arXiv subjects

Danylo Yakymenko

Publications and source records attributed to Danylo Yakymenko.

4 recordsLinked to original sources

arXiv2026

RotorMap and Quantum Fingerprints of DNA Sequences via Rotary Position Embeddings

For strings of letters from a small alphabet, such as DNA sequences, we present a quantum encoding that empirically provides a strong correlation between the Levenshtein edit distance and the fidelity between quantum states defined by the encodings. It is based on the principles of Rotary Position Embeddings (RoPE), employed in modern large language models. Classically, this encoding yields RotorMap - a GPU-accelerated DNA mapping algorithm that achieves speedups of 50-700x over single-thread Minimap2 in proof-of-concept tests on human and maize genomes. For use on quantum devices, we introduce the Angular encoding, which is built from RoPE and directly outputs state preparation circuits. To verify its properties and utility on NISQ devices, we report results of experiments conducted on quantum computers from Quantinuum: the 56-qubit H2-1, H2-2 and the latest 98-qubit Helios-1. As a potential application, we consider a quantum DNA authentication problem and conjecture that a quantum advantage in one-way communication complexity could be achieved over any comparable classical solution.

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arXiv2024

SICs and the Triangle Group $(3,3,3)$

The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group $(3,3,3)$. We give a full description of such representations and show how it can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension $d>3$ is prime.

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arXiv2021

On the continuous Zauner conjecture

In a recent paper by S. Pandey, V. Paulsen, J. Prakash, and M. Rahaman, the authors studied the entanglement breaking quantum channels $Φ_t:\mathbb{C}^{d\times d} \to \mathbb{C}^{d \times d}$ for $t \in [-\frac{1}{d^2-1}, \frac{1}{d+1}]$ defined by $Φ_t(X) = tX+ (1-t)\text{Tr}(X) \frac{1}{d}I$. They proved that Zauner's conjecture is equivalent to the statement that entanglement breaking rank of $Φ_{\frac{1}{d+1}}$ is $d^2$. The authors made the extended conjecture that $\text{ebr}(Φ_t)=d^2$ for every $t \in [0, \frac{1}{d+1}]$ and proved it in dimensions 2 and 3. In this paper we prove that for any $t \in [-\frac{1}{d^2-1}, \frac{1}{d+1}] \setminus\{0\}$ the equality $\text{ebr}(Φ_t)=d^2$ is equivalent to the existence of a pair of informationally complete unit norm tight frames $\{|x_i\rangle\}_{i=1}^{d^2}, \{|y_i\rangle\}_{i=1}^{d^2}$ in $\mathbb{C}^d $ which are mutually unbiased in a certain sense. That is, for any $i\neq j$ it holds that $|\langle x_i|y_j\rangle|^2 = \frac{1-t}{d}$ and $|\langle x_i|y_i\rangle|^2 = \frac{t(d^2-1)+1}{d}$ (also it follows that $|\langle x_i|x_j\rangle\langle y_i|y_j\rangle|=|t|$). Though, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of $t$ other than $0$ or $\frac{1}{d+1}$.

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arXiv2019

Geometric properties of SIC-POVM tensor square

It's known that if $d^2$ vectors from $d$-dimensional Hilbert space $H$ form a SIC-POVM (SIC for short) then tensor square of those vectors form an equiangular tight frame on the symmetric subspace of $H\otimes H$. We prove that for any SIC of WH-type (Weyl-Heisenberg group covariant) this squared frame can be obtained as a projection of WH-type basis of $H\otimes H$ onto the symmetric subspace. We give a full description of the set of all WH-type bases, so this set could be used as a search space for SIC solutions. Also we show that a particular element of this set is close to a SIC solution in some structural sense. Finally we give a geometric construction of a SIC-related symmetric tight fusion frames that were discovered in odd dimensions.

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