Quantum algorithm for systems of linear equations with exponentially improved dependence on precision (1511.02306v2)

Abstract: Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\vec{x}=\vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time $\mathrm{poly}(\log N, 1/\epsilon)$, where $\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive.

• The paper introduces a quantum algorithm that achieves polylogarithmic scaling with 1/ε, markedly improving upon the polynomial dependence in HHL.
• It employs Fourier and Chebyshev series methods combined with quantum walks to efficiently represent the inverse of the matrix.
• The approach achieves query complexities of O(dκ log^(2.5)(κ/ε)) and O(dκ log^(2)(κd/ε)), significantly reducing the impact of the condition number.

A Quantum Algorithm for Solving Linear Systems with Enhanced Precision Dependence

The paper by Childs, Kothari, and Somma presents a quantum algorithm aimed at improving the solution of the Quantum Linear Systems Problem (QLSP) concerning precision dependence for quantum state outputs. The QLSP involves preparing a quantum state that is proportional to the solution of a linear system $Ax = b$, given an N x N sparse matrix $A$ and a vector $b$. The authors introduce an algorithm with complexity that scales polynomially with $\log(1/\epsilon)$, exponentially improving the dependence on the desired precision $\epsilon$ over the previously established Harrow-Hassidim-Lloyd (HHL) algorithm, which scales polynomially with $1/\epsilon$.

Methodology and Theoretical Basis

The proposed algorithm leverages a novel technique to implement quantum operators with Fourier or Chebyshev series representation, thereby circumventing the costly quantum phase estimation used in HHL's approach. This technique involves expressing the inverse of matrix A using a linear combination of unitaries that replaces the requirement for phase estimation with a more error-tolerant approach using quantum walks and linear combination of unitary operations. The Fourier approach approximates $1/x$ using a Fourier series that translates to Hamiltonian simulation forms, while the Chebyshev approach uses Chebyshev polynomials and quantum walks, achieving better error dependence on precision $\epsilon$.

Numerical Results and Complexity Analysis

The algorithm's notable improvement in precision manifests in its complexity: it is shown to be $O(d\kappa \log^{2.5}(\kappa/\epsilon))$ and $O(d\kappa \log^{2}(\kappa d/\epsilon))$ in query complexity for the Fourier and Chebyshev approaches, respectively, replacing the $O(d\kappa^2/\epsilon)$ complexity inherent in HHL's algorithm. Additionally, variable-time amplitude amplification (VTAA) aids in reducing the dependence on the condition number $\kappa$ from quadratic to nearly linear $\tilde{O}(d\kappa)$ by utilizing a gapped phase estimation strategy applied iteratively across different eigenvalue regions of $A$.

Practical and Theoretical Implications

The exponentially improved precision dependence presents considerable implications for quantum algorithms, particularly when they are utilized recursively. This advancement is significant when output precision must meet stringent inverse polynomial scalings to ensure effectiveness across the entire computation process, especially in higher-dimensional applications such as finite element methods using quantum algorithms. Furthermore, the approach provides complexity-theoretic insights, possibly promising new directions for exploiting QMA protocols considering small soundness-completeness gaps.

Speculations on Future Developments

The authors’ techniques in decomposition for linear systems could stimulate new frameworks in approximation theory applicable to broader quantum computing tasks. As quantum technologies evolve, the capacity to handle real-world sparse matrices and complex Hamiltonian systems using quantum resources with lowered precision and condition number dependency might see increasingly practical applications, ultimately pushing towards resolving computationally intense problems more efficiently compared to their classical counterparts.

In essence, the paper advances the state-of-the-art in quantum linear algebra algorithms by substantially addressing the limiting factor of precision dependence, laying a groundwork with potential ripple effects on the development of future quantum algorithms that require high precision with manageable resource bounds.