ARTICLE IN PRESS

Physica E 42 (2010) 449–453

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/physe

Riemannian curvature in the differential geometry of quantum computation

Howard E. Brandt

US Army Research Laboratory, Adelphi, MD, USA

a r t i c l e i n f o

Available online 24 June 2009

Keywords:

Quantum computing

Quantum circuits

Riemannian geometry

77/$ - see front matter Published by Elsevier

016/j.physe.2009.06.016

ail address: hbrandt@arl.army.mil

a b s t r a c t

In recent developments in the differential geometry of quantum computation, the quantum evolution is

described in terms of the special unitary group SUð2nÞ of n-qubit unitary operators with unit

determinant. To elaborate on one aspect of the methodology, the Riemann curvature and sectional

curvature are explicitly derived using the Lie algebra suð2nÞ. This is germane to investigations of the

global characteristics of geodesic paths and minimal complexity quantum circuits.

Published by Elsevier B.V.

1. Introduction

A new innovative approach to the differential geometry ofquantum computation and quantum circuit complexity wasrecently introduced by Nielsen et al. [1–4]. A Riemannian metricwas formulated on the manifold of multi-qubit unitary transfor-mations, such that the metric distance between the identity andthe desired unitary operator, representing the quantum computa-tion, is equivalent to the number of quantum gates needed torepresent that unitary operator, thereby providing a measure ofthe complexity associated with the corresponding quantumcomputation. The Riemannian metric was defined as a positive-definite bilinear form defined in terms of the multi-qubitHamiltonian. The analytic form of the metric was chosen topenalize all directions on the manifold not easily simulated bylocal gates. In this way, basic differential geometric concepts suchas the Levi–Civita connection, geodesic path, Riemannian curva-ture, Jacobi fields, and conjugate points can be associated withquantum computation. The Hamiltonian can be expressed interms of tensor products of the Pauli matrices which act on thequbits. The Riemann curvature tensor can then be constructedfrom the Christoffel symbols and their ordinary derivatives.The geodesic equation on the manifold follows from the connec-tion and determines the local optimal Hamiltonian evolutioncorresponding to the unitary transformation representing thedesired quantum computation. The optimal unitary evolution maybe sought by solving the geodesic equation. Useful upper andlower bounds on the associated quantum circuit complexity maybe obtained. Such differential geometric approaches to quantumcomputation are currently preliminary and many details remainto be worked out.

B.V.

The present work is based on an invited talk presented at theconference, Frontiers of Quantum and Mesoscopic Thermodynamics,held in Prague in 2008, and in which an introduction was offeredon the differential geometry of quantum computation. Because ofpage limitations, the present work is limited to the elaboration ofone aspect of the methodology (other aspects are addressed inRefs. [5,6]). The Riemann curvature and sectional curvature areexplicitly derived with much greater detail and clarity than hasappeared in the literature [4,7–9]. The Lie algebra suð2n

Þ isexploited, expressed in terms of the tensor products of Paulimatrices appearing in the Hamiltonian and representing gateoperations. The curvature is germane to investigations of theglobal characteristics of geodesic paths [10] and the determina-tion of minimal complexity quantum circuits [4,11].

2. Metric

A Riemannian metric is first chosen on the manifold of the Liegroup SUð2n

Þ (special unitary group) of n-qubit unitary operatorswith unit determinant [12–24]. The traceless Hamiltonian servesas a tangent vector to a point on the group manifold of the n-qubitunitary transformation U. The Hamiltonian H is an element of theLie algebra suð2n

Þ of traceless 2n� 2n Hermitian matrices [22–24]

and is taken to be tangent to the evolutionary curve e�iHtU att ¼ 0. (Here and throughout, units are chosen such that Planck’sconstant divided by 2p is ‘ ¼ 1.)

Independent of U, the Riemannian metric (inner product) /.,.Sis taken to be a right-invariant positive definite bilinear form/H; JS defined on tangent vectors (Hamiltonians) H and J. Rightinvariance of the metric means that all right translations areisometries. Following Ref. [4], the n-qubit Hamiltonian H can bedivided into two parts PðHÞ and Q ðHÞ, where PðHÞ contains onlyone and two-body terms, and Q ðHÞ contains more than two-bodyterms. Parts of the Hamiltonian containing n-body terms require

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H.E. Brandt / Physica E 42 (2010) 449–453450

n-qubit gates for their implementation, and more than two-qubitgates are generally more difficult and costly to implement thanone- and two-body gates. Thus it is useful to make the followingseparation:

H ¼ PðHÞ þ Q ðHÞ; ð1Þ

in which P and Q are superoperators (matrices) acting on H, andobey the following relations:

P þ Q ¼ I; PQ ¼ QP ¼ 0; P2 ¼ P; Q2 ¼ Q ; ð2Þ

where I is the identity.The Hamiltonian can be expressed in terms of tensor products

of the Pauli matrices. The Pauli matrices are defined by [11]

s0 � I �1 0

0 1

� �; s1 � X �

0 1

1 0

� �;

s2 � Y �0 �i

i 0

� �; s3 � Z �

1 0

0 �1

� �: ð3Þ

They are Hermitian,

si ¼ syi ; i ¼ 1;2;3 ð4Þ

and, except for s0, they are traceless,

Trsi ¼ 0; ia0: ð5Þ

Their products are given by

s2i ¼ I ð6Þ

and

sisj ¼ ieijksk; i; j; ka0; ð7Þ

expressed in terms of the totally antisymmetric Levi–Civitasymbol with e123 ¼ 1.

As an example of Eq. (1) in the case of a 3-qubit Hamiltonian,one has

PðHÞ ¼ x1s1 � I� I þ x2s2 � I � I þ x3s3 � I � I þ x4I � s1 � Iþ x5I � s2 � I þ x6I � s3 � I þ x7I � I � s1 þ x8I � I� s2 þ x9I � I � s3 þ x10s1 � s2 � I þ x11s1 � I � s2

þ x12I � s1 � s2 þ x13s2 � s1 � I þ x14s2 � I � s1

þ x15I � s2 � s1 þ x16s1 � s3 � I þ x17s1 � I � s3

þ x18I � s1 � s3 þ x19s3 � s1 � I þ x20s3 � I � s1

þ x21I � s3 � s1 þ x22s2 � s3 � I þ x23s2 � I � s3

þ x24I � s2 � s3 þ x25s3 � s2 � I þ x26s3 � I � s2

þ x27I � s3 � s2 þ x28s1 � s1 � I þ x29s2 � s2 � Iþ x30s3 � s3 � I þ x31s1 � I � s1 þ x32s2 � I � s2

þ x33s3 � I� s3 þ x34I � s1 � s1 þ x35I� s2 � s2

þ x36I � s3 � s3 ð8Þ

in which � denotes the tensor product [25], and

Q ðHÞ ¼ x37s1 � s2 � s3 þ x38s1 � s3 � s2 þ x39s2 � s1 � s3

þ x40s2 � s3 � s1 þ x41s3 � s1 � s2 þ x42s3 � s2

� s1 þ x43s1 � s1 � s2 þ x44s1 � s2 � s1 þ x45s2

� s1 � s1 þ x46s1 � s1 � s3 þ x47s1 � s3 � s1

þ x48s3 � s1 � s1 þ x49s2 � s2 � s1 þ x50s2 � s1

� s2 þ x51s1 � s2 � s2 þ x52s2 � s2 � s3 þ x53s2

� s3 � s2 þ x54s3 � s2 � s2 þ x55s3 � s3 � s1

þ x56s3 � s1 � s3 þ x57s1 � s3 � s3 þ x58s3 � s3

� s2 þ x59s3 � s2 � s3 þ x60s2 � s3 � s3 þ x61s1

� s1 � s1 þ x62s2 � s2 � s2 þ x63s3 � s3 � s3: ð9Þ

Here, all possible tensor products of one and two-qubit Paulimatrix operators on three qubits appear in PðHÞ, and analogously,

all possible tensor products of three-qubit operators appear inQ ðHÞ. Tensor products including only the identity are excludedbecause the Hamiltonian is taken to be traceless. Each of theterms in Eqs. (8) and (9) is an 8� 8 matrix. The various tensorproducts of Pauli matrices such as those appearing in Eqs. (8) and(9) are referred to as generalized Pauli matrices. In the case of ann-qubit Hamiltonian, there are 4n

� 1 possible tensor products(corresponding to the dimension of SUð2n

Þ), and each term is a2n� 2n matrix.The right-invariant [12–15,23,24] Riemannian metric for

tangent vectors H and J is given by [4]

/H; JS �1

2n Tr½HPðJÞ þ qHQ ðJÞ�: ð10Þ

Here q is a large penalty parameter which taxes more than two-body terms. The length l of an evolutionary path on the SUð2n

Þ

manifold is given by the integral over time t from an initial time ti

to a final time tf , namely,

l ¼

Z tf

ti

dtð/HðtÞ;HðtÞSÞ1=2ð11Þ

and is a measure of the cost of applying a control Hamiltonian HðtÞ

along the path [4]. Justification for the form of the metric, Eq. (10),is given in Refs. [1,4].

