Noise classification in small quantum networks by Machine Learning

Shreyasi Mukherjee^1, Dario Penna^2, Fabio Cirinnà^2, Mauro Paternostro^3,4, Elisabetta Paladino^1,5,6, Giuseppe Falci^1, 5, 6, and Luigi Giannelli^1,5

$^1$ Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, Via S. Sofia 64, 95123 Catania, Italy
$^2$ Leonardo S.p.A., Cyber & Security Solutions, 95121, Catania, Italy
$^3$ Università degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segrè, via Archirafi 36, I-90123 Palermo, Italy
$^4$ Centre for Theoretical Atomic, Molecular, and Optical Physics, School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom
$^5$ INFN, Sezione di Catania, 95123, Catania, Italy
$^6$ CNR-IMM, UoS Università, 95123, Catania, Italy

We investigate machine learning-based  noise classification aimed at the  recognition of the Markovian character of a dynamics and the identification of correlations of classical noise, as well as their interplay in small quantum networks. We operate control based on Coherent Tunneling by Adiabatic Passage (CTAP) or Stimulated Raman Adiabatic Passage (STIRAP) in a three-level system using different pulse configurations as inputs to train a feedforward neural network. Our results show that supervised learning can classify distinct types of classical diagonal noise affecting the system. Three non-Markovian (quasistatic correlated, anti-correlated, and uncorrelated) and Markovian noise mechanisms are classified with $99\%$ accuracy. Instead, correlations of Markovian noises cannot be classified with our method. The approach is robust against statistical measurement errors keeping its effectiveness even for physical measurements where a limited number of samples is available.

§ INTRODUCTION

Quantum systems can nowadays be controlled with impressive accuracy <cit.> to perform new tasks leveraging their coherence properties, such as superposition and entanglement <cit.>. However, quantum behavior fades away because of environmental noise leading to loss of fidelity in quantum operations and decoherence <cit.>. Devising strategies to counteract these effects is thus paramount for advances in quantum technologies and in this context, machine learning (ML) emerges as an innovative and powerful diagnostic tool <cit.>. Approaches based on ML have been applied to various quantum control protocols <cit.>, to quantum state tomography <cit.>, to characterize quantum systems <cit.>, to simulate open quantum systems <cit.>, to investigate non-Markovian dynamics <cit.> and, in a spirit similar to this work, to characterize system-environment interactions <cit.> and perform noise spectroscopy <cit.>.

Mitigating decoherence effects is necessary to achieve quantum advantage. Several strategies have been developed in the last two decades both for quantum computation and in the broader perspective of quantum control. Error-avoidance (or passive stabilization) strategies consist of storing and processing information in suitably designed subspaces of the Hilbert space, which are protected from the interaction with the environment <cit.>. Examples of error-correcting (or active stabilization) strategies are Quantum Error Correction which relies on non-local information encoding <cit.>, and Dynamical Decoupling <cit.> consisting in repeated application of pulsed or switched control which has been used in solid-state coherent nanoscience to counteract $1/f$ noise <cit.>.
Specific strategies rely on the availability of information about the environment, whose characterization is thus a very important step towards the design of optimized systems and protocols. For individual qubits, this program has been carried out successfully in several platforms such as trapped ions <cit.>, photonic qubits <cit.>, nuclear magnetic resonance <cit.>, and nitrogen-vacancy centres in diamond <cit.>. Superconducting systems are the forefront examples in this context, with decoherence times that have improved from a few nanoseconds to milliseconds in about 15 years <cit.>.

On the contrary, characterization of more complex systems such as multilevel nodes and multi-qubit architectures
is still a challenge. In particular, the need of characterizing  correlations between noises affecting different transition <cit.> or different nodes of the quantum architecture <cit.> is an emerging issue. A strong motivation of principle is that spatially correlated noise between physical qubits may spoil one of the pillars of digital quantum computation namely error correction in logical qubits, besides the fact that correlated errors are indeed observed in noisy intermediate-scale quantum structures <cit.>.
Effects of spatially-correlated low-frequency noise between two quantum devices have been observed since several decades <cit.> and their impact on decoherence was investigated <cit.> since the first two-qubit superconducting gates were demonstrated. Recently, low-frequency noise correlations between pairs of qubits have been characterized by direct measurement of the noise power spectra <cit.>.
However, the full characterization of noise acting on a controllable quantum system is a difficult task requiring exceedingly large resources as the systems scale up. In this work, we seek to pursue a different avenue passing through
the classification of the multivariable noise acting on a  quantum architecture. Our primary focus is to find a ML-based procedure that recognizes the presence of noise correlations. We will see that this possibility is intimately related to the (non-)Markovian character of the dynamics, and might thus be
useful in the investigation of the effect that non-Markovianity has in the establishment of many-body phenomena  <cit.>.

For the sake of simplicity, we illustrate our ML method to identify correlations in the simplest case of a three-level network affected by diagonal classical noise. Physically, the model describes a qutrit operated by two AC classical electromagnetic fields or a system of three single-level quantum dots with tunable tunneling rate. In the latter case, we exploit a control protocol yielding coherent tunneling by adiabatic passage (CTAP) <cit.>, which is conventionally meant to be used for population transfer, as a tool for noise classification. We use supervised learning to classify five types of noise:

* Non-Markovian correlated noise;

* Non-Markovian anti-correlated noise;

* Non-Markovian uncorrelated noise;

(4a) Markovian correlated noise;

(4b) Markovian anti-correlated noise.

We show that, by measuring the efficiency of CTAP under three different combinations of pulse amplitudes, we can train a neural network to classify noise belonging to  four of the five classes listed above. In particular, our model can accurately classify noise of the first three classes and Markovian noise (classes 4a and 4b together) while it cannot distinguish between correlated and anti-correlated Markovian noise. Referring to actual experiments, we analyze in detail how to extract this information also in the case of a finite number of samples.

The paper is structured as follows: in Sec. <ref> we define the models of system and noise that we address in our study. In Sec. <ref>
we introduce the CTAP protocol and the figures of merits we use in the training phase. In Sec. <ref>
we give a brief overview of supervised learning and neural networks (NN) <cit.> and describe how we classify among the noises introduced in Sect. <ref>.
Sec. <ref>  presents the results on the classification for a finite number of samples and imperfect measurements. The conclusions are drawn in
Sec. <ref>, together with a physical interpretation of the results.

§ NOISY THREE-LEVEL SYSTEM

To fix the ideas we consider a system of three single-level quantum nodes with eigenbasis $\{\ket{i},\; i=0,1,2 \}$ and on-site energies $\mathrm{\epsilon}_i$. This model may describe a system of three quantum dots with an electron
tunneling between them <cit.>. The tunneling rates between the first and the second dot, $\Op(t)$, and between the second and the third dot, $\Os(t)$, can be controlled by time-dependent external gates. The same model describes a three-level atom driven by two external fields in a multiple rotating frame <cit.>.

The Hamiltonian of the system is $\Hsys = H_0 + H_\mathrm{c}$, where $H_0$ and $H_\mathrm{c}(t)$ are the free energy and the control term, respectively (throughout the manuscript we use units such that $\hbar=1$), reading

\begin{align}
& H_0 = \delp\ketbra{1}{1}+ \delta\ketbra{2}{2},                                                                                \\
& \Hc = \frac{\Op(t)}{2}\left(\ketbra{0}{1} + \ketbra{1}{0}\right) + \frac{\Os(t)}{2}\left(\ketbra{1}{2} + \ketbra{2}{1}\right),
\end{align}

where, in the language of the three-dot network, we defined the detunings $\delp = \epsilon_1-\epsilon_0$ and $\delta = \epsilon_2-\epsilon_0$.

We consider diagonal noise, noise affecting the energy levels of the three dots, which may arise from  charge noise, for instance <cit.>. We model it as the following classical stochastic process added to the diagonal entries of the Hamiltonian

\begin{equation}
\label{eq:Hnoise}
\Hnoise = \tilde x_1(t)\ketbra{1}{1} + \tilde x_2(t)\ketbra{2}{2}.
\end{equation}

Here, $\tilde x_i(t)$ ($i=1,2$) is a random variable <cit.> with normalized probability density $p(x_i, t)$.

The total Hamiltonian $H(t)$ in the basis $\{\ket{0}, \ket{1}, \ket{2}\}$ thus reads

\begin{equation}
\label{eq:totalH}
H(t) = \Hsys(t) + \Hnoise(t) = \begin{pmatrix}
0        & \Op(t)/2              & 0                      \\
\Op(t)/2 & \delp + \tilde x_1(t) & \Os(t)/2               \\
0        & \Os(t)/2              & \delta + \tilde x_2(t)
\end{pmatrix}.
\end{equation}

We now link the five classes of noise listed in Sec. <ref> to the features of the stochastic mechanism introduced here. We thus consider

* Non-Markovian types of noise: When addressing non-Markovian noise, we will make the assumption of quasistatic processes where the noise mechanism has a long correlation time and can thus be considered constant over the evolution of the system. The  random variables $\tilde x_i(t)$ are assumed to be picked from Gaussian distributions with zero mean, thus describing the cumulative effects of independent  microscopic sources. Needless to say, while $\tilde x_i(t)$ remains constant throughout a single realization of the protocol, it varies between different realizations. We identify the following three classes

* Correlated: $x_2 (t)= \eta x_1(t)$ with $\eta>0$;

* Anti-correlated: $x_2(t) = \eta x_1(t)$ with $\eta<0$;

* Uncorrelated: $x_2(t)$ and $x_1(t)$, are independent of each other.

* Markovian types of noise: The associated dynamics will be ruled by zero-mean, delta-correlated

stochastic processes $\tilde x_i$ making the dynamics of the system dependent only on its current state rather than its past history. We will thus set

\begin{equation}
\label{eq:markovian_correlations}
\avg{\tilde x_i(t)}=0,\quad \avg{\tilde x_i(t) \tilde x_i(t')} = \gamma\delta(t-t')
\end{equation}

and consider the two classes

(4a) Correlated: $x_2 (t)= \eta x_1(t)$, with $\eta>0$;

(4b) Anti-correlated: $x_2(t) = \eta x_1(t)$, with $\eta<0$.

§ DYNAMICS

Our method is based on population transfer by CTAP and the closely related stimulated Raman adiabatic passage (STIRAP) scheme, which has found ample use in atomic physics <cit.>.
We shortly review the CTAP/STIRAP approaches and discuss how we solve the equations and then calculate the classification performances in the presence of the different classes of noise considered.

§.§ CTAP/STIRAP

CTAP <cit.> (and its spin equivalent spin-CTAP <cit.>) is a protocol that ideally achieves population transfer from $\ket{0}$ to $\ket{2}$
by adiabatically following a “trapped state" (often referred to as a dark state) that contains no contribution from the intermediate state $\ket{1}$, which is thus never populated during the protocol. Under ideal conditions, CTAP/STIRAP yields $\sim 100\%$-efficiency population transfer
with remarkable robustness against parametric fluctuations and state-selectivity.

For the transfer process to be successful it is essential to work at small
detuning, $\delta \ll \Omega_{p,s}$.  In particular,
at the so-called two-photon resonace condition $\delta=0$, the trapped state is an instantaneous eigenstate of the Hamiltonian $\Hsys(t)$ of the form

\begin{equation}
\label{eq:darkstate}
\ket{\mathrm{\phi}_\mathrm{D}(t)} = \cos\theta(t)\ket{0}-\sin\theta(t)\ket{2}
\end{equation}

with $\theta(t) = \tan^{-1}\big[\Op(t)/\Os(t)\big]$.

The protocol is operated in the time interval $[\ti,\tf]$ by applying pulses $\Omega_{p,s}(t)$ in a counterintuitive manner <cit.>: $\Os$ is applied before $\Op$, while ensuring that the two pulses overlap in a fraction of the duration of the sequence. In this case, the trapped state $\ket{\phi_\mathrm{D}(t)}$ at the initial time $\ti$ coincides with $\ket{0}$ and at the final time $\tf$ with the target state $\ket{2}$. If the evolution is adiabatic and the system is prepared in $\ket{0}=\ket{\phi_\mathrm{D}(\ti)}$, the system evolves following the trapped state $\ket{\phi_\mathrm{D}(t)}$ throughout the evolution as prescribed by the adiabatic theorem <cit.>.
The requested adiabaticity of the process can be cast in the form of the global adiabaticity condition <cit.>

\begin{equation}
\label{eq:globaladiabaticcondition}
\Opsmax \tau \geq 10,
\end{equation}

where $\tau$ is the characteristic time scale of the pulses overlap and $\Opsmax = \max_t\Ops(t)$.

Under two-photon resonance condition, a counterintuitive sequence satisfying the global adiabaticity request is robust against the actual shape of $\Op$ and $\Os$. In fact, several pulse shapes <cit.> have been used in literature. In this work, we consider Gaussian pulses (see Fig. <ref>) of the form

\begin{equation}
\label{eq:gaussianpulses}
%    \begin{align}
\Op(t) = \Opmax e^{-(\frac{t-\tau}{T})^2},\qquad \Os(t) = \Osmax e^{-(\frac{t+\tau}{T})^2},
%   \end{align}
\end{equation}

and we let the system evolve in the time interval $t\in[-5T,5T]$ with $\tau=0.7T$.

