Thermalization in Nature and on a Quantum Computer

Arnau Riera; Christian Gogolin; Jens Eisert

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Thermalization in Nature and on a Quantum Computer

Arnau Riera, Christian Gogolin, Jens Eisert

arXiv:1102.2389v4Feb 11, 2011quant-phcond-mat.stat-mechmath-ph

TL;DR

The paper examines when thermal states emerge and how Gibbs states can be prepared with controlled error. It derives conditions for thermalization, bounds the quantum algorithm’s error, and identifies the low-temperature complexity boundary.

Problem
The analysis must determine which energy distributions produce Gibbs-like states and how difficult generic Gibbs-state preparation becomes at low temperatures.
Method
The paper analyzes thermalization conditions and bounds the one-norm error between the algorithm’s relevant states, including finite-size bath effects.
Results
Thermal states can emerge for energy distributions with bounded variation and sufficiently sharp high-energy cutoffs, while the algorithm’s error has an exponentially small bath-size contribution.
Takeaways & Limitations
Weak interactions are characterized relative to the thermal energy k_BT, and the framework supports Gibbs-state preparation with a certified error bound.
Takeaways & Limitations
The algorithm is exponentially hard to run at low temperatures when no Hamiltonian structure is exploited.

Abstract

from arXiv · show

In this work, we show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, building on the program of dynamical typicality. We introduce a novel perturbation theorem for physically relevant weak system-bath couplings that is applicable even in the thermodynamic limit. We identify conditions under which thermalization happens and discuss the underlying physics. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime and error bound. This complements quantum Metropolis algorithms, which are expected to be efficient but have no known runtime estimates and only work for local Hamiltonians.

A Reduced density matrix of the dephased rectangular decoupled state – the general case

The reduced state of a dephased rectangular state is determined by the bath’s state-counting function, and its distance from a Gibbs state depends on how closely that function is locally exponential.

General reduction: The subsystem probabilities are proportional to the number of bath states available in the energy interval shifted by each subsystem energy.The bath-state count is evaluated over [E−E_Sk, E−E_Sk+∆].
General reduction: The bath spectrum is approximated through the logarithm of its state count, represented by a twice-differentiable function for sufficiently large baths.The continuous approximation introduces errors described as exponentially small in bath size for natural systems and reasonable energy ranges.
Gibbs approximation: The resulting subsystem probabilities take a Gibbs-like exponential form when the bath state count is locally approximated by an exponential curve.The normalization is expressed through ZP = Σ_i e^−βE_Si.
Gibbs approximation: The trace distance between the reduced state and the Gibbs state is bounded by the variation in the bath’s effective inverse-temperature parameter.The bound includes a correction C that is exponentially small in bath size.

B Reduced density matrix of the dephased rectangular decoupled state – a specific model

For an uncoupled-spin bath, level degeneracies obstruct a sufficiently accurate smoothed density-of-states approximation, so a suitable perturbation is introduced before bounding the Gibbs-state error.

Bath model: The specific bath consists of m non-interacting spin-1/2 particles with energies 0 and η, producing integer-spaced, highly degenerate levels.The number of states at energy ηk follows a binomial distribution.
Bath model: The degeneracy prevents a sufficiently accurate smoothed approximation to the bath density of states at widths scaling as d_B^-κ.The resulting approximation error does not decrease exponentially with bath size.
Perturbed bath: A suitable perturbation that lifts level degeneracies is therefore required, and essentially any adequately strong weak interaction can suffice when only the bath spectrum matters.The argument does not require preserving the bath eigenstates, only obtaining an appropriate spectrum.
Perturbed bath: Random local-field perturbations can produce, with overwhelmingly high probability, a bath state count that is well approximated over the relevant energy interval.The perturbation keeps the average local excitation energy at η.
Error bound: For the general case, the reduced-state error is controlled by the curvature of the bath entropy function and a correction exponentially small in bath size.Larger energy windows reduce the curvature contribution in the stated general bound.

C Rectangular states

Thermalization extends beyond rectangular energy distributions when the distribution varies moderately and then sharply cuts off toward higher energies, provided the bath density of states grows exponentially.

General energy distributions: The exponential growth of the bath density of states makes the upper edge of a rectangular energy interval dominate the reduced state.This motivates considering broader energy distributions with a similar high-energy cutoff.
General energy distributions: A state can remain close to Gibbs when its energy distribution has a bounded-variation region followed by a sufficiently sharp cutoff toward higher energies.The cutoff must fall from much larger than 1/d to much smaller than 1/d over an interval small compared with (∂^2s/∂E^2)^−1/2.
General energy distributions: Bounded variation is needed because, for systems violating the eigenstate thermalization hypothesis, details of the energy distribution can strongly affect some state properties.The condition guarantees thermalization despite this sensitivity.

