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Barrier relaxations of the classical and quantum optimal transport problems

Shmuel Friedland

TL;DR

The paper reframes both classical and quantum optimal transport in barrier-relaxation terms and applies interior-point methods to compute barrier-approximations of the primal and dual problems. By constructing self-concordant barrier functions (−log det for quantum, and log-based barriers for classical), it derives explicit optimality structures (e.g., rescaled exponential forms) and shows that the barrier problems admit efficient IPM-based solutions. It establishes rigorous complexity bounds across bipartite and multipartite settings, with classical BPOTP achieving $O(n^4\log(n^2 r/δ))$ iterations (and $O(n^4)$ per iteration) and quantum bipartite OT reaching $O(n^7\log n)$, plus scalable extensions to tensor-structured MPOTP and quantum MPOTP with bounds like $O(n^{3d/2}\log(n^d r/δ))$ and $O(n^{7d/2}\log(n^d r/δ))$ respectively. The framework provides a principled alternative to entropic/Sinkhorn relaxations, offering dual-trajectory advantages and broad applicability to high-dimensional OT problems in classical and quantum domains.

Abstract

In the last fifteen years a significant progress was achieved by considering an entropic relaxation of the classical multi-partite optimal transport problem (MPOTP). The entropic relaxation gives rise to the rescaling problem of a given tensor. This rescaling can be achieved fast with the Sinkhorn type algorithms. Recently, it was shown that a similar approach works for the quantum MPOTP. However, the analog of the rescaling Sinkhorn algorithm is much more complicated than in the classical MPOTP. In this paper we show that the interior point method (IPM) for the primary and dual problems of classical and quantum MPOTP problems can be considered as barrier relaxations of the optimal transport problems (OTP). It is well known that the dual of the OTP are advantageous as it has much less variables than the primary problem. The IPM for the dual problem of the classical MPOTP are not as fast as the Sinkhorn type algorithm. However, IPM method for the dual of the quantum MPOTP seems to work quite efficiently.

Barrier relaxations of the classical and quantum optimal transport problems

TL;DR

The paper reframes both classical and quantum optimal transport in barrier-relaxation terms and applies interior-point methods to compute barrier-approximations of the primal and dual problems. By constructing self-concordant barrier functions (−log det for quantum, and log-based barriers for classical), it derives explicit optimality structures (e.g., rescaled exponential forms) and shows that the barrier problems admit efficient IPM-based solutions. It establishes rigorous complexity bounds across bipartite and multipartite settings, with classical BPOTP achieving iterations (and per iteration) and quantum bipartite OT reaching , plus scalable extensions to tensor-structured MPOTP and quantum MPOTP with bounds like and respectively. The framework provides a principled alternative to entropic/Sinkhorn relaxations, offering dual-trajectory advantages and broad applicability to high-dimensional OT problems in classical and quantum domains.

Abstract

In the last fifteen years a significant progress was achieved by considering an entropic relaxation of the classical multi-partite optimal transport problem (MPOTP). The entropic relaxation gives rise to the rescaling problem of a given tensor. This rescaling can be achieved fast with the Sinkhorn type algorithms. Recently, it was shown that a similar approach works for the quantum MPOTP. However, the analog of the rescaling Sinkhorn algorithm is much more complicated than in the classical MPOTP. In this paper we show that the interior point method (IPM) for the primary and dual problems of classical and quantum MPOTP problems can be considered as barrier relaxations of the optimal transport problems (OTP). It is well known that the dual of the OTP are advantageous as it has much less variables than the primary problem. The IPM for the dual problem of the classical MPOTP are not as fast as the Sinkhorn type algorithm. However, IPM method for the dual of the quantum MPOTP seems to work quite efficiently.
Paper Structure (19 sections, 24 theorems, 128 equations)

This paper contains 19 sections, 24 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Discrete classical optimal transport problem
  3. Our contribution
  4. Brief survey of the contents of the paper
  5. Classical bi-partite optimal control and its relaxations
  6. An entropic relaxation of the classical bi-partite optimal transport problem
  7. A barrier relaxation of the classical bi-partite optimal transport problem
  8. The interior point method for finding a barrier approximation of $\tau(C,P)$
  9. Generalized Sinkhorn algorithm
  10. Classical multi-partite optimal transport problem and its relaxations
  11. An entropic relaxation of the classical multi-partite optimal transport problem
  12. A barrier relaxation of the classical multi-partite optimal transport problem
  13. Quantum bi-partite optimal control and its relaxations
  14. A barrier relaxation of the quantum bi-partite optimal transport problem
  15. The interior point method for finding a barrier approximation of $\kappa(C,R)$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(\mathbf{p}_1,\mathbf{p}_2)\in (R^{n_1}\times\mathbb{R}^{n_2})_{++,b}$, and define The barrier function for $\hat{\mathrm{D}}$ is The function is concave in $\hat{\mathrm{D}}$ and strictly concave in $\mathrm{D}$. Furthermore, where $\beta$ and $\tau_{\beta}(C,P,\varepsilon)$ are defined by bVP and betBOTP respectively. The function $\varphi$ achieves its maximum exactly on the line whic

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 29 more