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In this talk, and the corresponding slides on page 24/44, Brandao talks about the max eigenvalue problem which is: Given a Hermitian $n\times n$ matrix $H$, approximate its largest eigenvalue. (Note that this is basically the problem of finding the ground state energy for Hamiltonian $-H$) He then suggests the following algorithm to solve this problem with a quantum computer:

  1. Prepare the Gibbs state $\rho=e^{\beta H}/\text{tr}(e^{\beta H})$
  2. Cool down the system, i.e. take the limit $\beta \to \infty$

I can see, how in this limit, the Gibbs state tends to the ground state of $-H$, i.e. the maximum eigenvalue state of $H$ since in the eigenbasis with eigenvalues of $H$ given by $E_1\geq E_2 \geq \dots \geq E_n$, we have \begin{align} \rho=e^{\beta H}/\text{tr}(e^{\beta H})&=\frac{1}{\sum_{i=1}^n e^{\beta E_i}}\text{diag}(e^{\beta E_1}, \dots,e^{\beta E_n} )\\ &=\frac{1}{1+\sum_{i=2}^n e^{\beta (E_i-E_1)}}\text{diag}(1, e^{\beta (E_2-E_1)} \dots,e^{\beta (E_n-E_1)} )\\ & \overset{\beta\to\infty}{\to} \text{diag}(1, 0, \dots, 0) \end{align} and thus the state ends up in the pure state $\rho=|\psi_{max}\rangle \langle \psi_{max}|$ where $|\psi_{max}\rangle $ is the eigenstate to the maximum eigenvalue $E_1$.

So far, so good. Now, I have two questions:

  1. Brandao has on his slides the result that cooling down to $\beta = O(\log (n) /\delta)$ one can approximate the max eigenvalue to precision $\delta$. How do you derive this? Does one need to solve $|E_1-\text{tr}(\rho H)|<\delta$ for $\beta$?
  2. In this abstract algorithm, how does one actually get the eigenvalue from the cooled down Gibbs state? One still has to measure the system somehow as far as I understand.
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