arXiv subjects

Changpeng Shao

Publications and source records attributed to Changpeng Shao.

At least 37 records · Page 2Linked to original sources

arXiv2023

Faster quantum-inspired algorithms for solving linear systems

We establish an improved classical algorithm for solving linear systems in a model analogous to the QRAM that is used by quantum linear solvers. Precisely, for the linear system $A\x = \b$, we show that there is a classical algorithm that outputs a data structure for $\x$ allowing sampling and querying to the entries, where $\x$ is such that $\|\x - A^{+}\b\|\leq ε\|A^{+}\b\|$. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is $\widetilde{O}(κ_F^4 κ^2/ε^2 )$, where $κ_F = \|A\|_F\|A^{+}\|$ and $κ= \|A\|\|A^{+}\|$. This improves the previous best algorithm [Gily{é}n, Song and Tang, arXiv:2009.07268] of complexity $\widetilde{O}(κ_F^6 κ^6/ε^4)$. Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that when $A$ is row sparse, this method already returns an approximate solution $\x$ in time $\widetilde{O}(κ_F^2)$, while the best quantum algorithm known returns $\ket{\x}$ in time $\widetilde{O}(κ_F)$ when $A$ is stored in the QRAM data structure. As a result, assuming access to QRAM and if $A$ is row sparse, the speedup based on current quantum algorithms is quadratic.

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arXiv2021

Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than the standard one based on quantum phase estimation. Moreover, it works for a wider case than symmetric: $B$ is invertible, $B^{-1}A$ is diagonalizable and all the eigenvalues are real.

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arXiv2021

Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance

Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.

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arXiv2021

Quantum algorithms for learning a hidden graph and beyond

We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al.\ for a "gapped" version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.

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arXiv2020

Computing eigenvalues of diagonalizable matrices in a quantum computer

Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix $M$ is $s$-sparse, the complexity is $\widetilde{O}(sρ^2 κ^2/ε^2)$ for diagonalizable matrices that only have real eigenvalues, and $\widetilde{O}(sρ\|M\|_{\max} /ε^2)$ for normal matrices. Here $ρ$ is an upper bound of the eigenvalues, $κ$ is the conditioning of the eigenvalue problem, and $ε$ is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.

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arXiv2020

Data classification by quantum radial basis function networks

Radial basis function (RBF) network is a third layered neural network that is widely used in function approximation and data classification. Here we propose a quantum model of the RBF network. Similar to the classical case, we still use the radial basis functions as the activation functions. Quantum linear algebraic techniques and coherent states can be applied to implement these functions. Differently, we define the state of the weight as a tensor product of single-qubit states. This gives a simple approach to implement the quantum RBF network in the quantum circuits. Theoretically, we prove that the training is almost quadratic faster than the classical one. Numerically, we demonstrate that the quantum RBF network can solve binary classification problems as good as the classical RBF network. While the time used for training is much shorter.

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arXiv2020

Quantum vs. classical algorithms for solving the heat equation

Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension $d \ge 2$, there is an at most quadratic quantum speedup using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative approach based on a quantum algorithm for linear equations is never faster than the best classical algorithms.

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arXiv2019

Randomized Row and Column Iterative Methods with a Quantum Computer

We consider the quantum implementations of the two classical iterative solvers for a system of linear equations, including the Kaczmarz method which uses a row of coefficient matrix in each iteration step, and the coordinate descent method which utilizes a column instead. These two methods are widely applied in big data science due to their very simple iteration schemes. In this paper we use the block-encoding technique and propose fast quantum implementations for these two approaches, under the assumption that the quantum states of each row or each column can be efficiently prepared. The quantum algorithms achieve exponential speed up at the problem size over the classical versions, meanwhile their complexity is nearly linear at the number of steps.

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arXiv2019

An Improved Algorithm for Quantum Principal Component Analysis

Principal component analysis is an important dimension reduction technique in machine learning. In [S. Lloyd, M. Mohseni and P. Rebentrost, Nature Physics 10, 631-633, (2014)], a quantum algorithm to implement principal component analysis on quantum computer was obtained by computing the Hamiltonian simulation of unknown density operators. The complexity is $O((\log d)t^2/ε)$, where $d$ is the dimension, $t$ is the evolution time and $ε$ is the precision. We improve this result into $O((\log d)t^{1+\frac{1}{k}}/ε^{\frac{1}{k}})$ for arbitrary constant integer $k\geq 1$. As a result, we show that the Hamiltonian simulation of low-rank dense Hermitian matrices can be implemented in the same time.