(a) Example of the pump $\Op(t)$ and Stokes $\Os(t)$ pulses, Eqs. (<ref>), as a function of time $t$. The parameters used are $\Opmax=\Osmax=50/T$, $\tau = 0.7T$ and $t\in[-5T, 5T]$. (b) Instaneous eigenvalues for $\delta=0$: CTAP/STIRAP achieves 100% population transfer by adiabatically following the trapped state $\ket{\Phi_D(t)}$ (red line). (c) Instaneous eigenvalues for $\delta \neq 0$: The trapped state is not an instantaneous eigestate and an adiabatic evolution produces only partial population transfer.

More details about CTAP/STIRAP can be found in <ref>.

§.§ Figures of merit

The efficiency $\xi$ of the population transfer is defined as the population of the target state $\ket{2}$
at the final time $\tf$, averaged over all the possible realizations of the noise <cit.>

\begin{equation}
\label{eq:efficiency_statavg}
\xi = \lim_{N\to\infty}\frac{1}{N} \sum_{r=1}^N \abs{\braket{2}{\psii(\tf)}}^2 = \lim_{N\to\infty}\frac{1}{N} \sum_{r=1}^N \xii
\end{equation}

with $\xii = \abs{\braket{2}{\psii(\tf)}}^2$  the population of $\ket{2}$ for a single trajectory $\ket{\psii(t)}$ identified by an individual realization of noise, a choice of $\{x_1^{(r)}(t),x_2^{(r)}(t)\}$. The
index $r=1,\dots,N$ identifies the  trajectory being considered.

The quasistatic noise is constant during an individual trajectory
thus the index $r$ can be unambiguously mapped to the two real values $x_1^{(r)}$ and $x_2^{(r)}$ of the random processes $\tilde{x}_1$ and $\tilde{x}_2$. Therefore $\xii = \xi(x_1^{(r)},x_2^{(r)})$. In this case, the efficiency in Eq. (<ref>) is obtained by averaging over the random processes

\begin{equation}
\label{eq:efficiency_uncorrelated}
\xi = \int \xi(x_1,x_2) p(x_1,x_2) \de x_1 \de x_2.
\end{equation}

If the noise is (anti-)correlated, then $x_2^{(r)} = \eta x_1^{(r)}$
and Eq. <ref> simplifies to

\begin{equation}
\label{eq:efficiency_correlated}
\xi = \int \xi(x_1,\eta x_1) p_1(x_1) \de x_1.
\end{equation}

For Markovian noise, the average is calculated via the density matrix
$\rho(t)$ of the system which solves a master equation in the Lindblad form (cf. <ref> for the derivation)

\begin{equation}
\label{eq:lindbaleq}
\dot\rho(t) = -\I\mathrm[H_0 + H_\mathrm{c}(t), \rho(t)] -\frac{\gamma}{2}(O^2\rho(t) + \rho(t)O^2-2O\rho(t)O),
\end{equation}

where $O = \ketbra{1}{1} + \eta\ketbra{2}{2}$, and $\gamma$ is the dephasing rate given by $\avg{\tilde x_1(t)\tilde x_1(t^\prime)} = \gamma\delta(t-t^\prime)$. The efficiency is given by

\begin{equation}
\label{eq:efficiencydm}
\xi =  \bra{2}\rho(\tf) \ket{2}.
\end{equation}

§ CLASSIFICATION WITH NEURAL NETWORKS

§.§ Neural Networks

Supervised learning requires a labeled dataset, $\{(\vect{x}_i, \vect{\hat{y}}_i)\}_{i=1,\dots,N}$, where the input data $\vect{x}_i$ (features, also called input feature vector) are paired with the correct output $\vect{\hat{y}}_i$ (labels or target values) the index $i$ iterating over all the $N$ data points (also called samples).
To perform classification, we employ a feedforward neural network, specifically a multilayer perceptron (MLP), as our machine learning model. An MLP consists of three primary parts: the  input layer, the hidden layer(s), and the output layer. Each layer is composed of neurons, which are the fundamental units of a neural network. In a fully connected neural network, each neuron in a layer receives an input from each neuron of the preceding layer and calculates a real-valued output which is passed on to the next layer.

Schematic representation of a Multi Layer Perceptron (MLP) with $L-1$ hidden layers. $(x_1,x_2,\dots,x_{D^{(0)}})$ are the inputs to the MLP, $\vect{w}^{(l)}$ are the weights connecting layer $l-1$ with layer $l$ and $(y_1,y_2,\dots,y_{D^{(L)}})$ are the outputs. Details on its working are given in the text.

To illustrate the concept, consider a MLP consisting of $L+1$ layers (as depicted in Fig. <ref>), with the input layer identified by the index $l=0$ and the output layer by the index $l=L$. The value of the $D^{(l)}$ neurons in layer $l$ are calculated from the values of the $D^{(l-1)}$ neurons in layer $l-1$, with

\begin{equation}
\label{eq:yl}
\vect{y}^{(l)} = f^{(l)}(\vect{z}^{(l)}),
\end{equation}

where $f^{(l)}$ is a nonlinear funtion called the activation function, and

\begin{equation}
\label{eq:zl}
\vect{z}^{(l)} = \vect{w}^{(l)}\vect{y}^{(l-1)} + \vect{b}^{(l)}.
\end{equation}

Here $\vect{w}^{(l)}\in\mathbb{R}^{D^{(l)} \times D^{(l-1)}}$ is the weight matrix, $\vect{b}^{(l)}\in\mathbb{R}^{D^{(l)}}$ is called the bias vector and $\vect{y}^{(l-1)}\in\mathbb{R}^{D^{(l-1)}}$ are the values of the neurons in layer $l-1$. The values of the input layer $l=0$ are set to be the input feature vector $\vect{y}^{(0)}=\vect{x}$, while the values of the output layer $\vect{y}^{(L)}$ is the output that the model associates to the given input: it is called predicted output and we indicate it simply as $\vect{y}=\vect{y}^{(L)}$. The weight matrices, together with the biases and the activation functions, govern the collective influence of the neurons from one layer to the next.

The training process consists in finding weights $\vect{W}=\{\vect{w}^{(l)}\}_{l=1,2,\dots,L}$ and biases $\vect{B} = \{\vect{b}^{(l)}\}_{l=1,2,\dots,L}$ such that the model predicts the correct output given an input, and is able to generalize to unseen inputs. This is done by minimizing a cost function $C(\{\vect{y}_i, \vect{\hat{y}}_i\}_i | \vect{W},\vect{B})$ which measures the average distance (“error”) between the NN predicted output $\vect{y}_i$ and the target output $\vect{\hat{y}}_i$. Typically, this minimization process is executed using algorithms such as stochastic gradient descent or its variants. The  gradients needed for these optimization algorithms are computed through backpropagation, a highly efficient algorithm that leverages the chain rule to propagate errors backward through the network.

We use a neural network consisting of two hidden layers and the leaky rectified linear unit (LeakyReLU) <cit.> as an activation function for both the hidden layers, for details see Tab. <ref>. LeakyReLU is defined as

\begin{equation}
\label{eq:LeakyRELU}
\mathit{f}_{\text{LReLU}}(z) =
\begin{cases}
z        & \text{if} \quad z\geq 0, \\
\alpha z & \text{if} \quad z<0,     \\
\end{cases}
\end{equation}

where $\alpha$ is a constant that we set to the value $0.01$. This activation function helps to avoid the “dying ReLU” problem <cit.>, that is problem of obtaining always the same output (often the null one), regardless of the input.

Layer     $\#$ neurons     Activation function
input                     $3$
hidden 1                  $128$                            LeakyReLU
hidden 2                  $100$                            LeakyReLU
output                    $4$ or $5$                       Softmax

Structure of the neural network used for classification of the 4 or 5 classes of noise.

As we are performing a classification task, the output of the NN represents the possible classes (labels). This is accomplished with one-hot encoding: each class is represented by a binary vector with a length $D^{(L)}$ equal to the number of classes, where all elements are zero except for the one corresponding to the specific class, which is set to one. This representation ensures that categorical values are treated as distinct entities by the model, avoiding any implicit ordering that might be inferred from numerical encoding. The output of the network for a given input $\vect{x}_i$ is then a vector where each element corresponds to the probability that $\vect{x}_i$ belongs to one of the possible classes.

This is achieved by the use of the softmax activation function for the output layer

\begin{equation}
\label{eq:softmax}
\vect{y} = f_\textrm{softmax}(\vect{z}^{(L)}) = \frac{e^{\vect{z}^{(L)}}}{\sum_{k=1}^{D^{(L)}} e^{z_k^{(L)}}},
\end{equation}

which ensures the normalization of $\vect{y}$ (the exponentiation is intended to act component-wise).
Finally, the class with the highest probability is taken as the model's prediction for the input's class.

We employ the Categorical cross-entropy as cost function defined as

\begin{equation}
\label{eq:categorical_cross-entropy}
C(\{\vect{y}_i, \vect{\hat{y}}_i\}_i) = -\frac{1}{N}\sum_{i=1}^N\sum_{j=1}^{D^{(L)}} {\hat{y}_{ij}} \ln(y_{ij}),
\end{equation}

where $y_{ij}$ denotes the components of the predicted output $\vect{y}_i$, i.e. the probability assigned by the model for $\vect{x}_i$ to belong to the class $j=1,\dots, D^{(L)}$ while
$\hat{y}_{ij}$ gives the components of the
one-hot encoded target vector $\vect{\hat{y}}_i$.

§.§ Data generation

In order to efficientlys classify between the different noise types it is imperative to identify features which are sensitive to the distinct characteristics of the classes introduced in Sec. <ref>.
To this end, we choose as features the efficiency of the CTAP/STIRAP protocol under three different driving conditions. Referring to Eqs. (<ref>) we consider:

* $\Opmax = \Osmax$,

* $\Opmax > \Osmax$,

* $\Opmax < \Osmax$.

For each type of noise,
we solve numerically the dynamics of the system under the three driving conditions and compute the efficiency as described in Sec. <ref>.
We denote the efficiencies as $\xieq$, $\xigr$, and $\xilo$, for the driving conditions (i), (ii), and (iii) respectively,  and we use $\vect{x} = (\xieq, \xigr, \xilo)$ as input feature vector to the NN, see Fig. <ref>.

Schematic representation of our model. The efficiencies obtained with the three driving conditions $\Opmax = \Osmax$, $\Opmax > \Osmax$, and $\Opmax < \Osmax$ are given as input to the neural network. The Neural Network is composed of 2 hidden layers with $128$ and $100$ neurons respectively, and activation function LeakyReLU, Eq. (<ref>). The output layer composed of $4$ or $5$ neurons represent the probability of assigning each data to the relative noise class. This is achieved with the softmax activation function, Eq. (<ref>), on the output layer.

The parameters we use are (i) $\Opmax = \Osmax = 50/T$, (ii) $\Opmax=20\sqrt{10}/T > \Osmax=10\sqrt{10}/T$, and (iii) $\Opmax=10\sqrt{10}/T < \Osmax=20\sqrt{10}/T$ with evolution time $t\in[-5T,5T]$, $\tau=0.7T$ and $\sigma_1\approx 17.6/T$ being the standard deviation of the Gaussian distribution $p_1(x_1,t)$.

For the correlated and anti-correlated noise, data are generated by randomly sampling the correlation parameter $\eta$ in the intervals $[0.1,5]$ and $[-5,-0.1]$, respectively. For each randomly selected $\eta$ we calculate three efficiencies, one for each of the pulse conditions (i), (ii), and (iii), thus generating the input feature vector $\vect{x}$. The efficiencies are calculated using Eq. (<ref>) for non-Markovian quasistatic noise (with $p_1(x_1,t)$ being a Gaussian distribution) and Eq. (<ref>) for Markovian noise.

For uncorrelated, non-Markovian quasistatic noise, the values of $\tilde x_1$ and $\tilde x_2$ are independently sampled from two Gaussian probability distributions $p_1(x_1,t)$ and $p_2(x_2,t)$ with mean $\mu_1=\mu_2=0$. The standard deviation $\sigma_1$ of the first distribution is kept fixed, while the standard deviation $\sigma_2$ of the distribution $p_2(x_2,t)$ is randomly sampled in the interval $[-5\sigma_1, 5\sigma_1]$. As mentioned in Sec. <ref>, the classes are one-hot encoded, thus the output of the neural network is a layer with four or five neurons. For each noise class, we generate $500$ samples such that the total data consists of $2000$ or $2500$ data points $\{\vect{x}, \vect{\hat{y}}\}$. We then split the data in a training, validation and test set with a ratio of $0.6:0.2:0.2$, respectively.

