A system equilibrates when diagonalizing its hamiltonian is sufficiently hard

Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and the complexity of their eigenvectors, provided that a conjecture about the incompressibility of quantum circuits holds. We quantify the complexity by the size of the smallest quantum circuit mapping the local basis onto the energy eigenbasis. Specifically, we consider the set of all Hamiltonians having complexity $C$ , and show that almost all such Hamiltonians equilibrate if $C$ is superquadratic in the system size, which includes the fully random Hamiltonian case in the limit $C\ensuremath{\rightarrow}\ensuremath{\infty}$, and do not equilibrate if $C$ is sublinear. We also provide a simple formula for the equilibration time scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet they establish a fundamental link between equilibration and complexity theory.Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and the complexity of their eigenvectors, provided that a conjecture about the incompressibility of quantum circuits holds. We quantify the complexity by the size of the smallest quantum circuit mapping the local basis onto the energy eigenbasis. Specifically, we consider the set of all Hamiltonians having complexity $C$, and show that almost all such Hamiltonians equilibrate if $C$ is superquadratic in the system size, which includes the fully random Hamiltonian case in the limit $C\ensuremath{\rightarrow}\ensuremath{\infty}$, and do not equilibrate if $C$ is sublinear. We also provide a simple formula for the equilibration time scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet they establish a fundamental link between equilibration and complexity theory.

arXiv:1108.0374Physical Review EMar 18, 201372 citations

Equilibration of Measurement Statistics Under Complex Dynamics

Cozmin Ududec, Nathan Wiebe, Joseph Emerson

There has been considerable recent progress in clarifying the conditions under which quantum systems equilibrate. We demonstrate that effective equilibration of an isolated system can occur under much weaker conditions than previously considered, by requiring only an assumption of sufficiently complex dynamics (which encodes a condition on the complexity of the system eigenvectors) and natural information-theoretic constraints resulting from the infeasibility of collecting an astronomically large amount of measurement data. In particular, we prove that, for generic Hamiltonian systems, the measurement statistics for maximally fine-grained measurements on pure states are effectively indistinguishable from the micro-canonical distribution after a finite equilibration time. We demonstrate that our assumptions correspond to physically realistic conditions by numerically confirming their validity for both generic many-body systems limited to two-body interactions and the quantum kicked-top in a globally chaotic regime.There has been considerable recent progress in clarifying the conditions under which quantum systems equilibrate. We demonstrate that effective equilibration of an isolated system can occur under much weaker conditions than previously considered, by requiring only an assumption of sufficiently complex dynamics (which encodes a condition on the complexity of the system eigenvectors) and natural information-theoretic constraints resulting from the infeasibility of collecting an astronomically large amount of measurement data. In particular, we prove that, for generic Hamiltonian systems, the measurement statistics for maximally fine-grained measurements on pure states are effectively indistinguishable from the micro-canonical distribution after a finite equilibration time. We demonstrate that our assumptions correspond to physically realistic conditions by numerically confirming their validity for both generic many-body systems limited to two-body interactions and the quantum kicked-top in a globally chaotic regime.

Aug 16, 20122 citations

Statistical complexity and the road to equilibrium in many-body chaotic quantum systems

Manuel H. Muñoz-Arias

In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.

arXiv:2205.11777Physical review. EOct 3, 20223 citations

Relaxation of non-integrable systems and correlation functions

Jonathon Riddell, Luis Pedro García-Pintos, Alvaro Martin Alhambra

We investigate early-time equilibration rates of observables in closed many-body quantum systems and compare them to those of two correlation functions, first introduced by Kubo and Srednicki. We explore whether these different rates coincide at a universal value that sets the timescales of processes at a finite energy density. We find evidence for this coincidence when the initial conditions are sufficiently generic, or typical. We quantify this with the effective dimension of the state and with a state-observable effective dimension, which estimate the number of energy levels that participate in the dynamics. Our findings are confirmed by proving that these different timescales coincide for dynamics generated by Haar-random Hamiltonians. This also allows to quantitatively understand the scope of previous theoretical results on equilibration timescales and on random matrix formalisms. We approach this problem with exact, full spectrum diagonalization. The numerics are carried out in a non-integrable Heisenberg-like Hamiltonian, and the dynamics are investigated for several pairs of observables and states.We investigate early-time equilibration rates of observables in closed many-body quantum systems and compare them to those of two correlation functions, first introduced by Kubo and Srednicki. We explore whether these different rates coincide at a universal value that sets the timescales of processes at a finite energy density. We find evidence for this coincidence when the initial conditions are sufficiently generic, or typical. We quantify this with the effective dimension of the state and with a state-observable effective dimension, which estimate the number of energy levels that participate in the dynamics. Our findings are confirmed by proving that these different timescales coincide for dynamics generated by Haar-random Hamiltonians. This also allows to quantitatively understand the scope of previous theoretical results on equilibration timescales and on random matrix formalisms. We approach this problem with exact, full spectrum diagonalization. The numerics are carried out in a non-integrable Heisenberg-like Hamiltonian, and the dynamics are investigated for several pairs of observables and states.

