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Recently it was shown that the so-called guided local Hamiltonian problem -- estimating the smallest eigenvalue of a $k$-local Hamiltonian when provided with a description of a quantum state ('guiding state') that is guaranteed to have substantial overlap with the true groundstate -- is BQP-complete for $k \geq 6$ when the required precision is inverse polynomial in the system size $n$, and remains hard even when the overlap of the guiding state with the groundstate is close to a constant $\left(\frac12 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$. We improve upon this result in three ways: by showing that it remains BQP-complete when i) the Hamiltonian is 2-local, ii) the overlap between the guiding state and target eigenstate is as large as $1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$, and iii) when one is interested in estimating energies of excited states, rather than just the groundstate. Interestingly, iii) is only made possible by first showing that ii) holds.

We introduce $\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of Euclidean vector bundle when $\varepsilon = 0$. In particular, we study \v{C}ech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon$-approximate vector bundles can be used to represent classical vector bundles when $\varepsilon > 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

We consider the problem of determining the asymptotics of the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$. We close this gap and extend the existing lower bounds to all $1\leq p,q\leq \infty$. In the process, we show that the Bernstein widths, which are typically used to lower bound the manifold widths, may decay asymptotically slower than the manifold widths.

Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime power composite dimensions ($d = k\times s$) is a long standing open problem, which leads to different construction methods for the class Approximate MUBs (AMUBs) by relaxing the criterion that the absolute value of the dot product between two vectors chosen from different bases should be $\leq \frac{\beta}{\sqrt{d}}$. In this chapter, we consider a more general class of AMUBs (ARMUBs, considering the real ones too), compared to our earlier work in [Cryptography and Communications, 14(3): 527--549, 2022]. We note that the quality of AMUBs (ARMUBs) constructed using RBD$(X,A)$ with $|X|= d$, critically depends on the parameters, $|s-k|$, $\mu$ (maximum number of elements common between any pair of blocks), and the set of block sizes. We present the construction of $\mathcal{O}(\sqrt{d})$ many $\beta$-AMUBs for composite $d$ when $|s-k|< \sqrt{d}$, using RBDs having block sizes approximately $\sqrt{d}$, such that $|\braket{\psi^l_i|\psi^m_j}| \leq \frac{\beta}{\sqrt{d}}$ where $\beta = 1 + \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1}) \leq 2$. Moreover, if real Hadamard matrix of order $k$ or $s$ exists, then one can construct at least $N(k)+1$ (or $N(s)+1$) many $\beta$-ARMUBs for dimension $d$, with $\beta \leq 2 - \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1})< 2$, where $N(w)$ is the number of MOLS$(w)$. This improves and generalizes some of our previous results for ARMUBs from two points, viz., the real cases are now extended to complex ones too. The earlier efforts use some existing RBDs, whereas here we consider new instances of RBDs that provide better results. Similar to the earlier cases, the AMUBs (ARMUBs) constructed using RBDs are in general very sparse, where the sparsity $(\epsilon)$ is $1 - \mathcal{O}(d^{-\frac{1}{2}})$.

We tackle estimating sparse coefficients in a linear regression when the covariates are sampled from an $L$-subexponential random vector. This vector belongs to a class of distributions that exhibit heavier tails than Gaussian random vector. Previous studies have established error bounds similar to those derived for Gaussian random vectors. However, these methods require stronger conditions than those used for Gaussian random vectors to derive the error bounds. In this study, we present an error bound identical to the one obtained for Gaussian random vectors up to constant factors without imposing stronger conditions, when the covariates are drawn from an $L$-subexponential random vector. Interestingly, we employ an $\ell_1$-penalized Huber regression, which is known for its robustness against heavy-tailed random noises rather than covariates. We believe that this study uncovers a new aspect of the $\ell_1$-penalized Huber regression method.

A subset $\mathcal{C}\subseteq\{0,1,2\}^n$ is said to be a $\textit{trifferent}$ code (of block length $n$) if for every three distinct codewords $x,y, z \in \mathcal{C}$, there is a coordinate $i\in \{1,2,\ldots,n\}$ where they all differ, that is, $\{x(i),y(i),z(i)\}$ is same as $\{0,1,2\}$. Let $T(n)$ denote the size of the largest trifferent code of block length $n$. Understanding the asymptotic behavior of $T(n)$ is closely related to determining the zero-error capacity of the $(3/2)$-channel defined by Elias'88, and is a long-standing open problem in the area. Elias had shown that $T(n)\leq 2\times (3/2)^n$ and prior to our work the best upper bound was $T(n)\leq 0.6937 \times (3/2)^n$ due to Kurz'23. We improve this bound to $T(n)\leq c \times n^{-2/5}\times (3/2)^n$ where $c$ is an absolute constant.

