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The $\Sigma$-QMAC problem is introduced, involving $S$ servers, $K$ classical ($\mathbb{F}_d$) data streams, and $T$ independent quantum systems. Data stream ${\sf W}_k, k\in[K]$ is replicated at a subset of servers $\mathcal{W}(k)\subset[S]$, and quantum system $\mathcal{Q}_t, t\in[T]$ is distributed among a subset of servers $\mathcal{E}(t)\subset[S]$ such that Server $s\in\mathcal{E}(t)$ receives subsystem $\mathcal{Q}_{t,s}$ of $\mathcal{Q}_t=(\mathcal{Q}_{t,s})_{s\in\mathcal{E}(t)}$. Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is $\sum_{t\in[T]}\sum_{s\in\mathcal{E}(t)}\log_d|\mathcal{Q}_{t,s}|$ qudits, where $|\mathcal{Q}|$ is the dimension of $\mathcal{Q}$. The states and measurements of $(\mathcal{Q}_t)_{t\in[T]}$ are required to be separable across $t\in[T]$ throughout, but for each $t\in[T]$, the subsystems of $\mathcal{Q}_{t}$ can be prepared initially in an arbitrary (independent of data) entangled state, manipulated arbitrarily by the respective servers, and measured jointly by the receiver. From the measurements, the receiver must recover the sum of all data streams. Rate is defined as the number of dits ($\mathbb{F}_d$ symbols) of the desired sum computed per qudit of download. The capacity of $\Sigma$-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data and entanglement distributions $\mathcal{W}, \mathcal{E}$. Coding based on the $N$-sum box abstraction is optimal in every case.

Qubit Mapping is an essential step in realizing quantum circuits on actual hardware devices. However, due to the high complexity of this problem, current solutions can only work on circuits in fairly small scales (i.e. $<50$ qubits). In this paper, we propose a qubit mapping methodology which, to the best of our knowledge, is the first framework to handle very large quantum circuits (i.e. thousands of qubits) towards the quantum advantage. Our novel routing algorithm, Duostra, can efficiently identify the optimal routing path for a given two-qubit gate to operate on physical qubits through swap-gate insertions, and our scheduling heuristic offers the flexibility to strike the balance in optimizing the performance and pursuing the scalability. Experimental results show that our method runs $10$ times faster than the state-of-the-art approaches, while on average can still outperform them by over $5\%$ in terms of the execution time of the quantum circuits. More specifically, our proposed algorithm can complete the qubit mapping of an $11,969$-qubit Quantum Fourier Transform circuit within five hours.

We consider \emph{Gibbs distributions}, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The \emph{partition function} is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions.

Subset selection tasks, arise in recommendation systems and search engines and ask to select a subset of items that maximize the value for the user. The values of subsets often display diminishing returns, and hence, submodular functions have been used to model them. If the inputs defining the submodular function are known, then existing algorithms can be used. In many applications, however, inputs have been observed to have social biases that reduce the utility of the output subset. Hence, interventions to improve the utility are desired. Prior works focus on maximizing linear functions -- a special case of submodular functions -- and show that fairness constraint-based interventions can not only ensure proportional representation but also achieve near-optimal utility in the presence of biases. We study the maximization of a family of submodular functions that capture functions arising in the aforementioned applications. Our first result is that, unlike linear functions, constraint-based interventions cannot guarantee any constant fraction of the optimal utility for this family of submodular functions. Our second result is an algorithm for submodular maximization. The algorithm provably outputs subsets that have near-optimal utility for this family under mild assumptions and that proportionally represent items from each group. In empirical evaluation, with both synthetic and real-world data, we observe that this algorithm improves the utility of the output subset for this family of submodular functions over baselines.

