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In semiconductor hole spin qubits, low magnetic field ($B$) operation extends the coherence time ($T_\mathrm{2}^*$) but proportionally reduces the gate speed. In contrast, singlet-triplet (ST) qubits are primarily controlled by the exchange interaction ($J$) and can thus maintain high gate speeds even at low $B$. However, a large $J$ introduces a significant charge component to the qubit, rendering ST qubits more vulnerable to charge noise when driven. Here, we demonstrate a highly coherent ST hole spin qubit in germanium, operating at both low $B$ and low $J$. By modulating $J$, we achieve resonant driving of the ST qubit, obtaining an average gate fidelity of $99.68\%$ and a coherence time of $T_\mathrm{2}^*=1.9\,\mu$s. Moreover, by applying the resonant drive continuously, we realize a dressed ST qubit with a tenfold increase in coherence time ($T_\mathrm{2\rho}^*=20.3\,\mu$s). Frequency modulation of the driving signal enables universal control, with an average gate fidelity of $99.64\%$. Our results demonstrate the potential for extending coherence times while preserving high-fidelity control of germanium-based ST qubits, paving the way for more efficient operations in semiconductor-based quantum processors.

Semiconductor spin qubits have emerged as a promising platform for quantum computing, following a significant improvement in their control fidelities over recent years. Increasing the qubit count remains challenging, beginning with the fabrication of small features and complex fanouts. A particular challenge has been formed by the need for individual barrier gates to control the exchange interaction between adjacent spin qubits. Here, we propose a method to vary two-qubit interactions without applying pulses on individual barrier gates while also remaining insensitive to detuning noise in first order. By changing plunger gate voltages over 300 mV we tune the exchange energy $J$ from 100 kHz to 60 MHz. This allows us to perform two-qubit operations without changing the barrier gate voltage. Based on these findings we conceptualize a spin qubit architecture without individual barrier gates, simplifying the fabrication while maintaining the control necessary for universal quantum computation., Comment: 13 pages, 7 figures

We study the identifiability of nonlinear network systems with partial excitation and partial measurement when the network dynamics is linear on the edges and nonlinear on the nodes. We assume that the graph topology and the nonlinear functions at the node level are known, and we aim to identify the weight matrix of the graph. Our main result is to prove that fully-connected layered feed-forward networks are generically locally identifiable by exciting sources and measuring sinks in the class of analytic functions that cross the origin. This holds even when all other nodes remain unexcited and unmeasured and stands in sharp contrast to most findings on network identifiability requiring measurement and/or excitation of each node. The result applies in particular to feed-forward artificial neural networks with no offsets and generalizes previous literature by considering a broader class of functions and topologies., Comment: 9 pages, 2 figures

Disorder in the heterogeneous material stack of semiconductor spin qubit systems introduces noise that compromises quantum information processing, posing a challenge to coherently control large-scale quantum devices. Here, we exploit low-disorder epitaxial strained quantum wells in Ge/SiGe heterostructures grown on Ge wafers to comprehensively probe the noise properties of complex micron-scale devices comprising of up to ten quantum dots and four rf-charge sensors arranged in a two-dimensional array. We demonstrate an average charge noise of $\sqrt{S_{0}}=0.3(1)$ $\mu\mathrm{eV}/\sqrt{\mathrm{Hz}}$ at 1 Hz across different locations on the wafer, providing a benchmark for quantum confined holes. We then establish hole-spin qubit control in these heterostructures and extend our investigation from electrical to magnetic noise through spin echo measurements. Exploiting dynamical decoupling sequences, we quantify the power spectral density components arising from the hyperfine interaction with $^{73}$Ge spinful isotopes and identify coherence modulations associated with the interaction with the $^{29}$Si nuclear spin bath near the Ge quantum well. We estimate an integrated hyperfine noise amplitude $\sigma_f$ of 180(8) kHz from $^{73}$Ge and of 47(5) kHz from $^{29}$Si, underscoring the need for full isotopic purification of the qubit host environment.

