Experimental Implementation of an Adiabatic Quantum Optimization Algorithm

Abstract

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.Since the discovery of Shor's[1] and Grover's [2] algorithms, the quest of finding new quantum algorithms proved a formidable challenge. Recently however, a novel algorithm was proposed, using adiabatic evolution [3,4]. Despite the uncertainty in its scaling behavior, this algorithm remains a remarkable discovery because it offers new insights into the potential usefulness of quantum resources for computational tasks.Experimental realizations of quantum algorithms in the past demonstrated Grover's search algorithm [5,6], the Deutsch-Jozsa algorithm [7,8,9], order-finding [10], and Shor's algorithm [11]. Recently, Hogg's algorithm was implemented using only one computational step [12], however a demonstration of an adiabatic quantum algorithm thus far has remained beyond reach.Here, we provide the first experimental implementation of an adiabatic quantum optimization algorithm using three qubits and nuclear magnetic resonance (NMR) techniques [13,14]. NMR techniques are especially attractive because several tens of qubits may be accessible, which is precisely the range that could be crucial in determining the scaling behavior of adiabatic quantum algorithms [15]. Compared to earlier implementations of search problems [5,6], this experiment is a full implementation of a true optimization problem, which does not require a black box function or ancilla bits.This experiment was made possible by overcoming two experimental challenges. First, an adiabatic evolution requires a smoothly varying Hamiltonian over time, but the terms of the available Hamiltonian in our system cannot be smoothly varied and may even have fixed values. We developed a method to approximately smoothly vary a Hamiltonian despite the given restrictions by extending NMR average Hamiltonian techniques [16]. Second, general instances of the optimization algorithm may require the application of Hamiltonians that are not easily accessible. We developed methods to implement general instances of a well known classical NP-complete optimization problem given a fixed natural system Hamiltonian.We provide a concrete procedure detailing these methods. We then apply the results to our optimization problem which is known as Maximum Cut or maxcut [17]. Our experimental results indicate there exists an optimal total running time which can be predicted using a decoherence model that is based on independent stochastic relaxation of the spins. The evolut...

Colloquium: Quantum annealing and analog quantum computation

“…1). An experimental realization of quantum adiabatic annealing for 3-bit instances of MAXCUT problem using NMR technique has also been reported (see Steffen et al 2003). Here, the the smoothly varying time dependent Hamiltonian was realized by the technique of quantum simulation (see Nielsen and Chuang 2000), where a smooth time evolution was achieved approximately (Trotter approximation) through the application of a series of discrete unitary operations.…”

Thermally assisted quantum annealing of a 16-qubit problem

“…QA has been experimentally demonstrated in non-programmable bulk solid state systems 19,20 and also using 3-qubit nuclear magnetic resonance 21 . Recently, we have designed, fabricated and tested a programmable superconducting QA processor 22,23 .…”

Ramsey Numbers and Adiabatic Quantum Computing

“…For a given Hamiltonian H(t), two approaches have been demonstrated to experimentally implement AQE [19][20][21]. Refs.…”

“…Refs. [19], [20] partitioned the full evolution into N subintervals of duration ∆t = T /N which are sufficently short that the propagator U l for each subinterval l can be factored via a Trotter expansion. This approach was applied to three-qubit systems, though it can be used for arbitrary size qubit systems.…”