Abstract
We investigate anisotropic XXZ Heisenberg spin-1 / 2 chains with control fields acting on one of the end spins, with the aim of exploring local quantum control in arrays of interacting qubits. In this work, which uses a recent Lie-algebraic result on the local controllability of spin chains with “always-on” interactions, we determine piecewise-constant control pulses corresponding to optimal fidelities for quantum gates such as spin-flip (NOT), controlled-NOT (CNOT), and square-root-of-SWAP (). We find the minimal times for realizing different gates depending on the anisotropy parameter Δ of the model, showing that the shortest among these gate times are achieved for particular values of Δ larger than unity. To study the influence of possible imperfections in anticipated experimental realizations of qubit arrays, we analyze the robustness of the obtained results for the gate fidelities to random variations in the control-field amplitudes and finite rise time of the pulses. Finally, we discuss the implications of our study for superconducting charge-qubit arrays.
Similar content being viewed by others
References
S. Lloyd, A.J. Landahl, J.J.E. Slotine, Phys. Rev. A 69, 012305 (2004)
D. D’Alessandro, Introduction to Quantum Control and Dynamics (Taylor & Francis, Boca Raton, 2008)
V. Jurdjevic, H.J. Sussmann, J. Differ. Equ. 12, 313 (1972)
For a recent review, see C. Brif, R. Chakrabarti, H. Rabitz, New. J. Phys. 12, 075008 (2010).
See, e.g., S. Bose, Phys. Rev. Lett. 91, 207901 (2003)
A. Romito, R. Fazio, C. Bruder, Phys. Rev. B 71, 100501(R) (2005)
A.O. Lyakhov, C. Bruder, Phys. Rev. B 74, 235303 (2006)
D. Burgarth, Eur. Phys. J. Special Top. 151, 147 (2007)
T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, G.E. Santoro, Phys. Rev. Lett. 103, 240501 (2009)
K. Maruyama, T. Iitaka, F. Nori, Phys. Rev. A 75, 012325 (2007)
S.G. Schirmer, I.C.H. Pullen, P.J. Pemberton-Ross, Phys. Rev. A 78, 062339 (2008)
D. Burgarth, S. Bose, C. Bruder, V. Giovannetti, Phys. Rev. A 79, 060305(R) (2009)
A. Kay, P.J. Pemberton-Ross, Phys. Rev. A 81, 010301(R) (2010)
D. Burgarth, K. Maruyama, M. Murphey, S. Montangero, T. Calarco, F. Nori, M.B. Plenio, Phys. Rev. A 81, 040303(R) (2010)
X. Wang, A. Bayat, S.G. Schirmer, S. Bose, Phys. Rev. A 81, 032312 (2010)
R. Heule, C. Bruder, D. Burgarth, V.M. Stojanović, Phys. Rev. A 82, 052333 (2010)
Y. Makhlin, G. Schön, A. Shnirman, Rev. Mod. Phys. 73, 357 (2001)
L.S. Levitov, T.P. Orlando, J.B. Majer, J.E. Mooij, e-print arXiv:cond-mat/0108266v2 (2001)
J.Q. You, F. Nori, Phys. Today 58, 42 (2005)
T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, 2004)
S.G. Schirmer, H. Fu, A.I. Solomon, Phys. Rev. A 63, 063410 (2001)
W. Pfeifer, The Lie Algebras su(N): An Introduction (Birkhäuser, Basel, 2003)
T. Polack, H. Suchowski, D.J. Tannor, Phys. Rev. A 79, 053403 (2009)
U. Sander, T. Schulte-Herbrüggen, e-print arXiv: 0904.4654
G. Burkard, D. Loss, D.P. Di Vincenzo, J.A. Smolin, Phys. Rev. B 60, 11404 (1999)
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 77 and 90: The Art of Scientific and Parallel Computing (Cambridge University Press, Cambridge, 1997)
A. Carlini, A. Hosoya, T. Koike, Y. Okudaira, Phys. Rev. A 75, 042308 (2007)
C. Bruder, R. Fazio, G. Schön, Phys. Rev. B 47, 342 (1993)
R. Fazio, H. van der Zant, Phys. Rep. 355, 235 (2001)
Y. Makhlin, Quantum Inf. Process 1, 243 (2002)
S. Montangero, T. Calarco, R. Fazio, Phys. Rev. Lett. 99, 170501 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Heule, R., Bruder, C., Burgarth, D. et al. Controlling qubit arrays with anisotropic XXZ Heisenberg interaction by acting on a single qubit. Eur. Phys. J. D 63, 41–46 (2011). https://doi.org/10.1140/epjd/e2010-10623-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjd/e2010-10623-y