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arXiv:2609.08432

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

Yuhei Ikeda, Hokuto Iwakiri, Soichiro Nishio, Kentaro Matsumoto

quant-phphysics.chem-phSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
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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.

Comments
30 pages, 11 figures

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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 (r1r2≃1r_1 r_2 \simeq 1r1​r2​≃1) and blocky (r1r2>1r_1 r_2 > 1r1​r2​>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)O(\log N)O(logN) qubits in the matrix dimension NNN (an exponential memory saving over the classical state space), with gate counts growing gradually, reaching 10410^4104 to 10510^5105 gates at 10310^3103 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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Quantum circuits naturally implement unitary operations on input quantum states. However, non-unitary operations can also be implemented through "block encodings", where additional ancilla qubits are introduced and later measured. While block encoding underlies a host of well-established theoretical applications, its implementation in practice has been prohibitively expensive for current quantum hardware. In this paper, we present practical and explicit block encoding circuits implementing matrix polynomial transformations of a target matrix. With standard approaches, block-encoding a degree-d matrix polynomial requires a circuit depth scaling as d times the depth for block-encoding the original matrix alone. By leveraging the recently introduced Fast One-Qubit Controlled Select Linear Combination of Unitaries (FOQCS-LCU) framework, we show that the additional circuit-depth overhead required for encoding matrix polynomials can be reduced to scale linearly in d with no dependence on system size or the cost of block encoding the original matrix. Moreover, we demonstrate that the FOQCS-LCU circuits and their associated matrix polynomial transformations can be controlled with negligible overhead, enabling efficient applications such as Hadamard tests. Finally, we provide explicit hardware-efficient circuits for representative non-unitary operators, namely quantum spin models, together with detailed non-asymptotic gate counts and circuit depths.
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We develop a block-encoding compiler that treats matrix product operators as compressed, virtual-path linear combination of unitaries programs. The compiler constructs conditional PREP and local SELECT stages directly from a parent matrix product operator, establishing tensor networks as structured quantum intermediate representations that can be efficiently compiled to block-encoded circuits. We apply the construction to real-time evolution in the Heisenberg chain and two perturbed Heisenberg-family models. Across the regimes studied, the compressed, approximately unitary propagator MPOs retain mild bond dimension and LCU normalization. Relative to an LCU that explicitly lists the O(NK)\mathcal{O}(N^K)O(NK) Pauli-product branches of an order-K truncated Taylor polynomial, our virtual-transition implementation replaces combinatorial branch enumeration by a circuit complexity scaling as O(αMPONχ2)\mathcal{O}(α_{\rm MPO}Nχ^2)O(αMPO​Nχ2), approaching O(Nχ2)\mathcal{O}(Nχ^2)O(Nχ2) when αMPOα_{\rm MPO}αMPO​ remains mild. We numerically characterize how truncation order, bond-dimension budget, and system size affect approximation error, normalization, and compiler cost. These results demonstrate how classically compressed tensor-network representations can serve as quantum compiler intermediate representations for block encoding and opens new avenues to accelerate quantum algorithms.
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To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.