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REVIEW 4 major objections 5 minor 58 references

First-Quantized Quantum Simulation of Non-Relativistic QED with Emergent Topologically Protected Coulomb Interactions

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Coulomb repulsion can emerge from a Gauss' law constraint in quantum simulation, and can cost less than explicit Coulomb simulation in the right regime.

desk verdict A genuinely novel simulation construction with detailed cost analysis, but the central claim that Coulomb dynamics emerge from Gauss' law is not proven: the topological-protection argument drops the matter-field coupling, and the M=O(eta) discretization bound is open. read the letter →

arxiv 2508.19343 v1 pith:TGHANIY6 submitted 2025-08-26 quant-ph

Torin Stetina , Nathan Wiebe This is my paper
classification quant-ph PACS 03.67.Ac
keywords quantumsimulationemergentCoulombinteractionGauss'lawconstraintconstrainedPauli-FierzHamiltoniantopologicalprotectionpicturefirst-quantizedQEDnon-relativisticelectrodynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that non-relativistic quantum electrodynamics can be simulated without writing down any 1/|x_i - x_j| interaction. Instead it introduces the constrained Pauli–Fierz Hamiltonian, H = H_f + λH_c, where H_c penalizes any field configuration violating Gauss' law; in the non-relativistic limit, Coulomb repulsion then emerges from the field constraint itself, mediated by field modes rather than by pairwise potentials. The authors prove a simulation cost of O~(N^{2/3}η^{4/3} t log^5(1/ε)) non-Clifford gates in the thermodynamic limit, versus O~(N^{1/3}η^{8/3} t log^2(1/ε)) for first-quantized Coulomb-only simulation, so when N ∈ o~(η^4) the field-based algorithm wins asymptotically. They also argue a toric-code-like topological protection prevents simulation-induced electric-field errors—both contractable loops and, at large volume, non-contractable loops—from breaking Coulomb's law.

What carries the argument

The load-bearing object is the constraint Hamiltonian H_c, a projector that gives unit energy penalty to any electric-field configuration whose discrete divergence disagrees with the enclosed charge. The algorithm lives in the interaction frame of λH_c plus the electric-field energy, using the interaction-picture truncated Dyson series so the query count is independent of the large penalty λ. A coherent quantum merge-sort subroutine identifies which field cells contain charges, reducing the cost of the Gauss' law check from O(Mη) to O(M+η) via the replica trick for charges, while Newton–Cotes flux integrals over cubes of side (2b+1)h control discretization error through the quadrature order

What would settle it

Take a small periodic cubic lattice, prepare a state in the kernel of the Gauss' law constraint with no contractable loops, evolve under the full constrained Pauli–Fierz Hamiltonian for time t with finite penalty λ, cutoff Λ, and speed c, and measure the expectation of a contractable loop integral ∮C E·ds and the probability of a non-contractable loop. If either quantity grows with t and does not decrease as λ, c, and the box size are increased, the claimed topological protection and emergent Coulomb law fail. The unproved M=O(η) discretization assumption can be settled by computing the L² dif

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the Coulomb interaction is not primitive but emergent: if Gauss' law is imposed as a hard constraint through a large penalty λ on a Pauli–Fierz Hamiltonian describing charged particles coupled to a discretized electromagnetic field, then the low-energy, large-volume, large-c dynamics of the field reproduces the Coulomb force law. The work further claims that this dynamics can be simulated in first quantization with a truncated Dyson-series interaction-picture algorithm at the stated gate counts, because the Gauss' law check can be implemented by coherent quantum merge sort in near-linear rather than quadratic time. Finally, it claims a topo

Load-bearing premise

Everything rests on the assumption that electric-field loops evolve independently of the charged particles: the proof treats each plaquette loop as a free oscillator and omits the particle–field coupling, while a separate unproved assumption fixes the field-mode count at O(η) as enough to resolve Coulomb's law to error ε.

