We give new dequantization and hardness results for estimating spectral sums of matrices, such as the log-determinant. Recent quantum algorithms have demonstrated that the logarithm of the determinant of sparse, well-conditioned, positive matrices can be approximated to $\varepsilon$-relative accuracy in time polylogarithmic in the dimension $N$, specifically in time $\mathrm{poly}(\mathrm{log}(N), s, \kappa, 1/\varepsilon)$, where $s$ is the sparsity and $\kappa$ the condition number of the input matrix. We provide a simple dequantization of these techniques that preserves the polylogarithmic dependence on the dimension. Our classical algorithm runs in time $\mathrm{polylog}(N)\cdot s^{O(\sqrt{\kappa}\log \kappa/\varepsilon)}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes. We complement our classical upper bound with $\mathsf{DQC1}$-completeness results for estimating specific spectral sums such as the trace of the inverse and the trace of matrix powers for log-local Hamiltonians, with parameter scalings analogous to those of known quantum algorithms. Assuming $\mathsf{BPP}\subsetneq\mathsf{DQC1}$, this rules out classical algorithms with the same scalings. It also resolves a main open problem of Cade and Montanaro (TQC 2018) concerning the complexity of Schatten-$p$ norm estimation. We further analyze a block-encoding input model, where instead of a classical description of a sparse matrix, we are given a block-encoding of it. We show $\mathsf{DQC}1$-completeness in a very general way in this model for estimating $\mathrm{tr}[f(A)]$ whenever $f$ and $f^{-1}$ are sufficiently smooth. We conclude our work with $\mathsf{BQP}$-hardness and $\mathsf{PP}$-completeness results for high-accuracy log-determinant estimation.

Spectral gaps play a fundamental role in many areas of mathematics, computer science, and physics. In quantum mechanics, the spectral gap of Schrödinger 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 Schrödinger 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.

Quantum machine learning has the potential for a transformative impact across industry sectors and in particular in finance. In our work we look at the problem of hedging where deep reinforcement learning offers a powerful framework for real markets. We develop quantum reinforcement learning methods based on policy-search and distributional actor-critic algorithms that use quantum neural network architectures with orthogonal and compound layers for the policy and value functions. We prove that the quantum neural networks we use are trainable, and we perform extensive simulations that show that quantum models can reduce the number of trainable parameters while achieving comparable performance and that the distributional approach obtains better performance than other standard approaches, both classical and quantum. We successfully implement the proposed models on a trapped-ion quantum processor, utilizing circuits with up to $16$ qubits, and observe performance that agrees well with noiseless simulation. Our quantum techniques are general and can be applied to other reinforcement learning problems beyond hedging.

Quantum machine learning techniques have been proposed as a way to potentially enhance performance in machine learning applications. In this paper, we introduce two new quantum methods for neural networks. The first one is a quantum orthogonal neural network, which is based on a quantum pyramidal circuit as the building block for implementing orthogonal matrix multiplication. We provide an efficient way for training such orthogonal neural networks; novel algorithms are detailed for both classical and quantum hardware, where both are proven to scale asymptotically better than previously known training algorithms. The second method is quantum-assisted neural networks, where a quantum computer is used to perform inner product estimation for inference and training of classical neural networks. We then present extensive experiments applied to medical image classification tasks using current state of the art quantum hardware, where we compare different quantum methods with classical ones, on both real quantum hardware and simulators. Our results show that quantum and classical neural networks generates similar level of accuracy, supporting the promise that quantum methods can be useful in solving visual tasks, given the advent of better quantum hardware.

We introduce fast algorithms for solving $\ell_{p}$ regression problems using the iteratively reweighted least squares (IRLS) method. Our approach achieves state-of-the-art iteration complexity, outperforming the IRLS algorithm by Adil-Peng-Sachdeva (NeurIPS 2019) and matching the theoretical bounds established by the complex algorithm of Adil-Kyng-Peng-Sachdeva (SODA 2019, J. ACM 2024) via a simpler lightweight iterative scheme. This bridges the existing gap between theoretical and practical algorithms for $\ell_{p}$ regression. Our algorithms depart from prior approaches, using a primal-dual framework, in which the update rule can be naturally derived from an invariant maintained for the dual objective. Empirically, we show that our algorithms significantly outperform both the IRLS algorithm by Adil-Peng-Sachdeva and MATLAB/CVX implementations.

