[Paper Review] Near-term quantum algorithms for linear systems of equations

Hsin-Yuan Huang, Kishor Bharti+1 more|arXiv (Cornell University)|Sep 16, 2019

Quantum Computing Algorithms and ArchitectureComputer Science1 references58 citations

TL;DR

The paper studies near-term quantum algorithms for solving Ax=b, introducing Classical Combination of Variational Quantum States (CQS) with an Ansatz tree to guarantee solutions and exploring variational landscapes and potential plateaus.

ABSTRACT

Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum resources are not immediately available on near-term quantum devices. In this work, we study near-term quantum algorithms for linear systems of equations of the form $Ax = b$. We investigate the use of variational algorithms and analyze their optimization landscapes. There exist types of linear systems for which variational algorithms designed to avoid barren plateaus, such as properly-initialized imaginary time evolution and adiabatic-inspired optimization, suffer from a different plateau problem. To circumvent this issue, we design near-term algorithms based on a core idea: the classical combination of variational quantum states (CQS). We exhibit several provable guarantees for these algorithms, supported by the representation of the linear system on a so-called Ansatz tree. The CQS approach and the Ansatz tree also admit the systematic application of heuristic approaches, including a gradient-based search. We have conducted numerical experiments solving linear systems as large as $2^{300} imes 2^{300}$ by considering cases where we can simulate the quantum algorithm efficiently on a classical computer. These experiments demonstrate the algorithms' ability to scale to system sizes within reach in near-term quantum devices of about $100$-$300$ qubits.

Motivation & Objective

• Motivate solving linear systems as a practical near-term quantum computation task.
• Evaluate variational quantum algorithms and their optimization landscapes for Ax=b.
• Identify plateau phenomena in variational approaches and propose the CQS framework as a remedy.
• Introduce the Ansatz tree concept and prove guarantees for recovering the solution under certain conditions.
• Demonstrate scalability through numerical simulations on large effective system sizes and compare to existing quantum methods.

Proposed method

• Analyze basic variational algorithms and two Ansätze: Agnostic (hardware-efficient) and Alternating Operator (A− and A+b dependent) Ansatz.
• Define loss functions: L_R(x)=||Ax-|b>||^2, L_T(x)=1/2||x||^2+||Ax-|b>||^2, and a Hamiltonian-based loss L_H(|x>) involving a constructed H(1).
• Discuss optimization strategies: VQE, imaginary-time propagation, adiabatic-assisted optimization (AAVQE).
• Introduce Classical Combination of Variational Quantum States (CQS) and the Ansatz tree to combine quantum states classically for improved solution coverage.
• Provide a gradient-expansion heuristic to prune and expand the Ansatz tree.
• Show how the CQS framework yields provable guarantees and relates to Tikhonov regularized regression.

Experimental results

Research questions

• RQ1 Can near-term quantum devices solve Ax=b efficiently for practically large, structured linear systems?
• RQ2 Do variational landscapes for Ax=b exhibit barren plateaus or other flat regions that hinder optimization on NISQ hardware?
• RQ3 Does combining multiple variational states classically (CQS) with an Ansatz tree provide guarantees and practical advantages over standard VQE approaches?
• RQ4 What are the guarantees and limits of the CQS/Ansatz-tree approach in recovering the true solution for A and b under the given assumptions?
• RQ5 How do these methods scale in practice to large effective system sizes, and how do they compare to existing quantum linear-system approaches?

Key findings

• Variational approaches can encounter plateau-like optimization landscapes for certain structured linear systems, hindering progress with standard Ansätze.
• Properly initialized imaginary-time evolution and adiabatic-inspired optimization do not universally solve the plateau issue; alternative routes are needed.
• A Classical Combination of Variational Quantum States (CQS) with an Ansatz tree can provide provable guarantees and enable gradient-based heuristics to select useful state combinations.
• Numerical experiments simulate solving linear systems as large as 2^300 × 2^300 by classical simulation, indicating potential scalability to near-term devices with ~100–300 qubits.
• A variation of CQS achieves provable guarantees similar to existing quantum algorithms while reducing gate counts by a factor of up to 1/ε and uses only one ancilla qubit.
• The framework systematically allows heuristic pruning/expansion of the Ansatz tree via gradient expansion and related methods.