Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification of any experiment capable of refuting hypercomputation. We consider the implications of relativistic algorithms capable of solving the (Turing) Halting Problem. We also reject as a fallacy the argument that hypercomputation has no relevance because non-computable values are indistinguishable from sufficiently close computable approximations. In addition to

We study the problem of compressing a block of symbols (a block quantum state) emitted by a memoryless quantum Bernoulli source. We present a simple-to-implement quantum algorithm for projecting, with high probability, the block quantum state onto the typical subspace spanned by the leading eigenstates of its density matrix. We propose a fixed-rate quantum Shannon-Fano code to compress the projected block quantum state using a per symbol code rate that is slightly higher than the von Neumann entropy limit. Finally, we propose quantum arithmetic codes to efficiently implement quantum Shannon-Fano codes. Our arithmetic encoder/decoder have a cubic circuit and a cubic computational complexity in the block size. The encoder and decoder are quantum-mechanical inverses of each other, and constitute an elegant example of reversible quantum computation. Keywords: quantum computation, quantum information theory, quantum measurement, noiseless coding, reversible computation, Schumacher coding, a...

In the framework of a model for quantum computer media, a nondigital implementation of the arithmetic of the real numbers is described. For this model, an elementary storage “cell” is an ensemble of qubits (quantum bits). It is found that to store an arbitrary real number it is sufficient to use four of these ensembles and the arithmetical operations can be implemented by fixed quantum circuits.

The continuous minituarization of integrated circuits is going to affect the underlying physics of the future computers. This new physics first came into play as the effect of Coulomb blockade in electron transport through small conducting islands. Then, as the size of the island L continued to shrink further, the quantum phase coherence length became larger than L leading to mesoscopic fluctuations – fluctuations of the island’s quantum mechanical properties upon small external perturbations. Quantum coherence of the mesoscopic systems is essential for building reliable quantum computer. Unfortunately, one can not completely isolate the system from the environment and its coupling to the environment inevitably leads to the loss of coherence or decoherence. All these effects are to be thoroughly investigated as the potential of the future applications is enormous. In this thesis I find an analytic expression for the conductance of a single electron transistor in the regime when temperature, level spacing, and charging energy of an island are all of the same order. I also study the correction to the spacing between Coulomb blockade peaks due to finite dot-lead tunnel couplings. I find analytic

Abstract—In the framework of a model for quantum computer media, a nondigital implementation of the arithmetic of the real numbers is described. For this model, an elementary storage “cell ” is an ensemble of qubits (quantum bits). It is found that to store an arbitrary real number it is sufficient to use four of these ensembles and the arithmetical operations can be implemented by fixed quantum circuits. Key words: quantum media, quantum computer, arithmetic, qubit, q-ensemble.