arXiv subjects

Ricardo Rivera Cardoso

Publications and source records attributed to Ricardo Rivera Cardoso.

3 recordsLinked to original sources

arXiv2026

Local Relaxation Hierarchies for Quantum Ground State Energies: Convergence Guarantees and Message Passing Algorithms

Convex relaxation hierarchies provide lower bounds to the ground state energy of quantum many-body systems that can be computed in polynomial time on a classical computer, at any fixed hierarchy level. However, scaling these methods to large systems and accurate approximations remains challenging due to the computational cost of traditional solvers and the scarcity of efficiency guarantees. In this work, we develop local relaxation hierarchies and efficient, highly parallelisable message passing algorithms for estimating the relaxed ground state energies. We show that the first level of the hierarchy---based on local consistency of pairwise reduced density matrices---is exact for commuting Hamiltonians on trees. We further establish that another hierarchy, based on consistent intervals, converges exponentially fast in the interval size to the ground state energy for weak perturbations of separable Hamiltonians on a chain, thereby providing an efficient classical algorithm for these systems. Then, we introduce two variants of message passing algorithms that run in $\mathcal{O}(n/ε^2)$ and $\mathcal{O}(n/ε)$ time for any fixed level of the local hierarchy on bounded-degree graphs, where $ε$ is the precision for the relaxed ground state energy per site. This assumes that the optimal messages have $\mathcal{O}(1)$ norm---a condition we observe in practical settings in our experiments. These algorithms are based on the subgradient method and the Nesterov-type accelerated gradient descent method applied to an entropy-smoothed objective. Finally, we benchmark the message passing algorithms across different quantum Hamiltonians, lattice geometries, and relaxation levels, validating the theoretical predictions and their potential to surpass standard convex optimisation solvers for this problem. We release the resulting library at github.com/rick1924/gse-message-passing.

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arXiv2026

Observable Estimation in the Absence of Classical Verification

The predictive success of quantum mechanics underpins many areas of modern science, even as the exact simulation of large, interacting quantum systems remains beyond the reach of classical computation. This success has been enabled by the remarkable advancement of scalable numerical approximation methods, which often demonstrate practical accuracy despite the absence of formal guarantees. As quantum simulation pushes into regimes where these approximations struggle, a fundamental challenge arises: How can quantum outcomes be trusted when reliable classical benchmarks are unavailable? Here, we establish a framework for the independent validation of quantum estimates in this setting and present evidence that they provide the most credible result among several considered methods, in the absence of an immediately accessible ground-truth solution. We apply our framework to the semi-scrambling dynamics of a physical model that strains several leading classical simulation methods yet remains experimentally accessible, in part through our introduction of the \textit{operator Loschmidt echo}. We systematically design a series of experiments using quantum heuristics that, taken together, test the underlying assumptions and provide strong confidence in the observable estimates obtained from the quantum computer. We then show how this framework can be extended to place accuracy bounds on quantum estimates via careful characterization and manipulation of the device noise, transforming the problem of validating the observable estimation to validating the noise model. These results establish a route towards trusted quantum computation for scientific discovery, independent of classical verification.

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arXiv2025

Quantum SAT Problems with Finite Sets of Projectors are Complete for a Plethora of Classes

Previously, all known variants of the Quantum Satisfiability (QSAT) problem, i.e. deciding whether a $k$-local ($k$-body) Hamiltonian is frustration-free, could be classified as being either in $\mathsf{P}$; or complete for $\mathsf{NP}$, $\mathsf{MA}$, or $\mathsf{QMA_1}$. Here, we demonstrate new qubit variants of this problem that are complete for $\mathsf{BQP_1}$, $\mathsf{coRP}$, $\mathsf{QCMA}$, $\mathsf{PI(coRP,NP)}$, $\mathsf{PI(BQP_1,NP)}$, $\mathsf{PI(BQP_1,MA)}$, $\mathsf{SoPU(coRP,NP)}$, $\mathsf{SoPU(BQP_1,NP)}$, and $\mathsf{SoPU(BQP_1,MA)}$. Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer's dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial $\mathsf{BQP_1}$-complete problem. We first prove there are qudit QSAT problems that are complete for $\mathsf{BQP_1}$, $\mathsf{coRP}$, and $\mathsf{QCMA}$ by re-defining elements of the circuit-to-Hamiltonian transformation. We then show that any QCSP can be reduced to a problem in qubits while maintaining the same complexity - something believed not to be possible classically. The remaining six problems are obtained by considering "sums" and "products" of the first seven QSAT problems. Before this work, the QSAT problems generated in this way resulted in complete problems for $\mathsf{PI}$ and $\mathsf{SoPU}$ classes that were trivially equal to other known classes. We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for the first three classes, we note that his constructions are flawed. Here, we rework them and obtain improvements on the required qudit dimensionality.

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