Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

Abstract

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

Full text

Page 1

Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions
A. T. Rezakhani(1,4), D. F. Abasto(2,4), D. A. Lidar(1,2,3,4), and P. Zanardi(2,4,5)
(1)Departments of Chemistry,(2)Physics, and(3)Electrical Engineering,
and(4)Center for Quantum Information Science & Technology,
University of Southern California, Los Angeles, California 90089, USA
(5)Institute for Scientific Interchange, Viale Settimio Severo 65, I-10133 Torino, Italy
We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity
we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a
unified geometric description of quantum adiabatic evolution and quantum phase transitions, which generalizes
previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum infor-
mation processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold
of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with
examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of
a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum
phase transition. In particular, we show that in the vicinity of a critical point describing a second order quantum
phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the
product of the spatial and scaling dimensions.
PACS numbers: 03.67.Lx, 02.30.Xx, 02.30.Yy, 02.40.-k
I.INTRODUCTION
Geometric and topological concepts have long played use-
ful roles in both classical and quantum physics [1]. Important
applications where the use of geometry has led to new insights
include quantum evolutions [2], distance measures in quan-
tum information theory [3, 4], circuit-based quantum compu-
tation [5], and holonomic quantum computation [6]. More
recently quantum phase transitions (QPTs) [7] and adiabatic
quantum computation [8, 9] have also been been explored
from a geometric perspective [10, 11]. While geometry can
be seen as an underlying unifying theme in these applications,
an explicit geometry-based connection between them is not
always apparent. The central theme of this work is to elu-
cidate the geometry of adiabatic evolution. In particular, we
describe an all-geometric connection between QPTs and adi-
abatic quantum evolution. We do this by showing how the
Riemannian metric tensor that describes transitions through
quantum critical points [10] also arises in adiabatic quantum
evolution. Morespecifically, weexplainhow themetricwhich
provides an information-geometric framework for QPTs can
also provide a geometry for the control manifold arising in
adiabatic evolutions. That QPTs and adiabatic quantum evo-
lution should be so intimately related was previously under-
stood in terms of the role of ground state evolution in adiabatic
quantum computation, and in particular the basic observation
that those points where ground state properties undergo dras-
tic changes, i.e., quantum critical points, are bottlenecks for
adiabaticity [8, 12, 13].
The metric tensor we identify is a natural extension of the
metric found in Ref. [10] to systems with degenerate ground
states. In this sense we go beyond adiabatic quantum com-
putation, which is typically concerned with nondegenerate
ground states, and find results with applications to holonomic
quantum computation, where quantum gates are performed
as holonomies in the degenerate ground eigensubspace of the
system Hamiltonian. We analyze the relevance of the metric
tensor we identify for determining paths with minimum com-
putational error, in the sense of deviation from the desired
final adiabatic state. In addition, we find a prescription for
adiabatic passage through quantum critical regions by solving
the corresponding geodesic equations derived from the metric
tensor. As a result we are able to identify universal charac-
teristics of adiabatic passage through a critical point. Namely,
we find that in the vicinity of a critical point the geodesic ex-
hibits power-law scaling with an exponent given by twice the
inverse of the product of the spatial and scaling dimensions.
The structure of this paper is as follows. In Sec. II we for-
mulate our geometric picture. Specifically, after defining the
model in subsection IIA, in subsection IIB we introduce the
adiabatic error and show how to upper bound it as a sum of
two components, one of which encodes the geometric aspects
of the evolution. We obtain a Riemannian metric by mini-
mizing this error. Next, in subsection IIC we demonstrate
the emergence of the same geometry from the concept of adi-
abatic operator fidelity. In subsection IID we demonstrate
how our metric arises from three more (interrelated) natural
origins: Grassmannian geometry, Uhlmann parallel transport,
and the Bures metric. In subsection IIE we compare our
metric with another, related metric for adiabatic evolutions
which we proposed in earlier work [11]. We briefly discuss
strategies for further making the adiabatic error small in sub-
section IIF. We make the connection to QPTs in section III.
Specifically, in subsection IIIA we establish the relevance of
our metric in the sense of QPTs, by showing that the same
metric is responsible for signaling quantum criticality. Then,
in subsection IIIB we derive the quantum critical scaling of
the metric tensor. Switching gears, we define the notion of
an adiabatic geodesic in Sec. IV. In subsection IVA we an-
alyze three examples, namely the Deutsch-Jozsa algorithm,
projective Hamiltonians (including Grover’s algorithm), and
the transverse field Ising model, for which we analytically find
the adiabatic metric and the corresponding geodesics. In sub-
section IVB we analyze the properties of geodesics when the
arXiv:1004.0509v2 [quant-ph] 10 Apr 2010

Page 2

2
adiabatic evolution passes through a quantum critical point. It
is here that we identify the universal characteristics of such
geodesics. We summarize our results and conclude in Sec. V.
Several appendices provide detailed proofs omitted from the
main text so as not to interrupt the presentation.
II.GEOMETRY OF ADIABATIC QUANTUM EVOLUTION
A.Model
Considerann-bodysystemwiththeN-dimensionalHilbert
space H. The Hamiltonian family {H(x)} for this system,
which depends on the (time-dependent) coupling strengths or
“controlknobs”x, canbeidentifiedbypointsovertherealM-
dimensional manifold M ? x. Given a total evolution time
T and rescaled time s = t/T, a path x : s ∈ [0,1] ?→ M
then represents the dynamics in this time interval, starting
from x0 ≡ x(0) and ending at x1 ≡ x(1). We shall use
the notation xs ≡ x(s) interchangeably, or sometimes drop
the s-dependence entirely to lighten the notation.
low for a g0(x)-fold degenerate ground-state eigensubspace
of {H(x)}, with eigenstates {|Φα
can be identified by the projector
We al-
0(x)?}. Thus this subspace
P0(x) =
g0(x)
?
α=1
|Φα
0(x)??Φα
0(x)|,
(1)
with Tr[P0(x)] = g0(x) ≥ 1. We assume that for all finite n
the ground-state energy E0(x) is separated by a nonvanishing
gap ∆(x) from the rest of the spectrum. In the thermody-
namic limit n → ∞ we allow the gap to vanish at some finite
set of points {xc≡ x(sc)}, or a segment of the path. These
are the critical points where a QPT takes place. Although our
results would hold if we picked any other eigensubspace satis-
fying the previous requirements, rather than the ground state,
for specificity we shall henceforth consider the ground state
and the initialization |ψ(0)? =?g0(x0)
dence on x(s) hereafter where possible).
α=1
aα|Φα
0(x0)? (where
|ψ(s)? ≡ |ψ (xs)?, and we similarly drop the explicit depen-
B.Adiabatic Error
1.Degenerate case
We wish to compare the desired, “ideal” adiabatic evolution
to the actual evolution induced by the the Hamiltonian family.
To this end we shall define an appropriate “adiabatic error”
which measures the deviation between the two. The state of
the system,
|ψ(s)? = V (s)|ψ(0)?,
(2)
at any rescaled time s, is given in ? = 1 units [adopted here-
after], in terms of the propagator V (s) which is the solution
to the time-dependent Schr¨ odinger equation
i∂sV (s) = TH(s)V (s).
(3)
We can similarly associate an adiabatic propagator Vad(s) and
an adiabatic Hamiltonian Had(s) to the ideal adiabatic evolu-
tion, where the two are related via the Schr¨ odinger equation
i∂sVad(s) = THad(s)Vad(s).
(4)
What defines the adiabatic propagator is the “intertwining
property”
Vad(s)P0(0)V†
ad(s) = P0(s),
(5)
which means that Vad(s) preserves the band structure of
the ground eigensubspace of H(s).
the intertwining property is equivalent to i∂sP0(s)
T[Had(s),P0(s)], and when it holds we have
By differentiation
=
|ψad(s)? = Vad(s)|ψ(0)? =
g0
?
αα?=0
aαV[0]
αα?(s)|Φα?
0(s)?, (6)
where V[0]
Wilczek-Zee holonomy [14]—usually expressed as the path-
ordered exponential
αα?(s) = ?Φα
0(s)|Vad(s)|Φα?
0(0)? is the (non-Abelian)
V[0](s) = P exp
?
−
?s
0
A(s?)ds??
,
(7)
with the gauge connection
Aαα? ≡ ?Φα
0|∂s|Φα?
0?.
(8)
We prove Eq. (7) in Appendix A (see also Ref. [15]).
The adiabatic Hamiltonian can be expressed in terms of the
original Hamiltonian plus a “correction” term [16, 17]:
Had(s) = H(s) + i[∂sP0(s),P0(s)]/T,
(9)
Clearly then, the actual state |ψ(s)? need not be the same as
the adiabatic state |ψad(s)?. Our objective is to find the path
xsthat minimizes the adiabatic error ?|ψ(xs)?−|ψad(xs)?? =
?{V (xs)−Vad(xs)}|ψ(x0)??, where the norm is the standard
Euclidean norm: ?|φ?? ≡
a result which does not depend on the initial state |ψ(x0)? we
shall adopt a state-independent error measure, and define the
adiabatic error to be
??φ|φ?. However, so as to obtain
δ[x(s)] ≡ ?V (xs) − Vad(xs)?.
Since ?(V − Vad)|ψ?? ≤ ?V − Vad?, where the norm on the
right-hand side is the standard sup-operator norm (often de-
noted ? · ?∞) [18]
?
where {σi(X)} are the singular values of X (eigenvalues of
√X†X), an upper bound on δ[x(s)] is then also an upper
bound on ?|ψ(xs)? − |ψad(xs)??.
(10)
?X? ≡
sup
|v?: ?|v??=1
?v|X†X|v? = max
i
σi(X),
(11)

