Srinivasan Arunachalam
An alternative formula is presented for the evaluation of the zeta function values $ζ(2k)$ without the need for Bernoulli numbers. Our formula is recursive, and improves the efficiency with which we can calculate large values of the zeta function.
Srinivasan Arunachalam
We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature. We have also evaluated an expression for the zeta function on the plane $\Re(s) = 1$.
Srinivasan Arunachalam, Arkopal Dutt, Alexandru Gheorghiu et al.
We initiate the study of quantum agnostic learning of phase states with respect to a function class $\mathsf{C}\subseteq \{c:\{0,1\}^n\rightarrow \{0,1\}\}$: given copies of an unknown $n$-qubit state $|ψ\rangle$ which has fidelity $\textsf{opt}$ with a phase state $|φ_c\rangle=\frac{1}{\sqrt{2^n}}\sum_{x\in \{0,1\}^n}(-1)^{c(x)}|x\rangle$ for some $c\in \mathsf{C}$, output $|φ\rangle$ which has fidelity $|\langle φ| ψ\rangle|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: (i) Size-$t$ decision trees which runs in time $\textsf{poly}(n,t,1/\varepsilon)$. This also implies $k$-juntas can be agnostically learned in time $\textsf{poly}(n,2^k,1/\varepsilon)$. (ii) $s$-term DNF formulas in time $\textsf{poly}(n,(s/\varepsilon)^{\log \log (s/\varepsilon) \cdot \log(1/\varepsilon)})$. Our main technical contribution is a quantum agnostic boosting protocol which converts a weak agnostic learner, which outputs a parity state $|φ\rangle$ such that $|\langle φ|ψ\rangle|^2\geq \textsf{opt}/\textsf{poly}(n)$, into a strong learner which outputs a superposition of parity states $|φ'\rangle$ such that $|\langle φ'|ψ\rangle|^2\geq \textsf{opt} - \varepsilon$. Using quantum agnostic boosting, we obtain a $n^{O(\log \log n \cdot \log(1/\varepsilon))}$-time algorithm for learning $\textsf{poly}(n)$-sized depth-$3$ circuits (consisting of $\textsf{AND}$, $\textsf{OR}$, $\textsf{NOT}$ gates) in the uniform $\textsf{PAC}$ model given quantum examples, which is near-polynomial time for constant $\varepsilon$. Classically, obtaining an algorithm with a similar complexity has been an open question in the $\textsf{PAC}$ model and our work answers this given quantum examples.
Srinivasan Arunachalam, Alex B. Grilo, Tom Gur et al.
We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - γ$ by a time $T$ quantum algorithm. We prove that if $γ^2 \cdot T \ll 2^n/n$, then $\mathsf{BQE} \nsubseteq \mathfrak{C}$, where $\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$ is an exponential-time analogue of $\mathsf{BQP}$. This result is optimal in both $γ$ and $T$, since it is not hard to learn any class $\mathfrak{C}$ of functions in (classical) time $T = 2^n$ (with no error), or in quantum time $T = \mathsf{poly}(n)$ with error at most $1/2 - Ω(2^{-n/2})$ via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.