[Paper Review] Dissipative mean-field theory of IBM utility experiment

Emanuele G. Dalla Torre, Mor M. Roses|arXiv (Cornell University)|Aug 2, 2023

Quantum many-body systemsPhysics and Astronomy3 citations

TL;DR

This paper resolves the paradox of high-fidelity results from a noisy 127-qubit quantum computer experiment by introducing a dissipative mean-field theory that maps the complex many-body dynamics to a single qubit undergoing rotations and dephasing. The theory explains the experimental success despite overwhelming gate errors by showing that effective decoherence from the many-body circuit suppresses noise, enabling accurate simulation of the kicked Ising model with quantitative agreement to exact solutions.

ABSTRACT

In spite of remarkable recent advances, quantum computers still lack useful applications. A promising direction for such utility is offered by the simulation of the dynamics of many-body quantum systems, which cannot be efficiently computed classically. Recently, IBM used a superconducting quantum computer to simulate a kicked quantum Ising model with large numbers of qubits and time steps. These results were later reproduced using numerical techniques based on tensor networks and Clifford expansion. In this work, we analyze the experiment in the eyes of a simple-minded mean-field approximation. We treat neighboring qubits as a self-consistent source of dephasing and express them in terms of Kraus operators. Although our approach completely disregards entanglement between qubits, it captures the overall dependence of physical observables as a function of time and external magnetic field. This observation can help rationalize the success of the quantum computer in solving this specific problem.

Motivation & Objective

• To resolve the paradox of high-fidelity results from a noisy 127-qubit quantum computer experiment with over 2,800 two-qubit gates and estimated fidelity ~5×10⁻⁷.
• To understand why error mitigation techniques like zero-noise extrapolation succeeded where standard error bounds suggest failure.
• To develop a simplified effective theory that captures the essential dynamics of the many-body system under experimental conditions.
• To identify conditions under which noisy quantum computers can outperform classical simulations despite high error rates.

Proposed method

• Formulate a spatially uniform, self-consistent mean-field approximation where each qubit evolves under an effective single-qubit Hamiltonian with averaged magnetization.
• Introduce a dissipative mean-field map using Kraus operators to model decoherence effects arising from many-body interactions.
• Derive an effective single-qubit evolution with time-dependent dephasing, where the dephasing rate is determined by the effective coupling strength $ J_{\text{eff}} $, which vanishes at $ h = \pi/4 $.
• Use the effective model to compute expectation values of local and non-local observables, such as $ \langle X(t) \rangle_{\text{eff}} $, via a classical probabilistic process.
• Validate the effective theory by comparing its predictions to exact solutions and experimental data from IBM’s 127-qubit device.
• Establish a connection between the dissipative dynamics and the observed robustness, showing that physical decoherence is subdominant to engineered dephasing.

Figure 1: Magnetization $\expectationvalue{Z}$ after $t=5$ Trotter steps. The experimental data and exact solution is reproduced from Ref. [ 10 ] . The mean field approach $\ket{\psi}$ describes the unitary dynamics of a single qubit affected by a magnetic field determined by the average magnetizati

Experimental results

Research questions

• RQ1 Why did the IBM 127-qubit quantum computer achieve high-fidelity results in simulating the kicked Ising model despite an estimated fidelity of only ~5×10⁻⁷?
• RQ2 What effective dynamical description explains the resilience of the quantum computer to noise in this specific experiment?
• RQ3 How does the interplay between unitary evolution and decoherence lead to a simplified single-qubit effective theory?
• RQ4 In what regime does the dissipative mean-field approximation accurately reproduce both local and non-local observables?
• RQ5 What conditions make a quantum circuit both classically simulable and robust to noise, and how does this relate to quantum utility?

Key findings

• The many-body dynamics of the kicked Ising model is effectively described by a single qubit undergoing rotations and dephasing, with the dephasing rate determined by the effective coupling $ J_{\text{eff}} $.
• At $ h = \pi/4 $, the effective coupling vanishes ($ J_{\text{eff}} = 0 $), leading to a classical probabilistic flip process with survival probability $ p_0 = \exp(-2\xi J_{\text{eff}}^2) $, which matches the exact solution.
• The effective model predicts $ \langle X(t) \rangle_{\text{eff}} = \left( \frac{1}{2} + \frac{1}{2} e^{-2\xi J_{\text{eff}}^2} \right)^t $, which quantitatively matches the exact and experimental results.
• The theory explains the success of zero-noise extrapolation: the dominant noise is not independent but is instead correlated and suppressed by the many-body dynamics.
• The dissipative mean-field framework reveals that noise resilience arises not from low error rates, but from the effective dynamics being less sensitive to physical decoherence.
• The work identifies a class of quantum circuits—those amenable to local-in-time and space mean-field theories—that are both classically simulable and robust to noise, suggesting a frontier for future quantum advantage.

Figure 2: Expectation value of a 17-qubit stabilizer after $t=5$ Trotter steps. The experimental data and the exact solution is reproduced from Ref. [ 10 ] . The unitary mean-field $\ket{\psi}$ and $\rho$ theories respectively predict $\expectationvalue{X\quantity(t)}_{\mathrm{eff}}=1$ and Eq. 11 .