Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Scaling-up ESOP Synthesis for Quantum Compilation

Bruno de O. Schmitt, Mathias Soeken, Giovanni De Micheli, Alan Mishchenko

Published May 1, 2019• 8 citations
Publisher

Abstract

Today's rapid advances in quantum computing hardware call for scalable synthesis methods to map combinational logic represented as multi-level Boolean networks (e.g., an and-inverter graph, AIG) to quantum circuits. Such synthesis process must yield reversible logic function since quantum circuits are reversible. Thus, logic representations using exclusive sum-of-products (ESOP) are advantageous because of their natural relation to Toffoli gates, one of the primitives in reversible logic. This motivates developing effective methods to collapse AIG logic networks into ESOPs. In this work, we present two state-of-the-art methods to collapse an AIG into an ESOP expression, describe their shortcomings and introduce a new approach based on the divide-and-conquer paradigm. We demonstrate the effectiveness of our method in collapsing IEEE-compliant half precision floating point networks. Results show that our method can collapse designs-which were previously not solvable within a week-in less than 5 minutes. We also describe a technique capable of taking advantage of this new method to generate quantum circuits with up to 50% fewer T gates compared to state-of-the-art methods.

DOI
10.1109/ismvl.2019.00011

Related papers

A pre-optimization technique to generate initial reversible circuits with low quantum cost

Nurul Ain Binti Adnan, Kouhei Kushida, Shigeru Yamashita

In order to generate an initial reversible/quantum circuit to realize a given Boolean function, one of the major approaches is to find a small Exclusive-or Sum-Of-Products (ESOP) expression for the function; each product term in the ESOP expression naturally corresponds to a single Mixed Polarity Multiple-Control Toffoli (MPMCT) gate. In this paper, we propose a technique to perform pre-optimization before generating an ESOP expression. Our approach is unique in a sense that instead of finding a small ESOP expression for a given function, we try to change it by adding MPMCT gates so that the modified function has a smaller ESOP expression. We expect that our approach can generate a better initial circuit compared to using ESOP minimization techniques only. Indeed our preliminary experiments for small functions confirm this expectation. Thus, we expect that our approach would be a good pre-optimization technique that can be used with most existing reversible circuit synthesis techniques.

IEEE International Symposium on Circuits and Systems proceedings•May 1, 2016•2 citations

Quantum algorithm for ESOP minimization

Hrithik Ketineni, Marek A. Perkowski

The Exclusive-OR Sum-of-Product (ESOP) minimization problem has long been of interest to the research community because of its importance in classical logic design (including low-power design and design for test), reversible logic synthesis, and knowledge discovery, among other applications. However, no exact minimal minimization method has been presented for more than seven variables on arbitrary functions. This paper presents a novel quantum-classical hybrid algorithm for the exact minimal ESOP minimization of incompletely specified Boolean functions. This algorithm constructs oracles from sets of constraints and leverages the quantum speedup offered by Grover's algorithm to find solutions to these oracles, thereby improving over classical algorithms. Improved encoding of ESOP expressions results in substantially fewer decision variables compared to many existing algorithms for many classes of Boolean functions. This paper also extends the idea of exact minimal ESOP minimization to additionally minimize the cost of realizing an ESOP expression as a quantum circuit. To the extent of the authors' knowledge, such a method has never been published. This algorithm was tested on completely and incompletely specified Boolean functions via quantum simulation.

Quantum Information and Computation•Jan 29, 2023•2 citations

Optimization of Reversible Logic Networks with Gate Sharing

Yung‐Chih Chen, Feng-Jie Chao

Logic synthesis for quantum computing aims to transform a Boolean logic network into a quantum circuit. A conventional two-stage flow first synthesizes the given Boolean logic network into a reversible logic network composed of reversible logic gates. Then, it maps each reversible logic gate into quantum gates to generate a quantum circuit. The state-of-the-art method for the first stage takes advantage of the lookup-table (LUT) mapping technology for FPGAs to decompose the given Boolean logic network into sub-networks, and then maps the sub-networks into reversible logic networks. Although every sub-network is well synthesized, we observe that the reversible logic networks could be further optimized by sharing the reversible logic gates belonging to different sub-networks. Thus, in this paper, we propose a new optimization method for the reversible logic networks by sharing gates. We translate the problem of extracting shareable gates to the exclusive-sums-of-product term optimization problem. The experimental results show that the proposed method successfully optimizes the reversible logic networks generated by the LUT-based method. It is able to reduce an average of approximately 4% of quantum gate cost without increasing the number of ancilla lines for a set of IWLS 2005 benchmarks.

