[Paper Review] Quantum Simulation of Phylogenetic Trees

Demosthenes Ellinas, Peter Jarvis|arXiv (Cornell University)|May 9, 2011

Genomics and Phylogenetic Studies3 citations

TL;DR

This paper proposes a quantum simulation framework for phylogenetic trees using quantum walks and positive trace-preserving maps to model evolutionary processes and statistical inference. It establishes a correspondence between phylogenetic tree likelihood computation and quantum measurement of likelihood operator observables, enabling quantum-enhanced maximum likelihood estimation through state-observable duality and decoherent quantum circuits.

ABSTRACT

Quantum simulations constructing probability tensors of biological multi-taxa in phylogenetic trees are proposed, in terms of positive trace preserving maps, describing evolving systems of quantum walks with multiple walkers. Basic phylogenetic models applying on trees of various topologies are simulated following appropriate decoherent quantum circuits. Quantum simulations of statistical inference for aligned sequences of biological characters are provided in terms of a quantum pruning map operating on likelihood operator observables, utilizing state-observable duality and measurement theory.

Motivation & Objective

• To develop a quantum information-theoretic framework for simulating multi-taxa evolutionary processes in phylogenetics.
• To map classical phylogenetic models, including group-based and Felsenstein-type models, onto quantum circuits using unitary operations and decoherence.
• To simulate statistical inference for aligned biological sequences using quantum pruning maps and likelihood operator observables.
• To establish a quantum mechanical analog of the iterative pruning algorithm used in maximum likelihood phylogenetic tree estimation.
• To enable quantum computation of tree likelihoods via duality between quantum states and observables, leveraging trace cyclic properties and unitary dilations.

Proposed method

• Models phyletic evolution using local unitary operations $U = igotimes_{i=1}^{s} U_i$ acting on $s$-taxon density matrices, followed by a decoherence map ${ m f E}_d^{igotimes s}$ to project onto diagonal density matrices.
• Constructs tree topologies via controlled-not gates $U_{cn}$ applied in tensor product structures, enabling splitting (cladogenesis) through adjoint action $\Delta\rho = U_{cn}(\rho \otimes \widehat{P}_0)U_{cn}^\dagger$.
• Represents site-specific likelihoods as quantum operators $\widehat{L}_{tr}^{(l)}$, obtained by recursively applying a quantum pruning map $\mu$ to likelihood operators of terminal cherries.
• Employs a dual map formalism: $L^{(l)} = \nu_f^{-1} \operatorname{Tr}(\widehat{L}^{C_f} {\mathcal{E}}_{B_f}^*(\rho^\pi))$, where $\mathcal{E}^*$ is the dual of a stochastic map acting on the stationary state $\rho^\pi$.
• Extends the likelihood computation to full alignments using tensor products $\otimes_{l=1}^\Lambda \widehat{L}_{tr}^{(l)}$ and collective maps $\otimes_{l=1}^\Lambda {\mathcal{E}}_{B_{f;l}}^*$, realizable via unitary dilations.
• Utilizes state-observable duality and the trace cyclic property to convert the Heisenberg-like evolution of observables into a Schrödinger-like evolution of states for computational equivalence.

Experimental results

Research questions

• RQ1 Can quantum walks and unitary evolution simulate the stochastic processes underlying classical phylogenetic models?
• RQ2 How can the iterative pruning process used in maximum likelihood phylogenetic inference be mapped onto quantum measurement operations?
• RQ3 What is the quantum mechanical analog of the likelihood operator in phylogenetic trees, and how can it be measured?
• RQ4 Can the duality between quantum states and observables be leveraged to compute alignment likelihoods via quantum circuits?
• RQ5 How can decoherence and trace-preserving maps be used to model the transition probabilities in group-based and Felsenstein-type phylogenetic models?

Key findings

• Group-based phylogenetic models are mapped to quantum walks via unitary evolution and decoherence, with Markov matrices derived from Hadamard products of unitary matrices.
• The Felsenstein model is realized as a post-measurement state map, with the pruning map $\mu$ expressed as a stochastic map $\mathcal{E}_B$ acting on likelihood operators.
• The likelihood of a single site is computed as $L^{(l)} = \left\langle \widehat{L}_{tr}^{(l)}, \rho^\pi \right\rangle = \operatorname{Tr}(\widehat{L}_{tr}^{(l)} \rho^\pi)$, where $\rho^\pi$ is the stationary state.
• For full alignments, the total log-likelihood is expressed as $\log \operatorname{Tr}\left( \bigotimes_{l=1}^\Lambda \widehat{L}_{tr}^{(l)} \right) \rho_\Lambda$, with $\rho_\Lambda = (\rho^\pi)^{\otimes \Lambda}$.
• The final likelihood computation is equivalent to a Schrödinger-like evolution via the dual map $\mathcal{E}_{B_f}^*$, which acts on the stationary density matrix $\rho^\pi$.
• The collective map $\otimes_{l=1}^\Lambda {\mathcal{E}}_{B_{f;l}}^*$ can be realized via unitary dilation, suggesting a Hamiltonian-based quantum implementation of the likelihood computation.