Editorial extensions

If this is right

  • Chemistry and condensed-matter simulations could run without O(η²) pairwise Coulomb terms; all inter-particle interactions would be mediated locally by the electromagnetic field.
  • In the thermodynamic limit η ∝ Ω, the algorithm's cost O~(N^{2/3}η^{4/3} t log^5(1/ε)) beats the Coulomb-only cost O~(N^{1/3}η^{8/3} t log^2(1/ε)) whenever N ∈ o~(η^4), a regime the authors associate with dilute systems.
  • Simulation-induced violations of Gauss' law are suppressed by choosing λ ∈ Θ(‖H_f‖² t/ε), and the topological argument says the remaining errors cannot grow into wrong Coulomb physics in the limits c, Λ, Ω → ∞.
  • The same loop operators used in the topological protection argument can be measured during the simulation, offering a built-in error-detection signal for field-sector faults.
  • Because the model includes the field explicitly, the simulation captures finite-speed, retarded, and magnetic effects that the Coulomb Hamiltonian omits.

Reading between the lines

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

  • If a concrete bound on M—the number of field modes needed to resolve the emergent Coulomb potential to error ε—were established, the crossover regime N ∈ o~(η^4) could become an explicit resource estimate rather than an asymptotic comparison.
  • The loop viewpoint suggests a stabilizer-like code: measuring plaquette and loop observables during the simulation might allow active correction or post-selection of field errors, reducing the need for full quantum error correction.
  • The same 'constraint instead of potential' mechanism is transferable to other lattice gauge theory simulations where Gauss' law projectors already play a role, such as compact QED or Schwinger-model dynamics.
  • If the rotational-symmetry argument is made quantitative on the lattice, the method would also handle non-Coulomb, retarded interactions, so its advantage over Coulomb-only simulation may be larger than the stated comparison indicates.
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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

4 major / 5 minor

Summary. The paper proposes a first-quantized quantum simulation algorithm for the constrained Pauli-Fierz Hamiltonian H = Hf + λHc (Definition 8), in which Gauss' law is imposed as a large penalty instead of inserting a Coulomb interaction by hand. The authors present an interaction-picture truncated-Dyson-series implementation, with a quantum-merge-sort-based circuit for the constraint evolution (Lemma 12, Appendix D), block encodings for the matter-field coupling Hπ and the magnetic term Hf2 (Lemmas 16–17), and a gate count O~(N^{2/3} η^{4/3} t log^5(1/ε)) in the thermodynamic limit (Theorem 19, Eq. (60)). They claim an asymptotic advantage over Coulomb-only simulation when N ∈ õ(η^4), and argue in Section IV that contractable and non-contractable electric-field loops are topologically suppressed, so a Coulombic initial field remains Coulombic.

Significance. If the central claims were rigorously established, this would be a significant conceptual and algorithmic result: it would remove the O(η^2) pairwise Coulomb summation from first-quantized electronic structure simulation and would connect Gauss' law, topological protection, and quantum simulation cost in a new way. The paper's concrete strengths are its detailed circuit-level constructions: the merge-sort-based Gauss-law oracle with near-linear cost in M + Σ|ζ_i| (Appendix D), the explicit block encodings for Hπ and Hf2, and the careful query-count accounting leading to Theorem 19. These are substantive technical contributions that go beyond a complexity sketch. However, the two load-bearing pillars of the paper — the emergent-Coulomb derivation and the M = O~(η) discretization assumption — are not established at the same level of rigor, and the manuscript itself acknowledges the latter as open. The failure of these pillars materially undermines the advertised scaling advantage.

