Current address:] London Mathematical Laboratory, 18 Margravine Gardens,
London W6 8RH, United Kingdom.

# Noise Effects on the Wilczek-Zee Geometric Phase

Pedro Aguilar<EMAIL_ADDRESS>Chryssomalis Chryssomalakos
<EMAIL_ADDRESS>Edgar Guzmán-González
<EMAIL_ADDRESS>[  Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de México
PO Box 70-543, 04510, CDMX, México

###### Abstract

Non-abelian geometric phases have been proposed as an essential ingredient in
logical gate implementation — their geometric nature guarantees their
invariance under reparametrizations of the associated cyclic path in parameter
space. However, they are still dependent on deformations of that path, due to,
_e.g._ , noise. The first question that we tackle in this work is how to
quantify in a meaningful way this effect of noise, focusing, for concreteness,
on the nuclear quadrupole resonance hamiltonian — other systems of this nature
can clearly be treated analogously. We consider a precessing magnetic field
that drives adiabatically a degenerate doublet, and is subjected to noise, the
effects of which on the Wilczek-Zee holonomy are computed analytically. A
critical review of previous related works reveals a series of assumptions,
like sudden jumps in the field, or the presence of white noise, that might
violate adiabaticity. We propose a state-independent measure of the effect,
and then consider sinusoidal noise in the field, of random amplitude and
phase. We find that all integer noise frequencies $m\neq 2$ behave similarly,
in a manner reminiscent of the abelian case, but that noise of frequency $m=2$
has a very different, and, at the same time, very pronounced effect, that
might well affect robustness estimations.

##  I Introduction

Recent interest in applications of geometric phases to quantum computing
Zanardi and Rasetti (1999); Pachos and Zanardi (2001); Ekert  _et al._ (2000),
based on the seminal work of Berry Berry (1984), Wilczek and Zee Wilczek and
Zee (1984), and Aharonov and Anandan Aharonov and Anandan (1987); Anandan
(1988), has prompted a flurry of related activity in the last decade or so.

A natural division of the above work is into the abelian and non-abelian case.
In the first, which is the only one where the term “phase” applies literally,
the wavefunction of a non-degenerate hamiltonian eigenstate acquires a phase
factor, additional to the expected dynamical one, every time it traces
adiabatically a loop in the projective Hilbert space $P(\mathcal{H})$ of the
system. In the second, the eigenstate is assumed degenerate, and the above
phase factor gets promoted to a unitary matrix operating in the degenerate
subspace. In both cases, the “phase” only depends on the loop traced in
$P(\mathcal{H})$, and is insensitive to its parametrization, hence the term
“geometric”. Generalizations exist that drop both the eigenstate and the
adiabatic requierement Samuel and Bhandari (1988); Moore and Stedman (1990);
Wang  _et al._ (2007); Xu  _et al._ (2012), but we work with the original
formulation as the distinction between the two is not essential for our
purposes.

A second possible division, transversal to the above, is into experimental,
numerical, and theoretical approaches. On the experimental front, a multitude
of devices, including Josephson junctions and other superconducting circuits
Möttönen  _et al._ (2008); Brosco  _et al._ (2008); Faoro  _et al._ (2003);
Solinas  _et al._ (2010); Pirkkalainen  _et al._ (2010), NMR systems Feng  _et
al._ (2013); Li  _et al._ (2017), electronic harmonic oscillators Pechal  _et
al._ (2012), trapped ions Duan  _et al._ (2001), multi-level atoms Unanyan
_et al._ (1999); Abdumalikov Jr  _et al._ (2013), solid state defects Arroyo-
Camejo  _et al._ (2014); Zu  _et al._ (2014) and electron spins in quantum
dots Golovach  _et al._ (2010) or interacting with light Sekiguchi  _et al._
(2017), among others, have already been considered as candidates for quantum
logical gates. On the other hand, much theoretical effort has been expended
evaluating the potential performance of such devices, with particular
attention to their robustness. Potential degrading factors for quantum
computing are generally divided in enviromental ones, that lead, typically, to
decoherence Carollo  _et al._ (2003); Whitney and Gefen (2003); Li and Li
(2011); Fuentes-Guridi  _et al._ (2005); Thunström  _et al._ (2005); Sarandy
and Lidar (2006); Møller  _et al._ (2008), and parametric ones, related to the
finite accuracy of parameter manipulation Shenvi  _et al._ (2003); De Chiara
and Palma (2008, 2003); Solinas  _et al._ (2004, 2012). The seminal work of
Shor Shor (1994) and Grover Grover (1996), whose algorithms catapulted quantum
computing to the forefront of research, has provided a concrete testing ground
for the effect of gate degradation on algorithmic complexity — detailed
analyses can be found in Long  _et al._ (2000); Guo  _et al._ (2001). Lying
somewhere in between and bridging the above two fronts, numerical simulations
have also been carried out, modelling both decoherence Niwa  _et al._ (2002);
Salas (2008) and parametric noise Niwa  _et al._ (2002); Shenvi  _et al._
(2003); Filipp (2008), followed by experiments designed to test their results
Filipp  _et al._ (2009); Berger  _et al._ (2013).

Our own motivation to contribute to this line of research stemmed from the
analysis of parametric noise effects in the archetypical system of a spin-1/2
coupled to a precessing magnetic field contained in De Chiara and Palma
(2003). In that work, a stochastic component, simulating noise, is added to
the field, and the resulting average geometric phase is computed, along with
its standard deviation, to first order in the noise amplitude. The answer for
the average phase is, predictably, null, as the noise is assumed to have zero
mean, and the analysis is linear in the noise components. Two natural
questions that we posed were to calculate, on the one hand, the first non-
vanishing correction to the phase, expected to be quadratic in the noise, and,
in a more conceptual direction, consider a _quantum_ vector noise, and compare
to the classical case — our conclusions are contained in Aguilar  _et al._
(2016). The next natural step, it seems to us, is to explore what happens in
the non-abelian case. Here, too, one may consider classical stochastic noise,
and inquire about the statistics of the resulting holonomies, to the first
non-vanishing order in the noise, or, more ambitiously, one may envisage
quantum fluctuations of the “parameters”, which would now be themselves
quantum dynamical variables, and compute corrections to the geometric phase
due to quantum fluctuations in the “parameters”. The present work deals with
the first of these problems, in the particular example of nuclear quadrupole
resonance, that has been studied both theoretically Zee (1988) and
experimentally Tycko (1987) (note though that in this last reference, the
treatment is essentially abelian). Although this system is not usually
associated to quantum computing, it serves well as a prototype for this kind
of analysis — we expect our methods and conclusions to be easily adaptable to
more realistic scenarios.

Some related work has appeared before, in Solinas  _et al._ (2004, 2012). In
the former, a numerical approach is adopted. The control parameters
$\Omega_{i}$, $i=1,2,3$, of a hamiltonian are perturbed by piecewise constant
noise, that changes after a fixed time interval $\tau$, so that the (3D)
vector parameter $\vec{\Omega}(t)$ takes the value
$\vec{\Omega}_{0}+\delta\vec{\Omega}_{k}$ for $k\tau\leq t<(k+1)\tau$,
$k\in\mathbb{Z}$. The associated holonomy only depends on the area enclosed by
the curve the direction of $\vec{\Omega}$ traces on the unit sphere. The
presence of noise modifies this curve, and the modulus of the inner product of
the resulting state with the unperturbed one (known as _fidelity_) is used as
an indicator of the effect of noise. The result is averaged over a sample of
18 states, and over five “runs” of the noise, each run producing a particular
value for $\delta\vec{\Omega}_{k}$, according to a fixed gaussian
distribution.

There are two aspects of the above approach that we find worrisome: first, the
sudden jumps of the noise at integer multiples of $\tau$ might well violate
the adiabaticity condition. There is, in fact, ample evidence that this indeed
happens, in Figure 7 of that reference, where the non-logical states become
heavily populated, a clear sign of non-adiabatic transitions taking place.
Second, for the averaging over states a particular (somewhat symmetric, but
otherwise arbitrary) grid on the unit sphere was chosen, without any
quantification of the dependence of the results on this choice. These
considerations added two items to our to-do list: (i) work with smooth
sinusoidals as noise, limiting their frequency so that adiabaticity is
preserved, and (ii) find a way to quantify the effect of noise that does not
depend on any choice of states.

Ref. Solinas  _et al._ (2012), on the other hand, takes an analytical
approach, which would make it, in principle, more directly comparable with our
intended analysis. After working out the holonomy in the presence of noise,
the authors of Solinas  _et al._ (2012) turn their attention to its
statistics, choosing, in their Eq. (50), a flat-spectrum white noise. It is
not clear to us, however, whether this choice respects adiabaticity, as the
high frequencies (with respect to the scale dictated by the spectrum of the
hamiltonian) are not adequately suppressed: no physical system can lock
adiabatically onto white noise. On the positive side, instead of dubiously
averaging over arbitrarily selected states, the authors of Solinas  _et al._
(2012) choose to focus on the holonomy, and quantify noise effects in a state-
independent way. Accordingly, our to-do list gets two more entries: (iii)
define the noise _statistics_ in a way consistent with adiabaticity, and (iv)
focus on the holonomy itself, rather than on its output acting on particular
states.

