Quantum Chaos Simulations Boosted by Algorithm with a Cubic Scaling Advantage

Researchers are continually seeking efficient methods to generate the random thermal states essential for investigating thermalisation, chaos, and phase transitions in complex quantum systems. Jiyu Jiang, Mingrui Jing, and Jizhe Lai, from the Thrust of Artificial Intelligence, Information Hub at The Hong Kong University of Science and Technology (Guangzhou), alongside Xin Wang and Lei Zhang et al., have developed a novel approach called thermal-drift sampling to address the computational cost of preparing these states individually. Their work introduces a measurement-based operation and algorithm that generates thermal states alongside Hamiltonian labels, scaling favourably with system size and offering a trade-off between accuracy and range. Validated through simulations on a 2D Heisenberg model, this technique provides a practical and scalable pathway for chaos diagnostics, demonstrated by the successful application to a 2D transverse-field Ising model and its alignment with Wigner, Dyson predictions, potentially advancing studies on near-term quantum hardware.

Scientists introduce the thermal-drift channel, a measurement-based operation that implements a tunable nonunitary drift along a chosen Pauli term. Based on this channel, they present a measurement-controlled sampling algorithm that generates thermal states together with their Hamiltonian “labels” for general physical models.
They prove that the total gate count of their algorithm scales cubically with system size, quadratically with inverse temperature, and as the inverse error tolerance to the two-thirds power, with logarithmic dependence on the allowed failure probability. Furthermore, they show that the induced label distribution approaches a normal distribution reweighted by the.

Haar and Thermal State Generation for Quantum System Characterisation

Scientists are investigating the generation of random quantum states, offering a resource analogous to random numbers in classical stochastic theories. Ensembles of random pure states underpin our understanding of quantum complexity, chaos, and quantum metrology, with direct relevance to quantum circuits, black-hole physics, and quantum supremacy experiments.

Haar random states exhibit universal entanglement and spectral properties predicted by random matrix theory, making them indispensable for benchmarking quantum devices and probing the limits of classical simulability. As quantum processors scale up, the controlled generation of such random pure states has become both a theoretical testbed for many-body physics and a practical diagnostic for near-term quantum hardware.

While pure states are idealized, random mixed states effectively describe realistic quantum many-body systems, elucidating thermalization and phase structures. Within this domain, random thermal states are essential due to their connection to statistical mechanics, encoding equilibrium physics and enabling critical tests of both the eigenstate thermalization hypothesis and random matrix theory predictions.

Importantly, thermal states of generic many-body Hamiltonians exhibit highly nontrivial correlations and often volume-law entanglement, making them representative of the “typical” mixed states encountered in quantum chaos and finite-temperature phases. This positions random thermal states as a minimal yet physically grounded ensemble for probing universal many-body phenomena, motivating the development of quantum algorithms that can efficiently generate them without relying on detailed spectral knowledge or classical simulation of the underlying dynamics.

The practical generation of random thermal states remains costly with existing algorithms. Current quantum methods typically rely on quantum phase estimation combined with imaginary-time evolution, fluctuation theorems, or quantum Metropolis and Lindbladian thermalization algorithms that couple the system to an effective bath.

These system-dependent methods are highly effective for preparing one thermal state, but do not directly address the task of sampling thermal states from an ensemble of Hamiltonians. Extending these approaches to generate random thermal states requires repeatedly reconfiguring the algorithm for different Hamiltonian instances, with the randomness largely produced by classical preprocessing steps.

The overall pipeline thus lacks global optimization and incurs substantial classical computational overhead, limiting scalability and diminishing the advantages of quantum sampling. Researchers bridge this gap by introducing a dedicated quantum algorithmic framework for the scalable generation of random thermal states, as illustrated in Figure.

They assume a setup where the device is programmed solely with a target set of L Pauli operators Σ = {σ1, . . . , σL} representing the interaction topology. Their protocol autonomously generates the thermal state Gβ,H = e−βH/ Tr e−βH together with its label H = P cjσj, with coefficients cj drawn from a random distribution, effectively functioning as a stochastic quantum oracle.

