Quantum search with non-orthogonal entangled states

We propose a classical to quantum information encoding system using non–orthogonal states and apply it to the problem of searching an element in a quantum list. We show that the proposed encoding scheme leads to an exponential gain in terms of quantum resources and, in some cases, to an exponential gain in the number of runs of the protocol. In the case where the output of the search algorithm is a quantum state with some particular physical property, the searched state is found with a single query to the introduced oracle. If the obtained quantum state must be converted back to classical information, our protocol demands a number of repetitions that scales polynomially with the number of qubits required to encode a classical string. PACS numbers: Using quantum mechanical tools to design algorithms solving problems initially stated in a classical context is one goal goal of quantum information theory. In some cases, quantum algorithms outperform known classical ones. This is the case of Shor’s factorization algorithm [1], that provides an exponential gain on the number of operations required to factorize an arbitrary number. Another great achievement of quantum information is Grover’s search algorithm that shows a quadratic gain, when compared to its classical analog, on the number of steps required to find a given element in an unsorted list [2, 3]. These algorithms are based on encoding classical information (arrays of bits) in quantum states (qubits) and using quantum mechanical allowed operations to manipulate these qubits. The output of the algorithm is thus a quantum state, that depending on the considered application, can be converted back to a classical bit string [4, 5] or not. Examples of the latter are algorithms looking for properties of solutions of systems of linear equations [6] and quantum simulations [7]. In the present Letter we propose an original encoding system using entangled non–orthogonal states and apply it to quantum search problems. The proposed encoding and search algorithm lead, in some cases, to an exponential speed–up in the number of required oracle queries when compared to their classical analog. They also allow for an exponential reduction of space complexity [8] when compared to the usual encoding of quantum and classical protocols. The studied search protocols use a single application of the Grover search operator to a redundant quantum list of arbitrary size composed of bit strings encoded in non–orthogonal states. The outcome of this operation is a quantum state that can be directly used for some quantum application using entangled states (such as teleportation [9]) or that can be assigned to the expectation value of some physical property [6]. Alternatively, the quantum outcome can be univocally associated to one classical bit string out of a list of 2 2n possible strings after a set of O(nlogn) measurements (or repetitions of the protocol). We start by recalling the principles of the Grover search algorithm, that is the basic tool used in the protocol exposed in the present Letter. The Grover search algorithm applies to the following problem: given a set of all possible N = 2 2n elements (bit strings) that can be encoded in an unsorted way in an array of 2n bits, we would like to find a given (known) element with close to unit probability. In 1996, L. Grover proposed a quantum algorithm solving this problem in a number of steps ns ≃ O(N 1/2 ) [2, 3] which works as follows.