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2010

Preparing thermal states of quantum systems by dimension reduction

Bilgin, Ersen, Boixo, Sergio

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We present an algorithm that prepares thermal Gibbs states of one dimensional quantum systems on a quantum computer without any memory overhead, and in a time significantly shorter than other known alternatives.

Specifically, the time complexity is dominated by the quantity $N^{\|h\|/ T}$, where $N$ is the size of the system, $\|h\|$ is a bound on the operator norm of the local terms of the Hamiltonian (coupling energy), and $T$ is the temperature.
Given other results on the complexity of thermalization, this overall scaling is likely optimal.
For higher dimensions, our algorithm lowers the known scaling of the time complexity with the dimension of the system by one.

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We can truncate the tails of the Gaussian distribution to ensure a worst time cost of the order of the average cost, and the resulting error is already dominated by the imperfect dephasing

Cited in the paper.

An analogous bound holds between ρ \rho and the Gibbs state of \mathaccentV t i l d e 07 E H − ϵ h \mathaccentV{tilde}07EH-\epsilon h . We also need a bound on the difference between the partition functions of ρ ( ϵ ) \rho^{(\epsilon)} and \mathaccentV t i l d e 07 E ρ ( ϵ ) \mathaccentV{tilde}07E{\rho}^{(\epsilon)} , which can be found in [ 11 ]

Cited in the paper.

This is also known from the theory of success runs. We give the average cost, but the tail has an exponential decay rate, so the worst case cost is similar (see, for instance [ 26 ] )

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E. Bilgin and D. Poulin, Phys. Rev. B 81

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