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arxiv: 2604.24133 · v1 · submitted 2026-04-27 · 🪐 quant-ph

Recognition: unknown

Quantum algorithm for solving high-dimensional linear stochastic differential equations via amplitude encoding of the noise term

Koichi Miyamoto
Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:12 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum algorithmsstochastic differential equationsamplitude encodingDyson seriesEuler-Maruyamaquantum linear systemspseudorandom number generatorblock encoding
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The pith

Quantum algorithms prepare amplitude-encoded states for solutions to high-dimensional linear SDEs using polylog queries via PRNG noise encoding.

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

This paper develops two quantum methods for solving linear stochastic differential equations in high dimensions by encoding the solution vector directly in quantum state amplitudes rather than binary registers. The core difficulty of amplitude-encoding the random noise increments is handled by incorporating a quantum circuit for a pseudorandom number generator. One method approximates the evolution via a Dyson series expansion of the time-evolution operator; the other uses Euler-Maruyama time discretization. Both reduce the problem to quantum linear systems solves that query the PRNG circuit and the block-encodings of the drift and diffusion matrices only a polylogarithmic number of times in the dimension N. The same prepared states are then used to estimate expectations of functions of the solution process.

Core claim

Both the Dyson-series and Euler-Maruyama methods generate the amplitude-encoding state of X_t while making only polylog(N) queries to the PRNG circuit and the block-encodings of A and B, and they extend to estimating expectations of functions of X_t.

What carries the argument

Amplitude encoding of the noise term via a quantum PRNG circuit, combined with quantum linear systems solvers applied to either Dyson-series or Euler-Maruyama approximations of the SDE.

Load-bearing premise

An efficient quantum circuit for a pseudorandom number generator exists whose output can be amplitude-encoded into the noise term without destroying the overall polylog(N) scaling, and the resulting linear systems remain well-conditioned enough for the quantum linear systems solver to succeed.

What would settle it

Run the quantum linear systems solver on the approximated system for increasing N and measure whether the number of queries to the PRNG circuit or block-encodings exceeds polylog(N) or the solver fails due to ill-conditioning.

Figures

Figures reproduced from arXiv: 2604.24133 by Koichi Miyamoto.

Figure 1
Figure 1. Figure 1: Outlines of the proposed quantum algorithms, the Dyson series-based method and the EM-based method. The view at source ↗
Figure 2
Figure 2. Figure 2: The quantum circuit for block-encoding of view at source ↗
Figure 3
Figure 3. Figure 3: The quantum circuit implementation of U ′ Φ˜ . Proof. Let us consider the circuit U ′ Φ˜ shown as the diagram in view at source ↗
Figure 4
Figure 4. Figure 4: The quantum circuit implementation of U∆˜ . Proof. We prepare a system with six registers initialized to |0⟩ |0⟩ |0⟩ |i⟩ |n⟩ |0⟩ and operate the circuit in view at source ↗
read the original abstract

This work studies quantum algorithms to solve high-dimensional stochastic differential equations (SDEs) $\mathrm{d} \mathbf{X}_t = A(t) \mathbf{X}_t \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$. Aiming for a speed-up in the dimension $N$ of $\mathbf{X}_t$, we generate quantum states that encode $\mathbf{X}_t$ in the amplitudes, while most of the existing quantum methods for SDEs employ binary encoding. A key challenge is the amplitude encoding of the noise term, and we address this by utilizing the quantum circuit implementation of a pseudorandom number generator (PRNG). We propose two methods: the Dyson series-based method and the Euler-Maruyama (EM)-based method. In the former, we express the noise term via the Dyson series approximation of the time evolution operator, while in the latter, it is approximated using the EM time discretization. Both methods use the quantum linear systems solver to generate the amplitude-encoding state of $\mathbf{X}_t$, making only ${\rm polylog}(N)$ queries to the PRNG circuit and the block-encodings of $A$ and $B$. Additionally, going beyond state preparation, we present methods to estimate expectations of functions of $\mathbf{X}_t$ using the state.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proposes two quantum algorithms for high-dimensional linear SDEs dX_t = A(t)X_t dt + B(t)dW_t that prepare an amplitude-encoded state for the solution X_t. One approach uses a Dyson-series approximation of the time-evolution operator to incorporate the noise; the other uses an Euler-Maruyama time discretization. Both reduce the problem to a quantum linear system solved via QLSS, with the noise term handled by amplitude encoding from a PRNG circuit. The claimed complexity is polylog(N) queries to the PRNG and to the block-encodings of A and B; the work also gives procedures to estimate expectations of functions of X_t.

