arxiv: 2604.26098 · v1 · submitted 2026-04-28 · 🪐 quant-ph

Recognition: unknown

Solving a Linear System of Equations on a Quantum Computer by Measurement

Alain Giresse Tene , Thomas Konrad

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:07 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum linear systemsvariational quantum algorithmsphase estimationfault tolerant quantum computingeigenvector preparationmeasurement based quantum algorithmsdense matrix solvers

0 comments

The pith

A variational quantum algorithm solves linear systems by iteratively maximizing the fidelity to a phase-estimated eigenvector state.

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

The paper presents a measurement test algorithm for fault-tolerant quantum computers that prepares the solution to a linear system as the eigenvector of a self-adjoint matrix. It implements a von Neumann measurement of the corresponding observable using phase estimation, then measures the projection probability (which equals the target fidelity) via relative frequencies across repeated trials. Parameters are optimized iteratively to maximize this probability until the target state is obtained. This approach applies to any task whose solution is such an eigenvector. A reader would care because it addresses limitations in earlier variational methods for a core computational primitive used across physics, engineering, and data analysis.

Core claim

The measurement test algorithm prepares the solution state of a linear system of equations as the eigenvector of a self-adjoint matrix through a von Neumann measurement implemented by the phase estimation algorithm; the probability of projecting into this unknown target state equals the target fidelity, which is estimated from relative frequencies and iteratively optimized to read out the state. The algorithm applies to any computational task whose solution is represented as an eigenvector of a self-adjoint matrix.

What carries the argument

The measurement test algorithm, which uses phase estimation to realize a von Neumann measurement of an observable whose target eigenvector encodes the linear-system solution, then optimizes the measured projection probability (fidelity) over repeated trials.

If this is right

The algorithm computes eigenvectors of dense matrices without any decomposition into Pauli strings.
Solution accuracy is independent of the matrix condition number κ when O(log κ) qubits are allocated to encode eigenvalues.
Target fidelity F_T = 1-ε is reached with accuracy ε that improves proportionally to 1/N for N measurements per iteration.
The method extends directly to any problem whose solution is the eigenvector of a self-adjoint matrix.

Where Pith is reading between the lines

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

Hybrid quantum-classical loops could use the measured frequencies to guide parameter updates on classical hardware without needing full state tomography.
The method might reduce the effective qubit overhead for fault-tolerant linear solvers in applications where condition numbers grow with system size.
Extension to non-Hermitian or nonlinear problems would require generalizing the observable whose measurement yields the solution state.

Load-bearing premise

The iterative optimization of fidelity parameters will converge to the global maximum corresponding to the desired eigenvector state.

What would settle it

An experiment or simulation on a known dense matrix where the fidelity optimization plateaus below the claimed 1/N scaling or fails to reach the target state despite using O(log κ) qubits for eigenvalue encoding.

Figures

Figures reproduced from arXiv: 2604.26098 by Alain Giresse Tene, Thomas Konrad.

**Figure 1.** Figure 1: FIG. 1: Ansatz with 36 view at source ↗

**Figure 2.** Figure 2: FIG. 2: Quantum circuit of measurement test view at source ↗

**Figure 3.** Figure 3: FIG. 3: Numerical simulation show convergence of view at source ↗

**Figure 4.** Figure 4: FIG. 4: Target fidelity for simulation of a single run view at source ↗

**Figure 5.** Figure 5: FIG. 5: Subsequent section to the one shown in Fig. view at source ↗

**Figure 7.** Figure 7: FIG. 7: The median relative standard deviation view at source ↗

**Figure 8.** Figure 8: FIG. 8: The median relative standard deviation view at source ↗

read the original abstract

We present a variational algorithm for fault tolerant quantum computing to solve a system of linear equations which directly maximises the parameters of the target fidelity. This so-called measurement test algorithm can be applied to any computational task with a solution that is represented as eigenvector of a self-adjoint matrix. The solution is prepared as state of a register in the quantum computer by a von Neumann measurement of a corresponding observable, which is implemented using the phase estimation algorithm. The probability to project the system thus into the unknown target state, which equals the target fidelity, is measured in terms of relative frequencies and iteratively optimised to read out the target state. The new algorithm overcomes three issues of previous variational quantum algorithms: i) It does not rely on a decomposition in terms of Pauli strings and therefore can compute eigenvectors of dense matrices. ii) The accuracy is not limited by the condition number $\kappa$ of the matrix, provided a logarithmic number ($O(\log\kappa)$) of qubits is used to encode the eigenvalues and iii) the target fidelity $F_T = 1-\epsilon$ can be reached with an accuracy $\epsilon$ that scales with $1/N$ for $N$ measurements per iteration. We demonstrate this by numerical simulations for dense random real-valued $16\times 16$ matrices with non-vanishing determinant.

