Quantum Walks with Encrypted Data

Abstract

In the setting of networked computation, data security can be a significant concern. Here we consider the problem of allowing a server to remotely manipulate client supplied data, in such a way that both the information obtained by the client about the server's operation and the information obtained by the server about the client's data are significantly limited. We present a protocol for achieving such functionality in two closely related models of restricted quantum computation -the Boson sampling and quantum walk models. Due to the limited technological requirements of the Boson scattering model, small scale implementations of this technique are feasible with present-day technology.Introduction -Quantum information processing [1] allows certain key problems, which are believed to be classically hard, to be efficiently solved. Well known examples with real world applications include Shor's algorithm for integer factorisation [2] and Grover's search algorithm [3]. One of the more promising approaches to implementing quantum algorithms is linear optics quantum computation (LOQC) [4,5], where information is encoded into single photons and the their wave properties are manipulated using linear optics elements. Photons are ideally suited to communication, leading naturally to models of distributed quantum computation.A key consideration in any distributed computation scheme is security. Consider two parties, Alice and Bob. Alice has some data to which she would like to apply a computation, whilst Bob has a quantum computer and an algorithm with which he can process the data. However both sides have proprietary knowledge. Alice wants to keep her data secret from others, and Bob wants to keep his algorithm secret. This is related to the problem of homomorphic encryption which allows data to be manipulated without decrypting, so Bob can perform a universal set of operations on Alice's data without ever learning Alice's input state. Universal classical homomorphic encryption was only first discovered in 2009 [6] and subsequently simplified [7]. Closely related is blind computing, where Alice possesses both the data and the algorithm, and Bob owns the computer [8][9][10], as is the quantum private queries protocol [11], which is used to query a database while keeping the query secret.In this paper we describe a technique for solving the above problem, and hence achieving a limited quantum homomorphic encryption using the Boson sampling and multi-walker quantum walk models for quantum computation.The Boson sampling model -A first protocol for universal LOQC was introduced by Knill, Laflamme & Milburn (KLM) [4]. While universal for quantum computation, their protocol is extremely demanding, requiring fast-feedforward and quantum memory, which are technologically challenging and well beyond the capabilities of present-day experiments. Since then numerous simplifications have been proposed, most notably approaches based on cluster states [12][13][14], which significantly reduce physical resource requirements. However they...

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