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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

I.INTRODUCTION

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. [4] (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: zeier@eecs.harvard.edu

†Electronic address: haidong@mit.edu

‡Electronic address: navin@hrl.harvard.edu

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

by ωI

r.

r, where ωI

r

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The paper is organized as follows. In Sec. II, we recall

the physical details of our model system exemplified by a

coupled electron-nuclear spin system. The Lie-algebraic

structure of our model is described in Sec. III, which

is used to derive control algorithms (pulse sequences)

for synthesizing arbitrary unitary transformations in our

coupled spin system. In Sec. IV, we present examples.

We prove the time-optimality of our control algorithms

in Sec. V, and some details of the proof are given in Ap-

pendix A.

Our work draws some results from the theory of Lie

groups, which are explained as needed. We refer to [43,

44] for general reference. To make the paper broadly

accessible, we work with explicit matrix representations

of Lie groups and Lie algebras.

II.PHYSICAL MODEL

As our model system, we consider two coupled qubits.

We introduce the operators Sµand Iνwhich correspond

to operators on the first and second qubit, respectively.

In particular, these operators are defined by Sµ= (σµ⊗

id2)/2 and Iν= (id2⊗σν)/2 (see [45]), where σx:= (0 1

σy :=

?0 −i

and id2:= (1 0

In the remaining text, let µ,ν ∈ {x,y,z} and γ ∈ {x,y}.

In an experimental setting using an electron-nuclear

spin system, the first qubit is represented by the electron

spin (of spin 1/2). Similarly, the second qubit is rep-

resented by the nuclear spin (of spin 1/2). We assume

that in the presence of a static magnetic field pointing in

the z-direction, the free evolution is governed in the lab

frame by a Hamiltonian of the form

1 0),

i 0

?, and σz :=

?1 0

0 −1

?

are the Pauli matrices

0 1) is the 2×2-dimensional identity matrix.

Hlab

0

= ωSSz+ ωIIz+ J(2SzIz),(2)

where ωS and ωI represents the natural precession fre-

quency of, respectively, the first qubit and second qubit

and J is the coupling strength. We assume that

ωS≫ ωI≫ J.(3)

This assumption is motivated by coupled electron-nuclear

spin system occurring in EPR experiments at high fields

(see, e.g., Sect. 3.5 of [46]). The time scales in Eq. (3)

insure that the hyperfine coupling Hamiltonian between

the spins averages to the Ising Hamiltonian 2SzIz, as in

Eq. (2). This is the so-called high field limit.

The first and second qubit are controlled by transverse

oscillating fields, which result in the corresponding con-

trol Hamiltonian given by Hlab

S

+ Hlab

I , where

Hlab

S

= 2ωS

r(t)cos[ωS

ct + φS(t)]Sx

is the control Hamiltonian of the first qubit and

Hlab

I

= 2ωI

r(t)cos[ωI

ct + φI(t)]Ix

(4)

is the control Hamiltonian of the second qubit.

amplitude, frequency, and phase of the control func-

tion w.r.t. the first qubit are represented by ωS

and φS = φS(t) respectively. Similarly, ωI

φI = φI(t) represents the amplitude, frequency, and

phase of the control function w.r.t. the second qubit. We

use ωI

rto denote the maximal possible values of

ωI

r(t). In our model system, we assume that

The

r(t), ωS

c, and

c,

r(t), ωI

rand ωS

r(t) and ωS

ωI

r≪ J ≪ ωS

r.(5)

Therefore, we refer to the first qubit as the fast qubit and

the second qubit as the slow qubit.

We choose ωS

rotating frame, rotating with the first and second qubit

at frequency ωS

c, the transformations Ulab(t) and

Urot(t) describe, respectively, a unitary transformation in

the lab frame and the double rotating frame. We have

c= ωS and ωI

c= ωI− J. In a double

cand ωI

Ulab(t) = exp(−itωS

cSz)exp(−itωI

cIz)Urot(t).

Using the rotating wave approximation, the Hamiltoni-

ans Hlab

I

transform, respectively, to

0 , Hlab

S, and Hlab

H0=JIz+ J(2SzIz),

HS=ωS

(6)

r(t)[SxcosφS(t) + SysinφS(t)],

and

HI=ωI

r(t)[IxcosφI(t) + IysinφI(t)].