In order to evaluate the efficacy of the model we use the accuracy $A$ defined as the number of correct predictions divided by the number of total predictions $N$

\begin{equation}
\label{eq:accuracy}
A = \frac{1}{N}\sum_{i=1}^N \delta (\argmax{\vect{y}_i}, \argmax{\vect{\hat y}_i})
%= \frac{1}{N}\sum_{i=1}^N \delta(\argmax_j{y_{ij}}, \argmax_j{\hat y_{ij}})
\end{equation}

where the $\argmax$ function yields the index $j$ of the maximal component of $\vect{y}$ and $\delta(\cdot)$ is the Kronecker delta.

§ RESULTS

The ML model and the training are performed with TensorFlow <cit.> and the results are shown in Fig. <ref>. The accuracy $A$ for the training and validation sets is shown in Fig. <ref>(a). After training on the test set the accuracy of the model is $A\approx0.81$ and varies within the values of approx $0.79$ and $0.81$ depending on the random initialization of the NN weights and the random shuffling of the data for the splitting in the three sets. For the same test set, we report in Fig. <ref>(b) how the value of the cost function $C$ changes during training.

(a) Accuracy, Eq. (<ref>), and (b) value of the cost function, Eq. (<ref>), for the training (blue for four noises and green for five noises) and validation (orange for four noises and red for five noises) sets versus the number of epochs of training. The accuracy on the test set is $A\approx1$ for four noise classes and $A\approx0.81$ for five noise classes.

In Fig. <ref>, we show the confusion matrix for the trained model, a tabular representation used to evaluate the performance of a classification model. It cross-tabulates the actual class labels with the model's predictions, providing insight into the correct and incorrect classifications made by the model. Thus it helps to identify which classes are not easily distinguishable from each other posing a challenge to the classification. Each row of the matrix represents the instances of the actual classes, while each column represents the instances of the predicted classes. It is apparent from Fig. <ref>(a) that classes $4a$ and $4b$ are not distinguishable from each other with the input features of our choice and almost all the samples collapse in class $4a$.

Confusion matrix for the MLP model for the classification of the classes of noise comsidered in this work. Each row of the matrix represents the instances of the true classes, while each column represents the instances of the predicted classes. It is apparent that classes $4a$ and $4b$, Markovian correlated and anti-correlated, are not easily distinguishable.

We then repeat the same analysis considering the Markovian noise as a single class, by grouping together correlated and anti-correlated noise and taking $\eta\in[-5,5]$. As expected, the effectiveness of the model increases reaching an accuracy $A\approx1$ that varies between $\simeq0.97$ and $1$ depending on the random initialization. The accuracy $A$ and the value of the cost function $C$ during the training are shown in Fig. <ref>. The confusion matrix for this model is reported in Fig. <ref>(b) showing that now the four classes are clearly distinguishable with the chosen input features.

§.§ Results for a finite number of measurement

The analysis performed in the previous section describes the ideal physical situation that the data are obtained from an infinite number of $100\%$-efficiency of projective quantum measurements. While very large efficiency can be achieved by quantum non-demolition measurements <cit.>, in this section we analyze how
the training and the accuracy of the model are affected by the number of measurements $M$ being finite.

Accuracy of the classification task for four classes of noise versus the number $M$ of projective measurements on quantum trajectories. The shaded orange region corresponds to the accuracy obtained with ideal measurements.

To this end, we employ the same method described in Sec. <ref> but compute the efficiency $\xi_M$ as the one resulting from
a set of individual “quantum" trajectories. The result
of unravelling each trajectory is simulated
by extracting each time the value $1$ with probability equal to the population of the target state $\ket{2}$.  The output $\vect{y}$ is produced by
averaging over the $M$ quantum trajectories. We produce $500$ distinct datasets, each corresponding to a different measurement count $M=10,\dots,5000$ (in steps of $10$). For each $M$, we train the model and evaluate its accuracy using the same methodology as previously described.

The results are presented in Fig. <ref>, where we report the accuracy versus the number of measurements. As expected, the accuracy improves for increasing $M$, approaching the value obtained in the ideal case,
which lies in the range $A\approx[0.97,1]$.

§ PHISICAL IMPLEMENTATION AND INTERPRETATION OF THE RESULTS

§.§ Physical implementations of the model

Spatially correlated noise in solid-state quantum information processing is a topic of current experimental investigation in superconducting <cit.> and semiconducting <cit.> quantum devices. Hardware CTAP has been demonstrated in arrays of quantum dots <cit.>. In superconducting devices STIRAP has been demonstrated in Vee configuration <cit.> whereas the Lambda configuration we study in this work could be implemented either directly <cit.> or by a detuning-modulated protocol <cit.> generalizing hyper-STIRAP <cit.> which bypasses parity selection rules enforced at noise-protected operating points.

Diagonal noise such as the one addressed here describes the effect of sources producing a fluctuating electrical polarization of charges. In principle, noise and correlations may also affect the tunneling amplitudes, thus giving rise to fluctuations of the non-diagonal entries of the Hamitlonian. However, as long as the noise is smaller than $(\delta,\,\delta_p)$,  as in our case, off-diagonal noise affects the dynamics only at second order and gives rise to a higher-order effect that can be neglected. For STIRAP in multilevel artificial atoms, noise correlations stem due to the fact that each noise source is coupled to the device via a single operator. In particular, noise coupled by an operator commuting with the undriven Hamiltonian (referred as “longitudinal”) yields stochastic fluctuations of the energy splittings of the devices which are related to each other. The main effect is diagonal noise, the off-diagonal entries being negligible if noise is smaller than the Autler-Townes splitting $\propto \sqrt{\Omega_{p}^2 + \Omega_s^2}$ (see <ref>). Notice that in this case, the parameter $\eta$ is fully determined by the first derivatives of the band structure at the point where the device is biased.

§.§ Interpretation of the results

We now seek an explanation of the ability to classify among differently correlated non-Markovian quasistatic noises. To this end, we analyze the stability plots (efficiency versus the detunings $\delta$ and $\delp$) for the three driving conditions, shown in Fig. <ref>

Efficiency of CTAP for the three driving conditions (i) $\Opmax = \Osmax = 50/T$ (red, solid line), (ii) $\Opmax=20\sqrt{10}/T > \Osmax=10\sqrt{10}/T$ (blue, dotted line), and (iii) $\Opmax=10\sqrt{10}/T < \Osmax=20\sqrt{10}/T$ (green dashed line). The shaded areas within the boundary lines highlight the regions, in the parameter space, where the efficiency is $>0.7$ (the efficiency being exactly 0.7 along the boundary curves).
The efficiencies are calculated with Eqs. (<ref>) and (<ref>). The dashed gray lines represent correlation ($\eta>0$) or anti-correlation ($\eta<0$) between the detunings $\delta$ and $\delp$. The other parameter is $\tau=0.7T$.

Ideally, the system is operated at zero detunings, $\delta=\delta_p=0$.
The effect of quasistatic noise is to move, at each individual repetition of the protocol, the point associated with the operations of the system to somewhere else in the space of parameters [cf. Fig.<ref>]. The efficiency for correlated and anti-correlated noise types, is calculated as a weighed average over the line $\delta = \eta \delp$, see Eq. (<ref>). The  corresponding efficiency depends on the ratio between $\Opmax$ and $\Osmax$, this dependence being different for $\eta>0$ or $\eta<0$. Fig. <ref> reports as an example the lines $\delta=\eta\delp$ for $\eta\gtrless0$: it is clear that for $\eta>0$ the efficiency of CTAP with $\Opmax<\Osmax$ is higher than the one with $\Opmax>\Osmax$, and vice versa for $\eta<0$. For a more extensive analysis of this behavior we refer to Ref. <cit.>. For uncorrelated noise, instead, the average is over the whole $\delta-\delp$ plane – cf. Eq. (<ref>) – and its dependence on the three different driving conditions is not as explicit.

Such analysis cannot be replicated for Markovian noise, as individual trajectories cannot be represented as single points in Fig. <ref>. Instead, the noise assumes all values along the line $\delta=\eta\delp$ during each repetition of the protocol.

§  CONCLUSIONS

We have proposed a ML-based method for categorizing correlated classical noise affecting a three-level quantum system. We have considered various instances of   non-Markovian and Markovian noise, including(anti-)correlated and uncorrelated mechanisms.

We have shown that exploiting the sensitivity of a CTAP/STIRAP protocol
to  correlation and Markovianity,
it is possible to successfully classify with great detail the noise mechanisms. Specifically, the efficiency of population transfer under three different driving conditions is sufficient to distinguish among the correlations of non-Markovian quasistatic noise, and between the latter and Markovian noise. On the contrary, this approach is inadequate to differentiate between the correlations in Markovian noise. As input to the ML analysis, we have used the solution of the Stochastic Schrödinger equation and the Markovian Lindblad Master Equation which yield the result of averaging over an infinite number of projective quantum measurements.
Our study has been complemented by the analysis of the impact of statistical measurement errors, showing that our approach is robust against those arising from finite statistics.

We remark that the implications of our work extend beyond noise classification. The methodology developed herein offers a novel perspective on the use of quantum protocols, such as CTAP/STIRAP and their multilevel/multiqubit
extensions, not just for quantum control but also as diagnostic tools
for system-characterization.

Future directions of this research include investigating the potential of incorporating alternative features of the input and extending this approach to other classes of noise, and the use of unsupervised learning. Additionally, we aim to address the interplay with relaxation <cit.> and explore the application of this approach to more complex quantum structures, such as interacting multiqubit  systems <cit.> as well as strongly <cit.> and ultrastrongly coupled <cit.> architectures exploiting new STIRAP protocols controlled by detunings <cit.>. This work and the foreseen advances could provide important elements for the development of integrated robust quantum control and error correction techniques.

§ ACKNOWLEDGMENTS

The authors acknowledge Dario Allegra, Lorenzo Catania, Vincenzo Minissale and Giuliano Chiriacò for useful discussions.  SM acknowledges support from the project PON Ricerca e Innovazione 2014-2020, React EU, Axis IV, Action IV.4 (CUP: E69J21011430006), within the PhD course in Physics of the Catania University. LG and EP acknowledge support from the PNRR MUR project PE0000023-NQSTI; GF is supported by the  ICSC – Centro Nazionale di Ricerca in High-Performance Computing, Big Data and Quantum Computing. GF and EP acknowledge support from the University of Catania, Piano Incentivi Ricerca di Ateneo 2020-22, project Q-ICT. EP acknowledges the COST Action  SUPERQUMAP (CA 21144). MP acknowledges support from the European Union's Horizon Europe EIC-Pathfinder project QuCoM (grant no. 101046973), the Royal Society Wolfson Fellowship (grant no. RSWF/R3/183013), the UK EPSRC (grant no. EP/T028424/1), and the Department for the Economy Northern Ireland under the US-Ireland R&D Partnership Programme.

§ DERIVATION OF THE LINDBLAD MASTER EQUATION FOR MARKOVIAN DIAGONAL CLASSICAL NOISE

Here we derive the Lindblad master equation introduced in Eq. (<ref>) for the description of the dynamics of the system when it is subject to Markovian noise. As mentioned in Sec. <ref>, the total Hamiltonian of the system is $H = \Hsys + \Hnoise$, where $\Hnoise$ is given by Eq. (<ref>). Using the correlation $x_2(t) = \eta x_1(t)$, we can represent $\Hnoise$ as

\begin{equation}
\Hnoise = \tilde x_1(t)\left(\ketbra{1}{1} + \eta\ketbra{2}{2}\right) = \tilde x_1(t)O,
\end{equation}

where $O = \ketbra{1}{1} + \eta\ketbra{2}{2}$ is the noise operator. For Markovian noise, the noise has zero mean $\avg{\tilde x_1(t)} = 0$ and is delta-correlated as $\avg{\tilde x_1(t) \tilde x_1(t')} = \hbar^2\gamma\delta(t-t')$. Since the noise is $\delta$ correlated, the variation of the tunneling amplitudes $\Op(t)$ and $\Os(t)$ is adiabatic, thus we consider an instantaneous interaction picture

\begin{equation}
\tilde{\rho}(t) = e^{\frac{\I}{\hbar}\Hsys t}\rho(t)e^{-\frac{\I}{\hbar}\Hsys t},
\end{equation}

where $\rho(t)$ is the density operator in the Schrödinger picture. The time evolution of the system is thus given by

\begin{equation}
\label{eq:Vonneumann_intpic}
\dot{\tilde{\rho}}(t)= -\frac{\I}{\hbar}[\Hnoisetilde (t), \tilde{\rho}(t)],
\end{equation}

with $\Hnoisetilde(t) = e^{\frac{\I}{\hbar}\Hsys t}\Hnoise (t)e^{-\frac{\I}{\hbar}\Hsys t}$.
Integrating the above equation within the time limits $[0, t]$, we get

\begin{equation}
\tilde{\rho}(t)= \tilde{\rho}(0)-\frac{\I}{\hbar}\int_0^{t}[\Hnoisetilde (t'), \tilde{\rho}(t')]dt'.
\end{equation}