Given this state of affairs, we revisit the subject here, benefiting, on the
one hand, from all that _is_ done correctly in the above two references, but
trying, at the same time, to improve on the above mentioned aspects of the
problem. A final point that we felt deserved some attention was that the
hamiltonian considered in both of the above works gives rise to a rather
mildly non-abelian holonomy: as the authors of the second reference point out
(right after their Eq. (35)), their Wilczek-Zee connection matrix
$\mathcal{A}(t)$ is such that $[\mathcal{A}(t_{1}),\mathcal{A}(t_{2})]=0$, for
all $t_{1}$, $t_{2}$, so that the time evolution is given by simple
exponentiation. This is reflected in the fact that the resulting holonomy is
proportional to the area on the unit sphere of $\hat{\Omega}$ described above,
a hallmark feature of the abelian case, that has no _a priori_ reason to
persist in a genuinely non-abelian case (in fact, it does not, as we show in
what follows). To avoid such oversimplifications, we consider, as mentioned
above, nuclear quadrupole resonance, in which a spin-3/2 couples to a magnetic
field in such a way that two degenerate doublets form. The Wilczek-Zee
connection in one of them is diagonal, but turns out truly non-abelian in the
other — this is the space our qubit lives in, and our single-qubit gate
operates on. Our aim in this paper is to properly formulate the problem of
noise effects on holonomies, and present reliable results of the resulting
statistics, with an eye towards eventual applications in quantum logical
gates. For this reason, we do not consider two-qubit gates, which, despite
being necessary for universal quantum computing, present little novelty from
our present point of view. Relevant general references that we have found
helpful include Messiah (1962); Peres (1995); Bohm  _et al._ (2003);
Chruściński and Jamiołkowski (2004); Jacobs (2010); Nakahara (1990); Bengtsson
and Zyczkowski (2008).

We conclude this introduction with an overview of the rest of the paper. In
section II, we offer a short presentation of adiabatic non-abelian geometric
phases and of their most relevant properties for the following sections, as
well as the standard treatment of nuclear quadrupole resonance. In section III
we present our treatment of the nuclear quadrupole resonance when the driving
magnetic field is fluctuating randomly, to the first non-trivial order in the
noise amplitude. We employ a measure of these effects that is defined in the
space of unitary matrices themselves, unlike the commonly employed Bures or
Fubini-Study metrics. We study the effects of individual Fourier modes of
fluctuations and find the corresponding probability distribution of the
holonomy. Finally, in section IV we present conclusions, final remarks and
ideas for future research.

##  II Wilczek-Zee geometric phases

We restrict ourselves to the adiabatic cyclic formulations of the non-abelian
geometric phase, namely, we present here the relevant aspects of the Wilczek-
Zee Wilczek and Zee (1984) geometric phase.

Consider a physical system described by a hamiltonian $H(\xi)$ that depends on
external parameters $\xi^{i}$, the spectrum of which presents degeneracy. Let
$\mathcal{H}_{n}^{D}[\xi]$ be the $D$-dimensional degenerate subspace
corresponding to the $n$-th energy level of the hamiltonian $H(\xi)$, with
basis elements $|a,\xi\rangle$ for $a=1,\ldots,D$. Suppose that the $\xi$’s
vary in time adiabatically, tracing a curve
$C\colon\tau\rightarrow\xi^{i}(\tau)$ in the parameter space $M$ — through
these parameters, the hamiltonian acquires a time dependence. When the initial
state is degenerate, and this degeneracy is preserved under adiabatic
evolution, nothing prevents the state from mixing with other states with the
same energy — this mixing is described by a connection matrix $A^{(D)}$, with
1-form entries given by

$A^{(D)}_{ab}=i\langle a,\xi|\text{d}|b,\xi\rangle\,,$ (1)

where $a$, $b$ range over degenerate eigenstates. $A^{(D)}$ is commonly
referred to as the Wilczek-Zee connection because it transforms as a non-
abelian $\text{U}(n)$ connection under local unitary transformations of the
degenerate subspace. It is also evident from (1) that $A^{(D)}$ is
parameterization independent. The initial state, after a complete cycle $C$ of
the parameters, will acquire a dynamical phase $\gamma_{d}$ and will get mixed
with elements belonging to its degenerate subspace via the unitary matrix

$U_{C}^{(D)}=\mathcal{P}\text{e}^{i\oint_{C}A^{(D)}}\,,$ (2)

_i.e._ , the path ordered integral of the Wilczek-Zee connection along the
curve $C$. Note that if in some basis of the degenerate subspace the Wilczek-
Zee connection is diagonal, then, in that basis, no mixing in the subspace
occurs and each degenerate state just accumulates its corresponding Berry’s
phase.

A physical system exhibiting non-trivial Wilczek-Zee geometric phases is
nuclear quadrupole resonance Zee (1988). Consider the hamiltonian
$H=\mu(\mathbf{B}\cdot\mathbf{J})^{2}$ describing the coupling of a spin-3/2
particle with an external magnetic field $\mathbf{B}$. The eigenstates of this
hamiltonian are spin-3/2 states with spin projection $m$ along the
$\hat{\mathbf{B}}$ direction, which we denote by $|\hat{\mathbf{B}},m\rangle$.
The pair of states with projections $+m$ and $-m$ are degenerate, therefore
there exist two 2-dimensional degenerate subspaces, corresponding to $m=\pm
1/2$ and $m=\pm 3/2$. When the magnetic field traces adiabatically some loop,
the evolution of degenerate eigenstates will correspond to some non-trivial
mixing within their subspaces, given by the path ordered exponential of the
Wilczek-Zee connection. The space of parameters in this case is the 2-sphere
on which $\hat{\mathbf{B}}$ lives. The connection in the $m=\pm 3/2$ subspace
is abelian, that is, there is no mixing between states because $\mathbf{J}$
can only produce transitions with $\Delta m=0,\pm 1$. Only in the $m=\pm 1/2$
subspace a non-trivial mixing occurs.

For the computation of the Wilczek-Zee connection we will need the
instantaneous eigenstates of $H$, parameterized by coordinates on the
2-sphere, the local character of the latter giving rise to subtleties that
need special care Zee (1988). We obtain the eigenstates of $H$ by rotation of
the eigenstates of $J_{z}$, written as vectors $(1,0,0,0)$, and so on, in
column vector form, along the geodesic connecting the north pole to the point
with standard spherical coordinates $(\Theta,\Phi)$ defining the
$\hat{\mathbf{B}}$ direction. The corresponding rotation axis and angle are
$(-\sin\Phi,\cos\Phi,0)$ and $\Theta$, respectively, so that the rotation in
state space is effected by
$\exp(-i\Theta(-\sin\Phi,\cos\Phi,0)\cdot\mathbf{J})$ and the resulting
$H$-eigenstates are

$|\hat{\mathbf{B}},+1/2\rangle=\begin{pmatrix}-\frac{\sqrt{3}}{4}\text{e}^{-i\Phi}\sin^{2}\Theta\csc(\Theta/2)\\\
\frac{1}{4}\left[\cos(\Theta/2)+3\cos(3\Theta/2)\right]\\\
-\frac{1}{4}\text{e}^{i\Phi}\left[\sin(\Theta/2)-3\sin(3\Theta/2)\right]\\\
\frac{\sqrt{3}}{2}\text{e}^{2i\Phi}\sin\Theta\sin(\Theta/2)\end{pmatrix},\quad|\hat{\mathbf{B}},-1/2\rangle=\begin{pmatrix}\frac{\sqrt{3}}{2}\text{e}^{-2i\Phi}\sin\Theta\sin(\Theta/2)\\\
\frac{1}{4}\text{e}^{-i\Phi}\left[\sin(\Theta/2)-3\sin(3\Theta/2)\right]\\\
\frac{1}{4}\left[\cos(\Theta/2)+3\cos(3\Theta/2)\right]\\\
\frac{\sqrt{3}}{4}\text{e}^{i\Phi}\sin^{2}\Theta\csc(\Theta/2)\end{pmatrix}.$
(3)

Using (1), the Wilczek-Zee connection in this gauge is

$A_{N}=\left(-\sin\Phi\sigma_{1}+\cos\Phi\sigma_{2}\right)\text{d}\Theta+\left(-\sin\Theta\cos\Phi\sigma_{1}-\sin\Theta\sin\Phi\sigma_{2}+\frac{1}{2}(\cos\Theta-1)\sigma_{3}\right)\text{d}\Phi.$
(4)

The non-abelian character of the Wilczek-Zee holonomy shows itself in that
integrating this connection along a closed path on the sphere gives rise, in
general, to a nontrivial unitary matrix.

We note that for a constant-$\Theta$ loop close to the the north pole,
$A_{N}\approx 0$, while near the south pole we have
$A_{N}\approx-\sigma_{3}\text{d}\Phi$, so that, even for an infinitesimal
loop, the resulting holonomy is a $2\pi$ rotation around $\hat{z}$. This
connection is then defined on the sphere excluding a region around the south
pole (the north patch). A good connection $A_{S}$ for the south patch is
obtained after a gauge transformation
$A_{S}=\rho_{1}^{\dagger}A_{N}\rho_{1}+i\rho_{1}^{\dagger}\text{d}\rho_{1}$
with $\rho_{1}=\text{e}^{-i\Phi\sigma_{3}}$, resulting in

$A_{S}=\left(\sin\Phi\sigma_{1}+\cos\Phi\sigma_{2}\right)\text{d}\Theta+\left(-\sin\Theta\cos\Phi\sigma_{1}+\sin\Theta\sin\Phi\sigma_{2}+\frac{1}{2}(\cos\Theta+1)\sigma_{3}\right)\text{d}\Phi.$
(5)

For calculational purposes, the form of these connections is not convenient.
Restricting to the north patch, through the gauge transformation
$A_{E}=\rho_{2}^{\dagger}A_{N}\rho_{2}+i\rho_{2}^{\dagger}\text{d}\rho_{2}$
with a locally defined $\rho_{2}=\text{e}^{-i\Phi\sigma_{3}/2}$, we obtain the
connection

$A_{E}=\sigma_{2}\text{d}\Theta+\left(-\sin\Theta\sigma_{1}+\frac{1}{2}\cos\Theta\sigma_{3}\right)\text{d}\Phi,$
(6)

which is constant along segments with constant $\Phi$ or $\Theta$ and
therefore generates rotations along fixed axes that can be immediately
integrated. The path ordered exponentials of $A_{N}$ and $A_{E}$, which we
call $U_{N}$ and $U_{E}$, respectively, are related by
$U_{N}=\rho_{2}U_{E}=\text{e}^{-i\Phi\sigma_{3}/2}\mathcal{P}\text{e}^{i\int
A_{E}}$. Therefore, for the closed loop $C_{0}$ consisting of $\Theta$ fixed
at some value $\Theta_{0}$ and $0\leq\Phi<2\pi$, the holonomy is

$U_{\text{N}}(C_{0})\equiv
U_{0}=\text{e}^{-i\pi}\exp\left(2i\pi\left(-\sin\Theta_{0}\sigma_{1}+\frac{1}{2}\cos\Theta_{0}\sigma_{3}\right)\right).$
(7)

In the following sections, we will apply these results for nuclear quadrupole
resonance when the driving parameters have a fluctuating component.

