In recent years, quantum technology has been rapidly developed. As security analyses for symmetric ciphers continue to emerge, many require an evaluation of the resources needed for the quantum circuit implementation of the encryption algorithm. In this regard, we propose the quantum circuit decision problem, which requires us to determine whether there exists a quantum circuit for a given permutation f using M ancilla qubits and no more than K quantum gates within the circuit depth D. Firstly, we investigate heuristic algorithms and classical SAT-based models in previous works, revealing their limitations in solving the problem. Hence, we innovatively propose an improved SAT-based model incorporating three metrics of quantum circuits. The model enables us to find the optimal quantum circuit of an arbitrary 3 or 4-bit S-box under a given optimization goal based on SAT solvers, which has proved the optimality of circuits constructed by the tool, LIGHTER-R. Then, by combining different criteria in the model, we find more compact quantum circuit implementations of S-boxes such as RECTANGLE and GIFT. For GIFT S-box, our model provides the optimal quantum circuit that only requires 8 gates with a depth of 31. Furthermore, our model can be generalized to linear layers and improve the previous SAT-based model proposed by Huang et al. in ASIACRYPT 2022 by adding the criteria on the number of qubits and the circuit depth.In recent years, quantum technology has been rapidly developed. As security analyses for symmetric ciphers continue to emerge, many require an evaluation of the resources needed for the quantum circuit implementation of the encryption algorithm. In this regard, we propose the quantum circuit decision problem, which requires us to determine whether there exists a quantum circuit for a given permutation f using M ancilla qubits and no more than K quantum gates within the circuit depth D. Firstly, we investigate heuristic algorithms and classical SAT-based models in previous works, revealing their limitations in solving the problem. Hence, we innovatively propose an improved SAT-based model incorporating three metrics of quantum circuits. The model enables us to find the optimal quantum circuit of an arbitrary 3 or 4-bit S-box under a given optimization goal based on SAT solvers, which has proved the optimality of circuits constructed by the tool, LIGHTER-R. Then, by combining different criteria in the model, we find more compact quantum circuit implementations of S-boxes such as RECTANGLE and GIFT. For GIFT S-box, our model provides the optimal quantum circuit that only requires 8 gates with a depth of 31. Furthermore, our model can be generalized to linear layers and improve the previous SAT-based model proposed by Huang et al. in ASIACRYPT 2022 by adding the criteria on the number of qubits and the circuit depth.

IACR Communications in CryptologyApr 9, 20249 citations

Performance Gains in Quantum SAT Solvers Using ESOP Encoding

Majd Assaad, Abhoy Kole, Rolf Drechsler

The Boolean Satisfiability (SAT) problem is a canonical NP-complete problem and a natural candidate for quantum acceleration via search-based algorithms. In Grover-based quantum SAT solvers, the dominant computational cost stems from the construction of a reversible oracle that evaluates the Boolean formula, rendering the choice of SAT encoding crucial for overall quantum resource efficiency. Although SAT instances are conventionally expressed in Conjunctive Normal Form (CNF), such encodings typically translate into quantum circuits with significant qubit overhead and high non-Clifford gate complexity. In this work, we investigate an Exclusive-Sum-of-Products (ESOP)-based CNF (e-CNF) representation tailored for quantum SAT solving and analyze its impact on oracle construction. We derive tighter upper bounds on qubit requirements and Clifford+ $T$ gate counts for Grover-based SAT solvers when e-CNF encodings are employed in place of standard CNF. In addition, we propose a scalable transformation from Boolean formulas to e-CNF and present a systematic procedure for interpreting e-CNF representations as reversible quantum circuits suitable for oracle implementation. Experimental evaluation on representative SAT benchmarks demonstrates that the proposed e-CNF-based approach yields substantial and consistent reductions in quantum resources, including qubit count, T-gate complexity, and circuit depth, when compared to CNF-based oracle constructions. These results establish e-CNF as an effective quantum-aware SAT encoding that significantly improves the practicality of oracle-based quantum SAT solving.The Boolean Satisfiability (SAT) problem is a canonical NP-complete problem and a natural candidate for quantum acceleration via search-based algorithms. In Grover-based quantum SAT solvers, the dominant computational cost stems from the construction of a reversible oracle that evaluates the Boolean formula, rendering the choice of SAT encoding crucial for overall quantum resource efficiency. Although SAT instances are conventionally expressed in Conjunctive Normal Form (CNF), such encodings typically translate into quantum circuits with significant qubit overhead and high non-Clifford gate complexity. In this work, we investigate an Exclusive-Sum-of-Products (ESOP)-based CNF (e-CNF) representation tailored for quantum SAT solving and analyze its impact on oracle construction. We derive tighter upper bounds on qubit requirements and Clifford+$T$ gate counts for Grover-based SAT solvers when e-CNF encodings are employed in place of standard CNF. In addition, we propose a scalable transformation from Boolean formulas to e-CNF and present a systematic procedure for interpreting e-CNF representations as reversible quantum circuits suitable for oracle implementation. Experimental evaluation on representative SAT benchmarks demonstrates that the proposed e-CNF-based approach yields substantial and consistent reductions in quantum resources, including qubit count, T-gate complexity, and circuit depth, when compared to CNF-based oracle constructions. These results establish e-CNF as an effective quantum-aware SAT encoding that significantly improves the practicality of oracle-based quantum SAT solving.

