Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians

TL;DR

This paper extends Kitaev's Hamiltonian construction to include non-unitary evolution and post-selection, exploring the complexity and spectral gap implications of these modifications in quantum Hamiltonian problems.

Contribution

It generalizes Kitaev's construction to non-unitary and post-selected quantum processes, analyzing their impact on computational complexity and spectral gaps.

Findings

Post-selection probability can be exponentially small while the spectral gap decreases polynomially.

The generalized construction remains consistent under tame post-selection.

Numerical evidence shows no immediate relation between post-selection probability and spectral gap size.

Abstract

A central result in the study of Quantum Hamiltonian Complexity is that the k-Local hamiltonian problem is QMA-complete. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian is bounded below some value, or above another, promised one of these is true. Given the ground state of the Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently quantum preparable. Kitaev's proof of QMA-completeness encodes a unitary quantum circuit in QMA into the ground space of a Hamiltonian. However, we now have quantum computing models based on measurement instead of unitary evolution, furthermore we can use post-selected measurement as an additional computational tool. In this work, we generalise Kitaev's construction to allow for non-unitary evolution including post-selection. Furthermore, we consider a type of post-selection under which the construction is consistent, which we call tame post-selection. We consider the computational complexity consequences of this construction and then consider how the probability of an event upon which we are post-selecting affects the gap between the ground state energy and the energy of the first excited state of its corresponding Hamiltonian. We provide numerical evidence that the two are not immediately related, by giving a family of circuits where the probability of an event upon which we post-select is exponentially small, but the gap in the energy levels of the Hamiltonian decreases as a polynomial.

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