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REVIEW 3 major objections 5 minor 75 references

Randomized Benchmarking in the Analogue Setting

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Randomized benchmarking can be extended to analogue quantum simulators using disordered Hamiltonian evolutions, yielding a scalable average error rate per unit time that is independent of state-preparation and measurement errors.

desk verdict A genuine adaptation of RB to analogue simulators with an honest but unproven 2-design assumption; worth refereeing for the analogue community. read the letter →

arxiv 1909.01295 v2 pith:LSDUTJRH submitted 2019-09-03 quant-ph

Ellen Derbyshire , Jorge Yago Malo , Andrew Daley , Elham Kashefi , Petros Wallden This is my paper · ORCID
classification quant-ph
keywords randomizedbenchmarkinganaloguequantumsimulationunitary2-designdepolarisingchannelSPAMerrorsdisorderedHamiltoniansnoisecharacterisationLoschmidtecho
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 sets out to bring randomized benchmarking, the standard digital tool for measuring average gate error, to analogue quantum simulators. It claims that by replacing gates with native Hamiltonian evolutions $U_k=e^{-iH_k dt}$, adding disorder to generate a large family of unitaries, and inverting each step, one can measure the average error rate per unit time for that family. The metric is read from the decay of the survival probability, $P_T=A+Bf^T$, with SPAM errors absorbed into $A$ and $B$. If the claim holds, analogue simulators can be tested scalably, independently of state-preparation and measurement errors, and across devices running the same family of Hamiltonians.

What carries the argument

The load-bearing mechanism is the unitary 2-design twirl. When a noisy channel is conjugated and averaged over an exact 2-design, a finite set whose averages match the Haar measure for polynomials up to degree two, it becomes a depolarising channel, so the entire noise reduces to one number. ARB uses an $\epsilon$-approximate 2-design in the diamond-norm sense, samples long sequences of disordered evolutions, and inverts each step; the paper proves $|P^\alpha_l - P^\mu_l| \le l\epsilon$ and, with no SPAM errors and $l=1$, $r-\epsilon \le r' \le r+\epsilon$. The candidate 2-designs are generated by adding symmetry-breaking disorder terms to $H_s$, and convergence is diagnosed by how well survival data follow the exponential decay.

What would settle it

Compute the frame potential $F(\{U_k\}) = \frac{1}{K^2}\sum_{k,k'} |\mathrm{tr}(U_k^\dagger U_{k'})|^4$ for the specific sets used in the fits; if it is far from the Haar value 2 over the sequence lengths where the decay is fit, the sets are not approximate 2-designs and the fitted $r$ is not the average error rate. Alternatively, run ARB with a deliberately correlated, non-depolarising noise model on a small system: if the survival probabilities do not follow $P_T=A+Bf^T$ for long $T$, the twirling mechanism is not doing the work.

Watch

Extended reading notes

Core claim

Analogue randomized benchmarking (ARB) claims that an analogue quantum simulator can be characterized by a single average error rate per unit time for a family of Hamiltonian evolutions, with state-preparation and measurement errors absorbed into fit parameters. The protocol replaces digital gates with time-evolution unitaries $U_k = e^{-iH_k dt}$ built from a base Hamiltonian $H_s$ plus disorder terms, and replaces the single inversion gate of standard RB by systematic inversion of each unitary in an echo-style sequence. Under the assumption that the set $\{U_k\}$ forms an $\epsilon$-approximate 2-design, the average survival probability obeys $P_T = A + B f^T$, with $f$ related to the average error rate by $r = (d-1)(1-f)/d$. Simulations on a six-spin XY model with nearest-neighbour and all-to-all couplings show fits to this curve for several noise models, with the locally disordered all-to-all set fitting best; the globally disordered all-to-all set does not fit, which the authors interpret as failure to converge to a 2-design.

Load-bearing premise

The whole interpretation rests on the disordered set $\{U_k\}$ being an $\epsilon$-approximate 2-design over the sequence lengths used; if the set does not scramble enough, the noise is not depolarised and the fitted decay curve cannot be read as an average error rate, and the paper itself notes that this has not been formally proven.

Editorial extensions

If this is right

  • ARB turns a noisy analogue simulator into a single number $r$ per unit time for a family of Hamiltonians, so device quality can be compared without knowing SPAM errors.
  • Because it uses only native time evolutions, it avoids the compilation overhead that limits digital RB on hardware without native Clifford gates.
  • The exponential decay form gives a built-in diagnostic: when a disordered set fails to scramble enough to approximate a 2-design, the data visibly depart from $A+Bf^T$, as seen for the globally disordered all-to-all set.
  • The bound $r-\epsilon \le r' \le r+\epsilon$ lets experimenters quote the average error rate with an explicit uncertainty coming from how close the set is to a 2-design.

