Sunyeop Kim, Insung Kim, Seonggyeom Kim, Seokhie Hong
{"title":"用于二进制场乘法的托福利门计数优化空间效率量子电路","authors":"Sunyeop Kim, Insung Kim, Seonggyeom Kim, Seokhie Hong","doi":"10.1007/s11128-024-04536-1","DOIUrl":null,"url":null,"abstract":"<div><p>Shor’s algorithm solves the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. To optimize Shor’s algorithm for binary elliptic curves, reducing the cost of binary field multiplication is essential because it is the most cost-critical arithmetic operation. In this paper, we propose Toffoli gate count-optimized, space-efficient (i.e., no ancilla qubits are used) quantum circuits for binary field (<span>\\((\\mathbb {F}_{2^{n}})\\)</span>) multiplication. To achieve this, we leverage the Karatsuba-like formulae and demonstrate that its application can be implemented without the need for ancillary qubits. We optimize these circuits in terms of CNOT gate count and depth. Building upon the Karatsuba-like formulae, we develop a space-efficient CRT-based multiplication technique utilizing two types of out-of-place multiplication algorithms to reduce the CNOT gate count. Our quantum circuits exhibit an extremely low Toffoli gate count of <span>\\(O(n2^{\\log {2}^{*}n})\\)</span>, where <span>\\(\\log _{2}^{*}\\)</span> represents the iterative logarithmic function that grows very slowly. When compared to recent Karatsuba-based space-efficient quantum circuit, our approach requires only (10–25 %) of the Toffoli gate count and Toffoli depth for cryptographic field sizes in the range of <i>n</i> = 233–571. To the best of our knowledge, this represents the first successful utilization of the Karatsuba-like formulae and CRT-based multiplication in quantum circuits.</p></div>","PeriodicalId":746,"journal":{"name":"Quantum Information Processing","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Toffoli gate count optimized space-efficient quantum circuit for binary field multiplication\",\"authors\":\"Sunyeop Kim, Insung Kim, Seonggyeom Kim, Seokhie Hong\",\"doi\":\"10.1007/s11128-024-04536-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Shor’s algorithm solves the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. To optimize Shor’s algorithm for binary elliptic curves, reducing the cost of binary field multiplication is essential because it is the most cost-critical arithmetic operation. In this paper, we propose Toffoli gate count-optimized, space-efficient (i.e., no ancilla qubits are used) quantum circuits for binary field (<span>\\\\((\\\\mathbb {F}_{2^{n}})\\\\)</span>) multiplication. To achieve this, we leverage the Karatsuba-like formulae and demonstrate that its application can be implemented without the need for ancillary qubits. We optimize these circuits in terms of CNOT gate count and depth. Building upon the Karatsuba-like formulae, we develop a space-efficient CRT-based multiplication technique utilizing two types of out-of-place multiplication algorithms to reduce the CNOT gate count. Our quantum circuits exhibit an extremely low Toffoli gate count of <span>\\\\(O(n2^{\\\\log {2}^{*}n})\\\\)</span>, where <span>\\\\(\\\\log _{2}^{*}\\\\)</span> represents the iterative logarithmic function that grows very slowly. When compared to recent Karatsuba-based space-efficient quantum circuit, our approach requires only (10–25 %) of the Toffoli gate count and Toffoli depth for cryptographic field sizes in the range of <i>n</i> = 233–571. To the best of our knowledge, this represents the first successful utilization of the Karatsuba-like formulae and CRT-based multiplication in quantum circuits.</p></div>\",\"PeriodicalId\":746,\"journal\":{\"name\":\"Quantum Information Processing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-09-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quantum Information Processing\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11128-024-04536-1\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Information Processing","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11128-024-04536-1","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Toffoli gate count optimized space-efficient quantum circuit for binary field multiplication
Shor’s algorithm solves the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. To optimize Shor’s algorithm for binary elliptic curves, reducing the cost of binary field multiplication is essential because it is the most cost-critical arithmetic operation. In this paper, we propose Toffoli gate count-optimized, space-efficient (i.e., no ancilla qubits are used) quantum circuits for binary field (\((\mathbb {F}_{2^{n}})\)) multiplication. To achieve this, we leverage the Karatsuba-like formulae and demonstrate that its application can be implemented without the need for ancillary qubits. We optimize these circuits in terms of CNOT gate count and depth. Building upon the Karatsuba-like formulae, we develop a space-efficient CRT-based multiplication technique utilizing two types of out-of-place multiplication algorithms to reduce the CNOT gate count. Our quantum circuits exhibit an extremely low Toffoli gate count of \(O(n2^{\log {2}^{*}n})\), where \(\log _{2}^{*}\) represents the iterative logarithmic function that grows very slowly. When compared to recent Karatsuba-based space-efficient quantum circuit, our approach requires only (10–25 %) of the Toffoli gate count and Toffoli depth for cryptographic field sizes in the range of n = 233–571. To the best of our knowledge, this represents the first successful utilization of the Karatsuba-like formulae and CRT-based multiplication in quantum circuits.
期刊介绍:
Quantum Information Processing is a high-impact, international journal publishing cutting-edge experimental and theoretical research in all areas of Quantum Information Science. Topics of interest include quantum cryptography and communications, entanglement and discord, quantum algorithms, quantum error correction and fault tolerance, quantum computer science, quantum imaging and sensing, and experimental platforms for quantum information. Quantum Information Processing supports and inspires research by providing a comprehensive peer review process, and broadcasting high quality results in a range of formats. These include original papers, letters, broadly focused perspectives, comprehensive review articles, book reviews, and special topical issues. The journal is particularly interested in papers detailing and demonstrating quantum information protocols for cryptography, communications, computation, and sensing.