Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.
Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.
Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases—such as for low-rank matrices—dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems if there is no restriction on the rank. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols, and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. We feel this will be helpful in developing efficient quantum protocols for many other problems.
Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.
Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ∈Rd of coefficients is constrained in either ℓ1-norm (for Lasso) or in ℓ2-norm (for Ridge). We study the complexity of quantum algorithms for finding ε-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of d by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in d, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.
Motivated by recent progress in quantum technologies and in particular quantum software, research and industrial communities have been trying to discover new applications of quantum algorithms such as quantum optimization and machine learning. Regardless of which hardware platform these novel algorithms operate on, whether it is adiabatic or gate based, from theoretical point of view, they are performing drastically better than their classical counterparts. Two promising areas of quantum algorithms quantum machine learning and quantum optimization. These are based on performing matrix operations using quantum states and operation, in order to speed up data analysis where quantum computing can efficiently work with high dimensional vectors. Motivated by that, quantum inspired algorithms (e.g. for recommendation systems and principal component analysis) are developed to cope with high dimensionality using probabilistic techniques that are inspire from quantum computing. In this paper we review recent progress in the area of quantum inspired algorithms for low rank matrix approximation. We further explore the possibility of using parametrized complexity for such algorithms to refine practical complexity analysis. Finally, we conjecture that quantum inspired algorithms that use low rank approximation and also sample and query technique for input representations are Fixed Parameter Tractable (FPT).
Quantum algorithms for linear algebra that access a matrix A through quantum random access memory (QRAM) prepare quantum states proportional to the solutions of linear systems and related problems. Dequantized classical algorithms, when given sampling and query access to A, return sampling and query access to an approximate solution. Both kinds of algorithms use a number of queries that is polynomial in the normalized Frobenius norm ρ=∥A∥F/∥A∥ and does not directly depend on the dimension. For linear systems at fixed condition number and accuracy, quantum algorithms use O(ρ) queries, whereas the best dequantized algorithms use O(ρ4). We prove that this quartic gap is tight in the worst case, up to a logarithmic factor. For every constant κ0>1, we construct dense linear systems with condition number at most κ0 for which sampling from the solution within a sufficiently small constant total variation distance requires Ω(ρ4/logρ) classical queries but only Θ(ρ) quantum queries. The same separation holds, for errors below smaller thresholds, for positive-definite systems, least squares, ridge regression, matrix powers and square roots, imaginary-time and real-time Hamiltonian evolution, and sampling from the top principal component. Our results therefore show that many quantum algorithms for linear algebra, achieve a near-quartic speedup in the worst case, over classical algorithms with sampling and query access. All of these classical lower bounds follow from the same framework, which is based on a lower bound for sampling from distributions that depend on a fixed set of Fourier coefficients of an unknown Boolean function.
Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ ∈ R^d of coefficients is constrained in either ℓ1-norm (for Lasso) or in ℓ2-norm (for Ridge). We study the complexity of quantum algorithms for finding ε-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of d by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in d, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.