arXiv:0709.4484v1 [quant-ph] 27 Sep 2007
Time-optimal synthesis of unitary transformations
in coupled fast and slow qubit system
Robert Zeier,1, ∗Haidong Yuan,2, †and Navin Khaneja1, ‡
1Harvard School of Engineering and Applied Sciences,
33 Oxford Street, Cambridge, Massachusetts 02138, USA
2Department of Mechanical Engineering, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
(Dated: September 27, 2007)
In this paper, we study time-optimal control problems related to system of two coupled qubits
where the time scales involved in performing unitary transformations on each qubit are significantly
different. In particular, we address the case where unitary transformations produced by evolutions
of the coupling take much longer time as compared to the time required to produce unitary trans-
formations on the first qubit but much shorter time as compared to the time to produce unitary
transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the
subgroup SU(2)×SU(2)×U(1), which is natural in understanding the time-optimal control problem
of such a coupled qubit system with significantly different time scales. A typical setting involves
dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance ex-
periments at high fields. Using the proposed canonical decomposition, we give time-optimal control
algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and
quantum information processing.
PACS numbers: 03.67.Lx
The synthesis of unitary transformations using time-
efficient control algorithms is a well studied problem
in quantum information processing and coherent spec-
troscopy. Time-efficient control algorithms can reduce
decoherence effects in experimental realizations, and the
study of such control algorithms is related to the com-
plexity of quantum algorithms (see, e.g., [1, 2, 3]). Signif-
icant literature in this subject treat the case where uni-
tary transformations on single qubits take negligible time
compared to transformations interacting between differ-
ent qubits. This particular assumption is very realistic
for nuclear spins in nuclear magnetic resonance (NMR)
spectroscopy. Under this assumption, Ref.  (see also
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]) presents
time-optimal control algorithms to synthesize arbitrary
unitary transformations on a system of two qubits. Fur-
ther progress in the case of multiple qubits is reported in
[5, 10, 16, 18, 19, 20, 21, 22, 23, 24, 25].
In this work, we consider a coupled qubit system where
local unitary transformations on the first qubit take sig-
nificantly less time than local transformations on the sec-
ond one. In addition, we assume that the coupling evo-
lution is much slower than transformations on the first
qubit but much faster than transformations on the sec-
ond one. We present a canonical decomposition of SU(4)
in terms of the subgroup SU(2) × SU(2) × U(1) reflect-
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
‡Electronic address: firstname.lastname@example.org
ing the significantly different time scales immanent in the
system. Employing this canonical decomposition, we de-
rive time-optimal control algorithms to synthesize various
unitary transformations. Our methods are applicable to
coupled electron-nuclear spin systems occurring in pulsed
electron paramagnetic resonance (EPR) experiments at
high fields, where the Rabi frequency of the electron is
much larger than the hyperfine coupling which is fur-
ther much larger than the Rabi frequency of the nucleus.
In the context of quantum computing similar electron-
nuclear spin systems appear in the Refs. [26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].
The main results of this paper are as follows. Let Sµ
and Iνrepresent spin operators for the fast (electron spin)
and slow (nuclear spin) qubit, respectively. Any unitary
transformation G ∈ SU(4) on the coupled spin system
can be decomposed as
G = K1exp(t1SβIx+ t2SαIx)K2,(1)
where SαIx and SβIx correspond to x-rotations of the
slow qubit, conditioned, respectively, on the up or down
state of the fast qubit. The elements K1and K2are rota-
tions synthesized by rapid manipulations of the fast qubit
in conjunction with the evolution of the natural Hamil-
tonian. The elements K1and K2belong to the subgroup
SU(2) × SU(2) × U(1), and in appropriately chosen ba-
sis correspond to block-diagonal special unitary matrices
with 2×2-dimensional blocks of unitary matrices.
The minimum time to produce any unitary transfor-
mation G is the smallest value of (|t1|+|t2|)/ωI
is the maximum achievable Rabi frequency of the nucleus
and (t1,t2)Tis a pair satisfying Eq. (1). Synthesizing K1
and K2takes negligible time on the time scale governed
r, where ωI
 H. Yuan and N. Khaneja, in Proceedings of the 44th IEEE
Conference on Decision and Control, 2005 and 2005 Eu-
ropean Control Conference (Seville, Spain, 2005), pp.
 J. Swoboda (2006), arXiv:quant-ph/0601131v1.
 H. Yuan and N. Khaneja, Systems & Control Letters 55,
 H. Yuan, Ph.D. thesis, Harvard University (2006).
 G. Dirr, U. Helmke, K. H¨ uper, M. Kleinsteuber, and
Y. Liu, J. Global Optim. 35, 443 (2006).
 R. M. Zeier, Lie-theoretischer Zugang zur Erzeugung
unit¨ arer Transformationen auf Quantenrechnern (Uni-
versit¨ atsverlag Karlsruhe, Karlsruhe, 2006), Ph.D. thesis,
Universit¨ at Karlsruhe, 2006.
