Unsupervised Clustering Using Quantum Similarity Matrix in Quantum Feature Space

Abstract

A method for performing unsupervised clustering of data points includes determining the number of qubits to include in a quantum processor based on the feature dimensions of each data point. The method includes, for each pair of data points, executing a quantum circuit on a quantum processor having the determined number of qubits. The quantum circuit includes a feature map template circuit parameterized to have a first plurality of rotations, a backward feature map template circuit parameterized to have a second plurality of rotations, and a metric circuit that outputs a similarity metric. The method includes creating a similarity matrix based on the similarity metric for each pair of data points and inputting the similarity matrix into a classical clustering algorithm to cluster the data points. The feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor.

Description

Technical Field

[0001] The present invention relates to unsupervised clustering, and more particularly, to unsupervised clustering using a quantum similarity matrix in a quantum feature space. Background Art

[0002] Unsupervised learning is a key component of many big data workflows, especially in pharmaceuticals and the biosciences. When data contains a large number of measurement data, it is usually not possible to label it proportionally, and clustering is relied upon to process the data into interpretable groups.

[0003] Therefore, there is a need in the art to solve the above problems. Summary of the Invention

[0004] From a first aspect, the present invention provides a method for performing unsupervised clustering of a plurality of data points, comprising: determining the number of qubits to be included in a quantum processor based on a plurality of feature dimensions of each data point of the plurality of data points; for each data point pair of the plurality of data points, executing a quantum circuit on a quantum processor having the determined number of qubits, wherein the quantum circuit comprises: a feature map template circuit parameterized with a first plurality of rotations, wherein the first plurality of rotations is based on the feature values of a first data point of the data point pair; a backward feature map template circuit parameterized with a second plurality of rotations, wherein the second plurality of rotations is based on the feature values of a second data point of the data point pair; and a metric circuit that outputs a similarity metric for the data point pair; creating a similarity matrix of the plurality of data points based on the similarity metric of each data point pair; and inputting the similarity matrix into a classical clustering algorithm to cluster the plurality of data points, wherein the feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor.

[0005] On the other hand, the present invention provides a hybrid quantum-classical system, comprising: a quantum processor having a plurality of qubits corresponding to a plurality of feature dimensions of each of a plurality of data points to be clustered, the quantum processor being configured to: for each pair of the plurality of data points, execute a quantum circuit, the quantum circuit comprising: a feature map template circuit parameterized to have a first plurality of rotations, wherein the first plurality of rotations are based on the feature values of a first data point of the pair of data points; a backward feature map template circuit parameterized to have a second plurality of rotations, wherein the second plurality of rotations are based on the feature values of a second data point of the pair of data points; and a metric circuit that outputs a similarity metric for the pair of data points, wherein the feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor; and a classical processor in communication with the quantum processor, the classical processor being configured to: receive the similarity metric for each pair of the plurality of data points; create a similarity matrix of the plurality of data points based on the similarity metric for each pair of data points; and perform a classical clustering algorithm on the similarity matrix to cluster the plurality of data points.

[0006] According to an embodiment of the present invention, a method for unsupervised clustering of a plurality of data points includes determining the number of qubits to be included in a quantum processor based on a plurality of feature dimensions of each of the plurality of data points. The method includes, for each pair of the plurality of data points, executing a quantum circuit on a quantum processor having the determined number of qubits. The quantum circuit includes a feature map template circuit parameterized to have a first plurality of rotations, wherein the first plurality of rotations are based on the feature values of a first data point of the pair of data points. The quantum circuit further includes a backward feature map template circuit parameterized to have a second plurality of rotations, wherein the second plurality of rotations are based on the feature values of a second data point of the pair of data points. The quantum circuit further includes a metric circuit that outputs a similarity metric for the pair of data points. The method includes creating a similarity matrix of the plurality of data points based on the similarity metric for each pair of data points and inputting the similarity matrix into a classical clustering algorithm to cluster the plurality of data points. The feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor.

