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arXiv:0802.1639 · Physical Review A

Noise gates for decoherent quantum circuits

Angelo Bassi, Dirk-André Deckert

Published Mar 14, 2008• 18 citations
arXivHTMLPDF

Abstract

A major problem in exploiting microscopic systems for developing a new technology based on the principles of quantum information is the influence of noise, which tends to work against the quantum features of such systems. It thus becomes crucial to understand how noise affects the evolution of quantum circuits: several techniques have been proposed, among which stochastic differential equations (SDEs) can represent a very convenient tool. We show how SDEs naturally map any Markovian noise into a linear operator, which we call a noise gate, acting on the wave function describing the state of the circuit, and we discuss some examples. We show that these gates can be manipulated like any standard quantum gate, thus simplifying in certain circumstances the task of computing the overall effect of the noise at each stage of the protocol. This approach yields equivalent results to those derived from the Lindblad equation; yet, as we show, it represents a handy and fast tool for performing computations, and, moreover, it allows for fast numerical simulations and generalizations to non-Markovian noise. In detail we review the depolarizing channel and the generalized amplitude damping channel in terms of this noise gate formalism and show how these techniques can be applied to any quantum circuit.

DOI
10.1103/physreva.77.032323

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A major problem in exploiting microscopic systems for developing a new technology based on the principles of quantum information is the influence of noise, which tends to work against the quantum features of such systems. It thus becomes crucial to understand how noise affects the evolution of quantum circuits: several techniques have been proposed, among which stochastic differential equations (SDEs) can represent a very convenient tool. We show how SDEs naturally map any Markovian noise into a linear operator, which we call a noise gate, acting on the wave function describing the state of the circuit, and we discuss some examples. We show that these gates can be manipulated like any standard quantum gate, thus simplifying in certain circumstances the task of computing the overall effect of the noise at each stage of the protocol. This approach yields equivalent results to those derived from the Lindblad equation; yet, as we show, it represents a handy and fast tool for performing computations, and, moreover, it allows for fast numerical simulations and generalizations to non-Markovian noise. In detail we review the depolarizing channel and the generalized amplitude damping channel in terms of this noise gate formalism and show how these techniques can be applied to any quantum circuit.
Quantum computing is an emergent field promising the improvement of processing speed in key algorithms by reducing their exponential scaling to polynomial, thus enabling solutions to problems that exceed classical computational capabilities. Gate-based quantum computing is the most common approach but still faces high levels of noise and decoherence. Gates play the role of probability mixers codifying information settled in quantum systems. However, they are deviated from their programmed behaviour due to those decoherent effects as a hidden source modifies the desired probability flux. Their quantification of such unavoidable behaviours becomes crucial for quantum error correction or mitigation. This work presents an approach to decoherence in quantum circuits using the Lindblad master equation to model the impact of noise and thermalisation underlying the ideal programmed behaviour expected for processing gates. The Lindblad approach then provides a comprehensive tool to model both probability fluxes being present in the process, thus regarding the gate and the environment. It analyses the deviation of resulting noisy states from the ideal unitary evolution of some gates considered as universal, setting some operating regimes. Thermalisation considers a radiation bath where gates are immersed as a feasible model of decoherence. Numerical simulations track the information loss as a function of the decay rate magnitude. It also exhibits the minimal impact on decoherence coming from particular quantum states being processed, but a higher impact on the number of qubits being processed by the gate. The methodology provides a unified framework to characterise the processing probability transport in quantum gates, including noise or thermalisation effects.
Quantum noise or decoherence is a major factor impacting the performance of quantum technologies. On the qubit, an important quantum noise, often relevant in practice, is the thermal noise or generalized amplitude damping noise, describing the interaction with a thermal bath at an arbitrary temperature. A qubit thermal noise however cannot be modeled nor directly simulated with a few elementary Pauli operators, but instead requires specific operators. Our main goal here is to construct a circuit model for simulating the thermal noise from standard elementary qubit operators. Starting from a common quantum-operation model based on Kraus operators and an associated qubit-environment model, we derive a proper Stinespring dilated representation for the thermal noise. This dilated unitary model is then decomposed in terms of simple elementary qubit operators, and converted into a circuit based on elementary quantum gates. We arrive at our targeted simulator circuit for the thermal noise, coming with built-in easy control on the noise parameters. The noise simulator is then physically implemented and tested on an IBM-Q quantum processor. The simulator represents a useful addition to existing libraries of quantum circuits for quantum processors, and it offers a new tool for investigating quantum signal and information processing having to cope with thermal noise.
We present a novel method for simulating the noisy behavior of quantum computers, which allows to efficiently incorporate environmental effects in the driven evolution implementing the gates acting on the qubits. We show how to modify the noiseless gate executed by the computer to include any Markovian noise, hence resulting in what we will call a noisy gate. We compare our method with the IBM qiskit simulator, and show that it follows more closely both the analytical solution of the Lindblad equation as well as the behavior of a real quantum computer, where we ran algorithms involving up to 18 qubits; as such, our protocol offers a more accurate simulator for NISQ devices. The method is flexible enough to potentially describe any noise, including non-Markovian ones. The noise simulator based on this work is available as a python package at the link, https://pypi.org/project/quantum-gates.
The advent of noisy intermediate-scale quantum (NISQ) devices has not only opened up new possibilities to studying quantum computational benefits, but it has also presented new challenges associated with noise and short coherence times. Entanglement is one of the inherent resources that can be used to perform quantum computation and implement the performance and reliability of quantum algorithms. This paper looks at entanglement dynamics in NISQ machines subjected to natural noises. The study focuses on the problem of examining the role of various noise channels in the creation, dynamics, and decay of the entanglement during quantum operations, such as depolarizing, amplitude damping, and phase damping. The study is based on theoretical modeling and numerical simulations of multi-qubit quantum circuits to determine the stability of entangled states with respect to circuit depth and intensity of noise. Concurrence and von Neumann entropy are examples of entanglement measures that are used to characterize correlations between qubits as well as to trace the dynamics of these correlations across computational interactions. The results show that noise has major impacts on entanglement distribution, which tends to cause rapid decays in highly entangled states with increase in circuit complexity. The findings, however, also show that there are some circuit architectures and error-mitigation schemes that can somewhat sustain entanglement with moderate levels of noise. The paper also emphasizes the relevance of minimalizing sequence of gates, and reduced circuit depth in an effort to preserve useful entanglement in present-day quantum hardware. These observations help in understanding the behaviour of quantum correlations better in real world quantum devices and provide a way of developing more resilient quantum algorithms that would work on NISQ systems. In general, the article offers a systematic view on entanglement behavior in disordered quantum systems and highlights its value in the development of quantum computing applications in the near term.
A major challenge in exploiting the principles of quantum information is the influence of noise which tends to work against the quantum features of a system. The traditional quantum gate error model provides a general framework to study noisy quantum circuits, but may fail to capture intricate microscopic interactions between qubits in addition to the environment. This can significantly overestimate the feasibility of implementing useful quantum algorithms on physical quantum hardware. In this paper, we study a noise model driven by random matrix theory that captures an effective microscopic interaction, and assess the suitability of the quantum gate error model. Our approach assumes noise arises due to interactions with many physical sources, and modeled using random matrix theory and applied to the quantum circuit model.
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.