Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

A new Pauli-measurement protocol certifies Clifford-enhanced product states, the resource states behind magic-state injection, with sample complexity scaling polynomially in the number of qubits.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 23:04 UTC pith:CAYZBQRS

load-bearing objection The gap is real and the idea is plausible, but the central proof misses the Pauli normalization and the protocol's sampling distribution isn't a distribution. the 3 major comments →

arxiv 2511.07300 v2 pith:CAYZBQRS submitted 2025-11-10 quant-ph

Efficient certification of intractable quantum states with few Pauli measurements

Sami Abdul Sater , Maxime Garnier , Thierry Martinez , Harold Ollivier , Ulysse Chabaud This is my paper
classification quant-ph MSC 81P6881P45 PACS 03.67.-a03.67.Lx
keywords state certificationPauli measurementsClifford-enhanced product statesmagic-state injectionfidelity witnessquantum verificationsample complexityClifford back-propagation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to close a gap in quantum state certification: until now, resource states for magic-state injection (states of the form a Clifford circuit applied to a product of single-qubit states, some possibly 'magic') could only be verified by protocols requiring classically hard information about the target state, like #P-hard amplitude oracles. The authors claim that these Clifford-enhanced Product States can be certified with only single-qubit Pauli measurements, using a number of copies that scales as O(n^2/epsilon^2) in the i.i.d. setting and polynomially in n, epsilon, delta in the adversarial setting. If correct, this would be the first Pauli-only, oracle-free certification method for the states that drive universal Pauli-based quantum computation, making verification practical on near-term fault-tolerant architectures. The protocol combines single-qubit fidelity estimation with a robust fidelity witness, using Clifford back-propagation to reduce measurements on the entangled target to measurements on the original product state.

Core claim

The central claim is that for any target state |Psi> = C(⊗_i |psi_i>), where C is a Clifford circuit and each |psi_i> is an arbitrary known single-qubit state, Protocol 1 certifies the state from N ∈ O(n^2/epsilon^2 log(1/delta)) copies using only single-qubit Pauli measurements. It rejects if the fidelity F(rho,|Psi>) < 1−epsilon and accepts if F(rho,|Psi>) ≥ 1−epsilon/(3n), with failure probability at most delta. Protocol 2 extends this to the adversarial (non-i.i.d.) setting with a polynomial sample overhead. The protocol estimates a robust fidelity witness W = 1−n+m·X, where X is an empirically estimated average of signed Pauli measurement outcomes, and back-propagation of Pauli observab

What carries the argument

The central object is the Clifford-enhanced Product State (CPS) combined with a robust fidelity witness built from single-qubit fidelities. The protocol uses three ingredients: (1) measurement back-propagation — measuring a Pauli P on C†ρC is equivalent in expectation to measuring CPC† on ρ, and since C is Clifford, this remains a Pauli measurement; (2) single-qubit Direct Fidelity Estimation via the Pauli-basis expansion |ψ⟩⟨ψ| = Σ_P χ_ψ(P) P, where χ_ψ(P)=Tr[|ψ⟩⟨ψ|P]; and (3) importance sampling over qubit indices and Pauli operators according to the distribution D(i,P)=|χ_{ψ_i}(P)|/m, with m=Σ_i ||χ_{ψ_i}||_1. The key identity is that the expected value of the signed outcome is (1/m) Σ_i

Load-bearing premise

The proof depends on the Pauli-basis expansion |ψ⟩⟨ψ| = Σ_P χ_ψ(P) P holding with the paper's definition of χ_ψ(P) = Tr[|ψ⟩⟨ψ|P]; with standard unnormalized Pauli operators this identity is missing a factor of 1/d, and the related distribution D(i,P)=|χ_{ψ_i}(P)|/m is not normalized for the stated norm, so if either premise fails the estimator X=(1/m)Σ_i F_i does not follow from the measurement outcomes.

What would settle it

Compute the left- and right-hand sides of Eq. (8) for a single-qubit state, say |0⟩, using the standard Pauli matrices I, X, Y, Z: the identity requires |0⟩⟨0| = (1/2)(I + Z), but the paper's formula gives I + Z without the 1/2. A protocol simulation on a simple CPS target (e.g., one qubit in |0⟩ and a Clifford identity) should show that the empirical estimator W is biased, and the acceptance threshold cannot be met at the claimed sample complexity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the protocol works as claimed, experimental groups can certify the resource states for magic-state injection using only single-qubit Pauli measurements, with no need to compute classically intractable amplitudes of the target state.
  • The protocol makes verification of universal Pauli-based quantum computation practical under minimal assumptions: non-adaptive, single-qubit measurements and efficient classical post-processing.
  • The sample complexity in the i.i.d. setting is O(n^2/epsilon^2), which is polynomial and comparable to existing Pauli-based certification protocols for stabilizer and graph states, but now covering non-stabilizer (magic) states.
  • In the adversarial setting, the same guarantees hold with a polynomial increase in sample complexity, O~(n^5/(delta^2 epsilon^6)), allowing verification even when the prover is untrusted and may send correlated states.
  • The protocol can serve as a subroutine to verify the output of a magic-state-injection computation: if the resource state is certified with fidelity at least 1−epsilon, the trace distance between the actual and ideal output distributions is at most sqrt(epsilon).

Where Pith is reading between the lines

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

  • The back-propagation idea may extend to other quantum computational models where the resource state is obtained by a Clifford circuit followed by measurements; one could probe whether coherence-based (non-magic) frameworks admit similar certification.
  • The use of importance sampling over qubits suggests that the sample complexity might be improvable by optimizing the sampling distribution based on the specific single-qubit states, potentially reducing the n^2 factor for states with 'less magic'.
  • The protocol's reliance on a fidelity witness that lower-bounds fidelity might be tightened: a sharper witness could yield a smaller acceptance gap (epsilon/(3n)) and reduce the required copies, as hinted by the authors' footnote on choosing different constants.
  • A direct experimental test would be to prepare a known CPS state (e.g., a small hypergraph-like state with a T gate) and run Protocol 1; the estimator's empirical distribution should show the predicted acceptance/rejection behavior exactly at the claimed sample counts.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a certification protocol for Clifford-enhanced Product States (CPS), i.e. states |Ψ⟩=C(⊗_i |ψ_i⟩) obtained by applying a Clifford circuit to a product of arbitrary single-qubit states. Protocol 1 estimates a robust fidelity witness from single-qubit Pauli measurements, with claimed sample complexity O(n²/ε² log(1/δ)) in the i.i.d. setting (Theorem 1). Protocol 2 extends this to the adversarial setting using a random-partition argument from [41], with claimed sample complexity O~(n⁵/(δ²ε⁶)) (Theorem 2), and Protocol 3 applies the result to verification of magic-state-injection computations. The central technical step is the Pauli-basis expansion of a pure state in Eq. (8), from which the estimator and sampling distribution are derived. I find the overall approach promising, but as written the expansion and the sampling distribution are not correctly normalized. These errors invalidate the proof of the central theorems as stated.

Significance. If the technical issues were corrected, the result would be significant: it would give an oracle-free, Pauli-only certification method for a class of classically hard resource states relevant to universal quantum computation, with polynomial sample complexity and efficient classical post-processing. The use of measurement back-propagation and the non-i.i.d. lifting from [41] are appropriate and potentially valuable. However, because the central estimator and sampling distribution are not correctly defined, the advertised theorems are not established in the current manuscript. The errors appear repairable, so I do not regard the underlying idea as unsalvageable.

