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arxiv: 2412.18578 · v2 · submitted 2024-12-24 · 🪐 quant-ph

Randomized Benchmarking with Synthetic Quantum Circuits

Yale Fan , Riley Murray , Thaddeus D. Ladd , Kevin Young , Robin Blume-Kohout This is my paper

Pith reviewed 2026-05-23 06:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords randomized benchmarkingsynthetic circuitsquantum noise characterizationhigh-spin systemssample complexityrotational symmetrySU(2) representations
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The pith

Synthetic circuits with classical post-processing boost randomized benchmarking efficiency by over 100 times for rotationally symmetric quantum systems.

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

The paper develops a new framework for randomized benchmarking that uses synthetic quantum circuits and classical post-processing of data to exploit the structure of reducible representations in high-dimensional systems. This approach targets systems with rotational symmetry, such as high-spin qudits or bosonic modes, where control is limited. It shows that this method can measure rotationally invariant error rates with more than two orders of magnitude fewer samples than standard techniques like character RB. A sympathetic reader would care because current RB methods become inefficient for these experimentally relevant high-dimensional systems, limiting noise characterization in scalable quantum devices. The central mechanism leverages the full group representation structure without needing more experimental unitaries.

Core claim

The authors introduce synthetic RB protocols that apply to any benchmarking group and use classical post-processing on input and output data to leverage reducible superoperator representations. For SU(2)-symmetric systems, these protocols achieve a sample complexity advantage exceeding two orders of magnitude over character RB when measuring rotationally invariant error rates in high-spin systems.

What carries the argument

Synthetic quantum circuits with classical post-processing of input and output data, which leverage the full structure of any reducible superoperator representation generated by the benchmarking operations.

If this is right

  • RB becomes practical for high-dimensional systems like qudits and bosonic modes where only a small subset of unitaries are accessible.
  • Rotationally invariant error rates can be extracted more efficiently in systems with natural SU(2) action.
  • The framework extends RB beyond multiqubit settings to novel quantum error-correcting codes based on rotational symmetry.
  • Sample efficiency gains of over 100x relative to standard methods for these error rates.

Where Pith is reading between the lines

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

  • Applying this to other symmetry groups could further expand RB to additional physical systems.
  • Integration with existing quantum hardware might allow faster characterization without hardware changes.
  • Future work could test if the advantage holds when noise deviates from the assumed models.

Load-bearing premise

Classical post-processing of input and output data can fully leverage the structure of any reducible superoperator representation without requiring additional experimental unitaries beyond the small accessible subset.

What would settle it

An experiment on a high-spin system where the measured sample complexity for synthetic RB is not at least 100 times lower than character RB for the same rotationally invariant error rate precision.

Figures

Figures reproduced from arXiv: 2412.18578 by Kevin Young, Riley Murray, Robin Blume-Kohout, Thaddeus D. Ladd, Yale Fan.

Figure 1
Figure 1. Figure 1: A taxonomy of several RB protocols discussed here and [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An SU(2) superoperator has a characteristic “kite” structure, shown here for j = 7/2. To make this irrep decomposition manifest, note that any spin-j operator can be expanded in irreducible spherical tensors of rank k = 0, 1, . . . , 2 j (see Appendices A and B for details): T (k) q = s 2k + 1 2 j + 1 X j ℓ,ℓ′=−j C j,ℓ j,ℓ′ ;k,q |ℓ⟩⟨ℓ ′ |. (16) A rank-k spherical tensor operator is a homogeneous degree￾k p… view at source ↗
Figure 3
Figure 3. Figure 3: Estimated weight-2 SU(2) error rates p2 obtained by simu￾lation under various models of SPAM error and a common per-gate error of ρ 7→ UρU † for U = exp(−i0.04J 2 z ). For each ℓ ∈ {0, . . . , 7}, state preparation error is modeled by a random SU(2) rotation through an angle ϕ ∈ {0, 0.1, 0.2} (left to right). In the top row, measurement error is modeled by a single coherent rotation of all POVM effects thr… view at source ↗
Figure 5
Figure 5. Figure 5: Synthetic survival probabilities (and exponential fits thereof) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Simulated data for synthetic RB protocols with [PITH_FULL_IMAGE:figures/full_fig_p060_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Simulated data for synthetic RB protocols with [PITH_FULL_IMAGE:figures/full_fig_p061_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simulated data for synthetic RB protocols with [PITH_FULL_IMAGE:figures/full_fig_p062_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Simulated data for synthetic RB protocols with [PITH_FULL_IMAGE:figures/full_fig_p063_9.png] view at source ↗
read the original abstract

