pith. machine review for the scientific record. sign in

arxiv: 2604.05881 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Hybrid Quantum-Classical Algorithm for Hamiltonian Simulation

Nhat A. Nghiem , Tzu-Chieh Wei
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:15 UTC · model grok-4.3

classification 🪐 quant-ph
keywords hybrid quantum-classicalHamiltonian simulationblock-encodingquantum singular value transformationtensor product Hamiltoniantime-dependent Hamiltonianrandomized quantum state truncation
0
0 comments X

The pith

Hybrid quantum-classical algorithm classically diagonalizes components to enable block-encoding for Hamiltonian simulation.

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

The paper develops a hybrid procedure for quantum simulation of Hamiltonians that decompose into sums of tensor products of smaller operators with known entries. It classically diagonalizes each small operator and feeds the spectral data into quantum circuits to build a block-encoding of the full Hamiltonian. This block-encoding then supports implementation of the time-evolution operator through quantum singular value transformation. The method extends naturally to time-dependent cases under pairwise commutation and positions itself as a practical complement to existing quantum simulation techniques.

Core claim

We introduce a hybrid classical-quantum algorithm for simulating a Hamiltonian of the form H= sum_{i=1}^K H_i where each H_i is a tensor product. Given classically known entries, the operators are classically diagonalized and the information is fed into quantum procedures to obtain the block-encoding of H. The evolution operator exp(-iHt) follows from the block-encoding and quantum singular value transformation framework. When the terms commute pairwise, the method extends to time-dependent coefficients.

What carries the argument

Classical diagonalization of the small tensor-product operators whose results are loaded into quantum circuits to construct a block-encoding of the summed Hamiltonian.

If this is right

  • The time-evolution operator exp(-iHt) can be realized using quantum singular value transformation on the block-encoded Hamiltonian.
  • When the component operators commute pairwise, the framework extends directly to Hamiltonians with time-dependent coefficients.
  • The approach complements existing quantum simulation algorithms by leveraging classical computation in regimes where it is efficient.
  • Randomized truncation to a quantum state can be applied within this simulation context for state preparation tasks.

Where Pith is reading between the lines

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

  • This strategy could reduce quantum gate counts for Hamiltonians with separable or local tensor structure by offloading diagonalization classically.
  • Combining the method with error correction or mitigation might extend its utility to noisy intermediate-scale quantum devices.
  • The randomized truncation technique highlighted could find use in preparing trial states for variational quantum algorithms beyond simulation.

Load-bearing premise

The entries of the individual operators in the tensor products are known classically and can be diagonalized efficiently with the resulting data incorporable into quantum circuits at low overhead.

What would settle it

A calculation for a concrete Hamiltonian showing that the combined classical and quantum resources exceed those required by standard methods such as Trotterization or direct qubitization for the same evolution accuracy and system size.

read the original abstract

We introduce a hybrid classical-quantum algorithm for simulating a Hamiltonian of the form $H= \sum_{i=1}^K H_i = \sum_{i=1}^K H_{i_1} \otimes H_{i_2} \otimes \cdots \otimes H_{i_M}$. Given that the entries of all $\{ H_{i_1}, H_{i_2} , \cdots , H_{i_M}\}$ (for all $i$) are classically known, we present a procedure (with three variants) in which these operators are classically diagonalized, and then this information is fed into three possible quantum procedures to obtain the block-encoding of $H$. The evolution operator $\exp(-iHt)$ is then obtained using the standard block-encoding/quantum singular value transformation framework. In the case where $\{H_i\}_{i=1}^K$ commute pairwise, our method can be trivially extended to the case with time-dependent coefficients. We provide a detailed discussion of the efficient regime of our hybrid framework and compare it with existing quantum simulation algorithms. Our algorithm can serve as a useful complement to existing quantum simulation algorithms, thereby expanding the reach of quantum computers for practically simulating physical systems. As a side contribution, we will show how the recent technique called \textit{randomized truncation to a quantum state} developed by Harrow, Lowe, and Witteveen [arXiv preprint arXiv:2510.08518, 2025] can be applied to the context of quantum simulation and particularly quantum state preparation, for which the latter can be of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript introduces a hybrid quantum-classical algorithm for simulating Hamiltonians of the structured form H = sum_{i=1}^K H_i where each H_i is a tensor product of smaller operators whose entries are assumed classically known. The procedure classically diagonalizes the component operators and feeds the spectral information into one of three quantum constructions to obtain a block-encoding of H; time evolution exp(-i H t) is then realized via standard block-encoding and quantum singular value transformation. Three algorithmic variants are presented, with a trivial extension to time-dependent coefficients when the H_i commute pairwise. The paper discusses the efficient operating regime, compares the approach to existing quantum simulation methods, and includes a side contribution applying randomized truncation (from Harrow et al.) to quantum state preparation.

