Noise Hamiltonian

The noise Hamiltonian appearing in the total system Hamiltonian captures the effect of an imperfect, time-varying environment on the controlled evolution of a quantum system. Here the generic structure of the noise Hamiltonian is described.

Noise processes are modelled semi-classically in terms of stochastic time-dependent fluctuating classical noise fields with associated noise axis operators. Without further specificity, we allow the noise Hamiltonian to be expressed as a sum of $p$ noise channels, written

\[\begin{align} H_{n}(t) = \sum_{k=1}^{p}\beta_{k}(t)N_{k}(t). \end{align}\]

The noise axis operator $N_{k}$ is an operator defining the axis of the noise (possibly changing in time) in the Hilbert space of the system, rather than a stochastic variable.

The noise Hamiltonian generates uncontrolled rotations in the qubit dynamics, leading to errors in the evolution path (and hence the final state) relative to the target transformation intended under $H_{c}(t)$. An estimate for this error in terms of the average infidelity may be derived using a first-order expansion of operators, resulting in a generic description in terms of associated filter functions.

In our model the noise fields $\beta_{k}(t)$ are assumed to be classical random variables with zero mean and non-Markvovian power spectra. We also assume they are wide sense stationary (w.s.s.) and independent. The assumption of w.s.s. implies the autocorrelation functions

\[\begin{align} C_{k}(t_2-t_1)\equiv\langle\beta_{k}(t_1)\beta_{k}(t_2)\rangle, \hspace{1cm} k\in \{1,..,p\} \end{align}\]

depend only on the time difference $t_2-t_1$.

The assumption of independence implies the cross-correlation functions vanish. That is,

\[\begin{align} \langle\beta_{j}(t_1)\beta_{k}(t_2)\rangle = 0, \hspace{1cm} j \ne k \in\{1,..,p\}. \end{align}\]

The angle brackets here denote a time average of the stochastic variables. Finally, our model permits access to a wide range of parameter regimes, from quasistatic (noise slow compared to the control Hamiltonian $H_{c}(t)$) to the limit in which the noise fluctuates on timescales comparable to or faster than $H_{c}(t)$.

The assumption of independence is reasonable, for instance, in the case of a driving field where random fluctuations in frequency and amplitude arise from different physical processes. A general model including correlations between noise processes is possible, however, following the approach outlined by Green et al.

Noise power spectral density

Given noise noise fields $\beta_{k}(t)$ described above, the autocorrelation function for each noise channel may be related to the Fourier transform of the associated noise power spectral density, $S_{k}(\omega)$, using the Wiener-Khinchin Theorem. Namely

\[\begin{align} C_{k}(t_2-t_1) = \frac{1}{2\pi}\int_{-\infty}^{\infty}S_{k}(\omega) e^{i\omega(t_2-t_1)}d\omega. \end{align}\]

This is the physical quantity employed to capture the time-dependent behavior of various noise processes impacting qubit evolution. The definition of the noise power spectral density used to characterize all noise processes is consistent with standard spectrum analyzer measurements. The power spectral density (PSD), $S_y(\omega)$, of a real-valued, time-dependent stochastic noise field $y(t)$ is defined as

\[\begin{align} S_y(\omega) = \lim_{T\rightarrow\infty}\frac{1}{T}\left\langle\left|Y_{T}(\omega)\right|^2\right\rangle. \end{align}\]

Here the angle brackets $\langle\cdot\rangle$ denote an ensemble average over realizations, and

\[\begin{align} Y_{T}(\omega) = \int_{-\infty}^{\infty}y_{T}(t)e^{-i\omega t}dt = \int_{-T/2}^{T/2}y(t)e^{-i\omega t}dt \end{align}\]

is the Fourier transform of the time-gated noise field, $y_{T}(t)$, defined by

\[\begin{align} y_{T}(t) = \begin{cases} y(t)&-\frac{T}{2}\le t \le \frac{T}{2}\\ 0&\hspace{0.55cm}|{t}|>\frac{T}{2} \end{cases}. \end{align}\]

This definition avoids the difficulty that the full Fourier transform may not be well defined if the integral over $y(t)$ fails to converge in the limit $T\rightarrow\infty$. All noise processes $y(t)$ treated are assumed to satisfy a number of standard properties (outlined below) which, taken together, establish an important relationship between the PSD $S_y(\omega)$ of $y(t)$ in the frequency domain, and its autocorrelation function, $C_y(\tau)$, in the time domain. Specifically

\[\begin{align}\label{} S_y(\omega)& = \int_{-\infty}^{\infty} C_y(\tau) e^{-i\omega \tau} d\tau,\\ C_y(\tau) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} S_y(\omega) e^{i\omega \tau} d\omega \end{align}\]

implying that $S_y(\omega)$ and $C_y(\tau)$ form a Fourier transform pair. This is a statement of the Wiener-Khintchine theorem.

