$\mathbsf{1}/\mathbsfit{f}$ noise: Implications for solid-state quantum information

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

E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler

Rev. Mod. Phys. 86, 361 – Published 3 April 2014

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.

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Received 21 December 2012

DOI:https://doi.org/10.1103/RevModPhys.86.361

Authors & Affiliations

E. Paladino^*

Dipartimento di Fisica e Astronomia, Università di Catania, Via Santa Sofia 64, I-95123, Catania, Italy and
CNR-IMM-UOS Catania (Universitá), Via Santa Sofia 64, I-95123, Catania, Italy

Y. M. Galperin^†

Department of Physics, University of Oslo, PO Box 1048 Blindern, 0316 Oslo, Norway, Centre for Advanced Study, Drammensveien 78, 0271 Oslo, Norway, and
Ioffe Physical Technical Institute, 26 Polytekhnicheskaya, St. Petersburg 194021, Russian Federation

G. Falci^‡

Dipartimento di Fisica e Astronomia, Università di Catania, Via Santa Sofia 64, I-95123, Catania, Italy and
CNR-IMM-UOS Catania (Universitá), Via Santa Sofia 64, I-95123, Catania, Italy

B. L. Altshuler^§

Physics Department, Columbia University, New York, New York 10027, USA

^*epaladino@dmfci.unict.it
^†iouri.galperine@fys.uio.no
^‡gfalci@dmfci.unict.it
^§bla@phys.columbia.edu

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Vol. 86, Iss. 2 — April - June 2014

