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Quantum Algorithms for the Minimum Steiner Tree problem with application to Binary Near-Perfect Phylogenies

Lingfa Meng, David Salvador Novo, Albert H. Werner, Samir Bhatt

TL;DR

This work tackles quantum acceleration of Binary Near-Perfect Phylogeny (BNPP) and the Minimum Steiner Tree (MST) in bioinformatics. It introduces a QRAM-based BNPP algorithm with complexity $O(8.926^q + 8^q nm^2)$ and a polynomial-space quantum MST algorithm with circuit-model complexity $O^*(e^{(1+g(k,l))k})$, by embedding Grover-style search into a divide-and-conquer DP and using Dicke-state encoding. The BNPP algorithm leverages a quantum MST subroutine to speed up subproblems within a conflict-graph–driven framework, achieving a provable speedup over the best classical BNPP bound. The MST result provides a polynomial-space exact quantum method and a spacetime-tradeoff bound $O^*(e^{(1+g(k,l))k})$ that approaches $O^*(2^k)$ as levels $l$ increase; data encoding to QRAM is analyzed to preserve speedups. Together, these contributions widen the practical reach of quantum algorithms for phylogeny and combinatorial optimization on graphs, with potential pathways to QRAM-free variants and weighted parsimony applications.

Abstract

We present a quantum algorithm in bioinformatics for solving the Binary Near-Perfect Phylogeny Problem (BNPP) with a complexity bound of $O(8.926^q + 8^q nm2)$, where n is the number of input taxa and m is the sequence length for each taxon with each character in the sequence being a binary bit using the QRAM model. We give another polynomial space exact algorithm for the Minimum Steiner Tree (MST) problem with complexity $O^*(e^{(1+g(k,l))k})$ in the circuit model.

Quantum Algorithms for the Minimum Steiner Tree problem with application to Binary Near-Perfect Phylogenies

TL;DR

This work tackles quantum acceleration of Binary Near-Perfect Phylogeny (BNPP) and the Minimum Steiner Tree (MST) in bioinformatics. It introduces a QRAM-based BNPP algorithm with complexity and a polynomial-space quantum MST algorithm with circuit-model complexity , by embedding Grover-style search into a divide-and-conquer DP and using Dicke-state encoding. The BNPP algorithm leverages a quantum MST subroutine to speed up subproblems within a conflict-graph–driven framework, achieving a provable speedup over the best classical BNPP bound. The MST result provides a polynomial-space exact quantum method and a spacetime-tradeoff bound that approaches as levels increase; data encoding to QRAM is analyzed to preserve speedups. Together, these contributions widen the practical reach of quantum algorithms for phylogeny and combinatorial optimization on graphs, with potential pathways to QRAM-free variants and weighted parsimony applications.

Abstract

We present a quantum algorithm in bioinformatics for solving the Binary Near-Perfect Phylogeny Problem (BNPP) with a complexity bound of , where n is the number of input taxa and m is the sequence length for each taxon with each character in the sequence being a binary bit using the QRAM model. We give another polynomial space exact algorithm for the Minimum Steiner Tree (MST) problem with complexity in the circuit model.

Paper Structure

This paper contains 17 sections, 8 theorems, 34 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Binary Near-Perfect Phylogeny
  3. PP and BNPP
  4. Classical Algorithm for BNPP
  5. Asymptotic growth of the number of homoplasms with the number species and number sites $q(n, m)$
  6. Quantum algorithm for Binary Near Perfect Phylogeny
  7. Grover's algorithm for Dynamic Programming
  8. Implementation of the quantum dynamic programming
  9. State preparation for 3-level DW recursion
  10. Encoding complexity of classical data
  11. Polynomial space exact quantum algorithm for the Minimum Steiner Tree problem
  12. Conclusions and Future Directions
  13. Acknowledgements
  14. Asymptotic growth of $q(n,m)$
  15. Dreyfus-Wagner Recursion
  16. ...and 2 more sections

Key Result

Lemma 1

Given a MST algorithm running in time $O^*(C^k)$, there exists a BNPP algorithm running in time $O^*((2C^2\frac{2C^2+1}{2C^2-1})^q+8^qnm^2)$. k is the number of vertices to be connected for MST. q is the near-perfect number for the BNPP solution. n is number of taxa and m is the number of characters

Figures (4)

  • Figure 1: Dicke state preparation circuit over 4 qubits
  • Figure 2: Constructing solution space for graph index qubits with subset relations shown on the edges. Each node represents a superposition of a set of sub-solution states. The arrows are directed top-down with set relations constraining the lower level sets by the higher level sets.
  • Figure 3: Constructing weight qubits for solution space.
  • Figure 4: The final quantum state ready for quantum minimum finding. The quantum state shall be read as if it were a 3 line equation top to down, left to right.

Theorems & Definitions (18)

  • Definition 1: Phylogeny sridhar_algorithms_2007
  • Definition 2: Perfect Phylogeny
  • Definition 3: Binary Near-Perfect Phylogeny
  • Definition 4: Conflict Graph
  • Lemma 1: MST to BNPP
  • Theorem 1: Theorem 1 of 10.1007/978-3-030-58150-3_19
  • Theorem 2: Quantum Algorithm for BNPP
  • Definition 5: Minimum Steiner Tree
  • Definition 6: Minimum Steiner Tree on a hypercube
  • Theorem 3: Polynomial space exact quantum algorithm for MST
  • ...and 8 more