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S c And . k the , by , eans (1) ers rix m, nd k k e s - f ; l 2) Assign each data point to the cluster with the closest to 0. The smallest eigenvalues of both matrices are 0 and the centroid. elements of the associated eigenvector are equal to one. For 3) Assign the means of data in clusters as the new means: undirected graphs with non-negative weights, the multiplicity vi i.e., mc = . k of eigenvalue 0 gives the number of connected components Pvi Sc Sc 4) Repeat step 2 and∈ 3| until| there is no change in the means. in the graph. Clustering is generally done through the first The quantum version of this algorithm also has been intro- k-eigenvectors associated to the smallest eigenvalues of the duced in Ref.[3] and used for the nearest-neighbor classifi- . Here, k represents the number of clusters cation. Moreover, in a prior work [17], quantum subroutines to be constructed from the graph. This number describes the based on Grover’s algorithm [18] are presented to speed-up smallest eigenvalues λ1 ...λk such that γk = λk λk+1 | − | the classical clustering algorithm. gives the largest eigengap among all possible eigengaps. The clusters are obtained by applying k-means algorithm to B. Spectral Clustering the rows of the matrix V formed by using k column eigenvec- In this subsection, we mainly follow the related sections tors [20]. The same clustering methodology can also be used of Ref.[19] and try to briefly summarize the concept of the after the rows of V are normalized into unit vectors[21]. spectral clustering. Similarities between data points are most k-means clustering can also be done in the subspace of T commonly represented by similarity graphs: i.e., undirected principal components of XX , where X is the centered data matrix[22], [23]. Furthermore, for a given kernel matrix K weighted graphs in which the vertices vi and vj are connected and a diagonal weight matrix with , where if the data points, xi and xj represented by these vertices are W K = W AW A is the adjacency (or similarity) matrix and W is the diagonal similar. And the weights on the edges wij indicates the amount weight matrix; the leading k eigenvectors of W 1/2KW 1/2 of the similarity, sij between xi and xj . The construction of graph G(V, E) from a given data set can be used for graph partitioning (clustering) by minimizing V V T Y Y T = 2k 2trace(Y T V V T Y ) or maximizing x1 ...xN with pairwise similarities sij or distances dij can || −T T || − be{ done in} many different ways. Three of the famous ones are: trace(Y V V Y ). Here, V represents the k eigenvectors and Y is the orthonormal indicator matrix representing clusters: The undirected ǫ-neighborhood graph: The vertices vi and Y = [y , y ,..., y ] with vj are connected if the pairwise distance dij for xi and xj is 1 2 k greater than some threshold ǫ. The k-nearest neighborhood 1 T graph: The vertices v and v are connected if vertex v is yj = [0,..., 0, 1,..., 1, 0,..., 0] (4) i j i pNj one of the k-nearest neighbor of vj and vice versa. Note | N{zj } that there are some other definitions in the literature even Therefore, the clustering problem turns into the maximization though this is the most commonly used criteria to construct of the following: the k-nearest neighborhood graph as an undirected graph. k The fully connected graph: This describes a fully connected T T T T trace(Y V V Y )= X yj V V yj. (5) weighted graph where the weights, wij s, are determined from j=1 a similarity function sij : e.g., Gaussian similarity function xi xj sij = exp( || −2 || ), where σ is a control parameter. After finding the k eigenvectors, the maximization is done by 2σ 1 2 The clustering− problem can be described as finding a running the weighted k means on the rows of W / V [24], partition of the graph such that the sum of the weights on [15]. the edges from one group to another is very small. And the II. QUANTUM SPECTRAL CLUSTERING weights on the edges between vertices inside the same group is very high. In the quantum version of the spectral clustering, we will use T 1) Similarity Matrix and Its Laplacian: Let W be the y V V y as the similarity measure for the clustering. This givesh | a similar| i measure to the one in Eq.(5). Here V represents adjacency matrix with the matrix elements, wij representing the weight on the edge between vertex v and vertex v . The the eiegenvectors associated with the non-zero eigenvalues of i j T eigenvectors of a Laplacian matrix generated from W is the either a Laplacian L, or a data matrix XX , or a weighted kernel matrix W 1/2KW 1/2; and y represents a basis vector main instrument for the spectral clustering. The unnormalized | i Laplacian for the graph given by W is defined as: chosen from some arbitrary indicator matrix. In the following section, we first describe how to obtain V V T y through the L = D W, (2) phase estimation algorithm then measure the| outputi in the − basis Y consisting of y . where D is the diagonal degree matrix with diagonal elements | i N dii = Pj=1 wij . The matrix L is a symmetric matrix and A. Principal Component Analysis through Phase Estimation generally normalized as: Algorithm 1 1 1 1 The phase estimation algorithm [25], [26] finds the phase of L = D− 2 LD− 2 = I D− 2 WD− 2 , (3) e − the eigenvalue of a unitary matrix. If the unitary matrix cor- iH n which preserves the symmetry. L and L are semi-positive responds to the exponential, U = e , of a matrix H ⊗ definite matrices: i.e., their eigenvaluese are greater or equal which is either a Laplacian matrix or equal to XXT , then∈ R the

