arXiv subjects

Michael Walter

Publications and source records attributed to Michael Walter.

At least 19 recordsLinked to original sources

arXiv2026

Convex optimization on moment polytopes: Hadamard mirror descent and efficient algorithms for quantum functionals and other tensor parameters

Convex optimization on polytopes arises in many areas of science. When the polytope is given implicitly or has exponentially many vertices and facets, standard methods may not apply or be ineffective. This is the case for moment polytopes, such as the entanglement polytopes, which play a foundational role in quantum information and algebraic complexity. They give rise to important entanglement measures and tensor parameters such as the quantum functionals, yet general effective methods for computing these quantities have been elusive. In this paper we address this challenge. We develop a first-order framework called Hadamard mirror descent to optimize suitable convex functions over moment polytopes and, more generally, the gradient sets of geodesically convex functions. It operates locally and does not rely on any explicit description of the polytope. Our framework extends mirror descent, an effective and widely used framework for convex optimization, from the Euclidean setting to Hadamard manifolds, and is motivated by a recent work by Hirai, which we interpret as a Hadamard version of mirror flow. Applying the framework to entanglement polytopes yields the first efficient first-order algorithms to compute the quantum functionals, the symmetric quantum functional, and the G-stable ranks, as well as a new direct algorithm for the non-commutative rank.

cs.CC↗

arXiv2026

Optimal tomography of bosonic and fermionic Gaussian states

The sample complexity is the minimum number of copies required to learn an accurate classical description of a quantum state. Bosonic and fermionic Gaussian quantum states are families of quantum states that play a key role in quantum science and technology, from quantum optics and many-body physics to quantum chemistry, quantum computing, and quantum information theory. Despite their importance, their sample complexity had not been fully determined. We settle this open problem and show that both bosonic and fermionic Gaussian states can be learned using a number of copies that scales quadratically in the number of modes, regardless of whether the state is pure or mixed, and independently of any energy bound on the state. We derive these results by using the representation theory of Gaussian unitaries and by putting forth a generalization of the random purification channel to this setting and beyond.

quant-ph↗

arXiv2026

Negative curvature obstructs the existence of good barriers for interior-point methods

Interior-point methods (IPMs) are a cornerstone of Euclidean convex optimization, due to their strong theoretical guarantees and practical performance. Motivated by scaling problems, recent work by Hirai and the last two authors (FOCS'23) extended IPMs to geodesically convex optimization on Hadamard manifolds. Crucially, the complexity of IPMs (both in Euclidean and Hadamard spaces) is governed by the \emph{barrier parameter} of the domain. Here we prove that already in hyperbolic space, several natural domains -- including geodesic balls and triangles -- have a barrier parameter that grows polynomially with the domain's diameter. By extension, the same holds for the positive-definite matrices and other symmetric Hadamard spaces. This growth implies a fundamental limitation: interior-point methods relying on barriers for a ball cannot efficiently solve challenging scaling problems, such as tensor scaling, where the domain's diameter can be exponentially large in the input size. Our results are partially inspired by, and complement, lower bounds on the condition number of geodesically convex functions established by Hamilton and Moitra (NeurIPS'21).

math.OC↗

arXiv2026

Approximate QCAs in one dimension using approximate algebras

Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate $C^*$-algebras.

quant-ph↗

arXiv2026

Plethysm is in #BQP

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

quant-ph↗

arXiv2026

Strassen's support functionals coincide with the quantum functionals

Strassen's asymptotic spectrum offers a framework for analyzing the complexity of tensors. It has found applications in diverse areas, from computer science to additive combinatorics and quantum information. A long-standing open problem, dating back to 1991, asks whether Strassen's support functionals are universal spectral points, that is, points in the asymptotic spectrum of tensors. In this paper, we answer this question in the affirmative by proving that the support functionals coincide with the quantum functionals - universal spectral points that are defined via entropy optimization on entanglement polytopes. We obtain this result as a special case of a general minimax formula for convex optimization on entanglement polytopes (and other moment polytopes) that has further applications to other tensor parameters, including the asymptotic slice rank. Our proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai.

cs.CC↗

arXiv2026

Communication Technologies for Intelligent Transportation Systems: From Railways to UAVs and Beyond

This white paper aims to comprehensively analyze and consolidate the state of the art in communication technologies supporting modern and future Information and Communication Technology (ICT). Its primary objective is to establish a common understanding of how communication solutions enable automation, safety, and efficiency across multiple transport domains, including railways, road vehicles, aircraft, and unmanned aerial vehicles. The document seeks to identify key communication requirements and technological enablers necessary for interoperable and reliable ITS operation. It also assesses the limitations of current systems and proposes pathways for integrating emerging technologies such as 5G, Sixth Generation (6G), and Artificial Intelligence (AI)-driven network control. The white paper also intends to support harmonization between different transport modes through a unified framework for communication modeling, testing, and standardization. It highlights the importance of accurate channel modeling and empirical validation to design efficient, robust, and scalable systems. Another objective is to explore the use of reconfigurable intelligent surfaces, integrated sensing and communication, and digital twin concepts within ITS. The document emphasizes the role of spectrum management and standardization efforts in ensuring interoperability among diverse communication systems. Finally, the paper seeks to stimulate collaboration among academia, industry, and standardization bodies to advance the design of resilient and adaptive communication infrastructures for future transportation systems.

