使用HADAMARD硬幣和與自動圖完整圖相關聯的移位操作員，基於UCDTQW的搜索補充算法

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.

搜索算法是一個計算過程，用於在較大的數據集合中找到特定項目或一組項目。搜索算法既構成了科學和工業應用中使用的一組固體和廣泛研究的計算工具（例如，1,2,3,4,5,6），部分是由於出現了新的計算機平台（例如，移動和移動設備和分佈式系統）以及主動研究領域（例如，7,8）。經典計算中廣泛使用搜索算法的字段的具體示例包括數據庫管理9，自然語言處理10，卷積神經網絡11和計算機網絡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.

隨著量子計算的出現，已經創建了新的搜索算法。最著名的建議是Grover的搜索算法13，我們可以通過它擴大具有復雜性\（\ sqrt {n}）均勻分佈中甲骨文操作員在甲骨文運營商中編碼的特定狀態的概率幅度，或者問題14, 15。量子搜索算法的另一個示例是Shenvi，Kempe和Whaley提出的SKW算法，它使用統一造成的離散時間量子步行（UCDTQW）的框架也增加了在Oracle運營商中編碼的特定狀態的概率。

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.

根據17的說法，UCDTQW由三個元素組成：量子步行者的狀態，\（{| {\ psi} \ rangle} \），系統u的進化運算符和系統\（\（\）的一組測量操作員{m_k \} \）。沃克的量子狀態是由硬幣狀態組成的兩分狀態，\（{| {c_i} \ rangle} \），這是m二維Hilbert Space \（H_C \）的一部分，並且是位置狀態，\（{ | {v_j} \ rangle} \），這是n維Hilbert space \（h_c \）的一部分，因此\（{| {| {\ psi} \ rangle} \）是線性組合硬幣和位置狀態之間的張量產品，即\（{| {\ psi} \ rangle} = \ sum \ limits _ {i} \ sum \ limits _ {j} a_ {j} a_ {ij {ij} \ rangle} \ otimes {| {v_j} \ rangle}。\）系統的進化運算符是一個二級操作員，以表單\（u = sc \）為coin coin操作員，其中COIN是系統的coin操作員僅在\（{| {\ psi} \ rangle} \）中的硬幣狀態，即它具有form \（c = c'\ otimes i_n \），而s是系統的移位運算符，原則上可以可以通過任何兩部分操作員，它都會編碼有關在uCDTQW發生的圖形連接的信息。通常，S始終與定向的多編碼相關聯。測量運算符集的元素，\（\ {m_k \} \）具有form \（m_k = i_} 。 SKW算法執行UCDTQW，使量子助行器沿HyperCube圖的頂點移動並蒐索標記的節點。

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算法通過理論計算和數值模擬進行了詳細的研究。18,19,20,21,22，實際上，通過使用與\ \ \ \ \（2^n的移位器相關的移位運算符） ，量子電路的複雜性降低了（2^n \） - 用自寬的尺寸完整圖，\（\ mathscr {k} _ {2^n} \），而不是與HyperCube關聯的Shift Operator。但是，即使在此改進之後，鑑於該算法的量子電路形式使用多控制的GROVER操作員作為UCDTTQW的Coin操作員的一部分，因此從未據報導SKW算法在通用量子計算機中有效實現。在23和24提出的分解之後，將分解成多項式的量子門，這使得在NISQ計算機上有效運行起來具有挑戰性。

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

鑑於以前的說法，我們建議修改UCDTQW的硬幣操作員，以使SKW算法更有效。 N Qubit Hadamard操作員是一種自然的選擇，這是UCDTQW領域的常用操作員，因為它由N Sigle-Qubit-Qubit hadamard Gates組成，因此計算機運算符的角色確實不那麼昂貴。因此，這項工作的目的之一是使用Hadamard硬幣研究UCDTQW的行為。此外，我們的目標也是在IBM的通用量子計算機中有效實現搜索算法，因此我們決定將提出的想法在21中提出並執行搜索算法，並在圖上執行搜索算法\（\ Mathscr {k} _ { 2^n _ {2^n } \）。但是我們使用