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Figure 1
Qualitative representation of the noise spectrum $S (ω)$ [defined in Eq. (10)] on log-log scale. Lines correspond to the behavior expected in different frequency domains, and dots represent typical experimental observations in selected frequency ranges. Long-dashed lines indicate the $1 / f$ dependence extending between the intrinsic low- and high-frequency cutoffs $γ_{m}$ and $γ_{M}$ , which in general are not known. The thick lines indicate the $1 / f^{α}$ dependence, with $α \sim 0.9$ . $1 / f$ noise can be measured in different frequency windows using different detection techniques (different symbols and dots). For instance, $1 / f$ -type flux noise has been detected by a free-induction Ramsey interference experiment in the 0.01–100 Hz range ([324]) and by noise spectroscopy by means of dynamical decoupling in the 0.2–20 MHz range ([45]). Fluctuations of flux, critical current, and charge correspond usually to different $1 / f$ noise amplitudes and extend in different frequency ranges: top and bottom curves illustrate typical $1 / f$ measurements of different variables in the same setup. At frequencies corresponding to the qubit splittings, $Ω \sim 1 0^{10} Hz$ for superconducting qubits, the measured quantum noise is white or Ohmic (dotted and thin lines). The dash-dotted line indicates the linearly increasing Johnson-Nyquist noise.
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Figure 2
Bloch vector representing the state of a qubit in the rotating (with angular frequency $B_{0}$ ) frame of reference. In the laboratory frame of reference it precesses around the $z$ axis with (time-dependent) angular velocity $B$ . The fluctuation of this velocity is $b (t) .$
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Figure 3
Schematic diagram of the potential energy of a two-level tunneling system vs a configuration coordinate. $U$ is the distance between the single-well energy levels, characterizing asymmetry of the potential. $E$ is the difference between the lowest energy levels in a two-well potential with the account of tunneling below the barrier.
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Figure 4
Left panel: The energy diagram for the ground state of a superconducting grain with respect to charge for the case $Δ_{0} > E_{C}$ . For a certain bias voltage when $q = (2 n + 1) e$ , ground states differing by only one single Cooper pair become degenerate. Right panel: Energy of a Cooper-pair box with account of the Josephson tunneling.
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Figure 5
Schematic diagram of a single-Cooper-pair box. An island of superconducting material is connected to a larger superconducting lead via a weak link. This allows coherent tunneling of Cooper pairs between them. For a nanoscale system, such quantum fluctuations of the charge on the island are generally suppressed due to the strong charging energy associated with a small grain capacitance. However, by appropriate biasing of the gate electrode it is possible to make the two states $| 2 n ⟩$ and $| 2 (n + 1) ⟩$ , differing by one Cooper pair, have the same energy (degeneracy of the ground state). This allows the creation of a hybrid state $| Ψ ⟩ = α_{1} | 2 n ⟩ + α_{2} | 2 (n + 1) ⟩$ .
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Figure 6
Single-Cooper-pair box with a probe junction—micrograph of the sample. The electrodes were fabricated by electron-beam lithography and shadow evaporation of Al on a ${SiN}_{x}$ insulating layer (400 nm thick) above a gold ground plane (100 nm thick) on the oxidized Si substrate. The “box” electrode is a $700 \times 50 \times 15 nm$ Al strip containing $\sim 1 0^{8}$ conduction electrons. Adapted from [218].
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Figure 7
Decay of the normalized amplitude of the echo signal (filled circles) and the free induction decay signal (open circles) compared with estimated decoherence factors $⟨ e^{i φ} ⟩$ due to charge noise with the spectrum $α / ω$ . Here $\sqrt{α} \times 1 0^{3} e^{- 1}$ is 1.3 for line 1 and 0.3 for line 2. Adapted from [220].
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Figure 8
Energy relaxation rate $Γ_{1}$ (closed circles and open squares) and phase decoherence rate $Γ_{2} = T_{2}^{- 1}$ (open circles) vs gate induced charge $q$ . Adapted from [7].
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Figure 9
Temperature dependence of the amplitudes $α^{1 / 2}$ , which can be approximated as $α^{1 / 2} = (1.0 \times 1 0^{- 2} e / K) T$ . Adapted from [8].
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Figure 10
Three possible models for the fluctuating charges: model I, electrons jumping between a localized state and a normal metal, as discussed by [232], [120], and [1]; model II, electrons jumping between localized states; and model III, electrons jumping between localized states and a superconductor, as discussed by [91].
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Figure 11
