Pith. sign in

REVIEW 4 major objections 5 minor 48 references

Adaptive-depth randomized measurement for fermionic observables

T0 review · 4 major / 5 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 This paper introduces an adaptive-depth fermionic classical shadow protocol and claims that the required depth for approximating a fermionic observable $H$ scales as $\max\{d_{\mathrm{int}}(H)^2/\log n,\ d_{\mathrm{int}}(H)\}$ while…

desk verdict Good new idea about adaptive-depth fermionic shadows, but the sample-complexity guarantee is average-case and rests on unproven sign and factorization assumptions; the abstract overclaims. read the letter →

arxiv 2501.09574 v1 pith:ZWXO6JL7 submitted 2025-01-16 quant-ph

Kaiming Bian , Bujiao Wu This is my paper · ORCID
classification quant-ph
keywords fermionicclassicalshadowsadaptive-depthcircuitsmatchgateMajoranaobservablesinteractiondistanceshadowestimationtensornetworkcontractionsymmetrylazyrandomwalk
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

Fermionic classical shadows estimate expectation values of fermionic observables by applying random matchgate circuits, but the standard protocol needs polynomial-depth circuits. This paper proposes an adaptive-depth version (ADFCS) in which the depth of the brickwork matchgate circuit is chosen from the observable itself, and claims that a fermionic observable $H$ can be approximated with depth $\Theta(\max\{d_{\mathrm{int}}(H)^2/\log n,\ d_{\mathrm{int}}(H)\})$, where $d_{\mathrm{int}}(H)$ is the interaction distance of $H$, while keeping the same order of sample complexity as the full fermionic classical shadow protocol. The practical point is that local and short-ranged fermionic Hamiltonians, such as the Kitaev chain treated in the paper, can then be measured with constant or logarithmic-depth circuits rather than polynomial-depth ones.

What carries the argument

The load-bearing object is the tensor-network representation of the $d$-depth matchgate shadow channel. Each independent two-qubit matchgate is replaced by a fixed fourth-order tensor obtained from the second-moment twirl of the matchgate group, and connecting these tensors in the brickwork architecture produces a tensor network whose contraction gives $\alpha_{S,d} = \langle\langle 0,0|\mathcal{C}|\gamma_S,\gamma_S\rangle\rangle$. For two-local Majorana strings the contraction is then shown to evolve as a symmetry lazy random walk on a polynomial space; the walk's transition probabilities yield a closed-form Poisson-sum expression for the dominant part $\alpha^L_{S,d}$, and an auxiliary remainder bound relates $\alpha^L_{S,d}$ to $\alpha_{S,d}$. The adaptive depth rule follows from requiring $\alpha^L_{S,d}=\Omega(1/\mathrm{poly}(n))$.

What would settle it

Prepare a fixed computational-basis state or an adversarially chosen state and run ADFCS at depth $d^*$, measuring the empirical variance of $\mathrm{Tr}(\hat\rho\gamma_S)$; if the variance exceeds $1/\alpha_{S,d}$ by more than a constant factor, the sample-complexity claim as stated fails. A second check is to compute the remainder coefficients $R_{i,j}(\mu,\nu,t)$ for an adjacent Majorana pair at shallow depth: a positive diagonal remainder or a negative off-diagonal remainder violates the assumption behind the proof of the depth rule.

Watch

Extended reading notes

Core claim

The paper's central claim is that the depth needed for a random matchgate measurement is set by the observable's interaction distance, not by the system size. For a $k$-local Majorana string $\gamma_S$ with constant $k$ and interaction distance $d_{\mathrm{int}}(S)=O(\log n)$, a random circuit of depth $d=\Theta(\log n)$ already makes the shadow eigenvalue $\alpha_{S,d}$ polynomially large, i.e. $\alpha_{S,d}=\Omega(1/\mathrm{poly}(n))$; more generally the required depth is $d^*=\Theta(\max\{d_{\mathrm{int}}(S)^2/\log n,\ d_{\mathrm{int}}(S)\})$. Because the variance of the estimation is bounded by $1/\alpha_{S,d}$, this gives the same order of sample complexity as the original FCS protocol while using far shallower circuits.

Load-bearing premise

The depth guarantee and the matching sample complexity rest on the variance bound $\mathrm{Var}[v]\le 1/\alpha_{S,d}$, which is derived by averaging the unknown state over the uniform (Haar) distribution; a fixed adversarial state could require more samples than the stated order.

