[Paper Review] Towards quantum advantage via topological data analysis

Casper Gyurik, Chris Cade+1 more|arXiv (Cornell University)|May 6, 2020

Topological and Geometric Data Analysis4 citations

TL;DR

This paper demonstrates that the quantum algorithm for topological data analysis (TDA) by Lloyd, Garnerone, and Zanardi is classically intractable under widely accepted complexity-theoretic assumptions, as its generalized form is as hard as simulating the one clean qubit model—widely believed to require superpolynomial classical time. The authors present new quantum algorithms for rank estimation and complex network analysis, providing provable superpolynomial quantum speedups and analyzing their feasibility for near-term quantum devices.

ABSTRACT

Even after decades of quantum computing development, examples of generally useful quantum algorithms with exponential speedups over classical counterparts are scarce. Recent progress in quantum algorithms for linear-algebra positioned quantum machine learning (QML) as a potential source of such useful exponential improvements. Yet, in an unexpected development, a recent series of "dequantization" results has equally rapidly removed the promise of exponential speedups for several QML algorithms. This raises the critical question whether exponential speedups of other linear-algebraic QML algorithms persist. In this paper, we study the quantum-algorithmic methods behind the algorithm for topological data analysis of Lloyd, Garnerone and Zanardi through this lens. We provide evidence that the problem solved by this algorithm is classically intractable by showing that its natural generalization is as hard as simulating the one clean qubit model -- which is widely believed to require superpolynomial time on a classical computer -- and is thus very likely immune to dequantizations. Based on this result, we provide a number of new quantum algorithms for problems such as rank estimation and complex network analysis, along with complexity-theoretic evidence for their classical intractability. Furthermore, we analyze the suitability of the proposed quantum algorithms for near-term implementations. Our results provide a number of useful applications for full-blown, and restricted quantum computers with a guaranteed exponential speedup over classical methods, recovering some of the potential for linear-algebraic QML to become one of quantum computing's killer applications.

Motivation & Objective

• To investigate whether the quantum algorithm for topological data analysis (TDA) remains classically intractable despite recent dequantization results that undermined other quantum machine learning (QML) algorithms.
• To establish that the underlying linear-algebraic methods of the LGZ TDA algorithm are resilient to generic dequantization techniques.
• To develop new quantum algorithms for practical problems such as rank estimation and complex network analysis, grounded in the same quantum methods.
• To analyze the feasibility of implementing these quantum algorithms on near-term NISQ devices, focusing on resource reduction and practical constraints.
• To strengthen the case for quantum advantage in linear-algebraic QML by identifying problems that are provably hard for classical computers.

Proposed method

• Generalize the TDA problem to show it is as hard as simulating the one clean qubit model, a problem widely believed to require superpolynomial classical time.
• Use quantum algorithmic techniques from the LGZ TDA algorithm—specifically, quantum phase estimation and sparse matrix access—to solve rank estimation and complex network analysis tasks.
• Apply classical precompilation strategies to reduce the number of qubits required for sparse matrix access, a major bottleneck in near-term implementations.
• Leverage perturbation theory to analyze robustness of the quantum algorithm under noise, showing that small perturbations to the combinatorial Laplacian do not significantly alter low-lying spectral density.
• Use complexity-theoretic arguments to show that the generalized TDA and rank estimation problems are not amenable to dequantization via generic classical methods.
• Evaluate resource requirements and error-mitigation compatibility to assess suitability for NISQ-era quantum computers.

Experimental results

Research questions

• RQ1 Is the TDA problem solved by the LGZ algorithm classically intractable under widely accepted complexity-theoretic assumptions?
• RQ2 Can the quantum algorithmic framework of LGZ be extended to provide provable superpolynomial quantum speedups for practical problems like rank estimation and network analysis?
• RQ3 Are the quantum methods behind the LGZ algorithm resilient to generic dequantization techniques that have undermined other QML proposals?
• RQ4 What are the practical challenges and resource requirements for implementing these quantum algorithms on near-term quantum devices?
• RQ5 To what extent can noise in the input data or hardware be tolerated without compromising the correctness of the quantum TDA computation?

Key findings

• The generalized TDA problem is as hard as simulating the one clean qubit model, which is widely believed to require superpolynomial time on classical computers, implying classical intractability.
• The rank estimation problem derived from the LGZ framework is provably classically intractable, ensuring a superpolynomial quantum speedup for this specific task.
• The quantum algorithms for TDA, rank estimation, and complex network analysis are resilient to generic dequantization methods that have invalidated other QML proposals.
• Classical precompilation strategies can significantly reduce the qubit requirements for sparse matrix access, making the algorithms more suitable for near-term NISQ devices.
• The quantum algorithm demonstrates robustness to small perturbations in the combinatorial Laplacian, supporting its viability in noisy or approximate data settings.
• Error-mitigation techniques from quantum chemistry and many-body physics can be readily applied, enhancing reliability in near-term implementations.