See The Analysis of Time Series: An Introduction (Chatfield, 2003), Probability and Random Processes: With Applications to Signal Processing and Communications (Miller, 2012) and Generalized harmonic analysis (Wiener, 1930).

Mathematical details and assumptions

In this treatment we use the nonunitary angular frequency notation for the Fourier transform, specifically defining the Fourier transform pair

\[\begin{align} Y(\omega) & = \int_{-\infty}^{\infty}y(t) e^{-i\omega t}dt\\ y(t) & = \frac{1}{2\pi} \int_{-\infty}^{\infty}Y(\omega)e^{i\omega t} d\omega \end{align}\]

where $\omega = 2\pi f$ is the angular frequency in radians per second, and $f$ is the frequency in Hz.

Obtaining the above expressions requires a number of assumptions, outlined bellow following the treatment in The role of master clock stability in quantum information processing. For a set \(\mathcal{E}\) of individual realizations of this process, let $y_k(t)$, $k\in\mathcal{E}$, denote the $k$th realization. Further, let $Y_{k,t} \equiv y_k(t)$ denote the (scalar) random variable taking the value of the $k$th realization at fixed time $t$. We introduce the following notation to distinguish between the ensemble average and the time average:

\[\begin{align} \left\langle{y(t)}\right\rangle&\equiv\lim_{|\mathcal{E}|\rightarrow\infty}\frac{1}{|\mathcal{E}|}\sum_{k\in\mathcal{E}}Y_{k,t} &&\text{(ensemble average)},\\ \overline{y_k(t)}&\equiv \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}y_k(t)dt &&\text{(time average)}, \end{align}\]



is the number of elements in the set \(\mathcal{E}\), and $T$ is the duration of the measurement time spanned by the time window $[-T/2, T/2]$. We explicitly include the subscript $k$ in the expression for the time average to emphasize that this is computed over a single realization of $y(t)$. We assume the following properties of $y(t)$:

  1. $y(t)$ is an ergodic process.
  2. $y(t)$ is a wide-sense stationary process.
  3. $y(t)$ is a zero-mean process: $\left\langle{y(t)}\right\rangle = \overline{y_k(t)} =0$ for all $t$.

The assumption of ergodicity implies that, in the limit of infinite ensemble sizes, the ensemble mean of $y(t)$ - or the ensemble mean of a function of $y(t)$ - approaches the corresponding time-average over any given realization $y_k(t)$. That is,

\[\begin{align} \left\langle{f(y(t))}\right\rangle \longleftrightarrow \overline{f(y_k(t))}. \end{align}\]

The assumption of wide-sense stationarity carries with it two implications. First, the ensemble average is invariant with respect to translation in time, namely $\left\langle{y(t)}\right\rangle = \left\langle{y(t+\tau)}\right\rangle$ for all $\tau$. Second, the autocorrelation function $C_{y}(t_1,t_2) \equiv \langle y(t_1)y(t_2)\rangle$ describing correlations between $y(t)$ at times $t_1$ and $t_2$ depends only on the time difference, and consequently may be parameterized by the single variable $\tau = t_2 - t_1$. Thus,

\[\begin{align} C_y(\tau) &= \left\langle{y(t_{1})y(t_{1}+\tau)}\right\rangle = \overline{y_k(t_{1})y_k(t_{1}+\tau)} \end{align}\]

where the dependence on $t_{1}$ is averaged out, this being an arbitrary reference point, and the equivalence between the ensemble-averaged and time-averaged formulation in the second equality is permitted by ergodicity. We further assume $y(t)$ has finite temporal correlations, appropriate for realistic physical processes, such that the autocorrelation function $C_y(\tau)$ vanishes in the limit $\tau\rightarrow\infty$.

