818 research outputs found
Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
A method for synthesizing quantum gates is presented based on interpolation
methods applied to operators in Hilbert space. Starting from the diagonal forms
of specific generating seed operators with non-degenerate eigenvalue spectrum
one obtains for arity-one a complete family of logical operators corresponding
to all the one-argument logical connectives. Scaling-up to n-arity gates is
obtained by using the Kronecker product and unitary transformations. The
quantum version of the Fourier transform of Boolean functions is presented and
a Reed-Muller decomposition for quantum logical gates is derived. The common
control gates can be easily obtained by considering the logical correspondence
between the control logic operator and the binary propositional logic operator.
A new polynomial and exponential formulation of the Toffoli gate is presented.
The method has parallels to quantum gate-T optimization methods using powers of
multilinear operator polynomials. The method is then applied naturally to
alphabets greater than two for multi-valued logical gates used for quantum
Fourier transform, min-max decision circuits and multivalued adders
Fredkin Gates for Finite-valued Reversible and Conservative Logics
The basic principles and results of Conservative Logic introduced by Fredkin
and Toffoli on the basis of a seminal paper of Landauer are extended to
d-valued logics, with a special attention to three-valued logics. Different
approaches to d-valued logics are examined in order to determine some possible
universal sets of logic primitives. In particular, we consider the typical
connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As
a result, some possible three-valued and d-valued universal gates are described
which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram
An Holistic Extension for Classical Logic via Quantum Fredkin Gate
An holistic extension for classical propositional logic is introduced in the framework of
quantum computation with mixed states. The mentioned extension is obtained by applying the
quantum Fredkin gate to non-factorizable bipartite states. In particular, an extended notion of
classical contradiction is studied in this holistic framework
Eigenlogic: a Quantum View for Multiple-Valued and Fuzzy Systems
We propose a matrix model for two- and many-valued logic using families of
observables in Hilbert space, the eigenvalues give the truth values of logical
propositions where the atomic input proposition cases are represented by the
respective eigenvectors. For binary logic using the truth values {0,1} logical
observables are pairwise commuting projectors. For the truth values {+1,-1} the
operator system is formally equivalent to that of a composite spin 1/2 system,
the logical observables being isometries belonging to the Pauli group. Also in
this approach fuzzy logic arises naturally when considering non-eigenvectors.
The fuzzy membership function is obtained by the quantum mean value of the
logical projector observable and turns out to be a probability measure in
agreement with recent quantum cognition models. The analogy of many-valued
logic with quantum angular momentum is then established. Logical observables
for three-value logic are formulated as functions of the Lz observable of the
orbital angular momentum l=1. The representative 3-valued 2-argument logical
observables for the Min and Max connectives are explicitly obtained.Comment: 11 pages, 2 table
Fuzzy approach for CNOT gate in quantum computation with mixed states
In the framework of quantum computation with mixed states, a fuzzy
representation of CNOT gate is introduced. In this representation, the
incidence of non-factorizability is specially investigated.Comment: 14 pages, 2 figure
Model checking quantum protocols
This thesis describes model checking techniques for protocols arising in quantum information
theory and quantum cryptography. We discuss the theory and implementation of a practical
model checker, QMC, for quantum protocols. In our framework, we assume that the quantum
operations performed in a protocol are restricted to those within the stabilizer formalism; while
this particular set of operations is not universal for quantum computation, it allows us to develop
models of several useful protocols as well as of systems involving both classical and quantum
information processing. We detail the syntax, semantics and type system of QMC’s modelling
language, the logic QCTL which is used for verification, and the verification algorithms that have
been implemented in the tool. We demonstrate our techniques with applications to a number of
case studies
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