major comments (4)
  1. [IV, Proposition 23 (proof around Eqs. (50)–(53))] The no-contractable-loop argument is made for a per-plaquette oscillator H□ = h^3/(4π)(c^2P^2/2 + Q^2/(2h^2)) with P = (∇×A) and Q = hE□. The full dynamics is H = Hf + λHc (Definition 8); Hf contains Hπ of Eq. (16), including ∇·A and A^2 particle-field couplings. These terms are absent from H□, and no bound on [Hπ, E□] or on ∥H − H□∥ is provided. The invoked inequality ∥e^{−iHt} − e^{−iH̃t}∥ ≤ ∥H − H̃∥t is therefore not usable as stated. Hence the conclusion that contractable loops are not generated under the full dynamics is unproved, and Corollary 24 plus the emergent-Coulomb conclusion rest on this gap. The proof also specializes to coherent initial states while the proposition quantifies over all states in the kernel of the plaquette projectors.
  2. [IV, Lemma 22] Item 1 reads 'Pr(F_{P;m}=0) if m is non-constant or if P is not a closed loop' — this is not a well-formed probability statement. No proof of the lemma is supplied; the preceding paragraph gives only a heuristic energy/Chebyshev argument for non-contractable loops. Since Lemma 22 is used to suppress non-contractable loops in the thermodynamic limit and is invoked by Corollary 24, the formal statement and proof must be corrected before the topological-protection claim can be evaluated.
  3. [V, Eq. (57) and VI] The claimed advantage of Eq. (60) over Eq. (61) assumes M ∈ O~(η). Section VI explicitly states that the field-grid size and volume needed for an ε-approximation to the correct Coulomb dynamics are an open question and are 'essential for understanding the complexity tradeoffs'. Without a discretization bound connecting M to the target error ε, the complexity comparison is not established. This is not a minor caveat: the central algorithmic advantage over Coulomb-only simulation depends on M being small enough while still resolving the emergent interaction to accuracy ε.
  4. [IV, paragraph preceding Eq. (55)] The derivation that the field remains Coulombic uses the claim that the Hamiltonian is rotationally symmetric in the continuum limit. The manuscript explicitly leaves 'the precise role that discretization has on the rotational symmetry for future work'. On a finite cubic lattice with periodic boundary conditions, the discrete Hamiltonian is not rotationally symmetric, and the limiting argument from discrete dynamics to continuous rotational symmetry is not supplied. This is a further unproven step in the emergence argument.
minor comments (5)
  1. [Abstract] Typo: 'energetically dissallowed' should be 'energetically disallowed'.
  2. [Definition 7 and Eq. (12)] The function 'rect' is used without a formal definition of its argument and range; please specify it explicitly.
  3. [Eq. (40)] The projector Π_coul is introduced informally. If it is intended as a projector, define it with a clear domain and action.
  4. [Appendix B and Section III] The truncation error of the finite-difference kinetic operator is stated to be left to subsequent work; this assumption should also be stated in Section III where a is set to a constant, since it affects the claimed error scaling.
  5. [IV, Eq. (54)] The probability bound contains terms such as Ω^{2/3}/h^2; please check dimensional consistency and clarify the lattice units used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Coulomb-from-Gauss'-law claim is a non-circular limit argument, though it rests on unproven topological-protection and discretization bounds.

full rationale

The central derivation does not reduce to its inputs by construction. The Hamiltonian H = Hf + λHc (Definition 8) and the interaction-picture simulation (Definition 9, Lemma 11) are defined independently of the Coulomb target; no parameter is fitted to the quantity being predicted, and the asymptotic gate count in Theorem 19 follows from norm estimates and block-encoding costs, not from assuming the conclusion. The emergence argument in Section IV is conditional: initial states are restricted to the kernel of Hc and to states satisfying Coulomb's law (Πcoul), and the paper argues this subspace is approximately preserved. Even if the argument is incomplete, that is a correctness gap, not circularity: Gauss' law plus rotational symmetry is used to obtain E = ζr/|r|^3, and the Coulomb law is not inserted as a term in the simulated Hamiltonian. Self-citations to Refs. [3,9,11] supply block-encoding decompositions, the truncated Dyson / interaction-picture method, and a constraint-error lemma; these are external constructions or theorems with stated assumptions and do not presuppose the paper's main claim. Two in-scope limitations should be weighed as correctness risks, not circularity. First, Proposition 23 introduces a per-plaquette oscillator H□_{q,x} = h^3/(4π)(c^2P^2/2 + Q^2/(2h^2)) and then invokes ∥e^{-iHt} - e^{-iH̃t}∥ ≤ ∥H - H̃∥t without bounding ∥H - H□∥; in particular the matter-field coupling Hπ from Eq. (16) is absent from H□, so the no-contractable-loop conclusion is not derived for the full dynamics. Second, Section VI explicitly defers the discretization bound that would justify M = O~(η), which is needed for the claimed advantage of Eq. (60) over Eq. (61). These are missing proofs, not definitional or fitted-input circularity.

