Quantum games and quantum algorithms

A quantum algorithm for an oracle problem can be understood as a quantum strategy for a player in a two-player zero-sum game in which the other player is constrained to play classically. I formalize this correspondence and give examples of games (and hence oracle problems) for which the quantum player can do better than would be possible classically. The most remarkable example is the Bernstein-Vazirani quantum search algorithm which I show creates no entanglement at any timestep.Comment: 10 pages, plain TeX; to appear in the AMS Contemporary Mathematics volume: Quantum Computation and Quantum Information Science; revised remarks about other quantum games formalisms; for related work see http://math.ucsd.edu/~dmeyer/research.htm

Computational capacity of the universe

Merely by existing, all physical systems register information. And by evolving dynamically in time, they transform and process that information. The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The universe is a physical system. This paper quantifies the amount of information that the universe can register and the number of elementary operations that it can have performed over its history. The universe can have performed no more than $10^{120}$ ops on $10^{90}$ bits.Comment: 17 pages, TeX. submitted to Natur

Climbing Mount Scalable: Physical-Resource Requirements for a Scalable Quantum Computer

The primary resource for quantum computation is Hilbert-space dimension. Whereas Hilbert space itself is an abstract construction, the number of dimensions available to a system is a physical quantity that requires physical resources. Avoiding a demand for an exponential amount of these resources places a fundamental constraint on the systems that are suitable for scalable quantum computation. To be scalable, the effective number of degrees of freedom in the computer must grow nearly linearly with the number of qubits in an equivalent qubit-based quantum computer.Comment: LATEX, 24 pages, 1 color .eps figure. This new version has been accepted for publication in Foundations of Physic

Synthesis of Topological Quantum Circuits

Topological quantum computing has recently proven itself to be a very powerful model when considering large- scale, fully error corrected quantum architectures. In addition to its robust nature under hardware errors, it is a software driven method of error corrected computation, with the hardware responsible for only creating a generic quantum resource (the topological lattice). Computation in this scheme is achieved by the geometric manipulation of holes (defects) within the lattice. Interactions between logical qubits (quantum gate operations) are implemented by using particular arrangements of the defects, such as braids and junctions. We demonstrate that junction-based topological quantum gates allow highly regular and structured implementation of large CNOT (controlled-not) gate networks, which ultimately form the basis of the error corrected primitives that must be used for an error corrected algorithm. We present a number of heuristics to optimise the area of the resulting structures and therefore the number of the required hardware resources.Comment: 7 Pages, 10 Figures, 1 Tabl

Physical-resource demands for scalable quantum computation

The primary resource for quantum computation is Hilbert-space dimension. Whereas Hilbert space itself is an abstract construction, the number of dimensions available to a system is a physical quantity that requires physical resources. Avoiding a demand for an exponential amount of these resources places a fundamental constraint on the systems that are suitable for scalable quantum computation. To be scalable, the number of degrees of freedom in the computer must grow nearly linearly with the number of qubits in an equivalent qubit-based quantum computer.Comment: This paper will be published in the proceedings of the SPIE Conference on Fluctuations and Noise in Photonics and Quantum Optics, Santa Fe, New Mexico, June 1--4, 200