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INFORMS journal on computing

A New Multicommodity Network Flow Model and Branch and Cut for Optimal Quantum Boolean Circuit Synthesis

Jeenah Jung, In-Chan Choi

Published Nov 14, 2024• 3 citations
Publisher

Abstract

This study introduces a new optimization model and a branch-and-cut approach for synthesizing optimal quantum circuits for reversible Boolean functions, which are pivotal components in quantum algorithms. Although heuristic algorithms have been extensively explored for quantum circuit synthesis, research on exact counterparts remains relatively limited. However, the need to design quantum circuits with guaranteed optimality is increasing, especially for improving computational fidelity on noisy intermediate-scale quantum devices. This study presents mathematical optimization as a viable option for optimal synthesis, with the potential to accommodate practical considerations arising in fast-evolving quantum technologies. We set a demonstrative problem to implement reversible Boolean functions using high-level gates known as multiple control Toffoli gates while minimizing a technology-based proxy called quantum cost—the number of low-level gates used to realize each high-level gate. To address this problem, we propose a discrete optimization model based on a multicommodity network and discuss potential future variations at an abstract level to incorporate technical considerations. A customized branch and cut is then developed upon different aspects of our model, including polyhedron integrality, surrogate constraints, and variable prioritization. Our experiments demonstrate the robustness of the proposed approach in finding cost-optimal circuits for all benchmark instances within a two-hour time frame. Furthermore, we present interesting intuitions from these experiments and compare our computational results with relevant studies, highlighting newly discovered circuits with the lowest quantum costs reported in this paper. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing. This paper has been accepted for the INFORMS Journal on Computing Special Issue on Quantum Computing. Funding: This research was supported by the Ministry of Science and Information and Communication Technology, South Korea [Grants 2017R1E1A1A0307098814 and 2020R1A4A307986411]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0562 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0562 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

