# Holonomic quantum manipulation in the Weyl Disk

Victor Boogers  Janis Erdmanis  Yuli Nazarov  Kavli Institute of Nanoscience,
Delft University of Technology, 2628 CJ Delft, The Netherlands

###### Abstract

It has been shown that a Weyl point in a superconducting nanostructure may
give rise to a Weyl disk where two quantum states are almost degenerate in a
2D manifold in the parametric space. This opens up the possibility of a
holonomic quantum manipulation: a transformation of the wave function upon
adiabatic change of the parameters within the degenerate manifold. In this
paper, we investigate in detail the opportunities for holonomic manipulation
in Weyl disks.

We compute the connection at the manifold in quasiclassical approximation to
show it is Abelian and can be used for a phase gate. To provide a closed
example of quantum manipulation that includes a state preparation and read-
out, we augment the holonomic gate with a change of parameters that brings the
system out of the degenerate subspace. For numerical illustrations, we use a
finite value of quasiclassical parameter and exact quantum dynamics. We
investigate the fidelity of an example gate for different execution times.

##  I Introduction

A quantum computer promises unprecedented advantage over classical one for a
set of challenging problems like protein folding[1] and prime number
factorization[2]. However, building it is a challenge since it is necessary to
isolate a set of quantum states from the environment as well as to manipulate
it within the same environment. A particular obstacle is a decoherence whereby
the states pick up the fluctuations of the environment, which adds stochastic
dynamical phases and may induce dissipation[3, 4].

An alternative way to meet the challenge is the quantum manipulation within
degenerate subspaces that have gained considerable interest nowadays[5, 6, 7].
The resonant manipulation that can be applied in the most systems with well-
separated energy levels, does not work for degenerate subspaces. Instead, the
manipulation is performed within a degenerate manifold: a set of parameters
within which the states are degenerate and distinct. An adiabatic change of
parameters in time along a trajectory within this manifold corresponds to a
unitary operator in the degenerate subspace, that depends on the trajectory
rather then on the way it is traversed. Such mapping of a trajectory to a
unitary operator for manifold is often characterized with a connection that
sets how a wavefunction defined at a point is transported to other points in
an infinitesimally small neighbourhood. An important property of a connection
is the holonomy: a property that transporting a vector over different
trajectories with the same starting point and destination results in different
vectors. This is the manifestation of either curvature of the connection or a
singularity that separates the trajectories into topological classes, so that
the transportation results differ only if two trajectories belong to different
classes.

In quantum mechanics, the connection is characterized by a gauge potential[8],
and the unitary operators correspond to the path integrals involving the
potential. They are called holonomic transformations. The advantage of using
these transformations for wave function manipulation is that in adiabatic
limit the result is determined by the trajectory only, rather then by details
of the time dependence of the parameter evolution along the trajectory. This
provides the robustness against parametric noise and other decoherence
sources.

In quantum mechanics, the connection of a manifold can be either Abelian or
non-Abelian. The Abelian connection has been studied by Berry [9] while
Wilczek and Zee [8] addressed the general non-Abelian case specific for
degenerate manifolds. In general, a holonomic transformation is assigned to
any curve in the Hilbert manifold. The tasformations have been studied in the
context of geometric phase [10], and nonadiabatic geometric computation[11,
12, 13, 14, 15, 16, 17, 18, 19]. For an Abelian connection, the unitary
operators representing all trajectories with the same endpoints can be
simultaneously diagonalized. Such diagonalization is not possible for a non-
Abelian connection, and thus holonomy is irreducible. This in principle
permits an implementation of a complete set of quantum gates to achieve
universal holonomic quantum computation in a variety of systems[20]. However,
the experimental realization of these schemes appeared difficult due to long
execution times required to get rid of non-adiabatic corrections[11, 21, 22].

A particular example of holonomic manipulation involves a two-dimensional
system of indistinguishable anyons. Changing the positions of the anyons
provides a connection with a vanishing curvature[23]. The holonomic
transformations are the same for all trajectories from the same homotopy class
specified by number of windings around the anyons. This sets the paradigm of
topological quantum computation.[24] Within the paradigm, a quantum
manipulation is implemented as a braiding, a move of anyons along the
trajectories whereby all anyons return to starting points making loops around
each other. The holonomic transformations in this case are robust against the
fluctuations of the trajectory shapes, this promises to implement
manipulations of an exceptional fidelity[25]. These opportunities have raised
interest in anyonic excitations in solid-state systems such as Majorana
superconductor-semiconductor nanowires[26] and specific fractional quantum
Hall effect setups.[27, 28]

Many quantum systems exhibit topologically protected energy-level crossings in
three-dimensional parameter space that are commonly called Weyl points [29].
In solid-state physics, the parameter space is a space of wave vectors
confined to a Brillion zone of the crystal lattice. The Weyl points in solid-
state band structures are a subject of active theoretical and experimental
research[30]. Another realization of Weyl points concerns a multi-terminal
superconducting nanostructure, where the Weyl points appear as the crossings
of mirror-symmetric Andreev bond states at zero energy[31] in the parametric
space of three independent superconducting phases.

It has been shown recently that the interaction effects can cause a
substantial modification of Weyl points.[32] A generic interaction model
combines soft confinement and fluctuations in the parameter space. In the
quasiclassical limit, a Weyl disk is formed in the vicinity of a point: two
quantum states are (almost) degenerate in a finite two-dimensional region of
the three-dimensional parameter space. The residual level splitting is
exponentially small in the quasiclassical parameter. This makes the degeneracy
physically achievable for a variety of systems. For instance, in a multi-
terminal superconducting junction, the Weyl disk can be realized by placing
large inductances between each superconductor terminal and the nanostructure.
This makes the phase differences at the nanostructure softly constrained by
the phase differences at the terminals.

Figure 1:  A Weyl disk (shown in white) is a 2D elliptic region in the
parametric space where two quantum states are degenerate. A. A purely
holonomic transformation is achieved by adiabatic change of parameters along a
closed trajectory. B. To achieve more functionality, we consider adiabatic
passages beyond the degenerate maniold. A Hadamard gate describes the crossing
of the disk boundary. C. The crossings of the boundary do not have to be in
the same point. D. The gate from (C) is equivalent to the gate where the
trajectory in the disk is closed along the disk boundary.

The degenerate manifold at Weyl disk may be used for holonomic quantum
manipulation. We explore this opportunity in the present Article. Using the
quasiclassical approximation, we compute the connection at the Weyl disk to
show it is Abelian. A corresponding quantum gate (Fig. 1 a) is thus a phase
gate in a proper basis. We show the relation of the phase shift and the Berry
phase from the classic example of $1/2$ spin in magnetic field [9]. To
demonstrate richer opportunities for quantum manipulation, we augment the
purely holonomic transformations by adiabatic passages to the exterior of the
disk. (Fig. 1 b) We show that the crossing of the disk boundary corresponds to
a Hadamard gate. With this, we provide a closed example of quantum
manipulation that incudes an initialization in a superposition state,
holonomic manipulation, and subsequent readout. The crossings of the disk
boundary can occur in different points as well (Fig. 1 c). The resulting gates
are equivalent if a trajectory in the disk is closed along the boundary of the
disk (cf. Fig. 1 c and d). We investigate the work of these quantum gates
beyond adiabatic approximation with a full numerical simulation at a finite
and moderate value of the quasiclassical parameter. We evaluate the gate
fidelity as functio n of execution time and provide analysis of the dominant
nonadiabatic corrections.

