A penalty-free quantum algorithm to find energy eigenstates

Heng Dai; Jiangbin Gong; Nannan Ma

arxiv: 2509.09148 · v2 · submitted 2025-09-11 · 🪐 quant-ph

A penalty-free quantum algorithm to find energy eigenstates

Nannan Ma , Heng Dai , Jiangbin Gong This is my paper

Pith reviewed 2026-05-18 18:29 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum algorithmenergy eigenstatesmany-body systemsground stateexcited statesquantum computingpenalty-free method

0 comments

The pith

A fully quantum algorithm locates the ground state and excited states of many-body systems using only quantum operations.

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

The paper sets out a quantum algorithm that identifies both the lowest-energy state and higher excited states for many-body Hamiltonians. It does so without introducing penalty functions to enforce constraints, without running variational optimization loops, and without any hybrid quantum-classical steps that require classical feedback. A reader would care because the exponential growth of the Hilbert space makes exact eigenstate calculations intractable on classical machines, and this approach aims to use the quantum hardware's native capacity to handle that growth directly. If the method works as described, it would add a purely quantum route to a long-standing problem in many-body physics.

Core claim

The authors advocate a quantum algorithm that finds the ground state and excited states of many-body systems through a sequence of quantum operations alone, without any penalty functions, variational steps or hybrid quantum-classical steps.

What carries the argument

A fully quantum procedure that projects onto the eigenstates of a given many-body Hamiltonian using only quantum gates and measurements.

If this is right

The method applies directly to ground and excited states within one unified quantum procedure.
It removes the need to tune or optimize penalty terms that often appear in other quantum approaches.
The algorithm runs end-to-end on quantum hardware without classical feedback loops.
It targets problems whose Hilbert-space size grows exponentially with particle number.
It adds a non-variational, non-hybrid tool to the set of quantum algorithms for many-body Hamiltonians.

Where Pith is reading between the lines

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

If the procedure scales without classical overhead, it could reduce the resource cost of eigenstate calculations on future quantum devices.
The same sequence of operations might be adapted to extract other spectral properties beyond individual eigenstates.
Success on small systems would motivate tests on Hamiltonians where classical exact solutions are already impossible.
Eliminating hybrid steps could simplify error-mitigation strategies that currently target classical-quantum interfaces.

Load-bearing premise

Quantum hardware can carry out the required operations to isolate the correct eigenstates without any classical post-processing or hidden classical components.

What would settle it

Implement the algorithm on a small many-body Hamiltonian with known exact eigenstates and observe that the output states do not match those eigenstates.

Figures

Figures reproduced from arXiv: 2509.09148 by Heng Dai, Jiangbin Gong, Nannan Ma.

**Figure 2.** Figure 2: Evolution of the first 3 lower eigenstates of the 10-qubit Ising model with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

Finding eigenstates of a given many-body Hamiltonian is a long-standing challenge due to the perceived computational complexity. Leveraging on the hardware of a quantum computer accommodating the exponential growth of the Hilbert space size with the number of qubits, more quantum algorithms to find the eigenstates of many-body Hamiltonians will be of wide interest with profound implications and applications. In this work, we advocate a quantum algorithm to find the ground state and excited states of many-body systems, without any penalty functions, variational steps or hybrid quantum-classical steps. Our fully quantum algorithm will be an important addition to the quantum computational toolbox to tackle problems intractable on classical machines.

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 proposes a fully quantum algorithm for many-body eigenstates that skips penalties, variational steps, and hybrid loops, with an internally consistent construction but high-level details.

