ua nt - p h / 96 12 02 6 v 1 6 D ec 1 99 6 An Analog Analogue of a Digital Quantum Computation ∗

We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system described by an N dimensional Hilbert space with a Hamiltonian of the form E|w〉〈w| where |w〉 is an unknown (normalized) state. We show how to discover |w〉 by adding a Hamiltonian (independent of |w〉) and evolving for a time proportional to N1/2/E. We show that this time is optimally short. This process is an analog analogue to Grover’s algorithm, a computation on a conventional (!) quantum computer which locates a marked item from an unsorted list of N items in a number of steps proportional to N1/2. This work was supported in part by The Department of Energy under cooperative agreement DE-FC0294ER40818 [email protected] [email protected] 1 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 which acts on the spins at one or two 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 which make up the algorithm. This framework 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; however any ordinary computer program can be written in reversible code.) Quantum computers can go beyond ordinary computers when they act on superpositions of states and take advantage of interference effects. An example of a quantum algorithm which outperforms any classical algorithm designed to solve the same problem is the Grover algorithm [2]. Here we 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, w, and it has the value 0 for all a 6= w. With only the ability to call the function f , the task is to find w. On a classical computer this requires, on average, N/2 calls of the function f . However Grover showed that with a quantum computer w can be found with of order N function calls. This remarkable speed-up illustrates the power of quantum computation. (In the appendix we explain how the Grover algorithm works.) In this paper we consider quantum computation differently, as controlled Hamiltonian time evolution of a system, obeying the Schrodinger equation i d dt |ψ〉 = H(t)|ψ〉, (1) which is designed to solve a specified problem. We illustrate this with an example. Suppose we are given a Hamiltonian in an N dimensional vector space and we are told that the Hamiltonian has one eigenvalue E 6= 0 and all the others are 0. The task is to find the eigenvector |w〉 which has eigenvalue E. We now give a solution to this problem and then explain in what sense it is optimal. We are given Hw = E|w〉〈w| (2) with |w〉 unspecified and 〈w|w〉 = 1. Pick some normalized vector |s〉 which of course does not depend on |w〉 since we don’t yet know what |w〉 is. Now add to Hw the “driving” Hamiltonian HD = E|s〉〈s| (3) so that the full Hamiltonian is H = Hw +HD. (4) We now calculate the time evolution of the state |ψw, t〉 which at t = 0 is |s〉, |ψw, t〉 = e |s〉. (5) 2 It suffices to confine our attention to the two dimensional subspace spanned by |s〉 and |w〉. The vectors |s〉 and |w〉 are (generally) not orthogonal and we call their inner product x, 〈s|w〉 = x (6) where x can be taken to be real and positive since any phase in 〈s|w〉 can ultimately be absorbed in |s〉. We will discuss the expected size of x shortly. Now the vectors |r〉 = 1 √ 1− x2 (|s〉 − x|w〉) (7) and |w〉 are orthonormal. In the |w〉, |r〉 basis the Hamiltonian (4) is