Original software publication| Volume 12, 100260, May 2022
• PDF [631 KB]PDF [631 KB]
• Top

# Noise fingerprints in quantum computers: Machine learning software tools

Open AccessPublished:February 18, 2022

## Highlights

• Quantum noise leaves a fingerprint that can be identified.
• We develop Machine Learning models to classify different quantum noise fingerprints.
• We adopt Support Vector Machines as Machine Learning method.
• We propose a way to discern if quantum noise is machine dependent.
• We can evaluate if quantum noise exhibits a dependency on the execution time.

## Abstract

In this paper we present the high-level functionalities of a quantum–classical machine learning software, whose purpose is to learn the main features (the fingerprint) of quantum noise sources affecting a quantum device, as a quantum computer. Specifically, the software architecture is designed to classify successfully (more than 99% of accuracy) the noise fingerprints in different quantum devices with similar technical specifications, or distinct time-dependences of a noise fingerprint in single quantum machines.

## Keywords

Tabled 1
 Current code version v1 Permanent link to code/repository used for this code version https://github.com/SoftwareImpacts/SIMPAC-2022-9 Permanent link to Reproducible Capsule https://codeocean.com/capsule/8363708/tree/v1 Legal Code License GNU General Public License v3.0 Code versioning system used GIT Software code languages, tools, and services used python, IBM quantum services Compilation requirements, operating environments & dependencies numpy, qiskit, qiskit_terra, scikit_learn If available Link to developer documentation/manual Support email for questions [email protected]