3. Covariant derivative

In order to obtain the Levi–Civita connection, it is necessary toexploit the Lie algebra suð2n

Þ associated with the group SUð2nÞ.

Because of the right-invariance of the metric, meaning that righttranslations are isometries, it follows that if the connection iscalculated at the origin, the same expression applies everywhereon the manifold. Following [4], consider the unitary transforma-tion

U ¼ e�iX ð12Þ

in the neighborhood of the origin I � SUð2nÞ with

X ¼ x � s �Xs

xss; ð13Þ

which expresses symbolically terms like those in Eqs. (8) and (9)generalized to 2n dimensions. In Eqs. (12) and (13), X is definedusing the standard branch of the logarithm with a cut along thenegative real axis. In Eq. (13), for the general case of n qubits, x

represents the set of real (4n� 1) coefficients of the generalized

Pauli matrices s which represent all of the n-fold tensor products.It follows from Eq. (13) that the factor xs multiplying a particularterm s is given by

xs ¼1

2n TrðXsÞ: ð14Þ

Next, the right-invariant metric, Eq. (10), in the so-calledHamiltonian representation can be written as

/H; JS ¼1

2n Tr½HGðJÞ�; ð15Þ

in which the positive self-adjoint superoperator G is given by

G ¼ P þ qQ : ð16Þ

Recall that the penalty parameter q was introduced in Eq. (10).Using Eqs. (2) and (16) it follows that

G�1 ¼ P þ q�1Q ; ð17Þ

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H.E. Brandt / Physica E 42 (2010) 449–453 451

since one then has

GG�1¼ ðP þ qQ ÞðP þ q�1Q Þ ¼ 1: ð18Þ

A vector Y in the group tangent space can be written as

Y ¼Xs

yss ð19Þ

with so-called Pauli coordinates ys. Here s, as an index, is used torefer to a particular tensor product appearing in the generalizedPauli matrix s. This index notation, used throughout, is aconvenient abbreviation for the actual numerical indices (e.g. inEq. (8), the number 22 appearing in x22, the coefficient ofs2 � s3 � I). It can be shown that the Hamiltonian representationof the directional derivative of the vector Z along the vector Y isgiven by [4]:

ðrY ZÞ ¼ ysZ;s þi

2ð½Y ; Z� þ Fð½Y ;GðZÞ� þ ½Z;GðYÞ�ÞÞ ð20Þ

in which the superoperator F is defined by

FðrÞ � G�1ðrÞ ¼ ðP þ q�1Q ÞðrÞ; ð21Þ

where Eq. (17) has been used.Next consider a curve passing through the origin with tangent

vector Y and ys ¼ dxs=dt. Then according to Eq. (20) and the chainrule, the covariant derivative along the curve in the Hamiltonianrepresentation is given by

ðDtZÞ � ðrY ZÞ ¼dZ

dtþ

i

2ð½Y ; Z� þ Fð½Y ;GðZÞ� þ ½Z;GðYÞ�ÞÞ: ð22Þ

Because of the right-invariance of the metric, Eq. (22) is true onthe entire manifold. Furthermore, for a right-invariant vector fieldZ, one has

dZ

dt¼ 0 ð23Þ

and substituting Eq. (23) in Eq. (22), one obtains

ðrY ZÞ ¼i

2f½Y ; Z� þ Fð½Y ;GðZÞ� þ ½Z;GðYÞ�Þg; ð24Þ

which is also true everywhere on the manifold.

4. Riemann curvature

For a right-invariant vector field Z, one has after substituting

Z ¼Xt

ztt; Y ¼Xs

yss ð25Þ

in Eq. (24):

Xstrstyszt ¼

i

2

Xstð½s; t� þ Fð½s;GðtÞ� þ ½t;GðsÞ�ÞÞyszt ð26Þ

and therefore

rst ¼i

2ð½s; t� þ Fð½s;GðtÞ� þ ½t;GðsÞ�ÞÞ: ð27Þ

Next, denoting S0 as the set containing only tensor products of theidentity, and S12 as the set of terms in the Hamiltonian containingonly one and two-body terms, that is

