使用HADAMARD硬币和与自动图完整图相关联的移位操作员，基于UCDTQW的搜索补充算法

D Layera，Alba Cervera-Liera

A search algorithm is a computational process that is used to locate a specific item or group of items within a larger collection of data. Search algorithms constitute both a solid and widely studied set of computational tools used in scientific and industrial applications everyday (e.g.,1,2,3,4,5,6), partly due to the emergence of new computer platforms (e.g., mobile and distributed systems), as well as an active research area (e.g.,7,8). Concrete examples of fields that make extensive use of search algorithms in classical computing include Database Management9, Natural Language Processing10, Convolutional Neural Networks11, and Computer Networks12.

搜索算法是一个计算过程，用于在较大的数据集合中找到特定项目或一组项目。搜索算法既构成了科学和工业应用中使用的一组固体和广泛研究的计算工具（例如，1,2,3,4,5,6），部分是由于出现了新的计算机平台（例如，移动和移动设备和分布式系统）以及主动研究领域（例如，7,8）。经典计算中广泛使用搜索算法的字段的具体示例包括数据库管理9，自然语言处理10，卷积神经网络11和计算机网络11。

With the emergence of quantum computing, new search algorithms have been created. The most famous proposal is Grover’s search algorithm13, with which we can amplify the probability amplitude of a specific state encoded in an oracle operator within a uniform distribution of states with complexity \(O(\sqrt{N})\) or solve combinatorial optimization problems14,15. Another example of a quantum search algorithm is the SKW algorithm16, proposed by Shenvi, Kempe and Whaley, which uses the framework of Unitary Coined Discrete-Time Quantum Walks (UCDTQW) to also increase the probability amplitude of a specific state encoded in an oracle operator.

随着量子计算的出现，已经创建了新的搜索算法。最著名的建议是Grover的搜索算法13，我们可以通过它扩大具有复杂性\（\ sqrt {n}）均匀分布中甲骨文操作员在甲骨文运营商中编码的特定状态的概率幅度，或者问题14,15。量子搜索算法的另一个示例是Shenvi，Kempe和Whaley提出的SKW算法，它使用统一造成的离散时间量子步行（UCDTQW）的框架也增加了在Oracle运营商中编码的特定状态的概率。

According to17, a UCDTQW consists of three elements: the state of a quantum walker, \({|{\psi }\rangle }\), the evolution operator of the system U and the set of measurement operators of the system \(\{M_k\}\). The quantum state of a walker is a bipartite state that is composed of a coin state, \({|{c_i}\rangle }\), that is part of an m-dimensional Hilbert space \(H_C\), and a position state, \({|{v_j}\rangle }\), that is part of an n-dimensional Hilbert space \(H_C\), such that \({|{\psi }\rangle }\) is a linear combination of the tensor product between pairs of coin and position states, i.e. \({|{\psi }\rangle } = \sum \limits _{i}\sum \limits _{j} a_{ij}{|{c_i}\rangle }\otimes {|{v_j}\rangle }.\) The evolution operator of the system is a bipartite operator that takes the form \(U = SC\), where C is the coin operator of the system, which modifies only the coin state in \({|{\psi }\rangle }\), i.e. it has the form \(C=C'\otimes I_n\), and S is the shift operator of the system, which in principle could by any bipartite operator, and it codifies the information about connections of the graph where the UCDTQW takes place. In general, S is always associated to a directed multigraph. The elements of the set of measurement operators, \(\{M_k\}\), have the form \(M_k = I_{m}\otimes {|{v_k}\rangle }{\langle {v_k}|}\). The SKW algorithm performs a UCDTQW to let a quantum walker move along the vertices of a hypercube graph and search for a marked node.