## 1. Introduction

Quantum technologies are a fast developing scientific and industrial field [
• Dowling J.P.
• Milburn G.J.
Quantum technology: the second quantum revolution.
]. They have been already implemented in several different platforms, as for instance photonic circuits [
• O’Brien J.L.
• Furusawa A.
• Vučković J.
Photonic quantum technologies.
,
• Wang J.
• Sciarrino F.
• Laing A.
• Thompson M.G.
Integrated photonic quantum technologies.
], but also Rydberg atoms [
• Pritchard J.D.
• Shaffer J.P.
Rydberg atom quantum technologies.
], superconducting devices [
• Devoret Michel H.
• Wallraff Andreas
• Martinis John M.
Superconducting qubits: A short review.
] and others. Likely, the most promising quantum technology is represented by quantum computers, i.e., quantum devices for quantum computing, among which it is worth mentioning superconducting circuits [
• Clarke John
• Wilhelm Frank K.
Superconducting quantum bits.
,
• Wu Y.
• Bao W.
• Cao S.
• et al.
Strong quantum computational advantage using a superconducting quantum processor.
], trapped-ions quantum computers [
• Wineland David J
• Barrett Murray
• Britton Joseph
• Chiaverini J
• DeMarco B
• Itano Wayne M
• Jelenković B
• Langer Christopher
• Leibfried Dietrich
• Meyer V
• et al.
Quantum information processing with trapped ions.
,
• Pogorelov I.
• Feldker T.
• Marciniak Ch. D.
• Postler L.
• Jacob G.
• Krieglsteiner O.
• Podlesnic V.
• Meth M.
• Negnevitsky V.
• Höfer B.
• Wächter C.
• Lakhmanskiy K.
• Blatt R.
• Schindler P.
• Monz T.
Compact ion-trap quantum computing demonstrator.
], photonic chips [
• Spring Justin B
• Metcalf Benjamin J
• Humphreys Peter C
• Kolthammer W Steven
• Jin Xian-Min
• Barbieri Marco
• Datta Animesh
• Thomas-Peter Nicholas
• Langford Nathan K
• Kundys Dmytro
• et al.
Boson sampling on a photonic chip.
,
• Metcalf Benjamin J
• Spring Justin B
• Humphreys Peter C
• Thomas-Peter Nicholas
• Barbieri Marco
• Kolthammer W Steven
• Jin Xian-Min
• Langford Nathan K
• Kundys Dmytro
• Gates James C
• et al.
Quantum teleportation on a photonic chip.
] and topological qubits [
• Freedman Michael
• Kitaev Alexei
• Larsen Michael
• Wang Zhenghan
Topological quantum computation.
]. Both academic laboratories and industrial companies are devoting lots of effort and funding to boost research and technological improvements, towards the so-called quantum supremacy [
• Arute F.
• Arya K.
• Babbush R.
• et al.
Quantum supremacy using a programmable superconducting processor.
], i.e., a quantum advantage to solve (numerical) problems that no classical computer will never solve. The actual drawback of these devices is the absence of a standard hardware (and thus even software) architecture, on which research activities may be jointly coordinated. For each of these platforms, indeed, ad hoc solutions are proposed and then realized, and this makes such a technologies still very expensive and incompatible from a device to another.
However, in quantum computing, the main issue to be still solved is the unavoidable presence of external noise sources that dramatically limit the accuracy of quantum computations, as well as the large-scale realization of quantum circuits and algorithms. The negative impact of noise on quantum computing is so noticeable that the acronym Noisy Intermediate-Scale Quantum (NISQ) technology has been recently introduced [
• Preskill John
Quantum computing in the NISQ era and beyond.
]. Furthermore, commercial quantum devices as for example the quantum computers by the companies Q-IBM® [] and Rigetti® [], albeit they have been made available by anyone who creates a free account on their database, are not physically accessible and several specifications on the chip’s parameters are not made public.
In the paper [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
], we have recently observed on some IBM quantum computers that main features of the noise sources affecting the devices are specific of each single computer and have a clear time-dependence. For such a purpose, a testbed quantum circuit – composed by a fixed number of qubits – is designed, then made run for a sufficient number of times and finally locally measured in correspondence of each qubit. From the measurements of the qubits (the measurement observable was the Pauli matrix $σz$), a set of measurement outcomes is recorded, collected, and then used to train a machine learning (ML) algorithm [
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
,
• Hastie Trevor
• Tibshirani Robert
• Friedman Jerome
The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
]. However, it is worth noting that in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
] the features of the quantum noise are not reconstructed but just classified from a quantum device to another. Specifically, the classification task was successfully carried out by means of a support vector machine (SVM) [
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
,
• Hastie Trevor
• Tibshirani Robert
• Friedman Jerome
The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
], with a classification accuracy equal or greater than 99%. Hence, thanks to our procedure, one just needs to collect an informative statistics of quantum measurement outcomes (that are quantum data) from the testbed quantum circuit, and subsequently train ML (classical) algorithms. In fact, no quantum noise modelling is required nor, in principle, the testbed circuit has to be controlled by time-dependent pulses [
• Müller Matthias M.
• Gherardini Stefano
• Calarco Tommaso
• Montangero Simone
• Caruso Filippo
Information theoretical limits for quantum optimal control solutions: Error scaling of noisy channels.
]. Also for these reasons, the use of a ML technique is the most natural choice to perform classification, since it naturally provides a black-box model with predictive outcomes. In this regard, we recall that in the current literature ML has been already adopted to distinguish open quantum dynamics [
• Youssry Akram
• Paz-Silva Gerardo A.
• Ferrie Christopher
Beyond quantum noise spectroscopy: modelling and mitigating noise with quantum feature engineering.
,
• Luchnikov I.A.
• Vintskevich S.V.
• Grigoriev D.A.
• Filippov S.N.
Machine learning non-Markovian quantum dynamics.
,
• Fanchini Felipe F.
• Karpat Göktu ğ
• Rossatto Daniel Z.
• Norambuena Ariel
• Coto Raúl
Estimating the degree of non-Markovianity using machine learning.
] and to perform quantum sensing tasks [
• Niu Murphy Yuezhen
• Klimov Paul
• Boixo Sergio
• Barends Rami
• Kelly Julian
• Chen Yu
• Arya Kunal
• Burkett Brian
• Bacon Dave
• et al.
Learning non-Markovian quantum noise from moiré-enhanced swap spectroscopy with deep evolutionary algorithm.
,
• Harper Robin
• Flammia Steven T.
• Wallman Joel J.
Efficient learning of quantum noise.
,
• Martina Stefano
• Gherardini Stefano
• Caruso Filippo
Machine learning approach for quantum non-Markovian noise classification.
,
• Wise David F.
• Morton John J.L.
• Dhomkar Siddharth
Using deep learning to understand and mitigate the qubit noise environment.
], as for example the learning and classification of non-Markovian noise [
• Niu Murphy Yuezhen
• Klimov Paul
• Boixo Sergio
• Barends Rami
• Kelly Julian
• Chen Yu
• Arya Kunal
• Burkett Brian
• Bacon Dave
• et al.
Learning non-Markovian quantum noise from moiré-enhanced swap spectroscopy with deep evolutionary algorithm.
,
• Martina Stefano
• Gherardini Stefano
• Caruso Filippo
Machine learning approach for quantum non-Markovian noise classification.
] or the detection of qubits correlations [
• Harper Robin
• Flammia Steven T.
• Wallman Joel J.
Efficient learning of quantum noise.
].
As depicted in Fig. 1, our software architecture adopted in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
] has two distinct parts: The one on the left of the figure generates the testbed quantum circuit (see Section 2), while the other is designed for the implementation of the ML models that classify quantum noise fingerprints (refer to Section 3).

