# American Institute of Mathematical Sciences

December  2019, 1(4): 419-431. doi: 10.3934/fods.2019017

## Quantum topological data analysis with continuous variables

 Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200, USA

Published  December 2019

I introduce a continuous-variable quantum topological data algorithm. The goal of the quantum algorithm is to calculate the Betti numbers in persistent homology which are the dimensions of the kernel of the combinatorial Laplacian. I accomplish this task with the use of qRAM to create an oracle which organizes sets of data. I then perform a continuous-variable phase estimation on a Dirac operator to get a probability distribution with eigenvalue peaks. The results also leverage an implementation of continuous-variable conditional swap gate.

Citation: George Siopsis. Quantum topological data analysis with continuous variables. Foundations of Data Science, 2019, 1 (4) : 419-431. doi: 10.3934/fods.2019017
##### References:
 [1] R. N. Alexander, S. C. Armstrong, R. Ukai and N. C. Menicucci, Noise analysis of single-mode Gaussian operations using continuous-variable cluster states, Phys. Rev. A, 90 (2014), 062324. Google Scholar [2] S. Basu, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discret. Comput. Geom., 22 (1999), 1–18. doi: 10.1007/PL00009443.  Google Scholar [3] S. Basu, Different bounds on the different Betti numbers of semi-algebraic sets, Discret. Comput. Geom., 30 (2003), 65–85. doi: 10.1007/s00454-003-2922-9.  Google Scholar [4] S. Basu, Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math., 8 (2008), 45–80. doi: 10.1007/s10208-005-0208-8.  Google Scholar [5] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe and S. Lloyd, Quantum machine learning, Nature, 549 (2017), 195. Google Scholar [6] S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys., 77 (2005), 513–577. doi: 10.1103/RevModPhys.77.513.  Google Scholar [7] G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, Int. J. Shape Model, 11 (2005), 149. Google Scholar [8] F. Chazal and A. Lieutier, Stability and computation of topological invariants of solids in $\mathbb R^n$, Discret. Comput. Geom., 37 (2007), 601–617. doi: 10.1007/s00454-007-1309-8.  Google Scholar [9] D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discret. Comput. Geom., 37 (2007), 103–120. doi: 10.1007/s00454-006-1276-5.  Google Scholar [10] F. De Martini, V. Giovannetti, S. Lloyd, L. Maccone, E. Nagali, L. Sansoni and F. Sciarrino, Quantum random access memory, Phys. Rev. A, 80 (2009), 010302. Google Scholar [11] R. Dridi and H. Alghassi, Homology Computation of Large Point Clouds using Quantum Annealing, arXiv: 1512.09328. Google Scholar [12] H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological Persistence and Simplification, Discret. Comput. Geom., 28 (2002), 511–533. doi: 10.1007/s00454-002-2885-2.  Google Scholar [13] P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal., 9 (1999), 596. Google Scholar [14] R. Ghrist, Barcodes: The persistent topology of data, Bull. Am. Math. Soc. (N.S.), 45 (2008), 61–75. doi: 10.1090/S0273-0979-07-01191-3.  Google Scholar [15] V. Giovannetti, S. Lloyd and L. Maccone, Quantum random access memory, Phys. Rev. Lett., 100 (2008), 160501, 4pp. doi: 10.1103/PhysRevLett.100.160501.  Google Scholar [16] V. Giovannetti, S. Lloyd and L. Maccone, Architectures for a quantum random access memory, Phys. Rev. A, 78 (2008), 052310. Google Scholar [17] L. K. Grover, A fast quantum mechanical algorithm for database search, Annual ACM Symposium on the Theory of Computing, (1996), 212–219. doi: 10.1145/237814.237866.  