Page 3

3
Using the fact that the sup-operator norm is unitarily in-
variant (?V AW? = ?A? for any operator A and any pair of
unitaries V and W) we can rewrite δ as
δ[x(s)] = ?I − Ω(xs)?,
(12)
where the “wave operator”
Ω(s) ≡ V†
ad(s)V (s)
(13)
satisfies the Volterra equation
Ω(s) = I −
?s
0
KT(s?)Ω(s?)ds?,
(14)
with the kernel
KT(s) ≡ V†
ad(s)[∂sP0(s),P0(s)]Vad(s).
(15)
Considering Eq. (9), −iKT(s)/T is simply the interaction-
picture Hamiltonian which results from transforming H(s)
to the interaction picture with respect to Had(s), where
i[∂sP0(s),P0(s)]/T plays the role of the perturbation. There-
fore, in analogy to the Dyson series of time-dependent pertur-
bation theory, the Volterra equation can be solved by iteration,
which yields
Ω(s) =
∞
?
l=0
Ωl(s),
(16)
where
Ω0(s) = I,
(17)
Ωl>0(s) = −
?s
0
KT(s?)Ωl−1(s?)ds?.
(18)
As shown in Refs. [16, 17], ∀l ∈ {2k − 1,2k} (k ∈ N):
sup
s
?
Using the above results, ?I−Ω(s)? can be expressed in terms
of a 1/T series expansion, since
?Ωl(s)? = O(1/Tk),
(19)
sup
s
?Ω(s) −
l−1
j=0
Ωj(s)? = O(1/Tk).
(20)
?I − Ω(s)? = ?Ω1(s) −
?
l≥2
?s
0
KT(s?)Ωl−1(s?)ds??
≤ ?Ω1(s)? +
?s
= ?Ω1(s)? + ? ?(s)O(1/T),
?s
0
?KT(s?)?
?
l≥2
?Ωl−1(s?)?ds?
(21)
(22)
where
? ?(s) ≡
0
?[∂s?P0(s?),P0(s?)]?ds?.
(23)
Thus the error δ is upper-bounded as
δ[x(s)] ≤ δ1(s) + δ2[x(s)],
(24)
where
δ1(s) ≡ ?Ω1(s)? ∼ O(1/T)
δ2[x(s)] ≡ ? ?[x(s)]O(1/T).
ing a large T, while for a given T, δ2can additionally be made
small by choosing a path over the control manifold M with
small ? ?. Note that in addition to ?Ω1(s)? ∼ O(1/T) we also
form ?Ω1(s)? ≤ ? ?[x(s)]O(1/T). One can see from Ref. [19]
term of δ1does not appear to have a geometric significance,
and we shall therefore exclude δ1from our study of adiabatic
geometry in this paper.
In the following we shall make the upper bound on δ small
by finding a path which makes ? ?[x(s)] small. Finding the path
norm by the Frobenius norm, the problem of minimizing δ2
has a geometric solution in the sense that a Riemannian metric
tensor is encapsulated in ?[x(s)] [Eq. (23) with the modified
norm].
In Appendix B we prove that
?
(25)
(26)
Both error components can evidently be made small by choos-
have the bound ?Ω1(s)? ≤
from Eqs. (19) and (20) as such we do not have a bound of the
?s
0?KT(s?)?ds?= ? ?[x(s)], but
how δ1(s) depends on T, the gap, and the norm of the Hamil-
tonian or its derivatives. However, the coefficient of the 1/T
which minimizes? ? is, however, beyond the scope of this work.
Instead, as we show below, after replacing the sup-operator
?[∂sP0,P0]? =
?P0(∂sH)
?
1
H − E0
?2
(∂sH)P0?,
(27)
where [H − E0]−1is called the reduced resolvent and is a
shorthand for (I − P0)[H − E0]−1(I − P0).
For a different method of traversing eigenstate paths of
Hamiltonians, based on the use of evolution randomization
and a quantum phase estimation algorithm, see Ref. [20].
2.Nondegenerate case
When H has a discrete and nondegenerate spectrum, P0=
|Φ0??Φ0| and I − P0=?
In this case
?
Using the chain rule of differentiation to write ∂sH
(∂iH)˙ xi, where dot denotes ∂sand ∂idenotes ∂/∂xi, and
using the Einstein summation convention Eq. (27) is easily
simplified in the nondegenerate case to yield:
?s
n>0|Φn??Φn|, where {|Φn?}n>0
are the excited eigenstates of H with eigenvalues {En}n>0.
1
H − E0
=
n>0
1
En− E0|Φn??Φn|.
(28)
=
? ?[x(s)] =
0
?
2g(1)
ij(x)˙ xi˙ xjds?,
(29)