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arXiv2018

Quantum Regularized Least Squares Solver with Parameter Estimate

In this paper we propose a quantum algorithm to determine the Tikhonov regularization parameter and solve the ill-conditioned linear equations, for example, arising from the finite element discretization of linear or nonlinear inverse problems. For regularized least squares problem with a fixed regularization parameter, we use the HHL algorithm and work on an extended matrix with smaller condition number. For the determination of the regularization parameter, we combine the classical L-curve and GCV function, and design quantum algorithms to compute the norms of regularized solution and the corresponding residual in parallel and locate the best regularization parameter by Grover's search. The quantum algorithm can achieve a quadratic speedup in the number of regularization parameters and an exponential speedup in the dimension of problem size.

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arXiv2018

A Quantum Model for Multilayer Perceptron

Multilayer perceptron is the most common used class of feed-forward artificial neural network. It contains many applications in diverse fields such as speech recognition, image recognition, and machine translation software. To cater for the fast development of quantum machine learning, in this paper, we propose a new model to study multilayer perceptron in quantum computer. This contains the tasks to prepare the quantum state of the output signal in each layer and to establish the quantum version of learning algorithm about the weights in each layer. We will show that the corresponding quantum versions can achieve at least quadratic speedup or even exponential speedup over the classical algorithms. This provide us an efficient method to study multilayer perceptron and its applications in machine learning in quantum computer. Finally, as an inspiration, an exponential fast learning algorithm (based on Hebb's learning rule) of Hopfield network will be proposed.

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arXiv2018

Quantum Algorithm to Cubic Spline Interpolation

HHL algorithm \cite{harrow} to solve linear system is a powerful and efficient quantum technique to deal with many matrix operations (such as matrix multiplication, powers and inversion). It inspires many applications in quantum machine learning \cite{biamonte, dunjko}. However, due to the restrictions of HHL algorithm itself, many quantum machine learning algorithms also share one or two restrictions. The most common restrictions include quantum state preparation, condition number and Hamiltonian simulation. In this work, we first give an efficient quantum algorithm to achieve quantum state preparation, which actually achieves an exponential speedup than the algorithms given in \cite{clader,lloyd13}. Then we provide an application of HHL algorithm in cubic spline interpolation problem. We will show that in this problem, the condition number is small, the preparation of quantum state is efficient based on the new algorithm we proposed and the Hamiltonian simulation is efficiently implemented. So the quantum algorithm obtained by HHL algorithm towards this problem actually achieves an exponential speedup than any classical algorithm with no restrictions. This can be viewed as another application of HHL algorithm with no restrictions after the work of Clader et al \cite{clader} in studying electromagnetic scattering cross-section.

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arXiv2018

From linear combination of quantum states to Grover's searching algorithm

Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [Childs-linear-system,Clader,Long11], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. %Since in iteration algorithms, $m$ often refers to the iteration steps, which is small if the iteration algorithms are efficient. Moreover, as an application of the new LCU, three new quantum algorithms to the searching problem will be proposed, which will provide us new insights into Grover's searching algorithm. We also show that the problem of LCU is closely related to the problem of if we can efficiently implement $U^t$ for $0<t<1$ when $U$ is an efficiently implemented unitary operator? This problem is not hard to solve. However, it becomes inefficient when it contains a strict requirement on precision, such as in Grover's algorithm. Finally, as an application of the new LCU technique, we will show that the quantum state of any real classical vector can be prepared efficiently in quantum computer. So this solves the "input problem" in quantum computer efficiently.