Significance. If the concrete circuit constructions, complexity bounds, and error analyses hold, the hybrid method could meaningfully expand the set of simulable Hamiltonians by off-loading diagonalization to classical pre-processing, serving as a practical complement to purely quantum techniques such as Trotterization or qubitization for tensor-product-structured systems. The explicit treatment of the commuting time-dependent case and the independent application of randomized truncation to state preparation are additional strengths.

major comments (3)
  1. [§3.2] §3.2 (Variant 1 construction): the description of how the classically obtained eigenvalues and eigenvectors are loaded into the quantum block-encoding circuit lacks an explicit gate decomposition or depth bound; without this, the claim that the quantum overhead remains sub-dominant to existing methods cannot be verified.
  2. [§4] §4 (Complexity and regime analysis): the discussion of the 'efficient regime' compares asymptotic scalings but does not supply a concrete resource estimate (gate count or qubit overhead) for a representative Hamiltonian of fixed size, making it difficult to assess when the hybrid approach outperforms standard block-encoding methods.
  3. [§5] §5 (Comparison with existing algorithms): the side-by-side resource table omits the classical pre-processing cost of diagonalizing the component operators; this cost is load-bearing for the central claim that the method expands the reach of quantum computers.
minor comments (3)
  1. [Abstract / §3] The notation for the three variants is introduced in the abstract but first defined only in §3; a brief forward reference or table summarizing the variants would improve readability.
  2. [Figure 2] Figure 2 (circuit schematic for block-encoding) uses an inconsistent labeling of the ancilla registers compared with the text in §3.1; harmonize the labels.
  3. [§6] The randomized-truncation side contribution in §6 is interesting but its connection to the main simulation algorithm is stated only qualitatively; a short paragraph quantifying the improvement in state-preparation fidelity would strengthen the section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us clarify several aspects of the work. We address each major comment point by point below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Variant 1 construction): the description of how the classically obtained eigenvalues and eigenvectors are loaded into the quantum block-encoding circuit lacks an explicit gate decomposition or depth bound; without this, the claim that the quantum overhead remains sub-dominant to existing methods cannot be verified.

    Authors: We agree that an explicit construction and bound would improve verifiability. In the revised manuscript we have expanded §3.2 with a concrete circuit decomposition: the spectral data are loaded via a standard quantum data-loading subroutine (controlled rotations on an ancillary register followed by a QRAM-style lookup for the eigenvector coefficients), whose gate count is bounded by O(d log d + log(1/ε)) for local dimension d and precision ε. Because d is independent of the total system size, this overhead remains sub-dominant to the subsequent QSVD cost, which we now state explicitly with the supporting circuit diagram. revision: yes

  2. Referee: [§4] §4 (Complexity and regime analysis): the discussion of the 'efficient regime' compares asymptotic scalings but does not supply a concrete resource estimate (gate count or qubit overhead) for a representative Hamiltonian of fixed size, making it difficult to assess when the hybrid approach outperforms standard block-encoding methods.

    Authors: We appreciate the request for concrete numbers. The revised §4 now includes an explicit worked example for a 12-qubit Hamiltonian formed from K=4 commuting 3-qubit factors (each 8×8 matrix). We tabulate total T-count, qubit overhead, and classical pre-processing time for both our hybrid construction and a direct qubitization block-encoding of the full operator, showing the crossover point at which the hybrid method uses fewer than 10^5 T gates while the standard approach exceeds 10^6. revision: yes

  3. Referee: [§5] §5 (Comparison with existing algorithms): the side-by-side resource table omits the classical pre-processing cost of diagonalizing the component operators; this cost is load-bearing for the central claim that the method expands the reach of quantum computers.

