[Paper Review] Variational Quantum Algorithm for Schmidt Decomposition

Ranyiliu Chen, Benchi Zhao+1 more|arXiv (Cornell University)|Sep 22, 2021

Quantum Computing Algorithms and Architecture4 references4 citations

TL;DR

This paper proposes a variational quantum algorithm for Schmidt decomposition (VQASD) that efficiently extracts entanglement structure in bipartite pure states using a bi-local quantum neural network to maximize a cost function with only one expectation value estimate per iteration. The method achieves high hardware efficiency and enables experimental implementation on superconducting quantum processors, demonstrating practical entanglement analysis on near-term devices.

ABSTRACT

Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. In entanglement theory, Schmidt decomposition is a powerful tool to analyze the fundamental properties and structure of quantum entanglement. This work introduces a hybrid quantum-classical algorithm for Schmidt decomposition of bipartite pure states on near-term quantum devices. First, we show that the Schmidt decomposition task could be accomplished by maximizing a cost function utilizing bi-local quantum neural networks. Based on this, we propose a variational quantum algorithm for Schmidt decomposition (named VQASD) of which the cost function evaluation notably requires only one estimate of expectation with no extra copies of the input state. In this sense, VQASD outperforms existent approaches in resource cost and hardware efficiency. Second, by further exploring VQASD, we introduce a variational quantum algorithm to estimate the logarithm negativity, which can be applied to efficiently quantify entanglement of bipartite pure states. Third, we experimentally implement our algorithm on Quantum Leaf using the IoP CAS superconducting quantum processor. Both experimental implementations and numerical simulations exhibit the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.

Motivation & Objective

• To develop a hybrid quantum-classical algorithm for Schmidt decomposition of bipartite pure states suitable for near-term quantum devices.
• To minimize resource cost by requiring only a single expectation value estimate per iteration, avoiding state copies.
• To enable efficient quantification of entanglement via logarithmic negativity estimation using the same variational framework.
• To validate the algorithm experimentally on a superconducting quantum processor (IoP CAS Quantum Leaf).

Proposed method

• The algorithm employs a bi-local quantum neural network architecture to parameterize the unitary transformations required for Schmidt decomposition.
• A cost function is constructed to maximize the overlap with the Schmidt-decomposed form, enabling variational optimization via classical feedback.
• The cost function evaluation requires only one expectation value estimate per iteration, significantly reducing circuit repetitions and resource overhead.
• The same variational framework is extended to estimate logarithmic negativity, a measure of entanglement for bipartite pure states.
• The algorithm is implemented on the IoP CAS superconducting quantum processor, with results compared to numerical simulations.

Experimental results

Research questions

• RQ1 Can a variational quantum algorithm achieve Schmidt decomposition with minimal quantum resource overhead on near-term devices?
• RQ2 How can the cost function be designed to require only a single expectation value estimate without state copies?
• RQ3 Can the same variational framework be adapted to estimate logarithmic negativity for entanglement quantification?
• RQ4 What is the experimental feasibility and accuracy of the proposed algorithm on real superconducting hardware?

Key findings

• The VQASD algorithm successfully performs Schmidt decomposition using only one expectation value estimate per iteration, reducing resource demands compared to prior methods.
• The method enables accurate estimation of logarithmic negativity for bipartite pure states using the same variational circuit framework.
• Numerical simulations confirm the convergence and robustness of the algorithm under realistic noise conditions.
• Experimental implementation on the IoP CAS superconducting quantum processor demonstrates the practicality and viability of the approach on real hardware.