A quadratic Grassmann manifold optimization problem arising from quantum embedding methods

arXiv:2603.17080 · math.OC, quant-ph · Submitted 2026-03-17 · 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: We're looking at the paper: "A quadratic Grassmann manifold optimization problem arising from quantum embedding methods".

Mira: Here's the gist of the paper. The gist This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function J(P) = Tr(BP)−1/2Tr(AP AP), where A, B ∈ R M×M sym, with A ⪰ 0, over the Grassmann manifold Gr(m, R M).

Kai: Mathematical Analysis The core of the analysis involves understanding the optimization problem (1)–(3) defined by minimizing J(P) over the Grassmann manifold M:= Gr(m, R M), which is defined as P ∈ R M×M sym P 2 = P, Tr(P) = m <ref:2603.17080#pg3>.

Mira: The tangent space to M at a point P is given by TPM =

X, P: for X in R M×M sym antisym <ref:2603.17080#pg5>.

Kai: The Riemannian gradient of the restricted function J on M is computed as gradMJ(P) =

[G(P), P: , P], where G(P):= B − AP A <ref:2603.17080#pg11>.

Paper summary: Kai: So we've just heard a bit about this paper now "A quadratic Grassmann manifold optimization problem arising from quantum embedding methods." The main idea is that they are analyzing minimizing J(P) = Tr(BP) minus half of Tr(AP AP) over the Grassmann manifold Gr(m, R M).

Mira: They claim that this problem shows up naturally in quantum chemistry and physics when you look at building bath orbitals using Density-Matrix Embedding Theory, or DMET eighteen <ref:2603.17080#pg1>.

Kai: The core thesis is that this quadratic Grassmann optimization problem has not been studied in the mathematical literature before, even beyond its application in DMET.

Lev: It’s a problem of minimizing P in M where M is the set of rank-m orthogonal projectors in R M, and J is the quadratic function defined by Tr(BP) minus half of Tr(AP AP) = Tr(BP) minus half of the Frobenius norm squared two <ref:2603.17080#pg1>.

Mira: They are looking at how this generalized bath construction leads to this optimization problem. It’s an extension of the standard DMET method.

Kai: What they claim is that they find ways to solve it by introducing a convexification of the original problem into minimizing Je(D) over the convex hull K, which is CH(M).

Lev: This new functional Je(D) is defined as Tr(CD) plus one-half of

A, D: squared, where C is B minus half of A squared six <ref:2603.17080#pg1>.

Mira: They assert that J and Je have the same critical points on M, which means they share the same important solutions.

Kai: But more importantly for practical use, Theorem four states that if H star satisfies the gap condition mu m < mu m+one then D star is a global minimizer of the non-convex problem (one) through (three) eight.

Lev: That gap condition is key because it ensures that the minimizer of D over CH(M) is unique, belongs to M, and is the global minimizer over M three <ref:2603.17080#pg1>.

Mira: So if that condition holds, then they've essentially found a way to guarantee a global minimum for the original problem.

Kai: They also look at special cases like when A and B commute or when the commutator

A, B: is small, which gives them explicit solutions under those conditions.

Lev: And they analyze how A and B arise from a DMET bath construction by decomposing matrices into blocks derived from a larger matrix gamma six <ref:2603.17080#pg1>.

Mira: That decomposition involves A and B being defined through formulae involving gamma frag and gamma ext six, where gamma satisfies zero <= gamma <= I <ref:2603.17080#pg1>.

Kai: The paper sets up the groundwork for what they call full disentanglement, which is relevant to quantum embedding methods.

Conclusion: Kai: So we've covered the main points of this work on "A quadratic Grassmann manifold optimization problem arising from quantum embedding methods." The authors are looking at solving that quadratic function J(P) over the Grassmann manifold Gr(m, R M).

Mira: They’re essentially showing that for a specific condition involving eigenvalues of B, they can find the true global minimizer D star.

Kai: This result is important because it provides a mathematical tool to guarantee you get the best possible answer when constructing those quantum bath orbitals.

Lev: If you're trying to build hardware, you need that guarantee about whether your optimization method will actually land on the right spot.

Mira: The paper shows that using the convex functional Je as a starting point is effective for Riemannian optimization methods even when you don't have those perfect conditions.

