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arXiv:2609.40345

Gibbs Sampling in the Shattered Phase by Decoded Quantum Interferometry

Leo Zhou, Noah Shutty, Mark Sellke, Stephen P. Jordan

quant-phcond-mat.dis-nncs.CCcs.DSmath.PRSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $β_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=αk$ with fixed $α>1$, both Prange's algorithm and DQI can sample at any inverse temperature $β< \tanh^{-1}(1/α)$ for sufficiently large $k$, well beyond the dynamical threshold $β_{\rm dyn} \sim \sqrt{2\ln k / (αk)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.

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44 pages, 1 figure, 3 tables

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We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising kkk-spin glasses (or Max-kkk-XORSAT) on random Erdős-Rényi hypergraphs with average degree D≥kD\ge kD≥k. In a temperature range beginning asymptotically at the predicted dynamical phase transition, βdyn(k,D)=(2ln⁡k)/D×[1+ok→∞(1)]β_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]βdyn​(k,D)=(2lnk)/D​×[1+ok→∞​(1)], we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when D=αkD=αkD=αk with fixed α>1α>1α>1, both Prange's algorithm and DQI can sample at any inverse temperature β<tanh⁡−1(1/α)β< \tanh^{-1}(1/α)β<tanh−1(1/α) for sufficiently large kkk, well beyond the dynamical threshold βdyn∼2ln⁡k/(αk)β_{\rm dyn} \sim \sqrt{2\ln k / (αk)}βdyn​∼2lnk/(αk)​. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.
We propose a Gibbs state preparation algorithm that exploits algebraic structure in the Hamiltonian. Our starting point is Hamiltonian Decoded Quantum Interferometry (HDQI), a recent algorithm that reduces Gibbs state preparation for Pauli Hamiltonians to decoding error-correcting codes. We introduce quantum decoding HDQI (QD-HDQI), replacing HDQI's coherent classical decoder by a quantum decoder that can exploit interference between different errors. For commuting Pauli Hamiltonians, our reduction is efficient and the associated quantum decoding problem corresponds to transmitting the associated symplectic code over i.i.d. pure-state classical-quantum channels. For uniformly random Hamiltonian signs, the vanishing-error threshold of an efficient decoder directly determines the lowest temperature that our algorithm can reach. Even using only classical decoders, this improves upon HDQI's achievable inverse temperature from linear to square-root in the correctable error fraction; quantum decoders can improve further by tolerating still stronger noise. A similar picture extends to the noncommuting regime previously handled by HDQI, where now the noise is correlated within small, independent blocks. We study two such quantum decoders in our QD-HDQI Gibbs sampling framework. The first, based on unambiguous state discrimination, works efficiently for arbitrary binary linear codes. However, we "dequantize" this strategy for commuting Hamiltonians, contributing a classical rejection sampler that matches its temperature threshold. The second is belief propagation with quantum messages (BPQM), a quantum generalization of classical belief propagation. We show that for certain dilute spin glasses, BPQM's decoding threshold exceeds the Shannon limit for classical decoding, reaching temperatures inaccessible to any classical decoder within our framework.
We introduce Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm that utilizes coherent Bell measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian optimization to classical decoding. For a signed Pauli Hamiltonian HHH and any degree-ℓ\ellℓ polynomial P{P}P, HDQI prepares a purification of the density matrix ρP(H)∝P2(H)ρ_{P}(H) \propto {P}^2(H)ρP​(H)∝P2(H) by solving a combination of two tasks: decoding ℓ\ellℓ errors on a classical code defined by HHH, and preparing a pilot state that encodes the anti-commutation structure of HHH. Choosing P(x)P(x)P(x) to approximate exp⁡(−βx/2)\exp(-βx/2)exp(−βx/2) yields Gibbs states at inverse temperature βββ; other choices prepare approximate ground states, microcanonical ensembles, and other spectral filters. For local Hamiltonians, the corresponding decoding problem is that of LDPC codes. Preparing the pilot state is always efficient for commuting Hamiltonians, but highly non-trivial for non-commuting Hamiltonians. Nevertheless, we prove that this state admits an efficient matrix product state representation for Hamiltonians whose anti-commutation graph decomposes into connected components of logarithmic size. We show that HDQI efficiently prepares Gibbs states at arbitrary temperatures for a class of physically motivated commuting Hamiltonians -- including the toric code and Haah's cubic code -- but we also develop a matching efficient classical algorithm for this task. For a non-commuting semiclassical spin glass and commuting stabilizer Hamiltonians with quantum defects, HDQI prepares Gibbs states up to a constant inverse-temperature threshold using polynomial quantum resources and quasi-polynomial classical pre-processing. These results position HDQI as a versatile algorithmic primitive and the first extension of Regev's reduction to non-abelian groups.
