An Introduction to Quantum Computing for Non-Physicists

Transcription

1 An Introduction to Quantum Comuting for Non-hysicists Eleanor Rieffel F alo Alto Labratory and Wolfgang olak Consultant F alo Alto Laboratory, 3400 Hillview Avenue, alo Alto, CA Richard Feynman s observation that certain quantum mechanical effects cannot be simulated efficiently on a comuter led to seculation that comutation in general could be done more efficiently if it used these quantum effects. This seculation roved justified when eter Shor described a olynomial time quantum algorithm for factoring integers. In quantumsystems, the comutationalsace increases exonentiallywith the size of the system which enables exonential arallelism. This arallelism could lead to exonentially faster quantum algorithms than ossible classically. The catch is that accessing the results, which requires measurement, roves tricky and requires new non-traditional rogramming techniques. The aim of this aer is to guide comuter scientists through the barriers that searate quantum comuting from conventional comuting. We introduce basic rinciles of quantum mechanics to exlain where the ower of quantum comuters comes from and why it is difficult to harness. We describe quantum crytograhy, teleortation, and dense coding. Various aroaches to exloiting the ower of quantum arallelism are exlained. We conclude with a discussion of quantum error correction. Categories and Subject Descritors: A. [Introductory and Survey] General Terms: Algorithms, Security, Theory Additional Key Words and hrases: Quantum comuting, Comlexity, arallelism Name: Eleanor Rieffel Affiliation: F alo Alto Laboratory Address: 3400 Hillview Avenue, alo Alto, CA Name: Wolfgang olak Address: 0 Yorktown Drive, Sunnyvale, CA ermission to make digital or hard coies of art or all of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or direct commercial advantage and that coies show this notice on the first age or initial screen of a dislay along with the full citation. Coyrights for comonents of this work owned by others than ACM must be honored. Abstracting with credit is ermitted. To coy otherwise, to reublish, to ost on servers, to redistribute to lists, or to use any comonent of this work in other works, requires rior secific ermission and/or a fee. ermissions may be requested from ublications Det, ACM Inc., 55 Broadway, New York, NY 0036 USA, fax + () , or ermissions@acm.org.

2 E. Rieffel and W. olak. INTRODUCTION Richard Feynman observed in the early 980 s [Feynman 98] that certain quantum mechanical effects cannot be simulated efficiently on a classical comuter. This observation led to seculation that erhas comutation in general could be done more efficiently if it made use of these quantum effects. But building quantum comuters, comutational machines that use such quantum effects, roved tricky, and as no one was sure how to use the quantum effects to seed u comutation, the field develoed slowly. It wasn t until 994, when eter Shor surrised the world by describing a olynomial time quantum algorithm for factoring integers [Shor 994; Shor 997], that the field of quantum comuting came into its own. This discovery romted a flurry of activity, both among exerimentalists trying to build quantum comuters 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, oular ress accounts of exerimental successes in quantum teleortation and the demonstration of a three-bit quantum comuter. The aim of this aer is to guide comuter scientists and other non-hysicists through the concetual and notational barriers that searate quantum comuting from conventional comuting and to acquaint them with this new and exciting field. It is imortant for the comuter science community to understand these new develoments since they may radically change the way we have to think about comutation, rogramming, and comlexity. Classically, the time it takes to do certain comutations can be decreased by using arallel rocessors. To achieve an exonential decrease in time requires an exonential increase in the number of rocessors, and hence an exonential increase in the amount of hysical sace needed. However, in quantum systems the amount of arallelism increases exonentially with the size of the system. Thus, an exonential increase in arallelism requires only a linear increase in the amount of hysical sace needed. This effect is called quantum arallelism [Deutsch and Jozsa 99]. There is a catch, and a big catch at that. While a quantum system can erform massive arallel comutation, access to the results of the comutation is restricted. Accessing the results is equivalent to making a measurement, which disturbs the quantum state. This roblem makes the situation, on the face of it, seem even worse than the classical situation; we can only read the result of one arallel thread, and because measurement is robabilistic, we cannot even choose which one we get. But in the ast few years, various eole have found clever ways of finessing the measurement roblem to exloit the ower of quantum arallelism. This sort of maniulation has no classical analog, and requires non-traditional rogramming techniques. One technique maniulates the quantum state so that a common roerty of all of the outut values such as the symmetry or eriod of a function can be read off. This technique is used in Shor s factorization algorithm. Another technique transforms the quantum state to increase the likelihood that outut of interest will be read. Grover s search algorithm makes use of such an amlification technique. This aer describes quantum arallelism in detail, and the techniques currently known for harnessing its ower. Section, following this introduction, exlains of the basic concets of quantum mechanics that are imortant for quantum comutation. This section cannot give a comrehensive view of quantum mechanics. Our aim is to rovide the reader with tools in the form of mathematics and notation with which to work with the quantum mechanics involved in quantum comutation. We hoe that this aer will equi readers well enough that they