The paper is organized as follows. In Section section II we provide a
Hamiltonian description a multiterminal superconducting junction with a Weyl
point and discuss the soft constraints that enable the Weyl disk regime. In
Section section III we consider the Weyl disk manifold in quasiclassical
approximation and supplement it with a numerical example at a finite and
moderate quasiclassical parameter. The Section section IV is separated into
subsections where we (A) recall the concept of holonomic transformations,
(B)compute the connection in the quasiclassical limit, (C) consider the
adiabatic passages beyond the disk, (D) evaluate the connection beyond the
quasiclassical limit. In Section section V we analyze the deviations from
adiabatic approximation, and present the results of the full quantum dynamics
simulation evaluating the fidelity of the swap gate as function of the gate
execution time. We conclude in the Section section VI.

##  II The system

Weyl points in various physical systems have been a subject of an active
research[33, 29, 30]. An important property of a Weyl point is its topological
protection: the conservation of topological charge guarantees that small
perturbations of the system just shift rather than destoy the point and
associated conical singularity in energy spectrum.

Recently it was shown [31] that multiterminal superconducting junctions with
the leads of ordinary topologically trivial material can host Weyl points. In
other words, the lowest in energy Andreev bound state (ABS) can be tuned to
zero energy (Fig. 2). The tuning parameters are the superconducting phases of
the terminals. Owing to gauge invariance, only phase differences between
terminals matter and thus at least four terminals are required to achieve Weyl
points.

Figure 2: (Ref. [31]) Left: Multiterminal superconducting junction with
superconducting phases $\phi_{n}$. Middle: The Andreev bound state spectrum as
a function of one of the supeconductor terminal phases $\phi_{1}$ with the
other phases at general settings. Right: The same spectrum for the choice of
the other phases corresponding to a Weyl point.

The Hamiltonian describing the conical spectrum in the vicinity of a Weyl
point is a $2\times 2$ matrix in the basis of two singlet degenerate ground
states of the nanostructure [34],

$H_{WP}=(\hbar/2e)\sum_{n=x,y,z}I_{n}\phi_{n}\hat{\sigma}_{n}.$ (1)

here, $\hat{\sigma}_{n}$ are the Pauli matrices in the space of the singlet
states, $\phi_{n}$ are the superconducting phases counted from the positions,
and $I_{n}$ are the coefficients defining the energy slopes of the spectrum.
The corresponding energy levels are
$E=\pm(\hbar/2e)\sqrt{\sum_{n}I^{2}_{n}\phi_{n}^{2}}$.

In a realistic setup, the multiterminal superconducting junction is embedded
in a linear circuit (Fig. 3), and thus the phases determining the Weyl point
become dynamical variables rather than parameters,
$\phi_{n}\to\hat{\phi}_{n}$, and can deviate from the external phases
$\phi_{n}^{r}$ that play the role of parameters now. The linear circuit yet
constrains softly the $\hat{\phi}_{n}$ to $\phi_{n}^{r}$.

Figure 3: (Ref. [32]). Formation of the Weyl disk. Left: A four-terminal
Josephson junction embedded in a linear circuit. The linear circuit represents
the finite capacitances and inductances of the superconducting leads and the
surrounding electromagnetic environment. Middle The 3D region where the two
minima of quasiclassical potential are present. The two minima are of the same
energy at a 2D region shown in red: the Weyl disk. Right: The energy spectrum
along a line perpendicular to the disk (upper panel) and one in the disc plane
(lower panel). In upper panel, the energies of two lowest minima split upon
increasing the distance from the disk plane. In lower panel, two minima are
degenerate and merge in one at the disk edge (red dot in the plot).

The constraint in general can be implemented as a quadratic addition to the
energy. In particular, for our setup this is an inductive energy: each
inductance $L_{n}$ adds a term
$(\hbar/2e)^{2}(\hat{\phi}_{n}-\phi^{r}_{n})^{2}/2L_{n}$ constraining the
corresponding phase. The quantum fluctuations of the phase around this point
are determined by the corresponding capacitance of the system that produces
charging energy $(2e\hat{N}_{n})^{2}/2C_{n}$, $\hat{N}_{n}$ being a variable
canonically conjugated to $\hat{\phi}_{n}$. The full Hamiltonian of the system
with a soft constraint reads:

$\displaystyle H(\vec{\phi^{r}})$ $\displaystyle=$ $\displaystyle
H_{WP}+\left(\frac{\hbar}{2e}\right)^{2}\sum_{n}\frac{(\hat{\phi}_{n}-\phi_{n}^{r})^{2}}{2L_{n}}$
$\displaystyle+$ $\displaystyle\sum_{n}\frac{(2e\hat{N}_{n})^{2}}{2C_{n}}$

where the middle term accounts for the soft constraint and the last term for
the fluctuations. This is a minimum model of the embedding linear circuit,
more complex models involve general frequency-dependent response functions of
the circuit and do not change the qualitative conclusions.

The relevant scales in this Hamilonian can be understood when considering a
single-dimension version of it,

$\displaystyle H(\phi^{r})$ $\displaystyle=$
$\displaystyle(\hbar/2e)I\phi\hat{\sigma}_{z}+\left(\frac{\hbar}{2e}\right)^{2}\frac{(\hat{\phi}-\phi_{n}^{r})^{2}}{2L}$
$\displaystyle+$ $\displaystyle\frac{(2e\hat{N})^{2}}{2C}$

The diagonalization of this Hamiltonian is trivial since the quasi-spin and
$\phi$ separate and we have an oscillator centered at the positions that
depend on the eigenvalue of spin $\sigma=\pm 1$,
$\phi=\phi^{r}-\sigma\phi_{0}$, $\phi_{0}\equiv 2eIL/\hbar$. At $\phi^{r}=0$
these two positions correspond to degenerate minina separated by energy
barrier $E_{B}=LI^{2}/2$. The energy spectrum is given by ($m$ being the
number of quanta in the oscillator)

$E_{m,\sigma}=\hbar\omega(m+1/2)+\frac{\hbar}{2e}I\phi^{r}\sigma-E_{B}.$ (4)

where $\omega=\sqrt{LC}$ is the oscillator frequency.

We assess the significance of quantum fluctuations by comparing the barrier
height and the energy quantum in the oscillator. We introduce a quasiclassical
parameter:

$Q=\frac{E_{B}}{\hbar\omega}=\frac{1}{2}\left(\frac{LIe}{\hbar}\right)^{2}\frac{\hbar}{e^{2}Z},$
(5)

$Z=\sqrt{L/C}$ being the characteristic impedance of the oscillator. Since
$Ze^{2}/\hbar\simeq 10^{-2}$ for typical circuits, the parameter can be large
even for relatively small inductances. If $Q\gg 1$, the system is in the
quasiclassical regime and the overlap of the states in two minima is
exponentially small. In the opposite limit $Q\ll 1$ the overlap is big and the
effect of soft confinement can be treated perturbatively.