We analyze the identifiability of directed acyclic graphs in the case of partial excitation and measurement. We consider an additive model where the nonlinear functions located in the edges depend only on a past input, and we analyze the identifiability problem in the class of pure nonlinear functions satisfying $f(0)=0$. We show that any identification pattern (set of measured nodes and set of excited nodes) requires the excitation of sources, measurement of sinks and the excitation or measurement of the other nodes. Then, we show that a directed acyclic graph (DAG) is identifiable with a given identification pattern if and only if it is identifiable with the measurement of all the nodes. Next, we analyze the case of trees where we prove that any identification pattern guarantees the identifiability of the network. Finally, by introducing the notion of a generic nonlinear network matrix, we provide sufficient conditions for the identifiability of DAGs based on the notion of vertex-disjoint paths., Comment: 7 pages, 6 figures, to appear in IEEE Conference on Decision and Control (CDC 2024)

We propose a novel approach to obtain interpolation constraints for a wide range of function classes, i.e. necessary and sufficient constraints that a set of points, functions values and (sub)gradients must satisfy to ensure the existence of a global function of the class considered, consistent with this set. The derivation of such constraints is crucial for instance in the performance analysis of optimization methods, since obtaining a priori tight performance guarantees requires using a tight description of function classes of interest. Our method allows setting aside all analytic properties of the function class to work only at an algebraic level, and to easily obtain counterexamples when a condition characterizing a function class cannot serve as an interpolation constraint. As an illustration, we provide interpolation constraints for a class of non convex non smooth functions: weakly convex functions with bounded subgradients, and rely on these new interpolation constraints to outperform state of the art bounds on the performance of the subgradient method on this class.

We derive conditions for the identifiability of nonlinear networks characterized by additive dynamics at the level of the edges when all the nodes are excited. In contrast to linear systems, we show that the measurement of all sinks is necessary and sufficient for the identifiability of directed acyclic graphs, under the assumption that dynamics are described by analytic functions without constant terms (i.e., $f(0)=0$). But if constant terms are present, then the identifiability is impossible as soon as one node has more than one in-neighbor. In the case of general digraphs where cycles can exist, we consider additively separable functions for the analysis of the identifiability, and we show that the measurement of one node of all the sinks of the condensation digraph is necessary and sufficient. Several examples are added to illustrate the results., Comment: 12 pages, 5 figures, submitted to IEEE Transactions on Automatic Control

We show that, in many settings, the worst-case performance of a distributed optimization algorithm is independent of the number of agents in the system, and can thus be computed in the fundamental case with just two agents. This result relies on a novel approach that systematically exploits symmetries in worst-case performance computation, framed as Semidefinite Programming (SDP) via the Performance Estimation Problem (PEP) framework. Harnessing agent symmetries in the PEP yields compact problems whose size is independent of the number of agents in the system. When all agents are equivalent in the problem, we establish the explicit conditions under which the resulting worst-case performance is independent of the number of agents and is therefore equivalent to the basic case with two agents. Our compact PEP formulation also allows the consideration of multiple equivalence classes of agents, and its size only depends on the number of equivalence classes. This enables practical and automated performance analysis of distributed algorithms in numerous complex and realistic settings, such as the analysis of the worst agent performance. We leverage this new tool to analyze the performance of the EXTRA algorithm in advanced settings and its scalability with the number of agents, providing a tighter analysis and deeper understanding of the algorithm performance., Comment: submitted to Open Journal of Mathematical Optimization (OJMO)

Selecting the fastest algorithm for a specific signal/image processing task is a challenging question. We propose an approach based on the Performance Estimation Problem framework that numerically and automatically computes the worst-case performance of a given optimization method on a class of functions. We first propose a computer-assisted analysis and comparison of several first-order primal optimization methods, namely, the gradient method, the forward-backward, Peaceman-Rachford, and Douglas-Rachford splittings. We tighten the existing convergence results of these algorithms and extend them to new classes of functions. Our analysis is then extended and evaluated in the context of the primal-dual Chambolle-Pock and Condat-V\~u methods., Comment: 5 pages, 7 figures

Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results cover several types of assumptions on the summands, such as smoothness or strong convexity. Our main tool is the use of necessary and sufficient conditions for interpolating the considered function classes, which leads to shorter and more direct proofs in comparison with previous work. We also address the setting where each summand minimizer is assumed to lie in a unit ball, and prove a tight bound on the norm of any minimizer of the sum.

