Calculates a heuristic for robustness of the controls against noise.
Use this function to calculate a filter function, a heuristic that
quantifies robustness of controls as a function of the noise frequency.
You must express the control Hamiltonian in terms of piecewise-constant
complex controls (drives), piecewise-constant real controls (shifts),
and static terms (drifts); and you must select one of those terms to
represent the noise that this function uses to calculate the robustness.
Warning
You must provide at least one control (drives or shifts).
Exactly one of the drives, shifts, or drifts must represent the
noise. If you select a drive or a shift as the noise, the function
includes it in the control Hamiltonian and also in the noise Hamiltonian.
However, if you select a drift as the noise, the function doesn’t
include it in the control Hamiltonian (unless you add an identical
noiseless drift).
Notes
The filter function provides a measure of noise robustness: the lower the
value of the filter function, the less susceptible the controls are to
noise of that frequency. To calculate the filter function, consider a
system whose total Hamiltonian \(H(t)\) contains a control term
\(H_c(t)\) and a noise term \(H_n(t)\):
\[H(t) = H_c(t) + H_n(t).\]
The control Hamiltonian defines the intended unitary operation \(U_c(t)\)
as the solution of the Schrödinger equation. The noise Hamiltonian
describes the noise against which you want to measure robustness.
To define the control Hamiltonian \(H(t)\), and thus the intended unitary
operation \(U_c(t)\), provide a control Hamiltonian in the following
form:
\[H_c(t) = \sum_j^{N_\textrm{drives}} \left( \gamma_j (t) C_{j} +
\text{H.c.} \right) + \sum_k^{N_\textrm{shifts}} \alpha_k (t) A_{k} +
\sum_l^{N_\textrm{drifts}} D_l.\]
To define the noise Hamiltonian \(H_n(t)\), select one of the drives,
shifts, or drifts as the noise operator \(N(t)\). The noise operator is
identical to the term of the control Hamiltonian that you selected, and
it contributes to a noise Hamiltonian of the form:
\[H_n (t) = \beta(t) N(t),\]
where \(\beta(t)\) is a stochastic variable that represents the amplitude
of the noise. The filter function measures the robustness of the controls
as a function of the oscillation frequency of this stochastic variable.
You can express the information about the frequency distribution of the
stochastic variable in the form of its noise power spectrum
\(S(f)\):
\[C(t_1, t_2) \equiv \langle \beta(t_1) \beta(t_2) \rangle_\mathrm{t} =
\int_{-\infty}^\infty S(f)e^{i 2\pi f(t_2-t_1)} \mathrm{d}f.\]
Here, \(\langle\bullet\rangle_\mathrm{t}\) denotes a time average and
\(C(t_1, t_2)\) is the autocorrelation function.
Add noise to one of the drives or shifts to create a multiplicative
noise, which multiplies the controls. Selecting one of the drives or
shifts as a noise doesn’t affect its presence in the control
Hamiltonian, it only means it is also present in the noise Hamiltonian.
On the other hand, you can add noise to one of the drifts to generate
additive noise. Selecting one of the drifts as a noise means it is
not present in the control Hamiltonian, unless you add an identical
noiseless drift.
Given the intended operation \(U_c(t)\) and the noise operator \(N(t)\),
this algorithm broadly follows the procedure from Reference , where
you can find a full derivation of the formula for a filter function.
First, the algorithm calculates the toggling-frame noise operator with:
\[\tilde N(t) = U_c^\dagger(t) N(t) U_c(t).\]
The algorithm removes the trace of the noise operator in the subspace of
interest by applying the transformation:
\[\tilde N^\prime(t) = \tilde N(t) - \frac{\mathrm{Tr}\left( P \tilde N(t)
P \right)}{\mathrm{Tr}(P)} \mathbb{I},\]
where \({P}\) is the orthogonal projector into the subspace and
\(\mathbb{I}\) is the identity in the Hilbert space of the noise
operator. It then goes to the frequency domain through a Fourier
transform:
\[\mathcal{F} \left\{ \tilde N^\prime(t) \right\}(f) = \int_0^\tau
e^{-i 2\pi f t} \tilde N^\prime(t) \mathrm{d}t.\]
The algorithm can calculate this integral exactly, or efficiently approximate
it as the sum of \(M\) terms:
\[\mathcal{F} \left\{ \tilde N^\prime(t) \right\}(f) \approx \sum_{n=0}^{M-1}
\frac{\tau}{M} e^{-i 2\pi f n \tau/M}
\langle \tilde N^\prime (n\tau/M) \rangle,\]
where \(\langle \tilde N^\prime (t) \rangle\) represents an estimate of
the noise operator at \(t\).
Finally, the algorithm calculates the filter function \(F(f)\) using
the following formula:
\[F(f) = \frac{1}{\mathrm{Tr}(P)} \mathrm{Tr} \left( P
\mathcal{F} \left\{ \tilde N^\prime (t) \right\} \left[ \mathcal{F}
\left\{ \tilde N^\prime (t) \right\} \right]^\dagger P \right).\]
This formula for the filter function comes from the first-order term of
a Magnus expansion of the infidelity. It allows you to estimate the
average infidelity \(\mathcal{I}\) of the effective operation when both
control and noise Hamiltonians are present (\(U_\mathrm{tot}(t)\)) in
comparison to the intended operation (\(U_c(t)\)), where only the
control Hamiltonian is present:
\[\mathcal{I} = 1 - \left\langle \left| \frac{\mathrm{Tr}\left( P
U_c^\dagger (\tau) U_\mathrm{tot} (\tau) P \right)}{\mathrm{Tr}(P)}
\right|^2 \right\rangle.\]
In the weak noise regime, you can estimate the infidelity \(\mathcal{I}\)
as the overlap integral of the filter function \(F(f)\) and the
noise power spectrum \(S(f)\) :
\[\mathcal{I} \approx \int_{-\infty}^\infty S(f) F(f) \mathrm{d}f.\]
References
Examples
See the How to calculate and use filter functions for arbitrary controls
user guide.