Browsing by Author "Cleve, Richard E."
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Item Open Access Applying quantum information to fingerprinting schemes and algebraic structures(2004) de Beaudrap, Jonathan Robert Niel; Cleve, Richard E.This research consists of three topics in quantum information science. First is an examination of the advantages that quantum information gives over classical information in very small instances of the fingerprinting problem. The second topic presents the properties of quantum Fourier transforms (QFTs) with respect to finite abelian groups, extends these ideas to finite unitary rings, and demonstrates how these properties characterize QFTs. The final topic applies some of the properties of the QFT to solve an oracle problem defined in terms of scrambled or "hidden" polynomial functions. Understanding quantum information often requires a major change in perspective, and thus has a reputation of being very difficult. Muddled and vague "popular presentations" contribute to the problem. In reaction to these issues, an introduction is presented here which relies on mathematics and supplementary commentary to present the ideas of quantum information, suitable for any reader with basic knowledge of linear algebra.Item Open Access Improved algorithms for approximate quantum fourier transforms and sparse hamiltonian simulations(2004) Ahokas, Graeme Robert; Cleve, Richard E.We investigate and improve algorithms for computing the approximate quantum Fourier transform (QFT ) and simulating a sparse Hamiltonian. We present an improved near-linear time algorithm for the approximate QFT modulo 2™. We give a circuit for the arbitrary modulus version that matches the best currently-known bound. We then relate the difficulty of the arbitrary modulus QFT to classical integer multiplication and division. We also investigate the problem of simulating an n-qubit sparse Hamiltonian. As input we receive the system's start state, a black box which, via queries, provides the non-zero entries of each row of the Hamiltonian, and a time t. We present an improved polynomial-time quantum algorithm for computing an approximation of the state of the system at time t. Prior to this, we provide an introduction to quantum computing, and survey some quantum algorithms that are used in our improved algorithms.Item Open Access On the security of the BB84 quantum key distribution protocol(2004) Cannings, Richard; Scheidler, Renate; Cleve, Richard E.The BB84 quantum key distribution (QKD) protocol enables two authenticated parties to generate a secret key over an insecure quantum channel. Using a standardized security definition, we prove that BB84 is secure and include explicit bounds on its security. Furthermore, our use of quantum circuit diagrams simplify the Shor-Preskill proof. Namely, we can reduce the Modified Lo-Chau QKD to a practical version of BB84 using the observation from Shor and Preskill that one may ignore a correctable number of phase errors, and the fact that computational basis measurements commute with controls of CNOT operations. The first four chapters provide the required background material on quantum computing, information theory, cryptography, coding theory, and quantum error correcting codes. Chapter 5 presents protocols for entanglement purification. Chapter 6 reduces an entanglement purification protocol to the Modified Lo-Chau QKD, and proves that it is secure. Finally, a reduction from the Modified Lo-Chau QKD to BB84 establishes the security of the latter.Item Open Access Shared randomness and entanglement in communication complexity(2006) Gavinsky, Dmitry; Watrous, John; Cleve, Richard E.Item Open Access The classical and quantum complexity of the Goldreich-Levin problem with applications to bit commitment(2004) Adcock, Mark R.A.; Cleve, Richard E.The classical Goldreich-Levin (G-L) theorem is presented as a block-box query problem, referred to herein as the G-L problem. The query complexity of this problem is bounded in both classical and quantum settings. The well-known upper bound of the classical G-L problem is analyzed in a pedagogical manner. A proof of the lower bound of the classical G-L problem is given using the techniques of classical information theory. This classical analysis is then extended to the realm of quantum computin g by noting the similarity of the noiseless G-L problem to the inner-product query problem solved by the quantum circuit defined by Bernstein and Vazirani (B-V) . An upper bound of the query complexity of the quantum G-L problem is proven by extension of the B-V circuit to incorporate noisy inner-product queries. The lower bound of the query complexity of the quantum G-L problem is proven by adapting the proof of the optimaliy of the quantum search algorithm to include modified inner-product queries. Both the classical and quantum versions of the Goldreich-Levi n theorem have cryptographic applications in the area of bit commitment. A discussion of the impossibility of unconditional quantum bit commitment is followed by the presentation of both classical and quantum bit commitment protocols that are based on the assumption of the existence of classical and quantum one-way permutations. The relative security of the classical and quant m protocols are compared where it is shown that the quantum version is quantitatively more secure.