Synthesis and Optimization of Reversible Circuits - A Survey

Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search-based, cycle-based, transformation-based, and BDD-based --- as well as specific algorithms for reversible synthesis, both exact and heuristic. We conclude the survey by outlining key open challenges in synthesis of reversible and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table

A Library-Based Synthesis Methodology for Reversible Logic

In this paper, a library-based synthesis methodology for reversible circuits is proposed where a reversible specification is considered as a permutation comprising a set of cycles. To this end, a pre-synthesis optimization step is introduced to construct a reversible specification from an irreversible function. In addition, a cycle-based representation model is presented to be used as an intermediate format in the proposed synthesis methodology. The selected intermediate format serves as a focal point for all potential representation models. In order to synthesize a given function, a library containing seven building blocks is used where each building block is a cycle of length less than 6. To synthesize large cycles, we also propose a decomposition algorithm which produces all possible minimal and inequivalent factorizations for a given cycle of length greater than 5. All decompositions contain the maximum number of disjoint cycles. The generated decompositions are used in conjunction with a novel cycle assignment algorithm which is proposed based on the graph matching problem to select the best possible cycle pairs. Then, each pair is synthesized by using the available components of the library. The decomposition algorithm together with the cycle assignment method are considered as a binding method which selects a building block from the library for each cycle. Finally, a post-synthesis optimization step is introduced to optimize the synthesis results in terms of different costs.Comment: 24 pages, 8 figures, Microelectronics Journal, Elsevie

ROM-based quantum computation: Experimental explorations using Nuclear Magnetic Resonance, and future prospects

ROM-based quantum computation (QC) is an alternative to oracle-based QC. It has the advantages of being less ``magical'', and being more suited to implementing space-efficient computation (i.e. computation using the minimum number of writable qubits). Here we consider a number of small (one and two-qubit) quantum algorithms illustrating different aspects of ROM-based QC. They are: (a) a one-qubit algorithm to solve the Deutsch problem; (b) a one-qubit binary multiplication algorithm; (c) a two-qubit controlled binary multiplication algorithm; and (d) a two-qubit ROM-based version of the Deutsch-Jozsa algorithm. For each algorithm we present experimental verification using NMR ensemble QC. The average fidelities for the implementation were in the ranges 0.9 - 0.97 for the one-qubit algorithms, and 0.84 - 0.94 for the two-qubit algorithms. We conclude with a discussion of future prospects for ROM-based quantum computation. We propose a four-qubit algorithm, using Grover's iterate, for solving a miniature ``real-world'' problem relating to the lengths of paths in a network.Comment: 11 pages, 5 figure

Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging

The controlled-SWAP and controlled-controlled-NOT gates are at the heart of the original proposal of reversible classical computation by Fredkin and Toffoli. Their widespread use in quantum computation, both in the implementation of classical logic subroutines of quantum algorithms and in quantum schemes with no direct classical counterparts, have made it imperative early on to pursue their efficient decomposition in terms of the lower-level gate sets native to different physical platforms. Here, we add to this body of literature by providing several logically equivalent CNOT-count-optimal circuits for the Toffoli and Fredkin gates under all-to-all and linear qubit connectivity, the latter with two different routings for control and target qubits. We then demonstrate how these decompositions can be employed on near-term quantum computers to mitigate coherent errors via equivalent circuit averaging. We also consider the case where the three qubits on which the Toffoli or Fredkin gates act nontrivially are not adjacent, proposing a novel scheme to reorder them that saves one CNOT for every SWAP. This scheme also finds use in the shallow implementation of long-range CNOTs. Our results highlight the importance of considering different entanglement structures and connectivity constraints when designing efficient quantum circuits.Comment: Main text: 10 pages, 8 figures. Appendix: 4 sections, 5 figures. QASM files will be made available in open-source online platform upon next update of preprin

Balancing Error and Dissipation in Computing

Modern digital electronics support remarkably reliable computing, especially given the challenge of controlling nanoscale logical components that interact in fluctuating environments. However, we demonstrate that the high-reliability limit is subject to a fundamental error-energy-efficiency tradeoff that arises from time-symmetric control: Requiring a low probability of error causes energy consumption to diverge as logarithm of the inverse error rate for nonreciprocal logical transitions. The reciprocity (self-invertibility) of a computation is a stricter condition for thermodynamic efficiency than logical reversibility (invertibility), the latter being the root of Landauer's work bound on erasing information. Beyond engineered computation, the results identify a generic error-dissipation tradeoff in steady-state transformations of genetic information carried out by biological organisms. The lesson is that computation under time-symmetric control cannot reach, and is often far above, the Landauer limit. In this way, time-asymmetry becomes a design principle for thermodynamically efficient computing.Comment: 19 pages, 8 figures; Supplementary material 7 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/tsp.ht