[0007] According to an embodiment of the present invention, a hybrid quantum-classical system includes a quantum processor. The quantum processor has a plurality of qubits corresponding to a plurality of feature dimensions of each of a plurality of data points to be clustered. The quantum processor is configured to execute a quantum circuit for each pair of the plurality of data points. The quantum circuit includes a feature map template circuit parameterized to have a first plurality of rotations, where the first plurality of rotations are based on the feature values of a first data point in the data point pair. The quantum circuit includes a backward feature map template circuit parameterized to have a second plurality of rotations, where the second plurality of rotations are based on the feature values of a second data point in the data point pair. The quantum circuit includes a metric circuit that outputs a similarity metric for the data point pair. The feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of qubits of the quantum processor. The hybrid quantum-classical system includes a classical processor in communication with the quantum processor. The classical processor is configured to receive the similarity metric for each pair of the plurality of data points, create a similarity matrix of the plurality of data points based on the similarity metric for each data point pair, and perform a classical clustering algorithm on the similarity matrix to cluster the plurality of data points. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] The present invention will now be described, by way of example only, with reference to preferred embodiments, as shown in the following drawings:

[0009] Figure 1 is a flowchart showing a method 100 for performing unsupervised clustering of a plurality of data points according to an embodiment of the present invention.

[0010] Figure 2 is a schematic diagram of an example of a feature map template circuit according to an embodiment of the present invention.

[0011] Figure 3 is a schematic diagram of an example of a quantum circuit according to an embodiment of the present invention.

[0012] Figure 4 is a schematic diagram of a hybrid quantum-classical system according to an embodiment of the present invention.

[0013] Figure 5 shows simulation and experimental results of a data set. DETAILED DESCRIPTION

[0014] Clustering is the task of grouping a set of objects in such a way that objects in the same cluster are more similar to each other than to those in other clusters. Unsupervised clustering is unsupervised learning that classifies samples from a high-dimensional distribution into a discrete number of clusters without the need for training data to do so. Clustering algorithms operate on the concept of distance between samples. In the simplest case, the dot product is used.

[0015] Common large-scale examples include grouping ligands when searching for biotherapeutics or classifying cells by RNA sequencing or protein presence. Clustering is also used in marketing to identify customer groups with specific preferences, allowing companies to target specific customers for specific product advertisements.

[0016] Spectral clustering algorithms are clustering algorithms that utilize the eigenvalues of the provided similarity matrix of the data to allow clustering in a feature space that follows the more natural dimensions of the data. The performance of spectral clustering algorithms depends on the extent to which the provided similarity matrix encapsulates the distances between data points. For example, if two clusters are radially separated in three dimensions, an algorithm that uses a three-dimensional radial distance metric to calculate the similarity matrix will outperform a one-dimensional linear distance metric. In addition to spectral clustering, there are other clustering algorithms that use the provided similarity matrix to determine clusters. Examples of these include DBSCAN and agglomerative hierarchical clustering.

[0017] In the above biological and chemical cases, as well as many other cases, the ligands under discussion are more naturally described by their high-dimensional quantum characterizations and their quantum distances from each other. This distance is inaccessible to classical computers because it requires computing wave function exponents of molecular size, and the values of the ground truth classical data may be too messy and non-linear for meaningful distance metrics. This is the core of the difficulty in these fields. Therefore, if we want to accurately cluster these systems to be able to analyze or isolate important groups, we must use a quantum computer to generate the relationships between data points. The methods and systems described in this paper use quantum kernels to represent the distance between two samples in Hilbert space.

[0018] Even for non-biological data, using quantum distances can give us access to feature spaces that are inaccessible to classical computers, thus exponentially expanding the range of available clustering approaches. In the near term, we will only be able to use noisy intermediate-scale quantum (NISQ) systems, so when we use these devices to maximize the opportunity for quantum advantage, we must be very selective about how we design our quantum methods. Approaches that use shorter circuits in hybrid quantum-classical systems are the most promising frontiers.

