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Quantum Algorithm for Estimating Ollivier-Ricci Curvature

Nhat A. Nghiem, Linh Nguyen, Tuan K. Do, Tzu-Chieh Wei, Trung V. Phan

TL;DR

The paper tackles the problem of efficiently estimating the Ollivier–Ricci curvature on graphs from distance-based point-cloud data by formulating ORC via the Earth Mover distance $W_1(x,y)$. It introduces a quantum algorithm built on block-encoding and quantum singular value transformation (QSVT) to encode the geodesic distance information and solve the discrete optimal transport subproblem for $W_1(x,y)$ in two regimes: when $G$ is a tree and when $p=q$, achieving exponential speedup in the number of data points $N$ under favorable conditions. The key contributions include a projector-based approach to isolate the transport terms, a distance-operator block-encoding, and a minimum-eigenvalue method to obtain $W_1(x,y)$ and thus the curvature $\kappa(x,y)=1-\frac{W_1(x,y)}{d_G(x,y)}$. This work advances geometrical data analysis (GDA) by providing quantum-accelerated tools for discrete curvature computations with potential applications in finance, network science, and combinatorial quantum gravity.

Abstract

We introduce a quantum algorithm for computing the Ollivier Ricci curvature, a discrete analogue of the Ricci curvature defined via optimal transport on graphs and general metric spaces. This curvature has seen applications ranging from signaling fragility in financial networks to serving as basic quantities in combinatorial quantum gravity. For inputs given as a point cloud with pairwise distances, we show that our algorithm can achieve an exponential speedup over the best-known classical methods for two particular classes of problem. Our work is another step toward quantum algorithms for geometrical problems that are capable of delivering practical value while also informing fundamental theory.

Quantum Algorithm for Estimating Ollivier-Ricci Curvature

TL;DR

The paper tackles the problem of efficiently estimating the Ollivier–Ricci curvature on graphs from distance-based point-cloud data by formulating ORC via the Earth Mover distance . It introduces a quantum algorithm built on block-encoding and quantum singular value transformation (QSVT) to encode the geodesic distance information and solve the discrete optimal transport subproblem for in two regimes: when is a tree and when , achieving exponential speedup in the number of data points under favorable conditions. The key contributions include a projector-based approach to isolate the transport terms, a distance-operator block-encoding, and a minimum-eigenvalue method to obtain and thus the curvature . This work advances geometrical data analysis (GDA) by providing quantum-accelerated tools for discrete curvature computations with potential applications in finance, network science, and combinatorial quantum gravity.

Abstract

We introduce a quantum algorithm for computing the Ollivier Ricci curvature, a discrete analogue of the Ricci curvature defined via optimal transport on graphs and general metric spaces. This curvature has seen applications ranging from signaling fragility in financial networks to serving as basic quantities in combinatorial quantum gravity. For inputs given as a point cloud with pairwise distances, we show that our algorithm can achieve an exponential speedup over the best-known classical methods for two particular classes of problem. Our work is another step toward quantum algorithms for geometrical problems that are capable of delivering practical value while also informing fundamental theory.

Paper Structure

This paper contains 8 sections, 12 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Algorithm for Calculating the Ollivier-Ricci Curvature
  3. Quantum Algorithm
  4. The case when G is a tree
  5. The case when $p=q$
  6. Conclusion
  7. Calculating ORC
  8. Block-encoding and quantum singular value transformation

Key Result

Theorem 2.1

Provided the raw distances $d(\textbf{x}_i, \textbf{x}_j)$ between a pair $(\textbf{x}_i,\textbf{x}_j)$ among $N$ data points. Then: where in the above, $\kappa = \min_{i,j,q,k}\{ \frac{d_G(\textbf{x}_i,\textbf{x}_j)}{d_G(\textbf{x}_k,\textbf{x}_q)} \}$, and $\gamma$ is a randomized factor that could be of $\mathcal{O}(1)$ in the best case and of $\mathcal{O}(\exp (p))$ in the worst case.

Figures (1)

  • Figure 1: Illustration of the setup used to compute the Earth Mover distance associated with an edge in a graph. The graph's vertices are in blue, and its edges are in magenta. Here we consider the edge $(x,y)$, in which the neighbors of $x$ excluding $y$ are $\{ x_1, x_2, x_3\}$ ($p=3$) and the neighbors of $y$ excluding $x$ are $\{ y_1, y_2, y_3, y_4\}$ ($q=4$). We show the graph distance $d_G(x_2,y_3)$ between points $x_2$ and $y_3$ in green. By optimizing the expression given in Eq. \ref{['optimal_transport']}, we arrive at the Earth Mover distance $W_1(x,y)$, which then allows us to calculate the ORC $\gamma(x,y)$ associated with the edge $(x,y)$ using Eq. \ref{['orc']}.

Theorems & Definitions (14)

  • Theorem 2.1
  • Lemma 3.1: Appendix D of nghiem2025quantum
  • Lemma 3.2: Positive Power Exponent gilyen2019quantum,chakraborty2018power
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5: Block encoding of density matrix, see e.g. gilyen2019quantum
  • Lemma 3.6
  • Definition B.1: Block-encoding unitary, see e.g. low2017optimallow2019hamiltoniangilyen2019quantum
  • Remark B.1: Properties of block-encoding unitary
  • Lemma B.1: Informal, product of block-encoded operators, see e.g. gilyen2019quantum
  • ...and 4 more