We see that for one-dimensional version of the Hamiltonian the energy levels
retain conical singularity as far as its dependence on $\vec{\phi}^{r}$ is
concerned. Generally, one expects this to hold for 3D case as well: the
confinement would just renormalize the ”velocities” $\partial
E/\partial\phi_{n}$ defining the singularity. It has been discovered in [32]
that there is an important exception from this general rule: for one of the
directions — we will call it an easy axis and take it for z direction — the
velocity remains finite while vanishing in the limit of large $Q$for two
perpendicular direction. The easy direction is defined by the biggest energy
barrier: $E_{B}^{z}\equiv L^{z}I_{z}/2>E_{B}^{x},E_{B}^{y}$.

Thereby a Weyl point in the presence of confinement and in the quasiclassical
limit becomes a Weyl disk (see Fig. 3): two energy levels remain degenerate
within an ellipse in the $\phi^{r}_{x}-\phi^{r}_{y}$ plane, the semiaxes being
given by ($n=x,y$)

$A_{n}=\frac{4e}{\hbar I_{n}}(E^{z}_{B}-E^{n}_{B}).$ (6)

This is related to the existence of two potential minima in the vicinity of
the Weyl point corresponding to two spin directions. In quasiclassical
approximation, the corresponding wavefunctions do not overlap being localized
near the minima.

It has been shown that the residual level splitting is exponentially small in
the quasiclassical parameter. The approximate degeneracy in the Weyl disk is
potentially interesting for quantum manipilation: that could allow a
superposition of two degenerate states to evolve in time very slowly and
enables holomorphic manupulations of this superposition by changing
$\phi^{r}_{x},\phi^{r}_{y}$ along a trajectory. In the next Section, we will
discuss in detail the properties of the states at the Weyl disk.

##  III Properties of the states at the Weyl disk

In this Section, we complete and expand the analysis of the quantum states at
the Weyl disk that was started in Ref. [32]. We present the numerical results
and compare with the analytical ones in the quasiclassical limit. This
analysis is crucial for understanding the available holonomic transformations
and other manipulations at the manifold.

Figure 4: Numerical results for the energy spectrum and wave functions. For
numerical illustrations in this Article, we choose a moderate value of the
quasiclassical parameter $Q=E_{B}/\hbar\omega_{z}=5$. ($E_{B}\equiv
E_{B}^{z}$). Other parameters are $\hbar\omega_{x,y}/E_{B}=1/3$,
$E_{B}^{x,y}/E_{B}=1/3$. Left pane: Four lowest energy levels of the
Hamiltonian section II in the disk plane $\phi^{r}_{z}=0$ versus
$\phi^{r}_{x}$. The two lowest levels are degenerate deep in the disk. In the
limit $Q\to\infty$ the degeneracy persists till the disk edge
$(\phi^{r}_{x})^{2}/A_{x}^{2}+(\phi^{r}_{y})^{2}/A_{y}^{2}=1$ with $A_{x,y}$
given by Eq. eq. 6. Since $Q=5$ is taken, the residual splitting at the disk
edge is already comparable with $\omega_{x,y,z}$ which defines the energy
distance to the higher levels. However, the splitting is not visible at
$\phi^{r}_{x}<0.5A_{x}$. Inset: Four lowest energy levels versus
$\phi^{r}_{z}$ at $\phi_{x,y}=0$. The splitting is lifted upon a shift
$\phi^{r}_{z}$ in the easy direction and is proportional to the shift. Right
pane: The probability density of $\phi_{z}$,
$p(\phi_{z})=\sum_{\sigma}\int|\Psi_{\sigma}|^{2}d\phi_{x}d\phi_{y}$ in the
disk plane for even (solid line) and odd (dashed line) lowest energy states at
several values of $\phi^{r}_{x}$. The probability deep in the disk separates
in two almost non-overlapping peaks corresponding to two degenerate minima in
the effective potential. At the edge of the disk, the minima merge resulting
in a single peak.

Let us first consider a numerical illustration (Fig. 4). We choose a moderate
value of the quasiclassical parameter $Q=E^{z}_{B}/\hbar\omega_{z}=5$ in the
Hamiltonian (II). The barriers in $x,y$ directions are lower and equal,
$E_{B}^{x,y}/E_{B}=1/3$. The corresponding oscillator frequencies are also the
same, $\hbar\omega^{x,y}=E_{B}^{x,y}$, this suggest the circular symmetry of
the setup with respect to rotations about $z$. The effective $Q$ in this
direction is thus $\simeq 1$. We will use this set of parameters for all
numerical illustrations in the paper, since it proves the feasibility of
holonomic manipulations at moderate values of the quasiclassical parameter.

In the left pane and inset of the Figure, we present the energies of the four
lowest states. We will perform quantum manipulations in the basis of the two
lowest states. As we will see later in dynamical simulations, the wave
function from this basis mostly leaks to the 3rd and 4th state.

In the left pane, we plot the energies versus $\phi^{r}_{x}$ at
$\phi^{r}_{z,y}=0$, that is, in the disk plane. The energies of the two lowest
states are apparently degenerate up to $\phi^{r}_{x}\simeq 0.5A_{x}$. Deep in
quasiclassical limit, they remain degenerate up to $\phi^{r}_{x}=A_{x}$. The
moderate value of $Q$ results in the residual splitting that becomes
comparable with the energy distance between the 3rd and the 2nd state at the
edge of the disk.

It is important for further consideration to note a symmetry of the
Hamiltonian at this choice of the parameters: it is invariant with respect to
$180^{\circ}$ rotation about $x$-axis. Owing two this, the splitting is
diagonal in the basis of odd and even states with respect to the rotation. The
1st and the 4th state are even, while the 2nd and 3rd are odd.

If we change the phase in easy direction, $\phi^{r}_{z}$, the degeneracy is
immediately lifted (inset of the Figure), the splitting being proportional to
$\phi^{r}_{z}$. An instructive picture to comprehend this is that of a two-
well potential depending on $\phi_{z}$. The energies of the distinct potential
minima are aligned in the disk plane and are shifted by $\phi^{r}_{z}$ in
opposite directions. The distance between the mimima reaches maximim at the
center of the disk and decreases upon moving to the edge of the disk where two
minima merge into one. In the quasiclassical limit, the wave function is
localized at the minima. This is illustrated in the right pane of the Figure
where we plot the probability density $p(\phi_{z})$ for even and odd state. We
observe two distinct peaks at the center, slightly overlapping peaks at
$\phi^{r}_{x}\simeq 0.5A_{x}$ and a single peak at the disk edge.

