Quantum Computational Geodesics 

by Howard E. Brandt 

ARL-TR-5055 January 2010 

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Army Research Laboratory 

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ARL-TR-5055 January 2010 

Quantum Computational Geodesics 

Howard E. Brandt 

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14. ABSTRACT 

Recent developments in the differential geometry of quantum computation offer a new approach to the analysis of quantum 

computation. In the Riemannian geometry of quantum computation, the quantum evolution is described in terms of the special 

unitary group of n-qubit unitary operators with unit determinant. The group manifold is taken to be Riemannian. The 

objective of this report is to mathematically elaborate on characteristics of geodesics describing possible minimal complexity 

paths in the group manifold representing the unitary evolution associated with a quantum computation. For this purpose the 

Jacobi equation, generic lifted Jacobi equation, lifted Jacobi equation for varying penalty parameter, and the so-called 

geodesic derivative are reviewed. These tools are important for investigations of the global characteristics of geodesic paths in 

the group manifold, and the determination of optimal quantum circuits for carrying out a quantum computation. 

15. SUBJECT TERMS

Quantum computing, Riemannian geometry, group theory, quantum circuits 

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Contents

1. Objective 1

2. Approach 1

3. Results 8

4. Other Work 27

5. Conclusions 27

6. References 28

7. Transitions 31

Distribution 32

iii
iv 

INTENTIONALLY LEFT BLANK. 
1. Objective

The objective of this project is to mathematically investigate characteristics of geodesics,

describing possible minimal complexity paths in the group manifold representing the

unitary evolution associated with a quantum computation.

2. Approach

In the Riemannian geometry of quantum computation (1–11), a Riemannian metric can be

chosen on the manifold of the (4n − 1)-dimensional Lie Group SU(2n) (special unitary

group) of n-qubit unitary operators with unit determinant (1–27). The traceless

Hamiltonian of a quantum computational system is a tangent vector to a point on the

group manifold of the n-qubit unitary transformation describing the time evolution of the

system. The Hamiltonian H is an element of the Lie algebra su(2n) of traceless 2n × 2

n

Hermitian matrices (25–27) and is tangent at the n-qubit unitary operator U to the

evolutionary curve e

−iHtU at t = 0. (Here and throughout, units are chosen such that

Planck’s constant divided by 2π is ~ = 1.)

The Riemannian metric (inner product) h., .i is a positive definite bilinear form hH, Ji

defined on tangent vectors (Hamiltonians) H and J. The n-qubit Hamiltonian H can be

divided into two parts P(H) and Q(H), where P(H) contains only one and two-body

terms, and Q(H) contains more than two-body terms (1). Thus,

H = P(H) + Q(H), (1)

in which P and Q are superoperators acting on H, and obey the following relations:

P + Q = I, P Q = QP = 0, P2 = P, Q2 = Q, (2)

where I is the identity.

The Hamiltonian can be expressed in terms of tensor products of the Pauli matrices. The

Pauli matrices are given by (28)

σ0 ≡ I ≡



1 0

0 1 

, σ1 ≡ X ≡



0 1

1 0 

,

σ2 ≡ Y ≡



0 −i

i 0



, σ3 ≡ Z ≡



1 0

0 −1



. (3)

They are Hermitian,

σi = σ

†

i

, i = 0, 1, 2, 3, (4)

1
where † denotes the adjoint, and, except for σ0, they are traceless,

Trσi = 0, i 6= 0. (5)

Their products are given by

σ

2

i = I. (6)

Also,

σiσj = iεijkσk, i, j, k 6= 0, (7)

expressed in terms of the totally antisymmetric Levi-Civita symbol with ε123 = 1, and

using the Einstein sum convention.

An example of equation 1, in the case of a 3-qubit Hamiltonian, is

P(H) = x1σ1 ⊗ I ⊗ I +x2σ2 ⊗ I ⊗ I + x3σ3 ⊗ I ⊗ I

+ x4I ⊗ σ1 ⊗ I + x5I ⊗ σ2 ⊗ I + x6I ⊗ σ3 ⊗ I + x7I ⊗ I ⊗ σ1

+x8I ⊗ I ⊗ σ2 + x9I ⊗ I ⊗ σ3 + x10σ1 ⊗ σ2 ⊗ I + x11σ1 ⊗ I ⊗ σ2

+x12I ⊗ σ1 ⊗ σ2 + x13σ2 ⊗ σ1 ⊗ I + x14σ2 ⊗ I ⊗ σ1 + x15I ⊗ σ2 ⊗ σ1

+ x16σ1 ⊗ σ3 ⊗ I + x17σ1 ⊗ I ⊗ σ3 + x18I ⊗ σ1 ⊗ σ3 + x19σ3 ⊗ σ1 ⊗ I

+x20σ3 ⊗ I ⊗ σ1 + x21I ⊗ σ3 ⊗ σ1 + x22σ2 ⊗ σ3 ⊗ I + x23σ2 ⊗ I ⊗ σ3

+x24I ⊗ σ2 ⊗ σ3 + x25σ3 ⊗ σ2 ⊗ I + x26σ3 ⊗ I ⊗ σ2 + x27I ⊗ σ3 ⊗ σ2

+ x28σ1 ⊗ σ1 ⊗ I + x29σ2 ⊗ σ2 ⊗ I + x30σ3 ⊗ σ3 ⊗ I + x31σ1 ⊗ I ⊗ σ1

+x32σ2 ⊗ I ⊗ σ2 + x33σ3 ⊗ I ⊗ σ3 + x34I ⊗ σ1 ⊗ σ1 + x35I ⊗ σ2 ⊗ σ2

+x36I ⊗ σ3 ⊗ σ3, (8)

in which ⊗ denotes the tensor product, and

Q(H) = x37σ1 ⊗ σ2 ⊗ σ3 + x38σ1 ⊗ σ3 ⊗ σ2

+ x39σ2 ⊗ σ1 ⊗ σ3 + x40σ2 ⊗ σ3 ⊗ σ1 (9)

+ x41σ3 ⊗ σ1 ⊗ σ2 + x42σ3 ⊗ σ2 ⊗ σ1

+ x43σ1 ⊗ σ1 ⊗ σ2 + x44σ1 ⊗ σ2 ⊗ σ1

+ x45σ2 ⊗ σ1 ⊗ σ1 + x46σ1 ⊗ σ1 ⊗ σ3

+ x47σ1 ⊗ σ3 ⊗ σ1 + x48σ3 ⊗ σ1 ⊗ σ1

+ x49σ2 ⊗ σ2 ⊗ σ1 + x50σ2 ⊗ σ1 ⊗ σ2

+ x51σ1 ⊗ σ2 ⊗ σ2 + x52σ2 ⊗ σ2 ⊗ σ3

+ x53σ2 ⊗ σ3 ⊗ σ2 + x54σ3 ⊗ σ2 ⊗ σ2

+ x55σ3 ⊗ σ3 ⊗ σ1 + x56σ3 ⊗ σ1 ⊗ σ3

+ x57σ1 ⊗ σ3 ⊗ σ3 + x58σ3 ⊗ σ3 ⊗ σ2

+ x59σ3 ⊗ σ2 ⊗ σ3 + x60σ2 ⊗ σ3 ⊗ σ3

+ x61σ1 ⊗ σ1 ⊗ σ1 + x62σ2 ⊗ σ2 ⊗ σ2

+ x63σ3 ⊗ σ3 ⊗ σ3. (10)

2
Here, all possible tensor products of one and two-qubit Pauli matrix operators on three

qubits appear in P(H), and analogously, all possible tensor products of three-qubit

operators appear in Q(H). Tensor products including only the identity are excluded

because the Hamiltonian is taken to be traceless. Each of the terms in equations 8 and 9 is

an 8×8 matrix. The various tensor products of Pauli matrices such as those appearing in

equations 8 and 9 are referred to as generalized Pauli matrices. In the case of an n-qubit

Hamiltonian, there are 4n − 1 possible tensor products (corresponding to the dimension of

SU(2n)), and each term is a 2nx2n matrix.

