67 research outputs found
A Synthesis Method for Quaternary Quantum Logic Circuits
Synthesis of quaternary quantum circuits involves basic quaternary gates and
logic operations in the quaternary quantum domain. In this paper, we propose
new projection operations and quaternary logic gates for synthesizing
quaternary logic functions. We also demonstrate the realization of the proposed
gates using basic quantum quaternary operations. We then employ our synthesis
method to design of quaternary adder and some benchmark circuits. Our results
in terms of circuit cost, are better than the existing works.Comment: 10 page
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
Improved reversible and quantum circuits for Karatsuba-based integer multiplication
Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor\u27s algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba\u27s recursive method. The main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required from O(n^1.585) to O(n^1.427). This improvement is obtained in exchange for a small constant increase in the number of operations by a factor less than 2 and a small asymptotic increase in depth for the parallel version. The asymptotic improvement are obtained from analyzing pebble games on complete ternary trees
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