Decoherence in a dynamical quantum phase transition

Motivated by the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm, which displays a first-order quantum phase transition. For site-independent and site-dependent coupling strengths as well as different operator couplings, the results show (in contrast to first-order transitions) that the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins or qubits). This might limit the scalability of the corresponding adiabatic quantum algorithm.

Figure 1

Sketch of the lowest eigenvalues of a Hamiltonian $H(g)$ as a function of some external parameter $g$ for a first-order quantum phase transition. At the critical point $g=g_c$, the ground state changes from $|{\Psi }_{<}(g)\rangle$ to $|{\Psi }_{>}(g)\rangle$: (a) a level-crossing and (b) an avoided level-crossing.

Figure 2

(a) Sketch of the two lowest-energy eigenvalues of the Grover Hamiltonian. (b) In the continuum limit, this corresponds to the time evolution of the energy landscape for a first-order transition. The (green) dot in the energy landscape denotes the ground state.

Figure 3

The ground-state energy of the Grover Hamiltonian and its first-order derivative. The discontinuity in the first-order derivative of the ground state suggests the first-order quantum phase transition in adiabatic quantum search algorithm.

Figure 4

(a) Sketch of the two lowest energy levels of the Ising Hamiltonian given in Eq. (5) (b) The time evolution of the energy landscape for a second-order transition. A symmetry-breaking transition corresponds to the deformation of the energy landscape. The green dot in the energy landscape denotes the ground state.

Figure 5

The ground-state energy density for the Ising model and its first and second derivatives in the infinite size limit. The discontinuity in the second-order derivative of the ground state suggests a second-order quantum phase transition.

Figure 6

Sketch of the time evolution of the energy landscape for a symmetry-breaking quantum phase transition which is of first order. The (green) dot denotes the ground state.

Figure 7

Sketch of the lowest eigenvalues of Hamiltonian (7). One can clearly see that the spectrum displays an avoided level-crossing at the critical point, thus corresponding to a first-order transition [22].

Figure 8

Sketch of the excitation spectrum of the Ising chain $H_{\mathrm{sys}}$ as a function of $g$. For a given frequency $\omega >0$, real saddle points correspond to intersections of the (solid) energy-level curves (e.g., $\Delta E_1$) with the (dashed) vertical $\omega$-line, which occur shortly before ($g_{*}^{-}$) and after ($g_{*}^{+}$) the quantum phase transition at $g_{\mathrm{cr}}=1/2$. The saddle-point approximation can only be applied if the intersection angle is large enough; that is, for the $\omega >0$ line, it would work for $\Delta E_1$ but not for $\Delta E_3$, and so forth.

Figure 9

Sketch of the deformed integration contour. The original integration contour (blue bold line along the real axis) is shifted to the complex plane (curved line), where $t_{*}$ indicates the singular point, $t_{*}=T/2+iT/2\tan (\mathit{ka}/2)$. Only paths $a$ and $b$ contribute significantly to the integral.