In absence of any irradiation on qubits, the system

evolves under the free Hamiltonian −iH0. From the time

scales in Eq. (5), we can synthesize any unitary transfor-

mation of the form exp(−itSµ) in arbitrarily small time

as compared to the evolution under H0or H0+ HI.

Let us define the operators,

Sβ= (id4/2 + Sz) =

?id2 02

02 02

?

and

Sα= (id4/2 − Sz) =

?02 02

02 id2

?

,

where iddis the d×d-dimensional identity matrix and 02

is the 2×2-dimensional zero matrix. Note that H0 =

2JSβIz, and the system is described by the Hamiltonian

H0+HI= 2JSβIz+wI

r(t)(Sα+Sβ)(IxcosφI+IysinφI).

Since J ≫ wI

SβIz, the above Hamiltonian gets in the first order ap-

proximation truncated to

r(t), and SβIγ, does not commute with

Hα(φI) = 2JSβIz+ wI

r(t)Sα(IxcosφI+ IysinφI). (7)

Similarly, we can prepare an Hamiltonian

Hβ(φI) = 2JSαIz+ wI

r(t)Sβ(IxcosφI+ IysinφI) (8)

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αβ

ββ

αα

βα

ωS− J

ωS+ J

ωI− J

ωI+ J

FIG. 1: The eigenstates of the Hamiltonian Hlab

where the transitions αα ↔ αβ and βα ↔ ββ correspond re-

spectively to the orientation along and opposite to the static

magnetic field. The first and second index refer to the orien-

tation of the electron and nuclear spin, respectively. Refer to

the text for details.

0

are shown,

by using Hβ(φI) = exp(iπSx)Hα(φI)exp(−iπSx).

The Hamiltonians Hα(φI) and Hβ(φI), operate on the

slow qubit and induce transitions αα ↔ αβ and βα ↔ ββ

of the nuclear spin as shown in Fig. 1 (cp. Table 6.1.1 of

[46]). The α and β states of the spins denote their ori-

entation along and opposite to the static magnetic field,

respectively. For the electron spin, the β state has lower

energy than the α state as its gyromagnetic ratio is neg-

ative. Similarly, for the nuclear spin, the α state has

lower energy than the β state as its gyromagnetic ratio

is positive (as for a proton). We remark that the energy

eigenstates βα, ββ, αα, and αβ correspond, respectively,

to the basis states 00, 01, 10, and 11. In Fig. 1, the first

and second index in eigenstates refers to the orientation

of the electron and nuclear spin, respectively. In absence

of any irradiation on the two qubits, the system evolves

under the Hamiltonian −iH0. In this section, we have

shown how to synthesize generators of the form −iSµ,

−iHα(φI), and −iH0.

III.LIE-ALGEBRAIC STRUCTURE OF THE

MODEL SYSTEM

All transformations of our model system are contained

in the Lie group G = SU(4), which is the set of 4×4-

dimensional unitary transformations of determinant one.

The operators −iIµ, −iSν, and −i2IµSν, are infinitesi-

mal generators of the Lie group G, and they generate the

15-dimensional Lie algebra g = su(4) given by the (real)

vector space of 4×4-dimensional (traceless) skew Hermi-

tian matrices. We have shown how to synthesize gen-

erators of the form −iSµ, −iHα(φI), and −iH0. These

generators are sufficient to produce any unitary transfor-

mation on the coupled qubit system, as described below.

Lemma 1. The Lie algebra generated by the elements

−iSµ, −iHα(φI), and −iH0, is equal to g = su(4).

Therefore, a standard result on the controllability of

(Thm. 7.1 of Ref. [47]) implies that the system is com-

pletely controllable and any unitary transformation in

G = SU(4), can be synthesized by alternate evolution

under the above Hamiltonians.

Lemma 2. The Lie algebrak, generated by the elements

−iSµand −iH0consists of the elements −iSµ, −i2SνIz,

and −iIz.

The Lie algebra k represents a class of generators that

take significantly less time to be synthesized, as they only

involve controlled rotations of the fast qubit and evolu-

tion of the free Hamiltonian −iH0 (no controlled rota-

tions of the slow qubit are involved). We can decompose

g = k ⊕ p, (9)

where the subspace p (of g) consists of the elements −iIγ

and −i2SµIγ. The decomposition of Eq. (9) is a Cartan

decomposition (see, e.g., [43], p. 213) as

[k,k] ⊂ k, [k,p] ⊂ p, and [p,p] ⊂ k,(10)

where [g1,g2] = g1g2− g2g1is the commutator (gi∈ g).

