Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.35660

Quantum computation of energy level degeneracy and thermodynamical properties of spin systems

V. M. Tkachuk, M. I. Samar

quant-phSubmitted Oct 26, 2026• Announced Oct 27, 2026• 0 citations• Revised Oct 27, 2026 (v2)
v1 · Oct 26, 2026v2 · Oct 27, 2026
arXivHTMLPDF

Abstract

We propose a hybrid quantum-classical method for reconstructing energy-level degeneracies and thermodynamic properties of spin systems. A single ancilla qubit is used to measure the partition function at imaginary inverse temperatures. For spectra lying on a known equally spaced energy grid, the inverse discrete Fourier transform allows exact reconstruction of degeneracies in the absence of noise, without spectral leakage. The quantum circuits are optimized using commuting interactions and Eulerian ordering. We demonstrate the method for frustrated Ising antiferromagnets on periodic Kagome graphs with up to 36 spins, using the ibm_kingston quantum processor. Despite substantial circuit depth and hardware noise, the reconstructed spectra retain their dominant features, and the calculated heat capacities reproduce the main temperature dependence of the exact results. Keywords: partition function, imaginary temperature, Ising model, heat capacity, ancilla qubit, quantum computing.

AI Summary

The paper studies reconstruction of spin-system energy degeneracies and equilibrium thermodynamics from ancilla measurements of $Z(it)/2^N$. For an Ising Hamiltonian diagonal in the computational basis, a product-state register and one ancilla provide the real and imaginary parts of this normalized trace. If the spectrum lies on a known equally spaced grid containing $Q$ candidate energies, measurements at grid-matched times recover the spectral weights by an inverse DFT, exactly without leakage in the noiseless limit. Commuting bond interactions are ordered along Euler circuits to cancel parity gates, reducing the logical CNOT count from $4L$ to $2L+2$ for $L$ bonds. Experiments on `ibm_kingston` cover a five-spin benchmark and periodic kagome Ising antiferromagnets with $N=6,12,24,36$, using $Q=N+1$ sampling times for kagome. Comparisons with exact spectra and shot-based noiseless simulations show recovery of dominant spectral contributions and the broad heat-capacity dependence, with larger hardware deviations at larger sizes. The 36-spin circuits contain 710 native CZ gates after routing. For $M$ shots per ancilla component per time, the unprocessed weight estimator satisfies $\sigma_{w_k}\leq1/\sqrt{QM}$. At 36 spins and $M=6400$, this bound is approximately $2.1\times10^{-3}$, exceeding the ground-state weight of about $10^{-3}$. That reconstruction additionally imposes the known ground-state energy. Thresholding, normalization, symmetry constraints and readout mitigation stabilize reconstruction, but their nonlinear uncertainty propagation is not fully analyzed. Small spectral weights can require large measurement budgets; low-temperature thermodynamics remains sensitive to low-energy errors, and additional shots do not remove systematic hardware noise.

Little sign of AI-generated textIncrementalMissing comparisonStrong assumptions

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Comments
Author contributions updated

Related papers

Preparing thermal states of frustrated quantum spin systems using 139 qubits

Roland C. Farrell, Yongtao Zhan, Lucas Katschke +3 more

Finite-temperature properties of strongly correlated quantum matter are central to condensed matter, chemistry, and high-energy physics, yet are often inaccessible to classical methods such as quantum Monte Carlo (QMC). Here, we investigate dissipative thermal state preparation of frustrated spin systems using digital quantum computers. We focus on two paradigmatic models on the kagome lattice: the antiferromagnetic Heisenberg model (AFHM), whose finite-temperature properties are inaccessible to QMC due to a severe sign problem, and the antiferromagnetic Ising model (AFIM), which serves as a sign-problem-free benchmark. Using IBM quantum processors, we prepare approximate thermal states of the AFIM on kagome lattices with up to 79 spins coupled to 60 environment qubits. We observe the emergence of a robust steady state with an adjustable effective temperature that persists in circuits with over 1000 layers of two-qubit gates. We further study the scalability of the dissipative protocol through classical statevector simulations of the AFIM and AFHM. On lattices with up to 27 sites, we find that the circuit depth to reach thermal equilibrium is independent of system size and grows at most linearly with inverse temperature. These results establish engineered dissipation as a promising approach to finite-temperature quantum simulation of frustrated matter, and point toward regimes where quantum devices may outperform classical methods.

