Space-Efficient Error Reduction for Unitary Quantum Computations

Space-Efficient Error Reduction for Unitary Quantum Computations∗ Bill Fefferman1 , Hirotada Kobayashi2 , Cedric Yen-Yu Lin3 , Tomoyuki Morimae4 , and Harumichi Nishimura5 1 Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, USA Principles of Informatics Research Division, National Institute of Informatics, Tokyo, Japan Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, USA Advanced Scientific Research Leaders Development Unit, Gunma University, Kiryu, Gunma, Japan Department of Computer Science and Mathematical Informatics, Graduate School of Information Science, Nagoya University, Nagoya, Aichi, Japan 2 3 4 5 Abstract 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 con−p vert this computation into a new computation with error
at most 2 , the most space-efficient 1 method known requires extra workspace of O p log 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
p measurements) that requires extra workspace of just O log 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. 1998 ACM Subject Classification F.1.2 Modes of Computation, F.1.3 Complexity Measures and Classes Keywords and phrases space-bounded computation, quantum Merlin-Arthur proof systems, error reduction, quantum computing Digital Object Identifier 10.4230/LIPIcs.ICALP.2016.14 1 Introduction 1.1 Background A very basic topic in various models of quantum computation is whether computation error can be efficiently reduced within a given model. For polynomial-time bounded error quantum computation, the most standard model of quantum computation, the computation error can ∗ A full version [3] of this paper is available at arXiv.org e-Print archive, arXiv:1604.08192 [quant-ph]. EA TC S © Bill Fefferman, Hirotada Kobayashi, Cedric Yen-Yu Lin, Tomoyuki Morimae, and Harumichi Nishimura; licensed under Creative Commons License CC-BY 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Editors: Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi; Article No. 14; pp. 14:1–14:14 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 14:2 Space-Efficient Error Reduction for Unitary Quantum Computations be made exponentially small via a simple repetition followed by a threshold-value decision. This justifies the choice of 2/3 and 1/3 for the completeness and soundness parameters in the definition of the corresponding complexity class BQP. This is also the case for quantum Merlin-Arthur (QMA) proof systems, another central model of quantum computation that models a quantum analogue of NP (more precisely, MA), and the resulting class QMA may again be defined with completeness and soundness parameters 2/3 and 1/3. An undesirable feature of the simple repetition-based error reduction above is that the necessary workspace enlarges linearly with respect to the number of repetitions. More explicitly,
for a given p, the number of repetitions necessary to achieve an error of
2−p is p p O (c−s) times 2 , and thus both the workspace size and the witness size become O (c−s)2 larger. This implies that the simple repetition-based method is no longer useful when either the workspace size or the witness size is required to be logarithmically bounded. Marriott and Watrous [13] developed a more sophisticated method of error reduction for QMA proof systems that does not increase the witness size at all. For a given p, their
p method still requires O (c−s) calls of the original computation and its inverse to achieve 2 −p the computation error 2 , but the method reuses both the workspace and the witness every time it calls the original computation and its inverse. Hence, the witness size never increases in their method. This is a strong property that allows them to show the uselessness of logarithmic-size quantum witnesses in QMA proof systems (i.e., QMAlog = BQP, where QMAlog is the class of problems having QMA proof systems with logarithmic-size quantum witnesses). Their method is also more efficient in workspace size
than the simple repetitionp based method, but still requires extra workspace of size O (c−s) 2 , as it must record outcomes of all the calls of the original computation and its inverse.
p Nagaj, Wocjan, and Zhang [15] succeeded in reducing to O c−s the number of calls of the original computation and its inverse necessary to achieve the computation error 2−p for a given p, while keeping the witness size unchanged. Their method makes use of the phase-estimation algorithm, an essential component of many quantum algorithms including the celebrated factoring algorithm. To achieve error 2−p for a given p, their method
must 1 repeat O(p) times the phase-estimation algorithm with precision of at least O log c−s bits and record all these estimated
phases. Hence, this phase-estimation-based method uses extra 1 . workspace of size O p log c−s As can be seen from above, both of the Marriott-Watrous method and the phaseestimation-based method are still insufficient for the case where the workspace size must be logarithmically bounded. No efficient error-reduction method is known that keeps the size of additionally necessary workspace logarithmically bounded. This is not limited to the case of QMA proof systems, and in fact almost no efficient error-reduction method is known even in the case of logarithmic-space quantum computations, and in the case of spacebounded quantum computations in general. The study of general space-bounded quantum computations was initiated by Watrous [21] based on quantum Turing machines. Several models of space-bounded quantum computations have been proposed and investigated since then in the literature [22, 23, 24, 9, 14, 18], some considering only logarithmic-space quantum computations and others treating general cases. It is not known whether any of these models are computationally equivalent. It is also not known whether error reduction is possible for logarithmic-space quantum computation defined according to any of these models, except the only known (...truncated)