D Complexity of the algorithm at low temperatures

The algorithm’s repetition cost is governed by the probability of obtaining the desired energy window, and reaching low temperatures is exponentially hard without exploiting additional Hamiltonian structure.

Run probability: The expected number of algorithm runs is determined by the probability of producing a rectangular state in the energy window corresponding to the target temperature.The analysis assumes the generated rectangular state is exact and that the bath spectrum is dense enough for Gibbs-state closeness.
Run probability: The probability of obtaining energy E is evaluated using the total Hilbert-space dimension and a locally exponential approximation to the bath density of states.The total dimension is d = d_S d_B.
Run probability: The average number of runs required to obtain the target rectangular state is lower bounded by the inverse of its preparation probability.The bound is expressed using the system energy gap and the relevant energy position.
Low-temperature complexity: Low-temperature preparation is exponentially hard for the general algorithm because it does not use Hamiltonian structure.Specific Hamiltonians with additional structure may permit greater efficiency, while generic arbitrary cooling would relate to QMA-hard problems.

E Temperature and its error for the bath described in Appendix B

This section relates the measured register value and ancilla size to the prepared rectangular state's energy, width, inverse temperature, and temperature precision.

Energy and temperature: The ancilla-register size q determines how the energy measurement resolves the rectangular state's temperature.The section explicitly analyzes how β and its error δβ depend on the obtained energy and q.
Energy and temperature: The measured value s∗ determines the prepared state's energy E = ∥H∥∞2−qs∗ and thereby its inverse temperature β.The algorithm is repeated until the desired energy, and hence temperature, is obtained.
State resolution: The rectangular state's width is determined by the ancilla-register size r through the number of compatible states Δ∗.Choosing r fixes the width Δ of the prepared rectangular state.
Temperature precision: A sufficient choice of parameters can be made to reach a specified temperature precision δβ.The required choice is given after deriving the dependence of the temperature error on the measured energy and register size.

F Error and average number of runs of the quantum algorithm

The algorithm's total error is decomposed into deviations from the Gibbs state and from the ideal rectangular state, while its runtime reflects a trade-off between thermal accuracy and phase-estimation cost.

Error decomposition: The total algorithmic error is bounded by the sum of two errors.One contribution is the rectangular state's deviation from the Gibbs state, and the other is the circuit's deviation from the rectangular state.
Error trade-off: Increasing the dimensionless parameter λ improves the Gibbs-state approximation by making the bath's energy content larger than the system's.The same increase makes phase estimation harder because the spectrum becomes denser.
Runtime: Lower temperatures require a quadratic increase in ancilla qubits of order O(β2∥HS∥2∞).The phase-estimation accuracy can also be increased by linearly increasing the number of ancilla qubits r.
Runtime: Phase estimation uses 2^r controlled unitary operations, producing an exponential runtime cost in the number of ancilla qubits.The passage identifies this as a problem common to algorithms using phase estimation on many-particle systems.
Number of runs: The average number of runs needed to obtain a target temperature is derived for the bath model considered in Appendix B.The derivation uses the model's smoothed number of states and the parameter λ.

Deviation from the rectangular state for a general bath

For a general bath, the circuit prepares a state close to a rectangular energy-window state, and the resulting trace-distance error is bounded using spectral-density smoothing and the convergence of Fr,q to Gq.

Circuit output: After measuring q of r ancilla qubits and obtaining s∗, the circuit prepares a rectangular state with energy E = s∗2−q∥H∥∞ and width Δ = 2q−r∥H∥∞.The trace distance between the circuit output and this rectangular state is the quantity bounded in this section.
Error formulation: The circuit's diagonal-state trace distance can be expressed as the one-norm distance between the eigenvalue distributions of the circuit output and rectangular state.The eigenvalues are denoted {qk} and {pk}.
Error formulation: The functions Fr,q and Gq determine the eigenvalue distributions, with Gq representing the rectangular profile and Fr,q its finite-r approximation.As r increases, Fr,q tends to Gq.
Spectral approximation: A sufficiently smooth spectral density allows the relevant spectral sum to be approximated by an integral using bins of width ∥H∥∞/L.The smoothing assumption requires the number of states in each bin to be well approximated by its smoothed version.
Spectral approximation: For suitable baths, L can scale exponentially with bath size, making the binning approximation error O(L−1) exponentially small.The two endpoint bins contribute only an error of order 1/L.
Error bound: The circuit error is bounded from above by combining the binwise estimates and the resulting integral approximation.The bound is subsequently related to the one-norm distance between the relevant distributions.

