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

ACM/IEEE Design Automation Conference (DAC)

Embedding Functions Into Reversible Circuits

Niels Gleinig, Frances Ann Hubis, Torsten Hoefler

Published May 23, 2019• 3 citations
Publisher

Abstract

In order to compute a non-invertible function on a reversible circuit, one needs to "embed" the function into a larger function which has some garbage bits, corresponding to additional lines. The problem of determining the minimal number of garbage bits that are needed to embed a given function has attracted extensive research, largely motivated by quantum computing, where the number of lines equals the number of qubits. However, all approaches that are known have either no theoretical quality guarantees (bounds on approximation factors) or require exponential runtime. We present an efficient probabilistic approximation algorithm with theoretical bounds.

DOI
10.1145/3316781.3317814

Related papers

Determining the minimal number of lines for large reversible circuits

Robert Wille, Oliver Keszöcze, Rolf Drechsler

Synthesis of reversible circuits is an active research area motivated by its applications e.g. in quantum computation or low-power design. The number of used circuit lines is thereby a crucial criterion. In this paper, we introduce several methods (including a theoretical upper bound) for the efficient computation or at least approximation of the minimal number of lines needed to realize a given function in reversible logic. While the proposed exact approach requires a significant amount of run-time (exponential in the worst case), the heuristic methods lead to very precise approximations in very short run-time. Using this, it can be shown that current synthesis approaches for large functions are still far away from producing optimal circuits with respect to the number of lines.

Design, Automation, and Test in Europe•Mar 1, 2011•46 citations

One Additional Qubit is Enough: Encoded Embeddings for Boolean Components in Quantum Circuits

Alwin Zulehner, Philipp Niemann, Rolf Drechsler +1 more

Research on quantum computing has recently gained significant momentum since first physical devices became available. Many quantum algorithms make use of so-called oracles that implement Boolean functions and are queried with highly superposed input states in order to evaluate the implemented Boolean function for many different input patterns in parallel. To simplify or enable a realization of these oracles in quantum logic in the first place, the Boolean reversible functions to be realized usually need to be broken down into several non-reversible sub-functions. However, since quantum logic is inherently reversible, these sub-functions have to be realized in a reversible fashion by adding further qubits in order to make the output patterns distinguishable (a process that is also known as embedding). This usually results in a significant increase of the qubits required in total. In this work, we show how this overhead can be significantly reduced by utilizing coding. More precisely, we prove that one additional qubit is always enough to embed any non-reversible function into a reversible one by using a variable-length encoding of the output patterns. Moreover, we characterize those functions that do not require an additional qubit at all. The made observations show that coding often allows one to undercut the usually considered minimum of additional qubits in sub-functions of oracles by far.

arXiv:1906.02352•IEEE International Symposium on Multiple-Valued Logic (ISMVL)•May 1, 2019•5 citations

Synthesis of reversible circuits with minimal lines for large functions

Mathias Soeken, Robert Wille, Christoph Hilken +2 more

Reversible circuits are an emerging technology where all computations are performed in an invertible manner. Motivated by their promising applications, e.g. in the domain of quantum computation or in the low-power design, the synthesis of such circuits has been intensely studied. However, how to automatically realize reversible circuits with the minimal number of lines for large functions is an open research problem. In this paper, we propose a new synthesis approach which relies on concepts that are complementary to existing ones. While “conventional” function representations have been applied for synthesis so far (such as truth tables, ESOPs, BDDs), we exploit Quantum Multiple-valued Decision Diagrams (QMDDs) for this purpose. An algorithm is presented that performs transformations on this data-structure eventually leading to the desired circuit. Experimental results show the novelty of the proposed approach through enabling automatic synthesis of large reversible functions with the minimal number of circuit lines. Furthermore, the quantum cost of the resulting circuits is reduced by 50% on average compared to an existing state-of-the-art synthesis method.

Asia and South Pacific Design Automation Conference (ASP-DAC)•Jan 1, 2012•107 citations

Optimized Boolean expression embedding in quantum and reversible logic circuits

Zaid Al-Wardi

The quantum and reversible circuits represent a promising replacement of conventional computers in many applications in the future. The transition between conventional computing and this alternative is indeed a huge paradigm shift. During the transition, embedding of conventional Boolean logic functions and expressions within quantum and reversible logic circuits has to be considered, because conventional computing is generally based on Boolean algebra. This embedding comes with a serious drawback of additional increase in the problem complexity represented by additional constant inputs, the so-called ancillary lines. An algorithmic optimization technique is suggested in this paper to reduce the number of these ancillary lines. With this algorithm embedding of Boolean expressions within reversible/quantum circuits comes out with minimal increase in circuit cost and gate count in the optimized circuits.