quant-pharXiv:2605.26245•May 27, 2026•0 citations

Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor

Aaron Szasz, Ed Younis, Wibe Albert de Jong

Models of interacting many-body quantum systems that may realize new exotic phases of matter, notably quantum spin liquids, are challenging to study using even state-of-the-art classical methods such as tensor network simulations. Quantum computing provides a promising route for overcoming these difficulties to find ground states, dynamics, and more. In this paper, we argue that recently developed hybrid quantum-classical algorithms based on real-time evolution are promising methods for solving a particularly important model in the search for spin liquids, the antiferromagnetic Heisenberg model on the two-dimensional kagome lattice. We show how to construct efficient quantum circuits to implement time evolution for the model and to evaluate key observables on the quantum computer, and we argue that the method has favorable scaling with increasing system size. We then restrict to a 12-spin star plaquette from the kagome lattice and a related 8-spin system, and we give an empirical demonstration on these small systems that the hybrid algorithms can efficiently find the ground state energy and the magnetization curve. For these demonstrations, we use four levels of approximation: exact state vectors, exact state vectors with statistical noise from sampling, noisy classical emulators, and (for the 8-spin system only) real quantum hardware, specifically the Quantinuum H1-1 processor; for the noisy simulations and hardware demonstration, we also employ error mitigation strategies based on the symmetries of the Hamiltonian. Our results strongly suggest that these hybrid algorithms present a promising direction for studying quantum spin liquids and more generally for resolving important unsolved problems in condensed matter theory and beyond.

arXiv:2401.03015•Quantum•Apr 9, 2025•2 citations

Low depth quantum circuits for Ising models

S. Iblisdir, Mauro Cirio, Octavi Boada +1 more

A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield esti-mates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided and shown to be compatible with previous results. Interestingly, the kind of state prepa-rations and measurements involved in this application can in principle be made “instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accu-rately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allow to reconstruct scat-tering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that wavefunction overlaps for ground states of quantum Hamiltonians, which can serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetisations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently. 1

•Jan 1, 2014•12 citations

Detecting the purely imaginary Fisher zeros of an Ising spin system on a quantum computer

Andrij Romanovych Kuzmak, V. M. Tkachuk

We propose a protocol for studying the purely imaginary Fisher zeros of the Ising model on a quantum computer. Our protocol is based on the direct relation between the partition function for purely imaginary temperature and the evolution operator of the Ising model. In this case, the inverse temperature is equal to the time of evolution. This protocol allows one to measure the zeros only those localized on the imaginary axes. We test this protocol on the ibm-lagos quantum computer in the cases of a 3-spin chain and a triangle cluster in a purely imaginary magnetic field, as well as a 7-spin cluster in which the interaction between spins reproduces the architecture of the quantum computer.

arXiv:2311.14453•Scientific Reports•Sep 25, 2025•0 citations

Quantum Annealing Algorithms for Estimating Ising Partition Functions

Haowei Li, Zhiyuan Yao, Xingze Qiu

Estimating partition functions of Ising spin glasses is a cornerstone of statistical physics and computational science, yet it remains classically challenging due to its #P-hard complexity. While Jarzynski's equality offers a theoretical pathway, its practical application is crippled at low temperatures by rare, divergent statistical fluctuations. Here, we introduce a quantum protocol that overcomes this fundamental limitation by synergizing reverse quantum annealing with optimized nonequilibrium initial distributions. Our method dramatically suppresses the estimator variance, achieving saturation in the low-temperature regime where existing methods fail. Numerical benchmarks on the Sherrington-Kirkpatrick spin glass and the 3-SAT problem demonstrate that our protocol reduces computational scaling exponents by over an order of magnitude (e.g., from ∼8.5 to ∼0.5), despite retaining exponential system-size dependence. Crucially, our protocol circumvents stringent adiabatic constraints, making it feasible for near-term quantum devices like superconducting qubits, trapped ions, and Rydberg atom arrays. This Letter provides a methodological framework for quantum-enhanced estimation in spin glass thermodynamics and beyond by harnessing nonadiabatic quantum dynamics to address a classically difficult problem.

arXiv:2504.21666•Physical Review Letters•Feb 11, 2026•1 citation

Partition function and energy spectrum for a three-ring Ising chain with even number of spins- 1 2

