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We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the $p$-faces of the $n$-cube (for $n>p$) and stabilizer constraints with faces of dimension $(p\pm1)$. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into $N = 2^{n-p-1} \tbinom{n}{p}$ physical qubits and displays local testability with a soundness of $\Omega(1/\log(N))$ beating the current state-of-the-art of $1/\log^{2}(N)$ due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. We then extend this code family by considering the quotient of the $n$-cube by arbitrary linear classical codes of length $n$. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary $n$-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.

Finding balanced, highly nonlinear Boolean functions is a difficult problem where it is not known what nonlinearity values are possible to be reached in general. At the same time, evolutionary computation is successfully used to evolve specific Boolean function instances, but the approach cannot easily scale for larger Boolean function sizes. Indeed, while evolving smaller Boolean functions is almost trivial, larger sizes become increasingly difficult, and evolutionary algorithms perform suboptimally. In this work, we ask whether genetic programming (GP) can evolve constructions resulting in balanced Boolean functions with high nonlinearity. This question is especially interesting as there are only a few known such constructions. Our results show that GP can find constructions that generalize well, i.e., result in the required functions for multiple tested sizes. Further, we show that GP evolves many equivalent constructions under different syntactic representations. Interestingly, the simplest solution found by GP is a particular case of the well-known indirect sum construction.

Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel's capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture", which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In a companion paper, we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before.

In this paper, we study quantum algorithms for computing the exact value of the treewidth of a graph. Our algorithms are based on the classical algorithm by Fomin and Villanger (Combinatorica 32, 2012) that uses $O(2.616^n)$ time and polynomial space. We show three quantum algorithms with the following complexity, using QRAM in both exponential space algorithms: $\bullet$ $O(1.618^n)$ time and polynomial space; $\bullet$ $O(1.554^n)$ time and $O(1.452^n)$ space; $\bullet$ $O(1.538^n)$ time and space. In contrast, the fastest known classical algorithm for treewidth uses $O(1.755^n)$ time and space. The first two speed-ups are obtained in a fairly straightforward way. The first version uses additionally only Grover's search and provides a quadratic speedup. The second speedup is more time-efficient and uses both Grover's search and the quantum exponential dynamic programming by Ambainis et al. (SODA '19). The third version uses the specific properties of the classical algorithm and treewidth, with a modified version of the quantum dynamic programming on the hypercube. Lastly, as a small side result, we also give a new classical time-space tradeoff for computing treewidth in $O^*(2^n)$ time and $O^*(\sqrt{2^n})$ space.

We initiate the study of parameterized complexity of $\textsf{QMA}$ problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + $t$ $T$-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most $t$ qubits (independent of the system size). Furthermore, we derive new lower bounds on the $T$-count of circuit satisfiability instances and the $T$-count of the $W$-state assuming the classical exponential time hypothesis ($\textsf{ETH}$). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.

We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We also present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. The main application of our analysis is the description of the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massive, free QFT in two spacetime dimensions, thus complementing recent results in higher space dimensions. In particular, we have the formula for the local entropy of a one dimensional Klein-Gordon wave packet and Araki's vacuum relative entropy of a coherent state on a double cone von Neumann algebra. Besides, we derive the type III_1 factor property. Incidentally, we run across certain positive selfadjoint extensions of the Laplacian, with outer boundary conditions, seemingly not considered so far.

In this work, we consider the problem of jointly minimizing the average cost of sampling and transmitting status updates by users over a wireless channel subject to average Age of Information (AoI) constraints. Errors in the transmission may occur and a scheduling policy has to decide if the users sample a new packet or attempt for retransmission of the packet sampled previously. The cost consists of both sampling and transmission costs. The sampling of a new packet after a failure imposes an additional cost on the system. We formulate a stochastic optimization problem with the average cost in the objective under average AoI constraints. To solve this problem, we propose three scheduling policies; a) a dynamic policy, that is centralized and requires full knowledge of the state of the system, b) two stationary randomized policies that require no knowledge of the state of the system. We utilize tools from Lyapunov optimization theory in order to provide the dynamic policy, and we prove that its solution is arbitrary close to the optimal one. In order to provide the randomized policies, we model the system by utilizing Discrete Time Markov Chain (DTMC). We provide closed-form and approximated expressions for the average AoI and its distribution, for each randomized policy. Simulation results show the importance of providing the option to transmit an old packet in order to minimize the total average cost.

Nonbinary LDPC codes have shown superior performance close to the Shannon limit. Compared to binary LDPC codes of similar lengths, they can reach orders of magnitudes lower error rate. However, multitude of design freedoms of nonbinary LDPC codes complicates the practical code and decoder design process. Fast simulations are critically important to evaluate the pros and cons. Rapid prototyping on FPGA is attractive but takes significant design efforts due to its high design complexity. We propose a high-throughput reconfigurable hardware emulation architecture with decoder and peripheral co-design. The architecture enables a library and script-based framework that automates the construction of FPGA emulations. Code and decoder design parameters are programmed either during run time or by script in design time. We demonstrate the capability of the framework in evaluating practical code and decoder design by experimenting with two popular nonbinary LDPC codes, regular (2, dc) codes and quasi-cyclic codes: each emulation model can be auto-constructed within hours and the decoder delivers excellent error-correcting performance on a Xilinx Virtex-5 FPGA with throughput of up to hundreds of Mbps.

In this paper, we provide a framework for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from classical additive codes over a finite commutative local Frobenius ring. We derive a formula for the minimum number of entanglement qudits required to construct an EAQECC from a linear code over the ring $\mathbb{Z}_{p^s}$. This is used to obtain an exact expression for the minimum number of entanglement qudits required to construct an EAQECC from an additive code over a Galois ring, which significantly extends known results for EAQECCs over finite fields.

Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables stable single-player version of self-play training that helps the agent to escape local optima. The training on any problem instance can be accelerated by applying transfer learning from an agent trained on randomly generated problems. Our approach allows sampling high-quality solutions to the Ising problem with high probability and outperforms both baseline heuristics and a black-box hyperparameter optimization approach.