27,690 research outputs found
Multi-valued Logic Gates for Quantum Computation
We develop a multi-valued logic for quantum computing for use in multi-level
quantum systems, and discuss the practical advantages of this approach for
scaling up a quantum computer. Generalizing the methods of binary quantum
logic, we establish that arbitrary unitary operations on any number of d-level
systems (d > 2) can be decomposed into logic gates that operate on only two
systems at a time. We show that such multi-valued logic gates are
experimentally feasible in the context of the linear ion trap scheme for
quantum computing. By using d levels in each ion in this scheme, we reduce the
number of ions needed for a computation by a factor of log d.Comment: Revised version; 8 pages, 3 figures; to appear in Physical Review
Applications of Multi-Valued Quantum Algorithms
This paper generalizes both the binary Deutsch-Jozsa and Grover algorithms to
-valued logic using the quantum Fourier transform. Our extended
Deutsch-Jozsa algorithm is not only able to distinguish between constant and
balanced Boolean functions in a single query, but can also find closed
expressions for classes of affine logical functions in quantum oracles,
accurate to a constant term. Furthermore, our multi-valued extension of the
Grover algorithm for quantum database search requires fewer qudits and hence a
substantially smaller memory register, as well as fewer wasted information
states, to implement. We note several applications of these algorithms and
their advantages over the binary cases.Comment: 12 pages, 4 figures; updated version of paper for ISMVL 2007;
contains new proof of multi-valued Grover algorithm time complexity, with
typos correcte
Topos Quantum Logic and Mixed States
The topos approach to the formulation of physical theories includes a new
form of quantum logic. We present this topos quantum logic, including some new
results, and compare it to standard quantum logic, all with an eye to
conceptual issues. In particular, we show that topos quantum logic is
distributive, multi-valued, contextual and intuitionistic. It incorporates
superposition without being based on linear structures, has a built-in form of
coarse-graining which automatically avoids interpretational problems usually
associated with the conjunction of propositions about incompatible physical
quantities, and provides a material implication that is lacking from standard
quantum logic. Importantly, topos quantum logic comes with a clear geometrical
underpinning. The representation of pure states and truth-value assignments are
discussed. It is briefly shown how mixed states fit into this approach.Comment: 25 pages; to appear in Electronic Notes in Theoretical Computer
Science (6th Workshop on Quantum Physics and Logic, QPL VI, Oxford, 8.--9.
April 2009), eds. B. Coecke, P. Panangaden, P. Selinger (2010
A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations
The Kochen-Specker theorem asserts the impossibility of assigning values to
quantum quantities in a way that preserves functional relations between them.
We construct a new type of valuation which is defined on all operators, and
which respects an appropriate version of the functional composition principle.
The truth-values assigned to propositions are (i) contextual; and (ii)
multi-valued, where the space of contexts and the multi-valued logic for each
context come naturally from the topos theory of presheaves.
The first step in our theory is to demonstrate that the Kochen-Specker
theorem is equivalent to the statement that a certain presheaf defined on the
category of self-adjoint operators has no global elements. We then show how the
use of ideas drawn from the theory of presheaves leads to the definition of a
generalized valuation in quantum theory whose values are sieves of operators.
In particular, we show how each quantum state leads to such a generalized
valuation.Comment: Clarification of situation for situation for operators with
continuous spectr
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
Realization of Multi-Valued Logic Using Optical Quantum Computing
Quantum computing is a paradigm of computing using physical systems, which operate according to quantum mechanical principles. Since 2017, functioning quantum processing units with limited capabilities are available on the cloud. There are two models of quantum computing in the literature: discrete variable and continuous variable models. The discrete variable model is an extension of the binary logic of digital computing with quantum bits |0⟩ and |1⟩ . In the continuous variable model, the quantum state space is infinite-dimensional and the quantum state is expressed with an infinite number of basis elements.
In the physical implementation of quantum computing, however, the quantized energy levels of the electromagnetic field come in multiple values, naturally realizing the multi-valued logic of computing. Hence, to implement the discrete variable model (binary logic) of quantum computing, the temperature control is needed to restrict the energy levels to the lowest two to express the binary quantum states |0⟩ and |1⟩. The physical realization of the continuous variable model naturally implements the multi-valued logic of computing because any physical system always has the highest level of quantized energy observed i.e., the quantum state space is always finite dimensional.
In 2001, Knill, Laflamme, and Milburn proved that linear optics realizes universal quantum computing in the qubit-based model. Optical quantum computers by Xanadu, under the phase space representation of quantum optics, naturally realizes the multi-valued logic of quantum computing at room temperature. Optical quantum computers use optical signals, which are most compatible with the fiber optics communication network. They are easily fabricable for mass production, robust to noise, and have low latency.
Optical quantum computing provides flexibility to the users for determining the dimension of the computational space for each instance of computation. Additionally, nonlinear quantum optical effects are incorporated as nonlinear quantum gates. That flexibility of user-defined dimension of the computational space and availability of nonlinear gates lead to a faithful implementation of quantum neural networks in optical quantum computing. This dissertation provides a full description of a multi-class data quantum classifier on ten classes of the MNIST dataset.
In this dissertation, I provide the background information of optical quantum computing as an ideal candidate material for building the future classical-quantum hybrid internet for its numerous benefits, among which the compatibility with the existing communications/computing infrastructure is a main one. I also show that optical quantum computing can be a hardware platform for realizing the multi- valued logic of computing without the need to encode and decode computational problems in binary logic. I also derive explicit matrix representation of optical quantum gates in the phase space representation. Using the multi-valued logic of optical quantum computing, I introduce the first quantum multi-class data classifier, classifying all ten classes of the MNIST dataset
Exact Synthesis of 3-qubit Quantum Circuits from Non-binary Quantum Gates Using Multiple-Valued Logic and Group Theory
We propose an approach to optimally synthesize quantum circuits from non-permutative quantum gates such as Controlled-Square-Root–of-Not (i.e. Controlled-V). Our approach reduces the synthesis problem to multiple-valued optimization and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimization problem to a permutable representation. The transformation enables us to utilize group theory to exploit the symmetric properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we found all reversible circuits with quantum costs of 4, 5, 6, etc, and give another algorithm to realize these reversible circuits with quantum gates. The approach can be used for both binary permutative deterministic circuits and probabilistic circuits such as controlled random number generators and hidden Markov models
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