Pith. sign in

REVIEW 2 major objections 4 minor 31 references

Preparing low-variance states using a distributed quantum algorithm

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper claims that a distributed filtering algorithm with one auxiliary qubit per device and postselected joint measurement reduces energy variance faster than a single-device filter, shown numerically for the Ising model with up to…

desk verdict A solid analysis of a distributed filtering protocol; the core advantage claim needs one more baseline (single-device with postselection) before I'd take the 'distributed advantage' at face value. read the letter →

arxiv 2501.13097 v2 pith:DGHO6MVA submitted 2025-01-22 quant-ph

Xiaoyu Liu , Benjamin F. Schiffer , Jordi Tura This is my paper · ORCID
classification quant-ph
keywords distributedquantumcomputingeigenstatepreparationfilteringiterativephaseestimationpostselectionenergyvarianceswaptestcyclicpermutation
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 claims that a quantum filtering algorithm split across several devices can prepare low-energy-variance states in fewer rounds than running the same filter on a single device. The protocol runs identical iterative phase-estimation filters in parallel on each device, connects the devices' single auxiliary qubits by a swap test, and postselects on the joint measurement outcome. Postselection on the symmetric subspace amplifies the dominant eigenstate population faster than a single-device filter, at the cost of restarting failed runs. The paper proves that the weak-postselection overhead plateaus, analyzes the induced energy bias, and gives numerical evidence that two or three devices beat one device on a non-integrable Ising model for up to six qubits. If correct, the protocol offers a lower-circuit-depth route to eigenstate preparation suited to near-term and early fault-tolerant hardware.

What carries the argument

The carrying object is the joint controlled measurement of the auxiliary qubits: a swap-test circuit (control-SWAP between each device's single auxiliary qubit) for two devices, generalized to a cyclic permutation test in which an $s$-level qudit controls a permutation operator $D$ on the $s$ auxiliary registers. A single Bell pair per round teleports Bob's auxiliary qubit to Alice so the control-SWAP is applied locally, and the generalization uses $s-1$ Bell pairs. The filtering operation itself is inherited from iterative quantum phase estimation: a round applies $e^{-iH t_k}$ controlled on the auxiliary qubit, and the randomness of large $t_k$ makes the eigenphases nearly independent and uniform, so a postselected successful round multiplies the eigenstate populations by independent factors and sharpens the energy distribution. The analysis tracks the first two moments of the energy distribution, proving the plateauing success rate of weak postselection, the exponential decay of the strong one, and the bounded energy bias of the final state.

What would settle it

Run the two-device protocol on the symmetric Ising Hamiltonian $H = \sum_{j=1}^{n-1} Z_j Z_{j+1}$ starting from $|+\rangle^{\otimes n}$ and measure the average energy variance versus round number under weak postselection; Proposition 1 predicts no filtering, because the correlated eigenphases of the $\pm\lambda$ pairs prevent single-population amplification. A direct numerical check that the variance does not decrease would confirm the claim's scope.

Watch

Extended reading notes

Core claim

Starting from identical product states on every device, the protocol acts each round $k$ with the same controlled time evolution $e^{-iH t_k}$ on each device for a randomly drawn large time $t_k$, then applies a control-SWAP between the devices' auxiliary qubits, and measures all auxiliary qubits. Postselection keeps runs whose auxiliary outcomes meet the weak criterion (top qubit $0$) or the strong criterion ($000$ or $011$). Because the eigenphases $\varphi_j^{(k)} = -t_k \lambda_j \pmod{2\pi}$ are effectively i.i.d. uniform whenever the Hamiltonian's eigenvalue ratios are (almost) irrational, each accepted round multiplies eigenstate amplitudes by independent random filtering factors, concentrating the population on the dominant eigenstate faster than the single-device filter. The central analytical results are closed-form expressions for the ensemble energy and variance: weak postselection has a success probability that plateaus at $\sum_j |c_j^{(0)}|^4$ with overhead linear in the round number, strong postselection has exponentially decaying success but faster variance reduction, and the expected energy converges to $(\sum_j \lambda_j |c_j^{(0)}|^4)/(\sum_j |c_j^{(0)}|^4)$, bounding the postselection bias by at most $0.1716|\mu|$ for Gaussian spectra. The construction extends to $s \ge 2$ devices through a cyclic permutation test on an $s$-level qudit, with numerical evidence that three devices reduce variance faster than two.

