Data compression can be achieved by reducing the dimensionality of high-dimensional but approxima... more Data compression can be achieved by reducing the dimensionality of high-dimensional but approximately low-rank datasets, which may in fact be described by the variation of a much smaller number of parameters. It often serves as a preprocessing step to surmount the curse of dimensionality and to gain efficiency, and thus it plays an important role in machine learning and data mining. In this paper, we present a quantum algorithm that compresses an exponentially large high-dimensional but approximately low-rank dataset in quantum parallel, by dimensionality reduction (DR) based on principal component analysis (PCA), the most popular classical DR algorithm. We show that the proposed algorithm achieves exponential speedup over the classical PCA algorithm when the original dataset are projected onto a polylogarithmically low-dimensional space. The compressed dataset can then be further processed to implement other tasks of interest, with significantly less quantum resources. As examples, we apply this algorithm to reduce data dimensionality for two important quantum machine learning algorithms, quantum support vector machine and quantum linear regression for prediction. This work demonstrates that quantum machine learning can be released from the curse of dimensionality to solve problems of practical importance.
This paper presents the Maximum Amplification Optimisation Algorithm (MAOA), a novel quantum algo... more This paper presents the Maximum Amplification Optimisation Algorithm (MAOA), a novel quantum algorithm designed for combinatorial optimisation in the restricted circuit depth context of near-term quantum computing. The MAOA first produces a quantum state in which the optimal solutions to a problem are amplified to the maximum extent possible subject to a given restricted circuit depth. Subsequent repeated preparation and measurement of this maximally amplified state produces solutions of the highest quality as efficiently as possible. The MAOA performs considerably better than other near-term quantum algorithms, such as the Quantum Approximate Optimisation Algorithm (QAOA), as it amplifies optimal solutions significantly more and does so without the computationally demanding variational procedure required by these other algorithms. Additionally, a restricted circuit depth modification of the existing Grover adaptive search is introduced. This modified algorithm is referred to as the restricted Grover adaptive search (RGAS), and provides a useful comparison to the MAOA. The MAOA and RGAS are simulated on a practical vehicle routing problem, a computationally demanding portfolio optimisation problem, and an arbitrarily large problem with normally distributed solution qualities. In all cases, the MAOA and RGAS are shown to provide substantial speedup over classical random sampling in finding optimal solutions, while the MAOA consistently outperforms the RGAS. The speedup provided by the MAOA is quantified by demonstrating numerical convergence to a theoretically derived upper bound.
QuOp MPI is a Python package designed for parallel simulation of quantum variational algorithms. ... more QuOp MPI is a Python package designed for parallel simulation of quantum variational algorithms. It presents an object-orientated approach to quantum variational algorithm design and utilises MPI-parallelised sparse-matrix exponentiation, the fast Fourier transform and parallel gradient evaluation to achieve the highly efficient simulation of the fundamental unitary dynamics on massively parallel systems. In this article, we introduce QuOp MPI and explore its application to the simulation of quantum algorithms designed to solve combinatorial optimisation algorithms, including the Quantum Approximation Optimisation Algorithm, the Quantum Alternating Operator Ansatz, and the Quantum Walk-assisted Optimisation Algorithm.
In this paper, we present a general quantum computation compiler, which maps any given quantum al... more In this paper, we present a general quantum computation compiler, which maps any given quantum algorithm to a quantum circuit consisting a sequential set of elementary quantum logic gates based on recursive cosinesine decomposition. The resulting quantum circuit diagram is provided by directly linking the package output written in LaTex to Qcircuit.tex <http. We illustrate the use of the Qcompiler package through various examples with full details of the derived quantum circuits. Besides its generality and simplicity, Qcompiler produces quantum circuits which reflect the symmetry of the systems under study.
Journal of Computational and Theoretical Nanoscience, Jul 1, 2013
We present several families of graphs that allow both efficient quantum walk implementations and ... more We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a 'black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O( √ n log(n)) elementary 2-qubit quantum gate operations to find a marked node.
Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fr... more Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the 'glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.
We present a novel methodological framework for quantum spatial search, generalising the Childs &... more We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone (CG) algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain O( √ N ) search on a variety of graphs. One important class is the n × n 3 rook graph, which has N = n 4 vertices. On this class of graphs the CG algorithm performs suboptimally, achieving only O(N -1/8 ) overlap after time O(N 5/8 ). Using the new alternating phase-walk framework, we show that O(1) overlap is obtained in O( √ N ) phase-walk iterations.