Elixir is a dynamically-typed functional language running on the Erlang Virtual Machine, designed for building scalable and maintainable applications. Its characteristics have earned it a surging adoption by hundreds of industrial actors and tens of thousands of developers. Static typing seems nowadays to be the most important request coming from the Elixir community. We present a gradual type system we plan to include in the Elixir compiler, outline its characteristics and design principles, and show by some short examples how to use it in practice. Developing a static type system suitable for Erlang's family of languages has been an open research problem for almost two decades. Our system transposes to this family of languages a polymorphic type system with set-theoretic types and semantic subtyping. To do that, we had to improve and extend both semantic subtyping and the typing techniques thereof, to account for several characteristics of these languages -- and of Elixir in particular -- such as the arity of functions, the use of guards, a uniform treatment of records and dictionaries, the need for a new sound gradual typing discipline that does not rely on the insertion at compile time of specific run-time type-tests but, rather, takes into account both the type tests performed by the virtual machine and those explicitly added by the programmer. The system presented here is "gradually" being implemented and integrated in Elixir, but a prototype implementation is already available. The aim of this work is to serve as a longstanding reference that will be used to introduce types to Elixir programmers, as well as to hint at some future directions and possible evolutions of the Elixir language.

We describe a quantum algorithm based on an interior point method for solving a linear program with $n$ inequality constraints on $d$ variables. The algorithm explicitly returns a feasible solution that is $\varepsilon$-close to optimal, and runs in time $\sqrt{n} \cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$ which is sublinear for tall linear programs (i.e., $n \gg d$). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [FOCS~'14]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the \emph{spectral approximation} of $A^T A$ for a tall matrix $A \in \mathbb R^{n \times d}$. The algorithm uses leverage score sampling in combination with Grover search, and returns a $\delta$-approximation by making $O(\sqrt{nd}/\delta)$ row queries to $A$. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf~[FOCS~'20]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi and Jerbi [STOC '22]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by pre-conditioning our random variable using our quantum algorithm for spectral approximation.

A foundational theory of compositional categorical rewriting theory is presented, based on a collection of fibration-like properties that collectively induce and intrinsically structure the large collection of lemmata used in the proofs of theorems such as concurrency and associativity. The resulting highly generic proofs of these theorems are given. It is noteworthy that the proof of the concurrency theorem takes only a few lines and, while that of associativity remains somewhat longer, it would be unreadably long if written directly in terms of the basic lemmata. In essence, our framework improves the readability and ease of comprehension of these proofs by exposing latent modularity. A curated list of known instances of our framework is used to conclude the paper with a detailed discussion of the conditions under which the Double Pushout and Sesqui-Pushout semantics of graph transformation are compositional.

For a graph $G$, the parameter treedepth measures the minimum depth among all forests $F$, called elimination forests, such that $G$ is a subgraph of the ancestor-descendant closure of $F$. We introduce a logic, called neighborhood operator logic with acyclicity, connectivity and clique constraints ($\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{ACK}$ for short), that captures all NP-hard problems$\unicode{x2013}$like Independent Set or Hamiltonian Cycle$\unicode{x2013}$that are known to be tractable in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ and space $n^{\mathcal{O}(1)}$ on $n$-vertex graphs provided with elimination forests of depth $k$. We provide a model checking algorithm for $\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{ACK}$ with such complexity that unifies and extends these results. For $\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{k}$, the fragment of the above logic that does not use acyclicity and connectivity constraints, we get a strengthening of this result, where the space complexity is reduced to $\mathcal{O}(k\log(n))$. With a similar mechanism as the distance neighborhood logic introduced in [Bergougnoux, Dreier and Jaffke, SODA 2023], the logic $\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{ACK}$ is an extension of the fully-existential $\mathsf{MSO}_2$ with predicates for (1) querying generalizations of the neighborhoods of vertex sets, (2) verifying the connectivity and acyclicity of vertex and edge sets, and (3) verifying that a vertex set induces a clique. Our results provide $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ time and $n^{\mathcal{O}(1)}$ space algorithms for problems for which the existence of such algorithms was previously unknown. In particular, $\mathsf{NEO}_2[\mathsf{FRec}]$ captures CNF-SAT via the incidence graphs associated to CNF formulas, and it also captures several modulo counting problems like Odd Dominating Set.

