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Dynamical mean-field theory (DMFT) maps the local Green's function of the Hubbard model to that of the Anderson impurity model and thus gives an approximate solution of the Hubbard model from the solution of a simpler quantum impurity model. Accurate solutions to the Anderson impurity model nonetheless become intractable for large systems. Quantum and hybrid quantum-classical algorithms have been proposed to efficiently solve impurity models by preparing and evolving the ground state under the impurity Hamiltonian on a quantum computer that is assumed to have the scalability and accuracy far beyond the current state-of-the-art quantum hardware. As a proof of principle demonstration targeting the Anderson impurity model we, for the first time, close the DMFT loop with current noisy hardware. With a highly optimized fast-forwarding quantum circuit and a noise-resilient spectral analysis we observe both the metallic and Mott-insulating phases. Based on a Cartan decomposition, our algorithm gives a fixed depth, fast-forwarding, quantum circuit that can evolve the initial state over arbitrarily long times without time-discretization errors typical of other product decomposition formulas such as Trotter decomposition. By exploiting the structure of the fast-forwarding circuits we reduce the gate count (to 77 CNOTS after optimization), simulate the dynamics, and extract frequencies from the Anderson impurity model on noisy quantum hardware. We then demonstrate the Mott transition by mapping both phases of the metal-insulator phase diagram. Near the Mott phase transition, our method maintains accuracy where the Trotter error would otherwise dominate due to the long-time evolution required to resolve quasiparticle resonance frequency extremely close to zero. This work presents the first computation on both sides of the Mott phase transition using noisy digital quantum hardware, made viable by a highly optimized computation in terms of gate depth, simulation error, and runtime on quantum hardware. To inform future computations we analyze the accuracy of our method versus a noisy Trotter evolution in the time domain. Both algebraic circuit decompositions and error mitigation techniques adopted could be applied in an attempt to solve other correlated electronic phenomena beyond DMFT on noisy quantum computers.

Quantum algorithms on the noisy intermediate-scale quantum (NISQ) devices are expected to simulate quan- tum systems that are classically intractable to demonstrate quantum advantages. However, the non-negligible gate error on the NISQ devices impedes the conventional quantum algorithms to be implemented. Practical strategies usually exploit hybrid quantum-classical quantum algorithms to demonstrate potentially useful ap- plications of quantum computing in the NISQ era. Among the numerous hybrid quantum-classical algorithms, recent efforts highlight the development of quantum algorithms based upon quantum computed Hamiltonian moments, ?f|Hˆn|f? (n = 1, 2, · · · ), with respect to quantum state |f?. In this tutorial, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. Furthermore, we will present a tutorial to show how we can measure and compute the Hamiltonian moments of a four-site Heisenberg model, and compute the energy and magnetization of the model utilizing the imaginary time evolution in the real IBM-Q NISQ hardware environment. Along this line, we will further discuss some practical issues associated with these algorithms. We will conclude this tutorial review by overviewing some possible developments and applications in this direction in the near future.

The three key elements of a quantum simulation are state preparation, time evolution, and measurement. While the complexity scaling of time evolution and measurements are well known, many state preparation methods are strongly system-dependent and require prior knowledge of the system's eigenvalue spectrum. Here, we report on a quantum-classical implementation of the coupled-cluster Green's function (CCGF) method, which replaces explicit ground state preparation with the task of applying unitary operators to a simple product state. While our approach is broadly applicable to many models, we demonstrate it here for the Anderson impurity model (AIM). The method requires a number of T gates that grows as $$\mathcal{O}$$(N⁵) per time step to calculate the impurity Green's function in the time domain, where N is the total number of energy levels in the AIM. Since the number of T gates is analogous to the computational time complexity of a classical simulation, we achieve an order of magnitude improvement over a classical CCGF calculation of the same order, which requires $$\mathcal{O}$$(N⁶) computational resources per time step.

Abstract Quantum algorithms on noisy intermediate‐scale quantum (NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non‐negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum‐classical algorithms. One such hybrid quantum‐classical algorithm, is based upon quantum computed Hamiltonian moments , with respect to quantum state . In this tutorial review, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. We also present a computation of the Hamiltonian moments of a four‐site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM‐Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.