logo

SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 62 , Issue 12 : 120362(2019) https://doi.org/10.1007/s11433-019-9447-5

Topological dynamical decoupling

More info
  • ReceivedJun 14, 2019
  • AcceptedJul 1, 2019
  • PublishedSep 16, 2019
PACS numbers

Abstract


Acknowledgment

This work was supported by the National Basic Research Program of China (Grant Nos. 2017YFA0303700, and 2015CB921001), the National Natural Science Foundation of China (Grant Nos. 61726801, 11474168, and 11474181), and in part by the Beijing Advanced Innovative Center for Future Chip (ICFC). Jiang Zhang acknowledges support by the China Postdoctoral Science Foundation (Grant No. 2018M631437). Xiao-Dong Yu acknowledges support by the Deutsche Forschungs Gemeinschaft (DFG) and the European Research Council (ERC) (Consolidator Grant 683107/TempoQ).


Supplementary data

The supporting information is available online at phys.scichina.com and link.springer.com The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.


References

[1] DiVincenzo D. P.. Science, 1995, 270255-261 CrossRef ADS Google Scholar

[2] Ekert A., Jozsa R.. Rev. Mod. Phys., 1996, 68733-753 CrossRef ADS Google Scholar

[3] Shor P. W.. Phys. Rev. A, 1995, 52R2493-R2496 CrossRef ADS Google Scholar

[4] Steane A. M.. Phys. Rev. Lett., 1996, 77793-797 CrossRef PubMed ADS Google Scholar

[5] Laflamme R., Miquel C., Paz J. P., Zurek W. H.. Phys. Rev. Lett., 1996, 77198-201 CrossRef PubMed ADS Google Scholar

[6] Terhal B. M.. Rev. Mod. Phys., 2015, 87307-346 CrossRef ADS Google Scholar

[7] Yao X. C., Wang T. X., Chen H. Z., Gao W. B., Fowler A. G., Raussendorf R., Chen Z. B., Liu N. L., Lu C. Y., Deng Y. J., Chen Y. A., Pan J. W.. Nature, 2012, 482489-494 CrossRef PubMed ADS arXiv Google Scholar

[8] Barends R., Kelly J., Megrant A., Veitia A., Sank D., Jeffrey E., White T. C., Mutus J., Fowler A. G., Campbell B., Chen Y., Chen Z., Chiaro B., Dunsworth A., Neill C., O'Malley P., Roushan P., Vainsencher A., Wenner J., Korotkov A. N., Cleland A. N., Martinis J. M.. Nature, 2014, 508500-503 CrossRef PubMed ADS arXiv Google Scholar

[9] Nigg D., Muller M., Martinez E. A., Schindler P., Hennrich M., Monz T., Martin-Delgado M. A., Blatt R.. Science, 2014, 345302-305 CrossRef PubMed ADS arXiv Google Scholar

[10] Córcoles A. D., Magesan E., Srinivasan S. J., Cross A. W., Steffen M., Gambetta J. M., Chow J. M.. Nat Commun, 2015, 66979 CrossRef PubMed ADS arXiv Google Scholar

[11] Kelly J., Barends R., Fowler A. G., Megrant A., Jeffrey E., White T. C., Sank D., Mutus J. Y., Campbell B., Chen Y., Chen Z., Chiaro B., Dunsworth A., Hoi I. C., Neill C., O'Malley P. J. J., Quintana C., Roushan P., Vainsencher A., Wenner J., Cleland A. N., Martinis J. M.. Nature, 2015, 51966-69 CrossRef PubMed ADS arXiv Google Scholar

[12] Kitaev A. Y.. Russ. Math. Surv., 1997, 521191-1249 CrossRef ADS Google Scholar

[13] S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary,. arXiv Google Scholar

[14] Dennis E., Kitaev A., Landahl A., Preskill J.. J. Math. Phys., 2002, 434452-4505 CrossRef ADS Google Scholar

[15] Fowler A. G., Mariantoni M., Martinis J. M., Cleland A. N.. Phys. Rev. A, 2012, 86032324 CrossRef ADS arXiv Google Scholar

[16] Vijay S., Hsieh T. H., Fu L.. Phys. Rev. X, 2015, 5041038 CrossRef ADS arXiv Google Scholar

[17] Landau L. A., Plugge S., Sela E., Altland A., Albrecht S. M., Egger R.. Phys. Rev. Lett., 2016, 116050501 CrossRef PubMed ADS arXiv Google Scholar