To characterize the double-well potential, we resort to quasiclassical
approximation. If we neglect the quantum fluctuations completely, the quantum
states are localized in superconductor phase space. The states can be
decomposed as follows:

$\ket{\Psi}=\ket{\vec{\phi}}\otimes\ket{S}$ (7)

where $\ket{\vec{\phi}}$ is a wave function with definite values of the
superconducting phases and $\ket{S}$ is a 2-component wave function in the
spin space. This decomposition allows us to determine $\ket{S}$ from the
Hamiltonian (II). Since we are looking for the states of the minimum energy,
we set the spin to be antiparallel to the effective ”magnetic field” at the
point $\vec{\phi}$,

$\vec{W}\cdot\vec{\sigma}|S\rangle=-|S\rangle$ (8)

$\vec{W}$ being the normalized vector in the direction of the ”field”,

$W_{n}=\frac{I_{n}\phi_{n}}{\sqrt{\sum_{n}I_{n}^{2}\hat{\phi}^{2}_{n}}}.$ (9)

Figure 5:  The effective potential (Eq. 10) and comparison with the numerical
results. Left pane: colour contours of the quasiclassical effective potential
in the $\phi_{z}-\phi_{x}$ plane for the parameters at the Weyl disk:
$\phi^{r}_{z},\phi^{r}_{y}=0$ and $\phi^{r}_{x}/A_{x}=0.5,1,1.5$ from the
upper to lower plots. The minima of this double-well potential merge at the
disk edge. Right pane: The locations of the potential minima $\phi^{0}_{x,z}$
given by Eq. eq. 11 and the numerical averages $\langle\phi_{x,z}\rangle$. To
compute the averages, we take the wave functions of the two lowest states and
arrange their superposition corresponding to the state localized in the upper
minimum. The locations and averages are in reasonable agreement even for the
moderate $Q$ chosen.

With this, we evaluate the effective potential for the state $\ket{\psi}$
aweraging the Hamiltonian over the state eq. 8 and neglecting the charging
energy:

$V(\vec{\phi})=-\frac{\hbar}{2e}\sqrt{\sum_{n}I_{n}^{2}\phi_{n}^{2}}+\left(\frac{\hbar}{2e}\right)^{2}\sum_{n}\frac{(\phi_{n}-\phi_{n}^{r})^{2}}{2L_{n}}$
(10)

In Figure 5 (left) we plot the color contours of this effective potential in
the $\phi_{x}-\phi_{z}$ plane at three values of $\phi^{r}_{x}$ at
$\phi^{r}_{z},\phi^{r}_{y}=0$. We see the minima moving towards each other
upon increasing of $\phi^{r}_{x}$ and eventually merging at
$\phi^{r}_{x}=A_{x}$. We expect the wave functions to be localized in the
minima. To check for this, we compare the positions of the quasi-classical
mimima with the numerical averages $\langle\phi_{x,z}\rangle$ to find the
reasonable correspondence even for the moderate $Q$.

The positions of the minima are given by [32]:

$\displaystyle\langle\hat{\phi}_{x}\rangle=\frac{\phi_{0}I_{z}}{A_{x}I_{x}}\phi^{r}_{x}$
$\displaystyle\langle\hat{\phi}_{y}\rangle=\frac{\phi_{0}I_{z}}{A_{y}I_{y}}\phi^{r}_{y}$
$\langle\hat{\phi}_{z}\rangle=\pm\phi_{0}\sqrt{1-\rho^{2}}$ (11)

where $\phi_{0}=(2e/\hbar)I_{z}L_{z}$,
$\rho^{2}=(\phi^{r}_{x})^{2}/A_{x}^{2}+(\phi^{r}_{y})^{2}/A_{y}^{2}$
($\rho^{2}=1$ corresponds to the disk edge).

The locations of the minima determine the spin of the localized states. With
Eqs. 8 and 11 we evaluate the angle $\theta$ it makes with the easy direction:
$\cos\theta=\pm\langle\hat{\phi}_{z}\rangle/\phi_{0}$. The spins of two
localized states in the center of the disk are aligned with $z$ and are
antiparallel. At the disk we come closer to the disk boundary, the the spins
align with the disk plane and are parallel. (See Fig. 6. The dependence of
spin directions on $\vec{\phi}^{r}$ is essential for the holonomic
transformations as discussed in the next Section. To veryfy this numerically,
we compute numerically the average values of $\sigma_{x},\sigma_{z}$ for the
superposition of the two lowest states that corresponds to the localization in
the upper minimum. The results are presented in the right pane of Fig. 6. We
see a reasonable agreement with the quasicalssical results that is expectedly
gets worse near the edge of the disk where the overlap of the states localized
in different minima is significant.

Figure 6:  The spin directions of the localized states compared with the
numerical averages. Left pane: The countours of the even wave function. The
spin directions in the potential minima are shown by the arrows. We rescaled
the axes by $A_{x},A_{z}$ so that the spin directions point towards the
origin. Right pane: the quasiclassical predictions for $\sigma_{x},\sigma_{z}$
as compared with the numerical averages for the superposition of two lowest
eigenfunctions corresponding to the localization in the upper minimum. we show
the quasiclassical spin direction $\vec{n}$ with the spin direction obtained
numerically. We normalize $\langle\vec{\sigma}\rangle$ to
$\langle\vec{\sigma}\rangle^{2}=1$ to validate the decomposition given by Eq.
eq. 7. There is a good agreement even for the moderate $Q$ taken.

##  IV Quantum gates

###  IV.1 Geometric phase and holomonic transformations

Let us recall here the basics of geometric phase and holonomic
transformations. We consider a Hamiltonian $\hat{H}(\vec{x})$ that depends on
a set of parameters $\vec{x}$. We change the parameters in time along a
trajectory $\vec{x}(t)$, this results in the time-dependent Hamiltonian
$\hat{H}(\vec{x})$. We introduce a local basis that diagonalizes
$\hat{H}(\vec{x})$ in each point of parameter space,

$E_{n}(\vec{x})|n\rangle=\hat{H}(\vec{x})|n\rangle$ (12)

The Schrödinger equation in the local basis reads

$i\hbar\dot{\psi}_{n}=E_{n}-\hbar\dot{x}_{i}M^{i}_{nm}\psi_{m}$ (13)

where the effective vector potential $\hat{M}^{i}$

$M^{i}_{nm}=-i\langle n|\partial_{i}(|m\rangle)$ (14)

represents the connection of the bases. With this, an in general, the unitary
transformation of the wave function can be separated into two parts[10]: the
dynamical phase arising from the first term representing the time-dependent
energies and a geometric phase arising from the second term that depends
solely on the trajectory in the parametric space. The geometric phase is an
attractive phenonemon to use in quantum information processing. There are
setups where the dynamical phase can be neglected beyond the adiababic
approximation. Such examples have been actively studied in the field of
nonadiabatic geometric quantum manipulation[11, 12, 13, 14, 15, 16, 17, 18].
Alternatively, the trajectory in the parameter space can be passed
adiabatically. The resulting geometric phase then reduces to a holonomic phase
and is a basis of actively studied holonomic quantum manipulation[6, 7].

