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arXiv:1304.7925 · Reviews of Modern Physics

1 / f noise: Implications for solid-state quantum information

Elisabetta Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler

Published Apr 3, 2014• 611 citations
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Abstract

The efficiency of the future devices for quantum information processing is limited mostly by the finite decoherence rates of the individual qubits and quantum gates. Recently, substantial progress was achieved in enhancing the time within which a solid-state qubit demonstrates coherent dynamics. This progress is based mostly on a successful isolation of the qubits from external decoherence sources obtained by engineering. Under these conditions, the material-inherent sources of noise start to play a crucial role. In most cases, quantum devices are affected by noise decreasing with frequency $f$ approximately as $1/f$. According to the present point of view, such noise is due to material- and device-specific microscopic degrees of freedom interacting with quantum variables of the nanodevice. The simplest picture is that the environment that destroys the phase coherence of the device can be thought of as a system of two-state fluctuators, which experience random hops between their states. If the hopping times are distributed in an exponentially broad domain, the resulting fluctuations have a spectrum close to $1/f$ in a large frequency range. This paper reviews the current state of the theory of decoherence due to degrees of freedom producing $1/f$ noise. Basic mechanisms of such noises in various nanodevices are discussed and several models describing the interaction of the noise sources with quantum devices are reviewed. The main focus of the review is to analyze how the $1/f$ noise destroys their coherent operation. The start is from individual qubits concentrating mostly on the devices based on superconductor circuits and then some special issues related to more complicated architectures are discussed. Finally, several strategies for minimizing the noise-induced decoherence are considered.

