Quantum Linear System Solvers on the IBMQ

Recent investment from the private sector in quantum computing has led to several commercially available quantum computing devices. Among them, the first publicly available quantum gate computer is the IBMQ. There are two operational devices which are a 5 qubit and 16 qubit machines. IBM is planning in the near future to have an operational 50 qubit machine. Alongside their development of hardware devices they are developing a software package QISKit. This is completely open source and available to the public so that anybody can run a circuit on a quantum device. This package can initialize the quantum computer to any state, build a circuit with around 50 gates, convert this to a circuit that can be physically implemented, and output the probabilities of the final state.

In 2009, Harrow, Hassidim, and Llyod (HHL) proposed a quantum algorithm to solve linear-systems. The solution to linear-systems is an important and widespread numerical task in scientific computing. The HHL algorithm can be used for any sparse matrix. It scales approximately as O(log N) with N the size of the linear system, which is a factor of N better than the Conjugate Gradient (CG) method, the best known classical algorithm. The conditioning number and desired precision also play an important role in the overall performance of the HHL algorithm, and in certain cases the HLL algorithm may not outperform the classical algorithms. Recent improvements have been made to alleviate this constraint, and it is currently an active area of research.

This work investigates the feasibility of running the HHL algorithm on the IBMQ quantum computer. IBM is poised to release a 50 qubit machine in the near future, and this would allow very large linear-systems to be solved. However, the decoherence and noise in the IBMQ limits the number of quantum gates that can be used reliably. The number of gates needed to solve a linear-system is currently the limiting factor, not the number of qubits to store the linear system. The two main components to the HHL algorithm are the Quantum Fourier Transform and Phase Estimation. The results of implementing and running these algorithms on the IBMQ will be presented and discussed.