Load-bearing premise

The protocol filters only when the Hamiltonian's eigenvalue ratios are (almost) irrational, so that random evolution times scramble the eigenphases into effectively independent uniform variables; for a Hamiltonian with a symmetric spectrum such as $H = \sum_j Z_j Z_{j+1}$ the algorithm fails to converge.

Editorial extensions

If this is right

  • For any Hamiltonian with generic irrational-ratio spectrum, a two-device filter with weak postselection reaches a target energy variance in fewer rounds than the single-device filter, with a success probability that stabilizes instead of decaying.
  • Strong postselection converges even faster but has exponentially decaying success, so it fits short round budgets; the authors explicitly suggest a hybrid protocol that starts strong and switches to weak postselection.
  • Each successful run outputs $s$ identical filtered states, so the protocol is a heralded way to prepare multiple copies of an approximate eigenstate in parallel.
  • The postselection-induced energy bias is bounded and explicitly computable from the initial-state populations, so the final energy can be predicted and corrected.
  • Distributed filtering trades circuit depth for entanglement links and restart overhead; the paper quantifies this as the per-round Bell-pair cost and the number of controlled time evolutions, with per-state costs shown in the resource-counting figures.

Reading between the lines

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

  • The plateau value $\sum_j |c_j^{(0)}|^4$ of the weak-postselection success rate is the purity of the initial state in the eigenbasis; for local Hamiltonians the authors show it scales as $A e^{B/n}$, suggesting the protocol remains practical for system sizes beyond the numerics shown.
  • Because the circuit is a symmetrization onto the symmetric subspace of the two registers, one could test whether replacing the swap with other symmetric projections changes the filtering rate or the final bias.
  • After many rounds the two devices hold nearly identical eigenstates, so a direct swap-test comparison of the two output registers could certify convergence without measuring the energy.
  • The bounded bias formula implies that for variational or phase-estimation pipelines the filtered state's energy must be corrected by $\sum_j \lambda_j |c_j|^4 / \sum_j |c_j|^4$, which requires knowing the initial-state populations; estimating those populations is a natural extension.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript proposes a distributed quantum filtering algorithm for preparing low-variance eigenstate approximations. Two (or more) devices each hold a copy of the same initial state and run iterative-phase-estimation-style Hadamard-test filters with synchronized random evolution times; a joint measurement on the ancillary qubits, followed by weak or strong postselection, heralds successful runs. The central claim is that this distributed protocol reduces the energy variance faster per iteration than a single-device implementation. Analytical formulas are provided for the success rates, the energy bias, and the eigenstate spread, with a conjectured extension from weak to strong postselection. Numerical simulations on a non-integrable Ising model with n=4,5,6 and up to three devices support the variance reduction, and resource costs (controlled time evolutions and Bell pairs) are analyzed via cumulative success rates and Monte Carlo estimates.

Significance. If properly benchmarked, the protocol would be a useful tool for near-term distributed quantum processors, trading circuit depth for postselection overhead and entanglement links. The paper is transparent about several limitations: the generic-spectrum assumption underlying the random-phase randomization, the exponential overhead of strong postselection, and the conjectural status of the strong-postselection analytic formulas. It provides reproducible numerical procedures, error-bar plots in Appendix E, and a resource-cost analysis. However, the main comparative claim is currently confounded with the effect of postselection because the single-device baseline does not postselect. This is a significant gap that must be addressed before the central claim can be accepted.

