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Annual Meeting Optical Society of America

Encoding for error correction in optical computing

Scott A. Ellett, John F. Walkup, Thomas F. Krile

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Optical processors can perform complex calculations in parallel at high speeds. They commonly suffer, however, from low accuracy, which currently hinders their widespread use. Various models for the noise in optical processors have been developed.1 In addition, various techniques have been developed to improve their accuracy. One technique involves the use of error-detection and correction codes to improve the accuracy of optical linear algebra processors.2 This allows some of the errors that occur during computation to be detected and possibly corrected. These codes, developed earlier for data transmission and fault tolerance in electronic processors, can thus be used to improve accuracy. Many of the required encoding and decoding operations can be performed optically. Computer simulations of optical matrix-vector multipliers employing various error codes will be presented. They indicate that a significant improvement in accuracy can be obtained when they are compared with processors not employing error codes.

DOI
10.1364/oam.1990.ft3

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•Oct 22, 2018•0 citations
Optical processors can perform complex calculations in parallel at high speeds. They commonly suffer, however, from low accuracy, which currently hinders their widespread use. Various models for the noise in optical processors have been developed.1 In addition, various techniques have been developed to improve their accuracy. One technique involves the use of error-detection and correction codes to improve the accuracy of optical linear algebra processors.2 This allows some of the errors that occur during computation to be detected and possibly corrected. These codes, developed earlier for data transmission and fault tolerance in electronic processors, can thus be used to improve accuracy. Many of the required encoding and decoding operations can be performed optically. Computer simulations of optical matrix-vector multipliers employing various error codes will be presented. They indicate that a significant improvement in accuracy can be obtained when they are compared with processors not employing error codes.
Optical algebraic processors can perform complex calculations in parallel and at high speeds. However, they commonly suffer from a low analog accuracy which hinders their widespread application. Error detection and correction codes provide one technique for improving the accuracy of optical algebraic processors. The use of these codes would allow some of the errors that may occur during a computation to be detected and possibly corrected. This paper describes the results of various computer simulations of optical matrix-vector multipliers employing error-correction codes. It discusses the application of convolutional codes to optical matrix-vector multipliers along with several block codes. Both binary and nonbinary codes are considered. The results indicate that a significant improvement in performance can be obtained when compared with processors not employing error-correction codes. Also, the type of noise, whether signal-independent or signal-dependent noise, has a significant effect on the performance of a matrix-vector multiplier employing an error code. The encoding and decoding operations required for the error codes can be performed optically.
Optical algebraic processors can perform complex calculations in parallel and at high speeds. However, they commonly suffer from a low analog accuracy which hinders their widespread application. Error detection and correction codes provide one technique for improving the accuracy of optical algebraic processors. The use of these codes would allow some of the errors that may occur during a computation to be detected and possibly corrected. This paper describes the results of various computer simulations of optical matrix-vector multipliers employing error-correction codes. It discusses the application of convolutional codes to optical matrix-vector multipliers along with several block codes. Both binary and nonbinary codes are not employing error-correction codes. Also, the type of noise, whether signal-independent or signal-dependent, has a significant effect on the performance of a matrix-vector multiplier employing an error code. The encoding and decoding operations required for the error codes can be performed optically.
Many disciplines, such as antenna beam-forming and \nadaptive optics, require the ability to perform high \nspeed matrix operations, which often can be reduced to a \nseries of matrix-vector products. Analog optical matrixvector \nprocessors (OMVP's) take advantage of the high \nspeed and parallelism of optics to rapidly compute \nmatrix-vector products. However, OMVP's suffer from a \nlow accuracy which currently hinders their widespread \napplication. Over the past decade, various techniques \nhave been introduced to achieve high accuracy computation \nwith OMVP's. This dissertation studies the application \nof error-correcting codes (ECC's) and the union of ECC's \nwith digital partitioning to achieve high accuracy computation. \nThe ability of ECC's to improve the performance \nof OMVP's is studied through the use of computer simulations. \nThe effect ECC's have on system throughput is \nalso studied. Digital partitioning is a new technique \nfor achieving high accuracy computations on analog \nOMVP's. One potential drawback of digital partitioning \nis its sensitivity to errors. Error-correcting codes can \nenhance the performance of the digital partitioning algorithm \nby reducing this sensitivity to errors. Computer \nsimulation results are presented showing the benefits of \nusing ECC's with digital partitioning. In addition, an \nanalysis of the time requirements for encoding and decoding \nthe ECC's and the effect on system throughput is \npresented.
Optical matrix-vector multipliers are known to suffer from low computational accuracy which is the major drawback preventing them from becoming a practical force in real world applications. Error-correction coding is one technique proposed to overcome this problem of low level accuracy. In this thesis, the effects of the application of multiple error-correction codes to optical matrix-vector multipliers are studied. For this purpose, many different cases were simulated on a computer. Noting that an optical matrix-vector multiplier performs the product y = Ax, basically four cases were considered; signal independent and signal dependent noise in the matrix A, and signal independent and signal dependent noise in the vector x. Two different error-correcting codes were simulated; one being a binary, multiple error-correcting BCH code and other being a non-binary, multiple error-correcting Reed-Solomon code, which is basically an extension of the binary BCH code. The advantages and disadvantages of switching from binary BCH codes to non-binary Reed-Solomon codes were investigated. Based on the results obtained from the simulations, the conditions under which the use of error-correction coding is feasible in optical matrix-vector multipliers are discussed.
Information theory shows that one can communicate with arbitrarily low error probability over a noisy channel. By recognizing that some types of computations can be cast as communications problems, it may be possible to compute accurately with an inexact processor. Traditional analog optical processors, for example, offer advantages of parallelism and speed but suffer from significant inaccuracies. We propose an algorithm whereby the accuracy of matched filter processors can be improved significantly at the cost of a modest increase in computational resources. Errors that are due to noisy data and/or inexact computing can be detected and in some cases corrected.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.