major comments (3)
  1. [§2.3.3, Eq. (8)] The Pauli expansion is missing the normalization factor 1/d. With the standard unnormalized Pauli operators satisfying Tr(PQ)=dδ_{P,Q}, the identity is |ψ⟩⟨ψ|=(1/d)Σ_P χ_ψ(P)P, not Σ_P χ_ψ(P)P as written. This error propagates directly into Eq. (11), where the expectation value is d times the fidelity, and into the proof of Theorem 1. In particular, the chain Eqs. (24)–(28) gives X=(d/m)Σ_i F_i (with d=2 for a single qubit), not X=(1/m)Σ_i F_i. Consequently Eq. (29) does not define the robust witness (14), and the acceptance/rejection thresholds in Theorem 1 are not justified. Since Theorem 2 and Protocol 3 invoke Protocol 1, their conclusions inherit this flaw.
  2. [§3, Eqs. (17)–(19)] The joint distribution D(i,P)=|χ_{ψ_i}(P)|/m is not a probability distribution under the paper's own definitions. Using Eq. (9), for |ψ_i⟩=|0⟩ we have ||χ||₁=1/2, while |χ(I)|=|χ(Z)|=1, so D_i(I)=D_i(Z)=2 and Σ_P D_i(P)=4. Thus Protocol 1's instruction to sample (i,P)∼D is undefined. A consistent normalization — for example S_i=Σ_P |χ_i(P)| with D_i(P)=|χ_i(P)|/S_i and µ(i)=S_i/m — would change m and require a compensating factor in the witness estimator. The protocol and proof therefore need to be rewritten together, not merely Eq. (8) adjusted in isolation.
  3. [§2.3.3, Eq. (9)] The stated ℓ₁-norm and its examples are internally inconsistent. For the n-qubit stabilizer state |0…0⟩, χ(P)=1 for every tensor product of I and Z, so Eq. (9) gives ||χ||₁=(d−1)/d, not 1/d as claimed. More importantly, the norm in Eq. (9) is not the normalization that makes D in Eq. (10) a probability distribution. Since the sample complexity of Theorem 1 explicitly uses m=Σ_i ||χ_{ψ_i}||₁, the definition of this norm is load-bearing, not a purely notational issue.
minor comments (4)
  1. [§2.3.5, Eq. (16)] The displayed identity appears to have a typo: measuring P on C†ρC gives Tr[(C†ρC)P]=Tr[ρ C P C†], not Tr[ρ(C† P C†)]. The surrounding text says the back-propagated observable is CPC†, so this is likely a typographical error, but it should be fixed.
  2. [Eq. (28) and surrounding text] The notation C†ΨC is unclear; C†|Ψ⟩ is a product state, and the intended expression is probably F((C†ρC)_i, |ψ_i⟩). Please make the notation consistent.
  3. [Figure 2 caption] Typo: 'teh' should be 'the'.
  4. [Protocol 3, §5] Protocol 1 as defined consumes all N copies and does not leave a remaining register; the statement in Protocol 3 that Protocol 1 outputs a remaining subsystem needs to be reconciled, e.g. by reserving one copy before applying Protocol 1 or by always using the Protocol 2 formulation.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained apart from standard external results, and the flagged normalization problems are correctness errors rather than circular reductions.

full rationale

Walking the derivation chain of Protocol 1/Theorem 1: the estimator X is defined directly from the target single-qubit characteristic functions and the sampled Pauli outcome, and the proof (Eqs. (21)-(28)) evaluates its expectation; it is not defined in terms of the fidelities it later claims to recover, and no parameter is fitted to measured data. The robust witness W (Eq. (14)) and its robustness bound (Eq. (15)) are imported from [40]; despite an author overlap, that cited result is an independently published product-state witness and is not the CPS certification claim of this paper. Back-propagation is derived in Eq. (16) by trace cyclicity rather than assumed from [44]. The non-i.i.d. lifting (Lemma 2) is taken from [41], an external result with no author overlap, and the protocol satisfies its non-adaptive, incoherent-measurement hypotheses. The reader-flagged failures in Eqs. (8), (11), and (19) — the missing 1/d in the Pauli expansion and the non-normalization of D — are substantive mathematical errors that affect correctness, but they do not make the claimed result equivalent to its inputs by construction. No equation reduces to a fitted parameter or to a self-citation chain, so there is no circularity step to exhibit. The concluding speculation about improved non-i.i.d. complexity is a research suggestion, not a load-bearing circular appeal.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The protocol has no fitted free parameters and postulates no new physical entities. Its load-bearing assumptions are the correctness of the Pauli expansion (which is false as written), the product-state fidelity witness from [40], the non-i.i.d. lifting theorem from [41], and the verifier's classical knowledge of the target state.

