Sequential realization of Quantum Instruments

TL;DR

This paper explores the sequential implementation of quantum instruments in adaptive circuits, revealing optimal tradeoffs between steps and ancillary qubits, and demonstrating efficient realization of certain transformations with minimal resources.

Contribution

It introduces a mathematical framework for adaptive sequence of instruments (ASI), establishes bounds on resources, and shows how to implement certain quantum transformations efficiently.

Findings

Lower bound on the product of steps and ancillary qubits for ASI

Optimal resource tradeoffs depend on the nature of quantum instruments

Efficient implementation of transformations increasing qubit number with minimal ancillas

Abstract

In adaptive quantum circuits classical results of mid-circuit measurements determine the upcoming gates. This allows POVMs, quantum channels or more generally quantum instruments to be implemented sequentially, so that fewer qubits need to be used at each of the $N$ measurement steps. In this paper, we mathematically describe these problems via adaptive sequence of instruments (ASI) and show how any instrument can be decomposed into it. Number of steps $N$ and number of ancillary qubits $n_A$ needed for actual implementation are crucial parameters of any such ASI. We show an achievable lower bound on the product $N.n_A$ and we determine in which situations this tradeoff is likely to be optimal. Contrary to common intuition we show that for quantum instruments which transform $n$ to $m(>n)$ qubits, there exist $N$-step ASI implementing them just with $(m-n)$ ancillary qubits, which are remeasured $(N-1)$ times and finally used as output qubits.