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:2512.19685

Partition Function Estimation Using Analog Quantum Processors

Thinh Le, Elijah Pelofske

quant-phcond-mat.dis-nncond-mat.stat-mechphysics.comp-phSubmitted Dec 22, 2025• Announced Dec 23, 2025• 0 citations
arXivHTMLPDF

Abstract

We evaluate using programmable superconducting flux qubit D-Wave quantum annealers to approximate the partition function of Ising models. We propose the use of two distinct quantum annealer sampling methods: chains of Monte Carlo-like reverse quantum anneals, and standard linear-ramp quantum annealing. The control parameters used to attenuate the quality of the simulations are the effective analog energy scale of the J coupling, the total annealing time, and for the case of reverse annealing the anneal-pause. The core estimation technique is to sample across the energy spectrum of the classical Hamiltonian of interest, and therefore obtain a density of states estimate for each energy level, which in turn can be used to compute an estimate of the partition function with some sampling error. This estimation technique is powerful because once the distribution is sampled it allows thermodynamic quantity computation at arbitrary temperatures. On a $25$ spin $\pm J$ hardware graph native Ising model we find parameter regimes of the D-Wave processors that provide comparable result quality to two standard classical Monte Carlo methods, Multiple Histogram Reweighting and Wang-Landau. Remarkably, we find that fast quench-like anneals can quickly generate ensemble distributions that are very good estimates of the true partition function of the classical Ising model; on a Pegasus graph-structured QPU we report a logarithmic relative error of $7.6 \times 10^{-6}$, from $171,000$ samples generated using $0.2$ seconds of QPU time with an anneal time of $8$ nanoseconds per sample which is interestingly within the closed system dynamics timescale of the superconducting qubits.

Related papers

Simulating Heavy-Hex Transverse Field Ising Model Magnetization Dynamics Using Programmable Quantum Annealers

Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz

Recently, a Hamiltonian dynamics simulation was performed on a kicked ferromagnetic 2D transverse field Ising model with a connectivity graph native to the 127 qubit heavy-hex IBM Quantum architecture using ZNE quantum error mitigation. We demonstrate that one of the observables in this Trotterized Hamiltonian dynamics simulation, namely magnetization, can be efficiently simulated on current superconducting qubit-based programmable quantum annealing computers. We show this using two distinct methods: reverse quantum annealing and h-gain state encoding. This simulation is possible because the 127 qubit heavy-hex connectivity graph can be natively embedded onto the D-Wave Pegasus quantum annealer hardware graph and because there exists a direct equivalence between the energy scales of the two types of quantum computers. We derive equivalent anneal pauses in order to simulate the Trotterized quantum circuit dynamics for varying Rx rotations $θ_h \in (0, \fracπ{2}]$, using quantum annealing processors. Multiple disjoint instances of the Ising model of interest can be embedded onto the D-Wave Pegasus hardware graph, allowing for parallel quantum annealing. We report equivalent magnetization dynamics using quantum annealing for time steps of 20, 50 up to 10,000, which we find are consistent with exact classical 27 qubit heavy-hex Trotterized circuit magnetization dynamics, and we observe reasonable, albeit noisy, agreement with the existing simulations for single site magnetization at 20 Trotter steps. The quantum annealers are able to simulate equivalent magnetization dynamics for thousands of time steps, significantly out of the computational reach of the digital quantum computers on which the original Hamiltonian dynamics simulations were performed.

arXiv:2311.01657•Nov 3, 2023•0 citations

Boltzmann Sampling of Frustrated J1 - J2 Ising Models with Programmable Quantum Annealers

Elijah Pelofske

One of the surprising, and potentially very useful, capabilities of analog quantum computers, such as D-Wave quantum annealers, is sampling from the Boltzmann, or Gibbs, distribution defined by a classical Hamiltonian. In this study, we thoroughly examine the ability of D-Wave quantum annealers to sample from the Boltzmann distribution defined by a canonical type of competing magnetic frustration $J_1$-$J_2$ model; the 1-dimensional ANNNI (axial next-nearest-neighbor Ising) model. Boltzmann sampling error rate is quantified for standard linear-ramp anneals ranging from $5$ nanosecond annealing times up to $2000$ microseconds on two different D-Wave quantum annealing processors. Interestingly, we find some analog hardware parameters which result in a very high accuracy (down to a TVD of $0.0003$) and low temperature sampling (down to $β=32.2$) in a frustrated region of the ANNNI model magnetic phase diagram. This bolsters the viability of current analog quantum computers for thermodynamic sampling applications of highly frustrated magnetic spin systems.

