Estimating Fidelity to a Reference Quantum State

Qisheng Wang

arxiv: 2606.26034 · v1 · pith:MRDTHATZnew · submitted 2026-06-24 · 🪐 quant-ph · cs.CC· cs.IT· math.IT

Estimating Fidelity to a Reference Quantum State

Qisheng Wang This is my paper

Pith reviewed 2026-06-25 19:11 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCcs.ITmath.IT

keywords fidelity estimationquantum statessample complexityrank-r referenceadditive errorquantum certificationquantum query complexitytolerant certification

0 comments

The pith

Estimating fidelity to a known rank-r quantum state requires O(r²/ε²) samples to additive error ε.

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

The paper establishes that the number of independent copies needed to estimate the fidelity between an unknown quantum state and a known reference state of rank r to within additive error ε is O(r²/ε²). This improves the prior bound by removing logarithmic factors in 1/ε and achieving optimal dependence on ε. A matching-style lower bound of Ω(r/ε²) is proven, with consequences for quantum query complexity. The analysis also covers the setting where the unknown state itself has rank at most r, giving a bound of O(r²/ε⁴), and yields a protocol for tolerant quantum state certification.

Core claim

We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error ε. We show that the sample complexity is O(r²/ε²) with optimal ε-dependence when the reference state is of rank r, improving the previous best O(r² log²(1/ε)/ε⁴). We also provide a lower bound of Ω(r/ε²), improving the previous best Ω(r/ε + 1/ε²), with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most r while the reference state can be arbitrary, for which the sample complexity is shown to be O(r²/ε⁴). As an application, we present an approach to tolerant quantum state certification.

What carries the argument

Upper and lower bounds on sample complexity for fidelity estimation in the i.i.d. copy model when the reference state has known rank r.

If this is right

The ε-dependence is optimal, removing the log²(1/ε) and 1/ε⁴ factors from prior upper bounds.
The improved lower bound Ω(r/ε²) has direct consequences for quantum query complexity.
When the unknown state also has rank at most r the complexity becomes O(r²/ε⁴).
The method yields a protocol for tolerant quantum state certification that generalizes exact certification.

Where Pith is reading between the lines

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

The bound may translate to more efficient protocols for other low-rank quantum estimation tasks.
The lower bound could serve as a template for hardness results in related distance measures such as trace distance.
Small-scale experiments on low-rank states could directly test whether the O(r²/ε²) scaling is observed in practice.

Load-bearing premise

The unknown quantum state is provided through independent identical copies and the reference state is known exactly with known rank r.

What would settle it

An explicit algorithm achieving fidelity estimation to additive ε with o(r²/ε²) samples for arbitrary rank-r reference states, or a matching lower bound showing Ω(r²/ε²) samples are required for some states.

Figures

Figures reproduced from arXiv: 2606.26034 by Qisheng Wang.

read the original abstract

We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error $\varepsilon$. We show that the sample complexity is $O(r^2/\varepsilon^2)$ with optimal $\varepsilon$-dependence when the reference state is of rank $r$, improving the previous best $O(r^2\log^2(1/\varepsilon)/\varepsilon^4)$ due to Utsumi, Nakata, Wang, and Takagi (QIP 2026). We also provide a lower bound of $\Omega(r/\varepsilon^2)$, improving the previous best $\Omega(r/\varepsilon+1/\varepsilon^2)$, with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most $r$ while the reference state can be arbitrary, for which the sample complexity is shown to be $O(r^2/\varepsilon^4)$. As an application, we present an approach to tolerant quantum state certification, generalizing the exact certification studied in B\u{a}descu, O'Donnell, and Wright (STOC 2019).

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

Tighter O(r²/ε²) upper bound and Ω(r/ε²) lower bound for fidelity estimation to known rank-r states, with a direct extension to tolerant certification.

read the letter

The core advance is a sample-complexity upper bound of O(r²/ε²) for estimating fidelity to a known rank-r reference state, together with a matching lower bound of Ω(r/ε²) that improves the ε dependence over the prior O(r² log²(1/ε)/ε⁴) result. The paper also gives O(r²/ε⁴) when the unknown state itself has rank at most r, and sketches an application to tolerant state certification that generalizes the exact-certification result from STOC 2019.

The improvements are stated cleanly in the standard copy-access model with an exactly known reference. The ε scaling is now optimal in the upper bound, and the lower bound removes the additive 1/ε term from earlier work. These are concrete, usable tightenings for anyone running fidelity tests on low-rank states.

The r-gap between the O(r²) upper and Ω(r) lower bound is left open, which is fine since the paper does not claim tightness there. The second regime (unknown state low-rank) reverts to 1/ε⁴ dependence, so the gain is narrower. No circularity or hidden fitting appears in the stated model.

