Broadband spectroscopy of quantum noise

Since a key ingredient of our scheme is breaking the dephasing preserving symmetry, it is worth understanding some of the constraints that arise due to such strong symmetry:

(i) The qubit is insensitive to the $q$ component of the noise.- In the dephasing-preserving scenario one finds that $Y^{-}(t)=y(t)\in\{-1,1\}$ and $Y^{+}(t)=0,$ so that $I^{-}(T)=0$ and

$\displaystyle I^{+}(T)$

$\displaystyle=\frac{1}{2}\int_{0}^{T}dt\int_{0}^{T}dt^{\prime}y(t)y(t^{\prime})C^{+}(t,t^{\prime}),$

i.e., only the classical component of the bath contributes to the dynamics. This can be overcome in at least two ways: using multiple qubits or using non dephasing-preserving control. The former implies a considerable increase in physical resources and control complexity, e.g., entangled probes or entangling operations, but has the distinct advantage of admitting exact equations [15], while the latter can no longer be studied via exact equations and thus is limited to short-time/weak-coupling [16] or steady state [9] regimes.

This limitation in turn implies the inability to extract the information encoded in the $q$ spectra or in the relationship between the $c$ and $q$ spectra, such as the temperature of the thermal bath in the bosonic bath case.

(ii) The noise frequency which can be sampled is effectively lower bounded.- Even if one is only interested in the $c$ spectrum, the dephasing-preserving scenario leads to sampling limitations. To see this, let us expand the $c$ (stationary) correlation function in Fourier series so that

$$C^{+}(\tau=t-t^{\prime},0)=\sum_{k=0}^{\infty}c_{k}\cos(k\omega_{0}\tau),$$

(4)

for some small frequency resolution $0<\omega_{0}\ll 1$. Extracting the “low frequency” information implies the need to learn $\{c_{k}\}$ for $k\in\{0,1,\cdots,K_{\rm min}\}$. Even more, assume the values of $c_{k}$ for $k>K_{\min}$ and the function value of $C^{+}(\tau,0)$ from $\tau=0$ up to some $T_{\rm{s}}$ are known. The problem is then to generate a system of (linear) equations out of Eq. (4) from which the unknown $\{c_{k}\}_{k\leq K_{\rm min}}$ can be reliably estimated.

Analysing the relative difference between the “coefficients” associated to $c_{k}$ and $c_{k+1},$ i.e., the corresponding cosines, one finds that for small $k\omega_{0}\tau,$

	$\displaystyle\cos(k\omega_{0}\tau)-\cos[(k+1)\omega_{0}\tau]$	$\displaystyle=2\sin[(k+\frac{1}{2})\omega_{0}\tau]\sin\frac{\omega_{0}\tau}{2}$
		$\displaystyle\sim(k+\frac{1}{2})2\omega_{0}\tau\sin\frac{\omega_{0}\tau}{2}.$

This shows that differentiating between two subsequent $c_{k}$s becomes harder the smaller $k$ and $\omega_{0}$ are, unless $\tau$ can be made large, say $\tau>T_{\rm upper}.$

On the other hand, the contribution of $C^{+}(\tau>T_{\rm upper},0)$ to the dynamics is contained in an integral like $\bar{I}^{+}|_{T_{\rm upper}}\equiv\int_{T_{\rm upper}}^{T}dt\int_{0}^{\eta}dt^{\prime}Y^{-}(t)Y^{-}(t^{\prime})C^{+}(t-t^{\prime},0)$, for some small $\eta>0$, mixed with the full history of the probe $I^{+}(T)$. Indeed, measuring the expectation value of $\sigma_{x}$ at a time $T$ given an initial state $|+\rangle_{x}$ (the eigenstate of $\sigma_{x}$ corresponding to $+1$) leads to

	$\displaystyle E[\sigma_{x}(T)]_{|+\rangle_{x}}$	$\displaystyle=e^{-I^{+}(T)}$
		$\displaystyle=e^{-\bar{I}^{+}|_{T_{\rm upper}}}\times e^{-I^{+}(T)|_{\rm rest}},$

i.e., the signal associated to $C^{+}(\tau>T_{\rm upper},0)$ is obscured by the decay from the history of the dynamics $e^{-I^{+}(T)|_{\rm rest}}$ and its contribution is thus harder to discriminate experimentally the larger $T_{\rm upper}$ – the smaller the target frequency range – is.

Additionally, the inability to accurately and broadbandly sample also implies the inability to extract reliable information about physical parameters which lie within certain ranges. For example, characterizing low bath temperatures (high $\beta$ regime) in the bosonic example we alluded to earlier becomes impractical.

