Quantum Mass Production Theorems

TL;DR

This paper proves that for large quantum systems, multiple copies of any unitary transformation can be efficiently implemented with a number of gates comparable to a single copy, extending classical mass production concepts to quantum circuits.

Contribution

It introduces quantum mass production theorems showing efficient implementation of multiple copies of unitaries, states, and diagonal unitaries with near-optimal gate complexity.

Findings

Efficient quantum circuit construction for multiple copies of unitaries

Asymptotic gate complexity matches single-copy implementation

Extension of classical mass production theorems to quantum domain

Abstract

We prove that for any $n$-qubit unitary transformation $U$ and for any $r = 2^{o(n / \log n)}$, there exists a quantum circuit to implement $U^{\otimes r}$ with at most $O(4^n)$ gates. This asymptotically equals the number of gates needed to implement just a single copy of a worst-case $U$. We also establish analogous results for quantum states and diagonal unitary transformations. Our techniques are based on the work of Uhlig [Math. Notes 1974], who proved a similar mass production theorem for Boolean functions.