arXiv:2112.09475Dec 17, 20210 citations

Equilibration time scales in closed many-body quantum systems

Thiago R de Oliveira, Christos Charalambous, Daniel Jonathan +2 more

We show that the physical mechanism for the equilibration of closed quantum systems is dephasing, and identify the energy scales that determine the equilibration timescale of a given observable. For realistic physical systems (e.g those with local Hamiltonians), our arguments imply timescales that do not increase with the system size, in contrast to previously known upper bounds. In particular we show that, for such Hamiltonians, the matrix representation of local observables in the energy basis is banded, and that this property is crucial in order to derive equilibration times that are non-negligible in macroscopic systems. Finally, we give an intuitive interpretation to recent theorems on equilibration time-scales.We show that the physical mechanism for the equilibration of closed quantum systems is dephasing, and identify the energy scales that determine the equilibration timescale of a given observable. For realistic physical systems (e.g those with local Hamiltonians), our arguments imply timescales that do not increase with the system size, in contrast to previously known upper bounds. In particular we show that, for such Hamiltonians, the matrix representation of local observables in the energy basis is banded, and that this property is crucial in order to derive equilibration times that are non-negligible in macroscopic systems. Finally, we give an intuitive interpretation to recent theorems on equilibration time-scales.

arXiv:1704.06646New Journal of PhysicsFeb 16, 201866 citations

Sustained growth of quantum circuit complexity in many-body Hamiltonian dynamics

Wonjun Lee, Saúl Pilatowsky-Cameo, Soonwon Choi

The quantum circuit complexity of an evolving many-body quantum system is believed to exhibit a sustained growth, maintained for timescales much longer than the onset of thermalization. Most previous works have focused on models which violate energy conservation, such as random unitary circuits. Here we study generic, local time-independent Hamiltonian dynamics. We unconditionally prove that for generic local Hamiltonians and typical initial product states at high effective temperature, the robust quantum circuit complexity must grow over a very long period of time, attaining an exponentially large value at late times. Our approach relies on two structural properties that we prove rigorously: (i) generic local Hamiltonians satisfy generalized spectral no-resonance conditions of arbitrary order, and (ii) typical high-temperature product states are effectively supported in exponentially many energy eigenstates. These properties have been widely assumed without proof in prior works. As corollaries of this result, we show the late-time state displays robust volume-law entanglement that is irremovable by polynomial-size circuits, and we establish a no fast-forwarding result for generic local Hamiltonians. We also lower bound the complexity of preparing sufficiently large subsystems with local quantum channels, in contrast with sufficiently small thermalizing regions which we show retain low complexity at late times.The quantum circuit complexity of an evolving many-body quantum system is believed to exhibit a sustained growth, maintained for timescales much longer than the onset of thermalization. Most previous works have focused on models which violate energy conservation, such as random unitary circuits. Here we study generic, local time-independent Hamiltonian dynamics. We unconditionally prove that for generic local Hamiltonians and typical initial product states at high effective temperature, the robust quantum circuit complexity must grow over a very long period of time, attaining an exponentially large value at late times. Our approach relies on two structural properties that we prove rigorously: (i) generic local Hamiltonians satisfy generalized spectral no-resonance conditions of arbitrary order, and (ii) typical high-temperature product states are effectively supported in exponentially many energy eigenstates. These properties have been widely assumed without proof in prior works. As corollaries of this result, we show the late-time state displays robust volume-law entanglement that is irremovable by polynomial-size circuits, and we establish a no fast-forwarding result for generic local Hamiltonians. We also lower bound the complexity of preparing sufficiently large subsystems with local quantum channels, in contrast with sufficiently small thermalizing regions which we show retain low complexity at late times.

quant-pharXiv:2609.26885Sep 24, 20260 citations

C

, and show that almost all such Hamiltonians equilibrate if

C

is superquadratic in the system size, which includes the fully random Hamiltonian case in the limit $C\ensuremath{\rightarrow}\ensuremath{\infty}$, and do not equilibrate if

C

is sublinear. We also provide a simple formula for the equilibration time scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet they establish a fundamental link between equilibration and complexity theory.