Universal energy-space localization and stable quantum phases against time-dependent perturbations

TL;DR

This work identifies energy-space localization as a universal property of time-dependent $q$-local Hamiltonians, proving that an evolving state remains exponentially localized near its initial energy within the instantaneous spectrum and that leakage is bounded by bounds depending on the total variation of the perturbation. The main results include explicit exponential leakage bounds for general $q$-local and quasi-$q$-local perturbations, plus corollaries for static perturbations, all hinging on controlled growth of nested commutators. By coupling energy-space localization with a clustered energy landscape, the authors demonstrate robust stability and ergodicity breaking in spin-glass–like systems and LDPC codes with linear energy barriers, including exponentially long dynamical localization and infinitely long eigenstate localization under suitable detuning. Applications to LDPC codes show exponentially slow mixing of Gibbs samplers, while implications for hard optimization problems indicate that quantum Hamiltonian-based algorithms may be effectively trapped within local minima unless driving is sufficiently strong. Collectively, the results offer a versatile mathematical framework for analyzing non-equilibrium dynamics, stability proofs, and quantum algorithm design in systems with extensive energy barriers.

Abstract

Stability against perturbation is a highly nontrivial property of quantum systems and is often a requirement to define new phases. In most systems where stability can be rigorously established, only static perturbations are considered; whether a system is stable against generic time-dependent perturbations remains largely elusive. In this work, we identify a universal phenomenon in $q$-local Hamiltonians called energy-space localization and prove that it can survive under generic time-dependent perturbations, where the evolving state is exponentially localized in an energy window of the instantaneous spectrum. The property holds ubiquitously, and the leakage bounds remain invariant under arbitrarily monotonic rescaling of evolution time. This flexibility enables the energy-space localization to be a powerful tool in proving the stability of systems. For spin glass models where the configuration spaces are separated by large energy barriers, the localization in energy space can induce a true localization in the configuration space and robustly break ergodicity. We then demonstrate the applications of our results in several systems with such barriers. For certain LDPC codes, we show that the evolving state is localized near the original codeword for an exponentially long time even under generic time-dependent perturbations. We also extend the stability of LDPC codes against static $q$-local perturbations to quasi-$q$-local. In addition, we show that for some classical hard optimization problems with clustered solution space, the stability becomes an obstacle for quantum Hamiltonian-based algorithms to drive the system out of local minima. Our work provides a new lens for analyzing the non-equilibrium dynamics of generic quantum systems, and versatile mathematical tools for stability proving and quantum algorithm design.

Figures (4)

• Figure 1: An illustrative example of the energy-space Localization. $\Lambda(t)$ denotes the total variation of $H(t)$ from $0$ to $t$. The results are from simulating an 8-qubit $H(t)$ consisting of random $2$-local Pauli operators with all-to-all couplings, where the initial state is chosen to be one eigenstate of $H(0)$. The quantity $\mathbf{P}$ for $j$-th eigenstates of the instantaneous Hamiltonian $H(t)$ is its probability in expanding the evolving state $\ket{\psi(t)}$.
• Figure 2: An illustrative example of the clustering property in the energy window $[E_1,E_2]$, where the gray area denotes all the eigenstates. We compress the high-dimensional eigenstates to one dimension for visual convenience, and the distance along the x-axis roughly represents the distance $\mathbf{D}$ between two eigenstates.
• Figure 3: Illustration of the clustering property in classical/quantum LDPC codes with linear soundness, where the blue area denotes all the eigenstates.
• Figure 4: Illustration of the clustering property in hard optimization problems, where all eigenstates are simply $Z$-basis states. If the total variance of the algorithmic driving is $\Lambda$, then states localized below $E_B-2 \Lambda$ (red zones) cannot escape from their clusters. In most cases, the number of the shallower local minima, like $w_3$, is much larger than the deeper minima, like $w_1$ and $w_2$.

Theorems & Definitions (17)

• Theorem 1: Energy-space localization, informal
• Corollary 1: Bounds from nested commutators, informal
• Corollary 2: Bounds for general scalings, informal
• Theorem 2: Energy-space localization, static cases
• Definition 1: Clustering property
• Proposition 3.1: Exponentially long dynamical localization under time-dependent perturbations
• Proposition 3.2: Eigenstate localization with (quasi-)$q$-local perturbations, informal
• Proposition 3.3: Infinitely long dynamical localization with static (quasi-)$q$-local perturbation
• Proposition 3.4: Robust slow mixing of Gibbs sampler with (quasi-)$q$-local perturbations
• Proposition 4.1: Freezing of solutions