3 Introduction to Quantum Comuting 3 can freely exlore the theoretical realm of quantum comuting. Section 3 defines the quantum bit, or qubit. Unlike classical bits, a quantum bit can be ut in a suerosition state that encodes both 0 and. There is no good classical exlanation of suerositions: a quantum bit reresenting 0 and can neither be viewed as between 0 and nor can it be viewed as a hidden unknown state that reresents either 0 or with a certain robability. Even single quantum bits enable interesting alications. We describe the use of a single quantum bit for secure key distribution. But the real ower of quantum comutation derives from the exonential state saces of multile quantum bits: just as a single qubit can be in a suerosition of 0 and, a register of n qubits can be in a suerosition of all n ossible values. The extra states that have no classical analog and lead to the exonential size of the quantum state sace are the entangled states, like the state leading to the famous ER aradox (see section 3.4). We discuss the two tyes of oerations a quantum system can undergo: measurement and quantum state transformations. Most quantum algorithms involve a sequence of quantum state transformations followed by a measurement. For classical comuters there are sets of gates that are universal in the sense that any classical comutation can be erformed using a sequence of these gates. Similarly, there are sets of rimitive quantum state transformations, called quantum gates, that are universal for quantum comutation. Given enough quantum bits, it is ossible to construct a universal quantum Turing machine. Quantum hysics uts restrictions on the tyes of transformations that can be done. In articular, all quantum state transformations, and therefore all quantum gates and all quantum comutations, must be reversible. Yet all classical algorithms can be made reversible and can be comuted on a quantum comuter in comarable time. Some common quantum gates are defined in section 4. Two alications combining quantum gates and entangled states are described in section 4.: teleortation and dense coding. Teleortation is the transfer of a quantum state from one lace to another through classical channels. That teleortation is ossible is surrising since quantum mechanics tells us that it is not ossible to clone quantum states or even measure them without disturbing the state. Thus, it is not obvious what information could be sent through classical channels that could ossibly enable the reconstruction of an unknown quantum state at the other end. Dense coding, a dual to teleortation, uses a single quantum bit to transmit two bits of classical information. Both teleortation and dense coding rely on the entangled states described in the ER exeriment. It is only in section 5 that we see where an exonential seed-u over classical comuters might come from. The inut to a quantum comutation can be ut in a suerosition state that encodes all ossible inut values. erforming the comutation on this initial state will result in suerosition of all of the corresonding outut values. Thus, in the same time it takes to comute the outut for a single inut state on a classical comuter, a quantum comuter can comute the values for all inut states. This rocess is known as quantum arallelism. However, measuring the outut states will randomly yield only one of the values in the suerosition, and at the same time destroy all of the other results of the comutation. Section 5 describes this situation in detail. Sections 6 and 7 describe techniques for taking advantage of quantum arallelism insite of the severe constraints imosed by quantum mechanics on what can be measured. Section 6 describes the details of Shor s olynomial time factoring algorithm. The fastest ER = Einstein, odolsky and Rosen

4 4 E. Rieffel and W. olak known classical factoring algorithm requires exonential time and it is generally believed that there is no classical olynomial time factoring algorithm. Shor s is a beautiful algorithm that takes advantage of quantum arallelism by using a quantum analog of the Fourier transform. Lov Grover develoed a technique for searching an unstructured list of n items in O( n) stes on a quantum comuter. Classical comuters can do no better than O(n), so unstructured search on a quantum comuter is rovably more efficient than search on a classical comuter. However, the seed-u is only olynomial, not exonential, and it has been shown that Grover s algorithm is otimal for quantum comuters. It seems likely that search algorithms that could take advantage of some roblem structure could do better. Tad Hogg, among others, has exlored such ossibilities. We describe various quantum search techniques in section 7. It is as yet unknown whether the ower of quantum arallelism can be harnessed for a wide variety of alications. One tantalizing oen question is whether quantum comuters can solve N comlete roblems in olynomial time. erhas the biggest oen question is whether useful quantum comuters can be built. There are a number of roosals for building quantum comuters using ion tras, nuclear magnetic resonance (NMR), otical and solid state techniques. All of the current roosals have scaling roblems, so that a breakthrough will be needed to go beyond tens of qubits to hundreds of qubits. While both otical and solid state techniques show romise, NMR and ion tra technologies are the most advanced so far. In an ion tra quantum comuter [Cirac and Zoller 995; Steane 996] a linear sequence of ions reresenting the qubits are confined by electric fields. Lasers are directed at individual ions to erform single bit quantum gates. Two-bit oerations are realized by using a laser on one qubit to create an imulse that riles through a chain of ions to the second qubit where another laser ulse stos the riling and erforms the two-bit oeration. The aroach requires that the ions be ket in extreme vacuum and at extremely low temeratures. The NMR aroach has the advantage that it will work at room temerature, and that NMR technology in general is already fairly advanced. The idea is to use macroscoic amounts of matter and encode a quantum bit in the average sin state of a large number of nuclei. The sin states can be maniulated by magnetic fields and the average sin state can be measured with NMR techniques. The main roblem with the technique is that it doesn t scale well; the measured signal scales as = n with the number of qubits n. However, a recent roosal [Schulman and Vazirani 998] has been made that may overcome this roblem. NMR comuters with three qubits have been built successfully [Cory et al. 998; Vandersyen et al. 999; Gershenfeld and Chuang 997; Laflamme et al. 997]. This aer will not discuss further the hysical and engineering roblems of building quantum comuters. The greatest roblem for building quantum comuters is decoherence, the distortion of the quantum state due to interaction with the environment. For some time it was feared that quantum comuters could not be built because it would be imossible to isolate them sufficiently from the external environment. The breakthrough came from the algorithmic rather than the hysical side, through the invention of quantum error correction techniques. Initially eole thought quantum error correction might be imossible because of the imossibility of reliably coying unknown quantum states, but it turns out that it is ossible to design quantum error correcting codes that detect certain kinds of errors and enable the

