An alternate algorithm for directed quantum search

Andrew M. Childs, Jeffrey Goldstone

Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of $N$ items laid out in $d$ spatial dimensions can be searched in time of order $\sqrt{N}$ for $d>2$, and in time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$ for $d=2$. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that $\sqrt{N}$ speedup can also be achieved on the hypercube. We show that full $\sqrt{N}$ speedup can be achieved on a $d$-dimensional periodic lattice for $d>4$. In $d=4$, the quantum walk search algorithm takes time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$, and in $d<4$, the algorithm does not provide substantial speedup.

Tad Hogg

Xishuo Liu, Stark Christiaan Draper

Linear programming (LP) decoding using the alternating direction method of multipliers (ADMM) has been shown to be an efficient algorithm. A non-convex variation based on the ADMM LP decoder called the ADMM penalized decoder was introduced by Liu et al. (IEEE ITW, Sep. 2012) to close the signal-to-noise ratio (SNR) gap between LP decoding and classic belief propagation (BP) decoding. This algorithm was shown to achieve or outperform BP decoding at all SNRs, including high SNRs where BP decoding suffers from the error floor effect. In this paper, we study the behaviors of the ADMM penalized decoder at high SNRs where simulation is infeasible. We use a generic tool called instanton analysis and propose an instanton search algorithm for the ADMM penalized decoder. We then apply the algorithm to the [155, 64] Tanner code and a [1057, 813] LDPC code. We show that the instanton information we obtained provides good predictions for word-error-rate curve at high SNRs. In addition, our results suggest that the ADMM penalized decoder can suffer from trapping sets.

Pulak Ranjan Giri, V. E. Korepin

Gabriel Turinici, Claude Le Bris, HERSCHEL A. RABITZ

The laboratory closed-loop optimal control of quantum phenomena, expressed as minimizing a suitable cost functional, is currently implemented through an optimization algorithm coupled to the experimental apparatus. In practice, the most commonly used search algorithms are variants of genetic algorithms. As an alternative choice, a direct search deterministic algorithm is proposed in this paper. For the simple simulations studied here, it outperforms the existing approaches. An additional algorithm is introduced in order to reveal some properties of the cost functional landscape.

V. E. Korepin, Lov K. Grover