Quantum machine learning framework for virtual screening in drug discovery: a prospective quantum advantage

The general nature and the purpose of our classification workflow both require a realistic benchmarking dataset specifically dedicated to applied ML in VS. Specifically, the dataset should [21–24] (a) imitate real-world screening libraries and guide ML methods to discriminate active from inactive compounds; (b) contain molecules with unambiguous experimentally measured activity against a target, i.e. with either potent/moderate activity or inactivity; (c) have a realistic ratio between active and inactive molecules; (d) contain active and inactive compounds with comparable molecular properties; (e) be chemically unbiased.

The LIT-PCBA dataset [22] encompasses all the previous points for fifteen well-characterised biological targets, and it is a natural choice for this project. Furthermore, and with respect to prior efforts [21, 25–27], the LIT-PCBA dataset was created specifically to test the performance of ML tasks in VS of molecules. The dataset was prepared by removal of potential false positives as well as including experimentally confirmed inactive molecules rather than seeding the database with decoy molecules (i.e. molecules assumed to be inactive).

The targets in the LIT-PCBA datasets have a real-world active-to-inactive molecules ratio, meaning that in some cases the number of confirmed active molecules is strongly imbalanced. As ML methods can be severely affected by class imbalance in a dataset [28, 29], we balanced active-to-inactive ratios of each target dataset. However, for some targets in the LIT-PCBA dataset, the number of active molecules is astoundingly small (less than 30), with thousands of confirmed inactive molecules. In such cases, we padded the dataset with a 1:6 active-to-inactive ratio to balance the dataset (see appendix A).

Following the above procedure and considerations, we also choose to assess the performance of our method by screening a novel dataset containing known active and inactive COVID-19 inhibitors [20]. This dataset is particularly challenging for ML classification tasks, and specifically for those based on a purely structural basis (i.e. using a fingerprint-based approach), due to the fact that more than 30% of the active or inactive molecules do not share a common scaffold or other recurrent structural features, showing how different and diverse the active molecules are. In summary, while LIT-PCBA was chosen to provide a realistic benchmark of our methodology against a dataset specifically dedicated to applied ML in VS for very well known and studied diseases, the COVID-19 case study is instead meant as a less standardised but paradigmatic example in which lack of good quality experimental data and high class imbalance result in a difficult problem for classical support vector machine (SVM) kernel methods.

SVMs [1, 3–5] are among the most widely used supervised ML methods for LB-VS. In its basic version, an SVC [30], is an SVM-based linear model that, given a training set of the form $\{(\vec{x}_i,y_i) \vert \vec{x}_i \in \mathbb{R}^m, y_i = \pm 1 \}$, finds an optimal separating hyperplane in the data space to distinguish between two given classes, labelled by y_i . SVMs can be turned into non-linear classifiers by lifting the input data into a higher dimensional feature space $\mathcal{F}$ through a feature map $\phi(\vec{x})$. Indeed, while in its elementary form a SVC relies on the computation of similarities between data points through inner products $\langle\vec{x}_i,\vec{x}_j\rangle$ in $\mathbb{R}^m$, in the more general case this is replaced by the corresponding feature space quantity $\langle\phi(\vec{x}_i),\phi(\vec{x}_j)\rangle$, thus allowing for complex non-linear representations of the original inputs. The SVC is usually trained by solving a constrained quadratic optimization problem, with a target function of the form

$$\begin{equation} \mathcal{C}(\vec{\alpha}) = \sum_i \alpha_i - \sum_{ij}y_iy_j\alpha_i\alpha_j\langle\phi(\vec{x}_i),\phi(\vec{x}_j)\rangle \end{equation} \tag{ 1 }$$

