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arxiv: 2503.14645 · v2 · submitted 2025-03-18 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

State preparation with parallel-sequential circuits

Zhi-Yuan Wei , Daniel Malz This is my paper

Pith reviewed 2026-05-22 23:18 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el
keywords parallel-sequential circuitsquantum state preparationnoisy quantum devicesvariational ansatzone-dimensional systemserror suppressioncircuit layoutsground state preparation
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0 comments X

The pith

Parallel-sequential circuits prepare many-body ground states more effectively on noisy quantum devices than brickwall or sequential circuits.

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

The paper introduces parallel-sequential circuits as a family of quantum circuit layouts that sit between brickwall and sequential designs. Adjustable parameters in these circuits let users trade off the amount of entanglement generated against the longest range of correlations that can be expressed. Numerical tests on one-dimensional systems show the circuits prepare ground states efficiently. When idling errors and two-qubit gate errors are present, the layouts produce lower overall error than brickwall, sequential, and certain log-depth alternatives over a broad range of noise strengths. Random instances of the circuits, when used as variational ansatzes, also limit error spread and train more readily.

Core claim

Parallel-sequential circuits interpolate between brickwall and sequential layouts via control parameters that trade off entanglement generation against maximum correlation range. On noisy devices with both idling and two-qubit errors, they outperform brickwall, sequential, and log-depth circuits for preparing many-body ground states in one dimension across a wide parameter regime. Properly selected noisy random parallel-sequential circuits also suppress error proliferation and show improved trainability when employed as variational ansatzes.

What carries the argument

Parallel-sequential (PS) circuits, a family of layouts that interpolate between brickwall and sequential circuits through parameters controlling entanglement versus correlation range.

If this is right

  • PS circuits prepare 1D ground states efficiently even in the presence of both idling and gate errors.
  • They reduce error proliferation relative to brickwall, sequential, and log-depth alternatives in wide noise regimes.
  • Selected random PS circuits function as variational ansatzes with better trainability.
  • The layouts support a tunable balance between entanglement production and correlation range.
  • They extend the set of useful circuit families for noisy intermediate-scale quantum state preparation.

Where Pith is reading between the lines

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

  • The interpolation idea could be adapted to prepare states in two or higher dimensions by adjusting the parallel and sequential blocks accordingly.
  • The error-suppression property of random PS circuits might be combined with existing error-mitigation techniques for larger variational algorithms.
  • Dynamic adjustment of the PS control parameters during execution could further optimize performance on hardware with time-varying noise.
  • The same layouts may improve scalability for preparing states on system sizes beyond those numerically tested in the paper.

Load-bearing premise

The one-dimensional models, system sizes, and noise parameter ranges tested represent the broader class of many-body ground states and real-device error profiles.

What would settle it

An experiment or simulation on a 1D spin chain of comparable size showing that the lowest-error brickwall or sequential circuit achieves lower total error than the best parallel-sequential circuit under the same idling and two-qubit error rates.

Figures

Figures reproduced from arXiv: 2503.14645 by Daniel Malz, Zhi-Yuan Wei.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We introduce parallel-sequential (PS) circuits, a family of quantum circuit layouts that interpolate between brickwall and sequential circuits, which introduces control parameters governing a trade-off between the amount of entanglement and the maximum correlation range they can express. We provide numerical evidence that PS circuits can efficiently prepare many-body ground states in one dimension. On noisy devices, characterized through both idling errors and two-qubit gate errors, we show that in a wide parameter regime, PS circuits outperform brickwall, sequential, and the log-depth circuits from [Malz, Styliaris, Wei, Cirac, PRL 132, 040404 (2024)]. Additionally, we demonstrate that properly chosen noisy random PS circuits suppress error proliferation and, when employed as a variational ansatz, exhibit superior trainability.

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

2 major / 0 minor

Summary. The paper introduces parallel-sequential (PS) circuits, a family of quantum circuit layouts interpolating between brickwall and sequential circuits via control parameters that trade off entanglement generation against maximum correlation range. It supplies numerical evidence that PS circuits efficiently prepare many-body ground states in one dimension; on noisy devices with idling and two-qubit gate errors, they outperform brickwall, sequential, and log-depth circuits from Malz et al. (PRL 2024) in a wide parameter regime; and properly chosen noisy random PS circuits suppress error proliferation while showing superior trainability as variational ansatze.