This is a technical paper aimed at quantum information theorists who care about query complexity and state certification. The bounds are the kind of incremental but measurable progress that experimental groups can actually use to reduce shot counts. The thinking is clear and the claims line up with the cited priors.

I would send it to peer review. The improvements are specific enough that referees can check the proofs and the application without needing to invent new machinery.

Referee Report

0 major / 3 minor

Summary. The paper studies sample complexity for estimating fidelity F(ρ, σ) to additive error ε, where σ is a known reference state of rank r and ρ is accessed via iid copies. It claims an upper bound of O(r²/ε²) (optimal in ε) that improves on the prior O(r² log²(1/ε)/ε⁴), a matching lower bound Ω(r/ε²) that improves on Ω(r/ε + 1/ε²), an O(r²/ε⁴) bound when ρ itself has rank ≤ r, and an application to tolerant quantum state certification that generalizes exact certification results.

Significance. If the stated bounds hold, the work tightens the dependence on ε for a core primitive in quantum information processing and supplies a lower bound with direct implications for quantum query complexity. The extension to tolerant certification is a natural and useful application of the fidelity estimator.

minor comments (3)

Abstract: the claimed upper bound is stated as O(r²/ε²) but the manuscript should explicitly note whether the hidden constant depends on the dimension or other parameters of the reference state.
The lower-bound construction (presumably in the section deriving Ω(r/ε²)) should include a short remark on whether the Ω(r) factor is expected to be improvable or is an artifact of the reduction technique.
Notation: ensure that the rank-r assumption on the reference state is restated at the beginning of each technical section so that the two regimes (known-rank reference vs. bounded-rank unknown state) are not conflated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review, recognition of the significance of the improved sample-complexity bounds, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

Minor self-citation to prior baseline result by overlapping author; central bounds derived independently in standard model

full rationale

The paper derives new sample complexity upper bound O(r²/ε²) and lower bound Ω(r/ε²) for fidelity estimation to a known rank-r reference state, improving on the cited prior O(r² log²(1/ε)/ε⁴) result. The cited prior work includes the present author (Wang), constituting a self-citation, but this citation serves only as the baseline being improved upon rather than providing load-bearing justification for the new claims. The derivation relies on the standard iid copy-access model with exactly known reference state and known rank r; no steps reduce by construction to fitted inputs, self-definitions, or ansatzes imported via citation. The result is self-contained against external query models with no evident circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; no explicit free parameters, invented entities, or non-standard axioms stated. Relies on standard definitions of quantum fidelity and sample access.

axioms (2)

standard math Quantum states are density operators on finite-dimensional Hilbert space; fidelity is the standard trace formula.
Implicit background for any fidelity estimation problem in quantum information.
domain assumption Access to the unknown state is via independent copies (standard quantum state model).
Required for sample complexity statements; not stated explicitly but standard in the field.

pith-pipeline@v0.9.1-grok · 5722 in / 1302 out tokens · 35255 ms · 2026-06-25T19:11:45.122353+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 17 canonical work pages · 1 internal anchor

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3 [Rus94] Mary Beth Ruskai

doi:10.1103/PhysRevA.108.012409. 3 [Rus94] Mary Beth Ruskai. Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy. Reviews in Mathematical Physics, 6(5):1147–1161,

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doi:10.1142/S0129055X94000407. 8 [SSW25] ThiloScharnhorst, JackSpilecki, andJohnWright. Optimallowerboundsforquantum state tomography. ArXiv preprints, 2025.arXiv:2510.07699. 3 [TWZ25] Ewin Tang, John Wright, and Mark Zhandry. Conjugate queries can help. ArXiv preprints,

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0034-4877(76)90060-4 1976
[16]

5 [Wat02] John Watrous

doi: 10.1109/TIT.2024.3447915. 5 [Wat02] John Watrous. Limits on the power of quantum statistical zero-knowledge. In Pro- ceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, pages 459–468, 2002.doi:10.1109/SFCS.2002.1181970. 3 [Wat09] John Watrous. Zero-knowledge against quantum attacks. SIAM Journal on Comput- ing, 39(1):25–58,...

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3 [WSR23] Jinzhao Wang, Volkher B

doi: 10.1103/PhysRevLett.122.190401. 3 [WSR23] Jinzhao Wang, Volkher B. Scholz, and Renato Renner. Detecting high-dimensional entanglement in cold-atom quantum simulators.PRX Quantum, 4(4):040338,

work page doi:10.1103/physrevlett.122.190401
[18]

3 [WZ24] Qisheng Wang and Zhicheng Zhang

doi:10.1103/PRXQuantum.4.040338. 3 [WZ24] Qisheng Wang and Zhicheng Zhang. Fast quantum algorithms for trace distance estimation. IEEE Transactions on Information Theory, 70(4):2720–2733, 2024.doi: 10.1109/TIT.2023.3321121. 8 [WZ25a] Qisheng Wang and Zhicheng Zhang. Quantum lower bounds by sample-to-query lift- ing. SIAM Journal on Computing, 54(5):1294–1...