Hossein Fakhri, S. Bahareh Seyedein Ardebili

Abstract An even-site spins- linear chain of cyclic in the nearest-neighbour and next-nearest-neighbour interactions has necessarily a three-ring structure. In this study, the interactions are considered to be short-range through the coupling constants J and K , respectively. For an arbitrary even number of spins- , the exact classical solution of the partition function and the pair correlations between the nearest- and next-nearest-neighbours are calculated in terms of the constants J and K , without using the energy levels and their degeneracy factors. Then, this partition function is used to verify the validity of an ansatz on the allowed energies of the chain and their corresponding degeneracy factors. The influence of next-nearest-neighbour interactions K on the specific heat capacity per spin at the low-temperatures is analysed for the ferromagnetic and antiferromagnetic chains. We especially show that specific heat capacity exhibits the spin frustration when the ferromagnetic and antiferromagnetic orders are demanded via the coupling constants J and K , respectively. Also, the maximal frustration is obtained analytically by two distinct ground levels for even number of spins- .

Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2019•1 citation
We propose a hybrid quantum-classical method for reconstructing energy-level degeneracies and thermodynamic properties of spin systems. A single ancilla qubit is used to measure the partition function at imaginary inverse temperatures. For spectra lying on a known equally spaced energy grid, the inverse discrete Fourier transform allows exact reconstruction of degeneracies in the absence of noise, without spectral leakage. The quantum circuits are optimized using commuting interactions and Eulerian ordering. We demonstrate the method for frustrated Ising antiferromagnets on periodic Kagome graphs with up to 36 spins, using the ibm_kingston quantum processor. Despite substantial circuit depth and hardware noise, the reconstructed spectra retain their dominant features, and the calculated heat capacities reproduce the main temperature dependence of the exact results.
Keywords: partition function, imaginary temperature, Ising model, heat capacity, ancilla qubit, quantum computing.
The paper studies reconstruction of spin-system energy degeneracies and equilibrium thermodynamics from ancilla measurements of Z(it)/2NZ(it)/2^NZ(it)/2N. For an Ising Hamiltonian diagonal in the computational basis, a product-state register and one ancilla provide the real and imaginary parts of this normalized trace. If the spectrum lies on a known equally spaced grid containing QQQ candidate energies, measurements at grid-matched times recover the spectral weights by an inverse DFT, exactly without leakage in the noiseless limit.
Commuting bond interactions are ordered along Euler circuits to cancel parity gates, reducing the logical CNOT count from 4L4L4L to 2L+22L+22L+2 for LLL bonds. Experiments on ibm_kingston cover a five-spin benchmark and periodic kagome Ising antiferromagnets with N=6,12,24,36N=6,12,24,36N=6,12,24,36, using Q=N+1Q=N+1Q=N+1 sampling times for kagome. Comparisons with exact spectra and shot-based noiseless simulations show recovery of dominant spectral contributions and the broad heat-capacity dependence, with larger hardware deviations at larger sizes. The 36-spin circuits contain 710 native CZ gates after routing.
For MMM shots per ancilla component per time, the unprocessed weight estimator satisfies σwk≤1/QM\sigma_{w_k}\leq1/\sqrt{QM}σwk​​≤1/QM​. At 36 spins and M=6400M=6400M=6400, this bound is approximately 2.1×10−32.1\times10^{-3}2.1×10−3, exceeding the ground-state weight of about 10−310^{-3}10−3. That reconstruction additionally imposes the known ground-state energy. Thresholding, normalization, symmetry constraints and readout mitigation stabilize reconstruction, but their nonlinear uncertainty propagation is not fully analyzed. Small spectral weights can require large measurement budgets; low-temperature thermodynamics remains sensitive to low-energy errors, and additional shots do not remove systematic hardware noise.
Finite-temperature properties of strongly correlated quantum matter are central to condensed matter, chemistry, and high-energy physics, yet are often inaccessible to classical methods such as quantum Monte Carlo (QMC). Here, we investigate dissipative thermal state preparation of frustrated spin systems using digital quantum computers. We focus on two paradigmatic models on the kagome lattice: the antiferromagnetic Heisenberg model (AFHM), whose finite-temperature properties are inaccessible to QMC due to a severe sign problem, and the antiferromagnetic Ising model (AFIM), which serves as a sign-problem-free benchmark. Using IBM quantum processors, we prepare approximate thermal states of the AFIM on kagome lattices with up to 79 spins coupled to 60 environment qubits. We observe the emergence of a robust steady state with an adjustable effective temperature that persists in circuits with over 1000 layers of two-qubit gates. We further study the scalability of the dissipative protocol through classical statevector simulations of the AFIM and AFHM. On lattices with up to 27 sites, we find that the circuit depth to reach thermal equilibrium is independent of system size and grows at most linearly with inverse temperature. These results establish engineered dissipation as a promising approach to finite-temperature quantum simulation of frustrated matter, and point toward regimes where quantum devices may outperform classical methods.