subject to $\sum_i \alpha_iy_i = 0$ and $0\leqslant \alpha_i \leqslant 1/(2nC)$. Here n is the cardinality of the training set and C is a regularization parameter controlling the hardness of the margin. Moreover, the so-called kernel trick [31] allows one to replace all scalar products with a symmetric positive definite kernel function $k(\vec{x}_i,\vec{x}_j)$, such that, in general, the feature map φ is only specified in an implicit manner. Some popular choices for classical SVCs are represented by polynomial kernels $k(\vec{x}_i,\vec{x}_j) = (\vec{x}_i\cdot\vec{x}_j+r)^d$, which reduce to the linear case for d = 1, and the Gaussian radial basis function (RBF) kernel $k(\vec{x}_i,\vec{x}_j) = \exp{(-\gamma\vert\vec{x}_i-\vec{x}_j\vert^2)}$.

The family of QK methods [13, 14] employs a quantum computer to extend the class of available kernel functions, particularly to those instances which are believed to be hard to define and compute classically. Here, the complex-valued Hilbert space of quantum mechanical states plays the role of the feature space through a mapping of form $\vec{x}\mapsto \vert\phi(\vec{x})\rangle\langle \phi(\vec{x})\vert$ and a natural kernel choice is provided by $k(\vec{x}_i,\vec{x}_j) = \vert\langle\phi(\vec{x}_i)\vert\phi(\vec{x}_j)\rangle\vert^2$. The specific nature and properties of the QK depend on the explicit choice of the embedding function $\vert\phi(\vec{x})\rangle$, which is typically realized as $\vert\phi(\vec{x})\rangle = U(\vec{x})\vert 0\rangle$ with a unitary operation U acting on a reference state $\vert 0\rangle$ and depending parametrically on the classical input data.

Some important remarks are in order here: on the one hand, it is not difficult to see how carefully designed U can in principle lead to classically intractable kernels. On the other hand, it has also been argued that the mere computational hardness of U does not a priori guarantee quantum advantage in speed or accuracy for ML tasks. In fact, complexity considerations in this context are influenced by both the availability of training data [17] and the inherent structure of the data sets [16, 32]. Although a rigorous quantum speed-up in supervised learning with QK methods has been proven theoretically in at least one specific example [16], the quest for quantum advantage on naturally occurring data sets remains open. Here we provide strong empirical evidence towards such a goal, working from the assumption that good candidate models for practical quantum advantage are those that, using a classically hard QK, exhibit superior performances with respect to all classical counterparts in at least some instances of the problem under study.

In the following, we will make use of a unitary template, known as the ZZ feature map with linear entanglement [14, 33], to embed classical feature vectors into quantum states. For a 4-dimensional classical input $\vec{x} = (x_0,x_1,x_2,x_3)$ this reads

where $P(\theta)$ is the single-qubit phase gate, $\hat{x}_i\equiv\pi-x_i$ and $\ell$ is called the depth of the feature map. Notice that the number of qubits in the register matches the size of the classical inputs, and the generalization of the parameterized unitary transformation to larger classical inputs is straightforward. This design is derived from the one presented in [14] and yields computationally hard kernels under suitable complexity theory assumptions. In the following, we will set $\ell = 2$.

Zoom In Zoom Out Reset image size

For every pair of classical inputs $\vec{x}$, $\vec{x}^{^{\prime}}$, the QK is evaluated on a quantum processor according to the definition

$$\begin{equation} k(\vec{x},\vec{x}^{^{\prime}}) = \vert\langle\phi(\vec{x})\vert\phi(\vec{x}^{^{\prime}})\rangle\vert^2 = \vert\langle0\vert U^\dagger(\vec{x})U(\vec{x}^{^{\prime}})\vert 0\rangle\vert^2. \end{equation} \tag{ 3 }$$

In practice, after initializing the quantum register in the reference state $\vert 0\rangle$, one applies the unitary $U(\vec{x}^{^{\prime}})$ and the inverse $U^\dagger(\vec{x}) = U^{-1}(\vec{x})$, which is easily obtained by reversing the order of operations appearing in equation (2) while also replacing $P(\theta) \rightarrow P(-\theta)$ in all single-qubit phase gates. Finally, the value of the QK corresponds to the probability of observing the register in the reference state $\vert 0\rangle$ after a measure in the computational basis.

