Pith. sign in

REVIEW 2 major objections 6 minor 2 cited by

Fault-Tolerant Operation of a Quantum Error-Correction Code

T0 review · 2 major / 6 minor · reviewed 2026-08-27 · deepseek-v4-flash

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

Pith's one-line read This paper reports the first experimental demonstration of fully fault-tolerant operation of a quantum error-correcting code, on 13 trapped-ion qubits, with logical error rates at or below the physical hardware's error rates.

desk verdict First real FT-vs-nFT experimental win in a distance-3 code; ask for pre-crosstalk-flag counts before trusting the headline ratio. read the letter →

arxiv 2009.11482 v2 pith:6OPR3RBL submitted 2020-09-24 quant-ph

Laird Egan , Dripto M. Debroy , Crystal Noel , Andrew Risinger , Daiwei Zhu , Debopriyo Biswas , Michael Newman , Muyuan Li
show 3 more authors
Kenneth R. Brown Marko Cetina Christopher Monroe
This is my paper · ORCID
classification quant-ph
keywords fault-tolerantquantumcomputationBacon-Shorcodesubsystemtrapped-ioncomputererrorcorrectionlogicalqubitmagicstatedistillationstabilizermeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports the first experimental demonstration of fault-tolerant operation of a quantum error-correcting code that can correct any single-qubit error, using 13 trapped-ion qubits arranged as a Bacon-Shor subsystem code. The authors show that every primitive needed for fault-tolerant logic—state preparation, measurement, logical rotation, and stabilizer measurement—can be run with error rates at or below the level of the physical hardware. After error correction, average state preparation and measurement error is 0.6% and Clifford gate error is 0.3%, and in the Z basis the logical qubit's preparation-and-measurement error (0.30%) is lower than that of a single physical qubit (0.46%). The strongest evidence is a head-to-head comparison: fault-tolerant error detection fails 2 times in 13,288 shots, while a non-fault-tolerant circuit of similar complexity fails 197 times in 12,105 shots. These results matter because they show that fault-tolerant design can suppress errors in a real device with native noise, not just in principle.

What carries the argument

The central object is the Bacon-Shor subsystem code, a [[9,1,3]] code whose logical states are products of three GHZ states (e.g. $|0/1\rangle_L \propto (|{+}{+}{+}\rangle \pm |{-}{-}{-}\rangle)^{\otimes 3}$). This product structure is what makes fault-tolerance practical: because the logical information is redundant across three decoupled GHZ blocks, any single circuit fault can corrupt at most one block, and the two intact blocks let a decoder recover the logical value. Fault-tolerant preparation is a unitary circuit with no entangling operations between blocks; fault-tolerant measurement reads each data qubit and decodes stabilizer correlations in post-processing; fault-tolerant rotation is transversal $Y_L(\pi/2)$, implemented as physical $Y(\pi/2)$ on all nine data qubits followed by a relabeling of qubit indices; and fault-tolerant stabilizer measurement uses a carefully ordered sequence of ancilla-data interactions so that an ancilla error propagates to at most a single data-qubit fault plus a benign gauge transformation.

What would settle it

Inject a correlated error acting jointly on two data qubits (or raise two-qubit gate error while holding single-qubit error fixed) and measure the fault-tolerant error-detection failure rate: under the paper's locality assumption it should scale roughly cubically with the single-fault rate, and if the scaling exponent drops toward one the suppression is coming from low physical error rates rather than from fault-tolerant design.

Watch

Extended reading notes

Core claim

The central claim is that a Bacon-Shor [[9,1,3]] subsystem code, executed on 13 of 15 ytterbium ions with all-to-all connectivity, can be prepared, measured, rotated, and stabilized fault-tolerantly, protecting the logical qubit against any single circuit fault without postselection. The code's logical states factor into three independent GHZ states, so a single fault can corrupt at most one GHZ block and the information is recoverable from the other two. Fault-tolerant circuits are compared directly with non-fault-tolerant versions of the same primitives: error correction reduces logical errors substantially, and error detection shows a two-orders-of-magnitude gap in failure counts. After correction, the average state-preparation-and-measurement error is 0.6% and the Clifford rotation error is 0.3%, and the logical Z-basis preparation-and-measurement outperforms the physical qubit. The authors also prepare magic states with fidelities above the distillation threshold, completing all single-qubit ingredients for universal fault-tolerant operation.