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

The paper's new variational measurement-test algorithm for linear systems handles dense matrices but rests on a flawed 1/N accuracy scaling claim that contradicts binomial statistics.

read the letter

The main thing to know is that this paper offers a variational quantum algorithm for linear systems based on maximizing the fidelity to the target state via repeated phase estimation measurements, but its advertised accuracy scaling of 1/N is at odds with how binomial sampling works. The new element is the 'measurement test algorithm' that directly optimizes parameters to increase the probability of projecting onto the eigenvector using von Neumann measurement implemented by phase estimation. This lets it work with dense matrices since it skips Pauli decompositions, and it uses a logarithmic number of qubits to encode eigenvalues so the condition number does not limit precision in the usual way. The numerical simulations for 16x16 dense random matrices show that the method can recover the solution in these small instances. What the paper does well is present a clear alternative to existing variational solvers and back it with concrete small-scale tests. The avoidance of Pauli strings is a practical plus for dense problems. The soft spot is the third advantage listed: reaching fidelity 1-ε with accuracy ε scaling as 1/N using N measurements per iteration. Since the fidelity comes from the relative frequency of measurement successes, the estimator's standard error is on the order of 1/sqrt(N) when the probability is close to 1. This is a standard result from statistics and does not depend on the matrix properties. If the efficiency claims rest on this 1/N scaling, then that part of the argument does not stand up. The paper would need to explain how the iterative optimization handles this noise or if they have a different estimator in mind. Overall, this is for quantum information researchers exploring variational methods for linear algebra tasks. Someone working on quantum linear solvers might find the measurement-based optimization interesting to compare against other approaches. It deserves a serious referee because the core proposal is distinct and the simulations are there, even though the scaling claim needs scrutiny and likely revision. I recommend sending it out for peer review rather than desk rejecting it.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a variational 'measurement test algorithm' for solving linear systems of equations on a fault-tolerant quantum computer. The solution is encoded as an eigenvector prepared via phase estimation; the target fidelity is estimated from relative frequencies of successful projections and iteratively maximized over variational parameters. The approach is claimed to handle dense matrices without Pauli-string decompositions, to be independent of the matrix condition number when O(log κ) qubits encode the eigenvalues, and to achieve fidelity accuracy scaling as 1/N with N measurements per iteration. Numerical simulations on random 16×16 dense matrices are presented to support the claims.

Significance. If the central claims hold, the algorithm would provide a distinct variational route to linear-system solutions that avoids the overhead of Pauli decompositions and potentially relaxes the dependence on condition number. The reported simulations on modest-sized dense matrices constitute concrete evidence of implementability. However, the statistical scaling claim requires correction before the efficiency advantage can be assessed.

major comments (1)

Abstract: the claim that 'the target fidelity F_T = 1-ε can be reached with an accuracy ε that scales with 1/N for N measurements per iteration' is incorrect. The fidelity is estimated as the relative frequency of a von Neumann measurement outcome (implemented by phase estimation). For a binomial estimator with success probability p ≈ F_T near 1, the standard error is Θ(1/√N), not Θ(1/N). This statistical fact is independent of matrix density or the O(log κ) qubit count and directly undermines the third claimed improvement over prior variational quantum algorithms.

minor comments (1)

The manuscript should include an explicit section detailing the optimization loop, the precise implementation of the phase-estimation circuit, and the convergence criterion used in the numerical experiments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the statistical issue in our scaling claim. We address the major comment below and will make the necessary revisions.

read point-by-point responses

Referee: Abstract: the claim that 'the target fidelity F_T = 1-ε can be reached with an accuracy ε that scales with 1/N for N measurements per iteration' is incorrect. The fidelity is estimated as the relative frequency of a von Neumann measurement outcome (implemented by phase estimation). For a binomial estimator with success probability p ≈ F_T near 1, the standard error is Θ(1/√N), not Θ(1/N). This statistical fact is independent of matrix density or the O(log κ) qubit count and directly undermines the third claimed improvement over prior variational quantum algorithms.

