Predicting Gibbs-State Expectation Values with Pure Thermal Shadows

The preparation and computation of many properties of quantum Gibbs states is essential for algorithms such as quantum semidefinite programming and quantum Boltzmann machines. We propose a quantum algorithm that can predict $M$ linear functions of an arbitrary Gibbs state with only $\mathcal{O}(\log{M})$ experimental measurements. Our main insight is that for sufficiently large systems we do not need to prepare the $n$-qubit mixed Gibbs state explicitly but, instead, we can evolve a random $n$-qubit pure state in imaginary time. The result then follows by constructing classical shadows of these random pure states. We propose a quantum circuit that implements this algorithm by using quantum signal processing for the imaginary time evolution. We numerically verify the efficiency of the algorithm by simulating the circuit for a ten-spin-1/2 XXZ-Heisenberg model. In addition, we show that the algorithm can be successfully employed as a subroutine for training an eight-qubit fully connected quantum Boltzmann machine.