Asia and South Pacific Design Automation Conference (ASP-DAC)•Jan 16, 2023•2 citations

An Improved Factorization Approach to Reversible Circuit Synthesis Based on EXORs of Products of EXORs

Linh Hoang Tran, Addison Gronquist, Marek A. Perkowski +1 more

This paper introduces new algorithms to synthesize reversible functions using EXOR-sum of Products-of-EXOR-sums (EPOE) structures. The motivation for using these structures is to reduce the number of as well as the sizes of multiple controlled Toffoli gates, and thus the quantum cost. To achieve these reductions the paper generalizes from existing 2-level AND-EXOR structures (ESOP) commonly used in reversible logic to a mixture of 3-level EXOR-AND-EXOR structures and ESOPs. Our approach can be applied to reversible and permutative quantum circuits to synthesize single output functions on to an output line, with no additional ancilla bits. A comparison of the ESOP minimizer EXORCISM-4 and two variants of the EPOE minimizer, called EPOEM-1s and EPOEM-2, is presented. The results show that EPOE circuits do in fact achieve the above-stated cost reductions, in particular when expressed in terms of Maslov's quantum cost, the metric commonly used in quantum circuit synthesis.

IEEE International Symposium on Multiple-Valued Logic (ISMVL)•May 1, 2016•5 citations

Lowering the T-depth of Quantum Circuits via Logic Network Optimization

Thomas Häner, Mathias Soeken

The multiplicative depth of a logic network over the gate basis {∧ , ⊕ , ¬} is the largest number of ∧ gates on any path from a primary input to a primary output in the network. We describe a dynamic programming based logic synthesis algorithm to reduce the multiplicative depth of logic networks. It makes use of cut enumeration, tree balancing, and exclusive sum-of-products (ESOP) representations. Our algorithm has applications to cryptography and quantum computing, as a reduction in the multiplicative depth directly translates to a lower T -depth of the corresponding quantum circuit. Our experimental results show improvements in T -depth over state-of-the-art methods and over several hand-optimized quantum circuits, for instance, of AES, SHA, and floating-point arithmetic.

ACM Transactions on Quantum Computing•Mar 4, 2022•7 citations

Factorization of Exclusive-Sum-Of-Products Expressions with Rectangle Covering to Reduce Quantum Circuit Cost

Audrey Hou, Lucia Zhang, Ali Al-Bayaty +1 more

The implementation of quantum circuits is currently very expensive, especially due to the usage of large Toffoli gates. Therefore, it is critical to optimize circuit costs by factoring expressions as they become more complex. In the proposed algorithms to factor ESOP expressions, each product term is converted into a cell in a 2D matrix, and optimal factored AND/EXOR solutions are determined using Disjoint and Even-Odd Rectangle covering methods. Two Python programs implementing these algorithms were tested and evaluated using well-known benchmarks. The results showed that both the literal counts used in classical logic circuits as well as the Maslov cost used in quantum circuits was reduced by 20%-95% depending on expression size.

