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arXiv:2311.06573 · Qeios

A Generalized Space-Efficient Algorithm for Quantum Bit String Comparators

Khuram Shahzad, Omar Khan

Published Mar 14, 2024• 0 citations
arXivHTMLPDF

Abstract

Quantum Bit String Comparators (QBSC) operate on two sequences of n-qubits, enabling the determination of their relationships, such as equality, greater than, or less than. This is analogous to the way conditional statements are used in programming languages. Consequently, QBSCs play a crucial role in various algorithms that can be executed or adapted for quantum computers. The development of efficient and generalized comparators for any \(n\)-qubit length has long posed a challenge, as they have a high-cost footprint and lead to quantum delays. Comparators that are efficient are associated with inputs of fixed length. As a result, comparators without a generalized circuit cannot be employed at a higher level, though they are well-suited for problems with limited size requirements. In this paper, we introduce a generalized design for the comparison of two \(n\)-qubit logic states using just two ancillary bits. The design is examined on the basis of qubit requirements, ancillary bit usage, quantum cost, quantum delay, gate operations, and circuit complexity, and is tested comprehensively on various input lengths. The work allows for sufficient flexibility in the design of quantum algorithms, which can accelerate quantum algorithm development.

DOI
10.32388/nrq6w1

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Quantum Bit String Comparators (QBSC) operate on two sequences of n-qubits, enabling the determination of their relationships, such as equality, greater than, or less than. This is analogous to the way conditional statements are used in programming languages. Consequently, QBSCs play a crucial role in various algorithms that can be executed or adapted for quantum computers. The development of efficient and generalized comparators for any (n)-qubit length has long posed a challenge, as they have a high-cost footprint and lead to quantum delays. Comparators that are efficient are associated with inputs of fixed length. As a result, comparators without a generalized circuit cannot be employed at a higher level, though they are well-suited for problems with limited size requirements. In this paper, we introduce a generalized design for the comparison of two (n)-qubit logic states using just two ancillary bits. The design is examined on the basis of qubit requirements, ancillary bit usage, quantum cost, quantum delay, gate operations, and circuit complexity, and is tested comprehensively on various input lengths. The work allows for sufficient flexibility in the design of quantum algorithms, which can accelerate quantum algorithm development.
Abstract Efficient quantum comparators are a key building block in quantum algorithms, but their practical utilization during the Noisy Intermediate-Scale Quantum (NISQ) era is limited by high resource demands. In this paper, we present an automated and scalable approach focused on the synthesis of optimal n-bit quantum comparator circuits that significantly minimize key quantum parameters, namely quantum cost (QC), quantum delay (QD), and garbage outputs (GO). Our approach uses a dynamic algorithmic framework built with Python and Qiskit to adapt the circuit design based on the size of the input operands. This enables the automatic generation of resource-efficient, modular designs. Through systematic optimization of qubit assignment, gate ordering, and sub-circuit composition, our full circuit achieves substantial performance gains. Experimental validation using 4-bit comparators demonstrates 14.89% reduction in QC, 12.5% in QD, and 18.75% in GO over state-of-art designs, as validated through noise-free simulations under NISQ-era limitations. Although the algorithmic scalability of the produced circuits allows straightforward generalization to any n-bit operands without any loss in optimization quality. These findings confirm the effectiveness of our approach, suggesting that it may lead to further optimizations for related quantum circuits, including as Arithmetic Logic Units.
This paper presents a new technique of magnitudecomparator for quantum bit string comparison. In the proposedmethod, the comparison between two quantum bit strings isperformed with the optimum number of operations. We have alsoshown that the proposed technique has time complexityO(( + )), whereas the best known existing technique hasO(nlog n),where n is the number of quantum bits. In addition, wehave proposed another technique to produce a compact quantumcomparator circuit. We have also introduced three new quantumgates with unique unitary matrices which represent the proposedquantum circuit more compactly than the existing quantumgates. The proposed comparator circuit has been designed usingthe proposed quantum bit comparator circuit named MidpointQubits Comparison (MQC) Circuit and another proposedquantum bit comparator circuit named Rest Qubits Comparison(RQC) Circuit. The comparative study shows that the proposedcomparator circuit outperforms the existing comparators; e.g.,the proposed 64-qubit comparator improves 7.19% on number ofquantum gates, area and delay; and 50.39% on garbage outputsover the existing best one
Quantum computing is at the leading edge of advanced technology that uses quantum phenomena such as entanglement and super position to solve complex problems. One of the main application is to built comparator circuit to do arithmetic operations at stipulated time. Based on this our research is to propose the design and implementation of one-bit and two-bit quantum comparators using IBM Qiskit, which is formly compared with classical comparator. Also this work represents a comprehensive overview of the techniques required to optimize and design quantum comparator circuits using real quantum simulators. The results obtained from simulation promises the comparison accuracy with classical results. From the circuit designed it is inferred that the hardware complexity is very much reduced compared to classical comparator, which are tabulated in terms of number of quantum gates. The proposed method can further be extended to generate higher order comparator circuits. These results may form the basis for further research and development that will support the advancement of quantum computing and algorithms.
Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The solution requires the use of a quantum bit string comparator being used as oracle. This quantum circuit compares two quantum states and identifies if they are equal or, otherwise, which of them is the largest. Moreover, we also show the quantum bit string comparator allow us to implement conditional statements in quantum computation, a fundamental structure for designing of algorithms.
Abstract Quantum squaring circuits play a critical role in many quantum algorithms; however, most existing designs incur a significant qubit overhead due to the loss of input states and excessive use of ancillary qubits. In this work, we introduce a qubit-efficient quantum circuit for integer squaring that achieves a linear qubit cost of only 3 N qubits for an N -bit input, significantly outperforming state-of-the-art designs that scale quadratically in terms of qubits. Our approach reintegrates the input operand after computation, enabling the uncomputation of intermediate results and efficient recycling of ancilla qubits. This reversible strategy prevents the retention of redundant information, which is a common limitation of prior works. The comparative analysis confirms the scalability and practicality of our design for qubit-constrained quantum hardware, offering a promising solution for arithmetic operations in resource-limited quantum environments.
Abstract Multi-valued quantum systems have more significant advantages in computational performance than binary quantum system, since the circuit design is more compact and efficient in multi-valued quantum systems. Hence, synthesizing multi-valued logic circuits is a better choice for future quantum computing technology. A subcircuit in the Grover’s algorithm is called an oracle, which inputs a set of quantum qudits and gives an output according to the search conditions. Greater-than/less-than/equality comparators are widely used in search conditions. In this paper, the ternary Toffoli gate proposed by Monfared and Haghparast is extended to multi-valued under certain conditions, and the existing optimal quaternary n-qudit comparator is optimized using the multi-valued Toffoli gate. In addition, we design universal multi-valued equality/greater-than/less-than comparators and complementary component, which constitute a comprehensive multi-valued n-qudit comparator. Compared to existing comparator circuits, these comparator circuits are more efficient in terms of quantum cost, garbage output, hardware complexity, and constant input.