This approach bypasses the need for precise deterministic control of every coefficient, instead leveraging the device’s ability to sample from the typical subspace of the Hamiltonian ensemble. They present their measurement-controlled sampling algorithm as a randomized thermalization process that uses a sequence of thermal-drift channels to generate the thermal ensemble.

Compared with existing works, they provide the most explicit scaling of the resource complexity with respect to the problem parameters, as shown in Table I. This shift from Hamiltonian-specific preparation to ensemble-based sampling opens new avenues for studying thermal physics at scale. They also analyze the performance of their algorithm in terms of the β-dependence, the resource scaling, and the sampling behaviors through systematic benchmarking experiments.

Finally, they apply their algorithm to level-statistic analysis and observe a hallmark of quantum chaos and random matrix universality, distinguishing true quantum scrambling from decoherence or integrable dynamics. The observed Wigner, Dyson universality confirms that their sampler probes genuine many-body chaotic dynamics, connecting quantum algorithms to fundamental questions in random matrix theory.

This makes their protocol a robust tool for both thermodynamic simulation and the verification of quantum advantage. The sampling task is formalized as follows. Given a set of L linearly independent n-qubit Pauli words Σ and the inverse temperature β 0, the ensemble consists of any linear combination of the Pauli’s in Σ with bounded coefficients, H = n X j cjσj: |cj| ≤hj, cj ∈R, σj ∈Σ o.

The task is to sample from the thermal state Gβ,H with Hamiltonian H drawn from the ensemble. This framework departs from conventional random state sampling. Standard ensembles, such as Haar random states, unitary t-design, or random stabilizer states, are typically defined by their geometric properties within the Hilbert space.

These ensembles often decouple the state from a specific, physically motivated Hamiltonian. In contrast, their framework asks for generating a structured thermal equilibrium state, explicitly linking the thermodynamics and the random matrix theory. This distinct feature provides a controllable generating set essential for physical benchmarking and Hamiltonian learning.

Their numerical experiments consider two physical ensembles on a 3 × 3 two-dimensional grid, where two-local Pauli operators act on nearest-neighbor pairs. The first is a Heisenberg ensemble with Σ containing operators of the form XX, Y Y, and ZZ; the second is a transverse-field Ising ensemble with Σ containing operators of the form ZZ and X.

For both ensembles, the coefficient bounds hj are uniform, and they denote λ = P j hj, which upper bounds the largest eigenvalue of any H ∈H. To circumvent the high cost of standard ‘sample-then-prepare’ pipelines, they implement a randomized thermalization protocol directly on the quantum hardware. They define the thermal-drift channel Nσ for any Pauli operator σ, which is the core component of their algorithm.

For any input state ρ, the channel is written in, Nσ(ρ) = μ 2 |↑⟩⟨↑| ⊗E↑ σ(ρ) + |↓⟩⟨↓| ⊗E↓ σ(ρ), where μ The first register of Nσ(ρ) naturally generates a random branch. Measuring this flag register indicates which direction the state is thermalized toward, with probabilities proportional to the corresponding unnormalized traces.

The thermal-drift channel can be implemented using system-ancilla coupling in a dilated Hilbert space, followed by measuring and tracing out the flag register. They consider the following map N≈(ρ) = 1 1 + μNσ(ρ) + μ 1 + μ|⟲⟩⟨⟲| ⊗ρ. This map can be implemented within O(n) quantum elementary gates and 2n ancilla qubits.

Then, with failure probability at most δ, the thermal-drift channel is simulated by repeatedly applying N≈ and the associated measure-and-trace operations until the measurement outcome is not ‘⟲’, using at most log(1/δ) repetitions. In particular, the same state and ancillas can be reused without postselection.

