Complexity of quantum impurity problems (1609.00735v1)

Abstract: We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of $n$ fermionic modes and has a Hamiltonian $H=H_0+H_{imp}$, where $H_0$ is quadratic in creation-annihilation operators and $H_{imp}$ is an arbitrary Hamiltonian acting on a subset of $O(1)$ modes. We show that the ground energy of $H$ can be approximated with an additive error $2^{-b}$ in time $n^3 \exp{[O(b^3)]}$. Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of $\exp{[O(b^3)]}$ fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of $H_0$. A key ingredient of our proof is Zolotarev's rational approximation to the $\sqrt{x}$ function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model.

Analysis of Gapped and Gapless Phases of Frustration-Free Spin-1/2 Chains

The paper "Complexity of quantum impurity problems," authored by Sergey Bravyi and David Gosset, addresses the computational challenges associated with quantum impurity models. These models, critically important in condensed matter physics and material science, describe systems where an interacting impurity is coupled to a bath of non-interacting fermions. The paper presents a classical approach to estimate the ground state energy and compute low-energy states of such models, proposing a quasi-polynomial time algorithm.

Core Methodology and Results

The authors centralize their paper on Hamiltonians comprising a quadratic part for the bath, $H_0$, and an interacting impurity part, $H_{imp}$, with a focus on estimating the ground state energy $e_g$. The algorithm proposed operates in $O(n^3) \exp{[O(b^3)]}$ time for approximating $e_g$ within an additive error $2^{-b}$, where $n$ represents fermionic modes. The core computational strategy exploits a deformed impurity model, discretizing the spectrum of single-particle excitation energies and allowing a significant number of modes to decouple from the impurity through Gaussian transformations.

Mathematical and Computational Insights

A key contribution of this work is demonstrating the utility of Zolotarev's rational approximation of the square root function, a mathematical tool leveraged to achieve efficient approximation of $e_g$. The paper establishes a relation between the decay of eigenvalues of the ground state covariance matrix and the spectral gap of $H_0$, illuminating the effect of perturbations in quantum impurity problems.

This research further discerns the exponential decay in eigenvalues of the ground state covariance matrix in terms of the impurity size and bath gap, enabling an insightful decomposition of fermionic systems. They confirm that ground states can effectively be captured by manageable superpositions of Gaussian states, thus fostering a potential paradigm for addressing problem spaces extending into quantum chemistry and material sciences simulations.

Implications and Future Directions

From a theoretical standpoint, this work contributes to the broader understanding of Hamiltonian complexity, offering methodologies that persistently fracture the barriers posed by QMA-hard problems, albeit within quasi-polynomial bounds. Practically, implications extend towards enhancing hybrid quantum-classical simulation regimes, such as dynamical mean-field theory (DMFT), by mitigating the reliance on quantum-only approaches which are computationally expensive.

For future research, avenues include refining the time complexity bounds, potentially lowering the quasi-polynomial scaling relative to $\gamma^{-1}$, or improving the eigenvalue decay insights towards more generalized impurity settings. Additionally, extending the exact simulation capabilities for time-dependent impurity problems could open significant improvements in simulating strongly correlated electron systems.

Conclusion

Sergey Bravyi and David Gosset present a significant advancement in the computational techniques available for quantum impurity models. While highlighting computational challenges and methods to circumvent them, this paper sets a foundation upon which more precise and efficient algorithms can be developed, potentially synergizing with near-term quantum technologies in tractable physical simulations and expanding the understanding of intermediary fermionic complexity frameworks.