Abstract

Topological data analysis offers a robust way to extract useful information from noisy, unstructured data by identifying its underlying structure. Recently, an efficient quantum algorithm was proposed [Nat. Commun. 7, 10138 (2016) [CrossRef]  ] for calculating Betti numbers of data points—topological features that count the number of topological holes of various dimensions in a scatterplot. Here, we implement a proof-of-principle demonstration of this quantum algorithm by employing a six-photon quantum processor to successfully analyze the topological features of Betti numbers of a network including three data points, providing new insights into data analysis in the era of quantum computing.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

H.-L. Huang, Q. Zhao, X. Ma, C. Liu, Z.-E. Su, X.-L. Wang, L. Li, N.-L. Liu, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Experimental blind quantum computing for a classical client,” Phys. Rev. Lett. 119, 050503 (2017).
[Crossref]

Y.-M. He, J. Liu, S. Maier, M. Emmerling, S. Gerhardt, M. Davanço, K. Srinivasan, C. Schneider, and S. Höfling, “Deterministic implementation of a bright, on-demand single-photon source with near-unity indistinguishability via quantum dot imaging,” Optica 4, 802–808 (2017).
[Crossref]

H. Wang, Y. He, Y.-H. Li, Z.-E. Su, B. Li, H.-L. Huang, X. Ding, M.-C. Chen, C. Liu, J. Qin, J.-P. Li, Y.-M. He, C. Schneider, M. Kamp, C.-Z. Peng, S. Höfling, C.-Y. Lu, and J.-W. Pan, “High-efficiency multiphoton boson sampling,” Nat. Photonics 11, 361–365 (2017).
[Crossref]

2016 (4)

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

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

C. Giusti, R. Ghrist, and D. S. Bassett, “Two’s company, three (or more) is a simplex,” J. Comput. Neurosci. 41, 1–14 (2016).
[Crossref]

L.-D. Lord, P. Expert, H. M. Fernandes, G. Petri, T. J. Van Hartevelt, F. Vaccarino, G. Deco, F. Turkheimer, and M. L. Kringelbach, “Insights into brain architectures from the homological scaffolds of functional connectivity networks,” Front. Syst. Neurosci. 10, 85 (2016).

2015 (5)

C. Giusti, E. Pastalkova, C. Curto, and V. Itskov, “Clique topology reveals intrinsic geometric structure in neural correlations,” Proc. Natl. Acad. Sci. USA 112, 13455–13460 (2015).
[Crossref]

J. A. Perea and J. Harer, “Sliding windows and persistence: an application of topological methods to signal analysis,” Found. Comput. Math. 15, 799–838 (2015).
[Crossref]

X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantum computer,” Phys. Rev. Lett. 114, 110504 (2015).
[Crossref]

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

F. Kaneda, B. G. Christensen, J. J. Wong, H. S. Park, K. T. McCusker, and P. G. Kwiat, “Time-multiplexed heralded single-photon source,” Optica 2, 1010–1013 (2015).
[Crossref]

2014 (4)

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8, 801–807 (2014).
[Crossref]

P. Rebentrost, M. Mohseni, and S. Lloyd, “Quantum support vector machine for big data classification,” Phys. Rev. Lett. 113, 130503 (2014).
[Crossref]

S. Lloyd, M. Mohseni, and P. Rebentrost, “Quantum principal component analysis,” Nat. Phys. 10, 631–633 (2014).
[Crossref]

G. Petri, P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino, “Homological scaffolds of brain functional networks,” J. R. Soc. Interface 11, 20140873 (2014).
[Crossref]

2013 (2)

G. Petri, M. Scolamiero, I. Donato, and F. Vaccarino, “Topological strata of weighted complex networks,” PLoS One 8, e66506 (2013).
[Crossref]

X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, M. Gu, M.-J. Zhu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

2012 (1)

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

2010 (1)

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

2009 (4)

A. W. Harrow, A. Hassidim, and S. Lloyd, “Quantum algorithm for linear systems of equations,” Phys. Rev. Lett. 103, 150502 (2009).
[Crossref]

O. Gühne and G. Tóth, “Entanglement detection,” Phys. Rep. 474, 1–75 (2009).
[Crossref]

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

G. Carlsson, “Topology and data,” Bull. Am. Math. Soc. 46, 255–308 (2009).
[Crossref]

2008 (4)