Load-bearing premise

The observed error suppression assumes the device noise is dominated by low-weight, localized errors; if two-qubit gate errors or crosstalk produce correlated high-weight faults at rates comparable to single-qubit errors, the fault-tolerant circuits will not show the claimed quadratic suppression.

Editorial extensions

If this is right

  • A logical qubit can outperform the physical hardware it is built from: in the Z basis the logical preparation-and-measurement error is 0.30(3)% versus 0.46(2)% for a single physical qubit, and the fault-tolerant $Y_L(\pi/2)$ gate error of 0.3(1)% is below the native two-qubit gate error of 0.7–1.5%.
  • Fault-tolerant error detection provides a quantitative witness of fault-tolerance in practice: 2 failures in 13,288 shots versus 197 failures in 12,105 non-fault-tolerant shots.
  • A fault-tolerant stabilizer measurement leaves the logical error at baseline (0.20(13)% versus the 0.23(13)% encoding error), while the same-complexity non-fault-tolerant ordering raises it to 0.76(22)%.
  • Prepared magic states reach 97(1)% fidelity after correction, above the 92.4% distillation threshold, so all single-qubit ingredients for universal fault-tolerant computation are present.
  • With intermediate measurements or ion shuttling, repeated stabilization should extend these primitives into a robust logical memory.

Reading between the lines

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

  • A direct test of the locality assumption would be to inject a deliberate correlated two-qubit error into the final stabilizer measurement; the 2-versus-197 gap should shrink or disappear if such faults are the dominant noise.
  • The three-GHZ structure implies that treating each GHZ block as a unit, for example by placing it in a decoherence-free subspace, should push the logical $T_2^*$ toward the physical $T_2^*$ rather than one-third of it.
  • Because a four-parameter error model (coherent gate overrotation plus preparation and measurement errors) accounts for the data, the next order of suppression is most plausibly bought by improving two-qubit gates rather than by lowering SPAM errors.
  • The experiment's head-to-head FT-versus-nFT comparison of circuits with identical complexity is a portable benchmark: the same protocol could certify fault-tolerance in other qubit platforms without needing a theoretical noise model.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The manuscript reports experiments on a 13-qubit Bacon-Shor [[9,1,3]] code implemented in a 15-ion trapped-ion chain. The authors demonstrate fault-tolerant state preparation, measurement, a transversal Y_L(pi/2) rotation, and stabilizer measurement, and compare each against non-fault-tolerant counterparts. The central evidence is a large FT-vs-nFT reduction in logical failure rates: 2 failures in 13,288 FT error-detection shots versus 197 failures in 12,105 nFT shots, and an equal-complexity stabilizer-ordering comparison giving 0.20(13)% versus 0.76(22)% corrected error. After correction, the average logical SPAM error is 0.6% and the Clifford gate error is 0.3%. The paper also reports logical coherence times, magic-state fidelities, and a detailed error model in the Supplementary Information. The authors explicitly note that repeated stabilization with intermediate measurements is not demonstrated and is left to future work.

Significance. If the reported FT-versus-nFT comparisons are taken at face value, this is a significant experimental milestone: it is one of the first demonstrations of multiple fault-tolerant primitives in a code that can correct all single-qubit errors, with native device noise. The paper's strengths include direct empirical comparisons with binomial confidence intervals, a particularly clean equal-complexity stabilizer-ordering experiment, and a transparent Supplementary Information with error budgets, benchmarking, and a four-parameter simulation model. The crosstalk-flag postselection and the unmeasured |H_y>_L fidelity are important caveats, but both are addressable by additional analysis or by qualifying the claims, so they do not undermine the core experimental achievement.