5 Introduction to Quantum Comuting 5 reconstruction of the exact error-free quantum state. Quantum error correction is discussed in section 8. Aendices rovide background information on tensor roducts and continued fractions.. QUANTUM MECHANICS Quantum mechanical henomena are difficult to understand since most of our everyday exeriences are not alicable. This aer cannot rovide a dee understanding of quantum mechanics (see [Feynman et al. 965; Liboff 997; Greenstein and Zajonc 997] for exositions of 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 comuting. Quantum mechanics is a theory in the mathematical sense: it is governed by a set of axioms. The consequences of the axioms describe the behavior of quantum systems. The axioms lead to several aarent aradoxes: in the Comton effect it aears as if an action recedes its cause; the ER exeriment makes it aear as if action over a distance faster than the seed of light is ossible. We will discuss the ER exeriment in detail in section 3.4. Verification of most redictions is indirect, and requires careful exerimental design and secialized equiment. We will begin, however, with an exeriment that requires only readily available equiment and that will illustrate some of the key asects of quantum mechanics needed for quantum comutation.. hoton olarization hotons are the only articles that we can directly observe. The following simle exeriment can be erformed with minimal equiment: a strong light source, like a laser ointer, and three olaroids (olarization filters) that can be icked u at any camera suly store. The exeriment demonstrates some of the rinciles of quantum mechanics through hotons and their olarization... The Exeriment. A beam of light shines on a rojection screen. Filters A, B, and C are olarized horizontally, at 45 o, and vertically, resectively, and can be laced so as to intersect the beam of light. First, insert filter A. Assuming the incoming light is randomly olarized, the intensity of the outut will have half of the intensity of the incoming light. The outgoing hotons are now all horizontally olarized. A The function of filter A cannot be exlained as a sieve that only lets those hotons ass that haen to be already horizontally olarized. If that were the case, few of the randomly olarized incoming electrons would be horizontally olarized, so we would exect a much larger attenuation of the light as it asses through the filter. Next, when filter C is inserted the intensity of the outut dros to zero. None of the horizontally olarized hotons can ass through the vertical filter. A sieve model could exlain this behavior.

6 6 E. Rieffel and W. olak A C Finally, after filter B is inserted between A and C, a small amount of light will be visible on the screen, exactly one eighth of the original amount of light. A B C Here we have a nonintuitive effect. Classical exerience suggests that adding a filter should only be able to decrease the number of hotons getting through. How can it increase it?.. The Exlanation. A hoton s olarization state can be modelled by a unit vector ointing in the aroriate direction. Any arbitrary olarization can be exressed as a linear combination aj"i +bj!i of the two basis vectors j!i (horizontal olarization) and j"i (vertical olarization). Since we are only interested in the direction of the olarization (the notion of magnitude is not meaningful), the state vector will be a unit vector, i.e., jaj + jbj =. In general, the olarization of a hoton can be exressed as aj"i + bj!i where a and b are comlex numbers 3 such that jaj + jbj =. Note, the choice of basis for this reresentation is comletely arbitrary: any two orthogonal unit vectors will do (e.g. fj-i; j%ig). The measurement ostulate of quantum mechanics states that any device measuring a - dimensional system has an associated orthonormal basis with resect to which the quantum measurement takes lace. Measurement of a state transforms the state into one of the measuring device s associated basis vectors. The robability that the state is measured as basis vector jui is the square of the norm of the amlitude of the comonent of the original state in the direction of the basis vector jui. For examle, given a device for measuring the olarization of hotons with associated basis fj"i; jtoig, the state ψ = aj"i + bj!i is measured as j"i with robability jaj and as j!i with robability jbj (see Figure ). Note that different measuring devices with have different associated basis, and measurements using these devices will have different outcomes. As measurements are always made with resect to an orthonormal basis, throughout the rest of this aer all bases will be assumed to be orthonormal. Furthermore, measurement of the quantum state will change the state to the result of the measurement. That is, if measurement of ψ = aj"i + bj!i results in j"i, then the state ψ changes to j"i and a second measurement with resect to the same basis will return j"i with robability. Thus, unless the original state haened to be one of the basis vectors, measurement will change that state, and it is not ossible to determine what the original state was. The notation j!i is exlained in section.. 3 Imaginary coefficients corresond to circular olarization.

7 Introduction to Quantum Comuting 7 j"i a b j!i jψi Fig.. Measurement is a rojection onto the basis Quantum mechanics can exlain the olarization exeriment as follows. A olaroid measures the quantum state of hotons with resect to the basis consisting of the vector corresonding to its olarization together with a vector orthogonal to its olarization. The hotons which, after being measured by the filter, match the filter s olarization are let through. The others are reflected and now have a olarization erendicular to that of the filter. For examle, filter A measures the hoton olarization with resect to the basis vector j!i, corresonding to its olarization. The hotons that ass through filter A all have olarization j!i. Those that are reflected by the filter all have olarization j"i. Assuming that the light source roduces hotons with random olarization, filter A will measure 50% of all hotons as horizontally olarized. These hotons will ass through the filter and their state will be j!i. Filter C will measure these hotons with resect to j"i. But the state j!i = 0j"i + j!i will be rojected onto j"i with robability 0 and no hotons will ass filter C. Finally, filter B measures the quantum state with resect to the basis f (j"i + j!i); (j"i j!i)g which we write as fj%i; j-ig. Note that j!i = (j%i j-i) and j"i = (j%i + j-i). Those hotons that are measured as j%i ass through the filter. hotons assing through A with state j!i will be measured by B as j%i with robability = and so 50% of the hotons assing through A will ass through B and be in state j%i. As before, these hotons will be measured by filter C as j"i with robability =. Thus only one eighth of the original hotons manage to ass through the sequence of filters A, B, and C.. State Saces and Bra/Ket Notation The state sace of a quantum system, consisting of the ositions, momentums, olarizations, sins, etc. of the various articles, is modelled by a Hilbert sace of wave functions. We will not look at the details of these wave functions. For quantum comuting we need only deal with finite quantum systems and it suffices to consider finite dimensional comlex vector saces with an inner roduct that are sanned by abstract wave functions such as j!i. Quantum state saces and the tranformations acting on them can be described in terms of vectors and matrices or in the more comact bra/ket notation invented by Dirac [Dirac