Reading between the lines

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

  • Because the bound grows linearly with sequence length, the protocol's reliability may be improved by estimating $\epsilon$ directly, for example through frame-potential or second-moment comparisons, rather than only through fit quality.
  • If the inversion step can be implemented in Trotterised digital form on hybrid trapped-ion devices, ARB could benchmark the same hardware in both digital and analogue modes, giving a per-time error rather than per-gate error; the paper mentions this direction but does not develop it.
  • The observed failure signature for global all-to-all disorder suggests a practical protocol-development heuristic: local, site-dependent disorder produces richer scrambling and should be preferred when designing benchmark sets for long-range Hamiltonians.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes analogue randomized benchmarking (ARB), an adaptation of digital randomized benchmarking to programmable analogue quantum simulators. The protocol replaces discrete gates with unitaries U_k = exp(-i H_k dt) generated from a native Hamiltonian with added disorder, uses systematic time-inversion of each preceding unitary, and asserts that the average survival probability follows P_T = A + B f^T, from which an average error rate r = (d-1)(1-f)/d per unit time is extracted. The authors present classical simulations for nearest-neighbour and all-to-all XY models with several noise models (static fluctuations, weakly time-dependent errors, spontaneous emission, and noisy inversion), reporting fitted decay curves and 95% confidence intervals for r. They explicitly acknowledge that the unitary sets have not been proven to form epsilon-approximate 2-designs and that this is an assumption underlying the protocol.

Significance. If the central premise is established, ARB would fill a real gap: a scalable, SPAM-robust benchmarking method for analogue simulators that uses native operations rather than compiled gate sets. The paper is commendably transparent about its main assumption, and the numerical study is useful as a proof-of-principle for the protocol's curve-fitting machinery. However, the current evidence does not yet secure the interpretation of the fitted f as an average error rate: the 2-design property is unverified, the associated epsilon is never quantified, and the claimed SPAM robustness is not actually simulated. The contribution is therefore a promising protocol proposal with honest numerical illustrations, rather than a fully supported benchmarking method.

major comments (3)
  1. [Sec. III A, Assumption III.1, Sec. V] The load-bearing assumption that the sets {U_k = exp(-i H_k dt)} form an epsilon-approximate unitary 2-design is unproven, and the evidence offered for it is partly circular. The paper states in Sec. V that this has not been formally proven, and in Sec. IV A 1 it notes that the averaging during ARB, rather than twirling over an approximate 2-design, could be what makes the errors behave like a depolarising channel. The fact that simulated survival data fit an exponential decay does not distinguish these mechanisms. No frame potential or second-moment-operator distance (Defs. E.2 and E.3) is reported for the actual K=1000 sets, so the premise that connects the fitted f to average infidelity remains unsupported. I would ask the authors to compute and report a direct numerical estimate of epsilon for the unitary sets used, or otherwise provide a non-circular validation of the 2-design property.
  2. [Thms. III.1-III.2, Eq. (16), Sec. IV A 1] The stated bounds do not protect the reported r values. Theorem III.1 gives |P_alpha_l - P_mu_l| <= l*epsilon, which grows linearly with sequence length; for the long sequences actually fitted (e.g., up to T J = 150 with dt = 0.005, corresponding to l = 30000), this bound is vacuous unless epsilon is extraordinarily small. Theorem III.2 and Lemma F.5 give r' within epsilon of r only at l=1 and ps=1, but the protocol extracts f from a fit over all l, and Lemma F.3 shows the error in f grows with l. Since epsilon is never estimated, Eq. (16) is invoked without content, and the 95% confidence intervals reported in Eqs. (14)-(15) and (19)-(20) are purely statistical. The systematic error from the approximate design is therefore unquantified, and the claim that ARB produces a meaningful average error rate is not yet established.
  3. [Sec. IV A, Eq. (10), Abstract] The abstract claims that ARB incorporates SPAM errors, but the numerical simulations are run with no state-preparation or measurement errors: Sec. IV A states 'we run the protocol with no errors in state preparation or measurement', and Eq. (10) fixes A = 1/d, B = (d-1)/d, which assumes no SPAM. The robustness of the A + B f^T form to SPAM is never tested, so the central advertised advantage over fidelity-estimation methods is not demonstrated. I would ask for at least one simulation with explicit SPAM errors (e.g., imperfect initial-state preparation and measurement misclassification) showing that the decay curve retains the A + B f^T form and that r is unaffected.
minor comments (5)
  1. [Sec. IV A 2] In the comparison with directly computed average infidelities, the confidence interval for random product states is reported as 0.00150 (0.00145, 0.0000155); the upper endpoint 0.0000155 is almost certainly a typo and should likely be 0.00155.
  2. [Eq. (21), Sec. IV B 3] The exponent in P_T = A + B' f^{2(T-1*dt)} is notationally confusing: as written it mixes time T and time-step dt in a way that is dimensionally inconsistent. It should be expressed in terms of the integer sequence length l (e.g., f^{2(l-1)}) or defined explicitly.
  3. [Appendix D] The text describing Fig. 8 uses 'M = 6' where the system size is elsewhere denoted N; please use consistent notation.
  4. [References] References [19] and [20] spell the author name as 'Mageson'; the correct spelling is 'Magesan'.
  5. [Sec. V] The statement that 'with the standard error on our result we bound the unknown parameter epsilon' is not justified: standard errors quantify statistical fluctuations, not the systematic error epsilon entering through the approximate 2-design property. This sentence should be revised to reflect that epsilon remains unquantified.