major comments (2)
  1. [Section 2.2, Fig. 2; Appendix A (Algorithm 3)] The comparison between the distributed and single-device algorithms is not controlled for postselection. The single-device filter in Algorithm 3 accepts every measurement outcome (100% success, as noted in the Fig. 2 caption), whereas the distributed algorithms discard failing runs and report averages conditioned on surviving runs. The paper's own Fig. 2 shows that the distributed algorithm without postselection performs worse than the single-device case, confirming that the observed variance reduction is driven by postselection rather than by the distributed structure. A single-device Hadamard-test filter that postselects on the ancilla outcome |0> (applying (I+U)/2 per round and restarting on failure) is the natural equal-conditioning baseline; such a baseline is not benchmarked anywhere in the manuscript. Without it, the abstract's claim that the distributed algorithm 'reduces the energy variance faster compared to single-device implementations' is not established.
  2. [Section 2.4, Eq. (12); Appendix B.1.3, Conjecture 3] The analytical energy-bias and eigenstate-spread formulas for the strong-postselection case (Eqs. (8), (12), and the associated bounds) rely on the approximation E[N/D] ≈ E[N]/E[D] stated as Conjecture 3. The authors explicitly show in Eq. (84) that the variance of the denominator g(φ) grows faster than the square of its mean for large K, so the approximation is not justified by the same argument as Proposition 2. The main text presents Eq. (12) with 'one can show' without indicating that, for strong postselection, this is a numerically supported conjecture rather than a proven result. This should either be flagged at the point of use, proved, or replaced by a direct asymptotic argument for the K→∞ limit.
minor comments (4)
  1. [Abstract and Section 2.1 (Proposition 1)] The algorithm's filtering mechanism depends on the eigenphases being effectively i.i.d. uniform, which requires (almost) irrational eigenvalue ratios; for Hamiltonians with symmetric spectra (e.g., H = Σ Z_j Z_{j+1}) the protocol fails, as acknowledged in Appendix A.2. The numerical support is for a single non-integrable Ising model, yet the abstract and introduction state the variance-reduction claim without this 'generic spectrum' qualification. The scope should be stated explicitly.
  2. [Figure 2 caption] The caption reads 'The cumulative success rate after postselection at each iteration are shown in (b)'; 'are' should be 'is'.
  3. [Algorithm 1 and Figure 1(b)] The notation for the measurement outcome is inconsistent: the pseudocode writes |m_0 m_A m_B>, while Figure 1 labels the top qubit as 'aux', and the superscript k on the outcomes is not defined. Please align the notation.
  4. [Section 4, Discussion] The proposed hybrid protocol that starts with strong postselection and transitions to weak postselection is not numerically demonstrated. Either add a supporting simulation or present it only as a qualitative suggestion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variance-reduction claim is established by direct simulation and Born-rule analytics, with no prediction reducible to its inputs.

full rationale

The paper's central claim that the distributed filtering algorithm reduces energy variance faster than the single-device implementation is supported by numerical simulation of the actual circuits (Figs. 2 and 5c) and by analytical expressions derived from the Born rule (e.g., Eqs. (54), (78), and Appendix B.2). No fitted parameter is renamed as a prediction: the input state, Hamiltonian, and random evolution times are specified independently, and the reported variances and success rates are computed from the resulting measurement statistics. The analytical approximations, such as Proposition 2 and Conjecture 3, are explicitly identified as approximations or conjectures and are tested numerically rather than being used to define the outcome. The paper does cite prior work by co-authors ([16] for the two-device circuit and [29] for cyclic permutation tests), but these citations supply circuit ingredients and a peripheral asymptotic remark, not the load-bearing evidence for the main variance-reduction result. The one substantive caveat is that the distributed protocol uses postselection while the single-device baseline does not, so the comparison may reflect selection bias; however, this is a benchmarking concern, not a circularity, because no claim is equivalent to its inputs by construction. Therefore, the derivation chain is self-contained and no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivations rest on the phase-randomization assumption (Proposition 1), the Gaussian density-of-states model, and the Gaussian population model, plus two analytic approximations (Proposition 2 and the unproven Conjecture 3). No new physical entities are introduced. The free parameters are the time-sampling interval and, for the indicative extrapolation, the fitted exponential decay rates.