Rather than performing VS based on a similarity search of molecules in a database—using for example molecular fingerprints, like most conventional VS methods—we trained the classifier using generic features (chemical descriptors), which can be computed for every dataset, regardless of the target type and experimental features. Chemical descriptors exist in their thousands and can be quickly obtained using specific third-party cheminformatics software tools, both open-source and commercial. Our methodology can be coupled with any third-party package capable of producing such descriptors. Here, we use RDKit [34] to extract molecular properties starting from molecular structures in a SMILES format. As also described in [7, 35–38], the chosen features can be grouped as (a) atomic species; (b) structural properties; (c) physical-chemical properties; (d) basic electronic information; (e) molecular complexity - i.e. graph descriptors. The full list of descriptors can be found in appendix B.

As described in section 2.2, employing a ZZ feature map implies the use of a parametric type of encoding. Under this scheme, each classical data point must be represented as a vector of real numbers—in our case the chemical features or molecular descriptors—that, after a suitable normalization procedure, are used as angles in single-qubit quantum gates (see equation (2)). These operations, in combination with 2-qubit entangling operations, eventually create a quantum superposition state, i.e. a linear combination of multi-qubit basis states or strings, whose complex coefficients have a non-trivial dependence on the classical input and represent its embedding into the feature Hilbert space.

While in principle very powerful and versatile, this data encoding scheme is rather qubit-intensive, with typically one assigned feature per qubit. As a matter of fact, and despite continued and steady progress in quantum computing technology, implementing a quantum LB-VS workflow featuring hundreds of descriptors is currently demanding even for the largest publicly available quantum processors (127 qubits, IBM Eagle—late 2021 [39]). Similarly, this precludes the use of conventional 2048-bit molecular fingerprints, although reduced representations such as the ones proposed by Batra et al [40] may offer viable intermediate-scale solutions.

To achieve a meaningful yet manageable problem size, we therefore extracted, starting from SMILES representations, 47 chemical descriptors as described in section 2.3. First, each descriptor was normalised using standardization methods. Second, the descriptors were compressed into N variables using principal component analysis (PCA) [41] to match a set of N qubits used by the QSVC algorithm. Alternatively, the best N features were selected using the analysis of variance methodology (ANOVA) [42]. More in detail, PCA—which is widely adopted for feature reduction in drug discovery [36, 43] – was applied via singular value decomposition to obtain a lower-dimensional representation of the data as described in [44]. Conversely, the ANOVA f-test [44] enabled us to pick the most descriptive N features, as also suggested in [45], leveraging the numerical structure of the inputs and the binary categorical nature (i.e. active/inactive) of the classification output. Finally, the resulting feature vectors were used to parametrically initialise the corresponding quantum states on which the QK entries were evaluated.

The most intuitive way of assessing the performances of binary classification algorithms in both classical and quantum cases is to calculate the AUC-ROC (receiver operating characteristic) metric [44, 46–48]. The ROC score provides an indication of the capability of the method to distinguish accurately between true positives and false positives. The score ranges between 0 and 1, where 1 is a perfect classifier capable of retrieving true positives only.

Following the methodology described in section 2, all targets of the LIT-PCBA and the COVID-19 datasets were screened using our algorithm. To assess the average method performance, each simulation involving both classical and quantum SVC algorithms was run ten times for each feature selection method and per number of features. Similarly, the RMD and DL classical simulations were also run 10 times. In the interest of fairness, we quantified the standard deviation of the results using ten-fold cross-validation for all simulations. In addition, we used identical test and train subsets for all SVC methods, in order to have a true like-for-like comparison between quantum and classical algorithms. The full set of numerical results is reported in tables 2 and 3 of appendix C.

