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REVIEW 2 major objections 5 minor 101 references

A hardware-efficient variational ansatz with an exact diagonal metric for real- and imaginary-time evolution and Haar sampling

T0 review · 2 major / 5 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read A binary-tree quantum ansatz makes the Fubini–Study metric diagonal in closed form, so natural-gradient optimisation, real- and imaginary-time evolution, and sector-Haar sampling need no auxiliary metric circuits or matrix inversion.

desk verdict Solid, self-contained method paper: closed-form diagonal FS metric on a binary-tree ansatz is proved and used cleanly; pruning and dequantization boundary are honest; advantage language is prospective. read the letter →

arxiv 2607.07942 v1 pith:7M2M7EYT submitted 2026-07-08 quant-ph

Dario Picozzi This is my paper
classification quant-ph
keywords variationalquantumalgorithmsFubini-Studymetricnaturalgradientbinary-treeansatzhardware-efficientreal-timeevolutionHaarsamplingelectronicstructure
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

Variational quantum algorithms usually pay a heavy price for geometry: the Fubini–Study metric that steers natural gradient and time evolution is dense, must be estimated with many auxiliary circuits, and then inverted. This paper builds an n-qubit ansatz as a binary tree of controlled rotations whose pullback metric is exactly diagonal and readable from subtree and leaf probabilities. With that structure, quantum natural gradient, variational imaginary- and real-time evolution, and exact Haar sampling on a symmetry sector reduce to ordinary energy evaluations plus elementwise division—no metric circuits, no pseudoinverse. When the target lives on only k basis states, a pruning compiler turns gauge freedom into circuits whose two-qubit count grows linearly in k, and the same metric stays closed-form. On small-molecule electronic structure and Hubbard quenches the method matches reference accuracy with far fewer two-qubit gates than standard alternatives, while the bare tree remains classically simulable in k and becomes a candidate for quantum advantage only when composed with a classically hard dressing.

What carries the argument

Theorem 1 (diagonal chart metric): the binary-tree parametrisation makes g_μν = Re⟨∂_μψ|∂_νψ⟩ exactly diagonal, with entries equal to path-product subtree weights w_i and leaf probabilities p_j; the inverse is therefore the Hadamard reciprocal of those probabilities, and the same diagonal structure drives both imaginary-time descent and the real-time complex-structure rotation.

What would settle it

On a fixed sparse support with k active leaves, compile the pruned tree and count CNOTs after the inactive-node reordering: if the count systematically exceeds 2(A_s + k − 2) for many supports with k greater than 2, the claimed tight scaling fails; separately, if a classically contractible dressing still reproduces the dressed energy and correlators of the hard-dressed applications, the quantum-advantage boundary does not hold as stated.

Watch

Extended reading notes

Core claim

For the binary-tree ansatz the chart pullback of the Fubini–Study metric is block-diagonal between amplitude angles and leaf phases, and each block is itself diagonal with closed-form entries equal to subtree weights and leaf probabilities. Consequently the entire family of metric-aware updates—quantum natural gradient, imaginary-time flow, real-time Kähler flow, and exact sector-Haar sampling—runs from parameter-shift energy gradients alone, with no auxiliary metric measurements and no matrix inversion.

Load-bearing premise

The tight linear two-qubit bound that makes the pruned circuit especially cheap is proved only for two active leaves and checked numerically up to eight qubits; for larger sparse supports it is still a conjecture, and any claim of quantum advantage further assumes a dressing that is classically hard to simulate.

Editorial extensions

If this is right

  • Metric-aware VQE and variational real-time evolution on this ansatz cost only O(P) classical work per step on top of ordinary parameter-shift energy gradients, eliminating the usual O(P²) metric circuits and O(P³) inversion.
  • Any diagonally defined symmetry sector (particle number, S_z, point-group irrep, fixed Hamming weight) can be prepared exactly by choosing active leaves, with no penalty terms and with the closed-form metric intact.
  • Drawing tree parameters from the Fubini–Study volume measure yields exact, rejection-free Haar samples on the chosen sector at O(k) classical cost per sample, enabling unbiased process benchmarking and infinite-temperature correlators.
  • When the target is k-sparse, the pruned circuit’s two-qubit count is at most linear in k (unconditionally O(n²k)), and a routing-dressed variant reaches near-optimal O(nk/log n) while keeping the diagonal metric.
  • The bare tree is an exactly controllable, barren-plateau-free, classically simulable primitive; genuine quantum content appears only after composition with a dressing that destroys the sparse path-product structure.

Reading between the lines

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

  • The same diagonal-metric primitive could serve as a feasibility-preserving ansatz for constrained combinatorial optimisation, because any constraint that is diagonal in the computational basis becomes an active-leaf set with no penalty term.
  • Because the bare geometry is free of barren plateaus on polynomially sized sectors, the construction isolates trainability risk inside the dressing block alone—an architecture that may generalise to other ‘easy core + hard dressing’ variational schemes.
  • A hardware demonstration that reads the metric preconditioner from measured subtree probabilities (rather than Hadamard tests) would be a direct, device-level test of the paper’s central geometric claim at present-day depths.
  • If a polynomial-time qubit ordering can be proved to attain the tight CNOT bound for general k, the pruning compiler would match the best known sparse-state-preparation asymptotics while uniquely retaining closed-form geometry.
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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

2 major / 5 minor

Summary. The paper introduces a binary-tree variational ansatz of uniformly controlled Ry/Rz rotations whose Fubini–Study pullback metric is exactly diagonal in closed form (Theorem 1), with diagonal entries equal to subtree weights and leaf probabilities. This removes auxiliary metric circuits and matrix inversion for quantum natural gradient, variational imaginary- and real-time evolution, and exact sector-Haar sampling. A pruning compiler that classifies nodes as Active/Fixed/Inactive converts gauge freedom on k-sparse supports into circuits whose CNOT count is unconditionally O(n^{2}k) (Proposition 2) and conjecturally O(nk). Benchmarks on small-molecule VQE and Hubbard quenches report reference-level accuracy at substantially lower two-qubit depth than UCCSD, pVQD, VarQRTE and Trotter. The bare ansatz is proved classically simulable in k for a general class of sector-sparse families (Appendix S); quantum content is located in composition with a hard dressing, with spin-adapted and two-block constructions supplied.