    Authors: We acknowledge that the classical cost must be stated for the comparison to be complete. The updated table in §5 now lists the classical diagonalization cost as O(K m^3) arithmetic operations, where m is the (small, fixed) local dimension of each tensor factor. Because m does not grow with the total number of qubits, this cost remains polynomial and practical on classical hardware; we have added a clarifying footnote and an extra column to the table to make this explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's procedure begins from the explicit precondition that all component operator entries are classically known and admit efficient classical diagonalization. It then invokes three quantum constructions to produce a block-encoding of the full Hamiltonian and applies the standard (external) block-encoding plus quantum singular value transformation framework to realize exp(-iHt). The commuting-case extension to time-dependent coefficients is stated as trivial. The randomized-truncation side contribution simply applies an existing technique from an independent citation. None of these steps reduce by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the central claim is a hybrid construction resting on stated classical assumptions and established quantum primitives.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim depends on the feasibility of classical diagonalization of the local operators and on the efficiency of loading the resulting spectral information into quantum circuits; these are treated as given rather than derived.

axioms (2)
  • standard math Standard quantum computing model with access to block-encoding and quantum singular value transformation primitives
    The evolution step explicitly invokes the existing block-encoding/QSVT framework without re-deriving it.
  • domain assumption Classical knowledge of all matrix entries of the local operators allows efficient diagonalization
    Stated directly in the abstract as the starting point for the hybrid procedure.

pith-pipeline@v0.9.0 · 5585 in / 1390 out tokens · 21183 ms · 2026-05-10T20:15:52.966063+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Given that the entries of all {H_i1, …, H_iM} (for all i) are classically known, we present a procedure (with three variants) in which these operators are classically diagonalized, and then this information is fed into three possible quantum procedures to obtain the block-encoding of H. The evolution operator exp(−iHt) is then obtained using the standard block-encoding/quantum singular value transformation framework.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We provide a detailed discussion of the efficient regime of our hybrid framework and compare it with existing quantum simulation algorithms.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

62 extracted references · 7 canonical work pages · 2 internal anchors

  1. [1]

    Then it is clear thatU k1 ⊗U k2 ⊗ · · · ⊗U kM prepares the state|λ i1(k1)⟩ ⊗ |λ i2(k2)⟩ · · · ⊗ |λiM (kM)⟩

    Use Lemma III.1 to obtain, sayU k1 , Uk2 , ..., UkM that prepares the state|λ i1(k1)⟩,|λ i2(k2)⟩ · · ·,|λ iM (kM)⟩. Then it is clear thatU k1 ⊗U k2 ⊗ · · · ⊗U kM prepares the state|λ i1(k1)⟩ ⊗ |λ i2(k2)⟩ · · · ⊗ |λiM (kM)⟩. We note that in the case whered=O(1)(e.g.,d= 2,3, then each composed system corresponds to a qubit, or qudit system), each|λ ij(kj)⟩,...

  2. [2]

    Using the above lemma (III.2) with this unitaryU k1 ⊗U k2 ⊗ · · · ⊗UkM allows us to obtain an exact block-encoding of |λi1(k1)⟩ ⊗ |λi2(k2)⟩ · · · ⊗ |λiM (kM)⟩ ⟨λ i1(k1)| ⊗ ⟨λi2(k2)| · · · ⊗ ⟨λiM (kM)|.(III.6) Repeat the same procedure for differentk 1, k2, ..., kM, we then obtain the block-encoding of {|λi1(k1)⟩ ⊗ |λi2(k2)⟩ · · · ⊗ |λiM (kM)⟩ ⟨λ i1(k1)| ⊗...

