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Manuscript Title: Simulation of n-qubit quantum systems
1. Quantum registers and quantum gates.
Authors: T. Radtke, S. Fritzsche
Program title: FEYNMAN
Catalogue identifier: ADWE_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 173(2005)91
Programming language: Maple 9.5 (but should be compatible with 9.0 and 8.0, too).
Computer: All computers with a license for the computer algebra system Maple [1].
Operating system: Linux, MS Windows XP.
RAM: 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 for even small qubit numbers. However, most of the standard commands (see Sec. 4 for simple examples) react promptly for up to five qubits on a normal single processor machine (≥ 1GHz with 512MB memory) and use less than 10MB memory.
Keywords: Kronecker product, quantum computation, quantum logic gate, quantum register, qubit, reduced density matrix, unitary transformation.
PACS: 03.67.-a, 03.67.Lx.
Classification: 4.15.

Nature of problem:
During the last decade, quantum computing has been found to provide a revolutionary new form of computation. The algorithms by Shor [2] and Grover [3], 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, however, 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.

Solution method:
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 correction models in the future.

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 modelling 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:
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 [4], 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.

[1] Maple is a registered trademark of Waterlo Maple Inc.
[2] P. W. Shor, SIAM J. Sci. Statist. Comput. 26 (1997) 1484.
[3] L. K. Grover, Phys. Rev. Lett. 79 (1997) 325.
[4] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).