The right-invariant (16–18, 26, 27) Riemannian metric for tangent vectors H and J is

given by (1)

hH, Ji ≡ 1

2

n

Tr [HP(J) + qHQ(J)] . (11)

Here q is a large penalty parameter that taxes more than two-body terms. The length l of

an evolutionary path on the SU(2n) manifold is given by the integral over time t from an

initial time ti to a final time tf , namely,

l =

Z

tf

ti

dt(hH(t), H(t)i)

1/2

, (12)

and is a measure of the cost, in terms of quantum circuit complexity, of applying a control

Hamiltonian H(t) along the path (1).

In order to obtain the Levi-Civita connection on the group manifold, one exploits the Lie

algebra su(2n) associated with the group SU(2n). Because of the right-invariance of the

metric, if the connection is calculated at the origin, the same expression applies everywhere

on the manifold. Following reference 1, consider the unitary transformation

U = e

−iX (13)

in the neighborhood of the identity I ⊂ SU(2n) with

X = x · σ ≡

X

σ

xσσ. (14)

Equation 14 expresses symbolically terms like those in equations 8 and 9 generalized to 2n

dimensions. In equations 12 and 13, X is defined in terms of U using the standard branch

of the logarithm with a cut along the negative real axis. In equation 13, for the general

case of n qubits, x represents the set of real (4n − 1) coefficients of the generalized Pauli

matrices σ, which represent all of the n-fold tensor products. Taking the trace of equation

13, it follows that the factor x

σ multiplying a particular term σ is given by

x

σ =

1

2

n

Tr(Xσ). (15)

3
The right-invariant metric, equation 10, can be written as

hH, Ji =

1

2

n

Tr[HG(J)], (16)

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

G = P + qQ. (17)

Using equations 2 and 16, it follows that

F ≡ G

−1 = P + q−1Q. (18)

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

Y =

X

σ

y

σ

σ (19)

with so-called Pauli coordinates y

σ

. Here σ, as an index, is used to refer to a particular

tensor product appearing in the generalized Pauli matrix σ. This index notation, used

throughout, is a convenient abbreviation for the actual numerical indices (e.g., in equation

9, the number 57 appearing in x57, the coefficient of σ1 ⊗ σ3 ⊗ σ3).

Next consider a curve passing through the origin with tangent vector Y having

components y

σ = dxσ/dt. It can be shown that the covariant derivative of a right-invariant

vector field Z along the curve in the Hamiltonian representation is given by (1, 2)

(∇Y Z) = i

2

{[Y, Z] + F([Y, G(Z)] + [Z, G(Y )])}. (20)

Because of the right-invariance of the metric, equation 19 is true everywhere on the

manifold.

The Riemann curvature on the group manifold affects the behavior of geodesics and can be

obtained as follows. In the case of a right-invariant vector field Z, one has after substituting

Z =

X

τ

z

τ

τ, Y =

X

σ

y

σ

σ (21)

in equation 20,

∇στ =

i

2

([σ, τ ] + F([σ, G(τ )] + [τ, G(σ)])). (22)

Next denote S0 as a set containing only tensor products of the identity, and S12 as the set

of terms in the Hamiltonian containing only one and two body terms, that is

S0 ≡ {I ⊗ I ⊗ ...}, (23)

4
and

S12 = {I ⊗ I ⊗ ...σi ⊗ I.., ..}

∪{I ⊗ I ⊗ ...σi ⊗ I..σj ⊗ I.., ..}. (24)

Evidently then

[σ, G(τ )] = 

[σ, τ ], τ ∈ S12 ∪ S0

q[σ, τ ], τ /∈ S12 ∪ S0

, (25)

and therefore

F ([σ, G(τ )]) = 

F([σ, τ ]), τ ∈ S12 ∪ S0

qF([σ, τ ]), τ /∈ S12 ∪ S0

. (26)

Using equation 18 in equation 26, one obtains

F ([σ, G(τ )]) = ( 1

q[σ,τ]

[σ, τ ], τ ∈ S12 ∪ S0

q

q[σ,τ]

[σ, τ ], τ /∈ S12 ∪ S0

, (27)

where

q[σ,τ] = 1 if [σ, τ ] = 0, q[σ,τ] = qλ if [σ, τ ] ∝ λ, and q[σ,τ] = q[τ,σ], (28)

and qλ is defined by

qσ ≡







0, σ ∈ S0

1, σ ∈ S12

q, σ /∈ S0 ∪ S12

. (29)

Equation 27 can also be written as

F ([σ, G(τ )]) = qτ

q[σ,τ ]

[σ, τ ]. (30)

Next substituting equation 30 in equation 22, and using equation 28, one obtains

∇στ = icσ,τ [σ, τ ], (31)

where

cσ,τ =

1

2



1 +

qτ − qσ

q[σ,τ ]



. (32)

The Riemann curvature tensor with the inner-product (metric) equation 16 is given by (29)

R(W, X, Y, Z) = 
∇W ∇XY − ∇X∇W Y − ∇i[W,X]Y, Z. (33)

After substituting the vector fields,

W =

X

σ

w

ρ

ρ, X =

X

σ

z

σ

σ, Y =

X

τ

y

τ

τ, Z =

X

μ

z

μμ, (34)

equation 33 becomes

Rρστ μ =




∇ρ∇στ − ∇σ∇ρτ − ∇i[ρ,σ]τ, μ. (35)

5
Next, for three right-invariant vector fields X, Y , and Z, one has

0 = ∇Y hX, Zi = hX, ∇Y Zi + h∇Y X, Zi, (36)

or

hX, ∇Y Zi = − h∇Y X, Zi, (37)

and substituting equation 34 in equation 37, one then has

hσ, ∇τμi = − h∇τσ, μi. (38)

Then replacing the vector σ in equation 38 by the vector ∇στ , equation 31 (see equation

7), one has

h∇ρ∇στ, μi = − h∇στ, ∇ρμi, (39)

and interchanging indices ρ and σ, then

h∇σ∇ρτ, μi = − h∇ρτ, ∇σμi. (40)

Then substituting equations 39 and 40 in equation 35, and interchanging the first and

second terms, one obtains

Rρστ μ = h∇ρτ, ∇σμi − h∇στ, ∇ρμi − 
∇i[ρ,σ]τ, μ. (41)

Also clearly

∇iY Z = i∇Y Z, (42)

so equation 41 can also be written as

Rρστ μ = h∇ρτ, ∇σμi − h∇στ, ∇ρμi − i




∇[ρ,σ]τ, μ. (43)

Next substituting equation 31 in equation 43, one obtains the following useful form for the

Riemann curvature tensor (1):

Rρστ μ = cρ,τ cσ,μ hi[ρ, τ ], i[σ, μ]i

−cσ,τ cρ,μ hi[σ, τ ], i[ρ, μ]i

−c[ρ,σ],τ hi[i[ρ, σ], τ ], μi. (44)

The geodesic equation on the SU(2n) group manifold with the Riemannian metric,

equation 16, is obtained as follows. Consider a curve passing through the origin with

tangent vector Y having components y

σ = dxσ/dt. The covariant derivative along the

curve in the Hamiltonian representation is given by (1, 2)

(DtZ) ≡ (∇Y Z) = dZ

dt +

i

2

([Y, Z] + F ([Y, G(Z)] + [Z, G(Y )])). (45)

