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The Journal of Supercomputing

Fault-tolerant quantum algorithm for dual-threshold image segmentation

Luis O. López, Francisco José Orts, Gloria Ortega, Vicente González-Ruíz, Ester Martín Garzón

Published Mar 15, 2023• 10 citations
Publisher

Abstract

Abstract The intrinsic high parallelism and entanglement characteristics of quantum computing have made quantum image processing techniques a focus of great interest. One of the most widely used techniques in image processing is segmentation, which in one of their most basic forms can be carried out using thresholding algorithms. In this paper, a fault-tolerant quantum dual-threshold algorithm has been proposed. This algorithm is based on Clifford+T gates. Because T gates add fault tolerance but have the trade-off of having a much higher cost than other quantum gates,our focus has been on reducing the number of T gates. This has allowed adding noise tolerance, computational cost reduction and fault toleranceto the state-of-the-art dual-threshold segmentation circuits. Sincethe dual-threshold image segmentation involves the comparison operation,as part of this work we have implemented two comparator circuits.These circuits optimise the metrics T-count and T-depth with respect to the best circuit comparators currently available in the literature.

DOI
10.1007/s11227-023-05148-9

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Abstract The intrinsic high parallelism and entanglement characteristics of quantum computing have made quantum image processing techniques a focus of great interest. One of the most widely used techniques in image processing is segmentation, which in one of their most basic forms can be carried out using thresholding algorithms. In this paper, a fault-tolerant quantum dual-threshold algorithm has been proposed. This algorithm is based on Clifford+T gates. Because T gates add fault tolerance but have the trade-off of having a much higher cost than other quantum gates,our focus has been on reducing the number of T gates. This has allowed adding noise tolerance, computational cost reduction and fault toleranceto the state-of-the-art dual-threshold segmentation circuits. Sincethe dual-threshold image segmentation involves the comparison operation,as part of this work we have implemented two comparator circuits.These circuits optimise the metrics T-count and T-depth with respect to the best circuit comparators currently available in the literature.
Abstract Current quantum computers have a limited number of resources and are heavily affected by internal and external noise. Therefore, small, noise-tolerant circuits are of great interest. With regard to circuit size, it is especially important to reduce the number of required qubits. Concerning to fault-tolerance, circuits entirely built with Clifford+T gates allow the use of error correction codes. However, the T-gate has an excessive cost, so circuits with a high number of T-gates should be avoided. This work focuses on optimising in such terms an operation that is widely used in larger circuits and algorithms: the calculation of the absolute-value of two’s complement encoded integers. The proposed circuit halves the number of required T gates with respect to the best circuit currently available in the literature. Moreover, our circuit requires at least 2 qubits less than the other circuits for such an operation.
Abstract Quantum comparators hold substantial significance in the scientific community as fundamental components in a wide array of algorithms. In this research, we present an innovative approach where we explore the realm of comparator circuits, specifically focussing on three distinct circuit designs present in the literature. These circuits are notable for their use of T-gates, which have gained significant attention in circuit design due to their ability to enable the utilisation of error-correcting codes to ensure fault-tolerance in quantum circuits. However, it is important to note that T-gates come at a considerable computational cost. Our methodology revolves around a comprehensive study of these comparator circuits from a bottom-up perspective. We delve deep into the intricate details of these circuits, dissecting their underlying components and functions. This meticulous examination forms the foundation of our research, enabling us to identify areas where optimisations can be made to improve their performance. One of the key contributions of our work is the optimisation of the quantum gates used within these circuits. We have expressed the proposed circuits in Clifford+T gates to render them fault-tolerant while concurrently minimising the utilisation of T-gates, thereby mitigating the associated computational costs. By strategically reducing the number of T-gates used and improving their efficiency, we not only mitigate computational overhead, but also ensure the robustness of these circuits in the face of errors and environmental disturbances.