Authors: We agree with the referee that the claimed scaling is incorrect. The fidelity is estimated via the relative frequency of a successful projection outcome, which is a binomial random variable. The standard error of this estimator is indeed Θ(1/√N) for N independent measurements, not 1/N. This was an oversight in the presentation of the statistical analysis. We will revise the abstract and the corresponding discussion in the main text to state the correct Θ(1/√N) scaling. We will also add a brief explanation that achieving a target precision ε therefore requires O(1/ε²) measurements per iteration. This correction does not affect the validity of the numerical simulations or the other two claimed advantages of the algorithm (applicability to dense matrices without Pauli decompositions, and independence from κ with O(log κ) qubits). We will update the manuscript to reflect these changes and to clarify how the standard measurement scaling compares to other variational methods. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard quantum primitives and external measurements

full rationale

The paper describes a variational algorithm that prepares the solution state via phase estimation followed by iterative optimization of parameters to maximize the target fidelity, where fidelity is estimated directly from relative frequencies of measurement outcomes. This process uses externally obtained measurement statistics rather than any internal fit or self-referential definition. No steps reduce by construction to the paper's own inputs, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness theorem is smuggled in. The statistical scaling assertion in claim (iii), while open to correctness critique, is presented as a consequence of the measurement process and does not create a circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Limited information available from abstract only. The algorithm relies on standard quantum computing primitives and introduces a new optimization loop, but details on parameters are not specified.

free parameters (1)

variational parameters for fidelity maximization
The algorithm iteratively optimizes parameters to maximize target fidelity, but specific parameters and how they are chosen are not detailed in the abstract.

axioms (2)

domain assumption The matrix is self-adjoint and the solution is its eigenvector
The abstract states the solution is represented as eigenvector of a self-adjoint matrix.
domain assumption Fault-tolerant quantum computing with phase estimation is available
The algorithm is presented for fault tolerant quantum computing using phase estimation.

pith-pipeline@v0.9.0 · 5528 in / 1685 out tokens · 112439 ms · 2026-05-07T16:07:58.508945+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages · 1 internal anchor

[1]

and can be iterated to increase the number of angles for expressibility. It consists of two layers of general sin- gle qubit rotations with three variable Euler angles sand- wichingC-Xgates between next neighbours in a circular configuration with additional rotations. D. Implementation of ”textbook” quantum measurement The measurement of the observableAis...

2000
[2]

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

2009
[3]

Quantum gradi- ent descent and newton’s method for constrained polyno- mial optimization

Patrick Rebentrost, Maria Schuld, Leonard Wossnig, Francesco Petruccione, and Seth Lloyd. Quantum gradi- ent descent and newton’s method for constrained polyno- mial optimization. New Journal of Physics, 21(7):073023, jul 2019

2019
[4]

Cerezo, A

M. Cerezo, A. Arrasmith, and R. et al. Babbush. Vari- ational quantum algorithms. Nat Rev Phys, 3:625–644, 2021

2021
[5]

Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Al´ an Aspuru-Guzik

Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Al´ an Aspuru-Guzik. Noisy 8 intermediate-scale quantum algorithms. Rev. Mod. Phys., 94:015004, Feb 2022

2022
[6]

Love, Alan Aspuru- Guzik, and Jeremy L

Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man- Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alan Aspuru- Guzik, and Jeremy L. O’Brien. A variational eigen- value solver on a photonic quantum processor. Nature Communications, 5:4213, 2014

2014
[7]

Grimsley, E

Harper R. Grimsley, E. Economou, Sophia, Edwin Barnes, and Nicholas J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat Comm, 10:3007, 2019

2019
[8]

Tilly, H

J. Tilly, H. Chen, S. Cao, and et al. The variational quan- tum eigensolver: A review of methods and best practices. Physics Reports, 986:1–128, 2022

2022
[9]

Nakagawa, Yu-ya Ohnishi, and Wataru Mizukami

Nobuyuki Yoshioka, Takeshi Sato, Yuya O. Nakagawa, Yu-ya Ohnishi, and Wataru Mizukami. Variational quan- tum simulation for periodic materials. Phys. Rev. Res., 4:013052, Jan 2022

2022
[10]

Innan, M

N. Innan, M. A. Khan, and M. Bennai. Quantum com- puting for electronic structure analysis: Ground state energy and molecular properties calculations. Materials Today Commun, 38:107760, 2023

2023
[11]

X. Li, Y. Fan, J. Liu, Z. Li, and J. Yang. Adaptive variational quantum simulations of periodic materials us- ing qubit-encoded wave functions. Journal of Chemical Theory and Computation, 21:5973–5985, 2025. E-print cond-mat/0003225

work page arXiv 2025
[12]

A. M. Romero, J. Engel, H.-L. Tang, and S. E. Economou. Solving nuclear structure problems with the adaptive variational quantum algorithm. Phys Rev C, 105:064317, 2000

2000
[13]

A Quantum Approximate Optimization Algorithm

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm, 2014. E- print quant-ph/1411.4028

work page internal anchor Pith review arXiv 2014
[14]

Quantum optimization using variational algorithms on near-term quantum de- vices

Nikolaj Moll, Panagiotis Barkoutsos, Lev S Bishop, Jerry M Chow, Andrew Cross, Daniel J Egger, Ste- fan Filipp, Andreas Fuhrer, Jay M Gambetta, Marc Ganzhorn, Abhinav Kandala, Antonio Mezzacapo, Pe- ter M¨ uller, Walter Riess, Gian Salis, John Smolin, Ivano Tavernelli, and Kristan Temme. Quantum optimization using variational algorithms on near-term quant...

