Quantum Regularized Least Squares

Abstract

Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general 2 -regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result.To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variabletime quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

An improved quantum algorithm for low-rank rigid linear regressions with vector solution outputs

“…In this section, as an application of Theorem 12, we reconsider the problem of solving linear regressions -a problem of great concern to the classical and quantum communities. Regarding this problem, many quantum algorithms were proposed so far [1,2,9,10,16,28,49], to name a few here. These quantum algorithms, which only cost polylogarithmic in the dimension, usually return a quantum state of the solution.…”

“…The HHL algorithm implies that this problem can be solved on a quantum computer with complexity polylog in the dimension. So far, many improvements have been made to the HHL algorithm, ranging from the dependence on the precision, and the condition number to the sparsity of the matrix, e.g., see [1,9,10,16,27,40,56].…”