Page 4

4
where
g(1)
ij≡ Re
??
n>0
?Φ0|∂iH|Φn??Φn|∂jH|Φ0?
(En− E0)2
?
.
(30)
The manner in which g(1)
plays the role of a metric tensor. This metric tensor is identical
to the metric tensor which was identified in the differential-
geometric theory of QPTs [10]. We next consider how to gen-
eralize this result to the degenerate case.
ijappears in Eq. (29) suggests that it
3.Metric tensor for the degenerate case – moving to the
Hilbert-Schmidt norm
We would like to identify Eq. (27) with a metric tensor.
However, the appearance of the sup-operator norm presents a
problem, since this norm need not be differentiable. Hence
we replace the operator norm with the Frobenius (or Hilbert-
Schmidt) norm
?X?2≡
?
Tr[X†X] =
?
?
?
?
rank(X)
?
i=1
σ2
i(X),
(31)
which satisfies [18]
?X? ≤ ?X?2≤
?
rank(X)?X?.
?
(32)
Note that the operator P0(∂sH)
ing in Eq. (27) has support purely over the ground-state eigen-
subspace of H, due to the projections P0to the left and right.
Therefore its rank is at most g0, and as a consequence of
Eq. (32) the replacement of the operator norm by the Frobe-
nius norm does not alter ?[∂sP0,P0]? (hence ? ? or g) for the
times greater than the expression obtained with the operator
norm. Additionally, and this is our main reason for moving to
the Frobenius norm, it guarantees analyticity of the adiabatic
error and the metric tensor when H is analytic.
With these considerations in mind, let us now redefine the
adiabatic error using the Frobenius norm:
1
H−E0
?2
(∂sH)P0appear-
nondegenerate case (g0 = 1), while it enables a differential
geometric bound in the degenerate case, which is at most√g0
?(s) ≡
?s
0
?[∂s?P0(s?),P0(s?)]?2ds?.
(33)
Then ? ?(s) ≤ ?(s) ≤√g0? ?(s) and consequently
δ2(s) ≤ ?(s)O(1/T) ≤√g0δ2(s).
Minimization of ?(s) thus “squeezes” the error component δ2.
We show in Appendix C that
(34)
?(s) =
?s
0
?
2g0gij(x)˙ xi˙ xjds?,
(35)
where the metric tensor is defined as
gij ≡
1
2g0Tr[∂iP0∂jP0]
?
(36)
=
1
2g0TrP0(∂iH)
?
1
H − E0
?2
(∂jH)P0
?
+ i ↔ j.
(37)
It is simple to verify that gijreduces to g(1)
erate case, and similarly ?(s) reduces to ? ?(s) in this case.
which is the solution to the following Euler-Lagrange (EL)
equations:
ijin the nondegen-
Standard calculus of variations then tells us that minimiza-
tion of ?[x(s)] is tantamount to finding the geodesic path
¨ xi+ Γi
jk˙ xj˙ xk= 0,
(38)
where the connection Γ is
Γi
jk=1
2gil(∂kglj+ ∂jglk− ∂lgjk).
(39)
We have thus endowed the control manifold M with a Rie-
mannianstructure, givenbythemetrictensorg : TM⊗TM?→
R. That g really satisfies all the properties required of a met-
ric is shown in Appendix D. Other geometric functions such
as the curvature tensor R can be calculated from g [21].
C. Operator fidelity
Another approach to the adiabatic error is provided by the
“operator fidelity” [22] between V and Vad,
f?[x(s)] ≡ |Tr[Ω(xs)?]|,
(40)
where ? is an arbitrary density matrix of the system, which
here we take to be the totally mixed state I/N. The operator
fidelity derives its name from the fact that it quantifies the fi-
delity in the entire Hilbert space, and unlike our previous error
measures? ? and ?, which involve the ground state projector P0,
the two are obviously closely related. In Appendix E we show
that
is not restricted just to ground states. However neither is the
adiabatic error δ [Eq. (10)] restricted just to ground states, and
1 −
1
√N? ≤ f?≤ 1,
(41)
so that minimizing ? maximizes f?, and vice versa.
Let O be an arbitrary observable, and consider it in the ro-
tated bases associated with the actual or adiabatic dynamics:
O(s) ≡ V (s)OV†(s)
Oad(s) ≡ Vad(s)OV†
(42)
(43)
ad(s),
In addition to the bound (41) we show in Appendix E that
?O(s) − Oad(s)? ≤ ?O?(δ1(s) + δ2[x(s)])
[2 + O(1/T)],
(44)

Page 5

5
which is identical to the adiabatic error bound (24), apart from
the factor ?O?[2 + O(1/T)]. Thus our bound of the operator
distance ?O(s) − Oad(s)? also has the component δ1and the
component δ2with its apparent geometric contribution, which
can be squeezed by choosing a geodesic path, as in subsection
IIB3.
D. Natural geometric formulation
1.Grassmannian
An alternative, natural way to obtain a geometry for adi-
abatic evolutions employs the Grassmannian structure of the
dynamics [23]. As explained above, in the ideally adiabatic
case the eigensubspaces corresponding to the ground state and
the rest of the spectrum (P0and I − P0, respectively) do not
mix; each follows its own unitary dynamics determined by
its Wilczek-Zee holonomy, hence Vad= V[0]⊕ V[rest]. This
implies a Grassmannian manifold
GN,g0∼= U(N)/U(g0) × U(N − g0)
∼= {P0∈ D(H) | P2
where U(k) is the group of k ×k unitary matrices, and D(H)
is the convex space of all density operators (positive semidef-
inite, unit trace matrices) defined over H. A natural distance
(metric) over this space is given by [24, 25]
0= P0, Tr[P0] = g0}, (45)
d(P0,P?
0) ≡
1
√2g0?P0− P?
0?2,
(46)
whence, keeping only the lowest non-vanishing order, we
have
d2(P0(x),P0(x + dx)) =
1
2g0?P0(x + dx) − P0(x)?2
1
2g0Tr[(dP0(x) +1
1
2g0Tr[dP0(x)dP0(x)]
1
2g0Tr[∂iP0dxi∂jP0dxj]
= gijdxidxj,
2
=
2d2P0(x))2]
=
=
(47)
with the metric tensor as defined in Eq. (36). Thus the adia-
batic metric tensor is precisely the metric over the Grassman-
nian manifold defined by the ground state projectors.
2. Adiabatic parallel transport
In this subsection we wish to define a notion of adiabatic
parallel transport. We start with the standard purification [26,
27]
W = P0U
(48)
of P0, where U is an arbitrary unitary acting on H, so that
P0= WW†. Here W is considered a vector in a larger (ex-
tended) Hilbert space Hext, i.e., a pure state whose reduc-
tion yields (the density matrix) P0. The Hilbert space Hext
is equipped with the the Hilbert-Schmidt inner product
?A,B? := Tr[A†B].
(49)
Given P0, the fiber of all purifications sitting on the unit
sphere S(Hext) := {W ∈ Hext : ?W,W? = 1} of Hextis
the Stiefel manifold of orthonormal g0-frames of Hext, where
Tr[P0] = g0(i.e., the set of ordered g0-tuples of orthonormal
vectors in Hext). The gauge transformation (48) means that
the fiber admits the unitaries of H as right multipliers. Infor-
mally, the Us act as arbitrary “phases” associated with P0.
Starting with a curve of (unnormalized) density operators
s ?→ P0(s) and one of its purifications
s ?→ W(s)
the length ?U[s] =?s
equations for the variational problem ?[s] := infU?U[s], i.e.,
for the geodesic are [26, 27]
P0(s) = W(s)W†(s),
(50)
0?˙W(s?),˙W(s?)?ds?of the curve in Hext
is not invariant against gauge transformations (48). The Euler
W†dW = dW†W,
(51)
also known as the Uhlmann parallel transport condition. Sub-
stituting W†= U†P0and dW = (dP0)U + P0dU yields the
condition
U†P0((dP0)U + P0dU) = (U†dP0+ (dU†)P0)P0U, (52)
which, using UdU†= −(dU)U†, reduces to
P0(dU)U†+ (dU)U†P0= [dP0,P0]
(53)
on the vector bundle over the Grassmannian GN,g0. Here U =
U(s) is a general unitary undergoing parallel transport as s ?→
P0(s). We now seek those unitaries U which in addition to
parallel transport, also satisfy adiabaticity.
To this end let J(s) be the infinitesimal generator of U(s),
i.e.,
i∂sU(s) = TJ(s)U(s).
(54)
Substituting this expression into Eq. (53) we obtain
P0J + JP0 = i[∂sP0,P0]/T
= Had− H,
where in the second line we used Eq. (9). Thus, U satisfies
adiabatic parallel transport if in addition to being a solution
to the parallel transport condition (53) its generator also satis-
fies the adiabaticity condition
(55)
(56)
P0J + JP0= 0.
(57)
What is the generator J which satisfies this last condition?
Using Eqs. (B10) and (B12) for the nondegenerate case we