Overcoming these limitations will be the main contribution of this work, and we will do so by exploiting the ability to generate effective switching functions with zero values in set domains.

One way to do so, is to use incoherent mechanisms. This was done for the purely classical bath scenario in [17, 18, 19], by interleaving $n$ blocks of preparation, noisy evolution and measurement, with idle periods. Importantly, in this idle periods the qubit effectively does not interact with the bath or rather the effect of the noise on the qubit during those idle periods is deleted in the preparation of the following block. In this way, these protocols eventually allow one to sample the low frequency region of the (classical) noise spectrum.

In the quantum bath scenario, however, the situation is different. While a preparation would indeed delete the effect of the bath on the probe qubit, the opposite is not true. The presence of the qubit influences the bath, and the final expectation values will reflect this.

To overcome this limitation we propose to use instead coherent control and a detailed analysis of the system evolution in the presence of the quantum bath. That is, we exploit $H_{\rm ctrl}^{\neg\pi}(t),$ instead of sequential measurements. WE observe that when control is not dephasing-preserving $Y^{+}(t)$ need not vanish, and $Y^{\pm}(t)$ take values in $\{-1,0,1\}.$ This allows one to build combinations of expectation values of operators which are only sensitive to part of the history, in the sense that they only depend on integrals of the form

$$\mathcal{I}^{\pm}(\vec{T})=\int_{{T_{3}}}^{{T_{4}}}dt\int_{{T_{1}}}^{{T_{2}}}dt^{\prime}y(t)y^{\prime}(t^{\prime})C^{\pm}(t-t^{\prime},0),$$

for useful (but not quite arbitrary) values of $0\leq T_{i}\leq T$ and $y(t),y^{\prime}(t)\in\{-1,1\}.$ In other words, even in the quantum bath scenario, our protocol allows combinations of expectation values which yield quantities that mimic the ’idle’ behaviour in the protocols mentioned earlier. Section III focuses on deriving this result and on showing how these integrals can be isolated by appropriate combinations of observables, control sequences, and initial states. Importantly, we shall show how to do so exactly, i.e., without invoking any approximation, allowing us to characterize the effect of both $q$ and $c$ baths in a quantitatively accurate way, in contrast with recent results [11].

With this ability in hand, we achieve broadband characterization of the noise (both $c$ and $q$ components), i.e., even in the long-time & strong-coupling regime. In particular, we demonstrate how this ability allows us to characterize the very low frequency range of both the $c$ and $q$ spectra of a general stationary Gaussian bath – below the one set by the $\mathbb{T}_{2}$ time of the probe.

While moving away from the dephasing-preserving scenario is desirable from the point of view of allowing the qubit to be sensitive to more features of the bath, a considerable complication is added to the analysis: the ability to write exact analytical expressions for the response of the qubit to control and the noise is lost (or at the very least severely compromised). This leads to the need for perturbative approaches, which in turn restrict the ability to extract information which is encoded in the long-time behaviour of the qubit, e.g., low frequency features of the noise. In terms of going beyond $\pi$-pulses, for example, Ref. [20] proposed a protocol non-trivially employing non-$\pi$ pulses to reconstruct a $c$ spectrum with a simplified structure, which leverages many approximations in the deduction, but does not develop a systematic general solution. To overcome these sort of complications, we derive an exact solution for the reduced qubit dynamics subject to general non-$\pi$ pulses, laying a foundation for resolving both the $c$ and $q$ spectra. While the scaling of the complexity of the solution – which may be of independent interest²²2The expression derived here can be related to the process tensor approach [21] . This is an avenue that we will explore in a latter work. – is exponential in the number of non-$\pi$ pulses being considered and is thus impractical in many scenarios of interest, only a few non-$\pi$ pulses are necessary to achieve the results in this work.