2 The Probabilities of the States in the First Register 1 1110111 1100011 1001111 0.5 0111011 0100111 Probability 0010011 Fig. 1: Biased Phase Estimation Algorithm 0 0 2 4 6 8 10 12 14 16 18 Iteration The Total Probability and the Fidelity obtained phase represents the eigenvalue of H. Furthermore, 1 while the phase register in the algorithm holds the phase value, 0.8 the system register holds the associated eigenvector in the Probability Fidelity output. When given an arbitrary input, the phase estimation 0.6 Probability algorithm yields an output state which represents a superposi- 0.4 tion of the eigenvalues with the probabilities determined from 0 5 10 15 the overlaps of the input and the eigenvectors. In Ref.[12], we Iteration have introduced a quantum version of the principal component Fig. 2: The iterations of the amplitude amplification with the analysis and showed that one can obtain the eigenvalues standard phase estimation algorithm for a random 16 16 in certain desired region and the associated eigenvectors by matrix with six non-zero eigenvalues. × applying amplitude amplification algorithm [27], [28], [26] to the end of the phase estimation algorithm. As done in Ref.[13] for Widrow-Hoff learning rule, one can eliminate the extra where f1 =1/µ[κ, 1,..., 1] with κ being a coefficient and µ dimensions (zero eigenvalues and eigenvectors) and generate being ah normalization| constant. The number of iterations can T the output state equivalent to V V y for some input vector be minimized by engineering the value of κ. y by using the following amplification| i iteration: | i Using Uf1 in lieu of the quantum Fourier transform leads the algorithm drawn in Fig.1, which produces a biased su- Q = UPEAUsUPEAUf2 . (6) perposition of the eigenpairs in the output which speeds- Here, UPEA represents the phase estimation algorithm applied up the amplitude amplification: The amplification is done to H and Uf2 and Us are the marking and the amplification by inversing the marked item about the mean amplitude. operators described below: As the amplitude of the marked item becomes closer to m n U 2 = I⊗ 2 f2 f2 I⊗ (7) the mean amplitude, the amplification process gets slower f − | i h | ⊗ 1 (i.e., it requires more iterations). In fact, when κ = √M, with f2 = [0, 1,..., 1] and m represents the number √N 1 κ/µ ( P 0 + 1 P 0 )/2.: i.e. the mean amplitude. h | − p p of qubits in the phase register. And, In this≈ case,| noi increment− | i or decrement will be seen in m+n the amplitude amplification. Therefore, making the initial Us = I⊗ 2 0 0  . (8) − | i h | probability further away from the mean probability, we can In the output of the amplitude amplification process, when speed-up the amplitude amplification. As an example, we show the first register is in the equal superposition state, the second the iterations of the amplitude amplification with the standard T y register holds V V . phase estimation in Fig.2 and with the biased phase estimation 1) With a Biased| Phasedi Estimation Algorithm: In the using U 1 with κ = 1, and κ = 20, respectively in Fig.3 output of the phase estimation, the probability values for f and Fig.4: As seen from Fig.2, the probability and the fidelity different eigenpairs are determined from the overlaps of the in PEA with QFT are maximized after 12 iterations. Here, eigenvectors with the initial vector y : i.e. the inner product of the fidelity is defined as the overlap of the expected output the eigenvector and the initial vector.| i Our aim in the amplitude V V T y with the actual output. However, the maximum amplification is to eliminate eigenvectors corresponding to probability| i and the fidelity can be observed after 6 iterations the zero eigenvalues. The probability of the chosen states when we use U 1 with κ = 1, which generates a very low (the eigenvectors corresponding to the non-zero eigenvalues) f initial success probability but approaches to the maximum during the amplitude amplification process oscillates: i.e., faster. Using U 1 with κ = 20 to increase the initial success when the iteration operator is applied many times, depending f probability also decreases the required number of iterations on the iterations, the probabilities goes up and down with upto 2 iterations. some frequency. To speed up the amplitude amplification, to At the end of the phase estimation algorithm either the bi- decrease the frequency, we will use a biased phase estimation ased or the standard, we have V V T y , where a column of V algorithm which generates the same output but requires less represents a principal eigenvector (eigenvector| i corresponding number of iterations: In place of the quantum Fourier trans- to a non-zero eigenvalue). y V V T y provides a similarity form which creates an equal superposition of the states in the measure for the data point hy |to the |XXi T . y V V T y can first register, we use the following operator: be obtained either measuring the final stateh in| a such| i basis m n U 1 = I⊗ 2 f1 f1 I⊗ , (9) in which y is the first basis vector. Or one can apply a f − | i h | ⊗ | i