##  III Nuclear quadrupole resonance driven by a fluctuating magnetic field

In the standard example of Berry’s phase, involving a spin-1/2 coupled to a
precessing magnetic field, the effect of stochastic fluctuations of the field
has been analyzed in De Chiara and Palma (2003). The approach taken there was
to include a fluctuating component in the magnetic field, and compute Berry’s
phase as minus one half the solid angle subtended by the curve traced out by
the magnetic field. We propose a more economical treatment, based on the
geometrical nature of the Wilczek-Zee phase: since all that matters in the
calculation is the curve itself, not its particular parametrization, we opt to
skip the particulars of how it was produced by the fluctuating field, and
focus instead on its form, given as a function $\Theta(\Phi)$. In this way,
the considerable complexity introduced in the translation of fluctuating field
components into curve deformations is bypassed, and the calculation simplifies
considerably.

The noisy curves we consider are infinitesimal deformations of the
constant-$\Theta$ circles describing a precessing field, _i.e._ , we take
$\Theta(\Phi)=\Theta_{0}+\epsilon\theta(\Phi)$, where $\theta(\Phi)$ is a
periodic stochastic process, $\theta(0)=\theta(2\pi)$, with values of order 1.
We do not assume $\theta(0)=0$, meaning that while the initial degenerate
subspace coincides with the final one, in each realization of $\theta(\Phi)$,
they differ among themselves from one realization to another. This renders the
naive averaging of the corresponding holonomies meaningless, as each one of
them operates in a different Hilbert space. We remedy the situation by fixing
the base point $(\Theta_{0},0)$ on the unperturbed curve, and considering
infinitesimal segments that connect it to the initial (and final) point of the
perturbed curve. Analytically, we define

$(\Theta(t),\Phi(t))=\begin{cases}(\Theta_{0}+\epsilon\theta(0)(t+1),0)&\mbox{if
}-1\leq t<0\\\ (\Theta_{0}+\epsilon\theta(t),t)&\mbox{if }0\leq t<2\pi\\\
(\Theta_{0}+\epsilon\theta(0)-\epsilon\theta(0)(t-2\pi),2\pi)&\mbox{if
}2\pi\leq t<2\pi+1\\\ \end{cases}$ (8)

where the first and third segments are of order $\epsilon$, their purpose
being to make all realizations depart from and return to the same base point.
The curve describes a circle of fixed polar angle $\Theta_{0}$ subject to
random fluctuations given by $\theta(\Phi)$ at each value of the azimuthal
angle $\Phi$. The parameter $\epsilon$ characterizes the size of the
fluctuations and is assumed to be much smaller than one (the radius of the
sphere of magnetic field directions). To arrive at concrete numerical results,
one needs to specify the particular statistical properties of $\theta(\Phi)$,
which we do later on. An example of such statistics is used in Figure 1 to
produce a particular realization of $\theta(\Phi)$, and the curve it
corresponds to on the unit sphere.

Figure 1: Left: a particular realization of
$\theta(\Phi)=\theta_{0}/\sqrt{2\pi}+\sum_{k=1}^{100}(\theta_{k}^{*}e^{-ik\Phi}+\theta_{k}e^{ik\Phi})/\sqrt{2\pi}$,
where $\theta_{0}$, $\Re(\theta_{k})$, $\Im(\theta_{k})$ are taken to be
normally distributed, with zero average and standard deviation
$\sigma_{0}=\sqrt{2\pi}$, $\sigma_{k}=\sqrt{2\pi}/k$, $k=1,\ldots,100$. Note
that $\theta(\Phi)$ is periodic but does not necessarily vanish at its end
points. Right: the corresponding curve traced by the magnetic field direction
on the unit sphere — the relatively large value of $\epsilon=0.05$ was used
for visual clarity. The fluctuating curve
$\Theta(\Phi)=\Theta_{0}+\epsilon\theta(\Phi)$, with $\Theta_{0}=\pi/3$, is
drawn in black, the unperturbed curve $\Theta(\Phi)=\Theta_{0}$ in gray, and
the initial and final “closing” segments (shown slightly separated for
clarity) in blue.

Note that our assumption above of a periodic noise $\theta(\Phi)$, motivated
by the technical convenience of a discrete spectrum, and which, admittedly,
might seem somewhat artificial, could be easily relaxed. In that case, the
infinitesimal segments connecting $\Theta(0)$, $\Theta(2\pi)$ to the
unperturbed curve would be, in general, unequal — it is easily seen though
that this would not affect our results to leading order: in the non-periodic
case the curve traced out by the magnetic field would not close nicely onto
itself, but it could be modified smoothly, in a $\Phi$\- interval of order
$\epsilon$, $(2\pi-\epsilon,2\pi)$, so that it does — such modification, the
leading effect of which would vanish on the average, would only affect higher-
order corrections. Having said that, we also point out that _some_ (but
certainly not all) sources of noise are, by nature, periodic: for example, the
effects of any background static stray magnetic field present in the lab, that
simply adds to the rotating one, or those of mechanical imperfections in the
rotating apparatus, resulting, _e.g._ , in a slightly elliptical path for the
magnetic field, may be treated as periodic (albeit not necessarily stochastic)
noise.

We continue our calculation using $A_{E}$ in (6) as the Wilczek-Zee
connection. Using Eq. (8) we compute $\text{d}\Theta$, $\text{d}\Phi$ along
the three segments of the curve — expanding everything up to
$\mathcal{O}(\epsilon^{2})$ we obtain for the connection

$A_{E}(t)=\begin{cases}\epsilon\theta(0)\sigma_{2}\,\text{d}t&\mbox{if }-1\leq
t<0\\\ (A_{0}+\epsilon
A_{1}+\epsilon^{2}A_{2})\,\text{d}t+\mathcal{O}(\epsilon^{3})&\mbox{if }0\leq
t<2\pi\\\ -\epsilon\theta(0)\sigma_{2}\,\text{d}t&\mbox{if }2\pi\leq
t<2\pi+1\\\ \end{cases}$ (9)

with

$\displaystyle A_{0}$
$\displaystyle=-\sin\Theta_{0}\,\sigma_{1}+\frac{1}{2}\cos\Theta_{0}\sigma_{3}\,,$
(10) $\displaystyle A_{1}$
$\displaystyle=-\cos\Theta_{0}\theta(t)\,\sigma_{1}+\dot{\theta}(t)\,\sigma_{2}-\frac{1}{2}\sin\Theta_{0}\theta(t)\,\sigma_{3}\,,$
$\displaystyle A_{2}$
$\displaystyle=\frac{1}{2}\sin\Theta_{0}\,\theta^{2}(t)\,\sigma_{1}-\frac{1}{4}\cos\Theta_{0}\,\theta^{2}(t)\,\sigma_{3}\,.$

The time evolution operator for the degenerate subspace is determined by this
connection via

$\dot{U}(t)=iA_{E}(t)U(t),\qquad\qquad U(0)=1.$ (11)

Since $A_{E}(t)$ is constant along the closing segments, we may write for the
holonomy $U_{\text{f}}\equiv
U(2\pi+1)=\text{e}^{-i\epsilon\theta(0)\sigma_{2}}U_{\text{c}}\,\text{e}^{i\epsilon\theta(0)\sigma_{2}}$,
where $U_{\text{c}}$ is the holonomy associated to the middle (fluctuating)
segment of the curve, _i.e._ , for $0\leq t<2\pi$. For the latter, we note
that $A_{0}$ in (10) is time independent, so that
$W(t)\equiv\text{e}^{-iA_{0}t}U(t)$ satisfies

$\dot{W}(t)=iA^{I}(t)W(t),\qquad\qquad W(0)=1,$ (12)

where $A^{I}(t)=\epsilon A_{1}^{I}(t)+\epsilon^{2}A_{2}^{I}(t)$ and
$A_{m}^{I}(t)\equiv\text{e}^{-iA_{0}t}A_{m}(t)\text{e}^{iA_{0}t}$, $m=1,2$,
are given explicitly by

$\displaystyle A_{1}^{I}$
$\displaystyle=\frac{\cos\Theta_{0}}{\Omega^{2}}\left[\left(1-\Omega^{2}-\cos(\Omega
t)\right)\theta(t)-\Omega\sin(\Omega
t)\dot{\theta}(t)\right]\sigma_{1}+\left[\cos(\Omega
t)\dot{\theta}(t)-\frac{\sin(\Omega t)}{\Omega}\theta(t)\right]\sigma_{2}$
(13)
$\displaystyle\quad+\frac{\sin\Theta_{0}}{2\Omega^{2}}\left[\left(4-4\cos(\Omega
t)-\Omega^{2}\right)\theta(t)-4\sin(\Omega
t)\dot{\theta}(t)\right]\sigma_{3},$ $\displaystyle A_{2}^{I}$
$\displaystyle=\frac{1}{2}\sin\Theta_{0}\,\theta^{2}(t)\sigma_{1}-\frac{1}{4}\cos\Theta_{0}\,\theta^{2}(t)\sigma_{3},$

where $\Omega=\sqrt{1+3\sin^{2}\Theta_{0}}$. We can integrate this equation
perturbatively through a Dyson series up to second order in $\epsilon$,
arriving at $W(2\pi)$. Then, $U_{\text{c}}=e^{i2\pi A_{0}}W(2\pi)$,
$U_{\text{f}}$ follows by conjugation of $U_{\text{c}}$ by the closing
segments, and, finally, $U_{\text{N}}^{C}=e^{-i\pi}U_{\text{f}}$ is the
holonomy corresponding to our choice of gauge in (3). Putting everything
together, we get