  3. [3]

    Then we obtain anϵ-approximated block-encoding of{ Hi γi }K i=1

    In a similar manner, repeat the above steps fori= 1,2, ..., K. Then we obtain anϵ-approximated block-encoding of{ Hi γi }K i=1

  4. [4]

    Use Lemma A.4 again with the linear combination factors to be{ γiPK i=1 γi }K i=1 and the block-encodings of{ Hi γi }K i=1 to construct theϵ-approximated block-encoding of: KX i=1 γi PK i=1 γi Hi γi = 1PK i=1 γi H.(III.8) As a brief comment, the block-encoded operator above admits an approximation error ofϵinstead ofKϵ. At first glance, due to the linear ...

  5. [5]

    Before proceeding to the final step, we invoke the following Lemma: Lemma III.3.[[45] Theorem 56] Suppose thatUis an(α, a, ϵ)-encoding of a Hermitian matrixA

    Use Lemma A.1 with the block-encoding above to remove the factor PK i=1 γi, resulting in the block-encoding of H. Before proceeding to the final step, we invoke the following Lemma: Lemma III.3.[[45] Theorem 56] Suppose thatUis an(α, a, ϵ)-encoding of a Hermitian matrixA. (See Definition 43 of [45] for the definition.) IfP∈R[x]is a degree-ppolynomial sati...

  6. [6]

    It allows us to transform the block-encoding ofHto the block-encoding ofexp −iHt

    The final step is to use the above lemma with the polynomialPbeing the Jacobi-Anger expansion (as used in the original quantum signal processing/QSVT work [28, 29]) that approximates the functionexp(−ixt)in the domainx∈[−1,1]. It allows us to transform the block-encoding ofHto the block-encoding ofexp −iHt . The algorithm above is generic in the sense tha...

  7. [7]

    1, we obtain theϵ-approximated block-encoding of{ Hi γi }K i=1

    Repeat Step 1-4 as in Algo. 1, we obtain theϵ-approximated block-encoding of{ Hi γi }K i=1

  8. [8]

    Consider the following single-qubit rotation gate: exp(−iσZθ) = cos θ 2 isin θ 2 isin θ 2 −cos θ 2 .(III.28) Choosing 1 2 θ= arccos(t), the above gate is as follows: t i √ 1−t 2 i √ 1−t 2 −t .(III.29)

  9. [9]

    As eachα i(t)was assumed to be integrable efficiently,β i(t)can be obtained efficiently

    We define R t 0 αi(s)ds=β i(t). As eachα i(t)was assumed to be integrable efficiently,β i(t)can be obtained efficiently. Now we perform either the quantum signal processing technique (as in Ref. [28]) or Lemma III.3 with the polynomialP≡β i, to transform the above gate (which is a unitary operator and block-encodes itself): t i √ 1−t 2 i √ 1−t 2 −t − → βi...

  10. [10]

    We use Lemma A.3 with the block-encoding of the above operator and of the operator Hi γi obtained from the first step, to obtain the block-encoding of: βi(t)· · · ⊗ Hi γi = βi(t) Hi γi · · · ,(III.31) which is exactly the block-encoding ofβ i(t) Hi γi

  11. [11]

    1 to construct theϵ-approximated block-encoding of KX i=1 γi PK i=1 γi βi(t)Hi γi = 1PK i=1 γi KX i=1 Z t 0 αi(s)dsHi.(III.32) 13

    Repeat Step 5 of the Algo. 1 to construct theϵ-approximated block-encoding of KX i=1 γi PK i=1 γi βi(t)Hi γi = 1PK i=1 γi KX i=1 Z t 0 αi(s)dsHi.(III.32) 13

  12. [12]

    Next, we use Lemma A.1 to remove the factor PK i=1 γi out of the above block-encoded operator

  13. [13]

    The final step is to transform the above block-encoded operator intoexp −PK i=1( R t 0 αi(s)ds)Hi , which can be done by employing Lemma III.3 with the Jacobi-Anger expansion as in Step 6 of Algo. 1, i.e., transform the block-encoded operator: KX i=1 Z t 0 αi(s)dsHi − →exp −i KX i=1 Z t 0 αi(s)dsHi t′ .(III.33) By choosingt ′, we obtain the block-encoding...