Let us start by denoting $U(t_{a},t_{b})\equiv\mathcal{T}_{+}\exp\big{[}-{\rm i}\int_{t_{a}}^{t_{b}}y(t)\sigma_{z}\otimes B(t)dt\big{]}$. Let the non-$\pi$ pulse control $H_{\rm{ctrl}}^{\neg\pi}(t)=\sum_{j=0}^{N}\theta_{j}\sigma_{y}\delta(t-t_{j})$ implement $\theta_{j}$ angle pulse at $t_{j}$, where $t_{0}=0$ and $t_{N}=T$ are the preparation and measurement times, respectively. They generate unitaries $[\theta_{j}]_{y}\equiv e^{-{\rm i}\frac{\theta_{j}}{2}\sigma_{y}}$ which can be generically written as $[\theta_{j}]_{y}=\sum_{r=0}^{1}c^{(j)}_{r}({\rm i}\sigma_{y})^{r},$ with the constraint $\sum_{r=0}^{1}(c^{(j)}_{r})^{2}=1$. Concretely, the coefficients are related to the rotation angle via $c^{(j)}_{r}\equiv\cos(\frac{\theta_{j}}{2}+r\frac{\pi}{2}),$ for the unitary control we consider here³³3Notice that the decomposition can be made also if the operations/interventions are not unitary, e.g., measurements or projectors, albeit with a different constraints for the $c^{(j)}_{r}$.. Dividing the full-time propagator $U(T)\equiv\mathcal{T}_{+}\exp\big{[}-{\rm i}\int_{0}^{T}H_{I}(t)dt\big{]}$ into time intervals defined by the non-$\pi$ pulse timings, one obtains

	$\displaystyle\quad U(T)=[\theta_{N}]_{y}U(t_{N-1},t_{N})[\theta_{N-1}]_{y}\cdots[\theta_{1}]_{y}U(t_{0},t_{1})[\theta_{0}]_{y}$
	$\displaystyle\!=\!\sum_{\vec{r}}(\!\prod_{j=0}^{N}c_{r_{j}}^{(j)}\!){\rm i}^{|\vec{r}|}\sigma_{y}^{r_{N}}U(t_{N-1},t_{N})\sigma_{y}^{r_{N-1}}\cdots\sigma_{y}^{r_{1}}U(t_{0},t_{1})\sigma_{y}^{r_{0}}$
	$\displaystyle\!=\!\sum_{\vec{r}}(\prod_{j=0}^{N}c_{r_{j}}^{(j)}){\rm i}^{|\vec{r}|_{0}}U_{\vec{r}}(T)\sigma_{y}^{|\vec{r}|_{0}},$		(5)

with $\vec{r}\equiv(r_{0},\cdots,r_{N})$ and $|\vec{r}|_{k}\equiv\sum_{j=k}^{N}r_{j}$. The summation $\sum_{\vec{r}}$ in Eq. (5) ranges over all the combinations of $r_{j}\in\{0,1\}$, and for the last line we define

	$\displaystyle U_{\vec{r}}(T)$	$\displaystyle\equiv\bar{U}(t_{N-1},t_{N})\bar{U}(t_{N-2},t_{N-1})\cdots\bar{U}(t_{0},t_{1}),$
	$\displaystyle\bar{U}(t_{k-1},t_{k})$	$\displaystyle\equiv\mathcal{T}_{+}\exp\Big{[}-{\rm i}\int_{t_{k-1}}^{t_{k}}dt(-1)^{|\vec{r}|_{k}}y(t)\sigma_{z}\otimes B(t)\Big{]}.$

In our protocols, it will be sufficient to work with up to four non-$\pi$ pulses ($N=3$ in the above equations).

Any information we extract from the system comes in the form of system measurements. The expectation value of a system-only operator $\hat{O}$ is given by $E\big{[}\hat{O}(T)\big{]}_{\rho_{S}\otimes\rho_{B}}=\mathrm{Tr}\big{[}U(T)\rho_{S}\otimes\rho_{B}U(T)^{\dagger}\hat{O}\big{]}=\mathrm{Tr}\big{[}V_{\hat{O}}(T)\rho_{S}\hat{O}\big{]}$, where $V_{\hat{O}}(T)\equiv\langle\hat{O}^{-1}U(T)^{\dagger}\hat{O}U(T)\rangle$. Furthermore, the system operator $V_{\hat{O}}(T)$ can be calculated, by considering $\hat{O}\equiv\sigma_{\hat{o}}$ with $\hat{o}\in\{x,y,z\}$ and introducing the shorthand notation $f_{\hat{o}}^{\alpha}\equiv{\rm Tr}(\sigma_{\hat{o}}\sigma_{\alpha}\sigma_{\hat{o}}\sigma_{\alpha})/2\in\{-1,+1\}$ where $\alpha\in\{x,y,z\}$, as