[18] Brown B. J., Laubscher K., Kesselring M. S., Wootton J. R.. Phys. Rev. X, 2017, 7021029 CrossRef ADS arXiv Google Scholar

[19] Tuckett D. K., Bartlett S. D., Flammia S. T.. Phys. Rev. Lett., 2018, 120050505 CrossRef PubMed ADS arXiv Google Scholar

[20] Bombin H., Martin-Delgado M. A.. Phys. Rev. Lett., 2006, 97180501 CrossRef PubMed ADS Google Scholar

[21] Katzgraber H. G., Bombin H., Martin-Delgado M. A.. Phys. Rev. Lett., 2009, 103090501 CrossRef PubMed ADS arXiv Google Scholar

[22] Bombín H.. New J. Phys., 2015, 17083002 CrossRef ADS Google Scholar

[23] Litinski D., Kesselring M. S., Eisert J., von Oppen F.. Phys. Rev. X, 2017, 7031048 CrossRef ADS arXiv Google Scholar

[24] Li Y.. Phys. Rev. A, 2018, 98012336 CrossRef ADS arXiv Google Scholar

[25] Fowler A. G.. Phys. Rev. Lett., 2012, 109180502 CrossRef PubMed ADS arXiv Google Scholar

[26] Stephens A. M.. Phys. Rev. A, 2014, 89022321 CrossRef ADS arXiv Google Scholar

[27] Viola L., Lloyd S.. Phys. Rev. A, 1998, 582733-2744 CrossRef ADS Google Scholar

[28] Viola L., Knill E., Lloyd S.. Phys. Rev. Lett., 1999, 822417-2421 CrossRef ADS Google Scholar

[29] Viola L., Knill E., Lloyd S.. Phys. Rev. Lett., 2000, 853520-3523 CrossRef PubMed ADS Google Scholar

[30] Viola L., Knill E.. Phys. Rev. Lett., 2003, 90037901 CrossRef PubMed ADS Google Scholar

[31] Viola L., Knill E.. Phys. Rev. Lett., 2005, 94060502 CrossRef PubMed ADS Google Scholar

[32] Khodjasteh K., Lidar D. A.. Phys. Rev. Lett., 2005, 95180501 CrossRef PubMed ADS Google Scholar

[33] Khodjasteh K., Lidar D. A., Viola L.. Phys. Rev. Lett., 2010, 104090501 CrossRef PubMed ADS arXiv Google Scholar

[34] Uhrig G. S.. Phys. Rev. Lett., 2007, 98100504 CrossRef PubMed ADS Google Scholar

[35] Gordon G., Kurizki G., Lidar D. A.. Phys. Rev. Lett., 2008, 101010403 CrossRef PubMed ADS arXiv Google Scholar

[36] Yang W., Liu R. B.. Phys. Rev. Lett., 2008, 101180403 CrossRef PubMed ADS arXiv Google Scholar

[37] Uys H., Biercuk M., Bollinger J.. Phys. Rev. Lett., 2009, 103040501 CrossRef PubMed ADS Google Scholar

[38] West J. R., Fong B. H., Lidar D. A.. Phys. Rev. Lett., 2010, 104130501 CrossRef PubMed ADS arXiv Google Scholar

[39] Du J., Rong X., Zhao N., Wang Y., Yang J., Liu R. B.. Nature, 2009, 4611265-1268 CrossRef PubMed ADS Google Scholar

[40] Zhang J., Suter D.. Phys. Rev. Lett., 2015, 115110502 CrossRef PubMed ADS arXiv Google Scholar

[41] Wang F., Zu C., He L., Wang W. B., Zhang W. G., Duan L. M.. Phys. Rev. B, 2016, 94064304 CrossRef ADS arXiv Google Scholar

[42] Quantum Error Correction, edited by D. A. Lidar and T. A. Brun (Cambridge University Press, Cambridge, UK, 2013). Google Scholar

[43] Khodjasteh K., Viola L.. Phys. Rev. Lett., 2009, 102080501 CrossRef PubMed ADS arXiv Google Scholar

[44] Khodjasteh K., Viola L.. Phys. Rev. A, 2009, 80032314 CrossRef ADS arXiv Google Scholar

[45] West J. R., Lidar D. A., Fong B. H., Gyure M. F.. Phys. Rev. Lett., 2010, 105230503 CrossRef PubMed ADS arXiv Google Scholar