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3 [WZC+23] Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Fang, Junyi Liu, and Mingsheng Ying

doi:10.4230/LIPIcs.ICALP.2026.154. 3 [WZC+23] Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Fang, Junyi Liu, and Mingsheng Ying. Quantum algorithm for fidelity estimation.IEEE Transactions on Information Theory, 69(1):273–282, 2023.doi:10.1109/TIT.2022.3203985. 3, 5 [ZWY+25] Congcong Zheng, Kun Wang, Xutao Yu, Ping Xu, and Zaichen Zhang. Distribu...

work page doi:10.4230/lipics.icalp.2026.154 2026
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arXiv:2506.19574. 3 20

arXiv

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Distributed quantum inner product estimation

[ALL22] Anurag Anshu, Zeph Landau, and Yunchao Liu. Distributed quantum inner product estimation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 44–51, 2022.doi:10.1145/3519935.3519974. 3, 4 [AS25] Srinivasan Arunachalam and Louis Schatzki. Generalized inner product estimation with limited quantum communication. In Pr...

work page doi:10.1145/3519935.3519974 2022

[2] [2]

3 [BBC+01] Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf

doi:10.4230/LIPIcs.STACS.2025.11. 3 [BBC+01] Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials.Journal of the ACM, 48(4):778–797,

work page doi:10.4230/lipics.stacs.2025.11 2025

[3] [3]

5 [BCWdW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf

doi:10.1145/502090.502097. 5 [BCWdW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum finger- printing. Physical Review Letters, 87(16):167902,

work page doi:10.1145/502090.502097

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jstor.org/stable/25047882

URL:https://www. jstor.org/stable/25047882. 3 [BHMT02] Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. In Samuel J. Lomonaco, Jr. and Howard E. Brandt, editors, Quantum Computation and Information, volume 305 ofContemporary Math- ematics, pages 53–74. AMS, 2002.doi:10.1090/conm/305/05215. 5 [BOW...

work page doi:10.1090/conm/305/05215 2002

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5 [CHW07] Andrew M

doi:10.4230/LIPIcs.FSTTCS.2010.145. 5 [CHW07] Andrew M. Childs, Aram W. Harrow, and Pawel Wocjan. Weak Fourier-Schur sam- pling, the hidden subgroup problem, and the quantum collision problem. In Pro- ceedings of the 24th Annual Symposium on Theoretical Aspects of Computer Science, pages 598–609, 2007.doi:10.1007/978-3-540-70918-3_51. 7, 14 17 [CWZ25] Kea...

work page doi:10.4230/lipics.fsttcs.2010.145 2010

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4, 5, 6 [FL11] Steven T

ArXiv preprint.arXiv: 2512.01971. 4, 5, 6 [FL11] Steven T. Flammia and Yi-Kai Liu. Direct fidelity estimation from few pauli measure- ments. Physical Review Letters, 106(23):230501,

arXiv

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Villanueva, S

doi:10.1103/PhysRevLett. 106.230501. 3 [FR16] Philippe Faist and Renato Renner. Practical and reliable error bars in quantum to- mography.Physical Review Letters, 117(1):010404, 2016.doi:10.1103/PhysRevLett. 117.010404. 3 [FW25] Wang Fang and Qisheng Wang. Optimal quantum algorithm for estimating fidelity to a pure state. InProceedings of the 33rd Annual ...

work page doi:10.1103/physrevlett 2016

[8] [9]

3 [GL20] András Gilyén and Tongyang Li

doi:10.1103/ kh2j-jkn6. 3 [GL20] András Gilyén and Tongyang Li. Distributional property testing in a quantum world. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference, pages 25:1–25:19, 2020.doi:10.4230/LIPIcs.ITCS.2020.25. 5 [GP22] András Gilyén and Alexander Poremba. Improved quantum algorithms for fidelity estimation. ArX...

work page doi:10.4230/lipics.itcs.2020.25 2020

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Entanglement detection , journal =

doi:10.1016/j.physrep.2009.02.004. 3 [Hel67] Carl W. Helstrom. Detection theory and quantum mechanics. Information and Con- trol, 10(3):254–291, 1967.doi:10.1016/S0019-9958(67)90302-6. 8 [Hel69] Carl W. Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics, 1(2):231–252, 1969.doi:10.1007/BF01007479. 8 [HHJ+17] Jeongwan Haah, Ar...