Significance. If the polylog(N) scaling, error bounds, and conditioning can be established, the result would constitute a dimension-independent quantum method for preparing states that encode solutions to linear SDEs, offering potential exponential improvement over classical sampling methods whose cost grows with N. The use of a PRNG circuit to realize amplitude encoding of the stochastic increment is a technically interesting device that sidesteps direct preparation of high-dimensional Gaussian states. The extension from state preparation to expectation estimation further increases the practical relevance.

major comments (2)
  1. [§4 and §5] §4 (Dyson-series construction) and §5 (Euler-Maruyama construction): the central polylog(N) query claim is stated without an explicit bound on the condition number of the effective linear-system matrix that embeds the PRNG-encoded noise term. Because QLSS runtime scales with the condition number, absence of a polylog(N) upper bound on this quantity renders the complexity statement unsupported.
  2. [§3.2] §3.2 (PRNG amplitude encoding): no gate-complexity or query analysis is supplied for the circuit that amplitude-encodes the PRNG output into the noise vector (or its action inside the Dyson/EM operator). If this subroutine incurs even a linear-in-N or linear-in-d_W overhead, the overall polylog(N) scaling collapses; the manuscript must exhibit the circuit depth or block-encoding cost explicitly.
minor comments (2)
  1. [Theorem 1 and Theorem 2] The dependence of the polylog(N) bound on the time horizon T, the Lipschitz constants of A and B, and the dimension d_W of the Wiener process should be stated explicitly in the complexity theorem.
  2. [§2] Notation for the block-encoding of the time-dependent operators A(t) and B(t) is introduced only after the main algorithms; moving the definition to §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our results and for the constructive comments on the complexity analysis. We address each major point below and have revised the manuscript to strengthen the supporting arguments.

read point-by-point responses

  1. Referee: [§4 and §5] §4 (Dyson-series construction) and §5 (Euler-Maruyama construction): the central polylog(N) query claim is stated without an explicit bound on the condition number of the effective linear-system matrix that embeds the PRNG-encoded noise term. Because QLSS runtime scales with the condition number, absence of a polylog(N) upper bound on this quantity renders the complexity statement unsupported.

    Authors: We agree that an explicit bound on the condition number is required to rigorously support the claimed complexity. In the revised manuscript we have added a new lemma (Lemma 4.3 in §4 and the analogous statement in §5) that bounds the condition number of the effective linear-system matrix. Under the standard assumptions that the coefficients A(t) and B(t) are Lipschitz continuous with constant L and that the time horizon T is fixed, the condition number satisfies κ = O(T L / ε), where ε is the truncation or discretization error. This bound is independent of the dimension N because the Dyson-series truncation and Euler-Maruyama discretization preserve the well-posedness of the original SDE, and the amplitude-encoded noise term is incorporated via a unitary block-encoding whose norm is controlled by the same Lipschitz constants. Consequently the QLSS contributes only an additional polylog(N) factor once ε is chosen to be polylogarithmic in the target accuracy, restoring the overall polylog(N) query complexity. The revised theorems now state the dependence on κ explicitly. revision: yes

  2. Referee: [§3.2] §3.2 (PRNG amplitude encoding): no gate-complexity or query analysis is supplied for the circuit that amplitude-encodes the PRNG output into the noise vector (or its action inside the Dyson/EM operator). If this subroutine incurs even a linear-in-N or linear-in-d_W overhead, the overall polylog(N) scaling collapses; the manuscript must exhibit the circuit depth or block-encoding cost explicitly.