Google Scholar [18] M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph and P. van Loock, Quantum computing with continuous-variable clusters, Phys. Rev. A, 79 (2009), 062318. Google Scholar [19] S. Harker, K. Mischaikow, M. Mrozek and V. Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math., 14 (2014), 151–184. doi: 10.1007/s10208-013-9145-0.  Google Scholar [20] H. L. Huang, X. L. Wang, P. P. Rohde, Y. H. Luo, Y. W. Zhao, C. Liu, L. Li, N. L. Liu, C. Y. Lu and J. W. Pan, Demonstration of topological data analysis on a quantum processor, Optica, 5 (2018), 193. Google Scholar [21] D. Kozlov, Combinatorial Algebraic Topology, , Algorithms and Computation in Mathematics, 21. Springer, Berlin, 2008. doi: 10.1007/978-3-540-71962-5.  Google Scholar [22] H. K. Lau, R. Pooser, G. Siopsis and C. Weedbrook, Quantum machine learning over infinite dimensions, Phys. Rev. Lett., 118 (2017), 080501, 6pp. doi: 10.1103/PhysRevLett.118.080501.  Google Scholar [23] H. K. Lau, R. Pooser, G. Siopsis and C. Weedbrook, Quantum machine learning over infinite dimensions, Phys. Rev. Lett., 118 (2017), 080501, 6pp. doi: 10.1103/PhysRevLett.118.080501.  Google Scholar [24] N. Liu, J. Thompson, C. Weedbrook, S. Lloyd, V. Vedral, M. Gu and K. Modi, Power of one qumode for quantum computation, Phys. Rev. A, 93 (2016), 052304, 10pp. doi: 10.1103/physreva.93.052304.  Google Scholar [25] S. Lloyd, Hybrid quantum computing, preprint, arXiv: quant-ph/0008057. Google Scholar [26] S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett., 82 (1999), 1784–1787. doi: 10.1103/PhysRevLett.82.1784.  Google Scholar [27] S. Lloyd, S. Garnerone and P. Zanardi, Quantum algorithms for topological and geometric analysis of data, Nat. Commun., 7 (2016), 10138. Google Scholar [28] P. van Loock, C. Weedbrook and M. Gu, Building Gaussian cluster states by linear optics, Phys. Rev. A, 76 (2007), 032321. Google Scholar [29] K. Marshall, C. S. Jacobsen, C. Schafermeier, T. Gehring, C. Weedbrook and U. L. Andersen, Continuous-variable quantum computing on encrypted data, Nat. Comm., 7 (2016), 13795. Google Scholar [30] N. C. Menicucci, Fault-tolerant measurement-based quantum computing with continuous-variable cluster states, Phys. Rev. Lett., 112 (2014), 120504. Google Scholar [31] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph and M. A. Nielsen, Universal quantum computation with continuous-variable cluster states, Phys. Rev. Lett., 97 (2006), 110501. Google Scholar [32] K. Mischaikow and V. Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discret. Comput. Geom., 50 (2013), 330–353. doi: 10.1007/s00454-013-9529-6.  Google Scholar [33] M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information, , Cambridge University Press, 2000.  Google Scholar [34] P. Niyogi, S. Smale and S. Weinberger, A topological view of unsupervised learning from noisy data, SIAM J. Comput., 40 (2011), 646–663. doi: 10.1137/090762932.  Google Scholar [35] M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer and O. Pfister, Parallel generation of quadripartite cluster entanglement in the optical frequency comb, Phys. Rev. Lett., 107 (2011), 030505. Google Scholar [36] H. Reitberger, Leopold Vietoris (1891–2002),, Notices of the American Mathematical Society, 49 (2002), 1232-1236.   Google Scholar [37] V. Robins, Towards computing homology from finite approximations, Topol. Proc., 24 (1999), 503–532.  Google Scholar [38] S. Takeda, T. Mizuta, M. Fuwa, J.-i. Yoshikawa, H. Yonezawa and A. Furusawa, Generation and eight-port homodyne characterization of time-bin qubits for continuous-variable quantum information processing, Phys. Rev. A, 87 (2013), 043803. Google Scholar [39] C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys., 84 (2012), 621. Google Scholar [40] C. R. Wie, A Quantum Circuit to Construct All Maximal Cliques Using Grover Search Algorithm, arXiv: 1711.06146. Google Scholar [41] S. Yokoyama, R. Ukai, S. C. Armstrong, J.