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An Introduction to Quantum Computing for Non-Physicists Eleanor Rieffelrieffel@pal.xerox.com Wolfgang Polakpolak@pal.xerox.com August 14, 1998

Introduction

Richard Feynman observed in the early 1980’s Feynman-82 that certain quantum mechanical effects cannot be simulated efficiently on a classical computer. This observation led to speculation that perhaps computation in general could be done more efficiently if it made use of these quantum effects. But building quantum computers, computational machines that use such quantum mechanical effects, proved tricky, and as no one was sure how to use the quantum effects to speed up computation, the field developed slowly. It wasn’t until 1994, when Peter Shor surprised the world by describing a polynomial time quantum algorithm for factoring integers Shor-94,Shor-95, that the field of quantum computing came into its own. This discovery prompted a flurry of activity, both among experimentalists trying to build quantum computers and theoreticians trying to find other quantum algorithms. Additional interest in the subject has been created by the invention of quantum key distribution and, more recently, popular press accounts of experimental successes in quantum teleportation and the demonstration of a two-bit quantum computer.

The aim of this paper is to guide computer scientists and other non-physicists through the conceptual and notational barriers that separate quantum computing from conventional computing and to acquaint them with this new and exciting field. It is important for the computer science community to understand these new developments since they may radically change the way we have to think about computation, programming, and complexity.

The power of quantum computation comes from quantum parallelism. Classically, the time it takes to do certain computations can be decreased by using parallel processors. To achieve an exponential decrease in time requires an exponential increase in the number of processors, and hence an exponential increase in the amount of physical space needed. However, in quantum systems the amount of parallelism increases exponentially with the size of the system. Thus, an exponential increase in parallelism requires a linear increase in the amount of physical space needed.

There is a catch, and a big catch at that. While a quantum system can perform massive parallel computation, access to the results of the computation is restricted. Accessing the results is equivalent to making a measurement, which disturbs the quantum state. This problem makes the situation, on the face of it, seem even worse than the classical situation; we can only read the result of one parallel thread, and because measurement is probabilistic, we cannot even choose which one we get.

But in the past few years, various people have found clever ways of getting useful information out of quantum parallelism. One technique is to determine a common property of all of the output values such as the symmetry or period of a function. This is the technique used in Shor’s factorization algorithm. Another technique is to transform the state to increase the likelihood that the output of interest will be read. Grover’s search algorithm makes use of such an amplification technique. This paper describes the details of quantum parallelism, and the techniques currently known for harnessing its power.

The section following this introduction explains some of the basic concepts of quantum mechanics that are important for quantum computation. This section does not attempt to give a comprehensive view of quantum mechanics as this would be beyond the scope of this paper. Our aim is to provide the reader with tools in the form of mathematics and notation with which to work with quantum mechanics. We hope that this paper will equip computer scientists, and other non-physicists, well enough that they can freely explore the theoretical realm of quantum computing.

Section qubits defines the quantum bit, or qubit. Unlike classical bits, a quantum bit can be put in a superposition state that encodes both 0 and 1. There is no good classical explanation of superpositions: a quantum bit representing 0 and 1 can neither be viewed as “between” 0 and 1 nor can it be viewed as a hidden unknown state that represents either 0 or 1 with a certain probability. Even single quantum bits enable interesting applications. We show the use of a single quantum bit for secure key distribution.

But the real power of quantum computation derives from the exponential state spaces of multiple quantum bits: just as a single qubit can be in a superposition of 0 and 1, a register of n qubits can be in a superposition of all 2^{n} possible values. The famous EPREPR = Einstein, Podolsky and Rosenaradox (see section epr) is a result of entangled states that form a part of the quantum state space that do not exist in classical systems.

We will discuss the two types of operations a quantum system can undergo: measurement and quantum state transformations. Most quantum algorithms involve a sequence of quantum transformations followed by a measurement. For classical computers there are sets of gates that are universal in the sense that any classical computation can be performed using a sequence of these gates. Similarly, there are sets of primitive quantum transformations, called quantum gates, that are universal for quantum computation. Given enough quantum bits, it is possible to construct a universal quantum Turing machine.

Quantum physics puts restrictions on the types of transformations that can be done. In particular, all quantum transformations, and therefore all quantum gates and all quantum computations, must be reversible. Yet all classical algorithm can be computed on a quantum computer, i.e., in a manner that makes them reversible. Some common quantum gates are defined in section gates.

Two applications combining quantum gates and entangled states are described in section coding: teleportation and dense coding. Teleportation is the transfer of quantum state from one place to another through classical channels. That teleportation is possible is surprising since quantum mechanics tells us that it is not possible to clone quantum states or even measure them without disturbing the state. Thus, it is unclear what information could be sent through classical channels that could possibly enable the reconstruction of an unknown quantum state at the other end. Dense coding, a dual to teleportation, uses a single quantum bit to transmit two bits of classical information. Both teleportation and dense coding rely on the entangled states described in the EPR experiment.

