Exploiting many-body localization for scalable variational quantum simulation

TL;DR

Variational quantum algorithms face barren plateaus as system size grows. The authors propose Floquet-initialized circuits operated in the many-body localized (MBL) phase to preserve low entanglement and trainable gradients, enabling scalable VQAs, with diagnostics based on $IPR$, entanglement entropy, and the low-weight stabilizer Rényi entropy $M_{t,k}$ to characterize the MBL--thermal transition and its relation to $t$-designs. They show that, below a critical kick strength $W^\star$, the circuit remains far from a unitary $2$-design and maintains trainable landscapes, while above $W^\star$ it approaches Haar randomness and exhibits thermal behavior, corroborated by frame potentials and level statistics trending to the circular unitary ensemble (CUE). The approach enables efficient ground-state preparation for several model Hamiltonians and is experimentally validated on a 127-qubit device, where gradient restoration and phase-transition signatures are observed up to $n=31$ qubits. Overall, MBL-based initialization provides a viable path toward scalable, trainable variational quantum algorithms on near-term hardware, motivating the integration of localization phenomena into quantum algorithm design.

Abstract

Variational quantum algorithms (VQAs) represent a promising pathway toward achieving practical quantum advantage on near-term hardware. Despite this promise, for generic, expressive ansätze, their scalability is critically hindered by barren plateaus--regimes of exponentially vanishing gradients. We demonstrate that initializing a hardware-efficient, Floquet-structured ansatz within the many-body localized (MBL) phase mitigates barren plateaus and enhances algorithmic trainability. Through analysis of the inverse participation ratio, entanglement entropy, and a novel low-weight stabilizer Rényi entropy, we characterize a distinct MBL-thermalization transition. Below a critical kick strength, the circuit avoids forming a unitary 2-design, exhibits robust area-law entanglement, and maintains non-vanishing gradients. Leveraging this MBL regime facilitates the efficient variational preparation of ground states for several model Hamiltonians with significantly reduced computational resources. Crucially, experiments on a 127-qubit superconducting processor provide evidence for the preservation of trainable gradients in the MBL phase for a kicked Heisenberg chain, validating our approach on contemporary noisy hardware. Our findings position MBL-based initialization as a viable strategy for developing scalable VQAs and motivate broader integration of localization into quantum algorithm design.

Figures (14)

