[Paper Review] An area law and sub-exponential algorithm for 1D systems

Itai Arad, Alexei Kitaev+2 more|arXiv (Cornell University)|Jan 7, 2013

Quantum many-body systems6 references90 citations

TL;DR

This paper presents a new proof of the area law for one-dimensional (1D) gapped quantum systems, achieving an exponential improvement in the entanglement entropy bound to $ O(\log^3 d / \epsilon) $, where $ d $ is the local Hilbert space dimension and $ \epsilon $ is the spectral gap. The approach uses a Chebyshev-based Approximate Ground State Projector (AGSP) constructed directly from the Hamiltonian, avoiding reliance on the Detectability Lemma, enabling application to general (frustrated) 1D Hamiltonians. The method yields a subexponential-time algorithm for approximating the ground energy with bond dimension $ B = \tilde{O}(\exp(\log^{3/4} n / \epsilon^{1/4})) $, suggesting the problem is unlikely to be NP-hard unless 3-SAT admits subexponential algorithms.

ABSTRACT

Analog quantum simulation is a promising path towards solving classically intractable problems in many-body physics on near-term quantum devices. However, the presence of noise limits the size of the system and the length of time that can be simulated. In our work, we consider an error model in which the actual Hamiltonian of the simulator differs from the target Hamiltonian we want to simulate by small local perturbations, which are assumed to be random and unbiased. We analyze the error accumulated in observables in this setting and show that, due to stochastic error cancellation, with high probability the error scales as the square root of the number of qubits instead of linearly. We explore the concentration phenomenon of this error as well as its implications for local observables in the thermodynamic limit. Moreover, we show that stochastic error cancellation also manifests in the fidelity between the target state at the end of time-evolution and the actual state we obtain in the presence of noise. This indicates that, to reach a certain fidelity, more noise can be tolerated than implied by the worst-case bound if the noise comes from many statistically independent sources.

Motivation & Objective

• To establish a tighter, exponentially improved bound on entanglement entropy for ground states of 1D gapped Hamiltonians.
• To develop a general AGSP construction that works for frustrated Hamiltonians by avoiding the Detectability Lemma.
• To show that ground states can be approximated by matrix product states (MPS) with sublinear bond dimension, implying subexponential algorithms for ground energy approximation.
• To prove a novel 'random-walk-like' bound on the entanglement rank of powers of 1D Hamiltonians.

Proposed method

• Construct a truncated Hamiltonian $ H(t) $ by bounding eigenvalues of left and right subsystems, reducing norm while preserving spectral gap structure.
• Use Chebyshev polynomials to build a robust Approximate Ground State Projector (AGSP) on $ H(t) $, ensuring low entanglement rank and strong convergence to the ground state.
• Establish a new bound: $ \text{ER}(H^\ell) \leq (\ell d)^{O(\sqrt{\ell})} $, capturing 'random-walk-like' entanglement growth in 1D systems.
• Apply the AGSP to a product state, showing that the resulting state approximates the ground state with error $ \leq 1/\text{poly}(n) $.
• Use dynamic programming on the resulting low-bond-dimension MPS to compute a $ 1/\text{poly}(n) $-approximation of the ground energy in subexponential time.
• Leverage the truncation lemma to show that the projected state $ |\Gamma_t\rangle $ is exponentially close to the true ground state and to an eigenvector of both $ H $ and $ H(t) $.

Experimental results

Research questions

• RQ1 Can the entanglement entropy bound in the 1D area law be improved beyond Hastings' original $ \tilde{O}(\log d / \epsilon) $, especially for frustrated Hamiltonians?
• RQ2 Is it possible to construct an AGSP for general (frustrated) 1D Hamiltonians without relying on the Detectability Lemma?
• RQ3 Does the entanglement rank of powers of a 1D Hamiltonian grow in a 'random-walk-like' fashion, and can this be bounded?
• RQ4 Can a subexponential-time algorithm be constructed for approximating the ground energy of 1D gapped systems, implying it is not NP-hard?

Key findings

• The entanglement entropy of the ground state across any cut in a 1D gapped system is bounded by $ O(\log^3 d / \epsilon) $, representing an exponential improvement over Hastings' result.
• A new 'random-walk-like' bound is proven: $ \text{ER}(H^\ell) \leq (\ell d)^{O(\sqrt{\ell})} $, which may be of independent interest in quantum many-body theory.
• The ground state of a general 1D gapped Hamiltonian can be approximated by a matrix product state (MPS) with bond dimension $ B = \tilde{O}(\exp(\log^{3/4} n / \epsilon^{1/4})) $, which is sublinear in $ n $.
• A subexponential-time algorithm is constructed for approximating the ground energy within $ 1/\text{poly}(n) $, with runtime $ T \leq \exp(\tilde{O}(\log^{3/4} n / \epsilon^{1/4})) $.
• The existence of such an algorithm provides strong evidence that finding the ground energy of 1D gapped Hamiltonians is not NP-hard, unless 3-SAT can be solved in subexponential time.
• The truncated state $ |\Gamma_t\rangle $ is shown to be exponentially close to the true ground state, with $ \| |\Gamma\rangle - |\Gamma_t\rangle \| \leq 2^{-\Omega(t)} $, and is an approximate eigenvector of both $ H $ and $ H(t) $.