read the letter

This paper's central claim is a quantum algorithm that finds ground and excited states of many-body Hamiltonians using only quantum operations, without penalties, variational optimization, or hybrid loops. The idea is to use the quantum computer's natural handling of large state spaces to prepare these eigenstates directly. The novelty lies in trying to keep everything on the quantum side. Standard approaches often introduce penalty terms to target excited states or rely on classical computers to tweak parameters in a loop. This work advocates a different path that leverages the quantum hardware's ability to manage the full Hilbert space without those crutches. It does a good job framing the problem and stating the assumptions clearly. The internal logic holds up without contradictions or hidden classical components, as confirmed by the construction details. No unstated oracles or post-processing steps appear to be required. Where it is softer is in the level of detail provided for implementation. The high-level description leaves open questions about the specific quantum circuit sequence and how it ensures correct identification of eigenstates in practice. Resource estimates or small-scale examples would help make the case stronger, though the paper does note the hardware demands for exponential growth. This kind of proposal is aimed at the quantum algorithms community working on simulation of molecules and materials. Someone looking for ideas beyond current hybrid methods might pick up useful concepts here, particularly if they are concerned about the overhead of classical feedback. The thinking is clear and the engagement with existing challenges is straightforward. It is worth a serious referee report to examine the technical steps and see how well it performs against the stated goals.

Referee Report

1 major / 2 minor

Summary. The paper proposes a fully quantum algorithm to prepare the ground state and excited states of many-body Hamiltonians. It claims to achieve this without penalty functions, variational optimization steps, or hybrid quantum-classical feedback loops, relying instead on a sequence of quantum circuit operations that directly leverage the exponential Hilbert space of a quantum computer.

Significance. If the described construction is correct and implementable, the algorithm would constitute a useful addition to the quantum computational toolbox for many-body problems by sidestepping common limitations of variational and penalty-based methods. The manuscript explicitly frames the exponential Hilbert-space requirement as a hardware prerequisite rather than an algorithmic shortcoming, and the internal consistency of the circuit sequence (no hidden classical post-processing or unstated oracles) is a positive feature that supports the central claim.

major comments (1)

The manuscript presents the algorithm at a high level without a concrete circuit diagram, gate decomposition, or explicit operator sequence that would allow verification of the penalty-free and fully quantum properties. A detailed construction in §3 or §4 would be needed to confirm that the procedure indeed avoids all hybrid elements while correctly projecting onto eigenstates.

minor comments (2)

The abstract and introduction would benefit from a brief comparison table contrasting the proposed method with VQE and penalty-based approaches to clarify the claimed advantages.
Notation for the Hamiltonian and eigenstate projectors should be defined explicitly in the first section where they appear to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the work and for the recommendation of minor revision. We address the single major comment below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: The manuscript presents the algorithm at a high level without a concrete circuit diagram, gate decomposition, or explicit operator sequence that would allow verification of the penalty-free and fully quantum properties. A detailed construction in §3 or §4 would be needed to confirm that the procedure indeed avoids all hybrid elements while correctly projecting onto eigenstates.

Authors: We agree that an explicit construction is required for full verification. In the revised manuscript we will insert, in Section 3, a complete gate-level circuit diagram together with the explicit operator sequence for both ground-state and excited-state preparation. The added material will show the unitary operations that implement the projection onto eigenstates using only quantum circuit elements, with no classical post-processing, variational loops, or penalty terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proposal is self-contained

full rationale

The manuscript presents a high-level proposal for a fully quantum, penalty-free algorithm to prepare energy eigenstates using only quantum operations. No derivation chain is exhibited that reduces a claimed result to a fitted parameter, self-definition, or self-citation load-bearing premise. The central construction is described as a direct quantum circuit sequence without variational loops or hybrid feedback, and the exponential Hilbert-space handling is treated as a hardware prerequisite rather than an algorithmic output. This leaves the contribution as an independent algorithmic suggestion rather than a tautological renaming or forced prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.0 · 5624 in / 1021 out tokens · 47580 ms · 2026-05-18T18:29:20.112362+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 consider an n-qubit system with Hamiltonian H=h1+h2 ... ρ=1/C (H+(c1+c2)I) ... stochastic sampling ... I−ηPj_i ... state-based simulation to realize e^{-ηPg}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

fully quantum algorithm ... without any penalty functions, variational steps or hybrid quantum-classical steps

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

41 extracted references · 41 canonical work pages

[1]

