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Fault-tolerant Quantum Circuits and Computation Architectures for Image Processing Applications

Israel Olabisi Koiku

Published Jul 1, 2026• 0 citations
OpenAlex

Abstract

The classical computer is becoming inefficient to process the ever-increasing number of images generated on daily basis for various purposes. To address this, researchers have shifted attention to exploring the potential of quantum computing to realize image processing. As image pixels are represented as color intensities of integral points on the cartesian coordinates, there is need for interpolations of the color and positions of the pixels to restore the integral points locations after the image points have undergone some computational transformations. While color interpolation has received some attention in Quantum Image Processing, coordinate interpolation has not been explored. In this work, a novel quantum architecture that does image coordinate interpolation is proposed. Furthermore, the architecture generalizes the concept of nearest neighbor interpolation. By partitioning the distance between two adjacent pixels into 2𝑛 discrete levels, a computed point between them can be interpolated to either way depending on the difference between the computed point and a predetermined coefficient that determines where the user wishes to interpolate to. The proposed architecture is based on a quantum subtractor and a novel quantum conditional incrementor, which is another project that was completed in this study. Likewise, the proposed conditional incrementor is based on a novel circuit for incrementor which is also presented in this dissertation together with its associated decrementor. A novel construction of discrete time quantum walk (DTQW) along a closed ring was also proposed as another application of the proposed incrementor. DTQW has wide applications including "universal computation" and in image processing. Therefore, the contributions of this study include introduction of a coordinate interpolation circuit, design and construction of T-count optimized incrementor, decrementor, conditional incrementor, conditional decrementor, conditional incrementor/decrementor circuits, and a T-count efficient implementation of discrete-time quantum walk.

