Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

Abstract

Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the central issues in quantum computation, namely designing new quantum algorithms and constructing fault tolerant quantum computers. In particular, by translating the main open questions in the area of quantum algorithms to the language of spectral gaps of sparse matrices, the result makes these questions accessible to a wider scientific audience, acquainted with mathematical physics, expander theory and rapidly mixing Markov chains.

Thermally assisted quantum annealing of a 16-qubit problem

“…The error bars depict twice the s.e.m., assuming each sample is independent. (b) Total time required to achieve P GM ¼ 0.99 by annealing multiple times, as a function of the anneal time t f , computed using equation (3). In both (a and b), temperatures are coloured as in Fig.…”

Minor-embedding in adiabatic quantum computation: I. The parameter setting problem

“…Recall that according to the adiabatic theorem, the running time of an adiabatic algorithm depends on the minimum spectral gap of the system Hamiltonian, which however might be as hard as solving the original problem. Despite several serious investigations, the power of AQC remains an open question [24,25,22,16,1]. How does the embedding reduction effect the time complexity of an adiabatic algorithm?…”

“…Farhi et al in their original paper [15] showed that AQC can be efficiently simulated by standard quantum computers. In 2004, Aharonov et al [1] proved the more difficult converse statement that standard quantum computation can be efficiently simulated by AQC. In this paper, however, we will focus on a subclass of Hamiltonians, known as Ising models in a transverse field, that are NP-hard but not universal for quantum computation.…”

“…In particular, there is a degree-constraint in that each qubit can have at most a constant number of couplers dictated by hardware design. Therefore, besides the possible difficulty of the subgraph-embedding problem 1 , the graphs that can be solved on a given hardware graph U through subgraph-embedding must also be degree-bounded. Kaminsky et al [17,18] observed and proposed that one can embed G in U through ferromagnetic coupling "dummy vertices" to solve Maximum Independent Set (MIS) problem 2 of planar cubic graphs (regular graphs of degree-3) 3 on an adiabatic quantum computer.…”

Consistency of Local Density Matrices Is QMA-Complete

“…For instance, the fact that Local Hamiltonian is QMA-complete [2,3,4,5,6] is closely related to the fact that adiabatic quantum computation is equivalent to the standard quantum circuit model [7]. Other QMA-complete problems such as Identity Check involve properties of quantum circuits [8].…”