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arXiv 1106.4063cs.LOquant-ph
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量子程序的验证

Verification of Quantum Programs

Mingsheng Ying, Nengkun Yu, Yuan Feng, Runyao Duan

摘要:

本文发展了量子程序的验证方法学,其贡献有两个方面:1。Sharir,Pnueli and Hart [SIAM J. Comput. 13(1984)292 - 314]提出了一种证明概率程序性质的一般方法,该方法用马尔可夫链对概率程序进行建模,并将关于输出分布的断言扩展为关于所有中间分布的不变断言。他们的方法本质上是经典弗洛伊德归纳断言方法的概率推广。在本文中,我们考虑的量子程序模型的量子马尔可夫链的超算子定义。它表明,Sharir-Pnueli-Hart方法可以优雅地推广到量子程序,利用薛定谔-海森堡对偶量子态和可观测量。特别地,建立了量子程序的Sharir-Pnueli-Hart验证方法的完备性定理。2.正如完备性定理所指出的,Sharir-Pnueli-Hart方法原则上对于验证量子程序的所有性质都是有效的,这些性质可以用厄米算符(可观测量)来表示。但由于验证过程中涉及的计算复杂,在许多实际应用中并不可行。对于有限维状态空间的情形,我们利用超算子的矩阵表示和矩阵的Jordan分解,找到了一种比Sharir-Pnueli-Hart方法简单得多的量子程序验证方法.特别是,这种方法使我们能够很容易地计算平均运行时间,甚至可以分析量子程序在有限维状态空间中的一些有趣的长期行为。

英文摘要:

This paper develops verification methodology for quantum programs, and the contribution of the paper is two-fold: 1. Sharir, Pnueli and Hart [SIAM J. Comput. 13(1984)292-314] presented a general method for proving properties of probabilistic programs, in which a probabilistic program is modeled by a Markov chain and an assertion on the output distribution is extended into an invariant assertion on all intermediate distributions. Their method is essentially a probabilistic generalization of the classical Floyd inductive assertion method. In this paper, we consider quantum programs modeled by quantum Markov chains which are defined by super-operators. It is shown that the Sharir-Pnueli-Hart method can be elegantly generalized to quantum programs by exploiting the Schrödinger-Heisenberg duality between quantum states and observables. In particular, a completeness theorem for the Sharir-Pnueli-Hart verification method of quantum programs is established. 2. As indicated by the completeness theorem, the Sharir-Pnueli-Hart method is in principle effective for verifying all properties of quantum programs that can be expressed in terms of Hermitian operators (observables). But it is not feasible for many practical applications because of the complicated calculation involved in the verification. For the case of finite-dimensional state spaces, we find a method for verification of quantum programs much simpler than the Sharir-Pnueli-Hart method by employing the matrix representation of super-operators and Jordan decomposition of matrices. In particular, this method enables us to compute easily the average running time and even to analyze some interesting long-run behaviors of quantum programs in a finite-dimensional state space.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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