Quantum algorithms for Hamiltonian learning problems
- Publication Type:
- Thesis
- Issue Date:
- 2022
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Learning the dynamic Hamiltonian of a quantum system is a fundamental task in studying condensed matter physics and verifying quantum technologies. Many existing Hamiltonian learning methods require the ability to prepare quantum Gibbs states and estimate entropies of quantum states, which are not easy for classical computers. Recent experimental progress of quantum hardware has drawn much attention, motivating us to investigate the application of near-term quantum computers in Hamiltonian learning. In this dissertation, we study the Hamiltonian learning problems and propose algorithms to recover interaction coefficients of a many-body Hamiltonian, prepare quantum Gibbs states, and estimate the quantum entropies of quantum states. We employ quantum circuits that are expected to be implementable in the near-to-intermediate future.
First, we use the variational quantum algorithms to enable a hybrid quantum-classical algorithmic scheme to tackle the Hamiltonian learning problem. By transforming the Hamiltonian learning problem into an optimization problem using Jaynes' principle, we employ a gradient-descent method to give the solution and could reveal the interaction coefficients from the system's Gibbs state measurement results. In particular, the computation of the gradients relies on the Hamiltonian spectrum and the log-partition function. Hence, as the main subroutine, we develop a variational quantum algorithm to extract the Hamiltonian spectrum and utilize convex optimization to compute the log-partition function. We also apply the importance sampling technique to circumvent the resource requirements for large-scale Hamiltonians.
Second, we propose variational quantum algorithms for quantum Gibbs state preparation. Specifically, we take the loss function as the system's free energy and estimate it by a truncated version. Then train a parameterized quantum circuit to optimize the loss function so that it can learn the desired quantum Gibbs state. Furthermore, by performing numerical experiments, we show that shallow parameterized circuits with only one additional qubit can be trained to prepare the Ising chain and spin chain Gibbs states with a fidelity higher than 95%.
Third, we propose quantum algorithms to estimate the von Neumann and quantum alpha-R'enyi entropies of an n-qubit quantum state rho using independent copies of the input state. We show how to efficiently construct the quantum circuits of both methods using primitive single/two-qubit gates. We prove that the number of required copies scales polynomially in 1/epsilon and 1/Lambda, where epsilon denotes the additive precision and Lambda denotes the lower bound on all non-zero eigenvalues.
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