quant-pharXiv:2511.03796•Physical Review Research•Nov 7, 2025•0 citations

Neural and Tensor Networks in the Study of Quantum Annealing Processors

Tomasz Śmierzchalski

Quantum annealing targets low-energy solutions of Ising/QUBO problems, but reliable assessment requires more than best-energy comparisons. This dissertation develops a benchmarking framework for D-Wave quantum annealers that combines strong classical baselines, sampling and diversity metrics, and thermodynamic cost. Its first contribution, SpinGlassPEPS$.$jl, is a topology-aware tensor-network heuristic for optimization and sampling on Pegasus/Zephyr-like graphs. It maps Ising instances to local Potts clusters, represents the partition function with PEPS, and performs branch-and-bound search in probability space. Benchmarks show that it is a physically interpretable reference solver, though approximate contractions limit its competitiveness on the largest instances. The second contribution treats quantum annealers as effective thermal machines, relating success probability and solution quality to dissipation, entropy production, and effective temperature. Carefully placed pauses can improve performance while reducing thermodynamic cost, although longitudinal fields may become harmful in paused schedules. The thesis also introduces reinforcement-learning post-processing to improve returned samples and exact small-system simulations to probe annealing dynamics. Overall, it argues for quantum-annealing benchmarks that jointly measure algorithmic performance and physical expenditure.

quant-pharXiv:2604.26534•Apr 30, 2026•0 citations

Short-depth QAOA circuits and quantum annealing on higher-order ising models

Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz

Abstract The Quantum Alternating Operator Ansatz (QAOA) and Quantum Annealing (QA) are quantum algorithms that are both based on the adiabatic theorem and both have the goal of sampling the optimal solution(s) of combinatorial optimization problems. Quantum annealing has been physically instantiated on D-Wave devices using superconducting flux qubits, and QAOA can be programmed on digital gate-model quantum computers such as the programmable superconducting transmon qubit devices manufactured by IBM Quantum, for instance ibm_washington. QAOA and QA address the same types of problems, but it is unclear how they will scale to large problem sizes and to larger and higher-fidelity quantum computers. In this article, we present a direct comparison between QAOA, one and two rounds, run on all 127 qubits of ibm_washington and QA run on D-Wave Advantage_system4.1 and Advantage_system6.1. The problems which allow for this comparison are random Ising model problems whose connectivity matches the heavy hexagonal lattice topology of ibm_washington and the Pegasus graph connectivity of the two D-Wave devices. We create two classes of problem instances for this comparison: one with higher order terms (ZZZ variable interactions), linear terms, and quadratic terms, and a separate problem type with only linear and quadratic terms. The QAOA circuits are novel and extremely short depth, with a CNOT depth of 6 per round, which allows whole chip usage of ibm_washington's heavy hexagonal lattice and can be applied to future heavy-hex chips. We also test the effectiveness of the error suppression technique digital dynamical decoupling on the QAOA circuits, allowing one of the largest experimental evaluations of dynamical decoupling to date. The QAOA circuits compiled to ibm_washington are composed of approximately 3,000 gate instructions making these some the largest quantum circuits ever executed on a digital quantum processor. QAOA and QA are compared against the classical heuristic algorithm of simulated annealing and all problem instances are exactly solved using CPLEX in order to evaluate ground state success sampling probability. We find that (i) QA outperforms QAOA on all problem instances, (ii) QAOA samples the problems better than random sampling, and (iii) QAOA parameters exhibit clear parameter concentration across the ensemble of Ising models.

npj Quantum Information•Mar 12, 2024•47 citations

Numerical simulation of D-Wave's quantum advantage experiment with time-dependent variational Monte Carlo

Roeland Wiersema

Programmable quantum annealers can realize real-time dynamics of frustrated transverse-field Ising models on large, nontrivial graphs. Recent work by King et al. argued that the classical simulation of such experiments would require exponential computational resources for classical methods such as tensor networks and neural quantum states. Here, we numerically simulate the D-Wave spin-glass annealing protocol with time-dependent variational Monte Carlo (t-VMC) using a correlator state tailored to spin-glass dynamics. For the two-dimensional cylinder, three-dimensional dimer, diamond, and biclique instances considered, at annealing times of 7 and 20 ns, we show that systematically increasing the variational ansatz size enables t-VMC to approximate the final two-spin correlation errors of the quantum processing unit (QPU). We also perform an accurate large-scale simulation of a challenging biclique instance for which no other variational method is known to work, and we find close agreement with the quantum annealer. Through ablation studies, we identify poor Markov-chain mixing, high-variance local-energy estimators, and stochastic Runge-Kutta error estimates as the principal numerical failure modes. We address these numerical issues by using parallel tempering, blurred sampling and an importance-weighted differential equation solver, thereby clarifying the numerical requirements for stable, large-scale t-VMC simulations. Our results extend the frontier of classical simulation while providing a realistic assessment of the computational costs of simulating quantum dynamics at this scale.