Noise characterization methods such as randomized benchmarking (RB) are critical for the development of scalable quantum computers. Modern RB protocols for multiqubit systems extract physically relevant error rates by exploiting the structure of the group representation generated by the set of benchmarked operations. However, existing techniques become prohibitively inefficient for representations that are highly reducible yet decompose into irreducible subspaces of high dimension. These situations prevail when benchmarking high-dimensional systems such as qudits or bosonic modes, where experimental control is limited to implementing a small subset of all possible unitary operations. We introduce a broad framework for enhancing the sample efficiency of RB that is sufficiently powerful to extend the practical reach of RB beyond the multiqubit setting. Our strategy, which applies to any benchmarking group, uses "synthetic" quantum circuits with classical post-processing of both input and output data to leverage the full structure of reducible superoperator representations. To demonstrate the efficacy of our approach, we develop a detailed theory of RB for systems with rotational symmetry. Such systems carry a natural action of the group $\text{SU}(2)$, and they form the basis for several novel quantum error-correcting codes. We show that, for measuring rotationally invariant error rates of experimentally accessible high-spin systems, our synthetic RB protocols offer a sample complexity advantage of more than two orders of magnitude relative to standard approaches such as character RB.

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

1 major / 0 minor

Summary. The manuscript introduces a framework for randomized benchmarking (RB) that employs 'synthetic' quantum circuits together with classical post-processing of input and output data to exploit the full structure of reducible superoperator representations generated by the benchmarking group. The approach is claimed to apply to any benchmarking group and is illustrated for systems with SU(2) rotational symmetry; the central result is that the resulting protocols achieve a sample-complexity advantage of more than two orders of magnitude over standard methods such as character RB when measuring rotationally invariant error rates on experimentally accessible high-spin systems.

Significance. If the claimed advantage and the underlying representation-theoretic analysis are correct, the work would materially extend the practical reach of RB beyond multiqubit systems to qudits and bosonic modes where only a small subset of unitaries is experimentally accessible, addressing a recognized inefficiency of existing RB protocols for highly reducible representations.

major comments (1)
  1. [Abstract] Abstract: the manuscript asserts a sample-complexity advantage exceeding two orders of magnitude relative to character RB, yet supplies no derivations, explicit circuit constructions, representation-theory calculations, or numerical verification. Because the full text consists solely of the abstract, it is impossible to determine whether the synthetic-circuit construction and classical post-processing actually deliver the stated improvement or whether the assumptions about reducible representations hold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the manuscript asserts a sample-complexity advantage exceeding two orders of magnitude relative to character RB, yet supplies no derivations, explicit circuit constructions, representation-theory calculations, or numerical verification. Because the full text consists solely of the abstract, it is impossible to determine whether the synthetic-circuit construction and classical post-processing actually deliver the stated improvement or whether the assumptions about reducible representations hold.

    Authors: The referee correctly observes that only the abstract appears in the provided text. The complete manuscript, containing the full representation-theoretic derivations, explicit synthetic-circuit constructions, sample-complexity analysis, and numerical verification of the >100x advantage for SU(2)-symmetric high-spin systems, is available on arXiv:2412.18578. We will ensure the full manuscript is submitted for review. revision: no

Circularity Check

0 steps flagged

No circularity detectable from abstract alone

full rationale

Only the abstract is available, which presents the central claim of a sample-complexity advantage as a derived result without any equations, derivation steps, self-citations, or fitted parameters. No load-bearing step can be quoted or shown to reduce to its own inputs by construction, satisfying the rule that circularity requires explicit evidence of such a reduction. The paper is therefore scored as self-contained against external benchmarks with no circularity identified.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the framework implicitly relies on standard quantum mechanics group representation theory and the existence of experimentally accessible operations for high-spin systems.

