Ancillary entangling Floquet kicks for accelerating quantum algorithms

Quantum simulation with adiabatic annealing can provide insight into difficult problems that are impossible to study with classical computers. However, it deteriorates when the systems scale up due to the shrinkage of the excitation gap and thus places an annealing rate bottleneck for high success probability. Here, we accelerate quantum simulation using digital multi-qubit gates that entangle primary system qubits with the ancillary qubits. The practical benefits originate from tuning the ancillary gauge degrees of freedom to enhance the quantum algorithm’s original functionality in the system subspace. For simple but nontrivial short-ranged, infinite long-ranged transverse-field Ising models, and the hydrogen molecule model after qubit encoding, we show improvement in the time to solution by one hundred percent but with higher accuracy through exact state-vector numerical simulation in a digital-analog setting. The findings are further supported by time-averaged Hamiltonian theory.

💡 Research Summary

The paper tackles a fundamental bottleneck in quantum annealing: as the problem size grows, the minimum excitation gap (Δ_min) shrinks, forcing the annealing schedule to be exceedingly slow in order to remain adiabatic. To overcome this limitation, the authors introduce a protocol that couples the primary system qubits (S) with a set of ancillary qubits (A) via periodic entangling “Floquet kicks.” The combined Hilbert space S⊗A evolves under two concurrent unitaries: (i) an always‑on system Hamiltonian U_S(t) that encodes the target problem, and (ii) a kick unitary U_SA(Θ̂_A, Ŝ)=exp(−i Θ̂_A⊗Ŝ) applied at discrete times. Θ̂_A is a simple sum of Pauli operators on the ancilla with a single tunable angle θ, while Ŝ is a collective operator acting on all system qubits. By judiciously choosing the Pauli axes for Θ̂_A and Ŝ, the ancillary gauge degrees of freedom can be “tuned” to reshape the effective energy landscape of the reduced system state, suppressing diabatic transitions and guiding the system more rapidly toward its ground state.

The kicks are modeled as Dirac‑delta impulses occurring at regular intervals Δt_K (digital‑analog hybrid implementation). An analytical estimate for the optimal kick angle θ_opt is derived using a time‑averaged Hamiltonian (first‑order Magnus expansion). The resulting expression, θ_opt≈(1/N_A N_K)·(1−E_T^S/E_mix^S(τ)), depends only on the true ground‑state energy E_T^S, the mixer‑Hamiltonian energy E_mix^S(τ), and the numbers of ancilla qubits N_A and kicks N_K. Crucially, this formula is largely independent of the detailed problem Hamiltonian, making it broadly applicable whenever the mixer and kick terms dominate the dynamics.

The authors validate the protocol on three representative models: (1) a nearest‑neighbor transverse‑field Ising model (NN‑TFIM) on an open chain, (2) an infinite‑range long‑range TFIM (ILR‑TFIM) on a closed chain, and (3) a qubit‑encoded hydrogen molecule (H₂) after Bravyi‑Kitaev transformation. For the NN‑TFIM, they set the problem coupling θ_0^XX and the transverse‑field mixer amplitude θ_0^Z, initialize the system in a product state, and apply periodic kicks with θ≈0.004. Numerical state‑vector simulations show that, even when the annealing time τ is shorter than 1/Δ_min, the reduced system energy converges to the true ground‑state value roughly twice as fast as without kicks. Over‑strong kicks lead to excess excitations and energy overshoot, while too weak kicks provide no benefit; the optimal θ lies in a narrow window that can be predicted by the analytical formula. High‑frequency kicks (θ_0^Z Δt_K≪1, θ_0^XX Δt_K≪1) further suppress residual diabatic effects, yielding a clean speed‑up without loss of accuracy.

In the ILR‑TFIM, despite the all‑to‑all connectivity scaling as N_S(N_S−1)/2, the same kick protocol with the same θ_opt maintains its performance, demonstrating robustness against increased connectivity and system size. For the H₂ molecule, parity conservation prevents a Z‑type mixer from generating transitions between the two relevant parity sectors (|01⟩ and |10⟩). The authors therefore employ an X‑type mixer and choose the ancilla operator Θ̂_A = θ Σ_l Z_A^l, while the kick operator commutes with the H₂ Hamiltonian (X_0X_1+Y_0Y_1). Even with this less‑optimal choice, the protocol still reaches the exact ground‑state energy and halves the annealing time.

Overall, the work presents three key contributions: (i) a low‑overhead, single‑parameter Floquet‑kick design that requires only a few ancilla qubits, (ii) a theoretically grounded method for estimating the optimal kick angle via time‑averaged Hamiltonian theory, and (iii) extensive numerical evidence across short‑range, long‑range, and quantum‑chemical models that the protocol consistently reduces time‑to‑solution by ≈100 % while improving final accuracy. Importantly, the optimal kick angle shows negligible dependence on system size, suggesting that the method can be scaled to larger devices without re‑optimizing parameters for each problem instance. This approach offers a practical pathway to accelerate quantum annealing‑based simulations on near‑term noisy intermediate‑scale quantum (NISQ) hardware, potentially expanding the class of problems that can be tackled with current quantum technologies.