	$\displaystyle\quad V_{\hat{O}}(T)$
	$\displaystyle=\left\langle\hat{O}^{-1}\sum_{{\vec{r}^{\prime}}}(\prod_{j=0}^{N}c_{r^{\prime}_{j}}^{(j)}){\rm i}^{-|{\vec{r}^{\prime}}|_{0}}\sigma_{y}^{|{\vec{r}^{\prime}}|_{0}}U_{{\vec{r}^{\prime}}}(T)^{\dagger}\hat{O}\right.$
	$\displaystyle\quad\cdot\left.\sum_{\vec{r}}(\prod_{j=0}^{N}c_{r_{j}}^{(j)}){\rm i}^{|\vec{r}|_{0}}U_{\vec{r}}(T)\sigma_{y}^{|\vec{r}|_{0}}\right\rangle$		(6)
	$\displaystyle\equiv\sum_{\vec{r},{\vec{r}^{\prime}}}{\rm i}^{|\vec{r}|_{0}-|{\vec{r}^{\prime}}|_{0}}\prod_{j=0}^{N}c_{r^{\prime}_{j}}^{(j)}c_{r_{j}}^{(j)}\big{(}{f_{\hat{o}}^{y}}\big{)}^{|{\vec{r}^{\prime}}|_{0}}\sigma_{y}^{|{\vec{r}^{\prime}}|_{0}}V_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)\sigma_{y}^{|\vec{r}|_{0}},$

from which we see that the quantity of interest $V_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)=\langle{\hat{O}}^{-1}U_{{\vec{r}^{\prime}}}(T)^{\dagger}\hat{O}U_{\vec{r}}(T)\rangle$ contains the information about how the noise interacts which the unitaries associated to each $\vec{r},\vec{r}^{\prime}.$

The crucial observation is that each of the $V_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)$ can be written in terms of an effective dephasing Hamiltonian. Thus, given the Gaussian bath scenario we are considering, one can write an exact expression for each of them using the cumulant expansion technique [22, 23, 15, 16]. In more detail, $V_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)$ can be written in a time-ordered exponential form

	$\displaystyle V_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)=$	$\displaystyle\mathcal{T}_{+}\exp\Big{[}-{\rm i}\int_{-T}^{T}dtH_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(t)\Big{]}$		(7)
	$\displaystyle=$	$\displaystyle\exp\big{[}-{\rm i}\mathcal{C}^{(1)}_{\hat{o},\vec{r},\vec{r}^{\prime}}(T)-\frac{1}{2}\mathcal{C}^{(2)}_{\hat{o},\vec{r},\vec{r}^{\prime}}(T)\big{]}.$

In the first equality, we have used the effective Hamiltonian

	$\displaystyle\quad{H}_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(t)$
	$\displaystyle=\begin{cases}-y_{{\vec{r}^{\prime}}}(T-t)\hat{O}^{-1}\sigma_{z}\hat{O}\otimes B(T-t)&{\rm for}\ 0<t\leq T\vspace{1mm},\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y_{\vec{r}}(T+t)\sigma_{z}\otimes B(T+t)&{\rm for}\ -T\leq t\leq 0,\end{cases}$

with $y_{\vec{r}}(t)$ the piece-wise function

$\displaystyle y_{\vec{r}}(t)=\begin{cases}(-1)^{|\vec{r}|_{1}}y(t)&t_{0}\leq t<t_{1},\\ \quad\vdots\\ (-1)^{|\vec{r}|_{k}}y(t)&t_{k-1}\leq t<t_{k},\\ \quad\vdots\\ (-1)^{|\vec{r}|_{N}}y(t)&t_{N-1}\leq t\leq t_{N},\end{cases}$

(8)

which results from using the configuration vector $\vec{r}$ to modulate the original switching function $y(t)$. In turn, to reach the second equality of (7) we exploit the Gaussian character of the noise so that the cumulant expansion truncates exactly at order two. The two contributing cumulants (see Appendix A),

$\displaystyle\mathcal{C}^{(1)}_{\hat{o},\vec{r},\vec{r}^{\prime}}(T)=2\sigma_{z}\int_{0}^{T}dtY^{-,\hat{o}}_{\vec{r},\vec{r}^{\prime}}(t)\langle B(t)\rangle=0,$