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Reference graph

Works this paper leans on

134 extracted references · 134 canonical work pages · 1 internal anchor

  1. [1]

    Scal- able noise estimation with random unitary operators,

    Joseph Emerson, Robert Alicki, and Karol ˙Zyczkowski, “Scal- able noise estimation with random unitary operators,” Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005)

    work page 2005

  2. [2]

    Exact and approximate unitary 2-designs and their ap- plication to fidelity estimation,

    Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine, “Exact and approximate unitary 2-designs and their ap- plication to fidelity estimation,” Phys. Rev. A80, 012304 (2009)

    work page 2009

  3. [3]

    Symmetrized Characterization of Noisy Quantum Processes,

    Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme, “Symmetrized Characterization of Noisy Quantum Processes,” Science 317, 1893–1896 (2007)

    work page 2007

  4. [4]

    Randomized benchmarking of quantum gates,

    E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, 14 “Randomized benchmarking of quantum gates,” Phys. Rev. A77, 012307 (2008)

    work page 2008

  5. [5]

    Scal- able and Robust Randomized Benchmarking of Quantum Pro- cesses,

    Easwar Magesan, J. M. Gambetta, and Joseph Emerson, “Scal- able and Robust Randomized Benchmarking of Quantum Pro- cesses,” Phys. Rev. Lett.106, 180504 (2011)

    work page 2011

  6. [6]

    Char- acterizing quantum gates via randomized benchmarking,

    Easwar Magesan, Jay M. Gambetta, and Joseph Emerson, “Char- acterizing quantum gates via randomized benchmarking,” Phys. Rev. A 85, 042311 (2012)

    work page 2012

  7. [7]

    A new class of efficient randomized bench- marking protocols,

    Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, and Stephanie Wehner, “A new class of efficient randomized bench- marking protocols,” npj Quantum Information 5, 71 (2019)

    work page 2019

  8. [8]

    Character Ran- domized Benchmarking for Non-Multiplicity-Free Groups With Applications to Subspace, Leakage, and Matchgate Randomized Benchmarking,

    Jahan Claes, Eleanor Rieffel, and Zhihui Wang, “Character Ran- domized Benchmarking for Non-Multiplicity-Free Groups With Applications to Subspace, Leakage, and Matchgate Randomized Benchmarking,” PRX Quantum 2, 010351 (2021)

    work page 2021

  9. [9]

    Gen- eral Framework for Randomized Benchmarking,

    J. Helsen, I. Roth, E. Onorati, A.H. Werner, and J. Eisert, “Gen- eral Framework for Randomized Benchmarking,” PRX Quantum 3, 020357 (2022)

    work page 2022

  10. [10]

    A framework for randomized benchmarking over compact groups,

    Linghang Kong, “A framework for randomized benchmarking over compact groups,” arXiv preprint (2021), arXiv:2111.10357 [quant-ph]

    work page arXiv 2021

  11. [11]

    Randomized Benchmarking beyond Groups,

    J. Chen, D. Ding, and C. Huang, “Randomized Benchmarking beyond Groups,” PRX Quantum 3, 030320 (2022)

    work page 2022

  12. [12]

    Heinrich, M

    Markus Heinrich, Martin Kliesch, and Ingo Roth, “Randomized benchmarking with random quantum circuits,” arXiv preprint (2022), arXiv:2212.06181 [quant-ph]

    work page arXiv 2022

  13. [13]

    Qudits and High-Dimensional Quantum Computing,

    Yuchen Wang, Zixuan Hu, Barry C. Sanders, and Sabre Kais, “Qudits and High-Dimensional Quantum Computing,” Frontiers in Physics 8 (2020), 10.3389/fphy.2020.589504

    work page doi:10.3389/fphy.2020.589504 2020

  14. [14]

    Fermionic Quantum Computation,

    Sergey B. Bravyi and Alexei Yu. Kitaev, “Fermionic Quantum Computation,” Annals of Physics 298, 210–226 (2002)

    work page 2002

  15. [15]

    Quantum informa- tion with continuous variables,

    Samuel L. Braunstein and Peter van Loock, “Quantum informa- tion with continuous variables,” Rev. Mod. Phys.77, 513–577 (2005)

    work page 2005

  16. [16]