Qubits that can be efficiently controlled are essential for the development of scalable quantum hardware. While resonant control is used to execute high-fidelity quantum gates, the scalability is challenged by the integration of high-frequency oscillating signals, qubit crosstalk and heating. Here, we show that by engineering the hopping of spins between quantum dots with site-dependent spin quantization axis, quantum control can be established with discrete signals. We demonstrate hopping-based quantum logic and obtain single-qubit gate fidelities of 99.97\%, coherent shuttling fidelities of 99.992\% per hop, and a two-qubit gate fidelity of 99.3\%, corresponding to error rates that have been predicted to allow for quantum error correction. We also show that hopping spins constitute a tuning method by statistically mapping the coherence of a 10-quantum dot system. Our results show that dense quantum dot arrays with sparse occupation could be developed for efficient and high-connectivity qubit registers., Comment: main text with 18 pages and 3 figures, supplementary materials with 64 pages and 26 figures, in a single file

We establish that in distributed optimization, the prevalent strategy of minimizing the second-largest eigenvalue modulus (SLEM) of the averaging matrix for selecting communication weights, while optimal for existing theoretical performance bounds, is generally not optimal regarding the exact worst-case performance of the algorithms. This exact performance can be computed using the Performance Estimation Problem (PEP) approach. We thus rely on PEP to formulate an optimization problem that determines the optimal communication weights for a distributed optimization algorithm deployed on a specified undirected graph. Our results show that the optimal weights can outperform the weights minimizing the second-largest eigenvalue modulus (SLEM) of the averaging matrix. This suggests that the SLEM is not the best characterization of weighted network performance for decentralized optimization. Additionally, we explore and compare alternative heuristics for weight selection in distributed optimization., Comment: 7 pages. Submitted to MTNS

We consider the generalization error associated with stochastic gradient descent on a smooth convex function over a compact set. We show the first bound on the generalization error that vanishes when the number of iterations $T$ and the dataset size $n$ go to zero at arbitrary rates; our bound scales as $\tilde{O}(1/\sqrt{T} + 1/\sqrt{n})$ with step-size $\alpha_t = 1/\sqrt{t}$. In particular, strong convexity is not needed for stochastic gradient descent to generalize well.

Hole spins in Ge/SiGe heterostructure quantum dots have emerged as promising qubits for quantum computation. The strong spin-orbit coupling (SOC), characteristic of heavy-hole states in Ge, enables fast and all-electrical qubit control. However, SOC also increases the susceptibility of spin qubits to charge noise. While qubit coherence can be significantly improved by operating at sweet spots with reduced hyperfine or charge noise sensitivity, the latter ultimately limits coherence, underlining the importance of understanding and reducing charge noise at its source. In this work, we study the voltage-induced hysteresis commonly observed in SiGe-based quantum devices and show that the dominant charge fluctuators are localized at the semiconductor-oxide interface. By applying increasingly negative gate voltages to Hall bar and quantum dot devices, we investigate how the hysteretic filling of interface traps impacts transport metrics and charge noise. We find that the gate-induced accumulation and trapping of charge at the SiGe-oxide interface leads to an increased electrostatic disorder, as probed by transport measurements, as well as the activation of low-frequency relaxation dynamics, resulting in slow drifts and increased charge noise levels. Our results highlight the importance of a conservative device tuning strategy and reveal the critical role of the semiconductor-oxide interface in SiGe heterostructures for spin qubit applications.