free parameters (2)
  • Random time sampling interval T = not specified
    The algorithm requires t_k sampled from (0,T] with T sufficiently large so that phases are approximately uniform; the exact distribution or range used in the numerics is not reported, which affects reproducibility and the validity of the i.i.d.-phase assumption.
  • Exponential decay rate eta (extrapolation only) = 0.178 (single), 0.277 (2-device weak), 0.386 (3-device weak), 0.376 (2-device strong), 0.490 (3-device strong)
    Fitted to the average-variance curves in Fig. 5(c) over k in [4,12] to conjecture scaling behavior; the authors label this extrapolation as only indicative, so it is not load-bearing for the central claim.
assumptions (6)
  • domain assumption Eigenphases phi_j^(k) = -t_k lambda_j mod 2pi behave as i.i.d. uniform random variables (Proposition 1)
    Underpins the filtering analysis; requires large sampling interval T and (almost) irrational eigenvalue ratios; explicitly fails for symmetric spectra such as H = sum Z_j Z_{j+1}.
  • domain assumption Hamiltonian has a Gaussian density of states (Eq. 64)
    Used for the continuous-energy bias and eigenvalue-spread bounds; valid for local, traceless Hamiltonians in the thermodynamic limit [5].
  • domain assumption The initial product state has eigenstate populations with a Gaussian shape in energy (Eq. 65)
    Used to evaluate sum |c_j|^4 and derive scaling claims; the cited [11] concerns low-variance states, not generic product states, and the normalization in Eq. (68) appears inconsistent with the 2^n-dimensional Hilbert space.
  • ad hoc to paper E[numerator/denominator] approx E[numerator]/E[denominator] for the weak-postselection integrals (Proposition 2)
    Justified by showing the variance of the denominator is small relative to its mean for weak postselection; it is an approximation, not an identity.
  • ad hoc to paper Same quotient-of-integrals approximation for strong postselection (Conjecture 3)
    Explicitly unproven; the authors note the Proposition 2 argument does not apply, yet the analytic energy and variance formulas for strong postselection use it.
  • standard math Control-SWAP and cyclic permutation test circuit identities yield the stated Kraus operators
    Standard swap-test and permutation-test circuit identities [24,27-29]; used to derive the measurement probabilities and success rates.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Preparing low-variance states using a distributed quantum algorithm." pith.science (2026). https://pith.science/paper/DGHO6MVA

@misc{pith2026250113097,
  author       = {Pith},
  title        = {Pith review of: Preparing low-variance states using a distributed quantum algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGHO6MVA}},
  note         = {Machine review of arXiv:2501.13097}
}
read the original abstract

Quantum computers are a highly promising tool for efficiently simulating quantum many-body systems. The preparation of their eigenstates is of particular interest and can be addressed, e.g., by quantum phase estimation algorithms. The routine then acts as an effective filtering operation, reducing the energy variance of the initial state. In this work, we present a distributed quantum algorithm inspired by iterative phase estimation to prepare low-variance states. Our method uses a single auxiliary qubit per quantum device, which controls its dynamics, and a postselection strategy for a joint quantum measurement on such auxiliary qubits. In the multi-device case, the result of this measurement heralds the successful runs of the protocol. This allows us to demonstrate that our distributed algorithm reduces the energy variance faster compared to single-device implementations, thereby highlighting the potential of distributed algorithms for near-term and early fault-tolerant devices.

Figures

Figures reproduced from arXiv: 2501.13097 by the authors.

Figure 1
Figure 1. (a) Illustration of a filtering operation. Consider a local Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) Average variance of the pure state ensemble per iteration [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Eigenvalue spread V (k) for n = 6 and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Average energy of the pure state ensemble [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: (a) Proposed distributed filtering algorithm across [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Circuit for single-device filtering algorithm. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The quantum circuit per iteration to realize the distributed filtering protocol with shared Bell pairs and [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: Empirical average number of control time evolutions (control- [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]
Figure 9
Figure 9. Figure 9: Average variance and energy of the pure state ensemble per iteration [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]
Figure 10
Figure 10. Figure 10: The surviving rounds of repetitions after postselection at each iteration, for all four numerical instances [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]
Figure 11
Figure 11. Figure 11: The numerical experiment results with the same inputs in Fig. [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]
Figure 12
Figure 12. Figure 12: The surviving rounds of repetitions after postselection at each iteration for the numerical instances in [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 26 canonical work pages

  1. [1]

    Quantum Computation and Quantum In- formation: 10th Anniversary Edition

    Michael A. Nielsen and Isaac L. Chuang. “Quantum Computation and Quantum In- formation: 10th Anniversary Edition”. Cam- bridge University Press. (2010). url:https: //doi.org/10.1017/CBO9780511976667

    doi:10.1017/cbo9780511976667 2010

  2. [2]

    Quantum measurements and the Abelian Stabilizer Problem

    A. Yu Kitaev. “Quantum measurements and the Abelian Stabilizer Problem” (1995). arXiv:quant-ph/9511026

    arXiv 1995

  3. [3]

    Quantum phase es- timation of multiple eigenvalues for small- scale (noisy) experiments

    Thomas E. O’Brien, Brian Tarasinski, and Barbara M. Terhal. “Quantum phase es- timation of multiple eigenvalues for small- scale (noisy) experiments”. New Journal of Physics 21, 023022 (2019)

    work page 2019

  4. [4]

    Heisenberg-limited quantum phase estimation of multiple eigen- values with few control qubits

