Optimal transducers using symmetries

2026-10-01

AI Summary

This study addresses the prohibitive computational cost of constructing quantum transducers, which conventionally relies on complex semidefinite programming and explicit unitary matrix solutions. To overcome this, it exploits the symmetric group structure inherent in state transitions to simplify optimal transducer construction. Specifically, this work provides the first proof that optimal catalysts can be chosen as covariant representations and that transducers admit block-diagonalization under isotypic decomposition. By integrating group representation theory, symmetrization arguments, and semidefinite programming duality, an efficient algorithmic framework is developed. The primary contribution lies in extending representation-theoretic techniques from lower-bound analysis to constructive algorithms, thereby deriving optimal transducers with tight constants for fundamental quantum primitives, including unstructured search and amplitude amplification.

Abstract

Transducers (Belovs, Jeffery and Yolcu, 2024) are a quantum computing framework describing a quantum algorithm as a unitary converting an input state into a target state using a catalyst, an auxiliary vector that is left unchanged. They are a powerful tool in quantum algorithm design, especially in the context of quantum query complexity: feasible points of the (dual) adversary semidefinite program directly translate into transducers and the optimal transduction complexity is equal to the adversary bound, i.e. the Las Vegas complexity, which is known to characterize bounded-error quantum query complexity. Moreover, contrary to bounded-error algorithms, transducers compose exactly, which limits overheads due to controlling errors in algorithms constructed by composition. Constructing efficient, let alone optimal, transducers in terms of quantum query complexity nevertheless remains a hard task since it still requires solving the adversary SDP and constructing the unitary to obtain an explicit algorithm. In this paper, we show how using the symmetry group of state-conversion problems simplifies both steps. First, using a symmetrization argument, we prove an optimal catalyst can always be chosen covariant under a representation of the symmetry group. Second, we prove that the transducer intertwines two different representations of the group and can thus be chosen block diagonal in the isotypic decomposition of the Hilbert space. Using those methods, we then derive optimal transducers, with optimal constants, for different widely used quantum algorithmic primitives, such as unstructured search, amplitude amplification and amplitude estimation. Our approach extends previous work on the use of representation theory to compute adversary lower bounds (Høyer, Lee, and {\v S}palek, 2007; Ambainis, Magnin, Roetteler and Roland, 2011) to the systematic construction of optimal algorithms.

Problem

• quantum transducers
• quantum query complexity
• adversary semidefinite program
• state-conversion
• symmetry

Innovation

• Transducers
• Symmetry group
• Representation theory
• Quantum query complexity
• Adversary bound