8 8 E. Rieffel and W. olak 958]. Kets like jxi denote column vectors and are tyically used to describe quantum states. The matching bra, hxj, denotes the conjugate transose of jxi. For examle, the orthonormal basis fj0i; jig can be exressed as f(; 0) T ; (0; ) T g. Any comlex linear combination of j0i and ji, aj0i + bji, can be written (a; b) T. Note that the choice of the order of the basis vectors is arbitrary. For examle, reresenting j0i as (0; ) T and ji as (; 0) T would be fine as long as this is done consistently. Combining hxj and jyi as in hxjjyi, also written as hxjyi, denotes the inner roduct of the two vectors. For instance, since j0i is a unit vector we have h0j0i =and since j0i and ji are orthogonal we have h0ji =0. The notation jxihyj is the outer roduct of jxi and hyj. For examle, j0ihj is the transformation that mas ji to j0i and j0i to (0; 0) T since j0ihjji = j0ihji = j0i j0ihjj0i = j0ihj0i =0j0i = 0 : 0 Equivalently, j0ihj can be written in matrix form where j0i = (; 0) T, h0j = (; 0), ji =(0; ) T, and hj =(0; ). Then 0 j0ihj = (0; ) = : This notation gives us a convenient way of secifying transformations on quantum states in terms of what haens to the basis vectors (see section 4). For examle, the transformation that exchanges j0i and ji is given by the matrix = j0ihj + jih0j: In this aer we will refer the slightly more intuitive notation : j0i!ji ji!j0i that exlicitly secifies the result of a transformation on the basis vectors. 3. QUANTUM BITS A quantum bit, or qubit, is a unit vector in a two dimensional comlex vector sace for which a articular basis, denoted by fj0i; jig, has been fixed. The orthonormal basis j0i and ji may corresond to the j"i and j!i olarizations of a hoton resectively, or to the olarizations j%i and j-i. Orj0i and ji could corresond to the sin-u and sin-down states of an electron. When talking about qubits, and quantum comutations in general, a fixed basis with resect to which all statements are made has been chosen in advance. In articular, unless otherwise secified, all measurements will be made with resect to the standard basis for quantum comutation, fj0i; jig. For the uroses of quantum comutation, the basis states j0i and ji are taken to reresent the classical bit values 0 and resectively. Unlike classical bits however, qubits can be in a suerosition of j0i and ji such as aj0i + bji where a and b are comlex numbers such that jaj + jbj =. Just as in the hoton olarization case, if such a suerosition is measured with resect to the basis fj0i; jig, the robability that the measured value is j0i is jaj and the robability that the measured value is ji is jbj.

9 Introduction to Quantum Comuting 9 Even though a quantum bit can be ut in infinitely many suerosition states, it is only ossible to extract a single classical bit s worth of information from a single quantum bit. The reason that no more information can be gained from a qubit than in a classical bit is that information can only be obtained by measurement. When a qubit is measured, the measurement changes the state to one of the basis states in the way seen in the hoton olarization exeriment. As every measurement can result in only one of two states, one of the basis vectors associated to the given measuring device, so, just as in the classical case, there are only two ossible results. As measurement changes the state, one cannot measure the state of a qubit in two different bases. Furthermore, as we shall see in the section 4.., quantum states cannot be cloned so it is not ossible to measure a qubit in two ways, even indirectly by, say, coying the qubit and measuring the coy in a different basis from the original. 3. Quantum Key Distribution Sequences of single qubits can be used to transmit rivate keys on insecure channels. In 984 Bennett and Brassard described the first quantum key distribution scheme [Bennett and Brassard 987; Bennett et al. 99]. Classically, ublic key encrytion techniques, e.g. RSA, are used for key distribution. Consider the situation in which Alice and Bob want to agree on a secret key so that they can communicate rivately. They are connected by an ordinary bi-directional oen channel and a uni-directional quantum channel both of which can be observed by Eve, who wishes to eavesdro on their conversation. This situation is illustrated in the figure below. The quantum channel allows Alice to send individual articles (e.g. hotons) to Bob who can measure their quantum state. Eve can attemt to measure the state of these articles and can resend the articles to Bob. classical channel Alice quantum channel Bob Eve To begin the rocess of establishing a secret key, Alice sends a sequence of bits to Bob by encoding each bit in the quantum state of a hoton as follows. For each bit, Alice randomly uses one of the following two bases for encoding each bit: 0!j"i!j!i

