Explicit block encodings of rate matrices for simulating polymerization kinetics on quantum computers

@misc{ikeda2026explicit,
  title = {{Explicit block encodings of rate matrices for simulating polymerization kinetics on quantum computers}},
  author = {Yuhei Ikeda and Hokuto Iwakiri and Soichiro Nishio and Kentaro Matsumoto},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08432},
  eprint = {2609.08432},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08432},
}

TY  - UNPB
TI  - Explicit block encodings of rate matrices for simulating polymerization kinetics on quantum computers
AU  - Yuhei Ikeda
AU  - Hokuto Iwakiri
AU  - Soichiro Nishio
AU  - Kentaro Matsumoto
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08432
UR  - https://arxiv.org/abs/2609.08432
AB  - Predicting how molecular weight distribution and monomer sequence evolve during polymerization is central to polymer science, yet classical approaches face a trade-off between molecular resolution and computational cost: for copolymers, the number of distinguishable species grows exponentially with chain length. Quantum computing offers a potential alternative, provided the non-unitary rate matrices governing the kinetics can be embedded into unitary quantum circuits, a task known as block encoding. Here we construct explicit block-encoding circuits for two kinetic models of living polymerization: Model A, single-monomer polymerization, whose lower-bidiagonal rate matrix is encoded via a sparse-oracle construction and a two-term linear combination of unitaries (LCU) decomposition; and Model B, two-monomer copolymerization, where a bijective labeling of polymer species by an integer index (the m-index) yields a structured sparse matrix encoded via either a five-term LCU or a sparse-oracle construction. Numerical simulations with the sparse-oracle encodings reproduce the classical time evolution for reactivity ratios drawn from reported olefin copolymerization systems spanning near-random (r_1 r_2 ≃ 1) and blocky (r_1 r_2 > 1) microstructures, and the LCU encodings are verified by explicit reconstruction of the encoded matrix block. Resource estimation shows that both implementations require only O(log N) qubits in the matrix dimension N (an exponential memory saving over the classical state space), with gate counts growing gradually, reaching 10^4 to 10^5 gates at 10^3 system qubits. These results establish a concrete quantum circuit foundation for simulating polymerization kinetics on fault-tolerant quantum hardware, and a first step toward exploiting exponential state-space compression for high-dimensional polymer reaction networks.
ER  -