Qiang, X; Wang, Y; Xue, Y; Ge, R; Chen, L; Liu, Y; Huang, A; Fu, X; Xu, P; Yi, T; Xu, F; Deng, M; Wang, J; Meinecke, J; Matthews, J; Cai, X; Yang, X; Wu, J Implementing graphtheoretic quantum algorithms on a silicon photonic quantum walk processor Journal Article Science Advances, 7 , pp. eabb8375, 2021. Abstract  Links: @article{Qiang2021,
title = {Implementing graphtheoretic quantum algorithms on a silicon photonic quantum walk processor},
author = {X. Qiang and Y. Wang and Y. Xue and R. Ge and L. Chen and Y. Liu and A. Huang and X. Fu and P. Xu and T. Yi and F. Xu and M. Deng and J. Wang and J. Meinecke and J. Matthews and X. Cai and X. Yang and J. Wu},
url = {https://advances.sciencemag.org/content/7/9/eabb8375},
doi = {10.1126/sciadv.abb8375},
year = {2021},
date = {20210226},
journal = {Science Advances},
volume = {7},
pages = {eabb8375},
abstract = {Applications of quantum walks can depend on the number, exchange symmetry and indistinguishability of the particles involved, and the underlying graph structures where they move. Here, we show that silicon photonics, by exploiting an entanglementdriven scheme, can realize quantum walks with full control over all these properties in one device. The device we realize implements entangled twophoton quantum walks on any fivevertex graph, with continuously tunable particle exchange symmetry and indistinguishability. We show how this simulates singleparticle walks on larger graphs, with size and geometry controlled by tuning the properties of the composite quantum walkers. We apply the device to quantum walk algorithms for searching vertices in graphs and testing for graph isomorphisms. In doing so, we implement up to 100 sampled time steps of quantum walk evolution on each of 292 different graphs. This opens the way to largescale, programmable quantum walk processors for classically intractable applications.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Applications of quantum walks can depend on the number, exchange symmetry and indistinguishability of the particles involved, and the underlying graph structures where they move. Here, we show that silicon photonics, by exploiting an entanglementdriven scheme, can realize quantum walks with full control over all these properties in one device. The device we realize implements entangled twophoton quantum walks on any fivevertex graph, with continuously tunable particle exchange symmetry and indistinguishability. We show how this simulates singleparticle walks on larger graphs, with size and geometry controlled by tuning the properties of the composite quantum walkers. We apply the device to quantum walk algorithms for searching vertices in graphs and testing for graph isomorphisms. In doing so, we implement up to 100 sampled time steps of quantum walk evolution on each of 292 different graphs. This opens the way to largescale, programmable quantum walk processors for classically intractable applications. 
Matwiejew, Edric; Wang, Jingbo QSW_MPI: A framework for parallel simulation of quantum stochastic walks Journal Article Computer Physics Communications, 260 , pp. 107724, 2021, ISSN: 00104655. Abstract  Links: @article{MATWIEJEW2021107724,
title = {QSW_MPI: A framework for parallel simulation of quantum stochastic walks},
author = {Edric Matwiejew and Jingbo Wang},
url = {http://www.sciencedirect.com/science/article/pii/S0010465520303581},
doi = {https://doi.org/10.1016/j.cpc.2020.107724},
issn = {00104655},
year = {2021},
date = {20210101},
journal = {Computer Physics Communications},
volume = {260},
pages = {107724},
abstract = {QSW_MPI is a Python package developed for timeseries simulation of continuoustime quantum stochastic walks. This model allows for the study of Markovian open quantum systems in the Lindblad formalism, including a generalisation of the continuoustime random walk and continuoustime quantum walk. Consisting of a Python interface accessing parallelised Fortran libraries utilising sparse data structures, QSW_MPI is scalable to massively parallel computers, which makes possible the simulation of a wide range of walk dynamics on directed and undirected graphs of arbitrary complexity. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
QSW_MPI is a Python package developed for timeseries simulation of continuoustime quantum stochastic walks. This model allows for the study of Markovian open quantum systems in the Lindblad formalism, including a generalisation of the continuoustime random walk and continuoustime quantum walk. Consisting of a Python interface accessing parallelised Fortran libraries utilising sparse data structures, QSW_MPI is scalable to massively parallel computers, which makes possible the simulation of a wide range of walk dynamics on directed and undirected graphs of arbitrary complexity. 