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

The paper introduces no fitted free parameters: λ is set by the error bound, and a, b, Λ, M, N are discretization parameters with stated error targets. The load-bearing content is in the smoothness assumptions (L/Λ bounds) and the assumed decoupling of plaquette modes from the particle dynamics, which are ad hoc to this paper and not independently verified.

assumptions (6)
  • domain assumption The Pauli-Fierz Hamiltonian is the correct single-particle non-relativistic QED model and can be extended to multiple particles by adding a Gauss' law constraint.
    Section II takes the Pauli-Fierz Hamiltonian [8] as given and builds the multi-particle theory by adding λHc; the correctness of this extension for the multi-particle back-action is the foundation of the work.
  • domain assumption A finite strength-penalty λHc with λ ∈ Θ(||H0||²∞ t/ε) confines the dynamics to the kernel of Hc with error ε.
    Invoked in Section III via Lemma 10 from Ref. [11]; requires Hc ⪰ 0 with spectral gap at least 1, which the paper assumes holds for its discrete Gauss' law operator.
  • ad hoc to paper Input states are smooth in the sense that sup_{q,μ} ||(U_{q,μ} − I)|ψ⟩|| = O(L/Λ) for a constant L.
    Used in Proposition 21, Proposition 23 and Lemma 25 to control commutator and Taylor errors; no bound on L is derived from the physics, and the assumption is not stated as a testable condition.
  • ad hoc to paper The electric field plaquette modes behave as independent harmonic oscillators decoupled from the particle motion, so that H□ determines the loop dynamics by itself.
    In the proof of Proposition 23, the Hamiltonian H□ = (h³/4π)(c²P²/2 + Q²/(2h²)) omits the particle-field coupling term Hπ of Definition 8; the particles are the source of the fields, so this decoupling is a substantive unproven assumption.
  • ad hoc to paper In the continuum limit the discretized Hamiltonian is rotationally invariant, so a rotationally symmetric initial state remains symmetric.
    Section IV, paragraph after Eq. (55): 'We leave a discussion of the precise role that discretization has on the rotational symmetry for future work.' The derivation of E = ζr/|r|³ requires this symmetry.
  • domain assumption The initial state is prepared in the kernel of Hc and of the plaquette projector Π□, and its field satisfies Coulomb's law.
    Section IV defines the code space and projector Πcoul (Eq. 40); the paper proves preservation, not creation, of the Coulomb condition, so the input restriction is a stated assumption of the emergence result.

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Pith. "Pith review of First-Quantized Quantum Simulation of Non-Relativistic QED with Emergent Topologically Protected Coulomb Interactions." pith.science (2026). https://pith.science/paper/TGHANIY6

@misc{pith2026250819343,
  author       = {Pith},
  title        = {Pith review of: First-Quantized Quantum Simulation of Non-Relativistic QED with Emergent Topologically Protected Coulomb Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TGHANIY6}},
  note         = {Machine review of arXiv:2508.19343}
}
abstract