quant-pharXiv:2609.01719•Sep 3, 2026•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
We evaluate using programmable superconducting flux qubit D-Wave quantum annealers to approximate the partition function of Ising models. We propose the use of two distinct quantum annealer sampling methods: chains of Monte Carlo-like reverse quantum anneals, and standard linear-ramp quantum annealing. The control parameters used to attenuate the quality of the simulations are the effective analog energy scale of the J coupling, the total annealing time, and for the case of reverse annealing the anneal-pause. The core estimation technique is to sample across the energy spectrum of the classical Hamiltonian of interest, and therefore obtain a density of states estimate for each energy level, which in turn can be used to compute an estimate of the partition function with some sampling error. This estimation technique is powerful because once the distribution is sampled it allows thermodynamic quantity computation at arbitrary temperatures. On a 252525 spin ±J\pm J±J hardware graph native Ising model we find parameter regimes of the D-Wave processors that provide comparable result quality to two standard classical Monte Carlo methods, Multiple Histogram Reweighting and Wang-Landau. Remarkably, we find that fast quench-like anneals can quickly generate ensemble distributions that are very good estimates of the true partition function of the classical Ising model; on a Pegasus graph-structured QPU we report a logarithmic relative error of 7.6×10−67.6 \times 10^{-6}7.6×10−6, from 171,000171,000171,000 samples generated using 0.20.20.2 seconds of QPU time with an anneal time of 888 nanoseconds per sample which is interestingly within the closed system dynamics timescale of the superconducting qubits.
Recently, a Hamiltonian dynamics simulation was performed on a kicked ferromagnetic 2D transverse field Ising model with a connectivity graph native to the 127 qubit heavy-hex IBM Quantum architecture using ZNE quantum error mitigation. We demonstrate that one of the observables in this Trotterized Hamiltonian dynamics simulation, namely magnetization, can be efficiently simulated on current superconducting qubit-based programmable quantum annealing computers. We show this using two distinct methods: reverse quantum annealing and h-gain state encoding. This simulation is possible because the 127 qubit heavy-hex connectivity graph can be natively embedded onto the D-Wave Pegasus quantum annealer hardware graph and because there exists a direct equivalence between the energy scales of the two types of quantum computers. We derive equivalent anneal pauses in order to simulate the Trotterized quantum circuit dynamics for varying Rx rotations θh∈(0,π2]θ_h \in (0, \fracπ{2}]θh​∈(0,2π​], using quantum annealing processors. Multiple disjoint instances of the Ising model of interest can be embedded onto the D-Wave Pegasus hardware graph, allowing for parallel quantum annealing. We report equivalent magnetization dynamics using quantum annealing for time steps of 20, 50 up to 10,000, which we find are consistent with exact classical 27 qubit heavy-hex Trotterized circuit magnetization dynamics, and we observe reasonable, albeit noisy, agreement with the existing simulations for single site magnetization at 20 Trotter steps. The quantum annealers are able to simulate equivalent magnetization dynamics for thousands of time steps, significantly out of the computational reach of the digital quantum computers on which the original Hamiltonian dynamics simulations were performed.
One of the surprising, and potentially very useful, capabilities of analog quantum computers, such as D-Wave quantum annealers, is sampling from the Boltzmann, or Gibbs, distribution defined by a classical Hamiltonian. In this study, we thoroughly examine the ability of D-Wave quantum annealers to sample from the Boltzmann distribution defined by a canonical type of competing magnetic frustration J1J_1J1​-J2J_2J2​ model; the 1-dimensional ANNNI (axial next-nearest-neighbor Ising) model. Boltzmann sampling error rate is quantified for standard linear-ramp anneals ranging from 555 nanosecond annealing times up to 200020002000 microseconds on two different D-Wave quantum annealing processors. Interestingly, we find some analog hardware parameters which result in a very high accuracy (down to a TVD of 0.00030.00030.0003) and low temperature sampling (down to β=32.2β=32.2β=32.2) in a frustrated region of the ANNNI model magnetic phase diagram. This bolsters the viability of current analog quantum computers for thermodynamic sampling applications of highly frustrated magnetic spin systems.