Wu, Tong; Izaac, J A; Li, ZiXi; Wang, Kai; Chen, ZhaoZhong; Zhu, Shining; Wang, J B; Ma, XiaoSong Experimental ParityTime Symmetric Quantum Walks for Centrality Ranking on Directed Graphs Journal Article Phys. Rev. Lett., 125 , pp. 240501, 2020. Abstract  Links: @article{PhysRevLett.125.240501,
title = {Experimental ParityTime Symmetric Quantum Walks for Centrality Ranking on Directed Graphs},
author = {Tong Wu and J A Izaac and ZiXi Li and Kai Wang and ZhaoZhong Chen and Shining Zhu and J B Wang and XiaoSong Ma},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.125.240501},
doi = {10.1103/PhysRevLett.125.240501},
year = {2020},
date = {20201201},
journal = {Phys. Rev. Lett.},
volume = {125},
pages = {240501},
publisher = {American Physical Society},
abstract = {Quantum walks (QW) are of crucial importance in the development of quantum information processing algorithms. Recently, several quantum algorithms have been proposed to implement network analysis, in particular to rank the centrality of nodes in networks represented by graphs. Employing QW in centrality ranking is advantageous comparing to certain widely used classical algorithms (e.g. PageRank) because QW approach can lift the vertex rank degeneracy in certain graphs. However, it is challenging to implement a directed graph via QW, since it corresponds to a nonHermitian Hamiltonian and thus cannot be accomplished by conventional QW. Here we report the realizations of centrality rankings of both a threevertex and fourvertex directed graphs with paritytime (PT) symmetric quantum walks. To achieve this, we use highdimensional photonic quantum states, optical circuitries consisting of multiple concatenated interferometers and dimension dependent loss. Importantly, we demonstrate the advantage of QW approach experimentally by breaking the vertex rank degeneracy in a fourvertex graph. Our work shows that PTsymmetric quantum walks may be useful for realizing advanced algorithm in a quantum network.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Quantum walks (QW) are of crucial importance in the development of quantum information processing algorithms. Recently, several quantum algorithms have been proposed to implement network analysis, in particular to rank the centrality of nodes in networks represented by graphs. Employing QW in centrality ranking is advantageous comparing to certain widely used classical algorithms (e.g. PageRank) because QW approach can lift the vertex rank degeneracy in certain graphs. However, it is challenging to implement a directed graph via QW, since it corresponds to a nonHermitian Hamiltonian and thus cannot be accomplished by conventional QW. Here we report the realizations of centrality rankings of both a threevertex and fourvertex directed graphs with paritytime (PT) symmetric quantum walks. To achieve this, we use highdimensional photonic quantum states, optical circuitries consisting of multiple concatenated interferometers and dimension dependent loss. Importantly, we demonstrate the advantage of QW approach experimentally by breaking the vertex rank degeneracy in a fourvertex graph. Our work shows that PTsymmetric quantum walks may be useful for realizing advanced algorithm in a quantum network. 
Wang, Kunkun; Shi, Yuhao; Xiao, Lei; Wang, Jingbo; Joglekar, Yogesh N; Xue, Peng Experimental realization of continuoustime quantum walks on directed graphs and their application in PageRank Journal Article Optica, 7 (11), pp. 1524–1530, 2020. Abstract  Links: @article{Wang:20,
title = {Experimental realization of continuoustime quantum walks on directed graphs and their application in PageRank},
author = {Kunkun Wang and Yuhao Shi and Lei Xiao and Jingbo Wang and Yogesh N Joglekar and Peng Xue},
url = {http://www.osapublishing.org/optica/abstract.cfm?URI=optica7111524},
doi = {10.1364/OPTICA.396228},
year = {2020},
date = {20201101},
journal = {Optica},
volume = {7},
number = {11},
pages = {15241530},
publisher = {OSA},
abstract = {PageRank is an algorithm used by Google Search to rank web pages in their search engine results. An important step for quantum networks is to quantize the classical protocol as quantum mechanics provides computational resources that can be used to outperform classical algorithms. In this paper, we experimentally realize continuoustime quantum walks for directed graphs with nonHermitian adjacency matrices by using linear optical circuits and single photons. We find that the node classical centrality in a directed graph is correlated with the maximum node probability resulting from a continuoustime quantum walk and then demonstrate PageRank. Our work opens up an avenue of applications of quantum information in reallife tasks.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
PageRank is an algorithm used by Google Search to rank web pages in their search engine results. An important step for quantum networks is to quantize the classical protocol as quantum mechanics provides computational resources that can be used to outperform classical algorithms. In this paper, we experimentally realize continuoustime quantum walks for directed graphs with nonHermitian adjacency matrices by using linear optical circuits and single photons. We find that the node classical centrality in a directed graph is correlated with the maximum node probability resulting from a continuoustime quantum walk and then demonstrate PageRank. Our work opens up an avenue of applications of quantum information in reallife tasks. 
