We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level $n$ in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small $n$ ). The quantum walk shows a polynomial algorithmic speedup in $n$ over the classical walk, which we conjecture to be of the order $2/3$ , based on strong trends in data.We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level $n$ in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small $n$). The quantum walk shows a polynomial algorithmic speedup in $n$ over the classical walk, which we conjecture to be of the order $2/3$, based on strong trends in data.

arXiv:1107.4201Physical Review ASep 8, 201113 citations

Quantum Walks and Search Algorithms

Renato Portugal

Springer eBooksJan 1, 2013478 citations

Discrete-time quantum walks and graph structures

Chris D. Godsil, Hanmeng Zhan

arXiv:1701.04474Journal of Combinatorial Theory Series AMay 16, 201929 citations

Automated Quantum Circuit Synthesis and Cost Estimation for the Binary Welded Tree Oracle

Mrityunjay Ghosh, Amlan Chakrabarti, Niraj Kumar Jha

Quantum computing is a new computational paradigm that promises an exponential speed-up over classical algorithms. To develop efficient quantum algorithms for problems of a non-deterministic nature, random walk is one of the most successful concepts employed. In this article, we target both continuous-time and discrete-time random walk in both the classical and quantum regimes. Binary Welded Tree (BWT), or glued tree, is one of the most well-known quantum walk algorithms in the continuous-time domain. Prior work implements quantum walk on the BWT with static welding. In this context, static welding is randomized but case-specific. We propose a solution to automatically generate the circuit for the Oracle for welding. We implement the circuit using the Quantum Assembly Language, which is a language for describing quantum circuits. We then optimize the generated circuit using the Fault-Tolerant Quantum Logic Synthesis tool for any BWT instance. Automatic welding enables us to provide a generalized solution for quantum walk on the BWT.Quantum computing is a new computational paradigm that promises an exponential speed-up over classical algorithms. To develop efficient quantum algorithms for problems of a non-deterministic nature, random walk is one of the most successful concepts employed. In this article, we target both continuous-time and discrete-time random walk in both the classical and quantum regimes. Binary Welded Tree (BWT), or glued tree, is one of the most well-known quantum walk algorithms in the continuous-time domain. Prior work implements quantum walk on the BWT with static welding. In this context, static welding is randomized but case-specific. We propose a solution to automatically generate the circuit for the Oracle for welding. We implement the circuit using the Quantum Assembly Language, which is a language for describing quantum circuits. We then optimize the generated circuit using the Fault-Tolerant Quantum Logic Synthesis tool for any BWT instance. Automatic welding enables us to provide a generalized solution for quantum walk on the BWT.

ACM Journal on Emerging Technologies in Computing SystemsJun 29, 20176 citations

Applications of quantum walk algorithms

de Jan Vos

Aug 1, 20060 citations

n

in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small

n

). The quantum walk shows a polynomial algorithmic speedup in

n

over the classical walk, which we conjecture to be of the order

2/3

, based on strong trends in data.