[0019] Figure 1FIG. 0 is a flowchart showing a method 100 for performing unsupervised clustering of multiple data points according to an embodiment of the present invention. Method 100 includes determining the number of qubits to include in a quantum processor based on multiple feature dimensions of each of the multiple data points 102. Method 100 further includes, for each pair of data points among the multiple data points, executing a quantum circuit having a determined number of qubits 104 on the quantum processor. The quantum circuit includes a feature map template circuit parameterized to have a first plurality of rotations. The first plurality of rotations are based on the feature values of the first data point in the data point pair. The quantum circuit further includes a backward feature map template circuit parameterized to have a second plurality of rotations. The second plurality of rotations are based on the feature values of the second data point in the data point pair. The quantum circuit further includes a metric circuit that outputs a similarity metric for the data point pair. Method 100 includes creating a similarity matrix 106 of the data set based on the similarity metric for each pair of data points. Method 100 includes inputting the similarity matrix into a classical clustering algorithm to cluster the multiple data points 108. The feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor.

[0020] The eigenvalue of a data point having n feature dimensions can be represented by (eigenvalue_1, eigenvalue_2,..., eigenvalue_n). For example, a data point having two feature dimensions can have values (0.1, 0.2), where 0.1 is the eigenvalue of the first feature and 0.2 is the value of the second feature. It is assumed here that each data point among the multiple data points has the same number of feature dimensions.

[0021] According to one embodiment of the present invention, one qubit is included in the quantum processor for each feature dimension. However, embodiments of the present invention are not limited to a single qubit per feature. The number of qubits in the quantum processor can be greater than or less than the number of features.

[0022] According to an embodiment of the present invention, the feature map template circuit includes a plurality of entangled two-qubit quantum gates. For example, the feature map template circuit can include a plurality of CNOT gates.

[0023] Figure 2 FIG. 13 is a schematic diagram of an example of a feature map template circuit 200 according to an embodiment of the present invention. The feature map template circuit 200 operates on two qubits q0 202 and q1 204. The qubits q0 202 and q1 204 can be data qubits and can each be implemented using a plurality of physical qubits. The qubits q0 202 and q1 204 can be initialized to the ground state. |0> , for example, as Figure 2 shown, the feature map template circuit 200 includes a plurality of single-qubit gates and two-qubit gates. The gates can take one or more arguments. For example, Figure 2The U1 gates 206, 208, 210 in [[]] can be single-variable gates, while Figure 2 the U2 gates 212, 214 in [[]] can be two-variable gates. Higher-variable gates can also be included in the feature map template circuit.

[0024] For example, the gate can be a rotation operator. The rotation can be parameterized based on one or more eigenvalues of the data point. For example, Figure 2 the variable of the U1 gate 206 in [[]] can depend on the first eigenvalue of the data point. Figure 2 the variable of the U1 gate 208 in [[]] can depend on the second eigenvalue of the data point. Figure 2 the variable of the U1 gate 210 in [[]] can depend on the first eigenvalue and the second eigenvalue of the data point. For example, the variable can depend on the product of the first eigenvalue and the second eigenvalue.

[0025] The feature map template circuit 200 can include entanglement two-qubit quantum gates, such as the CNOT gates 216, 218. The feature map template circuit 200 can include a repeated series of gates. For example, the gates 206-218 can be repeated. As Figure 2 shown, the number of times the series of gates is repeated in the feature map template can depend on the complexity of the feature map. For a highly complex feature map, the series of gates can be repeated 10, 20, 50 or more times.

[0026] The feature map template circuit 200 is intended as a non-limiting example. Alternative types, combinations, and numbers of gates can be included in the feature map template. In addition, the gates can be included in an order different from Figure 2 that shown. The feature map template circuit can act on more than two qubits. For example, for a data point with n feature dimensions, the feature map template circuit can act on n qubits.