根据17的说法，UCDTQW由三个元素组成：量子步行者的状态，\（{| {\ psi} \ rangle} \），系统u的进化运算符和系统\（\（\）的一组测量操作员{m_k \} \）。沃克的量子状态是由硬币状态组成的两分状态，\（{| {c_i} \ rangle} \），这是m二维Hilbert Space \（H_C \）的一部分，并且是位置状态，\（{| {v_j} \ rangle} \），这是n维Hilbert space \（h_c \）的一部分，因此\（{| {| {\ psi} \ rangle} \）是线性组合硬币和位置状态之间的张量产品，即\（{| {\ psi} \ rangle} = \ sum \ limits _ {i} \ sum \ limits _ {j} a_ {j} a_ {ij {ij} \ rangle} \ otimes {| {v_j} \ rangle}。\）系统的进化运算符是一个二级操作员，以表单\（u = sc \）为coin coin操作员，其中COIN是系统的coin操作员仅在\（{| {\ psi} \ rangle} \）中的硬币状态，即它具有form \（c = c'\ otimes i_n \），而s是系统的移位运算符，原则上可以可以通过任何两部分操作员，它都会编码有关在uCDTQW发生的图形连接的信息。通常，S始终与定向的多编码相关联。测量运算符集的元素，\（\ {m_k \} \）具有form \（m_k = i_} 。 SKW算法执行UCDTQW，使量子助行器沿HyperCube图的顶点移动并搜索标记的节点。

The SKW algorithm has been studied in detail through theoretical calculations and numerical simulations18,19,20,21,22, and, in fact, in21 the complexity of the quantum circuit was reduced by using the shift operator associated to the \(2^n\)-dimensional complete graph with self-loops, \(\mathscr {K}_{2^n}\), instead of the shift operator associated to a hypercube. However, even after this improvement, the SKW algorithm has never been reported to be efficiently implemented in a general-purpose quantum computer given that the quantum circuit form of the algorithm uses a multi-control Grover operator as part of the coin operator of the UCDTQW, which decomposes into a polynomial number of quantum gates, following the decompositions proposed in23 and24, making it challenging to run efficiently on NISQ computers.

SKW算法通过理论计算和数值模拟进行了详细的研究。18,19,20,21,22，实际上，通过使用与\ \ \ \ \（2^n的移位器相关的移位运算符），量子电路的复杂性降低了（2^n \） - 用自宽的尺寸完整图，\（\ mathscr {k} _ {2^n} \），而不是与HyperCube关联的Shift Operator。但是，即使在此改进之后，鉴于该算法的量子电路形式使用多控制的GROVER操作员作为UCDTTQW的Coin操作员的一部分，因此从未据报道SKW算法在通用量子计算机中有效实现。在23和24提出的分解之后，将分解成多项式的量子门，这使得在NISQ计算机上有效运行起来具有挑战性。

In view of the former statement, we propose to modify the coin operator of the UCDTQW to make the SKW algorithm more efficient. A natural option arises with the n-qubit Hadamard operator, a commonly used operator in the field of UCDTQW for the role of the coin operator, which is indeed less computationally expensive, given that it consists of n sigle-qubit Hadamard gates. Thus one of the purposes of this work is to study the behaviour of the UCDTQW using the Hadamard coin. Moreover, it is also our goal to efficiently implement the search algorithm in IBM’s general-purpose quantum computers, thus we decided to take the idea proposed in21 and perform the search algorithm on the graph \(\mathscr {K}_{2^n}\). However we use the

鉴于以前的说法，我们建议修改UCDTQW的硬币操作员，以使SKW算法更有效。 N Qubit Hadamard操作员是一种自然的选择，这是UCDTQW领域的常用操作员，因为它由N Sigle-Qubit-Qubit hadamard Gates组成，因此计算机运算符的角色确实不那么昂贵。因此，这项工作的目的之一是使用Hadamard硬币研究UCDTQW的行为。此外，我们的目标也是在IBM的通用量子计算机中有效实现搜索算法，因此我们决定将提出的想法在21中提出并执行搜索算法，并在图上执行搜索算法\（\ Mathscr {k} _ {2^n _ {2^n } \）。但是我们使用