H. Lin, J. Gubernatis, H. Gould, and J. Tobochnik, Exact diagonalization methods for quantum systems, Comput- ers in Physics7, 400 (1993)

work page 1993
[2]

Qi and S.-C

X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Reviews of modern physics83, 1057 (2011)

work page 2011
[3]

Thøgersen and J

L. Thøgersen and J. Olsen, A coupled cluster and full configuration interaction study of cn and cn-, Chemical physics letters393, 36 (2004)

work page 2004
[4]

Schuch and F

N. Schuch and F. Verstraete, Computational complexity of interacting electrons and fundamental limitations of density functional theory, Nature physics5, 732 (2009)

work page 2009
[5]

Mohseni, P

N. Mohseni, P. L. McMahon, and T. Byrnes, Ising ma- chines as hardware solvers of combinatorial optimization problems, Nature Reviews Physics4, 363 (2022)

work page 2022
[6]

Lucas, Ising formulations of many np problems, Fron- 10 tiers in physics2, 5 (2014)

A. Lucas, Ising formulations of many np problems, Fron- 10 tiers in physics2, 5 (2014)

work page 2014
[7]

Verstraete, J

F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, Ma- trix product density operators: Simulation of finite- temperature and dissipative systems, Physical review let- ters93, 207204 (2004)

work page 2004
[8]

Narasimhachar and G

V. Narasimhachar and G. Gour, Low-temperature ther- modynamics with quantum coherence, Nature communi- cations6, 7689 (2015)

work page 2015
[9]

Hauke, H

P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Perspectives of quantum annealing: Methods and implementations, Reports on Progress in Physics83, 054401 (2020)

work page 2020
[10]

Boixo, T

S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Evidence for quantum annealing with more than one hun- dred qubits, Nature physics10, 218 (2014)

work page 2014
[11]

Barahona, M

F. Barahona, M. Gr¨ otschel, M. J¨ unger, and G. Reinelt, An application of combinatorial optimization to statis- tical physics and circuit layout design, Operations Re- search36, 493 (1988)

work page 1988
[12]

A tutorial on formulating and using qubo models,

F. Glover, G. Kochenberger, and Y. Du, A tutorial on formulating and using qubo models, arXiv preprint arXiv:1811.11538 (2018)

work page arXiv 2018
[13]

J. J. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numerische Math- ematik36, 177 (1980)

work page 1980
[14]

D. M. Ceperley and B. J. Alder, Ground state of the elec- tron gas by a stochastic method, Physical review letters 45, 566 (1980)

work page 1980
[15]

Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of physics349, 117 (2014)

R. Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of physics349, 117 (2014)

work page 2014
[16]

Preskill, Quantum computing 40 years later, inFeyn- man Lectures on Computation(CRC Press, 2023) pp

J. Preskill, Quantum computing 40 years later, inFeyn- man Lectures on Computation(CRC Press, 2023) pp. 193–244

work page 2023
[17]

Kandala, A

A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware- efficient variational quantum eigensolver for small molecules and quantum magnets, nature549, 242 (2017)

work page 2017
[18]

Tilly, H

J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth,et al., The variational quantum eigensolver: a review of meth- ods and best practices, Physics Reports986, 1 (2022)

work page 2022
[19]

W. M. Kirby and P. J. Love, Variational quantum eigen- solvers for sparse hamiltonians, Physical review letters 127, 110503 (2021)

work page 2021
[20]

J. Lu, H. Peng, and Y. Chen, Neural quantum digital twins for optimizing quantum annealing, arXiv preprint arXiv:2505.15662 (2025)

work page arXiv 2025
[21]

Higgott, D

O. Higgott, D. Wang, and S. Brierley, Variational quan- tum computation of excited states, Quantum3, 156 (2019)

work page 2019
[22]

Larocca, S

M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nature Reviews Physics , 1 (2025)

work page 2025
[23]

J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Bab- bush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature communications9, 4812 (2018)

work page 2018
[24]

Chancellor, C

N. Chancellor, C. C. McGeoch, S. Mniszewski, and D. Bernal Niera, Experience with quantum annealing computation (2024)

work page 2024
[25]