axioms (5)
  • standard math Pauli-basis expansion |ψ⟩⟨ψ| = Σ_P χ_ψ(P) P as stated in Eq. (8)
    The proof uses this expansion in Eq. (26). Under the paper's own definition χ(P)=Tr[|ψ⟩⟨ψ|P] with standard Pauli matrices, the expansion requires a factor 1/d and is therefore false as written.
  • domain assumption Robust fidelity witness lower bound W ≥ 1 − n(1−F) from [40], Eq. (15)
    The acceptance threshold and the completeness part of Theorem 1 rely on this external theorem relating the sum of local fidelities to the global fidelity.
  • domain assumption Non-i.i.d. lifting theorem, Lemma 2 from [41, Theorem 3]
    Protocol 2 and Theorem 2 are obtained by applying this external result to Protocol 1. The adversarial sample complexity depends entirely on this theorem.
  • domain assumption The verifier knows the full classical description of the target CPS state, including the Clifford circuit C and each single-qubit state |ψ_i⟩
    Protocol 1 requires classically sampling from D(i,P), which depends on the characteristic functions of the known single-qubit states, and requires computing C P C† for Clifford group elements.
  • domain assumption Protocol 1's measurements are non-adaptive and incoherent, so Lemma 2 applies
    Theorem 2 invokes Lemma 2 for the i.i.d. certification subroutine. If the Pauli observables could not be sampled in advance or were measured jointly, the non-i.i.d. lifting would not apply.

pith-pipeline@v1.3.0-alltime-deepseek · 13921 in / 23948 out tokens · 248275 ms · 2026-08-03T23:04:22.305495+00:00 · methodology

0 comments
read the original abstract

Efficient verification of quantum computational resources is crucial as experiments advance toward fault-tolerance. Universal quantum computation can be achieved by consuming resource states through simple Pauli measurements, yet a significant gap remains between states that are easy to certify and those required for universality. We focus on \emph{Clifford-enhanced Product States}, a class of resource states obtained by applying Clifford circuits to a product of single-qubit, potentially magic, states. While essential for universal computation, the certification of such states has previously relied on query oracles that are \#P-hard to implement, leaving their efficient, oracle-free verification an open challenge. In this work, we demonstrate that such classically intractable resource states can be efficiently verified using only Pauli measurements. Our protocol achieves sample- and time-efficiency in both i.i.d.\ and adversarial settings. This work fills a gap in Pauli-based certification, providing a new practical pathway to verify resource states that drive universal Pauli-based quantum computation.

Figures

Figures reproduced from arXiv: 2511.07300 by Harold Ollivier, Maxime Garnier, Sami Abdul Sater, Thierry Martinez, Ulysse Chabaud.