(9)

and

		$\displaystyle\quad{\mathcal{C}^{(2)}_{\hat{o},\vec{r},\vec{r}^{\prime}}}(T)$		(10)
		$\displaystyle=2I_{S}\int_{0}^{T}\!\!dt\!\int_{0}^{T}\!\!dt^{\prime}Y_{\vec{r},\vec{r}^{\prime}}^{-,\hat{o}}(t)Y_{\vec{r},\vec{r}^{\prime}}^{-,\hat{o}}(t^{\prime})C^{+}(t,t^{\prime})$
		$\displaystyle\quad+4I_{S}\int_{0}^{T}\!\!dt\!\int_{0}^{t}\!\!dt^{\prime}Y_{\vec{r},\vec{r}^{\prime}}^{-,\hat{o}}(t)Y_{\vec{r},\vec{r}^{\prime}}^{+,\hat{o}}(t^{\prime})C^{-}(t,t^{\prime}),$
		$\displaystyle\equiv 2I_{S}I^{+}_{\vec{r},\vec{r}^{\prime}}+4I_{S}I^{-}_{\vec{r},\vec{r}^{\prime}},$

can be compactly written in terms of the effective switching functions

$$Y_{\vec{r},\vec{r}^{\prime}}^{\pm,\hat{o}}(t)=\frac{y_{\vec{r}}(t)\pm f_{\hat{o}}^{z}y_{\vec{r}^{\prime}}(t)}{2}.$$

(11)

Since it will be enough for our purpose to use as observables $\sigma_{x}$ or $\sigma_{y},$ we can set $f_{\hat{o}}^{z}=-1$ and thus drop the superscript ${\hat{o}}$ in $Y^{\pm}(t)$, $I^{\pm}(T)$ and $v(T)$ below. Notice that because the non-vanishing cumulant is proportional to $I_{S}$, the qubit’s reduced dynamics is in essence a linear combination of the functions

$$v_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)\equiv e^{-I^{+}_{\vec{r},\vec{r}^{\prime}}-2\text{i}{\rm Im}I^{-}_{\vec{r},\vec{r}^{\prime}}},$$

(12)

with the specific linear coefficients depending on $\hat{O},$ $\rho_{S},$ and the control. Note that $I^{-}$ is purely imaginary, and we write it as above to emphasize the imaginary character of the exponential associated to it. In the same vein, $I^{+}$ is a real quantity. The above implies that $c$ noise generates decoherence, while $q$ noise is in charge of generating a phase in our expectation values. Notice that the periodic character of the complex exponential will complicate the task of inferring information about the $q$ component of the noise. We will call this the multi-value problem and show it can be overcome via a proper analysis in Section IV.4.2.

These equations encode the key ingredient of our result: the use of non-$\pi$ control in principle enables us to access functions $v_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)$ which depend on integrals (i) containing effective switching functions with zeros, i.e., as if the coupling could be effectively and exactly switched off during in a portion of the evolution, and, (ii) in the case of the $q$ contribution, containing two different switching functions, allowing us to exploit their constructive/destructive interference effects for noise characterization.

Before proceeding to show how the $v_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)$ can be isolated, and how useful doing this is, it will be convenient to explore the structure of the integrals involved.

Let us first note that from the definition of $Y^{\pm}_{\vec{r},\vec{r}^{\prime}}$ one has

$$|Y_{\vec{r},\vec{r}^{\prime}}^{\pm}(t)+Y_{\vec{r},\vec{r}^{\prime}}^{\mp}(t)|=|y_{\vec{r}}(t)|=1,$$

and

	$\displaystyle\quad Y_{\vec{r},\vec{r}^{\prime}}^{\pm}(t)\times Y_{\vec{r},\vec{r}^{\prime}}^{\mp}(t)$
	$\displaystyle=[y_{\vec{r}}(t)\pm f^{z}_{\hat{o}}y_{{\vec{r}}^{\prime}}(t)][y_{\vec{r}}(t)\mp f^{z}_{\hat{o}}y_{{\vec{r}}^{\prime}}(t)]/4$
	$\displaystyle=[y_{\vec{r}}(t)^{2}-y_{{\vec{r}}^{\prime}}(t)^{2}\mp f^{z}_{\hat{o}}(y_{\vec{r}}(t)y_{{\vec{r}}^{\prime}}(t)-y_{{\vec{r}}^{\prime}}(t)y_{{\vec{r}}}(t))]/4=0.$

We dub this the incompatibility condition, as it implies that at any time $|Y_{\vec{r},\vec{r}^{\prime}}^{\pm}(t)|=1$ while $|Y_{\vec{r},\vec{r}^{\prime}}^{\mp}(t)|=0.$ That is, it cannot be that both effective switching functions vanish or are non-zero simultaneously.

