Alwin Zulehner

Taking one-to-one mappings for granted: Advanced logic design of encoder circuits

Encoders play an important role in many areas such as memory addressing, data demultiplexing, or for interconnect solutions. However, design solutions for the automatic synthesis of corresponding circuits suffer from various drawbacks, e.g. they are often not scalable, do not exploit the full degree of freedom, or are applicable to realize certain codes only. All these problems are caused by the fact that existing design solutions have to explicitly guarantee a one-to-one mapping. In this work, we propose an alternative design approach which relies on dedicated description means for both, the specification of an encoder as well as its circuit. Based on that, synthesis can be conducted without the need to explicitly take care of guaranteeing one-to-one mappings. Experiments show that this indeed overcomes the drawbacks of current design solutions and leads to an improvement in the resulting number of gates by up to 92%.

Make it reversible: Efficient embedding of non-reversible functions

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.

Improving Synthesis of Reversible Circuits: Exploiting Redundancies in Paths and Nodes of QMDDs

In recent years, reversible circuits have become an established emerging technology through their variety of applications. Since these circuits employ a completely different structure from conventional circuitry, dedicated functional synthesis algorithms have been proposed. Although scalability has been achieved by using approaches based on decision diagrams, the resulting circuits employ a significant complexity measured in terms of quantum cost. In this paper, we aim for a reduction of this complexity. To this end, we review QMDD-based synthesis. Based on that, we propose optimizations that allow for a substantial reduction of the quantum costs by jointly considering paths and nodes in the decision diagram that employ a certain redundancy. In fact, in our experimental evaluation, we observe substantial improvements of up to three orders of magnitudes in terms of runtime and up to six orders of magnitudes (a factor of one million) in terms of quantum cost.

One-Pass Design of Reversible Circuits: Combining Embedding and Synthesis for Reversible Logic

Reversible computation is a heavily investigated emerging technology due to its promising characteristics in low-power design, its application in quantum computations, and several further application areas. The currently established functional synthesis flow for reversible circuits is composed of two distinct steps. First, an embedding process is conducted which makes nonunique output patterns distinguishable by adding further variables. Then, this function is passed to a synthesis method which eventually yields a reversible circuit. However, the separate consideration of the embedding and synthesis tasks leads to significant drawbacks. In fact, embedding is not necessarily conducted in a fashion which is suited for the following synthesis process. In addition, embedding adds further variables to the function to be synthesized which exponentially increases its corresponding representation in the worst case. In this paper, we propose one-pass design of reversible circuits, which combines embedding and synthesis. This allows for conducting synthesis with a high degree of freedom, since the embedding that suits best is inherently chosen during synthesis. We propose two solutions (an exact an a heuristic one) following this scheme that improve the currently established synthesis flow by magnitudes in terms of runtime—allowing to synthesize a reversible circuit with a minimum number of lines for some of the frequently considered benchmark functions for the first time. Furthermore, a significant reduction of the costs of the resulting circuits (up to several orders of magnitude) is achieved with this new design flow.

Exact Global Reordering for Nearest Neighbor Quantum Circuits Using A

Since for certain realizations of quantum circuits only adjacent qubits may interact, qubits have to be frequently swapped – leading to a significant overhead. Therefore, optimizations such as exact global reordering have been proposed, where qubits are reordered such that the overall number of swaps is minimal. However, to guarantee minimality all n! possible permutations of qubits have to be considered in the worst case – which becomes intractable for larger circuits. In this work, we tackle the complexity of exact global reordering using an A* search algorithm. The sophisticated heuristics for the search algorithm proposed in this paper allow for solving the problem in a much more scalable fashion. In fact, experimental evaluations show that the proposed approach is capable of determining the best order of the qubits for circuits with up to 25 qubits, whereas the recent state-of-the-art already reaches its limits with circuits composed of 10 qubits.

Efficient Construction of QMDDs for Irreversible, Reversible, and Quantum Functions

In reversible as well as quantum computation, unitary matrices (so-called transformation matrices) are employed to comprehensively describe the respectively considered functionality. Due to the exponential growth of these matrices, dedicated and efficient means for their representation and manipulation are essential in order to deal with this complexity and handle reversible/quantum systems of considerable size. To this end, Quantum Multiple-Valued Decision Diagrams (QMDDs) have shown to provide a compact representation of those matrices and have proven their effectiveness in many areas of reversible and quantum logic design such as embedding, synthesis, or equivalence checking. However, the desired functionality is usually not provided in terms of QMDDs, but relies on alternative representations such as Boolean Algebra, circuit netlists, or quantum algorithms. In order to apply QMDD-based design approaches, the corresponding QMDD has to be constructed first—a gap in many of these approaches. In this paper, we show how QMDD representations can efficiently be obtained for Boolean functions, both reversible and irreversible ones, as well as general quantum functionality.

Skipping Embedding in the Design of Reversible Circuits

Synthesis of reversible circuits finds application in many promising domains but has to deal with the fact that the underlying circuits require a unique mapping from the inputs to the outputs. Existing solutions addressed this problem by additionally performing a so-called embedding process prior to synthesis or by naively mapping building blocks of conventional logic to their corresponding reversible counterparts. This leads to solutions that either suffer from limited scalability or yield circuits with a huge number of additionally required circuit lines. In this work, we conduct investigations to overcome these problems. To this end, we simply ignore the fact that an arbitrary Boolean function to be synthesized might be non-reversible and deal with the resulting problem of ensuring a unique input/output mapping during the actual synthesis process. Experimental evaluations indicate that, following this approach, could provide the basis for an alternative synthesis scheme that allows for synthesizing arbitrary Boolean functions in reasonable time and without the need of a prior embedding process.

Identifying Reversible Circuit Synthesis Approaches to Enable IP Piracy Attacks

Reversible circuits are vulnerable to intellectual property and integrated circuit piracy. To show these vulnerabilities, a detailed understanding on how to identify the function embedded in a reversible circuit is crucial. To obtain the embedded function, one needs to know the synthesis approach used to generate the reversible circuit in the first place. We present a machine learning based scheme to identify the synthesis approach using telltale signs in the design.