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Workshop on Physics and Computation (PhysComp)

A Complex Logic For Computation With Simple Interpretations For Physics

Richard Shoup

Published Aug 24, 2005• 6 citations
Publisher

Abstract

This paper derives a simple Boolean logic represented as logic circuits. The logic is then extended to selfreferential circuits and shown to generate logic values, analogs to imaginary numbers in ordinary algebra. The generalized logic can represent both combinatorial and sequential circuits, and also demonstrates interesting properties suggestive of both classical and quantum physics, including superposition, interference, and quantum coherence. A solution is given to the Square Root of NOT problem, a function not realizable with isolated classical circuits, and it is shown how this circuit may also be interpreted to represent the equivalent quantum computer circuit. Possible interpretations of the logic for physical systems are discussed.

DOI
10.1109/phycmp.1992.615509

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This paper derives a simple Boolean logic represented as logic circuits. The logic is then extended to selfreferential circuits and shown to generate logic values, analogs to imaginary numbers in ordinary algebra. The generalized logic can represent both combinatorial and sequential circuits, and also demonstrates interesting properties suggestive of both classical and quantum physics, including superposition, interference, and quantum coherence. A solution is given to the Square Root of NOT problem, a function not realizable with isolated classical circuits, and it is shown how this circuit may also be interpreted to represent the equivalent quantum computer circuit. Possible interpretations of the logic for physical systems are discussed.
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal form of Boolean function, and quantum logic circuits is described. It is shown that quantum information approach provides simple algorithm to construct Zhegalkin polynomial using truth table. Developed methods and algorithms have arbitrary Boolean function generalization with multibit input and multibit output. Such generalization allows us to use many-valued logic (k-valued logic, where k is a prime number). Developed methods and algorithms can significantly improve quantum technology realization. The presented approach is the baseline for transition from classical machine logic to quantum hardware.
Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet { 0 , 1 } using projection operators to the binary alphabet { + 1 , - 1 } employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley-Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover's algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.
Abstract The fundamental relationship between quantum physics and discrete mathematics is examined. A method for representing Boolean functions in the form of unitary transformations is described. The question of the connection of Zhegalkin polynomials defining the algebraic normal form of a Boolean function with quantum circuits is considered. It is shown that the quantum information language provides a simple algorithm for constructing the Zhegalkin polynomial based on the truth table. The developed methods and algorithms are generalized to the case of an arbitrary Boolean function with a multibit domain of definition and a multibit set of values, as well as to the case of multivalued (kkk-value) logic when k=pk = pk=p is a prime number. The developed approach is important for the implementation of quantum computer technologies and is the foundation for the transition from classical computer logic to quantum hardware.
Abstract In the 1950s and 1960s it was causing interest among physicists that in the formalism of quantum mechanics (complex-valued) ‘probability amplitudes’ obeyed laws similar to the laws obeyed by probabilities in the ordinary probability calculus. But this did not then lead decisively to the claim that probabilities should be represented by complex numbers. It became fashionable instead to regard probability amplitudes as an abstract concept from which actual probabilities could be derived by taking the squared moduli of the amplitudes. In this monograph, however, we make another attempt to show how probability amplitudes might after all be identified with actual probabilities. To do this, probability itself is interpreted in a rational Bayesian manner (i.e. as an extension of logic) and a new (complex-valued) probability theory is formulated that incorporates the uncertainty principle (i.e. that takes account of the fact that acquisition of knowledge of a quantum mechanical process generally interferes with it). Taking this probability theory as the new logic of science, and assuming certain physical laws and properties of matter, an interpretation of non-relativistic quantum mechanics is built up. It is consistent with the usual quantum mechanical formalism but allows a clear distinction to be made between the physical world and our knowledge of it.
This paper uses a non-distributive system of Boolean fractions (a ib), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum con- cepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a lb) is said to be inapplicable (or undefined) in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3- valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a b) and (cid) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b = d, their conditions are equivalent. Since all aspects of quan- tum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the 'influence-at-a-distance', quantum entangle- ment phenomena. A quantum realist must still embrace non-local influences to hold that hidden variables are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics.