Variational quantum-algorithm based self-consistent calculations for the two-site DMFT model on noisy quantum computing hardware

arXiv:2311.10402 · cond-mat.str-el, quant-ph · Submitted 2023-11-17 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Variational quantum-algorithm based self-consistent calculations for the two-site DMFT model on noisy quantum computing hardware".

Mira: A variational quantum-algorithm based self-consistent calculation for the two-site DMFT model on noisy quantum computing hardware presents a method to solve complex many-body problems using near-term quantum devices.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: To recap, we're looking at how this paper tackles the challenge of solving Dynamical Mean Field Theory using a variational quantum algorithm on noisy hardware. The main thesis is that they successfully implemented a QC approach to solve the two-site DMFT model by mapping electronic orbitals to qubits and employing VQE simulations.

Mira: The paper claims that this method allows for obtaining self-consistent results for the two-site DMFT model using simulations conducted on hardware like IBMQ Ehningen QC, specifically focusing on analyzing how stochastic and device errors propagate through the process.

Lev: Essentially, the work is about testing if this hybrid quantum-classical approach is a viable pathway to solving complex many-body problems when we have these limited resources in mind.

Kai: They outline that the AIM is mapped onto a qubit register where fermionic spin-orbitals are assigned to qubits in an order like (d↑, c↑, d↓, c↓) corresponding to qubits (q0, q1, q2, q3).

Mira: They also detail the VQE algorithm they use; they use a parameterized state psi(theta i) and measure its expectation value with a quantum computer to guide a classical optimizer in finding the ground state energy.

Lev: And this is where my focus comes in: they systematically analyze the error propagation using different simulators, including a "noisy simulator" built specifically from IBMQ Ehningen device information, which is important for assessing real-world performance.

Kai: They test implementation across various simulators, from a noiseless state vector simulator to a probabilistic QASM simulator and that noisy device-based one we mentioned.

Mira: The results section highlights the struggle with statistical noise; they found an unphysical two-peak structure in the self-energy on the real frequency axis right around omega = zero which complicates extracting physical data like the quasi-particle weight <ref:2311.10402#pg2>.

Lev: That's a direct consequence of running limited shots, and it brings up whether error correction methods can overcome that statistical limitation when simulating this type of problem.

Kai: The paper then introduces a specific fitting approach to model the self-energy in that problematic region, which they found works for reproducing phase diagrams like the Mott transition even with at least 10k shots.

Mira: So, the core message is that while there are challenges with noise and statistics on NISQ hardware, this variational quantum-algorithm based approach provides a framework to get self-consistent results for a simplified DMFT model.

Lev: The implication here is that it shows the potential of using these hybrid methods to explore strongly correlated systems, even when the underlying hardware isn't fully fault-tolerant.

Conclusion: Kai: To wrap up, we're talking about the paper "Variational quantum-algorithm based self-consistent calculations for the two-site DMFT model on noisy quantum computing hardware." It really focuses on showing that this specific setup can yield results for the two-site DMFT model using VQE simulations despite running them on noisy quantum hardware.

Mira: The authors argue that mapping the AIM onto a qubit register and using a variational approach is a feasible way to solve this problem, even when considering the practical constraints of stochastic noise and device imperfections.

Lev: From my perspective, this work demonstrates that the concept of using hybrid algorithms to tackle DMFT is runnable on current NISQ systems, provided we acknowledge the limitations imposed by noise modeling and error mitigation strategies like SPAM or IC-ZNE.

Kai: The title itself really tells you what's important: it’s not just about solving a problem; it’s about doing that solution while explicitly accounting for the noise present in the hardware used.

Mira: The implication for condensed matter theory is that we can now use these quantum methods to get hints about electron correlation effects in solid-state materials, even if those hints require careful classical post-processing to clean up noise artifacts.

Lev: What this means for future work is that the next step needs to be about scaling beyond the two-site model and developing error mitigation techniques robust enough for larger systems on real hardware.

Kai: So, in essence, they've shown a proof of concept that this kind of variational approach can work on current noisy devices for problems like two-site DMFT.

Mira: It's a demonstration that the path forward involves combining quantum simulation with classical fitting techniques to extract physical meaning from the data gathered on noisy systems.

Lev: We need to see how we can make those noise modeling and error mitigation steps scalable so that this kind of calculation becomes more applicable for real, larger materials in the future.

