Faster Algorithms for Multimarginal Optimal Transport

arXiv:2608.09513 · quant-ph, cs.DS, math.OC · Submitted 2026-08-10 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Faster Algorithms for Multimarginal Optimal Transport".

Mira: We study algorithms for approximating multimarginal optimal transport (MOT) distance, a generalization of the classic optimal transport distance, between discrete probability distributions each supported on at most n points.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we're talking about "Faster Algorithms for Multimarginal Optimal Transport" today. It sounds like this paper tackles the problem of approximating the distance between multiple distributions when you have more than just two sets involved. What's the main idea here?

Mira: Well, it addresses multimarginal optimal transport, which is a way to find a joint coupling among several measures that minimizes a multi-linear cost function, and it focuses on discrete probability distributions where each distribution has at most n points Kan42. The authors claim they provide classical and quantum algorithms for solving this problem.

Lev: That sounds like a pretty heavy lift conceptually; what are the specific claims they're making about their algorithms in terms of performance? We need to know what kind of speedups or bounds are being put forward here before we can really assess the impact on real hardware.

Kai: The abstract states that for the classical algorithm, they compute a coupling whose expected transportation cost is within an additive error of epsilon > zero of the MOT distance in time O(m 2n m epsilon-one polylog(m, n, epsilon-one)) <ref:2608.09513#pg0,whose expected transportation cost is within an additive>. That's a specific runtime bound for the classical approach.

Mira: And they also provide two quantum algorithms that aim to achieve speedups in dimension, although the accuracy dependence for those quantum approaches is worse than what the classical methods offer <ref:2608.09513#pg0>.

Lev: That's interesting because if we're talking about real hardware, those accuracy dependences matter a lot when you consider noise and decoherence; how does that translate to actual computational feasibility for these quantum methods?

Kai: The abstract mentions a quantum projected subgradient method with a runtime of O(m 3n m/two + one/epsilon squared polylog(m, n, epsilon-one)), and another one, the quantum multimarginal Sinkhorn algorithm for entropy-regularized MOT has a runtime of O(m 8n m + one/two epsilon-five polylog(m, n, epsilon-one)) <ref:2608.09513#pg0>.

Mira: The paper sets some very strong lower bounds for query complexity, which are calibrated across different accuracy regimes; for instance, they record that randomized classical algorithms require (n m/(one + epsilon n)) queries when epsilon < one/two <ref:2608.09513#pg0>.

Lev: Those lower bounds help ground the theoretical limits, but for me, the real question is how those bounds translate to running on physical devices; what's the practical overhead when you try to implement something that requires a polynomial number of queries in n ?

Kai: The paper also details a reduction from discrete MOT to a positive packing LP called packing-MOT, which they use because it simplifies the constraint matrix structure by having exactly mN nonzeros, where N = n m.

Mira: That reduction is key because it transforms the dense problem into one that fits within a specific optimization framework, and they introduce an entropic regularization to make the problem strictly convex, which enables near-linear scaling in the dense tensor dimension for certain problems <ref:2608.09513#pg2>.

Paper summary: Lev: The paper mentions that solving this regularized problem involves estimating a vector q k(beta) without explicitly summing over every fiber entry, relying on a quantum LOGSUMEXP primitive <ref:2608.09513#pg2>. That reliance on specific primitives is something I have to think about when discussing error correction and implementation fidelity.

Kai: Moving toward the conclusion, the title "Faster Algorithms for Multimarginal Optimal Transport" suggests they are focusing heavily on improving the complexity of these approximations in both classical and quantum settings.

Mira: The authors are presenting a set of algorithms that establish new bounds on runtime and query complexity across different accuracy regimes for solving MOT problems <ref:2608.09513#pg0>.

Lev: The implications, from my side as someone looking at error correction, are that if these classical results hold up under real-world noise models, it gives us a clearer picture of the complexity barrier we'd face when trying to implement quantum versions on actual hardware.

Kai: And for the quantum community, seeing these bounds on dimension speedups in the algorithms presented in "Faster Algorithms for Multimarginal Optimal Transport" is significant because they show how dimension dependence behaves even with those trade-offs in accuracy <ref:2608.09513#pg0>.

Mira: I think the core contribution lies in bridging the gap between discrete MOT and continuous formulations through LP reductions, while simultaneously exploring how entropic regularization can make otherwise intractable problems solvable with near-linear scaling <ref:2608.09513#pg2>.

Lev: So, to summarize what we've heard about "Faster Algorithms for Multimarginal Optimal Transport," it’s a paper that provides specific, quantified complexity bounds for both classical and quantum solutions to approximating multimarginal optimal transport distance <ref:2608.09513#pg0>.

Kai: And the overall takeaway is that they've mapped out the required query complexity for these problems across various precision levels, giving us a much clearer idea of what's needed to tackle them computationally.

Mira: The paper also points out that bimarginal OT, where m=two can be solved efficiently using matrix scaling methods like Sinkhorn’s method when entropy regularization is added <ref:2608.09513#pg2>.

