circuit The equation for the heavy output probability for fixedNandTwith the noise model(Ω z)M z=1 is expressed as ̃hU = ∫C=(U 1,...,UM) dC⟨ΩM(UM . . .Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1). . . U † M),Π C⟩,(D1) whereCis a random circuit of the size(N, T)that containsTlayers of random subsystem permutations and random two-qubit gates...

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2N P(M) 0 −1 2N −1 M−1 ∏ z=1 az

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Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1)

Computing the estimator ̃hm U for parity preserving circuits Using similar notation as in previous section, we can express heavy output probability ̃hm U for a quantum volume circuit preserving parity onmqubits as ̃hm U = ∫Cm=(U1,...,UM) dCm⟨ΩM(UM . . .Ω1(U1∣0⊗N⟩⟨0⊗N∣U† 1). . . U † M),Π Cm⟩.(D6) Based on the scheme description and exponential distribution...

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Estimatingh U for a given noise model type(Ω z)M z=1 Having established concise expressions for heavy output frequency in different circuits we may leverage them, in particular scenarios, to obtain an estimator of original heavy output frequency. a. No estimation for general noise channelsΩ z Let us fix Ω 1 =Z ⊗N λ and Ω2, . . . ,ΩM =1 l withZ λ =∣0⟩⟨0∣+e...

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Meanwhile, for the Eq

2N P(M) 0 −1 2N−1 . Meanwhile, for the Eq. (D10) we get ̃hm U = 1 2 ∑ p tr((Π m,p⊗1 l⊗N−m 2 )Ω(s) M (σm,p⊗ρ⊗N−m ∗ )) = 1 2( N m) ∑ p,{q1,...,qm}⊂{1,...,N} tr(πq1,...,qm(Πm,p⊗1 l⊗N−m 2 )π † q1,...,qmΩM(πq1,...,qm(σm,p⊗ρ⊗N−m ∗ )π † q1,...,qm)) = 1 2N( N m) ∑ p,q1,...,qm ∑ i,j∶⊕m k=1iqk=p,⊕m k=1jqk=p tr(∣i⟩⟨i∣ΩM(∣j⟩⟨j∣)) = 1 2N( N m) ∑ p,q1,...,qm ∑ i,j w(M)...

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Precise estimation for the maximally depolarizing noises To calculate the estimator ̃hU for depolarizing noise we notice that⟨J Φϵ , J1 l⟩=4−3ϵwhich givesa z = (4−3ϵ)N−1 4N−1

2N⟨v ,̃h∗ U⟩−1 2N −1 .(D14) d. Precise estimation for the maximally depolarizing noises To calculate the estimator ̃hU for depolarizing noise we notice that⟨J Φϵ , J1 l⟩=4−3ϵwhich givesa z = (4−3ϵ)N−1 4N−1 . Moreover,P (M) 0 =(1−ϵ/2) N that gives ̃hU = 1 2 +(p ∗−1

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(D16) Using that to Eq

(2−ϵ)N −1 2N −1 ( (4−3ϵ)N −1 4N −1 ) M−1 .(D15) To calculate ̃hm U we notice that tr((Πm,pz ⊗1 l⊗N−m 2 )Ω(s) z (σm,pz−1⊗ρ⊗N−m ∗ )) =tr (Πm,pzΦ⊗m ϵ (σm,pz−1)) = 1 2m−1∑ i,j tr(∣i⟩⟨i∣Φ⊗m ϵ (∣j⟩⟨j∣))δ⊕j=pz−1δ⊕i=pz = 1 2m−1 m ∑ d=0 ∑ i,j∶ρH(i,j)=d (ϵ/2) d(1−ϵ/2)m−dδ⊕j=pz−1δ⊕i=pz = m ∑ d=0 ( m d)(ϵ/2) d(1−ϵ/2)m−dδpz⊕d=pz−1= 1+(−1)pz−pz−1(1−ϵ)m 2 . (D16) Using ...

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