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arXiv:2503.11616

Testing eigenstate thermalization hypothesis on small scale quantum processors

Maanav Srihari, Anil Shaji

Published Mar 14, 2025• 0 citations
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Abstract

The Eigenstate Thermalization Hypothesis (ETH) is a framework for discussing thermal behavior originating from chaotic dynamics in isolated many-body quantum systems. The PXP model, where certain states do not thermalize, has been compared with the Sachdev-Ye Kitaev (SYK) model, which is believed to be fully thermalizing. A gate-based quantum circuit approach is utilized to simulate time evolution and compute the Out-of-Time-Ordered Correlator (OTOC), a measure of the extent of chaos. Considering restrictions on implementing SYK on gate-based hardware, a simplified model called Spin-XY4 (SXY4), which has a thermal behavior similar to SYK, is tested. An alternate method, which optimizes control on an analog quantum device with the GRAPE (GRadient Ascent Pulse Engineering) algorithm, is also utilized to simulate the SYK model.

DOI
10.48550/arxiv.2503.11616

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The Eigenstate Thermalization Hypothesis (ETH) is a framework for discussing thermal behavior originating from chaotic dynamics in isolated many-body quantum systems. The PXP model, where certain states do not thermalize, has been compared with the Sachdev-Ye Kitaev (SYK) model, which is believed to be fully thermalizing. A gate-based quantum circuit approach is utilized to simulate time evolution and compute the Out-of-Time-Ordered Correlator (OTOC), a measure of the extent of chaos. Considering restrictions on implementing SYK on gate-based hardware, a simplified model called Spin-XY4 (SXY4), which has a thermal behavior similar to SYK, is tested. An alternate method, which optimizes control on an analog quantum device with the GRAPE (GRadient Ascent Pulse Engineering) algorithm, is also utilized to simulate the SYK model.
The eigenstate thermalization hypothesis (ETH) is a conjecture on the nature of isolated quantum systems that guarantees the thermal behavior of subsystems when it is satisfied. ETH has been tested in various forms on a number of local many-body interacting systems. Here we examine the validity of ETH in a class of nonlocal disordered many-body interacting systems --- the Sachdev-Ye-Kitaev Majorana (SYK) models --- that may be tuned from chaotic behavior to integrability. Our analysis shows that SYK4_44​ (with quartic couplings) which is maximally chaotic in the large system size limit satisfies the standard ETH scaling while SYK2_22​ (with quadratic couplings), which is integrable, does not. We show that the low-energy and high-energy properties are affected drastically differently when the two Hamiltonians are mixed.
The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its N \mathcal{N} N = 1 supersymmetric extension and find evidence that these models satisfy ETH. We discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.
The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its N = 1 supersymmetric extension and find evidence that these models satisfy ETH. In conclusion, we discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.
The Eigenstate Thermalization Hypothesis (ETH) is a standard tool to understand the thermalization properties of an isolated quantum system. Its generalization to higher order correlations of matrix elements of local operators, dubbed the full ETH, relates the ETH free cumulants to the corresponding thermal free cumulants in the thermodynamic limit. In this work, we numerically test the full ETH predictions using exact diagonalization of two integrable spin models: the Ising and the XXZ Heisenberg models. The differences from the behavior predicted by full ETH in chaotic systems are highlighted and contrasted along the way. We further study the out-of-time-ordered correlator (OTOC) through its approximate decomposition into contributions involving the second- and fourth-order ETH free cumulants. We find that, although the fourth-order contribution governs the late-time behavior of the OTOC in these integrable models, its dynamics differs significantly from that found in nonintegrable systems.
In an isolated quantum many-body system undergoing unitary evolution, we study the thermalization of a subsystem, treating the rest of the system as a bath. In this setting, the eigenstate thermalization hypothesis (ETH) was proposed to explain thermalization. Consider a nearly integrable Sachdev-Ye-Kitaev model obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. When the subsystem size is larger than the square root of but is still a vanishing fraction of the system size, we prove thermalization if the system is initialized in a random product state, while almost all eigenstates violate the ETH. In this sense, the ETH is not a necessary condition for thermalization.
The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS 2 , the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approximate closely their thermal counterparts and that information is scrambled in individual eigenstates. We study both the eigenstates of a single random realization of the model, as well as the model obtained after averaging of the random disordered couplings. We use our results to comment on the implications for thermal states of a putative dual theory, i.e. the AdS 2 black hole.