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arXiv:quant-ph/0302057 · Physical Review Letters

Experimental Implementation of an Adiabatic Quantum Optimization Algorithm

Matthias Steffen, Wim van Dam, Tad Hogg, Gregory Breyta, Isaac L. Chuang

Published Feb 14, 2003• 182 citations
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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.

DOI
10.1103/physrevlett.90.067903

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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.
Adiabatic quantum computation is of fundamental importance in the field of quantum computation as it offers an alternative approach to the gate-based model for the manipulation of quantum systems. Recently, an interesting work [arXiv:1805.10549] indicated that we can solve a linear equation system via an algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm in a four-qubit nuclear magnetic resonance system by determining the solution of an eight-dimensional linear equation Ax=bA\mathbf{x}=\mathbf{b}Ax=b. The result is by far the maximum-dimensional linear equation solution with a limited number of qubits in experiments, which include some ingenious simplifications. Our experiment provides the possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.
Quantum computers can potentially solve real and relevant mathematical and physical problems that are intractable using classical machines. However, the experimental realization of quantum computers represents a significant challenge, because several opposing experimental requirements must be met. A set of coupled quantum bits must be manipulated and measured while coherently retaining entangled quantum states. Yet the manipulation and measurement processes almost inevitably lead to the decay of these fragile states. This thesis work takes significant steps towards building a practical quantum computer using nuclear spins and liquid state nuclear magnetic resonance (NMR) techniques. I present experimental results for proof of principle of quantum computing in a series of small implementations of quantum algorithms, culminating in the implementation of an adiabatic quantum optimization algorithm. The performance of adiabatic algorithms compared with classical optimization methods is unknown, but small quantum computers could provide crucial insight into answering this question. Furthermore, adiabatic algorithms also shed new light on the usefulness of quantum resources for computational tasks and hence they represent an important class of algorithms. The second part of this thesis presents several methods developed for improving the control over NMR quantum computing experiments. Even though liquid state NMR quantum computers have well known and accepted scaling limitations, the developed tools are of general use. I will show how several tools are directly transferable to two other implementations of quantum computers; one using optical methods and the other using loops of superconducting material. This work provides insight into what is needed to build a practical quantum computer, and delivers several tools and techniques that will be useful in future large-scale implementations of quantum computers.
Having the advantages of inherent robustness and fault tolerance, adiabatic quantum algorithms have attracted much attention in recent years. In this paper, we report the NMR experimental realization of adiabatic algorithms for the modified Simon's problem in a NMR quantum information processor. We first realized an adiabatic quantum algorithm by Rao [Phys. Rev. A 67, 052306 (2003)] and then proposed a quantum algorithm by using the method of enhanced symmetry of the Hamiltonian given in Schaller and Schtzhold [Quantum Inf. Comput. 10, 109 (2010)]. We demonstrated this symmetry-enhanced algorithm in experiment. We studied the effects of the number of Trotter evolution steps on the validity of adiabatic evolution and decoherence. In practical applications, a trade-off between pulse imperfection and adiabatic requirement must be made in the Trotter evolution step number. An optimal evolution step number and run time were indicated by our results. The experimental demonstration also showed that the symmetry-enhanced algorithm gave a better performance.
We report the realization of a nuclear magnetic resonance quantum computer which combines the quantum Fourier transform with exponentiated permutations, demonstrating a quantum algorithm for order finding. This algorithm has the same structure as Shor's algorithm and its speed-up over classical algorithms scales exponentially. The implementation uses a particularly well-suited five quantum bit molecule and was made possible by a new state initialization procedure and several quantum control techniques.
A quantum computer is a device that processes information in a quantum-mechanically coherent fashion. In principle, it can exploit coherent quantum interference and entanglement to perform computations, such as factoring large numbers or searching an unsorted database, more rapidly than classical computers. Noise, decoherence, and manufacturing problems make constructing large scale quantum computers difficult. Ion traps and optical cavities offer promising experimental approaches, but no quantum algorithm has yet been implemented with those systems. On the other hand, because of their natural isolation from the environment, nuclear spins are particularly good 'quantum bits', and their use for quantum computation is possible by applying nuclear magnetic resonance (NMR) techniques in an unconventional manner. Here, we report on the experimental implementation of a quantum algorithm using NMR to solve a purely mathematical problem in fewer steps than is possible classically. Our simple quantum computer can determine the type of an unknown function using fewer function 'calls' than is possible using a classical computer.
The present work, briefly described below, consists of implementation of several quantum algorithms in an NMR Quantum Information Processor. Game theory gives us mathematical tools to analyze situations of conflict between two or more players who take decisions that influence their welfare. Classical game theory has been applied to various fields such as market strategy, communication theory, biological processes, foreign policies. It is interesting to study the behaviour of the games when the players share certain quantum correlations such as entanglement. Various games have been studied under the quantum regime with the hope of obtaining some insight into designing new quantum algorithms. Chapter 2 presents the NMR implementation of three such algorithms. Experimental NMR implementation given in this chapter are: (i) Three qubit ‘Dilemma’ game with corrupt sources’. The Dilemma game deals with the situation where three players have to choose between going/not going to a bar with a seating capacity of two. It is seen that in the players have a higher payoff if they share quantum correlations. However, the pay-off falls rapidly with increasing corruption in the source qubits. Here we report the experimental NMR implementation of the quantum version of the Dilemma game with and without corruption in the source qubits. (ii) Two qubit ‘Ulam’s game’. This is a two player game where one player has to find out the binary number thought by the other player. This problem can be solved with one query if quantum resources are used. This game has been implemented in a two qubit system in an NMR quantum information processor. (iii) Two qubit ‘Battle of Sexes’ game. This game deal with a situation where two players have conflicting choices but a deep desire to be together. This leads to a dilemma in the classical case. Quantum mechanically this dilemma is resolved and a unique solution emerges. The NMR implementation of the quantum version of this game is also given in this chapter. Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover’s search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, if one uses local adiabatic evolutions, their complexity is of the order √N (where N=2n) [37, 38]. In Chapter 3, the NMR implementation of (i) the Deutsch-Jozsa and the (ii) Grover’s search algorithm using local adiabatic evolution has been presented. In adiabatic algorithm, the system is first prepared in the equal superposition of all the possible states which is the ground state of the beginning Hamiltonian. The solution is encoded in the ground state of the final Hamiltonian. The system is evolved under a linear combination of the beginning and the final Hamiltonian. During each step of the…