## 2. Testbed quantum circuit

For our experiments of quantum noise classification in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
], we made use of the IBM Quantum cloud services to remotely run quantum circuits on different machines. In particular, to interact with the cloud services, one can use the Qiskit sdk [
• Abraham Héctor
• et al.
Qiskit: An open-source framework for quantum computing.
] that is an open-source python package, useful both to simulate quantum dynamics and to program a given set of operations on a real quantum computer. Currently, one has at disposal up to $11$ superconducting quantum computers, ranging from a single qubit up to $15$ qubits, with different topology and calibration routines. For all the available devices and their specifications, we direct the reader to the IBM documentation [].
In Algorithm 1 we provide the pseudo-code for the generation of the testbed quantum circuit in its baseline version (see also panel (c) in Fig. 2 for a pictorial representation of the circuit) realized in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
] to carry out the classification of noise fingerprints. For the quantum computation in the aforementioned circuit, we made use of standard gates whose mathematical definitions is given in terms of matrices that one can easily find in quantum computing textbooks [
• Nielsen Michael A.
• Chuang Isaac
Quantum Computation and Quantum Information.
]. The baseline version of the testbed quantum circuit is repeated 3 times overall for a total of 9 measurements (also denoted as measurement steps) of both qubits 2 and 3. Indeed, operationally, the 9 measurement steps are not performed all in the same run (i.e., sequentially), but on consecutive runs by implementing incremental parts of the quantum circuit. To clarify this aspect as much as possible, in Fig. 2 we have represented pictorially the first 4 measurement steps of the testbed quantum circuit, whose baseline version (returned by Algorithm 1) is depicted in the panel (c) of the figure. Specifically, first we execute the part of the circuit that is obtained by cutting the testbed quantum circuit after the first measurement step, i.e., after the measurement of qubit 2 and 3 following the Hadamard gates on the 0th and 1th qubits and the CNOT gate from qubit $0$ to $2$). Then, the measurement outcomes are recorded. Subsequently, we execute part of the testbed quantum circuit until the second measurement step (measurements of qubits 2 and 3 included), thus by also taking into account the $X$-gate on the 0th qubit and the CNOT gate from qubit $1$ to $3$, and again we record the measurement outcomes. The procedure is then repeated for all the 9 measurement steps. Before proceeding, it is worth stressing that, for each implemented quantum circuit, the measurements of the qubits 2 and 3 are performed only at the end of the circuits.
Overall, in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
], several experiments have been conducted on different IBM chips that have different physical specifications, as the architecture of the qubits or the quantum volume [
• Cross Andrew W
• Bishop Lev S
• Sheldon Sarah
• Nation Paul D
• Gambetta Jay M
Validating quantum computers using randomized model circuits.
]. Some quantum machines, indeed, are inherently noisier than other, and even single qubits inside a machine can have a distinctive noise profile. All these peculiar differences in noise and topology contribute to the fingerprint that we aim to classify using our ML method.

