Conserved Quantities in Linear and Nonlinear Quantum Search

Abstract: In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size $N$ using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian $H_0$ is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian $H(t) = H_0 + \lambda|\psi|^2$, which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive, $\lambda > 0$, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of $H_0 + \frac{1}{2} \lambda|\psi|^2$ is conserved. The third algorithm utilizes attractive interactions, corresponding to $\lambda < 0$. In this case there is a time-varying critical function for the jumping rate $\gamma_c(t)$ that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of $H(t)/[\gamma_c(t) N]$ is conserved.