$U_{\text{N}}(C)=\text{e}^{-i\pi}\text{e}^{-i\epsilon\theta(0)\sigma_{2}}\text{e}^{i2\pi
A_{0}}W(2\pi)\text{e}^{i\epsilon\theta(0)\sigma_{2}}\,,$ (14)

where

$\displaystyle W(2\pi)$
$\displaystyle=\exp\left(i\epsilon\int_{0}^{2\pi}\text{d}t_{1}A_{1}^{I}(t_{1})+i\epsilon^{2}\int_{0}^{2\pi}\text{d}t_{1}A_{2}^{I}(t_{1})+\frac{\epsilon^{2}}{2}\int_{0}^{2\pi}\text{d}t_{1}\int_{0}^{t_{1}}\text{d}t_{2}\left[A_{1}^{I}(t_{1}),\,A_{1}^{I}(t_{2})\right]\right)\,.$

Commuting the second factor in the r.h.s. of (14) past the third one, and
realizing that $e^{-i\pi}e^{i2\pi A_{0}}$ is just the unperturbed holonomy
$U_{0}$ of (7), we find

$U_{\text{N}}(C)=U_{0}\,\text{e}^{-i\epsilon\theta(0)\hat{\mathbf{m}}\cdot\bm{\sigma}}W(2\pi)\text{e}^{i\epsilon\theta(0)\sigma_{2}}\equiv
U_{0}\,U_{\epsilon}\,,$ (15)

where the last equation defines the “noise correction factor” $U_{\epsilon}$,
and

$\hat{\mathbf{m}}\cdot\bm{\sigma}=-\frac{\cos\Theta_{0}\sin(2\pi\Omega)}{\Omega}\,\sigma_{1}+\cos(2\pi\Omega)\,\sigma_{2}-\frac{2\sin\Theta_{0}\sin(2\pi\Omega)}{\Omega}\,\sigma_{3}\,.$
(16)

$U_{\text{N}}(C)$, as given in (14), (15), is the holonomy corresponding to a
single realization of $\theta(\Phi)$. In the following, we will consider an
appropriate averaging over all such realizations. Two questions that will have
to be answered before attempting that, are, first, which quantity exactly
should be averaged and, second, what type of statistics to assume for the
fluctuations.

###  III.1 Quantifying the effect of noise

In the implementation of quantum gates based on non-abelian holonomies, error
quantification relies typically on the concept of fidelity, itself deriving
from the Bures metric on density matrices. In our case, that only involves
pure states, the Bures metric reduces to the well-known Fubini-Study metric
Bengtsson and Zyczkowski (2008), which is defined on the projective Hilbert
space and, hence, ignores phase information. Such a concept of distance is
probably too coarse for our purposes, since the non-abelian holonomies we
study are generalizations of the abelian Berry’s phase, which the Fubini-Study
metric cannot detect. We are led then to look for a metric on unitary
matrices, rather than states, aiming at a unified geometrical treatment of
both the abelian and non-abelian cases.

For the case of $SU(2)$, a concept of distance between two elements

$U=x_{0}\mathbf{1}+i\mathbf{x}\cdot\bm{\sigma}\,,\qquad
V=y_{0}\mathbf{1}+i\mathbf{y}\cdot\bm{\sigma}\,,$ (17)

with $x_{0}^{2}+|\mathbf{x}|^{2}=1=y_{0}^{2}+|\mathbf{y}|^{2}$, is given by

$d(U,V)=\arccos\left(\frac{1}{2}\text{Tr}(UV^{\dagger})\right)=\arccos(x_{0}y_{0}+\mathbf{x}\cdot\mathbf{y})\,,$
(18)

which is evidently bi-invariant, and hence, invariant under gauge
transformations —geometrically, it gives the angle between the points
$(x_{0},\mathbf{x})$, $(y_{0},\mathbf{y})$, representing $U$, $V$,
respectively, on $S^{3}$ (_i.e._ , the underlying manifold of $SU(2)$).
Although the holonomies that we compute in the rest of the paper are in fact
elements of $SU(2)$, we would like, for completeness, to extend the above
definition to the whole of $U(2)$, maintaining its essential property of bi-
invariance. The trouble with using (18) for $U(2)$ matrices is that
$\text{Tr}UV^{\dagger}$ is no longer real, in general — we propose to stay as
close as possible to the $SU(2)$ case, by defining

$\tilde{d}(U,V)=\left|\arccos\left(\frac{1}{2}\text{Tr}(UV^{\dagger})\right)\right|\,.$
(19)

To get some idea about the $\tilde{d}$ distance, consider the general $U(2)$
matrix $U=e^{i\alpha}e^{i\theta\hat{\mathbf{n}}\cdot\bm{\sigma}}$. Its $SU(2)$
part (the second exponential factor) lies at an angle $\theta$ from the north
pole (identity) of $S^{3}$ ($SU(2)$). From that point, one has still to “climb
up” an angle $\alpha$ along the $U(1)$ circle, to get to $U$. The distance of
this matrix from the identity is
$\tilde{d}(U,I)\equiv\tilde{d}_{U}=|\arccos(e^{i\alpha}\cos\theta)|$ — we plot
it as a function of both $\theta$ and $\alpha$ in Figure 2.

Figure 2: Plot of the distance $\tilde{d}_{U}$ of the $U(2)$ element
$U=e^{i\alpha}e^{i\theta\hat{\mathbf{n}}\cdot\bm{\sigma}}$ from the identity,
as a function of $\theta$, $\alpha$. For $\alpha=0$ or $\pi$, $\tilde{d}_{U}$
is linear in $\theta$. For $\theta=\pi/2$, $\tilde{d}_{U}=\pi/2$, independent
of $\alpha$. Finally, for $\theta=\alpha=\pi$, $\tilde{d}_{U}=0$. This latter
behavior reflects the fact that from the south pole
$U_{\text{S}}=e^{i\pi\hat{\mathbf{n}}\cdot\bm{\sigma}}$ ($\hat{\mathbf{n}}$
arbitrary) of $S^{3}\sim SU(2)$, one gets to the identity by “climbing up”
$\pi$ radians along the $\alpha$-semicircle.

In our case, the unitary matrices corresponding to evolution with and without
noise are $U_{\text{N}}(C)=U_{0}U_{\epsilon}$ and $U_{0}$, respectively (both
in $SU(2)$), so that, writing
$U_{\epsilon}=e^{i\epsilon\bm{\Lambda}\cdot\bm{\sigma}}$, with $\epsilon\geq
0$, their distance comes out equal to

$d(U_{\text{N}}(C),U_{0})=\arccos\left(\frac{1}{2}\text{Tr}\,U_{\epsilon}\right)=\epsilon|\bm{\Lambda}|\equiv\epsilon\Lambda\,,$
(20)

_i.e._ , the effect of adding noise to the precessing magnetic field is to
displace the corresponding holonomy by an angle $\epsilon\Lambda$ on $S^{3}$.
The above holds true for each realization of $\theta(\Phi)$, _i.e._ ,
$\bm{\Lambda}$ itself is a vector of stochastic processes. Its explicit form
may be deduced from (15),

$U_{\epsilon}=e^{i\epsilon\bm{\Lambda}\cdot\bm{\sigma}}=\text{e}^{-i\epsilon\theta(0)\hat{\mathbf{m}}\cdot\bm{\sigma}}W(2\pi)\text{e}^{i\epsilon\theta(0)\sigma_{2}}\,,$
(21)

after combining the factors on the r.h.s. into a single exponential. Since we
are interested in the statistical properties of $\Lambda$, we defer further
calculations until after we discuss the statistics of the fluctuations, task
to which we now turn.

###  III.2 Noise statistics

The function $\theta(\Phi)$ describing curve fluctuations is formally a
stochastic process. We do not prescribe any particular statistics for it but
only some very general conditions guaranteeing adiabaticity and that no point
of the $\Theta_{0}$ curve is privileged. We denote averaging over all
realizations of fluctuations by $\langle\cdot\rangle$. Our minimalist
requirements are:

  1. i.

The autocorrelation function depends only on the azimuthal angle difference,
$\langle\theta(\Phi_{1})\theta(\Phi_{2})\rangle=R_{\theta\theta}(\Phi_{1}-\Phi_{2})$,
which implies that $\langle\theta^{2}(\Phi)\rangle=R(0)\equiv\sigma^{2}$ is a
$\Phi$-independent quantity. We also get information about the correlations of
the process with its derivatives,

$\displaystyle R_{\theta^{\prime}\theta}(\Phi_{1},\Phi_{2})$
$\displaystyle\equiv\left\langle\theta^{\prime}(\Phi_{1})\theta(\Phi_{2})\right\rangle=-\left\langle\theta(\Phi_{1})\theta^{\prime}(\Phi_{2})\right\rangle=R^{\prime}_{\theta\theta}(\Phi_{1}-\Phi_{2})$
(22) $\displaystyle R_{\theta^{\prime}\theta^{\prime}}(\Phi_{1},\Phi_{2})$
$\displaystyle\equiv\left\langle\theta^{\prime}(\Phi_{1})\theta^{\prime}(\Phi_{2})\right\rangle=-R^{\prime\prime}_{\theta\theta}(\Phi_{1}-\Phi_{2})\,,$
(23)

where primes denote derivatives w.r.t. the argument. Consider now a single,
nonzero, mode,
$\theta^{(m)}(\Phi)=(2\pi)^{-1/2}(\theta_{m}\text{e}^{im\Phi}+\theta_{m}^{*}\text{e}^{-im\Phi})$
and write $\theta_{m}=\theta_{m}^{R}+i\theta_{m}^{I}$. Then, the
autocorrelation function is

$\displaystyle R_{\theta\theta}^{(m)}(\Phi_{1}-\Phi_{2})$
$\displaystyle=\left\langle\theta^{(m)}(\Phi_{1})\theta^{(m)}(\Phi_{2})\right\rangle$
(24)
$\displaystyle=\frac{1}{\pi}\left[-2\langle\theta_{m}^{R}\theta_{m}^{I}\rangle\sin\left(m(\Phi_{1}+\Phi_{2})\right)+\langle(\theta_{m}^{R})^{2}+(\theta_{m}^{I})^{2}\rangle\cos\left(m(\Phi_{1}-\Phi_{2})\right)\right.$
$\displaystyle\quad\left.+\langle(\theta_{m}^{R})^{2}-(\theta_{m}^{I})^{2}\rangle\cos\left(m(\Phi_{1}+\Phi_{2})\right)\right]\,,$

from which we deduce that

$\left\langle\theta_{m}^{R}\theta_{m}^{I}\right\rangle=0\,,\qquad\left\langle(\theta_{m}^{R})^{2}\right\rangle=\left\langle(\theta_{m}^{I})^{2}\right\rangle\,,$
(25)

implying

$R^{(m)}_{\theta\theta}(\Phi_{1}-\Phi_{2})=\begin{cases}\frac{1}{\pi}\left\langle|\theta_{m}|^{2}\right\rangle\cos\left(m(\Phi_{1}-\Phi_{2})\right)&\text{for
$m$ nonzero integer}\\\
\frac{1}{2\pi}\left\langle\theta_{0}^{2}\right\rangle&\text{for
$m=0$}\end{cases}\,.$ (26)

  2. ii.