The most common example of adiabatic manipulation is the Berry phase [9]. In
this case, the energy levels are non-degenerate, and the adiabaticity implies
that the frequencies associated with the parameter change are much smaller
than the energy distances between the levels. One can neglect the non-diagonal
elements of $\hat{M}^{i}$ so the Shrödinger equation reduces to

$i\hbar\dot{\psi}_{n}=E_{n}(\vec{x}(t))\psi_{n}-\hbar\dot{x}_{i}M^{i}{n}\psi_{n}$
(15)

The dymanical phase separates from geometric phase. The latter depends on the
trajectory only and is given by the line integral over the trajectory,
$\vec{x}_{i},\vec{x}_{f}$ being the initial and final point of the trajectory,

$\beta_{n}=\int_{\vec{x}_{i}}^{\vec{x}_{f}}\vec{M}_{n}\cdot d\vec{l}$ (16)

The vector potential is not gauge invariant and changes upon a gauge
transformaton $|n\rangle\to U(\vec{x})|n\rangle$, $|U|^{2}=1$,

$M^{i}_{n}\to M_{n}^{i}+U^{*}\partial_{i}U$ (17)

The guage-invariant Berry phase is defined for closed trajectories
$\vec{x}_{i}=\vec{x}_{f}$ and, by virture of Stoke’s theorem, equals to the
surface intergal of the curl of $\vec{M}$ over the surface enclosed by the
trajectory.

A fascinating extension of this concept petrains the case where several levels
of the Hamiltonian are degenerate in a subspace of $\vec{x}$. The adiabaticity
implies that the frequencies of the change are much smaller than the energy
distance between the degenerate and non-degenerate levels. The adiabatic
motion along a trajectory results in a unitary transformation in the
degenerate subspace $\hat{S}(\vec{x}_{i},\vec{x}_{f})$,

$\hat{S}=\mathcal{P}\exp\left(i\int_{\vec{x}_{i}}^{\vec{x}_{f}}\vec{\hat{M}}(\vec{x})\cdot
d\vec{l}\right).$ (18)

$\mathcal{P}$ stands here for the ordering of $\hat{\vec{M}}$along the
trajectory. The operator vector potential is not invariant with respect to the
unitary transformations of the basis, $|n\rangle\to U(\vec{x})|n\rangle$,
$|U|^{2}=1$,

$M^{i}_{n}\to\hat{U}^{\dagger}M_{n}^{i}\hat{U}+\hat{U}^{\dagger}\partial_{i}\hat{U}.$
(19)

The gauge invariance is achieved for closed trajectories, and pertains the
eigenvalues of $\hat{S}$.

The holonomic transformations can be Abelian and non-Abelian. They are Abelian
if $\hat{M}^{i}(\vec{x})$ can be chosen to commute for all $\vec{x}$. As we
will see soon, this will be the case under consideration. The non-Abelian
connection permits to realize a universal set of quantum gates [6, 7] from
holonomic transformations over different trajectories. However, the
experimental implementation of them is difficult because of the nonadiabatic
corrections[11, 21, 22].

###  IV.2 Pure holonomic trasformation

Let us apply these general considerations to the manifold of nearly degenerate
wave functions at the Weyl disk. The first step is the parametrization of the
basis in the degenerate subspace. We restrict ourselfs to the deep
quasiclassical limit. The natural basis choice are that of wave functions
localized either in upper ($\phi_{z}>0$) or lower ($\phi_{z}>0$) minimum. As
discussed, those can be decomposed into spin and orbital part,

$|+\rangle=|S\rangle_{+}|O\rangle_{+};\;|+\rangle=|S\rangle_{-}|O\rangle_{-}$
(20)

Here, $|O\rangle_{\pm}$ are the normalized wave functions in $\vec{\phi}$
space located at the minima positions
$(\langle\hat{\phi}_{x}\rangle,\langle\hat{\phi}_{y}\rangle,\pm\sqrt{1-\rho^{2}})$
(see Eq. 11). The $|S\rangle_{\pm}$ are spinors representing the spin
antiparallel to the corresponding $\vec{w}_{\pm}$ (see Eq. 9),
$\vec{w}_{-}=(w^{x}_{+},w^{y}_{+},-w^{z}_{+})$.

The best choice of the coordinates in the elliptic disk corresponds to an
unambiguous mapping of $(\phi^{r}_{x},\phi^{r}_{y})$ to the upper hermisphere
of the vector $\vec{w}_{+}$. The two parametirzing angles $\theta$, $\alpha$,
$0<\theta<\pi/2,-\pi<\alpha<\pi$ are determined from

$\sin\theta=\rho;\;e^{i\alpha}=\left(\frac{\phi^{r}_{x}}{A_{x}}+i\frac{\phi^{r}_{y}}{A_{y}}\right)\rho^{-1};$
(21)

while $\vec{w}_{+}=(\cos\alpha\sin\theta,\sin\alpha\sin\theta,\cos\theta)$.

It is essential to choose $|S\rangle_{\pm}$ to insure the continuity over the
hemishere and the absence of singularity. This is achieved by setting

$|S\rangle_{+}=\begin{bmatrix}-e^{-i\alpha}\sin\frac{\theta}{2}\\\
\cos\frac{\theta}{2}\end{bmatrix};\;|S\rangle_{-}=\begin{bmatrix}\cos\frac{\theta}{2}\\\
-e^{i\alpha}\sin\frac{\theta}{2}\end{bmatrix}.$ (22)

We compute the connection from Eq. 14. We may neglect the overlap between
$|O\rangle_{+}$ and $|O\rangle_{-}$ in the quasiclassical limit, so the
connection is diagonal in this basis and therefore Abelian. Moreover, the
derivatives of $|O\rangle_{\pm}$ with respect to $\theta,\alpha$ may be
neglected as well. They give rise to the quantities proportional to the
expectation values of the momentum and angular momentum for these states,
those are zero since the states are localized. The connection is thus
determined by the derivatives of $|S\rangle_{+}$ and reads:

$\hat{M}^{\alpha}=-\hat{\tau}_{z}\sin^{2}\frac{\theta}{2};\;\hat{M}^{\theta}=0$
(23)

Thereby we reduce the situation to the classic example of Berry phase for an
electron spin in spin magnetic field of constant amplitude [9]. Any holonomic
transformation has a form of $\exp(-i\tau_{z}\beta)$. This is a phase gate,
whereby the states $|\pm\rangle$ acquire opposite phase shifts $\mp\beta$,
$\beta$ being the Berry phase from the example. For any closed trajectory,
$\beta$ is thus the half of a solid angle enclosed by the trajectory on the
hemisphere,

$\beta=\oint{\rm curl}\vec{M}dS=\frac{1}{2}\oint\sin\theta d\theta d\alpha.$
(24)

In original coordinates, the connection and the curl read as follows:

$\displaystyle M^{x}$ $\displaystyle=$
$\displaystyle\frac{\phi^{r}_{y}}{\rho^{2}A_{y}A_{x}}(1-\sqrt{1-\rho^{2}})$
(25) $\displaystyle M^{y}$ $\displaystyle=$
$\displaystyle-\frac{\phi^{r}_{x}}{\rho^{2}A_{y}A_{x}}(1-\sqrt{1-\rho^{2}})$
(26) $\displaystyle{\rm curl}\vec{M}$ $\displaystyle=$
$\displaystyle\frac{1}{2A_{x}A_{y}\sqrt{1-\rho^{2}}}$ (27)

We will consider the deviations due to finite $Q$ in the Subsection $D$. In
the next Subsection, we will present quantum gates that enable measuring of
the result of holonomic transformations.