Substituting $\tilde{\rho}(t)$ in Eq. (<ref>) we obtain

\begin{equation}
\dot{\tilde{\rho}}(t)= -\frac{\I}{\hbar}e^{\frac{\I}{\hbar}\Hsys t}[\tilde x_1(t)O\rho(0) - \rho(0)O \tilde x_1(t)]e^{-\frac{\I}{\hbar}\Hsys t} - \frac{1}{\hbar^2}\int_0^t[\Hnoisetilde (t), [\Hnoise (t'), \tilde{\rho}(t')]]dt'.
\end{equation}

Averaging over the stochastic variable at time $t$, we obtain

\begin{equation}
\langle \dot{\tilde{\rho}}(t) \rangle =  - \frac{1}{\hbar^2}\biggl< \int_0^t[\tilde x(t)\tilde O(t), [\tilde x(t')\tilde O(t'), \tilde{\rho}(t')]]dt' \biggr>,
\end{equation}

where, $\tilde O (t) = e^{\frac{\I}{\hbar}\Hsys t}Oe^{-\frac{\I}{\hbar}\Hsys t}$. Taking into account that $t^\prime < t$ and $\tilde \rho(t^\prime)$ cannot depend on $t$ except for $t^\prime =t$, we can write
$\avg{\tilde x(t)\tilde x(t')\tilde\rho(t')}=\avg{\tilde x(t)\tilde x(t')}\tilde\rho(t')$. We thus obtain

\begin{equation}
\langle \dot{\tilde{\rho}}(t) \rangle = -\frac{\gamma}{2}\left(\tilde O^2(t)\tilde{\rho}(t) + \tilde{\rho}(t)\tilde O^2(t) - 2 \tilde O(t)\tilde{\rho}(t)\tilde O(t)\right).
\end{equation}

Finally, the above equation in the Schrödinger picture can be represented in the Lindblad form as

\begin{equation}
\langle \dot{\rho}(t) \rangle = -\frac{\I}{\hbar}[\Hsys, \rho(t)] - \frac{\gamma}{2}\left(O^2(t)\rho(t) + \rho(t)O^2(t)-2O(t)\rho(t)O(t)\right).
\end{equation}

§ ADIABATIC PASSAGE IN A THREE-NODE NETWORK

Recall the Hamiltonian of the three-node network $\Hsys(t) = H_0 + H_\mathrm{c}(t)$, where $H_0$ and $H_\mathrm{c}(t)$ are the uncoupled and the control parts, respectively ($\hbar=1$):

\begin{align}
& H_0 = \delp\ketbra{1}{1}+ \delta\ketbra{2}{2},                                                                                \\
& \Hc = \frac{\Op(t)}{2}\left(\ketbra{0}{1} + \ketbra{1}{0}\right) + \frac{\Os(t)}{2}\left(\ketbra{1}{2} + \ketbra{2}{1}\right),
\end{align}

where $\delp = \epsilon_1-\epsilon_0$ and $\delta = \epsilon_2-\epsilon_0$ define the detunings.
CTAP (Coherent Tunneling by Adiabatic Passage) <cit.> (as well as its spin equivalent spin-CTAP <cit.>) is a protocol that achieves population transfer from $\ket{0}$ to $\ket{2}$ without populating the state $\ket{1}$ by adiabatic following a trapped state. The atomic analogue is called STIRAP <cit.> where high-fidelity population transfer occurs via a dark state. These protocol have remarkable robustness against parametric fluctuations.

The CTAP protocol is based on the counter-intuitive ordering of the pulses $\Op(t)$ and $\Os(t)$ and the condition $\delta\approx0$ is essential for the successful population transfer <cit.>. Considering thus $\delta=0$, the instantaneous eigenstates of $\Hsys(t)$, are

\begin{align}
& \ket{\mathrm{\phi}_\mathrm{D}(t)} = \cos\theta(t)\ket{0}-\sin\theta(t)\ket{2},                                  \\
& \ket{\mathrm{\phi}_-(t)} = \sin\theta(t)\cos\phi(t)\ket{0}-\sin\phi(t)\ket{1}+\cos\theta(t)\cos\phi(t)\ket{2},  \\
& \ket{\mathrm{\phi}_+(t)} = \sin\theta(t)\sin\phi(t)\ket{0}-\cos\phi(t)\ket{1} +\cos\theta(t)\sin\phi(t)\ket{2},
\end{align}

with corresponding eigenvalues

\begin{align}
& \lambda_\mathrm{D}(t) = 0,                                          \\
& \lambda_-(t) = -\frac{\hbar}{2}\sqrt{\Op(t)^2+\Os(t)^2}\tan\phi(t), \\
& \lambda_+(t) = \frac{\hbar}{2}\sqrt{\Op(t)^2+\Os(t)^2}\cot\phi(t),
\end{align}

\begin{align}
& \theta(t) = \tan^{-1}\left(\frac{\Op(t)}{\Os(t)}\right),                                                 \\
& \phi(t) = \tan^{-1}\left(\frac{\sqrt{\Op(t)^2+\Os(t)^2}}{\delp+\sqrt{\delp^2+\Op(t)^2+\Os(t)^2}}\right).
\end{align}

The eigenstate $\ket{\phi_\mathrm{D}(t)}$ corresponding to zero eigenvalue is called the trapped state (or also dark state) since it is trapped in the subspace $\{\ket{0},\ket{2}\}$ (and, in the case of an atom, cannot absorb or emit photons). It has no component along the state $\ket{1}$ and it is the state that the system follows during the population transfer <cit.>. The states $\ket{\phi_-(t)}$ and $\ket{\phi_+(t)}$ are the Autler-Townes states and $|\lambda_+-\lambda_-|$ the Autler-Townes splitting.

If the pulses are counter-intuitively ordered, i.e. if $\Os$ is applied before $\Op$:

\begin{equation}
\label{eq:counterintuitiveorder}
\lim_{t\to\ti} \frac{\Op(t)}{\Os(t)} = \lim_{t\to\tf} \frac{\Os(t)}{\Op(t)} = 0,
\end{equation}

then the dark state $\phi_\mathrm{D}(t)$ at the initial time $\ti$ will coincide with the initial state $\ket{0}$ and at the final time $\tf$ will coincide with the target state $\ket{2}$. If the evolution is adiabatic and the system is initially prepared in $\ket{0}=\ket{\phi_\mathrm{D}(\ti)}$, by the adiabatic theorem <cit.>, the system evolves following the dark state $\ket{\phi_\mathrm{D}(t)}$ throughout the time evolution. Thus, the population is transferred from state $\ket{0}$ to state $\ket{2}$ never populating the intermediate state $\ket{1}$. The adiabaticity condition for this protocol is given by <cit.>

\begin{equation}
\label{eq:adiabaticcondition}
\hbar\abs{\braket{\phi_{\pm}}{\mathrm{\dot{\phi}_\mathrm{D}}}}\ll\abs{\lambda_0-\lambda_{\pm}}.
\end{equation}

Using the expressions for the dressed bases, Eqs. (<ref>) and Eqs. (<ref>), in equation (<ref>) we have the condition for adiabatic evolution in terms of the pulses <cit.>

\begin{equation}
\label{eq:localadiabaticitycondition}
\abs{\dot{\theta}(t)} \ll \frac{1}{2}\absbr{\delp\pm\sqrt{\delp^2+\Op(t)^2+\Os(t)^2}},
\end{equation}

which is called “local adiabaticity condition” and has to hold for all times $t\in[\ti,\tf]$. By time averaging Eq. (<ref>) over $\tau$ and assuming $\delp\ll\Opmax,\Osmax$ one can obtain the less strict “global adiabaticity condition” which is often reported as <cit.>

\begin{equation}
\label{eq:globaladiabaticcondition2}
\Opsmax \tau \geq 10,
\end{equation}

where $\tau$ is the characteritic time scale of the pulses overlap and $\Opsmax = \max_t\Ops(t)$.

In this work, in order to perform the population transfer, we use Gaussian pulses of the form

\begin{align}
& \Op(t) = \Opmax e^{-(\frac{t-\tau}{T})^2}, \\
& \Os(t) = \Osmax e^{-(\frac{t+\tau}{T})^2},
\end{align}

and we let the system evolve in the time interval $t\in[-5T,5T]$ with $\tau=0.7T$, see Fig. <ref> for an example of pulses.

We point out that our aim is not achieving an efficient population transfer, but to exploit CTAP to get information about the noise affecting the system and deteriorating the efficiency.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Springer Berlin Heidelberg

abstractQuantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.
issn2662-4400, 2196-0763
journaltitleEPJ Quantum Technol.
titleQuantum Optimal Control in Quantum Technologies. Strategic Report on Current Status, Visions and Goals for Research in Europe

IOP Publishing

abstractWithin the last two decades, quantum technologies (QT) have made tremendous progress, moving from Nobel Prize award-winning experiments on quantum physics (1997: Chu, Cohen-Tanoudji, Phillips; 2001: Cornell, Ketterle, Wieman; 2005: Hall, Hänsch-, Glauber; 2012: Haroche, Wineland) into a cross-disciplinary field of applied research. Technologies are being developed now that explicitly address individual quantum states and make use of the `strange' quantum properties, such as superposition and entanglement. The field comprises four domains: quantum communication, where individual or entangled photons are used to transmit data in a provably secure way; quantum simulation, where well-controlled quantum systems are used to reproduce the behaviour of other, less accessible quantum systems; quantum computation, which employs quantum effects to dramatically speed up certain calculations, such as number factoring; and quantum sensing and metrology, where the high sensitivity of coherent quantum systems to external perturbations is exploited to enhance the performance of measurements of physical quantities. In Europe, the QT community has profited from several EC funded coordination projects, which, among other things, have coordinated the creation of a 150-page QT Roadmap (http://qurope.eu/h2020/qtflagship/roadmap2016). This article presents an updated summary of this roadmap.
journaltitleNew J. Phys.
shorttitleThe Quantum Technologies Roadmap
titleThe Quantum Technologies Roadmap: A European Community View

American Physical Society

abstractThe manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal “Schrödinger-cat states.” The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective “collapse of the wave packet.” The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand “classical reality” as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be “found out” without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their “fitness” in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born's rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.
journaltitleRev. Mod. Phys.
titleDecoherence, Einselection, and the Quantum Origins of the Classical

American Physical Society

abstractThe effects of the inevitable coupling to external degrees of freedom of a quantum computer are examined. It is found that for quantum calculations (in which the maintenance of coherence over a large number of states is important), not only must the coupling be small, but the time taken in the quantum calculation must be less than the thermal time scale $\hbar$/kBT. For longer times the condition on the strength of the coupling to the external world becomes much more stringent.
journaltitlePhys. Rev. A
titleMaintaining Coherence in Quantum Computers

American Association for the Advancement of Science

abstractIt is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L-digit number can be found in $\sim$L2 time [compared to $\sim$exp(L1/3) time] by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation.
titleQuantum Computers, Factoring, and Decoherence

abstractSciPost Journals Publication Detail SciPost Phys. Lect. Notes 29 (2021) Machine learning and quantum devices
journaltitleSciPost Physics Lecture Notes
titleMachine Learning and Quantum Devices

American Physical Society

abstractIn recent years the dramatic progress in machine learning has begun to impact many areas of science and technology significantly. In the present perspective article, we explore how quantum technologies are benefiting from this revolution. We showcase in illustrative examples how scientists in the past few years have started to use machine learning and more broadly methods of artificial intelligence to analyze quantum measurements, estimate the parameters of quantum devices, discover new quantum experimental setups, protocols, and feedback strategies, and generally improve aspects of quantum computing, quantum communication, and quantum simulation. We highlight open challenges and future possibilities and conclude with some speculative visions for the next decade.
journaltitlePhys. Rev. A
titleArtificial Intelligence and Machine Learning for Quantum Technologies

Nature Publishing Group

abstractThe future development of quantum technologies relies on creating and manipulating quantum systems of increasing complexity, with key applications in computation, simulation and sensing. This poses severe challenges in the efficient control, calibration and validation of quantum states and their dynamics. Although the full simulation of large-scale quantum systems may only be possible on a quantum computer, classical characterization and optimization methods still play an important role. Here, we review different approaches that use classical post-processing techniques, possibly combined with adaptive optimization, to learn quantum systems, their correlation properties, dynamics and interaction with the environment. We discuss theoretical proposals and successful implementations across different multiple-qubit architectures such as spin qubits, trapped ions, photonic and atomic systems, and superconducting circuits. This Review provides a brief background of key concepts recurring across many of these approaches with special emphasis on the Bayesian formalism and neural networks.
journaltitleNat Rev Phys
titleLearning Quantum Systems
Information theory and computation,Qubits

journaltitleNew J. Phys.
titleReinforcement Learning-Enhanced Protocols for Coherent Population-Transfer in Three-Level Quantum Systems

journaltitlePhysics Letters A
titleA Tutorial on Optimal Control and Reinforcement Learning Methods for Quantum Technologies