Circularity Check

1 steps flagged · score 1.0 of 10

Minor self-flagged circularity: curve fit used as heuristic evidence for the 2-design assumption; central r values remain independently anchored.

  1. other [Sec. IV A 1 (nearest-neighbour case); echoed in Sec. IV C and qualified in Sec. V]
    "The data fitting the curve could imply that the errors are depolarised by the process, which is conditioned on the set being a sufficiently good approximate 2-design. The fact that the data seems to fit to the curve at later times could indicate that the set of unitaries, both global and local, {U(g,l) k } converge to a 2-design at these longer sequences."

    The decay curve being fitted, PT = A + B f^T, was derived in Sec. III A precisely under the assumption that the sampled unitaries form an epsilon-approximate 2-design (Def. III.1, Thm. III.1). A good fit to this same curve is therefore a consistency check of the model, not an independent test of the 2-design assumption; using that fit as evidence for convergence to a 2-design affirms the consequent. The authors immediately qualify this ('the fit of just one noise model to the curve is not sufficient') and in Sec. V explicitly concede that the 2-design property is not formally proven. The reported r values are nonetheless independently anchored by direct average-infidelity calculations in Sec.

full rationale

The central ARB derivation is a standard RB argument: given an epsilon-approximate 2-design, twirling reduces the effective noise to a depolarising channel, yielding PT = A + B f^T with r = (d-1)(1-f)/d (Sec. III A, App. A). This conditional derivation is self-contained and does not import its conclusion as an input; f and r are extracted from the fit, not predefined. The one soft circle is the paper's use of the goodness of the fit as heuristic evidence that its unitary sets converge to a 2-design, even though the fitted curve was derived from that very assumption. The authors themselves flag this ('the fit of just one noise model to the curve is not sufficient...' and 'it is possible that the averaging during ARB rather than twirling over an approximate 2-design is what causes the errors to behave like a depolarising channel'), and their conclusion concedes the 2-design property is unproven. This is therefore an explicitly acknowledged limitation and correctness risk, not a masked circular derivation. The reported r for the all-to-all local-disorder set is checked against directly computed average gate infidelities (Sec. IV A 2: ARB rl = 0.005063 lies between direct values 0.00150 and 0.01046), providing an independent anchor. There is no load-bearing self-citation chain; citation [59] (Daley) is used only for the quantum-trajectory simulation method, and no fitted input is renamed as a prediction. Overall, the derivation is not circular by construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The protocol introduces no new physical entities. The free parameters listed are simulation and protocol-design choices that the reported numerical values of r depend on. The central axiomatic load is the unproven epsilon-approximate 2-design property of the disordered-Hamiltonian unitary sets, plus the standard RB noise and inversion assumptions.

free parameters (5)
  • disorder standard deviation delta = delta = J (=1)
    Chosen by hand as the scale of the normal-distributed disorder potentials in Eq. 4 (Sec III A). The claim that the unitary sets approximate a 2-design depends on this disorder scale; no optimization or physical derivation is given.
  • transverse-field noise sigma_B = 0.5
    Example noise amplitude on the field term used in all main simulations; the authors note these values are taken as an example (footnote in Sec IV A). Reported r values are conditional on this choice.
  • coupling noise sigma_J = 0.2
    Example noise amplitude on the J coupling in the simulated hardware; chosen for illustration only (Sec IV A).
  • unitary set size K = 1000
    Fixed as a large finite set of unitaries that is experimentally feasible (Sec III A). The approximation quality epsilon of the design depends on K, and no quantitative relation is provided.
  • time-step dt = 0.005
    Fixed time-step for each unitary, chosen to be above the minimum physical switching time and in the regime where numerical error is negligible (Sec IV B 4); the extracted per-time error rate depends on dt.
assumptions (4)
  • domain assumption The error channel is gate-independent and time-independent, trace-preserving, and memoryless.
    Invoked in Sec II-RB and Sec III C as the standard noise conditions; the numerical cases use noise that is mostly gate/time-independent (except the weakly time-dependent model, where robustness is tested).
  • ad hoc to paper The generated unitary sets {U_k = e^{-i H_k dt}} form an epsilon-approximate unitary 2-design (for long sequences).
    Load-bearing assumption introduced in Sec III A and unproven; the paper states we assume that our sets produce an epsilon-approximate 2-design for long sequences and in Conclusions we have not formally proven it. All decay-curve fits and r interpretations assume this.
  • domain assumption Systematic inversion operators are either perfect (Sec IV A) or subject to noise that is independent and uniformly distributed, with the twirl of two composed channels equal to the product of the individually twirled channels (App B).
    The main simulations take perfect inverses; the noisy-inversion analysis (Eq. 21) approximates (Lambda_e o Lambda_e)_t = (Lambda_e)_t^2, an assumption the authors note is not strictly satisfied.
  • standard math Randomized benchmarking theory: a unitary 2-design twirl maps any channel to a depolarizing channel, and average survival probability decays as A+B f^l.
    The whole ARB derivation builds on this textbook result (Sec II, App A). We take it as background.

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Cite this review

Pith. "Pith review of Randomized Benchmarking in the Analogue Setting." pith.science (2026). https://pith.science/paper/LSDUTJRH

@misc{pith2026190901295,
  author       = {Pith},
  title        = {Pith review of: Randomized Benchmarking in the Analogue Setting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSDUTJRH}},
  note         = {Machine review of arXiv:1909.01295}
}
read the original abstract

Current development in programmable analogue quantum simulators (AQS), whose physical implementation can be realised in the near-term compared to those of large-scale digital quantum computers, highlights the need for robust testing techniques in analogue platforms. Methods to properly certify or benchmark AQS should be efficiently scalable, and also provide a way to deal with errors from state preparation and measurement (SPAM). Up to now, attempts to address this combination of requirements have generally relied on model-specific properties. We put forward a new approach, applying a well-known digital noise characterisation technique called randomized benchmarking (RB) to the analogue setting. RB is a scalable experimental technique that provides a measure of the average error-rate of a gate-set on a quantum hardware, incorporating SPAM errors. We present the original form of digital RB, the necessary alterations to translate it to the analogue setting and introduce the analogue randomized benchmarking protocol (ARB). In ARB we measure the average error-rate per time evolution of a family of Hamiltonians and we illustrate this protocol with two case-studies of analogue models; classically simulating the system by incorporating several physically motivated noise scenarios. We find that for the noise models tested, the data fit with the theoretical predictions and we gain values for the average error rate for differing unitary sets. We compare our protocol with other relevant RB methods, where both advantages (physically motivated unitaries) and disadvantages (difficulty in reversing the time-evolution) are discussed.

Figures

Figures reproduced from arXiv: 1909.01295 by the authors.

Figure 1
Figure 1. FIG. 1. (Left) Survival probability as a function of total time per sequence, and ARB decay curve fit for a system with nearest [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Left) Survival probability as a function of total time per sequence, and ARB decay curve fit for a system with [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Survival probability for a system with nearest-neighbour interactions and a global disorder [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Survival probability for a system with nearest-neighbour interactions and a global disorder [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Survival probability for a system with nearest-neighbour interactions and a global disorder [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (Left) Survival probability as a function of total time per sequence, and ARB decay curve fit for a system with nearest [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Survival probability for a system with nearest-neighbour interactions and a global disorder [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (Top) Comparison of ARB decay curve for a system with [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (Top) Comparison of ARB decay curve for a system with [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]

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Works this paper leans on

75 extracted references · 46 canonical work pages

  1. [1]