Appendix B

Appendix B bounds the circuit error for a specific bath by relating total-system and bath densities of states and lower-bounding the smoothed number of states.

Bath-specific bound: The Appendix B error analysis bounds the one norm ∥h ϱ∥1 appearing in the circuit-error expression.The bound is obtained for the bath model described in Appendix B.
Density of states: The total system's density of states is expressed in terms of the bath's density of states.The bath density is then inserted and rewritten using the inverse temperature.
Density of states: For positive temperatures β ≥ 0, the total density of states reaches its maximum at β = 0.The total Hilbert-space dimension is d = dS dB.
Smoothed state count: The smoothed number of states Ξ∆(E) is lower-bounded at spectral positions corresponding to positive temperature.This lower bound is used alongside the density-of-states estimate in the error analysis.

One norm between Fr,q and Gq

This section bounds the one-norm distance between the functions Fr,q and Gq, then uses that bound to control the distance between FN and G.

Symmetry and periodicity of fr equate the relevant right-tail and left-tail contributions.
The one-norm distance between Fr,q and Gq is bounded by rewriting the integral over the support of the step function Gq.Gq is nonzero only on [0, 2^-q[, where Fr,q(ϕ) < Gq(ϕ) = 2^q.
The resulting sum is split at k = 2^(r−q), whose corresponding value is exactly 1/2, before being bounded in two steps.
The bound on the one-norm distance between Fr,q and Gq is inserted into the preceding expression to obtain a bound for FN and G.
The error bound is completed by noting that C is exponentially small in the bath size.

G Discussion of the argument presented in Ref. [17]

This discussion examines Ref. ’s iterative thermalization procedure and identifies a tension between the dephasing scale required by the Dyson expansion and the spectral-gap parameter ζ.

Discussion of the argument presented in Ref. [17]: Ref. assembles prethermalized parts iteratively to obtain a Gibbs state for a quantum system with a local Hamiltonian.
Discussion of the argument presented in Ref. [17]: Each merging step updates Gibbs weights by postselection and rotates the eigenbasis through dephasing, with target error O(ϵ^2).
Discussion of the argument presented in Ref. [17]: Finite-σ dephasing approximates perfect dephasing, while the second-order Dyson expansion supplies the corresponding asymptotic error analysis.
Discussion of the argument presented in Ref. [17]: The analysis states that larger σ improves dephasing accuracy but invalidates the second-order Dyson approximation.
Discussion of the argument presented in Ref. [17]: The constructed Hamiltonian retains the eigenbasis of H + ϵh while grouping eigenvalues into bins separated by a minimum gap ζ.
Discussion of the argument presented in Ref. [17]: The required scaling includes both σ = O(1/ζ) and σ = Ω(1/ζ), constraining σ from above and below.
Discussion of the argument presented in Ref. [17]: The discussion argues that these requirements force ζ = Ω(ϵ), conflicting with the alternative scaling ζ = Θ(ϵ^2).

H Theorem 1 for an exponential density of states

This section extends the indistinguishability analysis of interacting and noninteracting microcanonical states to baths whose density of states is locally exponential.

Theorem 1 for an exponential density of states: The trace-distance bound compares microcanonical states for H0 and H = H0 + V over the energy interval [E, E + Δ].
Theorem 1 for an exponential density of states: The bound depends on the support dimensions of the two microcanonical states and on eigenstate counts near the interval boundaries.
Theorem 1 for an exponential density of states: The analysis replaces the approximately constant-density assumption with a locally exponential density of states, under which thermal states can emerge.
Theorem 1 for an exponential density of states: The two positive terms in the upper bound must be independently small, and the first requires ∥V∥∞ ≪ ε.
Theorem 1 for an exponential density of states: The necessary conditions include βε ≪ 1 together with a sufficiently small interaction norm.
Theorem 1 for an exponential density of states: Equation (96) ensures indistinguishability when Δ > kBT is not too small and its stated interaction condition is satisfied.
Theorem 1 for an exponential density of states: The physical interpretation is that an interaction is weak when it is small compared with the thermal energy kBT.