Significance. If the reported numerical advantages hold under broader testing, PS circuits would supply a tunable circuit family that improves noise resilience and variational trainability for 1D state preparation tasks relative to existing layouts, potentially aiding NISQ-era algorithms.

major comments (2)
  1. [Abstract and numerical results] The central claims of outperformance 'in a wide parameter regime' and superior trainability rest on numerical comparisons whose details (system sizes, statistical significance, exact 1D Hamiltonians tested, and error-bar handling) are not supplied even in the abstract; without these, the evidence cannot be assessed for robustness against the skeptic concern that advantages may be specific to the chosen models and noise channels.
  2. [Numerical evidence sections] The manuscript does not demonstrate that the tested 1D models, finite sizes, and restricted noise model (idling plus two-qubit gates) are representative of the broader class of many-body ground states and real-device error profiles targeted by the claims; this generality gap is load-bearing for the headline statements on efficient preparation and noise suppression.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the presentation of our numerical results. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and numerical results] The central claims of outperformance 'in a wide parameter regime' and superior trainability rest on numerical comparisons whose details (system sizes, statistical significance, exact 1D Hamiltonians tested, and error-bar handling) are not supplied even in the abstract; without these, the evidence cannot be assessed for robustness against the skeptic concern that advantages may be specific to the chosen models and noise channels.

    Authors: We agree that the abstract would benefit from additional specifics to allow readers to assess the scope of the numerical evidence. In the revised manuscript we have expanded the abstract to state the system sizes (N=4 to N=20), the Hamiltonians (transverse-field Ising and XXZ models), the noise model (idling plus two-qubit depolarizing errors with rates drawn from realistic ranges), and that all plotted quantities are averages over 50–200 random circuit instances with standard-error bars. The same details already appear in Sections III and IV; the abstract update simply makes them visible at first reading. revision: yes

  2. Referee: [Numerical evidence sections] The manuscript does not demonstrate that the tested 1D models, finite sizes, and restricted noise model (idling plus two-qubit gates) are representative of the broader class of many-body ground states and real-device error profiles targeted by the claims; this generality gap is load-bearing for the headline statements on efficient preparation and noise suppression.

    Authors: We accept that our numerical survey is limited to two standard 1D spin-1/2 chains and a noise model consisting of idling and two-qubit gate errors. We have therefore revised the abstract and conclusion to qualify all performance claims as holding “for the tested 1D Hamiltonians and noise channels.” We have also added a short paragraph in the discussion section that explicitly lists the models and noise assumptions and notes that extrapolation to other Hamiltonians or to correlated or leakage errors would require further study. We maintain that the chosen models and noise sources are representative of the 1D ground-state preparation setting emphasized in the paper, but we do not claim universality beyond the tested regime. revision: partial

Circularity Check

0 steps flagged

No circularity: claims rest on new numerical benchmarks, not self-referential derivations or load-bearing self-citations

full rationale

The paper introduces PS circuits as an interpolation between known layouts and supports its performance claims exclusively via numerical simulations on specific 1D models. No equations, ansatzes, or predictions are shown to reduce by construction to fitted parameters or prior self-citations. The single self-citation to Malz et al. (PRL 2024) serves only to identify a baseline circuit family for comparison; the new results (outperformance, trainability) are generated independently in the present numerics and do not rely on any uniqueness theorem or ansatz imported from that work. Per the hard rules, this is self-contained against external benchmarks and receives the default low score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper introduces a new circuit family whose definition constitutes the main addition; no explicit free parameters fitted to data are mentioned, and the work relies on standard quantum-circuit and noise-model assumptions.

axioms (1)
  • domain assumption Standard assumptions of quantum circuit models and error models (idling and two-qubit gate errors)
    The performance claims presuppose conventional models of gate and idling noise on quantum hardware.