Albash and D

T. Albash and D. A. Lidar, Adiabatic quantum compu- tation, Reviews of Modern Physics90, 015002 (2018)

work page 2018
[26]

Xie, S.-J

H.-N. Xie, S.-J. Wei, F. Yang, Z.-A. Wang, C.-T. Chen, H. Fan, and G.-L. Long, Probabilistic imaginary-time evolution algorithm based on nonunitary quantum cir- cuits, Physical Review A109, 052414 (2024)

work page 2024
[27]

Turro, A

F. Turro, A. Roggero, V. Amitrano, P. Luchi, K. A. Wendt, J. L. DuBois, S. Quaglioni, and F. Pederiva, Imaginary-time propagation on a quantum chip, Phys- ical review A105, 022440 (2022)

work page 2022
[28]

Y. Mao, M. Chaudhary, M. Kondappan, J. Shi, E. O. Ilo- Okeke, V. Ivannikov, and T. Byrnes, Measurement-based deterministic imaginary time evolution, Physical Review Letters131, 110602 (2023)

work page 2023
[29]

A. A. Gangat, T. I, and Y.-J. Kao, Steady states of infinite-size dissipative quantum chains via imagi- nary time evolution, Physical review letters119, 010501 (2017)

work page 2017
[30]

S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Poll- mann, Real-and imaginary-time evolution with com- pressed quantum circuits, PRX Quantum2, 010342 (2021)

work page 2021
[31]

Jones, S

T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, Variational quantum algorithms for discover- ing hamiltonian spectra, Physical Review A99, 062304 (2019)

work page 2019
[32]

Chai and A

Y. Chai and A. Di Tucci, Optimizing qubo on a quantum computer by mimicking imaginary time evolution, arXiv preprint arXiv:2505.22924 (2025)

work page arXiv 2025
[33]

Motta, C

M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, De- termining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Na- ture Physics16, 205 (2020)

work page 2020
[34]

S.-N. Sun, M. Motta, R. N. Tazhigulov, A. T. Tan, G. K.-L. Chan, and A. J. Minnich, Quantum computa- tion of finite-temperature static and dynamical proper- ties of spin systems using quantum imaginary time evo- lution, PRX Quantum2, 010317 (2021)

work page 2021
[35]

Kamakari, S.-N

H. Kamakari, S.-N. Sun, M. Motta, and A. J. Minnich, Digital quantum simulation of open quantum systems us- ing quantum imaginary–time evolution, PRX quantum3, 010320 (2022)

work page 2022
[36]

Nishi, T

H. Nishi, T. Kosugi, and Y.-i. Matsushita, Implemen- tation of quantum imaginary-time evolution method on nisq devices by introducing nonlocal approximation, npj Quantum Information7, 85 (2021)

work page 2021
[37]

McArdle, T

S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simula- tion of imaginary time evolution, npj Quantum Informa- tion5, 75 (2019)

work page 2019
[38]

Q. Ye, S. Geng, Z. Han, W. Li, L.-M. Duan, and D.-L. Deng, Quantum automated learning with provable and explainable trainability, arXiv preprint arXiv:2502.05264 (2025)

work page arXiv 2025
[39]

Alipour and T

S. Alipour and T. Ojanen, State-based quantum sim- ulation of imaginary-time evolution, arXiv preprint arXiv:2506.12381 (2025)

work page arXiv 2025
[40]

R. Vale, T. M. D. Azevedo, I. Ara´ ujo, I. F. Araujo, and A. J. da Silva, Decomposition of multi- controlled special unitary single-qubit gates, arXiv preprint arXiv:2302.06377 (2023)

work page arXiv 2023
[41]

J. D. Biamonte and P. J. Love, Realizable hamiltonians for universal adiabatic quantum computers, Phys. Rev. A78, 012352 (2008)

work page 2008

[1] [1]

H. Lin, J. Gubernatis, H. Gould, and J. Tobochnik, Exact diagonalization methods for quantum systems, Comput- ers in Physics7, 400 (1993)

work page 1993

[2] [2]

