Physics – Quantum Physics
Scientific paper
2008-11-04
Phys. Rev. A 80, 022340 (2009)
Physics
Quantum Physics
17 pages; v3: corrected typos, added a reference about efficient implementations of quantum walks
Scientific paper
We achieve a quantum speed-up of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover's fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed-up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speed-up with respect to accuracy.
Abeyesinghe Anura
Chiang Chen-Fu
Nagaj Daniel
Wocjan Pawel
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