    Alicja Dutkiewicz, Barbara M. Terhal, and Thomas E. O’Brien. “Heisenberg-limited quantum phase estimation of multiple eigen- values with few control qubits”. Quantum6, 830 (2022)

    work page 2022

  5. [5]

    Spectral Densities and Partition Functions of Modular Quantum Systems as Derived from a Central Limit Theorem

    Michael Hartmann, Günter Mahler, and Or- twin Hess. “Spectral Densities and Partition Functions of Modular Quantum Systems as Derived from a Central Limit Theorem”. Journal of Statistical Physics 119, 1139– 1151 (2005)

    work page 2005

  6. [6]

    Quan- tumAlgorithmProvidingExponentialSpeed Increase for Finding Eigenvalues and Eigen- vectors

    Daniel S. Abrams and Seth Lloyd. “Quan- tumAlgorithmProvidingExponentialSpeed Increase for Finding Eigenvalues and Eigen- vectors”. Physical Review Letters83, 5162– 5165 (1999)

    work page 1999

  7. [7]

    Prepar- ing Ground States of Quantum Many-Body Systems on a Quantum Computer

    David Poulin and Pawel Wocjan. “Prepar- ing Ground States of Quantum Many-Body Systems on a Quantum Computer”. Physical Review Letters102, 130503 (2009)

    work page 2009

  8. [8]

    Faster ground state preparation and high- precision ground energy estimation with fewer qubits

    Yimin Ge, Jordi Tura, and J. Ignacio Cirac. “Faster ground state preparation and high- precision ground energy estimation with fewer qubits”. Journal of Mathematical Physics 60, 022202 (2019)

    work page 2019

Show all 31 references
  1. [9]

    Algorithms for Quantum Simula- tion at Finite Energies

    Sirui Lu, Mari Carmen Bañuls, and J. Igna- cio Cirac. “Algorithms for Quantum Simula- tion at Finite Energies”. PRX Quantum2, 020321 (2021)

    2021

  2. [10]

    Efficient Quantum Algo- rithm for Filtering Product States

    Reinis Irmejs, Mari Carmen Bañuls, and J. Ignacio Cirac. “Efficient Quantum Algo- rithm for Filtering Product States”. Quan- tum 8, 1389 (2024)

    2024

  3. [11]

    Matrix product state approximations to quantum states of low energy variance

    Kshiti Sneh Rai, J. Ignacio Cirac, and Ál- varo M. Alhambra. “Matrix product state approximations to quantum states of low energy variance”. Quantum8, 1401 (2024)

    2024

  4. [12]

    Arbitrary accuracyiterativequantumphaseestimation algorithm using a single ancillary qubit: A two-qubit benchmark

    Miroslav Dobšíček, Göran Johansson, Vitaly Shumeiko, and Göran Wendin. “Arbitrary accuracyiterativequantumphaseestimation algorithm using a single ancillary qubit: A two-qubit benchmark”. Physical Review A 76, 030306 (2007)

    2007

  5. [13]

    Demon-likealgorithmicquantumcool- ing and its realization with quantum optics

    Jin-Shi Xu, Man-Hong Yung, Xiao-Ye Xu, Sergio Boixo, Zheng-Wei Zhou, Chuan-Feng Li, Alán Aspuru-Guzik, and Guang-Can Guo. “Demon-likealgorithmicquantumcool- ing and its realization with quantum optics”. Nature Photonics8, 113–118 (2014)

    2014

  6. [14]

    Resource-frugal Hamiltonian eigen- state preparation via repeated quantum phase estimation measurements

    Richard Meister and Simon C. Ben- jamin. “Resource-frugal Hamiltonian eigen- state preparation via repeated quantum phase estimation measurements” (2022). arXiv:2212.00846

    2022 arXiv

  7. [15]

    Quantum algorithm for spectral projection by measur- ing an ancilla iteratively

    YanzhuChenandTzu-ChiehWei. “Quantum algorithm for spectral projection by measur- ing an ancilla iteratively”. Physical Review A 101, 032339 (2020)

    2020

  8. [16]

    Quan- tum eigenstate preparation assisted by a coherent link

    Benjamin F. Schiffer and Jordi Tura. “Quan- tum eigenstate preparation assisted by a coherent link”. Physical Review A 111, 012445 (2025). Accepted inQuantum 2025-08-12, click title to verify. Published under CC-BY 4.0. 11

    2025

  9. [17]

