Role of single-qubit decoherence time in adiabatic quantum computation

We have studied numerically the evolution of an adiabatic quantum computer in the presence of a Markovian Ohmic environment by considering Ising spin-glass systems with up to 20 qubits independently coupled to this environment via two conjugate degrees of freedom. The required computation time is demonstrated to be of the same order as that for an isolated system and is not limited by the single-qubit decoherence time $T_2^{\ast }$, even when the minimum gap is much smaller than the temperature and decoherence-induced level broadening. For small minimum gap, the system can be described by an effective two-state model coupled only longitudinally to environment.

Figure 1

(Color online) Energy spectrum for a 20-qubit instance with a very small minimum gap: $g_m/E\approx 5\times {10}^{-4}$. Only the first 50 energy levels of the total $\sim {10}^6$ are shown. The dashed line in the bottom represents the ground-state entanglement calculated using Meyer and Wallach entanglement measure (12).

Figure 2

(Color online) Probability of success $P_{0f}$ as a function of $t_f$ for the 20-qubit instance of Fig. 1. The solid lines are calculated with $\left(\eta =0.2\right)$ and without $\left(\eta =0\right)$ coupling to the environment. Other parameters are $E=10\text{GHz}$ and $T=25\text{mK}$. The case without coupling to the environment is compared with the pure Landau-Zener behavior using Eq. (2) (see dashed blue line marked by LZ). The excellent agreement confirms that for the coherent evolution, the two-state model is sufficient to calculate the probability. The (black) dashed line, marked by TSM, is obtained using analytical formula (17) obtained using a two-state model. It asymptotically shows the same behavior as the numerical result (solid line marked by $\eta =0.2$). The dotted line is numerical calculation with $\eta =0.2\theta \left(T-g\right)$, which eliminates relaxation after the anticrossing (see the text for description).

Figure 3

(Color online) (a) The rms value of the matrix elements of the Pauli matrices between the first two states as defined in Eq. (13). Solid (blue) line is $M_z$; dashed-dotted (black) line is $M_x$. We have also plotted the matrix element $\left(M_z^{\text{TSM}}\right)$ of ${\tau }_z$ between the two energy levels of a two-state model described by Hamiltonian (15). (b) The same curves zoomed near the anticrossing show qualitative agreement (except for a prefactor) between all three curves.