• Figure 1: Schematic representation of the Floquet-initialized variational quantum circuit, the localization-thermalization phase transition, and the optimization landscape. (a) Diagram of a variational quantum circuit. The illustration includes four qubits as a segment of a larger circuit, resulting in some 2-qubit gates acting on adjacent pairs $Q_i-Q_j$ appearing truncated. The circuit begins with a low-complexity trial state. Steady rotation angles within $\hat{\mathcal{R}}_z$ and $\hat{\mathcal{R}}_{zz}$ are initialized randomly within $[-\pi,\pi)$, while the kick rotation angles in $\hat{\mathcal{R}}_x$, $\hat{\mathcal{R}}_{xx}$, and $\hat{\mathcal{R}}_{yy}$ are initialized within $[-W,W]$. (b) Transition of the variational quantum circuit between the many-body localized (MBL) phase and the thermal phase as a function of the kick strength $W$. Below a critical threshold $W^\star$, the circuit remains in the MBL phase, characterized by output state entanglement entropy adhering to the area law and trainable circuit parameters. Above $W^\star$, the circuit enters the thermal phase, where the output state's entanglement entropy follows the volume law, rendering the circuit parameters untrainable. (c) Depiction of the optimization landscape for an MBL-initialized circuit. The initial output state is localized to a trial state, ideally positioned in the same "valley" as the ground state, facilitating effective optimization.
• Figure 2: Characterization of the MBL--thermalization transition. Panels (a, c, e, g) represent results for the 1D ring topology, and panels (b, d, f, h) represent results for the $\mathrm{Ci}_{n}(1,2)$ topology. (a, b) Ratios of the inverse participation ratio, $\text{IPR}_2$ [Eq. \ref{['eq:IPR_2']}], to the Haar-random inverse participation ratio, $\text{IPR}_2^{(\mathrm{Haar})}$ [Eq. \ref{['eq:IPR_Haar']}], as a function of kick strength $W$. The grey dashed lines correspond to $1$. (c, d) Half-chain entanglement entropy versus kick strength $W$. The grey dashed lines correspond to $S^{(\mathrm{Page})}$ [Eq. \ref{['eq:S^Page']}]. Insets: Demonstration of the topologies of a 6-qubit ring and a $\mathrm{Ci}_{6}(1,2)$ graph. (e, f) Entanglement entropy fluctuations versus kick strength $W$. Insets: Entanglement entropy of a local region $L$ versus the size of $L$ for both MBL ($W = 1/5, 2/5$ for the 1D ring topology and $W = 1/10, 1/5$ for the $\mathrm{Ci}_{n}(1,2)$ topology) and thermal ($W = 7/5$ for 1D ring and $W = 7/10$ for $\mathrm{Ci}_{n}(1,2)$) phases in a 20-spin chain. Darker green markers indicate circuits with higher $W$. The grey dashed lines correspond to $S^{(\mathrm{Page})}$. (g, h) The low-weight stabilizer Rényi entropy of order $2$ and locality $2$ [Eq. \ref{['def:lowweight-sre']}] versus kick strength $W$. The grey dashed lines correspond to the lower bound $\widetilde{M}_{2,2}^{(\mathrm{Haar})}$ [Eq. \ref{['eq:lowerbound_SE']}].
• Figure 3: Visualization of the absence of barren plateaus in the MBL phase and their presence in the thermal phase after sampling the initial circuit parameters, analyzed using the target Hamiltonian $\hat{\mathcal{H}}_{\text{AA}}$ (defined in Eq. \ref{['Eq:AA Hamiltonian - qubit']}). (a)$\ell_\infty$-norm of the energy gradient plotted as a function of kick strength $W$ for a 1D ring topology. (b)$\ell_\infty$-norm of the energy gradient as a function of $W$ for a $\mathrm{Ci}_{n}(1,2)$ lattice topology. Insets illustrate the gradient scaling at various $W$ values, emphasizing the contrast between the MBL phase (e.g., $W = 1/5, 2/5$ for the 1D ring and $W = 1/10, 1/5$ for the $\mathrm{Ci}_{n}(1,2)$ topology) and the thermal phase (e.g., $W = 7/5$ for the 1D ring and $W = 7/10$ for the $\mathrm{Ci}_{n}(1,2)$ topology). Darker green markers in the insets denote circuits with higher $W$.
• Figure 4: Converged approximation ratios from the variational quantum eigensolver for system sizes $n=4,6,8,\dots,18$ across 10 samples of initial parameters. This comparison illustrates the superiority of the MBL initialization ($W=2/5$, triangles) over thermal ($W=7/5$, diamonds) and random (circles) strategies. Black dotted lines mark the trial state energy. (a)$\hat{\mathcal{H}}_{\text{AA}}$ in the extended phase ($\Gamma=0, V = 1$). (b)$\hat{\mathcal{H}}_{\text{AA}}$ in the non-interacting Anderson localized phase ($\Gamma=0, V = 4$). (c)$\hat{\mathcal{H}}_{\text{AA}}$ in the ergodic phase ($\Gamma=1, V = 2$). (d)$\hat{\mathcal{H}}_{\text{AA}}$ in the MBL phase ($\Gamma=3, V = 6$). The insets provide a zoomed view for large systems.
• Figure 5: Relative energy error during optimization at $n=18$, showing the rapid convergence of MBL initialization. (a)$\hat{\mathcal{H}}_{\text{AA}}$ in the non-interacting Anderson localized phase ($\Gamma=0, V = 4$). (b)$\hat{\mathcal{H}}_{\text{AA}}$ in the extended phase ($\Gamma=0, V = 1$). (c)$\hat{\mathcal{H}}_{\text{AA}}$ in the MBL phase ($\Gamma=3, V = 6$). (d)$\hat{\mathcal{H}}_{\text{AA}}$ in the ergodic phase ($\Gamma=1, V = 2$). Black dotted lines indicate the energy of the trial state prior to optimization. Insets show the low-weight stabilizer Rényi entropy $M_{2,2}$ throughout the optimization, emphasizing how initialization impacts energy convergence and entropy behaviors. Each curve corresponds to a random sample of initial parameters.

Theorems & Definitions (7)

• Theorem 1
• Definition 1: low-weight SRE
• Theorem 2
• Definition 2: Uniform kick scaling
• Theorem 3
• Proposition 1: Circuit universality
• Proposition 2