Schofield, Callum; Wang, Jingbo B; Li, Yuying Quantum walk inspired algorithm for graph similarity and isomorphism Journal Article Quantum Information Processing, 19 (9), pp. 281, 2020, ISSN: 15731332. Abstract  Links: @article{Schofield2020,
title = {Quantum walk inspired algorithm for graph similarity and isomorphism},
author = {Callum Schofield and Jingbo B Wang and Yuying Li},
url = {https://doi.org/10.1007/s11128020027587},
doi = {10.1007/s11128020027587},
issn = {15731332},
year = {2020},
date = {20200824},
journal = {Quantum Information Processing},
volume = {19},
number = {9},
pages = {281},
abstract = {Largescale complex systems, such as social networks, electrical power grid, database structure, consumption pattern or brain connectivity, are often modelled using network graphs. Valuable insight can be gained by measuring similarity between network graphs in order to make quantitative comparisons. Since these networks can be very large, scalability and efficiency of the algorithm are key concerns. More importantly, for graphs with unknown labelling, this graph similarity problem requires exponential time to solve using existing algorithms. In this paper, we propose a quantum walk inspired algorithm, which provides a solution to the graph similarity problem without prior knowledge on graph labelling. This algorithm is capable of distinguishing between minor structural differences, such as between strongly regular graphs with the same parameters. The algorithm has a polynomial complexity, scaling with $$O(n^9)$$O(n9).},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Largescale complex systems, such as social networks, electrical power grid, database structure, consumption pattern or brain connectivity, are often modelled using network graphs. Valuable insight can be gained by measuring similarity between network graphs in order to make quantitative comparisons. Since these networks can be very large, scalability and efficiency of the algorithm are key concerns. More importantly, for graphs with unknown labelling, this graph similarity problem requires exponential time to solve using existing algorithms. In this paper, we propose a quantum walk inspired algorithm, which provides a solution to the graph similarity problem without prior knowledge on graph labelling. This algorithm is capable of distinguishing between minor structural differences, such as between strongly regular graphs with the same parameters. The algorithm has a polynomial complexity, scaling with $$O(n^9)$$O(n9). 
Marsh, S; Wang, J B Combinatorial optimization via highly efficient quantum walks Journal Article Phys. Rev. Research, 2 , pp. 023302, 2020. Abstract  Links: @article{PhysRevResearch.2.023302,
title = {Combinatorial optimization via highly efficient quantum walks},
author = {S Marsh and J B Wang},
url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.023302},
doi = {10.1103/PhysRevResearch.2.023302},
year = {2020},
date = {20200601},
journal = {Phys. Rev. Research},
volume = {2},
pages = {023302},
publisher = {American Physical Society},
abstract = {We present a highly efficient quantum circuit for performing continuous time quantum walks (CTQWs) over an exponentially large set of combinatorial objects, provided that the objects can be indexed efficiently. CTQWs form the core mixing operation of a generalized version of the quantum approximate optimization algorithm, which works by “steering” the quantum amplitude into highquality solutions. The efficient quantum circuit holds the promise of finding highquality solutions to certain classes of NPhard combinatorial problems such as the Travelling Salesman Problem, maximum set splitting, graph partitioning, and lattice path optimization.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We present a highly efficient quantum circuit for performing continuous time quantum walks (CTQWs) over an exponentially large set of combinatorial objects, provided that the objects can be indexed efficiently. CTQWs form the core mixing operation of a generalized version of the quantum approximate optimization algorithm, which works by “steering” the quantum amplitude into highquality solutions. The efficient quantum circuit holds the promise of finding highquality solutions to certain classes of NPhard combinatorial problems such as the Travelling Salesman Problem, maximum set splitting, graph partitioning, and lattice path optimization. 
Slate, N; Matwiejew, E; Marsh, S; Wang, J B Quantum walkbased portfolio optimisation Miscellaneous 2020. Abstract  Links: @misc{slate2020quantum,
title = {Quantum walkbased portfolio optimisation},
author = {N Slate and E Matwiejew and S Marsh and J B Wang},
url = {https://arxiv.org/abs/2011.08057},
year = {2020},
date = {20200101},
abstract = {This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at nearterm noisy intermediatescale quantum computers. Recent work by Hodson et. al (2019) explored potential application of hybrid quantumclassical algorithms to the problem of financial portfolio rebalancing. In particular, they deal with the portfolio optimisation problem using the Quantum Approximate Optimisation Algorithm and the Quantum Alternating Operator Ansatz. In this paper, we demonstrate substantially better performance using a newly developed Quantum Walkbased Optimisation Algorithm in finding highquality solutions to the portfolio optimisation problem.
