Abstract

Quantum tomography is an essential method of the photonic technology toolbox and is routinely used for evaluation of experimentally prepared states of light and characterization of devices transforming such states. The tomography procedure consists of many different sequentially performed measurements. We present considerable tomography speedup by optimally arranging the individual constituent measurements, which is equivalent to solving an instance of the traveling salesman problem. As an example, we obtain solutions for photonic systems of up to five qubits, and conclude that already for systems of three or more qubits, the total duration of the tomography procedure can be halved. The reported speedup has been verified experimentally for quantum state tomography and also for full quantum process characterization up to six qubits, without resorting to any complexity reduction or simplification of the system of interest. Our approach is versatile and reduces the time of an input-output characterization of optical devices and various scattering processes as well.

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

Full Article  |  PDF Article
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2018 (7)

F. Flamini, N. Spagnolo, and F. Sciarrino, “Photonic quantum information processing: a review,” Rep. Prog. Phys. 82, 016001 (2018).
[Crossref] [PubMed]

D. Huber, M. Reindl, S. F. C. da Silva, C. Schimpf, J. Martín-Sánchez, H. Huang, G. Piredda, J. Edlinger, A. Rastelli, and R. Trotta, “Strain-tunable GaAs quantum dot: A nearly dephasing-free source of entangled photon pairs on demand,” Phys. Rev. Lett. 121, 033902 (2018).
[Crossref] [PubMed]

M. Prilmüller, T. Huber, M. Müller, P. Michler, G. Weihs, and A. Predojević, “Hyperentanglement of photons emitted by a quantum dot,” Phys. Rev. Lett. 121, 110503 (2018).
[Crossref] [PubMed]

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
[Crossref] [PubMed]

P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S. Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, and P. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).
[Crossref]

J. G. Titchener, M. Gräfe, R. Heilmann, A. S. Solntsev, A. Szameit, and A. A. Sukhorukov, “Scalable on-chip quantum state tomography,” npj Quantum Inf. 4, 19 (2018).
[Crossref]

R. Stárek, M. Miková, I. Straka, M. Dušek, M. Ježek, J. Fiurášek, and M. Mičuda, “Experimental realization of SWAP operation on hyper-encoded qubits,” Opt. Express 26, 8443–8452 (2018).
[Crossref] [PubMed]

2017 (3)

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simulation of entanglement growth and engineering after a spin chain quench,” Nat. Commun. 8, 1569 (2017).
[Crossref] [PubMed]

Y.-Y. Zhao, Z. Hou, G.-Y. Xiang, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Experimental demonstration of efficient quantum state tomography of matrix product states,” Opt. Express 25, 9010–9018 (2017).
[Crossref] [PubMed]

2016 (1)

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304(R) (2016).
[Crossref]

2015 (2)

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’Ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light Sci. Appl. 4, e354 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martin-Lopez, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref] [PubMed]

2013 (3)

M. Mičuda, M. Sedlák, I. Straka, M. Miková, M. Dušek, M. Ježek, and J. Fiurášek, “Efficient experimental estimation of fidelity of linear optical quantum Toffoli gate,” Phys. Rev. Lett. 111, 160407 (2013).
[Crossref]

N. K. Langford, “Errors in quantum tomography: diagnosing systematic versus statistical errors,” New J. Phys. 15, 035003 (2013).
[Crossref]

S. J. van Enk and R. Blume-Kohout, “When quantum tomography goes wrong: drift of quantum sources and other errors,” New J. Phys. 15, 025024 (2013).
[Crossref]

2012 (1)

C. R. Müller, B. Stoklasa, C. Peuntinger, C. Gabriel, J. Řeháček, Z. Hradil, A. B. Klimov, G. Leuchs, C. Marquardt, and L. L. Sánchez-Soto, “Quantum polarization tomography of bright squeezed light,” New J. Phys. 14, 085002 (2012).
[Crossref]

2011 (2)

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

S. T. Flammia and Y.-K. Liu, “Direct fidelity estimation from few Pauli measurements,” Phys. Rev. Lett. 106, 230501 (2011).
[Crossref] [PubMed]

2010 (2)

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

2009 (4)

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photon. 3, 346–350 (2009).
[Crossref]

B. J. Smith, D. Kundys, N. Thomas-Peter, P. G. R. Smith, and I. A. Walmsley, “Phase-controlled integrated photonic quantum circuits,” Opt. Express 17, 13516–13525 (2009).
[Crossref] [PubMed]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photon. 3, 687–695 (2009).
[Crossref]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

2007 (1)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

2006 (1)

A. Ling, K. P. Soh, A. Lamas-Linares, and C. Kurtsiefer, “Experimental polarization state tomography using optimal polarimeters,” Phys. Rev. A 74, 022309 (2006).
[Crossref]

2005 (1)

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At. Mol. Opt. Phy. 52, 105–159 (2005).
[Crossref]

2004 (1)

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

2003 (2)

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref] [PubMed]

M. Ježek, J. Fiurášek, and Z. Hradil, “Quantum inference of states and processes,” Phys. Rev. A 68, 012305 (2003).
[Crossref]

2001 (2)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

G. M. D’Ariano and P. Lo Presti, “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation,” Phys. Rev. Lett. 86, 4195–4198 (2001).
[Crossref]

1999 (1)

