Robust Variational Ground-State Solvers via Dissipative Quantum Feedback Models

We propose a variational framework for solving ground-state problems of open quantum systems governed by quantum stochastic differential equations (QSDEs). This formulation naturally accommodates bosonic operators, as commonly encountered in quantum chemistry and quantum optics. By parameterizing a dissipative quantum optical system, we minimize its steady-state energy to approximate the ground state of a target Hamiltonian. The system converges to a unique steady state regardless of its initial condition, and the design inherently guarantees physical realizability. To enhance robustness against persistent disturbances, we incorporate H-infinity control into the system architecture. Numerical comparisons with the quantum approximate optimization algorithm (QAOA) highlight the method’s structural advantages, stability, and physical implementability. This framework is compatible with experimental platforms such as cavity quantum electrodynamics (QED) and photonic crystal circuits.

💡 Research Summary

The paper introduces a novel variational framework for finding ground‑state energies of quantum Hamiltonians that naturally involve bosonic modes, such as those arising in quantum chemistry and quantum optics. Instead of mapping the problem onto qubits, the authors work directly with continuous‑variable quantum optical systems described by quantum stochastic differential equations (QSDEs). The open system is characterized by a triple (S, L, H) where the Hamiltonian H is quadratic in annihilation operators (a†Ωa) and the coupling operator L = C a models interaction with vacuum input fields. By employing the Hudson‑Parthasarathy formalism, the Heisenberg‑picture dynamics are cast into a linear stochastic system dν(t) = A ν(t) dt + B dW(t) using doubled‑up notation.

The target cost Hamiltonian, typical for electronic structure problems, contains both quadratic (∑ h_i a†_i a_i) and quartic (∑ g_ij a†_i a†_j a_j a_i) terms. The authors define a variational cost J(θ) = J₁ + J₂, where J₁ = Tr(Q₁ S₁) depends on the steady‑state first‑order covariance S₁, and J₂ = Tr(Q₂ S₂) depends on the second‑order covariance S₂. Both covariances are functions of the system matrices A(θ) and B(θ), which are parameterized by a real vector θ. Crucially, the parameterization is constructed from a set of basis matrices {Y_i, B_i} such that the physical realizability (PR) condition A + A♭ + B B♭ = 0 holds for every θ. This guarantees that the resulting linear quantum system can be implemented with actual optical components (e.g., cavities, waveguides, nonlinear crystals).

Steady‑state covariances are obtained by solving algebraic Lyapunov equations: A S₁ + S₁ A† + B F_w B† = 0 for the first order, and a more involved equation (A⊗I + I⊗A) S₂ + S₂ (A†⊗I + I⊗A†) + (B F_w B†)⊗S₁ + S₁⊗(B F_w B†) + M = 0 for the second order, where M captures higher‑order moment contributions. While solving these equations classically scales poorly with system size, the authors argue that in a laboratory the system naturally relaxes to its steady state, allowing direct experimental measurement of the required moments.

Optimization of J(θ) is performed with Simultaneous Perturbation Stochastic Approximation (SPSA). SPSA requires only two noisy evaluations of the cost per iteration, making it well‑suited for quantum hardware where each measurement is expensive. The algorithm iteratively updates θ using a stochastic gradient estimate g_t =