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.
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.
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.
Photonic processors have emerged as an attractive platform for fast and energy-efficient matrix-vector multiplication. However, they are susceptible to error due to their analog nature. Here, we present an error-correction technique that implements a correction offset to the optical en-/decoders of photonic processors. Our proposed method is general-purpose, does not require introducing any additional components to the photonic network, and can address errors stemming from unbalanced losses, 50/50 beamsplitter deviations, digital-to-analog conversion inaccuracies, and any unknown sources. In particular, we show that our method is highly effective in mitigating unbalanced-loss errors, a problem that has not previously been addressed by any error-correction technique. Using this approach, we achieve over 90% error reduction in large triangular meshes, overcoming a key obstacle to highly accurate photonic processors for information processing.
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.
The intrinsic probabilistic nature of quantum systems makes error correction or mitigation indispensable for quantum computation. While current error-correcting strategies focus on correcting errors in quantum states or quantum gates, these fine-grained error-correction methods can incur significant overhead for quantum algorithms of increasing complexity. We present a first step in achieving error correction at the level of quantum algorithms by combining a unified perspective on modern quantum algorithms via quantum signal processing (QSP). An error model of under- or over-rotation of the signal processing operator parameterized by ε<1 is introduced. It is shown that while Pauli Z-errors are not recoverable without additional resources, Pauli X and Y errors can be arbitrarily suppressed by coherently appending a noisy `recovery QSP.' Furthermore, it is found that a recovery QSP of length O(2kck2d) is sufficient to correct any length-d QSP with c unique phases to kth-order in error ε. Allowing an additional assumption, a lower bound of Ω(cd) is shown, which is tight for k=1, on the length of the recovery sequence. Our algorithmic-level error correction method is applied to Grover's fixed-point search algorithm as a demonstration.
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.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.