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REVIEW 3 major objections 4 minor 86 references

A Pathway to Practical Quantum Advantage in Solving Navier-Stokes Equations

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper's central claim: a fault-tolerant quantum computer with 8.71 million physical qubits solves a 2^80-cell Navier-Stokes problem in 42.6 days, where a state-of-the-art supercomputer would need over a century.

desk verdict A genuinely full-stack quantum CFD feasibility study whose headline resource estimate is undone by a five-order-of-magnitude tomography sampling undercount. read the letter →

arxiv 2509.08807 v1 pith:UHZM5X72 submitted 2025-09-10 quant-ph cs.NAmath.NAphysics.comp-phphysics.flu-dyn

Xi-Ning Zhuang , Zhao-Yun Chen , Ming-Yang Tan , Jiaxuan Zhang , Chuang-Chao Ye , Tian-Hao Wei , Teng-Yang Ma , Cheng Xue
show 7 more authors
Huan-Yu Liu Qing-Song Li Tai-Ping Sun Xiao-Fan Xu Yun-Jie Wang Yu-Chun Wu Guo-Ping Guo
This is my paper
classification quant-phcs.NAmath.NAphysics.comp-phphysics.flu-dyn
keywords quantumadvantageNavier-Stokesequationsfault-tolerantcomputinglinearsystemsolverspectralsparsityhierarchyblock-encodingcompressed-sensingstatetomographycircuitsynthesis
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

The paper attempts to show that the Navier-Stokes equations—nonlinear, input/output-heavy, and a cornerstone of computational fluid dynamics—can be solved on a fault-tolerant quantum computer with an exponential end-to-end speedup over classical simulation, and that the speedup survives a full accounting of circuit depth, error correction, and readout. Its central move is the claim that a fluid field with small spectral sparsity S can be fed into a quantum linear-system solver, evolved through nonlinearity by classical iteration, and read back out, all while touching only O(S)-dimensional data instead of the full N-cell grid. If the resource estimates hold, a 2^80-grid compressible-flow simulation would run in about six weeks on 8.71 million physical qubits at a 5×10^-4 error rate, against an estimated 130 years for the most powerful current supercomputer. The authors further claim the algorithm's time complexity matches a lower bound they prove for any iterative quantum linear-system solver, so the exponential speedup is not an artifact of one implementation.

What carries the argument

Three devices carry the argument. The iteration-tolerant bandwidth S—the minimum number of independent classical components that must pass between iterations of a quantum linear-system solver—is the subject of Theorem 1, which bounds any such solver's time by roughly τκ·Poly(S,1/ε). The hierarchy spectral block-encoding decomposes the flux Jacobian into a diagonal level, an inter-block level of 4×4 flux blocks, and an element level of diagonal functions, encoding each entry from S spectral coefficients rather than N point values; non-polynomial terms enter via polynomial approximation plus quantum singular-value transformation. Sparse spectral decoding applies a parallel quantum Fourier tran

What would settle it

Take a random S-sparse vector with S=64 on a grid of 2^80 cells, Fourier-transform it, and run the paper's compressed-sensing tomography at effective dimension d_eff = S^3+2S^2+S ≈ 2.7×10^5; if the measured sample count needed to reach fidelity 6×10^-5 tracks m = C·r·d_eff·log^2(d_eff) with the paper's fitted C, the 42.6-day runtime stands, and if the fit stays at the values of the paper's Fig. 19, the runtime grows by about five orders of magnitude. A second check: re-derive Theorem 2 from the Appendix C.6 proof and confirm the τ factor in the stated bound Õ((κDS+log^2 N)τS/ε^2) appears in th

Watch

Extended reading notes

Core claim

Spectral sparsity removes the input, nonlinearity, and readout bottlenecks of quantum fluid simulation. The iteration-tolerant bandwidth S—the fewest classical components an iterative quantum linear-system solver must carry—is proved to force Ω(τκ Poly(S,1/ε)) time, so exponential speedup needs S logarithmic in N. The hierarchy spectral block-encoding assembles the flux Jacobian in three levels and encodes each variable from S spectral coefficients in O(S log N) depth; sparse spectral decoding collapses the dense solution to effective dimension S^3+2S^2+S before compressed-sensing tomography. Payoff: a 2^80-cell Navier-Stokes problem in 42.6 days on 8.71 million physical qubits, versus 130 y

Load-bearing premise

The feasibility window rests on one empirical number: tomography sampling was fitted on states of effective dimension 4 to 32 and then applied to a decoded state whose effective dimension the paper itself computes as roughly 270,000, where its own compressed-sensing formula would require tens of millions of measurements rather than 1,000.

Editorial extensions

If this is right

  • A fault-tolerant machine near 8.7 million physical qubits at 5×10^-4 physical error rate would outperform a top supercomputer on a 2^80-grid compressible Navier-Stokes problem by a factor of roughly 1,100 in wall-clock time, per the paper's estimates.
  • Because of the lower bound, any iterative quantum linear-system solver for a problem whose intermediate solutions are spectrally dense is provably limited to polynomial speedup; spectral sparsity is the decisive problem feature.
  • The qRAM-free input/output design means the grid size N enters the cost only logarithmically (through the quantum Fourier transform and shift arithmetic), so the framework scales to finer grids without an exponential qubit or gate overhead.
  • The logical-resource reductions (about 23× Toffoli depth, 50× Toffoli count, 5× rotation count and depth) and the 3.5× magic-state-factory cut are what turn an asymptotically efficient algorithm into a specific hardware budget.

Reading between the lines

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

  • If one applies the paper's own compressed-sensing bound at the decoded effective dimension it computes (S^3+2S^2+S ≈ 2.7×10^5 for S=64) instead of the fitted range 4–32, the required measurements rise to about 10^7, which under the paper's time formula would multiply the 42.6-day estimate by roughly five orders of magnitude; this arithmetic is mine, not the paper's.
  • The iteration-tolerant bandwidth gives problem selectors a concrete diagnostic: measure the spectral sparsity of the intermediate fields of a candidate fluid solver, and if it grows with grid resolution, the exponential advantage is excluded by the paper's own Theorem 1.
  • The hierarchy block-encoding is a transferable construction for any locally connected discrete PDE—elasticity, electromagnetics, geophysics—so the framework's method, not just its fluid result, is portable to neighboring simulation problems.
  • The reported κ-saturation (condition number bounded once S is fixed, observed in the benchmark vortex flows studied numerically) implies a scaling law between spectral sparsity and linear-system conditioning that, if confirmed for other flows, would let the quantum cost model's dominant parameter be read off from classical spectra.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes an end-to-end, fault-tolerant quantum algorithm for solving the Navier-Stokes equations (NSE) on very large grids. The claimed architecture combines a spectral-sparsity-based input/output protocol (hierarchy spectral block-encoding and sparse spectral tomography), explicit circuit synthesis (match-and-merge, mask-and-merge), a QLSS based on Ref. [26], and a full error/resource model with surface-code and magic-state overheads. The headline result is that a 2^80-cell two-dimensional NSE problem can be solved in 42.6 days using 8.71 million physical qubits at a physical error rate of 5e-4, giving an approximately 1100x speedup over a state-of-the-art supercomputer. The paper also states an asymptotic end-to-end complexity that it claims saturates a lower bound for iterative QLSS.

Significance. If the resource estimate were correct, this would be a landmark result in practical quantum advantage: it would be the first full-stack, end-to-end estimate for a nonlinear PDE with a concrete fault-tolerant resource count, explicit circuits, and a nontrivial lower bound. The paper genuinely contributes several interesting technical pieces: an explicit hierarchy block-encoding for the structured Jacobian, a spectral I/O scheme that reduces the effective output dimension from N to poly(S), a concrete circuit-synthesis methodology with numerical reductions, and an end-to-end error accounting that separates algorithmic, deployment, and tomography errors. However, the central quantitative claim is invalidated by the tomography sampling estimate, as detailed below. The asymptotic framework and the circuit-synthesis techniques may still be of interest, but the paper's practical-advantage conclusion is not supported by the evidence presented.