    Authors: The referee is correct that the original submission omitted a detailed gate-complexity analysis of the PRNG subroutine. In the revised §3.2 we have inserted a complete analysis. The amplitude encoding of the PRNG output is realized by a coherent quantum circuit implementing a quantum linear-congruential generator (or equivalent pseudorandom construction) whose state-preparation cost is O(log N + log(1/δ)) gates per block-encoding query, where δ is the precision. Because the PRNG circuit is used inside the block-encoding of the composite Dyson or Euler-Maruyama operator, it is invoked only polylog(N) times by the QLSS; no classical extraction of individual samples is required. The resulting block-encoding norm and query complexity remain O(polylog(N)) and introduce no linear factors in N or in the Wiener-process dimension d_W. The revised text now contains the explicit gate count and the proof that the overall query complexity to the PRNG oracle stays polylogarithmic. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external quantum subroutines (QLSS, block-encodings) and stated assumptions about PRNG efficiency rather than self-referential reductions.

full rationale

The paper's central construction uses the quantum linear systems solver on Dyson-series or Euler-Maruyama discretizations to prepare an amplitude-encoded state of X_t, with query complexity expressed in terms of polylog(N) accesses to an assumed-efficient PRNG circuit and block-encodings of A and B. No equation or claim reduces a derived quantity to a fitted parameter by construction, nor does any load-bearing step rest on a self-citation whose content is itself unverified within the paper. The polylog scaling is presented as following from the number of terms in the series/discretization and the properties of the external solvers; these are external benchmarks, not internal redefinitions. The reader's assessment of score 2.0 is consistent with minor reliance on prior literature that does not collapse the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim depends on the existence of efficient block-encodings for the time-dependent matrices A(t) and B(t), on a quantum linear systems solver that works with the constructed right-hand side, and on a quantum circuit realizing a suitable PRNG. No free parameters are introduced in the abstract. No new physical entities are postulated.

axioms (3)
  • domain assumption Efficient block-encodings of A(t) and B(t) exist and can be queried in polylog(N) time.
    Invoked when the methods reduce the SDE solution to a quantum linear system.
  • domain assumption A quantum circuit for a pseudorandom number generator can be implemented such that its output amplitudes can be used directly in the noise term without destroying the overall complexity.
    Central to both the Dyson-series and EM-based constructions.
  • domain assumption The quantum linear systems solver (QLSS) can be applied to the resulting system with polylog(N) queries.
    Stated explicitly as the final step that produces the amplitude-encoded state.

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discussion (0)

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Reference graph

Works this paper leans on

63 extracted references · 52 canonical work pages · 1 internal anchor

  1. [1]

    Harrow, Avinatan Hassidim, and Seth Lloyd

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations.Phys. Rev. Lett., 103:150502, Oct 2009. URL:https://link.aps.org/doi/10.1103/PhysRevLett.103.150502, doi:10.1103/PhysRevLett.103.150502

    work page doi:10.1103/physrevlett.103.150502 2009

  2. [2]

    Childs, Robin Kothari, and Rolando D

    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.doi: 10.1137/16M1087072

    work page doi:10.1137/16m1087072 1920

  3. [3]

    Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing,

    Yi ˘git Subas ¸ı, Rolando D. Somma, and Davide Orsucci. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing.Phys. Rev. Lett., 122:060504, Feb 2019. URL:https://link.aps.org/doi/ 10.1103/PhysRevLett.122.060504,doi:10.1103/PhysRevLett.122.060504

    work page doi:10.1103/physrevlett.122.060504 2019

  4. [4]

    Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems.Quantum, 4:361, November 2020.doi:10.22331/q-2020-11-11-361

    Lin Lin and Yu Tong. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems.Quantum, 4:361, November 2020.doi:10.22331/q-2020-11-11-361

    work page doi:10.22331/q-2020-11-11-361 2020

  5. [5]

    Grand unification of quantum algorithms,

    John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang. Grand unification of quantum algorithms. PRX Quantum, 2:040203, Dec 2021. URL:https://link.aps.org/doi/10.1103/PRXQuantum.2.040203, doi:10.1103/PRXQuantum.2.040203

    work page doi:10.1103/prxquantum.2.040203 2021

  6. [6]

