广义量子干涉原理及对偶量子计算机

THE GENERAL QUANTUM INTERFERENCE PRINCIPLE AND THE DUALITY QUANTUM COMPUTER

我们综述最近提出的广义量子干涉原理及其在量子计算中的应用。广义量子干涉原理是对狄拉克单光子干涉原理的具体化和多光子推广,不但对像原子这样的紧致的量子力学体系适用,而且适用于几个独立的光子这样的松散量子体系。利用广义量子干涉原理,许多引起争议的问题都可以得到合理的解释,例如两个以上的单光子的干涉等问题。从广义量子干涉原理来看双光子或者多光子的干涉就是双光子和双光子自身的干涉,多光子和多光子自身的干涉。广义量子干涉原理可以利用多组分量子力学体系的广义Feynman积分表示,可以定量地计算。基于这个原理我们提出了一种新的计算机,波粒二象计算机,又称为对偶计算机。在原理上对偶计算机超越了经典的计算机和现有的量子计算机。在对偶计算机中,计算机的波函数被分成若干个子波并使其通过不同的路径,在这些路径上进行不同的量子计算门操作,而后这些子波重新合并产生干涉从而给出计算结果。除了量子计算机具有的量子平行性外,对偶计算机还具有对偶平行性。形象地说,对偶计算机是一台通过多狭缝的运动着的量子计算机,在不同的狭缝进行不同的量子操作,实现对偶平行性。目前已经建立起严格的对偶量子计算机的数学理论,为今后的进一步发展打下了基础。本文着重从物理的角度去综述广义量子干涉原理和对偶计算机。现在的研究已经证明,一台d狭缝的n比特的对偶计算机等同与一个n比特+一个d比特(qudit)的普通量子计算机,证明了对偶计算机具有比量子计算机更强大的能力。这样,我们可以使用一台具有n+log2d个比特的普通量子计算机去模拟一个d狭缝的n比特对偶计算机,省去了研制运动量子计算机的巨大的技术上的障碍。我们把这种量子计算机的运行模式称为对偶计算模式,或简称为对偶模式。利用这一联系反过来可以帮助我们理解广义量子干涉原理,因为在量子计算机中一切计算都是普通的量子力学所允许的量子操作,因此广义量子干涉原理就是普通的量子力学体系所允许的原理,而这个原理只是是在多体量子力学体系中才会表现出来。对偶计算机是一种新式的计算机,里面有许多问题期待研究和发展,同时也充满了机会。在对偶计算机中,除了幺正操作外,还可以允许非幺正操作,几乎包括我们可以想到的任何操作,我们称之为对偶门操作或者广义量子门操作。目前这已经引起了数学家的注意,并给出了广义量子门操作的一些数学性质。此外,利用量子计算机和对偶计算机的联系,可以将许多经典计算机的算法移植到量子计算机中,经过改造成为量子算法。由于对偶计算机中的演化是非幺正的,对偶量子计算机将可能在开放量子力学的体系的研究中起到重要的作用。

Here we review the general quantum interference principle and the duality quantum computer which was proposed recently. The general quantum interference principle is the concretization of Dirac＇s statement about single photon interference and the generalization into multi-photon quantum systems. It applies not only to compact quantum system such as an atom, but also to loose quantum system such as a few independent photons. Using the general quantum interference principle, many debated issues can be explained, for instance the interference in a quantum system of two independent photons. From the view point of general quantum interference principle, two photon or multi-photon interferences,are just the interference of the two photon system with the two photon system itself, the interference of multi-photon system with itself. The general quantum interference principle can be expressed in terms of a general Feynman path-integral of a multi-constituents system and be calculated quantitatively. Based on this principle, we proposed a new type of computer, the duality computer or duality quantum computer. In principle, the duality computer is superior to classical computer and the quantum computer in the sense that it can do more than them. In a duality computer,the wave function of the computer is split into many parts, and each part is processed independently through a series of quantum gates, and then the different parts are recombined to get the final results. In addition to quantum parallelism, there is also duality parallelism in which different gate operations can be performed on different parts of the wave function simultaneously in a duality computer. The mathematical theory of duality computer has been established,and this has paved the way for further research. Studies have shown that a n bits duality computer with d-slits can be simulated by a quantum computer with n＋log2d qubits. This relationship makes duality computing in an ordinary quantum computer possible, and also spares the seemingly insurmountable obstacle of building a moving quantum computer which a duality computer at all is. The duality computing in an ordinary quantum computer is called duality computing mode of a quantum computer. This relationship also helps us in understanding the general quantum interference principle. Because the underlying physics of an ordinary quantum computer is just quantum mechanics, the general quantum interference principle is also compatible with the usual quantum mechanics. However this principle exhibits itself only in a multipartite quantum system,thus it is difficult to understand it in simple single particle quantum systems. Duality computer is a new type of computer, and there are many unresolved issues and opportunity as well. In a duality computer,the gate operations need not be unitary and could be any operation that we may think of. The type of gates allowed in duality computing is called duality gate or generalized quantum gate. Generalized quantum gates have attracted the attention of mathematicians and some mathematical results have been obtained on the properties of the generalized quantum gates. Besides the mathematical properties of duality computing, duality computing may offer a bridge between quantum computing and classical computing so that classical algorithms may be transformed into quantum algorithms. Because the nonunitarity of generalized quantum gates,it may play important role in the study of open quantum systems.