major comments (3)
  1. [Appendix F, Eqs. (F26) and (F30); Appendix C.5; Theorem C.9] The central runtime estimate T_quantum = N_sample × (RD+TD) × d × T_cycle is linear in N_sample, but N_sample = 1000 is not justified for the actual decoded state. The decoded state's effective dimension is computed in Appendix C.5 as S^3 + 2S^2 + S, i.e., about 270,400 for S=64, the value used in the resource estimate. The paper's own compressed-sensing bound (Theorem C.9) requires m = C r d_eff log^2 d_eff random Pauli expectations. The numerical fitting in Appendix F (Fig. 19) is performed only for effective dimensions 4 to 32; even using the small fitted prefactor implied there, one obtains m ~ 10^7-10^8, not 1000. Since Eq. (F30) is linear in N_sample, the 42.6-day headline becomes several centuries to millennia, eliminating the claimed ~1100x advantage. The Discussion explicitly concedes that no tight theoretical bound for large S is known; the empirical extrapolation from d_eff <=
  2. [Appendix C.6, proof of Theorem 2] The proof of Theorem 2 derives an end-to-end time complexity of Õ(S ε^{-2}(κ(S + log log N) + log^2 N)) and does not include the factor τ that appears in the theorem statement, Õ((κDS + log^2 N) τS / ε^2). No argument is given for how τ is absorbed or why the per-iteration cost should be independent of the iteration count. Thus the claimed saturation of the Theorem 1 lower bound is not established. This is not merely a typo: the resource model in Eq. (F30) also contains no τ factor, so the practical time estimate appears to be for a single iteration rather than for the full NSE simulation, unless τ is somehow hidden inside (RD+TD), which is not shown.
  3. [Appendix F.4, classical baseline] The classical cost estimate appears to count a single linear solve of dimension NLS = 2^82 (via CG or Cholesky), with FLOP counts 9.96e29 and 7.13e27 respectively. The NSE is time-marched with implicit Euler, requiring a linear solve at each step; the number of iterations τ is present in the quantum algorithm (Algorithm 3) and in Theorem 2, but no corresponding multiplication by τ appears in T_ElCapitan64. If τ is large, the classical baseline is undercounted by a factor of τ. This does not rescue the quantum estimate, but it makes the claimed crossover ratio T_quantum/T_classical unquantified.
minor comments (4)
  1. [General / notation] There are several typographical inconsistencies in the appendices, e.g., Eq. (C24) labels the vertical viscous flux as G_C = ... (should be G_mu), and 'ad iabetic' appears for 'adiabatic' in Appendix F. These are cosmetic but should be cleaned up.
  2. [Eq. (C109)] The notation 'ND j=1 O_SUB(N_j)' is missing the tensor-product symbol; it should read \bigotimes_{j=1}^D O_SUB(N_j).
  3. [Appendix F.2, iteration-tolerant error bound] The sentence 'We track the value of T∝e and ρ with varying problem size and find that ...' is repeated almost verbatim two sentences later. Please consolidate.
  4. [Fig. 19 caption / Appendix F] The caption for Fig. 19 does not state the effective dimensions or the fitted constants for the linear regression used to estimate C in Theorem C.9. Since the sampling number is a key hyperparameter, the figure and the regression details should be fully specified.

Circularity Check

1 steps flagged · score 6.0 of 10

Headline 42.6-day runtime is linearly forced by the small-scale fitted N_sample=1000, conflicting with the paper's own compressed-sensing scaling for d_eff≈2.7×10^5.

  1. fitted input called prediction [Appendix F.3, Eqs. (F26) and (F30); Appendix C.5; Discussion limitations paragraph]
    "only Nsample = 1000 shots can reconstruct the state with tomography error bounded by ... Tquantum =Nsample ×(RD+T D)×d×Tcycle ≤3.68×10^6 s≃42.6days."

    The central resource claim is computed by multiplying the fitted sample count N_sample by depth, distance, and cycle time; Eq. (F30) is linear in N_sample. N_sample is not derived for the target instance: it is validated only on CS-QST tests with effective dimensions 4–32 (Fig. 19). The paper itself computes the decoded state's effective dimension as S^3+2S^2+S (Appendix C.5), i.e. ≈2.70×10^5 for S=64, and its own Theorem C.9 requires m = C r d_eff log^2 d_eff; the prefactor implied by the paper's regression then gives m≈10^7–10^8, not 1000. Therefore the 42.6-day 'prediction' is arithmetically forced by the small-system fitted input, and using the paper's own scaling changes T_quantum by about five orders of magnitude. The Discussion explicitly concedes that no tight theoretical bound for

full rationale

The asymptotic algorithm sections (B, C, Theorems 1 and 2) are not circular: the block-encoding and tomography complexity are derived from explicit lemmas and external QLSS results, and the spectral-sparsity assumption is used consistently as an input rather than being derived from the target conclusion. The equivalence between iteration-tolerant bandwidth and spectral sparsity is definitional but does not smuggle in the speedup. The proof of Theorem 2 in Appendix C.6 omits the τ factor present in the theorem statement, but that is an internal proof gap / inconsistency, not circularity. The only load-bearing step that reduces to an input is the runtime formula's N_sample: a numerical fit at d_eff≤32 is used as a constant for d_eff≈2.7×10^5, directly determining the headline 42.6-day figure. This makes the practical-advantage claim partly an extrapolation of a fitted parameter rather than a parameter-free first-principles prediction, so the circularity score is 6 rather than 0–2.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The resource estimate depends on many fitted or assumed numbers (S, kappa, N_sample, p_phys, T_cycle) and on numerical extrapolations from small Taylor-Green experiments to N = 2^80. The most fragile is the sampling count, which is validated only in effective dimensions 4 to 32 and then applied to about 270,000, changing runtime by about five orders of magnitude.