Figure 1
Figure 1. Figure 1: Relationship between Clifford￾enhanced Product States (CPS), Stabilizer States (STAB), Graph States (GS), and Hypergraph States (HGS). CPS reduces to STAB if the prod￾uct state contains only stabilizer states ; and it furthermore reduces to GS if in addition the Clif￾ford circuit is only made of CZ. A HGS is a GS if the underlying hypergraph is a graph. states. We refer to this family as Clifford￾enhanced … view at source ↗
Figure 2
Figure 2. Figure 2: Magic-State Injection model. The blue dashed box captures the fact that teh state prior to measurements has the structure of the CPS class introduced previously. The green box con￾sists of a layer of adaptive single-qubit Pauli mea￾surements to drive the computation. 2.3 Mathematical tools 2.3.1 Fidelity and trace distance To quantify the closeness between two quantum states ρ and σ, two fundamental quanti… view at source ↗
Figure 3
Figure 3. Figure 3: Measurement back-propagation: mea￾suring observable P on C †ρC yields the same out￾come distribution as measuring CP C† on ρ. By definition of the Clifford group, CP C† is also a Pauli observable if C ∈ Cn. operation before the measurement. Formally, let ρ be an n-qubit density operator, C ∈ Cn, and P ∈ Pn. Measuring P on C †ρC yields the expec￾tation value Tr h (C † ρC)P i = Tr h ρ (C †P C† ) i , (16) by … view at source ↗
Figure 4
Figure 4. Figure 4: Visual description of our protocol to certify [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the protocol to run against an adversarial prover. The verifier re￾ceives different n−qubit subsystems of a global system ρ 1...N ∈ (H⊗n ) ⊗N The verifier chooses a random partition, discards K systems, applies the certification protocol on N − K − 1 chosen samples, and leaves one for further computation. (H⊗n ) ⊗Nnon−iid , randomly permutes the subsys￾tems, discards Nnon−iid − Niid − 1 of … view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Sample- and Hardware-Efficient Fidelity Estimation by Stripping Phase-Dominated Magic

    quant-ph 2026-02 unverdicted novelty 6.0

    Phase stripping reduces target-state magic to enable O(poly(n)) or O(1) sample fidelity estimation for phase-dominated states using a single fan-out gate plus nonlinear Pauli post-processing.

Reference graph

Works this paper leans on

46 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Algorithms for Quantum Computation: Discrete Logarithms and Factoring,

    P. W. Shor, “Algorithms for Quantum Computation: Discrete Logarithms and Factoring,” inProceedings of the 35th Annual Symposium on Foundations of Computer Science, SFCS ’94, p. 124–134. IEEE Computer Society, USA, 1994. https: //doi.org/10.1109/SFCS.1994.365700

    arXiv 1994

  2. [2]

    A Fast Quantum Mechanical Algorithm for Database Search,

    L. Grover, “A Fast Quantum Mechanical Algorithm for Database Search,” inProc. 28th Annual ACM Symposium on the Theory of Computation, pp. 212–219. ACM Press, New York, New York, NY, 1996. quant-ph/9605043. 11

    Pith/arXiv arXiv 1996

  3. [3]

    Simulating Physics with computers,

    R. P. Feynman, “Simulating Physics with computers,”Int. J. of Theor. Phys.21, 467 (1982)

    1982

  4. [4]

    Quantum-Mechanical Computers,

    R. P. Feynman, “Quantum-Mechanical Computers,”Found. Phys.16, 507–531 (1986)

    1986

  5. [5]

    Quantum theory, the Church-Turing principle and the universal quantum computer,

    D. Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,”Proc. R. Soc. Lond. A 400, 97–117 (1985)

    1985

  6. [6]

    The vast world of quantum advantage,

    H.-Y. Huang, S. Choi, J. R. McClean, and J. Preskill, “The vast world of quantum advantage,”arXiv:2508.05720 [quant-ph]

    Pith/arXiv arXiv

  7. [7]

    Quantum computing in the NISQ era and beyond,

    J. Preskill, “Quantum computing in the NISQ era and beyond,”Quantum2, 79 (2018)

    2018

  8. [8]

    Mind the gaps: The fraught road to quantum advantage,

    J. Eisert and J. Preskill, “Mind the gaps: The fraught road to quantum advantage,” arXiv:2510.19928 [quant-ph]

    Pith/arXiv arXiv

  9. [9]

