Ground state preparation with shallow variational warm-start

Abstract

Preparing the ground states of a many-body system is essential for evaluating physical quantities and determining the properties of materials. This work provides a quantum ground state preparation scheme with shallow variational warm-start to tackle the bottlenecks of current algorithms, i.e., demand for prior ground state energy information and lack of demonstration of efficient initial state preparation. Particularly, our methods would not experience the instability for small spectral gap ∆ during pre-encoding the phase factors since our methods involve only O(1) factors while O(∆ −1 ) is requested by the near-optimal methods. We demonstrate the effectiveness of our methods via extensive numerical simulations on spin-1/2 Heisenberg models. We also show that the shallow warm-start procedure can process chemical molecules by conducting numerical simulations on the hydrogen chain model. Moreover, we extend research on the Hubbard model, demonstrating superior performance compared to the prevalent variational quantum algorithms.

Resource-efficient quantum principal component analysis

“…In this case, the eigenvalue estimate is output randomly. The output probability is analyzed in [33], which uses a binary tree to explain the search procedure. In the tree, each node represents the timing of measurement, and each path from the root to the leaf corresponds to a bitstring composed of measurement outcomes.…”

“…In the tree, each node represents the timing of measurement, and each path from the root to the leaf corresponds to a bitstring composed of measurement outcomes. The discussion in [33] establishes that the probability of sampling all paths that a single eigenvector moves is at least the projection of the initial state onto the eigen-subspace spanned by this eigenvector, i.e. Pr τ j ⩾ |⟨χ j |ψ⟩| 2 .…”

“…f(x) = −1 for all x ∈ (−π + ∆, −∆) and f(x) = 1 for all x ∈ (∆, π − ∆). Then we could construct a binary tree using the method in [33]. Each path leads to an interval of size smaller than 2ϵ, given the estimation precision ϵ.…”

“…Each path leads to an interval of size smaller than 2ϵ, given the estimation precision ϵ. As discussed in [33], all paths that the eigenvector moves would lead to only one or two intervals. The output probability is the probability that the intervals are sampled, which is at least…”