[0027] The feature map template circuit can be combined with a backward feature map template circuit and a metric circuit to form a quantum circuit. The backward feature map template circuit according to an embodiment of the present invention includes a series of gates that "undo" the rotation of the gates of the feature map template circuit. The rotation of the backward feature map template circuit can be parameterized by the eigenvalues of the second data point. Therefore, by performing a first series of rotations based on the eigenvalues of the first data point and then performing a second series of "undo" rotations based on the eigenvalues of the second data point, a quantum simulation of the distance between the two data points can be obtained. This quantum distance can be used to construct a similarity matrix.

[0028] Figure 3Schematic diagram of an example of a quantum circuit 300 according to an embodiment of the present invention. The quantum circuit 300 is used to provide a similarity measure for a pair of two-dimensional data points whose eigenvalues can be expressed as (0.1, 0.2) and (0.3, 0.4). For a data set including more than two data points, the quantum circuit is implemented for each pair of data points.

[0029] The quantum circuit 300 acts on a quantum processor having two qubits q0302 and q1304. The quantum circuit 300 includes a feature map template circuit 306, a backward feature map template circuit 308, and a metric circuit 310. The feature mapping template circuit 306 includes gates parameterized based on the eigenvalues of the first data point (0.1, 0.2).

[0030] According to an embodiment of the present invention, the feature map template circuit 306 includes rotations parameterized based on the eigenvalues of the first data point in the data point pair for each feature dimension. For example, in Figure 3 , the feature map template circuit 306 includes a U1 gate 312 parameterized based on the first eigenvalue 0.1 of the first data point. In Figure 3 the example shown, the first eigenvalue is multiplied by 2, and the U1 gate 312 is parameterized based on the product, i.e., 2 * 0.1 = 0.2. The feature map template circuit 314 also includes a U1 gate 308 parameterized based on the second eigenvalue 0.2 of the first data point. In Figure 3 the example shown, the second eigenvalue is multiplied by 2, and the U1 gate 314 is parameterized based on the product, i.e., 2 * 0.2 = 0.4.

[0031] According to an embodiment of the present invention, the feature map template circuit 306 includes multiple rotations parameterized based on the eigenvalues of the first data point in the multiple data points for each feature dimension. For example, in Figure 3 , the feature map template circuit 306 includes a first U1 gate 312 parameterized based on the first eigenvalue 0.1 of the first data point, and a second U1 gate 316 parameterized based on the first eigenvalue 0.1 of the first data point. As described above, the feature map template circuit may include a series of operations repeated two or more times.

[0032] According to an embodiment of the present invention, the feature map template circuit 306 includes rotations parameterized based on multiple eigenvalues of the first data point for the first data point in the data points. For example, the feature map template circuit 306 includes a U1 gate 318 parameterized based on the first eigenvalue 0.1 and the second eigenvalue 0.2 of the first data point. As Figure 3As shown, the U1 gate 318 is parameterized based on 17.89, which is calculated based on the first eigenvalue and the second eigenvalue: 17.89 = 2*(π - 0.1)(π - 0.2). This quantity is provided as a non-limiting example of how to parameterize rotation based on multiple eigenvalues of data points.

[0033] The backward feature map template circuit 308 includes gates parameterized based on the eigenvalues of the second data point (0.3, 0.4). According to an embodiment of the present invention, for each feature dimension, the backward feature map template circuit 308 includes at least one rotation parameterized based on the eigenvalue of the second data point in the data point pair of the feature dimension. For example, the backward feature map template circuit 308 includes the U1 gate 318 parameterized based on the first eigenvalue 0.3 of the second data point. The backward feature mapping template circuit 308 also includes the U1 gate 320 parameterized based on the second eigenvalue 0.4 of the second data point.