Editorial extensions

If this is right

  • A fermionic Hamiltonian whose terms have logarithmic interaction distance can be measured with $O(\log n)$-depth random matchgate circuits and polynomial sample complexity.
  • Shallow-depth estimation preserves the unbiasedness of the classical-shadow estimator whenever $\alpha_{S,d}\neq 0$.
  • For a 10-qubit Kitaev chain, depth $d=3$ already suffices to reach estimation errors comparable with the full FCS protocol.
  • For observables with interaction distance $d_{\mathrm{int}}=\omega(\log n)$, the required depth interpolates between logarithmic and polynomial scaling as the observable becomes more nonlocal.
  • The sample-complexity bound retains the form $O(1/(\alpha_{S,d}\epsilon^2))$, matching the original FCS order when $\alpha_{S,d}$ is polynomially large.

Reading between the lines

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

  • The same tensor-network-to-random-walk reduction could be re-derived for other circuit geometries, such as non-brickwork or open-boundary matchgate layouts, changing only the kernel of the walk and hence the constants in the depth rule.
  • A state-dependent version of the variance bound would be needed before using ADFCS for adversarial or worst-case inputs; the current protocol's practical guarantee is tied to the Haar-averaged variance.
  • The per-term depth selection suggests a practical variational-quantum-eigensolver-style measurement strategy: choose a different measurement depth for each Hamiltonian term according to its own $d_{\mathrm{int}}$, rather than one global depth.
  • The closed-form Poisson-sum formula could be turned into a lookup table or analytic depth selector, removing the numerical-fitting step and giving a fully deterministic protocol.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The paper's central claim is that the depth needed for a random matchgate measurement is set by the observable's interaction distance, not by the system size. For a $k$-local Majorana string $\gamma_S$ with constant $k$ and interaction distance $d_{\mathrm{int}}(S)=O(\log n)$, a random circuit of depth $d=\Theta(\log n)$ already makes the shadow eigenvalue $\alpha_{S,d}$ polynomially large, i.e. $

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces an adaptive-depth fermionic classical shadow (ADFCS) protocol. The protocol uses d-depth brickwork random matchgate circuits and adapts the depth to the observable H via its interaction distance d_int(H). The central claim is that, for k-local Majorana observables with d_int(H)=O(log n), depth d* = Θ(max{d_int(H)^2/log n, d_int(H)}) suffices to keep the key parameter α_{S,d} at order Ω(1/poly(n)), while maintaining the same order of sample complexity as the full matchgate FCS protocol. The analysis maps the tensor-network contraction for α_{S,d} to a random walk on a polynomial space, derives an analytical Poisson-summation formula for the 2-local case, and proposes a factorization ansatz for the k-local case. Numerical experiments on random states and on the Kitaev chain Hamiltonian support the depth-vs-accuracy tradeoff.

Significance. If the main claim holds, the paper addresses a concrete open question: whether a shallow-depth FCS protocol can match the sample complexity of the full-depth FCS protocol for a meaningful class of fermionic observables. The tensor-network-to-random-walk mapping and the closed-form Poisson approximation for α_{S,d} are valuable technical contributions, and the Kitaev-chain demonstration is a nice practical illustration. However, the central result is conditional: the 2-local lower bound rests on an unproven sign assumption, the k-local extension rests on a numerically validated ansatz rather than a proof, and the sample-complexity statement is derived for Haar-averaged states. These gaps limit the strength of the conclusions that can be drawn from the paper in its current form.

