Circuit complexity lower bounds for quantum spin glasses

2026-07-15

AI Summary

This work investigates whether near-ground states of the quantum p-spin glass model can be efficiently prepared by shallow quantum circuits, specifically addressing the existence of a circuit depth lower bound. By integrating tools from random matrix theory, uniform control of Gaussian processes, and quantum circuit complexity analysis, the study rigorously establishes—for the first time in this model—that shallow circuits cannot approximate the ground-state energy of random spin glass Hamiltonians, even when arbitrary ancillary qubits are permitted. In particular, when the average interaction degree grows with system size \(n\), the required circuit depth is at least \(\Omega_p(\log n)\); under fixed average degree, for any constant depth \(D\), there exists a sufficiently large degree parameter rendering depth-\(D\) circuits ineffective. These results provide new probabilistic evidence supporting the “No Low-Energy Trivial States” conjecture.

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 $Ω_p(\log n)$. 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.

Problem

• quantum spin glasses
• circuit complexity
• shallow circuits
• entanglement
• ground states

Innovation

• quantum spin glasses
• circuit complexity lower bounds
• shallow quantum circuits
• No Low-Energy Trivial States
• Gaussian processes