cs.NI↗

arXiv2025

A random purification channel for arbitrary symmetries with applications to fermions and bosons

The random purification channel maps n copies of any mixed quantum state to n copies of a random purification of the state. We generalize this construction to arbitrary symmetries: for any group G of unitaries, we construct a quantum channel that maps states contained in the algebra generated by G to random purifications obtained by twirling over G. In addition to giving a surprisingly concise proof of the original random purification theorem, our result implies the existence of fermionic and bosonic Gaussian purification channels. As applications, we obtain the first tomography protocol for fermionic Gaussian states that scales optimally with the number of modes and the error, as well as an improved property test for this class of states.

quant-ph↗

arXiv2025

Ab-initio force prediction for single molecule force spectroscopy made simple

Bond rupture under the action of external forces is induced by temperature fluctuations. We show that measured forces from single molecule force spectroscopy experiments can be predicted from two quantities describing the bond that are the barrier to break the bond in absence of force as well as the maximal force the bond can withstand. The former can be obtained by a force free transition state calculation and the latter is determined by a simple constrained ge- ometry simulates forces (COGEF) calculation. Considering experimental temperature and force loading rate allows the prediction of measured bond rupture forces from a closed expression with very good accuracy.

cond-mat.mtrl-sci↗

arXiv2025

Channel Estimation under Large Doppler Shifts and Channel Aging in NOMA-Based Air-Ground Communications

This paper investigates a multiple antenna system with non-orthogonal multiple access (NOMA) for the exchange of air traffic management data between commercial aircraft pilots and ground-based air traffic controllers. While NOMA techniques enhance spectral efficiency, their application to aircraft communications is challenged by the high speed of the aircraft (up to 214 m/s) and the long communication ranges (up to 250 km), resulting in significant Doppler shifts and low signal-to-noise ratios, respectively. To accurately assess these challenges, we employ a realistic geometry-based stochastic air-ground channel model, derived from dedicated flight measurement campaigns. In this paper, multiple aircraft simultaneously transmit data to the ground station. We focus on the channel estimation problem at the ground station under high carrier frequency offsets and the effects of channel aging due to channel's time-varying nature. For the channel estimation problem, we compare the Zadoff-Chu sequences with time-division approach under varying carrier frequency offset pre-compensation accuracies at the aircraft transmitter. For the channel aging problem and performance evaluation of channel estimators, we compute the outage probability for both the zero-forcing detector and the minimum mean squared error detector with successive interference cancellation. The results show that the favorable channel estimator-detector combinations differ between the takeoff & landing phase and the enroute cruise phase of the flight, due to the distinct channel propagation characteristics of each phase.

eess.SY↗

arXiv2025

Computing moment polytopes -- with a focus on tensors, entanglement and matrix multiplication

Tensors are fundamental in mathematics, computer science, and physics. Their study through algebraic geometry and representation theory has proved very fruitful in the context of algebraic complexity theory and quantum information. In particular, moment polytopes have been understood to play a key role. In quantum information, moment polytopes (also known as entanglement polytopes) provide a framework for the single-particle quantum marginal problem and offer a geometric characterization of entanglement. In algebraic complexity, they underpin quantum functionals that capture asymptotic tensor relations. More recently, moment polytopes have also become foundational to the emerging field of scaling algorithms in computer science and optimization. Despite their fundamental role and interest from many angles, much is still unknown about these polytopes, and in particular for tensors beyond $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ and $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ only sporadically have they been computed. We give a new algorithm for computing moment polytopes of tensors (and in fact moment polytopes for the general class of reductive algebraic groups) based on a mathematical description by Franz (J. Lie Theory 2002). This algorithm enables us to compute moment polytopes of tensors of dimension an order of magnitude larger than previous methods, allowing us to compute with certainty, for the first time, all moment polytopes of tensors in $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$, and with high probability those in $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$ (which includes the $2\times 2$ matrix multiplication tensor). We discuss how these explicit moment polytopes have led to several new theoretical directions and results.

math.RT↗

arXiv2025

Guess your neighbor's input: Quantum advantage in Feige's game

In this article, we study a nonlocal game with two questions and three answers per player, which was first considered by Feige in 1991, and show that there is quantum advantage in this game. We prove that the game is a robust self-test for the $3$-dimensional maximally entangled state. Furthermore, we show that the game can be seen as the "or" of two games that each do not have quantum advantage. Lastly, we investigate the behavior of the game with respect to parallel repetition in the classical, quantum and non-signalling case and obtain perfect parallel repetition of the non-signalling value if Feige's game is repeated an even amount of times.

