Simultaneous variances of Pauli strings, weighted independence numbers, and a new kind of perfection of graphs

TL;DR

The paper introduces $\\hbar$-perfect graphs as a principled bridge between graph theory and quantum information, linking the ground-state properties of Pauli-string Hamiltonians to graph topology via the weighted beta number $\\beta(G,w)$ and the stable-set polytope $\\mathrm{STAB}(G)$. It develops a geometric framework with the beta body $\\mathrm{BETA}(G)$, proves that all perfect and $h$-perfect graphs are $\\hbar$-perfect, and provides six structural properties plus numerical hierarchies (SDP, De Finetti, and see-saw) to test $\\hbar$-perfection in graphs up to nine vertices. The work then demonstrates broad applications: enhanced nonlinear entanglement witnesses, improved shadow tomography sample complexity bounds, linear-programmed uncertainty relations, and efficient encodings of the independence number, including memory-efficient quantum encodings and exact quantum/classical hierarchies for $\\alpha(G,w)$. Together these results offer practical tools for exploiting graph structure to tackle quantum information tasks, with several open questions about generalizations, scalability, and deeper connections to other graph parameters. The findings suggest that $\\hbar$-perfectness provides a versatile, scalable lens for translating graph-theoretic properties into quantum-information advantages.

Abstract

A set of Pauli stings is well characterized by the graph that encodes its commutatitivity structure, i.e., by its frustration graph. This graph provides a natural interface between graph theory and quantum information, which we explore in this work. We investigate all aspects of this interface for a special class of graphs that bears tight connections between the groundstate structures of a spin systems and topological structure of a graph. We call this class $\hbar$-perfect, as it extends the class of perfect and $h$-perfect graphs. Having an $\hbar$-perfect graph opens up several applications: we find efficient schemes for entanglement detection, a connection to the complexity of shadow tomography, tight uncertainty relations and a construction for computing good lower on bounds ground state energies. Conversely this also induces quantum algorithms for computing the independence number. Albeit those algorithms do not immediately promise an advantage in runtime, we show that an approximate Hamilton encoding of the independence number can be achieved with an amount of qubits that typically scales logarithmically in the number of vertices. We also we also determine the behavior of $\hbar$-perfectness under basic graph operations and evaluate their prevalence among all graphs.

Figures (11)

• Figure 1: The graph-theoretic framework centered on $\hbar$-perfect graphs and its various applications distributed in each section.
• Figure 2: The joint numerical range ${\cal J}$, the convex hull $\tilde{\cal J}$ of the flips of ${\cal J}$ along each axis, its square ${\cal Q}$ and beta body $\operatorname{BETA}$ for Pauli strings $\{X, Y, Z\}$ and $\{XX, YY, Z Z\}$, where the beta body $\operatorname{BETA}$ is the convex hull of the corner generated by ${\cal Q}$. For $\{X, Y, Z\}$, $\tilde{\cal J}$ coincides with ${\cal J}$ and $\operatorname{BETA}$ coincides with ${\cal Q}$. For $\{XX, YY, Z Z\}$, $\tilde{\cal J}$ (c, in gray, the cube) is strictly larger than ${\cal J}$ (c, in blue), and $\operatorname{BETA}$ (d, in gray, the cube) is strictly larger than ${\cal Q}$ (d, in blue, not convex, not containing the point $(1,1,0)$). Such a $\operatorname{BETA}$ coincides with the ${\cal Q}$ of $\{XX\mathbb{I}, YY\mathbb{I}, ZZZ\}$. In each of these two cases the frustration graph is $\hbar$-perfect, and $\tilde{J}$ is fully characterized by the intersections of the ellipsoids defined by the facets of $\operatorname{BETA}$.
• Figure 3: (a) The fully connected and (b) the fully unconnected unions of $\hbar$-perfect graphs are again $\hbar$-perfect. The same accounts for (c) the lexicographic product.
• Figure 4: (a) splitting and (b) copying of vertices of an $\hbar$-perfect graph creates a new graph that is still $\hbar$-perfect.
• Figure 5: The smallest three $h$-imperfect graphs.

Theorems & Definitions (57)

• Definition 1: Weighted beta number xu2023bounding
• Proposition 1
• proof
• Theorem 2
• proof
• Definition 3: $\hbar$-Perfectness
• Theorem 4: Perfect and $h$-perfect graphs
• Theorem 5
• proof
• Theorem 6