Quantum trajectories for time-dependent adiabatic master equations

We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension $N$ instead of a complex density matrix of dimension $N^2$, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor $N$ advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor $N^2$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.

Figure 1

Depiction of the stochastic evolution of the state $|\psi \rangle$ by an infinitesimal time state at time $t$.

Figure 2

Graphs of (a) the 8-qubit chain, (b) the 8-qubit Hamiltonian exhibiting a small gap, and (c) the 16-qubit “tunneling-probe” Hamiltonian of Ref. [25]. (a) Only the first qubit is subjected to an applied field and each qubit is ferromagnetically coupled with $J=-1$. (b) Solid lines correspond to ferromagnetic coupling and dashed lines correspond to antiferromagnetic coupling. The thickness denotes the strength of the coupling. Local fields are shown inside the circles. Full parameters are given in Eq. (32). (c) The left 8-qubit cell and right 8-qubit cell are each subjected to applied fields with opposite directions. Each qubit is ferromagnetically coupled to others as shown by the lines, with $J=-1$.

Figure 3

Evolution of the population in the instantaneous ground state for the 8-qubit problem in Eqs. (27) and (30) for a total time of $t_f=10\mu \text{s}$ and temperature $2.6\mathrm{GHz}$ (in $\hslash \equiv 1$ units), as a function of the normalized time $s=t/t_f$. The quantum trajectories results with ${10}^4$ trajectories (QT, blue circles) are in excellent agreement with the adiabatic master equation (ME, red solid line). Inset: the convergence of the ground-state population (averaged over quantum trajectories) towards the master equation result as a function of the number of trajectories, at $s=1$. The error bars represent $2\sigma$ confidence intervals, where $\sigma$ is the standard deviation of the mean generated by taking ${10}^3$ bootstraps over the number of trajectories.

Figure 4

Overlap squared of the (normalized) state with the instantaneous ground state of $H_S\left(t\right)$ for a typical single trajectory of the 8-qubit chain in Sec. 5a, with $t_f=10\mu \text{s}$ and temperature $2.62\mathrm{GHz}$, as a function of the normalized time $s=t/t_f$. The sudden changes in the overlap are due to the action of the jump operators $\left\{A_i\left(t\right)\right\}$, taking the state from one eigenstate to another. This is to be contrasted with the smooth behavior of Fig. 3 when we average over different trajectories. Inset: a histogram of the net number of jumps to the instantaneous ground state (GS). A negative number indicates a jump out of the ground state and a positive number indicates a jump into the ground state. The change from negative to positive net jumps occurs at the minimum gap point.

Figure 5

Evolution of the population in the instantaneous ground state for the 8-qubit problem in Eqs. (27) and (32) for a total time of $t_f=10\mu \text{s}$ and temperature $1.57\mathrm{GHz}$ (in units where $k_B=1$), as a function of the normalized time $s=t/t_f$. The quantum trajectories results with $5\times {10}^3$ trajectories (blue circles) are in excellent agreement with the master equation (red solid line). We also show the closed-system evolution (yellow dashed line) to highlight that the evolution is not adiabatic. Inset: the instantaneous energy gap between the first excited state and the ground state during the anneal. The minimum occurs at $s^{*}=0.46$, coinciding with the sharp discontinuity observed in the instantaneous ground-state population.

Figure 6

Overlap squared of the (normalized) state with the instantaneous ground state of $H_S\left(t\right)$ (blue solid curve) and the sum of the first three instantaneous excited states (red dashed curve) for a typical single trajectory of the 8-qubit problem in Sec. 5b, with $t_f=10\mu \text{s}$ and temperature $1.57\mathrm{GHz}$, as a function of the normalized time $s=t/t_f$. A diabatic transition occurs at $s^{*}=0.46$. The continuous decay immediately afterward and between the “incomplete” jumps is to be contrasted with the step-function-like single trajectories of the adiabatic case seen in Fig. 4.

Figure 7

Quantum trajectory results for a temperature of $12\text{m}\mathrm{K}$ and $20\text{m}\mathrm{K}$ using the DW2X annealing schedule. Results are averaged over 5000 trajectories. A revival of instantaneous ground-state population occurs after the minimum gap (shown by the dashed vertical line). $E_0$ and $E_1$ denote the ground state and the first excited state energies, respectively.

Figure 8

(a) Dotted lines: linear schedule; solid lines: D-Wave One schedule used in Fig. 3; (b) same as in Fig. 3 (the evolution of the population in the instantaneous ground state for the 8-qubit problem), but with a linear schedule. Inset: the convergence of the ground-state population towards the AME results. (c) A histogram of the net number of jumps to the instantaneous ground state (GS). It shares the same pattern as in the nonlinear schedule case (the inset of Fig. 4), but the net number of jumps out of the ground state is smaller due to the smaller energy scale in the linear schedule.