Quantum algorithm for solving linear systems of equations (0811.3171v3)

Abstract: Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

• The paper demonstrates exponential speedup for solving linear systems by leveraging quantum phase estimation and Hamiltonian simulation.
• It introduces a framework that prepares the state |b⟩ and approximates A⁻¹|b⟩ through selective eigenvalue inversion with polylogarithmic complexity in system size.
• The authors address error sources and optimality by incorporating amplitude amplification and preconditioning techniques for matrices with small condition numbers.

Quantum Algorithm for Linear Systems of Equations

The paper "Quantum Algorithm for Linear Systems of Equations" by Harrow, Hassidim, and Lloyd introduces a quantum algorithm that demonstrates exponential speedup over its classical counterparts for solving linear systems of equations. The problem at hand is a fundamental one across many disciplines, including science and engineering, involving the challenge of finding a vector $x$ such that $Ax = b$, for given matrix $A$ and vector $b$. Instead of obtaining the entire solution vector, the algorithm focuses on estimating useful properties of the solution.

Key Contributions and Methodology

1. Quantum Speedup: The presented quantum algorithm operates in a time complexity that is polylogarithmic in the size of $N$ and polynomial in the condition number $\kappa$ of $A$, providing an exponential improvement over the classical approaches, which require polylinear time in $N$.
2. Algorithmic Framework: The algorithm uses well-established quantum techniques including phase estimation and Hamiltonian simulation. Initially, a state $|b \rangle$, representing the vector $b$, is prepared. Subsequently, through phase estimation, the algorithm decomposes this state in the eigenbasis of the Hermitian matrix $A$. This decomposition enables the approximation of $A^{-1} |b \rangle$ by selectively inverting the eigenvalues.
3. Condition Number Consideration: A significant aspect affecting the performance of this quantum algorithm is the condition number $\kappa$ of the matrix $A$. The authors acknowledge that their algorithm is most advantageous when $\kappa$ is small. They discuss methods to handle ill-conditioned matrices, such as using preconditioning to improve $\kappa$.
4. Error and Success Probability: The paper carefully discusses sources of error, primarily stemming from phase estimation and the probabilistic nature of quantum measurement. The algorithm incorporates techniques like amplitude amplification to mitigate these issues, ensuring a polynomial dependence on $\kappa$ and the desired precision $\epsilon$.
5. Complexity and Optimality: The authors establish that their algorithm's performance is near optimal based on current complexity-theoretic assumptions. They argue that substantially improving the dependency on $\kappa$ or $\epsilon$ would imply solving known hard problems for classical algorithms with unexpected efficiency levels.

Implications and Future Directions

The implications of this algorithm are profound in areas where solving large systems of linear equations is critical and time-consuming, such as optimization, quantum physics simulations, and financial modeling. Beyond theoretical interest, this algorithm might someday offer practical advantages as quantum hardware further develops.

The quantum matrix inversion procedure introduced here is speculative yet promising, opening future research avenues in extending the algorithm towards more generalized operators and exploring efficient implementations of the required subroutines in quantum systems. Additionally, exploring different preconditioning strategies tailored for quantum systems presents a fertile ground for further improvement.

This paper marks a significant step in demonstrating quantum advantages in problems outside the field of quantum mechanics purely, suggesting paths forth for continued research both in refining quantum algorithms and optimizing them for various practical applications. As quantum hardware matures, the theoretical findings of this work could eventually translate into impactful computational tools.