S0 � fI� I � � � �g ð28Þ

and

S12 ¼ fI� I� � � � � si � I . . .g[ fI � I� � � � � si � I . . .� sj � I . . .g; ð29Þ

then evidently,

½s;GðtÞ� ¼½s; t�; t 2 S12 [ S0;

q½s; t�; t=2S12 [ S0

(ð30Þ

and therefore

Fð½s;GðtÞ�Þ ¼Fð½s; t�Þ; t 2 S12 [ S0;

qFð½s; t�Þ; t=2S12 [ S0:

(ð31Þ

Using Eq. (21) in Eq. (31), one obtains

F ½s;GðtÞ�ð Þ ¼

1

q½s;t�½s; t�; t 2 S12 [ S0;

q

q½s;t�½s; t�; t=2S12 [ S0;

8>>><>>>:

ð32Þ

where

q½s;t� ¼ 1 if ½s; t� ¼ 0;q

½s;t� ¼ ql if ½s; t�pl and q½s;t� ¼ q

½t;s� ð33Þ

and ql is defined by

qs �

0; s 2 S0;

1; s 2 S12;

q; s=2S0 [ S12:

8><>: ð34Þ

Eq. (32) can be written as

Fð½s;GðtÞ�Þ ¼ qtq½s;t�½s; t�: ð35Þ

Next substituting Eq. (35) in Eq. (27), one obtains

rst ¼i

21þ

qtq½s;t�

� �½s; t� þ qs

q½t;s�½t;s�

� �ð36Þ

or equivalently, using Eq. (33), this becomes

rst ¼i

21þ

qt � qsq½s;t�

� �½s; t� ð37Þ

or

rst ¼ ics;t½s; t�; ð38Þ

where

cs;t ¼1

21þ

qt � qsq½s;t�

� �: ð39Þ

The Riemann curvature tensor with the inner-product (metric)Eq. (15) is given by [26]

RðW ;X;Y ; ZÞ ¼ /rWrXY �rXrW Y �ri½W ;X�Y ; ZS ð40Þ

and after substituting the vector fields,

W ¼Xs

wrr; X ¼Xs

zss;Y ¼Xt

ytt; Z ¼Xm

zmm: ð41Þ

Eq. (40) becomes

Rrstm ¼ /rrrst�rsrrt�ri½r;s�t;mS: ð42Þ

Next, for three right-invariant vector fields X, Y, and Z, one has

0 ¼ rY/X; ZS ¼ /X;rY ZSþ/rY X; ZS ð43Þ

or

/X;rY ZS ¼ �/rY X; ZS ð44Þ

and substituting Eqs. (41) in Eq. (44), one then has

/s;rtmS ¼ �/rts;mS: ð45Þ

Therefore

/rrrst;mS ¼ �/rst;rrmS ð46Þ

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H.E. Brandt / Physica E 42 (2010) 449–453452

and

/rsrrt;mS ¼ �/rrt;rsmS: ð47Þ

Then substituting Eqs. (46) and (47) in Eq. (42), and interchangingthe first and second terms, one obtains

Rrstm ¼ /rrt;rsmS�/rst;rrmS�/ri½r;s�t;mS: ð48Þ

Also clearly

riY Z ¼ irY Z; ð49Þ

so Eq. (48) can also be written as

Rrstm ¼ /rrt;rsmS�/rst;rrmS� i/r½r;s�t;mS: ð50Þ

Next substituting Eq. (38) in Eq. (50), one obtains the followinguseful form for the Riemann curvature tensor [4]:

Rrstm ¼ cr;tcs;m/i½r; t�; i½s;m�S� cs;tcr;m/i½s; t�; i½r;m�S�c½r;s�;t/i½i½r;s�; t�;mS: ð51Þ

5. Sectional curvature

The sectional curvature spanned by orthonormal right-invar-iant vector fields X and Y is defined by [15]