Adiabatic evolution guarantees that a quantum state locks onto an energy
level. We decompose $\theta$ in discrete Fourier components as above. Then,
adiabaticity may be enforced by requiring a small amplitude for $\theta$, and
by suppressing high frequencies, _e.g._ , by taking
$\left\langle|\theta_{m}|^{2}\right\rangle\propto\epsilon m^{-(1+\alpha)}$,
with $\epsilon$ appropriately small and $\alpha\geq 0$. Specific choices
should take into account that the “smallness” mentioned is w.r.t. the scale
set by the energy difference between the state in question and the closest
neighbor connected with it via the time-derivative of the hamiltonian.

###  III.3 Noise effects in nuclear quadrupole resonance

We retake now our calculation, continuing from Eq. (21). Our first aim is to
get an idea of the size of the effect of noise on the holonomy by calculating
the r.m.s. value of $\Lambda$. The latter is of order $\epsilon^{0}$ (with
higher order corrections) since we factored out an $\epsilon$ explicitly in,
_e.g._ , (20), so, to keep our expressions reasonably complicated, we drop all
quadratic and higher terms. This simplifies things considerably, as the
leading term in the exponents of all three factors in the r.h.s. of (21) is of
order $\epsilon$, meaning that all the Baker-Campbell-Hausdorff corrections
can be neglected, being at least quadratic in $\epsilon$. Thus, we arrive at

$\bm{\Lambda}=-\theta(0)\mathbf{m}^{\prime}+\int_{0}^{2\pi}dt_{1}\mathbf{A}_{1}^{I}(t_{1})\,,$
(27)

where $\mathbf{m}^{\prime}\equiv\hat{\mathbf{m}}-\mathbf{e}_{2}$, and we have
written $A_{1}^{I}=\mathbf{A}_{1}^{I}\cdot\bm{\sigma}$. The quantity

$d_{\text{rms}}=\sqrt{\left\langle
d(U_{\text{N}}^{C},U_{0})^{2}\right\rangle}=\epsilon\sqrt{\left\langle\Lambda^{2}\right\rangle}$
(28)

is a measure of the spread of the distribution of points on the 3-sphere
(_i.e._ , the $SU(2)$ manifold) corresponding to all possible evolution
operators. Computing $\Lambda^{2}$ from (27) and taking the average over noise
realizations, we find

$\displaystyle
d_{\text{rms}}^{2}=\epsilon^{2}\left\langle\Lambda^{2}\right\rangle$
$\displaystyle=\epsilon^{2}\left[R_{\theta\theta}(0)||\mathbf{m}^{\prime}||^{2}+\int_{0}^{2\pi}\int_{0}^{2\pi}\text{d}\Phi_{1}\text{d}\Phi_{2}\left\langle\mathbf{A}_{1}^{I}(\Phi_{1})\mathbf{A}_{1}^{I}(\Phi_{2})\right\rangle\right.$
(29)
$\displaystyle\qquad\left.-2\mathbf{m}^{\prime}\cdot\int_{0}^{2\pi}\text{d}\Phi_{1}\left\langle\mathbf{A}_{1}^{I}(\Phi_{1})\theta(0)\right\rangle\right]\,.$

A somewhat lengthy but straightforward computation using (13), (16), and the
above mentioned statistical properties of $\theta(\Phi)$, yields

$\displaystyle d_{\text{rms}}^{2}=\epsilon^{2}$
$\displaystyle\frac{\Omega^{2}-1}{2\Omega^{2}}\int\limits_{0}^{2\pi}\\!\,\text{d}\sigma
R_{\theta\theta}(\sigma)\left[\sigma\Omega^{2}+4(2\pi-\sigma)\left(\Omega^{2}-1\right)\cos(\sigma\Omega)\right.$
(30)
$\displaystyle\left.\quad\quad\quad\quad{}-4\Omega\sin(\sigma\Omega)+4\Omega\sin(2\pi\Omega-\sigma\Omega)-4\sigma-2\pi\Omega^{2}+8\pi\right]\,.$

Note that, to arrive at the above expression, derivatives of
$R_{\theta\theta}(\sigma)$ have been integrated by parts — the extra terms
thus produced cancel among themselves.

###  III.4 Sample statistics

We consider autocorrelation functions of the form

$R_{\theta\theta}(\Phi_{1}-\Phi_{2})=\begin{cases}\frac{1}{\pi}\sigma_{m}^{2}\cos\left(m(\Phi_{1}-\Phi_{2})\right)&\text{for
}m\text{ nonzero integer}\\\ \frac{1}{2\pi}\sigma_{0}^{2}&\text{for
}m=0\end{cases}\,,$ (31)

corresponding to single-mode noise, with
$\sigma_{m}^{2}=\langle|\theta_{m}|^{2}\rangle$. Then, using (30), the average
distance $d_{\text{rms}}^{(m)}$ integrates to

$d_{\text{rms}}^{\,(m)}=\begin{cases}\epsilon\frac{\sigma_{m}}{\sqrt{\pi}}\frac{2\sqrt{m^{2}+\Omega^{2}}\left(1-\Omega^{2}\right)\sin(\pi\Omega)}{\Omega|m^{2}-\Omega^{2}|}&\text{for
}m\text{ nonzero integer}\\\
\epsilon\frac{\sigma_{0}}{\sqrt{2\pi}}\frac{\sqrt{\Omega^{2}-1}}{\Omega^{2}}\sqrt{4(\Omega^{2}-1)\sin^{2}(\pi\Omega)+\pi^{2}\Omega^{2}(4-\Omega^{2})}&\text{for
}m=0\end{cases}\,.$ (32)

In Figure 3 we plot $d_{\text{rms}}^{(m)}$ vs $\Theta_{0}$, for
$m=0,1,\ldots,7$. Notice how the $m=2$ case has a completely different
behavior, compared to the rest, as well as a different scale. We clarify the
origin of this behavior in the following subsection.

Figure 3: Plot of the $d_{\text{rms}}^{(m)}$ vs $\Theta_{0}$ for $m=0$ (blue),
$m=1$ (orange) and $m=2$ (green) (left), and $m=3$ (blue), $m=4$ (orange),
$m=5$ (green), $m=6$ (red) and $m=7$ (purple) (right). Note the difference in
scale between the two plots.

###  III.5 Understanding the results

A look at Figure 3 reveals several characteristics the origin of which is not
immediately obvious at an intuitive level (at least not to the authors): all
$m$-values, except $m=2$, behave similarly, having common zeros at the poles
and the equator, and sharing a more or less common form over the entire range
of $\Theta_{0}$ (the curve $m=0$ differs from the rest in the way it
approaches zero at the equator). For $m\geq 3$, the maximum amplitude of
$d_{\text{rms}}^{\,(m)}/\sigma_{m}$ diminishes with increasing frequency — a
quick check reveals good fit by the law
$d_{\text{rms}}^{\,(m)}|_{\text{max}}\sim m^{-1}$, which gets increasingly
accurate as $m$ increases. And then, the $m=2$ curve has a much larger
amplitude than all the rest, and reaches its maximum at the equator, where all
the other curves have a minimum.

We set out to derive these results in a way that makes them obvious, or at
least, more predictable. We begin with the expression for $\bm{\Lambda}$ in
(27), and perform two successive $\mathfrak{su}(2)$ basis transformations,
motivated by the following considerations: a look at the first of (10), shows
that $\mathbf{A}_{0}$ points in some awkward direction in the 13-plane, making
an angle $\eta$ with the 3-axis, where

$\cos\eta=\frac{\cos\Theta_{0}}{\Omega}\,,\qquad\sin\eta=\frac{2\sin\Theta_{0}}{\Omega}\,.$
(33)

Note that subsequent perturbative calculations involve rotating
$\mathbf{A}_{i}$, $i=1,2$ around $\mathbf{A}_{0}$, as in (13), operation that
would simplify considerably if $\mathbf{A}_{0}$ were pointing along, say, the
3-axis. Thus, our first basis transformation, from the original
$\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$-frame, to a primed one,
$\\{\sigma^{\prime}_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3}\\}$, consists
in a rotation by $\eta$ around the 2-axis, so that $\mathbf{A}_{0}$ only has a
$3^{\prime}$-component,
$\mathbf{A}_{0}\cdot\bm{\sigma}=\frac{\Omega}{2}\sigma^{\prime}_{3}$, the new
basis vectors being given by

$\bm{\sigma}^{\prime}=\left(\begin{array}[]{ccc}\sigma^{\prime}_{1},&\sigma^{\prime}_{2},&\sigma^{\prime}_{3}\end{array}\right)=\left(\begin{array}[]{ccc}\cos\eta\,\sigma_{1}+\sin\eta\,\sigma_{3},&\sigma_{2},&-\sin\eta\,\sigma_{1}+\cos\eta\,\sigma_{3}\end{array}\right)\,.$
(34)

The above mentioned rotations of $\mathbf{A}_{i}$ around $\mathbf{A}_{0}$ mix
the $1^{\prime}$, $2^{\prime}$-components of $\mathbf{A}_{i}$ — their action
simplifies if one works instead in the basis $\hat{\bm{\sigma}}$, given by