10 0 E. Rieffel and W. olak or 0!j-i!j%i: Bob measures the state of the hotons he receives by randomly icking either basis. After the bits have been transmitted, Bob and Alice communicate the basis they used for encoding and decoding of each bit over the oen channel. With this information both can determine which bits have been transmitted correctly, by identifying those bits for which the sending and receiving bases agree. They will use these bits as the key and discard all the others. On average, Alice and Bob will agree on 50% of all bits transmitted. Suose that Eve measures the state of the hotons transmitted by Alice and resends new hotons with the measured state. In this rocess she will use the wrong basis aroximately 50% of the time, in which case she will resend the bit with the wrong basis. So when Bob measures a resent qubit with the correct basis there will be a 5% robability that he measures the wrong value. Thus any eavesdroer on the quantum channel is bound to introduce a high error rate that Alice and Bob can detect by communicating a sufficient number of arity bits of their keys over the oen channel. So, not only is it likely that Eve s version of the key is 5% incorrect, but the fact that someone is eavesdroing will be aarent to Alice and Bob. Other techniques for exloiting quantum effects for key distribution have been roosed. See, for examle, Ekert [Ekert et al. 99], Bennett [Bennett 99] and Lo and Chau [Lo and Chau 999]. But none of the quantum key distribution techniques are substitutes for ublic key encrytion schemes. Attacks by eavesdroers other than the one described here are ossible. Security against all such schemes are discussed in both Mayers [Mayers 998] and Lo and Chau [Lo and Chau 999]. Quantum key distribution has been realized over a distance of 4 km using standard fiber otical cables [Hughes et al. 997] and over 0.5 km through the atmoshere [Hughes et al. 999]. 3. Multile Qubits Imagine a macroscoic hysical object breaking aart and multile ieces flying off in different directions. The state of this system can be described comletely by describing the state of each of its comonent ieces searately. A surrising and unintuitive asect of the state sace of an n article quantum system is that the state of the system cannot always be described in terms of the state of its comonent ieces. It is when examining systems of more than one qubit that one first gets a glimse of where the comutational ower of quantum comuters could come from. As we saw, the state of a qubit can be reresented by a vector in the two dimensional comlex vector sace sanned by j0i and ji. In classical hysics, the ossible states of a system of n articles, whose individual states can be described by a vector in a two dimensional vector sace, form a vector sace of n dimensions. However, in a quantum system the resulting state sace is much larger; a system of n qubits has a state sace of n dimensions. 4 It is this exonential growth of the state sace with the number of articles that suggests a ossible exonential seed-u of comutation on quantum comuters over classical comuters. 4 Actually, as we shall see, the state sace is the set of normalized vectors in this n dimensional sace, just as the state aj0i + bji of a qubit is normalized so that jaj + jbj =.

11 Introduction to Quantum Comuting Individual state saces of n articles combine classically through the cartesian roduct. Quantum states, however, combine through the tensor roduct. Details on roerties of tensor roducts and their exression in terms of vectors and matrices is given in Aendix A. Let us look briefly at distinctions between the cartesian roduct and the tensor roduct that will be crucial to understanding quantum comutation. Let V and W be two -dimensional comlex vector saces with bases fv ;v g and fw ;w g resectively. The cartesian roduct of these two saces can take as its basis the union of the bases of its comonent saces fv ;v ;w ;w g. Note that the order of the basis was chosen arbitrarily. In articular, the dimension of the state sace of multile classical articles grows linearly with the number of articles, since dim( Y ) =dim() + dim(y ). The tensor roduct of V and W has basis fv Ω w ;v Ω w ;v Ω w ;v Ω w g. Note that the order of the basis, again, is arbitrary 5. So the state sace for two qubits, each with basis fj0i; jig, has basis fj0iωj0i; j0iωji; jiωj0i; jiωjig which can be written more comactly as fj00i; j0i; j0i; jig. More generally, we write jxi to mean jb n b n :::b 0 i where b i are the binary digits of the number x. A basis for a three qubit system is fj000i; j00i; j00i; j0i; j00i; j0i; j0i; jig and in general an n qubit system has n basis vectors. We can now see the exonential growth of the state sace with the number of quantum articles. The tensor roduct Ω Y has dimension dim() dim(y ). The state j00i + ji is an examle of a quantum state that cannot be described in terms of the state of each of its comonents (qubits) searately. In other words, we cannot find a ;a ;b ;b such that (a j0i + b ji) Ω (a j0i + b ji) =j00i + ji since (a j0i + b ji) Ω (a j0i + b ji) =a a j00i + a b j0i + b a j0i + b b ji and a b =0imlies that either a a =0or b b =0. States which cannot be decomosed in this way are called entangled states. These states reresent situations that have no classical counterart, and for which we have no intuition. These are also the states that rovide the exonential growth of quantum state saces with the number of articles. Note that it would require vast resources to simulate even a small quantum system on traditional comuters. The evolution of quantum systems is exonentially faster than their classical simulations. The reason for the otential ower of quantum comuters is the ossibility of exloiting the quantum state evolution as a comutational mechanism. 3.3 Measurement The exeriment in section.. illustrates how measurement of a single qubit rojects the quantum state on to one of the basis states associated with the measuring device. The result of a measurement is robabilistic and the rocess of measurement changes the state to that measured. Let us look at an examle of measurement in a two qubit system. Any two qubit state can be exressed as aj00i+bj0i+cj0i+dji, where a, b, c and d are comlex numbers such that jaj + jbj + jcj + jdj =. Suose we wish to measure the first qubit with resect 5 It is only when we use matrix notation to describe state transformations that the order of basis vectors becomes relevant.