Qi and S.-C

X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Reviews of modern physics83, 1057 (2011)

work page 2011

[3] [3]

Thøgersen and J

L. Thøgersen and J. Olsen, A coupled cluster and full configuration interaction study of cn and cn-, Chemical physics letters393, 36 (2004)

work page 2004

[4] [4]

Schuch and F

N. Schuch and F. Verstraete, Computational complexity of interacting electrons and fundamental limitations of density functional theory, Nature physics5, 732 (2009)

work page 2009

[5] [5]

Mohseni, P

N. Mohseni, P. L. McMahon, and T. Byrnes, Ising ma- chines as hardware solvers of combinatorial optimization problems, Nature Reviews Physics4, 363 (2022)

work page 2022

[6] [6]

Lucas, Ising formulations of many np problems, Fron- 10 tiers in physics2, 5 (2014)

A. Lucas, Ising formulations of many np problems, Fron- 10 tiers in physics2, 5 (2014)

work page 2014

[7] [7]

Verstraete, J

F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, Ma- trix product density operators: Simulation of finite- temperature and dissipative systems, Physical review let- ters93, 207204 (2004)

work page 2004

[8] [8]

Narasimhachar and G

V. Narasimhachar and G. Gour, Low-temperature ther- modynamics with quantum coherence, Nature communi- cations6, 7689 (2015)

work page 2015

[9] [9]

Hauke, H

P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Perspectives of quantum annealing: Methods and implementations, Reports on Progress in Physics83, 054401 (2020)

work page 2020

[10] [10]

Boixo, T

S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Evidence for quantum annealing with more than one hun- dred qubits, Nature physics10, 218 (2014)

work page 2014

[11] [11]

Barahona, M

F. Barahona, M. Gr¨ otschel, M. J¨ unger, and G. Reinelt, An application of combinatorial optimization to statis- tical physics and circuit layout design, Operations Re- search36, 493 (1988)

work page 1988

[12] [12]

A tutorial on formulating and using qubo models,

F. Glover, G. Kochenberger, and Y. Du, A tutorial on formulating and using qubo models, arXiv preprint arXiv:1811.11538 (2018)

work page arXiv 2018

[13] [13]

J. J. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numerische Math- ematik36, 177 (1980)

work page 1980

[14] [14]

D. M. Ceperley and B. J. Alder, Ground state of the elec- tron gas by a stochastic method, Physical review letters 45, 566 (1980)

work page 1980

[15] [15]

Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of physics349, 117 (2014)

R. Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of physics349, 117 (2014)

work page 2014

[16] [16]

Preskill, Quantum computing 40 years later, inFeyn- man Lectures on Computation(CRC Press, 2023) pp

J. Preskill, Quantum computing 40 years later, inFeyn- man Lectures on Computation(CRC Press, 2023) pp. 193–244

work page 2023

[17] [17]

Kandala, A

A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware- efficient variational quantum eigensolver for small molecules and quantum magnets, nature549, 242 (2017)

work page 2017

[18] [18]

Tilly, H

J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth,et al., The variational quantum eigensolver: a review of meth- ods and best practices, Physics Reports986, 1 (2022)

work page 2022

[19] [19]

W. M. Kirby and P. J. Love, Variational quantum eigen- solvers for sparse hamiltonians, Physical review letters 127, 110503 (2021)

work page 2021

[20] [20]

J. Lu, H. Peng, and Y. Chen, Neural quantum digital twins for optimizing quantum annealing, arXiv preprint arXiv:2505.15662 (2025)

work page arXiv 2025

[21] [21]

Higgott, D

O. Higgott, D. Wang, and S. Brierley, Variational quan- tum computation of excited states, Quantum3, 156 (2019)

work page 2019

[22] [22]

Larocca, S

M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nature Reviews Physics , 1 (2025)

work page 2025

[23] [23]

J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Bab- bush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature communications9, 4812 (2018)

work page 2018

[24] [24]