$\hat{\bm{\sigma}}=\left(\begin{array}[]{ccc}\sigma_{+},&\sigma_{0},&\sigma_{-}\end{array}\right)=\left(\begin{array}[]{ccc}\sigma^{\prime}_{1}+i\sigma^{\prime}_{2},&\sigma^{\prime}_{3},&\sigma^{\prime}_{1}-i\sigma^{\prime}_{2}\end{array}\right)\,,$
(35)

in which the components $A_{i\pm}$ transform by a phase under a
$3^{\prime}$-rotation. In this latter basis,

$\mathbf{m}^{\prime}\cdot\bm{\sigma}=-i\left(e^{-i2\pi\Omega}-1\right)\sigma_{+}+i\left(e^{i2\pi\Omega}-1\right)\sigma_{-}\,,$
(36)

while

$\displaystyle A^{I}_{0}$
$\displaystyle=\mathbf{A}_{0}\cdot\bm{\sigma}=\frac{\Omega}{2}\sigma_{0},$
(37) $\displaystyle A^{I}_{1}$
$\displaystyle=\mathbf{A}_{1}\cdot\bm{\sigma}=\frac{1}{\Omega}\left(-e^{-i\Omega
t}f(t)\sigma_{+}+\frac{3}{2}\sin\Theta_{0}\cos\Theta_{0}\,\theta(t)\sigma_{0}-e^{i\Omega
t}f^{*}(t)\sigma_{-}\right)\,,$

where $f(t)\equiv\theta(t)+i\Omega\dot{\theta}(t)$ is a periodic function,
with period $2\pi$. Define now a new (non-periodic) function $\mathsf{f}(t)$
as follows

$\mathsf{f}(t)=f(t)u(t)=\begin{cases}f(t)&\text{if }0\leq t\leq 2\pi\\\
0&\text{otherwise}\end{cases}\,,$ (38)

where $u(t)$ is the unit pulse, equal to 1 for $0\leq t\leq 2\pi$, and zero
elsewhere. The Fourier transform of $\mathsf{f}(t)$ is given by

$\displaystyle\tilde{\mathsf{F}}(\omega)$
$\displaystyle=(2\pi)^{-1/2}\int_{-\infty}^{\infty}\text{d}t\,\mathsf{f}(t)e^{-i\omega
t}$ $\displaystyle=(2\pi)^{-1/2}\int_{0}^{2\pi}\text{d}t\,f(t)e^{-i\omega
t}\,,$

and is equal to the convolution of the discrete Fourier transform of $f(t)$,
with the Fourier transform of $u(t)$. The relevance of these definitions
becomes apparent when calculating the second term in the r.h.s. of (27),

$\int_{0}^{2\pi}\text{d}t_{1}A_{1}^{I}(t_{1})=\frac{\sqrt{2\pi}}{\Omega}\left(-\tilde{\mathsf{F}}(\Omega)\sigma_{+}+\frac{3\theta_{0}}{2}\sin\Theta_{0}\cos\Theta_{0}\,\sigma_{0}-\tilde{\mathsf{F}}(\Omega)^{*}\sigma_{-}\right)\,,$
(39)

where $\theta(t)$ has been expanded in its Fourier modes, as before (see,
_e.g._ , the caption of Figure 1). Using (27), (36) and (39), we may write
$\bm{\Lambda}\cdot\bm{\sigma}$ in the $\hat{\bm{\sigma}}$-basis,

$\displaystyle\bm{\Lambda}\cdot\bm{\sigma}$
$\displaystyle=\left[-\frac{\sqrt{2\pi}}{\Omega}\tilde{\mathsf{F}}(\Omega)+i\theta(0)(e^{-2i\pi\Omega}-1)\right]\sigma_{+}+\frac{3\sqrt{2\pi}\theta_{0}}{2\Omega}\sin\Theta_{0}\cos\Theta_{0}\,\sigma_{0}$
(40)
$\displaystyle\quad+\left[-\frac{\sqrt{2\pi}}{\Omega}\tilde{\mathsf{F}}(\Omega)^{*}-i\theta(0)(e^{2i\pi\Omega}-1)\right]\sigma_{-}\,,$

so that $\Lambda^{2}=\Lambda_{+}\Lambda_{-}+\Lambda_{0}^{2}$ becomes

$\Lambda^{2}=\frac{2\pi}{\Omega^{2}}\big{|}\tilde{\mathsf{F}}(\Omega)\big{|}^{2}+4\theta(0)^{2}\sin^{2}(\pi\Omega)-\frac{2\sqrt{2\pi}}{\Omega}\theta(0)\,\Im\left[\left(\text{e}^{i2\pi\Omega}-1\right)\tilde{\mathsf{F}}(\Omega)\right]+\frac{9\pi\theta_{0}^{2}}{2\Omega^{2}}\sin^{2}\Theta_{0}\cos^{2}\Theta_{0}\,,$
(41)

where $\Im[\cdot]$ denotes imaginary part. For $\tilde{\mathsf{F}}(\omega)$ we
find

$\tilde{\mathsf{F}}(\omega)=\frac{1}{\pi}e^{-i\pi\omega}\sin\pi\omega\left(\frac{\theta_{0}}{\omega}+\sum_{k=1}^{\infty}\left(\frac{k\,\Omega-1}{k-\omega}\theta_{k}+\frac{k\,\Omega+1}{k+\omega}\theta_{k}^{*}\right)\right)\,,$
(42)

so that the average, over the various $\theta(t)$, of its modulus squared,
evaluated at $\omega=\Omega$, becomes

$\langle|\tilde{\mathsf{F}}(\Omega)|^{2}\rangle=\frac{1}{\pi^{2}}\sin^{2}\pi\Omega\left(\frac{\sigma_{0}^{2}}{\Omega^{2}}+2\sum_{k=1}^{\infty}\frac{\Omega^{2}+k^{2}\left(\Omega^{4}+(k^{2}-4)\Omega^{2}+1\right)}{(k^{2}-\Omega^{2})^{2}}\sigma_{k}^{2}\right)\,.$
(43)

Single-mode plots of $|\tilde{\mathsf{F}}(\Omega)|_{\text{rms}}$ appear in
Figure 4.

Figure 4: Left: Plot of
$\sqrt{\langle|\tilde{\mathsf{F}}^{(m)}(\Omega)|^{2}\rangle}$, obtained from
(43) assuming a single mode $m$, (_i.e._ , $\sigma_{k}=\delta_{km}$) for
$m=0,1,2,3$. The shaded stripe corresponds to the interval $1\leq\Omega\leq
2$, over which $\Omega$ ranges (back and forth) as $\Theta_{0}$ goes from 0
(north pole) to $\pi$ (south pole). Right: Plot of
$\sqrt{\langle|\tilde{\mathsf{F}}^{(m)}(\Omega(\Theta_{0}))|^{2}\rangle}$, for
the same values of $m$.

Note that as $\Theta_{0}$ ranges through the values 0, $\pi/2$, $\pi$,
$\Omega$ ranges through 1, 2, and back to 1, respectively. Thus, the only part
of the plot on the left in Figure 4 relevant to our means is the shaded
stripe, which is plotted again in the same figure on the right, as a function
of $\Theta_{0}$. As expected,
$|\tilde{\mathsf{F}}^{(m)}(\Omega)|_{\text{rms}}$ has a maximum close to
$\Omega=m$. A curious exception occurs for $m=1$, for which
$|\tilde{\mathsf{F}}^{(1)}(\Omega=1)|_{\text{rms}}=0$, but a moment’s thought
reveals that, in this case, $f(t)=\sqrt{2/\pi}\theta_{1}^{*}e^{-it}$, _i.e._ ,
$f(t)$ has no $e^{it}$-component, so all its energy is at $\omega=-1$. In the
interval $[1,2]$, over which we sample
$|\tilde{\mathsf{F}}^{(m)}(\Omega)|_{\text{rms}}$, we find the near-maximum of
the $m=2$ curve, and the secondary maxima of all the other modes — this
explains the prominence of the $m=2$ curve in Figure 3, as well as several
other observed features of $d^{\,(m)}_{\text{rms}}(\Theta_{0})$ that space
does not allow us to detail here.

###  III.6 Statistics of the Wilczek-Zee holonomy

For the single-mode $\theta^{(m)}(t)$,

$\theta^{(m)}(t)=\begin{cases}\frac{1}{\sqrt{2\pi}}(\theta_{m}e^{imt}+\theta_{m}^{*}e^{-imt})&\text{for
$m$ positive integer}\\\ \frac{1}{\sqrt{2\pi}}\theta_{0}&\text{for
$m=0$}\end{cases}\,,$ (44)

we calculate, with the help of Eq. (40), the components of
$\bm{\Lambda}^{(m)}$ in the $\bm{\sigma}^{\prime}$-frame of Eq. (34),

$\bm{\Lambda}^{(m)}\cdot\bm{\sigma}\\!=\\!\begin{cases}-\frac{4\left(\Omega^{2}-1\right)s_{\pi\Omega}}{\sqrt{2\pi}\Omega(m^{2}-\Omega^{2})}\left((\Omega
c_{\pi\Omega}\theta_{m}^{R}+ms_{\pi\Omega}\theta_{m}^{I})\sigma^{\prime}_{1}+(\Omega
s_{\pi\Omega}\theta_{m}^{R}-mc_{\pi\Omega}\theta_{m}^{I})\sigma^{\prime}_{2}\right)&\text{for
$m>0$}\\\
\frac{2\theta_{0}}{\sqrt{2\pi}\Omega^{2}}\left(c_{\pi\Omega}s_{\pi\Omega}(\Omega^{2}-1)\sigma^{\prime}_{1}+s^{2}_{\pi\Omega}(\Omega^{2}-1)\sigma^{\prime}_{2}+\frac{3\pi\Omega}{2}s_{\Theta_{0}}c_{\Theta_{0}}\sigma^{\prime}_{3}\right)&\text{for
$m=0$}\end{cases}$ (45)

where $\theta_{m}^{R}$ and $\theta_{m}^{I}$ are the real and imaginary parts
of $\theta_{m}$, and we use the notation $s_{x}\equiv\sin x$, $c_{x}\equiv\cos
x$. Note that only for $m=0$ does $\bm{\Lambda}^{(m)}$ acquire a
$3^{\prime}$-component. As a check, one may compute the average modulus
squared of $\bm{\Lambda}^{(m)}$ from these expressions, recovering the results
in (32).