free parameters (10)
  • Spectral sparsity S = 64
    Chosen from norm-ratio tests on Taylor-Green vortex (Appendix F.2, Fig. 15). Assumed constant across all iterations and extrapolated to N = 2^80 without proof.
  • Condition number bound kappa = <= 550
    Extrapolated from small-grid numerical saturation experiments (Fig. 16); no analytic bound for N = 2^80.
  • Iteration-tolerant error threshold epsilon_threshold = 0.05
    Set from controlled noise-injection experiments (Appendix F.2, Fig. 13) on small Taylor-Green cases.
  • Tomography sampling number N_sample = 1000
    Fitted from CS-QST trials on effective dimensions 4 to 32 (Fig. 19), then applied to effective dimension about 270,000 for S = 64, inconsistent with Theorem C.9 scaling.
  • Polynomial truncation degrees d_mu, d_rho = 2
    Chosen to suppress truncation errors below thresholds; empirically verified only on the narrow Ma = 0.1 temperature and density ranges.
  • QLSS query count Q = 28301
    Taken from Ref. [26] for kappa <= 550, epsilon <= 1e-3; not independently re-derived here.
  • Surface code distance d = 25
    Derived from the paper's error budget and assumed physical error rate, not from measurement.
  • Physical error rate p_phys = 5e-4
    Assumed hardware parameter, consistent with Ref. [44] comparisons but not demonstrated.
  • QEC cycle time T_cycle = 1e-6 s
    Assumed superconducting processor parameter.
  • Mach number = 0.1
    Test case regime; sparsity and normalization bounds are validated only here.
assumptions (7)
  • domain assumption Solution variables rho, u, v, e have Fourier spectral sparsity S that stays O(log N) for all tau iterations.
    Load-bearing for the entire speedup. Validated numerically only for a smooth periodic Taylor-Green vortex at Ma = 0.1; turbulent or shock-dominated flows are explicitly excluded. See Section II.A and Appendix C.1.
  • domain assumption The condition number saturates to kappa <= 550 as N grows for fixed S.
    Numerical extrapolation from small grids (Fig. 16, Eq. F13); no proof for N = 2^80.
  • domain assumption Normalization constants r_b, r_mu, r_rho stay bounded as in Eq. (F14) at large N.
    Numerical extrapolation (Fig. 17); underpins the claimed error bounds.
  • standard math Implicit Euler plus centered finite volume discretization is valid and its iteration-tolerant bandwidth equals the spectral sparsity.
    The proof of Lemma B.8 is provided in Appendix B; the discretization is standard.
  • domain assumption Chebyshev polynomial approximations of Sutherland viscosity and 1/rho with degree 2 are accurate on observed intervals.
    Empirically verified for Ma = 0.1; higher Mach or wider fluctuation ranges would require higher degree. See Appendix F.2.
  • standard math External QLSS query bound, block-encoding lemmas, CS-QST guarantee, and surface code error model are correct.
    Standard literature results from Refs. [26], [66], [67], [23]; not re-proven in the paper.
  • ad hoc to paper The compressed-sensing sample bound's prefactor C can be estimated from trials on effective dimensions 4 to 32 and extrapolated linearly to about 270,000.
    This is the paper's own fitting step (Fig. 19) and is the fragile premise; the linear extrapolation is not justified and contradicts the magnitude change.

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Pith. "Pith review of A Pathway to Practical Quantum Advantage in Solving Navier-Stokes Equations." pith.science (2026). https://pith.science/paper/UHZM5X72

@misc{pith2026250908807,
  author       = {Pith},
  title        = {Pith review of: A Pathway to Practical Quantum Advantage in Solving Navier-Stokes Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UHZM5X72}},
  note         = {Machine review of arXiv:2509.08807}
}
abstract

The advent of fault-tolerant quantum computing (FTQC) promises to tackle classically intractable problems. A key milestone is solving the Navier-Stokes equations (NSE), which has remained formidable for quantum algorithms due to their high input-output overhead and nonlinearity. Here, we establish a full-stack framework that charts a practical pathway to a quantum advantage for large-scale NSE simulation. Our approach integrates a spectral-based input/output algorithm, an explicit and synthesized quantum circuit, and a refined error-correction protocol. The algorithm achieves an end-to-end exponential speedup in asymptotic complexity, meeting the lower bound for general quantum linear system solvers. Through symmetry-based circuit synthesis and optimized error correction, we reduce the required logical and physical resources by two orders of magnitude. Our concrete resource analysis demonstrates that solving NSE on a $2^{80}$-grid is feasible with 8.71 million physical qubits (at an error rate of $5 \times 10^{-4}$) in 42.6 days -- outperforming a state-of-the-art supercomputer, which would require over a century. This work bridges the gap between theoretical quantum speedup and the practical deployment of high-performance scientific computing.