Schematic layout and equivalent lumped circuit representation of proposed implementation of cavity QED using superconducting circuits. The 1D transmission line resonator consists of a full-wave section of a superconducting coplanar waveguide, which may be lithographically fabricated using conventional optical lithography. A Cooper-pair box qubit is placed between the superconducting lines and is capacitively coupled to the center trace at a maximum of the voltage standing wave, yielding a strong electric dipole interaction between the qubit and a single photon in the cavity. The box consists of two small ( $\approx 100 \times 100 {nm}^{2}$ ) Josephson junctions, configured in a $\approx 1 μ m$ loop to permit tuning of the effective Josephson energy by an external flux $Φ_{ext}$ . Input and output signals are coupled to the resonator, via the capacitive gaps in the center line, from $50 Ω$ transmission lines, which allow measurements of the amplitude and phase of the cavity transmission, and the introduction of dc and rf pulses to manipulate the qubit states. From [26].
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Figure 12
A superconducting loop with three Josephson junctions (crosses) encloses a flux $Φ$ that is supplied by an external magnet. Two junctions have a Josephson coupling energy $E_{J}$ , and the third junction has $α E_{J}$ , where $α = 0.75$ . This system has two (meta)stable states $| 0 ⟩$ and $| 1 ⟩$ with opposite circulating persistent current. The level splitting is determined by the offset of the flux from $Φ_{0} / 2$ . The barrier between the states depends on the value of $α$ . The qubit is operated by resonant microwave modulation of the enclosed magnetic flux by a superconducting control line $I_{c}$ . From [215].
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Figure 13
Plot $u (δ)$ for $α = 0.75$ . The inset shows the potential profile along the line $A$ for different $β \equiv (2 Φ - Φ_{0}) / Φ_{0}$ . The potential is symmetric for $β = 0$ .
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Figure 14
Cubic potential $U$ showing qubit states and a measurement scheme. Adapted from [204].
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Figure 15
(a) Schematic density of states. (b) MIGS at a perfect interface with energy in the band gap are extended in the metal and evanescent in the insulator. Adapted from [55].
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Figure 16
Flux modulation from vortices hopping into and out of a loop, and critical-current modulation from electrons temporarily trapped at defect sites in the junction barrier. Adapted from [300].
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Figure 17
Idealized circuit diagram of the quantronium, a quantum-coherent circuit with its tuning, preparation, and readout blocks. The circuit consists of a CPB island (black node) delimited by two small Josephson junctions (crossed boxes) in a superconducting loop. The loop also includes a third, much larger Josephson junction shunted by a capacitance $C$ . The Josephson energies of the box and the large junction are $E_{J}$ and $E_{J 0}$ . The Cooper pair number $N$ and the phases $δ$ and $γ$ are the degrees of freedom of the circuit. A dc voltage $U$ applied to the gate capacitance $C_{g}$ and a dc current $I_{Φ}$ applied to a coil producing a flux $Φ$ in the circuit loop tune the quantum energy levels. Microwave pulses $u (t)$ applied to the gate prepare arbitrary quantum states of the circuit. The states are read out by applying a current pulse $I_{b} (t)$ to the large junction and by monitoring the voltage $V (t)$ across it. From [305].
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Figure 18
Scanning electron microscopy (SEM) image of a double quantum dot device. A quantum point contact with conductance $g_{S}$ senses a charge on the double dot. From [243].
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Figure 19
Electric circuit model of a DQD containing $m$ and $n$ electrons in the left and right dots, respectively. The two QDs are coupled to each other via a tunnel junction, and each dot is connected to an electron reservoir via a tunnel junction. The electrostatic potential of the left (right) quantum dot is controlled by the gate voltage $V_{l}$ $(V_{r})$ through the capacitance $C_{l}$ $(C_{r})$ . Adapted from [103].
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Figure 20
Fragment from the stability diagram. Adapted from [103].
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Figure 21
Scanning electron microphotograph of a device similar to the one measured. Adapted from [241].
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Figure 22
Time dependence if the FID amplitude for $2 b / γ = 0.8$ and 5.
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Figure 23
Reduced $Γ (t)$ due to fluctuators prepared in a stable state ( $δ p_{0} = 1$ , solid lines and $δ p_{0} = - 1$ , dotted lines) and in a thermal mixture ( $δ p_{0} = 0$ , dashed lines) for the indicated values of $g = 2 b / γ$ . Inset: Longer time behavior for stable state preparation. The curves are normalized in such a way that the oscillator approximation for all of them coincides (thick dashed line). Adapted from [231].
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Figure 24
Comparison of the dephasing rate $T_{2}^{- 1}$ for a single random telegraph process and the corresponding Gaussian approximation. Adapted from [17].
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Figure 25