Let K = exp(k) denote the subgroup of G = SU(4)

which is infinitesimally generated by k.

of K can be synthesized only by the free evolution and

employing controlled transformations on the fast qubit.

Therefore, synthesizing transformations of K takes signif-

icantly less time as compared to general unitary trans-

formations not contained in K. In particular, controlled

transformations on the slow qubit are necessary to syn-

thesize general unitary transformations. The Lie group

K = exp(k) is equal to S[U(2)×U(2)], which is sometimes

referred as SU(2) × SU(2) × U(1).

Consider a maximal Abelian subalgebra a contained

in p. In our case, a is spanned by the operators −iSβIx

and −iSαIx. Any element a ∈ a can be represented as

a1(−iSβIx)+a2(−iSαIx), where a1,a2∈ R. As a matrix,

a takes the form

The elements

−i

2

0 a1

a1

0

0

0

0

0 a2

0

00

0

0 a2

0

.

We obtain the Lie group A = exp(a) corresponding to

the Abelian algebra a. From a Cartan decomposition

of a real semisimple Lie-algebra as satisfying Eqs. (9)-

(10), we obtain a decomposition of the compact Lie group

G = KAK (see, e.g., [43], Chap. V, Thm. 6.7):

Lemma 3. Any element G ∈ SU(4), can be written as

G = K1exp[t1(−iSβIx) + t2(−iSαIx)]K2,(11)

where t1,t2∈ R and K1,K2∈ K.

Remark 1. The computation of KAK decompositions

was analyzed in Refs [48, 49, 50, 51, 52]. In this work, we

consider the Cartan decomposition, which corresponds

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to the type AIII in the classification of possible Cartan

decompositions (see, e.g., pp. 451–452 of Ref. [43]).

Transforming all elements G ∈ G to SWAP·G·SWAP,

where

SWAP = exp(−iπS · I) =

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

,

S = (Sx,Sy,Sz)T, and I = (Ix,Iy,Iz)T, the KAK de-

composition is given in explicit matrices by

?U1 02

02 U2

?

exp

−i

2

0

0

a1

0 a2

0

c2

0

−is2

0 a1

0

0

0

0 a2

0

0

0

0

0

?U3 02

02 U4

?

=

?U1 02

02 U2

?

c1

0

−is1

0

c1

0

−is2

0

c2

−is1

0

?U3 02

02 U4

?

,

where sj= sin(aj/2) and cj= cos(aj/2). In particular,

the Lie group K is given in this basis by block-diagonal

unitary transformations, where 02is the 2×2-dimensional

zero matrix and U1,U2(and U3,U4) are 2×2-dimensional

unitary matrices such that the product of their deter-

minants is one. The considered KAK decomposition is

equivalent to the cosine-sine decomposition [53, 54, 55].

Remark 2. In Ref. [4], a different Cartan decompo-

sition is considered. In that case, the subalgebra k is

given by the elements −iSµ and −iIν and corresponds

to unitary transformations on single qubits of a coupled

two-qubit system. Synthesizing unitary transformations

on single qubits is assumed in Ref. [4] to take significantly

less time, as compared to unitary transformations which

interact between different qubits.

Since elements of K can be synthesized in negligible

time, we obtain as the main result of this paper that the

minimum time to synthesize any element G ∈ SU(4) is

the minimum value of (|t1|+|t2|)/ωI

a pair satisfying Eq. (11). We defer the proof of this fact

to Sec. V. Let us describe how to use the KAK decom-

position of G, to synthesize an arbitrary transformation

using only the generators −iSµ, −iHα(φI), and −iH0.

The Lie algebra k decomposes to k1⊕ p1, where k1 is

a subalgebra, composed of operators −iSµand −i2SνIz,

and p1is generated by −iIzwhich commutes with all ele-

ments of k1. The Lie algebra k1can be further subdivided

by a Cartan decomposition k1= k2⊕p2. The subalgebra

k2is generated by the operators −iSµ, and the subspace

p2consists of the operators −i2SµIz. Therefore, similar

as in Lemma 3, we obtain a decomposition of K:

Lemma 4. Each element Kj ∈ K can be decomposed

as Kj= exp(−iτ2j−1Iz)L2j−1exp(−iτ2j2SzIz)L2j=

rsuch that (t1,t2)Tis

exp[−i(τ2j−1− τ2j)Iz]L2j−1exp(−iτ2jH0/J)L2j, (12)

where τj∈ R and Lj∈ K2= exp(k2).