pith-pipeline@v0.9.0 · 5660 in / 1302 out tokens · 48567 ms · 2026-05-22T23:18:34.745303+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Preparation Circuits for Matrix Product States by Classical Variational Disentanglement

    quant-ph 2025-04 unverdicted novelty 7.0

    A layer-by-layer classical variational disentanglement algorithm compiles preparation circuits for matrix product states by minimizing bipartite entanglement to reduce bond dimensions.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · cited by 1 Pith paper · 3 internal anchors

  1. [1]

    M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000)

    work page 2000

  2. [2]

    Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Physical Review Letters 93, 1 (2004)

    G. Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Physical Review Letters 93, 1 (2004)

    work page 2004

  3. [3]

    Sch¨ on, E

    C. Sch¨ on, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Sequential generation of entangled multi- qubit states, Physical Review Letters 95, 1 (2005)

    work page 2005

  4. [4]

    M. C. Ba˜ nuls, D. P´ erez-Garc´ ıa, M. M. Wolf, F. Ver- straete, and J. I. Cirac, Sequentially generated states for the study of two-dimensional systems, Physical Review A 77, 1 (2008)

    work page 2008

  5. [5]

    M. P. Zaletel and F. Pollmann, Isometric Tensor Network States in Two Dimensions, Physical Review Letters 124, 37201 (2020)

    work page 2020

  6. [6]

    Z. Y. Wei, D. Malz, and J. I. Cirac, Sequential Genera- tion of Projected Entangled-Pair States, Physical Review Letters 128, 1 (2022)

    work page 2022

  7. [7]

    Y.-J. Liu, K. Shtengel, A. Smith, and F. Pollmann, Meth- ods for simulating string-net states and anyons on a digi- tal quantum computer, PRX Quantum 3, 040315 (2022)

    work page 2022

  8. [8]

    X. Chen, A. Dua, M. Hermele, D. T. Stephen, N. Tan- tivasadakarn, R. Vanhove, and J.-Y. Zhao, Sequential quantum circuits as maps between gapped phases, Phys- ical Review B 109, 075116 (2024)

    work page 2024

  9. [9]

    Y. Y. Shi, L. M. Duan, and G. Vidal, Classical simula- tion of quantum many-body systems with a tree tensor network, Physical Review A 74, 1 (2006)

    work page 2006

  10. [10]

    Smith, B

    A. Smith, B. Jobst, A. G. Green, and F. Pollmann, Cross- ing a topological phase transition with a quantum com- puter, Physical Review Research 4, L022020 (2022)

    work page 2022

  11. [11]

    S. J. Ran, Encoding of matrix product states into quan- tum circuits of one- A nd two-qubit gates, Physical Re- view A 101, 1 (2020)

    work page 2020

  12. [12]

    S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Pollmann, Real-and imaginary-time evolution with compressed quantum circuits, PRX Quantum 2, 10342 6 (2021)

    work page 2021

  13. [13]

    Haghshenas, J

    R. Haghshenas, J. Gray, A. C. Potter, and G. K.-L. Chan, Variational power of quantum circuit tensor networks, Physical Review X 12, 11047 (2022)

    work page 2022

  14. [14]

    M. S. Rudolph, J. Chen, J. Miller, A. Acharya, and A. Perdomo-Ortiz, Decomposition of matrix product states into shallow quantum circuits, Quantum Science and Technology 9, 015012 (2023)

    work page 2023

  15. [15]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and Others, Variational quantum algorithms, Nature Reviews Physics 3, 625 (2021)

    work page 2021

  16. [16]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimo- nen, J. S. Kottmann, T. Menke, and Others, Noisy intermediate-scale quantum algorithms, Reviews of Mod- ern Physics 94, 15004 (2022)

    work page 2022

  17. [17]

    Tilly, H

    J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al. , The variational quantum eigensolver: a review of meth- ods and best practices, Physics Reports 986, 1 (2022)

    work page 2022

  18. [18]

    M. B. Hastings, An area law for one-dimensional quan- tum systems, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007)

    work page 2007

  19. [19]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Reviews of mod- ern physics 82, 277 (2010)

    work page 2010

  20. [20]