Single-qubit noise sources

For a single qubit we consider two noise processes and write the noise Hamiltonian

\[\begin{align} H_{n}(t) = H_{n,\Omega}(t) + H_{n,z}(t) = \sum_{i=\Omega,z}^{}\beta_{i}(t)N_{i}(t) \end{align}\]

where the contributory terms correspond to ($i=\Omega$) control noise (multiplicative), denoted $H_{n,\Omega}(t)$, describing noise proportional to the control Hamiltonian, and ($i=z$) dephasing noise (ambient), denoted $H_{n,z}(t)$, describing noise proportional to $\sigma_z$.

Single-qubit control noise refers to noise associated with the driving field itself, and produces random rotations coaxial with the component of the control Hamiltonian proportional to the Rabi rate

\[\begin{align} \Omega(t)\rightarrow\Omega(t)(1+\beta_{\Omega}(t)), \end{align}\]

where $\beta_\Omega(t)$ is a fractional time-dependent fluctuation with corresponding noise power spectral density

\[\begin{align} S_{\Omega}(\omega) \iff \beta_{\Omega}(t), \end{align}\]

resulting in the absolute fluctuation $\Omega(t)\beta_{\Omega}(t)$ from the target value $\Omega(t)$. For the generic single-qubit control Hamiltonian

\[\begin{align} H_{c}(t)& = \frac{1}{2}\left( \Omega(t)\cos\phi(t)\sigma_x + \Omega(t)\sin\phi(t)\sigma_y + \Delta(t)\sigma_z\right) \end{align}\]

including non-zero detuning $\Delta(t)$, the noise Hamiltonian corresponding to multiplicative amplitude noise therefore takes the form

\[\begin{align} &H_{n,\Omega}(t) = \beta_{\Omega}(t)N_{\Omega}(t) \end{align}\]

where the noise operator is defined

\begin{align} N_{\Omega}(t) = \frac{1}{2}\Omega(t)\left(\cos{\phi(t)}\sigma_x+\sin{\phi(t)}\sigma_y\right), \end{align}

capturing only the Hamiltonian terms proportional to $\Omega(t)$.

Single-qubit ambient dephasing noise physically corresponds to a stochastic detuning relative to the resonant driving field, e.g. due to Zeeman splitting from ambient magnetic field fluctuations. This captured through an independent additive Hamiltonian term generating uncontrolled rotations about the quantization ($\sigma_z$) axis. In terms of noise operators we write

\[N_{z} = \sigma_z\]

in which case the noise Hamiltonian takes the form

\[H_{n, z}(t) = \beta_{z}(t)N_{z} = \beta_{z}(t)\sigma_z\]

where $\beta_{z}(t)$ captures the time dependent noise process with corresponding noise power spectral density

\[S_{z}(\omega) \iff \beta_{z}(t).\]

Parametric drive noise

Parametrically-driven two-qubit gates are implemented by applying a control flux drive to a tunable-frequency transmon. This modulates the transition frequency of the tunable-frequency qubit and generates an effective two-qubit coupling via the capacitive coupling to the fixed-frequency qubit. Under ideal conditions, a flux drive $\Phi(t)$ with frequency $\omega_{p}$ and phase offset $\theta_{p}$ results in the parametrically-modulated transition frequency

\[\begin{align} \omega_{T}(t) = \bar{\omega}_{T}+\tilde{\omega}_{T}\cos\left(2\omega_{p}t+2\theta_{p}\right) \end{align}\]

where \(\bar{\omega}_{T}\) is the average shift in qubit frequency and \(\tilde{\omega}_{T}\) is the amplitude of the modulation caused by the applied flux drive. To account for sources of error we assume the control parameters of the flux drive, and consequently the parametric drive, are implemented imperfectly. Specifically, we consider

\[\begin{align} &\text{modulation offset error:} && && \bar{\omega}_{T}\rightarrow\bar{\omega}_{T}+\bar{\epsilon}_{T},\\ &\text{modulation amplitude error:} && && \tilde{\omega}_{T}\rightarrow\tilde{\omega}_{T}+\tilde{\epsilon}_{T},\\ &\text{modulation frequency error:} && && \omega_{p}\rightarrow\omega_{p}+\epsilon_{p}. \end{align}\]

