Evidence for a two-dimensional quantum glass state at high temperatures

Abstract: Disorder in quantum many-body systems can drive transitions between ergodic and non-ergodic phases, yet the nature--and even the existence--of these transitions remains intensely debated. Using a two-dimensional array of superconducting qubits, we study an interacting spin model at finite temperature in a disordered landscape, tracking dynamics both in real space and in Hilbert space. Over a broad disorder range, we observe an intermediate non-ergodic regime with glass-like characteristics: physical observables become broadly distributed and some, but not all, degrees of freedom are effectively frozen. The Hilbert-space return probability shows slow power-law decay, consistent with finite-temperature quantum glassiness. In the same regime, we detect the onset of a finite Edwards-Anderson order parameter and the disappearance of spin diffusion. By contrast, at lower disorder, spin transport persists with a nonzero diffusion coefficient. Our results show that there is a transition out of the ergodic phase in two-dimensional systems.

• The paper presents clear experimental evidence for a quantum glass phase in a 2D superconducting qubit array using magnetization dynamics, Edwards-Anderson order, and Hilbert space diagnostics.
• It demonstrates power-law decay in the return probability with system-size dependent exponents and observes the breakdown of conventional magnetization diffusion at high disorder.
• The findings challenge the traditional ergodic versus many-body localized view by uncovering non-ergodic extended glassy dynamics without local integrals of motion.

Evidence for a Two-Dimensional Quantum Glass State at High Temperatures

Introduction

The phase structure and dynamical properties of disordered quantum many-body systems have remained central open issues for condensed matter and quantum information theory, particularly regarding the ergodic-to-nonergodic (thermal to glassy or many-body localized, MBL) transitions. While theoretical advances have suggested the possibility of non-ergodic extended (NEE) phases in higher dimensions, experimental evidence has been severely limited due to the exponential growth of Hilbert space and the computational hardness of simulating such systems. This paper (2601.01309) delivers a comprehensive experimental study of dynamics in a two-dimensional (2D) array of up to 70 superconducting qubits, providing clear evidence for an intermediate quantum glass phase.

Theoretical Framework

Disordered quantum systems with local interactions exhibit a rich interplay between thermalization, localization, and glassiness. At low disorder, the system is ergodic and thermalizes rapidly, displaying diffusion and fast relaxation of local observables. At high disorder, the MBL phase emerges, characterized by completely frozen dynamics and strictly localized states with local integrals of motion (LIOMs).

The intermediate regime, hypothesized in numerous random matrix and mean-field spin glass models, is the NEE or quantum glass phase. This phase exhibits wavefunction support over a large but non-extensive subset of Hilbert space, extensive configuration entropy, slow (power-law) relaxation, large sample-to-sample fluctuations, and the absence of conventional transport. While robust in mean-field and random matrix settings, the stability and physical evidence of this regime in finite-dimensional, experimentally accessible systems have been uncertain.

Experimental Approach

The experiments utilize a large, planar array of superconducting qubits with programmable disorder. The studied spin-1/2 Hamiltonian involves nearest-neighbor XY coupling and site-dependent random fields, placing the system in the regime classically analogous to hard-core bosons in a disordered environment. The protocol involves initializing the system in a product state, evolving under the Hamiltonian for variable times, and measuring both real-space magnetization profiles and the return probability in Hilbert space.

Key observable diagnostics include:

1. Magnetization Dynamics: Tracking the relaxation of local magnetization, revealing frozen or slowly relaxing degrees of freedom characteristic of non-ergodic behavior.
2. Spin Glass Correlation Function: Extraction of the Edwards-Anderson (EA) order parameter, the canonical marker of spin glass order.
3. Return Probability $R(t)$: The overlap of the time-evolved state with the initial state, reflecting ergodicity in Hilbert space—decaying to $\sim 1/N$ in the ergodic phase, saturating in MBL, and showing power-law decay in the NEE regime.
4. Spin Transport and Diffusion: Analysis of magnetization mode relaxation to probe diffusion, subdiffusion, or its breakdown.