They present the thermal state sampling algorithm, implemented as a circuit of N blocks applied to an initial state ρ0. In block k, a thermal-drift channel Nσjk is applied, where σjk is selected independently from the set Σ. The probability of choosing Nσj is weighted by the bound hj, so that the distribution is hj/λ.

The thermal strength is fixed across all blocks to τ = βλ/N. After the thermal-drift channel, a measure-and-trace operation Mmk is applied to the flag register, generating a measurement outcome mk. This outcome is recorded as mk ∈{ + 1, −1} corresponding to ↑ and ↓. The full circuit implementation is labeled by an ordered list of values j = {j1, · · · , jN} and an ordered list of directions m = {m1, · · · , mN}, that corresponds to the channel Ej,m = MmN ◦NσjN ◦. . . ◦Mm1 ◦Nσj1.

Cubic Scaling and Logarithmic Dependence in Thermal State Generation via Measurement Control

The research details a measurement-controlled sampling algorithm generating thermal states alongside Hamiltonian labels, achieving a cubic scaling of gate count with system size. Specifically, the total gate count scales as N³, where N represents system size, and inversely proportional to the square of the inverse temperature, alongside a two-thirds power dependence on inverse error tolerance.

This algorithm also exhibits logarithmic dependence on the allowed failure probability, demonstrating efficient resource utilization for generating thermal states. Numerical simulations performed on a 2D Heisenberg model successfully validated the predicted scaling behaviour and the resulting distribution of Hamiltonian labels.

The induced label distribution closely approaches a normal distribution reweighted by the thermal partition function, establishing a clear trade-off between accuracy and the effective range of the sampled states. This allows for controlled adjustment of the sampling process based on desired precision and computational cost.

Applying this method to a 2D transverse-field Ising model enabled the calculation of unfolding-free level-spacing ratio statistics from the sampled thermal states. Observations reveal a crossover towards the Wigner, Dyson prediction, confirming the practical scalability of this approach for chaos diagnostics and studies of random matrix universality on near-term quantum hardware.

This crossover provides a measurable indicator of quantum chaos within the system. Furthermore, the work efficiently generates random thermal states for Hamiltonians drawn from broad Pauli ensembles, offering access to level statistics, entanglement measures, and finite-temperature signatures without reliance on exact diagonalization or classical Monte Carlo methods.

This provides a novel pathway for benchmarking quantum devices using thermal randomness, differing from Haar randomness. The framework integrates quantum algorithms, statistical mechanics, and machine learning, offering tools for both near-term quantum applications and fundamental many-body physics investigations.

Efficient thermal state generation via the thermal-drift channel and Hamiltonian identification

Scientists have developed a new method for efficiently generating random thermal states, which are crucial for studying complex physical phenomena such as thermalization, chaos, and phase transitions. This technique utilizes a measurement-based operation called the thermal-drift channel to create these states alongside information identifying the Hamiltonian from which they originate.

The algorithm’s computational cost scales favourably with system size, inverse temperature, and desired accuracy, demonstrating a significant improvement over existing methods that often require individual preparation of each Hamiltonian. The research validates this approach through numerical simulations on a two-dimensional Heisenberg model, confirming the predicted scaling and distribution of generated states.

A practical application of this method involves analysing level-spacing ratio statistics from a two-dimensional transverse-field Ising model, revealing a transition towards behaviour predicted by random matrix theory and providing a robust diagnostic for quantum chaos. This unfolding-free analysis indicates that the generated states access a more chaotic region of the system’s possible states than the initial structured input.

The authors acknowledge limitations related to the algorithm’s scaling with error tolerance, noting an inverse two-thirds power dependence. Future research directions include exploring the method’s potential for benchmarking quantum devices by generating thermal randomness, and applying it to quantum simulation in scenarios where traditional methods struggle with the sign problem.

Furthermore, the framework could be extended to quantum machine learning, utilising thermal states as training data and their Hamiltonian labels for supervised learning of many-body properties. This work integrates quantum algorithms, statistical mechanics, and machine learning, offering tools for both near-term quantum applications and fundamental investigations into interacting quantum systems.