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arXiv:2010.10898 · Physical Review A

Promoting quantum correlations in deterministic quantum computation with a one-qubit model via postselection

Elisa I. Goettems, Thiago O. Maciel, Diogo O. Soares-Pinto, Eduardo I. Duzzioni

Published Apr 19, 2021• 11 citations
arXivHTMLPDF

Abstract

The deterministic quantum computation with one qubit (DQC1) model is a restricted model of quantum computing able to calculate efficiently the normalized trace of a unitary matrix. In this work, we analyze the quantum correlations called entanglement, Bell's nonlocality, quantum discord, and coherence generated by the DQC1 circuit considering only two qubits (auxiliary and control). For the standard DQC1 model, only quantum discord and coherence appear. By introducing a filter in the circuit we purify the auxiliary qubit, taking it out from the totally mixed state and consequently promoting other quantum correlations between the qubits, such as entanglement and Bell's nonlocality. Through the optimization of the purification process, we conclude that even a small purification is enough to generate entanglement and Bell's nonlocality. We obtain, that by applying the purification process repeatedly an average of 12 times, the auxiliary qubit becomes $99%$ pure. In this situation, almost maximally entangled states are achieved, which almost maximally violate Bell's inequality. This result suggests that with a simple modification, the DQC1 model can be promoted to a universal model of quantum computing.

DOI
10.1103/physreva.103.042416

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The deterministic quantum computation with one qubit (DQC1) model is a restricted model of quantum computing able to calculate efficiently the normalized trace of a unitary matrix. In this work, we analyze the quantum correlations called entanglement, Bell's nonlocality, quantum discord, and coherence generated by the DQC1 circuit considering only two qubits (auxiliary and control). For the standard DQC1 model, only quantum discord and coherence appear. By introducing a filter in the circuit we purify the auxiliary qubit, taking it out from the totally mixed state and consequently promoting other quantum correlations between the qubits, such as entanglement and Bell's nonlocality. Through the optimization of the purification process, we conclude that even a small purification is enough to generate entanglement and Bell's nonlocality. We obtain, that by applying the purification process repeatedly an average of 12 times, the auxiliary qubit becomes 9999%99 pure. In this situation, almost maximally entangled states are achieved, which almost maximally violate Bell's inequality. This result suggests that with a simple modification, the DQC1 model can be promoted to a universal model of quantum computing.
Deterministic quantum computation with one pure qubit (DQC1) is an efficient model of computation that uses highly mixed states. Unlike pure-state models, its power is not derived from the generation of a large amount of entanglement. Instead it has been proposed that other nonclassical correlations are responsible for the computational speedup, and that these can be captured by the quantum discord. In this Letter we implement DQC1 in an all-optical architecture, and experimentally observe the generated correlations. We find no entanglement, but large amounts of quantum discord-except in three cases where an efficient classical simulation is always possible. Our results show that even fully separable, highly mixed, states can contain intrinsically quantum mechanical correlations and that these could offer a valuable resource for quantum information technologies.
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We use quantum discord to characterize the correlations present in the model called deterministic quantum computation with one quantum bit (DQC1), introduced by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)10.1103/PhysRevLett.81.5672]. The model involves a collection of qubits in the completely mixed state coupled to a single control qubit that has nonzero purity. The initial state, operations, and measurements in the model all point to a natural bipartite split between the control qubit and the mixed ones. Although there is no entanglement between these two parts, we show that the quantum discord across this split is nonzero for typical instances of the DQC1 ciruit. Nonzero values of discord indicate the presence of nonclassical correlations. We propose quantum discord as figure of merit for characterizing the resources present in this computational model.
In this thesis, I look at the role of quantum entanglement in mixed-state quantum computation. The model we consider is the DQC1 or `power of one qubit' model. I show that there is minimal bipartite entanglement in a typical instance of the DQC1 circuit and even put an upper bound on the possible amount of entanglement. This limited amount of entanglement, however, does not imply that the system is classically simulatable. A matrix product state (MPS) algorithm for a typical instance of the DQC1 system requires exponential classical resources. This exposes a gap between the amount of entanglement and the amount of purely nonclassical correlations in a quantum system. This gap, I suggest, can be filled by quantum discord. I calculate it in a typical instance of the DQC1 circuit and find that the amount of discord is a constant fraction of the maximum possible discord for a system of that size. This allows an interpretation of quantum discord as the resource that drives mixed-state quantum computation. I also study quantum discord as a quantity of independent interest. Its role in the phenomenon of entanglement distribution is studied through an easily comprehensible example. This thesis also contains discussions on the relation between the complexity classes P, BQP and DQC1. Additional material is presented on the connections between the DQC1 model, Jones polynomials and statistical mechanics. The thesis concludes with a discussion of a few open problems related to the DQC1 model, the quantum discord and their scope in quantum information science.
This paper explores the potential benefits of quantum coherence and quantum discord in the non-universal quantum computing model called deterministic quantum computing with one qubit (DQC1) in supervised machine learning. We show that the DQC1 model can be leveraged to develop an efficient method for estimating complex kernel functions. We demonstrate a simple relationship between coherence consumption and the kernel function, a crucial element in machine learning. The paper presents an implementation of a binary classification problem on IBM hardware using the DQC1 model and analyzes the impact of quantum coherence and hardware noise. The advantage of our proposal lies in its utilization of quantum discord, which is more resilient to noise than entanglement.
The one-clean-qubit model of quantum computation (DQC1) efficiently implements a computational task that is not known to have a classical alternative. During the computation, there is never more than a small but finite amount of entanglement present, and it is typically vanishingly small in the system size. In this paper, we demonstrate that there is nothing unexpected hidden within the DQC1 model---Grover's search, when acting on a mixed state, provably exhibits a speedup over classical, with guarantees as to the presence of only vanishingly small amounts of quantum correlations (entanglement and quantum discord)---while arguing that this is not an artifact of the oracle-based construction. We also present some important refinements in the evaluation of how much entanglement may be present in the DQC1 and how the typical entanglement of the system must be evaluated.