(Note that the term dZ

dt in equation 45 does not appear in equation 20 because there the

vector field Z is taken to be right invariant, in which case dZ

dt = 0.). Equation 45 is true on

6
the entire manifold because of the right-invariance of the metric. Furthermore, a geodesic

in the SU(2n) manifold is a curve U(t) with tangent vector H(t) parallel transported along

the curve, namely,

DtH = 0. (46)

However, according to equation 45 with Y = Z = H, one has

DtH =

dH

dt +

i

2

([H, H] + F ([H, G(H)] + [H, G(H)])), (47)

which when substituting equation 46 becomes (1)

dH

dt = −iF ([H, G(H)]). (48)

One can rewrite equation 48 using the dual L of H (1, 2) and equation 18,

L ≡ G(H) = F

−1

(H), (49)

and then noting that

dL

dt =

d

dt



F

−1

(H)



= F

−1



dH

dt 

. (50)

Thus substituting equation 48 in equation 50, one obtains

dL

dt = −iF −1

(F([H, G(H)])), (51)

or

dL

dt = −i[H, G(H)], (52)

and again using equations 50 and 52 it becomes

dL

dt = −i[H, L] = i[L, H]. (53)

Furthermore, again using equation 50 in equation 53, one obtains the sought geodesic

equation (1):

dL

dt = i[L, F(L)]. (54)

This equation is a Lax equation, a well-known nonlinear differential matrix equation, and L

and iF(L) are Lax pairs (30–32). Some solutions to the geodesic equation, equation 54, are

given in references 1 and 7.

7
3. Results

Jacobi fields describe the divergence or convergence of neighboring geodesics and are useful

in determining conjugate points. Conjugate points are points on a geodesic at which the

Jacobi field is vanishing without vanishing in between these points. It is well-known that

past the first conjugate point, a geodesic ceases to be minimizing (15, 18). Jacobi fields are

first to be addressed here for a general Riemannian manifold. Following this, Jacobi fields

will be specialized to the SU(2n) group manifold germane to quantum computation.

Consider a one-parameter family of geodesics on a generic Riemannian manifold,

x

j = xj

(s, t), (55)

in which the parameter s distinguishes a particular geodesic in the family, and t is the

usual curve parameter, which can be taken to be time. (In this section, Latin indices are

used in the description of the Riemannian manifold. Also, the x

j

in equation 55 are not to

be confused with the xσ of sections 1 and 2.) The Riemannian geodesic equation in a

coordinate representation is given by (18)

∂

2xj

∂t2

+ Γj

kl(s)

∂xk

∂t

∂xl

∂t = 0, (56)

in which the Levi-Civita connection is given by

Γ

j

kl(s) = 1

2

g

jm(s)(gkm,l(s) + glm,k(s) − gkl,m(s)), (57)

for metric gij (x(s, t)) ≡ gij (s). (Partial derivatives are used in equation 56 to distinguish

the s from the t dependence.) The geodesic equation, equation 59, on the SU(2n) group

manifold can be shown to also follow from equation 56 (1, 2).

Let x

j

(0, t) be the base geodesic, and define the lifted Jacobi field along the base geodesic

by (1)

J

j

(t) = ∂

∂sx

j

(s, t)|s=0, (58)

describing how the base geodesic changes as the parameter s is varied. Using a Taylor

series expansion, one has for small ∆s in the neighborhood of the base geodesic,

x

j

(∆s, t) = x

j

(0, t) + ∆sJj(t) + O(∆s

2

). (59)

Here x

j

(∆s, t) satisfies the geodesic equation with the metric gij (∆s). Operating on the

geodesic equation, equation 56 with ∂s ≡ ∂

∂s and substituting equations 58 and 59, one

obtains for ∆s → 0,

0 =

∂

2

∂t2

Lim

∆s→0

∆sJj(t)

∆s

+ Γj

kl,m(s)|s=0 Lim

∆s→0

∆sJm(t)

∆s

∂xk

∂t

∂xl

∂t + ∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t

+Γj

kl(0) 

∂

∂t 

Lim

∆s→0

∆sJ k(t)

∆s



∂xl

∂t +

∂xk

∂t

∂

∂t Lim

∆s→0

∆sJl(t)

∆s



, (60)

8
in which gij (0) ≡ gij is the base metric and Γj

kl(0) ≡ Γ

j

kl is the base connection. Equation

60 then becomes

0 =

∂

2Jj

(t)

∂t2

+ Γj

kl,m(s)|s=0J

m(t)

∂xk

∂t

∂xl

∂t

+∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t + Γj

kl 

∂J k

∂t

∂xl

∂t +

∂xk

∂t

∂Jl

∂t 

. (61)

Taking account of dummy indices summed over, it is clearly true that

−Γ

j

lqΓ

q

ik

∂xi

∂t

∂xl

∂t J

k + Γj

kpΓ

p

mn

∂xk

∂t

∂xm

∂t J

n = 0. (62)

One also has

−Γ

j

ik,l

∂xi

∂t

∂xl

∂t J

k + Γj

kp,m

∂xm

∂t

∂xk

∂t J

p = 0. (63)

Also, using the geodesic equation, equation 56, one has

Γ

j

kp

∂

2xk

∂t2

J

p = −Γ

j

kpΓ

k

iq

∂xi

∂t

∂xq

∂t J

p

, (64)

or renaming dummy indices on the right hand side, it follows that

Γ

j

kp

∂

2xk

∂t2

J

p + Γj

qkΓ

q

il

∂xi

∂t

∂xl

∂t J

k = 0. (65)

Next adding equations 61–63 and 65, one obtains

0 =

∂

2Jj

(t)

∂t2

+ Γj

kl,mJ

m(t)

∂xk

∂t

∂xl

∂t

+∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t + Γj

kl 

∂J k

∂t

∂xl

∂t +

∂xk

∂t

∂Jl

∂t 

−Γ

j

lqΓ

q

lk

∂xi

∂t

∂xl

∂t J

k + Γj

kpΓ

p

mn

∂xk

∂t

∂xm

∂t J

n

−Γ

j

ik,l

∂xi

∂t

∂xl

∂t J

k + Γj

kp,m

∂xm

∂t

∂xk

∂t J

p + Γj

kp

∂

2xk

∂t2

J

p + Γj

qkΓ

q

il

∂xi

∂t

∂xl

∂t J

k

, (66)

or equivalently,

∂

2Jj

(t)

∂t2

= − Γ

j

kl,m

∂xk

∂t

∂xl

∂t J

m + Γj

lqΓ

q

ik

∂xi

∂t

∂xl

∂t J

k

−Γ

j

kpΓ

p

mn

∂xk

∂t

∂xm

∂t J

n − Γ

j

qkΓ

q

il

∂xi

∂t

∂xl

∂t J

k + Γj

ik,l

∂xi

∂t

∂xl

∂t J

k − Γ

j

kp

∂

2xk

∂t2

J

p

−Γ

j

kl 

∂J k

∂t

∂xl

∂t +

∂xk

∂t

∂Jl

∂t 

− ∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t − Γ

j

kp,m

∂xm

∂t

∂xk

∂t J

p

.(67)

9
Rearranging terms in equation 67, then

∂

2Jj

(t)

∂t2

= Γj

ik,l

∂xi

∂t

∂xl

∂t J

k − Γ

j

kl,m

∂xk

∂t

∂xl

∂t J

m + Γj

lqΓ

q

ik

∂xi

∂t

∂xl

∂t J

k

−Γ

j

kpΓ

p

mn

∂xk

∂t

∂xm

∂t J

n − Γ

j

kp,m

∂xm

∂t

∂xk

∂t J

p − Γ

j

kp

∂

2xk

∂t2

J

p

−Γ

j

kl

∂xk

∂t

∂Jl

∂t − Γ

j

kl

∂xl

∂t

∂J k

∂t

−Γ

j

qkΓ

q

il

∂xi

∂t

∂xl

∂t J

k − ∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t . (68)