This suggests that to keep track of the effect of non-$\pi$ pulses it is enough to track the vanishing/non-vanishing character of $Y^{\pm}_{\vec{r},\vec{r}^{\prime}}(t)$ in each interval. This follows from the observation that we can propose the representation

$$Y^{\pm}_{\vec{r},\vec{r}^{\prime}}(t)\rightarrow Y^{\pm}_{(a_{1},...,a_{N})}(t)=\begin{cases}a_{1}y(t),&{t\in[t_{0},t_{1})},\\ \quad\vdots\\ a_{N}y(t),&{t\in[t_{N-1},t_{N})},\end{cases}$$

where $y(t)\in\{-1,1\}$ is a purely $\pi$-pulse generated switching function, which suggests that multiple $(\vec{r},\vec{r}^{\prime})$ configurations can lead to the same $\vec{a}^{(N)}\equiv(a_{1},\cdots,a_{N}).$ This multiplicity follows from the fact that while $\pi$ pulses define $y(t)$, they can also implement a transformation $Y^{\pm}_{(a_{1},...,a_{N})}(t)\rightarrow Y^{\pm}_{(a^{\prime}_{1},...,a^{\prime}_{N})}(t)$ with $|a_{j}|=|a^{\prime}_{j}|,$ i.e., they can change the sign but not the vanishing/non-vanishing character of $Y^{\pm}$ in a given time interval. Concretely, a pair of $\pi$ pulses at $t_{j-1}$ and $t_{j}$ would enforce $a^{\prime}_{j}=-a_{j}$ and $a^{\prime}_{i}=a_{i}\ \forall i\neq j$. Clearly, this can also be interpreted as a change in $y(t)$, but thinking of it as a transformation of $\vec{a}^{(N)}$ is useful for our purposes. Considering $N=3$, for example, one can achieve a partial inversion of the $Y^{\pm}$’s by applying extra $\pi$ pulses at $t=t_{0},t_{1},t_{2},t_{3}$ so that $Y^{\pm}_{(a_{1},0,a_{3})}\rightarrow-Y^{\pm}_{(a_{1},0,a_{3})},$ while $Y^{\mp}_{(0,a_{2},0)}\rightarrow+Y^{\mp}_{(0,a_{2},0)}$. More over a full inversion, implementing $Y^{\pm}_{(a_{1},0,a_{3})}\rightarrow-Y^{\pm}_{(a_{1},0,a_{3})}$ and $Y^{\mp}_{(0,a_{2},0)}\rightarrow-Y^{\mp}_{(0,a_{2},0)}$, can be executed by applying $\pi$ pulses at $t=0,t_{3}$. Similar arguments follow for any $N$.

The usefulness of this “gauge-free” representation becomes apparent when one considers the effect of switching functions on integration domains and, in turn, of the $v_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T).$ One can rewrite a given integral in terms of the $\vec{a}^{(N)}$ by making the association $I_{\vec{r},\vec{r}^{\prime}}^{\pm}(T)\rightarrow I_{(\vec{a},\vec{a}^{\prime})}^{\pm}(T),$ with

$$I_{(\vec{a},\vec{a}^{\prime})}^{\pm}(T)\equiv\int_{0}^{T}\!dt\!\int_{0}^{T^{\prime}}\!dt^{\prime}Y^{-}_{(\vec{a})}(t)Y^{\mp}_{(\vec{a}^{\prime})}(t^{\prime})C^{\pm}(t,t^{\prime}).$$

Note we have dropped the superscript $N$ to avoid cumbersome notation, since in what follows we will specialize to $N=3$.

Furthermore, let us note that $I_{(\vec{a},\vec{a})}^{+}$ is invariant under a full inversion, namely $I_{(\vec{a},\vec{a})}^{+}=I_{(-\vec{a},-\vec{a})}^{+}.$ On the other hand, the time-ordered character of $I^{-}$ implies that

	$\displaystyle I_{((a_{1},0,a_{3}),(0,a^{\prime}_{2},0))}^{-}$	$\displaystyle=I_{((0,0,a_{3}),(0,a^{\prime}_{2},0))}^{-},$
	$\displaystyle I_{((a_{1},a_{2},0),(0,0,a^{\prime}_{3}))}^{-}$	$\displaystyle=0,$
	$\displaystyle I_{((0,a_{2},a_{3}),(-a^{\prime}_{1},0,0))}^{-}$	$\displaystyle=I_{((0,-a_{2},-a_{3}),(a^{\prime}_{1},0,0))}^{-}$
		$\displaystyle=-I_{((0,a_{2},a_{3}),(a^{\prime}_{1},0,0))}^{-},$
	$\displaystyle I_{((0,-a_{2},-a_{3}),(-a^{\prime}_{1},0,0))}^{-}$	$\displaystyle=I_{((0,a_{2},a_{3}),(a^{\prime}_{1},0,0))}^{-}.$