Significance. If the results hold, the work supplies a rare, fully explicit geometry-aware primitive: a complete, hardware-efficient chart on any chosen k-dimensional sector whose metric, gradients, Kähler real-time RHS and Haar measure are all closed-form. The dequantization boundary (Theorem 2) is a genuine contribution that cleanly separates what is classical from what can be quantum, and the open-source QuantumSymmetry implementation makes the claims checkable. The unconditional linear-in-k pruning, exact spin adaptation without penalties, and the four-term shift rule are concrete engineering advances. The quantum-advantage language for dressed circuits is prospective rather than demonstrated, but the paper already labels it as such and does not rest the central metric claim on it.

major comments (2)
  1. The structure-dependent tight CNOT bound N_cx ≤ 2(A_s + k − 2) (and the sparse form ≤ 2nk − 2) is proved only for k = 2 and verified numerically under inactive-node-maximising reordering through n = 8 (Appendix L, Eq. (L6) and Table III). The unconditional result is the weaker O(n^{2}k) of Proposition 2. The abstract and Sec. I H should state the proved versus conjectural status of the tight constant with equal prominence, so that the linear-scaling claim is not overstated.
  2. Quantum-advantage claims for the composed circuit (abstract final sentence; Sec. III) rest on a dressing that simultaneously escapes stabiliser, Gaussian, light-cone and tensor-network contraction (Corollary 1). No explicit hard instance is exhibited, and all numerical demonstrations remain classically checkable. The language should be tightened to “potential” / “prospective” throughout, matching the careful wording already present in parts of Sec. III, so that the central metric and pruning results are not yoked to an undemonstrated hardness claim.
minor comments (5)
  1. Figure 1 and the running n = 4, k = 5 example are excellent; a short caption cross-reference to the corresponding pruned gate list in Fig. 3 would help readers who land on the figure first.
  2. Appendix L Table IV reports empirical slopes only up to n = 8; a one-sentence note on why larger-n sampling was not feasible (branch-and-bound cost) would forestall questions.
  3. The four-term shift rule (Eq. 9) and the real-target two-term reduction (Eq. 11) are derived carefully in Appendix H; a forward pointer in Sec. I E to the shot-noise variance comparison with the Wierichs equispaced family would be useful for experimentalists.
  4. Typographical consistency: “ansätze” / “ansätze” and “Kähler” / “K¨ahler” appear in both forms; standardise.
  5. Code availability is exemplary; adding a one-line DOI or commit hash for the exact figure-generating scripts would improve long-term reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the diagonal metric, pruning bounds, and dequantization boundary are self-contained derivations validated against independent external benchmarks.

full rationale

Theorem 1 (closed-form diagonal chart metric) is proved by a direct Hilbert-space argument on subtree supports and path products (Appendix B), not by fitting or by importing a uniqueness claim. Metric invariance under fixed dressing (Proposition 1), the Kähler real-time RHS (Appendix F), the four-term shift rule (Appendix H), and the general sector-sparse dequantization theorem (Appendix S / Theorem 2) are likewise self-contained. Numerical claims are checked against independent references—FCI/CASCI, exact diagonalisation, UCCSD, pVQD, VarQRTE, and Trotter—not against quantities fitted from the same data. Self-citations (QuantumSymmetry package; prior symmetry encodings) supply software and encoding infrastructure and are not load-bearing for the metric theorem or the unconditional O(n²k) CNOT bound (Proposition 2). The tight CNOT constant and dressed quantum-advantage language are already labeled conjectural/prospective in the manuscript and do not force the central geometric claims by construction. No self-definitional loop, fitted-input-as-prediction, or uniqueness-from-authors pattern is present.

Assumptions & free parameters 5 free parameters · 5 assumptions · 3 invented entities

The central metric theorem rests on standard Hilbert-space geometry and the explicit tree amplitude map; no fitted physical constants enter the claim. Free parameters appear only in numerical optimisers and baseline depths. Invented entities are the ansatz structure and compiler classification, which are constructive circuit objects with independent mathematical handles (metric formulas, CNOT bounds), not postulated physical mediators.

free parameters (5)
  • Natural-gradient / imaginary-time step size (learning rate)
    Absorbs the overall 1/(2ℏ) factor and controls optimiser progress; chosen for numerics, not derived.
  • Real-time integrator step dt and step count N
    Euler/RK4 timesteps (e.g. dt=0.002, N=500 on Hubbard) are numerical choices that set truncation error.
  • HEA / HVA repetition count at 'sweep knee'
    Baseline depths fixed per molecule by a dedicated repetition sweep; affects reported CNOT and infidelity floors for pVQD and VarQRTE.
  • ADAPT gradient / cost-aware screening thresholds for dressing pools
    Selects which P→Q generators enter the two-block and decoupling dressings; matched parameter or CNOT budgets are design choices.
  • Dipole-kick strength κ and incomplete-subspace size k
    κ=0.05 (and κ=1 variant) and chosen incomplete supports control dynamics benchmarks and truncation floors.
assumptions (5)
  • standard math Pure states form CP^{N-1} with Fubini–Study metric g=Re⟨·|·⟩ and compatible Kähler structure J from multiplication by i (Provost–Vallée / standard geometric quantum mechanics).
    Used throughout Sec. I B–I C and Appendices A–D to identify natural gradient and real-time flow.
  • standard math Uniformly controlled R_y/R_z rotations synthesise via Gray-code CNOT cascades and Walsh–Hadamard angle maps (Möttönen et al.).
    Defines the hardware scaffold of the tree ansatz (Sec. I A, Appendix E).
  • domain assumption Molecular and Hubbard Hamiltonians in STO-3G / standard lattice encodings, with FCI/CASCI/ED as ground truth for small systems.
    Benchmark claims of reference-level accuracy depend on these standard electronic-structure setups (Sec. II A–II C).
  • domain assumption A dressing U(ϕ) that escapes stabiliser, Gaussian, light-cone, and polynomial-bond-dimension contraction is classically hard for the relevant dressed observables.
    Load-bearing for the quantum-advantage discussion (Sec. III, Corollary 1); not proved for a concrete deep U in the paper.
  • ad hoc to paper Inactive-node-maximising qubit reordering attains the tight combinatorial run-count inequality for general k (conjectural).
    Needed for the sharp N_cx ≤ 2nk−2 claim beyond the proved O(n²k) bound (Appendix L).
invented entities (3)
  • Binary-tree variational ansatz with closed-form diagonal FS metric independent evidence
    purpose: Parametrise states so metric-aware updates and Haar sampling need only path-product weights and shift-rule gradients.
    Constructive circuit family; independent handle is Theorem 1 and the explicit weight formulas (2)–(4).
  • Active/Fixed/Inactive node classification and pruning compiler independent evidence
    purpose: Convert gauge freedom on sparse support S into linear-in-k CNOT circuits while keeping the diagonal metric.
    Algorithmic object with proved integrality and unconditional CNOT scaling; independent of any physical postulate.
  • MinimalMetric (MM) unified method independent evidence
    purpose: Name the pruned ansatz plus diagonal-metric updates and shift-rule oracle used in all benchmarks.
    Branding of the composed algorithm, not a new physical entity.

