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Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer
David Poulin, Pawel Wocjan
TL;DR
The paper analyzes how Hamiltonian simulation error affects effective Hamiltonians, partition functions, and Gibbs states. It shows that a simulated unitary corresponds to a nearby self-adjoint operator and that sufficiently small Hamiltonian errors preserve partition-function estimates and Gibbs-state fidelity.
Problem
The analysis addresses how errors in simulated unitaries and Hamiltonians affect partition functions and Gibbs states.
Method
Hamiltonian time-evolution simulation realizes an approximate unitary, whose effective Hamiltonian is analyzed using logarithm continuity and Hamiltonian perturbation bounds.
Results
K ≤6π + 12, and sufficiently small Hamiltonian error permits estimating Z(β) with small relative error while preserving high Gibbs-state fidelity.
Takeaways & Limitations
The bounds connect simulation accuracy to reliable partition-function estimation and preparation of Gibbs states at a specified temperature.
Abstract
from arXiv · showhide
We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D^alpha on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and alpha < 1/2 is proportional to the Helmholtz free energy density of the system. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared.
APPENDIX A: SIMULATION ERROR FOR THE EFFECTIVE HAMILTONIAN
The appendix shows that a sufficiently accurate simulation of U yields an effective Hamiltonian close to H, using continuity of the matrix logarithm on a suitable spectral domain.
- Simulation-error bound: ∥U − ˜U∥≤ϵ implies the existence of a self-adjoint ˜H satisfying ˜U = e^{it˜H} and ∥H − ˜H∥≤κϵ/t.The construction assumes ∥H∥≤π/(4t) and uses the principal matrix logarithm.
- Matrix-logarithm construction: The logarithm is defined on Cπ and extended as a primary matrix function for matrices whose spectra lie in that domain.For unitary operators, the logarithm is defined through diagonalization, with LogU = itH.
- Spectral containment: For ϵ ≤ 1/4, the spectra of U and ˜U remain within I(γ), enabling the logarithm-continuity argument and the resulting effective-Hamiltonian control.The appendix also bounds the spectrum of ˜H using ∥˜H + c∥≤∥H + c∥+ϵ/t.
- Lipschitz continuity: For noncommuting unitary matrices with spectra in I(γ), continuity of the logarithm supplies a Lipschitz bound in the operator norm.The proof uses a contour integral over γ and bounds logarithms and resolvents on its arcs and line segments.
- Lipschitz continuity: R = 2 yields the explicit Lipschitz constant K ≤ 6π + 12 for the logarithm estimate.The contour length is |γ| = π(R+1/R)+2(R−1/R).
APPENDIX B: LIPSCHITZ CONTINUITY OF PARTITION FUNCTIONS WITH RESPECT TO HAMILTONIANS
This appendix establishes continuity of the partition function under Hamiltonian perturbations, allowing the partition function of an approximate effective Hamiltonian to estimate that of H.
- Partition-function continuity: If ˜H = H + E with ∥E∥≤ϵ/t, the corresponding partition functions can be compared through a multiplicative error controlled by δ = e^(βϵ/t).The comparison follows from Weyl’s perturbation theorem applied to −βH and −β˜H.
- Implication: A sufficiently small Hamiltonian error permits estimating Z(β) with small relative error by estimating ˜Z(β) instead.The required error tolerance depends on the inverse temperature β.
APPENDIX C: LIPSCHITZ CONTINUITY OF GIBBS STATES WITH RESPECT TO HAMILTONIANS
This appendix bounds the fidelity between Gibbs states generated by H and by a perturbed Hamiltonian ˜H, showing that controlled Hamiltonian error preserves thermal-state closeness.
- Gibbs-state continuity: For ˜H = H + E with ∥E∥≤ϵ/t, the appendix derives a lower bound on the fidelity between ρ(β) and ˜ρ(β).The states are defined as normalized exponentials of −βH and −β˜H.
- Proof strategy: The fidelity analysis uses a strengthened Golden–Thompson-type inequality for positive operator parameters.The cited inequality is monotonic in p and recovers the Golden–Thompson inequality at p = 1.
- Proof strategy: Setting p = 2 relates the fidelity to partition functions, with the midpoint Hamiltonian ˆH = (H + ˜H)/2 entering the bound.The midpoint satisfies ∥H − ˆH∥≤ϵ/(2t), so the partition-function continuity result applies.