31 research outputs found
Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning
We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tangâs breakthrough quantum-inspired algorithm for recommendation systems [STOCâ19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of GilyĂ©n et al. [STOCâ19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffices to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: âÂČ-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive
Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
Semidefinite programming (SDP) is a central topic in mathematical
optimization with extensive studies on its efficient solvers. In this paper, we
present a proof-of-principle sublinear-time algorithm for solving SDPs with
low-rank constraints; specifically, given an SDP with constraint matrices,
each of dimension and rank , our algorithm can compute any entry and
efficient descriptions of the spectral decomposition of the solution matrix.
The algorithm runs in time
given access to a sampling-based low-overhead data structure for the constraint
matrices, where is the precision of the solution. In addition, we
apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix
multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2)
sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to
compute the matrix exponential required in the MMW framework, we introduce two
new techniques that may be of independent interest:
Weighted sampling: assuming sampling access to each individual
constraint matrix , we propose a procedure that gives a
good approximation of .
Symmetric approximation: we propose a sampling procedure that gives
the \emph{spectral decomposition} of a low-rank Hermitian matrix . To the
best of our knowledge, this is the first sampling-based algorithm for spectral
decomposition, as previous works only give singular values and vectors.Comment: 37 pages, 1 figure. To appear in the Proceedings of the 45th
International Symposium on Mathematical Foundations of Computer Science (MFCS
2020
Quantum computing for finance
Quantum computers are expected to surpass the computational capabilities of
classical computers and have a transformative impact on numerous industry
sectors. We present a comprehensive summary of the state of the art of quantum
computing for financial applications, with particular emphasis on stochastic
modeling, optimization, and machine learning. This Review is aimed at
physicists, so it outlines the classical techniques used by the financial
industry and discusses the potential advantages and limitations of quantum
techniques. Finally, we look at the challenges that physicists could help
tackle
Quantum-Inspired Classical Algorithms for Singular Value Transformation
A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the
quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS
2017). The resulting algorithm, classical but "quantum-inspired", efficiently
computes a low-rank approximation of the users' preference matrix. Subsequent
works have shown how to construct efficient quantum-inspired algorithms for
approximating the pseudo-inverse of a low-rank matrix as well, which can be
used to (approximately) solve low-rank linear systems of equations. In the
present paper, we pursue this line of research and develop quantum-inspired
algorithms for a large class of matrix transformations that are defined via the
singular value decomposition of the matrix. In particular, we obtain classical
algorithms with complexity polynomially related (in most parameters) to the
complexity of the best quantum algorithms for singular value transformation
recently developed by Chakraborty, Gily\'{e}n and Jeffery (ICALP 2019) and
Gily\'{e}n, Su, Low and Wiebe (STOC19).Comment: 19 page
An Improved Classical Singular Value Transformation for Quantum Machine Learning
We study quantum speedups in quantum machine learning (QML) by analyzing the
quantum singular value transformation (QSVT) framework. QSVT, introduced by
[GSLW, STOC'19, arXiv:1806.01838], unifies all major types of quantum speedup;
in particular, a wide variety of QML proposals are applications of QSVT on
low-rank classical data. We challenge these proposals by providing a classical
algorithm that matches the performance of QSVT in this regime up to a small
polynomial overhead.
We show that, given a matrix , a vector , a bounded degree- polynomial , and linear-time
pre-processing, we can output a description of a vector such that in time. This improves upon the
best known classical algorithm [CGLLTW, STOC'20, arXiv:1910.06151], which
requires time, and narrows the gap with QSVT, which, after linear-time
pre-processing to load input into a quantum-accessible memory, can estimate the
magnitude of an entry to error in
time.