major comments (2)
  1. [Methods, 'Crosstalk Detection'; 'Encoding the Logical Qubit'] The headline FT-vs-nFT error-detection comparison (2 failures in 13,288 FT shots versus 197 failures in 12,105 nFT shots) and the reported corrected logical error rates are processed after a second postselection step: any shot in which an idle qubit is measured in |1> is discarded as a possible crosstalk flag, with '<4% of total data discarded' on average. The manuscript does not give the logical failure counts before this crosstalk-flag discard, nor the discard rates separately for the FT and nFT arms. If crosstalk-flagged shots are enriched for logical failures in one arm, the reported two-order-of-magnitude ratio could be biased, and the statement in the Introduction that the code protects 'without postselection' is not literally true of the reported data. I request a table of pre-flag counts, the failure rate among discarded shots, and a clear statement of whether all reported error rates include this postselection.
  2. [Supplementary Information, 'Magic State Fidelity'] The abstract states that magic states are prepared with fidelities exceeding the distillation threshold, but this is directly measured only for |H_x>_L (F = 0.972 +/- 0.012 after correction). For |H_y>_L, the paper explicitly gives only the bound 0.75 <= F <= 0.99 and then relies on the argument that the two preparation circuits differ only by a single-qubit phase controlled to about 400 microradians. That argument is plausible but is not a measurement, and the inferred fidelity is load-bearing for the universality claim in the abstract. I recommend either directly measuring or tightly bounding the |H_y>_L fidelity, or rephrasing the claim so that it applies only to |H_x>_L.
minor comments (6)
  1. [Abstract and 'Encoding the Logical Qubit'] The abstract's 0.6% average logical SPAM error is averaged over all four logical basis states, whereas the main text's comparison of 0.46(2)% physical versus 0.30(3)% logical SPAM error refers only to the Z-basis states; please state this distinction explicitly to avoid an apparent inconsistency.
  2. [Introduction/Outlook] The abstract says 'fully fault-tolerant operation,' but the paper itself notes that repeated stabilization and intermediate measurements are not demonstrated; I suggest qualifying 'fully' to 'all single-qubit primitives in a single round' or similar.
  3. [Methods, 'Crosstalk Detection'] Please report the false-positive rate of the crosstalk flag from state-preparation-and-measurement errors on idle qubits, since SPAM errors alone would discard valid shots and the fraction discarded may vary between circuits of different duration and gate count.
  4. [Data Availability / Code Availability] The statements that data and code are available 'upon request and with the permission of the US Government sponsors' are restrictive; please provide a public repository or a detailed data release plan so that the binomial confidence intervals and fits can be independently verified.
  5. [Supplementary Information, Fig. S7 caption] The word 'sinuosoud' in the caption for the Raman T2 measurement should be corrected to 'sinusoid.'
  6. [Fig. 4a and accompanying text] Please state which statistical test produces the p-value < 0.015 for the FT-versus-nFT stabilizer-ordering comparison, so that the reader can reproduce it from the reported binomial counts.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central claims are direct experimental comparisons, and the cited theory (including author-overlapping references) is independently tested by the measurements.

full rationale

The paper's central claims are experimental: it measures logical error rates for fault-tolerant versus non-fault-tolerant preparation, gates, and stabilizer measurements and reports raw counts, error rates, and fit parameters. None of these quantities is defined in terms of another claimed output. The fault-tolerant circuit designs come from published theory (refs 36, 38, 42, some with overlapping authors), but the paper does not derive fault-tolerance from those citations; it tests the circuits empirically against non-fault-tolerant counterparts. The error-detection comparison (2 failures in 13,288 fault-tolerant shots versus 197 failures in 12,105 non-fault-tolerant shots) is a measured outcome, not a fitted parameter renamed as a prediction. The four-parameter simulation in the Supplementary Information uses gate fidelities and SPAM errors measured by independent randomized benchmarking and parity experiments, so it is a genuine consistency check rather than a fit to the target logical data. The logical T2 analysis likewise predicts GHZ dephasing from an independently measured physical-qubit T2. The only caveat is methodological rather than circular: the 'Crosstalk Detection' postselection discards shots with idle qubits measured in |1> (on average <4% of data) and the paper does not report pre-flag fault-tolerant versus non-fault-tolerant failure counts; this could affect the comparison but does not make any claimed result equivalent by construction to its inputs. Author-overlapping citations are present, but they are not load-bearing for the reported empirical conclusions, so the circularity score is minimal.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central experimental comparison uses direct measurements, so the ledger is short. The main theoretical inputs are prior Bacon-Shor code properties and fault-tolerant circuit constructions. The only ad hoc assumption is the inferred |Hy>_L fidelity. No new physical entities are introduced.