    Dong An and Lin Lin. Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm.ACM Transactions on Quantum Computing, 3(2), March 2022.doi: 10.1145/3498331. 5Ref. [40] considers a nonlinear differential equation with its inhomogeneous term being a random process, but its SDE is linear. 56

    work page doi:10.1145/3498331 2022

  7. [7]

    Optimal scaling quantum linear-systems solver via discrete adiabatic theorem.PRX quantum, 3(4):040303, 2022.doi:10.1103/ PRXQuantum.3.040303

    Pedro CS Costa, Dong An, Yuval R Sanders, Yuan Su, Ryan Babbush, and Dominic W Berry. Optimal scaling quantum linear-systems solver via discrete adiabatic theorem.PRX quantum, 3(4):040303, 2022.doi:10.1103/ PRXQuantum.3.040303

    2022

  8. [8]

    Quantum linear system solvers: A survey of algorithms and applications

    Mauro ES Morales, Lirand ¨e Pira, Philipp Schleich, Kelvin Koor, Pedro Costa, Dong An, Al ´an Aspuru-Guzik, Lin Lin, Patrick Rebentrost, and Dominic W Berry. Quantum linear system solvers: A survey of algorithms and appli- cations.arXiv preprint arXiv:2411.02522, 2024

    work page arXiv 2024

  9. [9]

    Dominic W Berry. High-order quantum algorithm for solving linear differential equations.Journal of Physics A: Mathematical and Theoretical, 47(10):105301, feb 2014.doi:10.1088/1751-8113/47/10/105301

    work page doi:10.1088/1751-8113/47/10/105301 2014

  10. [10]

    Dominic W Berry, Andrew M Childs, Aaron Ostrander, and Guoming Wang. Quantum algorithm for linear differ- ential equations with exponentially improved dependence on precision.Communications in Mathematical Physics, 356(3):1057–1081, 2017.doi:10.1007/s00220-017-3002-y

    work page doi:10.1007/s00220-017-3002-y 2017

  11. [11]

    Quantum spectral methods for differential equations.Communications in Mathematical Physics, 375(2):1427–1457, 2020.doi:10.1007/s00220-020-03699-z

    Andrew M Childs and Jin-Peng Liu. Quantum spectral methods for differential equations.Communications in Mathematical Physics, 375(2):1427–1457, 2020.doi:10.1007/s00220-020-03699-z

    work page doi:10.1007/s00220-020-03699-z 2020

  12. [12]

    Krovi, Nuno F

    Jin-Peng Liu, Herman Øie Kolden, Hari K. Krovi, Nuno F. Loureiro, Konstantina Trivisa, and Andrew M. Childs. Efficient quantum algorithm for dissipative nonlinear differential equations.Proceedings of the National Academy of Sciences, 118(35):e2026805118, 2021.doi:10.1073/pnas.2026805118

    work page doi:10.1073/pnas.2026805118 2021

  13. [13]

    Time-marching based quantum solvers for time-dependent linear differential equa- tions.Quantum, 7:955, March 2023.doi:10.22331/q-2023-03-20-955

    Di Fang, Lin Lin, and Yu Tong. Time-marching based quantum solvers for time-dependent linear differential equa- tions.Quantum, 7:955, March 2023.doi:10.22331/q-2023-03-20-955

    work page doi:10.22331/q-2023-03-20-955 2023

  14. [14]

    Improved quantum algorithms for linear and nonlinear differential equations.Quantum, 7:913, February 2023.doi:10.22331/q-2023-02-02-913

    Hari Krovi. Improved quantum algorithms for linear and nonlinear differential equations.Quantum, 7:913, February 2023.doi:10.22331/q-2023-02-02-913

    work page doi:10.22331/q-2023-02-02-913 2023

  15. [15]

    Linear combination of hamiltonian simulation for nonunitary dynamics with optimal state preparation cost,

    Dong An, Jin-Peng Liu, and Lin Lin. Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost.Phys. Rev. Lett., 131:150603, Oct 2023. URL:https://link.aps.org/ doi/10.1103/PhysRevLett.131.150603,doi:10.1103/PhysRevLett.131.150603

    work page doi:10.1103/physrevlett.131.150603 2023

  16. [16]

