Counterdiabatic Optimized Local Driving

Adiabatic protocols are employed across a variety of quantum technologies, from implementing state preparation and individual operations that are building blocks of larger devices, to higher-level protocols in quantum annealing and adiabatic quantum computation. The problem of speeding up these processes has garnered a large amount of interest, resulting in a menagerie of approaches, most notably quantum optimal control and shortcuts to adiabaticity. The two approaches are complementary: optimal control manipulates control fields to steer the dynamics in the minimum allowed time, while shortcuts to adiabaticity aims to retain the adiabatic condition upon speed-up. We outline a new method that combines the two methodologies and takes advantage of the strengths of each. The new technique improves upon approximate local counterdiabatic driving with the addition of time-dependent control fields. We refer to this new method as counterdiabatic optimized local driving (COLD) and we show that it can result in a substantial improvement when applied to annealing protocols, state preparation schemes, entanglement generation, and population transfer on a lattice. We also demonstrate a new approach to the optimization of control fields that does not require access to the wave function or the computation of system dynamics. COLD can be enhanced with existing advanced optimal control methods and we explore this using the chopped randomized basis method and gradient ascent pulse engineering.

Popular Summary

Quantum devices require efficient and effective techniques to achieve coherent control. Adiabatic methods are a stable solution to this problem but can require long timescales. Alternative approaches, like optimal control methods and shortcuts to adiabaticity, aim to reproduce the same high fidelity as adiabatic passage but in significantly shorter times. We present an approach that combines these two methods: counterdiabatic optimized local driving (COLD).

Counterdiabatic driving (CD) is a powerful control method that compensates diabatic terms exactly, allowing long-time adiabatic dynamics to be realized in arbitrarily short times. However, CD suffers from two key flaws: the CD can be difficult to derive, particularly for many-body systems, and it may include highly nonlocal terms that are difficult to realize experimentally. Local CD is a successful approximate approach that tackles these issues by suppressing diabatic terms with a constrained locality of the CD, for example, by driving only on-site terms of a spin chain.

The method of COLD optimizes the dynamical Hamiltonian for a given order of local CD. The approach minimizes the corrections to the approximate local CD through the addition of control fields, like those used in optimal control problems. We demonstrate the strength of COLD for annealing protocols, many-body state preparation schemes, and population transfer both for spin chains and bosons trapped in lattices. The extended possibilities for COLD include its application to systems where the dynamics are not tractable or to the preparation of entangled quantum states.

Figure 1

Optimization of the annealing protocol for the two-spin Hamiltonian given by Eq. (11) for $h/J=2$. (a) Final fidelities of the annealing protocol with triangles (black) representing the case where no CD is applied and circles showing the case of first-order (FO) LCD (pink) as well as the combination of first- and second-order (SO) LCD (orange). (b) Final fidelities achieved when using the optimal control method BPO (red diamonds) and the new approach of COLD (blue circles), both with $N_k=1$.

Figure 2

Optimization of the annealing protocol for the Ising model given by Eq. (23) for $N=5$ spins. (a) A comparison of final state fidelities (plots show $1-F$) for different driving times using the optimal control technique BPO (blue diamonds), first-order LCD (pink dash-dot line), and COLD (red circles). The same is shown in (b) for CRAB (green diamonds) and COLDCRAB (purple circles). CD-enhanced techniques (COLD and COLDCRAB) introduced in this work show a clear convergence to good fidelities at short driving times. All results are for the best (lowest) fidelity achieved over $500$ optimizations.

Figure 3

Maximum amplitudes of CD terms in the Ising model annealing protocol for (a) first- and second-order LCD only with no additional optimal control fields and (b) the COLD approach optimized for the best final state fidelity implementing first-order LCD as shown in Fig. 2. The plot shows the maximum amplitude at each driving time for the first-order $\alpha$ (red circles) and the two second-order terms $\gamma$ (blue diamonds) and $\zeta$ (green triangles) as given in Eq. (26) (although the second-order LCD is not actually implemented in COLD). An inversion in the strength of the second-order and first-order LCD terms for (a) no additional optimal control fields and (b) the addition of optimal control fields shows that COLD implements a dynamical Hamiltonian that is favorable for the applied order of LCD (first order in this case).

Figure 4

Scaling of fidelities (plots show $1-F$) in the annealing protocol for the Ising model with (a) system size $N$ and (b) optimization parameters $N_k$ at driving time $\tau ={10}^{-2}{J}^{-1}$. Plots show a comparison between BPO (blue diamonds) and COLD (red circles). In (a) we see that the COLD fidelity decreases as a function of $N$ but remains quite high when compared to BPO, while (b) shows the nonexistent improvement for both BPO and COLD with an increasing number of parameters in the $N=5$ spin case. Once again, plotted best fidelities are obtained across 500 optimizations.

Figure 5

Optimization of the constrained annealing protocol for the Ising model for $N=5$ spins with a maximum amplitude limit on each term in the Hamiltonian of Eq. (23) of $10J$. (a) A comparison between BPO (blue diamonds) and COLD (red circles) that both give much lower fidelities (plots show $1-F$) than in the unconstrained case in Fig. 2, although COLD persists in giving better results. In (b) the comparison is between CRAB (green diamonds) and COLDCRAB (purple circles) that show orders of magnitude better fidelities than those in (a), with COLDCRAB eking out higher fidelities at short driving times. The plotted best results are obtained from 200 optimizations for each method.

