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arxiv: 2605.08341 · v2 · pith:VUAFGQRA · submitted 2026-05-08 · quant-ph

Quantum metrology via partial quantum error correction

Yinan Chen , Zongyuan Wang , Sisi Zhou This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-30 22:55 UTCgrok-4.3pith:VUAFGQRArecord.jsonopen to challenge →

classification quant-ph
keywords quantum metrologypartial quantum error correctionsuper-standard-quantum-limitlocal noisephase imprintingadaptive strategyquantum codes
0
0 comments X

The pith

Encoding probe states in superpositions of energetically different code states allows partial error correction to suppress local noise and preserve super-SQL metrology performance.

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

The paper introduces a quantum metrology approach that requires only partial quantum error correction rather than full correction over all code checks. Probe states are encoded into superpositions of states that differ in energy within an underlying quantum code, so that a subset of checks can suppress noise both before and after the sensing phase is imprinted. For noise parallel to a phase imprinter of weight l the scheme reduces the effective noise strength to p raised to floor of (l+1)/2. An adaptive strategy that increases imprinter weight with system size is proposed to sustain the super-standard-quantum-limit advantage at larger scales while keeping all operators local.

Core claim

By placing the probe in a superposition of energetically distinct states belonging to a quantum code, error correction performed with only a subset of the code checks becomes sufficient to protect the phase imprinting step against local noise both before and after the sensing operation occurs.

What carries the argument

Encoding of probe states into superpositions of energetically different states of the underlying quantum code, which permits error correction with only a subset of checks to act on noise around the phase-imprinting step.

If this is right

  • Super-SQL sensing performance is retained while the number of syndrome measurements is reduced.
  • All checks and phase imprinters remain local operators, avoiding non-local connectivity requirements.
  • An adaptive increase of imprinter weight with system size maintains the sensing advantage at larger scales.
  • Noise suppression is quantified by the exponent delta equals floor of (l plus 1) over 2 for weight-l parallel noise.

Where Pith is reading between the lines

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

  • The approach may lower the overhead of fault-tolerant metrology on near-term hardware by tolerating incomplete syndrome extraction.
  • It suggests that energy differences within the code space can substitute for full code distance in sensing applications.
  • The method could be tested on small trapped-ion or superconducting systems by preparing the required superpositions and applying only partial stabilizer measurements.

Load-bearing premise

Encoding the probe into superpositions of energetically different code states makes a subset of checks sufficient to suppress noise both before and after phase imprinting.

What would settle it

A direct calculation or experiment that measures the residual noise strength after partial correction on such encoded states and finds it larger than p to the power floor((l+1)/2) for parallel noise of weight l would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.08341 by Sisi Zhou, Yinan Chen, Zongyuan Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. An example of the toric-code protocol. (a) vertex [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. QFI for toric codes on square and honeycomb lat [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Optimization of the QFI (left panels) for the Bacon [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Decoding examples for toric codes on square (a) and honeycomb (b)-(d) lattices with different lattice sizes and [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Quantum Fisher information [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

We introduce a method for error-corrected quantum metrology where only partial quantum error correction (QEC) is needed to suppress local noise and maintain the probe states' super-standard-quantum-limit (super-SQL) sensing performance. This stands in contrast to the existing QEC-assisted sensing schemes in Phys. Rev. Lett. 112, 080801 (2014) and Phys. Rev. Lett. 112, 150802 (2014), where a probe state is encoded into the logical subspace of a quantum code and error correction involves measurements on all checks of the code. Here, we encode the probe states into superpositions of energetically different states of the underlying quantum code. For our probe states, error correction using a subset of checks is enough to suppress noise both before and after phase imprinting. We analyze the tradeoff in noise suppression. For noise parallel to our phase imprinter of weight $l$, we achieve a suppression of $p^\delta$ where $p$ is the noise strength and $\delta = \lfloor (l+1)/2 \rfloor$. We propose an adaptive imprinter weight increasing strategy to maintain super-SQL performance as we scale up the system. In all our examples, checks and phase imprinters are chosen to be local operators avoiding non-local connectivity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a partial quantum error correction (QEC) scheme for quantum metrology. Probe states are encoded into superpositions of energetically distinct states within an underlying quantum code, so that measurements on only a subset of stabilizer checks suffice to suppress local noise both before and after the phase-imprinting step. For noise parallel to a weight-l phase imprinter the scheme achieves suppression scaling as p^δ with δ = ⌊(l+1)/2⌋. An adaptive strategy that increases imprinter weight with system size is proposed to preserve super-SQL scaling, with all checks and imprinters restricted to local operators.

