115,260 research outputs found
Quantum Optimization Problems
Krentel [J. Comput. System. Sci., 36, pp.490--509] presented a framework for
an NP optimization problem that searches an optimal value among
exponentially-many outcomes of polynomial-time computations. This paper expands
his framework to a quantum optimization problem using polynomial-time quantum
computations and introduces the notion of an ``universal'' quantum optimization
problem similar to a classical ``complete'' optimization problem. We exhibit a
canonical quantum optimization problem that is universal for the class of
polynomial-time quantum optimization problems. We show in a certain relativized
world that all quantum optimization problems cannot be approximated closely by
quantum polynomial-time computations. We also study the complexity of quantum
optimization problems in connection to well-known complexity classes.Comment: date change
Approximating random quantum optimization problems
We report a cluster of results regarding the difficulty of finding
approximate ground states to typical instances of the quantum satisfiability
problem -QSAT on large random graphs. As an approximation strategy, we
optimize the solution space over `classical' product states, which in turn
introduces a novel autonomous classical optimization problem, PSAT, over a
space of continuous degrees of freedom rather than discrete bits. Our central
results are: (i) The derivation of a set of bounds and approximations in
various limits of the problem, several of which we believe may be amenable to a
rigorous treatment. (ii) A demonstration that an approximation based on a
greedy algorithm borrowed from the study of frustrated magnetism performs well
over a wide range in parameter space, and its performance reflects structure of
the solution space of random -QSAT. Simulated annealing exhibits
metastability in similar `hard' regions of parameter space. (iii) A
generalization of belief propagation algorithms introduced for classical
problems to the case of continuous spins. This yields both approximate
solutions, as well as insights into the free energy `landscape' of the
approximation problem, including a so-called dynamical transition near the
satisfiability threshold. Taken together, these results allow us to elucidate
the phase diagram of random -QSAT in a two-dimensional
energy-density--clause-density space.Comment: 14 pages, 9 figure
Readiness of Quantum Optimization Machines for Industrial Applications
There have been multiple attempts to demonstrate that quantum annealing and,
in particular, quantum annealing on quantum annealing machines, has the
potential to outperform current classical optimization algorithms implemented
on CMOS technologies. The benchmarking of these devices has been controversial.
Initially, random spin-glass problems were used, however, these were quickly
shown to be not well suited to detect any quantum speedup. Subsequently,
benchmarking shifted to carefully crafted synthetic problems designed to
highlight the quantum nature of the hardware while (often) ensuring that
classical optimization techniques do not perform well on them. Even worse, to
date a true sign of improved scaling with the number of problem variables
remains elusive when compared to classical optimization techniques. Here, we
analyze the readiness of quantum annealing machines for real-world application
problems. These are typically not random and have an underlying structure that
is hard to capture in synthetic benchmarks, thus posing unexpected challenges
for optimization techniques, both classical and quantum alike. We present a
comprehensive computational scaling analysis of fault diagnosis in digital
circuits, considering architectures beyond D-wave quantum annealers. We find
that the instances generated from real data in multiplier circuits are harder
than other representative random spin-glass benchmarks with a comparable number
of variables. Although our results show that transverse-field quantum annealing
is outperformed by state-of-the-art classical optimization algorithms, these
benchmark instances are hard and small in the size of the input, therefore
representing the first industrial application ideally suited for testing
near-term quantum annealers and other quantum algorithmic strategies for
optimization problems.Comment: 22 pages, 12 figures. Content updated according to Phys. Rev. Applied
versio
Efficiency of quantum versus classical annealing in non-convex learning problems
Quantum annealers aim at solving non-convex optimization problems by
exploiting cooperative tunneling effects to escape local minima. The underlying
idea consists in designing a classical energy function whose ground states are
the sought optimal solutions of the original optimization problem and add a
controllable quantum transverse field to generate tunneling processes. A key
challenge is to identify classes of non-convex optimization problems for which
quantum annealing remains efficient while thermal annealing fails. We show that
this happens for a wide class of problems which are central to machine
learning. Their energy landscapes is dominated by local minima that cause
exponential slow down of classical thermal annealers while simulated quantum
annealing converges efficiently to rare dense regions of optimal solutions.Comment: 31 pages, 10 figure
Multistart Methods for Quantum Approximate Optimization
Hybrid quantum-classical algorithms such as the quantum approximate
optimization algorithm (QAOA) are considered one of the most promising
approaches for leveraging near-term quantum computers for practical
applications. Such algorithms are often implemented in a variational form,
combining classical optimization methods with a quantum machine to find
parameters to maximize performance. The quality of the QAOA solution depends
heavily on quality of the parameters produced by the classical optimizer.
Moreover, the presence of multiple local optima in the space of parameters
makes it harder for the classical optimizer. In this paper we study the use of
a multistart optimization approach within a QAOA framework to improve the
performance of quantum machines on important graph clustering problems. We also
demonstrate that reusing the optimal parameters from similar problems can
improve the performance of classical optimization methods, expanding on similar
results for MAXCUT
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
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