Consistently with classical SVC methods, QSVC statevector results suggest that the overall performance of our method is influenced by (a) the class balance and the number of the actives in the dataset, (b) the feature selection method, (c) number of features and (d) the value of regularization parameter C. This trend is captured in figures 6 and 7 of appendix C. The class balance of the target is the first major factor to affect the performance of our method, similarly to other classical ML/DL approaches. A small number of active molecules in a dataset (e.g. 17 actives vs 312 483 inactives for ADBR2) leads to large standard deviation values (up to ca. ±0.3 of AUC ROC). On the other hand, a near negligible standard deviation for target ALDH1, which has the largest number of actives (5386 actives plus 103 474 inactives), confirms the importance of class balance in terms of stability and performance of all methods. The feature selection method is the second major factor to have an effect on the overall performance of all SVC methods described in this paper. Due to its very essence, feature selection involves the inevitable loss of information, and this may have a positive or negative effect depending on the dataset/target and the number of features selected. Throughout this work, we observe that our QSVC method tends to outperform CSVC and the other classical methods when then number of features is 8 or higher, which also corresponds to a higher number of qubits. In some cases, such as the ALDH1 target shown in figure 2(a), there is a near-linear correlation between performance and the number of features/qubits used. Finally, the performance of QSVC method is also affected by the choice of the regularization factor C. This is expected as SVC based methods have an inherent dependency on the C value to determine the misclassification rate of the method. Our results suggest that a better QSVC performance is achieved when using a default value C = 1.

Zoom In Zoom Out Reset image size

A more detailed overview is represented in figure 2, where the comparison of ANOVA and PCA QSVC(statevector)/CSVC, RMD and DL trends is reported for the ALDH1, COVID-19 (see section 2.1) and GBA (166 active vs 296 052 inactive molecules) targets. Notice that the trend lines of RMD and DL methods are flat since the methods do not make use of an explicit number of features or feature selection methods. Overall, the performances of both classical and quantum SVC classifiers tend to increase with an incremented number of features used for training, with the steepness of increase and maximum ROC values depending on the features selection method used. In other words, the amount of information used to train the algorithm has a major impact on determining the quality of the classifier. figure 2(a) shows that the QSVC tends to perform significantly better with 8+ features if compared to its classical equivalent (up to 13% better when using 16 features), suggesting that after this threshold the effect of dimensionality reduction starts to vanish. This results in a better performance of the QSVC algorithm, possibly due to the larger size of the quantum feature space and a better expressivity of the quantum feature map, leading to a more favourable separation hyperplane between active and inactive classes. Importantly, comparing the classical and quantum methodologies, we notice that the performance of QSVC either remains stable or keeps increasing with an increasing number of features, thus hinting towards potential for quantum advantage. As previously discussed, the stability of the method on datasets with optimal class balance such as ALDH1 is suggested by little standard deviation values. Conversely, class-imbalanced datasets such as COVID-19 and GBA (figures 2(c) and (d)) typically show higher standard deviations. Nevertheless, the QSVC method consistently keeps higher average accuracy with increasing number of features for at least one choice of the feature selection method (PCA for COVID-19 and ANOVA for GBA).

