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arXiv:1011.6336 · Physical Review A

Superoperator analysis of entanglement in a four-qubit cluster state

Yaakov S. Weinstein, Jay Feldman, Jacob Robins, Jason Zukus, Gerald N. Gilbert

Published Mar 19, 2012• 19 citations
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Abstract

In this paper we utilize superoperator formalism to explore the entanglement evolution of four-qubit cluster states in a number of decohering environments. A four-qubit cluster state is a resource for the performance of an arbitrary single-logical-qubit rotation via measurement-based cluster-state quantum computation. We are specifically interested in the relationship between entanglement evolution and the fidelity with which the arbitrary single-logical-qubit rotation can be implemented in the presence of decoherence as this will have important experimental ramifications. We also note the exhibition of entanglement sudden death (ESD) and ask how severely its onset affects the utilization of the cluster state as a means of implementing an arbitrary single-logical-qubit rotation.

DOI
10.1103/physreva.85.032324

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In this paper we utilize superoperator formalism to explore the entanglement evolution of four-qubit cluster states in a number of decohering environments. A four-qubit cluster state is a resource for the performance of an arbitrary single-logical-qubit rotation via measurement-based cluster-state quantum computation. We are specifically interested in the relationship between entanglement evolution and the fidelity with which the arbitrary single-logical-qubit rotation can be implemented in the presence of decoherence as this will have important experimental ramifications. We also note the exhibition of entanglement sudden death (ESD) and ask how severely its onset affects the utilization of the cluster state as a means of implementing an arbitrary single-logical-qubit rotation.
The highly entangled four qubit cluster state can be used to perform an arbitrary single logical qubit rotation via the techniques of measurement-based quantum computation. In this paper, we explore the relationship between the entanglement in the cluster state and the ability of the state to accurately perform the desired logical rotation. This has important experimental ramifications as realistic cluster state experiments will always be subject to decoherence. We also note the exhibition of entanglement sudden death (ESD) and ask how severely its onset affects the utilization of the cluster state as a means of implementing the arbitrary single logical qubit rotation.
I explore the entanglement evolution of a four-qubit cluster state in a dephasing environment concentrating on the phenomenon of entanglement sudden death (ESD). Specifically, I ask whether the onset of ESD has an effect on the utilization of this cluster state as a means of implementing a single-qubit rotation in the measurement-based cluster state model of quantum computation. To do this, I compare the evolution of the entanglement to the fidelity, a measure of how accurately the desired state (after the measurement-based operations) is achieved. I find that ESD does not cause a change in behavior or discontinuity in the fidelity but may indicate for certain states when the fidelity goes to 0.5.
In this paper I explore the entanglement evolution of qubits that are part of a five qubit quantum error correction code subject to various decohering environments. Specifically, I look for possible parallels between the entanglement degradation and the fidelity of the logical qubit of quantum information stored in the physical qubits. In addition, I note the possible exhibition of entanglement sudden death (ESD) due to decoherence and question whether ESD is actually a roadblock to successful quantum computation.
We explore basic necessary protocols to achieve fault tolerance for quantum computation in the cluster state and circuit models. For the cluster state model we simulate the implementation of an arbitrary rotation via only measurement on a decohered cluster state. Fidelity is used to quantify the accuracy of the initial cluster state and a gate fidelity for the arbitrary rotation is determined. In the circuit model we compare the accuracy of two methods that can be used to construct a logical zero state appropriate for the [7, 1, 3] Steane quantum error correction code in a non-equiprobable Pauli operator error environment: a fault tolerant method done by applying error correction on seven qubits all in the state zero, and a non-fault tolerant method done by implementing the encoding gate sequence. We find that the latter construction method, in spite of its lack of "fault tolerance," outputs a seven qubit state with a higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information exhibits almost equivalent values between the two construction methods.
We investigate the decoherence of maximally mixed multi-partite cluster states of a physical system of n qubits. We introduce the disconnected cluster states for an ensemble of non interacting qubits and we give the corresponding separable density matrices. The maximally entangled cluster states can be generated from disconnected cluster states, by applying the entanglig operators between the pair of qubits. When exposed to a local noisy interaction with the environment, the multi-qubit system evolves from its initial pure maximally entangled cluster state to a decohered mixed state. The deco-hered mixed states for multipartite systems are explicitly expressed and their bipartite entanglement is characterized by using the genuine multipartite neg-ativity as a quantifier of entanglement. Conclusion Entangled cluster states are obtained by a dynamical evolution of multi-qubit system by using an Hamiltonian of Ising type. Cluster states are of great interest in quantum computing and are suitable in various tasks of quantum computation. Decohered mixed states are derived via a local noisy evolution of the entangled multi-qubit system in contact with environment. More precisely , under the influence of the de-coherence, the multi-qubit system loses entanglement and evolves from it's initial maximally entangled cluster state to a decohered mixed state. More precisely we have given explicit expressions of decohered mixed states derived from n-qubit entangled cluster states. The separability for decohered mixed states is addressed with the concept of the genuine multipartite nega-tivity.
Measurement-based Quantum Computation(MBQC) utilize entanglement as resource for performing quantum computation. Generating cluster state using entanglement as resource is a key bottleneck for the adoption of MBQC. To generate cluster state with charge-qubit arrrays, we provide analytical derivations and numerical validations for 4-qubit cluster state. We compare our fidelities under ideal (noise-free) Hamiltonian evolution and due to effect of decoherence. We show incorporating energy relaxation (T1T_1T1​) yields $>$90\% fidelity while pure dephasing T2T_2T2​ show 70%70\%70% decays at fourth harmonics. We further show under noise T2T_2T2​ decays to 50\% within 15 time units, versus $>70%underrelaxationtimeunits(70\% under relaxation time units (70%underrelaxationtimeunits(T_1$)--only. This decay quantify degradation effect of T2T_2T2​ on preparing cluster--state preparation is more than T1T_1T1​. We highlight the critical need for targeted error-mitigation strategies in near-term MBQC implementations.