12 E. Rieffel and W. olak to the standard basis fj0i; jig. For convenience we will rewrite the state as follows: aj00i + bj0i + cj0i + dji = j0i Ω(aj0i + bji) + ji Ω(cj0i + dji) = uj0i Ω(a=uj0i + b=uji)+ vjiω(c=vj0i + d=vji): For u = jaj + jbj and v = jcj + jdj the vectors a=uj0i + b=uji and c=vj0i + d=vji are of unit length. Once the state has been rewritten as above, as a tensor roduct of the bit being measured and a second vector of unit length, the robabalistic result of a measurement is easy to read off. Measurement of the first bit will with robability u = jaj + jbj return j0i rojecting the state to j0i Ω(a=uj0i + b=uji) or with robability v = jcj + jdj yield ji rojecting the state to jiω(c=vj0i + d=vji). As j0i Ω(a=uj0i + b=uji) and ji Ω(c=vj0i + d=vji) are both unit vectors, no scaling is necessary. Measuring the second bit works similarly. For the uroses of quantum comutation, multi-bit measurement can be treated as a series of single-bit measurements in the standard basis. Other sorts of measurements are ossible, like measuring whether two qubits have the same value without learning the actual value of the two qubits. But such measurements are equivalent to unitary transformations followed by a standard measurement of individual qubits, and so it suffices to look only at standard measurements. In the two qubit examle, the state sace is a cartesian roduct of the subsace consisting of all states whose first qubit is in the state j0i and the orthogonal subsace of states whose first qubit is in the state ji. Any quantum state can be written as the sum of two vectors, one in each of the subsaces. A measurement of k qubits in the standard basis has k ossible outcomes m i. Any device measuring k qubits of an n-qubit system slits of the n -dimensional state sace H into a cartesian roduct of orthogonal subsaces S ;:::;S k with H = S ::: S k, such that the value of the k qubits being measured is m i and the state after measurement is in sace the sace S i for some i. The device randomly chooses one of the S i s with robability the square of the amlitude of the comonent of ψ in S i, and rojects the state into that comonent, scaling to give length. Equivalently, the robability that the result of the measurement is a given value is the sum of the squares of the the absolute values of the amlitudes of all basis vectors comatible with that value of the measurement. Measurement gives another way of thinking about entangled articles. articles are not entangled if the measurement of one has no effect on the other. For instance, the state (j00i + ji) is entangled since the robability that the first bit is measured to be j0i is = if the second bit has not been measured. However, if the second bit had been measured, the robability that the first bit is measured as j0i is either or 0, deending on whether the second bit was measured as j0i or ji resectively. Thus the robable result of measuring the first bit is changed by a measurement of the second bit. On the other hand, the state (j00i + j0i) is not entangled: since (j00i + j0i) =j0iω (j0i + ji), any measurement of the first bit will yield j0i regardless of whether the second bit was measured. Similarly, the second bit has a fifty-fifty chance of being measured as j0i regardless of whether the first bit was measured or not. Note that entanglement, in the sense that measurement of one article has an effect on measurements of another article,

13 Introduction to Quantum Comuting 3 is equivalent to our revious definition of entangled states as states that cannot be written as a tensor roduct of individual states. 3.4 The ER aradox Einstein, odolsky and Rosen roosed a gedanken exeriment that uses entangled articles in a manner that seemed to violate fundamental rinciles relativity. Imagine a source that generates two maximally entangled articles j00i + ji, called an ER air, and sends one each to Alice and Bob. Alice ER source Bob Alice and Bob can be arbitrarily far aart. Suose that Alice measures her article and observes state j0i. This means that the combined state will now be j00i and if now Bob measures his article he will also observe j0i. Similarly, if Alice measures ji, so will Bob. Note that the change of the combined quantum state occurs instantaneously even though the two articles may be arbitrarily far aart. It aears that this would enable Alice and Bob to communicate faster than the seed of light. Further analysis, as we shall see, shows that even though there is a couling between the two articles, there is no way for Alice or Bob to use this mechanism to communicate. There are two standard ways that eole use to describe entangled states and their measurement. Both have their ositive asects, but both are incorrect and can lead to misunderstandings. Let us examine both in turn. Einstein, odolsky and Rosen roosed that each article has some internal state that comletely determines what the result of any given measurement will be. This state is, for the moment, hidden from us, and therefore the best we can currently do is to give robabilistic redictions. Such a theory is known as a local hidden variable theory. The simlest hidden variable theory for an ER air is that the articles are either both in state j0i or both in state ji, we just don t haen to know which. In such a theory no communication between ossibly distant articles is necessary to exlain the correlated measurements. However, this oint of view cannot exlain the results of measurements with resect to a different basis. In fact, Bell showed that any local hidden variable theory redicts that certain measurements will satisfy an inequality, known as Bell s inequality. However, the result of actual exeriments erforming these measurements show that Bell s inequality is violated. Thus quantum mechanics cannot be exlained by any local hidden variable theory. See [Greenstein and Zajonc 997] for a highly readable account of Bell s theorem and related exeriments. The second standard descrition is in terms of cause and effect. For examle, we said earlier that a measurement erformed by Alice affects a measurement erformed by Bob. However, this view is incorrect also, and results, as Einstein, odolsky and Rosen recognized, in dee inconsistencies when combined with relativity theory. It is ossible to set u the ER scenario so that one observer sees Alice measure first, then Bob, while another