We provide a simulation algorithm that properly addresses light matter interaction between non-relativistic first-quantized charged particles and quantum electromagnetic fields. Unlike previous work, our Hamiltonian does not include an explicit Coulomb interaction between particles. Rather, the Coulomb interaction emerges from the imposition of Gauss' law as a constraint upon the system in an appropriate non-relativistic limit. Furthermore, a form of topological protection emerges in our formalism, analogous to that of the Toric code Hamiltonian. This mechanism prevents simulation-induced electric field errors that can be contracted to a point from causing any deviations from Coulomb's law in the non-relativistic limit and any error that forms a non-contractable loop is energetically dissallowed in the limit of large volume. We find that, under appropriate continuity assumptions, the number of non-Clifford gates required by our algorithm scales in the thermodynamic limit as $\widetilde{O}(N^{2/3}\eta^{4/3} t \log^5(1/\epsilon))$ for $\eta$ particles, $N$ spatial grid points, simulation time $t$ and error tolerance $\epsilon$. In comparison, the more specific problem of simulating the Coulomb interaction scales as $\widetilde{O}(N^{1/3} \eta^{8/3} t \log^2(1/\epsilon))$. This suggests that if $N \in \tilde{o}(\eta^4)$ that our non-relativistic electrodynamic simulation method could provide a computational advantage for electronic structure problems in the thermodynamic limit under appropriate continuity assumptions as it obviates the need to compute the $O(\eta^2)$ pairwise interactions in the Coulomb Hamiltonian.

Figures

Figures reproduced from arXiv: 2508.19343 by the authors.

Figure 1
Figure 1. FIG. 1. Visualization of three types of error patterns considered on top of a background field that satisfies Gauss’ law. Case [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

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Works this paper leans on

58 extracted references · 54 canonical work pages

  1. [1]

    Babbush, D

    R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, Quantum simulation of chemistry with sublinear scaling in basis size, npj Quantum Information 5, 92 (2019). 22

    work page 2019

  2. [3]

    Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian

    P. Mukhopadhyay, T. F. Stetina, and N. Wiebe, Quantum simulation of the first-quantized pauli-fierz hamiltonian, arXiv preprint arXiv:2306.11198 (2023)

    work page Pith review arXiv 2023

  3. [4]

    D. W. Berry, N. C. Rubin, A. O. Elnabawy, G. Ahlers, A. E. DePrince III, J. Lee, C. Gogolin, and R. Babbush, Quantum simulation of realistic materials in first quantization using non-local pseudopotentials, npj Quantum Information 10, 130 (2024)

    work page 2024

  4. [5]

    T. N. Georges, M. Bothe, C. S¨ underhauf, B. K. Berntson, R. Izs´ ak, and A. V. Ivanov, Quantum simulations of chemistry in first quantization with any basis set, npj Quantum Information 11, 55 (2025)

    work page 2025

  5. [6]

    W. J. Huggins, O. Leimkuhler, T. F. Stetina, and K. B. Whaley, Efficient state preparation for the quantum simulation of molecules in first quantization, PRX Quantum 6, 020319 (2025)

    work page 2025

  6. [7]

    Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Babbush, Fault-tolerant quantum simulations of chemistry in first quantization, PRX Quantum 2, 040332 (2021)

    work page 2021

  7. [8]

    Hiroshima, Self-adjointness of the pauli-fierz hamiltonian for arbitrary values of coupling constants, in Annales Henri Poincar´ e, Vol

    F. Hiroshima, Self-adjointness of the pauli-fierz hamiltonian for arbitrary values of coupling constants, in Annales Henri Poincar´ e, Vol. 3 (Springer, 2002) pp. 171–201

    work page 2002

  8. [9]

    G. H. Low and N. Wiebe, Hamiltonian simulation in the interaction picture, arXiv preprint arXiv:1805.00675 (2018)

    arXiv 2018

Show all 58 references
  1. [10]

    Kieferov´ a, A

    M. Kieferov´ a, A. Scherer, and D. W. Berry, Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series, Physical Review A 99, 042314 (2019)

    2019

  2. [11]

    Rajput, A

    A. Rajput, A. Roggero, and N. Wiebe, Hybridized methods for quantum simulation in the interaction picture, Quantum 6, 780 (2022)

    2022

  3. [12]

    Zlokapa and R

    A. Zlokapa and R. D. Somma, Hamiltonian simulation for low-energy states with optimal time dependence, arXiv preprint arXiv:2404.03644 (2024)

    2024 arXiv

  4. [13]

    Gily´ en, Y

    A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) pp. 193–204

    2019

  5. [14]

    A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Information & Computation 12, 901 (2012)

    2012

  6. [15]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear t complexity, Physical Review X 8, 041015 (2018)

    2018

  7. [16]