Models of interacting many-body quantum systems that may realize new exotic phases of matter, notably quantum spin liquids, are challenging to study using even state-of-the-art classical methods such as tensor network simulations. Quantum computing provides a promising route for overcoming these difficulties to find ground states, dynamics, and more. In this paper, we argue that recently developed hybrid quantum-classical algorithms based on real-time evolution are promising methods for solving a particularly important model in the search for spin liquids, the antiferromagnetic Heisenberg model on the two-dimensional kagome lattice. We show how to construct efficient quantum circuits to implement time evolution for the model and to evaluate key observables on the quantum computer, and we argue that the method has favorable scaling with increasing system size. We then restrict to a 12-spin star plaquette from the kagome lattice and a related 8-spin system, and we give an empirical demonstration on these small systems that the hybrid algorithms can efficiently find the ground state energy and the magnetization curve. For these demonstrations, we use four levels of approximation: exact state vectors, exact state vectors with statistical noise from sampling, noisy classical emulators, and (for the 8-spin system only) real quantum hardware, specifically the Quantinuum H1-1 processor; for the noisy simulations and hardware demonstration, we also employ error mitigation strategies based on the symmetries of the Hamiltonian. Our results strongly suggest that these hybrid algorithms present a promising direction for studying quantum spin liquids and more generally for resolving important unsolved problems in condensed matter theory and beyond.
A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield esti-mates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided and shown to be compatible with previous results. Interestingly, the kind of state prepa-rations and measurements involved in this application can in principle be made “instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accu-rately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allow to reconstruct scat-tering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that wavefunction overlaps for ground states of quantum Hamiltonians, which can serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetisations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently. 1
We propose a protocol for studying the purely imaginary Fisher zeros of the Ising model on a quantum computer. Our protocol is based on the direct relation between the partition function for purely imaginary temperature and the evolution operator of the Ising model. In this case, the inverse temperature is equal to the time of evolution. This protocol allows one to measure the zeros only those localized on the imaginary axes. We test this protocol on the ibm-lagos quantum computer in the cases of a 3-spin chain and a triangle cluster in a purely imaginary magnetic field, as well as a 7-spin cluster in which the interaction between spins reproduces the architecture of the quantum computer.
Estimating partition functions of Ising spin glasses is a cornerstone of statistical physics and computational science, yet it remains classically challenging due to its #P-hard complexity. While Jarzynski's equality offers a theoretical pathway, its practical application is crippled at low temperatures by rare, divergent statistical fluctuations. Here, we introduce a quantum protocol that overcomes this fundamental limitation by synergizing reverse quantum annealing with optimized nonequilibrium initial distributions. Our method dramatically suppresses the estimator variance, achieving saturation in the low-temperature regime where existing methods fail. Numerical benchmarks on the Sherrington-Kirkpatrick spin glass and the 3-SAT problem demonstrate that our protocol reduces computational scaling exponents by over an order of magnitude (e.g., from ∼8.5 to ∼0.5), despite retaining exponential system-size dependence. Crucially, our protocol circumvents stringent adiabatic constraints, making it feasible for near-term quantum devices like superconducting qubits, trapped ions, and Rydberg atom arrays. This Letter provides a methodological framework for quantum-enhanced estimation in spin glass thermodynamics and beyond by harnessing nonadiabatic quantum dynamics to address a classically difficult problem.
Abstract An even-site spins- linear chain of cyclic in the nearest-neighbour and next-nearest-neighbour interactions has necessarily a three-ring structure. In this study, the interactions are considered to be short-range through the coupling constants J and K , respectively. For an arbitrary even number of spins- , the exact classical solution of the partition function and the pair correlations between the nearest- and next-nearest-neighbours are calculated in terms of the constants J and K , without using the energy levels and their degeneracy factors. Then, this partition function is used to verify the validity of an ansatz on the allowed energies of the chain and their corresponding degeneracy factors. The influence of next-nearest-neighbour interactions K on the specific heat capacity per spin at the low-temperatures is analysed for the ferromagnetic and antiferromagnetic chains. We especially show that specific heat capacity exhibits the spin frustration when the ferromagnetic and antiferromagnetic orders are demanded via the coupling constants J and K , respectively. Also, the maximal frustration is obtained analytically by two distinct ground levels for even number of spins- .