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Commitment scheme is a central task in cryptography, where a party (typically called a prover) stores a piece of information (e.g., a bit string) with the promise of not changing it. This information can be accessed by another party (typically called the verifier), who can later learn the information and verify that it was not meddled with. Merkle tree is a well-known construction for doing so in a succinct manner, in which the verfier can learn any part of the information by receiving a short proof from the honest prover. Despite its significance in classical cryptography, there was no quantum analog of the Merkle tree. A direct generalization using the Quantum Random Oracle Model (QROM) does not seem to be secure. In this work, we propose the quantum Merkle tree. It is based on what we call the Quantum Haar Random Oracle Model (QHROM). In QHROM, both the prover and the verifier have access to a Haar random quantum oracle G and its inverse. Using the quantum Merkle tree, we propose a succinct quantum argument for the Gap-k-Local-Hamiltonian problem. We prove it is secure against semi-honest provers in QHROM and conjecture its general security. Assuming the Quantum PCP conjecture is true, this succinct argument extends to all of QMA. This work raises a number of interesting open research problems.

Analysis and use of stochastic models represented by a discrete-time Markov Chain require evaluation of performance measures and characterization of its stationary distribution. Analytical solutions are often unavailable when the system states are continuous or mixed. This paper presents a new method for computing the stationary distribution and performance measures for stochastic systems represented by continuous-, or mixed-state Markov chains. We show the asymptotic convergence and provide deterministic non-asymptotic error bounds for our method under the supremum norm. Our finite approximation method is near-optimal among all discrete approximate distributions, including empirical distributions obtained from Markov chain Monte Carlo (MCMC). Numerical experiments validate the accuracy and efficiency of our method and show that it significantly outperforms MCMC based approach.

A fundamental and still largely unsolved question in the context of Generative Adversarial Networks is whether they are truly able to capture the real data distribution and, consequently, to sample from it. In particular, the multidimensional nature of image distributions leads to a complex evaluation of the diversity of GAN distributions. Existing approaches provide only a partial understanding of this issue, leaving the question unanswered. In this work, we introduce a loop-training scheme for the systematic investigation of observable shifts between the distributions of real training data and GAN generated data. Additionally, we introduce several bounded measures for distribution shifts, which are both easy to compute and to interpret. Overall, the combination of these methods allows an explorative investigation of innate limitations of current GAN algorithms. Our experiments on different data-sets and multiple state-of-the-art GAN architectures show large shifts between input and output distributions, showing that existing theoretical guarantees towards the convergence of output distributions appear not to be holding in practice.

State-of-the-art noisy intermediate-scale quantum computers require low-complexity techniques for the mitigation of computational errors inflicted by quantum decoherence. Symmetry verification constitutes a class of quantum error mitigation (QEM) techniques, which distinguishes erroneous computational results from the correct ones by exploiting the intrinsic symmetry of the computational tasks themselves. Inspired by the benefits of quantum switch in the quantum communication theory, we propose beneficial techniques for circuit-oriented symmetry verification that are capable of verifying the commutativity of quantum circuits without the knowledge of the quantum state. In particular, we propose the spatio-temporal stabilizer (STS) technique, which generalizes the conventional quantum-domain stabilizer formalism to circuit-oriented stabilizers. The applicability and implementational strategies of the proposed techniques are demonstrated by using practical quantum algorithms, including the quantum Fourier transform (QFT) and the quantum approximate optimization algorithm (QAOA).

In this paper we prove upper and lower bounds on the minimal spherical dispersion. In particular, we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion is, for fixed $\varepsilon>0$, up to logarithmic terms linear in the dimension $d$. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere.

In this paper, we consider the problem of guessing a sequence subject to a distortion constraint. Specifically, we assume the following game between Alice and Bob: Alice has a sequence $\bx$ of length $n$. Bob wishes to guess $\bx$, yet he is satisfied with finding any sequence $\hat{\bx}$ which is within a given distortion $D$ from $\bx$. Thus, he successively submits queries to Alice, until receiving an affirmative answer, stating that his guess was within the required distortion. Finding guessing strategies which minimize the number of guesses (the \emph{guesswork}), and analyzing its properties (e.g., its $\rho$--th moment) has several applications in information security, source and channel coding. Guessing subject to a distortion constraint is especially useful when considering contemporary biometrically--secured systems, where the "password" which protects the data is not a single, fixed vector but rather a \emph{ball of feature vectors} centered at some $\bx$, and any feature vector within the ball results in acceptance. We formally define the guessing problem under distortion in \emph{four different setups}: memoryless sources, guessing through a noisy channel, sources with memory and individual sequences. We suggest a randomized guessing strategy which is asymptotically optimal for all setups and is \emph{five--fold universal}, as it is independent of the source statistics, the channel, the moment to be optimized, the distortion measure and the distortion level.

The sensitivity of a string compression algorithm $C$ asks how much the output size $C(T)$ for an input string $T$ can increase when a single character edit operation is performed on $T$. This notion enables one to measure the robustness of compression algorithms in terms of errors and/or dynamic changes occurring in the input string. In this paper, we analyze the worst-case multiplicative sensitivity of string compression algorithms, defined by $\max_{T \in \Sigma^n}\{C(T')/C(T) : ed(T, T') = 1\}$, where $ed(T, T')$ denotes the edit distance between $T$ and $T'$. For the most common versions of the Lempel-Ziv 77 compressors, we prove that the worst-case multiplicative sensitivity is only a small constant (2 or 3, depending on the version of the Lempel-Ziv 77 and the edit operation type). We strengthen our upper bound results by presenting matching lower bounds on the worst-case sensitivity for all these major versions of the Lempel-Ziv 77 factorizations. This contrasts with the previously known related results such that the size $z_{\rm 78}$ of the Lempel-Ziv 78 factorization can increase by a factor of $\Omega(n^{3/4})$ [Lagarde and Perifel, 2018], and the number $r$ of runs in the Burrows-Wheeler transform can increase by a factor of $\Omega(\log n)$ [Giuliani et al., 2021] when a character is prepended to an input string of length $n$. We also study the worst-case sensitivity of several grammar compression algorithms including Bisection, AVL-grammar, GCIS, and CDAWG. Further, we extend the notion of the worst-case sensitivity to string repetitiveness measures such as the smallest string attractor size $\gamma$ and the substring complexity $\delta$. We present some non-trivial upper and lower bounds of the worst-case multiplicative sensitivity for $\gamma$ and matching upper and lower bounds of the worst-case multiplicative sensitivity for $\delta$.

In this paper, we consider the ``Shortest Superstring Problem''(SSP) or the ``Shortest Common Superstring Problem''(SCS). The problem is as follows. For a positive integer $n$, a sequence of n strings $S=(s^1,\dots,s^n)$ is given. We should construct the shortest string $t$ (we call it superstring) that contains each string from the given sequence as a substring. The problem is connected with the sequence assembly method for reconstructing a long DNA sequence from small fragments. We present a quantum algorithm with running time $O^*(1.728^n)$. Here $O^*$ notation does not consider polynomials of $n$ and the length of $t$.

Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

Quantum Key Distribution (QKD) is an approach for establishing symmetrical binary keys between distant users in an information-theoretically secure way. In this paper we provide an overview of existing solutions that integrate QKD within the most popular architecture for establishing secure communications in modern IP (Internet Protocol) networks - IPsec (Internet Protocol security). The provided overview can be used to further design the integration of QKD within the IPsec architecture striving for a standardized solution.