    He, M.-X

    Y. He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, Decompositions of n-qubit toffoli gates with linear circuit complexity, International Journal of Theoretical Physics 56, 2350 (2017)

    2017

  8. [17]

    Di Matteo, V

    O. Di Matteo, V. Gheorghiu, and M. Mosca, Fault-tolerant resource estimation of quantum random-access memories, IEEE Transactions on Quantum Engineering 1, 1 (2020)

    2020

  9. [18]

    N. C. Rubin, D. W. Berry, F. D. Malone, A. F. White, T. Khattar, A. E. DePrince III, S. Sicolo, M. K¨ uehn, M. Kaicher, J. Lee, et al. , Fault-tolerant quantum simulation of materials using bloch orbitals, PRX Quantum 4, 040303 (2023)

    2023

  10. [19]

    Mukhopadhyay, A quantum random access memory (qram) using a polynomial encoding of binary strings, arXiv preprint arXiv:2408.16794 (2024)

    P. Mukhopadhyay, A quantum random access memory (qram) using a polynomial encoding of binary strings, arXiv preprint arXiv:2408.16794 (2024)

    2024 arXiv

  11. [20]

    M. Amy, D. Maslov, M. Mosca, and M. Roetteler, A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 32, 818 (2013)

    2013

  12. [21]

    Gheorghiu, M

    V. Gheorghiu, M. Mosca, and P. Mukhopadhyay, A (quasi-) polynomial time heuristic algorithm for synthesizing t-depth optimal circuits, npj Quantum Information 8, 110 (2022)

    2022

  13. [22]

    Kliuchnikov, D

    V. Kliuchnikov, D. Maslov, and M. Mosca, Practical approximation of single-qubit unitaries by single-qubit quantum clifford and t circuits, IEEE Transactions on Computers 65, 161 (2015)

    2015

  14. [23]

    N. J. Ross and P. Selinger, Optimal ancilla-free clifford+ t approximation of z-rotations., Quantum Inf. Comput. 16, 901 (2016)

    2016

  15. [24]

    Gheorghiu, M

    V. Gheorghiu, M. Mosca, and P. Mukhopadhyay, T-count and t-depth of any multi-qubit unitary, npj Quantum Information 8, 1 (2022)

    2022

  16. [25]

    Mukhopadhyay, Composability of global phase invariant distance and its application to approximation error management, Journal of Physics Communications 5, 115017 (2021)

    P. Mukhopadhyay, Composability of global phase invariant distance and its application to approximation error management, Journal of Physics Communications 5, 115017 (2021)

    2021

  17. [26]

    J. B. Kogut, An introduction to lattice gauge theory and spin systems, Reviews of Modern Physics 51, 659 (1979)

    1979

  18. [27]

    Li, General explicit difference formulas for numerical differentiation, Journal of Computational and Applied Mathematics 183, 29 (2005)

    J. Li, General explicit difference formulas for numerical differentiation, Journal of Computational and Applied Mathematics 183, 29 (2005)

    2005

  19. [28]

    I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik, Bounding the costs of quantum simulation of many-body physics in real space, Journal of Physics A: Mathematical and Theoretical 50, 305301 (2017)

    2017

  20. [29]

    A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, Quantum algorithms for simulating the lattice schwinger model, Quantum 4, 306 (2020)

    2020

  21. [30]

    Niemann, R

    P. Niemann, R. Datta, and R. Wille, Logic synthesis for quantum state generation, in 2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL) (IEEE, 2016) pp. 247–252

    2016

  22. [31]

    Cheng and C.-Y

    S.-T. Cheng and C.-Y. Wang, Quantum switching and quantum merge sorting, IEEE Transactions on Circuits and Systems I: Regular Papers 53, 316 (2006)

    2006

  23. [32]

    Hoyer, Neerbek, and Shi, Quantum complexities of ordered searching, sorting, and element distinctness, Algorithmica 34, 429 (2002)

    2002

  24. [33]

    Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018)

    C. Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018)

    2018

  25. [34]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010). 23

    2010

  26. [35]

    Cleve and J

    R. Cleve and J. Watrous, Fast parallel circuits for the quantum fourier transform, in Proceedings 41st Annual Symposium on Foundations of Computer Science (IEEE, 2000) pp. 526–536

    2000

  27. [36]

    Simon, M

    S. Simon, M. Degroote, N. Moll, R. Santagati, M. Streif, and N. Wiebe, Amplified amplitude estimation: Exploiting prior knowledge to improve estimates of expectation values, arXiv preprint arXiv:2402.14791 (2024). Appendix A: Discrete Couloumb Gauge T erm An interesting conseq...

