Accelerating Feedback-Based Quantum Algorithms through Time Rescaling

TL;DR

This work tackles the circuit-depth bottleneck of feedback-based quantum algorithms on NISQ devices by introducing time-rescaled variants TR-FQA and TR-FALQON, inspired by shortcuts to adiabaticity. By reparameterizing time with a function $f(\tau)$, the authors derive a rescaled Hamiltonian $\mathcal{H}(\tau)=H(f(\tau))\dot{f}(\tau)$ and adjust the control law to maintain a monotonically decreasing cost $J(\tau)=\langle\psi(\tau)|H_p|\psi(\tau)\rangle$, enabling faster convergence. They demonstrate significant improvements on two tasks: MaxCut optimization and ground-state preparation in the ANNNI model, with TR-FALQON showing superior performance in shallow circuits and TR-FQA achieving substantial depth reductions. The results indicate that time rescaling is a promising strategy to enhance the practicality of quantum optimization and state-preparation protocols on near-term hardware.

Abstract

This work investigates the impact of time rescaling on the performance of Feedback Quantum Algorithms (FQA) and their variant for optimization tasks, FALQON. We introduce TR-FQA and TR-FALQON, time-rescaled versions of FQA and FALQON, respectively. The method is applied to two representative problems: the MaxCut combinatorial optimization problem and ground-state preparation in the ANNNI quantum many-body model. The results show that TR-FALQON accelerates convergence to the optimal solution in the early layers of the circuit, significantly outperforming its standard counterpart in shallow-depth regimes. In the context of state preparation, TR-FQA demonstrates superior convergence, reducing the required circuit depth by several hundred layers. These findings highlight the potential of time rescaling as a strategy to enhance algorithmic performance on near-term quantum devices.

Figures (4)

• Figure 1: Illustrative diagram of the FQA PhysRevB.110.224422. The process begins with the state $\ket{\psi_0}$, and at each layer $k$, the unitary operators $e^{-iH_p \Delta t}$ and $e^{-iH_d \Delta t \beta_k}$ are applied sequentially. The parameter $\beta_k$ is adaptively adjusted at each iteration. This dynamic is repeated iteratively, guiding the evolution of the state $\ket{\psi}$ through the layers until the desired solution is reached.
• Figure 2: Illustrative diagram of the TR-FQA. The process starts with the initial state $\ket{\psi_0}$. In each layer $k$, the unitary operators $e^{-iH_p \dot{f}(k\Delta \tau) \Delta \tau}$ and $e^{-iH_d \dot{f}(k\Delta \tau) \Delta \tau \tilde{\beta}_k}$ are applied sequentially, adaptively adjusting the parameter $\tilde{\beta}_k$.
• Figure 3: Comparison of the performance of FALQON and TR-FALQON for different time rescaling functions, showing the probability of obtaining the solution to the problem from the prepared state for different layer depths. Panel (a) presents the results for a 16-vertex graph, where FALQON was executed with $\Delta t = 0.04$, 400 layers, and TR-FALQON was executed with $\Delta \tau = 0.04$, 400 layers, $a = 2$, and $t_f = 16$. Panel (b) presents the results for a 24-vertex graph, where FALQON was executed with $\Delta t = 0.03$, 600 layers, and TR-FALQON was executed with $\Delta \tau = 0.03$, 600 layers, $a = 2$, and $t_f = 18$.
• Figure 4: Numerical simulation comparing FQA and TR-FQA applied to the ANNNI model for different chain configurations and values of $\kappa$ and $g$. The TR-FQA simulations were performed using the function $f_1$, with the values of $a$ and $t_f$ specified in each panel. For chains with $L = 8$ sites, $\Delta t = \Delta \tau = 0.01$ was adopted, while for $L = 12$ sites, $\Delta t = \Delta \tau = 0.005$ was used. In the panels, the dashed line represents the ground state energy, while the curves show the convergence of the cost function $J = \langle\Psi_{k}|H|\Psi_{k}\rangle$ as a function of the number of layers $k$.