We investigate the decidability of the monadic second-order (MSO) theory of the structure \langle \mathbb{N};<,P_1, \ldots,P_d \rangle, for various unary predicates $P_1,\ldots,P_d \subseteq \mathbb{N}$. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers $k^{\mathbf{N}} = \{k^n : n \in \mathbb{N}\}$, $k$-th powers $\mathbf{N}^k = \{n^k : n \in \mathbb{N}\}$, and the set of terms of the Fibonacci sequence $\mathsf{Fib} = \{0,1,2,3,5,8,13,\ldots\}$ (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following: $\bullet$ The MSO theory of \langle \mathbb{N};<, 2^{\mathbf{N}}, \mathsf{Fib} \rangle is decidable; $\bullet$ The MSO theory of \langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 6^{\mathbf{N}} \rangle is decidable; $\bullet$ The MSO theory of \langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 5^{\mathbf{N}} \rangle is decidable assuming Schanuel's conjecture; $\bullet$ The MSO theory of \langle \mathbb{N};<, 4^{\mathbf{N}}, \mathbf{N}^2 \rangle is decidable; $\bullet$ The MSO theory of \langle \mathbb{N};<, 2^{\mathbf{N}}, \mathbf{N}^2 \rangle is Turing-equivalent to the MSO theory of \langle \mathbb{N};<,S \rangle, where $S$ is the predicate corresponding to the binary expansion of $\sqrt{2}$. (As the binary expansion of $\sqrt{2}$ is widely believed to be normal, the corresponding MSO theory is in turn expected to be decidable.) These results are obtained by exploiting and combining techniques from dynamical systems, number theory, and automata theory.

The increasing computational requirements of deep neural networks (DNNs) have led to significant interest in obtaining DNN models that are sparse, yet accurate. Recent work has investigated the even harder case of sparse training, where the DNN weights are, for as much as possible, already sparse to reduce computational costs during training. Existing sparse training methods are often empirical and can have lower accuracy relative to the dense baseline. In this paper, we present a general approach called Alternating Compressed/DeCompressed (AC/DC) training of DNNs, demonstrate convergence for a variant of the algorithm, and show that AC/DC outperforms existing sparse training methods in accuracy at similar computational budgets; at high sparsity levels, AC/DC even outperforms existing methods that rely on accurate pre-trained dense models. An important property of AC/DC is that it allows co-training of dense and sparse models, yielding accurate sparse-dense model pairs at the end of the training process. This is useful in practice, where compressed variants may be desirable for deployment in resource-constrained settings without re-doing the entire training flow, and also provides us with insights into the accuracy gap between dense and compressed models. The code is available at: this https URL .

We study networks of processes that all execute the same finite state protocol and that communicate through broadcasts. The processes are organized in a graph (a topology) and only the neighbors of a process in this graph can receive its broadcasts. The coverability problem asks, given a protocol and a state of the protocol, whether there is a topology for the processes such that one of them (at least) reaches the given state. This problem is undecidable. We study here an under-approximation of the problem where processes alternate a bounded number of times $k$ between phases of broadcasting and phases of receiving messages. We show that, if the problem remains undecidable when $k$ is greater than 6, it becomes decidable for $k=2$, and EXPSPACE-complete for $k=1$. Furthermore, we show that if we restrict ourselves to line topologies, the problem is in $P$ for $k=1$ and $k=2$.

The focus of these lecture notes is on abstract models and basic ideas and results that relate to the operational semantics of programming languages largely conceived. The approach is to start with an abstract description of the computation steps of programs and then to build on top semantic equivalences, specification languages, and static analyses. While other approaches to the semantics of programming languages are possible, it appears that the operational one is particularly effective in that it requires a moderate level of mathematical sophistication and scales reasonably well to a large variety of programming features. In practice, operational semantics is a suitable framework to build portable language implementations and to specify and test program properties. It is also used routinely to tackle more ambitious tasks such as proving the correctness of a compiler or a static analyzer.

