Efficient circuits for quantum walks
(pp0420-0434)
Chen-Fu
Chiang, Daniel Nagaj, and Pawel Wocjan
doi:https://doi.org/10.26421/QIC10.5-6-4
Abstracts:
We present an efficient general method for realizing a quantum walk
operator corresponding to an arbitrary sparse classical random walk. Our
approach is based on Grover and Rudolph’s method for preparing coherent
versions of efficiently integrable probability distributions [1]. This
method is intended for use in quantum walk algorithms with polynomial
speedups, whose complexity is usually measured in terms of how many
times we have to apply a step of a quantum walk [2], compared to the
number of necessary classical Markov chain steps. We consider a finer
notion of complexity including the number of elementary gates it takes
to implement each step of the quantum walk with some desired accuracy.
The difference in complexity for various implementation approaches is
that our method scales linearly in the sparsity parameter and
poly-logarithmically with the inverse of the desired precision. The best
previously known general methods either scale quadratically in the
sparsity parameter, or polynomially in the inverse precision. Our
approach is especially relevant for implementing quantum walks
corresponding to classical random walks like those used in the classical
algorithms for approximating permanents [3, 4] and sampling from binary
contingency tables [5]. In those algorithms, the sparsity parameter
grows with the problem size, while maintaining high precision is
required.
Key words:
Quantum Walks, Quantum Speedup, Circuit, Markov Chains |