Recalling that the Levi-Civita connection is symmetric, one has

Γ

j

qp = Γjpq, (69)

and renaming dummy indices, equation 68 becomes

∂

2Jj

∂t2

=



Γ

j

ik,l − Γ

j

il,k + ΓjlqΓ

q

ik − Γ

j

kpΓ

p

li ∂xi

∂t

∂xl

∂t J

k

−Γ

j

kp,m

∂xm

∂t

∂xk

∂t J

p − Γ

j

kp

∂

2xk

∂t2

J

p − Γ

j

kl

∂xk

∂t

∂Jl

∂t

−Γ

j

pk

∂xk

∂t 

∂Jp

∂t + Γp

mn

∂xm

∂t J

n



− ∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t . (70)

Using the well-known expression for the covariant derivative (22, 29), it follows that

D2J

j

Dt2

=

∂

∂t 

DJj

Dt 

+ Γj

kp

∂xk

∂t

DJp

Dt

=

∂

∂t 

∂Jj

∂t + Γj

kp

∂xk

∂t J

p



+ Γj

kp

∂xk

∂t

DJp

Dt , (71)

or

D2J

j

Dt2

=

∂

2Jj

∂t2

+ Γj

kp,m

∂xm

∂t

∂xk

∂t J

p + Γj

kp

∂

2xk

∂t2

J

p + Γj

kp

∂xk

∂t

∂Jp

∂t

+Γj

kp

∂xk

∂t 

∂Jp

∂t + Γp

mn

∂xm

∂t J

n



. (72)

Next the well-known Riemann curvature tensor is given by (29)

R

j

ikl = Γjil,k − Γ

j

ik,l + ΓjkpΓ

p

li − Γ

j

lqΓ

q

ik. (73)

Substituting equations 79 and 73 in equation 72, one obtains the so-called lifted Jacobi

equation (1):

D2J

j

Dt2

+ R

j

ikl

∂xi

∂t

∂xl

∂t J

k + ∂sΓ

j

kl(s)|s=0

∂xk

∂t

∂xl

∂t = 0. (74)

10
This equation is useful for investigations of the global behavior of geodesics and their

extrapolation to values of the parameter s characterizing neighboring geodesics on the

Riemannian manifold (1).

If gij is independent of s, one has

∂sΓ

j

kl(s)|s=0 = 0; (75)

the last term of equation 74 is then vanishing, and one obtains the standard Jacobi

equation for the Jacobi vector J

j

(18),

D2J

j

Dt2

+ R

j

ikl

∂xi

∂t

∂xl

∂t J

k = 0. (76)

Equation 76 is also known as the equation of geodesic deviation (33, 29), measuring the

local convergence or divergence of neighboring geodesics, and it is useful in the

determination of possible geodesic conjugate points (18, 1). (Again it is well to recall that

conjugate points are points on a geodesic at which the Jacobi field is vanishing without

vanishing in between those points. It is well-known that past the first conjugate point, a

geodesic ceases to be minimizing (15, 18).

Next consider the factor in the last term of the lifted Jacobi equation, equation 74,

L

j

kl ≡ ∂sΓ

j

kl(s)|s=0. (77)

Substituting equation 57 in equation 77, one has

L

j

kl ≡



∂s



1

2

g

jm(s)(gkm,l(s) + glm,k(s) − gkl,m(s)



|s=0

, (78)

or equivalently,

L

j

kl ≡

∂gjm(s)

∂s |s=0

Γmkl +

1

2

g

jm(g0

km,l + g

0

lm,k − g

0

kl,m), (79)

in which one defines

g

0

km ≡ ∂sgkm(s)|s=0. (80)

Using the well-known expression for the covariant derivative of a second rank tensor (29),

one has

g

0

km;l = g

0

km,l − g

0

kiΓ

i

ml − g

0

miΓ

i

kl. (81)

Then substituting equation 81 in equation 79, one obtains

L

j

kl ≡

∂gjm(s)

∂s |s=0

Γmkl +

1

2

g

jm(g0

km;l + g

0

kiΓ

i

ml + g

0

miΓ

i

kl

+g

0

lm;k + g

0

liΓ

i

mk + g

0

miΓ

i

kl

−g

0

kl;m − g

0

kiΓ

i

lm − g

0

liΓ

i

km), (82)

11
and using equation 69, then

L

j

kl ≡

1

2

g

jm(g0

km;l + g

0

lm;k − g

0

kl;m)

+

∂gjm(s)

∂s |s=0

Γmkl + g

jmg0

miΓ

i

kl. (83)

Next noting that

(g

jmgmi)0 = (δ

j

i

)

0 = 0, (84)

then

g

jm(0) 

∂

∂sgmi(s)



|s=0

= −



∂gjm(s)

∂s 

|s=0

gmi(0). (85)

Multiplying both sides of equation 85 by Γi

kl, and using equation 80, one obtains

g

jmg0

miΓ

i

kl = −



∂gjm(s)

∂s 

|s=0

Γmkl, (86)

so that equation 83 reduces to

L

j

kl ≡

1

2

g

jm(g0

km;l + g

0

lm;k − g

0

kl;m). (87)

Finally then combining equations 74, 77, and 87, one obtains

D2J

j

Dt2

+ R

j

ikl

∂xi

∂t

∂xl

∂t J

k +

1

2

g

jm(g0

km;l + g

0

lm;k − g

0

kl;m)

∂xk

∂t

∂xl

∂t . (88)

Next define the vector field,

C

j ≡

1

2

g

jm(g0

km;l + g

0

lm;k − g

0

kl;m)

∂xk

∂t

∂xl

∂t , (89)

which is independent of the Jacobi field J

j

. Equivalently, by symmetry, equation 89 can

also be written as

C

j ≡

1

2

g

jm(2g0

km;l − g

0

kl;m)

∂xk

∂t

∂xl

∂t . (90)

Substituting equation 81 in equation 88, one obtains the second-order differential equation,

D2J

j

Dt2

+ R

j

ikl

∂xi

∂t

∂xl

∂t J

k + Cj = 0, (91)

the so-called “lifted Jacobi equation” (1). Nielsen and Dowling used the lifted Jacobi

equation, equation 91, adapted to the SU(2n) group manifold, to deform geodesics by

varying the penalty parameter q (see section 4). This enabled them to define a so-called

“geodesic derivative” and to numerically deform a geodesic as the penalty parameter q is

varied without changing the fixed values U = 1 and U = Uf of the initial and final unitary

transformation corresponding to a quantum computation (1).

12
The generic lifted Jacobi equation, equation 91, can be solved. One first rewrites equation

72 as

D2J

j

Dt2

=

∂

2Jj

∂t2

+ 2Γj

kp

∂xk

∂t

∂Jp

∂t + Γj

kp,m

∂xm

∂t

∂xk

∂t J

p + Γj

kp

∂

2xk

∂t2

J

p

+Γj

kp

∂xk

∂t Γ

p

mn

∂xm

∂t J

n

, (92)

and renaming dummy indices in the last term, then

D2J

j

Dt2

=

∂

2Jj

∂t2

+



2Γj

kp

∂xk

∂t 

∂Jp

∂t

+



Γ

j

kp,m

∂xm

∂t

∂xk

∂t + Γj

kp

∂

2xk

∂t2

+ Γj

kq

∂xk

∂t Γ

q

mp

∂xm

∂t 

J

p

, (93)

or equivalently

D2J

j

Dt2

=

∂

2Jj

∂t2

+ A

j

p

∂Jp

∂t +

 X

3

n=1

(n)Bj

p

!

