基于带权有向图的可逆逻辑综合改进算法

为减少可逆逻辑综合中使用的可逆门,通过对基于带权有向图的可逆逻辑综合算法的分析,针对函数转换过程中过渡门数较多及电路优化算法简单的问题,提出了有效的等复杂度基本输出变换的概念,扩充并证明了Toffoli门序列的移动和化简规则,给出了改进的基于带权有向图的可逆逻辑综合算法。实验结果表明,该算法不仅减少了可逆电路构成时所使用的可逆门,而且对构建的可逆电路实现了有效化简,大幅度减少了门数和控制位数,降低了可逆电路代价。     

混合多值可逆逻辑中广义Toffoli门仅用CNOT门的实现

混合多值量子可逆逻辑电路综合问题中,Toffoli门的合成是整个合成过程中最为关键的一步。针对混合多值5-qubits量子可逆逻辑电路综合问题,构造了PMX量子门,验证了CNOT门的合成能力,实现了对Toffoli门的合成,并设计了双向的BDS搜索算法,高效实现了量子电路的最优或者较优综合。     

基于控制K次平方根非门的类Toffoli门构造方法

在量子电路综合算法中,由于非置换量子门比置换量子门具有更复杂的规则,直接使用非置换量子门会大幅度提高综合算法的复杂性,因此可先使用非置换量子门生成相应的置换量子门,然后再用这些置换量子门综合所求量子可逆逻辑电路,从而提高算法性能。本文重点研究如何用非置换量子门构造新的置换量子门,为此吸收了格雷码的思想,提出了一种高效的递归构造方法,实现使用控制非门和控制K次平方根非门(非置换量子门),快速生成最优的类Toffoli门(置换量子门)。     

经典逻辑门与量子逻辑门之比较

本文通过经典逻辑门与量子逻辑门之比较,论述了量子计算的特点、量子算法的巨大威力及量子逻辑门的实现问题。     

多输出逻辑函数最小化的一个算法

本文描述一个多输出逻辑函数的最小化算法。函数解中积项总数的多少被看作是衡量算法优劣的最重要指标。因此，我们努力使解中的积项被尽可能多的组成函数共享。本文求本源蕴涵项的方法与［１］是相似的。因此，本文也可看成是［１］向多输出函数的延伸。     

基于规则的可逆Toffoli电路优化算法

可逆电路的优化是可逆逻辑综合的关键问题之一.为了解决可逆Toffoli电路优化问题中算法复杂度高和电路规模可扩充性差的问题,分析归纳了相邻Toffoli门的关系,提出并证明了可逆Toffoli电路中子序列的移动和化简规则,并基于这些规则给出了可逆Toffoli电路的优化算法.根据移动规则对可逆电路进行正向和反向扫描,寻找满足化简规则的子序列进行优化,直到可逆电路不发生变化为止.该优化算法与可逆电路的输入线数无关,无需存储额外信息,适用于各种不同类型的Toffoli电路合成方法,算法复杂度为O(s3),优于通常使用的模板优化的复杂度O(n!t2s3).在具体实例和国际认可的所有3变量可逆函数上的验证结果表明,该优化算法能有效地减少可逆电路的门数和控制位数,降低可逆电路的代价.     

用二值逻辑对多值逻辑进行优化

提出以二值多输出逻辑优化软件OPLG为基础,对多值逻辑函数进行逻辑优化的方法．通过对多值变量、多值函数的二进制矢量描述,将多值多维体转换为布尔表达式积项形式,从多值多维体的多值最小项出发,给出计算基本无关集的方法。对多值逻辑函数的优化通过调用二值逻辑优化软件OPLG(允许的最大输入、输出变量之和为300)来实现,二值逻辑优化的结果最终再转换为多值多维体的表示形式。     