-i. Yoshikawa, P. van Loock and A. Furusawa, Demonstration of a fully tunable entangling gate for continuous-variable one-way quantum computation, Phys. Rev. A, 92 (2015), 032304. Google Scholar [42] S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. i. Yoshikawa, H. Yonezawa, N. C. Menicucci and A. Furusawa, Ultra-large-scale continuous-variable cluster states multiplexed in the time domain, Nature Photonics, 7 (2013), 982. Google Scholar [43] J. i. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino and A. Furusawa, Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing, Applied Phys. Lett. Photonics, 1 (2016), 060801. Google Scholar [44] J. Zhang and S. L. Braunstein, Continuous-variable Gaussian analog of cluster states, Phys. Rev. A, 73 (2006), 032318. Google Scholar [45] A. Zomorodian and G. Carlsson, Computing persistent homology, Discret. Comput. Geom., 33 (2005), 249–274. doi: 10.1007/s00454-004-1146-y.  Google Scholar [46] A. Zomorodian, Algorithms and Theory of Computation Handbook, 2nd edition, Ch. 3, section 2., Chapman and Hall/CRC, 2009. Google Scholar

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##### References:
 [1] R. N. Alexander, S. C. Armstrong, R. Ukai and N. C. Menicucci, Noise analysis of single-mode Gaussian operations using continuous-variable cluster states, Phys. Rev. A, 90 (2014), 062324. Google Scholar [2] S. Basu, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discret. Comput. Geom., 22 (1999), 1–18. doi: 10.1007/PL00009443.  Google Scholar [3] S. Basu, Different bounds on the different Betti numbers of semi-algebraic sets, Discret. Comput. Geom., 30 (2003), 65–85. doi: 10.1007/s00454-003-2922-9.  Google Scholar [4] S. Basu, Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math., 8 (2008), 45–80. doi: 10.1007/s10208-005-0208-8.  Google Scholar [5] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe and S. Lloyd, Quantum machine learning, Nature, 549 (2017), 195. Google Scholar [6] S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys., 77 (2005), 513–577. doi: 10.1103/RevModPhys.77.513.  Google Scholar [7] G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, Int. J. Shape Model, 11 (2005), 149. Google Scholar [8] F. Chazal and A. Lieutier, Stability and computation of topological invariants of solids in $\mathbb R^n$, Discret. Comput. Geom., 37 (2007), 601–617. doi: 10.1007/s00454-007-1309-8.  Google Scholar [9] D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discret. Comput. Geom., 37 (2007), 103–120. doi: 10.1007/s00454-006-1276-5.  Google Scholar [10] F. De Martini, V. Giovannetti, S. Lloyd, L. Maccone, E. Nagali, L. Sansoni and F. Sciarrino, Quantum random access memory, Phys. Rev. A, 80 (2009), 010302. Google Scholar [11] R. Dridi and H. Alghassi, Homology Computation of Large Point Clouds using Quantum Annealing, arXiv: 1512.09328. Google Scholar [12] H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological Persistence and Simplification, Discret. Comput. Geom., 28 (2002), 511–533. doi: 10.1007/s00454-002-2885-2.  Google Scholar [13] P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal., 9 (1999), 596. Google Scholar [14] R. Ghrist, Barcodes: The persistent topology of data, Bull. Am. Math. Soc. (N.S.), 45 (2008), 61–75. doi: 10.1090/S0273-0979-07-01191-3.  Google Scholar [15] V. Giovannetti, S. Lloyd and L. Maccone, Quantum random access memory, Phys. Rev. Lett., 100 (2008), 160501, 4pp. doi: 10.1103/PhysRevLett.100.160501.  