Abstract-This paper proposes an approach to optimally synthesize quantum circuits by symbolic reachability analysis, where the primary inputs and outputs are basis binary and the internal signals can be nonbinary in a multiple-valued domain. The authors present an optimal synthesis method to minimize quantum cost and some speedup methods with nonoptimal quantum cost. The methods here are applicable to small reversible functions. Unlike previous works that use permutative reversible gates, a lower level library that includes nonpermutative quantum gates is used here. The proposed approach obtains the minimum cost quantum circuits for Miller gate, half adder, and full adder, which are better than previous results. This cost is minimum for any circuit using the set of quantum gates in this paper, where the control qubit of 2-qubit gates is always basis binary. In addition, the minimum quantum cost in the same manner for Fredkin, Peres, and Toffoli gates is proven. The method can also find the best conversion from an irreversible function to a reversible circuit as a byproduct of the generality of its formulation, thus synthesizing in principle arbitrary multi-output Boolean functions with quantum gate library. This paper constitutes the first successful experience of applying formal methods and satisfiability to quantum logic synthesis.Index Terms-Formal verification, logic synthesis, model checking, quantum computing, reversible logic, satisfiability.

Background Computer-aided diagnosis (CAD) in the medical field has received more and more attention in recent years. One important CAD application is to detect and classify breast lesions in ultrasound images. Traditionally, the process of CAD for breast lesions classification is mainly composed of two separated steps: i) locate the lesion region of interests (ROI); ii) classify the located region of interests (ROI) to see if they are benign or not. However, due to the complex structure of breast and the existence of noise in the ultrasound images, traditional handcrafted feature based methods usually can not achieve satisfactory result. Methods With the recent advance of deep learning, the performance of object detection and classification has been boosted to a great extent. In this paper, we aim to systematically evaluate the performance of several existing state-of-the-art object detection and classification methods for breast lesions CAD. To achieve that, we have collected a new dataset consisting of 579 benign and 464 malignant lesion cases with the corresponding ultrasound images manually annotated by experienced clinicians. We evaluate different deep learning architectures and conduct comprehensive experiments on our newly collected dataset. Results For the lesion regions detecting task, Single Shot MultiBox Detector with the input size as 300×300 (SSD300) achieves the best performance in terms of average precision rate (APR), average recall rate (ARR) and F 1 score. For the classification task, DenseNet is more suitable for our problems. Conclusions Our experiments reveal that better and more efficient detection and convolutional neural network (CNN) frameworks is one important factor for better performance of detecting and classification task of the breast lesion. Another significant factor for improving the performance of detecting and classification task, which is transfer learning from the large-scale annotated ImageNet to classify breast lesion.

Reversible quantum logic plays an important role in quantum computing. In this paper, we propose an approach to optimally synthesize quantum circuits by symbolic reachability analysis where the primary inputs are purely binary. We present an exact synthesis method with optimal quantum cost and a speedup method with non-optimal quantum cost. Both our methods guarantee the synthesizeability of all reversible circuits. Unlike previous works which use permutative reversible gates, we use a lower level library which includes non-permutative quantum gates. Our approach obtains the minimum cost quantum circuits for Miller's gate, half-adder, and full-adder, which are better than previous results. In addition, we prove the minimum quantum cost (using our elementary quantum gates) for Fredkin, Peres, and Toffoli gates. Our work constitutes the first successful experience of applying satisfiability with formal methods to quantum logic synthesis.

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An efficient pairwise Boolean matching algorithm for solving the problem of matching single-output specified Boolean functions under input negation and/or input permutation and/or output negation (NPN) is proposed in this paper. We present the structural signature (SS) vector, which comprises a first-order signature value, two symmetry marks, and a group mark. As a necessary condition for NPN Boolean matching, the SS is more effective than the traditional signature. Two Boolean functions, f and g, may be equivalent when they have the same SS vector. A symmetry mark can distinguish symmetric variables and asymmetric variables and be used to search for multiple variable mappings in a single variable-mapping search operation, which reduces the search space significantly. Updating the SS vector via Shannon decomposition provides benefits in distinguishing unidentified variables, and the group mark and phase collision check can be used to discover incorrect variable mappings quickly, which also speeds up the NPN Boolean matching process. Using the algorithm proposed in this paper, we test both equivalent and non-equivalent matching speeds on the MCNC benchmark circuit sets and random circuit sets. In the experiment, our algorithm is shown to be 4.2 times faster than competitors when testing equivalent circuits and 172 times faster, on average, when testing non-equivalent circuits. The experimental results show that our approach is highly effective at solving the NPN Boolean matching problem.