14 4 E. Rieffel and W. olak observer sees Bob measure first, then Alice. According to relativity, hysics must equally well exlain the observations of the first observer as the second. While our terminology of cause and effect cannot be comatible with both observers, the actual exerimental values are invariant under change of observer. The exerimental results can be exlained equally well by Bob s measuring first and causing a change in the state of Alice s article, as the other way around. This symmetry shows that Alice and Bob cannot, in fact, use their ER air to communicate faster than the seed of light, and thus resolves the aarent aradox. All that can be said is that Alice and Bob will observe the same random behavior. As we will see in the section on dense coding and teleortation, ER airs can be used to aid communication, albeit communication slower than the seed of light. 4. QUANTUM GATES So far we have looked at static quantum systems which change only when measured. The dynamics of a quantum system, when not being measured, are governed by Schrödinger s equation; the dynamics must take states to states in a way that reserves orthogonality. For a comlex vector sace, linear transformations that reserve orthogonality are unitary transformations, defined as follows. Any linear transformation on a comlex vector sace can be described by a matrix. Let M Λ denote the conjugate transose of the matrix M. A matrix M is unitary (describes a unitary transformation) if MM Λ = I. Any unitary transformation of a quantum state sace is a legitimate quantum transformation, and vice versa. One can think of unitary transformations as being rotations of a comlex vector sace. One imortant consequence of the fact that quantum transformations are unitary is that they are reversible. Thus quantum gates must be reversible. Bennett, Fredkin, and Toffoli had already looked at reversible versions of standard comuting models showing that all classical comutations can be done reversibly. See Feynman s Lectures on Comutation [Feynman 996] for an account of reversible comutation and its relation to the energy of comutation and information. 4. Simle Quantum Gates The following are some examles of useful single-qubit quantum state transformations. Because of linearity, the transformations are fully secified by their effect on the basis vectors. The associated matrix, with fj0i; jig as the referred ordered basis, is also shown. I : j0i!j0i 0 ji!ji 0 : j0i!ji 0 ji!j0i 0 Y : j0i! ji 0 ji!j0i 0 Z : j0i!j0i 0 ji! ji 0 The names of these transformations are conventional. I is the identity transformation, is negation, Z is a hase shift oeration, and Y = Z is a combination of both. The transformation was discussed reviously in section.. It can be readily verified that these

15 Introduction to Quantum Comuting 5 gates are unitary. For examle YY Λ = = I: 0 The controlled-not gate, C not, oerates on two qubits as follows: it changes the second bit if the first bit is and leaves this bit unchanged otherwise. The vectors j00i, j0i, j0i, and ji form an orthonormal basis for the state sace of a two-qubit system, a 4- dimensional comlex vector sace. In order to reresent transformations of this sace in matrix notation we need to choose an isomorhism between this sace and the sace of comlex four tules. There is no reason, other than convention, to ick one isomorhism over another. The one we use here associates j00i, j0i, j0i, and ji to the standard 4- tule basis (; 0; 0; 0) T, (0; ; 0; 0) T, (0; 0; ; 0) T and (0; 0; 0; ) T, in that order. The C not transformation has reresentations C not : j00i!j00i j0i!j0i j0i!ji ji!j0i C A : The transformation C not is unitary since C Λ not = C not and C not C not = I. The C not gate cannot be decomosed into a tensor roduct of two single-bit transformations. It is useful to have grahical reresentations of quantum state transformations, esecially when several transformations are combined. The controlled-not gate C not is tyically reresented by a circuit of the form b : The oen circle indicates the control bit, and the indicates the conditional negation of the subject bit. In general there can be multile control bits. Some authors use a solid circle to indicate negative control, in which the subject bit is toggled when the control bit is 0. Similarly, the controlled-controlled-not, which negates the last bit of three if and only if the first two are both, has the following grahical reresentation. b b Single bit oerations are grahically reresented by aroriately labelled boxes as shown. Y Z

16 6 E. Rieffel and W. olak 4.. The Walsh-Hadamard Transformation. Another imortant single-bit transformation is the Hadamard Transformation defined by H : j0i! (j0i + ji) ji! (j0i ji): The transformation H has a number of imortant alications. When alied to j0i, H creates a suerosition state (j0i + ji). Alied to n bits individually, H generates a suerosition of all n ossible states, which can be viewed as the binary reresentation of the numbers from 0 to n. (H Ω H Ω :::Ω H)j00 :::0i = ((j0i + ji) Ω (j0i + ji) Ω :::Ω (j0i + ji)) n = n jxi: n x=0 The transformation that alies H to n bits is called the Walsh, or Walsh-Hadamard, transformation W. It can be defined as a recursive decomosition of the form W = H; W n+ = H Ω W n : 4.. No Cloning. The unitary roerty imlies that quantum states cannot be coied or cloned. The no cloning roof given here, originally due to Wootters and Zurek [Wootters and Zurek 98], is a simle alication of the linearity of unitary transformations. Assume that U is a unitary transformation that clones, in that U (ja0i) =jaai for all quantum states jai. Let jai and jbi be two orthogonal quantum states. Say U (ja0i) =jaai and U (jb0i) =jbbi. Consider jci =(= )(jai + jbi). By linearity, But if U is a cloning transformation then U (jc0i) = (U (ja0i) +U (jb0i)) = (jaai + jbbi): U (jc0i) =jcci ==(jaai + jabi + jbai + jbbi); which is not equal to (= )(jaai + jbbi). Thus there is no unitary oeration that can reliably clone unknown quantum states. It is clear that cloning is not ossible by using measurement since measurement is both robabalistic and destructive of states not in the measuring device s associated subsaces. It is imortant to understand what sort of cloning is and isn t allowed. It is ossible to clone a known quantum state. What the no cloning rincile tells us is that it is imossible to reliably clone an unknown quantum state. Also, it is ossible to obtain n articles in an entangled state aj00 :::0i + bj :::i from an unknown state aj0i + bji. Each of these articles will behave in exactly the same way when measured with resect to the standard basis for quantum comutation fj00 :::0i; j00 :::0i;:::; j :::ig, but not when measured with resect to other bases. It is not ossible to create the n article state (aj0i + bji) Ω :::Ω (aj0i + bji) from an unknown state aj0i + bji.

