Steering random spin systems to speed up the quantum adiabatic algorithm

A general time-dependent quantum system can be driven fast from its initial ground state to its final ground state without generating transitions by adding a steering term to the Hamiltonian. We show how this technique can be modified to improve on the standard quantum adiabatic algorithm by making a single-particle and cluster approximation to the steering term. The method is applied to a one-dimensional Ising model in a random field. For the limit of strong disorder, the correction terms significantly enhance the probability for the whole system to remain in the ground state for the proposed nonstoquastic annealing protocol. We demonstrate that even when transitions occur for stronger interaction between qubits, the most probable quantum state is one of the lower-energy states of the final Hamiltonian. Since the method can be applied to any model, and more sophisticated approximations to the steering term are possible, the alternative technique opens up an avenue for the improvement of the quantum adiabatic algorithm.

Figure 1

Average ground-state probability as a function of the annealing time $t_a$. The $h_k$ are chosen uniformly from the interval $[-W,W]$, where $W=1$. All energy variables are measured in units of $W$, and time variables are measured in units of $\hslash /W(\hslash =1)$ throughout the paper. (a) $L=1$. In the inset, the red magnetic field vector rotates from $x$ to $z$ direction in the standard quantum annealing process. The steering field applied in the $-y$ direction suppresses transitions to the excited states. (b) $L=3$, $J=0.1$. The green diamond curve is the result of the application of Eq. (2), the exact Berry formula. The inset shows the sketch of the open chain of three spins considered here.

Figure 2

(a) Average ground-state probability as a function of the annealing time $t_a$. $L=8$ (square), $L=10$ (circle), $L=12$ (diamond) compared for $J=0.1$. (b) Average ground-state probability as a function of the interaction parameter $J$ for a short annealing time $t_a=1$. The red (upper), blue (middle) and green (lower) dashed lines show the naive algorithm results for $L=8,10,12$, respectively. (c) Average ground-state probability as a function of the interaction parameter $J$ for a longer annealing time $t_a=100$.

Figure 3

Same markers are used in this figure as in Figs. 2 and 2 for the standard QAA and the steered QAA. For the naive algorithm, red (upper), blue (middle), and green (lower) “x” markers are used in the insets for $L=8,10,12$, respectively. In the insets, the naive algorithm is compared with the steered QAA. $t_a=1$, $J=0.3$. Several system sizes are shown. (a) The probability distribution over all final eigenstates $\left|n\right(t_a)\rangle$ as a function of the level index $n$, computed by comparing the results of the QAA to an exact calculation. $P_n={\left|\langle \psi \left(t_a\right)|n\left(t_a\right)\rangle \right|}^2$. The effect of steering is to squeeze the width of the probability distribution by two orders of magnitude and in the direction of the ground state. (b) Cumulative probability distribution. $S_N={\sum }_{n=1}^N\overline{P_n}$. With the steered algorithm, the chance to find one of the low-lying states is significantly enhanced.

Figure 4

Average infidelity as a function of $1/J$ and $t_a$. $L=12$. Plots for (a) the naive algorithm, (b) the standard QAA, and (c) the steered QAA. In the region covered by the white dashed lines, the steered QAA gives higher fidelity than the other two algorithms. (d) The colorbar shows the infidelity values.

Figure 5

Average ground-state probability as a function of the interaction parameter $J$ for the QAA without steering, with 1-spin steering, and with cluster steering. Cluster steering improves the results for $J\le 0.2t_a=128$, $L=12$.