[0034] Following the backward feature map template circuit 308 is the metric circuit 310. The metric circuit outputs the metric values for each qubit in the quantum processor. The metric for each qubit can be read out to a corresponding classical bit. For example, in Figure 3 the metric of qubit q0 302 can be read out to classical bit c0 322, and the metric of qubit q1 304 can be read out to classical bit c1 324. Each corresponding classical bit is combined to provide a similarity metric. For example, the two-digit number C1C0 can be read as the value of the similarity metric (e.g., 11 = 3). For two data points i and j, the similarity metric can be used as the i-th and j-th elements of the similarity matrix. The similarity metric can be directly input into the similarity matrix, or it can be used to determine the number to be input into the similarity matrix. For example, the number can be found by multiplying, dividing, or negating the similarity metric. The number can be found by inputting the similarity metric into a function that outputs the number to be used in the similarity matrix.

[0035] Figure 4Schematic diagram of a hybrid quantum-classical system 400 according to an embodiment of the present invention. The system 400 includes a quantum processor 402. The quantum processor 402 has a plurality of qubits 404, 406, and these qubits correspond to multiple feature dimensions of each of the multiple data points to be clustered. The quantum processor 402 is configured to execute a quantum circuit for each pair of the multiple data points. The quantum circuit includes a feature map template circuit parameterized with a first plurality of rotations, where the first plurality of rotations is based on the feature values of the first data point in the data point pair. The quantum circuit includes a backward feature map template circuit parameterized with a second plurality of rotations, where the second plurality of rotations is based on the feature values of the second data point in the data point pair. The quantum circuit includes a metric circuit that outputs a similarity metric for the data point pair. The feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor. The feature map template circuit, the backward feature map template circuit, and the metric circuit of the quantum processor 402 may have the attributes described above with reference to Figures 1 to 3 as described.

[0036] The hybrid quantum-classical system 400 includes a classical processor 408 in communication with the quantum processor 402. The classical processor is configured to receive the similarity metric for each pair of the multiple data points and create a similarity matrix for the multiple data points based on the similarity metric for each data point pair. The classical processor 408 is further configured to perform a classical clustering algorithm on the similarity matrix to cluster the multiple data points.

[0037] According to an embodiment of the present invention, the classical processor 408 is further configured to assign each of the multiple data points to a cluster and output an indication of the cluster for each of the multiple data points.

[0038] The traditional processor 408 can be a dedicated "hard-wired" device, or it can be a programmable device. For example, it can be, but is not limited to, a personal computer, a workstation, or any other suitable electronic device for a specific application. In some embodiments, it can be integrated into a unit, or it can be attachable, remote, and / or distributed.

[0039] According to an embodiment of the present invention, the hybrid quantum-classical system uses a quantum device to calculate a similarity matrix for some data and then inputs the similarity matrix into a classical spectral clustering algorithm. Compared with an algorithm using a similarity matrix calculated on a classical computer, the system can perform better on data points that can be separated using a quantum feature map. In particular, the system employs a quantum feature map that utilizes the quantum properties of superposition and entanglement. Achieving the same dimension in the feature space on a classical computer would require exponential time (unless the polynomial hierarchy collapses).

[0040] We incorporate the similarity matrix computed using a quantum device into the spectral clustering algorithm as follows. Starting from the dataset to be clustered, we use the number of feature dimensions to determine the number of qubits to be used in the computation of the similarity matrix. According to an embodiment of the present invention, the number of qubits is equal to the number of feature dimensions. After determining the number of qubits to be used, we use a quantum circuit acting on the determined number of qubits to compute the similarity matrix. We ensure that a quantum circuit including a feature map template circuit is selected, and the feature map template circuit uses the quantum properties of superposition and entanglement. The feature map template circuit is specifically designed to be exponentially difficult to simulate conventionally.

