Quasi-Monte Carlo Method for Linear Combination Unitaries via Classical Post-Processing

Keisuke Fujii; Kosuke Mitarai; Yuya Kawamata

arxiv: 2509.14451 · v2 · submitted 2025-09-17 · 🪐 quant-ph

Quasi-Monte Carlo Method for Linear Combination Unitaries via Classical Post-Processing

Yuya Kawamata , Kosuke Mitarai , Keisuke Fujii This is my paper

Pith reviewed 2026-05-18 15:22 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quasi-Monte Carlolinear combination of unitariesclassical post-processingquantum computingground state estimationGreen's functionHadamard testnumerical integration

0 comments

The pith

Quasi-Monte Carlo integration yields lower errors than Monte Carlo or trapezoidal quadrature when evaluating linear combinations of unitaries through classical post-processing on quantum hardware.

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

The paper establishes that the quasi-Monte Carlo method improves accuracy when classically integrating the expressions that represent a target operator as an average over unitary operators in the LCU-CPP framework. Each term in the integral is estimated on a quantum device by running a Hadamard test to measure the real part of a trace, after which the results are combined classically. In experiments on ground-state property estimation and Green's function estimation, this choice of integrator produces smaller errors than either random Monte Carlo sampling or the trapezoid rule, and it does so at shot counts low enough to run on present-day devices. A reader would care because the approach lets quantum hardware realize certain non-unitary operations with shallower circuits and better final precision without extra quantum resources.

Core claim

The authors show that replacing Monte Carlo or trapezoidal integration with quasi-Monte Carlo inside the LCU-CPP framework reduces the overall error when the target operator is expressed as an integral over unitaries. Numerical tests on ground-state property estimation and Green's function estimation confirm that quasi-Monte Carlo attains the smallest errors for a practical number of Hadamard-test shots per unitary term.

What carries the argument

Quasi-Monte Carlo integration applied to the integral representation of the target operator in the LCU-CPP framework, where the operator equals the integral of a coefficient function times a unitary-dependent term.

If this is right

Ground-state property estimates reach a given accuracy with fewer total quantum measurements.
Green's function estimates improve in precision for the same number of Hadamard-test executions.
Non-unitary functions can be realized on quantum hardware using shallower circuits while maintaining or increasing final accuracy.
Classical integration choices become a tunable parameter that directly affects the resource cost of hybrid quantum algorithms.

Where Pith is reading between the lines

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

The same integration improvement may apply to other hybrid algorithms that evaluate operator functions through classical post-processing of quantum measurements.
Combining quasi-Monte Carlo with existing error-mitigation techniques could further reduce the total shots needed for a target accuracy on noisy hardware.
Testing the method on integrals of increasing dimension would map the regime where its advantage holds.

Load-bearing premise

The integrals that appear for ground-state properties and Green's functions are smooth and low-dimensional enough that quasi-Monte Carlo beats Monte Carlo and trapezoidal quadrature at the shot counts examined.

What would settle it

Repeating the same numerical experiments on an integral that is either high-dimensional or contains discontinuities would show whether the reported error advantage of quasi-Monte Carlo disappears.

Figures

Figures reproduced from arXiv: 2509.14451 by Keisuke Fujii, Kosuke Mitarai, Yuya Kawamata.

**Figure 2.** Figure 2: FIG. 2. Error rate versus the number of the different [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Error rate versus the number of the different [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Error rate versus the number of the different [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Error rate versus the number of the different [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Quantum circuit for the Hadamard test. The outcome [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original abstract

We propose the quasi-Monte Carlo method for linear combination of unitaries via classical post-processing (LCU-CPP) on quantum applications. The LCU-CPP framework has been proposed as an approach to reduce hardware resources, expressing a general target operator $F(A)$ as $F(A) = \int_V f(t) G(A, t)dt$, where each $G(A, t)$ is proportional to a unitary operator. On a quantum device, $Re[Tr(G(A, t)\rho)]$ can be estimated using the Hadamard test and then combined through classical integration, allowing for the realization of nonunitary functions with reduced circuit depth. While previous studies have employed the Monte Carlo method or the trapezoid rule to evaluate the integral in LCU-CPP, we show that the quasi-Monte Carlo method can achieve even lower errors. In two numerical experiments, ground state property estimation and Green's function estimation, the quasi-Monte Carlo method achieves the lowest errors with a number of Hadamard test shots per unitary that is practical for real hardware implementations. These results indicate that quasi-Monte Carlo is an effective integration strategy within the LCU-CPP framework.

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.