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The classical computer is becoming inefficient to process the ever-increasing number of images generated on daily basis for various purposes. To address this, researchers have shifted attention to exploring the potential of quantum computing to realize image processing. As image pixels are represented as color intensities of integral points on the cartesian coordinates, there is need for interpolations of the color and positions of the pixels to restore the integral points locations after the image points have undergone some computational transformations. While color interpolation has received some attention in Quantum Image Processing, coordinate interpolation has not been explored. In this work, a novel quantum architecture that does image coordinate interpolation is proposed. Furthermore, the architecture generalizes the concept of nearest neighbor interpolation. By partitioning the distance between two adjacent pixels into 2𝑛 discrete levels, a computed point between them can be interpolated to either way depending on the difference between the computed point and a predetermined coefficient that determines where the user wishes to interpolate to. The proposed architecture is based on a quantum subtractor and a novel quantum conditional incrementor, which is another project that was completed in this study. Likewise, the proposed conditional incrementor is based on a novel circuit for incrementor which is also presented in this dissertation together with its associated decrementor. A novel construction of discrete time quantum walk (DTQW) along a closed ring was also proposed as another application of the proposed incrementor. DTQW has wide applications including "universal computation" and in image processing. Therefore, the contributions of this study include introduction of a coordinate interpolation circuit, design and construction of T-count optimized incrementor, decrementor, conditional incrementor, conditional decrementor, conditional incrementor/decrementor circuits, and a T-count efficient implementation of discrete-time quantum walk.
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.
Quantum circuits for mathematical functions such as division are necessary to use quantum computers for scientific computing. Quantum circuits based on Clifford+T gates can easily be made fault-tolerant but the T gate is very costly to implement. The small number of qubits available in existing quantum computers adds another constraint on quantum circuits. As a result, reducing T-count and qubit cost have become important optimization goals. The design of quantum circuits for integer division has caught the attention of researchers and designs have been proposed in the literature. However, these designs suffer from excessive T gate and qubit costs. Many of these designs also produce significant garbage output resulting in additional qubit and T gate costs to eliminate these outputs. In this work, we propose two quantum integer division circuits. The first proposed quantum integer division circuit is based on the restoring division algorithm and the second proposed design implements the non-restoring division algorithm. Both proposed designs are optimized in terms of T-count, T-depth and qubits. Both proposed quantum circuit designs are based on (i) a quantum subtractor, (ii) a quantum adder-subtractor circuit, and (iii) a novel quantum conditional addition circuit. Our proposed restoring division circuit achieves average T-count savings from 79.03 to 91.69 percent compared to the existing works. Our proposed non-restoring division circuit achieves average T-count savings from 49.22 to 90.03 percent compared to the existing works. Further, both our proposed designs have linear T-depth. We also illustrate the application of the proposed quantum division circuits in quantum image processing with a case study of quantum bilinear interpolation.
In this work, we propose a novel Quantum Generalized Neighbor Interpolation (QGNI) circuit which can be used to implement Nearest Neighbor Interpolation (NNI) as a special case of its operations. Nearest neighbor interpolation (NNI) is used in image processing tasks such as image rotation. The proposed quantum circuit takes as input (i) the perpendicular distance between a transformed pixel value and one of the four adjacent integral coordinate locations and (ii) a weighting factor which determines which of the four adjacent integral coordinate locations a transformed pixel value is projected to. corresponding T gate cost is provided.
Quantum circuits for arithmetic functions over Galois fields such as squaring are required to implement quantum cryptanalysis algorithms. Quantum circuits for integer arithmetic such as multiplication are required to implement scientific computing algorithms and quantum image processing algorithms on quantum computers. Reliable quantum circuits require error correcting codes and gates that are fault tolerant in nature. Quantum circuits of many qubits are challenging to implement making designs with low qubit cost desirable. In this work, we present quantum arithmetic circuits for applications in quantum cryptanalysis and quantum image processing. We present a proposed algorithm for synthesizing gate cost, qubit cost and depth optimized Galois field (GF(2m)) squaring circuits for quantum cryptanalysis applications. In addition, these squaring circuits are incorporated into a proposed quantum circuit for inversion in GF(2m). This work also presents a proposed quantum integer conditional addition circuit and a quantum integer multiplication circuit optimized for T-count and qubit cost. The quantum conditional addition circuit and quantum multiplier are incorporated into proposed quantum circuits for bilinear interpolation optimized for T-count cost that can be used in quantum image processing applications.
The quest of achieving higher computing performance is driving the research on quantum computing, which is reporting new milestones almost on a daily basis. For practical quantum circuit design, fault tolerance is an essential condition. This is achieved by mapping the target functions into the Clifford+T group of elementary quantum gates. Furthermore, the application of error-correcting codes in quantum circuits requires the quantum gates to be formed between adjacent Qubits. In this work, we improve the state-of-the-art quantum circuit design by addressing both of the above challenges. First, we propose a novel mapping of Multiple-Control Toffoli (MCT) gates to Clifford+T group gates, which achieves lower gate count compared to earlier work. Secondly, we show a generic way to convert any Clifford+T circuit into a nearest neighbor one. We validate the efficacy of our approach with detailed experimental studies.
Bilinear interpolation is widely used in classical signal and image processing. Quantum algorithms have been designed for efficiently realizing bilinear interpolation. However, these quantum algorithms have limitations in circuit width and garbage outputs, which block the quantum algorithms applied to noisy intermediate-scale quantum devices. In addition, the existing quantum bilinear interpolation algorithms cannot keep the consistency between the geometric centers of the original and target images. To save the above questions, we propose quantum bilinear interpolation algorithms based on geometric centers using fault-tolerant implementations of quantum arithmetic operators. Proposed algorithms include the scaling-up and scaling-down for signals (grayscale images) and signals with three channels (color images). Simulation results demonstrate that the proposed bilinear interpolation algorithms obtain the same results as their classical counterparts with an exponential speedup. Performance analysis reveals that the proposed bilinear interpolation algorithms keep the consistency of geometric centers and significantly reduce circuit width and garbage outputs compared to the existing works.