Quantum Algorithms for Matching and Network Flows

Sebastian Dörn

Andris Ambainis, Robert Špalek

We present quantum algorithms for the following graph problems: finding a maximal bipartite matching in time O(n sqrt{m+n} log n), finding a maximal non-bipartite matching in time O(n^2 (sqrt{m/n} + log n) log n), and finding a maximal flow in an integer network in time O(min(n^{7/6} sqrt m * U^{1/3}, sqrt{n U} m) log n), where n is the number of vertices, m is the number of edges, and U <= n^{1/4} is an upper bound on the capacity of an edge.

Vlatko Vedral

Rui Che, Ludvig af Klinteberg

Flow models generate trajectories from an initial distribution to a target distribution by solving an ordinary differential equation defined by a velocity field. Flow matching learns this velocity field by modeling the transport dynamics between the two distributions. Wavefunction flow establishes a formal connection between flow models and quantum dynamics by introducing a continuity Hamiltonian, which drives the Schrödinger evolution of quantum states. In this paper, we investigate accurate and efficient quantum simulation of the wavefunction flow, thereby realizing the efficient implementation of flow models on quantum computers. We first leverage a quantum read-only memory (QROM)-based phase kickback framework for the wavefunction flow simulation, generating probability densities that closely match those produced by the corresponding conventional flow model. To address the high circuit-resource cost, we further incorporate a trained quantum neural network (QNN) into the phase kickback framework, replacing QROM for data encoding. Numerical experiments demonstrate that our proposed method implements flow models on quantum computers more efficiently, since it maintains the accuracy of wavefunction flow simulation compared with the QROM-based framework, and significantly reduces the circuit resources.

John Kath, John K. Golden, Allon G. Percus +1 more

Abstract Uncertainty quantification plays a crucial role in the modeling of subsurface flow. For instance, uncertainties in the properties of geologic fracture networks significantly impact flow, requiring numerous simulations to accurately estimate quantities of interest. However, each simulation is computationally expensive because it requires solving a large linear system to capture features that involve both small and large fractures. An example is in percolation, where the interaction of many small fractures (which cumulatively can have a large surface area) with the rock matrix must be modeled precisely. Quantum computing is an emerging tool with the potential to address this issue. Quantum algorithms offer a significant speedup in solving linear systems, achieving efficiencies that are challenging to match with classical approaches. These classical approaches include direct solvers, such as LU decomposition, and iterative methods, notably preconditioned conjugate gradient, commonly used in subsurface modeling to solve large sparse systems. However, applying quantum algorithms to geologic fracture flow requires careful attention to algorithmic and problem‐specific constraints to fully realize this quantum advantage. In this work we describe a quantum algorithm for generalized Monte Carlo applications with a quadratic speedup over the classical approaches which can be combined with the quantum speedup, currently under investigation, for solving quantum linear systems for subsurface flow. We show that for quantum algorithms the computational cost of estimating a quantity of interest for a statistical ensemble of networks is roughly the same as that of a single realization, essentially implying that one can get uncertainty quantification for free.

Andreea-Iulia Lefterovici, Lara Lelakowski, Michael Perk

The maximum flow problem asks to find the largest possible flow from a source to a sink in a capacitated network. It arises frequently in scheduling, project selection, and as a core subroutine in broader optimisation tasks. Classically, it can be efficiently solved using Dinic's algorithm, which repeatedly performs breadth-first search (BFS) and blocking flow computations on the graph. As a potential candidate for quantum speedups, these BFS subroutines can be naturally replaced with quantum BFS (qBFS), an instantiation of Grover's search algorithm. In this paper, we evaluate the expected performance of qBFS on standard classical datasets. These instances are too large to be solved directly on current quantum hardware, so we adopt a hybrid benchmarking approach: (i) we run a classical implementation of Dinic's algorithm and isolate the runtime of its BFS subroutines; (ii) we analytically estimate the number of two-qubit gates required to implement qBFS, where we use the classically logged data. Our results indicate that, on every tested instance, matching classical BFS would require two-qubit gate operation times below the current physical record by one to two orders of magnitude for most instances, and by a factor of about 1.3 for the smallest, most favourable ones. Since this comparison uses physical rather than error-corrected gate times, the actual gap is larger still.