9 research outputs found
Anderson localization casts clouds over adiabatic quantum optimization
Understanding NP-complete problems is a central topic in computer science.
This is why adiabatic quantum optimization has attracted so much attention, as
it provided a new approach to tackle NP-complete problems using a quantum
computer. The efficiency of this approach is limited by small spectral gaps
between the ground and excited states of the quantum computer's Hamiltonian. We
show that the statistics of the gaps can be analyzed in a novel way, borrowed
from the study of quantum disordered systems in statistical mechanics. It turns
out that due to a phenomenon similar to Anderson localization, exponentially
small gaps appear close to the end of the adiabatic algorithm for large random
instances of NP-complete problems. This implies that unfortunately, adiabatic
quantum optimization fails: the system gets trapped in one of the numerous
local minima.Comment: 14 pages, 4 figure
New Developments in Quantum Algorithms
In this survey, we describe two recent developments in quantum algorithms.
The first new development is a quantum algorithm for evaluating a Boolean
formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This
provides quantum speedups for any problem that can be expressed via Boolean
formulas. This result can be also extended to span problems, a generalization
of Boolean formulas. This provides an optimal quantum algorithm for any Boolean
function in the black-box query model.
The second new development is a quantum algorithm for solving systems of
linear equations. In contrast with traditional algorithms that run in time
O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in
time O(\log^c N). It outputs a quantum state describing the solution of the
system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201
Unstructured Randomness, Small Gaps and Localization
We study the Hamiltonian associated with the quantum adiabatic algorithm with
a random cost function. Because the cost function lacks structure we can prove
results about the ground state. We find the ground state energy as the number
of bits goes to infinity, show that the minimum gap goes to zero exponentially
quickly, and we see a localization transition. We prove that there are no
levels approaching the ground state near the end of the evolution. We do not
know which features of this model are shared by a quantum adiabatic algorithm
applied to random instances of satisfiability since despite being random they
do have bit structure
Quantum annealing: the fastest route to quantum computation?
In this review we consider the performance of the quantum adiabatic algorithm
for the solution of decision problems. We divide the possible failure
mechanisms into two sets: small gaps due to quantum phase transitions and small
gaps due to avoided crossings inside a phase. We argue that the thermodynamic
order of the phase transitions is not predictive of the scaling of the gap with
the system size. On the contrary, we also argue that, if the phase surrounding
the problem Hamiltonian is a Many-Body Localized (MBL) phase, the gaps are
going to be typically exponentially small and that this follows naturally from
the existence of local integrals of motion in the MBL phase.Comment: 16 pages, 1 figur
Case studies in quantum adiabatic optimization
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 139-143).Quantum adiabatic optimization is a quantum algorithm for solving classical optimization problems (E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106). The solution to an optimization problem is encoded in the ground state of a "problem Hamiltonian" Hp which acts on the Hilbert space of n spin 1/2 particles and is diagonal in the Pauli z basis. To produce this ground state, one first initializes the quantum system in the ground state of a different Hamiltonian and then adiabatically changes the Hamiltonian into Hp. Farhi et al suggest the interpolating Hamiltonian [mathematical formula] ... where the parameter s is slowly changed as a function of time between 0 and 1. The running time of this algorithm is related to the minimum spectral gap of H(s) for s E (0, 11. We study such transverse field spin Hamiltonians using both analytic and numerical techniques. Our approach is example-based, that is, we study some specific choices for the problem Hamiltonian Hp which illustrate the breadth of phenomena which can occur. We present I A random ensemble of 3SAT instances which this algorithm does not solve efficiently. For these instances H(s) has a small eigenvalue gap at a value s* which approaches 1 as n - oc. II Theorems concerning the interpolating Hamiltonian when Hp is "scrambled" by conjugating with a random permutation matrix. III Results pertaining to phase transitions that occur as a function of the transverse field. IV A new quantum monte carlo method which can be used to compute ground state properties of such quantum systems. We discuss the implications of our results for the performance of quantum adiabatic optimization algorithms.by David Gosset.Ph.D