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Pith. "Pith review of A hardware-efficient variational ansatz with an exact diagonal metric for real- and imaginary-time evolution and Haar sampling." pith.science (2026). https://pith.science/paper/7M2M7EYT

@misc{pith2026260707942,
  author       = {Pith},
  title        = {Pith review of: A hardware-efficient variational ansatz with an exact diagonal metric for real- and imaginary-time evolution and Haar sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7M2M7EYT}},
  note         = {Machine review of arXiv:2607.07942}
}
abstract

Variational quantum algorithms depend on the geometry of their parametrised circuits: metric-aware optimisation and time evolution require the Fubini-Study metric, which has hitherto demanded costly auxiliary measurements and ill-conditioned inversions. This work introduces a hardware-efficient $n$-qubit ansatz, which parametrises states by a binary tree and whose Fubini-Study pullback metric is diagonal in closed form. Quantum natural gradient on the tree parameters, variational imaginary- and real-time evolution, and exact unitary-invariant (Haar) sampling on a symmetry sector run with no auxiliary metric circuits or matrix inversion. When the target state is supported on a subspace of $k$ computational-basis states, the redundant tree parameters carry a gauge freedom a pruning compiler converts into circuits whose two-qubit count provably grows linearly in $k$; a variant reaches near-optimal $O(nk/\log n)$ scaling with the closed-form metric intact. On electronic-structure calculations for small molecules and half-filled Hubbard quench dynamics, the method reaches reference-level accuracy with one to three orders of magnitude fewer two-qubit gates than leading alternatives. Interchangeable constructions (a Schur-transform dressing or internal reparameterisations) make the ansatz exactly spin-adapted, with fixed total spin at every parameter and no penalty terms. The bare ansatz is an exactly controllable, well-conditioned and barren-plateau-free primitive for preparing and sampling sector states: on its own, it is classically simulable in $k$ (a boundary proved for a general class of sector-sparse ans\"atze); composed with a classically hard dressing, it yields molecular ground states, sector-Haar benchmarking, thermal correlators, and exact effective Hamiltonians trained from energy measurements alone, with the composed circuit carrying the potential for quantum advantage.

Figures

Figures reproduced from arXiv: 2607.07942 by the authors.

Figure 1
Figure 1. FIG. 1: End-to-end pipeline, illustrated on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Parametrization, classification, and the resulting diagonal metric for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: End-to-end pruning walkthrough (Algorithm 1) for the running example [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Symmetry-adapted VQE benchmark in the STO-3G basis. Convergence at equilibrium geometry for five [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Real-time molecular dipole-kick trajectories for H [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Real-time Hubbard quench at [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Two-block active-space dressing along the symmetric stretch of water in the STO-3G symmetry-adapted [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Sector-averaged echo infidelity 1 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Infinite-temperature nearest-neighbour [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Exact block decoupling of lithium hydride trained from energy measurements alone, across Li–H bond [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: One-qubit picture of the projective state manifold. [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Incomplete-subspace approximate dynamics, dependence on subspace size (companion to the [PITH_FULL_IMAGE:figures/full_fig_p044_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Total-spin contamination [PITH_FULL_IMAGE:figures/full_fig_p046_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Spin-adapted classification tree, in the style of Fig. 2(a), on the smallest sector that ties a node: two [PITH_FULL_IMAGE:figures/full_fig_p049_14.png]

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Reference graph

Works this paper leans on

101 extracted references · 101 canonical work pages

  1. [1]

    Tilly et al

    author author J. Tilly et al. ,\ 10.1016/j.physrep.2022.07.003 journal journal Physics Reports \ volume 986 ,\ pages 1 ( year 2022 ) NoStop

    work page doi:10.1016/j.physrep.2022.07.003 2022

  2. [2]

    Cerezo et al

    author author M. Cerezo et al. ,\ 10.1038/s42254-021-00348-9 journal journal Nat. Rev. Phys. \ volume 3 ,\ pages 625 ( year 2021 ) NoStop

    work page doi:10.1038/s42254-021-00348-9 2021

  3. [3]

    author author J. R. \ McClean et al. ,\ @noop journal journal New J. Phys. \ volume 18 ,\ pages 023023 ( year 2016 ) NoStop

    work page 2016

  4. [4]

    McArdle , author S

    author author S. McArdle , author S. Endo , author A. Aspuru-Guzik , author S. C. \ Benjamin , \ and\ author X. Yuan ,\ 10.1103/RevModPhys.92.015003 journal journal Rev. Mod. Phys. \ volume 92 ,\ pages 015003 ( year 2020 ) NoStop

    work page doi:10.1103/revmodphys.92.015003 2020

  5. [5]