    Quantifying error and leakage in an encoded Si/SiGe triple-dot qubit,

    R. W. Andrews, C. Jones, M. D. Reed, A. M. Jones, S. D. Ha, M. P. Jura, J. Kerckho ff, M. Levendorf, S. Meenehan, S. T. Merkel, A. Smith, B. Sun, A. J. Weinstein, M. T. Rakher, T. D. Ladd, and M. G. Borselli, “Quantifying error and leakage in an encoded Si/SiGe triple-dot qubit,” Nature Nanotechnology 14, 747–750 (2019)

    work page 2019

  17. [17]

    Randomized benchmarking for qudit Clifford gates,

    M. Jafarzadeh, Y .-D. Wu, Y . R. Sanders, and B. C. Sanders, “Randomized benchmarking for qudit Clifford gates,” New Jour- nal of Physics 22, 063014 (2020)

    work page 2020

  18. [18]

    Randomized benchmarking in the analogue setting,

    E. Derbyshire, J. Yago Malo, A. J. Daley, E. Kashefi, and P. Wallden, “Randomized benchmarking in the analogue setting,” Quantum Science and Technology 5, 034001 (2020)

    work page 2020

  19. [19]

    Practical verification protocols for analog quantum simulators,

    Ryan Shaffer, Eli Megidish, Joseph Broz, Wei-Ting Chen, and Hartmut Häffner, “Practical verification protocols for analog quantum simulators,” npj Quantum Information 7, 46 (2021)

    work page 2021

  20. [20]

    Benchmarking bosonic modes for quantum information with randomized displacements,

    Christophe H. Valahu, Tomas Navickas, Michael J. Biercuk, and Ting Rei Tan, “Benchmarking bosonic modes for quantum information with randomized displacements,” arXiv preprint (2024), arXiv:2405.15237 [quant-ph]

    work page arXiv 2024

  21. [21]

    Benchmarking bosonic and fermionic dynamics,

    Jadwiga Wilkens, Marios Ioannou, Ellen Derbyshire, Jens Eis- ert, Dominik Hangleiter, Ingo Roth, and Jonas Haferkamp, “Benchmarking bosonic and fermionic dynamics,” arXiv preprint (2024), arXiv:2408.11105 [quant-ph]

    work page arXiv 2024

  22. [22]

    Bosonic randomized benchmarking with passive transformations,

    M. Arienzo, D. Grinko, M. Kliesch, and M. Heinrich, “Bosonic randomized benchmarking with passive transformations,” arXiv preprint (2024), arXiv:2408.11111 [quant-ph]

    work page arXiv 2024

  23. [23]

    Creation and manipulation of Schrödinger cat states of a nuclear spin qudit in silicon,

    Xi Yu, Benjamin Wilhelm, Danielle Holmes, Arjen Vaart- jes, Daniel Schwienbacher, Martin Nurizzo, Anders Kringhøj, Mark R. van Blankenstein, Alexander M. Jakob, Pragati Gupta, Fay E. Hudson, Kohei M. Itoh, Riley J. Murray, Robin Blume- Kohout, Thaddeus D. Ladd, Andrew S. Dzurak, Barry C. Sanders, David N. Jamieson, and Andrea Morello, “Creation and manipu...

    work page arXiv 2024

  24. [24]

    Synthetic high angular momentum spin dynamics in a microwave oscillator,

    Saswata Roy, Alen Senanian, Christopher S. Wang, Owen C. Wetherbee, Luojia Zhang, B. Cole, C. P. Larson, E. Yelton, Kartikeya Arora, Peter L. McMahon, B. L. T. Plourde, Baptiste Royer, and Valla Fatemi, “Synthetic high angular momentum spin dynamics in a microwave oscillator,” arXiv preprint (2024), arXiv:2405.15695 [quant-ph]

    work page arXiv 2024

  25. [25]

    Multi-frequency control and measurement of a spin-7/2 system encoded in a transmon qudit,

    Elizabeth Champion, Zihao Wang, Rayleigh Parker, and Ma- chiel Blok, “Multi-frequency control and measurement of a spin-7/2 system encoded in a transmon qudit,” arXiv preprint (2024), arXiv:2405.15857 [quant-ph]

    work page arXiv 2024

  26. [26]

    Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System,

    B. E. Anderson, H. Sosa-Martinez, C. A. Riofrío, I. H. Deutsch, and P. S. Jessen, “Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System,” Phys. Rev. Lett.114, 240401 (2015)

    work page 2015

  27. [27]

    Designing Codes around Interactions: The Case of a Spin,

    Jonathan A. Gross, “Designing Codes around Interactions: The Case of a Spin,” Phys. Rev. Lett.127, 010504 (2021)

    work page 2021

  28. [28]

    Multispin Clifford codes for angular momentum errors in spin systems,

    Sivaprasad Omanakuttan and Jonathan A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems,” Phys. Rev. A 108, 022424 (2023)

    work page 2023

  29. [29]

    Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes,

    Sivaprasad Omanakuttan, Vikas Buchemmavari, Jonathan A. Gross, Ivan H. Deutsch, and Milad Marvian, “Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes,” PRX Quantum 5, 020355 (2024)

    work page 2024

  30. [30]

    What Randomized Benchmark- ing Actually Measures,

    Timothy Proctor, Kenneth Rudinger, Kevin Young, Mohan Saro- var, and Robin Blume-Kohout, “What Randomized Benchmark- ing Actually Measures,” Phys. Rev. Lett.119, 130502 (2017)

    work page 2017

  31. [31]

    Randomized benchmarking with gate-depen- dent noise,

    Joel J. Wallman, “Randomized benchmarking with gate-depen- dent noise,” Quantum 2, 47 (2018)

    work page 2018

  32. [32]

    Ran- domized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors,

    Seth T. Merkel, Emily J. Pritchett, and Bryan H. Fong, “Ran- domized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors,” Quantum 5, 581 (2021)

    work page 2021

  33. [33]

    Direct Randomized Benchmarking for Multiqubit Devices,

    Timothy J. Proctor, Arnaud Carignan-Dugas, Kenneth Rudinger, Erik Nielsen, Robin Blume-Kohout, and Kevin Young, “Direct Randomized Benchmarking for Multiqubit Devices,” Phys. Rev. Lett. 123, 030503 (2019)

    work page 2019

  34. [34]

    A Theory of Direct Randomized Benchmarking,

    Anthony M. Polloreno, Arnaud Carignan-Dugas, Jordan Hines, Robin Blume-Kohout, Kevin Young, and Timothy Proctor, “A Theory of Direct Randomized Benchmarking,” arXiv preprint (2023), arXiv:2302.13853 [quant-ph]

    work page arXiv 2023

  35. [35]

    Multiqubit randomized benchmarking using few samples,

    Jonas Helsen, Joel J. Wallman, Steven T. Flammia, and Steph- anie Wehner, “Multiqubit randomized benchmarking using few samples,” Phys. Rev. A100, 032304 (2019)

    work page 2019

  36. [36]

    Efficient Estimation of Pauli Channels,

    Steven T. Flammia and Joel J. Wallman, “Efficient Estimation of Pauli Channels,” ACM Transactions on Quantum Computing 1 (2020), 10.1145/3408039

    work page doi:10.1145/3408039 2020

  37. [37]

    Averaged Circuit Eigenvalue Sampling,

    Steven T. Flammia, “Averaged Circuit Eigenvalue Sampling,” in 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), Leibniz Inter- national Proceedings in Informatics (LIPIcs), V ol. 232, edited by François Le Gall and Tomoyuki Morimae (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2022) p...

    work page 2022

  38. [38]

    Randomized bench- marking with confidence,

    Joel J. Wallman and Steven T. Flammia, “Randomized bench- marking with confidence,” New Journal of Physics16, 103032 (2014)

    work page 2014

  39. [39]

    Repre- sentations of the multi-qubit Clifford group,

    Jonas Helsen, Joel J. Wallman, and Stephanie Wehner, “Repre- sentations of the multi-qubit Clifford group,” Journal of Mathe- matical Physics 59, 072201 (2018)

    work page 2018

  40. [40]

    Coherent electrical control of a single high-spin nucleus in silicon,

    Serwan Asaad, Vincent Mourik, Benjamin Joecker, Mark A. I. Johnson, Andrew D. Baczewski, Hannes R. Firgau, Mateusz T. 15 M˛ adzik, Vivien Schmitt, Jarryd J. Pla, Fay E. Hudson, Kohei M. Itoh, Jeffrey C. McCallum, Andrew S. Dzurak, Arne Laucht, and Andrea Morello, “Coherent electrical control of a single high-spin nucleus in silicon,” Nature 579, 205–209 (2020)

    work page 2020

  41. [41]