AIP conference proceedings•Jan 1, 2023•0 citations

Synthesis of the Optimal 4-bit Reversible Circuits

Oleg Golubitsky, Sean M. Falconer, Maslov, Dmitri

Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16!~2^44 functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present an algorithm that synthesizes an optimal circuit for any 4-bit reversible specification. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions, and distribution of optimal circuits. Our results have important implications for the design and optimization of quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.

arXiv:1003.1914•Mar 9, 2010•0 citations

Make it reversible: Efficient embedding of non-reversible functions

Alwin Zulehner, Robert Wille

Reversible computation became established as a promising concept due to its application in various areas like quantum computation, energy-aware circuits, and further areas. Unfortunately, most functions of interest are non-reversible. Therefore, a process called embedding has to be conducted to transform a non-reversible function into a reversible one - a coNP-hard problem. Existing solutions suffer from the resulting exponential complexity and, hence, are limited to rather small functions only. In this work, an approach is presented which tackles the problem in an entirely new fashion. We divide the embedding process into matrix operations, which can be conducted efficiently on a certain kind of decision diagram. Experiments show that improvements of several orders of magnitudes can be achieved using the proposed method. Moreover, for many benchmarks exact results can be obtained for the first time ever.

Design, Automation, and Test in Europe•Mar 1, 2017•44 citations
In order to compute a non-invertible function on a reversible circuit, one needs to "embed" the function into a larger function which has some garbage bits, corresponding to additional lines. The problem of determining the minimal number of garbage bits that are needed to embed a given function has attracted extensive research, largely motivated by quantum computing, where the number of lines equals the number of qubits. However, all approaches that are known have either no theoretical quality guarantees (bounds on approximation factors) or require exponential runtime. We present an efficient probabilistic approximation algorithm with theoretical bounds.
Synthesis of reversible circuits is an active research area motivated by its applications e.g. in quantum computation or low-power design. The number of used circuit lines is thereby a crucial criterion. In this paper, we introduce several methods (including a theoretical upper bound) for the efficient computation or at least approximation of the minimal number of lines needed to realize a given function in reversible logic. While the proposed exact approach requires a significant amount of run-time (exponential in the worst case), the heuristic methods lead to very precise approximations in very short run-time. Using this, it can be shown that current synthesis approaches for large functions are still far away from producing optimal circuits with respect to the number of lines.
Research on quantum computing has recently gained significant momentum since first physical devices became available. Many quantum algorithms make use of so-called oracles that implement Boolean functions and are queried with highly superposed input states in order to evaluate the implemented Boolean function for many different input patterns in parallel. To simplify or enable a realization of these oracles in quantum logic in the first place, the Boolean reversible functions to be realized usually need to be broken down into several non-reversible sub-functions. However, since quantum logic is inherently reversible, these sub-functions have to be realized in a reversible fashion by adding further qubits in order to make the output patterns distinguishable (a process that is also known as embedding). This usually results in a significant increase of the qubits required in total. In this work, we show how this overhead can be significantly reduced by utilizing coding. More precisely, we prove that one additional qubit is always enough to embed any non-reversible function into a reversible one by using a variable-length encoding of the output patterns. Moreover, we characterize those functions that do not require an additional qubit at all. The made observations show that coding often allows one to undercut the usually considered minimum of additional qubits in sub-functions of oracles by far.
Reversible circuits are an emerging technology where all computations are performed in an invertible manner. Motivated by their promising applications, e.g. in the domain of quantum computation or in the low-power design, the synthesis of such circuits has been intensely studied. However, how to automatically realize reversible circuits with the minimal number of lines for large functions is an open research problem. In this paper, we propose a new synthesis approach which relies on concepts that are complementary to existing ones. While “conventional” function representations have been applied for synthesis so far (such as truth tables, ESOPs, BDDs), we exploit Quantum Multiple-valued Decision Diagrams (QMDDs) for this purpose. An algorithm is presented that performs transformations on this data-structure eventually leading to the desired circuit. Experimental results show the novelty of the proposed approach through enabling automatic synthesis of large reversible functions with the minimal number of circuit lines. Furthermore, the quantum cost of the resulting circuits is reduced by 50% on average compared to an existing state-of-the-art synthesis method.
The quantum and reversible circuits represent a promising replacement of conventional computers in many applications in the future. The transition between conventional computing and this alternative is indeed a huge paradigm shift. During the transition, embedding of conventional Boolean logic functions and expressions within quantum and reversible logic circuits has to be considered, because conventional computing is generally based on Boolean algebra. This embedding comes with a serious drawback of additional increase in the problem complexity represented by additional constant inputs, the so-called ancillary lines. An algorithmic optimization technique is suggested in this paper to reduce the number of these ancillary lines. With this algorithm embedding of Boolean expressions within reversible/quantum circuits comes out with minimal increase in circuit cost and gate count in the optimized circuits.
Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16!~2^44 functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present an algorithm that synthesizes an optimal circuit for any 4-bit reversible specification. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions, and distribution of optimal circuits. Our results have important implications for the design and optimization of quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.
Reversible computation became established as a promising concept due to its application in various areas like quantum computation, energy-aware circuits, and further areas. Unfortunately, most functions of interest are non-reversible. Therefore, a process called embedding has to be conducted to transform a non-reversible function into a reversible one - a coNP-hard problem. Existing solutions suffer from the resulting exponential complexity and, hence, are limited to rather small functions only. In this work, an approach is presented which tackles the problem in an entirely new fashion. We divide the embedding process into matrix operations, which can be conducted efficiently on a certain kind of decision diagram. Experiments show that improvements of several orders of magnitudes can be achieved using the proposed method. Moreover, for many benchmarks exact results can be obtained for the first time ever.