    Beating the break-even point with a discrete-variable-encoded logical qubit,

    Z. Ni, S. Li, X. Deng, Y. Cai, L. Zhang, W. Wang, Z.-B. Yang, H. Yu, F. Yan, S. Liu, C.-L. Zou, L. Sun, S.-B. Zheng, Y. Xu, and D. Yu, “Beating the break-even point with a discrete-variable-encoded logical qubit,”Nature 616, 56-60 (2023) 616, 56–60 (2022),arXiv:2211.09319 [quant-ph]

    Pith/arXiv arXiv 2023

  10. [10]

    Constructive interference at the edge of quantum ergodic dynamics,

    D. A. e. a. Abanin, “Constructive interference at the edge of quantum ergodic dynamics,”arXiv:2506.10191 [quant-ph]

    Pith/arXiv arXiv

  11. [11]

    Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate,

    S. Dasu, S. Burton, K. Mayer, D. Amaro, J. A. Gerber, K. Gilmore, D. Gresh, D. DelVento, A. C. Potter, and D. Hayes, “Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate,” arXiv:2506.14688 [quant-ph]

    Pith/arXiv arXiv

  12. [12]

    Quantum certification and benchmarking,

    J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, “Quantum certification and benchmarking,”Nature Reviews Physics2, 382–390 (2020), https://arxiv.org/abs/1910.06343. https: //doi.org/10.1038/s42254-020-0186-4

    Pith/arXiv arXiv 2020

  13. [13]

    Quantum Characterization, Verification, and Validation,

    R. Blume-Kohout, T. Proctor, and K. Young, “Quantum Characterization, Verification, and Validation,” arXiv:2503.16383 [quant-ph]

    Pith/arXiv arXiv

  14. [14]

    Benchmarking quantum computers,

    T. Proctor, K. Young, A. D. Baczewski, and R. Blume-Kohout, “Benchmarking quantum computers,”arXiv:2407.08828 [quant-ph]

    arXiv

  15. [15]

    Verification of Quantum Computation: An Overview of Existing Approaches,

    A. Gheorghiu, T. Kapourniotis, and E. Kashefi, “Verification of Quantum Computation: An Overview of Existing Approaches,”Theory of Computing Systems 63, 715–808 (2019). https://link.springer.com/content/ pdf/10.1007/s00224-018-9872-3.pdf

    work page doi:10.1007/s00224-018-9872-3.pdf 2019

  16. [16]

    A One-Way Quantum Computer,

    R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer,”Phys. Rev. Lett.86, 5188–5191 (2001). http://link.aps.org/doi/10.1103/ PhysRevLett.86.5188

    2001

  17. [17]

    The Measurement Calculus,

    V. Danos, E. Kashefi, and P. Panangaden, “The Measurement Calculus,”J. ACM54, (2007).http://doi.acm.org/10.1145/ 1219092.1219096

    arXiv 2007

  18. [18]

    Measurement-based quantum computation,

    H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,”Nature Physics5, 19–26 (2009)

    2009

  19. [19]

    Universal Quantum Computation with ideal Clifford gates and noisy ancillas,

    S. Bravyi and A. Kitaev, “Universal Quantum Computation with ideal Clifford gates and noisy ancillas,”Phys. Rev. A 71, 022316 (2005)71, 022316 (2004), arXiv:quant-ph/0403025 [quant-ph]

    Pith/arXiv arXiv 2005

  20. [20]

    Cleaner magic states with hook injection,

    C. Gidney, “Cleaner magic states with hook injection,”arXiv:2302.12292 [quant-ph]

    Pith/arXiv arXiv

  21. [21]

    Suppressing quantum errors by scaling a surface code logical qubit,

    R. e. a. Acharya, “Suppressing quantum errors by scaling a surface code logical qubit,”Nature 614 (2023) 678-681614, 676–681 (2022),arXiv:2207.06431 [quant-ph]

    Pith/arXiv arXiv 2023

  22. [22]