The distribution (53) for $2 b / γ = 1$ and $γ t / 2 = 1$ , $3$ , and $5$ (numbers at the curves). Only the part with positive $φ$ is shown; the function is symmetric. The arrows represent the delta functions (not to scale). Adapted from [17].
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Figure 26
Solid lines: The echo signal for different values of the ratio $2 b / γ$ , Eq. (54). Dashed lines: The calculations along the Gaussian approximation, Eq. (55). Adapted from [17].
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Figure 27
$M_{z} (\infty)$ vs the ratio $v / a$ for different switching rate $γ$ (shown in units of $a$ ). Adapted from [301].
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Figure 28
The saturation effect of slow fluctuators for a $1 / f$ spectrum and coupling distributed with $⟨ Δ b ⟩ / ⟨ b ⟩ = 0.2$ . Labels indicate the number of decades included. Adapted from [232].
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Figure 29
Ratio $K_{m} (t) / ⟨ φ^{2} ⟩ \equiv Γ (t) / Γ_{2} (t)$ for $1 / f$ spectrum between $γ_{m} / (2 π) = 2 \times 1 0^{7} Hz$ and $γ_{M} / (2 π) = 2 \times 1 0^{9} Hz$ with different numbers of fluctuators per decade: (a) $n_{d} = 1 0^{3}$ , (b) $n_{d} = 4 \times 1 0^{3}$ , (c) $n_{d} = 8 \times 1 0^{3}$ , (d) and (d1) $n_{d} = 4 \times 1 0^{4}$ , and (e) and (e1) $n_{d} = 4 \times 1 0^{5}$ . Solid lines correspond to $δ p_{0 j} = \pm 1$ , and dashed lines correspond to equilibrium initial conditions (FID). Adapted from [232].
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Figure 30
Different averages over $δ p_{0 j}$ for $1 / f$ spectrum reproduce the effect of repeated measurements. They are obtained by neglecting (dotted lines) or accounting for (solid lines) the strongly correlated dynamics of $1 / f$ noise. The noise level of [220] is used, by setting $| \bar{b} | / (2 π) = 9.2 \times 1 0^{6} Hz$ , $n_{d} = 1 0^{5}$ , $γ_{m} / (2 π) = 1 Hz$ , and $γ_{M} / (2 π) = 1 0^{9} Hz$ . Dashed lines are the oscillator approximation with a lower cutoff at $ω = \min {| \bar{b} |, 1 / t_{m}}$ . Adapted from [231].
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Figure 31
Sketch of localized charges near an electrode in a Cooper pair box. Induced image charges create local dipoles that interact with the qubit.
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Figure 32
(a) Low-frequency noise as a sum of the RT signals. The switching rates for the “up” and “down” states are comparable $γ_{+} \approx γ_{-}$ and $1 / f$ noise results from a superposition of RT signals distributed as $\propto 1 / γ$ . (b) The intermittent limit corresponds to the limit where the noise stays in the down states most of the time ( $γ_{+} ≫ γ_{-}$ ). The intermittent noise can be approximated by a spike field with independent random waiting times and spike eighths. It gives a nonstationary $1 / f^{μ}$ spectrum depending on the plateaus distribution function. Adapted from [268].
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Figure 33
Left panel: Qubit Bloch sphere. An isolated qubit defines the mixing angle $θ = \arctan {Δ / ε}$ , $H_{\pm}$ define $θ_{\pm} = \arctan {Δ / (ε \pm b)}$ . Right panel: Qubit energy bands $\pm \sqrt{E^{2} + Δ^{2}}$ : the energy splittings depend on the impurity state $Ω_{\pm} = \sqrt{(ε \pm b)^{2} + Δ^{2}}$ .
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Figure 34
The threshold value $γ_{c} / v \equiv (Ω_{+} - Ω_{-}) / (2 b c^{2})$ for a fluctuator behaving as weakly coupled depends on the operating point ( $v / Ω \equiv 2 b / Ω$ ). Adapted from [231].
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Figure 35
Relaxation $1 / T_{1}$ and dephasing $1 / T_{2}$ rates (rescaled for visibility) as functions of $θ$ . Triangles: $1 / T_{2}^{*} = \cos^{2} θ S (0) / 2$ for a (a) weakly and (b) strongly coupled fluctuator. $γ / Ω$ : (a) 0.5, (b) 0.1. $b / Ω$ : (a) 0.1, (b) 0.3. The solid (dashed) lines correspond to a symmetric (slightly asymmetric) fluctuator. Adapted from [50].
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Figure 36
The Fourier transform of $⟨ σ_{z} (t) ⟩$ for a set of weakly coupled fluctuators plus a single strongly coupled fluctuator (thick line). The separate effect of the coupled fluctuator ( $g = 8.3$ , thin line) and the set of weakly coupled fluctuators (dotted line) is shown for comparison. Inset: corresponding power spectra. In all cases the noise level at $Δ = E_{J}$ is fixed to the value $S (E_{J}) / E_{J} = 3.18 \times 1 0^{- 4}$ [from typical $1 / f$ noise amplitude in charge qubits ([339]; [67]; [220]) extrapolated at GHz frequencies]. Adapted from [232].
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Figure 37
The Fourier transform $⟨ σ_{z} ⟩_{ω}$ for a set of weakly coupled fluctuators plus a strongly coupled fluctuator ( $2 b / γ = 61.25$ ) prepared in the ground (dotted line) or in the excited state (thick line). Inset: corresponding power spectra (the thin line corresponds to the extra fluctuator alone). Adapted from [232].
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Figure 38
