Limitations of Quantum Simulation Examined by Simulating a Pairing Hamiltonian Using Nuclear Magnetic Resonance

Quantum simulation uses a well-known quantum system to predict the behavior of another quantum system. Certain limitations in this technique arise, however, when applied to specific problems, as we demonstrate with a theoretical and experimental study of an algorithm proposed by Wu, Byrd, and Lidar [Phys. Rev. Lett. 89, 057904 (2002).] to find the low-lying spectrum of a pairing Hamiltonian. While the number of elementary quantum gates required scales polynomially with the size of the system, it increases inversely to the desired error bound $ϵ$. Making such simulations robust to decoherence using fault tolerance requires an additional factor of $\sim 1/ϵ$ gates. These constraints, along with the effects of control errors, are illustrated using a three qubit NMR system.

Figure 1

Three qubit quantum circuit for the unitary $U_{XX}$, implemented using method $W1$ and simulating $H_2$ (see text). $U_{YY}$ is done by replacing rotations about $\widehat{y}$ with rotations about $\widehat{x}$. For $H_1$, $\frac{\pi }{2}$ pulses are applied to qubit $c$ in parallel with those on $a$ and $b$, and the decoupling $X$ pulse is omitted.

Figure 2

Frequency-domain spectra of Hamiltonian $H_2$ obtained using methods $W1$ (marked by circles) and $W2$ (diamonds). The solid lines are fits to the time dependent data. The width of the exact curve is taken to be the dephasing rate ($1/\tau$) of the ${}^{13}\mathrm{C}$ nucleus.