This work proposes a low-cost criterion for convergence monitoring that exploits the weak measurements inherent in quantum Gibbs samplers and their qubit-efficient variants and constructs a Hamiltonian-agnostic stopping criterion based solely on data already generated by the sampler.
Abstract
Recent progress in fully quantum Markov chain Monte Carlo methods enables efficient Gibbs-state sampling on quantum computers [Chen et al., Nature 646, 561 (2025)]. Although rigorous worst-case bounds on mixing times remain largely inaccessible for classically intractable systems, experience from classical Monte Carlo suggests that convergence of relevant observables may nevertheless be rapid. This raises the practical question of how to diagnose convergence efficiently, i.e., with at most polynomial overhead. We propose a low-cost criterion for convergence monitoring that exploits the weak measurements inherent in quantum Gibbs samplers and their qubit-efficient variants [Ding et al., arXiv:2508.05703 (2025)]. Our approach is based on the observation that, at thermal equilibrium, the net energy flow between system and environment vanishes and energy-exchange statistics satisfy a balance condition. This condition appears in the distribution of (quasi-)frequencies extracted from the weak-measurement record and we use it to construct a Hamiltonian-agnostic stopping criterion based solely on data already generated by the sampler. We provide a statistical analysis, along with numerical and analytical studies to understand its performance, assumptions, and limitations.
Markovian open-system dynamics have widespread applications throughout quantum information science, including algorithmic state preparation. Their convergence is commonly quantified using the worst case global trace distance between the evolving and stationary states. However, this criterion can be unnecessarily stringent when only physically relevant observables are of interest. Here we introduce and study observable-specific mixing times. We prove that, for quasi-local, rapidly mixing Lindbladians, sums of geometrically local observables equilibrate in a time independent of system size, in contrast to the logarithmic dependence of global state mixing. This separation reduces the runtime of dissipative quantum algorithms, including quantum Gibbs samplers, for estimating quantities such as the Gibbs state energy and local order parameters, yielding an overall scaling that is linear in system size. Complementing this quantum result, we develop a quantum-inspired classical algorithm for estimating the same quantities. Its runtime is likewise linear in system size, but scaling exponentially in $\mathcal{O}\big(\log(1/\epsilon)^D\big)$, where $D$ denotes the spatial dimension of the lattice. We further analyse non-interacting Lindbladians over qudits, fermions, and bosons, demonstrating that locality of observables is not always necessary for a qualitatively faster mixing. Small-scale simulations of quantum Gibbs samplers reveal no large hidden constants in our asymptotic analysis and show that the theoretical predictions closely capture the finite-size dynamics.
Štěpán Šmíd, Richard Meister, Mario Berta et al.· 0 citations
Optimization problems are among the leading candidates for industrially relevant quantum advantage. Decoded quantum interferometry (DQI) has been proposed to tackle approximate optimization, establishing a connection to classical decoding problems. While previous work has primarily focused on the theoretical complexity of DQI, comparatively little is known about its empirical performance relative to classical algorithms. In this work, we shed further light on the complexity of DQI and investigate numerically whether classical sampling methods can emulate the optimization capabilities of DQI. We first present a simplified analytical characterization of DQI that connects its expected performance to binomial statistics, and we identify concrete obstacles in further studying the complexity of DQI. Exploiting the fact that DQI output probabilities are efficiently computable, we apply Markov chain Monte Carlo (MCMC) techniques, particularly block-Gibbs sampling, to sample from the induced distribution. We study the runtime scaling of these methods for two optimization problems called max-XORSAT, where we reach beyond $1000$ effective qubits; and OPI, where we reach beyond $150$ effective qubits. Our results show that MCMC algorithms can reliably attain the approximation ratios expected from DQI across a broad range of problem sizes. In OPI, in the regime where a super-polynomial advantage is claimed for DQI, we observe an empirical runtime for MCMC that scales approximately as $1.1^{n}$, indicating exponential growth with a comparatively small base. Our findings do not refute existing quantum advantage claims but provide new empirical evidence that classical sampling algorithms can closely match DQI's optimization performance, offering a more nuanced perspective on the practical advantage of DQI.
Elies Gil-Fuster, Matan Ninio, Lennart Bittel et al.· 0 citations
At high temperature, quantum Gibbs states retain several classical features of the maximally mixed state: the absence of entanglement, the absence of magic, analyticity of the partition function, correlation decay, and algorithmic tractability. We prove new and sharp bounds showing that these features persist down to finite temperatures independent of system size, but fail at distinct inverse-temperature scales, forming a hierarchy of classical-to-quantum transitions. Our results hold for long-range Pauli interactions with bounded strength at every site. Despite such all-to-all interactions, we show that the death of entanglement occurs at constant temperature, resolving an open question of Rouze, Franca and Alhambra (STOC'25). We give a polynomial-time classical algorithm that prepares Gibbs states up to the death of entanglement transition. Notably, this is asymptotically colder than temperatures at which quantum Gibbs samplers are known to mix quickly, as well as the original separability temperature of Bakshi et al. (FOCS'24), which we improve to be tight up to constants. At asymptotically even colder temperatures, we show that the Gibbs state remains in the thermodynamic infinite-temperature phase. This leads to polynomial-time classical algorithms for estimating thermal expectations despite both entanglement and magic, and the resolution of a correlation decay conjecture of Harrow, Mehraban and Soleimanifar (STOC'20).
Harald Putterman, Alexander Zlokapa, Jordan Cotler· 1 citation
Quantum state ensembles are important in quantum information processing. For example, quantum $t$-designs model highly entangled states in complex systems, while projected ensembles appear in generative quantum machine learning and studies of thermalization. With their sample state accompanied by a classical label, these ensembles contain operational information beyond their average density operators. Yet an ensemble differs from a classical-quantum state because it is invariant under permutations of labels. We formulate binary hypothesis testing between finite quantum ensembles and derive fundamental limits on error probability. Given an observed label pattern, we show that the joint sampled state can be described by power-weighted ensemble moments. This yields the Bayes-optimal measurement and exact finite-sample error, revealing that discrimination is governed by the full moment hierarchy up to the number of samples. In the many-sample limit, we derive Chernoff bounds and obtain exact error exponents for finite uniform pure-state ensembles. We apply these results to optical communication and $t$-designs. For finite uniform pure-state $t$-designs with large $t$, the maximal discrimination exponent scales sharply as $\sim t^{-2}$, while equal-prior fixed-error testing requires $\sim t^2$ samples.
We introduce a new class of fully-quantum Metropolis walks in which both the proposal and acceptance steps are intrinsically quantum. Unlike standard quantum walks obtained by quantizing classically efficient Markov chains, our algorithm employs Hamiltonian simulation as a quantum-native proposal mechanism, enlarging the class of quantum walks beyond classical counterparts. We target the problem of sampling from the low-temperature Gibbs distribution of classical dense Ising models, within a fixed error in total variation distance. This approach achieves about a cubic polynomial asymptotic advantage over previous quantum-walks, resulting in a total sixth-degree polynomial queries speedup compared to the best classical walk. This shows that speedups beyond the widely assumed quadratic limit are possible within the quantum walk formalism. We perform a complete fault-tolerant compilation of all algorithmic primitives and benchmark against CPU, GPU, and FPGA implementations of the best classical Markov chain. Under identical hardware assumptions, the resulting advantage runtime crossover is reduced from approximately $10^3$ years for conventional quantum walks to less than one day. These results identify fully-quantum Markov chains as a promising route toward practical quantum advantage.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.