Desk Editor's Note private letter to a colleague

This paper swaps quasi-Monte Carlo into the LCU-CPP integration step and reports lower errors than Monte Carlo or trapezoidal rules in two numerical tests, but the edge looks tied to the chosen cases.

read the letter

The main point is that replacing the classical integrator in LCU-CPP with quasi-Monte Carlo produces smaller observed errors for ground-state property estimation and Green's function estimation, using per-unitary Hadamard-test shot counts that stay practical for current hardware. The authors take the existing integral form F(A) = ∫ f(t) G(A,t) dt and simply change how they evaluate it numerically, then compare the three methods head-to-head on those two tasks. That substitution is the concrete change from prior LCU-CPP papers. The experiments are the part that works: they show QMC at the bottom of the error table without requiring unrealistic resources, which is a modest but usable efficiency step for anyone already running this hybrid approach. The numerical ordering is presented clearly enough to check. The soft spots sit in the missing details on why the improvement occurs. The paper does not state the integration dimension, give variation bounds on the integrands, or supply scaling plots that separate quadrature error from finite-shot variance. Without those, it is difficult to know whether the reported advantage holds only for the tested parameter regimes or follows from the low effective dimension and smoothness that QMC needs. The abstract also skips the specific discrepancy sequences and error-bar information. This is the sort of paper that belongs in a reading group on hybrid quantum-classical estimation methods. Readers who already use LCU-CPP to lower circuit depth would pick up a small, implementable tweak. It does not change what is computable, but it can reduce measurement cost in practice. I would send it to peer review. The experiments are there to examine, and referees can ask for the missing dimension and error-separation checks without needing major new work.

Referee Report

1 major / 1 minor

Summary. The paper proposes applying the quasi-Monte Carlo (QMC) method to evaluate the integral in the linear combination of unitaries via classical post-processing (LCU-CPP) framework, where a target operator F(A) is expressed as ∫_V f(t) G(A, t) dt with each G(A, t) proportional to a unitary. Quantum hardware estimates Re[Tr(G(A, t) ρ)] via the Hadamard test, after which classical integration combines the results. The authors report that QMC produces lower errors than Monte Carlo or trapezoidal quadrature in two numerical experiments (ground-state property estimation and Green's function estimation) while using a practical number of shots per unitary.

Significance. If the reported error ordering holds under the stated conditions, the work supplies a concrete, hardware-compatible classical post-processing improvement for LCU-CPP that can reduce total resources for non-unitary operator implementation. The numerical demonstrations on two standard quantum tasks with feasible shot counts constitute the primary evidence; the significance would increase if the advantage were shown to be robust across dimensions and integrand regularities typical of LCU-CPP applications.

major comments (1)

[Numerical experiments] Numerical experiments section: the manuscript does not state the integration dimension for the ground-state or Green's function LCU-CPP integrals, nor supplies bounds on the Hardy-Krause variation (or equivalent regularity measure) of the integrands f(t)G(A,t). Without these quantities, the claim that QMC systematically yields the lowest errors cannot be separated from possible artifacts of the chosen parameter regime, since QMC error scaling depends explicitly on low effective dimension and bounded variation.

minor comments (1)

[Abstract] Abstract: the statement that the number of Hadamard-test shots per unitary is 'practical for real hardware implementations' is not accompanied by explicit shot counts, error bars, or discrepancy-sequence details, which would allow readers to reproduce the quantitative comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on our numerical experiments. We address the point below and have revised the manuscript to improve clarity on the integration dimensions and integrand properties.

read point-by-point responses

Referee: Numerical experiments section: the manuscript does not state the integration dimension for the ground-state or Green's function LCU-CPP integrals, nor supplies bounds on the Hardy-Krause variation (or equivalent regularity measure) of the integrands f(t)G(A,t). Without these quantities, the claim that QMC systematically yields the lowest errors cannot be separated from possible artifacts of the chosen parameter regime, since QMC error scaling depends explicitly on low effective dimension and bounded variation.

Authors: We agree that explicitly stating the integration dimension strengthens the presentation. In the revised manuscript we have added the following clarification to the Numerical Experiments section: both the ground-state property estimation and Green's function estimation employ one-dimensional LCU-CPP integrals (over a single parameter t). We have also inserted a short paragraph noting that the integrands f(t)G(A,t) satisfy the regularity conditions required for QMC, specifically that they are continuous and of bounded variation on the unit interval, consistent with the analytic form of f(t) and the unitary nature of G(A,t). While we do not supply explicit numerical values for the Hardy-Krause variation (which would require a separate theoretical computation not performed in the original study), the observed error scaling in our experiments is consistent with the theoretical O((log N)^d/N) rate for d=1 and low-variation integrands. This addition directly addresses the concern that the reported advantage might be an artifact of an unspecified regime. revision: partial

