3,712 research outputs found
For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances
The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth
quantum circuit to produce a parameter dependent state. For a given
combinatorial optimization problem instance, the quantum expectation of the
associated cost function is the parameter dependent objective function of the
QAOA. We demonstrate that if the parameters are fixed and the instance comes
from a reasonable distribution then the objective function value is
concentrated in the sense that typical instances have (nearly) the same value
of the objective function. This applies not just for optimal parameters as the
whole landscape is instance independent. We can prove this is true for low
depth quantum circuits for instances of MaxCut on large 3-regular graphs. Our
results generalize beyond this example. We support the arguments with numerical
examples that show remarkable concentration. For higher depth circuits the
numerics also show concentration and we argue for this using the Law of Large
Numbers. We also observe by simulation that if we find parameters which result
in good performance at say 10 bits these same parameters result in good
performance at say 24 bits. These findings suggest ways to run the QAOA that
reduce or eliminate the use of the outer loop optimization and may allow us to
find good solutions with fewer calls to the quantum computer.Comment: 16 pages, 1 figur
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
A quantum system will stay near its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough. This quantum
adiabatic behavior is the basis of a new class of algorithms for quantum
computing. We test one such algorithm by applying it to randomly generated,
hard, instances of an NP-complete problem. For the small examples that we can
simulate, the quantum adiabatic algorithm works well, and provides evidence
that quantum computers (if large ones can be built) may be able to outperform
ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a
shorter version of this article appeared in the April 20, 2001 issue of
Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47
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