In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how codes whose hulls are algebraic geometry codes may be defined using only rational places of Kummer extensions (and Hermitian function fields in particular). Our primary tool is explicitly constructing non-special divisors of degrees $g$ and $g-1$ on certain families of function fields with many rational places, accomplished by appealing to Weierstrass semigroups. We provide explicit algebraic geometry codes with hulls of specified dimensions, producing along the way linearly complementary dual algebraic geometric codes from the Hermitian function field (among others) using only rational places and an answer to an open question posed by Ballet and Le Brigand for particular function fields. These results complement earlier work by Mesnager, Tang, and Qi that use lower-genus function fields as well as instances using places of a higher degree from Hermitian function fields to construct linearly complementary dual (LCD) codes and that of Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraic geometry codes with the LCD property rather than obtaining codes via monomial equivalences.

The circuit class $\mathsf{QAC}^0$ was introduced by Moore (1999) as a model for constant depth quantum circuits where the gate set includes many-qubit Toffoli gates. Proving lower bounds against such circuits is a longstanding challenge in quantum circuit complexity; in particular, showing that polynomial-size $\mathsf{QAC}^0$ cannot compute the parity function has remained an open question for over 20 years. In this work, we identify a notion of the Pauli spectrum of $\mathsf{QAC}^0$ circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical $\mathsf{AC}^0$ circuits. We conjecture that the Pauli spectrum of $\mathsf{QAC}^0$ circuits satisfies low-degree concentration, in analogy to the famous Linial, Nisan, Mansour theorem on the low-degree Fourier concentration of $\mathsf{AC}^0$ circuits. If true, this conjecture immediately implies that polynomial-size $\mathsf{QAC}^0$ circuits cannot compute parity. We prove this conjecture for the class of depth-$d$, polynomial-size $\mathsf{QAC}^0$ circuits with at most $n^{O(1/d)}$ auxiliary qubits. We obtain new circuit lower bounds and learning results as applications: this class of circuits cannot correctly compute - the $n$-bit parity function on more than $(\frac{1}{2} + 2^{-\Omega(n^{1/d})})$-fraction of inputs, and - the $n$-bit majority function on more than $(\frac{1}{2} + O(n^{-1/4}))$-fraction of inputs. Additionally we show that this class of $\mathsf{QAC}^0$ circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for $\mathsf{QAC}^0$ circuits. More broadly, our results add evidence that "Pauli-analytic" techniques can be a powerful tool in studying quantum circuits.

This paper addresses the computational problem of deciding invertibility (or one to one-ness) of a Boolean map $F$ in $n$-Boolean variables. This problem has a special case of deciding invertibilty of a map $F:\mathbb{F}_{2}^n\rightarrow\mathbb{F}_{2}^n$ over the binary field $\mathbb{F}_2$. Further the problem can be extended and stated over a finite field $\mathbb{F}$ instead of $\mathbb{F}_2$. Algebraic condition for invertibility of $F$ in this special case over a finite field is well known to be equivalent to invertibility of the Koopman operator of $F$ as shown in \cite{RamSule}. In this paper a condition for invertibility is derived in the special case of Boolean maps $F:B_0^n\rightarrow B_0^n$ where $B_0$ is the two element Boolean algebra in terms of \emph{implicants} of Boolean equations. This condition is then extended to the case of general maps in $n$ variables. Hence this condition answers the special case of invertibility of the map $F$ defined over the binary field $\mathbb{F}_2$ alternatively, in terms of implicants instead of the Koopman operator. The problem of deciding invertibility of a map $F$ (or that of finding its $GOE$) over finite fields appears to be distinct from the satisfiability problem (SAT) or the problem of deciding consistency of polynomial equations over finite fields. Hence the well known algorithms for deciding SAT or of solvability using Grobner basis for checking membership in an ideal generated by polynomials is not known to answer the question of invertibility of a map. Similarly it appears that algorithms for satisfiability or polynomial solvability are not useful for computation of $GOE(F)$ even for maps over the binary field $\mathbb{F}_2$.

The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension $k$ and length $n\ge 2k$ the number of all $[n,k]_q$ linear codes with hull dimension $l$ decreases as $l$ increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length $n$ and dimension $k$, comparing the obtained numbers with the number of all linear codes for the given $n$ and $k$.