In figure 2 we also report the mean ROC values obtained following VS of the ALDH1, COVID-19 and GBA targets using the RMD and MPNN methodologies, trained using various fingerprint encoding. The performance of our quantum classifier seems to outperform the classical VS methodology of RMD with up to ca. 20% better mean quantum ROC values when compared to the best value obtained with PyRMD (trained with RDKIT or ECFP6 fingerprints). Conversely, the MPNN method generally provides better mean ROC values if compared to RMD, and it seems to be outperformed by our methodology only when using a larger number of features. The standard deviation values of both RMD and MPNN methodologies are in agreement with our CSVC and QSVC methods, confirming the role of class balance in defining the stability of the method. In figures 2(c) and (d) we include data points obtained for 8 features using two different quantum simulator backends as implemented in Qiskit, namely the noiseless qasm and the mock Montreal backend simulators. The former adds statistical noise to the simulation, mimicking the probabilistic quantum measurement process, while the latter simulates both the statistical and hardware noise that one would get on the actual IBM Quantum Montreal processor, whose calibration data are used as parameters in the noise model. Due to memory and time limitations, we run simulations with qasm and/or gate noise for 8 features/qubits. These results show that the addition of noise is not significantly detrimental to the performance of the QSVC method for the COVID-19 and GBA targets. Furthermore, the standard deviation for the simulations are also in line with the noiseless statevector case.

Summarizing, the simulation of our quantum LB-VS methodology provides, in a like-for-like comparison, a performance that either compares to other existing classical methodologies or that significantly outperforms them for a sufficiently large number of selected features, sometimes showing evidence of potential linear scale-up. This implies that an increase in the number of features/qubits used to train the algorithm can, in principle, further improve the performance of the method. Finally, it is important to mention that the PQA suggested by this study is also supported by the geometric test recently introduced in [17]. In fact, the data sets shown in figure 2 noticeably passed such test, yielding $g_{CQ} \propto \sqrt{n}$, where g_CQ is the geometric difference between the QK and the best classical CSVC counterpart, and n is the number of the data points. Overall, the observed win-rate of QSVC compared to classical counterparts is 43.7% (16 datasets), with variability depending on the features selection method and learning rate.

In this work, we have so far established that the QSVC method in some cases outperforms an equivalent binary classifier or other classical methods used in VS of databases of molecules, highlighting its dependencies from dataset size, number of features and so forth. We also showed that the addition of statistical and simulated noise to the quantum algorithm does not considerably reduce the performance of the method, and it lies within the standard deviation of statevector (i.e. noise-free) simulations. The next logical step is therefore to verify whether the quantum algorithm can indeed perform well on actual quantum devices and if the numerical performance is a reliable tool for assessing the potential benefits coming from our method applied to LB-VS on quantum computers. To this aim, we run simulations on IBM Quantum processors using 8 qubits for ADRB2 on IBM Quantum Montreal (figure 3(b)) and for the COVID-19 dataset on IBM Quantum Guadalupe (figure 3(a)).

Zoom In Zoom Out Reset image size

The two datasets underwent the same preparation and simulation methodology as the other datasets described in the previous section. In figure 4 we report the best results obtained with the optimal feature selection methods (ANOVA for ADRB2, PCA for COVID-19) along with qasm, mock hardware (Montreal and Guadalupe backend) and hardware results. Also in this case, QSVC statevector numerical simulations for both ADRB2 and COVID-19 datasets provided better results if compared to classical equivalents, including RMD and MPNN methods. Remarkably, in our experiments we found that training and testing our QSVC algorithm on IBM Quantum Montreal provided an AUC ROC result for ADRB2 comparable to the results of all the other (quantum) numerical simulation methods, with outcomes lying well within the error bar of statevector simulations (see table 2). Similarly, running the QSVC algorithm on IBM Quantum Guadalupe for the COVID-19 dataset also provided an AUC ROC that is comparable to the best (quantum) numerical simulations. Most importantly, quantum hardware results for both targets greatly outperform the other classical methodologies, thus confirming that QK methods are indeed a well suited tool for potential active/inactive molecules classification against ADRB2 and COVID-19 targets.

Zoom In Zoom Out Reset image size

We can conclude that, for these two instances, simulated backends have correctly predicted the respective hardware results, hence providing a truthful insight into what to expect from real quantum hardware. These results are encouraging, suggesting that although it might not yet be possible to perform large scale simulations on currently available quantum hardware, promising proof-of-principle demonstrations can be achieved, confirming the reliability and predictive power of numerical simulations.