G. Carlsson, T. Ishkhanov, V. De Silva, and A. Zomorodian, “On the local behavior of spaces of natural images,” Int. J. Comput. Vis. 76, 1–12 (2008).
[Crossref]

S. Basu, “Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time,” Found. Comput. Math. 8, 45–80 (2008).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum random access memory,” Phys. Rev. Lett. 100, 160501 (2008).
[Crossref]

R. Ghrist, “Barcodes: the persistent topology of data,” Bull. Am. Math. Soc. 45, 61–75 (2008).
[Crossref]

2007 (6)

C.-Y. Lu, D. E. Browne, T. Yang, and J.-W. Pan, “Demonstration of a compiled version of Shor’s quantum factoring algorithm using photonic qubits,” Phys. Rev. Lett. 99, 250504 (2007).
[Crossref]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

P. Scheiblechner, “On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety,” J. Complexity 23, 359–379 (2007).
[Crossref]

D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams,” Discrete Comput. Geom. 37, 103–120 (2007).
[Crossref]

V. De Silva and R. Ghrist, “Homological sensor networks,” Not. Am. Math. Soc. 54, 10–17 (2007).

V. De Silva and R. Ghrist, “Coverage in sensor networks via persistent homology,” Algebr. Geom. Topol. 7, 339–358 (2007).
[Crossref]

2005 (1)

A. Zomorodian and G. Carlsson, “Computing persistent homology,” Discrete Comput. Geom. 33, 249–274 (2005).
[Crossref]

2003 (1)

S. Basu, “Different bounds on the different Betti numbers of semi-algebraic sets,” Discrete Comput. Geom. 30, 65–85 (2003).
[Crossref]

2002 (1)

H. Edelsbrunner, D. Letscher, and A. Zomorodian, “Topological persistence and simplification,” Discrete Comput. Geom. 28, 511–533 (2002).
[Crossref]

1999 (1)

S. Basu, “On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets,” Discrete Comput. Geom. 22, 1–18 (1999).
[Crossref]

1998 (1)

J. Friedman, “Computing Betti numbers via combinatorial Laplacians,” Algorithmica 21, 331–346 (1998).
[Crossref]

1997 (2)

P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. 26, 1484–1509 (1997).
[Crossref]

L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325–328 (1997).
[Crossref]

1996 (1)

S. Lloyd, “Universal quantum simulators,” Science 273, 1073–1078 (1996).
[Crossref]

1982 (1)

R. P. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
[Crossref]

Almeida, M. P.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Aspuru-Guzik, A.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Barbieri, M.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Bassett, D. S.

C. Giusti, R. Ghrist, and D. S. Bassett, “Two’s company, three (or more) is a simplex,” J. Comput. Neurosci. 41, 1–14 (2016).
[Crossref]

Basu, S.

S. Basu, “Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time,” Found. Comput. Math. 8, 45–80 (2008).
[Crossref]

S. Basu, “Different bounds on the different Betti numbers of semi-algebraic sets,” Discrete Comput. Geom. 30, 65–85 (2003).
[Crossref]

S. Basu, “On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets,” Discrete Comput. Geom. 22, 1–18 (1999).
[Crossref]

S. Basu, “Algorithms in real algebraic geometry: a survey,” arXiv:1409.1534 (2014).

Biamonte, J. D.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2, 106–111 (2010).
[Crossref]

Browne, D. E.

C.-Y. Lu, D. E. Browne, T. Yang, and J.-W. Pan, “Demonstration of a compiled version of Shor’s quantum factoring algorithm using photonic qubits,” Phys. Rev. Lett. 99, 250504 (2007).
[Crossref]

Cai, X.-D.

X.-D. Cai, D. Wu, Z.-E. Su, M.-C. Chen, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Entanglement-based machine learning on a quantum computer,” Phys. Rev. Lett. 114, 110504 (2015).
[Crossref]

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, M. Gu, M.-J. Zhu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Experimental quantum computing to solve systems of linear equations,” Phys. Rev. Lett. 110, 230501 (2013).
[Crossref]

Carhart-Harris, R.

G. Petri, P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino, “Homological scaffolds of brain functional networks,” J. R. Soc. Interface 11, 20140873 (2014).
[Crossref]

Carlsson, G.