Page 6

6
obtain
− iT(P0J + JP0) = [˙P0,P0]
= −
1
H − E0
?
˙HP0+ P0˙H
1
H − E0
=
n>0
P0˙H|Φn??Φn| − |Φn??Φn|˙HP0
En− E0
.
(58)
Taking matrix elements we find ?Φ0|J|Φ0?
−iT?Φ0|J|Φk? =
ments of J between the excited states are unspecified, so that
=0 and
1
Ek−E0?Φ0|˙H|Φk?, while the matrix ele-
J =i
T
?
n>0
?Φ0|∂sH|Φn?
En− E0
|Φ0??Φn| + H.c. + J⊥,
(59)
where J⊥is an arbitrary operator satisfying J⊥= Q0J⊥Q0.
Instead of trying to obtain perfect adiabaticity (Had= H)
we can settle for an approximation. Noting that Eqs. (33) and
(55) imply
?s
=
0
?(s) = T
0
?
?P0(s?)J(s?) + J(s?)P0(s?)?2ds?,
?s
2g0gij(x)˙ xi˙ xjds?
(60)
it follows that minimizing ?, or equivalently finding the adia-
batic geodesic, endows the “phase” U of P0with an adiabatic
characteristic which is compatible with the Uhlmann paral-
lel transport condition. Thus, we have shown that the metric
tensor g emerges naturally also from the notion of adiabatic
parallel transport.
3. Bures metric
There is also a straightforward connection between our
metric and the Bures metric [28]. For two arbitrary density
matrices ρ1and ρ2, the Bures distance is defined as
d2
Bures(ρ1,ρ2) ≡ 1 − F(ρ1,ρ2),
where F(ρ1,ρ2) ≡ Tr[(ρ1/2
tween these two states [29, 30].
trices depend on a parameter x, the infinitesimal distance
d2
(61)
1 ρ2ρ1/2
1 )1/2] is the fidelity be-
When the density ma-
Bures(ρ(x),ρ(x + dx)) can be shown to be [4]
d2
Bures(ρ(x),ρ(x + dx)) = Tr[ρ(x)L2(x)],
where L(x) is the “symmetric logarithmic derivative,” (SLD)
defined via
(62)
dρ(x) =1
2
?L(x)ρ(x) + ρ(x)L(x)?.
0= P0we obtain
(63)
From the property P2
dP0(x) = dP0(x)P0(x) + P0(x)dP0(x).
(64)
and hence [see Eq. (A2)]
dg0= Tr[dP0] = 2Tr[P0dP0] = 2Tr[P0dP0P0] = 0, (65)
i.e., the degeneracy is constant. Thus if ρ(x) ≡ P0(x)/g0,
then d[P0(x)/g0] = P0(x)/g0dP0+ dP0P0(x)/g0, and the
definition of the SLD [Eq. (63)] yields
L(x) = 2dP0(x).
(66)
Inserting this back into Eq. (62) results in
d2
Bures(P0(x),P0(x + dx)) =
4
g0Tr[P0(x)(dP0(x))2]
≡ gBures
ij
(x)dxidxj,
(67)
where
gBures
ij
(x) =
4
g0Tr[∂iP0(x)∂jP0(x)].
(68)
By comparison with Eq. (36), we obtain
gBures
ij
= 8gij.
(69)
We note that the Bures metric is also connected to “quan-
tum Fisher information tensor,” which plays a principal role
in quantum estimation theory [4, 28, 31, 32]. In fact the Bures
metric is (up to an unimportant constant multiplicative factor)
equal to the Fisher tensor. Therefore, the adiabatic metric is
the quantum Fisher metric. The role of the metric g in quan-
tum estimation theory is thus highlighted naturally this way.
E.Comparison of adiabatic metrics
Inadiabaticevolution(aswellasinadiabaticquantumcom-
putation) δ and T are the primary objects of interest. Our
method for obtaining the metric g here is based on minimizing
an upper bound on the adiabatic error δ for a given evolution
time T. In Ref. [11] we pursued a complementary route and
proposed a different metric,
? gij(x) = Tr[∂iH(x)∂jH(x)]/∆4(x),
ditional adiabatic condition. We called this the “quantum adi-
abatic brachistochrone”.
The major difference between these two metrics is in their
distinctgapdependence. Thiscanbeunderstood, forexample,
by noting that
(70)
derived from minimizing a time functional inspired by the tra-
|gij| ≤?∂iH∂jH?1
mins∆2
,
(71)
whereas
|? gij| ≤?∂iH∂jH?1
mins∆4
,
(72)
where ?X?1 ≡ Tr[√X†X] =?
a quadratically less dependence on the inverse gap. This may
imply different behaviors for these metrics and their corre-
sponding curvatures; hence they are essentially distinct.
iσi(X) is the trace norm
[18] (see Appendix G for the proof). Thus, the metric g has

Page 7

7
F.Strategies for reducing the adiabatic error and their effect
on geometry
Considering that g is related to minimizing the upper bound
on δ, it is useful to briefly recall how δ scales with T and how
this scaling may be improved.
Rigorous proofs of the adiabatic theorem—based on suc-
cessive integration by parts of Ω—state that if {H(s)} is
a family of Ck(k times continuously differentiable) in-
terpolations/paths with bounded ?∂l
and compactly supported ∂sH over s ∈ (0,1), then δ =
O(1/T2(k−1)) [16, 17, 19]. If these assumptions are supple-
mented with that of analyticity of H(s ∈ C) in a small strip
around the real time axis, and if in addition
sH? (l ∈ {1,...,k})
∂l
sH(0) = ∂l
sH(1) = 0 ∀l ≤ k,
(73)
the result is an exponentially smaller error
δ = O(e−cT),
(74)
where c ≡ mins∆3/maxs?∂sH?2(up to an O(1) prefactor)
[33].
Our path—as the solution to the second-order differen-
tial equation (38)—minimizes ? rather than δ, which is
not necessarily compatible with the boundary conditions
∂l
further optimization of the path for δ beyond what is captured
by simply minimizing its upper bound ?(s)O(1/T) [16, 17].
Such finer optimizations, however, may not always result in a
Riemannian geometry because the corresponding functionals
andEuler-Lagrangeequationswoulddependonhigherderiva-
tives of H.
sH({0,1}) = 0. Thus, in principle, there remains room for
III.CONNECTION TO QUANTUM PHASE TRANSITIONS
The other physically-important aspect of our geometric for-
mulation emerges from the observation that the metric g also
arises naturally as the underlying geometry of QPTs. QPTs
take place at zero temperature [7], where the system is in prin-
ciple in its ground state. Such phase transitions exhibit pecu-
liar behaviors and “orders,” radically different from their ther-
mal counterparts. In particular, in contrast to thermal phase
transitions, the standard paradigm of the Landau-Ginzburg
symmetry-breaking mechanism [7, 34] fails to explain the un-
derlying physics of some QPTs. In fact, defining an appro-
priate local “order parameter”—an essential ingredient of the
Landau-Ginzburg theory—is not straightforward for a quan-
tum critical system; some QPTs, such as those involving
“topological order,” provably do not admit any local order pa-
rameter [35, 36]. Additionally, tracking singularities of the
ground-state energy cannot always foreshadow QPTs; quan-
tum systems with matrix-product states may elude this test
[37].
Notwithstanding the above subtleties with identifying
QPTs, it has recently been shown that the simple notion of
the “ground-state fidelity” is remarkably successful in signal-
ingQPTs[10,38]. Thiscanbeunderstoodbynotingthatsince
QPTs take place at zero temperature, in which the system is in
its ground state, quantum criticality should be identifiable by
ground state properties. Specifically, the ground states right
before and right after a quantum critical point are expected to
have very little overlap. In this manner ground-state fidelity
may be considered as a natural, fairly general order parameter
for quantum critical systems, irrespectively of their internal
symmetries [10]. We shall discuss this feature in more detail
below.
A.Metric tensor for QPTs
Here we derive the metric attributed to QPTs for the case of
degenerate ground states as a natural extension of the similar
metric proposed for the nondegenerate case [10].
In the degenerate case we should work with the ground-
state projector P0(x). A variation in the properties of P0(x),
caused by the change x → x + dx in the Hamiltonian pa-
rameters, can be captured by the order parameter chosen to be
the operator fidelity of P0(x) and P0(x+dx) relative to, e.g.,
? = Ig0/g0(Ig0is the g0× g0identity matrix),
?P0(x),P0(x + dx)?
= 1 − Gij(x)dxidxj,
in which the Hermitian matrix
f?
= ?P0(x),P0(x + dx)??
(75)
Gij≡
1
g0Tr[P0(∂iP0)(∂jP0)P0]
(76)
is the “geometric tensor” for the degenerate case (see Ap-
pendix F for the proof). Thus the information about the crit-
icality of the quantum system is contained in the G tensor.
Note that in the nondegenerate case (g0= 1) Gijreduces to
Gij = ?∂iΦ0|∂jΦj? − ?∂iΦi|Φ0??Φ0|∂jΦ0?.
Accordingly, a Riemannian QPT metric tensor can be de-
fined through
(77)
gQPT
ij(x) ≡ Re[Gij(x)] =
= gij,
1
2g0Tr[∂iP0∂jP0]
(78)
where we used the same trick as that used in arriving at
Eq. (C2). Therefore, the QPT metric tensor gQPTis the
same as the adiabatic quantum evolution metric g defined in
Eq. (36).
B.Quantum critical scaling of the QPT metric tensor
The critical behavior of a quantum system with a degen-
erate ground state can be characterized by the metric tensor
g. This is already evident from the fact that the divergence
of gQPTis a sufficient condition for signaling a quantum crit-
icality. To further elaborate on this connection, we follow