major comments (4)
  1. [Appendix G, Theorem 1 and Eq. (G30)] The 2-local lower bound α_{S,d} ≥ (47/72) max_{d'≤d} α^L_{S,d'} is derived under the sign-definiteness assumption R_{ij}(μ,μ,t) ≤ 0 and R_{ij}(μ,ν,t) ≥ 0 for μ≠ν, which is only supported by the numerical sign plot in Fig. 8. If this assumption fails for some S,d in the target regime, Lemma 7 no longer applies and the depth formula d* = Θ(max{d_int²/log n, d_int}) does not imply the claimed polynomial sample complexity. Since Theorem 1 is the main theoretical basis for the 2-local depth scaling, this conditional nature is load-bearing.
  2. [Section III.D and Appendix H, Eq. (22)] The k-local extension relies on the factorization ansatz α'_{S,d} = 1/(k-1)!! (3n/2)^{k/2} ... ∏ α^L_{{i,j},d}, which is validated only numerically for d > 2 log n. The proof sketch in Appendix H preserves branches with non-increasing d_near and discards the others, asserting that the remaining coefficients sum to at least 1/36^{k d_int/2}, but it does not show that the discarded branches cannot contribute to the contraction with ⟨⟨0,0|. Consequently, the statement that d = Θ(log n) suffices for d_int = O(log n) and constant k is not established; a rigorous bound on |α_{S,d} - α'_{S,d}| is needed.
  3. [Appendix C, Eq. (11)] The variance bound Var[v] ≤ 1/α_{S,d} is derived by averaging the state ρ over the Haar distribution, i.e., E_ρ[ρ] = I/2^n. For a fixed, adversarially chosen ρ, the variance can be larger, yet the abstract and conclusion present the O(1/(α_{S,d} ε²)) sample complexity as a general statement. The numerics in Section IV use only random states and therefore do not test worst-case behavior. This is load-bearing because the depth scaling alone does not imply efficient estimation for arbitrary input states.
  4. [Section III.C, Eq. (19) and Appendix F, Lemma 4] The definitions of the parameters in Eq. (19) disagree with those in Lemma 4: Eq. (19) uses a = |⌊(i-1)/4⌋ - ⌊(j-1)/4⌋| and b = ⌊(i-1)/4⌋ + ⌊(j-1)/4⌋ + 1, whereas Lemma 4 uses a = |i-j| and b = i+j-1. The subsequent Lemma 5, which connects the order of α^L_{S,d} to d_int = |i-j|, is based on the Lemma 4 definitions. The main text does not reconcile these two expressions, so it is unclear which formula is being used to derive the depth bound d* = Θ(max{d_int²/log n, d_int}).
minor comments (5)
  1. [Section III.A] Typo: 'bouned' should be 'bounded'.
  2. [Section IV and Fig. 4] The Hamiltonian is misspelled 'Kiteav' in several places; it should be 'Kitaev'.
  3. [Eq. (22)] The word 'production' should be 'product', and the notation for the product over pairs (i,j) ∈ Λ should be defined explicitly.
  4. [Fig. 7] The figure is hard to read: the edge probabilities are not clearly associated with the graph, and the printed numbers appear garbled. Please redraw with explicit labels.
  5. [Appendix F, Lemma 4] The error term in Eq. (19) is written O(e^{-π² d}), while Lemma 4 obtains O(e^{-π² t/2}) with d = 2t+1; the translation between these expressions and the relevant regime of t should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the depth scaling is derived from an independent random-walk analysis, and the numerical fitting is not used to set the target result.

full rationale

Walking the derivation chain: (i) the paper defines αS,d through the shadow channel and bounds the estimator variance by Var[v] ≤ 1/αS,d; (ii) αS,d is expressed as a tensor network and contracted exactly for small systems; (iii) for 2-local Majorana strings, the contraction is mapped to a random walk in a polynomial space, and the lazy-symmetry-random-walk component αL is evaluated by the Poisson-summation formula in Eq. (19); (iv) Lemma 5 lower-bounds αL at d* = Θ(max{dint^2/log n, dint}); and (v) Appendix G gives a conditional theorem that αS,d ≥ (47/72) max_{d'≤d} αL under the explicit sign assumption (G30). None of these steps defines the depth formula in terms of the quantity it is supposed to predict. No parameter is fitted to the final sample-complexity claim: the numerical fitting is used only as supporting evidence that αL dominates αS,d and that the k-local ansatz α′ approximates αS,d. The central 2-local depth formula rests on Lemma 4, Lemma 5, and Lemma 7, not on a fitted constant. The sign assumption (G30) and the average-over-ρ variance bound are genuine rigor gaps that affect the strength of the theorem, but they are not circularity: a conditional theorem is not a reduction of the conclusion to its premises. The k-local extension is partly numerical, but it is an empirically validated ansatz, not a fitted parameter disguised as a prediction. The self-citations in the reference list (Refs. [19] and [23]) concern CNOT synthesis and superconducting-qubit experiments and are not load-bearing for the ADFCS claim. External citations such as Ref. [34] for the SLRW propagation equation provide independent mathematical support. The paper therefore does not reduce, by its own equations or by self-citation, to its own inputs; its main weaknesses are matters of proof completeness and worst-case guarantees rather than circularity.

