Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order $N$, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component $\mathcal{H}_{j}$ of the decomposition.

💡 Research Summary

The paper addresses a fundamental obstacle in quantum computing: the inability of quantum hardware to directly simulate nonlinear differential equations because quantum dynamics are intrinsically linear. To bypass this limitation, the authors focus on Carleman linearization, a technique that embeds a finite‑dimensional nonlinear system into an infinite‑dimensional linear one. By truncating the embedding at order N, the original nonlinear ODE is approximated by a large but finite linear system. The authors first review the Carleman procedure, showing how the state vector is lifted to a hierarchy of tensor‑product variables y_j = Φ^{⊗j}. The resulting linear dynamics are governed by a block‑structured sparse matrix whose dimension grows roughly as O(N²) while the number of non‑zero entries grows only linearly (≈ 2N‑1).

Even though the truncated system is linear, loading its Hamiltonian onto a quantum processor is non‑trivial. The standard approach is to express the Hamiltonian as a linear combination of unitaries (LCU) using the Pauli basis. However, because each matrix element must be mapped to a Pauli tensor product, the number of decomposition terms scales quadratically with the matrix size, leading to prohibitive ancilla overhead and circuit depth on near‑term devices.

To overcome this bottleneck, the authors introduce a new set of non‑unitary operators called the “Sigma basis.” Each Sigma operator corresponds directly to a non‑zero matrix entry, and the Hamiltonian is written as a weighted sum of these operators. Consequently, the total number of terms in the decomposition is proportional to the number of non‑zero entries (≈ 2N‑1) rather than the square of the matrix dimension. This yields an exponential reduction in the number of LCU terms compared with the Pauli decomposition.

Because quantum hardware can only implement unitary gates, the authors employ a “unitary completion” technique. For each weighted Sigma component 𝓗_j, they construct a unitary U_j acting on the system plus an ancilla such that U_j|0⟩ = 𝓗_j|ψ⟩/‖𝓗_j|ψ‖. This embeds the non‑unitary operation into a larger unitary transformation, allowing exact implementation of each term using standard gate sets.

The paper validates the approach on a quadratic model of Bernoulli’s equation. Numerical experiments demonstrate that increasing the truncation order N improves solution accuracy while the matrix dimension grows quadratically, yet the Sigma basis keeps the decomposition size linear. This confirms that the proposed representation captures the sparsity structure efficiently.

Beyond data loading, the authors discuss solving the truncated linear system with hybrid quantum‑classical algorithms. They outline a variational quantum eigensolver (VQE) framework where the cost function comprises linear, potential, and interaction contributions (⟨K⟩+⟨P⟩+⟨I⟩). Classical optimizers adjust variational parameters, while the quantum circuit evaluates expectation values of each 𝓗_j using the Sigma‑based circuits. This reduces the number of required measurements and circuit depth, making the approach more amenable to noisy intermediate‑scale quantum (NISQ) devices.

Finally, the authors acknowledge remaining challenges: (1) selecting an optimal truncation order that balances accuracy against resource demands; (2) extending the Sigma basis to more complex, possibly multi‑dimensional PDEs; (3) incorporating error mitigation and noise‑aware compilation to cope with hardware imperfections. They suggest that further theoretical analysis of truncation error propagation and experimental benchmarking on real quantum hardware are promising directions.

In summary, the work provides a comprehensive pipeline—from Carleman linearization through efficient Hamiltonian decomposition to unitary implementation and variational solution—addressing the critical post‑linearization bottlenecks that have limited quantum approaches to nonlinear dynamical systems. By dramatically reducing the number of LCU terms and offering concrete circuit constructions, the paper moves the field closer to practical quantum simulation of nonlinear phenomena.