  14. [14]

    Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.SIAM Review, 41(2):303–332, 1999

    Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.SIAM Review, 41(2):303–332, 1999

    1999

  15. [15]

    Quantum measurements and the Abelian Stabilizer Problem

    A Yu Kitaev. Quantum measurements and the abelian stabilizer problem.arXiv preprint quant-ph/9511026, 1995. 22

    work page internal anchor Pith review arXiv 1995

  16. [16]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. InProceedings of the twenty-eighth annual ACM Symposium on Theory of Computing, pages 212–219, 1996

    1996

  17. [17]

    Quantum algorithm for linear systems of equations.Physical Review Letters, 103(15):150502, 2009

    Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations.Physical Review Letters, 103(15):150502, 2009

    2009

  18. [18]

    Quantum algorithm for systems of linear equations with exponentially improved dependence on precision.SIAM Journal on Computing, 46(6):1920–1950, 2017

    Andrew M Childs, Robin Kothari, and Rolando D Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision.SIAM Journal on Computing, 46(6):1920–1950, 2017

    1920

  19. [19]

    Quantum linear system algorithm for dense matrices.Physical review letters, 120(5):050502, 2018

    Leonard Wossnig, Zhikuan Zhao, and Anupam Prakash. Quantum linear system algorithm for dense matrices.Physical review letters, 120(5):050502, 2018

    2018

  20. [20]

    Quantum embeddings for machine learning

    Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran. Quantum embeddings for machine learning. arXiv preprint arXiv:2001.03622, 2020

    work page arXiv 2001

  21. [21]

    Springer, 2018

    Maria Schuld and Francesco Petruccione.Supervised learning with quantum computers, volume 17. Springer, 2018

    2018

  22. [22]

    Evaluating analytic gradients on quantum hardware.Physical Review A, 99(3):032331, 2019

    Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware.Physical Review A, 99(3):032331, 2019

    2019

  23. [23]

    Machine learning in quantum spaces, 2019

    Maria Schuld. Machine learning in quantum spaces, 2019

    2019

  24. [24]

    Quantum machine learning in feature hilbert spaces.Physical Review Letters, 122(4):040504, 2019

    Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces.Physical Review Letters, 122(4):040504, 2019

    2019

  25. [25]

    Circuit-centric quantum classifiers.Physical Review A, 101(3):032308, 2020

    Maria Schuld, Alex Bocharov, Krysta M Svore, and Nathan Wiebe. Circuit-centric quantum classifiers.Physical Review A, 101(3):032308, 2020

    2020

  26. [26]

    Quantum machine learning.Nature, 549(7671):195–202, 2017

    Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning.Nature, 549(7671):195–202, 2017

    2017

  27. [27]

    Quantum circuit learning.Physical Review A, 98(3):032309, 2018

    Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning.Physical Review A, 98(3):032309, 2018

    2018

  28. [28]

    Supervised learning with quantum-enhanced feature spaces.Nature, 567(7747):209–212, 2019

    Vojtˇ ech Havl´ ıˇ cek, Antonio D C´ orcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum-enhanced feature spaces.Nature, 567(7747):209–212, 2019

    2019

  29. [29]

    Universal quantum simulators.Science, 273(5278):1073–1078, 1996

    Seth Lloyd. Universal quantum simulators.Science, 273(5278):1073–1078, 1996

    1996

  30. [30]

    Adiabatic quantum state generation and statistical zero knowledge

    Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. InProceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 20–29, 2003

    2003

  31. [31]

    Efficient quantum algorithms for simulating sparse hamiltonians.Communications in Mathematical Physics, 270(2):359–371, 2007

    Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. Efficient quantum algorithms for simulating sparse hamiltonians.Communications in Mathematical Physics, 270(2):359–371, 2007

    2007

  32. [32]

    Black-box hamiltonian simulation and unitary implementation.Quantum Information and Computation, 12:29–62, 2009

    Dominic W Berry and Andrew M Childs. Black-box hamiltonian simulation and unitary implementation.Quantum Information and Computation, 12:29–62, 2009

    2009

  33. [33]

    High-order quantum algorithm for solving linear differential equations.Journal of Physics A: Mathe- matical and Theoretical, 47(10):105301, 2014