R4

τ4

J

R3

Hα(τ)

t2

ωIr

t4

J

(π)x

Hα(τ+t3)

t1

ωIr

˜R2

τ2

J

R1

w

2J

(π)−x

w

2J

˜R0

I

S

FIG. 2: The figure shows a canonical pulse sequence for syn-

thesizing unitary transformations in the coupled qubit system.

Let˜R2 = R2exp(iπSx) and˜R0 = exp(−iv0Sz)exp(−iπSx).

Since 1/J ≪ 1/wI

larger as depicted. Refer to the text for details.

r, the length of the time intervals tj/wI

ris

Using an Euler angle decomposition (see, e.g., pp. 454–

455 of Ref. [56]), the elements Lj∈ K2are given as

Lj=exp(−iθj,1Sz)exp(−iθj,2Sx)exp(−iθj,3Sz)

=exp[−i(θj,1+ θj,3)Sz]exp[−iθj,2R(θj,3)], (13)

where R(θj,3) = Sxcosθj,3− Sysinθj,3.

Similarly, any element A of the subgroup A can be

written as A = exp[t1(−iSβIx) + t2(−iSαIx)] =

exp

?

eit3Ize−it1Hβ(t3)/wI

−it1

wI

r

Hβ(0)

?

eit3Ize−it4H0/Jexp

?

−it2

wI

r

Hα(0)

?

=

re−it4H0/Je−it2Hα(0)/wI

r,(14)

for t3 = 2Jt1/wI

mod 2π ≥ 0. This follows by substituting for expressions

of H0, Hα(φI), and Hβ(φI) (see Eqs. (6)-(8)). Com-

bining Eqs. (12)-(14), a complete decomposition of an

element G ∈ SU(4), can be written as K1AK2=

r mod 4π and t4 = J(t1 − t2)/wI

r

e−iv0Sze−iwIzR1e−iτ2H0/JR2exp

?

−it1

wI

r

Hβ(t3+ τ)

?

×e−it4H0/Jexp

?

−it2

wI

r

Hα(τ)

?

R3e−iτ4H0/JR4,

where all the transformations Rj operate on the fast

qubit. In particular, we have R4 = exp[−iθ4,2R(θ4,3)],

R3 = exp[−iθ3,2R(v3)], R2 = exp[−iθ2,2R(v2)], R1 =

exp[−iθ1,2R(v1)], v3= θ3,3+θ4,1+θ4,3, v2= θ2,3+θ3,1+

v3, v1= θ1,3+ θ2,1+ v2, v0= θ1,1+ v1, τ = τ4− τ3, and

w = τ1− τ2+ τ3− τ4− t3. The time to produce G is

essentially (t1+ t2)/wI

r. Note that exp(−iwIz) =

e−iπSxexp[−iwH0/(2J)]eiπSxexp[−iwH0/(2J)].

Transformations on the fast qubit such as exp(−iv0Sz)

are significantly faster.Figure 2 shows the canonical

pulse sequence realizing any unitary transformation as a

sequence of rotations under −iH0, −iHβ(φI), and −iSµ.

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IV.EXAMPLES

We introduce the unitary transformations CNOT[1,2],

CNOT[2,1], and SWAP which are given as follows

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

,

1 0 0 0

0 0 0 1

0 0 1 0

0 1 0 0

, and

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

.

Let c ∈ {1,3,−1,−3}. The elements of SU(4) corre-

sponding to the transformation CNOT[2,1] are given

by exp[cπ(−i2SxIz+ iSx+ iIz)/2], which is equal to

exp(icπ/4)CNOT[2,1]. For CNOT[1,2] and SWAP we

obtain the elements exp[cπ(−i2SzIx+ iSz+ iIx)/2] and

exp[cπ(i2SxIx+ i2SyIy+ i2SzIz)/2], which are equal

to exp(icπ/4)CNOT[1,2] and exp(icπ/4)SWAP, respec-

tively. These different instances of unitary transforma-

tions result from the irrelevance of the global phase in

quantum mechanics and can be described mathemati-

cally by multiplying with elements of the (finite) center

of G. The center consists of those elements which com-

mute with all elements of G. To find the time-optimal

control algorithm, we may have to consider multiplying

with different elements of the center.