3 The Probabilities of the States in the First Register of unitary matrices. 1 1110111 2) Then, prepare the input state 0 0 , where 0 1100011 | i | i | i 1001111 represents the first vector in the standard basis. 0.5 0111011 The phase estimation uses two registers: the phase 0100111 Probability 0010011 and the system register. In our case, the final state

0 of the system register becomes equivalent to the 0 2 4 6 8 10 12 14 16 18 T Iteration state obtained from V V y . The Total Probability and the Fidelity | i 1 3) To prepare y on the system register, apply Probability an input preparation| i gate, , to the system Fidelity Uinput register. 0.5 n 4) Then, apply Uf1 = I 2 f1 f1 I⊗ to put Probability the phase register into− a biased| i h superposition.|⊗ j 0 2 0 5 10 15 5) Then, apply controlled U s to the system regis- Iteration ter. † 6) Finally, apply Uf1 to get the eigenpairs on the Fig. 3: The iterations of the amplitude amplification with the output with some probability. biased phase estimation algorithm for the same matrix as in

Fig.2: in the construction of Uf1 , the value of κ =1 and the initial success probability is around 0.028. The steps of the amplitude amplification process are as follows: The Probabilities of the States in the First Register 1) After applying U to the initial state 0 0 , 1 BPEA | i | i 1110111 apply the following iteration operator: 1100011 1001111 0.5 0111011 Q = UBPEAUsUBPEAUf2 . (10) 0100111 Probability 0010011 2) Measure one of the qubits in the phase register: 0 0 2 4 6 8 10 12 14 16 18 if it is close to the equal superposition state, Iteration • The Total Probability and the Fidelity then stop the algorithm. 1 if it is not in the superposition state, then apply Probability • Fidelity the iteration operator given above again.

0.5 Repeat this step until one of the qubits in the •

Probability phase register is in the equal superposition state (Individual qubits present the same behavior as 0 0 5 10 15 the whole state of the register. If one of the Iteration qubits approaches to the equal superposition Fig. 4: The iterations of the amplitude amplification with the state, it generally indicates all qubit approaches biased phase estimation algorithm for the same matrix as in to the equal superposition state.).