    2024 arXiv

  28. [37]

    supq,µ,Sϕ ∥Aq,µ |ϕ⟩ ∥ ∈O(Λγ/Emax) for universal constant γ ∈ [0, 1),

  29. [38]

    the discrete Coulomb Gauge integral term yields sup|ϕ⟩∈Sϕ,Dq;b | ⟨ϕ| ‚ Dq;b A·dS |ϕ⟩ |2 = 0 where the integral refers to an integral over the cube Dq;b using a 2b + 1-point Newton-Cotes formula,

  30. [39]

    there exists ℵ ∈R such that for any q, µin the cubic lattice ∥ ⟨ϕ| Uq,µ − 11 |ϕ⟩ ∥ ≤ ℵ/Λ. The expectation value of the Hamiltonian and the approximate Coulomb gauge constraint within the set of states Sϕ ⟨ϕ| [H, HAc(q)] |ϕ⟩ := ⟨ϕ|  H, h2 2 2X µ=0 X u∈Sq,µ;b βu11 ⊗ (Aq+beµ+u,...

  31. [40]

    where Rr j,x∈Dq is the reflection operator with a control on the state of qubit |r⟩

    The ancillae preparation routine is, PREPj,µ 23π = 1√ M M −1X q=0 |q⟩ ! ⊗ 1√ 2 1X r=0 |r⟩ ! ⊗   r 2λ1 A1 |0⟩ + s λ′ 1 A1 |1⟩   ⊗PREPj,q,µ 2π ⊗ PREPj,q,µ 3π ; (C12) 28 and the unitary selection routine is, SELECTj,µ 23π : |q, r,0⟩ |k1, k2, k3⟩ |ϕ⟩ 7→ |q, r,0⟩ |k3⟩ Rr j,x∈Dq...

  32. [41]

    Now, again we use Theorem 15 in order to block encode Hπ using the block encodings of H j,µ 23 A′ 1 and H j,µ 1π λ2

    (C13) Therefore, we have that A′ 1 = 4πM ln(2a2) h∆ + 4πM h2 (C14) and ⟨0|(PREPj,µ 23π)† · SELECTj,µ 23π · PREPj,µ 23π|0⟩ = H j,µ 23π A′ 1 . Now, again we use Theorem 15 in order to block encode Hπ using the block encodings of H j,µ 23 A′ 1 and H j,µ 1π λ2 . Let A2 = λ2 + A′

  33. [42]

    Thus the entire ancilla preparation routine can be described as follows. PREPπ =   1√η η−1X j=0 |j⟩   ⊗ 1√ 3 2X µ=0 |µ⟩ ! ⊗   r λ2 A2 |0⟩ + s A′ 1 A2 |1⟩   ⊗ 1√ M M −1X q=0 |q⟩ ! ⊗ 1√ 2 1X r=0 |r⟩ ! ⊗   r 2λ1 A1 |0⟩ + s λ′ 1 A1 |1⟩   ⊗ PREPj,µ 1π ⊗ PREPj,q,µ 2π ⊗ P...

  34. [43]

    ⌈log2(2a + 1)⌉ qubits (PREPj,µ 1π )

  35. [44]

    ⌈log2(2a + 1)⌉ qubits (PREPj,q,µ 2π first register)

  36. [45]

    ⌈log2(log2(d))⌉ qubits (PREPj,q,µ 2π second register)

  37. [46]

    ⌈log2 log2 2(d)+log2(d) 2 ⌉ qubits (PREPj,q,µ 3π ) The following break down of gate costs is computed using the synthesis approach in Ref. [30]. Both Clifford and T-gates are included for completeness but only T-gates are included in the final asymptotic cost. The first regist...

