[Paper Review] Dynamical mean field theory algorithm and experiment on quantum computers

Ivan Rungger, Nathan Fitzpatrick+13 more|arXiv (Cornell University)|Oct 10, 2019

Quantum Computing Algorithms and Architecture50 references50 citations

TL;DR

The paper presents a VQE-based quantum-classical hybrid algorithm to perform 2-site DMFT calculations on current quantum hardware, with demonstrations on superconducting and trapped-ion devices and classical simulations for benchmarking.

ABSTRACT

The developments of quantum computing algorithms and experiments for atomic scale simulations have largely focused on quantum chemistry for molecules, while their application in condensed matter systems is scarcely explored. Here we present a quantum algorithm to perform dynamical mean field theory (DMFT) calculations for condensed matter systems on currently available quantum computers, and demonstrate it on two quantum hardware platforms. DMFT is required to properly describe the large class of materials with strongly correlated electrons. The computationally challenging part arises from solving the effective problem of an interacting impurity coupled to a bath, which scales exponentially with system size on conventional computers. An exponential speedup is expected on quantum computers, but the algorithms proposed so far are based on real time evolution of the wavefunction, which requires high-depth circuits and hence very low noise levels in the quantum hardware. Here we propose an alternative approach, which uses the variational quantum eigensolver (VQE) method for ground and excited states to obtain the needed quantities as part of an exact diagonalization impurity solver. We present the algorithm for a two site DMFT system, which we benchmark using simulations on conventional computers as well as experiments on superconducting and trapped ion qubits, demonstrating that this method is suitable for running DMFT calculations on currently available quantum hardware.

Motivation & Objective

• Motivate DMFT as a correction to DFT for strongly correlated solids.
• Propose a quantum-classical hybrid algorithm using VQE as an exact diagonalization impurity solver within DMFT.
• Enable DMFT self-consistency by computing impurity Green's function quantities on quantum hardware and updating bath parameters classically.

Proposed method

• Map the Anderson impurity Hamiltonian to qubits via Jordan-Wigner transform, enabling a 2(N_imp+N_b) qubit representation.
• Use a hardware-efficient ansatz and VQE to obtain ground and excited state energies for N0, N0±1 electrons.
• Compute pole energies ω_p,n and spectral weights λ_p/λ_h using total-energy based VQE and circuit-based measurements.
• Enforce particle-number by a penalty term or circuit reduction to obtain required eigenstates without extra qubits.
• Regularize the DMFT loop to restore cancellation of divergences in the self-energy and ensure convergence.
• Demonstrate 2-site DMFT with ph-symmetric and off-symmetric cases, benchmarking against exact results and analytic solutions.

Experimental results

Research questions

• RQ1 Can a VQE-based quantum impurity solver achieve self-consistent DMFT loops for a minimal two-site model?
• RQ2 How accurately can ω_p,n and λ_p,h,α,n be obtained on current quantum devices to drive DMFT self-consistency?
• RQ3 Does regularization suffice to mitigate quantum hardware noise and ensure DMFT convergence in the 2-site model?
• RQ4 How do hardware results compare to exact numerical and analytic DMFT solutions in the ph-symmetric and non-symmetric regimes?

Key findings

• The authors successfully implement a VQE-based quantum impurity solver for a 2-site DMFT system on current quantum hardware.
• Regularization of λ_p,h,α,n is necessary to achieve convergence of the DMFT loop and to obtain physically meaningful self-energies.
• Hardware experiments (IBM) show bath parameter V converging toward the analytic value under regularization, with close agreement to simulations.
• For ph-symmetric cases, the method reproduces energy levels E0, E3,0, E3,2 and the spectral weight λ with high fidelity in simulations and within ~2% energy discrepancy at 5000 shots on hardware.
• The 2-site DMFT results reproduce the known analytic relation V = sqrt(z) and the DMFT self-consistency conditions, validating the approach for near-term quantum devices.
• The work demonstrates that DMFT calculations for correlated materials can be executed on currently available quantum hardware using a hybrid quantum-classical approach.