Kai: So even if you can't satisfy the strict gap condition, you get a good place to start searching for that global minimum of J on M.

Lev: And in short, this paper gives us a solid mathematical basis for using these optimization strategies in real-world quantum hardware simulations where things aren't always perfect.

Thomas Ayral, Eric Cancǩs, Fabian M. Faulstich, Lin Lin, Alicia Negre

CPHT, CNRS, Ecole Polytechnique, IP Paris · CERMICS, ENPC, IP Paris, CNRS · MATHERIALS team-project, Inria Paris · Department of Mathematical Sciences at Rensselaer Polytechnic Institute · Department of Mathematics at University of California Berkeley · Applied Mathematics and Computational Research Division at Lawrence Berkeley National Laboratory

math.OC, quant-ph

Submitted: 2026-03-17

Updated: 2026-10-20

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function J(P) = Tr(BP)−1/2Tr(AP AP), where A, B ∈

Key concepts

Grassmann Manifold Gr(m, R^M): This is the space where the optimization takes place. It represents all possible $m$-dimensional subspaces within a larger $M$-dimensional real vector space. In quantum chemistry applications, these subspaces often correspond to important orbital spaces.
Riemannian Gradient and Hessian: These are mathematical tools used to calculate how much the objective function changes as you move across the curved surface of the Grassmann manifold. The gradient tells you the steepest direction downhill, and the Hessian describes the curvature of that landscape, which is crucial for understanding optimization paths.
Convexification Je(D): Since minimizing directly on a non-convex space is difficult, this method transforms it into minimizing a simpler convex function $Je(D)$ over the convex hull of the manifold. This allows researchers to find a guaranteed global minimum, which then helps guide searches on the original non-convex problem.

Terminology

Riemannian Optimization — 2× in this paper · explained in 3 episodes

Summary

The gist This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function J(P) = Tr(BP)−1/2Tr(AP AP), where A, B ∈ R M×M sym, with A ⪰ 0, over the Grassmann manifold Gr(m, R M).

Mathematical Analysis

The core of the analysis involves understanding the optimization problem (1)–(3) defined by minimizing J(P) over the Grassmann manifold M:= Gr(m, R M), which is defined as P ∈ R M×M sym P 2 = P, Tr(P) = m <ref:2603.17080#pg3>. The tangent space to M at a point P is given by TPM = [X, P] for X in R M×M sym antisym <ref:2603.17080#pg5>. The Riemannian gradient of the restricted function J on M is computed as gradMJ(P) = [[G(P), P], P], where G(P):= B − AP A <ref:2603.17080#pg11>. The Riemannian Hessian HessMJ(P) is decomposed into two terms, HessMJ(P) = omega(P) + K(P), where omega(P): TPM → TPM X 7→ [[G(P), X], P] and K(P): TPM → TPM X 7→ −[[AXA, P], P] <ref:2603.17080#pg16>.

Convexification and Global Minimization

The problem is addressed by introducing a convexification of the original problem (1)–(3) into minimizing Je(D) over the convex hull K:= CH(M), where Je(D):= Tr(CD) + 1/4 [A, D] squared with C:= B − 1/2 A squared <ref:2603.17080#pg6>. It is noted that J and Je have the same critical points on M <ref:2603.17080#pg8>. Theorem 4 states that if H⋆ satisfies the gap condition µm < µm+1, then D⋆ is a global minimizer of the non-convex problem (1)–(3) <ref:2603.17080#pg8>. This property extends to cases where there is a gap between eigenvalues of B, ensuring that the minimizer on CH(M) lies on M and is the global minimizer over M <ref:2603.17080#pg8>.

Special Cases and Properties

The paper explores several special cases. When A and B commute, an explicit solution can be derived from the eigenmodes of matrix C <ref:2603.17080#pg9>. Perturbation analysis examines cases where [A, B] is small, leading to real-analytic solutions for P ϵ when the gap condition holds <ref:2603.17080#pg11>. The DMET bath-construction problem is analyzed by decomposing matrices A and B into blocks derived from a larger matrix γ <ref:2603.17080#pg12>. Full disentanglement in this context requires m ≥ dim(X) − l, where X is the smallest γ-invariant subspace of R L containing Hfrag <ref:2603.17080#pg15>.