    Verification of Many-Qubit States,

    Y. Takeuchi and T. Morimae, “Verification of Many-Qubit States,”Phys. Rev. X 8, 021060 (2018)8, 021060 (2017), arXiv:1709.07575 [quant-ph]

    Pith/arXiv arXiv 2018

  23. [23]

    Resource-efficient verification of quantum computing using Serfling’s bound,

    Y. Takeuchi, A. Mantri, T. Morimae, A. Mizutani, and J. F. Fitzsimons, “Resource-efficient verification of quantum computing using Serfling’s bound,”npj Quantum Information 5, 27 (2019)5, (2018),arXiv:1806.09138 [quant-ph]

    Pith/arXiv arXiv 2019

  24. [24]

    Efficient Verification of Pure Quantum States in the Adversarial Scenario,

    H. Zhu and M. Hayashi, “Efficient Verification of Pure Quantum States in the Adversarial Scenario,”Phys. Rev. Lett. 123, 12 260504 (2019)123, 260504 (2019), arXiv:1909.01900 [quant-ph]

    Pith/arXiv arXiv 2019

  25. [25]

    Efficient Verification of Hypergraph States,

    H. Zhu and M. Hayashi, “Efficient Verification of Hypergraph States,”Phys. Rev. Applied 12, 054047 (2019)12, 054047 (2018),arXiv:1806.05565 [quant-ph]

    Pith/arXiv arXiv 2019

  26. [26]

    Robust and efficient verification of graph states in blind measurement-based quantum computation,

    Z. Li, H. Zhu, and M. Hayashi, “Robust and efficient verification of graph states in blind measurement-based quantum computation,” npj Quantum Inf. 9, 115 (2023)9, (2023), arXiv:2305.10742 [quant-ph]

    Pith/arXiv arXiv 2023

  27. [27]

    How IBM Will Build the World’s First Large-Scale, Fault-Tolerant Quantum Computer,

    R. Mandelbaum, J. Gambetta, J. Chow, T. Mittal, T. J. Yoder, A. Cross, and M. Steffen, “How IBM Will Build the World’s First Large-Scale, Fault-Tolerant Quantum Computer,” June, 2025. https://www.ibm.com/quantum/blog/ large-scale-ftqc. IBM Quantum Blog

    2025

  28. [28]

    Magic state cultivation: growing T states as cheap as CNOT gates,

    C. Gidney, N. Shutty, and C. Jones, “Magic state cultivation: growing T states as cheap as CNOT gates,”arXiv:2409.17595 [quant-ph]

    Pith/arXiv arXiv

  29. [29]

    Quantum State Designs with Clifford Enhanced Matrix Product States,

    G. Lami, T. Haug, and J. De Nardis, “Quantum State Designs with Clifford Enhanced Matrix Product States,” arXiv:2404.18751 [quant-ph]

    Pith/arXiv arXiv

  30. [30]

    Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements,

    Y. Takeuchi, T. Morimae, and M. Hayashi, “Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements,”Sci. Rep. 9, 13585 (2019) (2018),arXiv:1809.07552 [quant-ph]

    Pith/arXiv arXiv 2019

  31. [31]

    Direct Fidelity Estimation from Few Pauli Measurements,

    S. T. Flammia and Y.-K. Liu, “Direct Fidelity Estimation from Few Pauli Measurements,”Physical Review Letters 106, (2011)

    2011

  32. [32]

    Optimal Fidelity Estimation from Binary Measurements for Discrete and Continuous Variable Systems,

    O. Fawzi, A. Oufkir, and R. Salzmann, “Optimal Fidelity Estimation from Binary Measurements for Discrete and Continuous Variable Systems,”arXiv:2409.04189 [quant-ph]

    Pith/arXiv arXiv

  33. [33]

    Practical characterization of quantum devices without tomography,

    M. P. da Silva, O. Landon-Cardinal, and D. Poulin, “Practical characterization of quantum devices without tomography,” Phys. Rev. Lett. 107, 210404 (2011)107, 210404 (2011),arXiv:1104.3835 [quant-ph]