free parameters (1)
  • Logical gate error rate Gamma for the FT YL(pi/2) gate = 0.0027(7) per pi/2 after error correction
    Fitted from the decaying sinusoid <Z>_L = A cos(theta) exp(-Gamma theta/(pi/2)) in Fig. 3d; this fitted value is the reported 0.3% Clifford gate error. It is an output measurement, not an input used to derive the fault-tolerant advantage.
assumptions (4)
  • standard math Bacon-Shor [[9,1,3]] is a distance-3 subsystem code that corrects all single-qubit errors, with logical states decomposing into three GHZ states as in Eq. 1.
    Taken from Refs 35-36 and used throughout; if false, the encoding and decoder would not work.
  • domain assumption The fault-tolerant preparation and stabilizer-measurement circuits from Refs 38 and 42 ensure that any single circuit fault causes at most a correctable error or a benign gauge transformation.
    The paper relies on these prior theoretical constructions for the claim that the blue circuits are fault-tolerant; the experimental comparison supports but does not re-derive them.
  • domain assumption Native noise in the 15-ion chain is sufficiently local and Pauli-like for the minimum-weight decoder to correct single errors.
    The paper states that fault-tolerant error suppression requires high-fidelity components and localized errors; the decoder assumes errors are correctable by the stabilizer rules in the Methods.
  • ad hoc to paper The fidelity of |Hy>_L is close to that of |Hx>_L because the preparation circuits differ only by the phase of one single-qubit gate, controlled to about 400 microradians.
    Stated in Supplementary Information 'Magic State Fidelity'; used to extend the claim that both magic states exceed the distillation threshold, while |Hy>_L itself is only bounded to 0.75-0.99.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fault-Tolerant Operation of a Quantum Error-Correction Code." pith.science (2026). https://pith.science/paper/6OPR3RBL

@misc{pith2026200911482,
  author       = {Pith},
  title        = {Pith review of: Fault-Tolerant Operation of a Quantum Error-Correction Code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OPR3RBL}},
  note         = {Machine review of arXiv:2009.11482}
}
read the original abstract

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.

Figures

Figures reproduced from arXiv: 2009.11482 by the authors.

Figure 1
Figure 1. The Bacon-Shor subsystem code implemented on a 15 ion chain. Bacon-Shor is a [[9,1,3]] subsystem code that encodes 9 data qubits into 1 logical qubit. Four weight-6 stabilizers are mapped to ancillary qubits 10, 11, 12, and 13, for measuring errors in the X and Z basis. We demonstrate encoding of the logical qubit, with subsequent logical gate operations or error syndrome extraction. we compare non-fault-tolerant (n… view at source ↗
Figure 2
Figure 2. Fault-tolerant logical qubit state preparation. a, Encoding circuit for creating logical qubit states. The right subcircuit (blue) is used for FT preparation of Z-logical basis states. X-logical basis states can be created by omitting the final Hadamard gates. The left subcircuit (red, dashed) can be optionally prepended for nFT preparation of arbitrary logical states. b, Errors for the key basis states of the encod… view at source ↗
Figure 4
Figure 4. Detection of arbitrary single-qubit errors. a Expectation value of the logical Z operator after encoding |0iL (Baseline, grey/black line), and then performing the nFT (red) or FT measurement (blue) of a single X-type stabilizer with a Z(θ) error inserted on the ancilla during measurement. b After encoding |0iL, different Pauli errors are purposely introduced on a selected data qubit in the code. To detect the error,… view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: Manipulating logical states. a, A schematic depicting different logical operations. A FT discrete logical rotation (blue) operating on |0iL is a transversal operation, YL(π/2) = Y(π/2) ⊗9 , that leaves the code subspace (gray planes) and returns via a permutation of qu…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Feasibility of Logical Bell State Generation in Memory Assisted Quantum Networks

    quant-ph 2024-12 conditional novelty 6.0 of 10

    Logical Bell-state protocols with d=3/5 surface and Bacon-Shor codes in ion traps need gate error rates around 10^-4 to 10^-5 to beat unencoded Bell states at 1 km, and the non-local protocol reaches about 33 Hz over ...

  2. Surface-code Superconducting Quantum Processors: From Calibration To Logical Performance

    quant-ph 2025-04 conditional novelty 5.0 of 10

    A compilation of experiments showing that small surface codes on superconducting transmons can realize logical initialization, measurement, and gates, with fault-tolerant variants outperforming non-fault-tolerant ones...