Key Results

Observation of Glassy Regime

The experiments reveal that upon increasing disorder, the system transitions from rapid, nearly complete magnetization relaxation to a regime where a significant, finite remanent magnetization persists even at long times. The spin-glass correlation function transitions from power-law decay (with exponent $\alpha$ decreasing towards a critical threshold $w_c$) to saturation, with the Edwards-Anderson parameter showing a sharp kink at $w_c \sim 10$. This is the dynamical signature of glass formation and marks departure from canonical ergodicity.

Hilbert Space Structure: Power-law Dynamics and Clustering

Analysis of the return probability $R(t)$ exhibits three distinct regimes:

• At small disorder, $R(t)$ rapidly decays to values near the inverse Hilbert-space volume, with an initial "correlation hole"—a signature of ergodic spectral correlations.
• At intermediate disorder, $w_c < w < w_{MBL}$, $R(t)$ decays slowly as a power law, $R_{\mathrm{typ}}(t) \sim t^{-\eta}$, with the decay exponent superlinear in system size, $\eta \sim n^{2.4}$. Moreover, $R(t)$ exhibits a broad (nearly log-normal) distribution, indicating highly nonthermal, glassy dynamics.
• At very strong disorder, the decay exponent $\eta$ drops rapidly, consistent with convergence to an MBL phase.

Breakdown of Diffusion

Transport measurements identify the point at which conventional diffusion of magnetization breaks down. At weak disorder, the relaxation rates of the lowest diffusion modes scale linearly with eigenmode wavenumber, establishing Fickian diffusion. Above $w \sim 10$, this relationship collapses, with vanishing diffusion constant and growing non-relaxing (glassy) fraction, in full correspondence with glass transition phenomenology.

Absence of Local Integrals of Motion in the Glassy Regime

Unlike MBL phases, the glassy state does not support LIOMs and instead features dominant dephasing rates exceeding relaxation rates. This dynamical structure is reflected in the persistence of quantum coherence (slow dephasing, weak relaxation) and the absence of complete thermalization or localization.

Power Spectrum and Universal Glassy Noise

Numerical simulations and analysis indicate $1/f$ noise and a wide spectrum of local relaxation rates (as expected for a system with a broad distribution of two-level systems), in line with phenomenological expectations for both structural and spin glasses.

Implications

The experimental observation of a quantum glass phase in 2D disordered systems establishes that the dichotomy between thermal (ergodic) and localized (MBL) phases is insufficient. It provides direct evidence that, even in two dimensions and at high temperatures, quantum systems can exhibit intermediate non-ergodic extended glassy dynamics with slow relaxation, Hilbert space clustering, and vanishing diffusion without full localization.

This regime is theoretically distinct from both conventional ergodic and localized phases, as highlighted by:

• Finite Edwards-Anderson Order: Signaling frozen-in degrees of freedom without full localization.
• Power-Law Decay, Not Exponential or Complete Freeze: Long-lived memory effects and lack of full ergodicity.
• System-Size-Dependent Decay Exponents: Reflecting the effect of glassy Hilbert space fragmentation.

From a practical perspective, this phase presents novel challenges for quantum simulation and computation, as its non-universal dynamics and the absence of LIOMs substantially increase the classical hardness of simulating such systems. The results also have implications for quantum memory, decoherence, and noise in quantum devices, as glassy relaxation directly affects device performance.

Future Directions

This work motivates several lines of future inquiry:

• Scaling studies in larger 2D and 3D arrays to probe the ultimate fate of the glassy regime in the thermodynamic limit.
• Exploration of aging, rejuvenation, and memory effects in quantum glasses, paralleling classical glass phenomenology.
• Investigation of glassy physics in other architectures (e.g., Rydberg atom arrays, trapped ions, quantum dots).
• Studies of quantum information scrambling and operator hydrodynamics in glassy versus ergodic and MBL regimes.
• Analytical development of nonergodic extended phase theory for sparse and realistic local Hamiltonians.

Conclusion

This paper (2601.01309) delivers statistically robust, system-size-resolved evidence for a quantum glass phase in a two-dimensional array of superconducting qubits subject to disorder. The demonstrated existence of a phase with power-law relaxation, glassy Hilbert space structure, finite EA order, and breakdown of diffusion—distinct from both thermal and MBL regimes—extends the experimentally accessible non-ergodic quantum phase landscape. The results stimulate both experimental and theoretical investigations into high-dimensional quantum glassiness and its ramifications for quantum control, simulation, and computation.