Dynamical mean field theory with quantum computing

TL;DR

The tools and methods of quantum computing are introduced that could be used to overcome the limitations of these classical impurity solvers, either in the long term -- with fully quantum algorithms, or in the short term -- with hybrid quantum-classical algorithms.

Abstract

Near-term quantum processors are limited in terms of the number of qubits and gates they can afford. They nevertheless give unprecedented access to programmable quantum systems that can efficiently, although imperfectly, simulate quantum time evolutions. Dynamical mean field theory, on the other hand, maps strongly-correlated lattice models like the Hubbard model onto simpler, yet still many-body models called impurity models. Its computational bottleneck boils down to investigating the dynamics of the impurity upon addition or removal of one particle. This task is notoriously difficult for classical algorithms, which has warranted the development of specific classical algorithms called "impurity solvers" that work well in some regimes, but still struggle to reach some parameter regimes. In these lecture notes, we introduce the tools and methods of quantum computing that could be used to overcome the limitations of these classical impurity solvers, either in the long term -- with fully quantum algorithms, or in the short term -- with hybrid quantum-classical algorithms.

Figures (8)

• Figure 1: DMFT self-consistency cycle: a lattice model (here the Hubbard model) is self-consistently mapped to an impurity model, defined by its hybridization function $\Delta(\omega)$. This impurity model can be represented with an Anderson impurity Hamiltonian describing a correlated impurity site coupled to noninteracting bath sites (right). Quantum processors can be used to solve the impurity model, namely compute its Green's function $G_{\mathrm{imp}}(\omega)$.
• Figure 2: Matrix product states and the density matrix renormalization group method. (a) MPS representation of $|\Psi\rangle$. (b) MPO representation of $H$. (c) Tensor-network representation of the energy functional of DMRG. (d) TN representation of a partial derivative. (e) TN representation of the eigenproblem (thick lines denote groupings of tensor legs).
• Figure 3: Circuit model: a quantum circuit (on the right) describes a sequence of gates, which are individually implemented in an analog fashing with time-dependent terms in the hardware Hamiltonian. Reproduced from Ayral2023.
• Figure 4: Circuits for $R_{P_{i}}(\theta)$. (a) Circuit for $R_{\sigma_{z}^{(1)}\sigma_{z}^{(2)}}(\theta)$. (b) Circuit for $R_{\sigma_{x}^{(1)}\sigma_{z}^{(2)}}(\theta)$. (c) Circuit for $R_{\sigma_{z}^{(1)}\sigma_{z}^{(2)}\sigma_{z}^{(3)}\sigma_{z}^{(4)}}(\theta)$.
• Figure 5: Hadamard test: (a) Computation of $\langle\psi|U|\psi\rangle$ by the Hadamard test circuit. The $S$ gate is used only to compute the imaginary part. (b) Application to the computation of $\langle\Psi_{0}|U^{\dagger}P_{i}UP_{j}|\Psi_{0}\rangle$ terms.