Nature Publishing Group

abstractEmerging reinforcement learning techniques using deep neural networks have shown great promise in control optimization. They harness non-local regularities of noisy control trajectories and facilitate transfer learning between tasks. To leverage these powerful capabilities for quantum control optimization, we propose a new control framework to simultaneously optimize the speed and fidelity of quantum computation against both leakage and stochastic control errors. For a broad family of two-qubit unitary gates that are important for quantum simulation of many-electron systems, we improve the control robustness by adding control noise into training environments for reinforcement learning agents trained with trusted-region-policy-optimization. The agent control solutions demonstrate a two-order-of-magnitude reduction in average-gate-error over baseline stochastic-gradient-descent solutions and up to a one-order-of-magnitude reduction in gate time from optimal gate synthesis counterparts. These significant improvements in both fidelity and runtime are achieved by combining new physical understandings and state-of-the-art machine learning techniques. Our results open a venue for wider applications in quantum simulation, quantum chemistry and quantum supremacy tests using near-term quantum devices.
journaltitlenpj Quantum Inf
titleUniversal Quantum Control through Deep Reinforcement Learning
luigi/work/articles/Niu et al. - 2019 - Universal quantum control through deep reinforceme.pdf;/home/luigi/work/articles/Niu_et_al-npj_Quantum_Inf-2019-Universal_quantum_control_through_deep_reinforcement_learning.pdf;/home/luigi/Zotero/storage/SK2J4GNB/s41534-019-0141-3.html
Quantum information,Qubits

journaltitlePhysical Review Letters
titleReinforcement Learning Approach to Nonequilibrium Quantum Thermodynamics

IOP Publishing

abstractQuantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.
journaltitleNew J. Phys.
titleModelling Non-Markovian Quantum Processes with Recurrent Neural Networks

Nature Publishing Group

abstractThe experimental realization of increasingly complex synthetic quantum systems calls for the development of general theoretical methods to validate and fully exploit quantum resources. Quantum state tomography (QST) aims to reconstruct the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics1–3. However, exact brute-force approaches to QST place a high demand on computational resources, making them unfeasible for anything except small systems4,5. Here we show how machine learning techniques can be used to perform QST of highly entangled states with more than a hundred qubits, to a high degree of accuracy. We demonstrate that machine learning allows one to reconstruct traditionally challenging many-body quantities—such as the entanglement entropy—from simple, experimentally accessible measurements. This approach can benefit existing and future generations of devices ranging from quantum computers to ultracold-atom quantum simulators6–8.
journaltitleNature Phys.
titleNeural-Network Quantum State Tomography
Characterization and analytical techniques,Computational science,Matter waves and particle beams,Quantum information,Qubits

Nature Publishing Group

abstractQuantum tomography is currently ubiquitous for testing any implementation of a quantum information processing device. Various sophisticated procedures for state and process reconstruction from measured data are well developed and benefit from precise knowledge of the model describing state-preparation-and-measurement (SPAM) apparatus. However, physical models suffer from intrinsic limitations as actual measurement operators and trial states cannot be known precisely. This scenario inevitably leads to SPAM errors degrading reconstruction performance. Here we develop a framework based on machine learning which generally applies to both the tomography and SPAM mitigation problem. We experimentally implement our method. We trained a supervised neural network to filter the experimental data and hence uncovered salient patterns that characterize the measurement probabilities for the original state and the ideal experimental apparatus free from SPAM errors. We compared the neural network state reconstruction protocol with a protocol treating SPAM errors by process tomography, as well as to an SPAM-agnostic protocol with idealized measurements. The average reconstruction fidelity is shown to be enhanced by 10% and 27%, respectively. The presented methods apply to the vast range of quantum experiments which rely on tomography.
journaltitlenpj Quantum Inf
titleExperimental Neural Network Enhanced Quantum Tomography
Quantum information,Single photons and quantum effects

Multidisciplinary Digital Publishing Institute

abstractWe investigate the extent to which a two-level quantum system subjected to an external time-dependent drive can be characterized by supervised learning. We apply this approach to the case of bang-bang control and the estimation of the offset and the final distance to a given target state. For any control protocol, the goal is to find the mapping between the offset and the distance. This mapping is interpolated using a neural network. The estimate is global in the sense that no a priori knowledge is required on the relation to be determined. Different neural network algorithms are tested on a series of data sets. We show that the mapping can be reproduced with very high precision in the direct case when the offset is known, while obstacles appear in the indirect case starting from the distance to the target. We point out the limits of the estimation procedure with respect to the properties of the mapping to be interpolated. We discuss the physical relevance of the different results.
titleCharacterization of a Driven Two-Level Quantum System by Supervised Learning
optimal control,supervised learning,system characterization,two-level quantum systems

American Physical Society

abstractMachine learning (ML) is a promising approach for performing challenging quantum-information tasks such as device characterization, calibration, and control. ML models can train directly on the data produced by a quantum device while remaining agnostic to the quantum nature of the learning task. However, these generic models lack physical interpretability and usually require large datasets in order to learn accurately. Here we incorporate features of quantum mechanics in the design of our ML approach to characterize the dynamics of a quantum device and learn device parameters. This physics-inspired approach outperforms physics-agnostic recurrent neural networks trained on numerically generated and experimental data obtained from continuous weak measurement of a driven superconducting transmon qubit. This demonstration shows how leveraging domain knowledge improves the accuracy and efficiency of this characterization task, thus laying the groundwork for more scalable characterization techniques.
journaltitlePRX Quantum
titleQuantum-Tailored Machine-Learning Characterization of a Superconducting Qubit

abstractAbstract The ability to use quantum technology to achieve useful tasks, be they scientific or industry related, boils down to precise quantum control. In general it is difficult to assess a proposed solution due to the difficulties in characterizing the quantum system or device. These arise because of the impossibility to characterize certain components in situ, and are exacerbated by noise induced by the environment and active controls. Here, we present a general purpose characterization and control solution making use of a deep learning framework composed of quantum features. We provide the framework, sample datasets, trained models, and their performance metrics. In addition, we demonstrate how the trained model can be used to extract conventional indicators, such as noise power spectra.
journaltitlenpj Quantum Inf.
titleCharacterization and Control of Open Quantum Systems beyond Quantum Noise Spectroscopy
Quantum information,Qubits

abstractDespite neural networks' success, their applications to open-system dynamics are few. In this work, non-linear autoregressive neural networks are adopted to generalize time series of expectation values of observables of interest in open quantum systems. Using Dirac-Frenkel time-dependent variation with the multiple Davydov D2 Ansatz, we obtain first stages of dynamical states of both the spin-boson model and the dissipative Landau-Zener model. With calculated data, careful training of the non-linear neural networks is performed. It is shown that the training quality of the networks is sufficient to ensure a least mean square error of 1×10-11. Subsequently, the network is cross validated by testing with additional data. Successes of the network training demonstrate that initial data of simulated open-system dynamics contain sufficient knowledge regarding its future propagation. We use the first-stage information and the trained network to predict future values of target observables in the series, and succeed with considerable accuracy.
journaltitleChemical Physics
seriesUltrafast Photoinduced Processes in Polyatomic Molecules:Electronic Structure, Dynamics and Spectroscopy (Dedicated to Wolfgang Domcke on the Occasion of His 70th Birthday)
titleApplications of Neural Networks to the Simulation of Dynamics of Open Quantum Systems

American Physical Society

abstractThe theory of open quantum systems lays the foundation for a substantial part of modern research in quantum science and engineering. Rooted in the dimensionality of their extended Hilbert spaces, the high computational complexity of simulating open quantum systems calls for the development of strategies to approximate their dynamics. In this Letter, we present an approach for tackling open quantum system dynamics. Using an exact probabilistic formulation of quantum physics based on positive operator-valued measure, we compactly represent quantum states with autoregressive neural networks; such networks bring significant algorithmic flexibility due to efficient exact sampling and tractable density. We further introduce the concept of string states to partially restore the symmetry of the autoregressive neural network and improve the description of local correlations. Efficient algorithms have been developed to simulate the dynamics of the Liouvillian superoperator using a forward-backward trapezoid method and find the steady state via a variational formulation. Our approach is benchmarked on prototypical one-dimensional and two-dimensional systems, finding results which closely track the exact solution and achieve higher accuracy than alternative approaches based on using Markov chain Monte Carlo method to sample restricted Boltzmann machines. Our Letter provides general methods for understanding quantum dynamics in various contexts, as well as techniques for solving high-dimensional probabilistic differential equations in classical setups.
journaltitlePhys. Rev. Lett.
titleAutoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation

American Physical Society

abstractMachine learning methods have proved to be useful for the recognition of patterns in statistical data. The measurement outcomes are intrinsically random in quantum physics, however, they do have a pattern when the measurements are performed successively on an open quantum system. This pattern is due to the system-environment interaction and contains information about the relaxation rates as well as non-Markovian memory effects. Here we develop a method to extract the information about the unknown environment from a series of projective single-shot measurements on the system (without resorting to the process tomography). The method is based on embedding the non-Markovian system dynamics into a Markovian dynamics of the system and the effective reservoir of finite dimension. The generator of Markovian embedding is learned by the maximum likelihood estimation. We verify the method by comparing its prediction with an exactly solvable non-Markovian dynamics. The developed algorithm to learn unknown quantum environments enables one to efficiently control and manipulate quantum systems.
journaltitlePhys. Rev. Lett.
titleMachine Learning Non-Markovian Quantum Dynamics

American Physical Society

abstractIn the last few years, the application of machine learning methods has become increasingly relevant in different fields of physics. One of the most significant subjects in the theory of open quantum systems is the study of the characterization of non-Markovian memory effects that emerge dynamically throughout the time evolution of open systems as they interact with their surrounding environment. Here we consider two well-established quantifiers of the degree of memory effects, namely, the trace distance and the entanglement-based measures of non-Markovianity. We demonstrate that using machine learning techniques, in particular, support vector machine algorithms, it is possible to estimate the degree of non-Markovianity in two paradigmatic open system models with high precision. Our approach can be experimentally feasible to estimate the degree of non-Markovianity, since it requires a single or at most two rounds of state tomography.
journaltitlePhys. Rev. A
titleEstimating the Degree of Non-Markovianity Using Machine Learning
family=de Vega,

American Physical Society

abstractThe exact microscopic structure of the environments that produce 1/f noise in superconducting qubits remains largely unknown, hindering our ability to create robust simulations and harness the noise. In this paper we show how it is possible to infer information about such an environment based on a single measurement of the qubit coherence, circumventing any need for separate spectroscopy experiments. Similarly to other spectroscopic techniques, the qubit is used as a probe which interacts with its environment. The complexity of the relationship between the observed qubit dynamics and the impurities in the environment makes this problem ideal for machine learning methods—more specifically neural networks. With our algorithm we are able to reconstruct the parameters of the most prominent impurities in the environment, as well as differentiate between different environment models, paving the way towards a better understanding of 1/f noise in superconducting circuits.
journaltitlePhys. Rev. A
titleNeural-Network-Based Qubit-Environment Characterization

American Physical Society

abstractUnderstanding the spectrum of noise acting on a qubit can yield valuable information about its environment, and crucially underpins the optimization of dynamical decoupling protocols that can mitigate such noise. However, extracting accurate noise spectra from typical time-dynamics measurements on qubits is intractable using standard methods. Here, we propose to address this challenge using deep-learning algorithms, leveraging the remarkable progress made in the field of image recognition, natural language processing, and more recently, structured data. We demonstrate a neural-network-based methodology that allows for extraction of the noise spectrum associated with any qubit surrounded by an arbitrary bath, with significantly greater accuracy than the current methods of choice. The technique requires only a two-pulse echo decay curve as input data and can further be extended either for constructing customized optimal dynamical decoupling protocols or for obtaining critical qubit attributes such as its proximity to the sample surface. Our results can be applied to a wide range of qubit platforms, and provide a framework for improving qubit performance with applications not only in quantum computing and nanoscale sensing but also in material characterization techniques such as magnetic resonance.
journaltitlePRX Quantum
titleUsing Deep Learning to Understand and Mitigate the Qubit Noise Environment

IOP Publishing

abstractSpectral densities encode the relevant information characterizing the system–environment interaction in an open-quantum system problem. Such information is key to determining the system's dynamics. In this work, we leverage the potential of machine learning techniques to reconstruct the features of the environment. Specifically, we show that the time evolution of a system observable can be used by an artificial neural network to infer the main features of the spectral density. In particular, for relevant examples of spin-boson models, we can classify with high accuracy the Ohmicity parameter of the environment as either Ohmic, sub-Ohmic or super-Ohmic, thereby distinguishing between different forms of dissipation.
journaltitleMach. Learn.: Sci. Technol.
titleSpectral Density Classification for Environment Spectroscopy

abstractNoise sources unavoidably affect any quantum technological device. Noise's main features are expected to strictly depend on the physical platform on which the quantum device is realized, in the form of a distinguishable fingerprint. Noise sources are also expected to evolve and change over time. Here, we first identify and then characterize experimentally the noise fingerprint of IBM cloud-available quantum computers, by resorting to machine learning techniques designed to classify noise distributions using time-ordered sequences of measured outcome probabilities.
journaltitlearXiv:2109.11405 [quant-ph]
titleLearning the Noise Fingerprint of Quantum Devices
81-04 81-08 81-05 81-11,Computer Science - Artificial Intelligence,Computer Science - Machine Learning,I.2.1,J.2,Quantum Physics

