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Complexity of optical computing paradigms: Computational implications and a suggested improvement

Arthur David Post

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

Optical computing has been suggested as a means of achieving a high degree of parallelism for both scientific and symbolic applications. While a number of implementations of logic operations have been forwarded, all have some characteristic which prevents their direct extension to functions of a large number of input bits. This paper will analyze several of these implementations and demonstrate that all these implementations require some measure of the system (area, space-bandwidth product, or time) to grow exponentially with the number of inputs. We will then suggest an implementation whose complexity is not greater than the best theoretical realization of a boolean function. We will demonstrate the optimality of the realization, to within a constant multiple, for digital optical computing systems realized by bulk space-variant elements.

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Optical computing has been suggested as a means of achieving a high degree of parallelism for both scientific and symbolic applications. While a number of implementations of logic operations have been forwarded, all have some characteristic which prevents their direct extension to functions of a large number of input bits. This paper will analyze several of these implementations and demonstrate that all these implementations require some measure of the system (area, space-bandwidth product, or time) to grow exponentially with the number of inputs. We will then suggest an implementation whose complexity is not greater than the best theoretical realization of a boolean function. We will demonstrate the optimality of the realization, to within a constant multiple, for digital optical computing systems realized by bulk space-variant elements.
Optical computing has been suggested as a means of achieving a high degree of parallelism for both scientific and symbolic applications. While a number of implementations of logic operations have been forwarded, all have some characteristic that prevents their direct extension to functions of a large number of input bits. We analyze several of these implementations and demonstrate that all these implementations require that some measure of the system (area, space-bandwidth product, or time) grow exponentially with the number of inputs. We then suggest an implementation whose complexity is no greater than the best theoretical realization of a Boolean function. We demonstrate the optimality of that realization, to within a constant multiple, for digital optical-computing systems realized by bulk spatially variant elements.
In this paper, we study the query complexity of Boolean functions in the presence of uncertainty, motivated by parallel computation with an unlimited number of processors where inputs are allowed to be unknown. We allow each query to produce three results: zero, one, or unknown. The output could also be: zero, one, or unknown, with the constraint that we should output ''unknown'' only when we cannot determine the answer from the revealed input bits. Such an extension of a Boolean function is called its hazard-free extension. - We prove an analogue of Huang's celebrated sensitivity theorem [Annals of Mathematics, 2019] in our model of query complexity with uncertainty. - We show that the deterministic query complexity of the hazard-free extension of a Boolean function is at most quadratic in its randomized query complexity and quartic in its quantum query complexity, improving upon the best-known bounds in the Boolean world. - We exhibit an exponential gap between the smallest depth (size) of decision trees computing a Boolean function, and those computing its hazard-free extension. - We present general methods to convert decision trees for Boolean functions to those for their hazard-free counterparts, and show optimality of this construction. We also parameterize this result by the maximum number of unknown values in the input. - We show lower bounds on size complexity of decision trees for hazard-free extensions of Boolean functions in terms of the number of prime implicants and prime implicates of the underlying Boolean function.
We present a novel and simple theoretical model of computation that captures what we believe are the most important characteristics of an optical Fourier transform processor. We use this abstract model to reason about the computational properties of the physical systems it describes. We define a grammar for our model's instruction language, and use it to write algorithms for well-known filtering and correlation techniques. We also suggest suitable computational complexity measures that could be used to analyze any coherent optical information processing technique, described with the language, for efficiency. Our choice of instruction language allows us to argue that algorithms describable with this model should have optical implementations that do not require a digital electronic computer to act as a master unit. Through simulation of a well known model of computation from computer theory we investigate the general-purpose capabilities of analog optical processors.
Optimizing Boolean function components to have the minimum number of inputs in order to reduce the memory space required during these functions in computing devices is a significant demand. This paper proposes a quantum computation approach based on the degree-of-entanglement quantum computation model to estimate the number of junta variables of an unknown Boolean function presented through an oracle. The time complexity of the developed quantum approach is independent of the number of inputs and depends on an allowable assigned error ϵ. Thus, the time complexity of the developed algorithm is O(ϵ−2), compared to O(2n+1) in the traditional approach. Also, the memory space of the developed approach is linear, O(2n+4), in terms of the number of inputs compared to the exponential memory space O(2n+1) using the traditional approach. Therefore, the developed quantum approach has exponential supremacy in comparison to the traditional approach. The developed approach was implemented practically using both the Qiskit simulator and the IBM real quantum computer. The obtained results expose high statistical fidelities between the empirical and theoretical results.
Until recently, the paradigm of computing was the Turing machine and its numerous equivalents. All operations are made with state machines. Each operation is irreversible and brings about an increase of entropy and, therefore, an expenditure of energy of at least kTln2 per bit. We show that optics allows the first practical quantum mechanical digital processors. These processors require digital input and produce digital output but need no intermediate state machines. We show that, in the sense of not requiring material devices which would otherwise be required, nothing is superior to something for computing.