    9 for the case of α→∞ that we consider to be well-described by: Hs = N∑ j J(σ+ j σ− j+1 +σ− j σ+ j+1) +B N∑ j σz j

    XY Hamiltonian with transverse field (Nearest-Neighbours) Here, we present the ARB fits for the model described in Eq. 9 for the case of α→∞ that we consider to be well-described by: Hs = N∑ j J(σ+ j σ− j+1 +σ− j σ+ j+1) +B N∑ j σz j . (11) We generate a set of Hamiltonians {Hk} for both local (ζl k) and global (ζg k) disorder terms: Hg k =Hs + ∆k ∑ j σx j⊗...

    work page

  2. [2]

    9 for the case of α∼ 0, i.e

    All-to-all spin model Hamiltonian with transverse field Here, we present the ARB fits for a spin system governed by Eq. 9 for the case of α∼ 0, i.e. an all-to-all coupled spin system with the same system parameters as in Sec. IV A 1: Hs = N∑ ij Jij(σ+ i σ− j +σ− i σ+ j ) +B N∑ j σz j . (17) Again, we generated a set of Hamiltonians Hk for both local and glo...

    work page

  3. [3]

    We therefore analyse whether some of the physical system parameters, of our specific tested system, may impact the measured values of r

    Impact of Field-term Obtaining the average error-rates ( r) from the ARB protocol provides a characterisation of the hardware, and how it copes with a specific set of operations (gate-set). We therefore analyse whether some of the physical system parameters, of our specific tested system, may impact the measured values of r. In Fig. 3, we present the fidelit...

    work page

  4. [4]

    4 we study how robust our characterization is in the presence of some uncertainty in the time dt for which these random Hamiltonians are evolved and applied on the state

    Weakly time-dependent noise First, in Fig. 4 we study how robust our characterization is in the presence of some uncertainty in the time dt for which these random Hamiltonians are evolved and applied on the state. This could arise physically from a limited time resolution in the quantum hardware. We consider the case where every unitary is applied for a g...

    work page

  5. [5]

    We therefore consider (again, on our globally disordered nearest-neighbour model) in Fig

    Spontaneous Emission As mentioned previously, the affect of the ARB process on one noise model is not enough to indicate that our unitaries depolarise that channel. We therefore consider (again, on our globally disordered nearest-neighbour model) in Fig. 5 noise from spontaneous emission , an example of coupling of the quantum device to its environment. We...

    work page

  6. [6]

    Namely, in the previous results and those from Sec

    Noisy time-inversion Now that we have analysed how the ARB protocol is affected by both weak time-dependent noise and dissipation, it is necessary to address one of the assumptions of the implementation. Namely, in the previous results and those from Sec. IV B we model the systematic time inversion as perfect. This choice was motivated by the fact that eli...

    work page

  7. [7]

    Impact of time-step Finally, we consider how the numerical time-step ( dt) can impact the ARB curve. In Fig. 7, we present the survival probabilityPT for different values of the time-step (dt) used to create the unitaries for the same system as in Sec. IV A 1; again the nearest-neighbour HXY with global disorder. The values of dt were chosen in the regime ...

    work page 2020

  8. [8]

    Kokail, C

    C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, et al., arXiv:1810.03421 [cond-mat, physics: quant-ph] (2018), arXiv: 1810.03421, URL http://arxiv.org/abs/1810. 03421

    arXiv 2018

Show all 75 references
  1. [9]

    D. Yang, A. Grankin, L. M. Sieberer, D. V. Vasilyev, and P. Zoller, arXiv:1905.06444 (2019), URL arXiv:1905.06444. 20

    2019 arXiv

  2. [10]

    Hangleiter, M

    D. Hangleiter, M. Kliesch, M. Schwarz, and J. Eisert, Quantum Science and Technology 2, 015004 (2017), URL https: //doi.org/10.1088

    2017

  3. [11]

    Bermejo-Vega, D

    J. Bermejo-Vega, D. Hangleiter, M. Schwarz, R. Raussendorf, and J. Eisert, Phys. Rev. X 8, 021010 (2018), URL https://link.aps.org/doi/10.1103/PhysRevX.8.021010

    2018 doi

  4. [12]

    Cramer, M

    M. Cramer, M. B. Plenio, S. T. Flammia, D. Gross, S. D. Bartlett, R. Somma, O. Landon-Cardinal, Y.-K. Liu, and D. Poulin, Nature Communications 1, 149 (2010), ISSN 2041-1723, arXiv: 1101.4366, URL http://arxiv.org/abs/ 1101.4366