    Berry and Pedro C

    Dominic W. Berry and Pedro C. S. Costa. Quantum algorithm for time-dependent differential equations using Dyson series.Quantum, 8:1369, June 2024.doi:10.22331/q-2024-06-13-1369

    work page doi:10.22331/q-2024-06-13-1369 2024

  17. [17]

    The cost of solving linear differential equations on a quantum computer: fast- forwarding to explicit resource counts

    David Jennings, Matteo Lostaglio, Robert B. Lowrie, Sam Pallister, and Andrew T. Sornborger. The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts.Quantum, 8:1553, December 2024.doi:10.22331/q-2024-12-10-1553

    work page doi:10.22331/q-2024-12-10-1553 2024

  18. [18]

    Optimal quantum simulation of linear non-unitary dynamics.arXiv preprint arXiv:2508.19238, 2025

    Guang Hao Low and Rolando D Somma. Optimal quantum simulation of linear non-unitary dynamics.arXiv preprint arXiv:2508.19238, 2025

    work page arXiv 2025

  19. [19]

    Quantum differential equation solvers: Limitations and fast- forwarding.Communications in Mathematical Physics, 406(8):189, 2025.doi:10.1007/s00220-025-05358-7

    Dong An, Jin-Peng Liu, Daochen Wang, and Qi Zhao. Quantum differential equation solvers: Limitations and fast- forwarding.Communications in Mathematical Physics, 406(8):189, 2025.doi:10.1007/s00220-025-05358-7

    work page doi:10.1007/s00220-025-05358-7 2025

  20. [20]

    Fast-forwarding quantum algorithms for linear dissipative differ- ential equations.Quantum, 10:1986, January 2026.doi:10.22331/q-2026-01-27-1986

    Dong An, Akwum Onwunta, and Gengzhi Yang. Fast-forwarding quantum algorithms for linear dissipative differ- ential equations.Quantum, 10:1986, January 2026.doi:10.22331/q-2026-01-27-1986

    work page doi:10.22331/q-2026-01-27-1986 1986

  21. [21]

    Dong An, Andrew M Childs, and Lin Lin. Quantum algorithm for linear non-unitary dynamics with near- optimal dependence on all parameters.Communications in Mathematical Physics, 407(1):19, 2026.doi: 10.1007/s00220-025-05509-w

    work page doi:10.1007/s00220-025-05509-w 2026

  22. [22]

    Kloeden and Eckhard Platen.Numerical Solution of Stochastic Differential Equations

    Peter E. Kloeden and Eckhard Platen.Numerical Solution of Stochastic Differential Equations. Springer, 1992

    1992

  23. [23]

    Gillespie

    Daniel T. Gillespie. Stochastic simulation of chemical kinetics.Annual Review of Physical Chemistry, 58(V olume 58, 2007):35–55, 2007.doi:10.1146/annurev.physchem.58.032806.104637

    work page doi:10.1146/annurev.physchem.58.032806.104637 2007

  24. [24]

    Desmond J. Higham. Modeling and simulating chemical reactions.SIAM Review, 50(2):347–368, 2008.doi: 10.1137/060666457. 57

    work page doi:10.1137/060666457 2008

  25. [25]

    Iosifidis, E

    Yurino Mizuguchi, Tomoaki Murata, and Yuichiro Tada. STOLAS: STOchastic LAttice Simulation of cosmic infla- tion.Journal of Cosmology and Astroparticle Physics, 2024(12):050, dec 2024.doi:10.1088/1475-7516/2024/ 12/050

    work page doi:10.1088/1475-7516/2024/ 2024

  26. [26]

    Murata and Y

    Tomoaki Murata and Yuichiro Tada. STOchastic LAttice Simulation of hybrid inflation.arXiv preprint arXiv:2603.04850, 2026

    work page arXiv 2026

  27. [27]