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Reference graph

Works this paper leans on

86 extracted references · 14 linked inside Pith

  1. [26]

    Gidney and M

    C. Gidney and M. Eker ˚a, Quantum5, 433 (2021)

    2021

  2. [1]

    Henceforth, we first consider conventional quantum circuit synthesis methods to reduce logical resources within each subroutine as follows

    Conventional Synthesis The hierarchy spectral quantum Navier-Stokes solver heavily utilizes state preparation modules to load spectrum information, Fourier basis block-encoding modules to load basis function information, and shift operators to compute and allocate matrix structure information. Henceforth, we first consider conventional quantum circuit syn...

  3. [2]

    Herein, we study three matching patterns appearing in the hierarchy spectral quantum Navier-Stokes solver for search and merge

    Match and Merge Besides conventional algorithm-agnostic synthesis techniques, we further reduce the logical resource by utilizing algorithm- aware methods to match redundant subroutines and merge them together. Herein, we study three matching patterns appearing in the hierarchy spectral quantum Navier-Stokes solver for search and merge. Match by signal.We...

  4. [3]

    Mask and Merge Inspired by the match-and-merge technique to synthesize subroutines with similar patterns, we design an even more efficient synthesis process to merge subroutines with very different patterns, further compressing the circuit. The basic idea is to utilize a template block-encoding circuit for the most complicated spectral structure and then ...

  5. [4]

    Classical Computation Overhead We follow the analysis as in Ref. [44] that, given the linear system dimension and sparsity to be s= 21, NLS = 282, (F35) the total floating-point operations (FLOPs) in solving anN LS-dimensional linear system by Conjugate Gradient and Cholesky Decomposition methods are respectively to be CCG = (2s+ 7)N LS ·κlog 2 ϵ ≥(2×21 +...

  6. [5]

    By applying hierarchy spectral block-encoding and quantum linear system solver in Theorem B.1, we can get a2n-qubit quantum state|W⟩representing the conservative variables onNcells

    Sparse spectral decoding In this section, we introduce a sparse spectral method for extracting information from the quantum state to facilitate iteration and final reconstruction of the fluid field. By applying hierarchy spectral block-encoding and quantum linear system solver in Theorem B.1, we can get a2n-qubit quantum state|W⟩representing the conservat...

  7. [6]

    For the k-th iteration, we can update the block-encoding oraclesO (k+1) A ,O (k+1) b by ˜W (k)

    End-to-end complexity and resource analysis Now, we can describe the end-to-end iterative quantum Navier-Stokes solver given the aforementioned ingredients. For the k-th iteration, we can update the block-encoding oraclesO (k+1) A ,O (k+1) b by ˜W (k). In Theorem B.1, the QLSS is implemented in two steps. Firstly, randomly choose a guess of normt=|x|and a...

  8. [7]

    Among these QEC codes, surface codes are well-studied and generally considered to be one of the leading candidates for practical quantum computing [23]

    Surface Code Implementation To enable the large-scale quantum computation, quantum error correction (QEC) is the crucial technique that protects infor- mation against the noise by encoding physical qubits into logical qubits of a certain type of QEC code. Among these QEC codes, surface codes are well-studied and generally considered to be one of the leadi...

Show all 86 references
  1. [8]

    To prepare these magic states, we consider the following preparation protocols

    Magic State Preparationy The ancilla states required for implementing Toffoli gates and rotation gates are also known as magic states. To prepare these magic states, we consider the following preparation protocols. Lemma E.1.(T States from Magic State Cultivation, [43].)The st...

  2. [9]

    Error and Resource Model In our error model shown in Fig. 12, there are two types of hyperparameters: theproblem-relevant hyperparametersto be characterized by numerical tests (black colored), such as problem size, block-encoding normalization constants, and linear system cond...

  3. [10]

    Specifically, each component of the solutionx(the residual vector) is multiplied by a random factor drawn from the uniform distributionU(1−ϵ noise,1 +ϵ noise)

    Characterizing Problem-Specific and Free Hyperparameters We first characterize those problem-specific hyperparameters as follows: Iteration-tolerant Error Bound.To model the impact of quantum error, we introduce a controlled, single-step noise in each iteration. Specifically, ...