Notice that the above also implies that the integrals of the form $\int_{T_{1}}^{T_{2}}dt\int_{T_{1}}^{t}dt^{\prime}y(t)y(t^{\prime})C^{-}(t-t^{\prime},0)$ do not contribute to the probe dynamics, and are thus in principle not learnable. This does not imply, however, that $C^{-}(0<t-t^{\prime}<\Delta,0)$ does not contribute to the dynamics. In fact, $C^{-}(t-t^{\prime},0)$ is in principle learnable for any $t-t^{\prime}\in[0,T].$

Finally, as with the dynamical integrals, the representation $v_{\hat{o},\vec{r},{\vec{r}^{\prime}}}(T)\rightarrow v_{\hat{o},(\vec{a},\vec{a}^{\prime})}(T),$ where

$\displaystyle v_{\hat{o},(\vec{a},\vec{a}^{\prime})}(T)=e^{-I^{+}_{(\vec{a},\vec{a})}}e^{-2{\rm iIm}I^{-}_{(\vec{a},\vec{a}^{\prime})}}\equiv A^{+}_{(\vec{a})}A^{-}_{(\vec{a},\vec{a}^{\prime})},$

allows us to make explicit the different ways in which $c$ and $q$ noise influence the dynamics. Namely, $c$ noise induces decay via $A^{+}_{(\vec{a})}$, while $q$ noise induces a phase via $A^{-}_{(\vec{a},\vec{a}^{\prime})}.$

For exposition purposes, we will often represent the former as $A^{+}_{(\vec{a})}=A^{+}{\tiny{\begin{pmatrix}\cdot&\cdot&a_{3}a^{\prime}_{3}\\ \cdot&a_{2}a^{\prime}_{2}&a_{2}a^{\prime}_{3}\\ a_{1}a^{\prime}_{1}&a_{1}a^{\prime}_{2}&a_{1}a^{\prime}_{3}\end{pmatrix}}},$ where the $\cdot$ makes explicit the symmetry with respect to the diagonal $(a_{1}a_{1}^{\prime},a_{2}a_{2}^{\prime},a_{3}a_{3}^{\prime})$, and the latter by $A^{-}_{(\vec{a},\vec{a}^{\prime})}=A^{-}{\tiny{\begin{pmatrix}\cdot&\cdot&a_{3}a^{\prime}_{3}\\ \cdot&a_{2}a^{\prime}_{2}&a_{2}a^{\prime}_{3}\\ a_{1}a^{\prime}_{1}&a_{1}a^{\prime}_{2}&a_{1}a^{\prime}_{3}\end{pmatrix}}},$ where now the $\cdot$ makes explicit the $t\geq t^{\prime}$ time ordering in the relevant integral and the matrix form facilitates keeping track of the generated equivalence between different vector pairs of $(\vec{a},\vec{a}^{\prime})$. We will say that $A^{\pm}(\cdot)$ is a function of the integral

	$\displaystyle{\tiny{\begin{pmatrix}\cdot&\cdot&a_{3}a^{\prime}_{3}\\ \cdot&a_{2}a^{\prime}_{2}&a_{2}a^{\prime}_{3}\\ a_{1}a^{\prime}_{1}&a_{1}a^{\prime}_{2}&a_{1}a^{\prime}_{3}\end{pmatrix}}}^{+}\equiv$	$\displaystyle I^{+}_{(\vec{a},\vec{a}^{\prime})},$
	$\displaystyle{\tiny{\begin{pmatrix}\cdot&\cdot&a_{3}a^{\prime}_{3}\\ \cdot&a_{2}a^{\prime}_{2}&a_{2}a^{\prime}_{3}\\ a_{1}a^{\prime}_{1}&a_{1}a^{\prime}_{2}&a_{1}a^{\prime}_{3}\end{pmatrix}}}^{-}\equiv$	$\displaystyle\text{Im}I^{-}_{(\vec{a},\vec{a}^{\prime})}.$

Our task will be to infer the value of $I^{\pm}_{(\vec{a},\vec{a}^{\prime})}$ for all those configurations such that $\sum_{i\leq j}|a_{i}a^{\prime}_{j}|=1$ and $a_{i},a^{\prime}_{j}\in\{-1,0,1\},$ i.e., where only one of the matrix entries is non-vanishing (for $A^{+}$, two entries are non-zero, one of which is hidden among the dots).