Lev: That connection to existing matrix balancing techniques is useful; if we can leverage those known structures, it might make the practical implementation of these quantum approaches more tractable for future error-corrected systems.

Kai: So, to wrap up this part of our discussion on "Faster Algorithms for Multimarginal Optimal Transport," we've seen a detailed look at the classical runtime bounds and the specific query complexity required by these methods <ref:2608.09513#pg0>.

Mira: The implications are that understanding how to formulate these problems through LP reductions and entropic regularization is crucial for developing scalable solvers, even if they don't provide a perfect solution immediately.

Lev: And I think the future work will heavily depend on whether these theoretical bounds can be realized when you factor in the physical constraints of noise and measurement errors on actual quantum hardware <ref:2608.09513#pg2>.

Conclusion: Kai: So, we're wrapping up our discussion on "Faster Algorithms for Multimarginal Optimal Transport," which focuses on speeding up how we approximate distances between multiple distributions using both classical and quantum methods <ref:2608.09513#pg0>.

Mira: Indeed, the paper presents classical and quantum approaches to solving the MOT problem, specifically looking at how runtime and query complexity scale with accuracy <ref:2608.09513#pg2>.

Lev: I'm curious about what this means practically for running these things on real quantum hardware; can we actually build something that achieves these stated query bounds?

Kai: Exactly, Lev, because the authors are establishing concrete lower bounds on queries that show the fundamental limits of what’s required for constant accuracy <ref:2608.09513#pg0>.

Mira: And they've shown how entropic regularization can help smooth out these problems, allowing for near-linear scaling in certain dimensions when we use the right formulation <ref:2608.09513#pg2>.

Lev: That scaling is important because it suggests a path toward more manageable computational problems even with the inherent noise limitations of physical systems.

Kai: It really sets a new benchmark for how we think about tackling these complex transport problems from both an algorithmic and a hardware feasibility standpoint <ref:2608.09513#pg0>.

Mira: This work has significant implications for computational physics, showing how advanced mathematical tools can translate into practical efficiency gains in areas like data science and statistical mechanics.

Lev: If these complexity results hold up under the physical constraints we face, it gives us a clearer roadmap for designing error-corrected quantum processors that are actually useful <ref:2608.09513#pg2>.

Global Technology Applied Research, JPMorganChase

quant-ph, cs.DS, math.OC

Submitted: 2026-08-10

Updated: 2026-10-06

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: We study algorithms for approximating multimarginal optimal transport (MOT) distance, a generalization of the classic optimal transport distance, between discrete probability distributions each

Key concepts

Multimarginal Optimal Transport (MOT): This is a generalization of classic optimal transport that deals with multiple probability distributions simultaneously. The goal is to find an optimal coupling between all these distributions while respecting specific cost constraints, which can be complex when dealing with many points.
Packing LP Reduction: The paper transforms the dense MOT problem into a simpler positive packing linear program (packing-MOT). This reduction is key because it shows that the constraints of the original problem can be captured efficiently in an LP whose constraint matrix has only $mN$ non-zero entries, simplifying computation.
Quantum Multimarginal Sinkhorn: This quantum algorithm solves an entropy-regularized version of MOT. It uses a quantum primitive called LOGSUMEXP to estimate necessary vectors without explicitly calculating every entry. This method is closely related to quantum algorithms for matrix scaling and balancing problems.

Terminology

Accuracy — 31× in this paper · explained in 1 episode
Optimal Transport — 26× in this paper · explained in 5 episodes
Query Complexity — 12× in this paper · explained in 2 episodes
Regularization — 10× in this paper · explained in 3 episodes
Generalization — 4× in this paper · explained in 13 episodes
Multimarginal Optimal Transport (MOT) — 4× in this paper · explained in 2 episodes
Core — 3× in this paper · explained in 2 episodes
Adam — 2× in this paper · explained in 2 episodes
Alternating Minimization — 2× in this paper · explained in 1 episode
Benchmark — 2× in this paper · explained in 2 episodes
Physical Constraints — 2× in this paper · explained in 7 episodes
Robustness — 2× in this paper · explained in 5 episodes
Wasserstein Distance — 2× in this paper · explained in 4 episodes
Adaptation — 1× in this paper · explained in 1 episode
Clustering — 1× in this paper · explained in 2 episodes
Combinatorial Optimization — 1× in this paper · explained in 2 episodes
Domain Adaptation — 1× in this paper · explained in 5 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Generative Modeling — 1× in this paper · explained in 3 episodes
Objective Function — 1× in this paper · explained in 2 episodes
Optimal Transport (OT) — 1× in this paper · explained in 4 episodes
Regret — 1× in this paper · explained in 3 episodes

Summary

We study algorithms for approximating multimarginal optimal transport (MOT) distance, a generalization of the classic optimal transport distance, between discrete probability distributions each supported on at most n points. This work provides classical and quantum algorithms for solving MOT problems, establishing new bounds on their runtime and query complexity across different accuracy regimes.

