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Complete Problem for Perfect Zero-Knowledge Quantum Proof
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Jun Yan

2012
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Lecture Notes in Computer Science
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The main purpose of this paper is to prove that (promise) problem Quantum State Identicalness (abbreviated QSI) is essentially complete for perfect zero-knowledge quantum interactive proof (QPZK). Loosely speaking, problem QSI is to decide whether two efficiently preparable quantum states (captured by quantum circuit of polynomial size) are identical or far apart (in trace distance). It is worthy noting that our result does not have classical counterpart yet; natural complete problem for
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... zero-knowledge interactive proof (PZK) is still unknown. Our proof generalizes Watrous' completeness proof for statistical zeroknowledge quantum interactive proof (QSZK), with an extra idea inspired by Malka to deal with completeness error. With complete problem at our disposal, we can immediately prove (and reprove) several interesting facts about QPZK. 1 Some researchers may use term "quantum zero-knowledge proof", but we choose to follow Watrous [27] . Complete Problem for Perfect Zero-Knowledge Quantum Proof 3 among others. Using complete problems, many interesting facts about statistical zero-knowledge (interactive and non-interactive) proof are proved unconditionally (in contrast to those proved based on complexity assumption such as existence of one-way function). Refer to [23] for a survey on the study of statistical zeroknowledge proof via complete problem. In quantum case, we can also study zero-knowledge quantum proof via both transformation and complete problem. Interested readers are referred to [13] , [18] , [15] , et al., for the first approach. With respect to complete problem, Watrous [25] was the first to extend the idea of [20] to study statistical zeroknowledge quantum interactive proof (QSZK). Specifically, in [25] two promise problems, Quantum State Distinguishability (QSD) and its complement Quantum State Closeness (QSC), were shown to be QSZK-complete, where problem QSD can be viewed as the quantum analog of SZK-complete problem SD. Later, in the same spirit, Kobayashi [16] (implicitly) found a problem named Quantum State Closeness to Identity (QSCI) that is complete for statistical zero-knowledge quantum non-interactive proof (NIQSZK). More recently, using quantum extractor, complete problems for QSZK and NIQSZK about (von Neumann) entropy difference were found; see [3, 5] . Thus far, almost all complete problems for statistical zero-knowledge (classical) proof find their quantum counterparts. Motivation and related work

doi:10.1007/978-3-642-27660-6_34
fatcat:y67fxbrhivaaxcohx327kq3l4i