We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

翻译：考虑一个有限图 $(X,E)$ 的逆问题，其中给定顶点子集 $B\subset X$ 以及所有顶点 $b_1,b_2\in B$ 之间的距离 $d_{(X,E)}(b_1,b_2)$。两点 $x_1,x_2\in X$ 的距离定义为连接两顶点所需的最少边数，因此所有边长度均为1。该逆问题是黎曼几何中的边界刚性问题和地球物理学中的逆走时问题的离散版本。我们将证明该问题在特定条件下具有唯一解，并开发量子计算方法来求解。我们证明了如下唯一性结果：当 $(X,E)$ 是树且 $B$ 是该树的叶子集时，该图 $(X,E)$ 可在所有具有固定顶点数的图类中被唯一确定。我们提出一种量子计算算法，该算法可生成一个（或其中之一）具有给定顶点数且满足 $B$ 中顶点间所需距离的图 $(X,E)$。为此，我们开发了一种算法，该算法接收图的量子比特表示，并将其与 Grover 搜索算法相结合。该算法仅需 $O(|X|^2)$ 个量子比特即可实现，该数量级与 $(X,E)$ 的邻接矩阵元素数量相同。与经典算法相比，其计算成本具有二次改进。最后，我们考虑计算理论中的应用，并证明上述逆问题的一个简单变体是 NP 完全的：所有 NP 问题均可归约为我们所考虑的离散逆问题。