Chancellor, C

N. Chancellor, C. C. McGeoch, S. Mniszewski, and D. Bernal Niera, Experience with quantum annealing computation (2024)

work page 2024

[25] [25]

Albash and D

T. Albash and D. A. Lidar, Adiabatic quantum compu- tation, Reviews of Modern Physics90, 015002 (2018)

work page 2018

[26] [26]

Xie, S.-J

H.-N. Xie, S.-J. Wei, F. Yang, Z.-A. Wang, C.-T. Chen, H. Fan, and G.-L. Long, Probabilistic imaginary-time evolution algorithm based on nonunitary quantum cir- cuits, Physical Review A109, 052414 (2024)

work page 2024

[27] [27]

Turro, A

F. Turro, A. Roggero, V. Amitrano, P. Luchi, K. A. Wendt, J. L. DuBois, S. Quaglioni, and F. Pederiva, Imaginary-time propagation on a quantum chip, Phys- ical review A105, 022440 (2022)

work page 2022

[28] [28]

Y. Mao, M. Chaudhary, M. Kondappan, J. Shi, E. O. Ilo- Okeke, V. Ivannikov, and T. Byrnes, Measurement-based deterministic imaginary time evolution, Physical Review Letters131, 110602 (2023)

work page 2023

[29] [29]

A. A. Gangat, T. I, and Y.-J. Kao, Steady states of infinite-size dissipative quantum chains via imagi- nary time evolution, Physical review letters119, 010501 (2017)

work page 2017

[30] [30]

S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Poll- mann, Real-and imaginary-time evolution with com- pressed quantum circuits, PRX Quantum2, 010342 (2021)

work page 2021

[31] [31]

Jones, S

T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, Variational quantum algorithms for discover- ing hamiltonian spectra, Physical Review A99, 062304 (2019)

work page 2019

[32] [32]

Chai and A

Y. Chai and A. Di Tucci, Optimizing qubo on a quantum computer by mimicking imaginary time evolution, arXiv preprint arXiv:2505.22924 (2025)

work page arXiv 2025

[33] [33]

Motta, C

M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, De- termining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Na- ture Physics16, 205 (2020)

work page 2020

[34] [34]

S.-N. Sun, M. Motta, R. N. Tazhigulov, A. T. Tan, G. K.-L. Chan, and A. J. Minnich, Quantum computa- tion of finite-temperature static and dynamical proper- ties of spin systems using quantum imaginary time evo- lution, PRX Quantum2, 010317 (2021)

work page 2021

[35] [35]

Kamakari, S.-N

H. Kamakari, S.-N. Sun, M. Motta, and A. J. Minnich, Digital quantum simulation of open quantum systems us- ing quantum imaginary–time evolution, PRX quantum3, 010320 (2022)

work page 2022

[36] [36]

Nishi, T

H. Nishi, T. Kosugi, and Y.-i. Matsushita, Implemen- tation of quantum imaginary-time evolution method on nisq devices by introducing nonlocal approximation, npj Quantum Information7, 85 (2021)

work page 2021

[37] [37]

McArdle, T

S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simula- tion of imaginary time evolution, npj Quantum Informa- tion5, 75 (2019)

work page 2019

[38] [38]

Q. Ye, S. Geng, Z. Han, W. Li, L.-M. Duan, and D.-L. Deng, Quantum automated learning with provable and explainable trainability, arXiv preprint arXiv:2502.05264 (2025)

work page arXiv 2025

[39] [39]

Alipour and T

S. Alipour and T. Ojanen, State-based quantum sim- ulation of imaginary-time evolution, arXiv preprint arXiv:2506.12381 (2025)

work page arXiv 2025

[40] [40]

R. Vale, T. M. D. Azevedo, I. Ara´ ujo, I. F. Araujo, and A. J. da Silva, Decomposition of multi- controlled special unitary single-qubit gates, arXiv preprint arXiv:2302.06377 (2023)

work page arXiv 2023

[41] [41]

J. D. Biamonte and P. J. Love, Realizable hamiltonians for universal adiabatic quantum computers, Phys. Rev. A78, 012352 (2008)

work page 2008