    Pith/arXiv arXiv 2011

  34. [34]

    Certifying almost all quantum states with few single-qubit measurements,

    H.-Y. Huang, J. Preskill, and M. Soleimanifar, “Certifying almost all quantum states with few single-qubit measurements,”arXiv:2404.07281 [quant-ph]

    Pith/arXiv arXiv

  35. [35]

    Few Single-Qubit Measurements Suffice to Certify Any Quantum State,

    M. Gupta, W. He, and R. O’Donnell, “Few Single-Qubit Measurements Suffice to Certify Any Quantum State,” arXiv:2506.11355 [quant-ph]

    Pith/arXiv arXiv

  36. [36]

    Quantum Computing, Postselection, and Probabilistic Polynomial-Time,

    S. Aaronson, “Quantum Computing, Postselection, and Probabilistic Polynomial-Time,” arXiv:quant-ph/0412187 [quant-ph]

    Pith/arXiv arXiv

  37. [37]

    Quantum Supremacy and the Complexity of Random Circuit Sampling,

    A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, “Quantum Supremacy and the Complexity of Random Circuit Sampling,” Nature Physics15, 159–163 (2018), arXiv:1803.04402 [quant-ph]

    Pith/arXiv arXiv 2018

  38. [38]

    Quantum advantage of unitary Clifford circuits with magic state inputs,

    M. Yoganathan, R. Jozsa, and S. Strelchuk, “Quantum advantage of unitary Clifford circuits with magic state inputs,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences475, 20180427 (2018), arXiv:1806.03200 [quant-ph]

    Pith/arXiv arXiv 2018

  39. [39]

    Quantum Computation and Quantum Information: 10th Anniversary Edition,

    M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information: 10th Anniversary Edition,”. Cambridge University Press, 2000

    2000

  40. [40]

    Efficient verification of Boson sampling,

    U. Chabaud, F. Grosshans, E. Kashefi, and D. Markham, “Efficient verification of Boson sampling,”Quantum5, 578 (2021)

    2021

  41. [41]

    Learning Properties of Quantum States Without the I.I.D. Assumption,

    O. Fawzi, R. Kueng, D. Markham, and A. Oufkir, “Learning Properties of Quantum States Without the I.I.D. Assumption,”Nature Communications, 15, Article number: 9677 (2024)15, (2024), arXiv:2401.16922 [quant-ph]

    Pith/arXiv arXiv 2024

  42. [42]

    Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases,

    Y. Li and H. Zhu, “Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases,”arXiv:2506.19809 [quant-ph]

    arXiv

  43. [43]

    Cryptographic Distinguishability Measures for Quantum Mechanical States,

    C. A. Fuchs and J. van de Graaf, “Cryptographic Distinguishability Measures for Quantum Mechanical States,” arXiv:quant-ph/9712042 [quant-ph]

    Pith/arXiv arXiv

  44. [44]

    An efficient quantum state verification framework and its application to bosonic systems,

    V. Upreti and U. Chabaud, “An efficient quantum state verification framework and its application to bosonic systems,” arXiv:2411.04688 [quant-ph]

    Pith/arXiv arXiv

  45. [45]

    Unconditionally verifiable blind quantum computation,

    J. F. Fitzsimons and E. Kashefi, “Unconditionally verifiable blind quantum computation,”Phys. Rev. A96, 012303 (2017).https://link.aps.org/doi/10. 1103/PhysRevA.96.012303

    2017

  46. [46]

    Unifying Quantum Verification and Error-Detection: Theory and Tools for Optimisations,

    T. Kapourniotis, E. Kashefi, D. Leichtle, 13 L. Music, and H. Ollivier, “Unifying Quantum Verification and Error-Detection: Theory and Tools for Optimisations,” Quantum Science and Technology9, 035036 (2022),arXiv:2206.00631 [quant-ph]. 14

    Pith/arXiv arXiv 2022