### 2.1 Data acquisition

The pipeline designed for the creation of the dataset, set as input of the ML models, is constituted of several scripts that can be customized according to the needs of the user. First, for each implemented quantum circuit, the script createCircuit.py adopts Qiskit to interact with the IBM quantum services for the measurement of a predefined number of circuit executions. Specifically, such a script is parameterized to launch the runs of the circuits on several quantum machines with a specific amount of parallel tasks. The runs are executed in two different modalities that generates the datasets that we called FAST and SLOW in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
]. In the former dataset, the aim was to collect as many runs as possible in the shortest time interval. For this purpose, the script launches 20 parallel processes, each of them adds to the IBM queue a predefined number of runs with 8000 execution-shots. After that, each batch of 8000 execution-shots is split into 8 batches of 1000 shots that are then employed to compute the outcomes’ probabilities. Instead, for the dataset named SLOW, we collect a sequence of measurement outcomes that are uniformly distributed over time. To obtain such dataset, the script launches only one run at a time with 1000 execution-shots and waits two minutes from a run to another.
The second script in the pipeline is extractExecutions.py, whose objective is to compute the probabilities of the measurement outcomes from the raw data returned by the calls to the IBM quantum services (this is ensured by the previous script). After that, either createDataset.py and createDatasetTimeSeries.py pack the probability distributions in datasets. The difference between such scripts is the following. The former builds binary or multiclass classification dataset using data from at least two quantum machines, while the latter builds classification datasets with data collected in a single machine and labelled by the time interval in which the testbed quantum circuit is executed.
In the github repository at https://github.com/trianam/learningQuantumNoiseFingerprint and on CodeOcean at https://codeocean.com/capsule/fa6e1d85-c99f-4a38-9c16-ac204da85040/, we release the source code of all the scripts and all the data obtained from the execution of createCircuit.py on each quantum machine we employed.

## 3. Machine learning models

### 3.1 Support vector machine

In [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
] we have successfully classified the noise fingerprints on several IBM quantum computers by training Support Vector Machine models [
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
,
• Hastie Trevor
• Tibshirani Robert
• Friedman Jerome
The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
]. SVM is a machine learning technique that is usually used to solve binary classification tasks. Generally speaking, a SVM model is trained on a dataset composed of pairs $(xi,yi)$, where $xi$ are points in a certain space $Rn$ of dimension $n$, and $yi$ is equal to $1$ or $−1$ depending the corresponding point belongs to one or the other class. The Support Vector Machine is trained to find the hyperplane that divides the space representation of the two classes, by ensuring the maximum distance from the points. When the points of the two classes are not linearly separable, a common solution is to resort to the so-called “kernel trick”, i.e., the points $xi$ are mapped to a larger dimension space until they become linearly separable. The most common kernels are polynomial functions with varying degree number and the so-called Radial Basis Functions (RBF) [
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
,
• Hastie Trevor
• Tibshirani Robert
• Friedman Jerome
The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
]. Finally, SVM models can also be extended to multiclass classification tasks using the strategies One-Versus-All (OVA), or One-Versus-One (OVO) [
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
,
• Hastie Trevor
• Tibshirani Robert
• Friedman Jerome
The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
].
In our work, to implement and train the SVM models, we leveraged the scikit-learn python library [
• Pedregosa F.
• Varoquaux G.
• Gramfort A.
• Michel V.
• Thirion B.
• Grisel O.
• Blondel M.
• Prettenhofer P.
• Weiss R.
• Dubourg V.
• Vanderplas J.
• Passos A.
• Cournapeau D.
• Brucher M.
• Perrot M.
• Duchesnay E.
Scikit-learn: Machine learning in python.
].

### 3.2 Data interpretation

The code that implements and train the SVM is defined by the functions in the file runSVM.py. Specifically, the main function is called runSVM: it requires a configuration object that (i) identifies what model has to be used, and (ii) set optional arguments to tune the number of hyperparameters (mask) and to control if the method is verbose (verbose) and if the results have to be written in an output file (writeToFile). Practically, the function runSVM first calls extractData whose purpose is to load the dataset file, extract the data in the desired time steps and split them in training, validation and test sets. After that, runSVM proceeds to train a set of possible SVM models on the training set, by then evaluating them on the validation set and computing on the test set the resulting accuracy of the model that performed better on the validation set. The possible models that can be employed are: (i) Standard linear SVM (using two different libraries), (ii) SVM with polynomial kernel with degree from 2 to 4, and (iii) SVM with RBF kernel.
The file runSVM.py is provided with a main method. Thus, it can be directly called as a script using the configuration name as argument. We have also designed some useful methods that call runSVM, build directly the latex table with the results and calculate the points for the figures shown in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
].

