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Quantum topological data analysis with continuous variables
Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200, USA |
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.
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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. Google Scholar |
[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. 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. |
[46] |
A. Zomorodian, Algorithms and Theory of Computation Handbook, 2nd edition, Ch. 3, section 2., Chapman and Hall/CRC, 2009. Google Scholar |
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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. Google Scholar |
[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. 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. |
[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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