Page 8

8
Ref. [39] and obtain the scaling of the geometric tensor (76)
?
?
and via Eq. (78) also for gij. For simplicity we restrict our-
selves only to gapped quantum systems with second-order
QPTs. Thus, in a critical region x ≈ xc, the correlation length
ξ and the gap ∆ exhibit the following scalings
Gij =
1
g0TrP0(∂iH)
?
0|∂iH|Φη
1
H − E0
?2
(∂jH)P0
?
(79)
=
1
g0
n>0
g0,gn
?
α,η=1
?Φα
n??Φη
n|∂jH|Φα
0?
(En− E0)2
,(80)
ξ ∼ |x − xc|−ν, ∆ ∼ |x − xc|zν,
with the critical exponents ν > 0 and zν, where z > 0 is the
dynamical exponent [7]. The geometric tensor G has an inte-
gral representation which not only facilitates the derivation of
the scaling relation for G, but also enables an interpretation
for G in terms of correlation (or response) functions. Indeed,
as shown in Appendix H, Eq. (80) can be expressed as
?∞
−1
(81)
Gij =
1
g0
0
dτ τe−pτ?
Tr[P0∂iHτ∂jH]
????
g0Tr[P0∂iH]Tr[P0∂jH]
p=0,
(82)
with ∂iHτ≡ eτH∂iHe−τH.
Now we make some generic assumptions about the Hamil-
tonian H. First, let ∂iH be a local operator; that is, one can
write
?
in which y labels the spatial region over which the local op-
erator hi(y) has support. Second, the hi(y) operators have
well-defined scaling dimensions αinear the quantum critical
point xc, such that if
∂iH =
y
hi(y),
(83)
y → ay, τ → azτ,
(84)
for a > 0, we obtain
hi(y) → a−αihi(y).
(85)
Under these transformations, Eq. (82) yields the following
scaling for the rescaled geometric tensor in the thermody-
namic limit
1
LdGij→ a−κij1
LdGij,
(86)
where
κij≡ αi+ αj− 2z − d.
(87)
Here, L is the linear size of the system and d is its spatial
dimension. From Eq. (81), we obtain |x − xc| ∼ ξ−1/ν;
i.e., the scaling dimension of the Hamiltonian parameter x is
1/ν. Following standard scaling analysis arguments, the scal-
ing behavior of the metric tensor (recall that g = Re[G]) in
the off-critical limit ξ ? L is
gij(x ≈ xc) ≈ Ld|x − xc|νκij.
Moreover, in the critical region, where ξ ? L ? the spacing
between adjacent particles on the system lattice, in addition to
the regular extensive scaling Ld, the finite-size scaling of the
metric is gij∼ Ld−κij, which could be extensive, subexten-
sive, or superextensive [κ = 0, positive, or negative, respec-
tively]. We also remark that there exist models, exhibiting
quantum topological order, in which the critical g scales log-
arithmically, e.g., g ∼ ln|x − xc| [40–42].
(88)
IV.ADIABATIC GEODESICS
In this section we solve the geodesic equation (38) an-
alytically for some specific examples. Note that since the
eigenprojections do not depend on Tr[H], Eq. (38) corre-
sponds to an underdetermined system of coupled second-
order differential equations. This can be seen more clearly
by adopting a new parametrization (i.e., coordinate system)
y(x) for the Hamiltonian such that H?y(x)?
H − Tr[H]1 1/N. Since P0(y) does not depend on y1, the
metric g(y) does not depend on this parameter either. Inde-
pendence from y1translates in terms of x into the statement
that only M −1 equations in the system (38) are independent.
= y1(x)1 1 +
H??y2(x),...,yM(x)?, in which y1= Tr[H]/N and H?=
A.Examples
1. Deutsch-Jozsa algorithm
In the Deutsch-Jozsa algorithm [43] one is given an oracle
that calculates a function f : {0,1}n?→ {0,1}. The promise
is that f is either “constant” or “balanced,” meaning respec-
tively that, f(i) = f(i?) ∀i,i?or f(half of all i’s) = 0 [29].
Theobjectiveistoconcludewhetherf isconstantorbalanced.
The Deutsch-Jozsa algorithm finds the answer by querying the
oracle only once, while classical deterministic algorithms re-
quire a number of queries that is exponential in n.
An adiabatic version of this algorithm was introduced in
Ref. [44]. We consider the unitary interpolation Hamiltonian
[45]
H?x(s)?=?V (x(s)?H0?V†(x(s)?,
operator is defined by G|i? = (−1)f(i)|i?. Here H0is chosen
such that |Φ0(0)? = |+?⊗n= 2−n/2?2n−1
H0= h0
where |±? = (|0? ± |1?)/√2, σz = |0??0| − |1??1| is a
Pauli matrix, and h0 > 0 is an energy scale. The bound-
ary conditions are chosen as (x0,x1) = (0,1) such that
(89)
where?V?x(s)?= eiπ
state, e.g.,
2x(s)G, inwhichtheHermitian/unitaryG
i=0
|i? is its ground
?n
k=1|−?k?−|,
(90)

