Nonunitary quantum computation in the ground space of local Hamiltonians

Abstract

A central result in the study of Quantum Hamiltonian Complexity is that the k-local hamiltonian problem is QMA-complete [1]. 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.

Translationally Invariant Universal Quantum Hamiltonians in 1D

“…Note that U t need not necessarily be a gate in the quantum circuit model. It could also, e.g., be one time-step of a quantum Turing machine, or even a time-step in some more exotic model of quantum computation [12], or an isometry [13]. In the particular constructions, we make use of in this work, U t will be a time-step of a quantum Turing machine.…”

Analysis and limitations of modified circuit-to-Hamiltonian constructions

“…In the context of universal adiabatic computation Ganti and Somma [22] derived a limitation on improving the spectral gap of circuit Hamiltonians by applying the lower bound on the Gover search problem. In [16], the authors go beyond unitary circuit include certain projective measurements {Π, 1−Π} (ones whose outcome probabilities are independent of all valid input state at that computational step).…”

“…Based on ideas by Feynman [9] and cast into its current form by Kitaev [1], the construction remains relatively little changed, having undergone only some gradual evolutions since its modern introduction [6,[10][11][12][13][14][15]. Only recently steps have been undertaken to analyse modifications in depth [16], and revisit the spectral properties of Kitaev's original construction [17].…”

Unextendible product operator basis

“…For example, unextendible product bases (UPBs) are locally indistinguishable [16,17]. For operators, a series of them are the bases of quantum computation or algorithms [18][19][20]. It has been proved that any discrete finite-dimensional unitary operators can be constructed in the laboratory using optical devices [21].…”