quant-ph↗

arXiv2025

Complete Relational Logic for Infinite-Dimensional Quantum Programs with Unbounded Assertions

We present sound and complete relational program logics for infinite-dimensional quantum and classical-quantum programs. The logics model assertions as self-adjoint unbounded linear relations, which simultaneously support quantitative and qualitative reasoning. Our main theoretical results include new convergence theorems and infinite-dimensional duality theorems for infinite-dimensional quantum states, which we use to establish completeness.

quant-ph↗

arXiv2025

Is it Gaussian? Testing bosonic quantum states

Gaussian states are widely regarded as one of the most relevant classes of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.

quant-ph↗

arXiv2025

Self-concordant Schr\"odinger operators: spectral gaps and optimization without condition numbers

Spectral gaps play a fundamental role in many areas of mathematics, computer science, and physics. In quantum mechanics, the spectral gap of Schr\"odinger operators has a long history of study due to its physical relevance, while in quantum computing spectral gaps are an important proxy for efficiency, such as in the quantum adiabatic algorithm. Motivated by convex optimization, we study Schr\"odinger operators associated with self-concordant barriers over convex domains and prove non-asymptotic lower bounds on the spectral gap for this class of operators. Significantly, we find that the spectral gap does not display any condition-number dependence when the usual Laplacian is replaced by the Laplace--Beltrami operator, which uses second-order information of the barrier and hence can take the curvature of the barrier into account. As an algorithmic application, we construct a novel quantum interior point method that applies to arbitrary self-concordant barriers and shows no condition-number dependence. To achieve this we combine techniques from semiclassical analysis, convex optimization, and quantum annealing.

quant-ph↗

arXiv2025

Computational Monogamy of Entanglement and Non-Interactive Quantum Key Distribution

Quantum key distribution (QKD) enables Alice and Bob to exchange a secret key over a public, untrusted quantum channel. Compared to classical key exchange, QKD achieves everlasting security: after the protocol execution the key is secure against adversaries that can do unbounded computations. On the flip side, while classical key exchange can be achieved non-interactively (with two simultaneous messages between Alice and Bob), no non-interactive protocol is known that provides everlasting security, even using quantum information. In this work, we make progress on this problem. Our main technical contribution is a computational variant of the celebrated monogamy of entanglement game, where the secret is only computationally hidden from the players, rather than information-theoretically. In these settings, we prove a negligible bound on the maximal winning probability over all strategies. As a direct application, we obtain a non-interactive (simultaneous message) QKD protocol from any post-quantum classical non-interactive key exchange, which satisfies everlastingly secure assuming Alice and Bob agree on the same key. The protocol only uses EPR pairs and standard and Hadamard basis measurements, making it suitable for near-term quantum hardware. We also propose how to convert this protocol into a two-round protocol that satisfies the standard notion of everlasting security. Finally, we prove a no-go theorem which establishes that (in contrast to the case of ordinary multi-round QKD) entanglement is necessary for non-interactive QKD, i.e., the messages sent by Alice and Bob cannot both be unentangled with their respective quantum memories if the protocol is to be everlastingly secure.

quant-ph↗

arXiv2025

Atomic diffraction by patterned holes in hexagonal boron nitride: a comparison between semi-classical and quantum computational models

The diffraction of atoms and molecules through tiny, sub-nanometre holes in atomically thin membranes is a promising approach for advancing atom interferometry sensing and atomic holography. However, dispersion interactions, such as the Casimir-Polder force, pose a significant challenge by attracting diffracting particles to the membrane, limiting the minimum hole size. This paper presents a numerical simulation of helium matter-wave diffraction through sub-nanometre holes in hexagonal boron nitride by solving the time-dependent Schr\"odinger equation. Our results show that the transmission rates in the quantum approach are significantly higher than those predicted by the commonly used semi-classical approach. This suggests that significantly smaller holes can be used in the design of diffractive masks, provided that fabrication techniques can meet the atomic-level precision to realise such holes. Furthermore, we observe notable differences in diffraction patterns, even for atom velocities that are much greater than the expected convergence threshold between semi-classical and quantum computational models.

quant-ph↗

arXiv2025

Traq: Estimating the Quantum Cost of Classical Programs

Predicting practical speedups offered by future quantum computers has become a major focus of the quantum community. Typically, such predictions involve numerical simulations supported by lengthy manual analyses and are carried out for one specific algorithm at a time. In this work, we present Traq, a principled approach towards estimating the quantum speedup of classical programs fully automatically. It consists of a classical language that includes high-level primitives amenable to quantum speedups, a compilation to low-level quantum programs, and a source-level cost analysis with provable guarantees. Our cost analysis upper bounds the complexity of the resulting quantum program and is sensitive to the input data of the program (in addition to providing worst-case costs). Traq is implemented as a Haskell package with an extensive evaluation.

quant-ph↗