2,206 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
Constructing all qutrit controlled Clifford+T gates in Clifford+T
For a number of useful quantum circuits, qudit constructions have been found
which reduce resource requirements compared to the best known or best possible
qubit construction. However, many of the necessary qutrit gates in these
constructions have never been explicitly and efficiently constructed in a
fault-tolerant manner. We show how to exactly and unitarily construct any
qutrit multiple-controlled Clifford+T unitary using just Clifford+T gates and
without using ancillae. The T-count to do so is polynomial in the number of
controls , scaling as . With our results we can construct
ancilla-free Clifford+T implementations of multiple-controlled T gates as well
as all versions of the qutrit multiple-controlled Toffoli, while the analogous
results for qubits are impossible. As an application of our results, we provide
a procedure to implement any ternary classical reversible function on trits
as an ancilla-free qutrit unitary using T gates.Comment: 14 page
Entanglement as a semantic resource
The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly "Hilbert-space dependent". This certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic
Ternary Logic Design in Topological Quantum Computing
A quantum computer can perform exponentially faster than its classical
counterpart. It works on the principle of superposition. But due to the
decoherence effect, the superposition of a quantum state gets destroyed by the
interaction with the environment. It is a real challenge to completely isolate
a quantum system to make it free of decoherence. This problem can be
circumvented by the use of topological quantum phases of matter. These phases
have quasiparticles excitations called anyons. The anyons are charge-flux
composites and show exotic fractional statistics. When the order of exchange
matters, then the anyons are called non-Abelian anyons. Majorana fermions in
topological superconductors and quasiparticles in some quantum Hall states are
non-Abelian anyons. Such topological phases of matter have a ground state
degeneracy. The fusion of two or more non-Abelian anyons can result in a
superposition of several anyons. The topological quantum gates are implemented
by braiding and fusion of the non-Abelian anyons. The fault-tolerance is
achieved through the topological degrees of freedom of anyons. Such degrees of
freedom are non-local, hence inaccessible to the local perturbations. In this
paper, the Hilbert space for a topological qubit is discussed. The Ising and
Fibonacci anyonic models for binary gates are briefly given. Ternary logic
gates are more compact than their binary counterparts and naturally arise in a
type of anyonic model called the metaplectic anyons. The mathematical model,
for the fusion and braiding matrices of metaplectic anyons, is the quantum
deformation of the recoupling theory. We proposed that the existing quantum
ternary arithmetic gates can be realized by braiding and topological charge
measurement of the metaplectic anyons
Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes
We present two methods for the construction of quantum circuits for quantum
error-correcting codes (QECC). The underlying quantum systems are tensor
products of subsystems (qudits) of equal dimension which is a prime power. For
a QECC encoding k qudits into n qudits, the resulting quantum circuit has
O(n(n-k)) gates. The running time of the classical algorithm to compute the
quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC
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