where the $\epsilon$ are assumed to be small. These errors generate additional Hamiltonian terms which, performing a Taylor expansion in the small offset parameters and moving to the interaction picture, result in the noise Hamiltonian

\[H_n(t) = \beta(t) N\]

where $\beta(t)$ captures the effective noise strength, the form of which depends on the participating offset errors, and the noise operator, represented in the same operator basis used to express the parametric-drive control Hamiltonian, takes the form

\[\begin{align} N = \mathbb{I}\otimes\left(\Pi_{1}+ 2\Pi_{2}\right) =\Big(|01\rangle \langle 01| + |11\rangle \langle11| + |21\rangle \langle 21|\Big) + 2\Big(|02\rangle \langle 02| + |12\rangle \langle 12| + |22\rangle \langle 22|\Big). \end{align}\]

Here \(\Pi_{i} = |i\rangle\hspace{-0.07cm}\langle i|\) defines the projection operator onto the $i$th eigenstate of the tuneable-frequency transmon, and $\mathbb{I}$ is identity on fixed-frequency transmon.

iSWAP subspace

For iSWAP operations the relevant \((4\times 4)\) subspace is spanned by the eigenstates

\[\begin{align} \left|00\right\rangle, \hspace{0.25cm} \left|10\right\rangle, \hspace{0.25cm} \left|01\right\rangle, \hspace{0.25cm} \left|11\right\rangle. \end{align}\]

In this subspace the noise operator reduces to

\[N = |01\rangle \langle 01| + |11\rangle \langle 11| = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}\]

For performing filter function evaluations the noise operator is assumed to be traceless. This may be ensured by performing the transformation

\[\begin{align} N\rightarrow N - \frac{\text{Tr}\left(N\right))}{\text{dim}\left(N\right)}\mathbb{I} \end{align}\]

where $\mathbb{I}$ is the identity operator of equal dimension to $N$, namely $\text{dim}(N)$. This transformation introduces only a global phase on the system evolution (restricted to the subspace on which $N$ is defined), and is therefore permitted. The noise operator for the parametric drive used to generate filter-function evaluations therefore takes the form

\[\begin{align} N = \begin{pmatrix} -\frac{1}{2} & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & \frac{1}{2} \\ \end{pmatrix} \end{align}\]

where we have incorporated a factor of $\frac{1}{2}$. We observe this is identical to the operator

\[\begin{align} N =\frac{1}{2}\mathbb{I}\otimes \sigma_z \end{align}\]

resembling dephasing on the subspace of the tunable-frequency qubit. For this reason parametric drive noise, treated here, and dephasing noise may be used interchangeably.

Mølmer-Sørensen drive noise

The Mølmer-Sørensen control Hamiltonian takes the form

\[\begin{aligned} H_{c}(t) = \Omega(t)e^{+i\phi(t)} \sum_{\mu=1}^{N} \sum_{k=1}^{M} \eta_{k}^{(\mu)}e^{+i\delta_{k} t} \hat{S}_{\varphi}^{(\mu)} \otimes \hat{a}_{k} +\text{H.C.} \end{aligned}\]

where $\delta_{k}$ is the detuning of the $k$th mode from the laser; $\eta_{k}^{(\mu)}$ is Lamb-Dicke parameter capturing the mode-laser coupling; and $\Omega(t)$ and $\phi(t)$ are the Rabi rate and drive phase for resonantly-driven qubit transitions. To account for sources of error we assume the Mølmer-Sørensen control parameters may be imperfectly imperfectly. Specifically, we consider

\[\begin{align} &\text{amplitude noise:} && && \Omega(t) &&\longrightarrow&& \Omega(t)\left(1+\epsilon_{\Omega}(t)\right)\\ &\text{detuning noise:} && && \delta_{k} &&\longrightarrow&& \delta_{k}+\epsilon_{k}(t) \end{align}\]

where the $\epsilon_{\Omega}$ is the relative error on the Rabi rate, and $\epsilon_{k}$ is the absolute error on the detuning $\delta_{k}$. We assume these errors are small, specifically $\epsilon_{\Omega}\ll 1$ and $\epsilon_{k}/\delta_{k}\ll 1$. The impact of these noise sources on a given control drive may be assessed using filter functions.