Onset of Ergodicity Across Scales on a Digital Quantum Processor

Abstract

Understanding how isolated quantum many-body systems thermalize remains a central question in modern physics. We study the onset of ergodicity in a two-dimensional disordered Heisenberg Floquet model using digital quantum simulation on IBM's Nighthawk superconducting processor, reaching system sizes of up to $10\times10$ qubits. We probe ergodicity across different length scales by coarse-graining the system into spatial patches of varying sizes and introducing a measure based on the collision entropy of each patch, enabling a detailed study of when ergodic behavior emerges across scales. The high sampling rate of superconducting quantum processing units, together with an optimal sample estimator, allow us to access patches of sizes up to $3\times3$. We observe that as the Heisenberg coupling $J$ increases, the noiseless system undergoes a smooth crossover from subergodic to ergodic behavior, with smaller patches approaching their random-matrix-theory values first, thereby revealing a hierarchy across scales. In the region of parameter space where classical tensor-network simulations are reliable, small patches or small values of $J$, we find excellent agreement with the error-mitigated quantum simulation. Beyond this regime, volume-law entanglement and contraction complexity growth causes the cost of classical methods to rise sharply. Our results open new directions for the use of quantum computers in the study of quantum thermalization.

Figures (31)

• Figure 1: (a) The periodic quantum circuits we consider consist of a 2D brickwork of gates. Each blue slab covering all qubits represents one application of $U_F$, a single Floquet layer. The sequence of underlying two-qubit gates is shown for one such layer, forming a densely packed brickwork. Each gate consists of the entangling unitary $e^{iJ(X_iX_j+Y_iY_j+Z_iZ_j)}$ followed by two random single-qubit rotations $e^{i (h_iZ_i+ h_j Z_j)}$, with $h_i$ drawn uniformly from $[-\pi/2,\pi/2]$ on the first Floquet layer and then repeated periodically in time. (b) A visualization created from the results of an $8\times 8 = 64$ qubit simulation on the IBM Quantum's Nighthawk-family device, illustrating the crossover from non-ergodic to ergodic behavior in the Heisenberg Floquet system as a function of the coupling $0 \le J < \pi/4$. At each of the three values of $J$ we exhibit the collision entropy of the $64$ single qubit marginals, as well as the collision entropy of the $16$ non-overlapping $2\times 2$ subsystems. All patches of up to size $3\times3$ can be similarly obtained but are not shown here.
• Figure 2: Marginal 2-collision entropies $S_{2,Z}[A]$ for central connected marginals in the $5\times5$ system at $n_F=3$ cycles, computed via exact state vector simulation, averaged over $256$ disorder realizations. The stars ($\bigstar$) indicate the couplings $J^\star$ at which $S_{2,Z}[A]$ reaches a plateau within $0.1$ of the corresponding Haar values.
• Figure 3: (a)Single instance entropies: Experimental results from ibm_miami, mitigated using LEC (see \ref{['eq:LECdefinitionmain']}), for a range of system sizes beyond exact diagonalization. See \ref{['sec:Quantum simulation results']} in the Supplementary Material for the raw data and a more detailed discussion of mitigation techniques. In each case, we compute collision entropies of central marginals of various shapes for a fixed disorder instance. (b)Spatially averaged entropies:. For each system size, we fix the disorder instance from panel (a) and average the collision entropy over 16 marginals of a given shape. The symbol $\bigstar$ marks the smallest $J$ for which the measured entropy lies within $\varepsilon=0.1$ of the Haar value; since the entropy appears to vary smoothly with $J$, there is no sharply defined onset of ergodicity, and these markers are shown only to indicate trends.
• Figure 4: Depiction of a set of quantum mechanical degrees of freedom, e.g., qubits or spins and various patches $A$ on which one can define the marginal collision entropies.
• Figure 5: Sequence of gate operation implementing one Floquet cycle. The two-qubit gate \ref{['GateuijSM']} is sampled freshly in each application. Other orderings could be chosen but we do not expect the physics to depend on this choice.