Weighted Nested Commutators for Scalable Counterdiabatic State Preparation

Abstract

Counterdiabatic (CD) driving enables efficient quantum state preparation, but it requires implementing highly nonlocal adiabatic gauge potentials (AGP) that are impractical to compute and realize in large many-body systems. We introduce a \textit{weighted nested-commutator} (WNC) ansatz to approximate AGP using local operators. The WNC ansatz generalizes the standard nested-commutator ansatz by assigning independent variational weights to commutators of local Hamiltonian terms, thereby enlarging the variational space while preserving a fixed operator range. We show that the WNC ansatz can be efficiently optimized using a local optimization scheme. Moreover, it systematically outperforms the nested-commutator ansatz in preparing one-dimensional matrix product states (MPS) and the ground state of a nonintegrable quantum Ising model. We then numerically demonstrate that CD driving based on the WNC ansatz significantly accelerates the preparation of 1D MPS for system sizes up to $N = 1000$ qubits, as well as the two-dimensional Affleck-Kennedy-Lieb-Tasaki state on a hexagonal lattice with up to $N = 3 \times 10$ sites.

Figures (5)

• Figure 1: (a) 1D geometrically local Hamiltonian (blue, illustrated for the nearest-neighbor case) [Eq. \ref{['eq:k_general']}] and the corresponding first-order ($\ell = 1$) WNC ansatz (yellow) [Eq. \ref{['eq:wnc']}]. In the global optimization scheme, all coefficients $\{\alpha\}$ are optimized simultaneously, whereas in the local optimization scheme the coefficients within local regions $\{{\cal B}_\mu\}$ (red boxes) are optimized independently. (b,c) PEPS [Eq. \ref{['psi_f_targ']}] are constructed by contracting local tensors $\{Q_v\}$ (blue circles) with maximally entangled virtual-qudit pairs (connected dots), for both a 1D system of size $N=2N_p$ [panel (b)] and a 2D hexagonal lattice [panel (c)], illustrated here with size $N=3\times 5$. The red boxes indicate the corresponding parent Hamiltonian terms [Eq. \ref{['eq:parent_hamiltonian']}].
• Figure 2: Preparation of the MPS family [Eq. \ref{['cls_mps']}] of $N=30$ using adiabatic evolution and CD driving based on the NC and WNC ansatze. (a) Infidelity as a function of the total evolution time $T$ for a fixed target-state correlation length $\xi \approx 3.8$. (b) Infidelity as a function of $\xi$ for a fixed $T=6$.
• Figure 3: (a),(b) Extracted error density $\kappa(T)$ [cf. Eq. \ref{['Eq:fid_n_scale']}] versus the evolution time $T$ for the preparation of the MPS family with correlation lengths $\xi\approx 1.6,3.8$. (c),(d) Runtime $T_p$ required to reach the target fidelity $\mathcal{F}=0.95$ for the 1D MPS family preparation as a function of system size $N$. Solid green (adiabatic) and red (WNC) lines show the prediction from Eq. \ref{['Eq:fid_n_scale']}, and symbols denote TDVP simulation results.
• Figure 4: (a) Infidelity of preparing the 2D AKLT state on the hexagonal lattice ($N=3\times3$) as a function of the total evolution time $T$. (b) The evolution time $T_p$ required to prepare the 2D AKLT state on a hexagonal lattice with fidelity $\mathcal{F} = 0.99$ for system sizes $N = 3 \times L$$(L=3-10)$ for adiabatic evolution and the CD driving based on the WNC ansatz ($\ell =1$).
• Figure 5: Infidelity for preparing the ground state of \ref{['Eq:TFIM']} along the path \ref{['path:tfim']} as a function of total time $T$. Parameters: $h_x=2$, $h_z=J=1$, $N=15$.