Quantum annealing targets low-energy solutions of Ising/QUBO problems, but reliable assessment requires more than best-energy comparisons. This dissertation develops a benchmarking framework for D-Wave quantum annealers that combines strong classical baselines, sampling and diversity metrics, and thermodynamic cost. Its first contribution, SpinGlassPEPS...jl, is a topology-aware tensor-network heuristic for optimization and sampling on Pegasus/Zephyr-like graphs. It maps Ising instances to local Potts clusters, represents the partition function with PEPS, and performs branch-and-bound search in probability space. Benchmarks show that it is a physically interpretable reference solver, though approximate contractions limit its competitiveness on the largest instances. The second contribution treats quantum annealers as effective thermal machines, relating success probability and solution quality to dissipation, entropy production, and effective temperature. Carefully placed pauses can improve performance while reducing thermodynamic cost, although longitudinal fields may become harmful in paused schedules. The thesis also introduces reinforcement-learning post-processing to improve returned samples and exact small-system simulations to probe annealing dynamics. Overall, it argues for quantum-annealing benchmarks that jointly measure algorithmic performance and physical expenditure.
Abstract The Quantum Alternating Operator Ansatz (QAOA) and Quantum Annealing (QA) are quantum algorithms that are both based on the adiabatic theorem and both have the goal of sampling the optimal solution(s) of combinatorial optimization problems. Quantum annealing has been physically instantiated on D-Wave devices using superconducting flux qubits, and QAOA can be programmed on digital gate-model quantum computers such as the programmable superconducting transmon qubit devices manufactured by IBM Quantum, for instance ibm_washington. QAOA and QA address the same types of problems, but it is unclear how they will scale to large problem sizes and to larger and higher-fidelity quantum computers. In this article, we present a direct comparison between QAOA, one and two rounds, run on all 127 qubits of ibm_washington and QA run on D-Wave Advantage_system4.1 and Advantage_system6.1. The problems which allow for this comparison are random Ising model problems whose connectivity matches the heavy hexagonal lattice topology of ibm_washington and the Pegasus graph connectivity of the two D-Wave devices. We create two classes of problem instances for this comparison: one with higher order terms (ZZZ variable interactions), linear terms, and quadratic terms, and a separate problem type with only linear and quadratic terms. The QAOA circuits are novel and extremely short depth, with a CNOT depth of 6 per round, which allows whole chip usage of ibm_washington's heavy hexagonal lattice and can be applied to future heavy-hex chips. We also test the effectiveness of the error suppression technique digital dynamical decoupling on the QAOA circuits, allowing one of the largest experimental evaluations of dynamical decoupling to date. The QAOA circuits compiled to ibm_washington are composed of approximately 3,000 gate instructions making these some the largest quantum circuits ever executed on a digital quantum processor. QAOA and QA are compared against the classical heuristic algorithm of simulated annealing and all problem instances are exactly solved using CPLEX in order to evaluate ground state success sampling probability. We find that (i) QA outperforms QAOA on all problem instances, (ii) QAOA samples the problems better than random sampling, and (iii) QAOA parameters exhibit clear parameter concentration across the ensemble of Ising models.
Programmable quantum annealers can realize real-time dynamics of frustrated transverse-field Ising models on large, nontrivial graphs. Recent work by King et al. argued that the classical simulation of such experiments would require exponential computational resources for classical methods such as tensor networks and neural quantum states. Here, we numerically simulate the D-Wave spin-glass annealing protocol with time-dependent variational Monte Carlo (t-VMC) using a correlator state tailored to spin-glass dynamics. For the two-dimensional cylinder, three-dimensional dimer, diamond, and biclique instances considered, at annealing times of 7 and 20 ns, we show that systematically increasing the variational ansatz size enables t-VMC to approximate the final two-spin correlation errors of the quantum processing unit (QPU). We also perform an accurate large-scale simulation of a challenging biclique instance for which no other variational method is known to work, and we find close agreement with the quantum annealer. Through ablation studies, we identify poor Markov-chain mixing, high-variance local-energy estimators, and stochastic Runge-Kutta error estimates as the principal numerical failure modes. We address these numerical issues by using parallel tempering, blurred sampling and an importance-weighted differential equation solver, thereby clarifying the numerical requirements for stable, large-scale t-VMC simulations. Our results extend the frontier of classical simulation while providing a realistic assessment of the computational costs of simulating quantum dynamics at this scale.
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.