work page doi:10.1016/j.physrep.2009.02.004 2009

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Nielsen and Isaac L

doi:10.1038/ s41534-025-00973-7. 3, 5 [NC10] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum In- formation. Cambridge University Press, 2010.doi:10.1017/CBO9780511976667. 3 18 [NW99] Ashwin Nayak and Felix Wu. The quantum query complexity of approximating the median and related statistics. InProceedings of the 31st Annual ACM Sympo...

work page doi:10.1017/cbo9780511976667 2010

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3, 6, 9 [OW21] Ryan O’Donnell and John Wright

doi:10.1145/2897518.2897544. 3, 6, 9 [OW21] Ryan O’Donnell and John Wright. Quantum spectrum testing. Communications in Mathematical Physics, 387(1):1–75, 2021.doi:10.1007/s00220-021-04180-1. 3, 4, 7, 14 [PSTW25] Angelos Pelecanos, Jack Spilecki, Ewin Tang, and John Wright. Mixed state tomog- raphy reduces to pure state tomography. ArXiv preprints, 2025.a...

work page doi:10.1145/2897518.2897544 2021

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3 [Rus94] Mary Beth Ruskai

doi:10.1103/PhysRevA.108.012409. 3 [Rus94] Mary Beth Ruskai. Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy. Reviews in Mathematical Physics, 6(5):1147–1161,

work page doi:10.1103/physreva.108.012409

[13] [14]

8 [SSW25] ThiloScharnhorst, JackSpilecki, andJohnWright

doi:10.1142/S0129055X94000407. 8 [SSW25] ThiloScharnhorst, JackSpilecki, andJohnWright. Optimallowerboundsforquantum state tomography. ArXiv preprints, 2025.arXiv:2510.07699. 3 [TWZ25] Ewin Tang, John Wright, and Mark Zhandry. Conjugate queries can help. ArXiv preprints,

work page doi:10.1142/s0129055x94000407 2025

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Conjugate queries can help

arXiv:2510.07622. 6 [Uhl76] Armin Uhlmann. The “transition probability” in the state space of a *-algebra.Reports on Mathematical Physics, 9(2):273–279, 1976.doi:10.1016/0034-4877(76)90060-4. 3 [UNWT25] Takeru Utsumi, Yoshifumi Nakata, Qisheng Wang, and Ryuji Takagi. Quantum algo- rithms for Uhlmann transformation. ArXiv preprints, 2025.arXiv:2509.03619. ...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0034-4877(76)90060-4 1976

[15] [16]

5 [Wat02] John Watrous

doi: 10.1109/TIT.2024.3447915. 5 [Wat02] John Watrous. Limits on the power of quantum statistical zero-knowledge. In Pro- ceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, pages 459–468, 2002.doi:10.1109/SFCS.2002.1181970. 3 [Wat09] John Watrous. Zero-knowledge against quantum attacks. SIAM Journal on Comput- ing, 39(1):25–58,...

work page doi:10.1109/tit.2024.3447915 2024

[16] [17]

3 [WSR23] Jinzhao Wang, Volkher B

doi: 10.1103/PhysRevLett.122.190401. 3 [WSR23] Jinzhao Wang, Volkher B. Scholz, and Renato Renner. Detecting high-dimensional entanglement in cold-atom quantum simulators.PRX Quantum, 4(4):040338,

work page doi:10.1103/physrevlett.122.190401

[17] [18]

3 [WZ24] Qisheng Wang and Zhicheng Zhang

doi:10.1103/PRXQuantum.4.040338. 3 [WZ24] Qisheng Wang and Zhicheng Zhang. Fast quantum algorithms for trace distance estimation. IEEE Transactions on Information Theory, 70(4):2720–2733, 2024.doi: 10.1109/TIT.2023.3321121. 8 [WZ25a] Qisheng Wang and Zhicheng Zhang. Quantum lower bounds by sample-to-query lift- ing. SIAM Journal on Computing, 54(5):1294–1...

work page doi:10.1103/prxquantum.4.040338 2024

[18] [19]

3 [WZC+23] Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Fang, Junyi Liu, and Mingsheng Ying

doi:10.4230/LIPIcs.ICALP.2026.154. 3 [WZC+23] Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Fang, Junyi Liu, and Mingsheng Ying. Quantum algorithm for fidelity estimation.IEEE Transactions on Information Theory, 69(1):273–282, 2023.doi:10.1109/TIT.2022.3203985. 3, 5 [ZWY+25] Congcong Zheng, Kun Wang, Xutao Yu, Ping Xu, and Zaichen Zhang. Distribu...

work page doi:10.4230/lipics.icalp.2026.154 2026

[19] [20]

arXiv:2506.19574. 3 20

arXiv