Continuous-time quantum control across an exponentially small bottleneck in a frustrated Ising ring model

The gist: Optimized continuous-time control using a dressed-CRAB approach successfully bypasses an exponentially small spectral gap bottleneck in a frustrated Ising ring model by exploiting controlled diabatic transitions, leading to ground-state preparation times that grow linearly with system size rather than exponentially > ref:2606.07168

Model and Bottleneck

The study focuses on a frustrated Ising ring model which is characterized by an exponentially small avoided crossing close to the end of the annealing path (Page 1). This bottleneck arises from the specific couplings where a single antiferromagnetic bond induces frustration, leading to two closing gaps between the ground state and the first excited state, one decaying polynomially and another being exponentially small (Page 3). The model is mapped into a quadratic fermionic Hamiltonian using Jordan-Wigner transformation, which allows for efficient simulation at large system sizes (Page 3).

Optimization Strategy

To overcome the limitations of standard fixed schedules, the researchers implemented continuous-time quantum control by optimizing smooth annealing schedules using a dressed-CRAB approach (Page 2). This involves expanding the schedule over a finite set of Fourier components with randomized frequencies, iteratively improving the schedule by minimizing the average final energy EP(C) (Page 4). The optimized protocol bypasses the bottleneck through a strongly nonadiabatic mechanism: during the evolution, population weight is transferred away from the instantaneous ground state and is later brought back to the target ground state near the end of the protocol (Page 4).

Performance Results

The optimized dressed-CRAB schedules strongly outperform standard fixed schedules, including the traditional linear ramp, but also cubic and sinusoidal ramps with vanishing time-derivatives at the beginning and at the end of the annealing (Page 4). The comparison shows that reaching vanishing residual energies with such fixed protocols would require much longer, indeed exponentially large, annealing times [11, 25, 26] (Page 6). Furthermore, for a threshold residual energy of epsilonthreshold = 10−6, the threshold annealing time τ∗ is shown to grow linearly with N in the explored range (Page 6).

Physical Mechanism

Analysis of the populations of the instantaneous eigenstates reveals that the dynamics is "strongly nonadiabatic: the system leaves the instantaneous ground state during the early stage of the evolution, developing a sizable population in the first excited state, and returns almost completely to the target ground state at the end of the evolution (Page 7). This transfer allows for bypassing the bottleneck by exploiting controlled diabatic transitions across it (Page 4). The study also investigated adding a lowest-order variational counter-diabatic correction, which was found does not produce any improvement over the optimized schedule alone" because the optimized schedule already relies on a diabatic mechanism where the instantaneous ground-state population is appropriately initially depleted (Page 7).

Conclusion and Scaling

The main conclusion is that continuous-time schedule optimization can reduce the final residual energy by several orders of magnitude with respect to fixed schedules (Page 8). The optimized protocol generates a strongly nonadiabatic dynamics, and the annealing time required to reach a fixed residual-energy threshold grows approximately linearly with the system size over the range investigated, rather than following the exponential scaling expected from a purely adiabatic passage through the bottleneck gap (Page 8). This scaling is consistent with results for minimum QAOA steps Pcr necessary to reach controllability in the digital setting, as Pcr ∝ N2 (Page 8). The work also suggests that boundary bang-bang features might emerge below the controllability threshold, indicating a direction for future investigation into hybrid protocols (Page 9).

Appendix Details

The analysis involves representing the system using Dirac fermions and deriving an explicit expression for the optimal function α(s) that defines the lowest-order counter-diabatic term (Page 15). The comparison between digitized and continuous-time dynamics shows that the discrepancy decreases approximately as a power law compatible with P−2 in the explored range, justifying the choice of time discretization P = 10⋆int(τ) (Page 6). The final energy EP(θ) is expressed as Tr U†ev(θ) Az Uev(θ) G, where G is a symmetric matrix derived from the symmetry-adapted fermionic operators (Page 13). The analysis confirms that the optimized schedules generate a strongly nonadiabatic dynamics: "the state leaves the instantaneous ground state, develops significant population in the first excited instantaneous eigenstate, and is finally transferred back to the target ground state near the end of the evolution using the avoided crossing at sb ≈ 0.9 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35" (Page 7). The optimized schedules generate a strongly nonadiabatic dynamics: the state leaves the instantaneous ground state, develops significant population in the first excited instantaneous eigenstate, and is finally transferred back to the target ground state near the end of the evolution using the avoided crossing at sb ≈ 0.9 (Page 7). The final ground-state infidelities are 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be 1 − p0(τ) ≃ 3.8 × 10−8 for N = 13 and 1 − p0(τ) ≃ 9.6 × 10−7 for N = 35 (Page 7). The final ground-state infidelities are found to be "1 − p0(τ) ≃ 3.

1. Continuous-Time Quantum Control for Bottleneck Bypassing: The system can be equipped with an optimized continuous-time annealing schedule that bypasses the bottleneck through a strongly nonadiabatic mechanism, leading to efficient ground-state preparation where the annealing time required to reach a fixed residual-energy threshold is found to grow linearly with system size rather than exponentially.

2. Nonadiabatic Dynamics Exploitation: The improved AI system can exploit the dynamics by ensuring population weight is transferred away from the instantaneous ground state and is later brought back to the target ground state near the end of the protocol, effectively using controlled diabatic transitions across it to overcome exponentially small spectral gaps.

3. Schedule-Independent Control Verification: The system can verify that optimal control yields superior performance over fixed schedules, as demonstrated by showing that cubic and sinusoidal ramps have vanishing derivatives at the boundaries and are therefore expected, on the basis of the adiabatic theorem, to reduce boundary-induced diabatic errors but performing even worse than the linear schedule in this regime.

4. Counter-Diabatic Term Assessment: The system can determine when counter-diabatic driving is redundant, as shown by finding that once schedule optimization is allowed, it does not lead to any improvement, suggesting that the optimized protocol relies on a mechanism where the instantaneous ground-state population is appropriately initially depleted.

5. Scaling Prediction for Quantum Annealing: The improved system can predict the required annealing time scaling with high confidence, as the results indicate that "the annealing time required to reach a fixed residual-energy threshold grows approximately linearly with the system size over the range investigated, rather than following the exponential scaling expected from a purely adiabatic passage through the bottleneck gap."