    Peruzzo , author J

    author author A. Peruzzo , author J. McClean , author P. Shadbolt , author M.-H. \ Yung , author X.-Q. \ Zhou , author P. J. \ Love , author A. Aspuru-Guzik , \ and\ author J. L. \ O'Brien ,\ 10.1038/ncomms5213 journal journal Nat. Commun. \ volume 5 ,\ pages 4213 ( year 2014 ) NoStop

    work page doi:10.1038/ncomms5213 2014

  6. [6]

    Stokes , author J

    author author J. Stokes , author J. Izaac , author N. Killoran , \ and\ author G. Carleo ,\ @noop journal journal Quantum \ volume 4 ,\ pages 269 ( year 2020 ) NoStop

    work page 2020

  7. [7]

    McArdle et al

    author author S. McArdle et al. ,\ @noop journal journal npj Quantum Inf. \ volume 5 ,\ pages 75 ( year 2019 ) NoStop

    work page 2019

  8. [8]

    Yuan , author S

    author author X. Yuan , author S. Endo , author Q. Zhao , author Y. Li , \ and\ author S. C. \ Benjamin ,\ 10.22331/q-2019-10-07-191 journal journal Quantum \ volume 3 ,\ pages 191 ( year 2019 ) NoStop

    work page doi:10.22331/q-2019-10-07-191 2019

Show all 101 references
  1. [9]

    Gacon , author C

    author author J. Gacon , author C. Zoufal , author G. Carleo , \ and\ author S. Woerner ,\ 10.22331/q-2021-10-20-567 journal journal Quantum \ volume 5 ,\ pages 567 ( year 2021 ) NoStop

    2021 doi

  2. [10]

    Kandala et al

    author author A. Kandala et al. ,\ @noop journal journal Nature \ volume 549 ,\ pages 242 ( year 2017 ) NoStop

    2017

  3. [11]

    author author J. R. \ McClean et al. ,\ @noop journal journal Nat. Commun. \ volume 9 ,\ pages 4812 ( year 2018 ) NoStop

    2018

  4. [12]

    Holmes , author K

    author author Z. Holmes , author K. Sharma , author M. Cerezo , \ and\ author P. J. \ Coles ,\ @noop journal journal PRX Quantum \ volume 3 ,\ pages 010313 ( year 2022 ) NoStop

    2022

  5. [13]

    Sim , author P

    author author S. Sim , author P. D. \ Johnson , \ and\ author A. Aspuru-Guzik ,\ @noop journal journal Adv. Quantum Technol. \ volume 2 ,\ pages 1900070 ( year 2019 ) NoStop

    2019

  6. [14]

    Leone , author S

    author author L. Leone , author S. F. \ Oliviero , author L. Cincio , \ and\ author M. Cerezo ,\ 10.22331/q-2024-07-03-1395 journal journal Quantum \ volume 8 ,\ pages 1395 ( year 2024 ) ,\ http://arxiv.org/abs/arXiv:2211.01477 arXiv:2211.01477 NoStop

    2024 doi arXiv

  7. [15]

    Larocca , author S

    author author M. Larocca , author S. Thanasilp , author S. Wang , author K. Sharma , author J. Biamonte , author P. J. \ Coles , author L. Cincio , author J. R. \ McClean , author Z. Holmes , \ and\ author M. Cerezo ,\ 10.1038/s42254-025-00813-9 journal journal Nat. Rev. Phys....

    2025 doi

  8. [16]

    Möttönen , author J

    author author M. Möttönen , author J. J. \ Vartiainen , author V. Bergholm , \ and\ author M. M. \ Salomaa ,\ @noop journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 130502 ( year 2004 ) NoStop

    2004

  9. [17]

    Möttönen , author J

    author author M. Möttönen , author J. J. \ Vartiainen , author V. Bergholm , \ and\ author M. M. \ Salomaa ,\ https://arxiv.org/abs/quant-ph/0407010 journal journal Quantum Information and Computation \ volume 5 ,\ pages 467 ( year 2005 ) ,\ http://arxiv.org/abs/arXiv:quant-ph...

    2005 arXiv

  10. [18]

    Savage ,\ 10.1137/S0036144595295272 journal journal SIAM Review \ volume 39 ,\ pages 605 ( year 1997 ) NoStop

    author author C. Savage ,\ 10.1137/S0036144595295272 journal journal SIAM Review \ volume 39 ,\ pages 605 ( year 1997 ) NoStop

    1997 doi

  11. [19]

    Bergholm , author J

    author author V. Bergholm , author J. J. \ Vartiainen , author M. Möttönen , \ and\ author M. M. \ Salomaa ,\ 10.1103/PhysRevA.71.052330 journal journal Phys. Rev. A \ volume 71 ,\ pages 052330 ( year 2005 ) NoStop

    2005 doi

  12. [20]

    Amari ,\ @noop journal journal Neural Comput

    author author S. Amari ,\ @noop journal journal Neural Comput. \ volume 10 ,\ pages 251 ( year 1998 ) NoStop

    1998

  13. [21]

    author author S. L. \ Braunstein \ and\ author C. M. \ Caves ,\ 10.1103/PhysRevLett.72.3439 journal journal Phys. Rev. Lett. \ volume 72 ,\ pages 3439 ( year 1994 ) NoStop

    1994 doi

  14. [22]

    author author J. J. \ Meyer ,\ 10.22331/q-2021-09-09-539 journal journal Quantum \ volume 5 ,\ pages 539 ( year 2021 ) NoStop

    2021 doi

  15. [23]

    author author J. P. \ Provost \ and\ author G. Vallee ,\ 10.1007/BF02193559 journal journal Commun. Math. Phys. \ volume 76 ,\ pages 289 ( year 1980 ) NoStop

    1980 doi

  16. [24]

    Ashtekar \ and\ author T

    author author A. Ashtekar \ and\ author T. A. \ Schilling ,\ in\ 10.1007/978-1-4612-1422-9_3 booktitle On E instein's Path: Essays in Honor of Engelbert Schucking ,\ editor edited by\ editor A. Harvey \ ( publisher Springer ,\ year 1999 )\ pp.\ pages 23--65 ,\ note arXiv:gr-qc...