A minimum chain cover (MCC) of a $k$-width directed acyclic graph (DAG) $G = (V, E)$ is a set of $k$ chains (paths in the transitive closure) of $G$ such that every vertex appears in at least one chain in the cover. The state-of-the-art solutions for MCC run in time $\tilde{O}(k(|V|+|E|))$ [M\"akinen et at., TALG], $O(T_{MF}(|E|) + k|V|)$, $O(k^2|V| + |E|)$ [C\'aceres et al., SODA 2022], $\tilde{O}(|V|^{3/2} + |E|)$ [Kogan and Parter, ICALP 2022] and $\tilde{O}(T_{MCF}(|E|) + \sqrt{k}|V|)$ [Kogan and Parter, SODA 2023], where $T_{MF}(|E|)$ and $T_{MCF}(|E|)$ are the running times for solving maximum flow (MF) and minimum-cost flow (MCF), respectively. In this work we present an algorithm running in time $O(T_{MF}(|E|) + (|V|+|E|)\log{k})$. By considering the recent result for solving MF [Chen et al., FOCS 2022] our algorithm is the first running in almost linear time. Moreover, our techniques are deterministic and derive a deterministic near-linear time algorithm for MCC if the same is provided for MF. At the core of our solution we use a modified version of the mergeable dictionaries [Farach and Thorup, Algorithmica], [Iacono and \"Ozkan, ICALP 2010] data structure boosted with the SIZE-SPLIT operation and answering queries in amortized logarithmic time, which can be of independent interest.

We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on $n$ points. We first consider the $o(n)$-space regime and show that, when the input is a stream of all $\binom{n}{2}$ entries of the metric, for any $\alpha \ge 2$, both MST and TSP cost can be $\alpha$-approximated using $\tilde{O}(n/\alpha)$ space, and that $\Omega(n/\alpha^2)$ space is necessary for this task. Moreover, we show that even if the streaming algorithm is allowed $p$ passes over a metric stream, it still requires $\tilde{\Omega}(\sqrt{n/\alpha p^2})$ space. We next consider the semi-streaming regime, where computing even the exact MST cost is easy and the main challenge is to estimate TSP cost to within a factor that is strictly better than $2$. We show that, if the input is a stream of all edges of the weighted graph that induces the underlying metric, for any $\varepsilon > 0$, any one-pass $(2-\varepsilon)$-approximation of TSP cost requires $\Omega(\varepsilon^2 n^2)$ space; on the other hand, there is an $\tilde{O}(n)$ space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than $2$, when the algorithm is given access to a matrix that specifies pairwise distances between all points. For MST estimation in this model, it is known that a $(1+\varepsilon)$-approximation is achievable with $\tilde{O}(n/\varepsilon^{O(1)})$ queries. We design an algorithm that performs $\tilde{O}(n^{1.5})$ distance queries and achieves a strictly better than $2$-approximation when either the metric is known to contain a spanning tree supported on weight-$1$ edges or the algorithm is given access to a minimum spanning tree of the graph.

The continuous time stochastic process is a mainstream mathematical instrument modeling the random world with a wide range of applications involving finance, statistics, physics, and time series analysis, while the simulation and analysis of the continuous time stochastic process is a challenging problem for classical computers. In this work, a general framework is established to prepare the path of a continuous time stochastic process in a quantum computer efficiently. The storage and computation resource is exponentially reduced on the key parameter of holding time, as the qubit number and the circuit depth are both optimized via our compressed state preparation method. The desired information, including the path-dependent and history-sensitive information that is essential for financial problems, can be extracted efficiently from the compressed sampling path, and admits a further quadratic speed-up. Moreover, this extraction method is more sensitive to those discontinuous jumps capturing extreme market events. Two applications of option pricing in Merton jump diffusion model and ruin probability computing in the collective risk model are given.

Federated learning (FL) has received high interest from researchers and practitioners to train machine learning (ML) models for healthcare. Ensuring the trustworthiness of these models is essential. Especially bias, defined as a disparity in the model's predictive performance across different subgroups, may cause unfairness against specific subgroups, which is an undesired phenomenon for trustworthy ML models. In this research, we address the question to which extent bias occurs in medical FL and how to prevent excessive bias through reward systems. We first evaluate how to measure the contributions of institutions toward predictive performance and bias in cross-silo medical FL with a Shapley value approximation method. In a second step, we design different reward systems incentivizing contributions toward high predictive performance or low bias. We then propose a combined reward system that incentivizes contributions toward both. We evaluate our work using multiple medical chest X-ray datasets focusing on patient subgroups defined by patient sex and age. Our results show that we can successfully measure contributions toward bias, and an integrated reward system successfully incentivizes contributions toward a well-performing model with low bias. While the partitioning of scans only slightly influences the overall bias, institutions with data predominantly from one subgroup introduce a favorable bias for this subgroup. Our results indicate that reward systems, which focus on predictive performance only, can transfer model bias against patients to an institutional level. Our work helps researchers and practitioners design reward systems for FL with well-aligned incentives for trustworthy ML.

Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.

Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.