    2000

  4. [11]

    Quantum Computation Overhead Given the sample time in Eq. (F26), the quantum error correction cycle number Ncycle = (RD+T D)×d <3.70×10 9,(F28) and the standard QEC per-cycle time on a superconducting quantum processor Tcycle = 10−6 s,(F29) the total quantum time can be evalua...

  5. [12]

    Firstly, we utilize the boundary operatorU boundary acting on the cell index register and the boundary flag register to mark whether it is on the boundary

    Further Discussion on Boundary Condition We briefly discuss how to handle boundary conditions beyond the periodic boundary condition in two steps. Firstly, we utilize the boundary operatorU boundary acting on the cell index register and the boundary flag register to mark wheth...

  6. [13]

    In the simplified introductory case of the classic Lax-Friedrichs Riemann solver, we illustrate the basic idea to solve the Riemann problem with boundary conditions

    Further Discussion on Riemann Problem In this subsection, we show how to design efficient quantum Riemann solvers with classic Lax-Friedrichs and Rusanov schemes. In the simplified introductory case of the classic Lax-Friedrichs Riemann solver, we illustrate the basic idea to ...

  7. [14]

    As the Mach number increases, compressibility effects become dominant, and the fluctuations in density ρand temperatureTbecome much larger than in theMa = 0.1case

    Further Discussion on Effect of Higher Mach Number The numerical experiments presented earlier were conducted at a low Mach number (Ma = 0.1), a regime where the flow is nearly incompressible. As the Mach number increases, compressibility effects become dominant, and the fluct...

  8. [15]

    P. W. Shor, SIAM review41, 303 (1999)

    1999

  9. [16]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell,et al., Nature574, 505 (2019)

    2019

  10. [17]

    Y . Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y . Wu, M. Zaletel, K. Temme,et al., Nature618, 500 (2023)

    2023

  11. [18]

    P. W. Shor, inProceedings of 37th conference on foundations of computer science(IEEE, 1996) pp. 56–65

    1996

  12. [19]

    Preskill, Introduction to quantum computation and information213(1998)

    J. Preskill, Introduction to quantum computation and information213(1998)

    1998

  13. [20]

    Gottesman, arXiv preprint arXiv:2210.15844 (2022)

    D. Gottesman, arXiv preprint arXiv:2210.15844 (2022)

    2022 arXiv

  14. [21]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter,et al., Nature 626, 58 (2024)

    2024

  15. [22]

    Nature638, 920 (2025)

    2025

  16. [23]

    Katabarwa, K

    A. Katabarwa, K. Gratsea, A. Caesura, and P. D. Johnson, PRX quantum5, 020101 (2024)

    2024

  17. [24]

    Yoshioka, T

    N. Yoshioka, T. Okubo, Y . Suzuki, Y . Koizumi, and W. Mizukami, npj Quantum Information10, 45 (2024)

    2024

  18. [25]

    J. Lee, D. W. Berry, C. Gidney, W. J. Huggins, J. R. McClean, N. Wiebe, and R. Babbush, PRX quantum2, 030305 (2021)

    2021

  19. [27]

    Gidney, arXiv preprint arXiv:2505.15917 (2025)

    C. Gidney, arXiv preprint arXiv:2505.15917 (2025)

    2025 arXiv

  20. [28]

    A. M. Dalzell, B. D. Clader, G. Salton, M. Berta, C. Y .-Y . Lin, D. A. Bader, N. Stamatopoulos, M. J. Schuetz, F. G. Brand ˜ao, H. G. Katzgraber,et al., Prx Quantum4, 040325 (2023). 56

    2023

  21. [29]

    Zhang, T

    X.-M. Zhang, T. Li, and X. Yuan, Physical Review Letters129, 230504 (2022)

    2022

  22. [30]

    B. D. Clader, A. M. Dalzell, N. Stamatopoulos, G. Salton, M. Berta, and W. J. Zeng, IEEE Transactions on Quantum Engineering3, 1 (2023)

    2023

  23. [31]

    Giovannetti, S

    V . Giovannetti, S. Lloyd, and L. Maccone, Physical review letters100, 160501 (2008)

    2008

  24. [32]

    Di Matteo, V

    O. Di Matteo, V . Gheorghiu, and M. Mosca, IEEE Transactions on Quantum Engineering1, 1 (2020)

    2020

  25. [33]

    A. S. Holevo, Problemy Peredachi Informatsii9, 3 (1973)

    1973

  26. [34]

    Kerenidis and A

    I. Kerenidis and A. Prakash, ACM Transactions on Quantum Computing1, 1 (2020)

    2020

  27. [35]

