Variational Time Evolution Compression for Solving Impurity Models on Quantum Hardware

Abstract

Dynamical mean-field theory (DMFT) is a useful tool to analyze models of strongly correlated fermions like the Hubbard model. In DMFT, the lattice of the model is replaced by a single impurity site embedded in an effective bath. The resulting single impurity Anderson model (SIAM) can then be solved self-consistently with a quantum-classical hybrid algorithm. This procedure involves repeatedly preparing the ground state on a quantum computer and evolving it in time to measure the Greens function. We here develop an approximation of the time evolution operator for this setting by training a Hamiltonian variational ansatz. The parameters of the ansatz are obtained via a variational quantum algorithm that utilizes a small number of time steps, given by the Suzuki-Trotter expansion of the time evolution operator, to guide the evolution of the parameters. The resulting circuit has a fixed depth for the time evolution depending on the size of the bath and is significantly shallower than a comparable Suzuki-Trotter expansion.

Figures (12)

• Figure 1: Measuring the Green's function w.r.t. the ground state on a quantum device. Using the Hadamard test and measuring $\braket{Z_{\text{anc}}}$ returns the desired expectation values.
• Figure 2: One layer of the second order Hamiltonian variational ansatz using the same two qubit gate count as $\hat{U}_\text{trotter}(\Delta t)$ for a system with two bath sites ($B=2$, see appendix \ref{['appendix:trotterization']} for details)
• Figure 3: VQE circuit for preparing the ground state. After a sufficient number of initial VQE steps, the $R_Y$-layer is replaced by a fixed gate structure depending on the measured computational basis states to introduces the required excitations directly and symmetrically (if it is degenerate) in each spin sector. Afterwards the VQE algorithm is continued until convergency is reached.
• Figure 4: Circuit for measuring the global and local versions of the cost function
• Figure 5: Error in the ground state energy for $U = 4v$ for different fillings ($n=1$ left, $n = 0.5$ right). $L$ denotes the number of layers in the ansatz. In each case at most $L \leq B$ layers are necessary for an error below $10^{-4}$.