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Differentiable Quantum Computing for Large-scale Linear Control
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Proposes an end-to-end quantum algorithm for linear-quadratic regulator (LQR) control that combines a quantum Lyapunov-equation solver (block-encoded integral representation) with policy gradient descent, claiming O~(n/eps^1.5) cost.
Problem
Optimal feedback control (LQR) for large-scale linear dynamical systems, where classical methods scale as O(n^3) due to Riccati/Lyapunov equation solving.
Approach
A quantum-assisted differentiable simulator: a new quantum subroutine block-encodes the solution of the matrix Lyapunov equation via its integral representation, giving an analytic policy gradient estimate that feeds a classical policy gradient update on the feedback gain K, yielding linear convergence and super-quadratic speedup in state dimension n for m << n.
Key results
- Quantum policy gradient converges to K* within about 750 iterations versus 2x10^4 iterations for the classical model-free baseline on the mass-spring-damper example.
- Asymptotic end-to-end cost O~(n eps^-1.5) versus O~(n^3) for Schur and Newton-Kleinman classical methods.
- On a classical simulator, the method runs nearly three orders of magnitude faster than the baseline when scaling to g=64 masses.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the quantum policy-gradient algorithm instead of classical model-based policy gradient | LQR · LQR | time complexity: $\widetilde{\mathcal{O}}(n^3\cdot \poly(\varepsilon^{-1}))$ -> $\widetilde{\mathcal{O}}({n\varepsilon^{-1.5}})$ | Helped |
| Use the quantum policy-gradient algorithm instead of the classical model-free gradient-based method | Mass-spring-damper LQR · Mass-spring-damper, g=4 | iterations to converge: 750 vs. $2\times10^4$ | Helped |
| Use the proposed method instead of the classical method | Mass-spring-damper LQR · Mass-spring-damper, g=4 to 64 | runtime speedup: nearly 3 orders of magnitude | Helped |
Surprising findings
- The quantum gradient estimator is reported to be more robust and to converge faster in practice than zeroth-order stochastic estimators, though the numerical experiments are classically simulated.
Limitations
- Speedups are theoretical; experiments run on a classical simulator rather than quantum hardware.
- Analysis restricted to quadratic cost and linear dynamics; non-quadratic and nonlinear control not covered.
- Assumes sparse-access input model and quantum-accessible classical data structure (qRAM) for K.
Experimental setup
Datasets: mass-spring-damper system (synthetic LQR instance)
Benchmarks: mass-spring-damper LQR with g=4 and g=64 masses
Metrics: convergence iterations wall-clock runtime distance to K*
Baselines: Schur method Newton-Kleinman method model-based policy gradient model-free zeroth-order policy gradient
Builds on: policy gradient for LQR quantum linear system solver quantum eigenvalue transformation linear combination of unitaries
Closest papers by meaning
- Reinforcement Learning for Quantum Control under Physical Constraints (ICML 2025 · similarity 0.85)
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- Quantum speedup of non-linear Monte Carlo problems (NeurIPS 2025 · similarity 0.84)
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- PALQO: Physics-informed model for Accelerating Large-scale Quantum Optimization (NeurIPS 2025 · similarity 0.84)
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