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 100010001000 effective qubits; and OPI, where we reach beyond 150150150 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 1.1n1.1^{n}1.1n, 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.
For general max-k-XORSAT with k≥3k \geq 3k≥3, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless P=NP\mathsf{P} = \mathsf{NP}P=NP: approximating beyond the random-assignment value of 1/21/21/2 is NP\mathsf{NP}NP-hard. The picture changes when each variable appears in at most DDD constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order 1/D1/\sqrt{D}1/D​. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-Ekkk-LINSAT(q,r)(q,r)(q,r) with bounded degree DDD and over arbitrary finite fields Fq\mathbb{F}_qFq​, proving that it is NP\mathsf{NP}NP-hard to exceed r/q+Oq,r(1/D)r/q + \mathcal{O}_{q,r}(1/\sqrt{D})r/q+Oq,r​(1/D​). These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on (k,D)(k,D)(k,D)-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic 1/Dlog⁡D1/\sqrt{D \log D}1/DlogD​ barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the 1/D1/\sqrt{D}1/D​ scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.
Gibbs state preparation has emerged as a potential avenue for quantum advantage in the study of physical systems. Although there is known advantage for computing local properties of worst-case Gibbs states, recent results suggest a different picture for average-case local systems: at temperatures where their Gibbs states can be efficiently prepared quantumly, quasi-polynomial time classical algorithms can also compute local expectation values. Yet this leaves open the possibility of an advantage under a stronger notion of simulation: sampling. Gibbs sampling underlies Boltzmann machine-based quantum learning algorithms and quantum supremacy proposals, but its classical average-case complexity for physically relevant models remains unresolved. Here, we give strong evidence against an exponential quantum advantage in Gibbs sampling for an ensemble of local systems known as the quantum ppp-spin model. We describe a classical algorithm for sampling from the Gibbs state to any polynomially small error in total variation distance. Our algorithm is based on a meta-algorithm known as algorithmic stochastic localization (ASL). Our main technical result is a generalized convergence theorem for ASL which reduces approximate sampling from any distribution on the hypercube to estimating the mean of a related "tilted distribution" to sufficiently low additive error. We then extend known quasi-polynomial time algorithms for computing local expectation values of a related disordered model to compute these "tilted means" for the quantum ppp-spin model. Finally, we give a non-rigorous physics argument that this mean-estimation algorithm works down to the spin glass transition temperature. As efficient quantum algorithms also fail at this transition, this suggests they cannot Gibbs sample the quantum ppp-spin model at temperatures below that for which classical algorithms are also (almost) efficient.
Glassiness -- a phenomenon in physics characterized by a rough free-energy landscape -- implies hardness for stable classical algorithms. For example, it can obstruct constant-time Langevin dynamics and message-passing in random kkk-SAT and max-cut instances. We provide an analogous framework for average-case quantum complexity showing that a natural family of quantum algorithms (e.g., Lindbladian evolution) fails for natural Hamiltonian ensembles (e.g., random 3-local Hamiltonians). Specifically, we prove that the standard notion of quantum glassiness based on replica symmetry breaking obstructs stable quantum algorithms for Gibbs sampling, which we define by a Lipschitz temperature dependence in quantum Wasserstein complexity. Our proof relies on showing that such algorithms fail to capture a structural phase transition in the Gibbs state, where glassiness causes the Gibbs state to decompose into clusters extensively separated in quantum Wasserstein distance. This yields average-case lower bounds for constant-time local Lindbladian evolution and shallow variational circuits. Unlike mixing time lower bounds, our results hold even when dynamics are initialized from the maximally mixed state. We apply these lower bounds to non-commuting, non-stoquastic Hamiltonians by showing a glass transition via the replica trick. We find that the ensemble of all 3-local Pauli strings with independent Gaussian coefficients is average-case hard, while providing analytical evidence that the general ppp-local Pauli ensemble is non-glassy for sufficiently large constant ppp, in contrast to its classical (Ising ppp-spin, always glassy) and fermionic (SYK, never glassy) counterparts.