    Logical Randomized Benchmarking

    Joshua Combes, Christopher Granade, Christopher Ferrie, and Steven T. Flammia, “Logical Randomized Benchmarking,” arXiv preprint (2017), arXiv:1702.03688 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  42. [42]

    Spin qudit tomography and state reconstruction error,

    Michael A. Perlin, Diego Barberena, and Ana Maria Rey, “Spin qudit tomography and state reconstruction error,” Phys. Rev. A 104, 062413 (2021)

    work page 2021

  43. [43]

    D. A. Varshalovich, A. N. Moskalev, and V . K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific Pub- lishing Company, 1988)

    work page 1988

  44. [44]

    On the Condition of Reducibility of any Group of Linear Substitutions,

    W. Burnside, “On the Condition of Reducibility of any Group of Linear Substitutions,” Proceedings of the London Mathematical Society 3, 430–434 (1905)

    work page 1905

  45. [45]

    Characterization of Addressability by Simultaneous Random- ized Benchmarking,

    Jay M. Gambetta, A. D. Córcoles, S. T. Merkel, B. R. Johnson, John A. Smolin, Jerry M. Chow, Colm A. Ryan, Chad Rigetti, S. Poletto, Thomas A. Ohki, Mark B. Ketchen, and M. Ste ffen, “Characterization of Addressability by Simultaneous Random- ized Benchmarking,” Phys. Rev. Lett.109, 240504 (2012)

    work page 2012

  46. [46]

    129 (Springer, New York, NY , 2004)

    William Fulton and Joe Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, V ol. 129 (Springer, New York, NY , 2004)

    work page 2004

  47. [47]

    A simple formula for the average gate fidelity of a quantum dynamical operation,

    Michael A. Nielsen, “A simple formula for the average gate fidelity of a quantum dynamical operation,” Physics Letters A 303, 249–252 (2002)

    work page 2002

  48. [48]

    Rolling quantum dice with a superconducting qubit,

    R. Barends, J. Kelly, A. Veitia, A. Megrant, A. G. Fowler, B. Campbell, Y . Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, E. Jeffrey, C. Neill, P. J. J. O’Malley, J. Mutus, C. Quintana, P. Roushan, D. Sank, J. Wenner, T. C. White, A. N. Korotkov, A. N. Cleland, and John M. Martinis, “Rolling quantum dice with a superconducting qubit,” Phys. Rev. A90,...

    work page 2014

  49. [49]

    Atomic Coherent States in Quantum Optics,

    F. T. Arecchi, Eric Courtens, Robert Gilmore, and Harry Thom- as, “Atomic Coherent States in Quantum Optics,” Phys. Rev. A 6, 2211–2237 (1972)

    work page 1972

  50. [50]

    Sub-Planck structures: Analogies between the Heisenberg- Weyl and SU(2) groups,

    Naeem Akhtar, Barry C. Sanders, and Carlos Navarrete-Benll- och, “Sub-Planck structures: Analogies between the Heisenberg- Weyl and SU(2) groups,” Phys. Rev. A103, 053711 (2021)

    work page 2021

  51. [51]

    Permutation-Invariant Quan- tum Codes with Transversal Generalized Phase Gates,

    Eric Kubischta and Ian Teixeira, “Permutation-Invariant Quan- tum Codes with Transversal Generalized Phase Gates,” arXiv preprint (2023), arXiv:2310.17652 [quant-ph]

    work page arXiv 2023

  52. [52]

    Fault-tolerant qubit encoding using a spin-7/2 qudit,

    Sumin Lim, Junjie Liu, and Arzhang Ardavan, “Fault-tolerant qubit encoding using a spin-7/2 qudit,” Phys. Rev. A108, 062403 (2023)

    work page 2023

  53. [53]

    Reducing Quantum Computations to Elemen- tary Unitary Operations,

    George Cybenko, “Reducing Quantum Computations to Elemen- tary Unitary Operations,” Computing in Science & Engineering 3, 27–32 (2001)

    work page 2001

  54. [54]

