Two-parameter counter-diabatic driving in quantum annealing

We introduce a two-parameter approximate counter-diabatic term into the Hamiltonian of the transverse-field Ising model for quantum annealing to accelerate convergence to the solution, generalizing an existing single-parameter approach. The protocol is equivalent to unconventional diabatic control of the longitudinal and transverse fields in the transverse-field Ising model and thus makes it more feasible for experimental realization than an introduction of new terms such as nonstoquastic catalysts toward the same goal of performance enhancement. We test the idea for the $p$-spin model with $p=3$, which has a first-order quantum phase transition, and show that our two-parameter approach leads to significantly larger ground-state fidelity and lower residual energy than those by traditional quantum annealing and by the single-parameter method. We also find a scaling advantage in terms of the time-to-solution as a function of the system size in a certain range of parameters as compared to the traditional methods in the sense that an exponential time complexity is reduced to another exponential complexity with a smaller coefficient. Although the present method may not always lead to a drastic exponential speedup in difficult optimization problems, it is useful because of its versatility and applicability for any problem after a simple algebraic manipulation, in contrast to some other powerful prescriptions for acceleration such as nonstoquastic catalysts in which one should carefully study in advance if it works in a given problem and should identify a proper way to meticulously control the system parameters to achieve the goal, which is generally highly nontrivial.

Figure 1

Ground-state fidelity and residual energy. (a)–(c) Final ground-state fidelity and (d)–(f) residual energy for (i) traditional quantum annealing (diamonds, blue solid line), (ii) single-parameter CD drive (squares, orange dashed line), and (iii) two-parameter CD drive (circles, green dash-dotted line) as functions of annealing time $\tau$. The system sizes are (a),(d) $N=4$, (b),(e) $N=30$, and (c),(f) $N=50$, where ${\gamma }_{\mathrm{init}}=0.1$ for all panels. Time ranges are color coded as follows: short-time regime (green-shaded areas) where the fidelity is approximately $1/2^N$ for traditional quantum annealing, long-time regime (yellow-shaded areas) where transient behavior is observed and the two-parameter CD drive shows a clear advantage, and adiabatic regime (gray-shaded areas) where $F\left(\tau \right)>0.99$.

Figure 2

Time-to-solution. Time-to-solution $\mathrm{TTS}\left(\tau \right)$ for (i) traditional quantum annealing (diamonds, blue solid line), (ii) single-parameter CD drive (squares, orange dashed line), and (iii) two-parameter CD drive (circles, green dash-dotted line) for (a) $N=20$ and (b) $N=100$. For the latter, the data between $\tau =1$ to about 10 are not shown because the values are too large to achieve reasonable numerical precision. Minimal time-to-solution for (c) short-time region ($\tau \lesssim 1$) and (d) long-time region ($\tau \gtrsim 10$). Other parameters are the same as in Fig. 1.

Figure 3

Coefficients of two-parameter CD Hamiltonian. (a) Time dependence of coefficients of terms of the Hamiltonian as described in the text, ${\mathcal{H}}_{\mathrm{x}}\left(t\right)$ (upper blue solid line), ${\mathcal{H}}_{\mathrm{z}}\left(t\right)$ (middle green dash-dotted line), and ${\mathcal{H}}_{\mathrm{zzz}}\left(t\right)$ (lower orange dashed line), the last one being scaled by $N$ to fairly compare coefficients of extensive operator terms. (b) Time dependence of the coefficients of the adiabatic gauge potentials ${\mathcal{A}}_{\lambda }^{\prime }\left(t\right)$ (lower blue dash-dotted line) and ${\mathcal{A}}_{\gamma }^{\prime }\left(t\right)$ (upper green solid line). Inset depicts the corresponding coefficients $\alpha \left(t\right)$ and $\beta \left(t\right)$ under the two-parameter CD drive with $\gamma \left(t\right)=0.1+\lambda \left(t\right)$, annealing time $\tau =10$, and system size $N=30$. Other parameters are the same as in Fig. 1.

Figure 4

Energy spectrum. Instantaneous energy spectrum $E\left(t\right)$ for (a) traditional quantum annealing, (b) single-parameter CD driving, and (c) two-parameter CD driving for system size $N=30$, annealing time $\tau =300$, and ${\gamma }_{\mathrm{init}}=0.1$. Thickness of red curves indicates the occupation probability of each eigenstate in the dynamical processes of the three annealing protocols. Higher excited states have a neglectably small occupation probability and thus the corresponding very thin red lines cannot be seen.

Figure 5

Coefficient of two-parameter CD drive. Coefficients of the driver Hamiltonian, i.e., ${\mathcal{H}}_{\mathrm{LZ},\mathrm{x}}^{\lambda ,\gamma }\left(t\right)$ (blue solid line) and problem Hamiltonian ${\mathcal{H}}_{\mathrm{LZ},\mathrm{z}}^{\lambda ,\gamma }\left(t\right)$ (orange dashed line), as described in Eq. (B8) for two-parameter CD driving during annealing of $\tau =1$. Other parameter: $h=0.1$.

Figure 6

Ground-state fidelity and residual energy. (a)–(c) Final ground-state fidelity and (d)–(f) residual energy for (i) traditional quantum annealing (diamonds, blue solid line), (ii) single-parameter CD drive (squares, orange dashed line), and (iii) two-parameter CD drive with driving functions ${\gamma }_1\left(t\right)$, given by Eq. (D2) (circles, green dash-dotted line), ${\gamma }_2\left(t\right)$, given by Eq. (D3) (down triangles, magenta dotted line), and ${\gamma }_3\left(t\right)$, given by Eq. (D4) (up triangles, cyan widely dashed line), as functions of annealing time $\tau$. The system sizes are (a),(d) $N=4$, (b),(e) $N=30$, and (c),(f) $N=50$, where ${\gamma }_{\mathrm{init}}=1$ for all panels. Time ranges are color coded as follows: short-time regime (green-shaded areas) where the fidelity is approximately $1/2^N$ for traditional quantum annealing, long-time regime (yellow-shaded areas) where transient behavior is observed and the two-parameter CD drive shows a clear advantage, and adiabatic regime (gray-shaded areas) where $F\left(\tau \right)>0.99$.