Ancillary entangling Floquet kicks for accelerating quantum algorithms

Technical Summary: Ancillary Entangling Floquet Kicks for Accelerating Quantum Algorithms

Problem Statement
Quantum simulation via adiabatic annealing offers a pathway to solving complex many-body problems intractable for classical computers. However, this approach faces a fundamental bottleneck: as system size increases, the minimal excitation gap shrinks, leading to critical slowdown. To maintain high success probability, the annealing rate must remain slow enough to satisfy adiabatic conditions, which is often limited by the finite coherence times of current quantum hardware. While digital quantum simulations and hybrid variational quantum algorithms (HVQAs) have emerged as alternatives, they often suffer from optimization challenges or require extensive resources. Furthermore, counter-diabatic approaches designed to mitigate diabatic errors introduce nested terms that are difficult to implement practically.

Methodology
The authors propose a quantum protocol to accelerate simulation by introducing ancillary entangling Floquet kicks. This approach operates within a digital-analog framework, utilizing a primary system Hilbert space ($S$) and a secondary ancillary Hilbert space ($A$).

1. System Architecture: The protocol employs an "always-on" system unitary $U_S(t)$ that encodes the problem Hamiltonian (analog simulation) and periodic digital quantum kick unitaries $U_{SA}(\hat{\Theta}_A, \hat{S}) = \exp(-i\hat{\Theta}_A \otimes \hat{S})$.
2. Ancillary Design: The ancillary qubits are initialized in a product state. The kick operator $\hat{\Theta}_A$ acts on the ancillary space, while the collective system operator $\hat{S}$ acts on the system space. For simplicity and low-cost compilation, the authors utilize a single uniform parameter $\theta$ for the kick angle, selecting Pauli operators from $\{X, Y, Z\}$ to minimize compilation overhead.
3. Mechanism: The entangling gates are applied periodically (Floquet kicks) to deviate the state evolution from the adiabatic path early in the simulation and guide it back to the target quantum path shortly after. The choice of kick operators is guided by the requirement that they must not commute with the initial mixer Hamiltonian but should commute with the final target Hamiltonian to avoid disturbing the ground state at the end of the simulation.
4. Theoretical Basis: The optimal kick angle $\theta_{opt}$ is estimated using a non-perturbative time-averaged Hamiltonian approach (based on leading-order Magnus expansion). The derived expression relates the optimal angle to the ratio of the true system ground state energy ($E^T_S$) and the system mixer energy ($E^{Mixer}_S$) at a specific time cutoff $\tau$.

Key Contributions and Results
The authors validate this protocol through exact state-vector numerical simulations in a digital-analog setting across three distinct models:

• Nearest-Neighbor Transverse-Field Ising Model (NN-TFIM): In an open chain, the protocol successfully accelerates the passage through the quantum phase transition (QPT). For faster annealing rates (where diabatic transitions would normally limit success), the application of optimal Floquet kicks (e.g., $\theta \approx 0.004$) suppresses excess excitations. The results show a 100% reduction in time-to-solution compared to simulations without kicks, achieving the true ground state energy with higher accuracy.
• Infinite Long-Range Transverse-Field Ising Model (ILR-TFIM): The protocol demonstrates robustness in a closed chain with infinite-range interactions, maintaining performance even when the interaction strength is rescaled to match the gap size of the NN-TFIM.
• Hydrogen Molecule (H2) Model: After qubit encoding via the Bravyi-Kitaev transformation, the protocol is applied to the H2 molecule. The authors identify a mixer Hamiltonian that does not commute with the H2 system Hamiltonian to facilitate superposition. The kicks successfully drive the system to the exact ground state binding energy, showing convergence even with fast annealing rates and varying bond lengths.

Significance and Claims
The paper claims that this protocol offers a modular algorithm design principle that accelerates quantum simulations without altering the core algorithms operating in the system subspace. Key significance points include:

• Resource Efficiency: The method requires only a single parameter optimization (offline and insensitive to system size) and utilizes a limited, constant number of ancillary qubits, regardless of the complexity of the chained quantum subroutines.
• Hardware Compatibility: The approach is particularly useful for noisy quantum hardware with short coherence times, as it allows for faster annealing rates that would otherwise be prohibited by the excitation gap bottleneck.
• Universality: The protocol is applicable to any quantum application following Hamiltonian encoding, including quantum sensing, optimization, machine learning, and state preparation.
• Scalability: The authors note that the optimal kick angle is not sensitive to system size for the tested models, suggesting scalability for larger systems.

The work positions these "ancillary entangling Floquet kicks" as a practical primitive to enhance quantum utility on current and next-generation programmable digital-analog quantum devices.