Linear programming with unitary-equivariant constraints

TL;DR

This work develops a general framework for solving unitary-equivariant optimization problems by converting SDPs with symmetry into LPs whose size does not depend on the local dimension $d$. The core idea is to exploit mixed Schur--Weyl duality and the diagrammatic structure of the walled Brauer algebra to parametrize unitary-equivariant Choi matrices via primitive central idempotents in a GT basis, enabling diagonal or block-diagonal representations. By adapting the Doty–Lauda–Samelson construction to the matrix algebras $ ext{A}^d_{p,q}$, the authors derive a practical two-stage algorithm: pre-compute diagrammatic idempotents and then reduce the SDP to a dimension-free LP with $n$ variables, whose complexity scales with $(p+q)!$ and the sparsity $s$. They demonstrate the method on quantum information tasks such as principal eigenvalue decision, quantum majority vote, asymmetric cloning, and black-box unitary transformation, illustrating substantial computational gains and potential for broader applicability. The framework opens a path toward solving more general unitary-equivariant SDPs and provides a solid representation-theoretic toolbox for exploiting symmetry in quantum optimization problems.

Abstract

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $d^{p+q}$-dimensional matrix variable that commutes with $U^{\otimes p} \otimes \bar{U}^{\otimes q}$, for all $U \in \mathrm{U}(d)$. Solving such problems naively can be prohibitively expensive even if $p+q$ is small but the local dimension $d$ is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in $d$, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand-Tsetlin basis, which we obtain by adapting a general method arXiv:1606.08900 inspired by the Okounkov-Vershik approach. To illustrate potential applications, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

Figures (7)

• Figure 1: Bratteli diagram for the symmetric group algebras $\mathop{\mathrm{\mathbb{C}S}}\nolimits_0 \hookrightarrow \mathop{\mathrm{\mathbb{C}S}}\nolimits_1 \hookrightarrow \mathop{\mathrm{\mathbb{C}S}}\nolimits_2 \hookrightarrow \mathop{\mathrm{\mathbb{C}S}}\nolimits_3 \hookrightarrow \mathop{\mathrm{\mathbb{C}S}}\nolimits_4$, also known as Young's lattice. The Bratteli diagram for the permutation matrix algebras $\mathcal{A}^d_0 \hookrightarrow \mathcal{A}^d_1 \hookrightarrow \mathcal{A}^d_2 \hookrightarrow \mathcal{A}^d_3 \hookrightarrow \mathcal{A}^d_4$ defined in \ref{['eq:Ap']} is the same when $d \geqslant 4$. When $d = 2$ or $d = 3$, vertices with Young diagrams containing more than $d$ rows are removed.
• Figure 2: Bratteli diagram associated to the multiplicity-free family $\mathbb{C} \cong \mathcal{B}^\delta_{0,0} \hookrightarrow \mathcal{B}^\delta_{1,0} \hookrightarrow \mathcal{B}^\delta_{2,0} \hookrightarrow \mathcal{B}^\delta_{2,1} \hookrightarrow \mathcal{B}^\delta_{2,2}$ of walled Brauer algebras when they are semisimple.
• Figure 3: Bratteli diagram associated to the multiplicity-free family $\mathbb{C} \cong \mathcal{A}^d_{0,0} \hookrightarrow \mathcal{A}^d_{1,0} \hookrightarrow \mathcal{A}^d_{2,0} \hookrightarrow \mathcal{A}^d_{2,1} \hookrightarrow \mathcal{A}^d_{2,2}$ of partially transposed permutation matrix algebras for different values of the local dimension $d$. When $d \geqslant 4$, this diagram coincides with that of the walled Brauer algebras (see \ref{['fig:B22']}). However, for small values of $d$ (i.e., $d = 2$ and $d = 3$) the diagram has to be modified by removing the designated vertices. Note that removing the vertex $\lparen*\rparen{ , }$ when $d = 2$ eliminates the highlighted path from the root $\lparen*\rparen{\varnothing,\varnothing}$ to the leaf $\lparen*\rparen{ , }$, which decreases the dimension of the corresponding simple $\mathcal{A}^d_{2,2}$-module $V^{\lparen*\rparen{ , }}$ by one, i.e., $\dim\lparen V^{\lparen*\rparen{ , }}\rparen = 4$ if $d > 2$ while $\dim\lparen V^{\lparen*\rparen{ , }}\rparen = 3$ if $d = 2$.
• Figure 4: Plot of the success probability $p^3_{2,2}(c)$ from \ref{['eq:p222']} as a function of $c \in [1/2,1]$. The gray curves represent the trivial lower bound from \ref{['eq:trivial bound']} obtained by setting $n = 1$.
• Figure 5: Plots of $p^n_{2,2}(c)$ for $n = 1, \dotsc, 8$ (darker lines correspond to larger values of $n$). As $n$ gets larger, the curves move upwards and the number of their segments increases.

Theorems & Definitions (58)

• Theorem : Informal
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Theorem 3.1: Schur--Weyl duality
• Example 3.2: $p = 2$ and $d = 2$
• Example 3.3: Unfaithfulness of $\psi^d_3$
• Definition 3.4: brundan2012gradings
• Example 3.5