In this paper we present an efficiently scaling quantum algorithm which finds the size of the max... more In this paper we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for a pair of arbitrary graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with vanishingly small amplitude that even quantum algorithms such as Grover's search are not fast enough to distill the information efficiently. In order to extract the information we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires O(n 3 log 3 (n) log log(n)) elementary quantum gates and the nonlinear evolution under the Gross-Pitaevskii equation has a time scaling of O( g n 2 log 3 (n) log log(n)), where n is the number of vertices in each graph and g is the strength of the Gross-Pitaveskii non-linearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.
This paper examines the performance of spatial search where the Grover diffusion operator is repl... more This paper examines the performance of spatial search where the Grover diffusion operator is replaced by continuous-time quantum walks on a class of interdependent networks. We prove that for a set of optimal quantum walk times and marked vertex phase shifts, a deterministic algorithm for structured spatial search is established that finds the marked vertex with 100% probability. This improves on the Childs & Goldstone spatial search algorithm on the same class of graphs, which we show can only amplify to 50% probability. Our method uses π 2 √ 2 √ N marked vertex phase shifts for an N -vertex graph, making it comparable with Grover's algorithm for unstructured search. It is expected that this new framework can be readily extended to deterministic spatial search on other families of graph structures.
Quantum searching for one of N marked items in an unsorted database of n items is solved in O( n/... more Quantum searching for one of N marked items in an unsorted database of n items is solved in O( n/N ) steps using Grover's algorithm. Using nonlinear quantum dynamics with a Gross-Pitaevskii type quadratic nonlinearity, Childs and Young discovered an unstructured quantum search algorithm with a complexity O(min{1/g log(gn), √ n}), which can be used to find a marked item after o(log(n)) repetitions, where g is the nonlinearity strength . In this work we develop a structured search on a complete graph using a time dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime O(( where N ⊥ denotes the number of unmarked items and G is related to the time dependent nonlinearity. If N ⊥ ≤ N , we obtain a runtime O(1). We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 5.
In the noisy-intermediate-scale-quantum era, Variational Quantum Eigensolver (VQE) is a promising... more In the noisy-intermediate-scale-quantum era, Variational Quantum Eigensolver (VQE) is a promising method to study ground state properties in quantum chemistry, materials science, and condensed physics. However, general quantum eigensolvers are lack of systematical improvability, and achieve rigorous convergence is generally hard in practice, especially in solving strong-correlated systems. Here, we propose an Orbital Expansion VQE (OE-VQE) framework to construct an efficient convergence path. The path starts from a highly correlated compact active space and rapidly expands and converges to the ground state, enabling simulating ground states with much shallower quantum circuits. We benchmark the OE-VQE on a series of typical molecules including H6-chain, H10-ring and N2, and the simulation results show that proposed convergence paths dramatically enhance the performance of general quantum eigensolvers.
This paper presents the Maximum Amplification Optimisation Algorithm (MAOA), a novel quantum algo... more This paper presents the Maximum Amplification Optimisation Algorithm (MAOA), a novel quantum algorithm designed for combinatorial optimisation in the restricted circuit depth context of near-term quantum computing. The MAOA first produces a quantum state in which the optimal solutions to a problem are amplified to the maximum extent possible subject to a given restricted circuit depth. Subsequent repeated preparation and measurement of this maximally amplified state produces solutions of the highest quality as efficiently as possible. The MAOA performs considerably better than other near-term quantum algorithms, such as the Quantum Approximate Optimisation Algorithm (QAOA), as it amplifies optimal solutions significantly more and does so without the computationally demanding variational procedure required by these other algorithms. Additionally, a restricted circuit depth modification of the existing Grover adaptive search is introduced. This modified algorithm is referred to as the restricted Grover adaptive search (RGAS), and provides a useful comparison to the MAOA. The MAOA and RGAS are simulated on a practical vehicle routing problem, a computationally demanding portfolio optimisation problem, and an arbitrarily large problem with normally distributed solution qualities. In all cases, the MAOA and RGAS are shown to provide substantial speedup over classical random sampling in finding optimal solutions, while the MAOA consistently outperforms the RGAS. The speedup provided by the MAOA is quantified by demonstrating numerical convergence to a theoretically derived upper bound.