Fraunhofer Institute für Werkstoffmechanik IWM · Freiburger Materialforschungszentrum, Universität Freiburg

cond-mat.str-el, quant-ph

Submitted: 2023-11-17

Updated: 2026-10-05

Journal ref: J. Phys.: Condens. Matter 37 225901 (2025)

DOI: 10.1088/1361-648X/add4db

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 67/100

The gist: A variational quantum-algorithm based self-consistent calculation for the two-site DMFT model on noisy quantum computing hardware presents a method to solve complex many-body problems using near-term

Key concepts

Dynamical Mean Field Theory (DMFT): DMFT is a method used to study how electrons interact within solid-state materials. It helps calculate the complex many-body effects that determine material properties like conductivity and magnetism. The paper focuses on the simplest version, the two-site DMFT model, as a test case for quantum algorithms.
Variational Quantum Eigensolver (VQE): VQE is a hybrid quantum-classical algorithm used to find the lowest energy state of a system. A quantum computer prepares states using parameterized circuits, and a classical computer optimizes those parameters to minimize the calculated energy, effectively searching for the ground state of the material model.
Self-Consistency Loop: In DMFT, calculating material properties requires an iterative process where one step depends on the result of another. The self-consistency loop is this repeating cycle. The paper demonstrates how to run this cycle using VQE simulations on quantum hardware to obtain final, reliable results for the model.

Terminology

Quantum Computer — 21× in this paper
Dynamical Mean Field Theory — 11× in this paper
Accuracy — 10× in this paper · explained in 1 episode
Consistency — 10× in this paper · explained in 3 episodes
Self-consistency — 10× in this paper · explained in 2 episodes
Variational Quantum Eigensolver — 7× in this paper
Dynamical Mean Field Theory (DMFT) — 6× in this paper · explained in 2 episodes
Hilbert Space — 4× in this paper · explained in 2 episodes
Hubbard Model — 4× in this paper · explained in 3 episodes
Quantum Circuits — 4× in this paper · explained in 2 episodes
Regularization — 4× in this paper · explained in 3 episodes
Variational Quantum Eigensolver (VQE) — 4× in this paper · explained in 5 episodes
Fidelity — 3× in this paper · explained in 2 episodes
Density Matrix Renormalization Group — 2× in this paper
Quantum Monte Carlo — 2× in this paper
Adaptation — 1× in this paper · explained in 1 episode
Core — 1× in this paper · explained in 2 episodes
Entanglement — 1× in this paper · explained in 3 episodes
Generalizability — 1× in this paper · explained in 6 episodes
Simultaneous Perturbation Stochastic Approximation — 1× in this paper
State Preparation-and-Measurement — 1× in this paper

Summary

A variational quantum-algorithm based self-consistent calculation for the two-site DMFT model on noisy quantum computing hardware presents a method to solve complex many-body problems using near-term quantum devices. This work addresses the computational bottleneck in Dynamical Mean Field Theory (DMFT) by mapping its auxiliary Anderson impurity model onto a qubit register, demonstrating the feasibility of obtaining self-consistent results for the two-site DMFT model using Variational Quantum Eigensolver (VQE) simulations on hardware like IBMQ Ehningen QC, while systematically analyzing the propagation of stochastic and device errors.

The gist

This work demonstrates the feasibility to obtain self-consistent results of the two-site DMFT model based on VQE simulations with a finite number of shots.

Dynamical Mean Field Theory (DMFT) and its Bottleneck

Dynamical Mean Field Theory (DMFT) is a powerful approach to study electron correlation effects in solid-state materials, but its practical applicability is limited by the numerical resources required for solving the underlying auxiliary Anderson impurity model (AIM). The bottleneck in these calculations is the solution of the AIM, which has various classical methods with limitations: exact diagonalization is restricted to approximately 24 orbitals due to exponential scaling, Quantum Monte Carlo (QMC) struggles with low temperatures, and Density Matrix Renormalization Group (DMRG) is limited by a small bond dimension. The paper focuses on the minimal realization of DMFT: the two-site DMFT model, which serves as a test case for hybrid classical-quantum computer algorithms.

Quantum Mapping and Algorithm Choice

The AIM is mapped to a quantum computer where each fermionic spin-orbital of the original system is assigned one qubit. For the two-site AIM, this mapping assigns orbitals to qubits in the order (d↑, c↑, d↓, c↓) → (q0, q1, q2, q3). The AIM Hamiltonian is then written in terms of Pauli operators. To solve the AIM for the ground state and excited states required for the Green’s function (GF), a hybrid quantum-classical variational quantum eigensolver (VQE) algorithm is employed. This involves measuring the expectation value of the Hamiltonian with respect to a parameterized state ψ(θi)i with help of a quantum computer, which serves as the cost function for a classical optimizer to find the ground state energy. The choice of ansatz depends on various factors, and in this work, a particle- and Sz-number conserving ansatz based on Gard et al. [37] is chosen as a compromise.