quant-pharXiv:2608.03188•Aug 5, 2026•0 citations
Today's rapid advances in quantum computing hardware call for scalable synthesis methods to map combinational logic represented as multi-level Boolean networks (e.g., an and-inverter graph, AIG) to quantum circuits. Such synthesis process must yield reversible logic function since quantum circuits are reversible. Thus, logic representations using exclusive sum-of-products (ESOP) are advantageous because of their natural relation to Toffoli gates, one of the primitives in reversible logic. This motivates developing effective methods to collapse AIG logic networks into ESOPs. In this work, we present two state-of-the-art methods to collapse an AIG into an ESOP expression, describe their shortcomings and introduce a new approach based on the divide-and-conquer paradigm. We demonstrate the effectiveness of our method in collapsing IEEE-compliant half precision floating point networks. Results show that our method can collapse designs-which were previously not solvable within a week-in less than 5 minutes. We also describe a technique capable of taking advantage of this new method to generate quantum circuits with up to 50% fewer T gates compared to state-of-the-art methods.
In order to generate an initial reversible/quantum circuit to realize a given Boolean function, one of the major approaches is to find a small Exclusive-or Sum-Of-Products (ESOP) expression for the function; each product term in the ESOP expression naturally corresponds to a single Mixed Polarity Multiple-Control Toffoli (MPMCT) gate. In this paper, we propose a technique to perform pre-optimization before generating an ESOP expression. Our approach is unique in a sense that instead of finding a small ESOP expression for a given function, we try to change it by adding MPMCT gates so that the modified function has a smaller ESOP expression. We expect that our approach can generate a better initial circuit compared to using ESOP minimization techniques only. Indeed our preliminary experiments for small functions confirm this expectation. Thus, we expect that our approach would be a good pre-optimization technique that can be used with most existing reversible circuit synthesis techniques.
The Exclusive-OR Sum-of-Product (ESOP) minimization problem has long been of interest to the research community because of its importance in classical logic design (including low-power design and design for test), reversible logic synthesis, and knowledge discovery, among other applications. However, no exact minimal minimization method has been presented for more than seven variables on arbitrary functions. This paper presents a novel quantum-classical hybrid algorithm for the exact minimal ESOP minimization of incompletely specified Boolean functions. This algorithm constructs oracles from sets of constraints and leverages the quantum speedup offered by Grover's algorithm to find solutions to these oracles, thereby improving over classical algorithms. Improved encoding of ESOP expressions results in substantially fewer decision variables compared to many existing algorithms for many classes of Boolean functions. This paper also extends the idea of exact minimal ESOP minimization to additionally minimize the cost of realizing an ESOP expression as a quantum circuit. To the extent of the authors' knowledge, such a method has never been published. This algorithm was tested on completely and incompletely specified Boolean functions via quantum simulation.
Logic synthesis for quantum computing aims to transform a Boolean logic network into a quantum circuit. A conventional two-stage flow first synthesizes the given Boolean logic network into a reversible logic network composed of reversible logic gates. Then, it maps each reversible logic gate into quantum gates to generate a quantum circuit. The state-of-the-art method for the first stage takes advantage of the lookup-table (LUT) mapping technology for FPGAs to decompose the given Boolean logic network into sub-networks, and then maps the sub-networks into reversible logic networks. Although every sub-network is well synthesized, we observe that the reversible logic networks could be further optimized by sharing the reversible logic gates belonging to different sub-networks. Thus, in this paper, we propose a new optimization method for the reversible logic networks by sharing gates. We translate the problem of extracting shareable gates to the exclusive-sums-of-product term optimization problem. The experimental results show that the proposed method successfully optimizes the reversible logic networks generated by the LUT-based method. It is able to reduce an average of approximately 4% of quantum gate cost without increasing the number of ancilla lines for a set of IWLS 2005 benchmarks.
This paper introduces new algorithms to synthesize reversible functions using EXOR-sum of Products-of-EXOR-sums (EPOE) structures. The motivation for using these structures is to reduce the number of as well as the sizes of multiple controlled Toffoli gates, and thus the quantum cost. To achieve these reductions the paper generalizes from existing 2-level AND-EXOR structures (ESOP) commonly used in reversible logic to a mixture of 3-level EXOR-AND-EXOR structures and ESOPs. Our approach can be applied to reversible and permutative quantum circuits to synthesize single output functions on to an output line, with no additional ancilla bits. A comparison of the ESOP minimizer EXORCISM-4 and two variants of the EPOE minimizer, called EPOEM-1s and EPOEM-2, is presented. The results show that EPOE circuits do in fact achieve the above-stated cost reductions, in particular when expressed in terms of Maslov's quantum cost, the metric commonly used in quantum circuit synthesis.
The multiplicative depth of a logic network over the gate basis {∧ , ⊕ , ¬} is the largest number of ∧ gates on any path from a primary input to a primary output in the network. We describe a dynamic programming based logic synthesis algorithm to reduce the multiplicative depth of logic networks. It makes use of cut enumeration, tree balancing, and exclusive sum-of-products (ESOP) representations. Our algorithm has applications to cryptography and quantum computing, as a reduction in the multiplicative depth directly translates to a lower T -depth of the corresponding quantum circuit. Our experimental results show improvements in T -depth over state-of-the-art methods and over several hand-optimized quantum circuits, for instance, of AES, SHA, and floating-point arithmetic.
The implementation of quantum circuits is currently very expensive, especially due to the usage of large Toffoli gates. Therefore, it is critical to optimize circuit costs by factoring expressions as they become more complex. In the proposed algorithms to factor ESOP expressions, each product term is converted into a cell in a 2D matrix, and optimal factored AND/EXOR solutions are determined using Disjoint and Even-Odd Rectangle covering methods. Two Python programs implementing these algorithms were tested and evaluated using well-known benchmarks. The results showed that both the literal counts used in classical logic circuits as well as the Maslov cost used in quantum circuits was reduced by 20%-95% depending on expression size.