An Analog Analogue of a Digital Quantum Computation

Transcription

1 An Analog Analogue of a Digital Quantum Computation arxiv:quant-ph/ v1 6 Dec 1996 Edard Farhi Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA Sam Gutmann Department of Mathematics Northeastern University Boston, MA MIT-CTP-2593, quant-ph/ December 1996 Abstract We solve a problem, hich hile not fitting into the usual paradigm, can be vieed as a quantum computation. Suppose e are given a quantum system described by an N dimensional Hilbert space ith a Hamiltonian of the form E here is an unknon (normalized) state. We sho ho to discover by adding a Hamiltonian (independent of ) and evolving for a time proportional to N 1/2 /E. We sho that this time is optimally short. This process is an analog analogue to Grover s algorithm, a computation on a conventional (!) quantum computer hich locates a marked item from an unsorted list of N items in a number of steps proportional to N 1/2. This ork as supported in part by The Department of Energy under cooperative agreement DE-FC02-94ER40818 farhi@mitlns.mit.edu sgutm@nuhub.neu.edu 1

2 Although a quantum computer, beyond certain elementary gates, has not yet been constructed, a paradigm [1] for quantum computation is in place. A quantum computer is envisaged as acting on a collection of spin 1/2 particles sitting at specified sites. Each elementary operation is a unitary transformation hich acts on the spins at one or to sites. A quantum computer program, or algorithm, is a definite sequence of such unitary transformations. For a given initial spin state, the output of the program is the spin state after the sequence of transformations has acted. The length of the algorithm is equal to the number of elementary unitary transformations hich make up the algorithm. This frameork for quantum computation is general enough that any ordinary digital computer program can be turned into a quantum computer algorithm. (It is required that the ordinary program be reversible; hoever any ordinary computer program can be ritten in reversible code.) Quantum computers can go beyond ordinary computers hen they act on superpositions of states and take advantage of interference effects. An example of a quantum algorithm hich outperforms any classical algorithm designed to solve the same problem is the Grover algorithm [2]. Here e are given a function f(a) defined on the integers a from 1 to N. The function has the property that it takes the value 1 on just a single element of its domain,, and it has the value 0 for all a. With only the ability to call the function f, the task is to find. On a classical computer this requires, on average, N/2 calls of the function f. Hoever Grover shoed that ith a quantum computer can be found ith of order N 1/2 function calls. This remarkable speed-up illustrates the poer of quantum computation. (In the appendix e explain ho the Grover algorithm orks.) In this paper e consider quantum computation differently, as controlled Hamiltonian time evolution of a system, obeying the Schrodinger equation i d dt ψ = H(t) ψ, (1) hich is designed to solve a specified problem. We illustrate this ith an example. Suppose e are given a Hamiltonian in an N dimensional vector space and e are told that the Hamiltonian has one eigenvalue E 0 and all the others are 0. The task is to find the eigenvector hich has eigenvalue E. We no give a solution to this problem and then explain in hat sense it is optimal. We are given H = E (2) ith unspecified and = 1. Pick some normalized vector s hich of course does not depend on since e don t yet kno hat is. No add to H the driving Hamiltonian H D = E s s (3) so that the full Hamiltonian is H = H + H D. (4) We no calculate the time evolution of the state ψ, t hich at t = 0 is s, ψ, t = e iht s. (5) 2

3 It suffices to confine our attention to the to dimensional subspace spanned by s and. The vectors s and are (generally) not orthogonal and e call their inner product x, s = x (6) here x can be taken to be real and positive since any phase in s can ultimately be absorbed in s. We ill discuss the expected size of x shortly. No the vectors r = 1 ( s x ) (7) 1 x 2 and are orthonormal. In the, r basis the Hamiltonian (4) is H = E [ 1 + x 2 x 1 x 2 x 1 x 2 1 x 2 ] (8) and s = [ x 1 x 2 ] (9) No a simple calculation gives ψ, t = e iet [ x cos(ext) i sin(ext) 1 x2 cos(ext) ]. (10) Thus e see that at time t the probability of finding the state is P(t) = sin 2 (Ext) + x 2 cos 2 (Ext) (11) and that at a time t m given by the probability is one. t m = π 2Ex (12) Ho big do e expect x to be? In an N dimensional complex vector space, if you pick to normalized vectors at random (uniformly on the 2N-1 dimensional unit sphere), then the expected value of the inner product squared is 1/N so e kno that the expected value of x is of order N 1/2. Thus starting ith s, for the probability of finding to be appreciable e must ait a time of order N 1/2 /E. This is the analog analogue of the Grover algorithm result. Note that the eigenvalues of the Hamiltonian (8) are E(1 ± x). Thus the difference in eigenvalues is (2xE) hich is of order E/N 1/2. By the time-energy uncertainty principle, the time required to evolve substantially, that is from s to, must be of order N 1/2 /E hich is the time e found. You might think that by increasing the energy difference, that is for example, by using H D = E s s ith E E you could speed up the procedure for finding. Hoever the next result shos that this is not the case. We no sho that our procedure for finding, in a time hich gros like N 1/2 /E, is optimally short. The proof e give here is the analog analogue of the oracle proof [3] hich 3

