30,731 research outputs found
Two-Bit Gates are Universal for Quantum Computation
A proof is given, which relies on the commutator algebra of the unitary Lie
groups, that quantum gates operating on just two bits at a time are sufficient
to construct a general quantum circuit. The best previous result had shown the
universality of three-bit gates, by analogy to the universality of the Toffoli
three-bit gate of classical reversible computing. Two-bit quantum gates may be
implemented by magnetic resonance operations applied to a pair of electronic or
nuclear spins. A ``gearbox quantum computer'' proposed here, based on the
principles of atomic force microscopy, would permit the operation of such
two-bit gates in a physical system with very long phase breaking (i.e., quantum
phase coherence) times. Simpler versions of the gearbox computer could be used
to do experiments on Einstein-Podolsky-Rosen states and related entangled
quantum states.Comment: 21 pages, REVTeX 3.0, two .ps figures available from author upon
reques
Computable functions, quantum measurements, and quantum dynamics
We construct quantum mechanical observables and unitary operators which, if
implemented in physical systems as measurements and dynamical evolutions, would
contradict the Church-Turing thesis which lies at the foundation of computer
science. We conclude that either the Church-Turing thesis needs revision, or
that only restricted classes of observables may be realized, in principle, as
measurements, and that only restricted classes of unitary operators may be
realized, in principle, as dynamics.Comment: 4 pages, REVTE
Some Thoughts on Hypercomputation
Hypercomputation is a relatively new branch of computer science that emerged
from the idea that the Church--Turing Thesis, which is supposed to describe
what is computable and what is noncomputable, cannot possible be true. Because
of its apparent validity, the Church--Turing Thesis has been used to
investigate the possible limits of intelligence of any imaginable life form,
and, consequently, the limits of information processing, since living beings
are, among others, information processors. However, in the light of
hypercomputation, which seems to be feasibly in our universe, one cannot impose
arbitrary limits to what intelligence can achieve unless there are specific
physical laws that prohibit the realization of something. In addition,
hypercomputation allows us to ponder about aspects of communication between
intelligent beings that have not been considered befor
Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers
In this article we continue our investigations of one particle quantum
scattering theory for Schroedinger operators on a set of connected (idealized
one-dimensional) wires forming a graph with an arbitrary number of open ends.
The Hamiltonian is given as minus the Laplace operator with suitable linear
boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's
rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we
provided an explicit algebraic expression for the resulting (on-shell) S-matrix
in terms of the boundary conditions and the lengths of the internal lines and
we also proved its unitarity. Here we address the inverse problem in the
simplest context with one vertex only but with an arbitrary number of open
ends. We provide an explicit formula for the boundary conditions in terms of
the S-matrix at a fixed, prescribed energy. We show that any unitary matrix may be realized as the S-matrix at a given energy by choosing
appropriate (unique) boundary conditions. This might possibly be used for the
design of elementary gates in quantum computing. As an illustration we
calculate the boundary conditions associated to the unitary operators of some
elementary gates for quantum computers and raise the issue whether in general
the unitary operators associated to quantum gates should rather be viewed as
scattering operators instead of time evolution operators for a given time
associated to a quantum mechanical Hamiltonian.Comment: 16 page
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