    1999 doi arXiv

  17. [25]

    author author N. A. \ McMahon , author M. Pervez , \ and\ author C. Arenz ,\ 10.1103/ht2m-1j91 journal journal Phys. Rev. Research \ volume 8 ,\ pages 023024 ( year 2026 ) ,\ http://arxiv.org/abs/arXiv:2504.06123 arXiv:2504.06123 NoStop

    2026 doi arXiv

  18. [26]

    Wierichs , author J

    author author D. Wierichs , author J. Izaac , author C. Wang , \ and\ author C. Y.-Y. \ Lin ,\ 10.22331/q-2022-03-30-677 journal journal Quantum \ volume 6 ,\ pages 677 ( year 2022 ) NoStop

    2022 doi

  19. [27]

    Mitarai , author M

    author author K. Mitarai , author M. Negoro , author M. Kitagawa , \ and\ author K. Fujii ,\ 10.1103/PhysRevA.98.032309 journal journal Phys. Rev. A \ volume 98 ,\ pages 032309 ( year 2018 ) NoStop

    2018 doi

  20. [28]

    Schuld , author V

    author author M. Schuld , author V. Bergholm , author C. Gogolin , author J. Izaac , \ and\ author N. Killoran ,\ 10.1103/PhysRevA.99.032331 journal journal Phys. Rev. A \ volume 99 ,\ pages 032331 ( year 2019 ) NoStop

    2019 doi

  21. [29]

    author author G.-L. R. \ Anselmetti , author D. Wierichs , author C. Gogolin , \ and\ author R. M. \ Parrish ,\ 10.1088/1367-2630/ac2cb3 journal journal New Journal of Physics \ volume 23 ,\ pages 113010 ( year 2021 ) NoStop

    2021 doi

  22. [30]

    author author J. S. \ Kottmann , author A. Anand , \ and\ author A. Aspuru-Guzik ,\ 10.1039/D0SC06627C journal journal Chemical Science \ volume 12 ,\ pages 3497 ( year 2021 ) NoStop

    2021 doi

  23. [31]

    Mari , author T

    author author A. Mari , author T. R. \ Bromley , \ and\ author N. Killoran ,\ 10.1103/PhysRevA.103.012405 journal journal Physical Review A \ volume 103 ,\ pages 012405 ( year 2021 ) NoStop

    2021 doi

  24. [32]

    Barison , author F

    author author S. Barison , author F. Vicentini , \ and\ author G. Carleo ,\ 10.22331/q-2021-07-28-512 journal journal Quantum \ volume 5 ,\ pages 512 ( year 2021 ) NoStop

    2021 doi

  25. [33]

    Gacon , author J

    author author J. Gacon , author J. Nys , author R. Rossi , author S. Woerner , \ and\ author G. Carleo ,\ 10.1103/PhysRevResearch.6.013143 journal journal Phys. Rev. Research \ volume 6 ,\ pages 013143 ( year 2024 ) NoStop

    2024 doi

  26. [34]

    Farhi , author J

    author author E. Farhi , author J. Goldstone , \ and\ author S. Gutmann ,\ @noop journal journal arXiv preprint \ ( year 2014 ) ,\ http://arxiv.org/abs/arXiv:1411.4028 arXiv:1411.4028 NoStop

    2014 arXiv

  27. [35]

    Hadfield , author Z

    author author S. Hadfield , author Z. Wang , author B. O'Gorman , author E. G. \ Rieffel , author D. Venturelli , \ and\ author R. Biswas ,\ 10.3390/a12020034 journal journal Algorithms \ volume 12 ,\ pages 34 ( year 2019 ) NoStop

    2019 doi

  28. [36]

    Lucas ,\ 10.3389/fphy.2014.00005 journal journal Frontiers in Physics \ volume 2 ,\ pages 5 ( year 2014 ) NoStop

    author author A. Lucas ,\ 10.3389/fphy.2014.00005 journal journal Frontiers in Physics \ volume 2 ,\ pages 5 ( year 2014 ) NoStop

    2014 doi

  29. [37]

    Gottesman ,\ title Stabilizer codes and quantum error correction ,\ @noop Ph.D

    author author D. Gottesman ,\ title Stabilizer codes and quantum error correction ,\ @noop Ph.D. thesis ,\ school California Institute of Technology ( year 1997 ) NoStop

    1997

  30. [38]

    Sun , author G

    author author X. Sun , author G. Tian , author S. Yang , author P. Yuan , \ and\ author S. Zhang ,\ 10.1109/TCAD.2023.3244885 journal journal IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. \ volume 42 ,\ pages 3301 ( year 2023 ) NoStop

    2023 doi

  31. [39]

    Li \ and\ author J

    author author L. Li \ and\ author J. Luo ,\ @noop title Nearly optimal circuit size for sparse quantum state preparation , \ ( year 2025 ),\ note proc. ICALP 2025, arXiv:2406.16142 NoStop

    2025

  32. [40]

    Luo \ and\ author L

    author author J. Luo \ and\ author L. Li ,\ @noop title Optimal circuit size for fixed- H amming-weight quantum states preparation , \ ( year 2025 ),\ note arXiv:2508.17197 NoStop

    2025 arXiv

  33. [41]

    Picozzi \ and\ author J

    author author D. Picozzi \ and\ author J. Tennyson ,\ 10.1088/2058-9565/acd86c journal journal Quantum Sci. Technol. \ volume 8 ,\ pages 035026 ( year 2023 ) NoStop

    doi

  34. [42]

    author author D. Picozzi ,\ https://arxiv.org/abs/2606.05777 title Periodic symmetry-adapted encoding: Qubit reduction in crystalline electronic structure , \ ( year 2026 ),\ note arXiv:2606.05777 ,\ http://arxiv.org/abs/arXiv:2606.05777 arXiv:2606.05777 NoStop

    2026 arXiv

  35. [43]

    Bravyi , author J

    author author S. Bravyi , author J. M. \ Gambetta , author A. Mezzacapo , \ and\ author K. Temme ,\ @noop journal journal arXiv preprint \ ( year 2017 ) ,\ http://arxiv.org/abs/arXiv:1701.08213 arXiv:1701.08213 NoStop

    2017 arXiv

  36. [44]

    author author B. T. \ Gard , author L. Zhu , author G. S. \ Barron , author N. J. \ Mayhall , author S. E. \ Economou , \ and\ author E. Barnes ,\ 10.1038/s41534-019-0240-1 journal journal npj Quantum Inf. \ volume 6 ,\ pages 10 ( year 2020 ) NoStop

    2020 doi

  37. [45]

    Picozzi \ and\ author J

    author author D. Picozzi \ and\ author J. Tennyson ,\ https://arxiv.org/abs/2606.05865 title Symmetry-adapted qubit encoding with complete active space and B ravyi-- K itaev mapping for quantum chemistry on a quantum computer , \ ( year 2026 ),\ note arXiv:2606.05865 ,\ http:/...

