An area law and sub-exponential algorithm for 1D systems (1301.1162v1)

Abstract: We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3 d}{\eps}}$. Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the $1/\eps$ dependence of the bound in Arad et al. To achieve that, we establish a new, "random-walk like", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1/\poly(n) approximation of the ground energy with a subexponential running time T\le \exp(e^{O(\log^{3/4}n/\eps^{1/4})}).

• The paper presents an exponential improvement in entanglement entropy bounds over Hastings’ result by constructing an AGSP directly from the Hamiltonian with Chebyshev polynomials.
• The paper demonstrates that ground states of gapped 1D Hamiltonians can be effectively approximated by matrix product states with sublinear bond dimension.
• The paper introduces a sub-exponential time algorithm for ground energy approximation, challenging prior NP-hard assumptions for 1D quantum systems.

An Area Law and Sub-Exponential Algorithm for 1D Systems

This paper addresses two primary advancements in the paper of 1D systems governed by local Hamiltonians with a spectral gap, centered on improving the entanglement entropy bounds and developing computational methods to approximate ground states. The authors present a new proof of the area law specifically improving upon Hastings' original result by employing a direct utilization of the Hamiltonian to construct an AGSP, avoiding reliance on the Detectability lemma.

Exponential Improvement in Area Law Bounds

The paper makes significant progress by providing an exponential improvement over Hastings' bound on entanglement entropy. For a chain of $d$-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap greater than zero, the entanglement entropy of the ground state with respect to any cut in the chain is shown to be upper bounded by $O\left(\frac{\log^3 d}{\epsilon}\right)$. This result, applicable to general frustrated Hamiltonians as well, offers a more efficient entanglement scaling compared to prior bounds. The authors employ Chebyshev polynomials to construct an AGSP tailored to the structure of the Hamiltonian, avoiding the frustration-free assumption.

Matrix Product State Representation

The research demonstrates that the ground state of a gapped 1D Hamiltonian can be well-approximated by a matrix product state (MPS) with sublinear bond dimension. This substantiates the feasibility of computing the ground state to within $O\left(\frac{1}{\text{poly}(n)}\right)$ precision using classical algorithms, providing empirical evidence against the $NP$-hardness of such computation in gapped 1D systems.

Implications for Quantum Complexity Theory

This work has profound implications in quantum complexity theory and has considerably narrowed the gap between theoretical predictions and practical calculations of quantum ground states in 1D systems. The existence of an MPS with sub-exponential bond dimension not only sheds light on the structure of ground states but also offers a robust framework for developing efficient algorithms.

Sub-Exponential Algorithm for Ground Energy Approximation

A highlight of the paper is the discussion of an algorithm that approximates the ground energy with sub-exponential time complexity, $$T \le \exp\left(e^{\log^{3/4}n/\epsilon^{1/4}}\right)$$. This is achieved by leveraging the sublinear bond dimension MPS representation. The approach could fundamentally impact the landscape of solving ground state approximations, assuming constant spectral gaps.

Future Directions and Speculations

Given the significance of these results for 1D systems, future research could explore extensions to higher-dimensional systems where the complexity and interactions present further complications. The potential for similar breakthroughs suggests intriguing directions for applying these techniques or developing analogous methods that consider the unique characteristics of higher dimensions.

In conclusion, this paper contributes significantly to our understanding of the entanglement entropy in 1D systems and the computational feasibility of approximating ground states. The techniques and results discussed establish a foundation for further research aimed at tackling more generalized Hamiltonians and multidimensional quantum systems.