    2010 arXiv

  5. [13]

    Mohseni, A

    M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Physical Review A 77 (2008), ISSN 1050-2947, 1094-1622, arXiv: quant-ph/0702131, URL http://arxiv.org/abs/quant-ph/0702131

    2008 arXiv

  6. [14]

    S. T. Flammia and Y.-K. Liu, Physical Review Letters 106 (2011)

    2011

  7. [15]

    B. P. Lanyon, C. Maier, M. Hozpa..pfel, J. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. S. Buyskikh, A. J. Daley, M. Cramer, et al., Nature Physics (2016), URL https://www.nature.com/articles/nphys4244

    2016

  8. [16]

    Torlai, G

    G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Nature Physics 14, 447 (2018), URL https://doi.org/10.1038/s41567-018-0048-5

    2018 doi

  9. [17]

    G. Toth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Phys. Rev. Lett. 105, 250403 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.105.250403

    2010 doi

  10. [18]

    S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 79, 4790 (1997)

    1997

  11. [19]

    Gorin, T

    T. Gorin, T. Prosen, T. H. Seligman, and M. nidari, Physics Reports 435, 33 (2006), ISSN 0370-1573

    2006

  12. [20]

    Emerson, R

    J. Emerson, R. Alicki, and K. Z . yczkowski, Journal of Optics B: Quantum and Semiclassical Optics 7 (2005), URL https://iopscience.iop.org/article/10.1088/1464-4266/7/10/021/pdf

    2005 doi

  13. [21]

    Knill, D

    E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Phys. Rev. A 77, 012307 (2008), URL https://journals.aps.org/pra/abstract/10.1103/PhysRevA.77. 012307

    2008 doi

  14. [22]

    A. M. Meier, Ph.D. thesis, University of Colorado (2013)

    2013

  15. [23]

    Onorati, A

    E. Onorati, A. H. Werner, and J. Eisert, Phys. Rev. Lett. 123, 060501 (2019), URL https://journals.aps.org/prl/ abstract/10.1103/PhysRevLett.123.060501

    2019 doi

  16. [24]

    Magesan, Ph.D

    E. Magesan, Ph.D. thesis, University of Waterloo (2008)

    2008

  17. [25]

    J. J. Wallman (2017)

    2017

  18. [26]

    Mageson, J

    E. Mageson, J. M. Gambetta, and J. Emerson, Phys. Rev. Lett. 106 (2011), URL https://journals.aps.org/prl/ abstract/10.1103/PhysRevLett.106.180504

    2011 doi

  19. [27]

    Mageson, J

    E. Mageson, J. M. Gambetta, and J. Emerson, Phys. Rev. A 85 (2012), URL https://journals.aps.org/pra/ abstract/10.1103/PhysRevA.85.042311

    2012 doi

  20. [28]

    J. J. Wallman and S. T. Flammia, New Journal of Physics 16 (2014)

    2014

  21. [29]

    D. S. Franca and A. L. Hashagen, Journal of Physics A: Mathematical and Theoretical p. 215508 (2018), URL arXiv: 1803.03621[quant-ph]

    2018 arXiv

  22. [30]

    T. J. Proctor, A. Carignan-Dugas, K. Rudinger, E. Nielsen, R. Blume-Kohout, and K. Young, arXiv:1807.07975 [quant- ph] (2018), arXiv: 1807.07975, URL http://arxiv.org/abs/1807.07975

    2018 arXiv

  23. [31]

    S. T. Merkel, E. J. Pritchett, and B. H. Fong, arXiv preprint quant-ph/1804.05951 (2018), URL https://arxiv.org/ abs/1804.05951

    2018 arXiv

  24. [32]

    A. W. Harrow and R. A. Low, Communications in Mathematical Physics 291, 257 (2009), URL https://arxiv.org/ abs/0802.1919

    2009 arXiv

  25. [33]

    Gottesman, Talk at the International Conference on Group Theoretic Methods in Physics (1998)

    D. Gottesman, Talk at the International Conference on Group Theoretic Methods in Physics (1998)

    1998

  26. [34]

    Helsen, X

    J. Helsen, X. Xue, L. M. K. Vandersypen, and S. Wehner, Nature Physics Journals: Quantum Information 5, 71 (2019), URL https://doi.org/10.1038/s41534-019-0182-7

    2019 doi

  27. [35]

