Spectral Gap Optimization for Enhanced Adiabatic State Preparation

TL;DR

The paper tackles the bottleneck of adiabatic state preparation arising from the minimal spectral gap along a Hamiltonian path. It proposes maximizing the gap for tensor-network states by optimizing the local terms of the parent Hamiltonian, leveraging injectivity and symmetry to keep a fixed locality. An explicit ordinary differential equation governs the evolution of the optimal Hamiltonian parameters $S_{ ext{opt}}(\lambda)$ along a tensor deformation $A(\lambda)$, and the method is demonstrated for random injective MPS, the AKLT state, and the GHZ state, with symmetry sectors providing further gains. The approach promises reduced adiabatic runtimes for TN-state preparation and offers a pathway to scalable control in higher dimensions, including extensions via lower-bounded gap estimates when exact spectral data are unavailable.

Abstract

The preparation of non-trivial states is crucial to the study of quantum many-body physics. Such states can be prepared with adiabatic quantum algorithms, which are restricted by the minimum spectral gap along the path. In this letter, we propose an efficient method to adiabatically prepare tensor networks states (TNSs). We maximize the spectral gap leveraging degrees of freedom in the parent Hamiltonian construction. We demonstrate this efficient adiabatic algorithm for preparing TNS, through examples of random TNS in one dimension, AKLT, and GHZ states. The Hamiltonian optimization applies to both injective and non-injective tensors, in the latter case by exploiting symmetries present in the tensors.

Figures (9)

• Figure 1: Sketch of the optimization algorithm. The blue path is the optimized path through the space of Hamiltonians, i.e. it has the maximal gap. The convex optimization in the parameters $\boldsymbol{S}$ is illustrated by grey lines. For comparison, the purple curve is an initial, non-optimal guess for a path. The actual gap along the path is visualized as a projection on the light-grey $\Delta-\lambda$ plane.
• Figure 2: Gap optimization for the preparation of many random injective MPS on $N=24$ sites. In all cases, the gap along the path changes from exponentially decaying ($\Delta_{\mathrm{canonical}}$) to roughly constant ($\Delta_{\mathrm{optimized}}$). In the inset, we show the improvement in the gap at $\lambda=1$ for random seeds and system sizes from $N=16$ to $N=40$.
• Figure 3: Depiction of AKLT state preparation. (a) $N=30$ sites: the blue dots show gap of canonical Hamiltonian with $S_{11}=S_{22}=0.2$ in Eq. \ref{['eq:aklt_S_mat']} and orange dots show the gap after intermediate optimization. Dashed lines indicate the minimum gap along each path. (b) $N=8$ sites: preserving symmetries enhances the gap. The labels full and symm refer to the full Hilbert space and symmetric subspace with the ground state respectively. $\Delta_{\mathrm{can}}$ and $\Delta_{\mathrm{opt}}$ denote canonical and optimized gaps respectively.
• Figure 4: (a) Illustration of gaps along the canonical $(\mathrm{can})$ and optimized $(\mathrm{opt})$ adiabatic path for $N=8$ sites sweeping from product state $(\lambda=-1)$ to cluster state $(\lambda=1)$, crossing the GHZ state at $(\lambda=0)$; $\mathrm{full}$ refers to the full Hilbert space, $\mathrm{symm}$ to the symmetric subspace with the ground state. (b) Minimum gaps along the path.
• Figure S5: Preparation of random injective MPS: first prepare entangled pairs and then apply positive maps $\mathcal{P}$ on adjacent sites from two consecutive pairs. The map $\mathcal{P}$ is parametrized as the exponential of a random Hermitian matrix, giving a positive matrix $\mathcal{P}$, thus ensuring injectivity.

Theorems & Definitions (1)

• Lemma 1