Figure 6

Optimization of state transfer in a synthetic lattice. In (a) we compare the fidelities (plots show $1-F$) obtained via the bare ARP protocol (pink dashed line) and first-order LCD previously implemented in Ref. [43] (purple dash-dot line) to BPO (blue diamonds) and the COLD method (red circles). (c) Maximum amplitude of the tunneling term at each driving time for LCD (green diamonds) as given by Eq. (38) as well as COLD (red triangles) that includes additional control parameters as shown in Eq. (42) and BPO (blue triangles) that omits the modifications due to CD but retains the control terms $\mathit{\beta }$. In both (a) and (c) we simulate $N=7$ lattice sites and use the $N_k=1$ parameter for optimization of BPO and COLD. (b) Scaling of fidelities with an increasing number of lattice sites (where $N_k=1$) for both COLD (red circles) and BPO (blue diamonds), noting that the latter performs very poorly for $N>9$. Panel (d) shows the same for the number of parameters while keeping $N=7$, with the trend indicating that increasing $N_k$ does not lead to better fidelities in either the BPO or COLD case. Note that both (b) and (d) are simulated for driving time $\tau =0.5J^{-1}$ and the best fidelities are obtained across 500 optimizations.

Figure 7

GHZ state preparation in systems of frustrated spins. Spins are arranged in triangular formations as depicted in (a) for (i) three, (ii) five, and (iii) seven spins, with spins on the vertices and edges representing couplings. In the case of corner optimization, three separate optimizable drives are applied: one for the yellow corner spin, one for the red corner spin, and then a third drive for all of the blue spins in between. (b) Density matrix plots of the final state of a three-spin triangle after an evolution time $\tau =0.1J^{-1}$ when optimized using (i) BDA, (ii) first-order COLD, and (iii) second-order COLD and corner optimization. (c) Final fidelities (plots show $1-F$) of the GHZ state for the five-spin configuration depicted in (a)(ii) for an optimized global drive (red crosses) and locally driven corner spins (blue rings). (d) Final fidelities at driving time $\tau =0.1J^{-1}$ for different system sizes $N$. In the global case we use ten total optimizable parameters with $N_k=1$ drive and $N_m=10$ time intervals, while in the corners case there are 30 total parameters as we increase to $N_k=3$ separate drives. The plotted fidelities are the best results of five optimizations for each data point.

Figure 8

Optimization of $\beta$ via minimizing second-order LCD terms for the Ising model (all plots show $1-F$). In (a) we plot the final state fidelities after minimizing the integral from Eq. (22) of the $|\zeta (t)|$ drive [${\mathcal{I}}_2(|\zeta (t)|)$] and then apply the result at each driving time to different system sizes $N$. The results of minimizing the maximum amplitude ${\max }_{\beta }|\zeta (t)|$ instead are plotted in (c). The black crosses in both (a) and (c) are the results of optimizing $\beta$ by maximizing the final state fidelity $F(\tau )$ and are the same as the red circle plot in Fig. 2. Plotted in (b) and (d) are final state fidelities for $\tau =0.1J^{-1}$ for different system sizes $N$ when optimizing $\mathit{\beta }$ either by minimizing the maximum amplitude of a drive ${\max }_{\beta }(\cdot )$ or its integral ${\mathcal{I}}_2(\cdot )$. In (b) only first-order COLD is applied postoptimization, while in (d) one of the second-order drives is also applied after minimizing the other, e.g., if ${\max }_{\beta }|\zeta |$ is minimized to determine the optimal $\mathit{\beta }$ then both the first-order drive $\alpha$ and the other second-order drive $\gamma$ are applied. For system sizes above $N=10$, we use ITensor [60] matrix product state calculations that are converged with a truncation level of ${10}^{-14}$ per time step and at each site reaching a maximum bond dimension of $4$. In all cases, a single optimizable parameter is used ($N_k=1$).

Figure 9

Plot of the standard deviations from the mean for fidelities after 500 optimizations in the case of COLD and BPO for the Ising spin chain as discussed in Sec. 4. Panel (a) depicts the unconstrained case while (b) shows the constrained case. These correspond to the best result plots in Figs. 2 and 5, respectively. The plot in (c) gives the standard deviation for increasing lengths of the chain for spin number $N$ while (d) shows the same for an increasing number of parameters $N_k$. As in Fig. 4, both (b) and (c) are plotted for driving time $\tau ={10}^{-2}{J}^{-1}$. In all plots, results for COLD are depicted with red crosses while those for BPO are depicted with blue crosses.

Figure 10

Plots of the mean fidelities (diamonds) obtained over 500 optimization runs for the Ising spin chain as discussed in the main text. The error bars represent the interquartile range of the data while the shaded region encompasses the minimum and maximum fidelities obtained at each driving time. Panel (a) shows the case of COLDCRAB for the constrained instance, (b) plots the same for CRAB with (c) and (d) showing results for the unconstrained Ising chain case.