Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

TL;DR

The paper investigates a discrete boundary- or travel-time type inverse problem on finite graphs, proving a rigidity result for trees with leaves as observations and presenting a quantum algorithm to reconstruct graphs from leaf distances. The quantum approach encodes graphs with $O(n^2)$ qubits and employs Grover search, augmented by a quantum oracle that tests leaf-distance constraints, to achieve a quadratic speedup over classical search. It also establishes computational hardness for a restricted version by proving NP-completeness via a SAT reduction. Together, these results connect graph inverse problems, rigidity theory, and quantum algorithms, with implications for graph reconstruction and complexity theory. The combination of a general uniqueness result in the tree-leaf setting and a practical quantum procedure for reconstruction highlights both theoretical insight and potential computational advantages for discrete inverse problems on graphs.

Abstract

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.

Figures (3)

• Figure 1: From $F=(u_1\lor u_2) \land (\bar{u}_1\lor\bar{u}_3) \land (u_2\lor\bar{u}_3) \land u_3$ we construct an instance of the restricted inverse travel time problem (Problem \ref{['prob:restricted-inverse-travel-time']}). The set $E_0$ of edges that must be present in the solution is taken to be the empty set; the set $E_1$ of edges that are allowed to appear in the solution consists of the solid and the dotted lines. The distance from $u_j$ to $\bar{u}_j$ and from $\mathsf{TRUE}$ to $C_i$ is required to be $3$ for all $i$ and $j$. A solution for this instance is the subgraph drawn with solid lines; in the corresponding assignment that satisfies $F$ the literals connected with a solid line to $\mathsf{TRUE}$ are true.
• Figure 2: Figures (A)--(D) depict the four instances (labeled A--D, respectively) of Problem \ref{['prob:inverse-travel-time']} that we consider. The boundary vertices $b_j\in B$ are drawn in blue; the boundary distances are tabulated in (E). The dashed edges in (C) are edges that do not affect the boundary distances, thus this instance has a nonunique solution.
• Figure 3: Plots (A)--(D) display simulation results for instances A--D, respectively. In (A) the probability of measurement for each value of $\vec{e}$ (the adjacency matrix of a $3$-vertex graph) is shown. In (B)--(D) the probabilities of those values of $\vec{e}$ that correspond to a solution of the instance are shown; "rest" is the sum of the probabilities of those values that are not a solution. The bitstring denoting the measurement result consists of the elements of $e(j,k)$, $1\le j<k\le n$, of $\vec{e}$ in reverse lexicographical order so that, e.g., in the case $n=3$ the string corresponds to $e(2,3)\,e(1,3)\,e(1,2)$. Table (E) shows the value of parameter $L$ (number of Grover iterations), the number of qubits (including ancillae), and the dimension of the state space of the qubit register, for each of the instances A--D.

Theorems & Definitions (17)

• Theorem 1: Boundary rigidity for trees
• Theorem 2
• proof
• Theorem 3
• Proposition 5.1
• proof
• proof : Proof of Theorem \ref{['tree-rigidity']}
• Lemma 5.2: Uniqueness part of the classical Tree-metric Theorem
• Proposition 5.3
• proof