    Onorati, O

    E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A. H. Werner, and J. Eisert, Communications in Mathematical Physics 355, 905 (2017), ISSN 0010-3616, 1432-0916, arXiv: 1606.01914, URL http://arxiv.org/abs/1606.01914

    2017 arXiv

  28. [36]

    F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, Communications in Mathematical Physics 346, 397 (2016), URL https://arxiv.org/abs/1208.0692

    2016 arXiv

  29. [37]

    Vermersch, A

    B. Vermersch, A. Elben, M. Dalmonte, J. I. Cirac, and P. Zoller, Physical Review A 97, 023604 (2018), URL https: //link.aps.org/doi/10.1103/PhysRevA.97.023604

    2018 doi

  30. [38]

    D. A. Roberts and B. Yoshida, Journal of High Energy Physics 2017, 121 (2017), URL https://doi.org/10.1007/ JHEP04(2017)121

    2017

  31. [39]

    Srednicki, Physical Review E 50, 888 (1994), URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE

    M. Srednicki, Physical Review E 50, 888 (1994), URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE. 50.888

    1994 doi

  32. [40]

    W. G. Brown, L. F. Santos, D. J. Starling, and L. Viola, Physical Review E 77, 021106 (2008), URL https://journals. aps.org/pre/abstract/10.1103/PhysRevE.77.021106

    2008 doi

  33. [41]

    Lashkari, D

    N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, JHEP 2013:22, 2013 (2013), URL https://link. springer.com/article/10.1007%2FJHEP04%282013%29022. 21

    2013

  34. [42]

    Guhr and H

    T. Guhr and H. A. Weidenm¨ uller, Annals of Physics199, 412 (1990), URL https://doi.org/10.1016/0003-4916(90) 90383-Y

    1990 doi

  35. [43]

    Swingle, G

    B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Physical Review A 94, 040302(R) (2016), URL https: //journals.aps.org/pra/pdf/10.1103/PhysRevA.94.040302

    2016 doi

  36. [44]

    Hayden and J

    P. Hayden and J. Preskill, Journal of High Energy Physics 2007, 120 (2007), URL https://iopscience.iop.org/ article/10.1088/1126-6708/2007/09/120

    2007 doi

  37. [45]

    Marchildon, Quantum Mechanics (Elsevier, 2002), chap

    L. Marchildon, Quantum Mechanics (Elsevier, 2002), chap. Symmetry in Hamiltonians

    2002

  38. [46]

    Abd El-Hady, A

    A. Abd El-Hady, A. Y. Abul-Magd, and M. H. Simbel, Journal of Physics A: Mathematical and General 35 (2002)

    2002

  39. [47]

    T. Guhr, A. M¨ uller-Groaling, and H. A. Weidenm¨ uller, Physics Report299, 4 (1998), URL https://doi.org/10.1016/ S0370-1573(97)00088-4

    1998

  40. [48]

    Bl¨ umel and U

    R. Bl¨ umel and U. Smilansky, Physical Review Letters69, 217 (1992), URL https://journals.aps.org/prl/abstract/ 10.1103/PhysRevLett.69.217

    1992 doi

  41. [49]

    A. Y. Kitaev, A. H. Shen, and M. N. Vyali, Classical and Quantum Computation, Graduate Studies in Mathematics (American Mathematical Society, 2002), vol. 47

    2002

  42. [50]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, Physical Review A. 80 (2009), URL https://arxiv.org/abs/ quant-ph/0606161

    2009 arXiv

  43. [51]

    Dai and Y

    F. Dai and Y. Xu, Cubature Formulas on Spheres (Springer New York, New York, NY, 2013), p. 127153, ISBN 978-1- 4614-6660-4, URL https://doi.org/10.1007/978-1-4614-6660-4_6

    2013 doi

  44. [52]

    A. W. Harrow and S. Mehraban, arXiv:1809.06957v1 [quant-ph] (2018)

    2018 arXiv

  45. [53]

    Garttner, J

    M. Garttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Nature Physics 13, 781 (2017), URL https://doi.org/10.1038/nphys4119

    2017 doi

  46. [54]

    J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Phys. Rev. X 7, 031011 (2017), URL https: //link.aps.org/doi/10.1103/PhysRevX.7.031011

    2017 doi

  47. [55]

    G. B. Lesovik, I. A. Sadovskyy, M. V. Suslov, A. V. Lebedev, and V. M. Vinokur, Nature: Scientific Reports 9 (2019), URL https://doi.org/10.1038/s41598-019-40765-6

    2019 doi

  48. [56]

