Concetti Chiave
Quantum algorithms are developed to efficiently estimate quantities of interest such as kinetic energy in realistic classical mechanical systems with dissipation and forcing, as well as to solve the Riccati equation and the linear quadratic regulator problem in optimal control.
Sintesi
The content presents quantum algorithms for simulating realistic classical mechanical systems and solving optimal control problems.
Key highlights:
- Quantum algorithms are developed to estimate the kinetic energy of classical mechanical systems with dissipation and forcing, extending prior work on ideal coupled oscillators.
- It is shown that estimating the kinetic energy of damped coupled oscillators remains BQP-hard, indicating a quantum advantage even in the presence of dissipation.
- Quantum algorithms are presented to solve the Riccati equation, a nonlinear differential equation ubiquitous in optimal control theory, for regimes where the strength of the nonlinearity is asymptotically greater than the dissipation.
- The solution to the Riccati equation is then used to solve the linear quadratic regulator problem, an example of the Hamilton-Jacobi-Bellman equation.
- The algorithms leverage techniques from quantum linear systems, Hamiltonian simulation, and block encoding of classical data to enable efficient quantum solutions to these problems of practical relevance.
Statistiche
Quantum algorithms for simulating classical mechanical systems with dissipation and forcing scale polynomially with the logarithm of the system dimension.
Estimating the kinetic energy of damped coupled oscillators is BQP-hard when the strength of the damping term is bounded by an inverse polynomial in the number of qubits.
The quantum algorithm for the Riccati equation can handle nonlinearity that is asymptotically greater than the dissipation, going beyond prior limitations.
Citazioni
"Our results show that quantum algorithms for differential equations, especially for ordinary linear differential equations, can go quite far in solving problems of practical relevance."
"We show that even in the presence of damping, approximating the kinetic energy of an arbitrary sparse oscillator network is BQP hard (provided that the strength of damping is inverse polynomial in the number of qubits)."
"To our knowledge, this is the first example of any nonlinear differential equation that can be solved when the strength of the nonlinearity is asymptotically greater than the amount of dissipation."