    Criteria for exact qudit universality,

    Gavin K. Brennen, Dianne P. O’Leary, and Stephen S. Bullock, “Criteria for exact qudit universality,” Phys. Rev. A71, 052318 (2005). 16 APPENDICES A. Preliminaries 17

    work page 2005

  55. [55]

    Hilbert-Schmidt Space 17

    work page

  56. [56]

    Spherical Tensor Basis 18

    work page

  57. [57]

    Derivations 21

    Parametrizations of SU(2) 19 B. Derivations 21

    work page

  58. [58]

    SU(2) “Fourier” Transform 22

    work page

  59. [59]

    Review of Randomized Benchmarking 27

    Completeness Relations 24 C. Review of Randomized Benchmarking 27

    work page

  60. [60]

    Standard Randomized Benchmarking 27

    work page

  61. [61]

    Character Randomized Benchmarking 28

    work page

  62. [62]

    SU(2) Character Randomized Benchmarking 30

    Filtered Randomized Benchmarking 29 D. SU(2) Character Randomized Benchmarking 30

    work page

  63. [63]

    Synthetic Randomized Benchmarking 31

    Discrete Benchmarking Groups 30 E. Synthetic Randomized Benchmarking 31

    work page

  64. [64]

    Infinite Frame 32 b

    Synthetic Gates 31 a. Infinite Frame 32 b. Finite Frame 32

    work page

  65. [65]

    Sample Complexity: Bounds 35

    (Synthetic) SPAM 34 F. Sample Complexity: Bounds 35

    work page

  66. [66]

    Infinite Frame 36 b

    Synthetic-Gate RB 36 a. Infinite Frame 36 b. Finite Frame 37

    work page

  67. [67]

    Synthetic-SPAM RB 40

    work page

  68. [68]

    Synthetic-SPAM Character RB 42 b

    Synthetic-Circuit RB 42 a. Synthetic-SPAM Character RB 42 b. Synthetic-SPAM Rank-1 RB 44 c. Synthetic-SPAM Finite-Frame RB 45 G. Sample Complexity: Exact Results 47

    work page

  69. [69]

    χRB 51 b

    Non-Synthetic SPAM 51 a. χRB 51 b. R1RB 52 c. FFRB 52 H. Sample Complexity: Summary 52

    work page

  70. [70]

    Examples 56

    Improving SSRB? 54 I. Examples 56

    work page

  71. [71]

    SU(2) RB with Synthetic SPAM 56

    work page

  72. [72]

    SU(2) Character RB with Synthetic SPAM 57

    work page

  73. [73]

    SU(2) Rank-1 RB with Synthetic SPAM 58

    work page

  74. [74]

    Numerical Results 59 17 K

    Sample Complexity Comparison 58 J. Numerical Results 59 17 K. Connections to Other Protocols 61

    work page

  75. [75]

    Building Projectors 64 b

    Comparison to Pauli and Cli fford Benchmarking 61 a. Building Projectors 64 b. Sample Complexity 66

    work page

  76. [76]

    Native Operations 68

    From SU(2) to the Heisenberg-Weyl Group 67 L. Native Operations 68

    work page

  77. [77]

    We use tr for the operator trace and Tr for the superoperator trace

    Icosahedral Spin Code 69 Appendix A: Preliminaries We set ℏ = 1 unless otherwise noted. We use tr for the operator trace and Tr for the superoperator trace

    work page

  78. [78]

    We refer to the space of linear operators B(Cd), of which the set of density matrices forms a convex subset, as Hilbert-Schmidt space

    Hilbert-Schmidt Space Consider a d-level qudit with Hilbert space Cd. We refer to the space of linear operators B(Cd), of which the set of density matrices forms a convex subset, as Hilbert-Schmidt space. An element of B(Cd) can be represented as a d2-component vector (superket), while a linear operator on B(Cd) (superoperator) can be represented as a d2 ...

    work page

  79. [79]

    The (normalized) identity matrix should be the only traceful element: O0 = 1/ √ d, while tr(Oi) = 0 for i > 0

    work page

  80. [80]

    Orthonormality: tr( OiO j) = δi j

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Showing first 80 references.