Entangling Power in the Deterministic Quantum Computation with One Qubit

TL;DR

This paper explores how the entangling power of quantum circuits in DQC1 correlates with its computational complexity and ability to evaluate the trace of unitaries, revealing that entanglement is fundamental to its power.

Contribution

It establishes a direct link between entangling power and the normalized trace evaluation in DQC1, highlighting entanglement's role in quantum computational complexity.

Findings

Nontrivial DQC1 always has non-vanishing entangling power.

Larger entangling power indicates higher computational complexity.

Non-vanishing entangling power exists in similar DQC1 tasks.

Abstract

The deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of the DQC1, so the nontrivial DQC1 is always accompanied with the non-vanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., the larger entangling power corresponds to higher complexity. Besides, it is also shown that the non-vanishing entangling power does always exist in other similar tasks of DQC1.