[0041] For each pair of data points, we create and run a quantum circuit to measure the distance between the two data points. The quantum circuit includes a feature map template circuit parameterized with rotations equal to the eigenvalues of the first data point. The quantum circuit also includes a backward feature map template circuit parameterized with rotations equal to the eigenvalues of the second data point. The quantum circuit includes a measurement circuit that outputs a similarity measure for the pair of data points. The measurement circuit can measure each of the determined number of qubits. The values of the qubits can be read out onto classical bits indicating the similarity measure for the pair of data points. We use the similarity measures for each pair of data points to create a similarity matrix. We input the similarity matrix into a classical clustering algorithm to compute the clusters in the provided dataset. According to an embodiment of the present invention, the classical clustering algorithm is a spectral clustering algorithm. However, embodiments of the present invention are not limited to spectral clustering algorithms. Other types of clustering algorithms can be used, such as k-Means clustering and density clustering. Additionally, the quantum circuit can be constructed based on the specific classical clustering algorithm to be used.

[0042] We have implemented and tested the system using a dataset whose cluster identities are known, so the accuracy of the system can be verified. The system is able to identify two clusters with 100% accuracy, while spectral clustering with a similarity matrix computed using traditional computing is completely unable to cluster the data. Figure 5 The simulation results 500 and experimental results 502 of a dataset are shown, where there are results with three different measures of clustering: (1) Fowles_mallows, (2) mutual information, and (3) rand_index. The three different measures of clustering are three different ways of evaluating how well a clustering algorithm performs. For the actual results 502, the quantum circuit is implemented in quantum hardware.

[0043] For each data set, simulations using a hybrid quantum-classical algorithm are performed for the quantum circuit in combination with one of the following three classical algorithms: "agglomerative", "dbscan", and "spectral". A value of 1.0 indicates that each data point is accurately clustered, while a value below 1.0 indicates an inaccurate clustering. As Figure 5 shown, the value for each simulated hybrid quantum-classical algorithm is 1.0, indicating 100% accuracy.

[0044] For each data set, simulations using a pure classical algorithm are performed for each of the four classical algorithms: "linear", "poly", "rbf", and "sigmoid". As Figure 5 shown, the pure classical algorithms have a much lower accuracy than the hybrid system.

[0045] The experimental results for the three different metrics 502 each have a value equal to or very close to 1, indicating high accuracy even when implemented in physical hardware. Figure 5 The results in highlight the ability of the methods and systems disclosed herein to accurately cluster data sets that are very problematic for classical computers.

[0046] The description of the various embodiments of the invention has been presented for purposes of illustration, but is not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to a person of ordinary skill in the art without departing from the scope of the described embodiments. The terms used herein were chosen to best explain the principles of the embodiments, the practical application, or technical improvement over technologies found in the marketplace, or to enable other persons of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

1. A method for performing unsupervised clustering of multiple data points, comprising: Determining the number of qubits to include in a quantum processor based on multiple feature dimensions of each data point among the multiple data points; For each pair of data points among the multiple data points, executing a quantum circuit on a quantum processor having the determined number of qubits, wherein the quantum circuit comprises: A feature map template circuit parameterized to have a first plurality of rotations, wherein the first plurality of rotations are based on the feature values of a first data point of the data point pair; A backward feature map template circuit parameterized to have a second plurality of rotations, wherein the second plurality of rotations are based on the feature values of a second data point of the data point pair; and A metric circuit that outputs a similarity metric for the data point pair; Creating a similarity matrix of the multiple data points based on the similarity metric for each data point pair; and Inputting the similarity matrix into a classical clustering algorithm to cluster the multiple data points, wherein the feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor.

2. The method according to claim 1, wherein determining the number of qubits to include in the quantum processor based on multiple feature dimensions of each data point among the multiple data points comprises including one qubit in the quantum processor for each feature dimension.

3. The method according to claim 1 or 2, wherein the feature map template circuit comprises a plurality of entangled two-qubit quantum gates.

4. The method according to claim 1 or 2, wherein the feature map template circuit comprises, for each feature dimension, a rotation parameterized based on the feature value of the first data point of the data point pair for the feature dimension.

5. The method according to claim 4, wherein the feature map template circuit comprises, for each feature dimension, a plurality of rotations parameterized based on the feature values of the first data points of the multiple data points for the feature dimension.