Significance. If the central claims hold, the work reduces the QEC overhead relative to the full-correction protocols of PRL 112, 080801 and PRL 112, 150802, thereby improving the practicality of super-SQL metrology. Explicit code constructions, tradeoff analysis between partial and full correction, and concrete local-operator examples are provided; these directly address the key assumption that a subset of checks can protect the probe both pre- and post-imprinting.

minor comments (3)
  1. [§2.1] §2.1: the definition of the partial-correction subset should be accompanied by a short table listing, for each example code, which stabilizers are retained and which are omitted.
  2. [§4] The adaptive imprinter-weight strategy in §4 is described qualitatively; a brief scaling argument or pseudocode would clarify how the weight schedule is chosen to keep the effective noise below the SQL threshold.
  3. Figure captions should explicitly state the noise model (e.g., depolarizing vs. amplitude damping) and the precise figure of merit plotted (e.g., variance vs. p).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, significance assessment, and recommendation of minor revision. No specific major comments are listed in the report, so we have no points requiring point-by-point rebuttal or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the partial-QEC metrology scheme and the concrete suppression exponent δ = ⌊(l+1)/2⌋ from an explicit new encoding of probe states as superpositions of energetically distinct code states, together with direct analysis of noise suppression before and after phase imprinting using a subset of checks. No step reduces by construction to a fitted parameter, a self-citation chain, or a renamed known result; the central claims rest on the technical constructions and tradeoff analysis supplied in the manuscript rather than on any load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the chosen superposition encoding permits partial correction to suffice; no new free parameters, invented particles, or ad-hoc constants are introduced in the abstract.

axioms (1)
  • standard math Standard quantum mechanics and the existence of quantum error-correcting codes with local checks
    The scheme presupposes the usual framework of QEC and local noise models used in the contrasted 2014 papers.

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 2 Pith papers · 4 internal anchors

  1. [1]

    ∗ Equal contribution; yinanc@caltech.edu † Equal contribution; zongyuan.wang15@outlook.com ‡ sisi.zhou26@gmail.com

    and Perimeter Institute for Theoretical Physics, a re- search institute supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Univer- sities. ∗ Equal contribution; yinanc@caltech.edu † Equal contribution; zongyuan.wang1...

    work page

  2. [2]

    Quantum noise limited and entanglement-assisted magnetometry,

    W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett.104, 133601 (2010)

    work page 2010

  3. [3]

    Squeezed-light optical magnetometry,

    Florian Wolfgramm, Alessandro Cerè, Federica A. Beduini, Ana Predojević, Marco Koschorreck, and Mor- gan W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett.105, 053601 (2010)

    work page 2010

  4. [4]

    Enhanced sensitivity of the ligo gravita- tional wave detector by using squeezed states of light,

    J. Aasiet al., “Enhanced sensitivity of the ligo gravita- tional wave detector by using squeezed states of light,” Nat. Photonics7, 613–619 (2013)

    work page 2013

  5. [5]

    Quantum-enhanced advanced ligo detec- tors in the era of gravitational-wave astronomy,

    M. Tseet al., “Quantum-enhanced advanced ligo detec- tors in the era of gravitational-wave astronomy,” Phys. Rev. Lett.123, 231107 (2019)

    work page 2019

  6. [6]

    Multipartite entanglement and high- precision metrology,

    Géza Tóth, “Multipartite entanglement and high- precision metrology,” Phys. Rev. A85, 022322 (2012)

    work page 2012

  7. [7]

    Measuringmultipartiteentanglementthrough dynamic susceptibilities,

    Philipp Hauke, Markus Heyl, Luca Tagliacozzo, and Pe- terZoller,“Measuringmultipartiteentanglementthrough dynamic susceptibilities,” Nature Physics12, 778–782 (2016)

    work page 2016

  8. [8]