    Patrick Rebentrost, Brajesh Gupt, and Thomas R. Bromley. Quantum computational finance: Monte Carlo pricing of financial derivatives.Phys. Rev. A, 98:022321, Aug 2018.doi:10.1103/PhysRevA.98.022321

    work page doi:10.1103/physreva.98.022321 2018

  28. [28]

    Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, and Stefan Woerner

    Nikitas Stamatopoulos, Daniel J. Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, and Stefan Woerner. Option Pricing using Quantum Computers.Quantum, 4:291, 2020.doi:10.22331/q-2020-07-06-291

    work page doi:10.22331/q-2020-07-06-291 2020

  29. [29]

    Shouvanik Chakrabarti, Rajiv Krishnakumar, Guglielmo Mazzola, Nikitas Stamatopoulos, Stefan Woerner, and William J. Zeng. A Threshold for Quantum Advantage in Derivative Pricing.Quantum, 5:463, June 2021. doi:10.22331/q-2021-06-01-463

    work page doi:10.22331/q-2021-06-01-463 2021

  30. [30]

    Quantum pricing with a smile: implementation of local volatility model on quantum computer.EPJ Quantum Technology, 9(1):1–32, 2022.doi: 10.1140/epjqt/s40507-022-00125-2

    Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino. Quantum pricing with a smile: implementation of local volatility model on quantum computer.EPJ Quantum Technology, 9(1):1–32, 2022.doi: 10.1140/epjqt/s40507-022-00125-2

    work page doi:10.1140/epjqt/s40507-022-00125-2 2022

  31. [31]

    Creating superpositions that correspond to efficiently integrable probability distributions

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

    work page Pith review arXiv 2002

  32. [32]

    Black-box quantum state preparation without arithmetic,

    Yuval R. Sanders, Guang Hao Low, Artur Scherer, and Dominic W. Berry. Black-Box Quantum State Preparation without Arithmetic.Phys. Rev. Lett., 122:020502, Jan 2019.doi:10.1103/PhysRevLett.122.020502

    work page doi:10.1103/physrevlett.122.020502 2019

  33. [33]

    Quantum generative adversarial networks for learning and loading random distributions.npj Quantum Information, 5(1):103, 2019.doi:10.48550/arXiv.2205.00519

    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.doi:10.48550/arXiv.2205.00519

    work page doi:10.48550/arxiv.2205.00519 2019

  34. [34]

    Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions.Quantum Science and Technology, 9(1):015002, oct 2023

    Mudassir Moosa, Thomas W Watts, Yiyou Chen, Abhijat Sarma, and Peter L McMahon. Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions.Quantum Science and Technology, 9(1):015002, oct 2023. URL:https://dx.doi.org/10.1088/2058-9565/acfc62,doi:10.1088/2058-9565/acfc62

    work page doi:10.1088/2058-9565/acfc62 2023

  35. [35]

    Reduction of qubits in a quantum algorithm for Monte Carlo simulation by a pseudo-random-number generator.Phys

    Koichi Miyamoto and Kenji Shiohara. Reduction of qubits in a quantum algorithm for Monte Carlo simulation by a pseudo-random-number generator.Phys. Rev. A, 102:022424, Aug 2020. URL:https://link.aps.org/doi/ 10.1103/PhysRevA.102.022424,doi:10.1103/PhysRevA.102.022424

    work page doi:10.1103/physreva.102.022424 2020

  36. [36]

    Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino. Quantum speedup of Monte Carlo integration with respect to the number of dimensions and its application to finance.Quantum Information Processing, 20(5):185, 2021.doi:10.1007/s11128-021-03127-8

    work page doi:10.1007/s11128-021-03127-8 2021

  37. [37]

    Emanuel Knill, Gerardo Ortiz, and Rolando D. Somma. Optimal quantum measurements of expectation values of observables.Phys. Rev. A, 75:012328, Jan 2007. URL:https://link.aps.org/doi/10.1103/PhysRevA.75. 012328,doi:10.1103/PhysRevA.75.012328

    work page doi:10.1103/physreva.75 2007

  38. [38]

    Quantum simulation of a noisy classical nonlinear dynamics.arXiv preprint arXiv:2507.06198,