Fig.2: in the construction of Uf1 , the value of κ = 20 and the 3) Measure the second register in the basis in which initial success probability is around 0.43. y is a basis vector. | i

Householder transformation I 2 y y to the final state and then measure this state in the− standard| i h basis.| In that case, the III. CIRCUIT IMPLEMENTATION AND COMPUTATIONAL probability for measuring 0 yields the nearness (similarity) COMPLEXITY measure. | i A. Circuit for XXT and Complexity Analysis Below, we summarize the steps of the biased phase estima- tion and then the amplitude amplification algorithms: Most of the Laplace matrices used in the clustering algo- rithm are generally very large but sparse. The sparse matrices As shown in Fig.1, the steps of the biased phase can be simulated in polynomial time (polynomial in the estimation, UBPEA, are as follows: number of qubits) on quantum computers(see e.g. Ref.[29], 1) First, assume we have a mechanism to produce [30]). When the matrix H is dense (from now on we will j U 2 used in the phase estimation algorithm. Here, use H to represent XXT or L), the phase estimation algo- U represents the circuit design for the data matrix. rithm requires eiH . This exponential is generally approximated In the next section, we will discuss how to find through the Trotter-Suzuki decomposition. However, the order this circuit by writing the data matrix as a sum of the decomposition increases the complexity dramatically. Recently in Ref.[31], we have showed that one can use a

4 matrix directly in the phase estimation by converting the C. Measurement of the Output matrix into the following form: If only tensor product of the eigenvectors of Pauli operators H I, σx, σy, σz are used, the number of nonzero elements can H = I i , (11) {be either one} of the followings: N,N/2,... 1 to produce − k e a disentangled quantum state y , which can be efficiently where k is a coefficient equal to H 1 10, which guarantees | i || || × implemented. We can also use an operator similar to Y = that the imaginary parts of the eigenvalues of H represents the σi , where i indicates a σ gate on the ith qubit of the e Pi x x eigenvalues of H/k and are less than 0.1 so that sin(λj /k) system, to approximate the best clustering (This operator ≈ λj /k. Here, λj is an eigenvalue of H. When phase estimation is also used in Ref.[32] in the description of a quantum algorithm is applied to H, the obtained phase gives the value optimization algorithm). In this case, the initial state is taken e of λj /k. as an equal superposition state, then multiplied with eiY . In The circuit implementation of H can be achieved by writing the end, the operator eiY is applied again. The measurement H as a sum of unitary matrices whiche can be directly mapped outcome yields the index of the column of Y which can be toe quantum circuits. This sum can be produced through the considered as an approximate clustering solution. Since the iY Householder transformation matrices formed by the data vec- circuit implementation of e requires only n single σx-gates tors. For the matrix H = XXT , this can be done as follows: and the solution is obtained only from one trial, the total [31]: complexity becomes O(2mLN). 1 L As mentioned before, the other measurements can be ap- H = X [(I 2 xj xj ) I] (12) proximated through the mapping to the standard basis by −2 − | i h | − j=1 applying the Householder transformation I 2 y y to the − | i h | In the above, H consists of (L + 1) number of Householder system register. In that case, the probability to see the system register in 0 represents the similarity measure. matrices each of which can be implemented by using O(N) | i number of quantum gates. H can be generated in an ancilla IV. CONCLUSION based circuit combining the circuits for each Householder Here, we have described how to do spectral clustering on matrix. The resulting circuit will require O(LN) quantum quantum computers by employing the famous quantum phase operations. In the phase estimation, if we use m number of estimation and amplitude amplification algorithms. We have qubits in the first register, the complexity of the whole process m shown the implementation steps and analyzed the complexity becomes O(2 LN). In the case of H written as a some of the whole process. In addition, we have shown that the of simple unitaries which can be represented by O(logN) required number of iterations in the amplitude amplification quantum gates, the complexity can be further reduced to m process can be dramatically decreased by using a biased O(2 Llog(N)) number of quantum gates. operator instead of the quantum Fourier transform in the phase estimation algorithm. B. Comparison to Classical Computers REFERENCES Spectral clustering on classical computers requires some form of the eigendecomposition of the matrix H or at least the [1] J. 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