Page 9

9
H(0) = H0and H(1) = GH0G†; the latter guarantees that
|Φ0(1)? = G|Φ0(0)? is the ground state of H(1).
From Eq. (89) it is seen that |Φ0(s)? = ?V (s)|Φ0(0)?,
P0
?x(s)?= iπ
A straightforward calculation then yields
g = Tr[?∂xP0(x)?2]
=
2
π2
2
whence we obtain
?x(s)?= 2−n?2n−1
i,i?=0eiπ
2x(s)[(−1)f(i)−(−1)f(i?)]|i??i?|,
2[G,P0
∂xP0
?x(s)?].
(91)
π2
?
?
Tr[P0(x)G2] − Tr[?P0(x)G?2]
1 − 2−2n?2n−1
?
=
?
i=0
eiπf(i)?2
?
.
(92)
Since g is independent of x(s), the geodesic equation (38)
reduces to ¨ x = 0, whence the geodesic is simply
x(s) = s,
(93)
which corresponds to a rotation of the initial Hamiltonian H0
at a constant rate.
2.Projective Hamiltonians
Consider the following Hamiltonian:
H?x(s)?= x1(s)P⊥
where P⊥
a
= 1 1 − |a??a|, for a given |a? ∈ H (similarly
for P⊥
conditions are x0 = (1,0) and x1 = (0,1). This Hamilto-
nian may represent the adiabatic preparation of an unknown
(“hard”) state |b? from the supposedly known (“simple”) ini-
tialization|a?, providedthatonehasaccesstothe“oracle”P⊥
[11]. An important instance of this class is Grover’s Hamilto-
nian for search of a “marked” item among N unsorted items
[46] (generalized to arbitrary initial amplitude distributions in
Refs. [47, 48]), where |a? =?N−1
this algorithm was first described in Ref. [49].
Since the Hamiltonian (94) is effectively two-dimensional
over the span of the vectors |a? and |b?, it can be diagonalized
analytically. Indeed, given |a?, we have the freedom to choose
N − 1 vectors {|a⊥
they constitute an orthonormal basis for H. I.e., ?a|a⊥
and ?a⊥
?N−1
such that αi>1= 0. In this case, we have
a+ x2(s)P⊥
b,
(94)
b), ?a|b? is a given function of N, and the boundary
b
k=0|k?/√N and |b? = |m?,
for m ∈ {0,...,N − 1}. A successful adiabatic version of
i?}N−1
i=1at will such that together with |a?
i? = 0
i|a⊥
j? = δij. Thus we can decompose |b? = α0|a? +
i=1αi|a⊥
to the orthonormality condition), we can always rotate them
i?. Utilizing the freedom in choosing {|a⊥
i?} (up
|b? = α0|a? + α1|a⊥
1?,
(95)
where α0 = ?a|b? and α1 = ?a⊥
α1= eiφ1?1 − |?a|b?|2, for some arbitrary φ1∈ [0,2π)).
1|b? (or more explicitly:
Expanding Eq. (94) in the {|a?,|a⊥
Eq. (95) yields
?x2(1 − |α0|2)
⊕(x1+ x2)I{2,...,N−1},
where we have used the completeness of the basis to write I =
|a??a| +?N−1
{|a?,|a⊥
spectrum of H consists of the two nondegenerate eigenvalues
i?}N−1
i=1basis and using
H(x) =
−x2α0¯ α1
x1+ x2|α0|2
−x2¯ α0α1
?
(96)
i=1|a⊥
i??a⊥
i|, I{2,...,N−1}≡?N−1
i=2|a⊥
i??a⊥
i|,
and the matrix on the right hand side is written in the
1?} (sub-)basis. It then follows from Eq. (96) that the
E± =
1
2(x1+ x2
?
±
(x1)2+ (x2)2+ 2(2|?a|b?|2− 1)x1x2), (97)
and the (N − 2)-fold degenerate eigenvalue
E>= x1+ x2.
(98)
Thus, the gap between the ground state (E−) and the first ex-
cited state (E+) becomes
∆(x) =?(x1)2+ (x2)2+ 2(2|?a|b?|2− 1)x1x2. (99)
The Hamiltonian (96) can be diagonalized by noting that
one can rewrite
H(x) =
1
2A(x)[∆(x)Σz− (x1+ x2)I{0,1}]A†(x)
+(x1+ x2)I,
(100)
where Σzis the Pauli matrix σz = diag(1,−1) ≡ |0??0| −
|1??1| padded with zeros to embed it trivially into the N-
dimensional representation (i.e., Σz = diag(σz,0,...,0)),
I{0,1}≡ diag(1,1,,0,...,0), and the 2 × 2 unitary matrix
A(x) is defined as
A(x) = e−iθ(x)σy,
(101)
(the extension to N dimensions is similar to that of Σz by
padding with sufficiently many zeros) with
?
cosθ = 2x2|?a|b?|
× (x2)2+?x1− [1 − 2|?a|b?|2]x2− ∆?2?−1/2.
After removing the energy shift (x1+ x2)I from Eq. (100), it
is evident that the ground-state projection is
1 − |?a|b?|2/?4|?a|b?|2(1 − |?a|b?|2)
(102)
P0(x) = A(x)|1??1|A†(x),
(103)
(padded with zeros). This yields
gij = Tr[∂iP0∂jP0]
= −∂iθ∂jθ Tr?[σy,P0]2?
= ∂iθ∂jθ.
(104)

Page 10

10
Obtaining the geodesic for the one-dimensional case x =
(1−x,x) turns out to be simple and can be performed analyt-
ically, yielding
x(s) =1
2−
|?a|b?|
2?1 − |?a|b?|2tan[(1 − 2s)arccos|?a|b?|].
(105)
It is interesting to note that this is exactly the solution obtained
in Ref. [11] from the different metric ? g [Eq. (70)].
3.One-dimensional transverse-field Ising chain
Consider a one-dimensional chain of spin-1/2 particles in-
teracting according to the following Hamiltonian:
H?x(s)?= −
with the boundary conditions x0 = (1,0), x1 = (0,1), and
σ(m+1)≡ σ(1)[50]. Exact diagonalization by the Jordan-
Wigner transformation [7] yields
?cosθ?(x)|0?−?|0??+ isinθ?(x)|1?−?|1??
m
?
?=−m
x1(s)σ(?)
z
+ x2(s)σ(?)
xσ(?+1)
x
, (106)
|Φ0(x)? = ⊗m
?=1
?,
(107)
where (cf. Ref. [10])
sin2θ?=
x2sin(
2π?
2m+1)
?
(x2cos
2π?
2m+1− x1)2+ (x2)2sin2
2π?
2m+1
.(108)
It is evident from Eq. (107) that
|˙Φ0? =
2
?
2
?
⊗ |Φ¯??,
i=1
˙ xi∂i|Φ0?
=
i=1
˙ xi
m
?
?=1
∂iθ?
?−sinθ?|0?−?|0??+ icosθ?|1?−?|1??
?
where |Φ¯?? is the same as |Φ0? [Eq. (107)] except that the term
with the label ? is absent. In addition it is easily verified that
?Φ0|˙Φ0? = 0. Thus, we obtain
?˙Φ0|˙Φ0? =
2
?
i,j=1
˙ xi˙ xj
m
?
?
∂iθ?∂jθ?.
(109)
After inserting these results into Eq. (36) we have
gij(x) =
m
?
?=1
∂iθ?(x)∂jθ?(x).
(110)
This, then, is the geometric tensor for the transverse field Ising
model.
To make further progress we focus on the one-parameter
cases: (i) x = (1 − x,x), (ii) x = (x,1), and (iii) x = (1,x),
all subject to the boundary conditions x(0) = 1 − x(1) = 0.
Let
p(x) =1
4
m
?
?=1
sin2(
2π?
2m+1)
2π?
[1 − 2(1 + cos
2m+1)(1 − x)x]2.
(111)
For a given finite lattice size m, the geodesic equation for case
(i) reads
2p(x)¨ x + ∂xp(x)(˙ x)2= 0.
(112)
This equation can be integrated to yield
?x(s)
We next consider the thermodynamic limit m → ∞, where
we can obtain a simple closed-form formula for the geodesic.
The expression in this limit follows from substituting?
count that the model exhibits a QPT at xc= 1/2 correspond-
ing to sc= 1/2. This yields
?
2
2s =
0
?
p(x?)dx?/
?1/2
0
?
p(x?)dx?,
(113)
?→
2m+1
2π
?π
0dz [with z? = 2π?/(2m + 1)] and taking into ac-
x(s) =
1
2
1
?1 − tan2[π
4(1 − 2s)]?, 0 ≤ s ≤1
2,
?1 + tan2[π
4(1 − 2s)]?,
1
2≤ s ≤ 1.
(114)
For details of the derivation see Appendix IX.
Similarly, for both cases (ii) and (iii) we obtain the geodesic
for a given finite m as
?x(s)
where
s =
0
?
q(x?)dx?/
?1
0
?
q(x?)dx?,
(115)
q(x) =1
4
m
?
?=1
sin2(
2π?
2m+1)
2m+1]2.
[1 − 2cos
2π?
(116)
In the thermodynamic limit a quantum critical point emerges
at xc= 1 (sc= 1), and a similar approach as in case (i) yields
the geodesic
x(s) = sin(πs/2).
(117)
For details of the derivation again see Appendix IX.
Figure 1 illustrates the geodesics obtained for the transverse
field Ising model subject to the three parametrizations we have
discussed.
B.Geodesic for passage through a quantum critical point
A limitation of our formalism is that, in principle, exact
knowledge of the ground state is required in order to ob-
tain the geodesic. Unfortunately, such knowledge is rarely
available, the exceptions being certain exactly solvable mod-
els such as those we treated in the previous subsection. With