J

p

, (94)

where

A

j

p ≡



2Γj

kp

∂xk

∂t 

, (95)

(1)Bj

p ≡ Γ

j

kp,m

∂xm

∂t

∂xk

∂t , (96)

(2)Bj

p ≡ Γ

j

kp

∂

2xk

∂t2

, (97)

and

(3)Bj

p ≡ Γ

j

kq

∂xk

∂t Γ

q

mp

∂xm

∂t . (98)

Next equation 91 can written as

D2J

j

Dt2

+

(4)Bj

pJ

p + Cj = 0, (99)

where

(4)Bj

p ≡ R

j

ipl

∂xi

∂t

∂xl

∂t . (100)

Next substituting equation 99 in equation 94, one obtains

∂

2Jj

∂t2

+ A

j

p

∂Jp

∂t + B

j

pJ

p + Cj = 0, (101)

where

B

j

p =

X

4

n=1

(n)Bj

p

. (102)

Next define the column vectors

J ≡



J

j



, (103)

13
C ≡



C

j



, (104)

and the matrices

A ≡



A

j

p



, (105)

B ≡



B

j

p



=

"X

4

n=1

(n)Bj

p

.

#

. (106)

Equation 101 then becomes

∂

2J

∂t2

+ A

∂J

∂t + BJ + C = 0. (107)

Furthermore, defining the column vector

K ≡



J1

J2



≡



J

∂J

∂t



, (108)

then equation 107 is equivalent to

∂K

∂t

≡



0 I

−B −A



K −



0

C



. (109)

The homogeneous part of equation 109 with C = 0 is equivalent to the Jacobi equation,

equation 76, and is given by

∂K0

∂t

≡ MK0, (110)

in which the matrix M is given by

M ≡



0 I

−B(t) −A(t)



, (111)

and the time dependence of A and B is indicated explicitly. The solution to the Jacobi

equation, equation 110, is given in terms of the time-ordered exponential (34, 25), namely,

K0(t) =



I +

X∞

n=1

1

n!

Z

t

0

dt1.. Z

t

0

dtnT(M(t1)...M(tn))



 K0(0), (112)

where T denotes the time ordering operator (not to be confused with the transpose of a

matrix, appearing below). Thus, equation 112 gives the Jacobi field and can be expressed

formally as

K0(t) = T exp





Z

t

0

dt0M(t

0

)



 K0(0), (113)

or defining the operator

Et ≡ T exp





Z

t

0

dt0M(t

0

)



 = I +

X∞

n=1

1

n!

Z

t

0

dt1.. Z

t

0

dtnT(M(t1)...M(tn)). (114)

14
Equation 113 can also be written as

K0(t) = EtK0(0). (115)

It follows from equation 114 that

∂Et

∂t = M(t) +X∞

n=2

1

n!

n

Z

t

0

dt1.. Z

t

0

dtn−1T(M(t1)...M(tn−1)M(t))

= M(t) + M(t)

X∞

n=2

1

(n − 1)! Z

t

0

dt1.. Z

t

0

dtn−1T(M(t1)...M(tn−1))

= M(t)



I +

X∞

n=1

1

n!

Z

t

0

dt1.. Z

t

0

dtnT(M(t1)...M(tn))



 , (116)

or equivalently then substituting equation 114, one obtains

∂Et

∂t = M(t)Et

. (117)

The solution to the inhomogeneous equation, equation 109 is given by

K(t) = EtK(0) − Et

Z

t

0

drE−1

r



0

C(r)



. (118)

This is the lifted Jacobi field. To see that equation 118 solves the inhomogeneous equation,

equation 109, one notes that using equations 118 and 117 one has

∂K(t)

∂t =

∂Et

∂t K(0) −

∂Et

∂t Z

t

0

drE−1

r



0

C(r)



− EtE

−1

t



0

C(t)



= M(t)EtK(0) − M(t)Et

Z

t

0

drE−1

r



0

C(r)



−



0

C(t)



. (119)

Next substituting equations 115, 118, and 111 in equation 119, then

∂K(t)

∂t = M(t)EtK(0) + M(t)K(t) − M(t)EtK(0) −



0

C(t)



=



0 I

−B(t) −A(t)



K(t) −



0

C(t)



, (120)

and thus equation 109 is, in fact, satisfied by equation 118.

The manifold of interest in the present work is the SU(2n) group manifold. For this case,

consider a base geodesic with coordinates γ

σ

(q, t) on the SU(2n) group manifold with

15
penalty parameter q, and a neighboring geodesic with coordinates γ

σ

(q + ∆, t) with penalty

parameter q + ∆. To first order in ∆, one has

γ

σ

(q + ∆, t) = γ

σ

(q, t) + ∆J

σ

(t), (121)

in which the Jacobi field coordinates J

σ

(t) are defined by

J

σ

(t) = ∂γσ

(q, t)

∂q . (122)

The Hamiltonian for a geodesic γ(t) with penalty parameter q is given by

Hq =

dγ

dt . (123)

Then one has

dHq

dq =

d

dq

dγ

dt , (124)

or equivalently,

dHq

dq =

d

dt

dγ

dq , (125)

and substituting equation 122 in equation 125, then

dHq

dq =

dJ

dt , (126)

in which the Jacobi field J is

J = J

σ

σ. (127)

The geodesic equation for the base geodesic with penalty parameter q is given by equation

53, namely,

dL

dt = i[L, H], (128)

where the dual L is given by

L = G(H). (129)

The geodesic equation for the nearby geodesic with penalty parameter q + ∆ is

dL

dt = i[L, H], (130)

where

L = G(H), (131)

and in accord with equation 17,

G = P + (q + ∆)Q = G + ∆dG

dq

≡ G + ∆G

0

, (132)

16
in which

G

0 ≡

dG

dq = Q. (133)

Next letting U(t) and U(t) denote the geodesics for penalty parameter q and q + ∆,

respectively, then for small ∆ one expects

U = Ue−i∆J, (134)

and it follows that to first order in ∆,

dU

dt =

dU

dt e

−i∆J + U

d

dt



1 − i∆J + 0(∆2)



=

dU

dt e

−i∆J + U(−i∆

dJ

dt + 0(∆2

))

=

dU

dt e

−i∆J − iU∆

dJ

dt . (135)

But according to the Schr ̋odinger equation, one has

dU

dt = −iHU (136)

and

dU

dt = −iHU, (137)

so substituting equations 136 and 137 in equation 135, one obtains

−iHU = −iHUe−i∆J − iU∆

dJ

dt , (138)

or substituting equation 134, then equation 138 for small ∆ becomes

−iHUe−i∆J = −iHUe−i∆J − iU∆

dJ

dt

, (139)

or equivalently, then to order ∆,

−iHU = −iHU − iU∆

dJ

dt e

i∆J = −iHU − iU∆

dJ

dt . (140)

Next multiplying equation 140 on the right by U

† and noting that unitarity requires

UU† = 1, (141)

then equation 140 becomes

H = H + ∆U

dJ

dt U

†

(142)

to first order in ∆. Next substituting equation 131 in equation 130, one obtains

d

dt(G(H)) = i[G(H, H]. (143)

17
Then substituting equations 132 and 142 in the left side of equation 143, the left side

becomes

d

dt(G(H)) = ddt((G + ∆G

0

)(H + ∆K)), (144)

where

K ≡ U

dJ

dt U

†

. (145)

Equivalently, equation 144 to first order in ∆ is

d

dt(G(H)) = ddtG(H) + ∆ ddt(G

0

(H) + G(K)). (146)

Next, using equations 132, 142, and 1.45, the right side of equation 143 becomes

i[G(H, H] = i[(G + ∆G

0

)(H + ∆K), H + ∆K], (147)

or equivalently to first order in ∆,

i[G(H, H] = i[G(H), H] + ∆(i[G(H), K] + i[G

0

(H), H] + i[G(K), H]). (148)