Noise Modeling and Error Mitigation Strategies

The paper systematically analyzes the propagation of stochastic and device errors through the algorithm. The analysis includes:

Testing implementation using different simulators, including a linear algebra-based noiseless simulator (state vector simulator), a probabilistic QASM simulator, and a noisy simulator constructed using IBMQ Ehningen device information.
Using optimization algorithms like L BFGS B for the state vector simulator and Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer for other cases.
Applying error mitigation techniques such as the State Preparation-and-Measurement (SPAM) error mitigation, which assumes a redistribution of counts only between similar bitstrings, and Inverted-Circuit Zero Noise Extrapolation (IC-ZNE), which uses randomly inserted identity operations to extrapolate to zero noise.

Results on Self-Consistency and Physical Features

The study investigates the self-consistent DMFT loop by performing simulations with increasing numbers of shots (100, 1k, 10k shots) and varying noise levels. Key findings include:

(i) Stochastic Noise:

(ii) Device Noise:

The limited accuracy of the expectation values of eigenenergies and transition rates due to statistical noise leads to an unphysical two-peak structure in the self-energy on the real frequency axis, explicitly around ω = 0, which prohibits the determination of the quasi-particle weight from its derivative. To cure this, a fitting approach is introduced to fit a function of the form a · tan(ω) + b · ω + c to the self-energy in the region between two physical peaks. This approach successfully reproduces phase diagrams including the Mott transition, even with shot noise when using at least 10k shots.

Computational Resources and Scalability Challenges

The paper estimates the computational costs for a complete self-consistent calculation on a QC using the Lehmann representation. For one expectation value, this requires running quantum circuits three times (for each basis). The execution of the ground state quantum circuit with 9 layers of CNOT gates requires about 4.3 µs, and waiting times between executions are on the order of 150 µs. This suggests that one expectation value with 10k shots requires about 4.

Improvements for AI systems

Here are specific improvements to AI systems based on the provided scientific paper, detailing what the improved system can achieve:

) The resulting AI system will be a specialized, hybrid quantum-classical solver for strongly correlated electron systems, specifically designed for materials science and condensed matter physics simulations.

) The improved AI system will be capable of performing self-consistent calculations of the two-site Dynamical Mean Field Theory (DMFT) model on noisy intermediate-scale quantum (NISQ) hardware.

) It will utilize a Variational Quantum Eigensolver (VQE) framework, augmented by sophisticated error mitigation techniques like State Preparation and Measurement (SPAM), Inverted-Circuit Zero Noise Extrapolation (IC-ZNE), and post-processing optimization based on fitting functions (the tan-fit approach).

) The system will overcome the limitations of purely stochastic noise by systematically analyzing and correcting unphysical features in the calculated self-energy, such as the artificial two-peak structure around zero frequency that plagues standard QASM simulations.

) It will be able to reliably determine key physical quantities for materials, including:

@ The Green’s Function (GF) of the two-site DMFT model.

@ The Self-Energy self-consistently, ensuring that the constraints derived from the AIM (like the condition on hybridization strength V) are met.

@ Quasiparticle weights (z), which are crucial for understanding electron correlations and identifying Mott metal-insulator transitions.

) The improved AI system can perform a comprehensive phase diagram study of correlated materials, specifically investigating the Mott transition as a function of interaction strength (U). It will be able to distinguish between metallic and insulating phases with high fidelity, even under realistic device noise conditions (simulated and real hardware).

) It can achieve quantitative predictions for the Mott transition boundary, demonstrating improved accuracy over classical methods or purely stochastic quantum simulations. For instance, the system can reproduce a sharp transition at a more accurate interaction strength range than currently possible with limited shot counts (e.g., 10k shots).

) The system will be robust against realistic hardware imperfections like gate errors and crosstalk, by incorporating noise models derived from specific device data (like IBMQ Ehningen) and applying error mitigation strategies that are tailored to the specific noise profile of the simulation.

) It can provide a reliable estimate of the required quantum resources (number of shots, circuit depth) needed for a full self-consistent DMFT calculation on current NISQ devices. This allows researchers to plan computationally feasible experiments or identify hardware improvements necessary for future, more complex simulations.