Quantum Algorithm for the Collision Problem

In this note, we give a quantum algorithm that finds collisions in arbitrary r-to-one functions after only O((N/r)^(1/3)) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way.Comment: 8 pages, LaTeX2

Nested quantum search and NP-complete problems

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^\alpha}$, with a constant $\alpha<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $\alpha$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.Comment: 18 pages RevTeX, 3 Postscript figure

Implementation of quantum search algorithm using classical Fourier optics

We report on an experiment on Grover's quantum search algorithm showing that {\em classical waves} can search a $N$-item database as efficiently as quantum mechanics can. The transverse beam profile of a short laser pulse is processed iteratively as the pulse bounces back and forth between two mirrors. We directly observe the sought item being found in $\sim\sqrt{N}$ iterations, in the form of a growing intensity peak on this profile. Although the lack of quantum entanglement limits the {\em size} of our database, our results show that entanglement is neither necessary for the algorithm itself, nor for its efficiency.Comment: 4 pages, 3 figures; minor revisions plus extra referenc

Approximate quantum counting on an NMR ensemble quantum computer

We demonstrate the implementation of a quantum algorithm for estimating the number of matching items in a search operation using a two qubit nuclear magnetic resonance (NMR) quantum computer.Comment: 4 pages LaTeX/RevTex including 4 figures (3 LaTeX, 1 PostScript). Submitted to Physical Review Letter

Adiabatic quantum computation and quantum phase transitions

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR

Randomized Benchmarking of Quantum Gates

A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized pi/2 pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.Comment: 13 page

Ultrahigh Transmission Optical Nanofibers

We present a procedure for reproducibly fabricating ultrahigh transmission optical nanofibers (530 nm diameter and 84 mm stretch) with single-mode transmissions of 99.95 $\pm$ 0.02%, which represents a loss from tapering of 2.6 $\,\times \,$ 10$^{-5}$ dB/mm when normalized to the entire stretch. When controllably launching the next family of higher-order modes on a fiber with 195 mm stretch, we achieve a transmission of 97.8 $\pm$ 2.8%, which has a loss from tapering of 5.0 $\,\times \,$ 10$^{-4}$ dB/mm when normalized to the entire stretch. Our pulling and transfer procedures allow us to fabricate optical nanofibers that transmit more than 400 mW in high vacuum conditions. These results, published as parameters in our previous work, present an improvement of two orders of magnitude less loss for the fundamental mode and an increase in transmission of more than 300% for higher-order modes, when following the protocols detailed in this paper. We extract from the transmission during the pull, the only reported spectrogram of a fundamental mode launch that does not include excitation to asymmetric modes; in stark contrast to a pull in which our cleaning protocol is not followed. These results depend critically on the pre-pull cleanliness and when properly following our pulling protocols are in excellent agreement with simulations.Comment: 32 pages, 10 figures, accepted to AIP Advance

Quantum Chinos Game: winning strategies through quantum fluctuations

We apply several quantization schemes to simple versions of the Chinos game. Classically, for two players with one coin each, there is a symmetric stable strategy that allows each player to win half of the times on average. A partial quantization of the game (semiclassical) allows us to find a winning strategy for the second player, but it is unstable w.r.t. the classical strategy. However, in a fully quantum version of the game we find a winning strategy for the first player that is optimal: the symmetric classical situation is broken at the quantum level.Comment: REVTEX4.b4 file, 3 table

A critical evaluation of the electronic surgical logbook

BACKGROUND: The Association of Surgeons of Great Britain and Ireland (ASGBI) devised the electronic surgical logbook (version 2.4) for higher trainees in General Surgery enabling trainees to compile a uniform data set of their operative and training experience. This is in use by higher surgical trainees (HST) in the United Kingdom. This logbook permits trainees to submit data centrally into a Regional Analysis Database (RAD). With the implementation of the European Working Time Directive (EWTD) there is need for reliable data to assess the effects of the directive on training. In order to draw meaningful conclusions from the database the quality of data needs to be validated. We critically analysed the RAD in the Yorkshire region for a one-year period. METHODS: The RAD from the ASGBI for the Yorkshire region was analysed. Data for the period 01/10/2002–30/09/2003 was identified and interrogated using Microsoft Excel (2000 version). The RAD was compared with information obtained from the Regional Surgical Advisor for Yorkshire with respect to hospitals, surgical consultants and HST's in the region during the study period. RESULTS: There were 13,755 operations entered for the study period. 579 corrections to the data had to be made (4.2%) and a further 1140 entries were deleted (8.2%). Following corrections and deletions 12,615 operative entries were available for analysis. Overall 12.5% of the data required either correction or exclusion from the database prior to analysis. CONCLUSION: The RAD has a large dataset useful to monitor and assess training. However, the quality of the data needs to be verified prior to use. Recommendations have been made to develop the ASGBI logbook, which would eventually translate to improved data reliability of the RAD