QuOp MPI is a Python package designed for parallel simulation of quantum variational algorithms. ... more QuOp MPI is a Python package designed for parallel simulation of quantum variational algorithms. It presents an object-orientated approach to quantum variational algorithm design and utilises MPI-parallelised sparse-matrix exponentiation, the fast Fourier transform and parallel gradient evaluation to achieve the highly efficient simulation of the fundamental unitary dynamics on massively parallel systems. In this article, we introduce QuOp MPI and explore its application to the simulation of quantum algorithms designed to solve combinatorial optimisation algorithms including the Quantum Approximation Optimisation Algorithm, the Quantum Alternating Operator Ansatz, and the Quantum Walk-assisted Optimisation Algorithm.
This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at n... more This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at near-term noisy intermediate-scale quantum computers. Recent work by Hodson et al. (2019) explored potential application of hybrid quantum-classical 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 Walk Optimisation Algorithm in finding high-quality solutions to the portfolio optimisation problem.
In this paper we investigate properties of continuous time chiral quantum walks, which possess co... more In this paper we investigate properties of continuous time chiral quantum walks, which possess complex valued edge weights in the underlying graph structure, together with an initial Gaussian wavefunction spread over a number of vertices. We demonstrate that, for certain graph topology and phase matching conditions, we are able to direct the flow of probability amplitudes in a specific direction inside the graph network. We design a quantum walk graph analogue of an optical circulator which is a combination of a cycle and semi-infinite chain graphs. Excitations input into the circulator from a semi-infinite chain are routed in a directionally biased fashion to output to a different semi-infinite chain. We examine in detail a two port circulator graph which spatially separates excitations flowing back in forth between the two semi-finite chains to directionally occupy the top or bottom half of the cycle portion of the circulator. This setup can be used, for example, to detect non-Mar...
We present a highly efficient quantum circuit for performing continuous time quantum walks (CTQWs... more 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 high-quality solutions. The efficient quantum circuit holds the promise of finding high-quality solutions to certain classes of NP-hard combinatorial problems such as the Travelling Salesman Problem, maximum set splitting, graph partitioning, and lattice path optimization.
Large scale complex systems, such as social networks, electrical power grid, database structure, ... more Large scale 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 ).
In this paper we show how using complex valued edge weights in a graph can completely suppress th... more In this paper we show how using complex valued edge weights in a graph can completely suppress the flow of probability amplitude in a continuous time quantum walk to specific vertices of the graph when the edge weights, graph topology and initial state of the quantum walk satisfy certain conditions. The conditions presented in this paper are derived from the so-called chiral quantum walk, a variant of the continuous time quantum walk which incorporates directional bias with respect to site transfer probabilities between vertices of a graph by using complex edge weights. We examine the necessity to break the time reversal symmetry in order to achieve zero transfer in continuous time quantum walks. We also consider the effect of decoherence on zero transfer and suggest that this phenomena may be used to detect decoherence in the system.
Circulant matrices are an important family of operators, which have a wide range of applications ... more Circulant matrices are an important family of operators, which have a wide range of applications in science and engineering-related fields. They are, in general, non-sparse and non-unitary. In this paper, we present efficient quantum circuits to implement circulant operators using fewer resources and with lower complexity than existing methods. Moreover, our quantum circuits can be readily extended to the implementation of Toeplitz, Hankel and block circulant matrices. Efficient quantum algorithms to implement the inverses and products of circulant operators are also provided, and an example application in solving the equation of motion for cyclic systems is discussed.
Various quantum-walk based algorithms have been proposed to analyse and rank the centrality of gr... more Various quantum-walk based algorithms have been proposed to analyse and rank the centrality of graph vertices. However, issues arise when working with directed graphs -the resulting non-Hermitian Hamiltonian leads to non-unitary dynamics, and the total probability of the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using PT-symmetric quantum walks, allowing probability conserving non-unitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set, and can be extended to cover interdependent networks of directed graphs. Previous work has shown centrality measures based on the CTQW provide an eigenvector-like quantum centrality; using the PT-symmetric framework, we extend these centrality algorithms to directed graphs with a significantly reduced Hilbert space compared to previous proposals. In certain cases, this centrality measure provides an advantage over classical algorithms used in network analysis, for example by breaking vertex rank degeneracy. Finally, we perform a statistical analysis over ensembles of random graphs, and show strong agreement with the classical PageRank measure on directed acyclic graphs.
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Papers by Jingbo B Wang