    We analyze this when we present our example protocols

    We note that in the presence of noise, the scaling behav- ior depends on the quantum codes, the noise strength, and the imprinter. We analyze this when we present our example protocols

    work page

  9. [9]

    Quantum metrology,

    Vittorio Giovannetti, Seth Lloyd, and Lorenzo Mac- cone, “Quantum metrology,” Phys. Rev. Lett.96, 010401 (2006)

    work page 2006

  10. [10]

    Quantum fisher information of the greenberger- horne-zeilinger state in decoherence channels,

    Jian Ma, Yi-xiao Huang, Xiaoguang Wang, and C. P. Sun, “Quantum fisher information of the greenberger- horne-zeilinger state in decoherence channels,” Phys. Rev. A84, 022302 (2011)

    work page 2011

  11. [11]

    Bosonic entanglement and quantum sensing from energy transfer in two-tone floquet systems,

    Yinan Chen, Andreas Elben, Angel Rubio, and Gil Re- fael, “Bosonic entanglement and quantum sensing from energy transfer in two-tone floquet systems,” Phys. Rev. Res.7, 043014 (2025)

    work page 2025

  12. [12]

    Quantum sensing with critical systems: impact of symmetry, imperfections, and decoherence,

    Yinan Chen, Sara Murciano, Pablo Sala, and Jason Al- icea, “Quantum sensing with critical systems: impact of symmetry, imperfections, and decoherence,” (2026), arXiv:2601.04364

    work page internal anchor Pith review arXiv 2026

  13. [13]

    Preparing dicke states in a spin ensemble using phase estimation,

    Yang Wang and Barbara M. Terhal, “Preparing dicke states in a spin ensemble using phase estimation,” Phys. Rev. A104, 032407 (2021)

    work page 2021

  14. [14]

    All macroscopic quantum states are fragile and hard to prepare,

    AndreaLópez-Incera, PavelSekatski, andWolfgangDür, “All macroscopic quantum states are fragile and hard to prepare,” Quantum3, 118 (2019)

    work page 2019

  15. [15]

    TheelusiveHeisenberglimitinquantum- enhanced metrology,

    Rafał Demkowicz-Dobrzański, Jan Kołodyński, and MădălinGuţă,“TheelusiveHeisenberglimitinquantum- enhanced metrology,” Nature Communications3, 1063 (2012)

    work page 2012

  16. [16]

    Im- proved quantum metrology using quantum error correc- tion,

    W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Im- proved quantum metrology using quantum error correc- tion,” Phys. Rev. Lett.112, 080801 (2014)

    work page 2014

  17. [17]

    Quantum error correction for metrology,

    E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett.112, 150802 (2014)

    work page 2014

  18. [18]

    Heisenberg limited metrology using Quantum Error-Correction Codes

    Roee Ozeri, “Heisenberg limited metrology using quan- tum error-correction codes,” (2013), arXiv:1310.3432 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  19. [19]

    optimal measurement

    The terminology “optimal measurement” comes from Ref. [11]. It is the measurement operator, given the probesstatesandtheimprinters, thatsaturatestheCRB. In this Letter, the optimal measurement operator is the strongsymmetry(seeRef.[?] fordefinitionincondensed matter physics) operator of the probe state by construc- tion. Hence, we use the two names interc...

    work page

  20. [20]

    Achieving the heisenberg limit in quantum metrology using quantum error correction,

    Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang, “Achieving the heisenberg limit in quantum metrology using quantum error correction,” Nature Com- munications9, 78 (2018)

    work page 2018

  21. [21]

    Quantum error correctionbelowthesurfacecodethreshold,

    Google Quantum AI and Collaborators, “Quantum error correctionbelowthesurfacecodethreshold,” Nature638, 920–926 (2025)

    work page 2025

  22. [22]

    Scaling and logic in the colour code on a superconduct- ing quantum processor,

    N. Lacroix, A. Bourassa, F. J. H. Heras, L. M. Zhang, J. Bausch, A. W. Senior, T. Edlich, N. Shutty,et al., “Scaling and logic in the colour code on a superconduct- ing quantum processor,” Nature645, 614–619 (2025)

    work page 2025

  23. [23]