6. The method according to claim 5, wherein the feature map template circuit comprises, for the first data point in the data point pair, a rotation parameterized based on the multiple feature values of the first data point.

7. The method according to claim 1 or 2, wherein the backward feature map template circuit comprises, for each feature dimension, at least one rotation parameterized based on the feature value of the second data point of the data point pair for the feature dimension.

8. The method according to claim 1 or 2, further comprising constructing the feature map template circuit and the backward feature map template circuit based on the classical clustering algorithm.

9. The method according to claim 1 or 2, wherein the classical clustering algorithm is a spectral clustering algorithm.

10. The method according to claim 1 or 2, wherein the metric circuit outputs a metric value for each qubit included in the quantum processor.

11. The method according to claim 10, wherein the metric value for each qubit is read out to a corresponding classical bit.

12. The method according to claim 11, wherein each classical bit in the corresponding classical bits is combined to provide the similarity metric.

13. A hybrid quantum-classical system, comprising: A quantum processor having a plurality of qubits corresponding to a plurality of feature dimensions of each of a plurality of data points to be clustered, the quantum processor being configured to: For each pair of the plurality of data points, execute a quantum circuit, the quantum circuit comprising: A feature map template circuit parameterized to have a first plurality of rotations, wherein the first plurality of rotations are based on the feature values of a first data point of the pair of data points; A backward feature map template circuit parameterized to have a second plurality of rotations, wherein the second plurality of rotations are based on the feature values of a second data point of the pair of data points; and A metric circuit that outputs a similarity metric for the pair of data points, wherein the feature map template circuit and the backward feature map template circuit each utilize the quantum properties of superposition and entanglement of the qubits of the quantum processor; and A classical processor in communication with the quantum processor, the classical processor being configured to: Receive the similarity metric for each pair of the plurality of data points; Create a similarity matrix for the plurality of data points based on the similarity metric for each pair of data points; and Execute a classical clustering algorithm on the similarity matrix to cluster the plurality of data points.

14. The hybrid quantum-classical system according to claim 13, wherein the classical processor is further configured to: Assign each of the plurality of data points to a cluster; and Output an indication of the cluster for each of the plurality of data points.

15. The hybrid quantum-classical system according to claim 13 or 14, wherein the feature map template circuit comprises a plurality of entangled two-qubit quantum gates.

16. The hybrid quantum-classical system according to claim 13 or 14, wherein the feature map template circuit comprises, for each feature dimension, a rotation parameterized based on the feature value of the first data point of the pair of data points for the feature dimension.

17. The hybrid quantum-classical system according to claim 16, wherein the feature map template circuit comprises, for each feature dimension, a plurality of rotations parameterized based on the feature value of the first data point of the pair of data points for the feature dimension.

18. The hybrid quantum-classical system according to claim 17, wherein the feature map template circuit comprises, for the first data point in the pair of data points, rotations parameterized based on the plurality of feature values of the first data point.

19. The hybrid quantum-classical system according to claim 13 or 14, wherein the backward feature map template circuit comprises, for each feature dimension, at least one rotation parameterized based on the feature value of the second data point of the pair of data points for the feature dimension.

20. The hybrid quantum-classical system according to claim 13 or 14, further comprising constructing the feature map template circuit and the backward feature map template circuit based on the classical clustering algorithm.

21. The hybrid quantum-classical system according to claim 13 or 14, wherein the classical clustering algorithm is a spectral clustering algorithm.

22. The hybrid quantum-classical system according to claim 13 or 14, wherein the metric circuit outputs metric values for each qubit included in the quantum processor.

23. The hybrid quantum-classical system according to claim 22, wherein the metric value of each qubit is read out to a corresponding classical bit in the classical processor.

24. The hybrid quantum-classical system according to claim 23, wherein each classical bit in the corresponding classical bits is combined to provide the similarity metric.

Images