Randomly Walking 1D Quantum Harmonic Oscillator. Averaged Transition Probabilities

Connectivity is a Poor Indicator of Fast Quantum Search

A randomly walking quantum particle evolving by Schrödinger's equation searches on $d$-dimensional cubic lattices in $O(\sqrt{N})$ time when $d \ge 5$, and with progressively slower runtime as $d$ decreases. This suggests that graph connectivity (including vertex, edge, algebraic, and normalized algebraic connectivities) is an indicator of fast quantum search, a belief supported by fast quantum search on complete graphs, strongly regular graphs, and hypercubes, all of which are highly connected. In this paper, we show this intuition to be false by giving two examples of graphs for which the op…

Completeness is Unnecessary for Fast Nonlinear Quantum Search

Although strongly regular graphs and the hypercube are not complete, they are "sufficiently complete" such that a randomly walking quantum particle asymptotically searches on them in the same $Θ(\sqrt{N})$ time as on the complete graph, the latter of which is precisely Grover's algorithm. We show that physically realistic nonlinearities of the form $f(|ψ|^2)ψ$ can speed up search on sufficiently complete graphs, depending on the nonlinearity and graph. Thus nonlinear (quantum) computation can retain its power even when a degree of noncompleteness is introduced.

Faster Quantum Walk Search on a Weighted Graph

A randomly walking quantum particle evolving by Schrödinger's equation searches for a unique marked vertex on the "simplex of complete graphs" in time $Θ(N^{3/4})$. In this paper, we give a weighted version of this graph that preserves vertex-transitivity, and we show that the time to search on it can be reduced to nearly $Θ(\sqrt{N})$. To prove this, we introduce two novel extensions to degenerate perturbation theory: an adjustment that distinguishes the weights of the edges, and a method to determine how precisely the jumping rate of the quantum walk must be chosen.

基于二维随机行走者的量子光子门

Quantum Photonic Gates with Two-Dimensional Random Walkers

本文研究利用连续时间二维随机行走光子在硅介质中通过逆向设计实现高保真概率性量子门，提出C-NOT门和X门，并分析其空间关联性，推动线性光学量子计算的实用化。

利用张量方法重建随机采样的量子波函数

Reconstruction of Randomly Sampled Quantum Wavefunctions using Tensor Methods

本研究提出多种基于张量网络的量子波函数重建算法，通过最小化局域块平均Renyi纠缠熵补全波函数，仅需少量随机采样振幅即可高保真重建一维局域哈密顿量基态。

Global Symmetry is Unnecessary for Fast Quantum Search

Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a \emph{three}-dimensional subspace, but most have no \emph{global} symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs nevertheless achieves the full quantum speedup o…