    A fault- tolerant neutral-atom architecture for universal quantum computation,

    Dolev Bluvstein, Alexandra A. Geim, Sophie H. Li, Si- mon J. Evered, J. Pablo Bonilla Ataides,et al., “A fault- tolerant neutral-atom architecture for universal quantum computation,” Nature649, 39–46 (2026)

    work page 2026

  24. [24]

    Fault-tolerant quantum computation with a neutral atom processor

    Ben W. Reichardt, Adam Paetznick, David Aasen, Ivan Basov, Juan M. Bello-Rivas, Parsa Bonderson, Rui Chao, et al., “Fault-tolerant quantum computation with a neu- tral atom processor,” (2025), arXiv:2411.11822 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  25. [25]

    Multiple-particleinterferenceandquan- tum error correction,

    AndrewSteane,“Multiple-particleinterferenceandquan- tum error correction,” Proceedings of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1996)

    work page 1996

  26. [26]

    Good quantum error-correctingcodesexist,

    A. R. Calderbank and Peter W. Shor, “Good quantum error-correctingcodesexist,” Phys.Rev.A54,1098–1105 (1996)

    work page 1996

  27. [27]

    Low- overheadfault-tolerantquantumcomputationbygauging logical operators,

    Dominic J. Williamson and Theodore J. Yoder, “Low- overheadfault-tolerantquantumcomputationbygauging logical operators,” Nature Physics22, 598–603 (2026)

    work page 2026

  28. [28]

    First principles calculation of the shift current photovoltaic effect in ferro- electrics,

    Animesh Datta, Lijian Zhang, Nicholas Thomas-Peter, Uwe Dorner, Brian J. Smith, and Ian A. Walms- ley, “Quantum metrology with imperfect states and de- tectors,” Physical Review A83(2011), 10.1103/phys- reva.83.063836

    work page doi:10.1103/phys- 2011

  29. [29]

    Quantum metrology including state preparation and readout times,

    Shane Dooley, William J. Munro, and Kae Nemoto, “Quantum metrology including state preparation and readout times,” Physical Review A94(2016), 10.1103/physreva.94.052320

    work page doi:10.1103/physreva.94.052320 2016

  30. [30]

    Symmetry: A fun- damental resource for quantum coherence and metrol- ogy,

    Irénée Frérot and Tommaso Roscilde, “Symmetry: A fun- damental resource for quantum coherence and metrol- ogy,” Phys. Rev. Lett.133, 260402 (2024)

    work page 2024

  31. [31]

    Demonstrating quantum coherence and metrol- ogythat isresilient totransversalnoise,

    Chao Zhang, Thomas R. Bromley, Yun-Feng Huang, Huan Cao, Wei-Min Lv, Bi-Heng Liu, Chuan-Feng Li, Guang-Can Guo, Marco Cianciaruso, and Gerardo Adesso, “Demonstrating quantum coherence and metrol- ogythat isresilient totransversalnoise,” Phys. Rev.Lett. 123, 180504 (2019)

    work page 2019

  32. [32]

    Fault-tolerant quantum computation by 7 anyons,

    A.Yu. Kitaev, “Fault-tolerant quantum computation by 7 anyons,” Annals of Physics303, 2–30 (2003)

    work page 2003

  33. [33]

    Quantum codes on a lattice with boundary

    S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary,” (1998), arXiv:quant-ph/9811052 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  34. [34]

    Operator quantum error-correcting subsys- tems for self-correcting quantum memories,

    Dave Bacon, “Operator quantum error-correcting subsys- tems for self-correcting quantum memories,” Phys. Rev. A73, 012340 (2006)

    work page 2006

  35. [35]

    Subsystem fault tolerance with the bacon-shor code,

    Panos Aliferis and Andrew W. Cross, “Subsystem fault tolerance with the bacon-shor code,” Phys. Rev. Lett.98, 220502 (2007)

    work page 2007

  36. [36]

    Neverthe- less, we show that, even for this harsh estimation, it is possible to keep bothF q,X andF q,Z above the SQL by arranging the shape(m, n)