G. Carlsson, “Topology and data,” Bull. Am. Math. Soc. 46, 255–308 (2009).
[Crossref]

G. Carlsson, T. Ishkhanov, V. De Silva, and A. Zomorodian, “On the local behavior of spaces of natural images,” Int. J. Comput. Vis. 76, 1–12 (2008).
[Crossref]

A. Zomorodian and G. Carlsson, “Computing persistent homology,” Discrete Comput. Geom. 33, 249–274 (2005).
[Crossref]

Carlsson, G. E.

V. De Silva and G. E. Carlsson, “Topological estimation using witness complexes,” in Proceedings of the First Eurographics Conference on Point-Based Graphics (SPBG) (2004), Vol. 4, pp. 157–166.

Chen, C.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Chen, L.-K.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
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C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a)  k -simplices (shown for k = 0,1 , 2,3 ) are fully connected sets of k + 1 data points. (b) Scatterplot of data points. (c) Using some arbitrary metric for quantifying distance ε between data points, data points within ε of one another receive an edge between them. (d) The simplicial complex is formed as the set of simplices. The colored regions indicate the different simplices within the complex. (e) Construction of the barcode. The horizontal axis represents the distance ε . The bars are constructed such that the number of bars that intersect the vertical line through any ε in the area of H k equals the Betti number β k .
Fig. 2.
Fig. 2. Quantum circuit for quantum TDA. (a) Outline of the original quantum circuit. (b) A scatterplot including three data points. (c) Graph representation of the 1-simplices state | ϕ 1 ε 1 = | 110 for 3 < ε 1 < 4 . The first and second data points are connected by an edge. (d) Graph representation of 1-simplices state | ϕ 1 ε 2 = ( | 110 + | 101 ) / 2 for 4 < ε 2 < 5 . The first data point is connected to the second and third points by two edges. (e) Optimized circuit with 5 qubits. The blocks with different colors represent the four basic stages.
Fig. 3.
Fig. 3. Experimental setup. Ultraviolet laser pulses with a central wavelength of 394 nm, pulse duration of 150 fs, and repetition rate of 80 MHz pass through three HWP-sandwiched β -barium borate (BBO) crystals [42] to produce three entangled photon pairs ( | H | V + | V | H ) / 2 (see Supplement 1 for details) in spatial modes 1-2, 3-4, and 5-6. Photons 2(3) and 4(5) are temporally and spatially superposed on a PBS. All photons are spectrally filtered with 3 nm bandwidth filters. C-BBO, sandwich-like BBO + HWP + BBO combination; QWP, quarter-wave plate; POL, polarizer; SC-YVO4, YVO 4 crystal for spatial compensation; TC-YVO4, YVO 4 crystal for temporal compensation.
Fig. 4.
Fig. 4. Final experimental results. The output is determined by measuring the eigenvalue register in the Pauli- Z basis. Measured expectation values (blue bars) and theoretically predicted values (gray bars) are shown for two different 1-simplices state inputs: (a)  | ϕ 1 ε 1 = | 110 , and (b)  | ϕ 1 ε 2 = ( | 110 + | 101 ) / 2 . Error bars represent one standard deviation, deduced from propagated Poissonian counting statistics of the raw detection events. (c) The barcode for 0 < ε < 5 . Since no k -dimensional holes for k 1 exist at these scales, only the 0th Betti barcode is given here. For 0 < ε < 3 , there is no connection between each point, so the 0th Betti number is equal to the number of points. That is, there are three bars at 0 < ε < 3 . At scales of 3 < ε 1 < 4 and 4 < ε 2 < 5 , the 0th Betti numbers are 2 and 1.

Equations (7)

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| ψ k ε = 1 | S k ε | s k S k ε | s k .
ρ k ε = 1 | S k ε | s k S k ε | s k s k | ,
k ε | s k = l ( 1 ) l | s k 1 ( l ) ,
β k ε = dim ( Ker k ε / Im k + 1 ε ) .
B k ε = ( 0 k ε k ε 0 ) .
β k ε = dim ( Ker k ε ) dim ( Im k + 1 ε ) = dim ( Ker k ε ) + dim ( Ker k + 1 ε ) | S k + 1 ε | .
ρ ε 1 = | V 3 | V 2 | H 1 V | 3 V | 2 H | 1 , ρ ε 2 = ( | V 3 | V 2 | H 1 V | 3 V | 2 H | 1 + | V 3 | H 2 | V 1 V | 3 H | 2 V | 1 ) / 2

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