Reference graph

Works this paper leans on

59 extracted references · 57 canonical work pages · cited by 2 Pith papers

  1. [1]

    Feynman, R. P. Quantum mechanical computers. F oundations Phys.16, 507–531 (1986)

    work page 1986

  2. [2]

    Abrams, D. S. & Lloyd, S. Simulation of many-body fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586 (1997)

    work page 1997

  3. [3]

    D., Love, P

    Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005). 21/23

    work page 2005

  4. [4]

    M., Wecker, D

    Reiher, M., Wiebe, N., Svore, K. M., Wecker, D. & Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Natl. Acad. Sci. 114, 7555–7560 (2017)

    2017

  5. [5]

    Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41, 303–332 (1999)

    work page 1999

  6. [6]

    Quantum computing enhanced computational catalysis

    V on Burg, V .et al. Quantum computing enhanced computational catalysis. Preprint at https://arxiv.org/abs/2007.14460 (2020)

    work page Pith review arXiv 2020

  7. [7]

    How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

    Gidney, C. & Eker ˚a, M. How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits. Preprint at https://arxiv.org/abs/1905.09749 (2019)

    work page Pith review arXiv 2019

  8. [8]

    & Ben-Or, M

    Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error rate. SIAM J. on Comput. (2008)

    work page 2008

Show all 59 references
  1. [9]

    & Zurek, W

    Knill, E., Laflamme, R. & Zurek, W. Threshold accuracy for quantum computation. Preprint at https://arxiv.org/abs/quant- ph/9610011 (1996)

    1996

  2. [10]

    Gottesman, D. E. Stabilizer Codes and Quantum Error Correction . Ph.D. thesis, California Institute of Technology (1997)

    1997

  3. [11]

    Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493 (1995)

    1995

  4. [12]

    & Laflamme, R

    Knill, E. & Laflamme, R. Theory of quantum error-correcting codes. Phys. Rev. A 55, 900 (1997)

    1997

  5. [13]

    C´orcoles, A. D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6, 1–10 (2015)

    2015

  6. [14]

    W., C´orcoles, A., Chow, J

    Takita, M., Cross, A. W., C´orcoles, A., Chow, J. M. & Gambetta, J. M. Experimental demonstration of fault-tolerant state preparation with superconducting qubits. Phys. Rev. Lett. 119, 180501 (2017)

    2017

  7. [15]

    Linke, N. M. et al. Fault-tolerant quantum error detection. Sci. Adv. 3, e1701074 (2017)

    2017

  8. [16]

    & Flammia, S

    Harper, R. & Flammia, S. T. Fault-tolerant logical gates in the ibm quantum experience. Phys. Rev. Lett. 122, 080504 (2019)

    2019

  9. [17]

    Andersen, C. K. et al. Repeated quantum error detection in a surface code. Nat. Phys. 1–6 (2020)

    2020

  10. [18]

    Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81, 2152 (1998)

    1998

  11. [19]

    Chiaverini, J. et al. Realization of quantum error correction. Nature 432, 602–605 (2004)

    2004

  12. [20]

    Experimental repetitive quantum error correction

    Schindler, P.et al. Experimental repetitive quantum error correction. Science 332, 1059–1061 (2011)

    2011

  13. [21]

    Reed, M. D. et al. Realization of three-qubit quantum error correction with superconducting circuits. Nature 482, 382–385 (2012)

    2012

  14. [22]

    Riste, D. et al. Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6, 1–6 (2015)

    2015

  15. [23]

    Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015)

    2015

  16. [24]

    Gong, M. et al. Experimental verification of five-qubit quantum error correction with superconducting qubits. Preprint at https://arxiv.org/abs/1907.04507 (2019)

    2019 arXiv

  17. [25]

    Nigg, D. et al. Quantum computations on a topologically encoded qubit. Science 345, 302–305 (2014)

    2014

  18. [26]

    Luo, Y .-H. et al. Quantum teleportation of physical qubits into logical code-spaces. Preprint at https://arxiv.org/abs/2009.06242 (2020)

    2020 arXiv

  19. [27]

    Heeres, R. W. et al. Implementing a universal gate set on a logical qubit encoded in an oscillator. Nat. communications 8, 1–7 (2017)

    2017

  20. [28]

    Fl¨uhmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019)

    2019

  21. [29]

    Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441–445 (2016)

    2016

  22. [30]

    Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature 584, 368–372 (2020)

    2020

  23. [31]

    & Kitaev, A

    Bravyi, S. & Kitaev, A. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005)

    2005

  24. [32]

    Maunz, P. L. W. High optical access trap 2.0. Sandia Natl. Lab. Rep. No. SAND2016-0796R (2016)

    2016

  25. [33]

    Debnath, S. et al. Demonstration of a small programmable quantum computer with atomic qubits. Nature 563, 63 (2016). 22/23

    2016

  26. [34]