Simulations of an adiabatic fluctuators $1 / f$ environment at $θ = π / 2$ . Relaxation studied via $⟨ σ_{x} ⟩$ is well approximated by the weak coupling theory $T_{2}$ (dots). Dephasing in repeated measurement damps the oscillations (thin black line). Part of the signal is recovered if the environment is recalibrated (thin gray line). Noise is produced by $n_{d} = 250$ fluctuators per decade, with $1 / t_{m} = 1 0^{5} rad / s \leq γ_{i} \leq γ_{M} = 1 0^{9} rad / s < Ω = 1 0^{10} rad / s$ . The coupling $\bar{v} = 0.02 Ω$ is appropriate to charge devices and corresponds to $S = 16 π A E_{C}^{2} / ω$ with $A = 1 0^{- 6}$ ([339]). The adiabatic approximation, Eq. (91), fully accounts for dephasing (dot-dashed line). The static-path approximation (SPA), Eq. (95) (solid line) and the first correction (dashed line) account for the initial suppression, and it is valid also for times $t ≫ 1 / γ_{M}$ . Inset: Ramsey fringes with parameters appropriate to the experiment ([305]) (thin black lines). The SPA (solid line), Eq. (95), is in excellent agreement with observations and also predicts the correct phase shift of the Ramsey signal (dots, compared with simulations for small detuning $δ = 5 MHz$ , line), which tends to $\approx π / 4$ for large times. Adapted from [89].
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Figure 39
Ramsey signals at the optimal point for $ω_{R 0} / 2 π = 106 MHz$ and $Δ ν = 50 MHz$ , as a function of the delay $Δ t$ between the two $π / 2$ pulses. Top and middle panels: Solid lines are two successive records showing the partial reproducibility of the experiment. Dashed lines are a fit of the envelope of the oscillations in the middle panel leading to $T_{2} = 300 ns$ . The dotted line shows for comparison an exponential decay with the same $T_{2}$ . Bottom panels: Zoom windows of the middle panel. The dots represent the experimental points, whereas the solid line is a fit of the whole oscillation with $Δ ω / 2 π = 50.8 MHz$ . The arrows point out a few sudden jumps of the phase and amplitude of the oscillation, attributed to strongly coupled charged TLFs. Adapted from [143].
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Figure 40
(a) $⟨ σ_{y} ⟩$ at $θ = π / 2$ , $Ω / (2 π) = 1 0^{10} Hz$ . The effect of weak adiabatic $1 / f$ noise (light gray line) ( $γ \in 2 π \times [1 0^{5}, 1 0^{9}] Hz$ , uniform $2 b = 0.002 Ω$ , $n_{d} = 250$ ) is strongly enhanced by adding a single slow ( $γ / Ω = 0.005$ ) more strongly coupled ( $2 b_{0} / Ω = 0.2$ ) fluctuator (black line), which alone would give rise to beats (dark gray line). (b) When the bistable fluctuator is present the Fourier transform of the signal may show a split-peak structure. Even if peaks are symmetric for the single bistable fluctuator alone (dashed line), $1 / f$ noise broadens them in a different way (solid line). Adapted from [89].
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Figure 41
(a) Two qubits $Q_{1}$ and $Q_{2}$ coupled to a fluctuator in the substrate between the two qubits, where the charge can tunnel between two sites. (b) The qubits are coupled to a charged impurity through its image charge on the metallic gate; the charge can tunnel between the gate and the impurity. The coupling strengths are given by $ξ_{1}$ and $ξ_{2}$ . The configuration demonstrated in (a) gives origin to anticorrelated noise, while (b) gives origin to correlated noise. Adapted from [37].
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Figure 42
Dispersion in the SWAP subspace $δ ω_{21}$ (thin continuous and dashed lines) and the orthogonal subspace $δ ω_{30}$ (thick gray) as a function of transverse fluctuations on qubit 1 $x_{1} \equiv δ q_{1}$ ( $x_{2} \equiv δ q_{2} = 0$ ) for resonant qubits with $ν_{z z} / Ω = 0.01$ . The exact splitting (thin continuous) is compared with a second-order expansion (dotted) and the single qubit dispersion (circles). Adapted from [233].
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Figure 43
Timing of the Carr-Purcell-Meilboom-Gill (CPMG), CP, and Uhrig DD (UDD) pulse sequences for $N = 10$ . Adapted from [45].
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Figure 44
Scaled decay rate of the qubit coherence $Γ_{N} (t) / g^{2}$ at fixed $t = 10 γ^{- 1}$ and DD (here $N$ enumerates an echo pair of pulses). $N = 0$ corresponds to FID. A Gaussian environment with the same power spectrum would give, for arbitrary $g$ , the curve labeled with $g = 0.1$ , since $Γ_{N} (t) \propto g^{2}$ . Inset: $Γ_{N} (t)$ for $g = 1.1$ for different intervals between pulses $Δ t$ (lines with dots, $Δ t = 5$ , 2, 0.2) are compared with the FID $Γ_{0} (t)$ (thick line) and with results obtained by numerical solution of the stochastic Schrödinger equation. Adapted from [88].
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Figure 45
(a)–(c) Modified filter function $F_{N} (ω \bar{t}) / ω^{2}$ , for the indicated pulse sequences and $N$ values. Dominant spectral contributions to the measured $Γ_{N} (t)$ appear as peaks in the modified filter functions. (a) Modified filter function for FID showing a large weight for low-frequency noise on a semilog scale, with arbitrary units. (b) Demonstration of even-odd parity through the modified filter function of PDD on a semilog scale. Adapted from [20].
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Reviews of Modern Physics

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

E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler

Rev. Mod. Phys. 86, 361 – Published 3 April 2014

Abstract

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