We turn now to the determination of the probability distribution of
$\bm{\Lambda}$, given the one for the Fourier amplitudes $\theta_{m}$ of the
noise $\theta(t)$.

####  III.6.1 Distribution of $\bm{\Lambda}^{(m)}$ for $m\neq 0$

Considering random Fourier amplitudes $\theta_{m}$, obeying a normal
distribution with zero mean and width that scales appropriately with
frequency, suffices to account for physically reasonable fluctuations
complying with adiabaticity. For $m\neq 0$ the real and imaginary components
of the Fourier amplitude $\theta_{m}$ must be statistically independent and
have the same standard deviation (see (25)). Suppose that the probability
distributions of both $\theta_{m}^{R}$ and $\theta_{m}^{I}$ are given by

$P\left(\theta_{m}^{R,I}=\theta\right)=\frac{1}{\sqrt{2\pi}\sigma_{m}}\text{e}^{-\frac{\theta^{2}}{2\sigma_{m}^{2}}}\,.$
(46)

Putting
$\bm{\Lambda}^{(m)}=(\Lambda_{1^{\prime}},\Lambda_{2^{\prime}},\Lambda_{3^{\prime}})$,
the first of (45) implies that

$\displaystyle\left(\begin{array}[]{c}\theta_{m}^{R}\\\
\theta_{m}^{I}\end{array}\right)$
$\displaystyle=-\frac{\sqrt{2\pi}(m^{2}-\Omega^{2})}{4ms_{\pi\Omega}\left(\Omega^{2}-1\right)}\left(\begin{array}[]{cc}mc_{\pi\Omega}&ms_{\pi\Omega}\\\
\Omega s_{\pi\Omega}&-\Omega
c_{\pi\Omega}\end{array}\right)\left(\begin{array}[]{c}\Lambda_{1^{\prime}}\\\
\Lambda_{2^{\prime}}\end{array}\right)\,,$ (53)

and $\Lambda_{3^{\prime}}=0$, which, in turn, leads to

$P\left(\bm{\Lambda}^{(m)}=(\Lambda_{1^{\prime}},\Lambda_{2^{\prime}},\Lambda_{3^{\prime}})\right)=\frac{m\Omega}{2\pi\bar{\sigma}^{2}_{m}}\text{e}^{-\frac{1}{2\bar{\sigma}_{m}^{2}}(m^{2}(c_{\pi\Omega}\Lambda_{1^{\prime}}+s_{\pi\Omega}\Lambda_{2^{\prime}})^{2}+\Omega^{2}(-s_{\pi\Omega}\Lambda_{1^{\prime}}+c_{\pi\Omega}\Lambda_{2^{\prime}})^{2})}\delta(\Lambda_{3^{\prime}})\,,$
(54)

where

$\bar{\sigma}_{m}=\frac{4ms_{\pi\Omega}(\Omega^{2}-1)}{{\sqrt{2\pi}(m^{2}-\Omega^{2})}}\sigma_{m}\,.$
(55)

Figure 5 shows equiprobability curves on the $\Lambda_{3^{\prime}}=0$ plane
for single-mode fluctuations with frequencies $m=1,2,3$. The angle
$\Theta_{0}$ of the unperturbed curve has been chosen so as to maximize the
standard deviation $\tilde{\sigma}_{m}$ of Eq. (55). All these curves are
ellipses (see the exponent in (54)), with their axes rotated by an angle
$\pi\Omega$ w.r.t. the $1^{\prime}$-$2^{\prime}$ frame. As $\Theta_{0}$
varies, the ellipses rotate and, at the same time, expand or stretch, as
described by (55).

Figure 5: Top row: contour plots of the probability density of
$\bm{\Lambda}^{(m)}$, for $m=1,2,3$, in the $1^{\prime}$-$2^{\prime}$ plane.
The plots are 2D since the $3^{\prime}$-component of $\bm{\Lambda}^{(m)}$
vanishes for $m\neq 0$. Bottom row: plots of the standard deviation
$\bar{\sigma}_{m}$ _vs._ $\Theta_{0}$ — the dots indicate the values of
$\Theta_{0}$ used for the corresponding top row plots. The overall scale of
the plots is determined by taking $\langle|\theta_{m}|^{2}\rangle\propto
m^{-1}$.

To get a better idea of the orientation of these ellipses, we define a further
rotation in $\mathfrak{su}(2)$, around the $3^{\prime}$-axis by an angle
$\pi\Omega$, taking the primed frame $\bm{\sigma}^{\prime}$ to the tilded one
$\tilde{\bm{\sigma}}$, under which the exponent in the r.h.s. of (54) becomes
proportional to
$m^{2}\Lambda_{\tilde{1}}^{2}+\Omega^{2}\Lambda_{\tilde{2}}^{2}$, in other
words, the $\tilde{1}$, $\tilde{2}$-axes coincide with the axes of the
ellipses shown in Figure 5. Combining this rotation with the one taking
$\bm{\sigma}$ to $\bm{\sigma}^{\prime}$, (see (34)), we find the composite
rotation $R(\Theta_{0})$ that takes the $\bm{\sigma}$-coordinates of
$\bm{\Lambda}$ to the $\tilde{\bm{\sigma}}$-ones,

$\left(\begin{array}[]{c}\Lambda_{\tilde{1}}\\\ \Lambda_{\tilde{2}}\\\
\Lambda_{\tilde{3}}\end{array}\right)=\frac{1}{\Omega}\left(\begin{array}[]{ccc}c_{\Theta_{0}}c_{\pi\Omega}&\Omega
s_{\pi\Omega}&2s_{\Theta_{0}}c_{\pi\Omega}\\\
-c_{\Theta_{0}}s_{\pi\Omega}&\Omega
c_{\pi\Omega}&-2s_{\Theta_{0}}s_{\pi\Omega}\\\
-2s_{\Theta_{0}}&0&c_{\Theta_{0}}\end{array}\right)\left(\begin{array}[]{c}\Lambda_{1}\\\
\Lambda_{2}\\\ \Lambda_{3}\end{array}\right)\,.$ (56)

Figure 6 shows a plot of the corresponding rotation angle $\alpha(\Theta_{0})$
and the path of $R(\Theta_{0})$ in $SO(3)$, for $0\leq\Theta_{0}\leq\pi$.

Figure 6: Left: plot of the rotation angle $\alpha(\Theta_{0})$ of
$R(\Theta_{0})$, for $0\leq\Theta_{0}\leq\pi$. Right: plot of the path of
$R(\Theta_{0})$ in $SO(3)$, in the angle-axis parametrization, for
$0\leq\Theta_{0}\leq\pi$. The curve starts at $(0,0,\pi)$ for $\Theta_{0}=0$,
passes through $(0,\pi/2,0)$ for $\Theta_{0}=\pi/2$, and ends at $(-\pi,0,0)$
for $\Theta_{0}=\pi$.

Note that if the value taken by $\theta_{m}$ in a particular realization of
$\theta(t)$ is real, the corresponding $\bm{\Lambda}^{(m)}$ is along the
$\tilde{1}$-direction, while if $\theta_{m}$ turns out purely imaginary,
$\bm{\Lambda}^{(m)}$ points along the direction $\tilde{2}$.

####  III.6.2 Distribution of $\bm{\Lambda}^{(0)}$

The second of (45) implies that $\bm{\Lambda}^{(0)}=\theta_{0}\mathbf{n}$,
where the $\Theta_{0}$-dependent $\mathbf{n}$ is given by

$\displaystyle\mathbf{n}\cdot\bm{\sigma}$
$\displaystyle=\frac{2}{\sqrt{2\pi}\Omega^{2}}\left((\Omega^{2}-1)s_{\pi\Omega}\,\sigma_{\tilde{1}}+\frac{3\pi\Omega}{2}c_{\Theta_{0}}s_{\Theta_{0}}\,\sigma_{\tilde{3}}\right)$
$\displaystyle=\frac{2}{\sqrt{2\pi}\Omega^{2}}\left((\Omega^{2}-1)c_{\pi\Omega}s_{\pi\Omega}\,\sigma_{1^{\prime}}+(\Omega^{2}-1)s_{\pi\Omega}^{2}\,\sigma_{2^{\prime}}+\frac{3\pi\Omega}{2}c_{\Theta_{0}}s_{\Theta_{0}}\,\sigma_{3^{\prime}}\right)$
(57)
$\displaystyle=\frac{1}{\sqrt{8\pi}\Omega^{3}}\\{2c_{\Theta_{0}}[-3\pi\Omega(1-c_{2\Theta_{0}})+(\Omega^{2}-1)s_{2\pi\Omega}]\sigma_{1}+4\Omega(\Omega^{2}-1)s^{2}_{\pi\Omega}\,\sigma_{2}$
$\displaystyle\qquad\qquad\qquad{}+s_{\Theta_{0}}[3\pi\Omega(1+c_{2\Theta_{0}})+4(\Omega^{2}-1)s_{2\pi\Omega}]\,\sigma_{3}\\}\,,$

the first equality above making it evident that $\mathbf{n}$ is orthogonal to
the $\tilde{2}$ direction — a plot of $\mathbf{n}$ _vs._ $\Theta_{0}$ appears
in Figure 7. $\bm{\Lambda}^{(0)}$ lies along the line spanned by $\mathbf{n}$,
with probability density that of $\theta_{0}$, scaled by
$|\mathbf{n}|=2/(\sqrt{2\pi}\Omega^{2})(9\pi^{2}\Omega^{2}s^{2}_{2\Theta_{0}}/16+(\Omega^{2}-1)^{2}s^{2}_{\pi\Omega})^{1/2}$.
In particular, if $\theta_{0}$ follows a normal distribution, with zero
average and standard deviation $\sigma_{0}$, $\bm{\Lambda}^{(0)}$ will also
follow a normal distribution along $\mathbf{n}$, with zero average and
standard deviation $|\mathbf{n}|\sigma_{0}$. In terms of its
$\tilde{\bm{\sigma}}$-components, the probability density of
$\bm{\Lambda}^{(0)}$ is given by