DOI
10.1287/ijoc.2024.0562

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This study introduces a new optimization model and a branch-and-cut approach for synthesizing optimal quantum circuits for reversible Boolean functions, which are pivotal components in quantum algorithms. Although heuristic algorithms have been extensively explored for quantum circuit synthesis, research on exact counterparts remains relatively limited. However, the need to design quantum circuits with guaranteed optimality is increasing, especially for improving computational fidelity on noisy intermediate-scale quantum devices. This study presents mathematical optimization as a viable option for optimal synthesis, with the potential to accommodate practical considerations arising in fast-evolving quantum technologies. We set a demonstrative problem to implement reversible Boolean functions using high-level gates known as multiple control Toffoli gates while minimizing a technology-based proxy called quantum cost—the number of low-level gates used to realize each high-level gate. To address this problem, we propose a discrete optimization model based on a multicommodity network and discuss potential future variations at an abstract level to incorporate technical considerations. A customized branch and cut is then developed upon different aspects of our model, including polyhedron integrality, surrogate constraints, and variable prioritization. Our experiments demonstrate the robustness of the proposed approach in finding cost-optimal circuits for all benchmark instances within a two-hour time frame. Furthermore, we present interesting intuitions from these experiments and compare our computational results with relevant studies, highlighting newly discovered circuits with the lowest quantum costs reported in this paper. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing. This paper has been accepted for the INFORMS Journal on Computing Special Issue on Quantum Computing. Funding: This research was supported by the Ministry of Science and Information and Communication Technology, South Korea [Grants 2017R1E1A1A0307098814 and 2020R1A4A307986411]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0562 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0562 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
Quantum computing is a newly emerging computing environment that has recently attracted intense research interest in improving the output fidelity, fully utilizing its high computing power from both hardware and software perspectives. In particular, several attempts have been made to reduce the errors in quantum computing algorithms through the efficient synthesis of quantum circuits. In this study, we present an application of an optimization model for synthesizing quantum circuits with minimum implementation costs to lower the error rates by forming a simpler circuit. Our model has a unique structure that combines the arc-subset selection problem with a conventional multi-commodity network flow model. The model targets the circuit synthesis with multiple control Toffoli gates to implement Boolean reversible functions that are often used as a key component in many quantum algorithms. Compared to previous studies, the proposed model has a unifying yet straightforward structure for exploiting the operational characteristics of quantum gates. Our computational experiment shows the potential of the proposed model, obtaining quantum circuits with significantly lower quantum costs compared to prior studies. The proposed model is also applicable to various other fields where reversible logic is utilized, such as low-power computing, fault-tolerant designs, and DNA computing. In addition, our model can be applied to network-based problems, such as logistics distribution and time-stage network problems.
As quantum technology advances, the efficient design of quantum circuits has become an important area of research. This paper provides an introduction to the MCT quantum circuit design problem for reversible Boolean functions with the necessary background in quantum computing to comprehend the problem. While this is a well-studied problem, optimization models that minimize the true objective have only been explored recently. This paper introduces a new optimization model and symmetry-breaking constraints that improve solving time by up to two orders of magnitude compared to earlier work when a Constraint Programming solver is used. Experiments with up to seven qubits and using up to 15 quantum gates result in several new best-known circuits, obtained by any method, for well-known benchmarks. Several in-depth analyses are presented to validate the effectiveness of the symmetry-breaking constraints from multiple perspectives. Finally, an extensive comparison with other approaches shows that optimization models may require more time but can provide superior circuits with optimality guarantees.
Abstract As quantum technology advances, the efficient design of quantum circuits has become an important area of research. This paper provides an introduction to the MCT quantum circuit design problem for reversible Boolean functions with the necessary background in quantum computing to comprehend the problem. While this is a well-studied problem, optimization models that minimize the true objective have only been explored recently. This paper introduces a new optimization model and symmetry-breaking constraints that improve solving time by up to two orders of magnitude compared to earlier work when a Constraint Programming solver is used. Experiments with up to seven qubits and using up to 15 quantum gates result in several new best-known circuits, obtained by any method, for well-known benchmarks. Several in-depth analyses are presented to validate the effectiveness of the symmetry-breaking constraints from multiple perspectives. Finally, an extensive comparison with other approaches shows that optimization models may require more time but can provide superior circuits with optimality guarantees.
The problem of synthesis and optimization of reversible and quantum circuits have drawn the attention of researchers for the last two decades due to increasing interest in quantum computing. Although lot of works have been done on the synthesis of binary reversible circuits, very less works have been reported on the synthesis of ternary reversible circuits. Ternary circuits have lower cost of implementation as compared to their binary counterparts. However, the synthesis approaches that exist for ternary reversible circuits either use too many circuit lines (qutrits) or too many gates. Only one prior work has discussed the problem of generating cost-optimal ternary reversible circuits, but for a very restrictive gate library, which limits the approach to a specific subset of ternary reversible functions and often the solution becomes sub-optimal due to the imposed restrictions. The present paper overcomes that restriction, and uses multiple control ternary Toffoli gates with all possible ternary target operations as the gate library. This gate library is functionally complete and can be used to synthesize any arbitrary function. The proposed SAT-based synthesis approach provides low cost solutions in terms of the number of gates for any arbitrary ternary reversible function. Experimental results on various randomly generated permutations as well as standard ternary benchmarks establish this claim. The results can be used as template for other synthesis approaches by observing how far they deviate from the optimal solutions.
Quantum computing is the latest computing concept that can substantially reduce the computational burden of some difficult NP problems. Boolean reversible logic is frequently used as a key component in many quantum algorithms. This study proposes an approach based on a mixed integer programming model for reversible circuit synthesis. The proposed optimization model maps a given reversible function into a multiple-control Toffoli (MCT) circuit with minimum quantum costs. Our model exploits the relationship between MCT gate patterns and the operations on qubit states. We also report our experience in computational experiments on the circuit synthesis of reversible benchmark problems.
Quantum computers, more specifically quantum circuits, take on the eyes with their computational promises such as reversibility. In this paper, we perform synthesis and optimization of quantum circuits. In the first part, we propose a fast synthesis algorithm that implements any given reversible Boolean function with quantum gates. Instead of an exhaustive search on every given function, our algorithm creates a library of essential functions and performs sorting. As an example, to implement 4 bit circuits we only use 120 essential functions out of all 20922789888000 functions. In the second part, we optimize our circuits by using new templates. The proposed templates mainly consist of Toffoli gates with negative and positive controlling lines. These templates also show us that optimum area solutions proposed in the literature are not actually optimum; they can be improved.