DOI
10.1103/revmodphys.86.361

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The efficiency of the future devices for quantum information processing is limited mostly by the finite decoherence rates of the individual qubits and quantum gates. Recently, substantial progress was achieved in enhancing the time within which a solid-state qubit demonstrates coherent dynamics. This progress is based mostly on a successful isolation of the qubits from external decoherence sources obtained by engineering. Under these conditions, the material-inherent sources of noise start to play a crucial role. In most cases, quantum devices are affected by noise decreasing with frequency fff approximately as 1/f1/f1/f. According to the present point of view, such noise is due to material- and device-specific microscopic degrees of freedom interacting with quantum variables of the nanodevice. The simplest picture is that the environment that destroys the phase coherence of the device can be thought of as a system of two-state fluctuators, which experience random hops between their states. If the hopping times are distributed in an exponentially broad domain, the resulting fluctuations have a spectrum close to 1/f1/f1/f in a large frequency range. This paper reviews the current state of the theory of decoherence due to degrees of freedom producing 1/f1/f1/f noise. Basic mechanisms of such noises in various nanodevices are discussed and several models describing the interaction of the noise sources with quantum devices are reviewed. The main focus of the review is to analyze how the 1/f1/f1/f noise destroys their coherent operation. The start is from individual qubits concentrating mostly on the devices based on superconductor circuits and then some special issues related to more complicated architectures are discussed. Finally, several strategies for minimizing the noise-induced decoherence are considered.
The efficiency of the future devices for quantum information processing will be limited mostly by the finite decoherence rates of the qubits. Recently, substantial progress was achieved in enhancing the time within which a solid-state qubit demonstrates coherent dynamics. This progress is based mostly on a successful isolation of the qubits from external decoherence sources. Under these conditions, the material-inherent sources of noise start to play a crucial role. In most cases, the noise that the quantum device demonstrates has a 1/ f spectrum. This suggests that the environment that destroys the phase coherence of the qubit can be thought of as a system of two-state fluctuators, which experience random hops between their states. In this short review, the current state of the theory of the decoherence due to the qubit interaction with the fluctuators is discussed. The effect of such an environment on two different protocols of the qubit manipulations, free induction and echo signal, is described. It turns out that in many important cases the noise produced by the fluctuators is non-Gaussian. Consequently, the results of the interaction of the qubit with the fluctuators are not determined by the pair correlation function alone. We describe the effect of the fluctuators using the so-called spin-fluctuator model. Being quite realistic, this model allows one to exactly evaluate the qubit dynamics in the presence of one fluctuator. This solution is found, and its features, including non-Gaussian effects, are analyzed in detail. We extend this consideration to systems of large numbers of fluctuators, which interact with the qubit and lead to the 1/ f noise. We discuss existing experiments on the Josephson qubit manipulation and try to identify non-Gaussian behavior.
Abstract. The efficiency of the future devices for quantum information processing is limited mostly by the finite decoherence rates of the qubits. Recently a substantial progress was achieved in enhancing the time, which a solid-state qubit demonstrates a coherent dynamics. This progress is based mostly on a successful isolation of the qubits from external decoherence sources. Under these conditions the material-inherent sources of noise start to play a crucial role. In most cases the noise that quantum device demonstrate has 1/f spectrum. This suggests that the environment that destroys the phase coherence of the qubit can be thought of as a system of two-state fluctuators, which experience random hops between their states. In this short review we discuss the current state of the theory of the decoherence due to the qubit interaction with the fluctuators. We describe the effect of such an environment on different protocols of the qubit manipulations- free induction and echo signal. It turns out that in many important cases the noise produced by the fluctuators is non-Gaussian. Consequently the results of the interaction of the qubit with the fluctuators are not determined by the pair correlation function only.
A high degree of quantum coherence is a crucial requirement for the implementation of quantum logic devices. Solid state nanodevices seem particularly promising from the point of view of integrability and flexibility in the design. However decoherence is a serious limitation, due to the presence of many types low energy excitations in the ``internal'' environment and of ``external'' sources due to the control circuitery. Here we study both kind of dephasing in a special implementation, the charge Josephson qubit, however many of our results are applicable to a large class of solid state qubits. This is the case of 1/f noise for which we introduce and study a model of an environment of bistable fluctuatiors. External sources of noise are analized in terms of a suitable harmonic oscillator environment and the explicit mapping on the spin boson model is presented. We perform a detailed investigation of various computation procedures (single shot measurements, repeated measurements) and discuss the problem of the information needed to characterize the effect of the environment. For a fluctuator environment with 1/f spectrum memory effects turn out to be important. Although in general information beyond the power spectrum is needed, in many situations this results in the knowledge of only one more microscopic parameter of the environment. This allows to determine which degrees of freedom of the environment are effective sources of decoherence in each different physical situation considered.
Low-frequency noise has been recognized as the main mechanism of decoherence in present-day solid state coherent nanodevices. Often low-frequency noise has a 1/f spectrum, thus the responsible degrees of freedom are almost static during the coherent time evolution of the device. Their effect is to reduce the amplitude of the measured signal by an effect analogous to in-homogeneous broadening in NMR. Here we present a way to characterize effects of adiabatic noise exploiting the tunability of nanodevices. The results apply in principle to many implementations of quantum bits, and are particularly suited for solid state devices. The accurate characterization of the physics of the noise sources represents the first step toward the design specific strategies to eliminate their effects.
The controlled generation of entangled states of two quantum bits is a fundamental step toward the implementation of a quantum information processor. In nano-devices this operation is counteracted by the solid-state environment, characterized by a broadband and non-monotonic power spectrum, often 1/ f , at low frequencies. For single-qubit gates, incoherent processes due to fluctuations acting on different time scales result in peculiar short- and long-time behavior. Markovian noise gives rise to exponential decay with relaxation and decoherence times, T 1 and T 2 , simply related to the symmetry of the qubit–environment coupling Hamiltonian. Noise with the 1/ f power spectrum at low frequencies is instead responsible for defocusing processes and algebraic short-time behavior. In this paper, we identify the relevant decoherence times of an entangling operation due to the different decoherence channels originating from solid-state noise. Entanglement is quantified by concurrence, which we evaluate in an analytic form employing a multi-stage approach. The 'optimal' operating conditions of reduced sensitivity to noise sources are identified. We apply this analysis to a superconducting gate for experimental noise spectra.
With the growing efforts in isolating solid-state qubits from external decoherence sources, the origins of noise inherent to the material start to play a relevant role. One representative example are charged impurities in the device material or substrate, which typically produce telegraph noise and can hence be modelled as bistable fluctuators. In order to demonstrate the possibility of the active suppression of the disturbance from a {\em single} fluctuator, we theoretically implement an elementary bang-bang control protocol. We numerically simulate the random walk of the qubit state on the Bloch sphere with and without bang-bang compensation by means of the stochastic Schrödinger equation and compare it with an analytical saddle point solution of the corresponding Langevin equation in the long-time limit. We find that the deviation with respect to the noiseless case is significantly reduced when bang-bang pulses are applied, being scaled down approximately by the ratio of the bang-bang period and the typical flipping time of the bistable fluctuation. Our analysis gives not only the effect of bang-bang control on the variance of these deviations, but also their entire distribution. As a result, we expect that bang-bang control works as a high-pass filter on the spectrum of noise sources. This indicates how the influence of 1/f1/f1/f-noise ubiquitous to the solid state world can be reduced.