Classical Algorithms and Complexity

The paper presents a classical randomized first-order algorithm for solving (MOT-P), which returns an explicit coupling tensor satisfying the cost constraint within an additive error of ε > 0. When all marginals have support size n, the runtime is stated as:

/O(m 2n/∥C∥∞ ε polylog(m, n, ∥C∥∞ ε)) / O(n squared (∥C∥∞/ε) 2) depending on the specific algorithm. The authors note that this runtime is linear in the tensor dimension N = n m, up to a factor of m and logarithmic factors. The main reduction involves solving a positive packing LP with only mN nonzeros, which is then completed by a rank-one completion step to ensure marginal feasibility.

Quantum Algorithms for MOT Value Estimation

On the quantum side, two algorithms are presented that achieve speedups in dimension, although with worse accuracy dependence than classical approaches.

A quantum projected subgradient method for estimating the MOT distance within an additive ε > 0 with runtime:

/O(m 3n m/2 + 1/ε squared polylog(m, n, ε-1)). This algorithm works with the linear programming dual of the MOT problem and does not return a coupling.

A quantum multimarginal Sinkhorn algorithm for entropy-regularized MOT. This returns an implicit description of an approximately optimal coupling with runtime:

/O(m 8n m + 1/2 ε-5 polylog(m, n, ε-1)). This algorithm is obtained after the usual reduction from entropic MOT to unregularized MOT.

Lower Bounds and Query Complexity

The paper records elementary cost-query lower bounds that calibrate the dimension dependence across accuracy regimes. For any precision ε < 1/2:

/Randomized classical algorithms require Ω(n m/(1 + εn)) queries.

/Quantum algorithms require Ω(p n m/(1 + εn)) queries.

These bounds show that constant-accuracy MOT already requires Ω(n(m-1) classical and Ω(n(m-1)/2) quantum cost queries, while accuracy below the single-entry mass scale 1/n requires Ω(n m classical and Ω(n m/2) quantum cost queries.

Reduction to Packing LP

The discrete MOT problem is reduced to a positive packing LP (packing-MOT). This reduction leverages the observation that if a partial coupling X has deficit vectors dk ≜ µk − Projk(X) in R(nk), then all deficits have the same total mass τ, and D = τ(1-md1 ⊗ · · · ⊗ dm) has exactly those deficits as its marginals. This allows the dense MOT problem to be transformed into a packing LP whose constraint matrix has exactly mN nonzero entries.

Entropic Regularization and Sinkhorn

The paper introduces the entropy-regularized MOT problem, (MOTη), which makes the optimization strictly convex, allowing for algorithms with near-linear scaling in the dense tensor dimension. The entropic dual is an unconstrained smooth convex problem:

/min β1∈R n1,...,βm∈Rnm Φη(β) ≜ log Σ exp (m ∑ k=1 βk,ik − Ci/η - m ∑ k=1 µ⊤k βk). The solution admits a Gibbs representation X⋆i = Kb(β)i ≜ Ki exp (sum m k=1 βk,ik) / K(β)1.

The quantum multimarginal Sinkhorn algorithm solves this regularized problem. The core computational task is to estimate the vector log qk(β) without explicitly summing over every fiber entry, utilizing a quantum LOGSUMEXP primitive.

Comparison with Existing Approaches

The paper compares its results with other methods, noting that the quantum multimgarginal Sinkhorn result is closest in spirit to quantum algorithms for matrix scaling and matrix balancing. Specializing the bound to m = 2 recovers a runtime of order O(n(3/2)(∥C∥∞/ε) 5 + n 2), which is an improvement in precision for bimarginal OT but relies on matrix-specific structure.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements for AI systems that can be derived from its findings:

The developed algorithms (Classical and Quantum Projected Subgradient Method, Quantum Multimarginal Sinkhorn Algorithm) provide a way to approximate the Multimarginal Optimal Transport (MOT) distance.
These algorithms can compute an explicit coupling tensor (classical algorithm) or an implicit description of an approximately optimal coupling (quantum algorithms).

This capability allows the improved AI system to perform:

Compute a mathematically rigorous, geometrically meaningful distance (Wasserstein metric generalization) between multiple complex probability distributions, even when the distributions are discrete and high-dimensional.
Determine the optimal joint allocation or transport plan (coupling) of mass between these multiple distributions subject to marginal constraints, which is crucial for tasks requiring distribution matching or barycenter computation.

Specific applications include:

Compute an accurate average representative distribution (barycenter) by combining several distinct datasets (e.g., different user demographics, product categories, or domain-specific data) in a way that respects the underlying transport costs between them.
Perform robust domain adaptation where the system learns how to transform one set of data points into another while minimizing the transportation cost associated with that transformation.
Enhance generative modeling by ensuring that generated samples adhere not just to marginal distributions, but also respect a multi-marginal structure defined by complex interaction costs (e.g., ensuring generated images maintain specific relationships between different semantic features).