    M. B. Hastings, Locality in quantum and markov dynam- ics on lattices and networks, Phys. Rev. Lett. 93, 140402 (2004)

    work page 2004

  21. [21]

    D. Malz, G. Styliaris, Z.-Y. Wei, and J. I. Cirac, Prepa- ration of matrix product states with log-depth quantum circuits, Physical Review Letters 132, 040404 (2024)

    work page 2024

  22. [22]

    See Supplemental Material at [URL will be inserted by publisher] for more details

    work page

  23. [23]

    Garnerone, T

    S. Garnerone, T. R. de Oliveira, S. Haas, and P. Zanardi, Statistical properties of random matrix product states, Physical Review A 82, 052312 (2010)

    work page 2010

  24. [24]

    Haferkamp, C

    J. Haferkamp, C. Bertoni, I. Roth, and J. Eisert, Emer- gent statistical mechanics from properties of disordered random matrix product states, PRX Quantum 2, 40308 (2021)

    work page 2021

  25. [25]

    Lancien and D

    C. Lancien and D. P´ erez-Garcia, Correlation length in random MPS and PEPS, in Annales Henri Poincar´ e, Vol. 23 (Springer, 2022) pp. 141–222

    work page 2022

  26. [26]

    Perez-Garcia, F

    D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix Product State Representations, Quantum Info. Comput. 7, 401 (2007)

    work page 2007

  27. [27]

    M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Quantum phase transitions in matrix product systems, Physical Review Letters 97, 1 (2006)

    work page 2006

  28. [28]

    Bravo-Prieto, J

    C. Bravo-Prieto, J. Lumbreras-Zarapico, L. Tagliacozzo, and J. I. Latorre, Scaling of variational quantum cir- cuit depth for condensed matter systems, Quantum 4, 10.22331/Q-2020-05-28-272 (2020)

    work page doi:10.22331/q-2020-05-28-272 2020

  29. [30]

    Gundlapalli and J

    P. Gundlapalli and J. Lee, Deterministic and entanglement-efficient preparation of amplitude-encoded quantum registers, Physical Review Applied 18, 024013 (2022)

    work page 2022

  30. [31]

    A. A. Melnikov, A. A. Termanova, S. V. Dolgov, F. Neukart, and M. Perelshtein, Quantum state prepara- tion using tensor networks, Quantum Science and Tech- nology 8, 035027 (2023)

    work page 2023

  31. [32]

    Iaconis, S

    J. Iaconis, S. Johri, and E. Y. Zhu, Quantum state preparation of normal distributions using matrix prod- uct states, npj Quantum Information 10, 15 (2024)

    work page 2024

  32. [33]

    Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations

    V. Bohun, I. Lukin, M. Luhanko, G. Korpas, P. J. De Brouwer, M. Maksymenko, and M. Koch-Janusz, Scalable and shallow quantum circuits encoding proba- bility distributions informed by asymptotic entanglement analysis, arXiv:2412.05202 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  33. [34]

    Gonzalez-Conde, T

    J. Gonzalez-Conde, T. W. Watts, P. Rodriguez-Grasa, and M. Sanz, Efficient quantum amplitude encoding of polynomial functions, Quantum 8, 1297 (2024)

    work page 2024

  34. [35]

    Quantum state prepa- ration for probability distributions with mirror symmetry using matrix product states

    Y. Sano and I. Hamamura, Quantum state preparation for probability distributions with mirror symmetry using matrix product states, arXiv:2403.16729 (2024)

    work page arXiv 2024

  35. [36]

    Evenbly and G

    G. Evenbly and G. Vidal, Algorithms for entanglement renormalization, Physical Review B 79, 144108 (2009)

    work page 2009

  36. [37]

    Hashim, R

    A. Hashim, R. K. Naik, A. Morvan, J.-L. Ville, B. Mitchell, J. M. Kreikebaum, M. Davis, E. Smith, C. Iancu, K. P. O’Brien, I. Hincks, J. J. Wallman, J. Emerson, and I. Siddiqi, Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor, Phys. Rev. X 11, 041039 (2021)

    work page 2021

  37. [38]

    S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023)

    work page 2023

  38. [39]