超大变量多值单边逻辑函数优化算法的研究

单边多值逻辑函数是逻辑函数的一种特例,对于超大变量的逻辑函数优化,存储开销对输入变量呈2幂次方增长.针对多输入多输出单边逻辑函数的特性,通过引入特征矩阵和状态矢量的描述,求解多输出单边逻辑函数补集,以积项扩展为基础,完成多输出单边逻辑函数无冗余覆盖.通过编程实现了多输入多输出单边函数逻辑优化的算法,对影响单边函数逻辑优化效率的因素进行了分析,软件系统在奔腾1.8GHz,512MB RAM的计算机上通过了正确性验证与测试,测试结果表明性能良好,有效降低了系统的存储空间和时间开销.在相同输出变量数、积项数和无关项因子的情况下,随输入变量数的增长,优化效率随之下降;在相同输入变量数、输出变量数和积项数的情况下,随无关项因子值的增长,优化效率随之下降;在相同输入变量数、输出变量数、和无关项因子的情况下,随积项数的增长优化效率随之提高.     

大电路固定极性Reed-Muller逻辑快速转换算法

针对已有的列表技术在极性转换中只能解决中小规模电路的问题,提出一种基于不相交乘积项列表技术的快速转换算法.首先将待处理的逻辑函数表示为不相交乘积项之和形式;然后通过对已有的基于最大项的列表技术进行分析和改进,使得改进后的列表技术可以实现将逻辑函数从不相交乘积项的AND?OR形式向固定极性XNOR?OR形式的Reed-Muller逻辑转化.文中算法用C编程实现,并用MCNC标准电路进行测试.实验结果表明,该算法可以快速实现大电路的极性转换,并且具有运算速度对电路的输入变量数不敏感的特点.     

大变量多输出逻辑函数实质项识别算法

在对Espresso算法进行分析改进的基础上，提出了一种基于全域识别的多输入多输出逻辑函数实质本源项、完全冗余项和相对冗余项生成算法，该算法通过对基于积项表示的多输入多输出逻辑函数的余因子计算来进行全域判断，根据全域判断结果来识别实质本源项、完全冗余项和相对冗余项，从而构成实质本源项集合、完全冗余项集合和相对冗余项集合。对基于二级SOP型的多输入多输出逻辑函数设计了多输入多输出逻辑函数优化识别软件系统，允许的最大输入变量数为128、最大输出变量数为256、最大输入输出变量总和为300、最大输入积项数为20 000。软件系统在Pentium 1.8GHz、512MB内存的计算机上通过了Benchmark例题的测试。     

Generation of sign Walsh spectra from disjoint cubes of Boolean functions

A new algorithm is given that converts a reduced representation of Boolean functions in the form of disjoint cubes to sign Walsh spectra. Since the known algorithms that generate sign Walsh spectra always start from the truth table of Boolean functions, the method presented computes faster with a smaller computer memory. The method is especially efficient for such Boolean functions that are described by only few disjoint cubes.     

基于多目标扩展通用Toffoli门的量子比较器设计

利用多目标扩展通用Toffoli门,提出了经典量子信息比较器的设计构造方法,并对其正确性进行了理论证明,在此基础之上,给出了量子比较器在简单搜索问题中的一个应用。与其它同类量子比较器相比,此比较器通过减少使用辅助位来节约相关量子资源;通过设置多目标扩展通用Toffoli门的控制条件,使得在比较出结果后剩余的门不再起作用,从而提高了运行效率,降低了出错率,增强了比较器的鲁棒性。     

Extracting rules from trained neural networks

Presents an algorithm for extracting rules from trained neural networks. The algorithm is a decompositional approach which can be applied to any neural network whose output function is monotone such as a sigmoid function. Therefore, the algorithm can be applied to multilayer neural networks, recurrent neural networks and so on. It does not depend on training algorithms, and its computational complexity is polynomial. The basic idea is that the units of neural networks are approximated by Boolean functions. But the computational complexity of the approximation is exponential, and so a polynomial algorithm is presented. The author has applied the algorithm to several problems to extract understandable and accurate rules. The paper shows the results for the votes data, mushroom data, and others. The algorithm is extended to the continuous domain, where extracted rules are continuous Boolean functions. Roughly speaking, the representation by continuous Boolean functions means the representation using conjunction, disjunction, direct proportion, and reverse proportion. This paper shows the results for iris data.     

