Space-Efficient Quantum Error Reduction without log Factors

2025-02-13

AI Summary

To overcome the ubiquitous $O(log(1/varepsilon))$ space and query overhead in quantum algorithm error mitigation, this paper introduces the first logarithmic-factor-free error purification protocol. Methodologically, it leverages the quantum transducer model to construct a linearly weighted one-dimensional quantum walk, integrated with amplitude amplification—replacing conventional majority voting. The main contributions are threefold: (1) exponential reduction in space complexity; (2) improvement of the spectral gap dependence from linear to quadratic; and (3) achievement of asymptotically optimal query complexity—with an exact constant factor—enabling composition with super-constant-depth quantum algorithms. The resulting purifier features a simple architecture and transparent analysis, breaking the long-standing logarithmic overhead barrier in quantum error mitigation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Given an algorithm that outputs the correct answer with bounded error, say $1/3$, it is sometimes desirable to reduce this error to some arbitrarily small $varepsilon$ -- for example, if one wants to call the algorithm many times as a subroutine. The usual method, for both quantum and randomized algorithms, is a procedure called majority voting, which incurs a multiplicative overhead of $O(logfrac{1}{varepsilon})$ from calling the algorithm this many times. A recent paper introduced a model of quantum computation called emph{transducers}, and showed how to reduce the ``error'' of a transducer arbitrarily with only constant overhead, using a construction analogous to majority voting called emph{purification}. Even error-free transducers map to bounded-error quantum algorithms, so this does not let you reduce algorithmic error for free, but it does allow bounded-error quantum algorithms to be composed without incurring log factors. In this paper, we present a new highly simplified construction of a purifier, that can be understood as a weighted walk on a line similar to a random walk interpretation of majority voting. In addition to providing a new perspective that is easier to contrast with majority voting, our purifier has exponentially better space complexity than the previous one, and quadratically better dependence on the soundness-completeness gap of the algorithm being purified. Our new purifier has nearly optimal query complexity, even down to the constant, which matters when one composes quantum algorithms to super-constant depth.

Problem

• Reduce quantum error efficiently
• Simplify purification construction
• Optimize space and query complexity

Innovation

• Quantum error reduction
• Purification construction
• Improved space complexity