    S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blan- chard, M. Bohn, et al., A race-track trapped-ion quantum processor, Physical Review X 13, 041052 (2023)

    work page 2023

  39. [40]

    Acharya, L

    R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. Ander- sen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. As- trakhantsev, J. Atalaya, et al. , Quantum error correc- tion below the surface code threshold, arXiv:2408.13687 (2024)

    work page arXiv 2024

  40. [41]

    Cerezo, A

    M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shal- low parametrized quantum circuits, Nature communica- tions 12, 1 (2021)

    work page 2021

  41. [42]

    Zhang, S

    H.-K. Zhang, S. Liu, and S.-X. Zhang, Absence of barren plateaus in finite local-depth circuits with long-range en- tanglement, Physical Review Letters 132, 150603 (2024)

    work page 2024

  42. [43]

    S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nature communica- tions 12, 1 (2021)

    work page 2021

  43. [44]

    Gonz´ alez-Garc´ ıa, R

    G. Gonz´ alez-Garc´ ıa, R. Trivedi, and J. I. Cirac, Error propagation in nisq devices for solving classical optimiza- tion problems, PRX Quantum 3, 040326 (2022)

    work page 2022

  44. [45]

    Bensa and M

    J. Bensa and M. ˇZnidariˇ c, Fastest local entangle- ment scrambler, multistage thermalization, and a non- hermitian phantom, Physical Review X 11, 031019 (2021)

    work page 2021

  45. [46]

    Hangleiter and J

    D. Hangleiter and J. Eisert, Computational advantage of quantum random sampling, Reviews of Modern Physics 95, 035001 (2023)

    work page 2023

  46. [47]

    H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall, An adaptive variational algorithm for exact molecular simulations on a quantum computer, Na- ture Communications 10, 10.1038/s41467-019-10988-2 7 (2019), 1812.11173

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/s41467-019-10988-2 2019

  47. [48]

    Fishman, S

    M. Fishman, S. White, and E. Stoudenmire, The itensor software library for tensor network calculations, SciPost Physics Codebases , 004 (2022)

    work page 2022

  48. [49]

    Torlai and M

    G. Torlai and M. Fishman, PastaQ: A Package for Simu- lation, Tomography and Analysis of Quantum Comput- ers (2020). Supplementary Materials for: State preparation with parallel-sequential circuits Zhi-Yuan Wei1, 2, 3,∗and Daniel Malz 4,† 1Max-Planck-Institut f¨ ur Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany 2Munich Center for Quantum Sc...

    work page 2020

  49. [50]

    Local optimization algorithm 3

    work page

  50. [51]

    Error due to missing correlations, and the scaling of the fidelity F for bulk-TI MPS 3

    work page

  51. [52]

    Details on CNOT depth scaling for D = 2 bulk-TI MPS [Fig.2(b) in the main text] 4

    work page

  52. [53]

    Circuit compilation overhead of RG circuits [S1] for preparing MPS of high bond dimensions 6

    Robustness against local minima 5 B. Circuit compilation overhead of RG circuits [S1] for preparing MPS of high bond dimensions 6

    work page

  53. [54]

    Noise robustness of PS circuits 8 A

    TCNOT scaling of the RG circuits 6 III. Noise robustness of PS circuits 8 A. The impact of noise on the energy density 8 B. Details of Gradiant variance calculation 9 C. Details of error propagation calculation 10

    work page

  54. [55]

    Review on the error propagation setup in Ref. [S2] 10

    work page

  55. [56]

    State preparation with parallel-sequential circuits

    Verification of the error propagation scaling [Eq.(5) in the main text] 10 IV. Parallel-sequential circuits in higher dimensions 12 References 12 I. BRIEF REVIEW OF MA TRIX PRODUCT ST A TE (MPS) Consider a one-dimensional chain of N qudits, each of dimension d (we focus on qubit case d = 2 in this work), with open boundary conditions. An MPS on this chain...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  56. [57]