IOP Publishing

abstractThe undesired interaction of a quantum system with its environment generally leads to a coherence decay of superposition states in time. A precise knowledge of the spectral content of the noise induced by the environment is crucial to protect qubit coherence and optimize its employment in quantum device applications. We experimentally show that the use of neural networks (NNs) can highly increase the accuracy of noise spectroscopy, by reconstructing the power spectral density that characterizes an ensemble of carbon impurities around a nitrogen-vacancy (NV) center in diamond. NNs are trained over spin coherence functions of the NV center subjected to different Carr–Purcell sequences, typically used for dynamical decoupling (DD). As a result, we determine that deep learning models can be more accurate than standard DD noise-spectroscopy techniques, by requiring at the same time a much smaller number of DD sequences.
journaltitleMach. Learn.: Sci. Technol.
titleDeep Learning Enhanced Noise Spectroscopy of a Spin Qubit Environment

American Physical Society

abstractThe 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.
journaltitleRev. Mod. Phys.
shorttitle$\mathbsf{1}/\mathbsfit{f}$ Noise
title1/F Noise: Implications for Solid-State Quantum Information
luigi/work/articles/Paladino_et_al-Rev._Mod._Phys.-2014-$-mathbsf 1 --mathbsfit f $_noise_-_Implications_for_solid-state_quantum.pdf;/home/luigi/Zotero/storage/C5ZC985L/RevModPhys.86.html

American Association for the Advancement of Science

abstractWe have designed and operated a superconducting tunnel junction circuit that behaves as a two-level atom: the “quantronium.” An arbitrary evolution of its quantum state can be programmed with a series of microwave pulses, and a projective measurement of the state can be performed by a pulsed readout subcircuit. The measured quality factor of quantum coherenceQ $\phi$ $\cong$ 25,000 is sufficiently high that a solid-state quantum processor based on this type of circuit can be envisioned.
titleManipulating the Quantum State of an Electrical Circuit

American Physical Society

abstractMaintaining quantum coherence is a crucial requirement for quantum computation; hence protecting quantum systems against their irreversible corruption due to environmental noise is an important open problem. Dynamical decoupling (DD) is an effective method for reducing decoherence with a low control overhead. It also plays an important role in quantum metrology, where, for instance, it is employed in multiparameter estimation. While a sequence of equidistant control pulses [the Carr-Purcell-Meiboom-Gill (CPMG) sequence] has been ubiquitously used for decoupling, Uhrig recently proposed that a nonequidistant pulse sequence [the Uhrig dynamic decoupling (UDD) sequence] may enhance DD performance, especially for systems where the spectral density of the environment has a sharp frequency cutoff. On the other hand, equidistant sequences outperform UDD for soft cutoffs. The relative advantage provided by UDD for intermediate regimes is not clear. In this paper, we analyze the relative DD performance in this regime experimentally, using solid-state nuclear magnetic resonance. Our system qubits are 13C nuclear spins and the environment consists of a 1H nuclear spin bath whose spectral density is close to a normal (Gaussian) distribution. We find that in the presence of such a bath, the CPMG sequence outperforms the UDD sequence. An analogy between dynamical decoupling and interference effects in optics provides an intuitive explanation as to why the CPMG sequence performs better than any nonequidistant DD sequence in the presence of this kind of environmental noise.
journaltitlePhys. Rev. A
titleOptimal Pulse Spacing for Dynamical Decoupling in the Presence of a Purely Dephasing Spin Bath

journaltitleContemporary Physics
titleQuantum error correction: an introductory guide

journaltitleNat. Rev. Phys.
titleA series of fast-paced advances in Quantum Error Correction

American Physical Society

abstractWe propose a novel dynamical method for beating decoherence and dissipation in open quantum systems. We demonstrate the possibility of filtering out the effects of unwanted (not necessarily known) system-environment interactions and show that the noise-suppression procedure can be combined with the capability of retaining control over the effective dynamical evolution of the open quantum system. Implications for quantum information processing are discussed.
journaltitlePhys. Rev. Lett.
titleDynamical Decoupling of Open Quantum Systems

abstractWe study dynamical decoupling of a qubit from non-Gaussian quantum noise due to discrete sources, as bistable fluctuators and 1/f noise. We obtain analytic and numerical results for generic operating points. For very large pulse frequency, where dynamic decoupling compensates decoherence, we found universal behavior. At intermediate frequencies noise can be compensated or enhanced, depending on the nature of the fluctuators and on the operating point. Our technique can be applied to a larger class of non-Gaussian environments.
journaltitlePhys. Rev. A
notePublisher: American Physical Society
titleDynamical suppression of telegraph and $1/f$ noise due to quantum bistable fluctuators

abstractQuantum control techniques represent one of the most efficient tools to attain high-fidelity quantum operations and a convenient approach for quantum sensing and quantum noise spectroscopy. In this work, we investigate dynamical decoupling while processing an entangling two-qubit gate based on an Ising-xx interaction, each qubit being affected by pure dephasing classical correlated 1/ f -noises. To evaluate the gate error, we used the Magnus expansion introducing generalized filter functions that describe decoupling while processing and allow us to derive an approximate analytic expression as a hierarchy of nested integrals of noise cumulants. The error is separated in contributions of Gaussian and non-Gaussian noise, the corresponding generalized filter functions being calculated up to the fourth order. By exploiting the properties of selected pulse sequences, we show that it is possible to extract the second-order statistics (spectrum and cross-spectrum) and to highlight non-Gaussian features contained in the fourth-order cumulant. We discuss the applicability of these results to state-of-the-art small networks based on solid-state platforms.
eprintclasscond-mat, physics:quant-ph
titleOpen-Loop Quantum Control of Small-Size Networks for High-Order Cumulants and Cross-Correlations Sensing
Condensed Matter - Other Condensed Matter,Quantum Physics

Academic Press

abstractFundamental issues of 1/f noise in quantum nanoscience are reviewed starting from basic statistical noise processes. Fundamental noise models based on two-level systems (TLSs) are described. We emphasize the importance of TLSs in materials parameter fluctuations, such as dielectric constant. The present understanding of 1/f noise in superconducting quantum interferometers and in single electron devices is summarized. For coherent quantum nanoscience, we introduce superconducting qubits and the relation between decoherence and 1/f noise using the filter function formulation. We also clarify the qubit noise spectroscopy and emphasize the importance of materials with reduced 1/f noise for future quantum coherent nanodevices.
booktitleEncyclopedia of Condensed Matter Physics (Second Edition)
title1/f noise in quantum nanoscience
arge fluctuations,Decoherence,Dephasing,Flux noise,Quantum nanoscience,Quantum sensing,Qubits,Stochastic processes,Two-level systems

abstractThe Special Issue on Electronic Noise—From Advanced Materials to Quantum Technologies presents reports on recent developments in experimental and theoretical aspects of electronic noise and fluctuation processes across a wide spectrum of scientific and technological fields. The topics covered in the Special Issue include thermal noise and shot noise in advanced materials and devices; low-frequency 1/f noise and generation–recombination noise in electronics; low-frequency 1/f noise in graphene and 2D materials; charge, flux, and critical current 1/f noise in superconducting systems; noise in semiconductors and in spin qubits; microscopic models for 1/f noise; decoherence in quantum devices due to 1/f noise; low-frequency noise spectroscopy as a reliability tool for electronics; noise in strongly correlated systems; low-frequency noise in magnonic and spintronic devices; noise spectroscopy as the material characterization tool; noise in microwave devices due to two-level systems; noise and correlations in quantum amplifiers; quantum squeezing of noise; measurement techniques beyond standard quantum limit; noise and entanglement in pumped systems; and low-frequency phase noise in spin Hall nano-oscillators.
journaltitleApplied Physics Letters
titleElectronic noise—From advanced materials to quantum technologies
:/home/pino/Zotero/storage/5K8AHT7P/Balandin et al. - 2024 - Electronic noise—From advanced materials to quantu.pdf:application/pdf;Snapshot:/home/pino/Zotero/storage/GWET9MAL/Electronic-noise-From-advanced-materials-to.html:text/html

abstractShort ballistic graphene Josephson junctions sustain superconducting current with a non-sinusoidal current-phase relation up to a critical current threshold. The current-phase relation, arising from proximitized superconductivity, is gate-voltage tunable and exhibits peculiar skewness observed in high quality graphene superconductors heterostructures with clean interfaces. These properties make graphene Josephson junctions promising sensitive quantum probes of microscopic fluctuations underlying transport in two-dimensions. We show that the power spectrum of the critical current fluctuations has a characteristic $$1/f$$dependence on frequency, $$f$$, probing two points and higher correlations of carrier density fluctuations of the graphene channel induced by carrier traps in the nearby substrate. Tunability with the Fermi level, close to and far from the charge neutrality point, and temperature dependence of the noise amplitude are clear fingerprints of the underlying material-inherent processes. Our results suggest a roadmap for the analysis of decoherence sources in the implementation of coherent devices by hybrid nanostructures.
journaltitleCommun Phys
notePublisher: Nature Publishing Group
title1/f critical current noise in short ballistic graphene Josephson junctions
:/home/pino/Zotero/storage/HY4LYH28/Pellegrino et al. - 2020 - 1f critical current noise in short ballistic grap.pdf:application/pdf
ysics,Graphene,Nanoscale devices,Nanoscience and technology

titleOptimized dynamical decoupling in a model quantum memory

journaltitlePhys. Rev. A
titleExperimental Uhrig dynamical decoupling using trapped ions

American Physical Society

journaltitlePhys. Rev. Lett.
titleExperimental Inhibition of Decoherence on Flying Qubits via “Bang-Bang” Control

journaltitlePhys. Rev. A
titlePerformance comparison of dynamical decoupling sequences for a qubit in a rapidly fluctuating spin bath

journaltitleJ. Chem. Phys.
titleOptimized, unequal pulse spacing in multiple echo sequences improves refocusing in magnetic resonance

American Physical Society

journaltitlePhys. Rev. B
titleDynamical decoupling of a single-electron spin at room temperature

journaltitleNew J. Phys.
titleRobust dynamical decoupling with concatenated continuous driving

abstractSuperconducting qubits are leading candidates in the race to build a quantum computer capable of realizing computations beyond the reach of modern supercomputers. The superconducting qubit modality has been used to demonstrate prototype algorithms in the noisy intermediate-scale quantum (NISQ) technology era, in which non-error-corrected qubits are used to implement quantum simulations and quantum algorithms. With the recent demonstrations of multiple high-fidelity, two-qubit gates as well as operations on logical qubits in extensible superconducting qubit systems, this modality also holds promise for the longer-term goal of building largerscale error-corrected quantum computers. In this brief review, we discuss several of the recent experimental advances in qubit hardware, gate implementations, readout capabilities, early NISQ algorithm implementations, and quantum error correction using superconducting qubits. Although continued work on many aspects of this technology is certainly necessary, the pace of both conceptual and technical progress in recent years has been impressive, and here we hope to convey the excitement stemming from this progress.
issn1947-5454, 1947-5462
journaltitleAnnu. Rev. Condens. Matter Phys.
shorttitleSuperconducting Qubits
titleSuperconducting Qubits: Current State of Play
jaergaard et al. - 2020 - Superconducting Qubits Current State of Play.pdf:application/pdf

abstractSystem noise identification is crucial to the engineering of robust quantum systems. Although existing quantum noise spectroscopy (QNS) protocols measure an aggregate amount of noise affecting a quantum system, they generally cannot distinguish between the underlying processes that contribute to it. Here, we propose and experimentally validate a spin-locking-based QNS protocol that exploits the multi-level energy structure of a superconducting qubit to achieve two notable advances. First, our protocol extends the spectral range of weakly anharmonic qubit spectrometers beyond the present limitations set by their lack of strong anharmonicity. Second, the additional information gained from probing the higher-excited levels enables us to identify and distinguish contributions from different underlying noise mechanisms.
journaltitleNat Commun
notePublisher: Nature Publishing Group
titleMulti-level quantum noise spectroscopy
:/home/pino/Zotero/storage/36IVPU4Z/Sung et al. - 2021 - Multi-level quantum noise spectroscopy.pdf:application/pdf

abstractWe study the effects of correlated low-frequency noise sources acting on a two-qubit gate in a fixed coupling scheme. A decoherence model for the spatial and cross-talk correlations is introduced. The efficiency inside the SWAP subspace is analyzed by combining analytic results based on the adiabatic approximation and numerical simulations. Results critically depend on amplitude of the low-frequency noise with respect to the qubits' coupling strength. Correlations between noise sources induce different qualitative behavior depending on the values of the above parameters. The possibility to reduce dephasing due to correlated low-frequency noise by a recalibration protocol is discussed.
journaltitleNew J. Phys.
titleEffects of low-frequency noise cross-correlations in coupled superconducting qubits
mpact of ionizing radiation on superconducting qubit coherence
:/home/pino/Zotero/storage/CUVR74RK/Vepsäläinen et al. - 2020 - Impact of ionizing radiation on superconducting qu.pdf:application/pdf