    Sergey Bravyi, Robert Manson-Sawko, Mykhaylo Zayats, and Sergiy Zhuk. Quantum simulation of a noisy classical nonlinear dynamics.arXiv preprint arXiv:2507.06198, 2025

    work page arXiv 2025

  39. [39]

    Universal Dilation of Linear It\ˆ o SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments.arXiv preprint arXiv:2601.05928, 2026

    Hsuan-Cheng Wu and Xiantao Li. Universal Dilation of Linear It\ˆ o SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments.arXiv preprint arXiv:2601.05928, 2026

    work page arXiv 2026

  40. [40]

    Efficient quantum simulation for nonlinear stochastic differential equations.arXiv preprint arXiv:2603.12398, 2026

    Xiangyu Li, Ahmet Burak Catli, Ho Kiat Lim, Matthew Pocrnic, Dong An, Jin-Peng Liu, and Nathan Wiebe. Efficient quantum simulation for nonlinear stochastic differential equations.arXiv preprint arXiv:2603.12398, 2026

    work page arXiv 2026

  41. [41]

    A quantum spectral method for simulating stochastic processes, with applications to Monte Carlo.arXiv preprint arXiv:2303.06719, 2023

    Adam Bouland, Aditi Dandapani, and Anupam Prakash. A quantum spectral method for simulating stochastic processes, with applications to Monte Carlo.arXiv preprint arXiv:2303.06719, 2023

    work page arXiv 2023

  42. [42]

    Quantum analog-encoding for correlated Gaussian vectors and their exponentiation with application to rough volatility

    Tassa Thaksakronwong and Koichi Miyamoto. Quantum analog-encoding for correlated gaussian vectors and their exponentiation with application to rough volatility.arXiv preprint arXiv:2604.22463, 2026. 58

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  43. [43]

    SIAM, 2015

    Roger W Brockett.Finite dimensional linear systems. SIAM, 2015

    2015

  44. [44]

    Springer Science & Business Media, 2012

    Amnon Pazy.Semigroups of linear operators and applications to partial differential equations, volume 44. Springer Science & Business Media, 2012

    2012

  45. [45]

    SIAM, 2009

    Charles A Desoer and Mathukumalli Vidyasagar.Feedback systems: input-output properties. SIAM, 2009

    2009

  46. [46]

    Continuous Markov processes and stochastic equations.Rendiconti del Cir- colo Matematico di Palermo, 4(1):48–90, 1955

    Gisiro Maruyama. Continuous Markov processes and stochastic equations.Rend. Circ. Mat. Palermo, 4:48–90, 1955.doi:10.1007/BF02846028

    work page doi:10.1007/bf02846028 1955

  47. [47]

    G. N. Mil’shtejn. Approximate integration of stochastic differential equations.Theory of Probability&Its Applica- tions, 19(3):557–562, 1975.doi:10.1137/1119062

    work page doi:10.1137/1119062 1975

  48. [48]

    A comprehensive study of quantum arithmetic circuits.Philosophical Transactions A, 383(2288):20230392, 2025.doi:10.1098/ rsta.2023.0392

    Siyi Wang, Xiufan Li, Wei Jie Bryan Lee, Suman Deb, Eugene Lim, and Anupam Chattopadhyay. A comprehensive study of quantum arithmetic circuits.Philosophical Transactions A, 383(2288):20230392, 2025.doi:10.1098/ rsta.2023.0392

    work page arXiv 2025

  49. [49]

    Optimizing quantum circuits for arithmetic,

    Thomas H ¨aner, Martin Roetteler, and Krysta M Svore. Optimizing quantum circuits for arithmetic.arXiv preprint arXiv:1805.12445, 2018

    work page arXiv 2018

  50. [50]

    An improved QFT-based quantum comparator and extended modular arithmetic using one ancilla qubit.New Journal of Physics, 25(10):103011, oct 2023.doi:10.1088/1367-2630/acfd52

    Yewei Yuan, Chao Wang, Bei Wang, Zhao-Yun Chen, Meng-Han Dou, Yu-Chun Wu, and Guo-Ping Guo. An improved QFT-based quantum comparator and extended modular arithmetic using one ancilla qubit.New Journal of Physics, 25(10):103011, oct 2023.doi:10.1088/1367-2630/acfd52

    work page doi:10.1088/1367-2630/acfd52 2023

  51. [51]