Google Scholar [16] V. Giovannetti, S. Lloyd and L. Maccone, Architectures for a quantum random access memory, Phys. Rev. A, 78 (2008), 052310. Google Scholar [17] L. K. Grover, A fast quantum mechanical algorithm for database search, Annual ACM Symposium on the Theory of Computing, (1996), 212–219. doi: 10.1145/237814.237866.  Google Scholar [18] M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph and P. van Loock, Quantum computing with continuous-variable clusters, Phys. Rev. A, 79 (2009), 062318. Google Scholar [19] S. Harker, K. Mischaikow, M. Mrozek and V. Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math., 14 (2014), 151–184. doi: 10.1007/s10208-013-9145-0.  Google Scholar [20] H. L. Huang, X. L. Wang, P. P. Rohde, Y. H. Luo, Y. W. Zhao, C. Liu, L. Li, N. L. Liu, C. Y. Lu and J. W. Pan, Demonstration of topological data analysis on a quantum processor, Optica, 5 (2018), 193. Google Scholar [21] D. Kozlov, Combinatorial Algebraic Topology, , Algorithms and Computation in Mathematics, 21. Springer, Berlin, 2008. doi: 10.1007/978-3-540-71962-5.  Google Scholar [22] H. K. Lau, R. Pooser, G. Siopsis and C. Weedbrook, Quantum machine learning over infinite dimensions, Phys. Rev. Lett., 118 (2017), 080501, 6pp. doi: 10.1103/PhysRevLett.118.080501.  Google Scholar [23] H. K. Lau, R. Pooser, G. Siopsis and C. Weedbrook, Quantum machine learning over infinite dimensions, Phys. Rev. Lett., 118 (2017), 080501, 6pp. doi: 10.1103/PhysRevLett.118.080501.  Google Scholar [24] N. Liu, J. Thompson, C. Weedbrook, S. Lloyd, V. Vedral, M. Gu and K. Modi, Power of one qumode for quantum computation, Phys. Rev. A, 93 (2016), 052304, 10pp. doi: 10.1103/physreva.93.052304.  Google Scholar [25] S. Lloyd, Hybrid quantum computing, preprint, arXiv: quant-ph/0008057. Google Scholar [26] S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett., 82 (1999), 1784–1787. doi: 10.1103/PhysRevLett.82.1784.  Google Scholar [27] S. Lloyd, S. Garnerone and P. Zanardi, Quantum algorithms for topological and geometric analysis of data, Nat. Commun., 7 (2016), 10138. Google Scholar [28] P. van Loock, C. Weedbrook and M. Gu, Building Gaussian cluster states by linear optics, Phys. Rev. A, 76 (2007), 032321. Google Scholar [29] K. Marshall, C. S. Jacobsen, C. Schafermeier, T. Gehring, C. Weedbrook and U. L. Andersen, Continuous-variable quantum computing on encrypted data, Nat. Comm., 7 (2016), 13795. Google Scholar [30] N. C. Menicucci, Fault-tolerant measurement-based quantum computing with continuous-variable cluster states, Phys. Rev. Lett., 112 (2014), 120504. Google Scholar [31] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph and M. A. Nielsen, Universal quantum computation with continuous-variable cluster states, Phys. Rev. Lett., 97 (2006), 110501. Google Scholar [32] K. Mischaikow and V. Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discret. Comput. Geom., 50 (2013), 330–353. doi: 10.1007/s00454-013-9529-6.  Google Scholar [33] M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information, , Cambridge University Press, 2000.  Google Scholar [34] P. Niyogi, S. Smale and S. Weinberger, A topological view of unsupervised learning from noisy data, SIAM J. Comput., 40 (2011), 646–663. doi: 10.1137/090762932.  Google Scholar [35] M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer and O. Pfister, Parallel generation of quadripartite cluster entanglement in the optical frequency comb, Phys. Rev. Lett., 107 (2011), 030505. Google Scholar [36] H. Reitberger, Leopold Vietoris (1891–2002),, Notices of the American Mathematical Society, 49 (2002), 1232-1236.   Google Scholar [37] V. Robins, Towards computing homology from finite approximations, Topol. Proc., 24 (1999), 503–532.  