# Lensing with generalized symmetrons

Christian Käding<EMAIL_ADDRESS>Technische Universität Wien,
Atominstitut, Stadionallee 2, 1020 Vienna, Austria

###### Abstract

Generalized symmetrons are models that have qualitatively similar features to
the archetypal symmetron, but have barely been studied. In this article, we
investigate for what parameter values the fifth forces induced by disformally
coupling generalized symmetrons can provide an explanation for the difference
between baryonic and lens masses of galaxies. While it is known that the
standard symmetron struggles with providing an alternative source for the
lensing otherwise attributed to particle dark matter, we show that some
generalized symmetron models are more suitable for complying with existing
constraints on disformal couplings. This motivates future studies of these
only little explored models.

##  I Introduction

Some of modern physics’ most prominent open problems can be found in
cosmology, i.e. the questions after the natures of dark energy (DE) and dark
matter (DM). In order to tackle these problems, a zoo of modifications of
general relativity have been considered. Amongst those, scalar-tensor theories
[1] are some of the most studied ones. An overview of models that address the
problems of DE and DM can be found in Refs. [2, 3].
Some of the scalar fields appearing in scalar-tensor theories are expected to
cause a fifth force of Nature, which, however, is in tension with Solar
System-based experiments [4, 5, 6]. This led to some of these models being
already ruled out [7]. Though, so-called screened scalar fields, see Refs. [8,
9] for reviews, have ways of circumventing Solar System constraints by
screening their fifth forces in environments of higher mass densities, such
that the forces are effectively rendered to be feeble. The most well-known
screened scalar field models include the chameleon [10, 11], the symmetron
[12, 13, 14, 15, 16, 17, 18, 19], the environment dependent dilaton [14, 20,
21, 22, 23, 24, 25], and the galileon [26, 27, 28]. In recent years these
models have been (proposed to be) tested in various experiments, see e.g.
Refs. [29, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45],
studied as quantum fields [46, 47, 48, 49], and proposed for investigations in
analogue gravity simulations [50].
Screened scalar field models have a rich phenomenology, which allows them to
serve as promising candidate theories for DM or DE. For example, in Ref. [51]
it was shown that the symmetron fifth force can explain the stability and
rotation curves of disk galaxies, while Ref. [52] demonstrated how the same
force can lead to the observed motion of stars perpendicular to the plane of
the Milky Way disk. This made the symmetron a promising alternative to
particle DM. Though, in Refs. [53] and [49] it was shown that, using the same
parameter values as in Ref. [51], the difference between the baryonic masses
of galaxies and the masses required for causing the observed (gravitational)
lensing cannot be explained by a symmetron fifth force alone, but potentially
by adding another scalar field or considering a hybrid model between modified
gravity and particle DM.
In the present article we revisit the idea of a symmetron fifth force being an
alternative to particle DM. However, instead of only considering the
archetypal symmetron, we discuss so-called generalized symmetrons, which were
discovered using tomographic methods [54, 55]. This class of models comprises
the standard symmetron, but also an, in principle, infinite number of
qualitatively similar generalizations. Generalized symmetrons have barely been
studied in the literature [8] and their parameter spaces are still
unconstrained. As we show in the present article, some generalized symmetron
models can actually explain the observed lensing otherwise attributed to
particle DM while complying with existing constraints [56, 57] on the
necessary disformal coupling [58]. Since the parameter spaces of those models
have yet not been constrained by experiments, they offer significantly more
freedom to discuss them as alternatives to particle DM than many other
screened scalar field models including the standard symmetron do. This
motivates future studies of these yet only little explored models.
The article is structured as follows: In Sec. II we review the generalized
symmetron models, while in Sec. III, based on the discussion in Ref. [49], we
derive the deviation of a galaxy’s lensing from general relativity’s
predictions due the presence of a generalized symmetron fifth force.
Subsequently, in Sec. IV, we study which parts of the generalized symmetron
parameter spaces are suitable for explaining the difference between baryonic
and lens masses of galaxies. Finally, we draw our conclusions in Sec. V.