  38. [47]

    For each particle in position |xi⟩ compute |xi⟩ |Bi⟩ where |Bi⟩ is an O(log(M )) qubit string corresponding to the cell number of the field that the particle is in

  39. [48]

    Using quantum merge sort, sort all particles by their value of |Bi⟩ and append a ⌈log(η)⌉ bit register, denoted Qi in the following, to each of these particles initialized to the state |1⟩. This yields for any computational basis state of particle positions a state of the form...

  40. [49]

    For i decreasing from η to 2 compare Bfi to Bfi−1 and if they are the same add ζfi to register Qi−1 and swap register Qi with an unentangled ancilla containing |0⟩

  41. [50]

    Sort all registers using quantum merge sort on based on B. Specifically, sort each register |xfi ⟩ |Bfi ⟩ |Qi⟩ by the value of Bfi (1 − δQi,0) + δQi,0 which will order the states such that any particle that has been counted with all the other ones will be sorted out of the list

  42. [51]

    ηsearch over its (2b + 1)3 − 1 neighbors in the list and if |Bj⟩ is in the cube create a fictitious charge of Qi at field location Bj of the form |Bj⟩ |Qi⟩

    For each particle i = 1 . . . ηsearch over its (2b + 1)3 − 1 neighbors in the list and if |Bj⟩ is in the cube create a fictitious charge of Qi at field location Bj of the form |Bj⟩ |Qi⟩

  43. [52]

    Sort each of the ((2 b + 1)3 − 1)η field / charge site |Bi⟩ |Qi⟩ by their cell number Bi using quantum merge sort

  44. [53]

    For i in decreasing order, if |Bi⟩ = |Bi−1⟩ then add M to Bi and then merge sort again the resulting array by Bi to remove the duplicates of the fictitious charges

  45. [54]

    For each of the ((2 b + 1)3 − 1)ν fictitious charges search merge sort the arrays based on coordinate µ of Bi and compute sum of the charges along each of the closest (2 b + 1)3η that are within an L1 distance b of Bi

  46. [55]

    Sum the results of the above steps of each of the ((2 b + 1)3 − 1)ν locations and add the result together in a new register |Qtot,i⟩

  47. [56]

    For each field cell, q, compute the divergence to prepare a state of the form | ⃗E⟩ |0⟩ |D0⟩ · · · |M − 1⟩ |DM −1⟩ , where each Dq := round h2 8π P2 µ=0 P u,v∈Sq,µ;b βu,vI ⊗ (Eq+beµ,µ − Eq−beµ,µ) is the electric flux computed for the cube of length (2 b + 1)h centered at cell ...

  48. [57]

    Use quantum merge sort to sort the |q⟩ |Dq⟩ vectors by q(1 − δDq,0) − δDq,0 which corresponds to q if Dq ̸= 0 and −1 otherwise. This yields for any computational basis state of position and field (after an irrelevant permutation) a state of the form | ⃗E⟩ |g1⟩ |Dg1 ⟩ |x1⟩ |Bf1...

  49. [58]

    For each |Bfi ⟩ and compare the integer to |gi⟩ and if equal add the register Qfi the register containing the corresponding divergence |Qfi ⟩ |Dgi ⟩ → |Qfi ⟩ |Dgi − Qfi ⟩. 34

  50. [59]

    Let us consider step 1

    Implement the isometry |Dg1 ⟩ · · · |Dgη ⟩ → ( |Dg1 ⟩ · · · |DgM ⟩ |0⟩ if Dgi = 0 ∀ i |Dg1 ⟩ · · · |DgM ⟩ |1⟩ otherwise This final qubit computes rect P q ∆2 2 P2 µ=0 I ⊗ (Eq+eµ,µ − Eq−eµ,µ) − 4π Pη−1 i=0 sign(ζi)Πq,i . Let us consider step 1. As the particles are in a grid, w...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.