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arXiv2018

Quantum Arnoldi and conjugate gradient iteration algorithm

Arnoldi method and conjugate gradient method are important classical iteration methods in solving linear systems and estimating eigenvalues. Their efficiency often affected by the high dimension of the space, where quantum computer can play a role in. In this work, we establish their corresponding quantum algorithms. To achieve high efficiency, a new method about linear combination of quantum states will be proposed. The final complexity of quantum Arnoldi iteration method is $O(m^{3+\log (m/ε)}(\log n)^2 /ε^4)$ and the final complexity of quantum conjugate gradient iteration method is $O(m^{1+\log m/ε} (\log n)^2 κ/ε)$, where $ε$ is precision parameter, $m$ is the iteration steps, $n$ is the dimension of space and $κ$ is the condition number of the coefficient matrix of the linear system the conjugate gradient method works on. Compared with the classical methods, whose complexity are $O(mn^2+m^2n)$ and $O(mn^2)$ respectively, these two quantum algorithms provide us more efficient methods to solve linear systems and to compute eigenvalues and eigenvectors of general matrices. Different from the work \cite{rebentros}, the complexity here is almost polynomial in the iteration steps. Also this work is more general than the iteration method considered in \cite{kerenidis}.

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arXiv2018

Quantum Algorithms to Matrix Multiplication

In this paper, we study quantum algorithms of matrix multiplication from the viewpoint of inputting quantum/classical data to outputting quantum/classical data. The main target is trying to overcome the input and output problem, which are not easy to solve and many quantum algorithms will encounter, to study matrix operations in quantum computer with high efficiency. And solving matrix multiplication will be the first step. We propose three quantum algorithms to matrix multiplication based on swap test, SVE and HHL. From the point of making fewer assumptions, swap test method works the best than the other two. We also show that the quantum algorithm of matrix multiplication with classical input and output data by swap test achieves the best complexity $\widetilde{O}(n^2/ε)$ with no assumptions. This is proved by giving an efficient quantum algorithm in polynomial time to solve the input problem, that is to prepare the quantum states of the classical data efficiently. Other contributions of this paper include: (1). Extending swap test to a more general form that is suitable to deal with quantum data in parallel, which will have further applications in other matrix operations. (2). Generalizing SVE technique such that it applies to any matrix (not just Hermitian) directly only with quantum data. (3). Proposing two new efficient quantum algorithms to prepare quantum states of classical data, which solves the input problem efficiently than other quantum algorithms.

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arXiv2018

Quantum Circulant Preconditioner for Linear System of Equations

We consider the quantum linear solver for $Ax=b$ with the circulant preconditioner $C$. The main technique is the singular value estimation (SVE) introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in ITCS 2017]. However, some modifications of SVE should be made to solve the preconditioned linear system $C^{-1} Ax = C^{-1} b$. Moreover, different from the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett., 2013], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of $C$ and $C^{-1} A$, as well as the Frobenius norm $\|A\|_F$.

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arXiv2018

Reconsider HHL algorithm and its related quantum machine learning algorithms

HHL quantum algorithm to solve linear systems is one of the most important subroutines in many quantum machine learning algorithms. In this work, we present and analyze several other caveats in HHL algorithm, which have been ignored in the past. Their influences on the efficiency, accuracy and practicability of HHL algorithm and several related quantum machine learning algorithms will be discussed. We also found that these caveats affect HHL algorithm much deeper than the already noticed caveats. In order to obtain more practical quantum machine learning algorithms with less assumptions based on HHL algorithm, we should pay more attention to these caveats.

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arXiv2018

An Improvement of Rota's Straightening Algorithm

In bracket algebra, the calculation of invariant division and invariant Gröbner basis proposed in \cite{li 2014} rely on straightening algorithm. Until now, there are at least three different types of straightening algorithms, among which Rota's straightening algorithm has the best efficiency. However, there exists a flaw in Rota's straightening algorithm, i.e., it needs find all the straight bracket monomials with the same content as the input beforehand, which is quite difficult. So in this paper, we will propose a new straightening algorithm based on dual bracket, which is a new concept of Young tableau. In this new straightening algorithm, we only need to find a few number of straight bracket monomials in each step instead of finding them all in one step. And so it is an improvement of Rota's straightening algorithm. According to our tests, this new straightening algorithm reflects more advantages when the dimension and the degree increase. Moreover, this straightening algorithm still works when Rota's straightening algorithm fails.

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