A memory and gate efficient algorithm for unitary mixed Schur sampling

TL;DR

This paper introduces a memory- and gate-efficient streaming algorithm for unitary mixed Schur sampling, significantly reducing resource requirements for quantum state measurement tasks like tomography and spectrum estimation.

Contribution

It presents a novel streaming algorithm for unitary mixed Schur sampling with exponential memory and polynomial gate complexity reductions, improving previous methods.

Findings

Achieves exponential memory reduction over naive algorithms.

Reduces gate complexity polynomially for the task.

Optimizes performance for states with limited rank.

Abstract

We formalize the task of unitary Schur sampling -- an extension of weak Schur sampling -- which is the process of measuring the Young label and the unitary group register of an input $m$ qudit state. Intuitively, this task is equivalent to applying the Schur transform, projecting onto the isotypic subspaces of the unitary and symmetric groups indexed by the Young labels, and discarding of the permutation register. As such unitary Schur sampling is the natural task in processes such as quantum state tomography or spectrum estimation. We generalize this task to unitary mixed Schur sampling to account for the recently introduced mixed Schur-Weyl transform. We provide a streaming algorithm which achieves an exponential reduction in the memory complexity and a polynomial reduction in the gate complexity over na\"ive algorithms for the task of unitary (mixed) Schur sampling. Further, we show that if the input state has limited rank, the gate and memory complexities of our streaming algorithm as well as the algorithms for the full Schur and mixed Schur transforms are further reduced. Our work generalizes and improves on the results in arXiv2309.11947.