    Demonstration of the rodeo algorithm on a quantum computer

    Zhengrong Qian, Jacob Watkins, Gabriel Given, Joey Bonitati, Kenneth Choi, and Dean Lee. “Demonstration of the rodeo algorithm on a quantum computer”. The EuropeanPhysicalJournalA 60, 151(2024)

    2024

  10. [18]

    Rodeo Algorithm for Quantum Com- puting

    Kenneth Choi, Dean Lee, Joey Boni- tati, Zhengrong Qian, and Jacob Watkins. “Rodeo Algorithm for Quantum Com- puting”. Physical Review Letters 127, 040505 (2021)

    2021

  11. [19]

    A polynomial quantum algorithm for approximating the Jones polynomial

    Dorit Aharonov, Vaughan Jones, and Zeph Landau. “A polynomial quantum algorithm for approximating the Jones polynomial”. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing. Pages 427–436. STOC ’06New York, NY, USA (2006). Association for Computing ...

    2006

  12. [20]

    Microwave Quan- tum Link between Superconducting Circuits Housed in Spatially Separated Cryogenic Systems

    P. Magnard, S. Storz, P. Kurpiers, J. Schär, F. Marxer, J. Lütolf, T. Walter, J.-C. Besse, M. Gabureac, K. Reuer, A. Akin, B. Royer, A. Blais, and A. Wallraff. “Microwave Quan- tum Link between Superconducting Circuits Housed in Spatially Separated Cryogenic Systems”. Physical...

    2020

  13. [21]

    Coherent Spin-Spin Coupling Mediated by Virtual Microwave Photons

    Patrick Harvey-Collard, Jurgen Dijkema, Guoji Zheng, Amir Sammak, Giordano Scappucci, and Lieven M. K. Vandersypen. “Coherent Spin-Spin Coupling Mediated by Virtual Microwave Photons”. Physical Re- view X12, 021026 (2022)

    2022

  14. [22]

    Distributed quantum computation over noisy channels

    J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello. “Distributed quantum computation over noisy channels”. Physical Review A59, 4249–4254 (1999)

    1999

  15. [23]

    Dis- tributed Quantum Computing

    Harry Buhrman and Hein Röhrig. “Dis- tributed Quantum Computing”. In Branislav Rovan and Peter Vojtáš, editors, Mathe- matical Foundations of Computer Science

  16. [24]

    Stabilization of Quantum Computations by Symmetriza- tion

    Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. “Stabilization of Quantum Computations by Symmetriza- tion”. SIAM Journal on Computing 26, 1541–1557 (1997)

    1997

  17. [25]

    Quantum Fingerprinting

    Harry Buhrman, Richard Cleve, John Wa- trous, and Ronald de Wolf. “Quantum Fingerprinting”. Physical Review Letters87, 167902 (2001)

    2001

  18. [26]

    Zur Quantenmechanik der Stoßvorgänge

    Max Born. “Zur Quantenmechanik der Stoßvorgänge”. Zeitschrift für Physik 37, 863–867 (1926)

    1926

  19. [27]

    The efficiency of quantum identity testing of multiple states

    Masaru Kada, Harumichi Nishimura, and Tomoyuki Yamakami. “The efficiency of quantum identity testing of multiple states”. Journal of Physics A: Mathematical and Theoretical 41, 395309 (2008)

    2008

  20. [28]

    Per- mutation tests for quantum state iden- tity

    Harry Buhrman, Dmitry Grinko, Philip Ver- duyn Lunel, and Jordi Weggemans. “Per- mutation tests for quantum state iden- tity” (2024). arXiv:2405.09626

    2024 arXiv

  21. [29]

    Generalized concen- tratable entanglement via parallelized per- mutation tests

    Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, and Jordi Tura. “Generalized concen- tratable entanglement via parallelized per- mutation tests”. Physical Review Research 7, L032022 (2025)

    2025

  22. [30]

    Über die Gleichverteilung von Zahlen mod. Eins

    Hermann Weyl. “Über die Gleichverteilung von Zahlen mod. Eins”. Mathematische Annalen 77, 313–352 (1916). Accepted inQuantum 2025-08-12, click title to verify. Published under CC-BY 4.0. 12 A Single-device filter We start from stating the single-device filter algorithm [6, 13,...

    1916

  23. [2003]

    Berlin, Heidelberg (2003)

    Pages 1–20. Berlin, Heidelberg (2003). Springer

    2003

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.