[Paper Review] Fast Computation of Many-Body Entanglement

Johnnie Gray|arXiv (Cornell University)|Sep 5, 2018

Quantum Computing Algorithms and Architecture2 references3 citations

TL;DR

This paper introduces Tensor Network Stochastic Lanczos Quadrature (TNSLQ), a method to efficiently compute the logarithmic negativity—a mixed-state entanglement measure—for large quantum many-body systems. By combining lazily evaluated tensor network representations of the partially transposed density matrix with stochastic Lanczos quadrature, the approach enables accurate computation of entanglement in up to 30 qubits for dense states and up to bond dimension ~180 for matrix product states, significantly extending prior reach beyond ~15 qubits.

ABSTRACT

Mixed state entanglement measures can act as a versatile probes of many-body systems. However, they are generally hard to compute, often relying on tricky optimizations. One measure that is straightforward to compute is the logarithmic negativity, yet done naively even this is still limited to small system sizes. Here, we introduce a method to compute the logarithmic negativity for arbitrary subsystems of a densely represented state, as well as block subsystems of matrix product states. The method combines lazily evaluated, tensor network representations of the partially transposed density matrix with stochastic Lanczos quadrature, and is easily extendible to other quantities and classes of many-body states. As examples, we compute the entanglement within random pure states for density matrices of up to 30 qubits, explore scrambling in a many-body quench, and match the results of conformal field theory in the ground-state of the Heisenberg model for density matrices of up to 1000 spins. An implementation of the algorithm has been made available in the open-source library extit{quimb}.

Motivation & Objective

• To overcome the computational bottleneck of calculating mixed-state entanglement measures like logarithmic negativity in large many-body quantum systems.
• To address the cubic scaling of naive computation, which limits practical use to systems of ~15 qubits.
• To develop a scalable, accurate, and generalizable method applicable to both densely represented pure states and matrix product states (MPS).
• To enable the study of entanglement in complex scenarios such as quantum quenches and conformal field theory ground states.

Proposed method

• The method uses tensor network representations to implicitly encode the partially transposed density matrix, avoiding explicit construction and storage.
• It applies stochastic Lanczos quadrature (SLQ) to estimate the trace norm of the partially transposed density matrix, which defines the logarithmic negativity.
• The approach treats the reduced density matrix as a linear operator defined via a tensor network, enabling efficient spectral sum estimation.
• For matrix product states, a compression step is applied before SLQ to reduce the effective bond dimension and maintain computational feasibility.
• The algorithm is implemented in the open-source quimb library, supporting parallelization and GPU acceleration via single-precision arithmetic.
• The method is extensible to other entanglement-related quantities such as von Neumann entropy and mutual information.

Experimental results

Research questions

• RQ1 Can the logarithmic negativity be computed efficiently for subsystems of large quantum states beyond the 15-qubit limit of naive methods?
• RQ2 How can tensor network representations be leveraged to avoid explicit storage of the partially transposed density matrix?
• RQ3 To what extent can stochastic Lanczos quadrature provide accurate and bounded-error estimates of the trace norm for large-scale mixed states?
• RQ4 Can the method be applied to matrix product states with high bond dimensions while maintaining accuracy and efficiency?
• RQ5 What is the impact of system size and purity on the error bounds of the stochastic estimation process?

Key findings

• The method enables computation of logarithmic negativity for density matrices of up to 30 qubits in densely represented pure states, extending the practical limit from ~15 qubits.
• For matrix product states, the method is feasible for bond dimensions up to approximately 180, allowing study of entanglement in larger systems with open or periodic boundary conditions.
• The error in the logarithmic negativity estimate is bounded by a system-size-independent constant and is well-controlled, with variance easily trackable.
• The method achieves accuracy within 0.1% to 1% for typical physical systems, making it suitable for high-precision studies.
• The algorithm is highly parallelizable and GPU-accelerated, enabling efficient computation using single-precision arithmetic.
• The approach has been successfully applied to study entanglement in random pure states, quantum quenches, and the ground state of the Heisenberg model, matching conformal field theory predictions.