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arXiv:2607.06847

How thermal is a filtered state?

Yilun Yang, J. Ignacio Cirac, Mari Carmen Bañuls

quant-phSubmitted Jul 7, 2026• Announced Jul 9, 2026• 0 citations
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Abstract

Quantum many-body states with sufficiently low energy variance can serve as approximations to thermal states, and they may be prepared by energy filtering simple pure states. In this work, we examine how narrow the filter width must be to guarantee thermal behavior. To this end, we analyze the problem in the Floquet regime, where filtered states are found to be equivalent to time averages. This equivalence allows us to reproduce the distinct Rényi-$α$ entropy scalings as reported in [Morettini et al., Physical Review Letters 133, 240401 (2024)]. Crucially, we show that under the Floquet eigenstate thermalization hypothesis, the trace distance between Floquet-filtered states and thermal states is bounded by the square root of filter width. We further demonstrate that these results extend naturally to the conventional Hamiltonian setting by mapping Hamiltonian-filtered states to their Floquet counterparts.

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Quantum many-body states with sufficiently low energy variance can serve as approximations to thermal states, and they may be prepared by energy filtering simple pure states. In this work, we examine how narrow the filter width must be to guarantee thermal behavior. To this end, we analyze the problem in the Floquet regime, where filtered states are found to be equivalent to time averages. This equivalence allows us to reproduce the distinct Rényi-ααα entropy scalings as reported in [Morettini et al., Physical Review Letters 133, 240401 (2024)]. Crucially, we show that under the Floquet eigenstate thermalization hypothesis, the trace distance between Floquet-filtered states and thermal states is bounded by the square root of filter width. We further demonstrate that these results extend naturally to the conventional Hamiltonian setting by mapping Hamiltonian-filtered states to their Floquet counterparts.
Energy-filtered quantum states are promising candidates for efficiently simulating thermal states. We explore a protocol designed to transition a product state into an eigenstate located in the middle of the spectrum; this is achieved by gradually reducing its energy variance, which allows us to comprehensively understand the crossover phenomenon and the subsequent convergence toward thermal behavior. We introduce and discuss three energy-filtering regimes (short, medium, and long), and we interpret them as stages of thermalization. We show that the properties of the filtered states are locally indistinguishable from those of time-averaged density matrices, routinely employed in the theory of thermalization. On the other hand, nonlocal quantum correlations are generated in the medium regimes and are witnessed by the Rényi entanglement entropies of subsystems, which we compute via replica methods. Specifically, two-point correlation functions break cluster decomposition, and the entanglement entropy of large regions scales as the logarithm of the volume during the medium filter time.
We propose a method to calculate finite-temperature properties of a quantum many-body system for a microcanonical ensemble by introducing a pure quantum state named here an energy-filtered random-phase state, which is also a potentially promising application of near-term quantum computers. In our formalism, a microcanonical ensemble is specified by two parameters, i.e., the energy of the system and its associated energy window. Accordingly, the density of states is expressed as a sum of Gaussians centered at the target energy with its spread corresponding to the width of the energy window. We then show that the thermodynamic quantities such as entropy and temperature are calculated by evaluating the trace of the time-evolution operator and the trace of the time-evolution operator multiplied by the Hamiltonian of the system. We also describe how these traces can be evaluated using random diagonal-unitary circuits appropriate to quantum computation. The pure quantum state representing our microcanonical ensemble is related to a state of the form introduced by Wall and Neuhauser for the filter diagonalization method [M. R. Wall and D. Neuhauser, J. Chem. Phys. 102, 8011 (1995)], and therefore we refer to it as an energy-filtered random-phase state. The energy-filtered random-phase state is essentially a Fourier transform of a time-evolved state whose initial state is prepared as a random-phase state, and the cut-off time in the time-integral for the Fourier transform sets the inverse of the width of the energy window. The proposed method is demonstrated numerically by calculating thermodynamic quantities for the one-dimensional spin-1/2 Heisenberg model on small clusters up to 28 qubits, showing that the method is most efficient for the target energy around which the dense distribution of energy eigenstates is found.
We identify several distinct phases of thermalization that describe regimes of behavior in isolated, periodically driven (Floquet), mesoscopic quantum chaotic systems. In doing so, we also identify a Floquet thermal ensemble, the ``ladder ensemble,'' that is qualitatively distinct from the ``featureless infinite-temperature'' state that has long been assumed to be the appropriate maximum-entropy equilibrium ensemble for driven systems. The phases we find can be coarsely classified by (i) whether or not the system irreversibly exchanges energy of order $\ensuremath{\omega}$ with the drive, i.e., Floquet thermalizes, and (ii) the Floquet thermal ensemble describing the final equilibrium in systems that do Floquet thermalize. These phases are representative of regimes of behavior in mesoscopic systems, but they are sharply defined in a particular large-system limit where the drive frequency $\ensuremath{\omega}$ scales up with system size NNN as the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is taken: we examine frequency scalings ranging from a weakly NNN-dependent $\ensuremath{\omega}(N)\ensuremath{\sim}logN$, to stronger scalings ranging from $\ensuremath{\omega}(N)\ensuremath{\sim}\sqrt{N}$ to $\ensuremath{\omega}(N)\ensuremath{\sim}N$. We show that the transition where Floquet thermalization breaks down happens at an extensive drive frequency and, beyond that, systems that do not Floquet thermalize are distinguished based on the presence or absence of rare resonances across Floquet zones. We produce a thermalization phase diagram that is relevant for numerical studies of Floquet systems and experimental studies on small-scale quantum simulators, both of which lack a clean separation of scales between NNN and $\ensuremath{\omega}$. A striking prediction of our work is that, under the assumption of perfect isolation, certain realistic quench protocols from simple pure initial states can show Floquet thermalization to a type of Schrodinger-cat state that is a global superposition of states at distinct temperatures. Our work extends and organizes the theory of Floquet thermalization, heating, and equilibrium into the setting of mesoscopic quantum systems.
It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved. For example, we show that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian heats up to infinite temperature at long times. We show that in agreement with the physical intuition, the only obstruction to thermalization is the existence of local observables which are periodic in time.
We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width δ . Given a local Hamiltonian on N qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to δN . We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in poly(N,1/δ) time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width δ is independent of the system size N . Furthermore, the adiabatic evolution can be implemented with circuit depth O(N2δ−4) . Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.
Conventional wisdom suggests that the long time behavior of isolated interacting periodically driven (Floquet) systems is a featureless maximal entropy state characterized by an infinite temperature. Efforts to thwart this uninteresting fixed point include adding sufficient disorder to realize a Floquet many-body localized phase or working in a narrow region of drive frequencies to achieve glassy non-thermal behavior at long time. Here we show that in clean systems the Floquet eigenstates can exhibit non-thermal behavior due to finite system size. We consider a one-dimensional system of spinless fermions with nearest-neighbor interactions where the interaction term is driven. Interestingly, even with no static component of the interaction, the quasienergy spectrum contains gaps and a significant fraction of the Floquet eigenstates, at all quasienergies, have non-thermal average doublon densities. We show that this non-thermal behavior arises due to emergent integrability at large interaction strength and discuss how the integrability breaks down with power-law dependence on system size.