KðX;YÞ � RðX;Y ;Y ;XÞ: ð52Þ

From Eqs. (48) and (41), it immediately follows that

RðW ;X;Y ; ZÞ ¼ /rW Y ;rXZS�/rXY ;rW ZS�/ri½W ;X�Y ; ZS ð53Þ

and substituting Eq. (53) in Eq. (52), one obtains

KðX;YÞ ¼ /rXY ;rY XS�/rY Y ;rXXS�/ri½X;Y �Y ;XS: ð54Þ

Next it is useful to define

BðX;YÞ ¼ Fði½GðXÞ;Y �Þ ð55Þ

and using Eqs. (15) and (55), one obtains

/BðX;YÞ; ZS ¼ /Fði½GðXÞ;Y�Þ; ZS ð56Þ

or equivalently, using Eq. (15), then

/BðX;YÞ; ZS ¼1

2n TrðFði½GðXÞ;Y �ÞGðZÞÞ: ð57Þ

Because the superoperator G is Hermitian, Eq. (57) can also bewritten as

/BðX;YÞ; ZS ¼1

2n TrðGFði½GðXÞ;Y �ÞZÞ; ð58Þ

but according to Eq. (21) one has

GF ¼ I ð59Þ

and therefore Eq. (58) becomes

/BðX;YÞ; ZS ¼1

2n Trði½GðXÞ;Y �ZÞ: ð60Þ

Next expanding the commutator, Eq. (60) becomes

/BðX;YÞ; ZS ¼i

2n TrðGðXÞYZ � YGðXÞZÞ ð61Þ

and using the cyclic property of the trace, one obtains

/BðX;YÞ; ZS ¼i

2n TrðGðXÞYZ � GðXÞZYÞ ð62Þ

or, equivalently,

/BðX;YÞ; ZS ¼i

2n TrðGðXÞ½Y ; Z�Þ: ð63Þ

But since the superoperator G is Hermitian, Eq. (63) can also bewritten as

/BðX;YÞ; ZS ¼i

2n TrðXGð½Y ; Z�ÞÞ ð64Þ

or equivalently, using Eq. (15), this becomes

/X; i½Y ; Z�S ¼ /BðX;YÞ; ZS: ð65Þ

But according to Eq. (15), it follows that for vectors X and Y onehas

/X;YS ¼1

2n TrðXGðYÞÞ ð66Þ

and because the superoperator G is Hermitian, this can also bewritten as

/X;YS ¼1

2n TrðGðXÞYÞ; ð67Þ

which by the cyclic invariance of the trace becomes

/X;YS ¼1

2n TrðYGðXÞÞ ð68Þ

or equivalently using Eq. (15), it follows that

/X;YS ¼ /Y ;XS; ð69Þ

consistent with the Riemannian symmetric metric.Next, for a right-invariant field Y, one has, using Eq. (24),

rXY ¼ 12ði½X;Y� þ Fði½X;GðYÞ�Þ þ Fði½Y ;GðXÞ�ÞÞ ð70Þ

or

rXY ¼ 12ði½X;Y� � Fði½GðYÞ;X�Þ � Fði½GðXÞ;Y �ÞÞ: ð71Þ

Then substituting Eq. (55) in Eq. (71), one obtains

rXY ¼ 12ði½X;Y� � BðX;YÞ � BðY ;XÞÞ: ð72Þ

Next, according to Eqs. (52) and (54), one has

KðX;YÞ ¼ RðX;Y ;Y ;XÞ¼ /rXY ;rY XS�/rY Y ;rXXS� i/r½X;Y �Y ;XS: ð73Þ

According to Eq. (72), one has for the last term in Eq. (73),

r½X;Y �Y ¼12ði½½X;Y �;Y � � Bð½X;Y �;YÞ � BðY ; ½X;Y �ÞÞ: ð74Þ

Also using Eq. (72), one obtains for the first term in Eq. (73),

/rXY ;rY XS ¼ 14ð/i½X;Y � � BðX;YÞ � BðY ;XÞ; i½Y ;X� � BðY ;XÞ� BðX;YÞSÞ ð75Þ

or, equivalently,

/rXY ;rY XS ¼ 14ð/i½X;Y � � BðX;YÞ � BðY ;XÞ;� i½X;Y �� BðX;YÞ � BðY ;XÞSÞ ð76Þ

or

/rXY ;rY XS ¼ 14 /� i½X;Y�; i½X;Y �S� 1

4 /i½X;Y �;BðX;YÞ

þ BðY ;XÞSþ 14 /BðX;YÞ þ BðY ;XÞ; i½X;Y �S

þ 14/BðX;YÞ þ BðY ;XÞ;BðX;YÞ þ BðY ;XÞS: ð77Þ

Next using Eq. (69) in Eq. (77), then

/rXY ;rY XS ¼ �14 /i½X;Y�; i½X;Y �Sþ 1

4/BðX;YÞþ BðY ;XÞ;BðX;YÞ þ BðY ;XÞS: ð78Þ

Also, one has for the second term in Eq. (73),

/rY Y ;rXXS ¼ 14ð/� i½Y ;Y �; i½X;X�S�/i½Y ;Y�;2BðX;XÞSþ/2BðY ;YÞ; i½X;X�Sþ 4/BðY ;YÞ;BðX;XÞSÞ: ð79Þ

