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arXiv:quant-ph/0312053 · Physical Review A

Stochastic Liouville equation approach for the effect of noise in quantum computations

Yuan‐Chung Cheng, R. Silbey

Published May 19, 2004• 41 citations
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Abstract

We propose a model based on a generalized effective Hamiltonian for studying the effect of noise in quantum computations. The system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. Treating these fluctuations as Gaussian Markov processes with zero mean and $\ensuremath{\delta}$-function correlation times, we derive an exact equation of motion describing the dissipative dynamics for a system of n qubits. We then apply this model to study the effect of noise on the quantum teleportation and a generic quantum controlled-NOT (CNOT) gate. For quantum teleportation, the effect of noise in the quantum channels is found to be additive, and the teleportation fidelity depends on the state of the teleported qubit. The effect of collective decoherence is also studied for the two-qubit entangled states. For the quantum CNOT gate, we study the effect of noise on a set of one- and two-qubit quantum gates, and show that the results can be assembled together to investigate the quality of a quantum CNOT gate operation. We compute the averaged gate fidelity and gate purity for the quantum CNOT gate and investigate phase, bit-flip, and flip-flop errors during the CNOT gate operation. The effects of direct interqubit coupling and fluctuations on the control fields are also studied. We find that the quality of the CNOT gate operation is sensitive to the strengths of the control fields and the strengths of the noise, and the effect of noise is additive regardless of its origin. We discuss the limitations and possible extensions of this model. In sum, we demonstrate a simple model that enables us to investigate the effect of noise in arbitrary quantum circuits under realistic device conditions.

DOI
10.1103/physreva.69.052325

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We propose a model based on a generalized effective Hamiltonian for studying the effect of noise in quantum computations. The system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. Treating these fluctuations as Gaussian Markov processes with zero mean and \ensuremathδ\ensuremath{\delta}\ensuremathδ-function correlation times, we derive an exact equation of motion describing the dissipative dynamics for a system of n qubits. We then apply this model to study the effect of noise on the quantum teleportation and a generic quantum controlled-NOT (CNOT) gate. For quantum teleportation, the effect of noise in the quantum channels is found to be additive, and the teleportation fidelity depends on the state of the teleported qubit. The effect of collective decoherence is also studied for the two-qubit entangled states. For the quantum CNOT gate, we study the effect of noise on a set of one- and two-qubit quantum gates, and show that the results can be assembled together to investigate the quality of a quantum CNOT gate operation. We compute the averaged gate fidelity and gate purity for the quantum CNOT gate and investigate phase, bit-flip, and flip-flop errors during the CNOT gate operation. The effects of direct interqubit coupling and fluctuations on the control fields are also studied. We find that the quality of the CNOT gate operation is sensitive to the strengths of the control fields and the strengths of the noise, and the effect of noise is additive regardless of its origin. We discuss the limitations and possible extensions of this model. In sum, we demonstrate a simple model that enables us to investigate the effect of noise in arbitrary quantum circuits under realistic device conditions.
Noise is a major challenge in quantum computing, affecting the reliability of quantum protocols. In this work, we analytically study the impact of various noise processes, such as depolarization, bit flip, and phase flip, on the quantum state teleportation protocol. Each noise process is modeled as a quantum channel and is applied individually to all qubits after the corresponding unitary operations to simulate realistic conditions. We evaluate the fidelity between the ideal and noisy teleported states to quantify the effect of noise. Our analysis shows that the fidelity decreases polynomially, in general, as the noise strength increases for all noise types, highlighting the sensitivity of state teleportation to different noise mechanisms. However, in the low noise regime, the fidelity decreases only linearly, indicating the robustness of the teleportation protocol. These results provide insight into error characterization and can inform strategies for noise mitigation in practical quantum computing applications.
Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three-qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC codes are calculated. In addition, the effects of imprecision in projective measurements, collective bath, fault-tolerant repetition protocols, and level of parallelism in circuit constructions on the threshold values are also studied with emphasis on determining the optimal design for the fault-tolerant QEC circuit. These results provide insights into the fault-tolerant QEC process as well as useful information for designing the optimal fault-tolerant QEC circuit for particular physical implementation of quantum computer.
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Treating the effects of a time-dependent classical dephasing environment during quantum logic operations poses a theoretical challenge, as the application of noncommuting control operations gives rise to both dephasing and depolarization errors that must be accounted for in order to understand total average error rates. We develop a treatment based on effective Hamiltonian theory that allows us to efficiently model the effect of classical noise on nontrivial single-bit quantum logic operations composed of arbitrary control sequences. We present a general method to calculate the ensemble-averaged entanglement fidelity to arbitrary order in terms of noise filter functions, and provide explicit expressions to fourth order in the noise strength. In the weak noise limit we derive explicit filter functions for a broad class of piecewise-constant control sequences, and use them to study the performance of dynamically corrected gates, yielding good agreement with brute-force numerics.
Quantum algorithms have been proposed to accelerate the simulation of the chaotic dynamical systems that are ubiquitous in the physics of plasmas. Quantum computers without error correction might even use noise to their advantage to calculate the Lyapunov exponent by measuring the Loschmidt echo fidelity decay rate. For the first time, digital Hamiltonian simulations of the quantum sawtooth map, performed on the IBM-Q quantum hardware platform, show that the fidelity decay rate of a digital quantum simulation increases during the transition from dynamical localization to chaotic diffusion in the map. The observed error per CNOT gate increases by 1.5×1.5{\times }1.5× as the dynamics varies from localized to diffusive, while only changing the phases of virtual RZ gates and keeping the overall gate count constant. A gate-based Lindblad noise model that captures the effective change in relaxation and dephasing errors during gate operation qualitatively explains the effect of dynamics on fidelity as being due to the localization and entanglement of the states created. Specifically, highly delocalized states that are entangled with random phases show an increased sensitivity to dephasing and, on average, a similar sensitivity to relaxation as localized states. In contrast, delocalized unentangled states show an increased sensitivity to dephasing but a lower sensitivity to relaxation. This gate-based Lindblad model is shown to be a useful benchmarking tool by estimating the effective Lindblad coherence times during CNOT gates and finding a consistent $2\unicode{x2013}3{\times }$ shorter T2T_2T2​ time than reported for idle qubits. Thus, the interplay of the dynamics of a simulation with the noise processes that are active can strongly influence the overall fidelity decay rate.
We study the dynamics of quantum systems under classical and quantum noise, focusing on decoherence in qubit systems. Classical noise is described by a random process leading to a stochastic temporal evolution of a closed quantum system, whereas quantum noise originates from the coupling of the microscopic quantum system to its macroscopic environment. We derive deterministic master equations describing the average evolution of the quantum system under classical continuous-time Markovian noise and two sets of master equations under quantum noise. Strikingly, these three equations of motion are shown to be equivalent in the case of classical random telegraph noise and proper quantum environments. Hence fully quantum-mechanical models within the Born approximation can be mapped to a quantum system under classical noise. Furthermore, we apply the derived equations together with pulse optimization techniques to achieve high-fidelity one-qubit operations under random telegraph noise, and hence fight decoherence in these systems of great practical interest.