Predicting Gibbs-State Expectation Values with Pure Thermal Shadows

• URL: http://arxiv.org/abs/2206.05302v4
• Date: Mon, 26 Jun 2023 13:03:25 GMT
• Title: Predicting Gibbs-State Expectation Values with Pure Thermal Shadows
• Authors: Luuk Coopmans, Yuta Kikuchi, and Marcello Benedetti
• Abstract summary: We propose a quantum algorithm that can predict $M$ linear functions of an arbitrary Gibbs state with only $mathcalO(logM)$ experimental measurements. We show that the algorithm can be successfully employed as a subroutine for training an eight-qubit fully connected quantum Boltzmann machine.
• Score: 1.4050836886292868
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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