American Physical Society

abstractWith the help of two single-electron tunneling transistors whose islands were positioned about 100 nm apart, a low-frequency charge noise generated in the Al2O3 substrate has been measured. The signals detected by these electrometers have shown a 10–20 % correlation in power in the 1–10-Hz range. Using a simple model we show that the charge noise sources (fluctuating traps) can be distributed either in thin dielectric layers (including the barriers) adjacent to the islands or, alternatively but more likely, in a volume of the substrate. © 1996 The American Physical Society.
journaltitlePhys. Rev. B
titleBackground Charge Noise in Metallic Single-Electron Tunneling Devices

abstractNoise that exhibits significant temporal and spatial correlations across multiple qubits can be especially harmful to both fault-tolerant quantum computation and quantum-enhanced metrology. However, a complete spectral characterization of the noise environment of even a two-qubit system has not been reported thus far. We propose and experimentally demonstrate a protocol for two-qubit dephasing noise spectroscopy based on continuous-control modulation. By combining ideas from spin-locking relaxometry with a statistically motivated robust estimation approach, our protocol allows for the simultaneous reconstruction of all the single-qubit and two-qubit cross-correlation spectra, including access to their distinctive nonclassical features. Only single-qubit control manipulations and state-tomography measurements are employed, with no need for entangled-state preparation or readout of two-qubit observables. While our experimental demonstration uses two superconducting qubits coupled to a shared, colored engineered noise source, our methodology is portable to a variety of dephasing-dominated qubit architectures. By pushing quantum noise spectroscopy beyond the single-qubit setting, our work heralds the characterization of spatiotemporal correlations in both engineered and naturally occurring noise environments.
journaltitlePRX Quantum
notePublisher: American Physical Society
titleTwo-Qubit Spectroscopy of Spatiotemporally Correlated Quantum Noise in Superconducting Qubits
JKT/von Lüpke et al. - 2020 - Two-Qubit Spectroscopy of Spatiotemporally Correla.pdf:application/pdf

abstractSemiconductor qubits have a small footprint and so are appealing for building densely integrated quantum processors. However, fabricating them at high densities raises the issue of noise correlated across different qubits, which is of practical concern for scalability and fault tolerance. Here, we analyse and quantify the degree of noise correlation in a pair of neighbouring silicon spin qubits around 100 nm apart. We observe strong interqubit noise correlations with a correlation strength as large as 0.7 at around 1 Hz, even in the regime where the spin–spin exchange interaction contributes negligibly. We find that fluctuations of single-spin precession rates are strongly correlated with exchange noise, showing that they have an electrical origin. Noise cross-correlations have thus enabled us to pinpoint the most influential noise in our device. Our work presents a powerful tool set to assess and identify the noise acting on multiple qubits and highlights the importance of long-range electric noise in densely packed silicon spin qubits.
journaltitleNat. Phys.
noteNumber: 12 Publisher: Nature Publishing Group
titleNoise-correlation spectrum for a pair of spin qubits in silicon
:/home/pino/Zotero/storage/KTNPG3YD/Yoneda et al. - 2023 - Noise-correlation spectrum for a pair of spin qubi.pdf:application/pdf

abstractCorrelated noise across multiple qubits poses a significant challenge for achieving scalable and fault-tolerant quantum processors. Despite recent experimental efforts to quantify this noise in various qubit architectures, a comprehensive understanding of its role in qubit dynamics remains elusive. Here, we present an analytical study of the dynamics of driven qubits under spatially correlated noise, including both Markovian and non-Markovian noise. Surprisingly, we find that, while correlated classical noise only leads to correlated decoherence without increasing the quantum coherence in the system, the correlated quantum noise can be exploited to generate entanglement. In particular, we reveal that, in the quantum limit, pure dephasing noise induces a coherent long-range two-qubit Ising interaction that correlates distant qubits. In contrast, for purely transverse noise when qubits are subjected to coherent drives, the correlated quantum noise induces both coherent symmetric exchange and Dzyaloshinskii-Moriya interaction between the qubits, as well as correlated relaxation, both of which give rise to significant entanglement. Remarkably, in this case, we uncover that the system exhibits distinct dynamical phases in different parameter regimes. Finally, we reveal the impact of spatio-temporally correlated 1/f noise on the decoherence rate, and how its temporal correlations restore lost entanglement. Our analysis not only offers critical insights into designing effective error mitigation strategies to reduce harmful effects of correlated noise, but also enables tailored protocols to leverage and harness noise-induced correlations for quantum information processing.
notearXiv:2308.03054 [cond-mat, physics:quant-ph]
shorttitleSpatially correlated classical and quantum noise in driven qubits
titleSpatially correlated classical and quantum noise in driven qubits: The good, the bad, and the ugly
ge/9EV3ZB2W/Zou et al. - 2023 - Spatially correlated classical and quantum noise i.pdf:application/pdf;arXiv.org Snapshot:/home/pino/Zotero/storage/MQJQWVHB/2308.html:text/html
ysics,Condensed Matter - Mesoscale and Nanoscale Physics

American Physical Society

abstractWe study the quantum dynamics of a many-body system subject to coherent evolution and coupled to a non-Markovian bath. We propose a technique to unravel the non-Markovian dynamics in terms of quantum jumps, a connection that was so far only understood for single-body systems. We develop a systematic method to calculate the probability of a quantum trajectory and formulate it in a diagrammatic structure. We find that non-Markovianity renormalizes the probability of realizing a quantum trajectory and that memory effects can be interpreted as a perturbation on top of the Markovian dynamics. We show that the diagrammatic structure is akin to that of a Dyson equation and that the probability of the trajectories can be calculated analytically. We then apply our results to study the measurement-induced entanglement transition in random unitary circuits. We find that non-Markovianity does not significantly shift the transition but stabilizes the volume law phase of the entanglement by shielding it from transient strong dissipation.
journaltitlePhys. Rev. B
shorttitleDiagrammatic Method for Many-Body Non-Markovian Dynamics
titleDiagrammatic Method for Many-Body Non-Markovian Dynamics: Memory Effects and Entanglement Transitions

journaltitleSciPost Phys. Core
titleMeasurement induced transitions in non-Markovian free fermion ladders

American Physical Society

abstractWe describe a scheme for using an all-electrical, rapid, adiabatic population transfer between two spatially separated dots in a triple-quantum dot system. The electron spends no time in the middle dot and does not change its energy during the transfer process. Although a coherent population transfer method, this scheme may well prove useful in incoherent electronic computation (for example quantum-dot cellular automata) where it may provide a coherent advantage to an otherwise incoherent device. It can also be thought of as a limiting case of type II quantum computing, where sufficient coherence exists for a single gate operation, but not for the preservation of superpositions after the operation. We extend our analysis to the case of many intervening dots and address the issue of transporting quantum information through a multi-dot system.
journaltitlePhys. Rev. B
titleCoherent Electronic Transfer in Quantum Dot Systems Using Adiabatic Passage

Andriy Burkov Quebec City, QC, Canada

titleThe Hundred-Page Machine Learning Book

Beijing Boston Farnham Sebastopol Tokyo

editionThird edition
shorttitleHands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow
titleHands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems

Cambridge, Massachusetts

The MIT Press

seriesAdaptive Computation and Machine Learning
titleDeep Learning
Machine learning

American Physical Society

abstractThe authors discuss the technique of stimulated Raman adiabatic passage (STIRAP), a method of using partially overlapping pulses (from pump and Stokes lasers) to produce complete population transfer between two quantum states of an atom or molecule. The procedure relies on the initial creation of a coherence (a population-trapping state) with subsequent adiabatic evolution. The authors present the basic theory, with some extensions, and then describe examples of experimental utilization. They note some applications of the technique not only to preparation of selected states for reaction studies, but also to quantum optics and atom optics.
journaltitleRev. Mod. Phys.
titleCoherent Population Transfer among Quantum States of Atoms and Molecules

Nature Publishing Group

abstractSemiconductor qubits have a small footprint and so are appealing for building densely integrated quantum processors. However, fabricating them at high densities raises the issue of noise correlated across different qubits, which is of practical concern for scalability and fault tolerance. Here, we analyse and quantify the degree of noise correlation in a pair of neighbouring silicon spin qubits around 100 nm apart. We observe strong interqubit noise correlations with a correlation strength as large as 0.7 at around 1 Hz, even in the regime where the spin–spin exchange interaction contributes negligibly. We find that fluctuations of single-spin precession rates are strongly correlated with exchange noise, showing that they have an electrical origin. Noise cross-correlations have thus enabled us to pinpoint the most influential noise in our device. Our work presents a powerful tool set to assess and identify the noise acting on multiple qubits and highlights the importance of long-range electric noise in densely packed silicon spin qubits.
journaltitleNat. Phys.
titleNoise-Correlation Spectrum for a Pair of Spin Qubits in Silicon
Quantum information,Qubits

Cambridge ; New York

Cambridge University Press

titleOptical Coherence and Quantum Optics
Coherence (Optics),Quantum optics

journaltitleReviews of Modern Physics
titleStimulated Raman Adiabatic Passage in Physics, Chemistry, and Beyond

American Physical Society

abstractWe introduce an adiabatic transfer protocol for spin states in large quantum dot arrays that is based on time-dependent modulation of the Heisenberg exchange interaction in the presence of a magnetic field gradient. We refer to this protocol as spin-CTAP (coherent transport by adiabatic passage) in analogy to a related protocol developed for charge-state transfer in quantum dot arrays. The insensitivity of this adiabatic protocol to pulse imperfections has potential advantages for reading out extended spin qubit arrays. When the static exchange interaction varies across the array, a quantum-controlled version of spin-CTAP is possible, where the transfer process is conditional on the spin states in the middle of the array. This conditional operation can be used to generate N-qubit entangled GHZ states. Using a realistic noise model, we analyze the robustness of the spin-CTAP operations and find that high-fidelity ($>$95%) spin eigenstate transfer and GHZ state preparation is feasible in current devices.
journaltitlePhys. Rev. B
titleCoherent Transport of Spin by Adiabatic Passage in Quantum Dot Arrays

journaltitleZeitschrift für Physik
titleBeweis Des Adiabatensatzes

North-Holland, Amsterdam

titleQuantum Mechanics, Vol. 1/2

issn1050-2947, 1094-1622
journaltitlePhys. Rev. A
titleThree-Level Superadiabatic Quantum Driving

Berlin Heidelberg

edition3. ed., [Nachdr.]
seriesSpringer Series in Synergetics
shorttitleQuantum Noise
titleQuantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics

abstractDeep neural network acoustic models produce substantial gains in large vocabulary continuous speech recognition systems. Emerging work with rectified linear (ReL) hidden units demonstrates additional gains in final system performance relative to more commonly used sigmoidal nonlinearities. In this work, we explore the use of deep rectifier networks as acoustic models for the 300 hour Switchboard conversational speech recognition task. Using simple training procedures without pretraining, networks with rectifier nonlinearities produce 2% absolute reductions in word error rates over their sigmoidal counterparts. We analyze hidden layer representations to quantify differences in how ReL units encode inputs as compared to sigmoidal units. Finally, we evaluate a variant of the ReL unit with a gradient more amenable to optimization in an attempt to further improve deep rectifier networks.
booktitleICML Workshop on Deep Learning for Audio, Speech, and Language Processing (WDLASL 2013)
titleRectifier Nonlinearities Improve Neural Network Acoustic Models