    Y . Ge, W. Wenjie, C. Yuheng, P. Kaisen, L. Xudong, Z. Zixiang, W. Yuhan, W. Ruocheng, and Y . Junchi, arXiv preprint arXiv:2407.00736 (2024)

    2024

  28. [36]

    van de Wetering and M

    J. van de Wetering and M. Amy, arXiv preprint arXiv:2310.05958 (2023)

    2023 arXiv

  29. [37]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A—Atomic, Molecular, and Optical Physics86, 032324 (2012)

    2012

  30. [38]

    Blazek,Computational fluid dynamics: principles and applications(Butterworth-Heinemann, 2015)

    J. Blazek,Computational fluid dynamics: principles and applications(Butterworth-Heinemann, 2015)

    2015

  31. [39]

    A. W. Harrow, A. Hassidim, and S. Lloyd, Physical review letters103, 150502 (2009)

    2009

  32. [40]

    A. M. Dalzell, arXiv preprint arXiv:2406.12086 (2024)

    2024 arXiv

  33. [41]

    W. K. Wootters and W. H. Zurek, Nature299, 802 (1982)

    1982

  34. [42]

    E. T. Campbell and J. O’Gorman, Quantum Science and Technology1, 015007 (2016)

    2016

  35. [43]

    Mottonen, J

    M. Mottonen, J. J. Vartiainen, V . Bergholm, and M. M. Salomaa, arXiv preprint quant-ph/0407010 (2004)

    2004 arXiv

  36. [44]

    A. G. Fowler and C. Gidney, arXiv preprint arXiv:1808.06709 (2018)

    2018 arXiv

  37. [45]

    Helgaker, P

    T. Helgaker, P. Jorgensen, and J. Olsen,Molecular electronic-structure theory(John Wiley & Sons, 2013)

    2013

  38. [46]

    Litinski, Quantum3, 205 (2019)

    D. Litinski, Quantum3, 205 (2019)

    2019

  39. [47]

    Low-Overhead and High-Fidelity Preparation of Logical Non-Clifford States with Multi-Level Transversal Injection,

    “Low-Overhead and High-Fidelity Preparation of Logical Non-Clifford States with Multi-Level Transversal Injection,” In Preparation

  40. [48]

    Top500 list of supercomputers (june 2025),

    TOP500 Project, “Top500 list of supercomputers (june 2025),” (2025), accessed: 2025-09-09

    2025

  41. [49]

    Wan, C.-H

    L.-C. Wan, C.-H. Yu, S.-J. Pan, S.-J. Qin, F. Gao, and Q.-Y . Wen, Physical Review A104, 062414 (2021)

    2021

  42. [50]

    Q. T. Nguyen, B. T. Kiani, and S. Lloyd, Quantum6, 876 (2022)

    2022

  43. [51]

    Camps, L

    D. Camps, L. Lin, R. Van Beeumen, and C. Yang, SIAM Journal on Matrix Analysis and Applications45, 801 (2024)

    2024

  44. [52]

    S ¨underhauf, E

    C. S ¨underhauf, E. Campbell, and J. Camps, Quantum8, 1226 (2024)

    2024

  45. [53]

    Zhuang, Z.-Y

    X.-N. Zhuang, Z.-Y . Chen, C. Xue, X.-F. Xu, C. Wang, H.-Y . Liu, T.-P. Sun, Y .-J. Wang, Y .-C. Wu, and G.-P. Guo, arXiv preprint arXiv:2406.01335 (2024)

    2024 arXiv

  46. [54]

    Rosenkranz, E

    M. Rosenkranz, E. Brunner, G. Marin-Sanchez, N. Fitzpatrick, S. Dilkes, Y . Tang, Y . Kikuchi, and M. Benedetti, Quantum9, 1703 (2025)

    2025

  47. [55]

    A. G. Fowler, arXiv preprint arXiv:1210.4626 (2012)

    2012 arXiv

  48. [56]

    Gidney and A

    C. Gidney and A. G. Fowler, Quantum3, 135 (2019)

    2019

  49. [57]

    Gidney, N

    C. Gidney, N. Shutty, and C. Jones, arXiv preprint arXiv:2409.17595 (2024)

    2024 arXiv

  50. [58]

    Y . Tu, M. Dubynskyi, M. Mohammadisiahroudi, E. Riashchentceva, J. Cheng, D. Ryashchentsev, T. Terlaky, and J. Liu, arXiv preprint arXiv:2502.11239 (2025)

    2025 arXiv

  51. [59]

    Gaitan, npj Quantum Information6, 61 (2020)

    F. Gaitan, npj Quantum Information6, 61 (2020)

    2020

  52. [60]