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

The central claim depends on three non-standard assumptions: the average-case variance bound, the sign-definiteness of the random-walk remainder, and the factorization ansatz for k-local observables. The first two are acknowledged in the paper (though underemphasized), while the third is presented as a numerical observation. No new physical entities are introduced.

assumptions (4)
  • domain assumption The variance of the ADFCS estimator is bounded by 1/αS,d for any state, as stated in Eq. (11).
    Appendix C derives the bound by averaging the input state ρ over the Haar distribution, so for a fixed state the variance can be larger. This is load-bearing for the sample complexity claim.
  • ad hoc to paper The true random walk on the polynomial space has less probability on diagonal sites and more on off-diagonal sites than the symmetry lazy random walk: R_{ij}(μ,μ,t)≤0 and R_{ij}(μ,ν,t)≥0 for μ≠ν.
    Theorem 1 (Appendix G) states the bound on αS,d/αL_S,d under this assumption; only numerical evidence for small systems (Fig. 8) is provided.
  • ad hoc to paper For k-local Majorana strings, the random walk probability factorizes into products of independent pair random walks, giving α′_S,d in Eq. (22) as a proxy for αS,d.
    Section III.D states 'numerical experiments indicate that the proposed expression α′S,d closely approximates αS,d for depths d > 2 log(n)'; no proof is given.
  • standard math Poisson summation formula and the exact propagator for the symmetry lazy random walk from Ref. [34] are applicable to the derived sums.
    Used in Lemma 4 and Appendix F to estimate αL_S,d; these are standard mathematical tools.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Adaptive-depth randomized measurement for fermionic observables." pith.science (2026). https://pith.science/paper/ZWXO6JL7

@misc{pith2026250109574,
  author       = {Pith},
  title        = {Pith review of: Adaptive-depth randomized measurement for fermionic observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZWXO6JL7}},
  note         = {Machine review of arXiv:2501.09574}
}
abstract

Accurate estimation of fermionic observables is essential for advancing quantum physics and chemistry. The fermionic classical shadow (FCS) method offers an efficient framework for estimating these observables without requiring a transformation into a Pauli basis. However, the random matchgate circuits in FCS require polynomial-depth circuits with a brickwork structure, which presents significant challenges for near-term quantum devices with limited computational resources. To address this limitation, we introduce an adaptive-depth fermionic classical shadow (ADFCS) protocol designed to reduce the circuit depth while maintaining the estimation accuracy and the order of sample complexity. Through theoretical analysis and numerical fitting, we establish that the required depth for approximating a fermionic observable $H$ scales as $\max\{d^2_{\text{int}}(H)/\log n, d_{\text{int}}(H)\}$ where $d_{\text{int}}$ is the interaction distance of $H$. We demonstrate the effectiveness of the ADFCS protocol through numerical experiments, which show similar accuracy to the traditional FCS method while requiring significantly fewer resources. Additionally, we apply ADFCS to compute the expectation value of the Kitaev chain Hamiltonian, further validating its performance in practical scenarios. Our findings suggest that ADFCS can enable more efficient quantum simulations, reducing circuit depth while preserving the fidelity of quantum state estimations, offering a viable solution for near-term quantum devices.

Figures

Figures reproduced from arXiv: 2501.09574 by the authors.

Figure 1
Figure 1. FIG. 1: Illustration of Adaptive-Depth Fermionic Classical Shadow (ADFCS). (a) The interaction distance of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Representation of the tensor contraction of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Estimation errors with increasing numbers of measurements for FCS and ADFCS protocols at different [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Application of ADFCS to the Kitaev chain Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Illustration for the odd-layer tensor [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Illustration for the isometric mapping to the polynomial space. Γ [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Illustration of lazy symmetry random walk. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Numerical evaluation of [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Values of [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 35 canonical work pages

  1. [1]

    Two soluble models of an antiferromagnetic chain

    Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407–466, 1961

    1961

  2. [2]

    Kosterlitz-thouless transition and short range spatial correlations in an extended hubbard model