    2026 arXiv

  38. [46]

    Ostaszewski , author E

    author author M. Ostaszewski , author E. Grant , \ and\ author M. Benedetti ,\ 10.22331/q-2021-01-28-391 journal journal Quantum \ volume 5 ,\ pages 391 ( year 2021 ) NoStop

    2021 doi

  39. [47]

    author author K. M. \ Nakanishi , author K. Fujii , \ and\ author S. Todo ,\ 10.1103/PhysRevResearch.2.043158 journal journal Physical Review Research \ volume 2 ,\ pages 043158 ( year 2020 ) NoStop

    2020 doi

  40. [48]

    Romero , author R

    author author J. Romero , author R. Babbush , author J. R. \ McClean , author C. Hempel , author P. J. \ Love , \ and\ author A. Aspuru-Guzik ,\ 10.1088/2058-9565/aad3e4 journal journal Quantum Sci. Technol. \ volume 4 ,\ pages 014008 ( year 2019 ) NoStop

    doi

  41. [49]

    Raghavachari , author G

    author author K. Raghavachari , author G. W. \ Trucks , author J. A. \ Pople , \ and\ author M. Head-Gordon ,\ 10.1016/S0009-2614(89)87395-6 journal journal Chem. Phys. Lett. \ volume 157 ,\ pages 479 ( year 1989 ) NoStop

    1989 doi

  42. [50]

    Bacon , author I

    author author D. Bacon , author I. L. \ Chuang , \ and\ author A. W. \ Harrow ,\ 10.1103/PhysRevLett.97.170502 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 170502 ( year 2006 ) ,\ http://arxiv.org/abs/arXiv:quant-ph/0601001 arXiv:quant-ph/0601001 NoStop

    2006 doi arXiv

  43. [51]

    Burkat \ and\ author N

    author author J. Burkat \ and\ author N. Fitzpatrick ,\ @noop title The quantum P aldus transform: efficient circuits with applications , \ ( year 2025 ),\ note arXiv:2506.09151 NoStop

    2025

  44. [52]

    author author A. D. \ McLachlan ,\ 10.1080/00268976400100041 journal journal Mol. Phys. \ volume 8 ,\ pages 39 ( year 1964 ) NoStop

    1964 doi

  45. [53]

    Suzuki ,\ 10.1063/1.529425 journal journal J

    author author M. Suzuki ,\ 10.1063/1.529425 journal journal J. Math. Phys. \ volume 32 ,\ pages 400 ( year 1991 ) NoStop

    1991 doi

  46. [54]

    author author A. M. \ Childs , author Y. Su , author M. C. \ Tran , author N. Wiebe , \ and\ author S. Zhu ,\ 10.1103/PhysRevX.11.011020 journal journal Phys. Rev. X \ volume 11 ,\ pages 011020 ( year 2021 ) NoStop

    2021 doi

  47. [55]

    author author A. M. \ Childs , author D. Maslov , author Y. Nam , author N. J. \ Ross , \ and\ author Y. Su ,\ 10.1073/pnas.1801723115 journal journal Proc. Natl. Acad. Sci. U.S.A. \ volume 115 ,\ pages 9456 ( year 2018 ) NoStop

    2018 doi

  48. [56]

    Wecker , author M

    author author D. Wecker , author M. B. \ Hastings , \ and\ author M. Troyer ,\ 10.1103/PhysRevA.92.042303 journal journal Phys. Rev. A \ volume 92 ,\ pages 042303 ( year 2015 ) NoStop

    2015 doi

  49. [57]

    Cade , author L

    author author C. Cade , author L. Mineh , author A. Montanaro , \ and\ author S. Stanisic ,\ 10.1103/PhysRevB.102.235122 journal journal Phys. Rev. B \ volume 102 ,\ pages 235122 ( year 2020 ) NoStop

    2020 doi

  50. [58]

    author author I. D. \ Kivlichan , author J. McClean , author N. Wiebe , author C. Gidney , author A. Aspuru-Guzik , author G. K.-L. \ Chan , \ and\ author R. Babbush ,\ 10.1103/PhysRevLett.120.110501 journal journal Phys. Rev. Lett. \ volume 120 ,\ pages 110501 ( year 2018 ) NoStop

    2018 doi

  51. [59]

    author author H. R. \ Grimsley , author S. E. \ Economou , author E. Barnes , \ and\ author N. J. \ Mayhall ,\ 10.1038/s41467-019-10988-2 journal journal Nat. Commun. \ volume 10 ,\ pages 3007 ( year 2019 ) NoStop

    2019 doi

  52. [60]

    \.Z yczkowski \ and\ author M

    author author K. \.Z yczkowski \ and\ author M. Ku \'s ,\ 10.1088/0305-4470/27/12/028 journal journal J. Phys. A: Math. Gen. \ volume 27 ,\ pages 4235 ( year 1994 ) NoStop

    1994 doi

  53. [61]

    Mezzadri ,\ @noop journal journal Notices Am

    author author F. Mezzadri ,\ @noop journal journal Notices Am. Math. Soc. \ volume 54 ,\ pages 592 ( year 2007 ) NoStop

    2007

  54. [62]

