Quantum Circuits for Quantum Channels

TL;DR

This paper explores efficient quantum circuit implementations of quantum channels, establishing lower bounds and providing near-optimal decompositions with minimal C-NOT gates across different models, including measurements and classical randomness.

Contribution

It introduces a MeasuredQCM model for quantum channels, achieving low C-NOT counts with minimal ancillas, and provides explicit low-cost circuit examples.

Findings

Lower bounds on C-NOT gate counts for quantum channels.

Near-optimal circuit decompositions in various models.

Explicit low-cost circuits for small qubit systems.

Abstract

We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-NOT (C-NOT) gates required (single-qubit gates are free). We consider three different models. In the first, the Quantum Circuit Model (QCM), we consider sequences of single-qubit and C-NOT gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of C-NOT gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most one ancilla and has a low C-NOT count. We give explicit examples for small numbers of qubits that provide the lowest known C-NOT counts.