    Qimiao Si and J Lleweilun Smith. Kosterlitz-thouless transition and short range spatial correlations in an extended hubbard model. Physical review letters, 77(16):3391, 1996

    work page 1996

  3. [3]

    Simulated quantum computation of molecular energies

    Al´ an Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309(5741):1704–1707, 2005

    2005

  4. [4]

    Simulation of electronic structure hamiltonians using quantum computers

    James D Whitfield, Jacob Biamonte, and Al´ an Aspuru-Guzik. Simulation of electronic structure hamiltonians using quantum computers. Molecular Physics, 109(5):735–750, 2011

    work page 2011

  5. [5]

    Interacting electrons and quantum magnetism

    Assa Auerbach. Interacting electrons and quantum magnetism. Springer Science & Business Media, 2012

    work page 2012

  6. [6]

    Quantum chemistry in the age of quantum computing

    Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, M´ aria Kieferov´ a, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al. Quantum chemistry in the age of quantum computing. Chemical reviews, 119(19):10856–10915, 2019

    work page 2019

  7. [7]

    The variational quantum eigensolver: a review of methods and best practices

    Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H Booth, et al. The variational quantum eigensolver: a review of methods and best practices. Physics Reports, 986:1–128, 2022

    work page 2022

  8. [8]

    Variational quantum algorithms

    Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3(9):625–644, 2021

    2021

Show all 48 references
  1. [9]

    Quantum monte carlo simulations of solids

    William MC Foulkes, Lubos Mitas, RJ Needs, and Guna Rajagopal. Quantum monte carlo simulations of solids. Reviews of Modern Physics, 73(1):33, 2001

    2001

  2. [10]

    Quantum monte carlo methods for nuclear physics

    Joseph Carlson, Stefano Gandolfi, Francesco Pederiva, Steven C Pieper, Rocco Schiavilla, Kevin E Schmidt, and Robert B Wiringa. Quantum monte carlo methods for nuclear physics. Reviews of modern physics, 87(3):1067–1118, 2015

    2015

  3. [11]

    Predicting many properties of a quantum system from very few measurements

    Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16(10):1050–1057, 2020

    2020

  4. [12]

    Nearly optimal measurement scheduling for partial tomography of quantum states

    Xavier Bonet-Monroig, Ryan Babbush, and Thomas E O’Brien. Nearly optimal measurement scheduling for partial tomography of quantum states. Physical Review X, 10(3):031064, 2020

    2020

  5. [13]

    Rubin, and Akimasa Miyake

    Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake. Fermionic partial tomography via classical shadows. Phys. Rev. Lett., 127:110504, Sep 2021

    2021

  6. [14]

    Classical shadows of fermions with particle number symmetry

    Guang Hao Low. Classical shadows of fermions with particle number symmetry. arXiv preprint arXiv:2208.08964, 2022

    2022 arXiv

  7. [15]

    Huggins, Joonho Lee, and Ryan Babbush

    Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush. Matchgate shadows for fermionic quantum simulation, 2022. 11

    2022

  8. [16]

    Group-theoretic error mitigation enabled by classical shadows and symmetries

    Andrew Zhao and Akimasa Miyake. Group-theoretic error mitigation enabled by classical shadows and symmetries. npj Quantum Information, 10(1):57, 2024

    2024

  9. [17]

    Unified framework for matchgate classical shadows

    Valentin Heyraud, H´ eloise Chomet, and Jules Tilly. Unified framework for matchgate classical shadows. arXiv preprint arXiv:2409.03836, 2024

    2024 arXiv

  10. [18]

    Optimal synthesis of linear reversible circuits

    Ketan N Patel, Igor L Markov, and John P Hayes. Optimal synthesis of linear reversible circuits. Quantum Inf. Comput., 8(3):282–294, 2008

    2008

  11. [19]

    Optimal space-depth trade-off of cnot circuits in quantum logic synthesis

    Jiaqing Jiang, Xiaoming Sun, Shang-Hua Teng, Bujiao Wu, Kewen Wu, and Jialin Zhang. Optimal space-depth trade-off of cnot circuits in quantum logic synthesis. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 213–229. SIAM, 2020

    2020

  12. [20]

    Quantum algorithms to simulate many-body physics of correlated fermions

    Zhang Jiang, Kevin J Sung, Kostyantyn Kechedzhi, Vadim N Smelyanskiy, and Sergio Boixo. Quantum algorithms to simulate many-body physics of correlated fermions. Physical Review Applied, 9(4):044036, 2018

    2018

  13. [21]

    Quantum supremacy using a programmable superconducting processor

    Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510, 2019

    2019

  14. [22]

    Strong quantum computational advantage using a superconducting quantum processor

    Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, Cheng Guo, Chu Guo, Shaojun Guo, Lianchen Han, Linyin Hong, He-Liang Huang, Yong- Heng Huo, Liping Li, Na Li, Shaowei Li, Yuan Li, Futian Li...

