4 research outputs found
On the Existence of Non-golden Signed Graphs
A signed graph is a pair Γ = (G,σ), where G = (V(G),E(G)) is a graph and σ : E(G)→{+1,−1} is the sign function on the edges of G. For a signed graph we consider the least eigenvector λ(Γ) of the Laplacian matrix defined as L(Γ) = D(G)−A(Γ), where D(G) is the matrix of vertices degrees of G and A(Γ) is the signed adjacency matrix.
An unbalanced signed bicyclic graph is said to be golden if it is switching equivalent to a
graph Γ satisfying the following property: there exists a cycle C in Γ and a λ(Γ)-eigenvector
x such that the unique negative edge pq of Γ belongs to C and detects the minimum of the
set Sx(Γ,C) = { |xrxs| | rs ∈ E(C) }.
In this paper we show that non-golden bicyclic graphs with frustration index 1 exist for each
n ≥ 5
Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians
We investigate the relationship between the energy spectrum of a local
Hamiltonian and the geometric properties of its ground state. By generalizing a
standard framework from the analysis of Markov chains to arbitrary
(non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap
can always be upper bounded by an isoperimetric ratio that depends only on the
ground state probability distribution and the range of the terms in the
Hamiltonian, but not on any other details of the interaction couplings. This
means that for a given probability distribution the inequality constrains the
spectral gap of any local Hamiltonian with this distribution as its ground
state probability distribution in some basis (Eldar and Harrow derived a
similar result in order to characterize the output of low-depth quantum
circuits). Going further, we relate the Hilbert space localization properties
of the ground state to higher energy eigenvalues by showing that the presence
of k strongly localized ground state modes (i.e. clusters of probability, or
subsets with small expansion) in Hilbert space implies the presence of k energy
eigenvalues that are close to the ground state energy. Our results suggest that
quantum adiabatic optimization using local Hamiltonians will inevitably
encounter small spectral gaps when attempting to prepare ground states
corresponding to multi-modal probability distributions with strongly localized
modes, and this problem cannot necessarily be alleviated with the inclusion of
non-stoquastic couplings