###  IV.3 Beyond the disk

The initialization and measurement of a quantum state at the degenerate
manifold is questionable if ever possible in principle. To check if holonomic
transformations work as supposed, we need to extend the quantum manipilation
schemes. A simple way to achieve this would be to depart from the disk in easy
direction. This leads to energy splitting $|\pm\rangle$ and enables the
measurement in this basis. However, with this measurement one cannot
characterise the work of the phase gate predicted, since it does not alter the
probabilities to be in $|\pm\rangle$. Besides, the states almost do not
overlap: this makes it difficult to arrange their superposition. We need to do
something different.

We propose to augment the purely holonomic transformations in the disk with
adiabatic passages in the same plane that go beyond the degenerate manifold.
(Fig. 1) This will bring us to the basis of the ground and first excited
states that is continuous an unambiguous in the exterior of the disk. The
adiabatic passages in the exterior change the phase difference between these
basis states (mostly this is the effect of dynamical phase) not affecting the
probabilities. Since the states are distinguishable (e.g. they correspond to
different currents in the superconducting leads given by energy derivatives
with respect to $\vec{\phi}$), these propabilities can be measured. Such
measurements can be done with Andreev bound state spectroscopy[35, 36, 37].
The resonant quantum manipulation is also possible since the energies are
spilt and the wave functions of the states overlap.

Let us find how the wave function is transformed between the interior and
exterior bases upon crossing the disk boundary in the point
$\theta=\pi/2,\alpha=\alpha_{0}$. We consider a transformation
$R(\alpha_{0})$: $180^{\circ}$ rotation about the axis
$(\cos\alpha_{0},\sin\alpha_{0},0)$that is in the direction of the spin-
orientation vectors $\vec{w}_{\pm}$ at this point. For a circular-symmetric
disk $L_{x}=L_{y},I_{x}=I_{y},C_{x}=C_{y}$ this is a true symmetry
transformation of the Hamiltonian. For an anisotropic disk, this symmetry
holds approximately in quasiclassical limit where the wave functions are
concentrated near a point in $\vec{\phi}$ space.

The transformation should be diagonal in exterior basis. The ground and
excited state are respectively even and odd upon $R(\alpha_{0})$. As to the
interior states, let us note that
$|0\rangle_{\pm}=R(\alpha_{0})|0\rangle_{\mp}$, so that

$R(\alpha_{0})|\pm\rangle=e^{\mp i\alpha_{0}}|\mp\rangle$ (28)

With this, we find that the wave functions in exterior and interior bases are
related by a generalized Hadamard gate

$\mathcal{H}(\alpha_{0})\equiv\frac{1}{\sqrt{2}}\begin{bmatrix}1&e^{i\alpha_{0}}\\\
e^{-i\alpha_{0}}&-1\end{bmatrix}$ (29)

Since $\mathcal{H}^{2}=1$, the same matrix relates the bases upon the reverse
passage.

Let us consider the quantum gate given in Fig. 1 b. We initialize the wave
function in a point $A$ beyond the disk: we can wait for the relaxation that
brings the system to the ground state. After this, we can bring it to a
superposition of the ground and excited state with a pulse of an oscillating
$\vec{\phi}$ with the frequency matching the level splitting in this point.
The adiabatic trajectory enters the disk, makes a loop there for a holonimic
manipulation, and returns to the same point. The resulting quantum gate reads

$\hat{S}=\mathcal{H}(\alpha_{0})e^{i\tau_{z}\beta}\mathcal{H}(\alpha_{0})$
(30)

$\beta$ being the Berry phase accumulated on the loop. This does not include
the phase shifts in the exterior basis that do not change the probability. If
we start in the ground/excited state, we end up in the excited/ground state
with the probability

$\mathcal{T}=\sin^{2}\beta$ (31)

Measuring these probabilities thus permits the characterization of the
holonomic transformation. The answer for the probability, as expected, does
not depend on the entrance point $\alpha_{0}$.

To measure the wave function in the exterior basis, one does not have to
return to the initial point (Fig. 1 c): the measurement can be performed upon
leaving the disk at some other point $(\pi/2,\alpha_{1})$. The resulting
quantum gate upon the phase factors in exterior basis is given by

$\hat{S}=\mathcal{H}(\alpha_{0})e^{-i\tau_{z}\beta}\mathcal{H}(\alpha_{1})$
(32)

so the probability ${\cal T}=\sin^{2}(\beta^{*}+(\alpha_{1}-\alpha_{0})/2)$,
$\beta^{*}$ being the Berry phase acquired upon the part of the trajectory
that connects the entrance and exit points. One may be surprised with the fact
that $\beta^{*}$ in principle is not guage invariant quantity. This is
resolved if we note that in our gauge $(\alpha_{1}-\alpha_{0})/2$ is the Berry
phase acquired upon a passage along the disk boundary from $\alpha_{1}$ to
$\alpha_{0}$. So the gate in Fig. 1 c is equivalent to that in Fig. 1 d where
the trajectory in the disk is closed. This restores the guage-invariant
expression ${\cal T}=\sin^{2}\beta$, $\beta$ being the Berry phase accumulated
along the closed trajectory.

More sophisticated gates can be arranged by entering and leaving the disk
repeatedly along an adiabatic passage. They are composed of Hadamard gates,
holomonic phase shifts in interior basis and dynamical phase shifts in
exterior basis.

###  IV.4 Connection beyond the quasiclassical limit

The simple expression for holonimic transformation obtained above is valid in
the deep quasiclassical limit only and relies on the localization of the wave
functions. One may wonder how accurate it is at finite values of $Q$. At first
site, this problem is superfluous since finite values of $Q$ give rise to the
splitting of degenerate values in the disk, this formally invalidates the
holonomic transformation. However, the splitting is exponentially small and
may be neglected when the deviations from the deep quasiclassical limit are
noticeable.

To investigate and illustrate this, in this Subsection we compute numerically
the connection at finite $Q$. We restrict ourselves to a simple particular
case when this computation is straightforward: we concentrate on the circular
trajectories at a circular-symmetric disk.

The circular symmetry of the disk implies $I_{x,y}=I_{r}$, $L_{x,y}=L_{r}$,
$C_{x,y}=C_{r}$. Let us concentrate on a family of the Hamilitonians
$\hat{H}(\alpha)$ along a circular trajectory
$\phi^{r}_{x}=r\cos\alpha,\phi^{r}_{y}=r\sin\alpha$, $r$ being the radius.
This family is obtained by rotations about $z$ axis by $\alpha$,

$\hat{H}(\alpha)=R^{-1}_{z}(\alpha)\hat{H}R_{z}(\alpha);$ (33)

where

$R_{z}(\alpha)=\exp(-i\alpha\hat{J}_{z})$ (34)

and $\hat{J}_{z}$ is the angular momentum operator. $R_{z}^{\alpha}$ represent
rotation around easy direction with an angle $\alpha$. The rotation thus
generates the family of the bases diagonalizing $\hat{H}(\alpha)$:

$|n\rangle_{\alpha}=\exp(-i\alpha\hat{J}_{z})|n\rangle_{\alpha}$ (35)

The connection $\hat{M}^{\alpha}$ is thus determined through the matrix
elements of $\hat{J}_{z}$ (see Eq. 14), is constant over the trajectory. One
needs to diagonalize the Hamiltonian only once per trajectory to compute the
connection.