    Wright, K. et al. Benchmarking an 11-qubit quantum computer. Nat. communications 10, 1–6 (2019)

    2019

  27. [35]

    Operator quantum error-correcting subsystems for self-correcting quantum memories

    Bacon, D. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73, 012340 (2006)

    2006

  28. [36]

    & Cross, A

    Aliferis, P. & Cross, A. W. Subsystem fault tolerance with the Bacon-Shor code. Phys. Rev. Lett. 98, 220502 (2007)

    2007

  29. [37]

    M., Li, M., Huang, S

    Debroy, D. M., Li, M., Huang, S. & Brown, K. R. Logical performance of 9 qubit compass codes in ion traps with crosstalk errors. Quantum Sci. Technol. 5, 034002 (2020)

    2020

  30. [38]

    & Brown, K

    Li, M., Miller, D. & Brown, K. R. Direct measurement of Bacon-Shor code stabilizers. Phys. Rev. A 98, 050301 (2018)

    2018

  31. [39]

    Fault-tolerant quantum computation

    Shor, P. Fault-tolerant quantum computation. Proc. 37th Conf. on F oundations Comput. Sci. (1996)

    1996

  32. [40]

    Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307 (2015)

    2015

  33. [41]

    & Preskill, J

    Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002)

    2002

  34. [42]

    & Brown, K

    Li, M., Miller, D., Newman, M., Wu, Y . & Brown, K. R. 2D compass codes.Phys. Rev. X 9, 021041 (2019)

    2019

  35. [43]

    Reichardt, B. W. Quantum universality from magic states distillation applied to CSS codes. Quantum Inf. Process. 4, 251–264 (2005)

    2005

  36. [44]

    A., Chuang, I

    Lidar, D. A., Chuang, I. L. & Whaley, K. B. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81, 2594 (1998)

    1998

  37. [45]

    Kielpinski, D. et al. A decoherence-free quantum memory using trapped ions. Science 291, 1013–1015 (2001)

    2001

  38. [46]

    & Knill, E

    Eastin, B. & Knill, E. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett. 102, 110502 (2009)

    2009

  39. [47]

    Cetina, M. et al. Quantum gates on individually-addressed atomic qubits subject to noisy transverse motion. Preprint at https://arxiv.org/abs/2007.06768 (2020)

    2020 arXiv

  40. [48]

    & Wineland, D

    Kielpinski, D., Monroe, C. & Wineland, D. J. Architecture for a large-scale ion-trap quantum computer. Nature 417, 709–711 (2002)

    2002

  41. [49]

    Home, J. P. et al. Complete methods set for scalable ion trap quantum information processing. Science 325, 1227–1230 (2009)

    2009

  42. [50]

    Pino, J. et al. Demonstration of the QCCD trapped-ion quantum computer architecture. Preprint at https://arxiv.org/abs/2003.01293 (2020)

    2020 arXiv

  43. [51]

    R., Harrow, A

    Brown, K. R., Harrow, A. W. & Chuang, I. L. Arbitrarily accurate composite pulse sequences. Phys. Rev. A 70, 052318 (2004)

    2004

  44. [52]

    & Sørensen, A

    Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835 (1999)

    1999

  45. [53]

    Basic circuit compilation techniques for an ion-trap quantum machine

    Maslov, D. Basic circuit compilation techniques for an ion-trap quantum machine. New J. Phys. 19, 023035 (2017)

    2017

  46. [54]

    Knill, E. et al. Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008)

    2008

  47. [55]

    Barends, R. et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500–503 (2014)

    2014

  48. [56]

    Sackett, C. A. et al. Experimental entanglement of four particles. Nature 404, 256–259 (2000). Number: 6775 Publisher: Nature Publishing Group

    2000

  49. [57]

    & Monroe, C

    Inlek, I., Vittorini, G., Hucul, D., Crocker, C. & Monroe, C. Quantum gates with phase stability over space and time. Phys. Rev. A 90, 042316 (2014)

    2014

  50. [58]

    Gullion, T., Baker, D. B. & Conradi, M. S. New, compensated carr-purcell sequences. J. Magn. Reson. (1969) 89, 479–484 (1990)

    1969

  51. [59]

    Single ion-qubit exceeding one hour coherence time

    Wang, P.et al. Single ion-qubit exceeding one hour coherence time. Preprint at https://arxiv.org/abs/2008.00251 (2020). 23/23

    2020 arXiv

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.