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arXiv:1404.1829 · Physical Review A

Fidelity of measurement-based quantum computation in a bosonic environment

Jian Wang, Ding Zhong, Liang-Zhu Mu, Heng Fan

Published Nov 6, 2014• 3 citations
arXivHTMLPDF

Abstract

We investigate the fidelity of measurement-based quantum computation (MBQC) when it is coupled with a boson environment, by measuring the cluster-state fidelity and gate fidelity. Two schemes of cluster-state preparation are studied. In the controlled-Z (CZ) creation scheme, cluster states are prepared by entangling all qubits in $|+\ensuremath{\rangle}$ state with CZ gates at all neighboring sites. The fidelity shows an oscillation pattern over time. The influence of the environmental temperature is evaluated, and suggestions are given to enhance the performance of MBQCs realized in this way. In the Hamiltonian creation scheme, cluster states are made by cooling a system with cluster Hamiltonians, of which the ground states are cluster states. A fidelity sudden-drop phenomenon is discovered. When the coupling is below a threshold, MBQC systems are highly robust against the noise. Our main environmental model is one with a single collective bosonic mode.

DOI
10.1103/physreva.90.052306

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We investigate the fidelity of measurement-based quantum computation (MBQC) when it is coupled with a boson environment, by measuring the cluster-state fidelity and gate fidelity. Two schemes of cluster-state preparation are studied. In the controlled-Z (CZ) creation scheme, cluster states are prepared by entangling all qubits in ∣+\ensuremath⟩|+\ensuremath{\rangle}∣+\ensuremath⟩ state with CZ gates at all neighboring sites. The fidelity shows an oscillation pattern over time. The influence of the environmental temperature is evaluated, and suggestions are given to enhance the performance of MBQCs realized in this way. In the Hamiltonian creation scheme, cluster states are made by cooling a system with cluster Hamiltonians, of which the ground states are cluster states. A fidelity sudden-drop phenomenon is discovered. When the coupling is below a threshold, MBQC systems are highly robust against the noise. Our main environmental model is one with a single collective bosonic mode.
Measurement-based quantum computation is an efficient model to perform universal computation. Nevertheless, theoretical questions have been raised, mainly with respect to realistic noise conditions. In order to shed some light on this issue, we evaluate the exact dynamics of some single-qubit-gate fidelities using the measurement-based quantum computation scheme when the qubits which are used as a resource interact with a common dephasing environment. We report a necessary condition for the fidelity dynamics of a general pure NNN-qubit state, interacting with this type of error channel, to present an oscillatory behavior, and we show that for the initial canonical cluster state, the fidelity oscillates as a function of time. This state fidelity oscillatory behavior brings significant variations to the values of the computational results of a generic gate acting on that state depending on the instants we choose to apply our set of projective measurements. As we shall see, considering some specific gates that are frequently found in the literature, the fast application of the set of projective measurements does not necessarily imply high gate fidelity, and likewise the slow application thereof does not necessarily imply low gate fidelity. Our condition for the occurrence of the fidelity oscillatory behavior shows that the oscillation presented by the cluster state is due exclusively to its initial geometry. Other states that can be used as resources for measurement-based quantum computation can present the same initial geometrical condition. Therefore, it is very important for the present scheme to know when the fidelity of a particular resource state will oscillate in time and, if this is the case, what are the best times to perform the measurements.
Measurement based quantum computation (MBQC) is an effective paradigm for universal quantum computation. In this scheme, the universal set of quantum gates are realized by only local measurements on the prior prepared cluster states. The inevitable decoherence is harmful to the realization of those quantum gates. Here, we investigate the performance of the quantum gates exposed to different type of noises. We find that some errors may not influence the success of the quantum gates, in contrast, some others may destroy their realization. We show that there is a controlling pattern that can protect quantum gates from certain types of noises and thus can improve the success probability of the gates implementation.
Quantum computers, leveraging the principles of quantum mechanics, hold the potential to surpass classical computers in numerous applications, with implications across various domains. Besides the well-known gate model, Measurement-based Quantum Computation (MBQC) is another promising computational approach to achieve universal quantum computation. In MBQC, large ensembles of qubits are prepared in a highly entangled cluster state, forming the basis for executing quantum computations through sequential measurements. Cluster states are realized using both continuous variables (CV) and discrete variables (DV) techniques. In the CV-based methods, Frequency Domain Multiplexing (FDM), Time Domain Multiplexing (TDM), Spatial Domain Multiplexing (SDM), and hybrid schemes are employed. This paper thoroughly discusses and compares these approaches, elucidating their strengths and limitations. Additionally, the generation of photonic cluster states in DV is explored and some recent results are reported. Some recent state-of-the-art advancements in photonic and superconducting qubits entanglement, which can potentially serve as cluster states, are also presented. Finally, we highlight the approach that exhibits the most promising characteristics for achieving efficient cluster state realization in the context of MBQC.
In measurement-based quantum computation (MBQC), quantum circuits are implemented using adaptive measurements on an entangled resource state. In practice, the resource state will always be prepared with some noise, and it is crucial to understand the effect of this noise on the operation of MBQC. Typically, one measures the fidelity of the noisy resource state with the assumption that a high fidelity state means a high fidelity computation. However, the precise relationship between these two fidelities is not known. Here, we derive an expression that equates the average fidelity of the MBQC output state to a certain correlation function evaluated on the noisy resource state. Using this expression, we show that state fidelity provides a tight lower bound on average MBQC fidelity. Conversely, we also find that state fidelity can greatly underestimate average MBQC fidelity, implying that state fidelity is not a good indicator of MBQC performance in general. In response, we formulate an efficient method to directly estimate average MBQC fidelity by measuring the aforementioned correlation function. These results therefore improve our ability to characterize noisy resource states in quantum computers and benchmark MBQC performance.
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.
We consider measurement-based quantum computation using the state of a spin-lattice system in equilibrium with a thermal bath and free to evolve under its own Hamiltonian. Any single qubit measurements disturb the system from equilibrium and, with adaptive measurements performed at a finite rate, the resulting dynamics reduces the fidelity of the computation. We show that it is possible to describe the loss in fidelity by a single quantum operation on the encoded quantum state that is independent of the measurement history. To achieve this simple description, we choose a particular form of spin-boson coupling to describe the interaction with the environment, and perform measurements periodically at a natural rate determined by the energy gap of the system. We found that an optimal cooling exists, which is a trade-off between keeping the system cool enough that the resource state remains close to the ground state, but also isolated enough that the cooling does not strongly interfere with the dynamics of the computation. For a sufficiently low temperature we obtain a fault-tolerant threshold for the couplings to the environment.