Figure 7:  Holonomic phase at various finite values of $Q$ for a circular
trajectory of radius $r$ at the circular-symmetric disk. We compare it with
the quasiclassical result (Eq.37) in the limit $Q\to\infty$.

We project $\hat{J}_{z}$ on the two lowest eigenstates
$|e\rangle$,$|g\rangle$, of the Hamiltonian,

$M^{\alpha}_{ab}=\langle a|\hat{J}_{z}|b\rangle$ (36)

$a,b=g,e$.The half-difference of the eigenvalues of this matrix defines the
holomic transformation phase accumulated over the circular trajectory
$\beta=\pi(M_{+}-M_{)}$.

One needs to take into account that the einvalues of $\hat{J}_{z}$ are half-
integer and the basis given by Eq. 35 is discontinious. Owing to this, a
formal calculation would give $\beta=\pi$ even at $r\to 0$ where no change of
the Hamiltonian takes place. So one has to subtract $\pi$ to make sure
$\beta\to 0$ at $r\to 0$.

The results are plotted in Fig. 7. In the deep quasiclassical limit, the
angular momentum operator can be replaced with $\sigma_{z}/2$ and the
holonomic phase is given by

$\beta=\pi\frac{1-\langle\hat{\sigma}_{z}\rangle}{2}=\pi\left(1-\sqrt{1-(r/A)^{2}}\right),$
(37)

$A$ being the disk radius. We observe significant corrections to the
quasiclassical limit at finite $Q$, those become bigger at the disk boundary
and at the smaller $Q$. However, the overall dependence of $\beta(r)$ is
preserved even at $Q=4$.

##  V Quantum dynamic

In this Section, we discuss the deviations from the ideal results of the
execution of the quantum gates described. The deviations come from the
residual level splitting in the disk and from the non-adiabatic excitations to
higher levels in the course of the execution at finite time. We illustrate
these sources with the numerical examples of the quantum dynamics of the full
Hamiltonian (II). The parameters of the Hamiltonian used for illustrations are
the same as in the previous Sections corresponding to the moderate
quasiclassical parameter $Q$ and circular symmetry. We show that the gates
work well even in this case.

Figure 8: The concrete example trajectories investigated in Section V.

We concentrate on the gate of the type given in Fig. 1c. We chose a simple
family of the trajectories (Fig. 8) consisting of two straight passages
(”arms”) in radial direction and an arc around the origin. The initial and
final points $A,B$ are beyond the disk at the same distance $R>A$ from the
origin, $A$ being the disk radius. The radius and the angle of the arc are
$r$, $\Theta$ respectively. We expect the holonomic phase $\beta$ to
accumulate upon the passage:

$\beta=\frac{1}{2}\Theta\sqrt{1-\left(\frac{r}{A}\right)^{2}}.$ (38)

The time dependence of $\vec{\phi}^{r}$ at the trajectory can be defined in
terms of the angular velocity $\Omega$ at the arc part and the linear velocity
$v\equiv A\dot{\rho}$ The overall execution time of the gate is thus
$T=\Theta/\Omega+2(R-r)/v$. In a realistic manipulation, it is easy to make
these velocities time-dependent preserving the total execution time, this
might be a possibility to reduce the non-adiabatic corrections and thus
improve the gate performance. However, the qualitative analysis made and our
attempts of such optimization did not show any substantial improvement. The
optimal time dependence of the velocities is close to constant.

We consider the deviations at the arc part and at the arms separately, and
conclude by combining both in an example of the gate fidelity versus the
execution time $T$.

The Schrödinger equation at the arc part is best expressed in the local basis
(35),

$i\hbar\dot{\psi}_{n}=E_{n}\psi_{n}-\hbar\Omega(t)J^{z}_{nm}\psi_{m}$ (39)

the effective Hamiltonian not depending on time if $\Omega(t)={\rm const}$.
The initial condition corresponds to the wave function localized in the two
lowest levels, $\psi_{1,2}\neq 0$, and the equation needs to be solved at the
time interval $(0,\Theta/\Omega)$.

The first source of the deviations is the residual level splitting
$E_{2}-E_{1}$. The proper work of the holonomic gate requires this splitting
to be smaller than the second term $\propto\Omega$, that is,
$|E_{2}-E_{1}|\ll\hbar\Omega$. If this condition is fulfilled, the deviations
in probabilities are proportional to $((E_{2}-E_{1})/\hbar\Omega)^{2}$.

The second source are the non-adiabatic corrections corresponding to the
excitations to higher levels $n>2$. The probabilities of the excitations from
the states $|1\rangle,|2\rangle$ can be estimated as

$\displaystyle P_{1,2\to n}\approx|\langle
n|\hat{J}_{z}|1,2\rangle|^{2}\left(\frac{\hbar\Omega}{E_{n}-E_{1,2}}\right)^{2}$
(40)

The small probabilities thus require
$\hbar\Omega\ll|E_{n}-E_{1,2}|\simeq\hbar\omega_{x,y,z}$. The execution time
thus should be

Figure 9:  The deviations coming from the arc part. We assume the ideal work
at the arms of the trajectory. The initial state is an excited state
$|2\rangle$ in the exterior basis. We plot the probabilities $P_{1,2}$ (blue
and red) versus $\Theta$ at three different angular velocities
$\Omega=10^{-3},10^{-2},10^{-1}\omega_{z}$ (dashed, solid and dotted lines).
At the intermediate $\Omega$, the results coincide with those for the ideal
gate upon numerical accuracy. At the smallest $\Omega$, the deviations are due
to the residual level splitting in the disk interior. At the highest $\Omega$,
the deviations are due to non-adiabatic excitiation to the higher levels.

We illustrate this with the quantum dynamics simulation at the arc part of the
trajectory. (Fig. reffig:arc). We simulate the work of the gate assuming its
ideal execution at the arms. The initial state corresponds to the excited
state in the exterior basis at the point $A$. We compute the probabilities
$P_{1,2}$ to end up in the ground/excited state at the point $B$. For an ideal
gate, those are given by $P_{1,2}=\sin^{2}\beta,\cos^{2}\beta$.

We plot the probabilities versus $\Theta$ for three different angular
velocities $\Omega=10^{-2},10^{-1},10^{0}\omega_{z}$. We choose $r=0.25$ where
the residual splitting $E_{2}-E_{1}=0.002\omega_{z}$. At the smallest
$\Omega$, the deviation owing to the residual splitting is noticeable. There
is no excitation to the higher levels, so that $P_{1}+P_{2}=1$. At the
intermediate $\Omega$, the results follow those of the ideal gate with the
numerical accuracy. The probabilities achieve minimum/maximum at
$\Theta\approx\pi 1.03$ corresponding to $\beta=\pi/2$. At the highest
$\Omega$, the non-adiabatic correction become noticeable. The probability
progressively leaks to higher levels, with only a half of it remaining in the
computational subspace for the longest gate $\Theta=2\pi$. Despite the
significant deviations, the curves at the lowest and at the highest angular
velocity still follow the oscillatory pattern. This demonstrates that the gate
works in a wide range of the angular velocities.