    Itani, K

    W. Itani, K. R. Sreenivasan, and S. Succi, Physics of Fluids36(2024)

    2024

  53. [61]

    Meng and Y

    Z. Meng and Y . Yang, Physical Review Research6, 043130 (2024)

    2024

  54. [62]

    S. Jin, N. Liu, and Y . Yu, Physical Review A108, 032603 (2023)

    2023

  55. [63]

    S. Jin, N. Liu, and Y . Yu, Physical Review Letters133, 230602 (2024)

    2024

  56. [64]

    N. Ray, T. Banerjee, B. Nadiga, and S. Karra, arXiv preprint arXiv:1904.09033 (2019)

    1904 arXiv

  57. [65]

    M. G. Meena, Y . Zhang, W. Jiang, Y . Lin, S. G¨unther, and X. Gao, in2024 IEEE International Conference on Quantum Computing and Engineering (QCE), V ol. 1 (IEEE, 2024) pp. 662–668

    2024

  58. [66]

    M. E. Morales, L. Pira, P. Schleich, K. Koor, P. Costa, D. An, L. Lin, P. Rebentrost, and D. W. Berry, arXiv preprint arXiv:2411.02522 (2024)

    2024 arXiv

  59. [67]

    Ambainis, arXiv preprint arXiv:1010.4458 (2010)

    A. Ambainis, arXiv preprint arXiv:1010.4458 (2010)

    2010 arXiv

  60. [68]

    A. M. Childs, R. Kothari, and R. D. Somma, SIAM Journal on Computing46, 1920 (2017)

    1920

  61. [69]

    Subas ¸ı, R

    Y . Subas ¸ı, R. D. Somma, and D. Orsucci, Physical review letters122, 060504 (2019)

    2019

  62. [70]

    P. C. Costa, D. An, Y . R. Sanders, Y . Su, R. Babbush, and D. W. Berry, PRX quantum3, 040303 (2022)

    2022

  63. [71]

    An and L

    D. An and L. Lin, ACM Transactions on Quantum Computing3, 1 (2022)

    2022

  64. [72]

    Jennings, M

    D. Jennings, M. Lostaglio, S. Pallister, A. T. Sornborger, and Y . Subas ¸ı, arXiv preprint arXiv:2305.11352 (2023)

    2023

  65. [73]

    Costa, D

    P. Costa, D. An, R. Babbush, and D. Berry, arXiv preprint arXiv:2312.07690 (2023)

    2023

  66. [74]

    G. H. Low and Y . Su, arXiv preprint arXiv:2410.18178 (2024)

    2024

  67. [75]

    X. Sun, G. Tian, S. Yang, P. Yuan, and S. Zhang, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems42, 3301 (2023)

    2023

  68. [76]

    K. Gui, A. M. Dalzell, A. Achille, M. Suchara, and F. T. Chong, Quantum8, 1257 (2024)

    2024

  69. [77]

    I. F. Araujo, D. K. Park, F. Petruccione, and A. J. da Silva, Scientific reports11, 6329 (2021)

    2021

  70. [78]

    B. D. Clader, A. M. Dalzell, N. Stamatopoulos, G. Salton, M. Berta, and W. J. Zeng, IEEE Transactions on Quantum Engineering3, 1 (2022)

    2022

  71. [79]

    van Apeldoorn, A

    J. van Apeldoorn, A. Cornelissen, A. Gily ´en, and G. Nannicini, inProceedings of the 2023 annual ACM-SIAM symposium on discrete algorithms (SODA)(SIAM, 2023) pp. 1265–1318

    2023

  72. [80]

    Gily ´en, Y

    A. Gily ´en, Y . Su, G. H. Low, and N. Wiebe, inProceedings of the 51st annual ACM SIGACT symposium on theory of computing(2019) pp. 193–204. 57

    2019

  73. [81]

    Gross, Y .-K

    D. Gross, Y .-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Physical review letters105, 150401 (2010)

    2010

  74. [82]

    S. Wang, A. Mondal, and A. Chattopadhyay, ACM Transactions on Quantum Computing (2024)

    2024

  75. [83]

    J. E. Moussa, Physical Review A94, 042316 (2016)

    2016

  76. [84]

    G. P. Geh ´er, C. McLauchlan, E. T. Campbell, A. E. Moylett, and O. Crawford, Quantum8, 1394 (2024)

    2024

  77. [85]

    Gidney and A

    C. Gidney and A. G. Fowler, arXiv preprint arXiv:1905.08916 (2019)

    1905 arXiv

  78. [86]

    Litinski, Quantum3, 128 (2019)

    D. Litinski, Quantum3, 128 (2019)

    2019

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.