quant-pharXiv:2605.16202May 18, 20260 citations

Constructing Quantum Combinational Logic Circuits

Jae‐Seung Lee, Jae‐Hyun Kim, Yongwook Chung +1 more

APSMar 1, 20000 citations

Generalized Quantum Turing Machine and Its Application for SAT Problem

Satoshi Iriyama, Masanori Ohya

IEEE International Symposium on Information Theory (ISIT)Nov 28, 20061 citation

Automated Theorem Proving for Quantum Circuit Verification using SAT Solvers

Jincheng Zhang

The verification of quantum circuits is a critical task in quantum computing, ensuring the fidelity and correctness of quantum computations. However, the exponential scaling of the problem with circuit size presents a significant challenge. This paper proposes a novel approach to automated theorem proving for quantum circuit verification by formulating the verification problem as a Boolean satisfiability (SAT) problem. We demonstrate a methodology for translating quantum circuit properties (e.g., output state, gate operations) into equivalent Boolean expressions and subsequently employing existing SAT solvers to determine the satisfiability of these expressions. This approach effectively bridges the gap between quantum circuit theory and SAT solving, offering a potentially scalable solution for verifying complex quantum circuits. The core of our system revolves around a translation module that converts quantum circuit specifications into SAT instances, and a solver interface that interacts with established SAT solvers. We explore the limitations of this approach and discuss future research directions, including hybrid approaches combining SAT solving with quantum simulation techniques. The effectiveness of this method is demonstrated through theoretical analysis and conceptual design, with a focus on the algorithmic efficiency and potential scalability of the proposed framework.The verification of quantum circuits is a critical task in quantum computing, ensuring the fidelity and correctness of quantum computations. However, the exponential scaling of the problem with circuit size presents a significant challenge. This paper proposes a novel approach to automated theorem proving for quantum circuit verification by formulating the verification problem as a Boolean satisfiability (SAT) problem. We demonstrate a methodology for translating quantum circuit properties (e.g., output state, gate operations) into equivalent Boolean expressions and subsequently employing existing SAT solvers to determine the satisfiability of these expressions. This approach effectively bridges the gap between quantum circuit theory and SAT solving, offering a potentially scalable solution for verifying complex quantum circuits. The core of our system revolves around a translation module that converts quantum circuit specifications into SAT instances, and a solver interface that interacts with established SAT solvers. We explore the limitations of this approach and discuss future research directions, including hybrid approaches combining SAT solving with quantum simulation techniques. The effectiveness of this method is demonstrated through theoretical analysis and conceptual design, with a focus on the algorithmic efficiency and potential scalability of the proposed framework.

Sep 5, 20260 citations

Constructing Quantum Circuits for Unitary Operations

Joanna Wiśniewska

Jan 1, 20110 citations

T

gate counts for Grover-based SAT solvers when e-CNF encodings are employed in place of standard CNF. In addition, we propose a scalable transformation from Boolean formulas to e-CNF and present a systematic procedure for interpreting e-CNF representations as reversible quantum circuits suitable for oracle implementation. Experimental evaluation on representative SAT benchmarks demonstrates that the proposed e-CNF-based approach yields substantial and consistent reductions in quantum resources, including qubit count, T-gate complexity, and circuit depth, when compared to CNF-based oracle constructions. These results establish e-CNF as an effective quantum-aware SAT encoding that significantly improves the practicality of oracle-based quantum SAT solving.