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On the de-randomization of space-bounded approximate counting problems. (English) Zbl 1329.68287
Summary: It was recently shown that \(\mathsf{SVD}\) and matrix inversion can be approximated in quantum log-space [A. Ta-Shma, in: Proceedings of the 45th annual ACM symposium on theory of computing, STOC ’13. Palo Alto, CA, USA, June 1–4, 2013. New York, NY: Association for Computing Machinery (ACM). 881–890 (2013; Zbl 1293.68129)] for well formed matrices. This can be interpreted as a fully logarithmic quantum approximation scheme for both problems. We show that if \(\mathsf{prBQL} = \mathsf{prBPL}\) then every fully logarithmic quantum approximation scheme can be replaced by a probabilistic one. Hence, if classical algorithms cannot approximate the above functions in logarithmic space, then there is a gap already for languages, namely, \(\mathsf{prBQL}\neq\mathsf{prBPL}\).
On the way we simplify a proof of O. Goldreich [Lect. Notes Comput. Sci. 6650, 191–232 (2011; Zbl 1343.68084)] for a similar statement for time bounded probabilistic algorithms. We show that our simplified algorithm works also in the space bounded setting (for a large set of functions) whereas Goldreich’s approach does not seem to apply in the space bounded setting.

MSC:
68W25 Approximation algorithms
68Q12 Quantum algorithms and complexity in the theory of computing
68W20 Randomized algorithms
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