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, postsynthesis 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. ACM Reference Format:Saeedi, M. and Markov, I. L. 2013. Synthesis and optimization of reversible circuits-a survey. ACM Comput.

While a couple of impressive quantum technologies have been proposed, they have several intrinsic limitations which must be considered by circuit designers to produce realizable circuits. Limited interaction distance between gate qubits is one of the most common limitations. In this paper, we suggest extensions of the existing synthesis flow aimed to realize circuits for quantum architectures with linear nearest neighbor (LNN) interaction. To this end, a template matching optimization, an exact synthesis approach, and two reordering strategies are introduced. The proposed methods are combined as an integrated synthesis flow. Experiments show that by using the suggested flow, quantum cost can be improved by more than 50% on average.

Reversible logic has applications in various research areas including signal processing, cryptography and quantum computation. In this paper, direct NCT-based synthesis of a given k-cycle in a cycle-based synthesis scenario is examined. To this end, a set of seven building blocks is proposed that reveals the potential of direct synthesis of a given permutation to reduce both quantum cost and average runtime. To synthesize a given large cycle, we propose a decomposition algorithm to extract the suggested building blocks from the input specification. Then, a synthesis method is introduced which uses the building blocks and the decomposition algorithm. Finally, a hybrid synthesis framework is suggested which uses the proposed cycle-based synthesis method in conjunction with one of the recent NCT-based synthesis approaches which is based on ReedMuller (RM) spectra. The time complexity and the effectiveness of the proposed synthesis approach are analyzed in detail. Our analyses show that the proposed hybrid framework leads to a better quantum cost in the worst-case scenario compared to the previously presented methods. The proposed framework always converges and typically synthesizes a given specification very fast compared to the available synthesis algorithms. Besides, the quantum costs of benchmark functions are improved about 20% on average (55% in the best case).

Reversible circuits for modular multiplication $Cx\%M$ with $x<M$ arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific $C$ and $M$ values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient $C$ values. When zero-initialized ancilla registers are available, we reduce the search for compact circuits to a shortest-path problem. Some of our modular-multiplication circuits are asymptotically smaller than previous constructions, but worst-case bounds and average sizes remain $\Theta(n^2)$. In the context of modular exponentiation, we offer several constant-factor improvements, as well as an improvement by a constant additive term that is significant for few-qubit circuits arising in ongoing laboratory experiments with Shor's algorithm.

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