This work focuses on the generic identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by transfer functions according to a known topology, some nodes are excited, some are measured, and only a part of the transfer functions are known. Our goal is to determine whether the unknown transfer functions can be generically recovered based on the input-output data collected from the excited and measured nodes. We introduce the notion of separable networks, for which global and so-called local identifiability are equivalent. A novel approach yields a necessary and sufficient combinatorial characterization for local identifiability for such graphs, in terms of existence of paths and conditions on their parity. Furthermore, this yields a necessary condition not only for separable networks, but for networks of any topology., Comment: 8 pages, 1 figure, article to appear in IEEE Conference on Decision and Control 2023

The Performance Estimation Problem (PEP) approach consists in computing worst-case performance bounds on optimization algorithms by solving an optimization problem: one maximizes an error criterion over all initial conditions allowed and all functions in a given class of interest. The maximal value is then a worst-case bound, and the maximizer provides an example reaching that worst case. This approach was introduced for optimization algorithms but could in principle be applied to many other contexts involving worst-case bounds. The key challenge is the representation of infinite-dimensional objects involved in these optimization problems such as functions, and complex or non-convex objects as linear operators and their powers, networks in decentralized optimization etc. This challenge can be resolved by interpolation constraints, which allow representing the effect of these objects on vectors of interest, rather than the whole object, leading to tractable finite dimensional problems. We review several recent interpolation results and their implications in obtaining of worst-case bounds via PEP., Comment: 8 pages, 4 figures, Tutorial session at IEEE CDC 2023

In gate-based dispersive sensing, the response of a resonator attached to a quantum dot gate is detected by a reflected radio-frequency signal. This enables fast readout of spin qubits and tune up of arrays of quantum dots, but comes at the expense of increased susceptibility to crosstalk, as the resonator can amplify spurious signals and induce fluctuations in the quantum dot potential. We attach tank circuits with superconducting NbN inductors and internal quality factors $Q_{\mathrm{i}}$>1000 to the interdot barrier gate of silicon double quantum dot devices. Measuring the interdot transition in transport, we quantify radio-frequency crosstalk that results in a ring-up of the resonator when neighbouring plunger gates are driven with frequency components matching the resonator frequency. This effect complicates qubit operation and scales with the loaded quality factor of the resonator, the mutual capacitance between device gate electrodes, and with the inverse of the parasitic capacitance to ground. Setting qubit frequencies below the resonator frequency is expected to substantially suppress this type of crosstalk., Comment: 7 pages, 4 figures, supplementary information

We analyze the problem of network identifiability with nonlinear functions associated with the edges. We consider a static model for the output of each node and by assuming a perfect identification of the function associated with the measurement of a node, we provide conditions for the identifiability of the edges in a specific class of functions. First, we analyze the identifiability conditions in the class of all nonlinear functions and show that even for a path graph, it is necessary to measure all the nodes except by the source. Then, we consider analytic functions satisfying $f(0)=0$ and we provide conditions for the identifiability of paths and trees. Finally, by restricting the problem to a smaller class of functions where none of the functions is linear, we derive conditions for the identifiability of directed acyclic graphs. Some examples are presented to illustrate the results., Comment: 6 pages, 3 figures, to appear in IEEE Conference on Decision and Control (CDC 2023)

Electrically driven spin resonance is a powerful technique for controlling semiconductor spin qubits. However, it faces challenges in qubit addressability and off-resonance driving in larger systems. We demonstrate coherent bichromatic Rabi control of quantum dot hole spin qubits, offering a spatially selective approach for large qubit arrays. By applying simultaneous microwave bursts to different gate electrodes, we observe multichromatic resonance lines and resonance anticrossings that are caused by the ac Stark shift. Our theoretical framework aligns with experimental data, highlighting interdot motion as the dominant mechanism for bichromatic driving., Comment: 5 pages, 4 figures

We analyze the absolute spectral gap of Markov chains on graphs obtained from a cycle of $n$ vertices and perturbed only at approximately $n^{1/\rho}$ random locations with an appropriate, possibly sparse, interconnection structure. Together with a strong asymmetry along the cycle, the gap of the resulting chain can be bounded inversely proportionally by the longest arc length (up to logarithmic factors) with high probability, providing a significant mixing speedup compared to the reversible version., Comment: 17 pages, 3 figures

