A tree tensor network approach to simulating Shor's algorithm

Abstract

Simulating quantum systems constructively furthers our understanding of qualitative and quantitative features which may be analytically intractable. In this letter, we directly simulate and explore the entanglement structure present in a paradigmatic example of quantum information: Shor's wavefunction. The methodology employed is a dynamical tensor network which is initially constructed as a tree tensor network, inspired by the modular exponentiation quantum circuit, and later efficiently mapped to a matrix product state. Utilizing the Schmidt number as a local entanglement metric, our construction explicitly captures the wavefunction's non-local entanglement structure and an entanglement scaling relation is discovered. Specifically, we see that entanglement across a bipartition grows exponentially in the number of qubits before saturating at a critical scale which is proportional to the modular periodicity.

4+ pages, 4 figures, reference added

References in corpus (8)

• The density-matrix renormalization group in the age of matrix product states
• Fast Quantum Modular Exponentiation
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Efficient classical simulation of the approximate quantum Fourier transform
• The quantum FFT can be classically simulated
• Solving search problems by strongly simulating quantum circuits
• Simulations of Shor's Algorithm using Matrix Product States