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Reversible logic

Alexis De Vos

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Reversible logic circuits are beneficial to both classical and quantum computer design. Present-day logic building-blocks (like OR gates and NAND gates) are logically irreversible and therefore cannot be used for designing reversible computers. Thus reversible computation needs an appropriate design methodology. In contrast to conventional digital logic circuits, reversible logic circuits (of a same logic width w) form a mathematical group. The reversible circuits of width w form a group isomorphic to the symmetric group S_{2^w}. Its Young subgroups allow systematic and efficient synthesis of an arbitrary reversible circuit. We can choose either a left coset, a right coset, or a double coset approach. The optimal design is reminiscent of the so-called banyan networks of telecommunication. As an illustration, three experimental prototypes (in c-MOS chip technology) of reversible computing devices are presented. Special care has been taken to avoid as much as possible the appearance of garbage bits. The examples illustrate how, in a near future, reversible computers will outperform conventional computers, in terms of power dissipation and heat generation.

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Reversible logic circuits are beneficial to both classical and quantum computer design. Present-day logic building-blocks (like OR gates and NAND gates) are logically irreversible and therefore cannot be used for designing reversible computers. Thus reversible computation needs an appropriate design methodology. In contrast to conventional digital logic circuits, reversible logic circuits (of a same logic width w) form a mathematical group. The reversible circuits of width w form a group isomorphic to the symmetric group S_{2^w}. Its Young subgroups allow systematic and efficient synthesis of an arbitrary reversible circuit. We can choose either a left coset, a right coset, or a double coset approach. The optimal design is reminiscent of the so-called banyan networks of telecommunication. As an illustration, three experimental prototypes (in c-MOS chip technology) of reversible computing devices are presented. Special care has been taken to avoid as much as possible the appearance of garbage bits. The examples illustrate how, in a near future, reversible computers will outperform conventional computers, in terms of power dissipation and heat generation.
The (2w)! reversible transformations on w wires, i.e. reversible logic circuits with w inputs and w outputs, together with the action of cascading, form a group, isomorphic to the symmetric group S2w. Therefore, we investigate the group Sn as well as one of its subgroups isomorphic to Sn/2 × Sn/2. We then consider the left cosets, the right cosets, and the double cosets generated by the subgroup. Each element of a coset can function as the representative of the coset. The coset can then be considered as the set of all group elements that differ from the representative by merely multiplying (either to the left or to the right or to both sides) by an arbitrary element of the subgroup. Different choices of the coset space and different choices of the coset representatives lead to six different syntheses for implementing an arbitrary reversible logic operation into hardware. Evaluation of all six methods, by means of three different cost functions (gate cost, switch cost, and quantum cost), leads to a best choice.
The (2w)! reversible transformations on w wires, i.e. reversible logic circuits with w inputs and w outputs, together with the action of cascading, form a group, isomorphic to the symmetric group S2w. Therefore, we investigate the group Snas well as one of its subgroups isomorphic to Sn/2times Sn/2. We then consider the left cosets, the right cosets, and the double cosets generated by the subgroup. Each element of a coset can function as the representative of the coset. Different choices of the coset space and different choices of the coset representatives lead to four different syntheses for implementing an arbitrary reversible logic operation into hardware. Comparison leads to a best choice: a single coset space, with representatives that are generalized TOFFOLI and FREDKIN gates.
Reversible logic is emerging as a promising computing paradigm with applications in ultralow power nanocomputing and emerging nanotechnologies such as quantum computing, quantum dot cellular automata (QCA), optical computing, etc. Reversible circuits are similar to conventional logic circuits except that they are built from reversible gates. In reversible gates, there is a unique, one-to-one mapping between the inputs and outputs, not the case with conventional logic. One of the primary motivations for adopting reversible logic lies in the fact that it can provide a logic design methodology for designing ultra-low power circuits beyond KTln2 limit for those emerging nanotechnologies in which the energy dissipated due to information destruction will be a significant factor of the overall heat dissipation. Further, logic circuits for quantum computers must be built from reversible logic components. Several important metrics need to be considered in the design of reversible circuits the importance of which needs to be discussed. Quantum computers of many qubits are extremely difficult to realize thus the number of qubits in the quantum circuits needs to be minimized. This sets the major objective of optimizing the number of ancilla inputs and the number of the garbage outputs in the reversible logic based quantum circuits. The constant input in the reversible quantum circuit is called the ancilla input, while the garbage output refers to the output which exists in the circuit just to maintain one-to-one mapping but is not a primary or a useful output. The reversible circuit has other important parameters of quantum cost and delay which need to be optimized.
After decades of continuous improvements and shrinking feature sizes, the development of conventional computing technologies faces enormous challenges. In particular, power dissipation in today's computer chips becomes crucial. Reversible computation is a promising alternative to these technologies, where power dissipation can be reduced or even eliminated. Furthermore, reversible logic builds the basis for quantum computation - a completely new way of processing which enables to solve certain problems exponentially faster compared to conventional methods. However, the design of reversible and quantum circuits is significantly different. Thus, new methods e.g. for synthesis, optimization, or verification are needed. This paper provides a brief introduction into reversible circuits and their respective design methods that have been proposed within the last years.