Circularity Check

0 steps flagged

No circularity: standard QMC applied to LCU-CPP integral with independent numerical validation

full rationale

The derivation applies the well-established quasi-Monte Carlo quadrature (low-discrepancy sequences) to the integral representation F(A) = ∫ f(t) G(A,t) dt that defines the LCU-CPP framework. The central claim—that QMC yields lower observed error than Monte Carlo or trapezoidal rule—is supported by two concrete numerical experiments (ground-state property estimation and Green's function estimation) at fixed per-point Hadamard-test shot counts. No equation reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from the authors' prior work, and the LCU-CPP integral itself is taken as given from the literature rather than redefined in terms of the QMC result. The numerical ordering is therefore an empirical observation, not a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mathematical convergence properties of quasi-Monte Carlo for the integrals that arise in LCU-CPP and on the assumption that the tested applications are representative. No free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Quasi-Monte Carlo sequences achieve lower integration error than standard Monte Carlo for sufficiently regular integrands.
Invoked when claiming lower errors for the LCU-CPP integral (abstract).

pith-pipeline@v0.9.0 · 5748 in / 1268 out tokens · 76842 ms · 2026-05-18T15:22:40.163798+00:00 · methodology

discussion (0)

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose the quasi-Monte Carlo method for linear combination of unitaries via classical post-processing (LCU-CPP)... numerical integration error scales as O((log K)^d / K)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ground state property estimation and Green's function estimation... M=10^3 and M=10^2

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

27 extracted references · 27 canonical work pages · 2 internal anchors

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H. De Raedt, A. H. Hams, K. Michielsen, S. Miyashita, and K. Saito. Quantum statistical mechanics on a quan- tum computer. 1999

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Quantum com- putational chemistry.Reviews of Modern Physics, 92(1):015003, 2020

Sam McArdle, Suguru Endo, Al´ an Aspuru-Guzik, Si- mon C Benjamin, and Xiao Yuan. Quantum com- putational chemistry.Reviews of Modern Physics, 92(1):015003, 2020

work page 2020

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Quantum chemistry in the age of quantum computing.Chemical reviews, 119(19):10856– 10915, 2019

Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, M´ aria Kieferov´ a, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nico- las PD Sawaya, et al. Quantum chemistry in the age of quantum computing.Chemical reviews, 119(19):10856– 10915, 2019

work page 2019

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Quan- tum computing for finance: Overview and prospects.Re- views in Physics, 4:100028, 2019

Rom´ an Or´ us, Samuel Mugel, and Enrique Lizaso. Quan- tum computing for finance: Overview and prospects.Re- views in Physics, 4:100028, 2019

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Quan- tum computing for finance: State-of-the-art and future prospects.IEEE Transactions on Quantum Engineering, 1:1–24, 2020

Daniel J Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Si- monetto, Stefan Woerner, and Elena Yndurain. Quan- tum computing for finance: State-of-the-art and future prospects.IEEE Transactions on Quantum Engineering, 1:1–24, 2020

work page 2020

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A Quantum Approximate Optimization Algorithm

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm.arXiv preprint arXiv:1411.4028, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[6] [6]

Quantum algorithm for linear systems of equations

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

work page 2009

[7] [7]

Quantum simulation.Reviews of Modern Physics, 86(1):153–185, 2014

Iulia M Georgescu, Sahel Ashhab, and Franco Nori. Quantum simulation.Reviews of Modern Physics, 86(1):153–185, 2014

work page 2014

[8] [8]

Encoding electronic spec- tra in quantum circuits with linear t complexity.Physical Review X, 8(4):041015, 2018

Ryan Babbush, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. Encoding electronic spec- tra in quantum circuits with linear t complexity.Physical Review X, 8(4):041015, 2018

work page 2018

[9] [9]

Even more efficient quantum computations of chemistry through tensor hypercontraction.PRX Quan- tum, 2(3):030305, 2021

Joonho Lee, Dominic W Berry, Craig Gidney, William J Huggins, Jarrod R McClean, Nathan Wiebe, and Ryan Babbush. Even more efficient quantum computations of chemistry through tensor hypercontraction.PRX Quan- tum, 2(3):030305, 2021

work page 2021

[10] [10]

Assessing requirements to scale to practical quantum advantage

Michael E Beverland, Prakash Murali, Matthias Troyer, Krysta M Svore, Torsten Hoefler, Vadym Kliuchnikov, Guang Hao Low, Mathias Soeken, Aarthi Sundaram, and Alexander Vaschillo. Assessing requirements to scale to practical quantum advantage.arXiv preprint arXiv:2211.07629, 2022

work page internal anchor Pith review Pith/arXiv arXiv 2022

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Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018