It is only in section computer that we see where an exponential speed up over classical computers might come from. The input to a quantum computation can be put in a superposition state that encodes all possible input values. Performing the computation on this initial state will result in superposition of all corresponding output values. Thus, in the same time it takes to compute the output for a single input state on a classical computer, a quantum computer can compute the values for all input states. This process is known as quantum parallelism. However, measuring the output states will randomly yield only one of the values in the superposition, and at the same time destroy all of the other results of the computation. Section computer describes this situation in detail.

Section shor describes the details of Shor’s polynomial time factoring algorithm. The fastest known classical factoring algorithm requires exponential time and it is generally believed that there are no classical polynomial time factoring algorithms. Shor’s is a beautiful algorithm that takes advantage of quantum parallelism by using a quantum analog of the Fourier transform.

Lov Grover developed a technique for searching an unstructured list of n items in O(\sqrt{n}) steps on a quantum computer. Classical computers can do no better than O(n/2), so unstructured search on a quantum computer is provably more efficient than search on a classical computer. However, the speed-up is only polynomial, not exponential, and it has been shown that Grover’s algorithm is optimal for quantum computers. It seems likely that search algorithms that could take advantage of some problem structure could do better. Tad Hogg, among others, has explored such possibilities. We describe various quantum search techniques in section search.

It is as yet unknown whether the power of quantum parallelism can be harnessed for a wide variety of applications. One tantalizing open question is whether quantum computers can solve NP complete problems in polynomial time.

Perhaps the biggest open question is whether useful quantum computers can be built. There are a number of proposals for building quantum computers most of which are based either on ion traps or on nuclear magnetic resonance (NMR) technology.

In an ion trap quantum computer Cirac-Zoller-95,Steane-96b a linear sequence of ions (qubits) are confined by electric fields. Lasers are directed at individual atoms to perform single bit quantum gates. Two-bit operations are realized by using a laser on one qubit to create an impulse that ripples through a chain of ions to the second qubit where another laser pulse stops the rippling and performs the two-bit operation. The ion trap approach requires extreme vacuum and extremely low temperatures.

EPR = Einstein, Podolsky and Rosen p

The NMR approach has the major advantage that it will work at room temperature. The idea is to use macroscopic amounts of matter and encode a quantum bit in the average spin state of a large number of nuclei. The spin states can be manipulated by magnetic fields and the average spin state can be measured with NMR techniques. The main problem with the technique is that it doesn’t scale well; the measured signal scales as 1/2^{n} with the number of qubits n. However, a recent proposal (Schulman-Vazirani-98) has been made that may overcome this problem. NMR computers with two qubits have been built successfully Gershenfeld-Chuang-97,NMR-GHZ. This paper will not discuss further the physical and engineering problems of building quantum computers.

The greatest problem for building quantum computers is decoherence, the distortion of the quantum state due to interaction with the environment. For some time it was feared that quantum computers could not be built because it would be impossible to isolate them sufficiently from the external environment. The breakthrough came from the algorithmic rather than the physical side, through the invention of quantum error correction techniques. It is possible to design error correcting codes that tolerate certain kinds of errors and allow reconstruction of the exact error-free quantum state. Quantum error correction is discussed in section qec.

Appendices provide the necessary background information on tensor products and continued fractions. *Acknowledgements The authors would like to thank Tad Hogg and Carlos Mochon for many enjoyable conversations about

quantum computing, and for their feedback on an earlier draft of this paper. We are also grateful to for the detailed comments we received on an earlier draft from Lee Corbin, David Goldberg, and Norman Hardy. Finally, we would like to thank FXPAL for enthusiastically supporting this work.

Quantum Mechanics

Quantum phenomena are difficult to understand since most of our everyday experiences are not applicable. This paper cannot provide a deep understanding of quantum mechanics (see Feynman-65, GZ97, or Liboff for details)quantum mechanics. Instead, we will give some feeling as to the nature of quantum mechanics and some of the mathematical formalisms needed to work with quantum mechanics to the extent needed for quantum computing.

Quantum mechanicsquantum mechanics is a theory in the mathematical sense: it is governed by a set axioms. The consequences of the axioms describe the behavior of quantum systems. The axioms lead to several apparent paradoxes: in the Compton effectCompton effect it appears as if an action precedes its cause; the EPREPR experiment makes it appear as if action over a distance faster than the speed of light is possible. We will discuss the EPREPR experiment in detail in section epr. Verification of most predictions is indirect, and requires careful experimental design and specialized equipment. We will begin, however, with an experiment that requires only readily available equipment and that will illustrate some of the key aspects of quantum mechanics.

Photon Polarization

Photons are the only particlesparticles that we can directly observe. The following simple experiment can be performed with minimal equipment: a laser pointer (or other strong light source) and three polaroids (polarization filters) that can be picked up at any camera supply store. The experiment demonstrates some of the principles of quantum mechanicsquantum mechanics through photons and their polarization.

The Experiment

A beam of light shines on a projection screen. Filters A, B, and C are polarized horizontally, at 45^{o}, and vertically, respectively, and can be placed so as to intersect the beam of light.

First, insert filter A. Assuming the incoming light is randomly polarized, the intensity of the output will have half of the intensity of the incoming light. The outgoing photons are now all horizontally polarized.

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