Google Scholar [38] S. Takeda, T. Mizuta, M. Fuwa, J.-i. Yoshikawa, H. Yonezawa and A. Furusawa, Generation and eight-port homodyne characterization of time-bin qubits for continuous-variable quantum information processing, Phys. Rev. A, 87 (2013), 043803. Google Scholar [39] C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys., 84 (2012), 621. Google Scholar [40] C. R. Wie, A Quantum Circuit to Construct All Maximal Cliques Using Grover Search Algorithm, arXiv: 1711.06146. Google Scholar [41] S. Yokoyama, R. Ukai, S. C. Armstrong, J.-i. Yoshikawa, P. van Loock and A. Furusawa, Demonstration of a fully tunable entangling gate for continuous-variable one-way quantum computation, Phys. Rev. A, 92 (2015), 032304. Google Scholar [42] S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. i. Yoshikawa, H. Yonezawa, N. C. Menicucci and A. Furusawa, Ultra-large-scale continuous-variable cluster states multiplexed in the time domain, Nature Photonics, 7 (2013), 982. Google Scholar [43] J. i. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino and A. Furusawa, Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing, Applied Phys. Lett. Photonics, 1 (2016), 060801. Google Scholar [44] J. Zhang and S. L. Braunstein, Continuous-variable Gaussian analog of cluster states, Phys. Rev. A, 73 (2006), 032318. Google Scholar [45] A. Zomorodian and G. Carlsson, Computing persistent homology, Discret. Comput. Geom., 33 (2005), 249–274. doi: 10.1007/s00454-004-1146-y.  Google Scholar [46] A. Zomorodian, Algorithms and Theory of Computation Handbook, 2nd edition, Ch. 3, section 2., Chapman and Hall/CRC, 2009. Google Scholar
The Betti numbers $\beta_{0,1,2}$ for four example shapes (point, cirlce, spherical shell, and torus). They are the number of connected components, one-dimensional holes (also called tunnels or handles), and two-dimensional voids, respectively
(a) Given data represented by points. (b) For a given distance $\varepsilon$, a circle is drawn around each point. (c) Between every two points with contacting circles a line is drawn. These connections are edges of $n$-dimensional shapes (simplices), and the space of simplices in (c) is called a simplicial complex. For two different values of $\varepsilon$, as in (b) i, ii, and (c) i, ii, one can get more or less connections between the data points resulting in different topologies. Therefore Betti numbers depend on the initial choice of $\varepsilon$. It is useful to vary $\varepsilon$ to find interesting structures
The $k$-simplices for $k = 0,1,2,3$. These are a vertex, an edge, a triangle, and a tetrahedron, respectively
The action of the boundary operator is shown on a $k = 2$ simplex. A visual representation of a simplex being broken down into its boundary is depicted above. Its boundary consists of simplices of $k-1 = 1$. Below is the encoded representation of the boundary operator acting on the 2-simplex. In this encoding a 1 represents a vertex in the corresponding position in the string of bits. The boundary sum is represented by a clockwise rotation around the original simplex, and the negative sign in the result alternates as in Eq. (5)
Consider the $k = 2$ complex on the left, for a given value of $\varepsilon$. In order to show that the striped area is a void, it itself must be boundary-less, and not a boundary for any part of the complex. Fulfillment of these two properties is equivalent to the combinatorial Laplacian (11) applied to the stripped area returning zero. Therefore this area would be part of the kernel of the combinatorial Laplacian for $k = 2$ contributing to the $\beta_{2}$ Betti number
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