凸动态规划的量子加速算法
Quantum speedups for convex dynamic programming
摘要：
提出了一种求解凸值函数动态规划问题的量子算法。对于具有大小为$N$的$d$维状态空间的线性离散时间系统，所提出的算法输出值函数在时间上的量子力学表示$O(T γ^{dT}\mathrm{polylog}(N,(T/\varepsilon)^{d}))$，其中$\varepsilon$是解的精度，$T$是时间范围，并且$γ$是取决于成本函数的条件数的问题特定参数。这使得我们能够在时间$O(T γ^{dT}\sqrt{N}\,\mathrm{polylog}(N,(T/\varepsilon)^{d}))$中的任何固定状态下评估值函数，并且可以通过求解凸规划来恢复相应的最优动作。类的优化问题，我们的算法可以适用于包括可证明硬随机动态程序。最后，我们表明，该算法获得了二次加速比（最多为多对数因子）相比，经典的贝尔曼方法在一些动态程序具有连续的状态空间，有$γ=1$。
英文摘要：
We present a quantum algorithm to solve dynamic programming problems with convex value functions. For linear discrete-time systems with a $d$-dimensional state space of size $N$, the proposed algorithm outputs a quantum-mechanical representation of the value function in time $O(T γ^{dT}\mathrm{polylog}(N,(T/\varepsilon)^{d}))$, where $\varepsilon$ is the accuracy of the solution, $T$ is the time horizon, and $γ$ is a problem-specific parameter depending on the condition numbers of the cost functions. This allows us to evaluate the value function at any fixed state in time $O(T γ^{dT}\sqrt{N}\,\mathrm{polylog}(N,(T/\varepsilon)^{d}))$, and the corresponding optimal action can be recovered by solving a convex program. The class of optimization problems to which our algorithm can be applied includes provably hard stochastic dynamic programs. Finally, we show that the algorithm obtains a quadratic speedup (up to polylogarithmic factors) compared to the classical Bellman approach on some dynamic programs with continuous state space that have $γ=1$.
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