Pacific Grove, CA, USA

booktitle2018 52nd Asilomar Conference on Signals, Systems, and Computers
shorttitleWhy RELU Units Sometimes Die
titleWhy RELU Units Sometimes Die: Analysis of Single-Unit Error Backpropagation in Neural Networks

titleTensorFlow: Large-scale Machine Learning on Heterogeneous Systems

Cambridge University Press

abstractThis book is an up-to-date introduction to the quantum theory of measurement. Although the main principles of the field were elaborated in the 1930s by Bohr, Schrödinger, Heisenberg, von Neuman, and Mandelstam, it was not until the 1980s that technology became sufficiently advanced to allow its application in real experiments. Quantum measurement is now central to many ultra-high technology developments, such as "squeezed light," single atom traps, and searches for gravitational radiation. It is also considered to have great promise for computer science and engineering, particularly for its applications in information processing and transfer. The book begins with a brief introduction to the relevant theory and goes on to discuss all aspects of the design of practical quantum measurement systems.
noteGoogle-Books-ID: 42xfQgAACAAJ
titleQuantum Measurement
Science / Physics / Quantum Theory

abstractSignificant advances have been made towards fault-tolerant operation of silicon spin qubits, with single qubit fidelities exceeding 99.9%, several demonstrations of two-qubit gates based on exchange coupling, and the achievement of coherent single spin-photon coupling. Coupling arbitrary pairs of spatially separated qubits in a quantum register poses a significant challenge as most qubit systems are constrained to two dimensions with nearest neighbor connectivity. For spins in silicon, new methods for quantum state transfer should be developed to achieve connectivity beyond nearest-neighbor exchange. Here we demonstrate shuttling of a single electron across a linear array of nine series-coupled silicon quantum dots in ~50 ns via a series of pairwise interdot charge transfers. By constructing more complex pulse sequences we perform parallel shuttling of two and three electrons at a time through the array. These experiments demonstrate a scalable approach to physically transporting single electrons across large silicon quantum dot arrays.
journaltitleNat Commun
notePublisher: Nature Publishing Group
titleShuttling a single charge across a one-dimensional array of silicon quantum dots
:/home/pino/Zotero/storage/XDR2QB3A/Mills et al. - 2019 - Shuttling a single charge across a one-dimensional.pdf:application/pdf

abstractThe adiabatic manipulation of quantum states is a powerful technique that opened up new directions in quantum engineering—enabling tests of fundamental concepts such as geometrical phases and topological transitions, and holding the promise of alternative models of quantum computation. Here we benchmark the stimulated Raman adiabatic passage for circuit quantum electrodynamics by employing the first three levels of a transmon qubit. In this ladder configuration, we demonstrate a population transfer efficiency >80% between the ground state and the second excited state using two adiabatic Gaussian-shaped control microwave pulses. By doing quantum tomography at successive moments during the Raman pulses, we investigate the transfer of the population in time domain. Furthermore, we show that this protocol can be reversed by applying a third adiabatic pulse, we study a hybrid nondiabatic–adiabatic sequence, and we present experimental results for a quasi-degenerate intermediate level.
journaltitleNat Commun
notePublisher: Nature Publishing Group
titleStimulated Raman adiabatic passage in a three-level superconducting circuit
:/home/pino/Zotero/storage/GFY7CR3Y/Kumar et al. - 2016 - Stimulated Raman adiabatic passage in a three-leve.pdf:application/pdf
ysics,Quantum mechanics,Superconducting properties and materials

abstractStimulated Raman adiabatic passage offers significant advantages for coherent population transfer between uncoupled or weakly coupled states and has the potential of realizing efficient quantum gate, qubit entanglement and quantum information transfer. Here we report on the realization of the process in the superconducting Xmon and phase qutrits—two ladder-type three-level systems in which the ground state population is coherently transferred to the second excited state via the dark state subspace. We demonstrate that the population transfer efficiency is no less than 96% and 67% for the two devices, which agree well with the numerical simulation of the master equation. Population transfer via stimulated Raman adiabatic passage is significantly more robust against variations of the experimental parameters compared with that via the conventional resonant π pulse method. Our work opens up a new venue for exploring the process for quantum information processing using the superconducting artificial atoms.
journaltitleNat Commun
notePublisher: Nature Publishing Group
titleCoherent population transfer between uncoupled or weakly coupled states in ladder-type superconducting qutrits
:/home/pino/Zotero/storage/ENI5TBNB/Xu et al. - 2016 - Coherent population transfer between uncoupled or .pdf:application/pdf

American Physical Society

abstractReaching a given target quantum state with high fidelity and fast operation speed close to the quantum limit represents an important goal in quantum information science. Here we experimentally demonstrate superadiabatic quantum driving to achieve population transfer in a three-level solid-state spin system at ambient conditions. Starting from traditional stimulated Raman adiabatic passage (STIRAP), our approach implements superadiabatic corrections to the STIRAP Hamiltonian with several pulse shapes, including the paradigmatic Gaussian shapes and trigonometric pulse shapes that guarantee an operation speed equal to the quantum limit. It requires no intense microwave pulses or long transfer times and shows an enhanced robustness against pulse imperfections. Thereby, these results might provide a useful tool for quantum information processing and the coherent manipulation of quantum systems.
journaltitlePhys. Rev. A
titleTwo-Photon-Transition Superadiabatic Passage in a Nitrogen-Vacancy Center in Diamond

abstractThe rapid experimental progress in the field of superconducting nanocircuits gives rise to an increasing quest for advanced quantum-control techniques for these macroscopically coherent systems. Here we demonstrate theoretically that stimulated Raman adiabatic passage should be possible with the quantronium setup of a Cooper-pair box. The scheme appears to be robust against decoherence and should be realizable even with the existing technology. As an application we present a method to generate single-phonon states of a nanomechanical resonator by vacuum-stimulated adiabatic passage with the superconducting nanocircuit coupled to the resonator.
journaltitlePhys. Rev. B
notePublisher: American Physical Society
shorttitleAdvanced control with a Cooper-pair box
titleAdvanced control with a Cooper-pair box: Stimulated Raman adiabatic passage and Fock-state generation in a nanomechanical resonator
dvanced control with a Cooper-pair box Stimulate.pdf:application/pdf

abstractThe implementation of a Lambda scheme in superconducting artificial atoms could allow detection of stimulated Raman adiabatic passage (STIRAP) and other quantum manipulations in the microwave regime. However, symmetries which on one hand protect the system against decoherence yield selection rules which may cancel coupling to the pump external drive. The tradeoff between efficient coupling and decoherence due to broad-band colored noise (BBCN), which is often the main source of decoherence, is addressed in the class of nanodevices based on the Cooper pair box (CPB) design. We study transfer efficiency by STIRAP, showing that substantial efficiency is achieved for off-symmetric bias only in the charge-phase regime. We find a number of results uniquely due to non-Markovianity of BBCN, namely (a) the efficiency for STIRAP depends essentially on noise channels in the trapped subspace; (b) low-frequency fluctuations can be analyzed and represented as fictitious correlated fluctuations of the detunings of the external drives; and (c) a simple figure of merit for design and operating prescriptions allowing the observation of STIRAP is proposed. The emerging physical picture also applies to other classes of coherent nanodevices subject to BBCN.
journaltitlePhys. Rev. B
notePublisher: American Physical Society
titleDesign of a Lambda system for population transfer in superconducting nanocircuits

abstractWe realize a Λ system in a superconducting circuit, with metastable states exhibiting lifetimes up to 8 ms. We exponentially suppress the tunneling matrix elements involved in spontaneous energy relaxation by creating a “heavy” fluxonium, realized by adding a capacitive shunt to the original circuit design. The device allows for both cavity-assisted and direct fluorescent readouts, as well as state preparation schemes akin to optical pumping. Since direct transitions between the metastable states are strongly suppressed, we utilize Raman transitions for coherent manipulation of the states.
journaltitlePhys. Rev. Lett.
notePublisher: American Physical Society
titleRealization of a $\mathrm{\ensuremath{\Lambda}}$ System with Metastable States of a Capacitively Shunted Fluxonium
HHHALMB/Earnest et al. - 2018 - Realization of a $mathrm ensuremath Lambda $ S.pdf:application/pdf;APS Snapshot:/home/pino/Zotero/storage/MIEX7BFK/PhysRevLett.120.html:text/html

abstractWe propose a protocol for the manipulation of a three-level artificial atom in Lambda (Λ) configuration. It allows faithful, selective and robust population transfer analogous to stimulated Raman adiabatic passage (Λ-STIRAP), in last-generation superconducting artificial atoms, where protection from noise implies the absence of a direct pump coupling. It combines the use of a two-photon pump pulse with suitable advanced control, operated by a slow modulation of the phase of the external fields, leveraging on the stability of semiclassical microwave drives. This protocol is a building block for manipulation of microwave photons in complex quantum architectures. Its demonstration would be a benchmark for the implementation of a class of multilevel advanced control procedures for quantum computation and microwave quantum photonics in systems based on artificial atoms.
journaltitlePhys. Rev. A
notePublisher: American Physical Society
titleCoherent manipulation of noise-protected superconducting artificial atoms in the Lambda scheme
.93.html:text/html;Submitted Version:/home/pino/Zotero/storage/SADZNR4C/Di Stefano et al. - 2016 - Coherent manipulation of noise-protected supercond.pdf:application/pdf

abstractWe study the effect of low-frequency noise in ac-driven two- or many-level coherent nanodevices. Fluctuations in the properties of the device are translated into equivalent fluctuations of the driving fields. The impact on Rabi oscillations can be modulated with the detuning and minimized at resonance. In three-level atoms slow noise produces qualitative changes for protocols as coherent population transfer. We propose a strategy allowing us to operate at parity symmetry points, where the device is well protected against noise, despite selection rules preventing direct couplings to external fields of involved transitions.
issn0031-8949, 1402-4896
journaltitlePhys. Scr.
titleEffects of Low-Frequency Noise in Driven Coherent Nanodevices

issn0587-4254, 1509-5770
journaltitleActa Phys. Pol. B
titleRelaxation Phenomena in classical and Quantum systems
S/Spagnolo et al. - 2012 - [No title found].pdf:application/pdf

abstractSince the first observation of coherent quantum behaviour in a superconducting qubit, now more than 20 years ago, there have been substantial developments in the field of superconducting quantum circuits. One such advance is the introduction of the concepts of cavity quantum electrodynamics (QED) to superconducting circuits, to yield what is now known as circuit QED. This approach realizes in a single architecture the essential requirements for quantum computation, and has already been used to run simple quantum algorithms and to operate tens of superconducting qubits simultaneously. For these reasons, circuit QED is one of the leading architectures for quantum computation. In parallel to these advances towards quantum information processing, circuit QED offers new opportunities for the exploration of the rich physics of quantum optics in novel parameter regimes in which strongly nonlinear effects are readily visible at the level of individual microwave photons. We review circuit QED in the context of quantum information processing and quantum optics, and discuss some of the challenges on the road towards scalable quantum computation.
journaltitleNat. Phys.
notePublisher: Nature Publishing Group
titleQuantum information processing and quantum optics with circuit quantum electrodynamics
:/home/pino/Zotero/storage/JBGRVHGQ/Blais et al. - 2020 - Quantum information processing and quantum optics .pdf:application/pdf

abstractLight-matter interaction, and the understanding of the fundamental physics behind, is the scenario of emerging quantum technologies. Solid state devices allow the exploration of new regimes where ultrastrong coupling strengths are comparable to subsystem energies, and new exotic phenomena like quantum phase transitions and ground-state entanglement occur. While experiments so far provided only spectroscopic evidence of ultrastrong coupling, we propose a new dynamical protocol for detecting virtual photon pairs in the dressed eigenstates. This is the fingerprint of the violated conservation of the number of excitations, which heralds the symmetry broken by ultrastrong coupling. We show that in flux-based superconducting architectures this photon production channel can be coherently amplified by Stimulated Raman Adiabatic Passage, providing a unique tool for an unambiguous dynamical detection of ultrastrong coupling in present day hardware. This protocol could be a benchmark for control of the dynamics of ultrastrong coupling architectures, in view of applications to quantum information and microwave quantum photonics.
journaltitleSci Rep
notePublisher: Nature Publishing Group
titleUltrastrong coupling probed by Coherent Population Transfer
:/home/pino/Zotero/storage/C8EUKHVK/Falci et al. - 2019 - Ultrastrong coupling probed by Coherent Population.pdf:application/pdf

abstractLight-matter interaction and understanding the fundamental physics behind is essential for emerging quantum technologies. Solid-state devices may explore new regimes where coupling strengths are “ultrastrong”, i.e., comparable to the energies of the subsystems. New exotic phenomena occur the common root of many of them being the fact that the entangled vacuum contains virtual photons. They herald the lack of conservation of the number of excitations which is the witness of ultrastrong coupling breaking the U(1) symmetry. Despite more than a decade of research, the detection of ground-state virtual photons still awaits demonstration. In this work, we recognize the “conspiring” set of experimental challenges and show how to overcome them, thus providing a solution to this long-standing problem. We find that combining a superinductor-based unconventional “light fluxonium” qudit and coherent control yields a highly efficient, faithful, and selective conversion of virtual photons into real ones. This enables their detection with resources available to present-day quantum technologies.
journaltitlePhys. Rev. Res.
notePublisher: American Physical Society
titleDetecting virtual photons in ultrastrongly coupled superconducting quantum circuits

abstractIn this paper we propose a protocol to achieve coherent population transfer between two states in a three-level atom by using two ac fields. It is based on the physics of stimulated Raman adiabatic passage (STIRAP), but it is implemented with the constraint of a reduced control, namely, one of the fields cannot be switched off. A combination of frequency chirps is used with resonant fields, allowing us to achieve approximate destructive interference, despite the fact that an exact dark state does not exist. This chirped STIRAP protocol is tailored for applications to artificial atoms, where architectures with several elementary units can be strongly coupled but where the possibility of switching on and off such couplings is often very limited. Demonstration of this protocol would be a benchmark for the implementation of a class of multilevel advanced control procedures for quantum computation and microwave quantum photonics in artificial atoms.
journaltitlePhys. Rev. B
notePublisher: American Physical Society
titlePopulation transfer in a Lambda system induced by detunings
APS Snapshot:/home/pino/Zotero/storage/86D2XU8K/PhysRevB.91.html:text/html;Submitted Version:/home/pino/Zotero/storage/N2QLRX3C/Di Stefano et al. - 2015 - Population transfer in a Lambda system induced by .pdf:application/pdf