Timed languages contain sequences of discrete events ("letters'') separated by real-valued delays, they can be recognized by timed automata, and represent behaviors of various real-time systems. The notion of bandwidth of a timed language defined in a previous paper characterizes the amount of information per time unit, encoded in words of the language observed with some precision {\epsilon}. In this paper, we identify three classes of timed automata according to the asymptotics of the bandwidth of their languages with respect to this precision {\epsilon}: automata are either meager, with an O(1) bandwidth, normal, with a {\Theta}(log (1/{\epsilon})) bandwidth, or obese, with {\Theta}(1/{\epsilon}) bandwidth. We define two structural criteria and prove that they partition timed automata into these three classes of bandwidth, implying that there are no intermediate asymptotic classes. The classification problem of a timed automaton is PSPACE-complete. Both criteria are formulated using morphisms from paths of the timed automaton to some finite monoids extending Puri's orbit graphs; the proofs are based on Simon's factorization forest theorem.

In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended. We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid $\mathcal{M} = (V, \mathcal{I})$ of rank $k$ on a ground set $V$ and a coverage function $f$ on $V$, the goal is to find an independent set $S \in \mathcal{I}$ maximizing $f(S)$. This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum $k$-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency $\mu$ (i.e., any element of the underlying universe of the coverage function appears in at most $\mu$ sets), we design a procedure, parameterized by some integer $\rho$, to extract in polynomial time an approximate kernel of size $\rho \cdot k$ that is guaranteed to contain a $1 - (\mu - 1)/\rho$ approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a $1 - \varepsilon$ approximation in time $(\mu/\varepsilon)^{O(k)} \cdot |V|^{O(1)}$. This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, because of its simplicity, the kernel construction can be performed in the streaming setting.

A recent breakthrough by Künnemann, Mazowiecki, Schütze, Sinclair-Banks, and Wegrzycki (ICALP, 2023) bounds the running time for the coverability problem in $d$-dimensional vector addition systems under unary encoding to $n^{2^{O(d)}}$, improving on Rackoff's $n^{2^{O(d\lg d)}}$ upper bound (Theor. Comput. Sci., 1978), and provides conditional matching lower bounds. In this paper, we revisit Lazić and Schmitz' "ideal view" of the backward coverability algorithm (Inform. Comput., 2021) in the light of this breakthrough. We show that the controlled strongly monotone descending chains of downwards-closed sets over $\mathbb{N}^d$ that arise from the dual backward coverability algorithm of Lazić and Schmitz on $d$-dimensional unary vector addition systems also enjoy this tight $n^{2^{O(d)}}$ upper bound on their length, and that this also translates into the same bound on the running time of the backward coverability algorithm. Furthermore, our analysis takes place in a more general setting than that of Lazić and Schmitz, which allows to show the same results and improve on the 2EXPSPACE upper bound derived by Benedikt, Duff, Sharad, and Worrell (LICS, 2017) for the coverability problem in invertible affine nets.

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of $H$-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains $NP$-hard for most graphs $H$, we study its fixed-parameter tractability and make progress towards a dichotomy between $FPT$ and $W[1]$-hard cases. We first show that MIS remains $W[1]$-hard in graphs forbidding simultaneously $K_{1, 4}$, any finite set of cycles of length at least $4$, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning $C_4$-free graphs. Then we extend the polynomial algorithm of Alekseev when $H$ is a disjoint union of edges to an $FPT$ algorithm when $H$ is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of \emph{iterative expansion} accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called \emph{iterative compression}. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs $H$.

Identification of defects or anomalies in 3D objects is a crucial task to ensure correct functionality. In this work, we combine Bayesian learning with recent developments in quantum and quantum-inspired machine learning, specifically orthogonal neural networks, to tackle this anomaly detection problem for an industrially relevant use case. Bayesian learning enables uncertainty quantification of predictions, while orthogonality in weight matrices enables smooth training. We develop orthogonal (quantum) versions of 3D convolutional neural networks and show that these models can successfully detect anomalies in 3D objects. To test the feasibility of incorporating quantum computers into a quantum-enhanced anomaly detection pipeline, we perform hardware experiments with our models on IBM's 127-qubit Brisbane device, testing the effect of noise and limited measurement shots.