    Tilma \ and\ author E

    author author T. Tilma \ and\ author E. C. G. \ Sudarshan ,\ 10.1088/0305-4470/35/48/316 journal journal J. Phys. A: Math. Gen. \ volume 35 ,\ pages 10467 ( year 2002 ) NoStop

    2002 doi

  55. [63]

    Dankert , author R

    author author C. Dankert , author R. Cleve , author J. Emerson , \ and\ author E. Livine ,\ 10.1103/PhysRevA.80.012304 journal journal Phys. Rev. A \ volume 80 ,\ pages 012304 ( year 2009 ) NoStop

    2009 doi

  56. [64]

    \ Huang , author R

    author author H.-Y. \ Huang , author R. Kueng , \ and\ author J. Preskill ,\ 10.1038/s41567-020-0932-7 journal journal Nat. Phys. \ volume 16 ,\ pages 1050 ( year 2020 ) NoStop

    2020 doi

  57. [65]

    Elben , author S

    author author A. Elben , author S. T. \ Flammia , author H.-Y. \ Huang , author R. Kueng , author J. Preskill , author B. Vermersch , \ and\ author P. Zoller ,\ 10.1038/s42254-022-00535-2 journal journal Nat. Rev. Phys. \ volume 5 ,\ pages 9 ( year 2023 ) NoStop

    2023 doi

  58. [66]

    Sugiura \ and\ author A

    author author S. Sugiura \ and\ author A. Shimizu ,\ 10.1103/PhysRevLett.108.240401 journal journal Phys. Rev. Lett. \ volume 108 ,\ pages 240401 ( year 2012 ) NoStop

    2012 doi

  59. [67]

    Goldstein , author J

    author author S. Goldstein , author J. L. \ Lebowitz , author R. Tumulka , \ and\ author N. Zangh \`i ,\ 10.1103/PhysRevLett.96.050403 journal journal Phys. Rev. Lett. \ volume 96 ,\ pages 050403 ( year 2006 ) NoStop

    2006 doi

  60. [68]

    Popescu , author A

    author author S. Popescu , author A. J. \ Short , \ and\ author A. Winter ,\ 10.1038/nphys444 journal journal Nat. Phys. \ volume 2 ,\ pages 754 ( year 2006 ) NoStop

    2006 doi

  61. [69]

    Prosen \ and\ author I

    author author T. Prosen \ and\ author I. Pi z orn ,\ 10.1103/PhysRevA.76.032316 journal journal Phys. Rev. A \ volume 76 ,\ pages 032316 ( year 2007 ) NoStop

    2007 doi

  62. [70]

    Schollw \"o ck ,\ 10.1016/j.aop.2010.09.012 journal journal Ann

    author author U. Schollw \"o ck ,\ 10.1016/j.aop.2010.09.012 journal journal Ann. Phys. \ volume 326 ,\ pages 96 ( year 2011 ) NoStop

    2010 doi

  63. [71]

    Bravyi , author D

    author author S. Bravyi , author D. P. \ DiVincenzo , \ and\ author D. Loss ,\ 10.1016/j.aop.2011.06.004 journal journal Ann. Phys. \ volume 326 ,\ pages 2793 ( year 2011 ) NoStop

    2011 doi

  64. [72]

    Monbroussou , author E

    author author L. Monbroussou , author E. Mamon , author J. Landman , author A. B. \ Grilo , author R. Kukla , \ and\ author E. Kashefi ,\ @noop journal journal Quantum \ volume 9 ,\ pages 1745 ( year 2025 ) ,\ note arXiv:2309.15547 NoStop

    2025 arXiv

  65. [73]

    Van den Nest ,\ @noop journal journal Quantum Inf

    author author M. Van den Nest ,\ @noop journal journal Quantum Inf. Comput. \ volume 11 ,\ pages 784 ( year 2011 ) ,\ note arXiv:0911.1624 NoStop

    2011 arXiv

  66. [74]

    author author M. L. \ Goh , author M. Larocca , author L. Cincio , author M. Cerezo , \ and\ author F. Sauvage ,\ 10.1103/3y65-f5w6 journal journal Phys. Rev. Res. \ volume 7 ,\ pages 033266 ( year 2025 ) ,\ note arXiv:2308.01432 NoStop

    2025 doi

  67. [75]

    Cerezo , author M

    author author M. Cerezo , author M. Larocca , author D. Garc \'i a-Mart \'i n , author N. L. \ Diaz , author P. Braccia , author E. Fontana , author M. S. \ Rudolph , author P. Bermejo , author A. Ijaz , author S. Thanasilp , author E. R. \ Anschuetz , \ and\ author Z. Holmes ...

    2025 doi arXiv

  68. [76]

    author author R. P. \ Feynman ,\ 10.1007/BF02650179 journal journal Int. J. Theor. Phys. \ volume 21 ,\ pages 467 ( year 1982 ) NoStop

    1982 doi

  69. [77]

    Lloyd ,\ 10.1126/science.273.5278.1073 journal journal Science \ volume 273 ,\ pages 1073 ( year 1996 ) NoStop

    author author S. Lloyd ,\ 10.1126/science.273.5278.1073 journal journal Science \ volume 273 ,\ pages 1073 ( year 1996 ) NoStop

    1996 doi

  70. [78]

    author author A. J. \ Daley , author I. Bloch , author C. Kokail , author S. Flannigan , author N. Pearson , author M. Troyer , \ and\ author P. Zoller ,\ 10.1038/s41586-022-04940-6 journal journal Nature \ volume 607 ,\ pages 667 ( year 2022 ) NoStop

    2022 doi

  71. [79]

    Lee , author J

    author author S. Lee , author J. Lee , author H. Zhai , author Y. Tong , author A. M. \ Dalzell , author A. Kumar , author P. Helms , et al. ,\ 10.1038/s41467-023-37587-6 journal journal Nat. Commun. \ volume 14 ,\ pages 1952 ( year 2023 ) NoStop

    1952 doi

  72. [80]

    author author G. K.-L. \ Chan \ and\ author S. Sharma ,\ 10.1146/annurev-physchem-032210-103338 journal journal Annu. Rev. Phys. Chem. \ volume 62 ,\ pages 465 ( year 2011 ) NoStop