$P\left(\bm{\Lambda}^{(0)}=(\Lambda_{\tilde{1}},\Lambda_{\tilde{2}},\Lambda_{\tilde{3}})\right)=\delta\left(\Lambda_{\tilde{1}}-\mu\Lambda_{\tilde{3}}\right)\delta(\Lambda_{\tilde{2}})\frac{1}{\sqrt{2\pi}\bar{\sigma}_{0}}e^{-\frac{\Lambda_{\tilde{3}}^{2}}{2\bar{\sigma}_{0}^{2}}}\,,$
(58)

where

$\bar{\sigma}_{0}=\frac{3\pi}{\sqrt{2\pi}\Omega}c_{\Theta_{0}}s_{\Theta_{0}}\sigma_{0}\,,\qquad\mu=\frac{2(\Omega^{2}-1)s_{\pi\Omega}}{3\pi\,\Omega\,c_{\Theta_{0}}s_{\Theta_{0}}}\,.$
(59)

Figure 7: Plot of $\mathbf{n}$ (see Eq. (57)) _vs._ $\Theta_{0}$, for
$0\leq\Theta_{0}\leq\pi$, in the $\bm{\sigma}$-frame. The curve starts at the
origin of $\mathfrak{su}(2)$ for $\Theta_{0}=0$, pointing upwards, and traces
a loop with $x>0$, returning to the origin for $\Theta_{0}=\pi/2$. It then
traces a second loop, for $\pi/2\leq\Theta\leq\pi$, obtained from the first
one by reflection in the $y$-$z$ plane.

####  III.6.3 Distribution of $\bm{\Lambda}$: the general case

We consider now the general case of a multi-mode noise $\theta(t)$. Since
$\bm{\Lambda}$ is linear in $\theta(t)$, we obtain

$\bm{\Lambda}=\bm{\Lambda}^{(0)}+\sum_{m=1}^{\infty}\bm{\Lambda}^{(m)}\,,$
(60)

where, in the tilded frame,

$\bm{\Lambda}^{(m)}\\!=\\!\begin{cases}-\frac{4\left(\Omega^{2}-1\right)s_{\pi\Omega}}{\sqrt{2\pi}\Omega(m^{2}-\Omega^{2})}\left(\Omega\theta_{m}^{R}\,\sigma_{\tilde{1}}-m\theta_{m}^{I}\,\sigma_{\tilde{2}}\right)&\text{for
$m$ positive integer}\\\
\frac{2\theta_{0}}{\sqrt{2\pi}\Omega^{2}}\left((\Omega^{2}-1)s_{\pi\Omega}\,\sigma_{\tilde{1}}+\frac{3\pi\Omega}{2}c_{\Theta_{0}}s_{\Theta_{0}}\,\sigma_{\tilde{3}}\right)&\text{for
$m=0$}\end{cases}\,.$ (61)

For the second term in the r.h.s. of (60) the statistical independence of the
various $\bm{\Lambda}^{(m)}$ implies

$P\left(\sum_{m=1}^{\infty}\bm{\Lambda}^{(m)}=(\Lambda_{\tilde{1}},\Lambda_{\tilde{2}},\Lambda_{\tilde{3}})\right)=\frac{1}{2\pi\alpha_{1}\alpha_{2}}\text{e}^{-\frac{1}{2}\left(\frac{\Lambda_{\tilde{1}}^{2}}{\alpha_{1}^{2}}+\frac{\Lambda_{\tilde{2}}^{2}}{\alpha_{2}^{2}}\right)}\delta(\Lambda_{\tilde{3}})\,,$
(62)

where

$\alpha_{1}=\sqrt{\sum_{m=1}^{\infty}\frac{\bar{\sigma}_{m}^{2}}{m^{2}}}\,,\qquad\qquad\alpha_{2}=\frac{1}{\Omega}\sqrt{\sum_{m=1}^{\infty}\bar{\sigma}_{m}^{2}}\,.$
(63)

Convolution of the above expression with the probability distribution of
$\bm{\Lambda}^{(0)}$ gives finally

$\displaystyle
P\left(\bm{\Lambda}=(\Lambda_{\tilde{1}},\Lambda_{\tilde{2}},\Lambda_{\tilde{3}})\right)$
$\displaystyle=\frac{1}{(2\pi)^{3/2}\alpha_{1}\alpha_{2}\bar{\sigma}_{0}}\text{e}^{-\frac{1}{2}\left[\frac{(\Lambda_{\tilde{1}}-\mu\Lambda_{\tilde{3}})^{2}}{\alpha_{1}^{2}}+\frac{\Lambda_{\tilde{2}}^{2}}{\alpha_{2}^{2}}+\frac{\Lambda_{\tilde{3}}^{2}}{\bar{\sigma}_{0}^{2}}\right]}\,.$
(64)

Equiprobability surfaces for $\bm{\Lambda}$ are ellipsoids with one axis along
$\tilde{2}$, and the other two rotated in the $\tilde{1}$-$\tilde{3}$ plane.
For each $\Lambda_{\tilde{3}}=\text{constant}$ cross-section we get a planar
ellipse with semiaxes $\alpha_{1}$, $\alpha_{2}$, shifted by
$\mu\Lambda_{\tilde{3}}$ along the $\tilde{1}$-axis.

Figure 8: Top row, left: Surfaces of constant probability density for
$\bm{\Lambda}$, in the $\tilde{\bm{\sigma}}$-frame, assuming the standard
deviation $\sigma_{m}$ of the mode with frequency $m$ proportional to
$m^{-1}$. Top row, right: The product of the three semiaxes of the gaussian
probability distribution of $\bm{\Lambda}$, as a function of the precession
angle $\Theta_{0}$ — the value of $\Theta_{0}$ used for the plot on the left
is the one that maximizes this product, denoted by a dot in the plot. Bottom
row: Plot of the variation of the individual semiaxes with $\Theta_{0}$.

##  IV CONCLUDING REMARKS

We studied the effect of noise on the Wilczek-Zee holonomy of the nuclear
quadrupole resonance hamiltonian. An arbitrary periodic noise was decomposed
into its Fourier components, and the effect of individual frequencies was
analyzed. We found that all frequencies $m\neq 2$ behave similarly: their
effect tends to vanish when the unperturbed precession takes place close to
the poles or the equator (_i.e._ , when $\Theta_{0}$ approaches 0, $\pi/2$, or
$\pi$), while it attains its maxima at two intermediate angles, close to
$\pi/4$ and $3\pi/4$, respectively. This behavior, as function of
$\Theta_{0}$, closely resembles the abelian case, where the geometric phase
accumulated during precession is proportional to the enclosed solid angle, so
that, close to the equator, the effect of noise on the solid angle cancels
out. If all frequencies produced, more or less, this same effect, then the
experimental physicist seeking to minimize the effects of noise would have a
clear-cut solution available: make the unperturbed magnetic field precess
close to the equator. But this simple scenario is upset by the presence of the
$m=2$ harmonic, which attains its maximum exactly at the equator, and,
moreover, affects the holonomy considerably more than the other frequencies,
_e.g._ , about seven times more than the $m=3$ component, assuming equal
amplitudes. Thus, for a specific noise spectrum, the precession angle(s)
$\Theta_{0}$ at which noise effects get minimized can very well be away from
the equator. This predominance of the $m=2$ harmonic seems to be reported here
for the first time, and may be considered as a signature of the truly non-
abelian nature of the system studied.

It would have been interesting to be able to compare our results with those in
Solinas  _et al._ (2004, 2012). However, our analysis in only first-order in
the noise amplitude, while the use of fidelity as indicator of gate
performance entails a quadratic effect. Thus, a meaningful comparison can only
be attempted if the logarithm $\bm{\Lambda}$ of the unitary correction to the
holonomy is known to second order in the noise amplitude, and we have advanced
enough in this calculation to know that the results are much more complicated
than the first-order ones reported here. On the other hand, it should be
relatively easy to repeat the numerical study using sinusoidal noise, aiming,
for example, to verify the exceptional role of the $m=2$ component, and the
consequent departure from the abelian behavior. Another question that could be
settled this way would be whether the (partial) failure of adiabaticity
reported in Solinas  _et al._ (2004) persists when the smoother sinusoidal
noise is used.

On the analytical front, it would be interesting to consider _quantum_ noise,
_i.e._ , replace the classical magnetic field $\mathbf{B}(t)$ with a quantum
vector operator $\mathbf{A}$, the quantum fluctuations of which would play the
role of noise. In the abelian case, such a treatment gives rise to a novel
effect: part of the correction to the geometric phase involves the commutator
of certain components of $\mathbf{A}$ Aguilar  _et al._ (2016). A different
(and earlier) approach, considers a quantized field, and predicts a vacuum-
induced geometric phase (see Fuentes-Guridi  _et al._ (2002) for the original
theoretical treatment, Liu  _et al._ (2011); Larson (2012); Wang  _et al._
(2015) for subsequent theoretical debate, and the recent experimental
observation reported in Gasparinetti  _et al._ (2016)) — we wonder what the
corresponding statements, for either approach, would be in the non-abelian
case. Another line of research, that we currently pursue, is a geometric
description of the Wilczek-Zee holonomy, based on the recent generalization of
the Majorana stellar representation to the case of multipartite antisymmetric
states Chryssomalakos  _et al._ (2021). The subset of the latter that are
$\wedge$-factorizable, _i.e._ , that can be written in the form of a Slater
determinant, are in 1-to-1 correspondence with linear subspaces of the Hilbert
space, of the type employed, as degenerate subspaces of a Hamiltonian, in the
Wilczek-Zee formalism. Cyclic evolution of the Hamiltonian results in the
degenerate subspace tracing out a closed curve in the corresponding
Grassmannian, which, in principle, completely determines the Wilczek-Zee
holonomy. It would be very interesting to see if rotational symmetries of the
degenerate subspaces lead to robust holonomies, as it has been shown recently
to happen in the abelian case Aguilar  _et al._ (2020).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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