    K. Kim, S. Korenblit, R. Islam, E. E. Edwards, M.-S. Chang, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C. C. J. Wang, et al., 13, 105003 (2011), URL https://iopscience.iop.org/article/10.1088/1367-2630/13/10/105003/ meta

    2011 doi

  49. [57]

    B. P. Lanyon, C. Hempel, D. Nigg, M. M¨ uller, R. Gerritsma, F. Z¨ ahringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, et al., Science 334, 57 (2011), https://science.sciencemag.org/content/334/6052/57.full.pdf, URL https: //science.sciencemag.org/content/334/6052/57

    2011

  50. [58]

    J. W. Britton, B. C. Sawyer, A. C. Keith, C. C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Nature 484, 489 (2012), URL https://doi.org/10.1038/nature10981

    2012 doi

  51. [59]

    Porras and J

    D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 (2004), URL https://journals.aps.org/prl/abstract/10. 1103/PhysRevLett.92.207901

    2004

  52. [60]

    Richerme, Z.-X

    P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Nature 511, 198 (2014), URL https://doi.org/10.1038/nature13450

    2014 doi

  53. [61]

    Jurcevic, B

    P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Nature 511, 202 (2014), URL https://doi.org/10.1038/nature13461

    2014 doi

  54. [62]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Nature 551, 579 (2017), article, URL https://doi.org/10.1038/nature24622

    2017 doi

  55. [63]

    Nonlinear least squares fitting tool, Matlab curve fitting toolbox (2018), the MathWorks, Natick, MA, USA

    2018

  56. [64]

    Proctor, K

    T. Proctor, K. Rudinger, K. Young, M. Sarovar, and R. Blume-Kohout, Phys. Rev. Lett. 119, 130502 (2017), URL https://arxiv.org/abs/1702.01853

    2017 arXiv

  57. [65]

    Brown and O

    W. Brown and O. Fawzi (2013), URL https://arxiv.org/abs/1210.6644

    2013 arXiv

  58. [66]

    A. J. Daley, Advances in Physics 63, 77 (2014)

    2014

  59. [67]

    Hunter-Jones, arXiv e-prints p

    N. Hunter-Jones, arXiv e-prints p. arXiv:1905.12053 (2019)

    2019 arXiv

  60. [68]

    R. A. Low, Ph.D. thesis, University of Bristol (2010)

    2010

  61. [69]

    M. A. Nielsen, Physical Review Letters A. 303, 249 (2002), URL arXiv:quant-ph/0205035

    2002 arXiv

  62. [70]

    Wan, Lie Algebras (Elsevier, 1975), vol

    Z.-X. Wan, Lie Algebras (Elsevier, 1975), vol. 104, chap. 9.2, 1st ed

    1975

  63. [71]

    Dankert, Ph.D

    C. Dankert, Ph.D. thesis, University of Waterloo (2005)

    2005

  64. [72]

    off-diagonal

    D. Gross, K. Audenaert, and J. Eisert, J. Math. Phys. 48 (2008), URL https://arxiv.org/abs/quant-ph/0611002. 22 Appendix A: Randomized Benchmarking Running a unitary gate U on a physical device corresponds to a quantum channel denoted as Λ U. The action of this quantum channel...

    2008 arXiv

  65. [73]

    A set of vectors{|φ1⟩,..., |φK⟩} is a spherical 2-design in Cd if and only if: ∑ k,k′ |⟨φk||φk′⟩| 4 K2 = 2 d4 +d2

    is a well-known metric for determining whether one has an exact spherical design or not, and is defined as follows: Definition E.1 (Spherical t-design). A set of vectors{|φ1⟩,..., |φK⟩} is a spherical 2-design in Cd if and only if: ∑ k,k′ |⟨φk||φk′⟩| 4 K2 = 2 d4 +d2 . (E1) The d...

  66. [74]

    III.1 we first present the following definitions: Definition F.1

    Proof of Theorem III.1 In order to prove Theorem. III.1 we first present the following definitions: Definition F.1. The trace-norm of a quantum channel E in terms of the input state density matrix ρ, that minimises the error probability on distinguishing between two quantum chann...

  67. [75]

    LetA andB be known quantities

    Proof of Theorem III.2 Lemma F.3. LetA andB be known quantities. Under Assumption III.1 and with a small ϵ, the error in determining fl from the RB method of an ϵ-approximate 2-design, is given by: δfl≈ ϵ fl−1B , (F17) where f is dependent on l, fl = f±δfl is the value for the...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.