Number Partitioning on a Quantum Computer

The discovery of quantum algorithms that, when executed on a quantum computer (QC), give significant speedup over their classical counterparts [1,2] has given strong impetus to recent developments in the field of quantum computation. In this contribution we present a new quantum algorithm that fully exploits the potential power of a QC. It solves a basic problem of combinatorial optimization: The number partitioning problem. The number partitioning problem (NPP) is defined as follows [3–5]: Does there exist a partitioning of the set A = {a1, . . . , an} of n positive integers aj into two disjoint sets A1 and A2 = A − A1 such that ∑ aj∈A1 aj = ∑ aj∈A2 aj ? The answer to this question is trivially no if the sum of all aj , B ≡ ∑ aj∈A aj , is odd. Therefore we could restrict our attention to cases where B is even but we could equally well ask if there exists a partition such that |∑aj∈A1 aj−∑aj∈A2 aj | ≤ ∆ where ∆ = 1 (0) if B is odd (even). In this paper we will use the latter formulation. Number partitioning is one of Garey and Johnson’s six basic NP-complete problems [4]. It is a key problem in the theory of computational complexity and has a number of important practical applications such as job scheduling, task distribution on multiprocessor machines, VLSI circuit design to name a few. The NPP can be solved by dynamic programming, in a time bounded by a low order polynomial in nB [4]. For a given instance of A = {a1, . . . , an}, we may encode the whole problem using only n log2B bits. As nB is not bounded by any polynomial of the input size n log2B, the dynamic programming algorithm does not solve the NPP in polynomial time [4]. In practice the computation time to solve a NPP depends on the number of bits b = log2B needed to represent the integers aj and B. Numerical simulations using random instances of A show that the solution time grows exponentially with n for n b and polynomially for n b [6–9]. For random instances A, the NPP can be mapped onto a hard problem of statistical mechanics, namely that of finding the ground state of an infiniterange Ising spin glass [10–12]. The transition from the computationally “hard” (exponential) to “easy” (polynomial) has been related to the phase transition in the statistical mechanical system [10,12]. For certain applications there may be additional constraints on the partitioning of the set A. A common one is to fix the difference C between the number of elements in A1 and A2: C ≡ ∑ aj∈A1 1 − ∑ aj∈A2 1. For instance, if C = 0 we ask if there is a partitioning such that the number of elements in A1 and A2 is the same. The potential power of a QC stems from the fact that a QC operates on superpositions of states [13–19]. The interference of these states allows exponentially many computations to be done in parallel [13–19]. A quantum algorithm consists of a sequence of unitary transformations that change the state of the QC [13–19]. Therefore to solve a NPP on a QC, we first have to develop an algorithm that can be expressed entirely in terms of unitary operations. A generic n-qubit QC can be modeled by a collection of n two-state systems, represented by n Pauli-spin matrices {~σ1, . . . , ~σn} [13–19]. The two eigenstates of σ j will be denoted by | ↑〉j and | ↓〉j , corresponding to the states |0〉j and |1〉j of the j-th qubit respectively. The eigenvalues corresponding to | ↑〉j and (| ↓〉j) are Sj = +1 and Sj = −1. They can be used to represent a partitioning of A: We assign aj to A1 (A2) if Sj = 1 (Sj = −1). If we can find a set {S1, . . . , Sn} such that E = ∆ − ∑nj=1 ajSj = 0 we have found one solution of the NPP. The numbers E are the eigenvalues of the Hamiltonian H = ∆−∑nj=1 ajσ j . Thus a solution of the NPP corresponds to an eigenstate of H with energy zero. This is one key to the construction of a polynomial-time quantum algorithm to solve NPP’s on a QC. It is known that the most simple class of spin system, i.e. those involving interactions of the Ising type only, can be used to build universal QC’s [14,18,20]. The Hamiltonian H = ∆ −∑nj=1 ajσ j describes n non-interacting spins in external fields represented by the aj ’s and is of