},
keywords = {},
pubstate = {published},
tppubtype = {misc}
}
This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at nearterm noisy intermediatescale quantum computers. Recent work by Hodson et. al (2019) explored potential application of hybrid quantumclassical algorithms to the problem of financial portfolio rebalancing. In particular, they deal with the portfolio optimisation problem using the Quantum Approximate Optimisation Algorithm and the Quantum Alternating Operator Ansatz. In this paper, we demonstrate substantially better performance using a newly developed Quantum Walkbased Optimisation Algorithm in finding highquality solutions to the portfolio optimisation problem.

Jay, Gareth; Debbasch, Fabrice; Wang, Jingbo A systematic method to building Dirac quantum walks coupled to electromagnetic fields Journal Article Quantum Information Processing , 19 , pp. 422, 2020. Abstract  Links: @article{Jay2020,
title = {A systematic method to building Dirac quantum walks coupled to electromagnetic fields},
author = {Gareth Jay and Fabrice Debbasch and Jingbo Wang},
doi = {10.1007/s1112802002933w},
year = {2020},
date = {20200101},
journal = {Quantum Information Processing },
volume = {19},
pages = {422},
abstract = {A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac quantum walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a 3D walk on the cubic lattice. Recently introduced DQWs on the triangular lattice are then rederived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method’s effectiveness, a new 3D walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed. Furthermore, we show the simulation of the established DQWs can be efficiently implemented on a quantum computer.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac quantum walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a 3D walk on the cubic lattice. Recently introduced DQWs on the triangular lattice are then rederived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method’s effectiveness, a new 3D walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed. Furthermore, we show the simulation of the established DQWs can be efficiently implemented on a quantum computer.

Ruan, Yue; Marsh, Samuel; Xue, Xilin; Liu, Zhihao; Wang, Jingbo The Quantum Approximate Algorithm for Solving Traveling Salesman Problem Journal Article Computers, Materials & Continua, 63 (3), pp. 1237–1247, 2020, ISSN: 15462226. Abstract  Links: @article{cmc.2020.010001,
title = {The Quantum Approximate Algorithm for Solving Traveling Salesman Problem},
author = {Yue Ruan and Samuel Marsh and Xilin Xue and Zhihao Liu and Jingbo Wang},
url = {http://www.techscience.com/cmc/v63n3/38872},
doi = {10.32604/cmc.2020.010001},
issn = {15462226},
year = {2020},
date = {20200101},
journal = {Computers, Materials & Continua},
volume = {63},
number = {3},
pages = {12371247},
abstract = {The Quantum Approximate Optimization Algorithm (QAOA) is an
algorithmic framework for finding approximate solutions to combinatorial optimization
problems. It consists of interleaved unitary transformations induced by two operators
labelled the mixing and problem Hamiltonians. To fit this framework, one needs to
transform the original problem into a suitable form and embed it into these two
Hamiltonians. In this paper, for the wellknown NPhard Traveling Salesman Problem
(TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional
approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges
(routes) connecting each pair of cities to qubits, which decreases the search space
significantly in comparison to other approaches. As a result, our method can achieve a
higher probability for the shortest roundtrip route with only half the number of qubits
consumed compared to IBM Q’s approach. We argue the formalization approach
presented in this paper would lead to a generalized framework for finding, in the context
of QAOA, highquality approximate solutions to NP optimization problems.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Quantum Approximate Optimization Algorithm (QAOA) is an
algorithmic framework for finding approximate solutions to combinatorial optimization
problems. It consists of interleaved unitary transformations induced by two operators
labelled the mixing and problem Hamiltonians. To fit this framework, one needs to
transform the original problem into a suitable form and embed it into these two
Hamiltonians. In this paper, for the wellknown NPhard Traveling Salesman Problem
(TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional
approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges
(routes) connecting each pair of cities to qubits, which decreases the search space
significantly in comparison to other approaches. As a result, our method can achieve a
higher probability for the shortest roundtrip route with only half the number of qubits
consumed compared to IBM Q’s approach. We argue the formalization approach
presented in this paper would lead to a generalized framework for finding, in the context
of QAOA, highquality approximate solutions to NP optimization problems. 