    2021

  15. [23]

    Generation of genuine entanglement up to 51 superconducting qubits

    Sirui Cao, Bujiao Wu, Fusheng Chen, Ming Gong, Yulin Wu, Yangsen Ye, Chen Zha, Haoran Qian, Chong Ying, Shaojun Guo, et al. Generation of genuine entanglement up to 51 superconducting qubits. Nature, 619(7971):738–742, 2023

    2023

  16. [24]

    Shallow shadows: Expectation estimation using low-depth random clifford circuits

    Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan. Shallow shadows: Expectation estimation using low-depth random clifford circuits. Physical Review Letters, 133(2):020602, 2024

    2024

  17. [25]

    Random unitaries in extremely low depth

    Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. Random unitaries in extremely low depth. arXiv preprint arXiv:2407.07754, 2024

    2024 arXiv

  18. [26]

    Optimal twirling depth for classical shadows in the presence of noise

    Pierre-Gabriel Rozon, Ning Bao, and Kartiek Agarwal. Optimal twirling depth for classical shadows in the presence of noise. Phys. Rev. Lett., 133:130803, Sep 2024

    2024

  19. [27]

    How to construct random unitaries

    Fermi Ma and Hsin-Yuan Huang. How to construct random unitaries. arXiv preprint arXiv:2410.10116, 2024

    2024 arXiv

  20. [28]

    Triply efficient shadow tomography

    Robbie King, David Gosset, Robin Kothari, and Ryan Babbush. Triply efficient shadow tomography. arXiv preprint arXiv:2404.19211, 2024

    2024 arXiv

  21. [29]

    Sung, Kostyantyn Kechedzhi, Vadim N

    Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo. Quantum algorithms to simulate many-body physics of correlated fermions. Phys. Rev. Appl., 9:044036, Apr 2018

    2018

  22. [30]

    Bosonic and fermionic gaussian states from k¨ ahler structures

    Lucas Hackl and Eugenio Bianchi. Bosonic and fermionic gaussian states from k¨ ahler structures. SciPost Physics Core, 4(3):025, 2021

    2021

  23. [31]

    Leslie G. Valiant. Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing, 31(4):1229–1254, 2002

    2002

  24. [32]

    Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia. Robust shadow estimation. PRX Quantum, 2:030348, Sep 2021

    2021

  25. [33]

    Turner, Frank Pollmann, and Erez Berg

    Ari M. Turner, Frank Pollmann, and Erez Berg. Topological phases of one-dimensional fermions: An entanglement point of view. Phys. Rev. B, 83:075102, Feb 2011

    2011

  26. [34]

    Exact spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions: a century after smoluchowski and p´ olya.Physical Review X, 10(2):021045, 2020

    Luca Giuggioli. Exact spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions: a century after smoluchowski and p´ olya.Physical Review X, 10(2):021045, 2020

    2020

  27. [35]

    Random walk: a modern introduction, volume 123

    Gregory F Lawler and Vlada Limic. Random walk: a modern introduction, volume 123. Cambridge University Press, 2010

    2010

  28. [36]

    Unpaired majorana fermions in quantum wires

    A Yu Kitaev. Unpaired majorana fermions in quantum wires. Physics-uspekhi, 44(10S):131, 2001. Appendix A: Introduction to Pauli-transfer matrix representation By employing the Jordan-Wigner transformation, we can express γS in terms of Pauli operators. The Pauli-transfer matri...