## 4. Impacts

In this paper, we have explained in great detail the software architecture of the ML method, introduced in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
], to carry out quantum noise classification. Such a tools are intended to be applied to quantum technologies, as e.g., quantum computers.
The main impact of our software lies in its ability in classifying the fingerprint left by quantum noise sources on devices that have identical technical specifications and are thus expected to provide the same outcomes. Unfortunately, in quantum machines, the influence of the environment is so relevant that different noise fingerprints can be identified depending on the type of quantum computer (as previously explained, quantum computers can differ, e.g., on the number of qubits and/or the quantum volume), on the time period in which the single machine has worked, and on environmental changes mainly due to temperature fluctuations. However, thanks to our quantum–classical machine learning method, one can (i) distinguish the noise fingerprints in different quantum devices; (ii) classify the noise fingerprint on the same quantum devices but in different times; (iii) learn if and how a given noise fingerprint changes over time.
In [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
] our method is proved to be very accurate (more than 99% of effectiveness) in classifying a clear machine-related noise fingerprint in each of the analysed IBM quantum computers, and even robust since any noise fingerprint is highly predictable over time in windows of consecutive runs. Also an evident time-dependence of the noise fingerprints has been classified, by observing changes over time after few hours from the first execution of the testbed quantum circuit.
Another important feature of our software architecture is that the ML models do not require a complete set of measurement outcomes as input data, but conversely the outcomes from a sequence of repeated measurements of a single observable. For an example, for the experiments in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
], the chosen observable was the tensor product of $sigmaz$ Pauli matrices locally applied on each qubit of the testbed circuit. Furthermore, the proposed method is able to distinguish and classify noise fingerprints, even without knowing the microscopic model that describes the (real or effective) interaction between the device and the external noise fields. This important aspect allows the user to employ our quantum–classical machine learning algorithm to classify the noise fingerprints of even inaccessible quantum machines.

### 4.1 Applications

The experimental evidences in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
] lead us to conclude that different quantum devices exhibit distinctive, and thus distinguishable, noise fingerprints that one can classify and predict. Therefore, in principle, our method could be adopted to identify from which specific quantum device certain data (a collection of measurement outcomes) are generated, just looking at the noise fingerprint of the device. Moreover, the proposed solution might be employed to certify the time-scheduling in which a given quantum computation is executed. Both these applications are expected to play a key role for diagnostics purposes – especially in all those contexts where quantum computations cannot be error-corrected [
• Deutsch Ivan H.
Harnessing the power of the second quantum revolution.
] – and to accomplish benchmarking and certification [
• Wright K.
• Beck K.M.
• Debnath S.
• et al.
Benchmarking an 11-qubit quantum computer.
,
• Eisert Jens
• Hangleiter Dominik
• Walk Nathan
• Roth Ingo
• Markham Damian
• Parekh Rhea
• Chabaud Ulysse
• Kashefi Elham
Quantum certification and benchmarking.
] of quantum noise sources within a default error threshold.

### 4.2 Outlook

The proposed methodology, aimed to learn the noise fingerprint of quantum devices from time-ordered measurements of a testbed quantum circuit, may be in principle applied to any quantum devices, and thus not only to the IBM quantum computers as done in [
• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
]. The possibility to predict on which device, and at which time, a given quantum operation (even time-varying) has been executed is expected to help the mitigation of quantum computational errors (e.g., by means of calibration routines), and to assist the application of ad-hoc error corrections.
Furthermore, instead of SVMs, one could employ deep learning techniques, as for example Recurrent Neural Networks (RNN) [
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
,
• Goodfellow Ian
• Bengio Yoshua
• Courville Aaron
Deep Learning.
,
• Schmidhuber Jürgen
Deep learning in neural networks: An overview.
], to make more efficient the classification of quantum noise fingerprints. In such a case, the software architecture should be modified a bit, but not in the part that concerns the generation of the quantum data placed in input to the ML model. What should be different, indeed, is the way the input data would be processed.
Finally, we are also confident that, thanks to specific modifications, it is possible to carry out even the reconstruction of some quantum noise features. However, for such a purpose, a minimal knowledge of the way noise sources affect the quantum device under investigation will be required.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was financially supported from Fondazione CR Firenze, Italy through the project QUANTUM-AI, from University of Florence through the project Q-CODYCES, and from the European Union’s Horizon 2020 research and innovation programme under FET-OPEN Grant Agreement No.828946 (PATHOS).

## References

• Dowling J.P.
• Milburn G.J.
Quantum technology: the second quantum revolution.
Philos. Trans. R. Soc. A. 2003; 361: 1655-1674
• O’Brien J.L.
• Furusawa A.
• Vučković J.
Photonic quantum technologies.
Nat. Photon. 2009; 3: 687-695
• Wang J.
• Sciarrino F.
• Laing A.
• Thompson M.G.
Integrated photonic quantum technologies.
Nat. Photon. 2020; 14: 273-284
• Pritchard J.D.
• Shaffer J.P.
Rydberg atom quantum technologies.
J. Phys. B At. Mol. Opt. 2020; 53012002
• Devoret Michel H.
• Wallraff Andreas
• Martinis John M.
Superconducting qubits: A short review.
2004 (arXiv preprint Cond-Mat/0411174)
• Clarke John
• Wilhelm Frank K.
Superconducting quantum bits.
Nature. 2008; 453: 1031-1042
• Wu Y.
• Bao W.
• Cao S.
• et al.
Strong quantum computational advantage using a superconducting quantum processor.
Phys. Rev. Lett. 2021; 127180501
• Wineland David J
• Barrett Murray
• Britton Joseph
• Chiaverini J
• DeMarco B
• Itano Wayne M
• Jelenković B
• Langer Christopher
• Leibfried Dietrich
• Meyer V
• et al.
Quantum information processing with trapped ions.
Phil. Trans. R. Soc. A. 2003; 361: 1349-1361
• Pogorelov I.
• Feldker T.
• Marciniak Ch. D.
• Postler L.
• Jacob G.
• Krieglsteiner O.
• Podlesnic V.
• Meth M.
• Negnevitsky V.
• Höfer B.
• Wächter C.
• Lakhmanskiy K.
• Blatt R.
• Schindler P.
• Monz T.
Compact ion-trap quantum computing demonstrator.
PRX Quantum. 2021; 2020343
• Spring Justin B
• Metcalf Benjamin J
• Humphreys Peter C
• Kolthammer W Steven
• Jin Xian-Min
• Barbieri Marco
• Datta Animesh
• Thomas-Peter Nicholas
• Langford Nathan K
• Kundys Dmytro
• et al.
Boson sampling on a photonic chip.
Science. 2013; 339: 798-801
• Metcalf Benjamin J
• Spring Justin B
• Humphreys Peter C
• Thomas-Peter Nicholas
• Barbieri Marco
• Kolthammer W Steven
• Jin Xian-Min
• Langford Nathan K
• Kundys Dmytro
• Gates James C
• et al.
Quantum teleportation on a photonic chip.
Nat. Photonics. 2014; 8: 770-774
• Freedman Michael
• Kitaev Alexei
• Larsen Michael
• Wang Zhenghan
Topological quantum computation.
Bull. Amer. Math. Soc. 2003; 40: 31-38
• Arute F.
• Arya K.
• Babbush R.
• et al.
Quantum supremacy using a programmable superconducting processor.
Nature. 2019; 574: 505-510
• Preskill John
Quantum computing in the NISQ era and beyond.
Quantum. 2018; 2: 79
1. 2022, https://quantum-computing.ibm.com/. Visited on 2022.

2. 2022, https://www.rigetti.com/. Visited on 2022.