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H.E. Brandt / Physica E 42 (2010) 449–453 453

But, according to Eq. (69), one has

/BðY ;YÞ;BðX;XÞS ¼ /BðX;XÞ;BðY ;YÞS ð80Þ

Then simplifying Eq. (79), one obtains

/rY Y ;rXXS ¼ /BðX;XÞ;BðY ;YÞS: ð81Þ

Next substituting Eqs. (78), (81), and (74) in Eq. (73), one has

KðX;YÞ ¼ �1

4/i½X;Y �; i½X;Y �Sþ

1

4/BðX;YÞ þ BðY ;XÞ;BðX;YÞ

þ BðY ;XÞS�i

2/i½½X;Y �;Y �;XSþ

i

2/Bð½X;Y �;YÞ;XS

þi

2/BðY ; ½X;Y �Þ;XS�/BðX;XÞ;BðY ;YÞS: ð82Þ

Expanding the third term of Eq. (82), one has, using Eq. (15),

�i

2/i½½X;Y �;Y �;XS ¼

1

2

1

2n

� �Trð½½X;Y �;Y�GðXÞÞ

¼1

2

1

2n

� �Trðð½X;Y�Y � Y½X;Y�ÞGðXÞÞ ð83Þ

and using the cyclic invariance of the trace, then

�i

2/i½½X;Y �;Y �;XS ¼

1

2

1

2n

� �Trð½X;Y �YGðXÞ � ½X;Y �ÞGðXÞYÞ

¼1

2

1

2n

� �Trði½X;Y �i½GðXÞ;Y�Þ: ð84Þ

Next using Eqs. (55) and (21) in Eq. (84), one obtains

�i

2/i½½X;Y �;Y �;XS ¼

1

2/i½X;Y �;BðX;YÞS: ð85Þ

Next, in the fourth term of Eq. (82) one has, using Eq. (65),

/Bð½X;Y �;YÞ;XS ¼ /½X;Y �; i½Y ;X�S ¼ i/i½X;Y �; i½X;Y �S: ð86Þ

In the fifth term of Eq. (82), using Eq. (65), one has

/BðY ; ½X;Y�Þ;XS ¼ /Y ; i½½X;Y �;X�S ð87Þ

or equivalently,

/BðY ; ½X;Y�Þ;XS ¼ /Y ; i½X; ½Y ;X��S ð88Þ

and using Eq. (65), this becomes

/BðY ; ½X;Y�Þ;XS ¼ �/BðY ;XÞ; ½X;Y�S: ð89Þ

Next using Eq. (69), Eq. (89) becomes

/BðY ; ½X;Y�Þ;XS ¼ �/½X;Y �;BðY ;XÞS: ð90Þ

Next, substituting Eqs. (85), (86), and (90) in Eq. (82), one has

KðX;YÞ ¼ �14 /i½X;Y �; i½X;Y �Sþ 1

4 /BðX;YÞ þ BðY ;XÞ;BðX;YÞ

þ BðY ;XÞSþ 12 /i½X;Y �;BðX;YÞS� 1

2 /i½X;Y �; i½X;Y �S

� 12/i½X;Y �;BðY ;XÞS�/BðX;XÞ;BðY ;YÞS ð91Þ

and combining terms, this becomes [4,8,9]

KðX;YÞ ¼ �34 /i½X;Y �; i½X;Y �Sþ 1

4 /BðX;YÞ þ BðY ;XÞ;BðX;YÞ

þ BðY ;XÞSþ 12/i½X;Y�;BðX;YÞ � BðY ;XÞS

�/BðX;XÞ;BðY ;YÞS: ð92Þ

6. Conclusion

To elaborate on one aspect of the methodology of thedifferential geometry of quantum computation, the Riemann

curvature and sectional curvature on the manifold of the SUð2nÞ

group of n-qubit unitary operators with unit determinant wereexplicitly derived using the Lie algebra suð2n

Þ. The Riemanncurvature is given by Eqs. (51), (39), (33), and (34). The sectionalcurvature is given by Eqs. (92) and (55). This is germane toinvestigations of the global characteristics of geodesic paths andminimal complexity quantum circuits.

Acknowledgments

The author wishes to thank Vaclav Spicka for inviting him topresent this work at the conference Frontiers of Quantum andMesoscopic Thermodynamics, 28 July–2 August 2008 in Prague,Czech Republic. This work was supported by the Director’sResearch Initiative at the US Army Research Laboratory.

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