[46] Stockburger J. T., Mak C. H.. Phys. Rev. Lett., 1998, 802657-2660 CrossRef ADS Google Scholar

[47] Stockburger J. T., Grabert H.. Phys. Rev. Lett., 2002, 88170407 CrossRef PubMed ADS Google Scholar

[48] Koch W., Gro?mann F., Stockburger J. T., Ankerhold J.. Phys. Rev. Lett., 2008, 100230402 CrossRef PubMed ADS arXiv Google Scholar

[49] Yan Y. A., Shao J.. Front. Phys., 2016, 11110309 CrossRef ADS Google Scholar

  • Figure 1

    (Color online) (a) Illustration of elements of $\mathcal{B}^1$ in a periodic $6\times6$ square lattice. When a chain has only one square, e.g., $f_1$, its boundary is the four surrounding edges (labeled by blue solid lines). (b) When a chain contains $f_3$ and $f_4$, the common edge (labeled by blue dashed line) shared by them is not included in the boundary since $e_3^d\!+\!e_4^u=0$. (c) When a set of squares ($f_5$ to $f_{10}$) forms a row in the lattice, they constitute a ring on the surface.

  • Figure 2

    (Color online) A periodic square lattice (a) and its dual (b). (a) The four edges ($e_1^u$, $e_1^l$, $e_1^d$, and $e_1^r$) connecting head-to-tail form the boundary of $f_1$. The four edges ($e_c^u$, $e_c^l$, $e_c^d$, and $e_c^r$) sharing the same vertex constitute a cross. (b) The square $f_1$ in the original lattice is transformed into a cross in the dual. Meanwhile, a cross in the original lattice is changed into a square in the dual.

  • Figure 3

    (Color online) (a) A pair of nearest-neighbor qubits $a$ and $b$ (labeled as red circles), and related squares. Qubit $a$ is on the common edge of $f_1$ and $f_2$ while qubit $b$ is on the common edge of $f_1$ and $f_3$. (b) Four cases where the corresponding elements in $\mathcal{B}^z$ are not commutative with $H_{ab}$.

  • Figure 4

    (Color online) (a) Schematic diagram for the group element $t_1$ and $t_2$ in the original lattice. (b) Schematic diagram for the corresponding group element $t_1^d$ and $t_2^d$ in the dual lattice.

  • Figure 5

    (Color online) (a) Cayley graph and Eulerian cycles for $\mathcal{T}^z$. One possible Eulerian cycle is $I\rightarrow~t_1\rightarrow~t_{12}\rightarrow~t_2\rightarrow~I\rightarrow~t_2\rightarrow~t_{12} \rightarrow~t_1\rightarrow~I$. (b) A modified Eulerian cycle and its Cayley graph to implement logical Pauli-$x$ and Pauli-$z$ operators. Three additional $I$ operators along with a logical operator generated with the same Hamiltonian are added to the original Eulerian cycles.

  • Figure 6

    (Color online) (a) A finite planar square lattice to implement surface codes. Each logical qubit is characterised by a pair of pink squares. When a logical operation on $L_1$ is performing, the physical qubits on the small lattice colored blue could be “idle” qubits. (b) The “idle” lattice shown in (a). The red solid circles on the edges are qubits $a$ and $b$, respectively. (c) The dual lattice of the $3\times3$ lattice in (b). (d) The $1\times2$ lattice considered in the numerical simulations.

  • Figure 7

    (Color online) Fidelities between the initial state $\vert~\psi^L~\rangle$ and the corresponding $\rho(t)$ obtained from the reduced dynamics related to $D^z$ or $D^{xz}$ versus evolution time. For each procedure, we consider three cases: (i) with no control Hamiltonian (blue curves); (ii) with bounded-strength control Hamiltonians (yellow curves); (iii) with arbitrary-strong control Hamiltonians (green curves). (a) Fidelities for $D^z$; (b) fidelities for $D^{xz}$.

  • Figure 8

    (Color online) Fidelities between the initial state $\vert~~\psi^L~\rangle$ and the corresponding $\rho(t)$ obtained from the reduced dynamics related to $D^z$ with $\omega_{a,b}=0.03\omega_0$ versus evolution time. We consider three cases: (i) with no control Hamiltonian (blue curve); (ii) with bounded-strength control Hamiltonians (yellow curve); (iii) with arbitrarily strong control Hamiltonians (green curve).

qqqq

Contact and support