2,660 research outputs found
Quantum Adiabatic Algorithms, Small Gaps, and Different Paths
We construct a set of instances of 3SAT which are not solved efficiently using the simplestquantum adiabatic algorithm. These instances are obtained by picking randomclauses all consistent with two disparate planted solutions and then penalizing one ofthem with a single additional clause. We argue that by randomly modifying the beginningHamiltonian, one obtains (with substantial probability) an adiabatic path thatremoves this difficulty. This suggests that the quantum adiabatic algorithm should ingeneral be run on each instance with many different random paths leading to the problemHamiltonian. We do not know whether this trick will help for a random instance of3SAT (as opposed to an instance from the particular set we consider), especially if theinstance has an exponential number of disparate assignments that violate few clauses.We use a continuous imaginary time Quantum Monte Carlo algorithm in a novel way tonumerically investigate the ground state as well as the first excited state of our system.Our arguments are supplemented by Quantum Monte Carlo data from simulations withup to 150 spins.United States. Dept. of Energy (Cooperative Research Agreement DE-FG02-94ER40818)W. M. Keck Foundation Center for Extreme Quantum Information TheoryU.S. Army Research Laboratory (Grant W911NF-09-1-0438)National Science Foundation (U.S.) (Grant CCF-0829421
Quantum Adiabatic Algorithms, Small Gaps, and Different Paths
We construct a set of instances of 3SAT which are not solved efficiently
using the simplest quantum adiabatic algorithm. These instances are obtained by
picking random clauses all consistent with two disparate planted solutions and
then penalizing one of them with a single additional clause. We argue that by
randomly modifying the beginning Hamiltonian, one obtains (with substantial
probability) an adiabatic path that removes this difficulty. This suggests that
the quantum adiabatic algorithm should in general be run on each instance with
many different random paths leading to the problem Hamiltonian. We do not know
whether this trick will help for a random instance of 3SAT (as opposed to an
instance from the particular set we consider), especially if the instance has
an exponential number of disparate assignments that violate few clauses. We use
a continuous imaginary time Quantum Monte Carlo algorithm in a novel way to
numerically investigate the ground state as well as the first excited state of
our system. Our arguments are supplemented by Quantum Monte Carlo data from
simulations with up to 150 spins.Comment: The original version considered a unique satisfying assignment and
one problematic low lying state. The revision argues that the algorithm with
path change will succeed when there are polynomially many low lying state
Adiabatic optimization without local minima
Several previous works have investigated the circumstances under which
quantum adiabatic optimization algorithms can tunnel out of local energy minima
that trap simulated annealing or other classical local search algorithms. Here
we investigate the even more basic question of whether adiabatic optimization
algorithms always succeed in polynomial time for trivial optimization problems
in which there are no local energy minima other than the global minimum.
Surprisingly, we find a counterexample in which the potential is a single basin
on a graph, but the eigenvalue gap is exponentially small as a function of the
number of vertices. In this counterexample, the ground state wavefunction
consists of two "lobes" separated by a region of exponentially small amplitude.
Conversely, we prove if the ground state wavefunction is single-peaked then the
eigenvalue gap scales at worst as one over the square of the number of
vertices.Comment: 20 pages, 1 figure. Journal versio
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs
In this paper we study the performance of the quantum adiabatic algorithm on
random instances of two combinatorial optimization problems, 3-regular 3-XORSAT
and 3-regular Max-Cut. The cost functions associated with these two
clause-based optimization problems are similar as they are both defined on
3-regular hypergraphs. For 3-regular 3-XORSAT the clauses contain three
variables and for 3-regular Max-Cut the clauses contain two variables. The
quantum adiabatic algorithms we study for these two problems use interpolating
Hamiltonians which are stoquastic and therefore amenable to sign-problem free
quantum Monte Carlo and quantum cavity methods. Using these techniques we find
that the quantum adiabatic algorithm fails to solve either of these problems
efficiently, although for different reasons.Comment: 20 pages, 15 figure
Fast Quantum Methods for Optimization
Discrete combinatorial optimization consists in finding the optimal
configuration that minimizes a given discrete objective function. An
interpretation of such a function as the energy of a classical system allows us
to reduce the optimization problem into the preparation of a low-temperature
thermal state of the system. Motivated by the quantum annealing method, we
present three strategies to prepare the low-temperature state that exploit
quantum mechanics in remarkable ways. We focus on implementations without
uncontrolled errors induced by the environment. This allows us to rigorously
prove a quantum advantage. The first strategy uses a classical-to-quantum
mapping, where the equilibrium properties of a classical system in spatial
dimensions can be determined from the ground state properties of a quantum
system also in spatial dimensions. We show how such a ground state can be
prepared by means of quantum annealing, including quantum adiabatic evolutions.
This mapping also allows us to unveil some fundamental relations between
simulated and quantum annealing. The second strategy builds upon the first one
and introduces a technique called spectral gap amplification to reduce the time
required to prepare the same quantum state adiabatically. If implemented on a
quantum device that exploits quantum coherence, this strategy leads to a
quadratic improvement in complexity over the well-known bound of the classical
simulated annealing method. The third strategy is not purely adiabatic;
instead, it exploits diabatic processes between the low-energy states of the
corresponding quantum system. For some problems it results in an exponential
speedup (in the oracle model) over the best classical algorithms.Comment: 15 pages (2 figures
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