arXiv subjects
Publications and source records attributed to Carlos Mochon.
Paper withdrawn. Results are similar to V. P. Belavkin, "Optimum distinction of non-orthogonal quantum signals," Radio Engineering and Electronic Physics, vol. 20, June 1975, p. 39-47.
"God does not play dice. He flips coins instead." And though for some reason He has denied us quantum bit commitment. And though for some reason he has even denied us strong coin flipping. He has, in His infinite mercy, granted us quantum weak coin flipping so that we too may flip coins. Instructions for the flipping of coins are contained herein. But be warned! Only those who have mastered Kitaev's formalism relating coin flipping and operator monotone functions may succeed. For those foolhardy enough to even try, a complete tutorial is included.
Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of finding shortest paths on a manifold. The study of these shortest paths leads to lower bounds of the original unitary oracle problem. A number of example Hamiltonian oracles are studied in this paper, including oracle interrogation and the problem of computing the XOR of the hidden bits. Both of these problems are related to the study of geodesics on spheres with non-round metrics. For the case of two hidden bits a complete description of the geodesics is given. For n hidden bits a simple lower bound is proven that shows the problems require a query time proportional to n, even in the continuum limit. Finally, the problem of continuous Grover search is reexamined leading to a modest improvement to the protocol of Farhi and Gutmann.
Each classical public-coin protocol for coin flipping is naturally associated with a quantum protocol for weak coin flipping. The quantum protocol is obtained by replacing classical randomness with quantum entanglement and by adding a cheat detection test in the last round that verifies the integrity of this entanglement. The set of such protocols defines a family which contains the protocol with bias 0.192 previously found by the author, as well as protocols with bias as low as 1/6 described herein. The family is analyzed by identifying a set of optimal protocols for every number of messages. In the end, tight lower bounds for the bias are obtained which prove that 1/6 is optimal for all protocols within the family.
A family of protocols for quantum weak coin-flipping which asymptotically achieve a bias of 0.192 is described in this paper. The family contains protocols with n+2 messages for all n>1. The case n=2 is equivalent to the protocol of Spekkens and Rudolph with bias of 0.207. The case n=3 achieves a bias of 0.199, and n=8 achieves a bias of 0.193. The analysis of the protocols uses Kitaev's description of coin-flipping as a semidefinite program. The paper constructs an analytical solution to the dual problem which provides an upper bound on the amount that a party can cheat.
Quantum protocols for coin-flipping can be composed in series in such a way that a cheating party gains no extra advantage from using entanglement between different rounds. This composition principle applies to coin-flipping protocols with cheat sensitivity as well, and is used to derive two results: There are no quantum strong coin-flipping protocols with cheat sensitivity that is linear in the bias (or bit-commitment protocols with linear cheat detection) because these can be composed to produce strong coin-flipping with arbitrarily small bias. On the other hand, it appears that quadratic cheat detection cannot be composed in series to obtain even weak coin-flipping with arbitrarily small bias.
Anyons obtained from a finite gauge theory have a computational power that depends on the symmetry group. The relationship between group structure and computational power is discussed in this paper. In particular, it is shown that anyons based on finite groups that are solvable but not nilpotent are capable of universal quantum computation. This extends previously published results to groups that are smaller, and therefore more practical. Additionally, a new universal gate-set is built out of an operation called a probabilistic projection, and a quasi-universal leakage correction scheme is discussed.
We present a constructive proof that anyonic magnetic charges with fluxes in a non-solvable finite group can perform universal quantum computations. The gates are built out of the elementary operations of braiding, fusion, and vacuum pair creation, supplemented by a reservoir of ancillas of known flux. Procedures for building the ancilla reservoir and for correcting leakage are also described. Finally, a universal qudit gate-set, which is ideally suited for anyons, is presented. The gate-set consists of classical computation supplemented by measurements of the X operator.
We present efficient implementations of a number of operations for quantum computers. These include controlled phase adjustments of the amplitudes in a superposition, permutations, approximations of transformations and generalizations of the phase adjustments to block matrix transformations. These operations generalize those used in proposed quantum search algorithms.