In terms of the dual, equation 129, equation 148 becomes

i[G(H, H] = i[L, H] + ∆(i[L, K] + i[G

0

(H), H] + i[G(K), H]). (149)

Next substituting equations 146, 129, and 149 in equation 143, one obtains

d

dtL + ∆(G

0

(

d

dtH) + G(

d

dtK)) = i[L, H] + ∆(i[L, K] + i[G

0

(H), H] + i[G(K), H]), (150)

and further substituting equation 128 in equation 150, one concludes that

G

0

(

d

dtH) + G(

d

dtK)) = i[L, K] + i[G

0

(H), H] + i[G(K), H]. (151)

Furthermore, multiplying equation 151 on the left by G−1, one obtains

G

−1G0

(

d

dtH) + ddtK = G

−1

(i[L, K] + i[G

0

(H), H] + i[G(K), H]). (152)

But according to equations 128, 129, and 18,

d

dtH = iG−1

[L, H] = iF([L, H]), (153)

so that, using equation 18, one has

G

−1G0

(

d

dtH) = F G0

(

d

dtH) = iF G0F([L, H]). (154)

Then substituting equations 154 and 18 in equation 152, one obtains

0 = iF G0F([L, H]) + d

dtK + F(i[K, L] + i[H, G0

(H)] + i[H, G(K)]), (155)

18
or

0 =

d

dtK + F(i[K, L] + i[H, G(K)] + G

0F(i[L, H]) + i[H, G0

(H)]). (156)

Equation 156 is the lifted Jacobi equation for penalty parameter varied from q to q + ∆

(1). It is an inhomogeneous first order differential equation in K.

If G0 = 0, equation 156 reduces effectively to the conventional Jacobi equation, assuming

the form,

0 =

d

dtK + F(i[K, L] + i[H, G(K)]). (157)

Equation 157 can be rewritten as follows using equations 2, 17, and 18:

d

dtK = −iF([P(K) + Q(K), P(H) + qQ(H)] + [P(H) + Q(H), P(K) + qQ(K)]). (158)

Expanding the commutators, then

d

dt

K = −iF([P(K), P(H)] + q[P(K), Q(H)]

+[Q(K), P(H)] + q[Q(K), Q(H)]

+[P(H), P(K)] + q[P(H), Q(K)]

+[Q(H), P(K)] + q[Q(H), Q(K)]). (159)

The first and fifth terms cancel, and also the forth and eighth terms cancel, so one obtains

d

dt

K = −i(q − 1)F([P(K), Q(H)] − [Q(K), P(H)]), (160)

or equivalently,

d

dt

K = −i(q − 1)F([P(K), Q(H)] + [P(H), Q(K)]), (161)

or

d

dtK = i(q − 1)F([Q(H), P(K)] − [P(H), Q(K)]). (162)

Solving equation 162 for K yields the conventional Jacobi field.

The inhomogeneous term in equation 156 is given by

C = F(G

0F(i[L, H]) + i[H, G0

(H)]). (163)

Substituting equations 133, 49, 17, and 2 in equation 163, one has

C = F QF i[(P + qQ)(H), P(H) + Q(H)] + F i[H, Q(H)]

= F QF i([P(H), Q(H)] + q[Q(H), P(H)]) + F i([P(H) + Q(H), Q(H)])

= F QF i(1 − q)[P(H), Q(H)] + F i[P(H), Q(H)]. (164)

19
Equation 164 can be rewritten as follows:

C = F(P + Q)F i(1 − q)[P(H), Q(H)] + F i[P(H), Q(H)]

−F P F i(1 − q)[P(H), Q(H)]. (165)

But using equations 2 and 18 one has

P F = P(P +

1

q

Q) = P

2 = P, (166)

and using equations 2, 17, and 18, then equation 165 becomes

C = F

2

((1 − q + F

−1

)i[P(H), Q(H)] − F

−1P i(1 − q)[P(H), Q(H)])

= F

2

(1 − q + P + qQ − P(1 − q))i[P(H), Q(H)]

= F

2

(1 − q + q(P + Q))i[P(H), Q(H)], (167)

or using equation 2, then

C = F

2

i[P(H), Q(H)]. (168)

This is a useful form for the inhomogeneous term in the lifted Jacobi equation, equation

156 (1).

Next combining equations 156, 163, and 168, the lifted Jacobi equation for varying penalty

parameter q is then given by

d

dtK = i(q − 1)F([Q(H), P(K)] − [P(H), Q(K)]) − F

2

i[P(H), Q(H)]. (169)

In terms of the solution for K(t), the lifted Jacobi field for varying penalty parameter can

first be written as

J(t) = J(0) + Z

t

0

dt0

dJ(t

0

)

dt0

. (170)

But according to equations 145 and 141, one has

dJ(t)

dt = U

†

(t)K(t)U(t), (171)

and substituting equation 171 in equation 170, one obtains

J(t) = J(0) + Z

t

0

dt0U

†

(t

0

)K(t

0

)U(t

0

). (172)

Next consider the case in which the Hamiltonian is constant along a geodesic. The geodesic

equation 52 then implies

[G(H), H] = 0. (173)

20
Also, using equations 54, 18, 2, 49, and 133, one has

0 = dH

dt

= G

−1

dL

dt

= iF[L, F(L)] = iF[L, P(L) + q

−1Q(L)]

= iF[L, P(L) + q

−1

(1 − P)(L))]

= i(1 − q

−1

)F[L, P(L)]

= i(1 − q

−1

)F[P(H) + qQ(H), P(P(H) + qQ(H)]

= i(1 − q

−1

)F[P(H) + qQ(H), P(H)]

= i(q − 1)F[Q(H), P(H)] = i(1 − q)F[P(H), Q(H)]

= i(1 − q)F[H, Q(H)]

= i(1 − q)F[H, G0(H)] = 0. (174)

It then follows from equations 156, 173, and 174 that if the Hamiltonian is constant, then

again one obtains the conventional Jacobi equation, equation 157. Thus if the Hamiltonian

is constant, and J(0) = 0 and dJ(0)/dt = 0, then in accordance with equation 206 below,

J(t) is proportional to dJ(0)/dt and therefore J(t) = 0. In this case, it then follows that

the geodesics for the lifted Jacobi equation for varying penalty parameter remain the same

as for the conventional Jacobi equation and are the same for all values of the penalty

parameter q.

The so-called geodesic derivative can be used to determine geodesics that evolve from the

identity to a chosen unitary transformation U (1). In quantum computation, one generally

wishes the quantum computation to evolve to some final unitary transformataion which

solves a given problem. One first chooses a Hamiltonian H(0), which produces

U = exp(−iH(0)T at some fixed time T along the geodesic for penalty parameter q = 1.

The parameter q can next be varied to produce a corresponding change in the initial

Hamiltonian, and this produces the so-called geodesic derivative dHq (0)/dq. Integration

then may produce a geodesic connecting the identity U(0) = I and the chosen unitary

transformation U(T ) for any penalty parameter q.

To proceed then, the general lifted Jacobi equation, equation 156, for varied penalty

parameter can be solved. (It is convenient to solve equation 156 directly, instead of

equation 169.) First substituting equation 103 in equation 156, one has

d

dt

K = −i F([K, L] + [H, G(K)]) − C. (175)

The corresponding homogeneous equation is then

d

dtKs = −i F([Ks, L] + [H, G(Ks)]), (176)

21
and it can be solved if it is first recast in vectorized form (36, 37). For any matrix

M =









a11 a12 ... a1n

a21 a21 ... a2n

.

.

.

. ...

.

