Qudit has emerged as a promising system for next-generation quantum computers because of its significant advantages in multi-phase problems and quantum error correction, while multi-qudit operations are yet to be implemented. As one of the basic operations, quantum addition has been widely applied in number factorization and discrete logarithms. Meanwhile, the physical constraints of noisy intermediate-scale quantum (NISQ) devices necessitate the development of fault-tolerant quantum adders and the exploration of quaternary operations on binary devices is still in its infancy. In this work, we propose the first approach for implementing quaternary quantum addition algorithms by employing primitive quantum gates. A library of quaternary quantum gates and quaternary quantum full adders (Q2FA) able to produce carry-first results, along with lower depth and fewer T-gates optimizations are proposed and evaluated, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments show that our proposed Q2FA design, together with the optimization techniques, reduces T-depth by up to 1.4× and T-count by 1.7× compared with baseline quantum circuits without depth and T-gate optimizations. Meanwhile, the scalability of the proposed Q2FA is demonstrated by constructing quantum carry-ripple adders. Under noisy conditions, our proposed design can achieve an overall fidelity increase by 1.4×.
Qudit has emerged as a promising system for next-generation quantum computers because of its significant advantages in multi-phase problems and quantum error correction, while multi-qudit operations are yet to be implemented. As one of the basic operations, quantum addition has been widely applied in number factorization and discrete logarithms. Meanwhile, the physical constraints of noisy intermediate-scale quantum (NISQ) devices necessitate the development of fault-tolerant quantum adders and the exploration of quaternary operations on binary devices is still in its infancy. In this work, we propose the first approach for implementing quaternary quantum addition algorithms by employing primitive quantum gates. A library of quaternary quantum gates and quaternary quantum full adders (Q2FA) able to produce carry-first results, along with lower depth and fewer T-gates optimizations are proposed and evaluated, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments show that our proposed Q2FA design, together with the optimization techniques, reduces T-depth by up to 1.4× and T-count by 1.7× compared with baseline quantum circuits without depth and T-gate optimizations. Meanwhile, the scalability of the proposed Q2FA is demonstrated by constructing quantum carry-ripple adders. Under noisy conditions, our proposed design can achieve an overall fidelity increase by 1.4×.
Quantum computing has been considered as an emerging approach in addressing problems which are not easily solvable using classical computers. In parallel to the physical implementation of quantum processors, quantum algorithms have been actively developed for real-life applications to show quantum advantages, many of which benefit from quantum arithmetic algorithms and their efficient implementations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm and quantum linear algebra algorithms. Although various least-significant digit-first quantum adders have been introduced in previous work, interest in investigating the efficient implementation of most-significant digit-first addition is growing. In this work, we propose a novel design method for most-significant digit-first addition with several quantum circuit optimisations to reduce the number of quantum bits (i.e. qubits), quantum gates, and circuit depth. An open-source library of different arithmetic operators based on our proposed method is presented, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments demonstrate that our proposed design, together with the optimisation techniques, reduces T-depth by up-to 4.0×, T-count by 3.5×, and qubit consumption by 1.2×.
Quantum Computing is making significant advancements toward creating machines capable of implementing quantum algorithms in various fields, such as quantum cryptography, quantum image processing, and optimization. The development of quantum arithmetic circuits for modulo addition is vital for implementing these quantum algorithms. While it is ideal to use quantum circuits based on fault-tolerant gates to overcome noise and decoherence errors, the current Noisy Intermediate Scale Quantum (NISQ) era quantum computers cannot handle the additional computational cost associated with fault-tolerant designs. Our research aims to minimize circuit depth, which can reduce noise and facilitate the implementation of quantum modulo addition circuits on NISQ machines. This work presents quantum carry-lookahead modulo (2n - 1) adder (QCLMA), which is designed to receive two n-bit numbers and perform their addition with an O(log n) depth. Compared to existing work of O(n) depth, our proposed QCLMA reduces the depth and helps increase the noise fidelity. In order to increase error resilience, we also focus on creating a tree structure based Carry path, unlike the chain based Carry path of the current work. We run experiments on Quantum Computer IBM Cairo to evaluate the performance of the proposed QCLMA against the existing work and define Quantum State Fidelity Ratio (QSFR) to quantify the closeness of the correct output to the top output. When compared against existing work, the proposed QCLMA achieves a 47.21% increase in QSFR for 4-qubit modulo addition showcasing its superior noise fidelity.
In recent years, quantum computers have attracted extensive research interests due to their potential capability of solving problems which are not easily solvable using classical computers. In parallel to the constant research aiming at the physical implementation of quantum processors, there is another branch of research developing quantum algorithms for real-life applications, many of which need to perform arithmetic operations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm, quantum linear algebra algorithms and various quantum machine learning applications. Since precision is always a non-trivial issue to determine during the computation, most-significant digit-first quantum addition can be a fundamental operation for variable precision computing. Therefore, this paper proposes the first quantum adder circuit that is able to compute from the most-significant digits, which demonstrates the advantages over the state-of-the-art quantum adders requiring carry propagation to produce results from least-significant digits. We first present a review of quantum addition circuits design, and then propose a novel method to implement quantum most-significant digit-first adders. Scalability and quantitative comparisons for different quantum full adder, quantum carry-ripple adder and quantum most-significant digit-first adder circuits have been investigated, where all circuits are implemented on IBM Qiskit SDK.
The "Noisy intermediate-scale quantum" NISQ machine era primarily focuses on mitigating noise, controlling errors, and executing high-fidelity operations, hence requiring shallow circuit depth and noise robustness. Approximate computing is a novel computing paradigm that produces imprecise results by relaxing the need for fully precise output for error-tolerant applications including multimedia, data mining, and image processing. We investigate how approximate computing can improve the noise resilience of quantum adder circuits in NISQ quantum computing. We propose five designs of approximate quantum adders to reduce depth while making them noise-resilient, in which three designs are with carryout, while two are without carryout. We have used novel design approaches that include approximating the Sum only from the inputs (pass-through designs) and having zero depth, as they need no quantum gates. The second design style uses a single CNOT gate to approximate the SUM with a constant depth of O(1). We performed our experimentation on IBM Qiskit on noise models including thermal, depolarizing, amplitude damping, phase damping, and bitflip: (i) Compared to exact quantum ripple carry adder without carryout the proposed approximate adders without carryout have improved fidelity ranging from 8.34% to 219.22%, and (ii) Compared to exact quantum ripple carry adder with carryout the proposed approximate adders with carryout have improved fidelity ranging from 8.23% to 371%. Further, the proposed approximate quantum adders are evaluated in terms of various error metrics.
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 faulttolerant 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 faulttolerant 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 faulttolerant 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.
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.