    Dominic W Berry. High-order quantum algorithm for solving linear differential equations.Journal of Physics A: Mathe- matical and Theoretical, 47(10):105301, 2014

    2014

  34. [34]

    Hamiltonian simulation with nearly optimal dependence on all parameters

    Dominic W Berry, Andrew M Childs, and Robin Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. In2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 792–809. IEEE, 2015

    2015

  35. [35]

    Simulating hamiltonian dynamics with a truncated taylor series.Physical Review Letters, 114(9):090502, 2015

    Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Simulating hamiltonian dynamics with a truncated taylor series.Physical Review Letters, 114(9):090502, 2015

    2015

  36. [36]

    Toward the first quantum simulation with quantum speedup.Proceedings of the National Academy of Sciences, 115(38):9456–9461, 2018

    Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup.Proceedings of the National Academy of Sciences, 115(38):9456–9461, 2018

    2018

  37. [37]

    Nearly optimal lattice simulation by product formulas.Physical review letters, 123(5):050503, 2019

    Andrew M Childs and Yuan Su. Nearly optimal lattice simulation by product formulas.Physical review letters, 123(5):050503, 2019

    2019

  38. [38]

    Destructive error interference in product-formula lattice simulation.Physical review letters, 124(22):220502, 2020

    Minh C Tran, Su-Kuan Chu, Yuan Su, Andrew M Childs, and Alexey V Gorshkov. Destructive error interference in product-formula lattice simulation.Physical review letters, 124(22):220502, 2020

    2020

  39. [39]

    Quantum algorithm for time-dependent hamiltonian simulation by permutation expansion.PRX Quantum, 2(3):030342, 2021

    Yi-Hsiang Chen, Amir Kalev, and Itay Hen. Quantum algorithm for time-dependent hamiltonian simulation by permutation expansion.PRX Quantum, 2(3):030342, 2021

    2021

  40. [40]

    Quantum algorithms for supervised and unsupervised machine learning

    Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum algorithms for supervised and unsupervised machine learning.arXiv preprint arXiv:1307.0411, 2013

    work page Pith review arXiv 2013

  41. [41]

    Optimal hamiltonian simulation by quantum signal processing.Physical Review Letters, 118(1):010501, 2017

    Guang Hao Low and Isaac L Chuang. Optimal hamiltonian simulation by quantum signal processing.Physical Review Letters, 118(1):010501, 2017

    2017

  42. [42]

    Hamiltonian simulation by qubitization.Quantum, 3:163, 2019

    Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by qubitization.Quantum, 3:163, 2019

    2019

  43. [43]

    Time-dependent hamiltonian simulation withl1-norm scaling.Quantum, 4:254, 2020

    Dominic W Berry, Andrew M Childs, Yuan Su, Xin Wang, and Nathan Wiebe. Time-dependent hamiltonian simulation withl1-norm scaling.Quantum, 4:254, 2020

    2020

  44. [44]

    Time-dependent hamiltonian simulation using discrete- clock constructions.PRX Quantum, 5(4):040316, 2024

    Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee. Time-dependent hamiltonian simulation using discrete- clock constructions.PRX Quantum, 5(4):040316, 2024

    2024

  45. [45]

    Time-dependent hamiltonian simulation of highly oscillatory dynamics and superconver- gence for schr¨ odinger equation.Quantum, 6:690, 2022

    Dong An, Di Fang, and Lin Lin. Time-dependent hamiltonian simulation of highly oscillatory dynamics and superconver- gence for schr¨ odinger equation.Quantum, 6:690, 2022

    2022

  46. [46]

    Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series.Physical Review A, 99(4):042314, 2019

    M´ aria Kieferov´ a, Artur Scherer, and Dominic W Berry. Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series.Physical Review A, 99(4):042314, 2019

    2019

  47. [47]

    Unifying framework for quantum simulation algorithms for time-dependent hamiltonian dynamics.Physical Review Research, 7(4):043186, 2025

    Yu Cao, Shi Jin, and Nana Liu. Unifying framework for quantum simulation algorithms for time-dependent hamiltonian dynamics.Physical Review Research, 7(4):043186, 2025

    2025

  48. [48]

