RotorMap and Quantum Fingerprints of DNA Sequences via Rotary Position Embeddings

Abstract

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.

Figures (17)

• Figure 1: For a random 20,000-long DNA sequence and its mutation, we compute the Levenshtein distance on the one hand and fidelity between their RoPE encodings on the other. In this case, a RoPE with 4096 complex dimensions is used, forming a quantum state on 12 qubits. In each pair, the introduced mutations are evenly split between random deletions, insertions and substitutions, where the mutation rate is selected evenly from the interval (0, 0.5). A total of n=2,000 sampled pairs are plotted.
• Figure 2: When sampling randomly a DNA sequence and its mutation, we can predict the mutation rate based on the observed fidelity between encodings. For the encoding shown in Fig. \ref{['fig:intro_corr']}, the RMSE is less than 1% if the mutation rate is less than 25%.
• Figure 3: 100 random DNA sequences of length 1 billion each were generated along with their mutations. Since it is infeasible to find the exact Levenshtein distance in this case, we plot the correlation between the mutation rate and the fidelity between respective RoPE encodings of dimension 2048.
• Figure 4: $c_{\textbf{mer}}$ is a sum of complex units that correspond to locations of $\textbf{mer}$ within the $\textbf{dna}$ sequence.
• Figure 5: For 1000 random 20,000-long DNA sequences and their mutations, we plot the mutation rate on one axis and fidelity between the default version of RoPE encodings with parameters $s=7, m=4$ (65536 complex dimensions or 16 qubits) on the other.