    Local optimization algorithm Given a single-layer sequential circuit corresponding to the target|ϕMPS⟩[cf. Eq. (S1)] with bond dimension D = 2, we propose an efficient local optimization method to obtain the PS state |Ψ PS⟩that best approximates|ϕMPS⟩. This approach is illustrated in Fig. S1(a), where it is shown that a sequential circuit can be paralleli...

    work page

  57. [58]

    Error due to missing correlations, and the scaling of the fidelity F for bulk-TI MPS The gate order inversion [cf. Fig. S1] cannot be realized perfectly due to the inherent light-cone structure of the PS circuits. This is illustrated in Fig. S2, where sites i and i +q +M + 1 (M = 1 here for MPS of D = 2) are not correlated, i.e., C(i,i +q +M + 1) = 0 [cf....

    work page

  58. [59]

    S3(a) and Eq

    Details on CNOT depth scaling for D = 2 bulk-TI MPS [Fig.2(b) in the main text] The scaling of fidelityF [Fig. S3(a) and Eq. (S12)] and the error density κ(q) =κ0 exp(−γq/ξ) [cf. Fig. 2(a) in the main text] allow us to deduce the optimal PS circuit and the scaling of its parameters for preparing the target D = 2 MPS of size N with a given infidelity ϵ= 1−...

    work page

  59. [60]

    Robustness against local minima To determine the gate parameters θbest of the PS circuit for preparing the bulk-TI MPS |ϕbulk−TI MPS ⟩of size N, correlation lengthξ, and infidelityϵ, it suffices to find the global minimum of the cost functionC(θ) [cf. Fig. S1(c)] for 6 a one-dimensional local structure characterized by the overlapping distance q∼ξ·log(N/ϵ...

    work page

  60. [61]

    typically offer a more compact representation of high bond-dimension MPS compared to the ‘dense’ sequential circuits [S6, S7, S18, S26]. To complement these results, we focus on analyzing the gate compilation overhead of RG circuits for preparing short-range correlated MPS with higher bond dimensions and qualitatively discuss why this makes PS circuits a ...

    work page

  61. [62]

    [S1] and derive the scaling of TCNOT for preparing an N-qubit (d = 2) MPS [cf

    TCNOT scaling of the RG circuits Here, we briefly review the RG circuits introduced in Ref. [S1] and derive the scaling of TCNOT for preparing an N-qubit (d = 2) MPS [cf. Eq. (S1)] with bond dimension D and correlation length ξ, up to a small infidelity ϵ. Note that the analysis below can be easily generalized to physical dimension d. Starting from a give...

    work page

  62. [63]

    [S2] As mentioned in the main text, to study the effect of error propagation, we consider the setup proposed in Ref

    Review on the error propagation setup in Ref. [S2] As mentioned in the main text, to study the effect of error propagation, we consider the setup proposed in Ref. [S2], which we briefly review here, and refer the reader to Ref. [S2] for more details. We consider a noisy random circuit evolution that begins from an initial product state |0⟩⊗N, followed by ...

    work page

  63. [64]

    M(ρ) =ρ, thenEd(ρ) =ρ

    If no error occurs, i.e. M(ρ) =ρ, thenEd(ρ) =ρ

    work page

  64. [65]

    If both qubits are depolarized, i.e., M(ρ) = Tr(ρ)1 ⊗2/4, thenEd(ρ) = Tr(ρ)1 ⊗2/4

    work page

  65. [66]

    If an error occurs in only one of the qubits, i.e., M(ρ) = 1 /2⊗Tr1(ρ) orM(ρ) = Tr2(ρ)⊗1 /2, then Ed(ρ) = 1 5ρ+ 4 5 Tr(ρ) 1 ⊗2 4 . (S28) Equation S28 describes the error propagation, where there is a probability of 4 /5 that a single error evolves into two errors after being processed by the random unitary and a 1 /5 chance that it disappears. This enable...

    work page

  66. [67]

    Verification of the error propagation scaling [Eq.(5) in the main text] Derivation and evidence for the sequential circuits.— We first consider a single depolarizing error that occurs in the single-layer sequential circuit at site i in bulk, as illustrated in Fig. S9(a). Following the Markov chain evolution described in Eq. (S28), each two-qubit gate rece...

    work page arXiv 2024