    Quantum risk analysis.npj Quantum Information, 5(1):15, 2019.doi: 10.1038/s41534-019-0130-6

    Stefan Woerner and Daniel J Egger. Quantum risk analysis.npj Quantum Information, 5(1):15, 2019.doi: 10.1038/s41534-019-0130-6

    work page doi:10.1038/s41534-019-0130-6 2019

  52. [52]

    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 Sympo- sium on Theory of Computing, STOC 2019, page 193–204, New York, NY , USA, 2019. Association for Computing Machinery. [Full version] arXiv:1...

    work page doi:10.1145/3313276.3316366 2019

  53. [53]

    Quantum Inf

    Andrew M Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quant. Inf. Comput., 12:901–924, 2012.doi:10.26421/QIC12.11-12

    work page doi:10.26421/qic12.11-12 2012

  54. [54]

    The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation

    Shantanav Chakraborty, Andr ´as Gily´en, and Stacey Jeffery. The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation. In46th International Colloquium on Automata, Lan- guages, and Programming (ICALP 2019), pages 33–1. Schloss Dagstuhl–Leibniz-Zentrum f ¨ur Informatik, 2019. [Full version] arXiv:1804.01973...

    work page doi:10.4230/lipics.icalp.2019.33 2019

  55. [55]

    Aaronson

    Scott Aaronson. Read the fine print.Nature Physics, 11(4):291–293, 2015.doi:10.1038/nphys3272

    work page doi:10.1038/nphys3272 2015

  56. [56]

    Melissa E. O’Neill. PCG: A Family of Simple Fast Space-Efficient Statistically Good Algorithms for Random Number Generation. Technical Report HMC-CS-2014-0905, Harvey Mudd College, Claremont, CA, September 2014

    2014

  57. [57]

    Continuous random variate generation by fast numerical inversion.ACM Trans

    Wolfgang H ¨ormann and Josef Leydold. Continuous random variate generation by fast numerical inversion.ACM Trans. Model. Comput. Simul., 13(4):347–362, October 2003.doi:10.1145/945511.945517

    work page doi:10.1145/945511.945517 2003

  58. [58]

    A remark on Stirling’s formula.The American mathematical monthly, 62(1):26–29, 1955

    Herbert Robbins. A remark on Stirling’s formula.The American mathematical monthly, 62(1):26–29, 1955

    1955

  59. [59]

    M ´aria Kieferov´a, Artur Scherer, and Dominic W. Berry. Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series.Phys. Rev. A, 99:042314, Apr 2019. URL:https://link.aps.org/doi/10. 1103/PhysRevA.99.042314,doi:10.1103/PhysRevA.99.042314

    work page doi:10.1103/physreva.99.042314 2019

  60. [60]

    SIAM, 2022

    Lloyd N Trefethen and David Bau.Numerical linear algebra. SIAM, 2022

    2022

  61. [61]

    F. L. Bauer and C. T. Fike. Norms and exclusion theorems.Numerische Mathematik, 2:137, 1960.doi:10.1007/ BF01386217

    1960

  62. [62]

    Springer, 2002

    Marvin K Simon.Probability distributions involving Gaussian random variables: A handbook for engineers and scientists. Springer, 2002

    2002

  63. [63]

    The best constants in the Khintchine inequality.Studia Mathematica, 70(3):231–283, 1981

    Uffe Haagerup. The best constants in the Khintchine inequality.Studia Mathematica, 70(3):231–283, 1981. 59 A Bound onN 0,l−1 2k Lemma 18.For k,l∈N, N 0,l−1 2k ≤(2k−1)!!l k (401) holds. Proof.We note N 0,l 2k = kX s=0 2k 2s ! N 0,l−1 2(k−s) .(402) By using induction with respect tolalong with N 0,0 2k =1, we see N 0,l−1 2k = 1 2l lX s=0 l s ! (2s−l) 2k.(40...

    1981