Numerical Methods

Three numerical approaches are detailed for solving the problem (1)–(3). First, Riemannian optimization methods reformulate the problem on M into an optimization on the Stiefel manifold N, where P⋆ is a minimizer if and only if V⋆ is a minimizer of JSt on N <ref:2603.17080#pg16>. Second, Self-Consistent Field (SCF) algorithms exploit the Aufbau principle (Theorem 2), where the Roothaan algorithm converges to a critical point of J on M <ref:2603.17080#pg17>. Third, methods solving the convexified problem (24) are employed; specifically, the Optimal Damping Algorithm (ODA) is used, which provides a global minimum of J on M if H⋆ is gapped <ref:2603.17080#pg18>.

Applications in Quantum Chemistry

The study is motivated by applications in quantum embedding methods, particularly the construction of bath orbitals <ref:2603.17080#pg3>. The paper presents numerical tests on low-dimensional examples and an example from the DMET bath construction for the benzene molecule <ref:2603.17080#pg19>. These tests show that while SCF algorithms always converge to a local minimizer of (1)–(3), they do not necessarily converge to a global minimum, as demonstrated by oscillations between states when minimizing Je on M <ref:2603.17080#pg20>. The results suggest that the convex functional Je provides a good starting point for Riemannian optimization methods to reach the assumed global minimum of J on M <ref:2603.17080#pg23>.

Conclusion

The quadratic Grassmann optimization problem (1)–(3) arising from the DMET bath-construction problem (42) has interesting mathematical and numerical properties <ref:2603.17080#pg7>. The convex quadratic function Je defined in (23) takes the same values as J on the Grassmann manifold M <ref:2603.17080#pg8>. Minimizing Je on the convex hull CH(M), using e.g. the optimal damping algorithm (ODA), yields a global minimizer D⋆ ∈ CH(M) <ref:2603.17080#pg23>. The results in Theorem 4 show that if H⋆ satisfies the gap condition µm < µm+1, then D⋆ is a global minimizer of the non-convex problem (1)–(3) <ref:2603.17080#pg23>. Otherwise, the numerical tests we have run so far seem to indicate that D⋆ provides a good starting point for Riemannian optimization methods to reach the assumed global minimum of J on M <ref:2603.17080#pg23>.

Improvements for AI systems

Confidence in optimization convergence for quantum embedding problems will be significantly enhanced by employing an optimal damping algorithm (ODA) to solve the convexified problem (24), which provides the global minimum of (1)–(3) if the matrix H⋆ is gapped.
The improved AI system can robustly find global minimizers for quadratic Grassmann optimization problems, especially when initial guesses are poor, by utilizing a convex relaxation approach that yields a solution "if H⋆ satisfies the gap condition µm < µm+1."
Self-consistent field (SCF) algorithms will exhibit more reliable convergence behavior on the manifold M by exploiting the Aufbau principle (Theorem 2), which ensures they converge to critical points satisfying the property that P⋆ is a critical point of J on M satisfying the Aufbau principle (19).
For non-quadratic cost functions, iterative algorithms will be prevented from oscillating between non-critical states by leveraging the analysis showing that the sequence (J(Pk))k∈N is non-increasing and converges to a critical value of J on M.
The system can identify situations where minimizing the convex functional Je on the convex hull K yields a global minimizer for J on M, as shown by Theorem 4 and the conclusion that D⋆ is a global minimizer of the non-convex problem (1)–(3) if H⋆ satisfies the gap condition.

Abstract

This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function J(P) = Tr(BP)- 1 over 2 Tr(A P A P), where A,B in R M times M sym, with A 0, over the Grassmann manifold Gr (m, R M). While this problem is non-convex and typically admits non-global local minima - posing challenges for Riemannian optimization and self-consistent field (SCF) algorithms - we identify cases where the global minimizer can be obtained by solving an auxiliary convex problem. When this approach is not directly applicable, the solution to the auxiliary problem still serves as an effective initialization for Riemannian optimization methods and SCF algorithms, significantly improving their performance. This work is motivated by applications in quantum embedding methods, particularly in the construction of bath orbitals, where such optimization problems naturally arise.