 N. Khaneja, S. J. Glaser, and R. Brockett, Phys. Rev. A
65, 032301 (2002).
 T. O. Reiss, N. Khaneja, and S. J. Glaser, J. Magn. Re-
son. 165, 95 (2003).
 N. Khaneja,T. O. Reiss,
Herbr¨ uggen, and S. J. Glaser, J. Magn. Reson. 172, 296
 T. Schulte-Herbr¨ uggen, A. Sp¨ orl, N. Khaneja, and S. J.
Glaser, Phys. Rev. A 72, 042331 (2005).
 H. Yuan and N. Khaneja, in Proceedings of the 45th IEEE
Conference on Decision and Control, 2006 (San Diego,
USA, 2006), pp. 3117–3120.
 N. Khaneja, B. Heitmann, A. Sp¨ orl, H. Yuan, T. Schulte-
Herbr¨ uggen, and S. J. Glaser, Phys. Rev. A 75, 012322
 H. Yuan, S. J. Glaser, and N. Khaneja, Phys. Rev. A 76,
 M. Mehring, J. Mende, and W. Scherer, Phys. Rev. Lett.
90, 153001 (2003).
 F. Jelezko,T. Gaebel,
J. Wrachtrup, Phys. Rev. Lett. 92, 076401 (2004).
 F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber,
and J. Wrachtrup, Phys. Rev. Lett. 93, 130501 (2004).
 M. Mehring, W. Scherer, and A. Weidinger, Phys. Rev.
Lett. 93, 206603 (2004).
 J. Mende, Ph.D. thesis, Universit¨ at Stuttgart (2005).
 A. Heidebrecht, J. Mende, and M. Mehring, Solid State
Nucl. Magn. Reson. 29, 90 (2006).
 R. Rahimi Darabad, Ph.D. thesis, Osaka University
 M. Mehring and J. Mende, Phys. Rev. A 73, 052303
 T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neu-
mann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D.
Greentree, S. Prawer, et al., Nat. Phys. 2, 408 (2006).
 N. B. Manson, J. P. Harrison, and M. J. Sellars, Phys.
Rev. B 74, 104303 (2006).
 A. Heidebrecht, J. Mende, and M. Mehring, Fortschr.
C. Kehlet, T. Schulte-
I. Popa,A. Gruber,and
Phys. 54, 788 (2006).
 F. Jelezko and J. Wrachtrup, Phys. Status Solidi A 203,
 J. Wrachtrup and F. Jelezko, J. Phys.: Condens. Matter
18, S807 (2006).
 L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S.
Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and
M. D. Lukin, Science 314, 281 (2006).
 A. Heidebrecht, Ph.D. thesis, Universit¨ at Stuttgart
 D. D. Awschalom and M. E. Flatt´ e, Nat. Phys. 3, 153
 M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan,
J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and
M. D. Lukin, Science 316, 1312 (2007).
 S. Helgason, Differential Geometry, Lie Groups, and
(American Mathematical Society,
Providence, 2001), reprinted with corrections.
 A. W. Knapp, Lie Groups Beyond an Introduction
(Birkh¨ auser, Boston, 2002), 2nd ed.
 R. R. Ernst, G. Bodenhausen, and A. Wokaun, Prin-
ciples of Nuclear Magnetic Resonance in One and Two
Dimensions (Clarendon Press, Oxford, 1997), reprinted
 A. Schweiger and G. Jeschke, Principles of pulse elec-
tron paramagnetic resonance (Oxford University Press,
 V. Jurdjevic and H. J. Sussmann, J. Diff. Eq. 12, 313
 S. S. Bullock, Quantum Inf. Comput. 4, 396 (2004).
 S. S. Bullock and G. K. Brennen, J. Math. Phys. 45, 2447
 S. S. Bullock, G. K. Brennen, and D. P. O’Leary, J. Math.
Phys. 46, 062104 (2005).
 H. N. S. Earp and J. K. Pachos, J. Math. Phys. 46,
 Y. Nakajima, Y. Kawano, and H. Sekigawa, Quantum
Inf. Comput. 6, 67 (2006).
 C. Van Loan, Numer. Math. 46, 479 (1985).
 G. H. Golub and C. F. Van Loan, Matrix computations
(The Johns Hopkins University Press, Baltimore, 1989),
 C. C. Paige and M. Wei, Linear Algebr. Appl. 208/209,
 M. D. Shuster, J. Astronaut. Sci. 41, 439 (1993).
 N. Khaneja, Phys. Rev. A 76, 032326 (2007).
 J. S. Hodges, J. C. Yang, C. Ramanathan, and D. G.
Cory (2007), arXiv:0707.2956v1.
 A. W. Marshall and I. Olkin, Inequalities:
majorization and its applications (Academic Press, New