.

am1 am2 ... amn









, (177)

one defines the vectorized form of the matrix M by the column vector,

vec M = [a11..am1, a12..am2...a1n...amn]

T

, (178)

with each column of the matrix M appearing beneath the previous one, arranged in a

column vector. If one has a matrix equation

C = AX + XB (179)

for matrices A, B, C, and X, then it can be shown that (36)

vec C =



(I ⊗ A) + (B

T ⊗ I)



vec X. (180)

It then follows that the homogeneous equation 176 can be written in vectorized form as

follows:

d

dt

(vec Ks) = −iFvec [KsL − LKs + HG(Ks) − G(Ks)H]

= iFvec [(LKs + Ks(−L)) − (HG(Ks) + G(Ks)(−H))]

= iF[(I ⊗ L − L

T ⊗ I)vec Ks − (I ⊗ H − HT ⊗ I)vec G(Ks)]

= iF[(I ⊗ L − L

T ⊗ I) + (HT ⊗ I − I ⊗ H)G]vec (Ks)]

= iAvec Ks, (181)

where

A = F (I ⊗ L − L

T ⊗ I) + (HT ⊗ I − I ⊗ H)G

, (182)

and F and G are the vectorized forms of the superoperators F and G, respectively (1). For

example, the superoperator F acting on the matrix X can clearly be written as

F(X) = X

j

AjXBj , (183)

for some matrices Aj and Bj . But one has (36)

vec AjXBj = (B

T

j ⊗ Aj )vec X, (184)

and using equation 184 in equation 183, then

vec F(X) = X

j

(B

T

j ⊗ Aj )vec X, (185)

22
and therefore the vectorized form F of the superoperator F is given by

F = X

j

(B

T

j ⊗ Aj )vec. (186)

Evidently the solution to equation 181 is

vec Ks(t) = T



exp



i

Z

t

0

A(t

0

)dt0







 vec Ks(0), (187)

or

vecKs(t) = κtvec Ks(0), (188)

where

κt = T



exp



i

Z

t

0

A(t

0

)dt0







 . (189)

It follows from equation 189 that

d

dtκt = iA(t)κt

. (190)

Here κtis the propagator for the homogeneous form of equation 175, namely, equation 176.

The solution to equation 175 is then given by

K(t) = unvec



κtvec K(0) − κt

Z

t

0

drκ−1

r vec C(r)



 , (191)

in which unvec unvectorizes (1), namely, for a matrix M,

unvec (vec M) = M. (192)

To see that equation 191 solves equation 175, one has

d

dtK(t) = unvec





d

dtκtvec K(0) − (

d

dtκt)

Z

t

0

drκ−1

r vec C(r) − κtκ

−1

t vec C(t)



 . (193)

Substituting equation 130 in equation 193, then

d

dtK(t) = unvec



iA(t)κtvec K(0) − iA(t)κt

Z

t

0

drκ−1

t vec C(r) − vec C(t)



 , (194)

and substituting equation 193 in equation 194, then

d

dtK(t) = unvec (iA(t)κtvec K(0) + iA(t)(vec K(t) − κtvec K(0)) − vec C(t)), (195)

23
or

d

dtK(t) = unvec (iA(t)vec K(t) − vec C(t)). (196)

Substituting equation 182 in equation 196, then

d

dtK(t) = unvec 

iF



(I ⊗ L − L

T ⊗ I) + (HT ⊗ I − I ⊗ H)G

vecK(t)



− C(t). (197)

But according to equations 176, 181, and 182,

unvec iF



(I ⊗ L − L

T ⊗ I) + (HT ⊗ I − I ⊗ H)G

vecK(t)



= −i F([K, L] + [H, G(K)],

(198)

and then substituting equation 198 in equation 197 one obtains equation 175

d

dt

K = −i F([K, L] + [H, G(K)] − C, (199)

as required.

Next, in order to obtain the propagator of the standard (unlifted) Jacobi field Js, using

equation 184, then equation 171 in vectorized form is

vec ( d

dtJs) = (U

T ⊗ U†

)vec Ks. (200)

Substituting equation 188 in equation 200, then

vec ( d

dt

Js) = (U

T ⊗ U†

)κtvec Ks(0). (201)

Next substituting equation 171 in equation 201,

vec ( d

dtJs) = (U

T ⊗ U†

)κtvec (U(0) d

dtJs(0)U(0)†

), (202)

then for U(0) = I, one has

vec ( d

dtJs) = (U

T ⊗ U†

)κtvec

d

dtJs(0)). (203)

Unvectorizing equation 203, then

d

dtJs = unvec [(U

T ⊗ U†

)κt(vec d

dtJs(0))]. (204)

But assuming Js(0) = 0, one has

Js(t) = Z

t

0

dt0

d

dtJs(t

0

), (205)

24
and substituting equation 204 in equation 205, then

Js(t) = Z

t

0

dt0unvec [(U

T ⊗ U†

)κt

0(vec d

dt0

Js(0))]. (206)

Next defining the propagator jT that generates the standard unlifted Jacobi field at time T

by

Js(T ) = jT (

d

dt0

Js(0)), (207)

then according to equation 206, one has

jT =

Z

T

0

dt0unvec (U

T ⊗ U†

)κt

0vec. (208)

It follows from equations 191, 171, 205 and the homogeneous term having the same form as

equation 207 that at time T the solution to the lifted Jacobi equation for varying penalty

parameter is given by

J(T ) = jT (

d

dtJ(0)) −

Z

T

0

dtU(t)

†



unvec κt





Z

t

0

drκ−1

r vec C(r)







 U(t). (209)

If J(T ) = 0 in equation 209, then

d

dtJ(0) = j

−1

T





Z

T

0

dtU(t)

†



unvec κt





Z

t

0

drκ−1

r vec C(r)







 U(t)



 . (210)

Next, according to equation 126, the Hamiltonian Hq for penalty parameter q is such that

d

dqHq (0) = ddtJ(0), (211)

and substituting equation 210 in equation 211, one obtains the so-called geodesic derivative

(1)

d

dq Hq(0) = j

−1

T





Z

T

0

dtU(t)

†

unvec



 κt





Z

t

0

drκ−1

r vec C(r)







 U(t)



 . (212)

For q = 1, one has, according to equations 2, 17, and 18,

F = P +

1

q

Q = P + Q = G = 1, q = 1, (213)

and then according to equation 48 the Hamiltonian is constant,

dH

dt = 0, q = 1. (214)

25
Then equation 168 becomes

C = i[P(H), Q(H)], q = 1, (215)

and then because of equation 214,

dC

dt = 0, q = 1. (216)

Therefore in equation 212 for q = 1 one has using equations 68 and 213,

Z ≡ κt





Z

t

0

drκ−1

r vec C(r)



 = κt





Z

t

0

drκ−1

r

ivec [P(H), Q(H)]



 , (217)

and therefore

dZ

dt

≡





dκt

dt





Z

t

0

drκ−1

r



 + 1



 ivec [P(H), Q(H)]. (218)

But for q = 1 in equation 182, one has, according to equations 213, 17, and 49, F = 1,

G = 1, and L = H, and therefore

A = [I ⊗ H − HT ⊗ I + HT ⊗ I − I ⊗ H] = 0, q = 1. (219)

Then substituting equation 219 in equation 190, one obtains

dκt

dt

= 0. (220)

Also, according to equation 189 for t = 0, one has

κt = 1. (221)

Then substituting equation 220 in equation 218, one obtains

dZ

dt

≡ ivec [P(H), Q(H)]. (222)

Also, according to equations 217 and 221,

Z(0) = 0. (223)

Then combining equations 222 and 223, one obtains for q = 1,

Z = itvec [P(H), Q(H)], q = 1, (224)

or using equations 217 and 224, then

κt





Z

t

0

drκ−1

r vec C(r)



 = itvec [P(H), Q(H)], q = 1. (225)

26
Substituting equation 225 in equation 212, then for q = 1, one has

d

dqHq (0) = j

−1

T





Z

T

0

dtU(t)

†

it[P(H), Q(H)]U(t)



, q = 1. (226)

For q 6= 1, one has, substituting equation 168 in equation 212,

d

dq Hq(0) = j

−1

T





Z

T

0

dtU(t)

†

unvec



κt





Z

t

0

drκ−1

r vec F

2

i[P(H), Q(H)]







 U(t)



 , q 6= 1.