Our key insight is to combine the Clenshaw recurrence, an iterative method
for computing matrix polynomials, with sketching techniques to simulate QSVT
classically. We introduce several new classical techniques in this work,
including (a) a non-oblivious matrix sketch for approximately preserving
bi-linear forms, (b) a new stability analysis for the Clenshaw recurrence, and
(c) a new technique to bound arithmetic progressions of the coefficients
appearing in the Chebyshev series expansion of bounded functions, each of which
may be of independent interest.Comment: 62 pages, v3: fixed bug, runtime exponent now 11 instead of 9; v2:
revised abstract to clarify result
Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms
Several quantum algorithms for linear algebra problems, and in particular
quantum machine learning problems, have been "dequantized" in the past few
years. These dequantization results typically hold when classical algorithms
can access the data via length-squared sampling. In this work we investigate
how robust these dequantization results are. We introduce the notion of
approximate length-squared sampling, where classical algorithms are only able
to sample from a distribution close to the ideal distribution in total
variation distance. While quantum algorithms are natively robust against small
perturbations, current techniques in dequantization are not. Our main technical
contribution is showing how many techniques from randomized linear algebra can
be adapted to work under this weaker assumption as well. We then use these
techniques to show that the recent low-rank dequantization framework by Chia,
Gily\'en, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework
for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based
on the Quantum Singular Value Transformation, can be generalized to the case of
approximate length-squared sampling access to the input. We also apply these
results to obtain a robust dequantization of many quantum machine learning
algorithms, including quantum algorithms for recommendation systems, supervised
clustering and low-rank matrix inversion.Comment: 55 page
Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension
We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired âsmoothâ function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [15], our results indicate that more low-rank quantum algorithms can be effectively âdequantisedâ into classical length-square sampling algorithms
Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture
The Quantum Singular Value Transformation (QSVT) is a recent technique that
gives a unified framework to describe most quantum algorithms discovered so
far, and may lead to the development of novel quantum algorithms. In this paper
we investigate the hardness of classically simulating the QSVT. A recent result
by Chia, Gily\'en, Li, Lin, Tang and Wang (STOC 2020) showed that the QSVT can
be efficiently "dequantized" for low-rank matrices, and discussed its
implication to quantum machine learning. In this work, motivated by
establishing the superiority of quantum algorithms for quantum chemistry and
making progress on the quantum PCP conjecture, we focus on the other main class
of matrices considered in applications of the QSVT, sparse matrices.
We first show how to efficiently "dequantize", with arbitrarily small
constant precision, the QSVT associated with a low-degree polynomial. We apply
this technique to design classical algorithms that estimate, with constant
precision, the singular values of a sparse matrix. We show in particular that a
central computational problem considered by quantum algorithms for quantum
chemistry (estimating the ground state energy of a local Hamiltonian when
given, as an additional input, a state sufficiently close to the ground state)
can be solved efficiently with constant precision on a classical computer. As a
complementary result, we prove that with inverse-polynomial precision, the same
problem becomes BQP-complete. This gives theoretical evidence for the
superiority of quantum algorithms for chemistry, and strongly suggests that
said superiority stems from the improved precision achievable in the quantum
setting. We also discuss how this dequantization technique may help make
progress on the central quantum PCP conjecture.Comment: 32 pages, accepted to STOC 2022; v2: minor corrections and revisions
(especially in Section 4.2
An improved quantum-inspired algorithm for linear regression
We give a classical algorithm for linear regression analogous to the quantum
matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review
Letters'09] for low-rank matrices [Wossnig et al., Physical Review Letters'18],
when the input matrix is stored in a data structure applicable for
QRAM-based state preparation.
Namely, given an with minimum singular value
and which supports certain efficient -norm importance sampling
queries, along with a , we can output a description of an
such that in
time, improving on previous "quantum-inspired" algorithms in this line of
research by a factor of [Chia et
al., STOC'20]. The algorithm is stochastic gradient descent, and the analysis
bears similarities to those of optimization algorithms for regression in the
usual setting [Gupta and Sidford, NeurIPS'18]. Unlike earlier works, this is a
promising avenue that could lead to feasible implementations of classical
regression in a quantum-inspired setting, for comparison against future quantum
computers.Comment: 16 pages, bug fixe