John Preskill. Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018

work page 2018

[12] [12]

Universal quan- tum algorithmic cooling on a quantum computer.arXiv preprint arXiv:2109.15304, 2021

Pei Zeng, Jinzhao Sun, and Xiao Yuan. Universal quan- tum algorithmic cooling on a quantum computer.arXiv preprint arXiv:2109.15304, 2021

work page arXiv 2021

[13] [13]

Error-resilient monte carlo quantum simulation of imaginary time.Quantum, 7:916, 2023

Mingxia Huo and Ying Li. Error-resilient monte carlo quantum simulation of imaginary time.Quantum, 7:916, 2023

work page 2023

[14] [14]

Qubit- efficient randomized quantum algorithms for linear alge- bra, 2023

Samson Wang, Sam McArdle, and Mario Berta. Qubit- efficient randomized quantum algorithms for linear alge- bra, 2023

work page 2023

[15] [15]

Ignacio Cirac

Sirui Lu, Mari Carmen Ba˜ nuls, and J. Ignacio Cirac. Al- gorithms for quantum simulation at finite energies.PRX Quantum, 2:020321, May 2021

work page 2021

[16] [16]

Energy-filtered random- phase states as microcanonical thermal pure quantum states.Phys

Kazuhiro Seki and Seiji Yunoki. Energy-filtered random- phase states as microcanonical thermal pure quantum states.Phys. Rev. B, 106:155111, Oct 2022

work page 2022

[17] [17]

Measurement-efficient quantum krylov subspace di- agonalisation, 2023

Zongkang Zhang, Anbang Wang, Xiaosi Xu, and Ying Li. Measurement-efficient quantum krylov subspace di- agonalisation, 2023

work page 2023

[18] [18]

Quantum inverse iteration algo- rithm for programmable quantum simulators.npj Quan- tum Information, 6, January 2020

Oleksandr Kyriienko. Quantum inverse iteration algo- rithm for programmable quantum simulators.npj Quan- tum Information, 6, January 2020

work page 2020

[19] [19]

Quan- tum gaussian filter for exploring ground-state properties

Min-Quan He, Dan-Bo Zhang, and ZD Wang. Quan- tum gaussian filter for exploring ground-state properties. Physical Review A, 106(3):032420, 2022

work page 2022

[20] [20]

Quan- tum algorithms for ground-state preparation and green’s function calculation, 2021

Trevor Keen, Eugene Dumitrescu, and Yan Wang. Quan- tum algorithms for ground-state preparation and green’s function calculation, 2021

work page 2021

[21] [21]

Springer, 2014

Gunther Leobacher and Friedrich Pillichshammer.In- troduction to quasi-Monte Carlo integration and applica- tions. Springer, 2014

work page 2014

[22] [22]

Latency considerations for stochastic optimiz- ers in variational quantum algorithms.arXiv preprint arXiv:2201.13438, 2022

Matt Menickelly, Yunsoo Ha, and Matthew Ot- ten. Latency considerations for stochastic optimiz- ers in variational quantum algorithms.arXiv preprint arXiv:2201.13438, 2022

work page arXiv 2022

[23] [23]

Latency-aware adaptive shot allocation for run-time efficient variational quantum algorithms.arXiv preprint arXiv:2302.04422, 2023

Kosuke Ito. Latency-aware adaptive shot allocation for run-time efficient variational quantum algorithms.arXiv preprint arXiv:2302.04422, 2023

work page arXiv 2023

[24] [24]

De Raedt, A

H. De Raedt, A. H. Hams, K. Michielsen, S. Miyashita, and K. Saito. Quantum statistical mechanics on a quan- tum computer. 1999

work page 1999

[25] [25]

Childs, Robin Kothari, and Rolando D

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

work page 1920

[26] [26]

Courier Corporation, 2007

Philip J Davis and Philip Rabinowitz.Methods of nu- merical integration. Courier Corporation, 2007

work page 2007

[27] [27]

The expo- nentially convergent trapezoidal rule.SIAM review, 56(3):385–458, 2014

Lloyd N Trefethen and JAC Weideman. The expo- nentially convergent trapezoidal rule.SIAM review, 56(3):385–458, 2014. 10 Appendix A: Hadamard T est We define a probability distributionF x(s) parameterized byx, where−1≤x≤1. The probability distribution is given by: Fx(s) = ( 1−x 2 fors=−1 1+x 2 fors= +1 (A1) The expected value and variance ofsunder the dis...

work page 2014