论文标题
离散时间量子步行在Qudit系统中
Discrete-time Quantum Walks in Qudit Systems
论文作者
论文摘要
量子步行对开发量子算法和量子模拟有显着贡献。在这里，我们在$ d $维量子域中介绍了第一个一维量子步行，其中$ d> 2 $，并在使用任意有限尺寸量子逻辑的情况下显示出其对电路实现的等价性，以利用更大的状态空间的优势，这有助于减少与常规量子系统相比的量子量的运行时间。我们为在任何有限维量子系统中在一维位置空间中实施离散时间量子步行（DTQW）提供有效的量子电路，当尺寸使用适当的逻辑映射奇数时，步行者在多Qudit状态上演变的位置空间的逻辑映射是奇数的。对于各种Qudit状态空间的示例电路，我们还以$ n $ qudit $ d $ d $ -ary量子系统探索可扩展性。此外，已经研究了使用$ 2D $维二维硬币空间在$ d $ d $维晶格上扩展到$ d $ d $二维的DTQW，其中$ d> = 2 $。此后，已经描绘了用于实施可扩展$ d $二维DTQW的电路设计。最后，我们展示了在各种搜索空间上使用不同的硬币实现DTQW的电路设计。
Quantum walks contribute significantly to developing quantum algorithms and quantum simulations. Here, we introduce a first of its kind one-dimensional quantum walk in the $d$-dimensional quantum domain, where $d>2$, and show its equivalence for circuit realization in an arbitrary finite-dimensional quantum logic for utilizing the advantage of larger state space, which helps to reduce the run-time of the quantum walks as compared to the conventional binary quantum systems. We provide efficient quantum circuits for the implementation of discrete-time quantum walks (DTQW) in one-dimensional position space in any finite-dimensional quantum system when the dimension is odd using an appropriate logical mapping of the position space on which a walker evolves onto the multi-qudit states. With example circuits for various qudit state spaces, we also explore scalability in terms of $n$-qudit $d$-ary quantum systems. Further, the extension of one-dimensional DTQW to $d$-dimensional DTQW using $2d$-dimensional coin space on $d$-dimensional lattice has been studied, where $d>=2$. Thereafter, the circuit design for the implementation of scalable $d$-dimensional DTQW in $d$-ary quantum systems has been portrayed. Lastly, we exhibit the circuit design for the implementation of DTQW using different coins on various search spaces.