As exp(iπ/4)CNOT[2,1] is an element of K, it takes

negligible time to synthesize CNOT[2,1]. In strong con-

trast, exp(iπ/4)CNOT[1,2] is not contained in K. Using

the KAK decomposition, both exp(iπ/4)CNOT[1,2] and

exp(iπ/4)SWAP correspond to the same generator of A,

given by π(−iSβIx)+0(−iSαIx), and the minimum time

to synthesize each of them is equal to tmin= π. This is

still the optimal time if we consider to multiply with dif-

ferent elements of the center.

We explicitly state the control algorithms: The unitary

transformation exp(iπ/4)CNOT[1,2] is given by

exp(iπSz/2)exp(iπIz)exp(−iπSαIx)exp(−iπIz)

= exp(iπSz/2)exp(−it′H0/J)exp?−iπHα(π)/wI

where t′= −πJ/wI

r mod 2π ≥ 0. Similarly, the unitary

transformation exp(iπ/4)SWAP is given by

r

?,

eiπ/4CNOT[2,1]eiπ/4CNOT[1,2]e−iπ/4CNOT[2,1]

= eiπSz/2e−iπSx/2e−i3πH0/(2J)eiπSy/2e−it′H0/J

×exp?−iπHα(π)/wI

The corresponding pulse sequences are given in Fig. 3.

r

?e−iπSx/2e−iπH0/(2J)e−iπSy/2.

V.PROOF OF TIME-OPTIMALITY

In this section, we prove the time-optimality of the

given control algorithms in order to synthesize unitary

transformations in coupled fast and slow qubit system.

As expected, the maximal amplitude ωI

determines the optimal time.

r(see Eq. (5))

Hα(π)

π

ωIr

t′

J

?π

2

?

−z

I

S

?π

2

?

y

π

2J

?π

2

?

x

Hα(π)

π

ωIr

t′

J

?π

2

?

−y

3π

2J

˜R5

I

S

(a) (b)

FIG. 3: The figure shows the pulse sequences for synthesiz-

ing the unitary transformations (a) exp(iπ/4)CNOT[1,2] and

(b) exp(iπ/4)SWAP, where˜R5 = exp(iπSz/2)exp(−iπSx/2).

Since 1/J ≪ 1/wI

larger as depicted. Refer to the text for details.

r, the length of the time intervals π/wI

ris

A. The simple case

All control algorithms, synthesizing a unitary transfor-

mation in time t =?

K′

jtj, can be written in the form

n+1exp[−it′

nHβ(ψn)]K′

n···K′

2exp[−it′

1Hβ(ψ1)]K′

1,

(15)

where K′

compared to the evolution under Hβ, tj,ψj ∈ R, and

t′

r. We can rewrite Eq. (15) as

j∈ K take negligible time to be synthesized as

j= tj/wI

Kn+1exp[−itnSβIx]Kn···K2exp[−it1SβIx]K1,(16)

where Kj∈ K. Equation (16) can be rewritten as

˜Kn+1exp(˜ pn)···exp(˜ p1),(17)

where ˜ pj =˜Kj(−itjSβIx)˜K−1

ments of K. Observe that the elements ˜ pj are contained

in p. This follows from the Campbell-Baker-Hausdorff

formula (see, e.g., Appendix B.4 of Ref. [44]) and the

fact that [k,p] ∈ p (see Eq. 10). It was shown in Ref. [4]

that for all time-optimal control algorithms the elements

˜Kj can be chosen such that all ˜ pj commute. Therefore,

all ˜ pj belong to a maximal Abelian subalgebra inside p,

and we can find one K0∈ K such that K0˜ pjK−1

all j. Using this result and results of Eq. (20) below, we

can rewrite Eq. (17) in the form

j

and˜Kj are suitable ele-

0

∈ a for

¯K2exp(tnpn)···exp(t1p1)¯K1,(18)

where pj= βj(−iSβIx) + αj(−iSαIx),

(βj,αj)T∈ {(−1,0)T,(1,0)T,(0,−1)T,(0,1)T},

and¯K1,¯K2∈ K. Equation (18) can be simplified to

¯K2exp[¯β(−iSβIx) + ¯ α(−iSαIx)]¯ K1,(19)

where ¯ α =

the unitary transformation to be synthesized is given

?

jαjtj and¯β =

?

jβjtj.Assume that