To achieve this, one must first determine which among $\{v_{\hat{o},(\vec{a},\vec{a}^{\prime})}(T)\}_{\vec{a},\vec{a}^{\prime}}$ (or linear combinations of them) can be extracted via linear combinations of expectation values. Let us first recall one is interested in expectation values of $\hat{O}\in\{\sigma_{x},\sigma_{y}\}$ after applying a set of $\sigma_{y}$ pulses of angles $\{\theta_{i}\}$ at times $\{t_{i}\}.$ We denote the explicit pulse dependence by $E\big{[}\hat{O}(t_{1},...,t_{N-1},T)\big{]}_{\rho_{S}\otimes\rho_{B}}|_{\theta_{0},...,\theta_{N}}.$ From Eq. (6), one sees that each of these expectation values can be written as a sum of terms – each containing a specific linear combination of $v_{\hat{o},(\vec{a},\vec{a}^{\prime})}(T)$’s – with coefficients of the form $\prod_{j=0}^{N}\cos(\frac{\theta_{j}}{2})^{s_{j}}\sin(\frac{\theta_{j}}{2})^{2-s_{j}}$ for $s_{j}\in\{0,1,2\}.$ This is a linear system which can be solved by cycling over an appropriate set of angles $\theta_{j}$ for any $N$, e.g., for $N=3$ there are 81 $\{\theta_{j}\}_{j=0}^{N}$ configurations resulting from cycling $\theta_{j}\in\{0,\pi/2,\pi\}\,\forall\,j$, which yields the aforementioned linear combinations of interest⁴⁴4Note that this is a raw recipe but later we will be interested in minimizing the number of experiments, i.e., the number of configuration sets $\{\theta_{j}\}^{\prime}s$, needed to extract a given expectation value.. Building a linear system using $E\big{[}\hat{O}(t_{1},...,t_{N-1},T)\big{]}_{\rho_{S}\otimes\rho_{B}}|_{\theta_{0},...,\theta_{N}}$ for $\hat{O}\in\{\sigma_{x},\sigma_{y}\}$ and $\rho_{S}\in\{\frac{I_{S}\pm\sigma_{y}}{2},\frac{I_{S}\pm\sigma_{x}}{2}\},$ one finds that the 81 linear combinations reduce to 18 quantities that can now be accessed. Namely,

	$\displaystyle Q_{1}^{\pm}$	$\displaystyle=(A^{+}_{(1,0,-1)}\pm A^{+}_{(1,0,1)})\!\left(A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&1\\ 0&0&0\end{pmatrix}}+A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&-1\\ 0&0&0\end{pmatrix}}\right)\!,$
	$\displaystyle Q_{2}^{\pm}$	$\displaystyle=A^{+}_{(0,1,-1)}\left(A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 0&1&-1\end{pmatrix}}\pm A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 0&-1&1\end{pmatrix}}\right),$
	$\displaystyle Q_{3}^{\pm}$	$\displaystyle=A^{+}_{(0,1,1)}\left(A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 0&1&1\end{pmatrix}}\pm A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 0&-1&-1\end{pmatrix}}\right),$
	$\displaystyle Q_{4}^{+}$	$\displaystyle=A^{+}_{(0,1,0)}\left(A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 0&1&0\end{pmatrix}}+A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 0&-1&0\end{pmatrix}}\right),$
	$\displaystyle Q_{5}^{\pm}$	$\displaystyle=A^{+}_{(0,0,1)}\left(A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&1\\ 0&0&1\end{pmatrix}}\pm A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&-1\\ 0&0&-1\end{pmatrix}}\right),$
	$\displaystyle Q_{6}^{\pm}$	$\displaystyle=A^{+}_{(0,0,1)}\left(A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&1\\ 0&0&-1\end{pmatrix}}\pm A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&-1\\ 0&0&1\end{pmatrix}}\right),$

involving both $c$ and $q$ spectra, and those in the set $Q|_{\pi}=\{A^{+}_{(s_{1},0,0)},A^{+}_{(s_{1},s_{2},0)},A^{+}_{(s_{1},s_{2},s_{3})}\}$ for $s_{j}=\pm 1,$ involving only the $c$ noise contribution. Also note that

$$A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&0&0\\ 1&0&0\end{pmatrix}},A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&0\\ \cdot&1&0\\ 0&0&0\end{pmatrix}},A^{-}\tiny{\begin{pmatrix}\cdot&\cdot&1\\ \cdot&0&0\\ 0&0&0\end{pmatrix}}$$

are always zero and thus do not need to be learned.