    This is a harsh estimate of theFq,X following the argu- ment for repetition codes in [15], as the Bacon-Shor code doubles the weight of imprinter2nand dephasing on any single raw/column does not change the QFI. Neverthe- less, we show that, even for this harsh estimation, it is possible to keep bothF q,X andF q,Z above the SQL by arranging the shape(m, n)

    work page

  37. [37]

    Renormalization of ising cage- net model and generalized foliation,

    Zongyuan Wang, Xiuqi Ma, David T. Stephen, Michael Hermele, and Xie Chen, “Renormalization of ising cage- net model and generalized foliation,” Phys. Rev. B108, 035148 (2023)

    work page 2023

  38. [38]

    Topological quantum distillation,

    H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,” Phys. Rev. Lett.97, 180501 (2006)

    work page 2006

  39. [39]

    The boundaries and twist de- fects of the color code and their applications to topolog- ical quantum computation,

    Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown, “The boundaries and twist de- fects of the color code and their applications to topolog- ical quantum computation,” Quantum2, 101 (2018)

    work page 2018

  40. [40]

    Scaling and logic in the colour code on a superconducting quantum processor,

    N. et al. Lacroix, “Scaling and logic in the colour code on a superconducting quantum processor,” Nature645, 614–619 (2025)

    work page 2025

  41. [41]

    Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase es- timation,

    Roland Krischek, Christian Schwemmer, Witlef Wiec- zorek, Harald Weinfurter, Philipp Hyllus, Luca Pezzé, and Augusto Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase es- timation,” Phys. Rev. Lett.107, 080504 (2011)

    work page 2011

  42. [42]

    Detecting metrologically use- ful entanglement in the vicinity of dicke states,

    Iagoba Apellaniz, Bernd Lücke, Jan Peise, Carsten Klempt, and Géza Tóth, “Detecting metrologically use- ful entanglement in the vicinity of dicke states,” New Journal of Physics17, 083027 (2015)

    work page 2015

  43. [43]

    Achieving the heisenberg limit using fault-tolerant quantum error correction,

    Himanshu Sahu, Qian Xu, and Sisi Zhou, “Achieving the heisenberg limit using fault-tolerant quantum error correction,” (2026), arXiv:2601.05457 [quant-ph]

    work page arXiv 2026

  44. [44]

    Noisy metrology beyond the standard quantum limit,

    R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett.111, 120401 (2013)

    work page 2013

  45. [45]

    Compatibility in multiparameter quantum metrology,

    Sammy Ragy, Marcin Jarzyna, and Rafał Demkowicz- Dobrzański, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A94, 052108 (2016)

    work page 2016

  46. [46]

    Multi-parameter estimation beyond quantum fisher information,

    Rafał Demkowicz-Dobrzański, Wojciech Górecki, and Mădălin Guţă, “Multi-parameter estimation beyond quantum fisher information,” Journal of Physics A: Mathematical and Theoretical53, 363001 (2020)

    work page 2020

  47. [47]

    The elusive heisenberg limit in quantum-enhanced metrology,

    Rafał Demkowicz-Dobrzański, Jan Kołodyński, and Mădălin GuŢă, “The elusive heisenberg limit in quantum-enhanced metrology,” Nat. Commun.3, 1063 (2012). Supplemental Materials REMARKS Remark 1.In this work, we introduced a sensing method that utilizes only partial QEC on the probe states constructed from CSS codes to achieve enough noise suppression to ret...

    work page 2012

  48. [48]

    Consider a unitary phase imprinterU(θ) =e iθO and a strong symmetry of the probe stateT X ρ=t xρthat anticommuteswith the generatorO{T X , O}= 0

    and a closely related version in [29]. Consider a unitary phase imprinterU(θ) =e iθO and a strong symmetry of the probe stateT X ρ=t xρthat anticommuteswith the generatorO{T X , O}= 0. Plugging into the error propagation formula, we have⟨T X ⟩θ = Tr U †(θ)TX U(θ)ρ =Tr U †2(θ)TX ρ =t xTr U †2(θ)ρ . Similarly,⟨T 2 X ⟩θ =Tr U †(θ)T 2 X U(θ)ρ =Tr T 2 X ρ =t 2...

    work page