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.

[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.

[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.

[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.

[5]

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe and S. Lloyd, Quantum machine learning, Nature, 549 (2017), 195.

[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.

[7]

G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, Int. J. Shape Model, 11 (2005), 149.

[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.

[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.

[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.

[11]

R. Dridi and H. Alghassi, Homology Computation of Large Point Clouds using Quantum Annealing, arXiv: 1512.09328.

[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.

[13]

P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal., 9 (1999), 596.

[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.

[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.

[16]

V. Giovannetti, S. Lloyd and L. Maccone, Architectures for a quantum random access memory, Phys. Rev. A, 78 (2008), 052310.

[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.

[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.

[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.

[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.

[21]

D. Kozlov, Combinatorial Algebraic Topology, , Algorithms and Computation in Mathematics, 21. Springer, Berlin, 2008. doi: 10.1007/978-3-540-71962-5.

[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.

[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.

[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.

[25]

S. Lloyd, Hybrid quantum computing, preprint, arXiv: quant-ph/0008057.

[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.

[27]

S. Lloyd, S. Garnerone and P. Zanardi, Quantum algorithms for topological and geometric analysis of data, Nat. Commun., 7 (2016), 10138.

[28]

P. van Loock, C. Weedbrook and M. Gu, Building Gaussian cluster states by linear optics, Phys. Rev. A, 76 (2007), 032321.

[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.

[30]

N. C. Menicucci, Fault-tolerant measurement-based quantum computing with continuous-variable cluster states, Phys. Rev. Lett., 112 (2014), 120504.

[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.

[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.

[33]

M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information, , Cambridge University Press, 2000.

[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.

[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.

[36]

H. Reitberger, Leopold Vietoris (1891–2002),, Notices of the American Mathematical Society, 49 (2002), 1232-1236. 

[37]

V. Robins, Towards computing homology from finite approximations, Topol. Proc., 24 (1999), 503–532.

[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.

[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.

[40]

C. R. Wie, A Quantum Circuit to Construct All Maximal Cliques Using Grover Search Algorithm, arXiv: 1711.06146.

[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.

[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.

[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.

[44]

J. Zhang and S. L. Braunstein, Continuous-variable Gaussian analog of cluster states, Phys. Rev. A, 73 (2006), 032318.

[45]

A. Zomorodian and G. Carlsson, Computing persistent homology, Discret. Comput. Geom., 33 (2005), 249–274. doi: 10.1007/s00454-004-1146-y.

[46]

A. Zomorodian, Algorithms and Theory of Computation Handbook, 2nd edition, Ch. 3, section 2., Chapman and Hall/CRC, 2009.

show all references

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.

[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.

[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.

[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.

[5]

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe and S. Lloyd, Quantum machine learning, Nature, 549 (2017), 195.

[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.

[7]

G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, Int. J. Shape Model, 11 (2005), 149.

[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.

[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.

[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.

[11]

R. Dridi and H. Alghassi, Homology Computation of Large Point Clouds using Quantum Annealing, arXiv: 1512.09328.

[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.

[13]

P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal., 9 (1999), 596.

[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.

[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.

[16]

V. Giovannetti, S. Lloyd and L. Maccone, Architectures for a quantum random access memory, Phys. Rev. A, 78 (2008), 052310.

[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.

[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.

[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.

[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.

[21]

D. Kozlov, Combinatorial Algebraic Topology, , Algorithms and Computation in Mathematics, 21. Springer, Berlin, 2008. doi: 10.1007/978-3-540-71962-5.

[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.

[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.

[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.

[25]

S. Lloyd, Hybrid quantum computing, preprint, arXiv: quant-ph/0008057.

[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.

[27]

S. Lloyd, S. Garnerone and P. Zanardi, Quantum algorithms for topological and geometric analysis of data, Nat. Commun., 7 (2016), 10138.

[28]

P. van Loock, C. Weedbrook and M. Gu, Building Gaussian cluster states by linear optics, Phys. Rev. A, 76 (2007), 032321.

[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.

[30]

N. C. Menicucci, Fault-tolerant measurement-based quantum computing with continuous-variable cluster states, Phys. Rev. Lett., 112 (2014), 120504.

[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.

[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.

[33]

M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information, , Cambridge University Press, 2000.

[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.

[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.

[36]

H. Reitberger, Leopold Vietoris (1891–2002),, Notices of the American Mathematical Society, 49 (2002), 1232-1236. 

[37]

V. Robins, Towards computing homology from finite approximations, Topol. Proc., 24 (1999), 503–532.

[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.

[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.

[40]

C. R. Wie, A Quantum Circuit to Construct All Maximal Cliques Using Grover Search Algorithm, arXiv: 1711.06146.

[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.

[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.

[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.

[44]

J. Zhang and S. L. Braunstein, Continuous-variable Gaussian analog of cluster states, Phys. Rev. A, 73 (2006), 032318.

[45]

A. Zomorodian and G. Carlsson, Computing persistent homology, Discret. Comput. Geom., 33 (2005), 249–274. doi: 10.1007/s00454-004-1146-y.

[46]

A. Zomorodian, Algorithms and Theory of Computation Handbook, 2nd edition, Ch. 3, section 2., Chapman and Hall/CRC, 2009.

Figure 1.  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
Figure 2.  (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
Figure 3.  The $ k $-simplices for $ k = 0,1,2,3 $. These are a vertex, an edge, a triangle, and a tetrahedron, respectively
Figure 4.  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)
Figure 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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