    2011 doi

  73. [81]

    author author A. A. \ Holmes , author N. M. \ Tubman , \ and\ author C. J. \ Umrigar ,\ 10.1021/acs.jctc.6b00407 journal journal J. Chem. Theory Comput. \ volume 12 ,\ pages 3674 ( year 2016 ) NoStop

    2016 doi

  74. [82]

    Begu s i \'c , author J

    author author T. Begu s i \'c , author J. Gray , \ and\ author G. K.-L. \ Chan ,\ 10.1126/sciadv.adk4321 journal journal Sci. Adv. \ volume 10 ,\ pages eadk4321 ( year 2024 ) ,\ note arXiv:2308.05077 NoStop

    2024 doi arXiv

  75. [83]

    Angrisani , author A

    author author A. Angrisani , author A. Schmidhuber , author M. S. \ Rudolph , author M. Cerezo , author Z. Holmes , \ and\ author H.-Y. \ Huang ,\ 10.1103/lh6x-7rc3 journal journal Phys. Rev. Lett. \ volume 135 ,\ pages 170602 ( year 2025 ) ,\ note arXiv:2409.01706 NoStop

    2025 doi

  76. [84]

    author author D. Picozzi ,\ 10.5281/zenodo.7724696 title QuantumSymmetry : Quantum computing research package , \ howpublished https://github.com/dariopicozzi/quantumsymmetry ( year 2023 ),\ note concept DOI (all versions); source on GitHub NoStop

    2023 doi

  77. [85]

    Kolodrubetz , author D

    author author M. Kolodrubetz , author D. Sels , author P. Mehta , \ and\ author A. Polkovnikov ,\ @noop journal journal Phys. Rep. \ volume 697 ,\ pages 1 ( year 2017 ) NoStop

    2017

  78. [86]

    author author M. V. \ Berry ,\ 10.1098/rspa.1984.0023 journal journal Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences \ volume 392 ,\ pages 45 ( year 1984 ) NoStop

    1984 doi

  79. [87]

    Shapere \ and\ author F

    author author A. Shapere \ and\ author F. Wilczek ,\ @noop title Geometric Phases in Physics \ ( publisher World Scientific Publishing Company ,\ year 1989 ) NoStop

    1989

  80. [88]

    Xiao , author M.-C

    author author D. Xiao , author M.-C. \ Chang , \ and\ author Q. Niu ,\ 10.1103/RevModPhys.82.1959 journal journal Reviews of Modern Physics \ volume 82 ,\ pages 1959 ( year 2010 ) NoStop

    1959 doi

  81. [89]

    The shift rule is most naturally derived from the controlled form

    note Equivalently, the elementary R_y angles produced by the Walsh--Hadamard expansion of the cascade are integer-affine combinations of the _i , but the resulting circuit is unitarily equivalent to the multi-controlled rotation. The shift rule is most naturally derived from t...

  82. [90]

    author author V. V. \ Shende , author S. S. \ Bullock , \ and\ author I. L. \ Markov ,\ @noop journal journal IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. \ volume 25 ,\ pages 1000 ( year 2006 ) NoStop

    2006

  83. [91]

    author author V. V. \ Shende , author I. L. \ Markov , \ and\ author S. S. \ Bullock ,\ @noop journal journal Phys. Rev. A \ volume 69 ,\ pages 062321 ( year 2004 ) NoStop

    2004

  84. [92]

    Plesch \ and\ author C

    author author M. Plesch \ and\ author C . Brukner ,\ @noop journal journal Phys. Rev. A \ volume 83 ,\ pages 032302 ( year 2011 ) NoStop

    2011

  85. [93]

    Iten , author R

    author author R. Iten , author R. Colbeck , author I. Kukuljan , author J. Home , \ and\ author M. Christandl ,\ @noop journal journal Phys. Rev. A \ volume 93 ,\ pages 032318 ( year 2016 ) NoStop

    2016

  86. [94]

    Gleinig \ and\ author T

    author author N. Gleinig \ and\ author T. Hoefler ,\ in\ @noop booktitle 2021 58th ACM/IEEE Design Automation Conference (DAC) \ ( year 2021 )\ pp.\ pages 433--438 NoStop

    2021

  87. [95]

    Paldus ,\ 10.1063/1.1681883 journal journal J

    author author J. Paldus ,\ 10.1063/1.1681883 journal journal J. Chem. Phys. \ volume 61 ,\ pages 5321 ( year 1974 ) NoStop

    1974 doi

  88. [96]

    Shavitt ,\ 10.1002/qua.560120819 journal journal Int

    author author I. Shavitt ,\ 10.1002/qua.560120819 journal journal Int. J. Quantum Chem. \ volume 12 ,\ pages 131 ( year 1977 ) NoStop

    1977 doi

  89. [97]

    author author A. W. \ Harrow ,\ @noop title The C hurch of the S ymmetric S ubspace , \ ( year 2013 ),\ note arXiv:1308.6595 NoStop

    2013 arXiv

  90. [98]

    Aaronson \ and\ author D

    author author S. Aaronson \ and\ author D. Gottesman ,\ 10.1103/PhysRevA.70.052328 journal journal Phys. Rev. A \ volume 70 ,\ pages 052328 ( year 2004 ) NoStop

    2004 doi

  91. [99]

    author author L. G. \ Valiant ,\ 10.1137/S0097539700377025 journal journal SIAM J. Comput. \ volume 31 ,\ pages 1229 ( year 2002 ) NoStop

    2002 doi

  92. [100]

    author author B. M. \ Terhal \ and\ author D. P. \ DiVincenzo ,\ 10.1103/PhysRevA.65.032325 journal journal Phys. Rev. A \ volume 65 ,\ pages 032325 ( year 2002 ) NoStop

    2002 doi

  93. [101]

    Schwarz \ and\ author M

    author author M. Schwarz \ and\ author M. Van den Nest ,\ @noop title Simulating quantum circuits with sparse output distributions , \ ( year 2013 ),\ note arXiv:1310.6749 NoStop

    2013 arXiv

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