Circuit complexity lower bounds for quantum spin glasses

Abstract:A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum $p$-spin glasses, random Hamiltonians whose interactions act on $p$-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement. We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with $n$, we prove that, for all sufficiently large fixed $p$, any circuit preparing an $n$-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth $\Omega_p(\log n)$. In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed $D$, a sufficiently large degree prefactor rules out depth-$D$ preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits. Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.

Quantum Physics (quant-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Computational Complexity (cs.CC); Probability (math.PR)

arXiv:2607.14384 [quant-ph]

(or arXiv:2607.14384v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.14384 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.