Abstract Quantum image segmentation algorithm is crucial for quantum image processing. In this paper, a dual-threshold quantum image segmentation algorithm is designed and simulated in IBM Quantum Experience (IBM Q) platform, which can segment a complex image into three parts using fewer quantum bits. In our algorithm, given a high threshold and a low threshold, grayscale values larger than the high threshold are set to the high threshold and grayscale values smaller than the low threshold are set to the low threshold, with no change for the part between the two thresholds. Then we use a low-cost quantum comparator and design a complete and scalable quantum image segmentation circuit. Analysis of the circuit cost shows that the quantum gates required for the circuit are only related to the grayscale range q and are independent of the image size. The feasibility of the algorithm and the correctness of the quantum circuit are verified by simulation in IBM Q platform, and finally the MSE, PSNR AND SSIM value of the image is analyzed to prove the effectiveness of the segmentation algorithm.
Quantum circuits for basic image processing functions such as bilinear interpolation are required to implement image processing algorithms on quantum computers. In this work, we propose quantum circuits for the bilinear interpolation of NEQR encoded images based on Clifford+T gates. Quantum circuits for the scale up operation and scale down operation are illustrated. The proposed quantum circuits are based on quantum Clifford+T gates and are optimized for T-count. Quantum circuits based on Clifford+T gates can be made fault tolerant but the T gate is very costly to implement. As a result, reducing T-count is an important optimization goal. The proposed quantum bilinear interpolation circuits are based on (i) a quantum adder, (ii) a proposed quantum subtractor, and (iii) a quantum multiplication circuit. Further, both designs are compared and shown to be superior to existing work in terms of T-count. The proposed quantum bilinear interpolation circuits for the scale down operation and for the scale up operation each have a 92.52% improvement in terms of T-count compared to the existing work.
Progress in quantum hardware design is progressing toward machines of sufficient size to begin realizing quantum algorithms in disciplines such as encryption and physics. Quantum circuits for addition are crucial to realize many quantum algorithms on these machines. Ideally, quantum circuits based on fault-tolerant gates and error-correcting codes should be used as they tolerant environmental noise. However, current machines called Noisy Intermediate Scale Quantum (NISQ) machines cannot support the overhead associated with fault-tolerant design. In response, low depth circuits such as quantum carry lookahead adders (QCLA)s have caught the attention of researchers. The risk for noise errors and decoherence increase as the number of gate layers (or depth) in the circuit increases. This work presents an out-of-place QCLA based on Clifford+T gates. The QCLAs optimized for T gate count and make use of a novel uncomputation gate to save T gates. We base our QCLAs on Clifford+T gates because they can eventually be made fault-tolerant with error-correcting codes once quantum hardware that can support fault-tolerant designs becomes available. We focus on T gate cost as the T gate is significantly more costly to make fault-tolerant than the other Clifford+T gates. The proposed QCLAs are compared and shown to be superior to existing works in terms of T-count and therefore the total number of quantum gates. Finally, we illustrate the application of the proposed QCLAs in quantum image processing by presenting quantum circuits for bilinear interpolation.
Arithmetic operations are an important component of many quantum algorithms. Optimizing quantum circuits for these operations therefore leads to more efficient implementations of the corresponding algorithms. In this paper, we develop new fault-tolerant quantum circuits for various integer division algorithms (both reversible and non-reversible). These circuits, when implemented in the Clifford+T gate set, achieve improvements in the asymptotic T-count and CNOT-count from O(n2)O(n^2)O(n2) to O(nm)O(nm)O(nm), and in the asymptotic T-depth from O(n2)O(n^2)O(n2) to O(nlog⁡m)O(n \log m)O(nlogm), where nnn and mmm are the bit lengths of the dividend and divisor, respectively. The qubit counts are significantly lower than those in previous works. We achieve this through careful modifications of the division algorithms and by expressing them in terms of a primitive we call COMP-N-SUB that compares two integers and conditionally subtracts them. We show that this primitive can be implemented at a cost, in terms of both Clifford and non-Clifford gates, comparable to that of one addition. By contrast, performing comparison and conditional subtraction separately would cost roughly as much as a controlled addition plus a regular addition.