pyRiemann-qiskit facilitates the creation and parameterisation of a quantum backend and is fully compliant with scikit-learn’s transformers, estimators and classifiers, so it becomes easy to integrate quantum classification into existing pipelines. It also supports docplex for the specification of convex optimisation problems, with the limitation of using binary and unconstrained variables. pyRiemann-qiskit is built on top of pyRiemann (Barachant et al. 2020), a machine learning library based on RG, thereby enabling the manipulation of covariance matrices (and, to a larger extent, semi-positive definite matrices) within the Riemannian manifold and which allows the projection of these matrices into a tangent space at a specific point on the Riemannian manifold. Classification is performed either in the tangent space or by directly utilising the Riemannian distance between each sample (represented by its covariance matrix) and a class prototype in the manifold.

pyRiemann-qiskit also includes examples to guide practitioners, as well as a complete test suite. We will briefly describe below the functionalities provided by the software.

Support for quantum classifiers

The software supports QSVC and VQC classifiers. The first concern regarding quantum classifiers is the encoding of classical data into quantum states. This operation is known as feature mapping. To obtain an advantage over classical computing, feature mapping must implement quantum circuits which are difficult to emulate on a classical computer. Feature mapping is common in VQCs and QSVCs. Both are SVM-like classifiers in the sense that they generate a separating hyperplane. The difference between them is that VQCs uses a variational quantum circuit (also known as a variational form) for this task, whereas QSVCs uses a quantum-enhanced kernel with a conventional SVM. The software also supports the Pegasos implementation of QSVC, which offers a speed-up compared to QSVC (Gentinetta et al. 2022).

Table 1 demonstrates how to instantiate the VQC or QSVC classifier in pyRiemann-Qiskit.

By default, the backend will be a local quantum simulator. However, it is possible to register on IBM quantum and request a free token to use one of the publicly available quantum computers. All classifiers accept a q_account_token parameter which, if valid, provides access to one of several real quantum computers available with IBM.

However, note that, at the time of writing, the number of qubits (and, therefore, the feature dimension) is limited to:

• 36 on a local quantum simulator;
• 5000 on a remote quantum simulator;
• 5–7 on a real free quantum computer;
• 127 on exploratory quantum computers (not available for public use).

Support for convex optimisation problems

The MDM algorithm (Barachant et al. 2012) consists in finding the minimum distance between a trial and a class prototype before labelling the trial with the prototype which is the closest to the trial. Alternatively, this task can be defined as a convex optimisation problem and executed as a quantum circuit.

pyRiemann-qiskit supports the docplex library for the definition of convex optimisation problems. Specifically, this is useful in these two cases:

• in training: computing the barycenter of the covariance matrices, i.e. a matrix is used as a point in the manifold and thus an average – called a class prototype – of all these points can be calculated;
• in classification: finding the minimum distance between a trial and a class prototype can also be defined as a quadratic optimisation problem (Zhao et al. 2019) – see section Direct classification of covariance matrices under Additional Information.

To calculate the class prototype, we need to select a distance and provide an optimiser. We model the distance as a convex problem, based on frobenius distance (python method fro_mean_convex). The model objective is to find the matrix which is at minimum distance to all covariance matrices. This matrix is then designated as the barycenter or class prototype of the covariance matrices. The optimiser is the component which is responsible for finding the coefficients of this barycenter, based on the model objective and the covariance matrices used as inputs. For the optimiser, pyRiemann-qiskit relies on Qiskit implementation of QAOA (Quantum Approximate Optimisation Algorithm). However, QAOA is limited to solving only QUBO (Quadratic Unconstrained Binary Optimisation) problems, that is, problems with unconstrained and binary variables only. In other words, the mean matrix, being the subject of optimisation, can only accept binary variables and, thus, the input covariance matrices can only contain zeros and ones as coefficients. For this reason, pyRiemann-qiskit uses a wrapper around Qiskit’s QAOA optimiser (class NaiveQAOAOptimizer) that rounds covariance matrices to a certain precision and transforms each resulting integer coefficient to binary coefficients. This transformation is based on Qiskit’s IntegerToBinary, a bounded coefficient encoding method.

The complexity of the QAOA optimiser rises as a function of the size of the covariance matrices and the upper-bound coefficient. The size of the covariance matrices depends on the number of input channels in the input time epoch, as well as the dimension reduction method that is used. The upper-bound coefficient also has an impact on the final size of the covariance matrices. The upper-bound coefficient is simply the maximal value that can possibly take a coefficient in a matrix. If all coefficients inside a matrix are integers ranging from 0 to 3, then each of them can be represented by only 2 bits (00, 01, 10, 11). In this case, each integer coefficient inside the matrix can be replaced by two binary coefficients and, thus, the size of the resulting matrices will be "only" twice as large. As the mean matrix to optimise has the exact same size as the input covariance matrices, it means that there will be two times more variables to optimise inside the mean matrix. Therefore, this implies a higher number of qubits to hold the variables inside the matrix. As QAOA is computationally expensive, we also provide a wrapper over the classical optimiser Cobyla that allows for easier testing.

Table 2 demonstrates how to use the fro_mean_convex method.

If the MDM classifier is supplied with the "convex" metric, it will automatically use the fro_mean_convex method for computing the mean – the class prototype. The default optimiser for the fro_mean_convex method is the Cobyla optimiser.

Classification of vectorised covariances matrices

The number of qubits is limited (especially on real hardware) and this puts a limit on the length of the feature vectors that can be used with quantum programming. In pyRiemannQiskit, we use the method of tangent space vectorisation. It consists of creating covariance matrices from the signal then projecting them into the tangent space of the Riemannian manifold. The result is a feature vector with reduced dimensionality. The dimensionality can be further reduced using PCA, for example, in order to match the number of available qubits.

The code snippet below (Table 3) demonstrates how to perform a dimension reduction of the signal epoch using Xdawn (Rivet and Souloumiac 2013), apply the tangent space method, then further reduce the size of the feature space to match the capability of the quantum backend in charge of running our quantum classifier (here QuanticSVM).

For convenience, the library provides the QuantumClassifierWithDefaultRiemannianPipeline class, which implements the above pipeline.

pyRiemann-qiskit is a sandbox for experimenting with quantum computing in conjunction with RG. It unifies within the same library a quantum computational library and RG tools to ultimately allow for the creation of better Brain-Computer Interfaces (BCI). RG tools are proved to perform very well on the classification of EEG ERP signals (Lotte et al. 2018). Therefore, the primary audience we target is practitioners coming from the BCI field, willing to experiment with quantum computing for the classification of electroencephalography (EEG) or magnetoencephalography signals. An initial study on this topic is available in Cattan and Andreev (2022). It can be downloaded from pyRiemann-qiskit. However, note that the tools provided by the library are also relevant to other domains, such as classification of other biometric signals, image recognition or detection of fraud transactions (e.g. Grossi et al. (2022)).

In summary, we seek to encourage the use of quantum computing together with RG for concrete (BCI) applications, opening new and interesting research paths. For example, it would be interesting to investigate BCI illiteracy, a situation in which classical classifiers usually fail to generalise well using EEG data.