Quantum Glassiness From Efficient Learning

TL;DR

It is proved that preparing low-energy states of systems which exhibit the QOGP is intractable for quantum algorithms whose outputs are stable under perturbations of their inputs, giving the first known algorithmic hardness-of-approximation result for quantum algorithms in finding the ground state of a non-stoquastic, noncommuting quantum system.

Abstract

We show a relation between quantum learning theory and algorithmic hardness. We use the existence of efficient, local learning algorithms for energy estimation -- such as the classical shadows algorithm -- to prove that finding near-ground states of disordered quantum systems exhibiting a certain topological property is impossible in the average case for Lipschitz quantum algorithms. A corollary of our result is that many standard quantum algorithms fail to find near-ground states of these systems, including time-$T$ Lindbladian dynamics from an arbitrary initial state, time-$T$ quantum annealing, phase estimation to $T$ bits of precision, and depth-$T$ variational quantum algorithms, whenever $T$ is less than some universal constant times the logarithm of the system size. To achieve this, we introduce a generalization of the overlap gap property (OGP) for quantum systems that we call the quantum overlap gap property (QOGP). We prove that preparing low-energy states of systems which exhibit the QOGP is intractable for quantum algorithms whose outputs are stable under perturbations of their inputs. We then prove that the QOGP is satisfied for a sparsified variant of the quantum $p$-spin model, giving the first known algorithmic hardness-of-approximation result for quantum algorithms in finding the ground state of a non-stoquastic, noncommuting quantum system. Inversely, we show that the Sachdev--Ye--Kitaev (SYK) model does not exhibit the QOGP, consistent with previous evidence that the model is rapidly mixing at low temperatures.

Figures (2)

• Figure 1: An illustration of the low-energy space of a classical spin system satisfying the overlap gap property. No two configurations (small, solid-bordered circles) achieving an approximation ratio $\mu$ have a normalized Hamming distance in the closed set $\left[\nu_1,\nu_2\right]$. Larger circles with dashed borders are an aid to the eye.
• Figure 2: An illustration of the near-optimum space of a classical spin system satisfying (a) the $m$-overlap gap property ($m$-OGP) and (b) the chaos property with $m=2$. Small, solid-bordered circles (white) represent near-optimal solutions from one problem instance, and solid-bordered squares (blue) from another problem instance. The $m$-OGP is a statement that correlated problem instances have no near-optimal configurations with normalized Hamming distance in the closed set $\left[\nu_1,\nu_2\right]$. The chaos property only requires that near-optimal states of one problem instance are at least a normalized Hamming distance $\nu_2$ away from near-optimal states of an independent problem instance.

Theorems & Definitions (106)

• Theorem 2: The QOGP implies hardness for stable quantum algorithms, informal statement of Theorem \ref{['thm:m_qogp_implies_alg_hardness']}
• Proposition 3: Standard quantum algorithms are Lipschitz, informal statement of Corollaries \ref{['cor:p_trott_qa']}, \ref{['cor:phase_est_stab']}, and \ref{['cor:lind_ev_stab']}
• Corollary 4: Lipschitz quantum algorithms fail to optimize sparse quantum spin glass models, informal statement of Corollary \ref{['cor:stab_algs_fail_sparse_q_spin_glass']}
• Proposition 5: Self-averaging
• proof
• Definition 6: Stability, informal statement of Definition \ref{['def:stable_qas']}
• Definition 7: Quantum algorithm
• Definition 8: Associated pure quantum algorithm
• Definition 9: Stable quantum algorithm
• Definition 10: Near-optimal quantum algorithm, informal statement of Definition \ref{['def:no_qas']}