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

• URL: http://arxiv.org/abs/2609.08432v1
• Date: Tue, 08 Sep 2026 08:37:00 GMT
• Title: Explicit block encodings of rate matrices for simulating polymerization kinetics on quantum computers
• Abstract summary: We study block-encoding circuits for two kinetic models of living polymerization.<n>We prove that both implementations require only $O(log N)$ qubits in the matrix dimension $N$.<n>These results establish a concrete quantum circuit foundation for simulating polymerization kinetics on fault-tolerant quantum hardware.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 ($r_1 r_2 \simeq 1$) 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.

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