(227)

Equations 226 and 227 give the so-called geodesic derivative (1) which is useful in the

numerical determination of optimal geodesics representing a particular quantum

computation.

4. Other Work

1. Solutions to the geodesic equation were addressed, describing possible minimum

complexity paths in the SU(2n) group manifold and representing the unitary evolution

associated with a quantum computation (7).

2. A known obstruction to numerically solving the geodesic equation is the so-called

Razborov-Rudich theorem (1). It is possible that the obstruction, if it occurs, can be

circumvented by introducing and tweaking an additional parameter (other than q) in the

Hamiltonian. This parameter could be one multiplying only two-body terms.

5. Conclusions

In this work on the Riemannian geometry of quantum computation, the Riemann

curvature and geodesic equation were reviewed, and the Jacobi equation and the lifted

Jacobi equation on the manifold of the SU(2n) group of n-qubit unitary operators with

unit determinant were explicitly derived using the Lie algebra su(2n). The Riemann

curvature is given by equation 44. The geodesic equation is given by equation 54. The

generic Jacobi equation and its solution are given by equations 76 and 112, respectively.

The generic lifted Jacobi equation is given by equation 91, and the solution is given by

equation 118. The lifted Jacobi equation on the SU(2n) manifold, for a varying penalty

parameter, is given by equations 156 or 169, respectively, and the solution is given by

equations 191 and 168. Also, the geodesic derivative is given by equation 212. These

equations are germane to investigations of the global characteristics of geodesic paths (15,

18) and minimal complexity quantum circuits (1, 28, 2).

27
6. References

1. Dowling, M. R.; Nielsen, M. A. The Geometry of Quantum Computation. Quantum

Information and Computation 2008, 8, 0861–0899.

2. Brandt, H. E. Riemannian Geometry of Quantum Computation, to appear in

Nonlinear Analysis 2008, vol 71, pp 3474–3486, 2009; [Online early access

DOI: 10.1016/j.na.2008.11.013. Published Online: Nov 11, 2008].

3. Brandt, H. E. Riemannian Curvature in the Differential Geometry of Quantum

Computation, to appear in Physica E (2009); [Online early access DOI: 10.1016/

j.physe.2009.06.016. Published Online: Jan 24, 2009].

4. Brandt, H. E. Riemannian Geometry of Quantum Computation, AMS Short Course

Lecture, to appear in Proc. Symposia in Applied Mathematics., American

Mathematical Society, 2009.

5. Brandt, H. E. Quantum Computational Curvature and Jacobi Fields. submitted to

Contemporary Mathematics 2009,

6. Brandt, H. E. Differential Geometry of Quantum Computation. J. Mod. Optics 2008

55, 3403–3412.

7. Brandt, H. E. Quantum Computational Geodesics. J. Mod. Optics 2009, 56, 2112–2117.

8. Brandt, H. E. Aspects of Quantum Computational Geodesics. in Quantum Information

and Computation VII 2009, SPIE Vol. 7342, 734205-1-11.

9. Brandt, H. E. Jacobi Fields in the Riemannian Geometry of Quantum Computation.

to appear in Quantum Information and Computation VIII 2010, SPIE.

10. Brandt, H. E. Lifted Jacobi Equation for Varying Penalty Parameter in the

Riemannian Geometry of Quantum Computation. to appear in Quantum Information

and Computation VIII 2010, SPIE.

11. Brandt, H. E. Quantum Computational Jacobi Fields. submitted to Acta Mathematica

Academiae Paedagogicae Nyiregyhaziensis, 2009.

12. Milnor, J. Curvatures of Left Invariant Metrics on Lie Groups. Advances in Mathematics

1976 it 21, 293–329.

13. Arnold, V. I. Mathematical Methods of Classical Mechanics; 2nd Edition, Springer-Verlag,

New York, 1989.

28
14. Arnold, V. I.; Khesin, B. A. Topological Methods in Hydrodynamics; Springer,

New York, 1999.

15. Berger, M. A Panoramic View of Riemannian Geometry; Springer-Verlag, Berlin,

2003.

16. Sagle, A. A.; Walde, R. E. Introduction to Lie Groups and Lie Algebras; Academic

Press, New York, 1973.

17. Conlon, L. Differentiable Manifolds; 2nd Edition, Birkh ̈auser, Berlin, 2001).

18. Lee, J. M. Riemannian Manifolds: An Introduction to Curvature; Springer,

New York, 1997.

19. Milnor, J. Morse Theory; Princeton University Press, Princeton, NJ, 1973.

20. Hall, B. C. Lie Groups, Lie Algebras, and Representations; Springer, New York,

2004.

21. Postnikov, M. M. Geometry VI: Riemannian Geometry, Encyclopedia of

Mathematical Sciences; vol. 91, Springer-Verlag, Berlin, 2001.

22. Petersen, P. Riemannian Geometry; 2nd Edition, Springer, New York, 2006.

23. Naimark, M. A.; Stern, A. I. Theory of Group Representations; Springer-Verlag,

New York, 1982.

24. Sepanski, Mark R. Compact Lie Groups; Springer, 2007.

25. Pfeifer, Walter The Lie Algebras su(N), Birkh ̈auser, Basel, 2003.

26. Cornwell, J. F. Group Theory in Physics: An Introduction; Academic Press,

San Diego, CA, 1997.

27. Cornwell, J. F. Group Theory in Physics; Vols. I and II, Academic Press, London,

1984.

28. Nielsen, M. A.; Chuang, I. L. Quantum Information and Computation; Cambridge

University Press, 2000.

29. Misner, C. W.; Thorne, K. S.; Wheeler, J. A. Gravitation; pp. 224, 310, W. H.

Freeman and Company, New York, 1973.

30. Lax, P. D. Integrals of Nonlinear Equations of Evolution and Solitary Waves.

Communications on Pure and Applied Math. 1968, 21, 467–490.

29
31. Zeidler, E. Nonlinear Functional Analysis and its Applications IV: Applications to

Mathematical Physics; Springer-Verlag, New York, NY, 1997.

32. Zwillinger, D. Handbook of Differential Equations; Third Edition, Academic Press,

San Diego, CA, 1998.

33. Wasserman, R. Tensors and Manifolds; 2nd Edition, Oxford University Press,

Oxford, UK, 2004.

34. Greiner, W.; Reinhardt, J. Field Quantization; 219, Springer-Verlag, Berlin, 1996.

35. Weinberg, S. The Quantum Theory of Fields; I, 143, Cambridge University Press,

1995.

36. Horn, R.; Johnson, C. Topics in Matrix Analysis; 255, Cambridge University Press,

1991.

37. Nielsen, M. A. A Geometric Approach to Quantum Computation. Quantum

Information and Computation 2006, 6, 213–262.

30
7. Transitions

1. This work will supplement the small U.S. Army Research Laboratory (ARL)/Sensors

and Electron Devices Directorate (SEDD) Mission Program on Quantum Computing.

2. I initiated collaborative research with Dr. John Myers and Prof. Tai Tsun Wu of

Harvard University.

3. I wrote 10 papers (references 2–11).

4. I presented a American Mathematical Society (AMS) Short Course lecture on this work

at American Mathematical Society Meeting in Washington DC.

5. I gave nine invited talks at international meetings.

6. I periodically informed a classified government group of progress on this research.

31
32 

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