On Preparing Ground States of Gapped Hamiltonians: An Efficient Quantum Lovász Local Lemma

Abstract

Abstract-A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) -the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science.The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Moser's "compression argument" which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the "compression argument", so that it provides a simplified version of the previous quantum results, and improves on some classical results as well.More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is "uniformly" gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms. I. INTRODUCTION Frustration-free Hamiltonians and quantum satisfiability:Most physical systems and models are described by a local Hamiltonian H = i H i where each k-local term H i acts non-trivially only on at most k of its subsystems. Such a Hamiltonian is called frustration-free if its ground state is also the ground state of each of the local terms H i . Frustration-free Hamiltonians appear in various areas, for example: quantum error correcting codes [1], parent Hamiltonians for PEPS (a 2-D generalisation of matrixproduct-states) [2], and various models in many-body quantum physics.

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

“…This improves the ε dependence exponentially and improves the ς dependence quadratically, while solves a qualitatively stronger problem. These improvements also greatly improve the final complexity of the main algorithm presented in [GS17].…”

“…The singular value decomposition is not unique if there are multiple singular values with the same value. However, the singular value projectors are uniquely determined, see, e.g., Gilyén and Sattath [GS17].…”

“…As shown by Sattath and Arad [SA15] and Schwarz et al [SCV13] under same conditions the problem becomes efficiently solvable when the local Hamiltonian terms commute. The non-commuting case is more difficult, but under a gappromise Gilyén and Sattath [GS17] showed that a ground state can be efficiently prepared.…”

“…Gilyén and Sattath [GS17] essentially reduced the state preparation problem to solving the following task: Given two (non-commuting) orthogonal projectors Π F and Π c and quantum state |ψ ∈ img Π F perform a "non-commutative" measurement in the following sense. If |ψ ∈ img Π F ∩ ker(Π c ) then output 0 and leave the state intact, otherwise if |ψ is a right singular vector of Π c Π F with singular value greater than 0, then output 1 and "rotate" the state |ψ to the corresponding left singular vector of Π c Π F which in turn lies in img(Π c ).…”

“…Gilyén and Sattath [GS17] called exact quantum channel the procedure which solves the above problem, but also preserves coherence between singular vectors with different singular values. In their paper it was unclear how to efficiently implement such a "non-commutative" measurement.…”