Page 11

11
0.00.20.4 0.60.81.0
0.0
0.2
0.4
0.6
0.8
1.0
s
x?s?
0.00.20.40.60.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
s
x?s?
FIG. 1:
dimensional transverse-field Ising model, corresponding to the pa-
rameterizations x = (1 − x,x) (left), x = (x,1) and x = (1,x)
(right). The red dashed lines represent the thermodynamic limit,
while the solid blue lines correspond to m = 1,4,10,30,100, ap-
proaching the dashed line as m increases.
(Color online) Optimal adiabatic paths for the one-
partial knowledge or an approximation for the gap, one should
solve Eq. (38) on a case by case basis, possibly numerically.
However, while these observations apply in a setting where
one wishes to obtain the geodesic over the entire parame-
ter manifold, the situation in the vicinity of a quantum crit-
ical point is rather different. Indeed, the most interesting
physics usually happens in the vicinity of the quantum crit-
icality. In addition, the behavior of a quantum adiabatic algo-
rithm is essentially governed by how the system approaches
and/or passes through a quantum critical region. These con-
siderations suggest that knowledge of the geodesic around the
quantum critical region should suffice for most algorithmic or
physically relevant applications, thus obviating the need for
knowing P0everywhere.
Computation of the critical behavior of other geometric
functions, such as Γ and R, is straightforward.
the one-parameter case, where x = (x), the Euler-Lagrange
(geodesic) equation (38) in the critical region slightly before
and after the critical point reduces to ¨ x + νκ˙ x2/2x = 0,
whence
E.g., in
x(s ≈ sc) ≈ xc+ A(s − sc)χ.
After using α = d + z − 1/ν [51], where α is the scaling
dimension [recall Eq. (85)], we obtain
(118)
χ = 2/(2 + νκ) = 2/dν > 0,
(119)
with A constant (derivation details are given in Appendix IX).
This is a remarkable result as it characterizes the optimal adi-
abatic passage through a quantum critical point in terms of
the universality class of the system. Moreover, this result con-
firms that the critical geodesic has a power-law dependence on
s, as first reported in Ref. [52], although away from the crit-
ical region the dependence can be different. References [52–
55] report critical behaviors of the metric tensor and related
parameters obtained using different methods, such as mini-
mizing exact expressions for transition probability in thermo-
dynamic limit. In contrast to the result of Ref. [52], in our
analysis the exponent χ of the critical geodesic depends on the
dimensionality d, whereas it is independent of the total time
T. In adiabatic evolution the dependence on T is of course
expected; however, note that our scaling result depends only
upon the geometry of the control manifold, which does not
depend on T.
V.SUMMARY AND CONCLUSIONS
In this work we set out to elucidate the role of geometry
in adiabatic quantum evolution. By splitting the “adiabatic
error”, i.e., the norm of the difference between the ideal adi-
abatic evolution operator and the actual propagator, into two
components, one of which is endowed with a geometric mean-
ing, we were able to derive a Riemannian metric tensor which
encodes the geometry of adiabatic evolution. This metric is
capable of describing evolution over both nondegenerate and
degenerate subspaces. We then showed that this same metric
tensor arises naturally from a number of different but comple-
mentary viewpoints, including a minimization of the operator
fidelity, and a focus on the Grassmannian structure of the dy-
namics.
Our second major goal in this work was to establish a firm
connection between adiabatic evolution and quantum phase
transitions. By analyzing the infinitesimal variation in the op-
erator fidelity we showed that, in fact the same metric tensor,
arises in both cases. We further derived the quantum critical
scaling of this metric tensor.
Having established a unified geometric framework for adi-
abatic quantum evolution and quantum phase transitions, we
proceeded to find the geodesics on the manifold described by
the unifying Riemannian metric tensor. Such geodesics are
of particular interest in adiabatic quantum computing, where
they correspond to paths which minimize the geometric com-
ponent of the deviation between the actual and desired final
states. We analytically determined the geodesics in three ex-
amples of interest: the Deutsch-Jozsa algorithm, a general-
ization of Grover’s algorithm, and a model described by the
transverse field Ising model. While such examples are im-
portant as proofs of principle, one cannot in general hope to
analytically find the geodesics. For this reason we focused on
the passage through the quantum critical point, and showed
that in general, for second order QPTs, the geodesic in this
case obeys a universal scaling relation.
Among other applications, we expect that the formalism we
have developed will lead to further developments in adiabatic
quantumcomputing, wheretheroleofcriticalityiswellappre-
ciated. We expect additional applications in holonomic quan-
tum computing, where degeneracy plays an essential role, and
whereadifferentialgeometricanalysisofgateerrorminimiza-
tion has not yet been carried out.
VI. ACKNOWLEDGMENTS
Supported by NSF under Grants No. PHY-802678 and
CCF-726439 (to D.A.L.), and PHY-803304 (to P.Z. and
D.A.L.). D.F.A. acknowledges support by a John Stauffer
fellowship from the University of Southern California.

Page 12

12
Note added.—While this work was being finalized for sub-
mission, a related manuscript appeared [56], which similarly
proposes a generalized quantum geometric tensor related to
adiabatic evolution of quantum many-body systems.
Appendix A: Proof of the Wilczek-Zee holonomy formula
Notice that from the fact that P0(s) is a projector, i.e., P0(s) = P2
0(s), we obtain
˙P0(s) =˙P0(s)P0(s) + P0(s)˙P0(s),
(A1)
(where˙P0(s) ≡ ∂sP0(s)), so that
P0(s)˙P0(s)P0(s) = 0,
(A2)
and
[˙P0(s),P0(s)] = 2˙P0(s)P0(s) −˙P0(s).
(A3)
Let Q0(s) denote the projector orthogonal to P0(s), i.e., P0(s) + Q0(s) = I. Then we have
P0(s)Q0(s) = Q0(s)P0(s) = 0.
(A4)
The differential equation for V[0]
αα?(s) [Eq. (7)] can be obtained as follows:
∂sV[0]
αα?(s) = ?˙Φα
0(s)|Vad(s)|Φα?
0(0)? + ?Φα
0(s)?,
H(s) + 2i˙P0(s)P0(s)/T − i˙P0(s)/T
= E0(s)|Φα
0(s)??˙Φβ
g0
?
0(s)|˙Vad(s)|Φα?
0(0)?.
(A5)
In addition, consider the action of Had(s) [Eq. (9)] on |Φα
Had(s)|Φα
0(s)? =
??
|Φα
0(s)?
0(s)? + i˙P0(s)|Φα
0(s)|, we have
0(s)?/T.
(A6)
Since˙P0(s) =?g0
β=1|˙Φβ
0(s)??Φβ
0(s)| + |Φβ
˙P0(s)|Φα
0(s)? = |˙Φα
0(s)? +
β=1
?˙Φβ
0(s)|Φα
0(s)?|Φβ
0(s)?.
(A7)
Using Eq. (A6) and (A7), we can rewrite Eq. (A5) as
∂sV[0]
αα?(s) = ?˙Φα
0(s)|Vad(s)|Φα?
= −iTE0(s)?Φα
0(0)? − iT?Φα
0(s)|Vad(s)|Φα?
0(s)|Had(s)Vad(s)|Φα?
g0
?
0(0)?
0(0)? −
β=1
?Φα
0(s)|˙Φβ
0(s)??Φβ
0(s)|Vad(s)|Φα?
0(0)?
(A8)
Without loss of generality, after setting E0(s) = 0, we obtain the following differential equation for V[0]
αα?(s):
∂sV[0]
αα?(s) = −
g0
?
g0
?
β=1
?Φα
0(s)|˙Φβ
0(s)??Φβ
0(s)|Vad(s)|Φα?
0(0)?
= −
β=1
Aαβ(s)V[0]
βα?(s),
(A9)
whose solution is
V[0](s) = P exp
?
−
?s
0
A(s?)ds?
?
,
(A10)
with
Aαβ≡ ?Φα
0|∂s|Φβ
0?.
(A11)