• Martina Stefano
• Buffoni Lorenzo
• Gherardini Stefano
• Caruso Filippo
Learning the noise fingerprint of quantum devices.
2021 (arXiv preprint arXiv:2109.11405)
• Bishop Christopher M.
Pattern Recognition and Machine Learning.
first ed. Springer-Verlag, New York2006
• Hastie Trevor
• Tibshirani Robert
• Friedman Jerome
The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
second ed. Springer series in statistics, New York2009
• Müller Matthias M.
• Gherardini Stefano
• Calarco Tommaso
• Montangero Simone
• Caruso Filippo
Information theoretical limits for quantum optimal control solutions: Error scaling of noisy channels.
2020 (arXiv preprint arXiv:2006.16113)
• Youssry Akram
• Paz-Silva Gerardo A.
• Ferrie Christopher
Beyond quantum noise spectroscopy: modelling and mitigating noise with quantum feature engineering.
Npj Quantum Inf. 2020; 6: 95
• Luchnikov I.A.
• Vintskevich S.V.
• Grigoriev D.A.
• Filippov S.N.
Machine learning non-Markovian quantum dynamics.
Phys. Rev. Lett. 2020; 124140502
• Fanchini Felipe F.
• Karpat Göktu ğ
• Rossatto Daniel Z.
• Norambuena Ariel
• Coto Raúl
Estimating the degree of non-Markovianity using machine learning.
Phys. Rev. A. 2021; 103022425
• Niu Murphy Yuezhen
• Klimov Paul
• Boixo Sergio
• Barends Rami
• Kelly Julian
• Chen Yu
• Arya Kunal
• Burkett Brian
• Bacon Dave
• et al.
Learning non-Markovian quantum noise from moiré-enhanced swap spectroscopy with deep evolutionary algorithm.
2019 (arXiv preprint arXiv:1912.04368)
• Harper Robin
• Flammia Steven T.
• Wallman Joel J.
Efficient learning of quantum noise.
Nat. Phys. 2020; : 1-5
• Martina Stefano
• Gherardini Stefano
• Caruso Filippo
Machine learning approach for quantum non-Markovian noise classification.
2021 (arXiv preprint arXiv:2101.03221)
• Wise David F.
• Morton John J.L.
• Dhomkar Siddharth
Using deep learning to understand and mitigate the qubit noise environment.
PRX Quantum. 2021; 2010316
• Abraham Héctor
• et al.
Qiskit: An open-source framework for quantum computing.
2019
• Nielsen Michael A.
• Chuang Isaac
Quantum Computation and Quantum Information.
American Association of Physics Teachers, 2002
• Cross Andrew W
• Bishop Lev S
• Sheldon Sarah
• Nation Paul D
• Gambetta Jay M
Validating quantum computers using randomized model circuits.
Phys. Rev. A. 2019; 100032328
• Pedregosa F.
• Varoquaux G.
• Gramfort A.
• Michel V.
• Thirion B.
• Grisel O.
• Blondel M.
• Prettenhofer P.
• Weiss R.
• Dubourg V.
• Vanderplas J.
• Passos A.
• Cournapeau D.
• Brucher M.
• Perrot M.
• Duchesnay E.
Scikit-learn: Machine learning in python.
J. Mach. Learn. Res. 2011; 12: 2825-2830
• Deutsch Ivan H.
Harnessing the power of the second quantum revolution.
PRX Quantum. 2020; 1020101
• Wright K.
• Beck K.M.
• Debnath S.
• et al.
Benchmarking an 11-qubit quantum computer.
Nature Commun. 2019; 10: 5464
• Eisert Jens
• Hangleiter Dominik
• Walk Nathan
• Roth Ingo
• Markham Damian
• Parekh Rhea
• Chabaud Ulysse
• Kashefi Elham
Quantum certification and benchmarking.
Nat. Rev. Phys. 2020; 2: 382-390
• Goodfellow Ian
• Bengio Yoshua
• Courville Aaron
Deep Learning.
MIT press Cambridge, 2016 (http://www.deeplearningbook.org)
• Schmidhuber Jürgen
Deep learning in neural networks: An overview.
Neural Netw. 2015; 61: 85-117