4 can be used to sho that the Grover algorithm is optimal for the problem it sets out to solve. Again e are given the Hamiltonian H = E and e ish to add some Hamiltonian H D (t) to it hich drives the system to a state hich allos us to determine. In an N dimensional vector space, there are N linearly independent choices for. We can pick these to be a basis for the vector space and e then have H = E = E. (13) The idea of the proof is this: Start ith some initial -independent state i and evolve it ith the Hamiltonian H = H + H D (t). (14) After a time t the state e get must be substantially different from hat e ould have gotten using H + H D (t) or else e can not tell from. Let ith i d dt ψ, t = (H + H D (t)) ψ, t (15) ψ, 0 = i. In order for ψ, t to differ sufficiently from ψ, t it is certainly necessary that, for all (but one), ψ, t differs sufficiently from any -independent vector. (If some of the ψ, t ere very close to a particular -independent vector, e could not tell them apart.) Let ψ, t evolve ith H D (t), that is, ith i d dt ψ, t = H D(t) ψ, t (16) ψ, 0 = i. We ill use ψ, t as a -independent vector hich the ψ, t must differ from. We require t to be large enough that ψ, t ψ, t 2 ε for some fixed ε hich implies ψ, t ψ, t 2 Nε. (17) No consider d ψ, t ψ, t 2 = 2Re d dt dt ψ, t ψ, t (18) hich upon using (15) and (16) gives d ψ, t ψ, t 2 = 2 Im ψ, t H ψ, t dt 2 ψ, t H ψ, t (19) 2H ψ, t. 4

5 We no sum on and use the fact that if N i=1 a i 2 = 1 then N i=1 a i N 1/2 along ith (13) to obtain d ψ, t ψ, t 2 2EN 1/2. (20) dt Since ψ, 0 = ψ, 0 e have Therefore in order to satisfy (17) e must have ψ, t ψ, t 2 2EN 1/2 t. (21) t εn1/2 2E. (22) This shos that the H D e have chosen allos us to determine as quickly as possible in terms of N. Appendix: The Grover Algorithm We are given a function f(a) ith a = 1,...N such that f() = 1 and f(a) = 0 for a. We assume that the function f(a) can be calculated using ordinary (reversible) computer code. The goal is to find. Classically this requires, on average, N/2 evaluations of the function f. We no explain ho the Grover algorithm solves this problem; see also [4]. The quantum computer acts on a vector space hich has an orthonormal basis a ith a = 1,...N. It is possible to rite a quantum computer algorithm hich implements the unitary transformation U f a = ( 1) f(a) a. (A1) Equivalently e can rite U f = 1 2 (A2) The quantum computer algorithm hich implements U f requires to evaluations of the function f because it is necessary to erase certain ork bits hich e have supressed. It is also assumed that the ordinary code hich is used to evaluate f has a length hich does not gro like N to a positive poer. Then the number of to bit quantum computer steps required to evaluate f ill also not gro as fast as N to a poer. No consider the vector s = 1 a. N 1/2 It is also possible to rite quantum computer code hich implements the unitary operator a (A3) U s = 2 s s 1. (A4) The number of to bit operations required to implement U s gros more sloly than N to any positive poer. 5

6 The Grover algorithm consists of letting the operator U s U f act k times on the vector s. To see hat happens e can restrict our attention to the to dimensional subspace spanned by s and. Let 1 r = a (A5) N 1 so that and r form an orthonormal basis for the relevant subspace. In the, r basis the operator U s U f takes the form [ ] cos θ sin θ U s U f = (A6) sin θ cos θ here cos θ = 1 2/N. This implies that [ ] cos(kθ) sin(kθ) (U s U f ) k =. (A7) sin(kθ) cos(kθ) No for N large θ 2N 1/2 so each application of U s U f is a rotation by an angle 2N 1/2. In the, r basis, the initial state s is [ N s = 1/2 ] (1 1 (A8) N )1/2 hich is very close to r. Hoever after k steps here kθ = π/2 the algorithm has rotated the initial state to lie (almost) along. This requires k πn 1/2 /4 steps. Each step actually requires to evaluations of f so the number of evaluations of f required to find gros like N 1/2. Accordingly the number of to bit operations required to implement the algorithm also gros like N 1/2. Acknoledgement E. F. ould like to thank the theory group at Università di Roma 1 for their hospitality and discussions as ell as the INFN for partial support. a References [1] For a summary and references see A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A 52, 3457 (1995). [2] L. K. Grover, A fast quantum mechanical algorithm for database search, quantph/ [3] C. H. Bennett, E. Bernstein, G. Brassard, and U. V. Vazirani, Strengths and eaknesses of quantum computing, SIAM Journal on Computing, to appear. [4] M. Boyer, G. Brassard, P. Hoeyer, and A. Tapp, Tight bounds on quantum searching, quant-ph/