Let us analyse the deviations coming from the radial arms of the trajectory.
The residual splitting in this case only modifies the phase factor in the
exterior basis and can be disregarded. To quantify the non-adiabatic
corrections, we represent the Schrödinger equation in the instantaneous basis
$H(t)|n(t)\rangle=E(t)|n(t)\rangle$ and compute the probabilities to be in the
excited states $n>2$ in the second order perturbation theory in terms of the
non-diagonal elements of ${\partial H}{\partial t}$.

Owing to circular symmetry, the results do not depend on the direction of the
arm. We can compute the amplitudes for the motion in $x$ direction with the
velocity depending on the actual value of $\phi^{r}_{x}$,
$\dot{\phi}^{r}_{x}(\phi^{r}_{x})$ The amplitude in the excited state $n$
accumulated in the course of motion from $r$ to $R$ then reads:

$\psi_{n}(R)=-\frac{\hbar I_{x}}{2e}\int_{r}^{R}ds_{2}\frac{|\langle
n(s_{2})|\sigma_{x}|a(s_{2})\rangle|}{E_{n}(s_{2})-E_{a}(s_{2})}\\\
\times\exp\left[\frac{i}{\hbar}\int_{r}^{s_{2}}\frac{1}{\dot{\phi}^{r}_{x}(s_{1})}(E_{n}(s_{1})-E_{a}(s_{1}))ds_{1}\right]$
(41)

where $|a\rangle$ is a state from the computational subspace, either ground
one or excited one.

To evaluate this integral numerically, we use instantaneous eigenstates
obtained by diagonalization of the Hamiltonian at various $\phi^{r}_{x}$. We
summarize the results in Fig. 10 where we present the probability $1\to 3$
versus the average velocity at the arm. The non-diagonal matrix element
energizing the transition is plotted versus $\phi^{r}_{x}$ in Fig. 10 a. We
try a family of $\phi^{r}_{x}$ dependencies of this velocity parametrized by
$\gamma$:

$\frac{\dot{\phi}^{r}_{x}(\phi^{r}_{x})}{\langle\dot{\phi}^{r}_{x}\rangle}=\frac{\gamma}{2\tanh(\gamma/2)}(1-\rho^{2}\tanh^{2}(\gamma/2))$
(42)

where $\rho=(2\phi^{r}_{x}-R-r)/(R-r)$. These velocity profiles are plotted in
Fig. 10 b for several values of $\gamma$. In the main Figure, we plot the
excitation probability versus the average velocity for several $\gamma$. The
overall dependence is qualitatively consistent with exponential suppression of
the transitions at low velocities, $\ln P\simeq v^{-1}$. The detailed
dependence is not smooth, the probabilities oscillate showing interference due
to finite length of the arm. At all velocity profiles checked, the
probabilities are $<5\cdot 10^{3}$ for $\dot{\phi}^{r}_{x}\approx
A\omega_{z}/4\pi$ and grow rapidly at higher velocities. We find that the
time-independent velocity profile $\gamma=0$ eventually provides smaller
probabilities and is thus advantageous for the gate design.

Figure 10:  The deviations coming from the radial part of the trajectory. The
probability of the dominant excitation $P_{1\to 3}$. For this examle, the
trajectory starts at $r=0.25A$ and ends at $R=2A$. a. The intensity of the
matrix element energizing the transition versus the distance from the origin
$\phi_{x}^{r}$. b. The velocity profiles corresponding to Eq. 42. The curves
correspond to various $\gamma$ shown in the labels. Main figure: we plot the
probability for various velocity profiles ($\gamma$ is shown in the curve
labels) as function of the average velocity. The probabilities are lower than
$0.005$ for lower velocities $<0.06A\omega_{z}$ and increase rapidly at higher
velocities. The slowest growth corresponds to the constant velocity profile
$\gamma=0$.

We illustrate the deviations on all parts of the trajectory by computing the
fidelity of the swap gate versus the execution time. (Fig. 11). A swap gate
implements the transformation $|1\rangle\to|2\rangle;|2\rangle\to|1\rangle$ in
the exterior basis, that is, $\cal{T}=1$. The actual example differs a bit
from the swap gate: the trajectory parameters are $r=0.25A$, $R=2A$,
$\Theta=\pi$, this corresponds to $\beta\approx 1.5209$ that differs slightly
from the swap gate value of $\pi/2$. However, ${\cal T}\approx 0.9975$ for
this value of $\beta$ so the difference with the ideal swap gate is
negligible. The velocity in $\phi^{r}_{x}-\phi^{r}_{y}$ is constant along all
parts of the trajectory. The fidelity is almost zero at $T\omega_{z}<10$,
increases non-monotonically till $T\omega_{z}<80$ and is close to ideal value
at bigger $T$. We thus expect the gate to work good at $T\omega_{z}\simeq 100$
and longer. This looks parametrically bigger than a naive ad-hoc estimation
$T\omega_{z}\simeq 1$. However, we have to take into account that the typical
energy differences along the path are $\simeq 0.2\omega_{z}$ and the better
estimation for time is a period rather than the inverse frequency. So that
$T\omega_{z}\simeq 100$ corresponds to $3-4$ typical oscillation periods.
Eventually, this corresponds to the number of peaks seen in the time
dependence of the fidelity.

Figure 11:  The fidelity of the holonomic swap gate as function of the
execution time $T$. The gate corresponts to the trajectory with $r=0.25A$,
$R=2A$, $\Theta=\pi$ and is executed with constant velocity along the
trajectory.

In the figure, we show a full numerical simulation of the swap gate for
different runtimes. As expected from small runtimes, the execution is diabatic
and thus all state probabilities after execution are close to zero. On the
other hand, at long runtime limit, we see that the gate is relatively stable a
desirable property for holonomic computation.

##  VI Conclusions

In conclusion, we have investigated holonomic manipulations that can be
performed utilizing approximate degeneracy at the Weyl disk: a rather counter-
intuitive example of 2D finite degenerate manifold in 3D parameter space. The
Weyl disks can be realized by soft confinement of parameters in the
superconducting nanostructures hosting the Weyl points in the spectrum of
Andreev bound states, which is the example considered here. The resulting
Hamiltonian is however generic and can be realized in many quantum systems
with the Weyl like crossings in the spectrum of discrete energy states, so our
results are of general nature.

We have computed the connection in the Weyl disk manifold in quasi-classical
limit and found it Abelian: it realizes a phase gate, the phase difference
being related to the Berry phase in its classic example. This may seem a
rather discouraging result. However, we propose to augment the purely
holonomic transformations with the adiabatic passages beyond the degenerate
manifold. With this, we can propose the realization of more sophisticated
gates and practical measurement of the results of the holonomic
transformations.

We did quantum dynamic simulation of the gates proposed for realistic
Hamiltonians and find they can work properly at rather short execution times.
We did not consider decoherence in this respect: the point is that our
exemplary setup provides no projection against the fluctuations of the
parameter along the easy axis that can cause the decoherence in the degenerate
2D subspace.

A natural continuation of this research line would include the consideration
of several Weyl points brought into soft confinement and interaction with each
other. They would give rise to degenerate manifolds of higher dimensions with
richer holonomic transformations and perhaps provide the protection against
decoherence.

To support open science and open software initiatives and to comply with
institutional policies, we have published all relevant code and instructions
for running it on the Zenodo repository[38].

###### Acknowledgements.

This research was supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694272).

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