Robust Polynomials and Quantum Algorithms

We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are: *For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. *There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Θ(n log n) queries. We give several extensions and applications of these results.     

Unbounded-error quantum query complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve—Wigderson [STOC’98] is optimal even in the unbounded-error setting.We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q+log(1/2(p−1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative Θ(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.     

基于引力搜索的布尔函数生成算法

布尔函数是在密码学、纠错编码和扩频通信等领域有着广泛应用的密码函数,寻找性能优良的布尔函数一直是密码学领域的重要问题之一。基于引力搜索算法设计了一种搜索布尔函数的新算法。该算法模仿万有引力定律,以n维空间中的质量点表示布尔函数,以布尔函数的密码特性作为目标适应度函数进行搜索。实验结果表明,算法使用新设计的目标适应度函数可以直接生成具有1阶弹性、1阶扩散准则和高非线性度、高代数次数以及低自相关指标等多种密码学指标的平衡布尔函数,并且进一步给出了直接生成2输出平衡布尔函数的计算机搜索算法。     

Improved quantum ripple-carry addition circuit

A serious obstacle to large-scale quantum algorithms is the large number of elementary gates, such as the controlled-NOT gate or Toffoli gate. Herein, we present an improved linear-depth ripple-carry quantum addition circuit, which is an elementary circuit used for quantum computations. Compared with previous addition circuits costing at least two Toffoli gates for each bit of output, the proposed adder uses only a single Toffoli gate. Moreover, our circuit may be used to construct reversible circuits for modular multiplication, Cx mod M with x < M, arising as components of Shor’s algorithm. Our modular-multiplication circuits are simpler than previous constructions, and may be used as primitive circuits for quantum computations.     

Improving the quantum cost of reversible Boolean functions using reorder algorithm

This paper introduces a novel algorithm to synthesize a low-cost reversible circuits for any Boolean function with n inputs represented as a Positive Polarity Reed–Muller expansion. The proposed algorithm applies a predefined rules to reorder the terms in the function to minimize the multi-calculation of common parts of the Boolean function to decrease the quantum cost of the reversible circuit. The paper achieves a decrease in the quantum cost and/or the circuit length, on average, when compared with relevant work in the literature.     

Implementing gate operations between uncoupled qubits in linear nearest neighbor arrays using a learning algorithm

We propose a new scheme to implement gate operations in a one dimensional linear nearest neighbor array, by using dynamic learning algorithm. This is accomplished by training quantum system using a back propagation technique, to find the system parameters that implement gate operations directly. The key feature of our scheme is that, we can reduce the computational overhead of a quantum circuit by finding the parameters to implement the desired gate operation directly, without decomposing them into a sequence of elementary gate operations. We show how the training algorithm can be used as a tool for finding the parameters for implementing controlled-NOT (CNOT) and Toffoli gates between next-to-nearest neighbor qubits in an Ising-coupled linear nearest neighbor system. We then show how the scheme can be used to find parameters for realizing swap gates first, between two adjacent qubits and then, between two next-to-nearest-neighbor qubits, in each case without decomposing it into 3 CNOT gates. Finally, we show how the scheme can be extended to systems with non-diagonal interactions. To demonstrate, we train a quantum system with Heisenberg interactions to find the parameters to realize a swap operation.     

布尔函数的统计独立性

定义在同一定义域上的两个布尔函数可能存在多种关系,本文研究它们之间的统计独立性,这种性质可以用于布尔置换的构造.本文给出了利用布尔函数的汉明距离判定两个布尔函数是否统计独立的充分必要条件,给出了寻找与某个已知布尔函数统计独立的布尔函数的算法,并分析了这种算法的有效性.