Page 13

13
Appendix B: Proof of Eq. (27)
Equation (A2) yields
[˙P0,P0]2= −(˙P0P0˙P0+ P0˙P2
0P0).
(B1)
Using Eq. (A1) to write P0˙P0=˙P0−˙P0P0and substituting this into the first term of Eq. (B1) we then have
[˙P0,P0]2= −(˙P2
= −˙P2
The second term vanishes, as can be seen by using Eq. (A1) to write˙P2
0−˙P2
0+ Q0˙P2
0P0+ P0˙P2
0P0)
0P0.
(B2)
0= (˙P0P0+ P0˙P0)2:
Q0˙P2
0P0= Q0(˙P0P0˙P0P0+˙P0P0˙P0+ P0˙P2
0P0+ P0˙P0P0˙P0)P0= 0,
where we used Eq. (A2) on the first two summands and Eq. (A4) on the last two. Thus we conclude that
[˙P0,P0]2= −˙P2
0| is Hermitian and that therefore [˙P0,P0] is anti-Hermitian. Thus both˙P0
0.
(B3)
Note that˙P0 =
and [˙P0,P0] are unitarily diagonalizable: −˙P0 = V DV†, [˙P0,P0] = WEW†, where V and W are unitary, while D and E
are the diagonal matrices of eigenvalues. Therefore it follows from Eq. (B2) that ?V D2V†? = ?WE2W†?, and from the
unitary invariance of the operator norm that ?D2? = ?E2?. From here we conclude that the maximum absolute values of their
eigenvalues are equal, i.e.:
?g0
α=1|˙Φα
0??Φα
0| + |Φα
0??˙Φα
?[˙P0,P0]? = ?˙P0?.
(B4)
It also follows that ?˙P2
0? = ?[˙P0,P0]2? = ?D2? = ?D?2= ?[˙P0,P0]?2, i.e.,
?[˙P0,P0]? =
?
?˙P2
0?.
(B5)
Next we wish to show that
˙P0= −
?
P0˙H
1
H − E0
+
1
H − E0
˙HP0
?
.
(B6)
To prove this note first that the Hamiltonian can be decomposed as
H = E0P0+ Q0HQ0.
(B7)
Then
˙H =˙E0P0+ E0˙P0−˙P0HQ0+ Q0˙HQ0− Q0H˙P0,
(B8)
and multiplying this equation by P0from the right while using Eqs. (A2) and (A4) and the fact that H commutes with P0, yields
˙HP0 =
˙E0P0+ E0˙P0P0− (I − P0)H˙P0P0
˙E0P0+ E0˙P0P0− H˙P0P0.
=
(B9)
The operator H − E0I is invertible when its domain excludes the spectrum of H (and is then called the “reduced resolvent;”
see, e.g., Ref. [33]). That is, the inverse is defined as Q0[H − E0]−1Q0(but for brevity and when there is no risk of confusion,
we simply write [H − E0]−1henceforth). With this restriction in mind we then have
1
H − E0(˙H −˙E0)P0= −
where in the last step we used
˙P0P0= −
1
H − E0
˙HP0,
(B10)
1
H − E0P0= P0
1
H − E0
= 0,
(B11)

Page 14

14
which is due to the fact that the range of [H − E0]−1is the range of Q0[recall also Eq. (28)]. Similarly, by multiplying
Eq. (B8) from the left by P0we obtain:
P0˙P0= −P0˙H
1
H − E0.
(B12)
Adding Eqs. (B10) and (B12), and using Eq. (A1) again then yields Eq. (B6).
As a corollary, we can also calculate˙E0(s) from Eq. (B9)
˙E0(s) = Tr[˙HP0]/g0.
(B13)
Calculation of¨P0or higher order derivatives of P0follows similar logic (see, for example, Ref. [33]). For example, we obtain
?
¨P0= −
?
˙P0˙H
1
H − E0
+ P0¨H
1
H − E0
+ P0˙H∂s
1
H − E0
?
+ ∂s
?
1
H − E0
?
˙HP0+
1
H − E0
¨HP0+
1
H − E0
˙H˙P0
?
. (B14)
.
This relation can be simplified further after replacing˙P0[Eq. (B6)], using the identity
?
and inserting˙E0[Eq. (B13)]. However, we do not need the final explicit form here.
We are now ready to prove Eq. (27). Let
∂s
1
H − E0
?
= −
1
H − E0(˙H −˙E0)
1
H − E0,
(B15)
A ≡
1
H − E0
˙HP0,B ≡ P0˙H
1
H − E0.
(B16)
Then, using Eqs. (B5), (B6), and (B11) yields
?[˙P0,P0]? =
?
?A†A + B†B?.
(B17)
Note that A†A and B†B are both positive operators and that they have orthogonal support. Therefore ?A†A + B†B? =
max{?A†A?,?B†B?}. Moreover, we have A†A = BB†, and it is a basic property of the operator norm that ?BB†? = ?B†B?
for any operator B. Thus
??A†A + B†B? =
Appendix C: Proof of the error formula in the Frobenius norm
??A†A?, which is Eq. (27).
Starting from the definition of the adiabatic error, Eq. (33), we have, by using Eq. (A2) together with P2
invariance of the trace:
?s
=
0
?s
0= P0and cyclic
?(s) =
0
?
?
?
Tr[(P0˙P0−˙P0P0)(˙P0P0− P0˙P0)]ds?
?s
Tr[P0˙P0˙P0+˙P0P0˙P0]ds?
=
0
Tr[P0(∂iP0)(∂jP0) + (∂iP0)P0(∂jP0)]˙ xi˙ xjds?
(C1)
where˙P0= ∂iP0˙ xi. Using P2
0= P0once more to obtain P0(∂iP0) + (∂iP0)P0= ∂iP0we have:
?s
=
0
?(s) =
0
?
?
Tr[{∂iP0− (∂iP0)P0}(∂jP0) + (∂iP0)P0(∂jP0)]˙ xi˙ xjds?
(C2)
?s
2g0gij(x)˙ xi˙ xjds?
(C3)
where the metric tensor is defined as gij≡ Tr[∂iP0∂jP0]/2g0, which is Eq. (36).

Page 15

15
Next let us derive Eq. (37). From Eq. (B6) we have
∂iP0= −
?
P0(∂iH)
1
H − E0
+
1
H − E0(∂iH)P0
?
.
(C4)
Inserting this into Tr[∂iP0∂jP0] and expanding the product while using Eq. (B11), we obtain:
??
?
?
H − E0
as desired.
Tr[∂iP0∂jP0] = TrP0(∂iH)
1
H − E0
?
?
+
??
?2
1
H − E0(∂iH)P0
1
H − E0
??
P0(∂jH)
1
H − E0
+
1
H − E0(∂jH)P0
??
?
= TrP0(∂iH)
1
H − E0
1
?
?
(∂jH)P0+
?
1
H − E0(∂iH)P0P0(∂jH)
?
1
H − E0
?
= TrP0(∂iH)(∂jH)P0
+ TrP0(∂jH)
1
H − E0
?2
(∂iH)P0
,
(C5)
Appendix D: Proof that g is a metric
By definition, a metric must satisfy three properties [1]: it must be positive, real, and symmetric.
(1) Positive: For any nonzero α(x) ∈ TM(x) we have
α(x) · g(x) · α(x) = gij(x)αi(x)αj(x)
=
??
≡ Tr[C†(α,x)C(α,x)] ≥ 0,
where
1
2g0Tr[?∂iP0(x)??∂jP0(x)?]αi(x)αj(x)
= Tr
√2g0αi(x)∂iP(x)
1
?
lk
?
1
√2g0αj(x)∂jP(x)
?
kl
?
(D1)
C(α,x) ≡
1
√2g0αi(x)∂iP(x).
(D2)
Note that although Tr[(dP0)2] is always positive, when we move to a coordinate x the resulting pull-back metric g(x) might
become singular (non-invertible) at some points or even identically zero. In this strict sense g(x) is not a metric.
(2) Real: This is obvious from the very construction of g = Re[G].
(3) Symmetric: This is obvious from the definition and cyclic invariance of the trace: gij ≡ Tr[∂iP0∂jP0]/2g0 =
Tr[∂jP0∂iP0]/2g0= gji.
Appendix E: Proof of the operator fidelity inequalities
We start by proving Eq. (41). From the definition of the operator fidelity, Eq. (40), with ? = I/N, we have, using Eq. (14):
????Tr
=
N
0
≥ 1 −1
N
0
= 1 −1
N
0
f(s) =
?I
NΩ(s)
?s
?s
?s
?????=
Tr[KTΩ]ds?
????Tr
?I
N−1
????
N
?s
0
KT(s?)Ω(s?)ds?
?????
????1 −1
|Tr[KTΩ]|ds?
???Tr([∂s?P0,P0]V V†
ad)
???ds?,
(E1)
where in the last line we used the definitions of Ω(s) [Eq. (13)] and KT(s) [Eq. (15)], and cyclic invariance of the trace. Now
recall the Cauchy-Schwartz inequality for operators [18]:
?A?2?B?2≥ |?A,B?| :=??Tr[A†B]??.
(E2)