Simulation of n-qubit quantum systems. I. Quantum registers and quantum gates

Abstract: During recent years, quantum computations and the study of n-qubit quantum systems have attracted a lot of interest, both in theory and experiment. Apart from the promise of performing quantum computations, however, these investigations also revealed a great deal of difficulties which still need to be solved in practice. In quantum computing, unitary and non-unitary quantum operations act on a given set of qubits to form (entangled) states, in which the information is encoded by the overall system often referred to as quantum registers. To facilitate the simulation of such n-qubit quantum systems, we present the Feynman program to provide all necessary tools in order to define and to deal with quantum registers and quantum operations. Although the present version of the program is restricted to unitary transformations, it equally supports—whenever possible—the representation of the quantum registers both, in terms of their state vectors and density matrices. In addition to the composition of two or more quantum registers, moreover, the program also supports their decomposition into various parts by applying the partial trace operation and the concept of the reduced density matrix. Using an interactive design within the framework of Maple, therefore, we expect the Feynman program to be helpful not only for teaching the basic elements of quantum computing but also for studying their physical realization in the future. Program summary: Title of program: Feynman Catalogue number:ADWE Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWE Program obtainable from:CPC Program Library, Queen''s University of Belfast, N. Ireland Licensing provisions:None Computers for which the program is designed:All computers with a license of the computer algebra system Maple [Maple is a registered trademark of Waterlo Maple Inc.] Operating systems or monitors under which the program has been tested:Linux, MS Windows XP Programming language used: Maple 9.5 (but should be compatible with 9.0 and 8.0, too) Memory and time required to execute with typical data:Storage and time requirements critically depend on the number of qubits, n, in the quantum registers due to the exponential increase of the associated Hilbert space. In particular, complex algebraic operations may require large amounts of memory even for small qubit numbers. However, most of the standard commands (see Section 4 for simple examples) react promptly for up to five qubits on a normal single-processor machine ( with 512 MB memory) and use less than 10 MB memory. No. of lines in distributed program, including test data, etc.: 8864 No. of bytes in distributed program, including test data, etc.: 493 182 Distribution format: tar.gz Nature of the physical problem:During the last decade, quantum computing has been found to provide a revolutionary new form of computation. The algorithms by Shor [P.W. Shor, SIAM J. Sci. Statist. Comput. 26 (1997) 1484] and Grover [L.K. Grover, Phys. Rev. Lett. 79 (1997) 325. ], for example, gave a first impression how one could solve problems in the future, that are intractable otherwise with all classical computers. Broadly speaking, quantum computing applies quantum logic gates (unitary transformations) on a given set of qubits, often referred to a quantum registers. Although, the theoretical foundation of quantum computing is now well understood, there are still many practical difficulties to be overcome for which (classical) simulations on n-qubit systems may help understand how quantum algorithms work in detail and what kind of physical systems and environments are most suitable for their realization. Method of solution:Using the computer algebra system Maple, a set of procedures has been developed to define and to deal with n-qubit quantum registers and quantum logic gates. It provides a hierarchy of commands which can be applied interactively and which is flexible enough to incorporate non-unitary quantum operations and quantum error corrections models in the future. Restrictions on the complexity of the problem:The present version of the program facilitates the set-up and manipulation of quantum registers by a large number of (predefined) quantum logic gates. In contrast to such idealized unitary transformations, however, less attention has been paid so far to non-unitary quantum operations or to the modeling of decoherence phenomena, although various suitable data structures are already designed and implemented in the code. A further restriction concerns the number of qubits, n, due to the exponentially growing time and memory requirements. Up to now, most of the complex commands are restricted to quantum registers with about 6 to 8 qubits, if use has to be made of a standard single-processor machine. Unusual features of the program:The Feynman program has been designed for interactive simulations on n-qubit quantum registers with no other restriction than given by the size and time resources of the computer. Apart from the standard quantum gates, as discussed in the literature [M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000], it provides all the necessary tools to generalize these gates for n-qubits (in any given order of the individual qubits). Both common representations of the quantum registers in terms of their state vectors and/or density matrices are equally supported by the program whenever possible. In addition, the program also facilitates the composition of two or more quantum registers into a combined one as well as their decomposition into subsystems by using the partial trace and the use of the reduced density matrix for the individual parts. [Copyright &y& Elsevier]