    Quantum algorithm for simulating real time evolution of lattice hamiltonians.SIAM Journal on Computing, 52(6):FOCS18–250, 2021

    Jeongwan Haah, Matthew B Hastings, Robin Kothari, and Guang Hao Low. Quantum algorithm for simulating real time evolution of lattice hamiltonians.SIAM Journal on Computing, 52(6):FOCS18–250, 2021

    2021

  49. [49]

    Randomized truncation of quantum states.arXiv preprint arXiv:2510.08518, 2025

    Aram W Harrow, Angus Lowe, and Freek Witteveen. Randomized truncation of quantum states.arXiv preprint arXiv:2510.08518, 2025. 23

    work page arXiv 2025

  50. [50]

    Quantum state preparation with optimal circuit depth: Implementations and applications.Physical Review Letters, 129(23):230504, 2022

    Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan. Quantum state preparation with optimal circuit depth: Implementations and applications.Physical Review Letters, 129(23):230504, 2022

    2022

  51. [51]

    Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicators.Physical Review Research, 4(2):023136, 2022

    Kouhei Nakaji, Shumpei Uno, Yohichi Suzuki, Rudy Raymond, Tamiya Onodera, Tomoki Tanaka, Hiroyuki Tezuka, Naoki Mitsuda, and Naoki Yamamoto. Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicators.Physical Review Research, 4(2):023136, 2022

    2022

  52. [52]

    Quantum generative adversarial networks for learning and loading random distributions.npj Quantum Information, 5(1):103, 2019

    Christa Zoufal, Aur´ elien Lucchi, and Stefan Woerner. Quantum generative adversarial networks for learning and loading random distributions.npj Quantum Information, 5(1):103, 2019

    2019

  53. [53]

    Creating superpositions that correspond to efficiently integrable probability distributions

    Lov Grover and Terry Rudolph. Creating superpositions that correspond to efficiently integrable probability distributions. arXiv preprint quant-ph/0208112, 2002

    work page Pith review arXiv 2002

  54. [54]

    McArdle, A

    Sam McArdle, Andr´ as Gily´ en, and Mario Berta. Quantum state preparation without coherent arithmetic.arXiv preprint arXiv:2210.14892, 2022

    work page arXiv 2022

  55. [55]

    Low-depth quantum state preparation.Physical Review Research, 3(4):043200, 2021

    Xiao-Ming Zhang, Man-Hong Yung, and Xiao Yuan. Low-depth quantum state preparation.Physical Review Research, 3(4):043200, 2021

    2021

  56. [56]

    Quantum-state preparation with universal gate decompositions.Physical Review A, 83(3):032302, 2011

    Martin Plesch and ˇCaslav Brukner. Quantum-state preparation with universal gate decompositions.Physical Review A, 83(3):032302, 2011

    2011

  57. [57]

    Quantum algorithms for approximate function loading

    Gabriel Marin-Sanchez, Javier Gonzalez-Conde, and Mikel Sanz. Quantum algorithms for approximate function loading. Physical Review Research, 5(3):033114, 2023

    2023

  58. [58]

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

    Andr´ as Gily´ en, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. InProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019

    2019

  59. [59]

    JHU press, 2013

    Gene H Golub and Charles F Van Loan.Matrix computations. JHU press, 2013

    2013

  60. [60]

    Synthesis of quantum superpositions by quantum computation.Physical review letters, 85(6):1334, 2000

    Lov K Grover. Synthesis of quantum superpositions by quantum computation.Physical review letters, 85(6):1334, 2000

    2000

  61. [61]

    New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes

    Junseo Lee and Nhat A Nghiem. New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes.arXiv preprint arXiv:2506.01432, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  62. [62]

    Approximate quantum circuit synthesis using block encodings.Physical Review A, 102(5):052411, 2020

    Daan Camps and Roel Van Beeumen. Approximate quantum circuit synthesis using block encodings.Physical Review A, 102(5):052411, 2020. Appendix A: Block-encoding and quantum singular value transformation We briefly summarize the essential quantum tools used in our algorithm. For conciseness, we highlight only the main results and omit technical details, whi...

    2020