Dequantization Barriers for Guided Stoquastic Hamiltonians

Abstract

We construct a probability distribution, induced by the Perron--Frobenius eigenvector of an exponentially large graph, which cannot be efficiently sampled by any classical algorithm, even when provided with the best-possible warm-start distribution. In the quantum setting, this problem can be viewed as preparing the ground state of a stoquastic Hamiltonian given a guiding state as input, and is known to be efficiently solvable on a quantum computer. Our result suggests that no efficient classical algorithm can solve a broad class of stoquastic ground-state problems. Our graph is constructed from a class of high-degree, high-girth spectral expanders to which self-similar trees are attached. This builds on and extends prior work of Gilyén, Hastings, and Vazirani [Quantum 2021, STOC 2021], which ruled out dequantization for a specific stoquastic adiabatic path algorithm. We strengthen their result by ruling out any classical algorithm for guided ground-state preparation.

Figures (2)

• Figure 1: Pictorial representation of the graph $G$, with the expander $E$ shown inside the gray ellipse (circular vertices) and details of the self-similar trees attached to a vertex $u$ (square vertices). The expander is locally tree-like within a ball of radius equal to half the girth around $u$ (shown by the green boundary). The graph $G$ locally resembles a self-similar tree -- see \ref{['Lem:locss']}. The simplified view depicts the local tree structure around $u$, drawing the incident edges within the expander, but omitting the decorations arising from the blue and red edges.
• Figure 2: Pictorial representation of the local structure of the graph $G_n$ with girth $g_n = 12$ (only expander vertices are shown). The orange diamonds represent the vertices in a set $U$. The solid edges connect pairs of vertices in $U$ at distance $< g_n/2$ from one another, so that the set $\br{U}$ corresponds to the greyed vertices. The triangles depict the local tree structure within distance $< g_n/4$ around $\br{U}$. The white circles represent the vertices identified with the $0$-level leaves of $\treh_{n,K}$. The dashed lines indicate other paths in the expander graph that may create cycles but lie too far from $\br{U}$ to be detected by a local exploration.

Theorems & Definitions (55)

• definition 1: Informal definition of GGSP
• definition 2
• proposition 1: Quantum algorithm for the GGSP problem
• proof
• proposition 2: High-degree, high-girth expanders $E_n$
• proof
• definition 3: Self-similar trees $\treh_{n,k}$
• definition 4: Main graph $G_n$
• lemma 1: Spectral properties of the adjacency matrix $A_n$
• proof