17 Introduction to Quantum Comuting 7 4. Examles The use of simle quantum gates can be studied with two simle examles: dense coding and teleortation. Dense coding uses one quantum bit together with an ER air to encode and transmit two classical bits. Since ER airs can be distributed ahead of time, only one qubit (article) needs to be hysically transmitted to communicate two bits of information. This result is surrising since, as was discussed in section 3, only one classical bit s worth of information can be extracted from a qubit. Teleortation is the oosite of dense coding, in that it uses two classical bits to transmit a single qubit. Teleortation is surrising in light of the no cloning rincile of quantum mechanics, in that it enables the transmission of an unknown quantum state. The key to both dense coding and teleortation is the use of entangled articles. The initial set u is the same for both rocesses. Alice and Bob wish to communicate. Each is sent one of the entangled articles making u an ER air, ψ 0 = (j00i + ji): Say Alice is sent the first article, and Bob the second. So until a article is transmitted, only Alice can erform transformations on her article, and only Bob can erform transformations on his. 4.. Dense Coding Alice Bob Encoder Decoder ER source Alice. Alice receives two classical bits, encoding the numbers 0 through 3. Deending on this number Alice erforms one of the transformations fi;;y;zg on her qubit of the entangled air ψ 0. Transforming just one bit of an entangled air means erforming the identity transformation on the other bit. The resulting state is shown in the table. Value Transformation New state 0 ψ 0 =(I Ω I)ψ 0 (j00i + ji) Alice then sends her qubit to Bob. ψ =( Ω I)ψ 0 (j0i + j0i) ψ =(Y Ω I)ψ 0 ( j0i + j0i) 3 ψ 3 =(Z Ω I)ψ 0 (j00i ji) Bob. Bob alies a controlled-not to the two qubits of the entangled air.

18 8 E. Rieffel and W. olak Initial state Controlled-NOT First bit Second bit ψ 0 = (j0i + ji) j0i ψ = (j0i + j0i) (ji + j0i) (ji + j0i) ji ψ = ( j0i + j0i) ( ji + j0i) ( ji + j0i) ji ψ 3 = j0i Note that Bob can now measure the second qubit without disturbing the quantum state. If the measurement returns j0i then the encoded value was either 0 or 3, if the measurement returns ji then the encoded value was either or. Bob now alies H to the first bit: Initial state First bit H(First bit) ψ 0 (j0i + ji) (j0i + ji) + (j0i ji) = j0i ψ (ji + j0i) ψ ( ji + j0i) ψ 3 (j0i ji) (j0i ji) + (j0i ji) + (j0i + ji) = j0i (j0i + ji) = ji (j0i + ji) (j0i ji) = ji Finally, Bob measures the resulting bit which allows him to distinguish between 0 and 3, and and. 4.. Teleortation. The objective is to transmit the quantum state of a article using classical bits and reconstruct the exact quantum state at the receiver. Since quantum state cannot be coied, the quantum state of the given article will necessarily be destroyed. Single bit teleortation has been realized exerimentally [Bouwmeester et al. 997; Nielsen et al. 998; Boschi et al. 998]. Alice Bob Decoder Encoder ER source Alice. Alice has a qubit whose state she doesn t know. She wants to send the state of ths qubit ffi = aj0i + bji to Bob through classical channels. As with dense coding, Alice and Bob each ossess one qubit of an entangled air ψ 0 = (j00i + ji):

19 Introduction to Quantum Comuting 9 Alice alies the decoding ste of dense coding to the qubit ffi to be transmitted and her half of the entangled air. The starting state is quantum state ffi Ω ψ 0 = aj0iω(j00i + ji) +bjiω(j00i + ji) = aj000i + aj0i + bj00i + bji ; of which Alice controls the first two bits and Bob controls the last one. Alice now alies C not Ω I and H Ω I Ω I to this state: (H Ω I Ω I)(C not Ω I)(ffi Ω ψ 0 ) = (H Ω I Ω I)(C not Ω I) aj000i + aj0i + bj00i + bji = (H Ω I Ω I) aj000i + aj0i + bj0i + bj0i = a(j000i + j0i + j00i + ji) +b(j00i + j00i j0i j0i) = j00i(aj0i + bji) +j0i(aji + bj0i)+j0i(aj0i bji) +ji(aji bj0i) Alice measures the first two qubits to get one of j00i, j0i, j0i, orji with equal robability. Deending on the result of the measurement, the quantum state of Bob s qubit is rojected to aj0i + bji, aji + bj0i, aj0i bji, oraji bj0i resectively. Alice sends the result of her measurement as two classical bits to Bob. Note that when she measured it, Alice irretrievably altered the state of her original qubit ffi, whose state she is in the rocess of sending to Bob. This loss of the original state is the reason teleortation does not violate the no cloning rincile. Bob. When Bob receives the two classical bits from Alice he knows how the state of his half of the entangled air comares to the original state of Alice s qubit. bits received state decoding 00 aj0i + bji I 0 aji + bj0i 0 aj0i bji Z aji bj0i Y Bob can reconstruct the original state of Alice s qubit, ffi, by alying the aroriate decoding transformation to his art of the entangled air. Note that this is the encoding ste of dense coding. 5. QUANTUM COMUTERS This section discusses how quantum mechanics can be used to erform comutations and how these comutations are qualitatively different from those erformed by a conventional comuter. Recall from section 4 that all quantum state transformations have to be reversible. While the classical NOT gate is reversible, AND, OR and NAND gates are not. Thus it is not obvious that quantum transformations can carry out all classical comutations. The first subsection describes comlete sets of reversible gates that can erform any