2018
[15]

Multiob- jective variational quantum optimization for constrained problems: an application to cash handling

Pablo D´ ıez-Valle, Jorge Luis-Hita, Senaida Hern´ andez- Santana, Fernando Mart´ ınez-Garc´ ıa, ´Alvaro D´ ıaz- Fern´ andez, Eva Andr´ es, Juan Jos´ e Garc´ ıa-Ripoll, Es- col´ astico S´ anchez-Mart´ ınez, and Diego Porras. Multiob- jective variational quantum optimization for constrained problems: an application to cash handling. Quantum Science and Tec...

2023
[16]

Childs, Jin-Peng Liu, and Aaron Ostrander

Andrew M. Childs, Jin-Peng Liu, and Aaron Ostrander. High-precision quantum algorithms for partial differen- tial equations. Quantum, 5:574, November 2021

2021
[17]

Review and perspectives in quantum computing for partial differ- ential equations in structural mechanics

Giorgio Tosti Balducci, Boyang Chen, Matthias M¨ oller, Marc Gerritsma, and Roeland De Breuker. Review and perspectives in quantum computing for partial differ- ential equations in structural mechanics. Frontiers in Mechanical Engineering, Volume 8 - 2022, 2022

2022
[18]

Quantum simulation of partial differential equations: Applications and detailed analysis

Shi Jin, Nana Liu, and Yue Yu. Quantum simulation of partial differential equations: Applications and detailed analysis. Phys. Rev. A, 108:032603, Sep 2023

2023
[19]

Quantum algorithms for partial differen- tial equations: A performance review and future trajec- tories

Thanh Nguyen. Quantum algorithms for partial differen- tial equations: A performance review and future trajec- tories. In Nagar Atulya K., Jat Dharm Singh, Mishra Durgesh Kumar, and Amit Joshi, editors, Intelligent Sustainable Systems, pages 18–37, Cham, 2025. Springer Nature Switzerland

2025
[20]

Variational quantum algo- rithms for nonlinear problems

Michael Lubasch, Jaewoo Joo, Pierre Moinier, Martin Kiffner, and Dieter Jaksch. Variational quantum algo- rithms for nonlinear problems. Phys. Rev. A, 101:010301, Jan 2020

2020
[21]

Jonas J¨ ager and Roman V. Krems. Universal expres- siveness of variational quantum classifiers and quan- tum kernels for support vector machines. Nature Communications, 14(1):576, 2023

2023
[22]

Nielsen and I.L

M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge and New York, 2009

2009
[23]

Bravo-Prieto, R

C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P.J. Coles. Variational quantum linear solver. Quantum, 7:1188, 2023

2023
[24]

Busch, P.J

P. Busch, P.J. Lahti, and P. Mittelstaedt. The Quantum Theory of Measurement. Springer-Verlag, Berlin, 1991

1991
[25]

Random unitaries in extremely low depth

Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. Random unitaries in extremely low depth. Science, 389(6755):92–96, 2025

2025
[26]

Structure optimization for parameterized quantum circuits

Mateusz Ostaszewski, Edward Grant, and Marcello Benedetti. Structure optimization for parameterized quantum circuits. Quantum, 5:391, January 2021

2021
[27]

Liu, and Yuan H

M Hayashi, Z.W. Liu, and Yuan H. Global heisenberg scaling in noisy and practical phase estimation.Quantum Sci. Technol., 7:025030, 2022

2022
[28]

E. Kreysig. Statistische Methoden und ihre Anwendungen (in german). Vandenhoek & Ruprecht, Goettingen, Germany, 1970

1970
[29]

Al-Mohy and N.J

A.H. . Al-Mohy and N.J. Higham. A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Appl., 31:970–989, 2009

2009
[30]

Higham and Tisseur F

N.J. Higham and Tisseur F. A block algorithm for ma- trix 1-norm estimation, with an application to 1-norm pseudospectra. SIAM J. Matrix Anal. Appl., 21:025030, 2000

2000
[31]

Nelson and Andrew D

Jacob S. Nelson and Andrew D. Baczewski. Assessment of quantum phase estimation protocols for early fault- tolerant quantum computers. Phys. Rev. A, 110:042420, Oct 2024

2024