Papers for

quantum information teams

Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.

Delsarte bound rates for binary and spherical codes solved exactly

The Rate of the Delsarte Bound for Binary Codes and Spheres

Abstract: We give the asymptotic rate of Delsarte's linear programming bound for binary codes and spherical codes. This extends previous work of the author which formulated and proved an analogue of the conjecture of Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini for binary codes, obtaining the rate of the Delsarte linear program at distance $1/2-1/π$. The lower bounds introduce ideas from potential theory into the study of Delsarte's bounds, whereby the measures underlying a system of orthogonal polynomials give an obstacle problem. We solve the resulting obstacle problem in the case of binary codes and spherical codes, obtaining a closed form for the lower bound in terms of elliptic integrals. For the upper bounds, we relate the $N$-qubit construction for binary codes used in previous work of the author to the constructions in OpenAI's Chapter 2 for spherical codes and observe that they both involve interlacing spectra. This leads to a variational problem involving measures constructed from these spectra under the Markov-Krein correspondence. This variational problem admits an exact solution and matches the lower bound.

Thu 15 OctInformation Theory
The gist
The paper finds an exact formula for how well a mathematical limit, called the Delsarte bound, performs for two important types of codes used in communication and data storage. This extends the author's earlier work on binary codes at a specific distance, now covering a broader range and also spherical codes, which correspond to points on a sphere. To achieve this, the author blends tools from potential theory and spectral analysis, solving a complex problem involving orthogonal polynomials and measures. The results provide matching lower and upper limits, giving a precise understanding of these bounds' rates.
Open → 2610.21263v1

Fusion frame approach enables faster phase retrieval with projections

The Fusion Frame Phase Retrieval

Abstract: The phase retrieval problem involves reconstructing a function or signal solely from the magnitude of linear measurements. Most theoretical analyses of phase retrieval algorithms rely on i.i.d. Gaussian random measurements or sub-Gaussian random measurements. In this paper, our focus is on the fusion frame phase retrieval problem, where the sampling matrices are i.i.d. rank-$r$ orthogonal projections drawn from the Haar measure. We present concentration inequalities for functions on the set of rank-$r$ orthogonal projection matrices. These inequalities are crucial for the theoretical analysis of the fusion frame phase retrieval problem. Based on these inequalities, we demonstrate that gradient descent, combined with a two-stage initialization, achieves linear convergence to the target signal up to a global phase with a measurement complexity of $O(d\log^2 d)$ when the rank $r = O(1)$. We verify this convergence through numerical results.

Tue 8 SeptInformation Theory
The gist
Phase retrieval is about figuring out a signal from measurements where only the size is known, not the direction. This paper studies a version where measurements come from special projection matrices instead of typical random ones. The authors develop new mathematical tools to show that a gradient descent method quickly finds the original signal with fewer measurements. They also confirm their approach works with computer experiments.
Open → 2609.08531v1