Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing

We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state $|x⟩$ that is proportional to the solution of the system of linear equations $A\overrightarrow{x}=\overrightarrow{b}$. The time complexities of our algorithms are $O\mathbf{(}{\kappa }^2\log (\kappa )/\epsilon \mathbf{)}$ and $O\mathbf{(}\kappa \log (\kappa )/\epsilon \mathbf{)}$, where $\kappa$ is the condition number of $A$ and $\epsilon$ is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of $A$, the projector onto the initial state $|b⟩$, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of $\kappa$. Like previous methods, our techniques yield an exponential quantum speed-up under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.

Figure 1

The inverse of the error for the two quantum algorithms as a function of $q$, the number of steps in the RM. Subscript $Q$ refers to the quantum algorithm with complexity that is almost quadratic in $\kappa$ and $L$ to the quantum algorithm with complexity almost linear in $\kappa$. $n_{\mathrm{rep}}$ is the number of repetitions of the algorithm. The results are for two randomly generated matrices $A$ with $N=16$, $\kappa \approx 10$, and $N=32$, $\kappa \approx 50$.