Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing

Adiabatic quantum computation is of fundamental importance in the field of quantum computation as it offers an alternative approach to the gate-based model for the manipulation of quantum systems. Recently, an interesting work [arXiv:1805.10549] indicated that we can solve a linear equation system via an algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm in a four-qubit nuclear magnetic resonance system by determining the solution of an eight-dimensional linear equation $A\mathbf{x}=\mathbf{b}$. The result is by far the maximum-dimensional linear equation solution with a limited number of qubits in experiments, which include some ingenious simplifications. Our experiment provides the possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.

Figure 1

(a) Molecule structure of ${}^{13}\mathrm{C}$-labeled crotonic acid. C1–C4 are used as four qubits in the experiment, whereas all ${}^1\mathrm{H}$'s are decoupled throughout the experiment. (b) Molecule parameters of the sample: the chemical shifts and $J$ couplings (in hertz) are listed by the diagonal and off-diagonal elements, respectively. T2's(in seconds) are also shown at the bottom.

Figure 2

Experimental results for the first algorithm. (a) (Energy and fidelity). The light and dark blue solid lines are theoretical values of the first-two energy levels of the time-dependent Hamiltonian, respectively. The solid points (black below) represent the experimental energy results, and the corresponding experimental fidelities are shown by the circle points above. (b) (Final solution). The real and imaginary parts of the theoretical (light blue bars) and experimental (dark blue bars) final quantum states of solution $|\mathbf{x}\rangle$ are shown. The numbers labeled present the corresponding differences between experimental and theoretical values.

Figure 3

Experimental results for the second algorithm. The solid lines are theoretical values of the energy spectrum of the time-dependent Hamiltonian. The light blue line is the middle level of the spectrum, and the other lines represent the closest eight energy levels. The black points represent experimental energy results in each step of our experiment.