In this work we present a comprehensive theory of spin physics in planar Ge hole quantum dots in an in-plane magnetic field, where the orbital terms play a dominant role in qubit physics, and provide a brief comparison with experimental measurements of the angular dependence of electrically driven spin resonance. We focus the theoretical analysis on electrical spin operation, phonon-induced relaxation, and the existence of coherence sweet spots. We find that the choice of magnetic field orientation makes a substantial difference for the properties of hole spin qubits. Furthermore, although the Schrieffer-Wolff approximation can describe electron dipole spin resonance (EDSR), it does not capture the fundamental spin dynamics underlying qubit coherence. Specifically, we find that: (i) EDSR for in-plane magnetic fields varies non-linearly with the field strength and weaker than for perpendicular magnetic fields; (ii) The EDSR Rabi frequency is maximized when the a.c. electric field is aligned parallel to the magnetic field, and vanishes when the two are perpendicular; (iii) The Rabi ratio $T_1/T_\pi$, i.e. the number of EDSR gate operation per unit relaxation time, is expected to be as large as $5{\times}10^5$ at the magnetic fields used experimentally; (iv) The orbital magnetic field terms make the in-plane $g$-factor strongly anisotropic in a squeezed dot, in excellent agreement with experimental measurements; (v) The coherence sweet spots do not exist in an in-plane magnetic field, as the orbital magnetic field terms expose the qubit to all components of the defect electric field. These findings will provide a guideline for experiments to design ultrafast, highly coherent hole spin qubits in Ge.

The Performance Estimation Problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving linear operators. This extension requires an explicit formulation of interpolation conditions for those linear operators. We consider the class of linear operators $\mathcal{M}:x \mapsto Mx$ where matrix $M$ has bounded singular values, and the class of linear operators where $M$ is symmetric and has bounded eigenvalues. We describe interpolation conditions for these classes, i.e. necessary and sufficient conditions that, given a list of pairs $\{(x_i,y_i)\}$, characterize the existence of a linear operator mapping $x_i$ to $y_i$ for all $i$. Using these conditions, we first identify the exact worst-case behavior of the gradient method applied to the composed objective $h\circ \mathcal{M}$, and observe that it always corresponds to $\mathcal{M}$ being a scaling operator. We then investigate the Chambolle-Pock method applied to $f+g\circ \mathcal{M}$, and improve the existing analysis to obtain a proof of the exact convergence rate of the primal-dual gap. In addition, we study how this method behaves on Lipschitz convex functions, and obtain a numerical convergence rate for the primal accuracy of the last iterate. We also show numerically that averaging iterates is beneficial in this setting., Comment: Proof of the main interpolation theorem is streamlined. Added results about the Chambolle-Pock algorithm including the exact convergence rate of the primal-dual gap in a standard setting

Mechanistic and tractable mathematical models play a key role in understanding how social influence shapes public opinions. Recently, a weighted-median mechanism has been proposed as a new micro-foundation of opinion dynamics and validated via experimental data. Numerical studies also indicate that this new mechanism recreates some non-trivial real-world features of opinion evolution. In this paper, we conduct a thorough theoretical analysis of the weighted-median opinion dynamics. We fully characterize the set of all equilibria, and we establish the almost-sure finite-time convergence for any initial condition. Moreover, we prove a necessary and sufficient graph-theoretic condition for the almost-sure convergence to consensus, as well as a sufficient graph-theoretic condition for almost-sure persistent dissensus. It turns out that the weighted-median opinion dynamics, despite its simplicity in form, exhibit rich dynamical behavior that depends on some delicate network structures. To complement our sufficient conditions for almost-sure dissensus, we further prove that, given the influence network, determining whether the system almost surely achieves persistent dissensus is NP-hard, which reflects the complexity the network topology contributes to opinion evolution.