This paper presents a general space-efficient method for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness c and soundness s, either with or without a witness (corresponding to QMA and BQP, respectively). To convert this computation into a new computation with error at most 2^{-p}, the most space-efficient method known requires extra workspace of O(p*log(1/(c-s))) qubits. This space requirement is too large for scenarios like logarithmic-space quantum computations. This paper shows an errorreduction method for unitary quantum computations (i.e., computations without intermediate measurements) that requires extra workspace of just O(log(p/(c-s))) qubits. This in particular gives the first method of strong amplification for logarithmic-space unitary quantum computations with two-sided bounded error. This also leads to a number of consequences in complexity theory, such as the uselessness of quantum witnesses in bounded-error logarithmic-space unitary quantum computations, the PSPACE upper bound for QMA with exponentially-small completeness-soundness gap, and strong amplification for matchgate computations.
@InProceedings{fefferman_et_al:LIPIcs.ICALP.2016.14, author = {Fefferman, Bill and Kobayashi, Hirotada and Yen-Yu Lin, Cedric and Morimae, Tomoyuki and Nishimura, Harumichi}, title = {{Space-Efficient Error Reduction for Unitary Quantum Computations}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.14}, URN = {urn:nbn:de:0030-drops-62975}, doi = {10.4230/LIPIcs.ICALP.2016.14}, annote = {Keywords: space-bounded computation, quantum Merlin-Arthur proof systems, error reduction, quantum computing} }
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