    2001

  29. [37]

    And then (−1) |S|(|S|−1) 2 (−1)⌊|S|/2⌋ = (−1)q(2q+1)+q = 1

    |S| is an odd number.Let |S| = 2q + 1, q ∈ N, q ≥ 0. And then (−1) |S|(|S|−1) 2 (−1)⌊|S|/2⌋ = (−1)q(2q+1)+q = 1. (D13)

  30. [38]

    representation

    |S| is an even number.Let |S| = 2q, q ∈ N, q ≥ 0. And then (−1) |S|(|S|−1) 2 (−1)⌊|S|/2⌋ = (−1)(2q−1)q+q = 1. (D14) Thus, we conclude that the αS,d can be expressed by αS,d =(−1) |S|(|S|−1) 2 (−1)⌊|S|/2⌋22n⟨⟨0, 0| Z dUQd U ⊗2 Qd |PS, PS⟩⟩ (D15) = ⟨⟨0, 0| Z dUQd U ⊗2 Qd |PS, PS...

  31. [39]

    According to Table I, Tinit(Γ′

    Reduce the calculation to polynomial space We represent C|γS, γS⟩⟩ in a polynomial space to allow key operations, such as multiplication and addition, that are essential for our analysis. According to Table I, Tinit(Γ′

  32. [40]

    (E3) 16 ℬ ℬ𝒫𝑛 ℬ𝒫𝑁 𝒯init(Γ′

    is spanned by operators of the form    |ψij⟩⟩ = 1 4 |Xi j−1Y k=i+1 Zk ! Yj⟩⟩⊗2 + 1 4 |Xi j−1Y k=i+1 Zk ! Yj⟩⟩⊗2 + 1 4 |Xi j−1Y k=i+1 Zk ! Yj⟩⟩⊗2 + 1 4 |Xi j−1Y k=i+1 Zk ! Yj⟩⟩⊗2, i < j, |ψii⟩⟩ = |Zi, Zi⟩⟩. (E3) 16 ℬ ℬ𝒫𝑛 ℬ𝒫𝑁 𝒯init(Γ′

  33. [41]

    6: Illustration for the isometric mapping to the polynomial space

    𝒫𝑛 𝒫𝑁 FIG. 6: Illustration for the isometric mapping to the polynomial space. Γ ′ 2 represents the vectorized double supervector space. Horizontal arrows indicate transitions between spaces, while vertical arrows correspond to the operators B in different representations. Ulti...

  34. [42]

    This map is a linear isomorphism and a homomorphism with an inverse, which means Tinit(Γ′

    → Pn by setting ϕ(|ψij⟩⟩) := xixj and ϕ(|ψii⟩⟩) := x2 i . This map is a linear isomorphism and a homomorphism with an inverse, which means Tinit(Γ′

  35. [43]

    lose information

    and Pn are isomorphic. We then define the action of C on Pn by CPn (·) := ϕ ◦ C ◦ϕ−1(·). (E4) Consequently, the action of a d-depth matchgate circuit on γ{i,j} is equivalent to the action of CPn on xixj. By Lemma 3, we can further simplify this representation by identifying a ...

  36. [44]

    1 N + 2 N N −1X k=1 cos i − 1 2 πk N cos µ − 1 2 πk N cos2t πk 2N # ×

    Mapping the spread of polynomials to random walk We can calculate the output of BPN (yiyj) by BPN (yiyj) := φ′ ◦ ϕ−1 ◦ B ◦ϕ ◦ φ(yiyj), and the results are shown in Table II. The definition of function L is L (yi) =    3 4 y1 + 1 4 y2, i = 1 1 4 yi−1 + 1 2 yi + 1 4 yi+1, 1...

  37. [45]

    preserves |PS′, PS′⟩⟩ if dnear(S′) ≤ dnear(S)

  38. [46]

    Table I tells us two facts:

    discards |PS′, PS′⟩⟩ (let ξS′ = 0) if dnear(S′) > dnear(S) where dnear(S) := max{i2j − i2j−1 | j ∈ [n]}. Table I tells us two facts:

  39. [47]

    There exist a S′ such that dnear(S′) = dnear(S) − m for S with dnear(S) ≥ m, m ≤ 2

  40. [48]

    Thus, with dnear(S)/2 steps, there exists a S′ such that the ξS′ is non-zero

    the summation of ξS′ for remaining branches is greater than 1 36k . Thus, with dnear(S)/2 steps, there exists a S′ such that the ξS′ is non-zero. Since dint(S) ≥ dnear(S), the summation of coefficients of the remaining branches is greater than 1 36kdint(S)/2 after dint(S)/2 st...

Pith tools

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