Preparing thermal states of frustrated quantum spin systems using 139 qubits

Abstract: 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 24 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.

• The paper introduces a robust dissipative quantum algorithm that prepares thermal states for frustrated spin systems via engineered quantum channels.
• It presents classical simulations and experimental results showing size-independent mixing times and effective thermalization even with circuits exceeding 1000 two-qubit gates.
• Key findings include linear scaling of circuit depth with temperature for AFHM and accelerated thermalization in AFIM's ice regime, highlighting practical quantum simulation advantages.

Preparing Thermal States of Frustrated Quantum Spin Systems Using 139 Qubits

Introduction and Motivation

The simulation of quantum many-body systems at finite temperature is a critical open problem in condensed matter, chemistry, and high-energy physics, where classical computational techniques frequently fail—especially for regimes governed by strong correlations and geometric frustration. This breakdown is most pronounced in systems with the quantum Monte Carlo (QMC) sign problem, which renders many relevant models (e.g., the antiferromagnetic Heisenberg model on the kagome lattice) intractable via stochastic approaches. The emergence of programmable digital quantum computers has provoked significant interest in alternative protocols for thermal state preparation.

This work leverages dissipative quantum algorithms for thermalization, focusing on the transverse-field antiferromagnetic Ising model (AFIM) and the antiferromagnetic Heisenberg model (AFHM) on kagome lattices. The AFHM, central in the study of quantum spin liquids, is inaccessible to QMC below a threshold temperature due to exponentially decaying signal-to-noise. The AFIM, which retains geometric frustration but remains sign-problem free, is used for benchmarking scalability and accuracy.

Dissipative Quantum Gibbs Sampling Protocol

Thermal state preparation is performed using the dissipative algorithm introduced in Ref. [Ding:2025ulc], which relies on repeated applications of a quantum channel $\Phi$ engineered to have the Gibbs state $\rho_S(\beta)$ as its unique fixed point, up to systematically improvable error. The algorithm partitions $N_S$ qubits into system and $N_E$ environment registers. Environment qubits are initialized from the desired temperature, interact with system qubits via two-qubit random Pauli couplings, and are consistently measured and reset, implementing explicit dissipation.

Figure 1: Schematic of the dissipative thermal state preparation process, including reset cycles and system-environment couplings.

The protocol, critically, requires no prior structural knowledge of the system Hamiltonian. It avoids deep circuits associated with imaginary time evolution, phase estimation, or adiabatic approaches, and exhibits inherent robustness to device noise. The channel's convergence is governed by the mixing time $\tau_{\text{mix}}$; polynomial scaling in $N_S$ and $\beta$ is rigorously established only for high-temperature and weakly interacting regimes, but not analytically for frustrated lattices.

Classical Simulation Results and Mixing Time Scaling

To probe scalability, the quantum channel is simulated classically for AFIM and AFHM on kagome lattices up to $N_S=24$. The AFIM spectrum is characterized by an exponentially large ice manifold separated from excited states by a substantial gap. Mixing time analyses confirm:

• For AFHM, the circuit depth required to reach thermal equilibrium scales at most linearly in $\beta$ and, remarkably, is largely independent of system size.
• For AFIM, mixing time exhibits non-monotonic dependence, with a local maximum at the ice manifold crossover temperature $\beta_c$, and a plateau deep into the ice regime; surprisingly, thermalization accelerates at lower temperatures due to block-diagonal jump operator connectivity.

  Figure 2: AFIM spectrum and mixing times, illustrating both size and temperature scaling in classical simulations.

Jump operator connectivity analysis shows thermalization bottlenecks stem from weak transitions between ice and non-ice sectors in AFIM, or lack of $\rho_S(\beta)$0 symmetry-preserving jumps in AFHM, which are necessary for efficiency in low-energy singlet-dominated regimes.

Figure 3: Jump operator connectivity matrix between low-energy eigenstates of AFIM and AFHM, demonstrating bottleneck structure and necessity of symmetry-adapted couplings.

Experimental Quantum Computer Implementation

The protocol was executed on IBM's superconducting quantum processors, preparing AFIM thermal states on kagome lattices up to $\rho_S(\beta)$1, coupled to $\rho_S(\beta)$2 environment qubits. Circuits were optimized for depth, leveraging heavy-hex connectivity and custom Trotterization routines.

Figure 4: Embedding of kagome lattice and circuit structure on IBM's heavy-hex topology, including Trotter step layout.

Measured energy densities as a function of reset cycles, bath temperature, and environment density show:

• Convergence to robust, temperature-dependent steady states—even with substantial circuit depths ($\rho_S(\beta)$3 two-qubit gates)—confirming the protocol's noise-resilience.
• Deviations from ideal thermal distributions scale with mixing time and environment density, aligning with theoretical bounds derived for depolarizing noise. Bulk observables (magnetization, connected triangle correlators) retain qualitative temperature sensitivity expected from QMC.

  Figure 5: Steady-state energy densities, site-resolved magnetization, and triangle correlators for AFIM on $\rho_S(\beta)$4, compared to QMC.

Spatial inhomogeneities appear in systems with reduced environment density or hot environment qubits due to imperfect resets, which have analyzable effects on local temperature.

Figure 6: Partitioning and pairing of system and environment qubits for optimal circuit design and fast mixing on the $\rho_S(\beta)$5 lattice.

Numerical and Empirical Results

Strong empirical results are:

• Steady state circuit depth required for local observables is constant in $\rho_S(\beta)$6, at fixed $\rho_S(\beta)$7, up to the largest simulated size ($\rho_S(\beta)$8), and plateaus rapidly in experiment ($\rho_S(\beta)$9).
• Experimental energy densities for AFIM on $N_S$0 are only modestly higher than noiseless simulation, with effective temperature degradation explained by mixing time and environment density scaling.
• Correlation and magnetization measurements show boundary sensitivity and thermal trends matching QMC, despite spatial noise.

  Figure 7: Mixing time scaling and convergence dynamics as a function of environment qubit number, circuit depth, and $N_S$1.

Implications and Future Directions

The observed rapid mixing and noise robustness of the dissipative protocol for frustrated models suggest practical quantum simulation utility in regimes intractable to classical methods. If size-independent mixing persists asymptotically, an exponential quantum runtime advantage is achievable for models with severe sign problems (e.g., AFHM at low $N_S$2).

Key practical implications include:

• Extensive scaling of environment qubits ($N_S$3) simultaneously improves mixing and noise resilience; resource-efficient layouts are now feasible on state-of-the-art devices (~139 qubits).
• States prepared by this protocol provide a foundation for simulating non-equilibrium dynamics in frustrated systems, potentially unlocking quantum advantage for both dynamical and equilibrium observables.
• Achieving global thermal equilibrium at ultra-low temperatures will ultimately necessitate fault-tolerant hardware and improved mid-circuit reset fidelity.

Theoretically, future research should examine whether rapid mixing is generic for strongly frustrated systems, probe the interplay between many-body entanglement and channel convergence, and explore quantum spin liquid phases beyond current classical reach.

Conclusion

This study establishes engineered dissipation as a scalable, robust algorithm for quantum thermal state preparation in frustrated spin models. Classical and quantum computer experiments demonstrate that mixing time remains constant with system size up to $N_S$4, and that noise-induced steady states preserve nontrivial thermal signatures on quantum hardware even in circuits exceeding 1000 gates. These results point toward regimes where quantum devices, using extensive environment coupling, may outperform classical methods in accessing equilibrium and dynamical properties of strongly correlated, geometrically frustrated quantum matter.