K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, “Maximum-likelihood estimation of the density matrix,” Phys. Rev. A 61, 010304 (1999).
[Crossref]

1997 (2)

Z. Hradil, “Quantum-state estimation,” Phys. Rev. A 55, R1561–R1564 (1997).
[Crossref]

I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt. 44, 2455–2467 (1997).
[Crossref]

1996 (1)

P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, and W. K. Wootters, “Classical information capacity of a quantum channel,” Phys. Rev. A 54, 1869–1876 (1996).
[Crossref] [PubMed]

1994 (1)

K. R. W. Jones, “Fundamental limits upon the measurement of state vectors,” Phys. Rev. A 50, 3682–3699 (1994).
[Crossref] [PubMed]

1991 (1)

G. Reinelt, “TSPLIB—a traveling salesman problem library,” INFORMS J Comput. 3, 376 (1991).
[Crossref]

1989 (1)

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[Crossref]

Almeida, M. P.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Altepeter, J. B.

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At. Mol. Opt. Phy. 52, 105–159 (2005).
[Crossref]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref] [PubMed]

Banaszek, K.

K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, “Maximum-likelihood estimation of the density matrix,” Phys. Rev. A 61, 010304 (1999).
[Crossref]

Banchi, L.

I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simulation of entanglement growth and engineering after a spin chain quench,” Nat. Commun. 8, 1569 (2017).
[Crossref] [PubMed]

Bannerman, R. H. S.

Bartlett, S. D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Becker, S.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

Bentivegna, M.

I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simulation of entanglement growth and engineering after a spin chain quench,” Nat. Commun. 8, 1569 (2017).
[Crossref] [PubMed]

Blatt, R.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

Blume-Kohout, R.

S. J. van Enk and R. Blume-Kohout, “When quantum tomography goes wrong: drift of quantum sources and other errors,” New J. Phys. 15, 025024 (2013).
[Crossref]

Bose, S.

I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simulation of entanglement growth and engineering after a spin chain quench,” Nat. Commun. 8, 1569 (2017).
[Crossref] [PubMed]

Branning, D.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref] [PubMed]

Broome, M. A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Burgwal, R.

Caprara, D.

I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, and F. Sciarrino, “Photonic simulation of entanglement growth and engineering after a spin chain quench,” Nat. Commun. 8, 1569 (2017).
[Crossref] [PubMed]

Carolan, J.

C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O’Brien, D. P. Tew, and A. Laing, “Simulating the vibrational quantum dynamics of molecules using photonics,” Nature 557, 660–667 (2018).
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Figures (5)

Fig. 1
Fig. 1 (a) Complete characterization of a quantum circuit based on probing (P) the input qubits by all possible combinations of quantum states selected from a quorum and performing full analysis (A) of the output qubits. The analysis of the output qubit consists of projections of the qubit state onto quorum states. (b) For photonic circuits the analysis (A) is often polarization-encoded and formed by a sequence of wave plates (half-wave, HWP; quarter-wave, QWP), a polarization beam splitter (PBS), and a single-photon detector (SPD). The wave plates are rotated to perform the projections to particular polarization states. (c) When both outputs of the PBS are detected by the SPDs, two orthogonal projections are measured at the same time, reducing the duration of the measurement at the expense of a higher number of SPDs employed. The preparation stage (P) is constructed in a similar way using a proper single-photon source, a polarizer, and a sequence of wave plates.
Fig. 2
Fig. 2 The reduction of the total angular duration of tomography achieved when TSP-optimized sequence of measurements is used instead of the conventional one, for the six-state scheme (blue square) and the three-base scheme (orange circle). The temporal duration reduction, shown on the right-hand scale, assumes rotation mounts with 10 deg/s speed. The conventional one-qubit, three-base tomography is already optimal, and the corresponding data point of zero reduction is not shown in the plot.
Fig. 3
Fig. 3 The speedup factor. The six-state scheme (solid blue square) achieves greater values of speedup than the three-base scheme (solid orange circle). Empty blue squares represent the values measured in the three-qubit state tomography and the two-qubit process characterization, respectively.
Fig. 4
Fig. 4 The scheme of on-chip tomography characterization of path-encoded qubit in quantum state ρ. The phase φ of qubit projection measurement is set by an heating element. The second heater affects the phase θ in the Mach-Zehnder interferometer acting as a variable ratio coupler.
Fig. 5
Fig. 5 The speedup factor (a) for temporal optimization of the path-encoded state tomography using heater elements practically saturates at the value of 1.8 for more than two qubits. The total heat reduction (b) for heat-optimized tomography, assuming that power of 0.5 W applied over 1 second is required for a 2π phase change.

Tables (5)

Tables Icon

Table 1 One-qubit polarization tomography (the six-state scheme). The H, V, D, A, R, and L projections are shown with their corresponding wave plate angles in degrees.

Tables Icon

Table 2 The TSP specification using the adjacency matrix consists of the maximal angles rotated by any wave plate during transitions between projections.

Tables Icon

Table 3 The optimal projection measurement sequence for one-qubit tomography.

Tables Icon

Table 4 The TSP-optimized projection measurement sequence for two-qubit tomography, to be read left-to-right, top-to-bottom.

Tables Icon

Table 5 One-qubit path-encoded tomography (the six-state scheme). The 0, 1, +, −, i, and −i projections are shown with the corresponding phase settings.

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