##  II Generalized symmetrons

The standard symmetron model is a scalar-tensor modification of gravity and
introduces a new type of scalar field - the symmetron. It was first mentioned
in Refs. [12, 13, 14, 15, 16, 17] and then introduced with its current name in
Refs. [18, 19].
The symmetron $\varphi$ couples conformally to the metric tensor via a factor

$\displaystyle A(\varphi)$ $\displaystyle=$ $\displaystyle
1+\frac{\varphi^{2}}{\mathcal{M}^{2}}+\mathcal{O}\left(\frac{\varphi^{4}}{\mathcal{M}^{4}}\right)\,\,\,,\,\,\,\varphi\ll\mathcal{M}\,\,\,,$
(1)

which leads to a universal coupling to the trace of the matter energy-momentum
tensor $T^{\mu}_{\,\,\,\mu}$. Due to this coupling to matter, an additional
force of Nature, a so-called fifth force, is expected to be mediated by the
symmetron. Though, in order to avoid Solar System constraints on fifth forces
[4, 5, 6], the symmetron is equipped with a screening mechanism, which renders
the fifth force to be feeble in regions where the trace of the energy-momentum
tensor is sufficiently large. This screening is achieved by the behavior of
the symmetron’s non-linear effective potential given, setting
$T^{\mu}_{\,\,\,\mu}=-\rho$ for dust, by

$\displaystyle V(\varphi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\rho}{\mathcal{M}^{2}}-\mu^{2}\right)\varphi^{2}+\frac{\lambda}{4}\varphi^{4}\,\,\,.$
(2)

As long as $\mu^{2}\mathcal{M}^{2}>\rho$, this potential has minima at

$\displaystyle\varphi_{0}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{1}{\lambda}\left(\mu^{2}-\frac{\rho}{\mathcal{M}^{2}}\right)}\,\,\,.$
(3)

However, if this condition is not fulfilled, the vacuum expectation value
(vev) of the symmetron can only be $\varphi_{0}=0$. Both possible scenarios
are depicted in Fig. 1. Separating the symmetron into its vev and a small
fluctuation $\delta\varphi$, such that $\varphi=\varphi_{0}+\delta\varphi$, it
can be shown that the symmetron fifth force at leading order goes like
$F_{\varphi}\sim\varphi_{0}\nabla\delta\varphi$. Consequently, in situations
where $\mu^{2}\mathcal{M}^{2}<\rho$, the symmetron-induced fifth force is
screened since its leading term vanishes, leaving only small corrections of
higher order in $\delta\varphi$.

Figure 1: Potential $V(\varphi)$ of the (generalized) symmetron: if the
density $\rho$ is sufficiently small, then the potential takes on the shape of
the blue curve and the field has a non-vanishing vev. However, if
$\rho>\mu^{2}\mathcal{M}^{2}$ (orange curve), then the potential can only have
a minimum at $\varphi=0$ , which results in the fifth force being screened.

Besides this standard symmetron model, a radiatively-stable symmetron has been
developed [59], and generalizations were discovered by using tomographic
methods [54, 55]. The latter are the main subject of this article and
discussed in what follows. Generalized symmetrons couple to the metric tensor
via

$\displaystyle A(\varphi)$ $\displaystyle=$ $\displaystyle
1+\frac{\varphi^{2\alpha}}{\mathcal{M}^{2\alpha}}+\mathcal{O}\left(\frac{\varphi^{4\alpha}}{\mathcal{M}^{4\alpha}}\right)\,\,\,,\,\,\,\varphi\ll\mathcal{M}\,\,\,,$
(4)

and have potentials [8]

$\displaystyle V(\varphi)$ $\displaystyle=$
$\displaystyle\left(\frac{\rho}{\mathcal{M}^{2\alpha}}-\mu^{4-2\alpha}\right)\varphi^{2\alpha}+\frac{\varphi^{2\beta}}{\Lambda^{2\beta-4}}\,\,\,.$
(5)

The numbers $\alpha,\beta\in\mathbb{Z}^{+}$ label each model and are, in
principle, arbitrary, but must adhere to $\beta>\alpha$. Choosing the smallest
possible pair $(\alpha,\beta)=(1,2)$ recovers the standard symmetron model as
given in Eq. (2). Each choice for $(\alpha,\beta)$ leads to a model that has
similar qualitative features to the standard symmetron. This means, the
potential in Eq. (5) is also represented by Fig. 1, a generalized symmetron
fifth force is screened in regimes where
$\rho>\mu^{4-2\alpha}\mathcal{M}^{2\alpha}$ since there $\varphi_{0}=0$, and
in unscreened regimes the field’s vev takes on the form

$\displaystyle\varphi_{0}$ $\displaystyle=$
$\displaystyle\pm\sqrt[2(\beta-\alpha)]{\frac{\alpha}{\beta}\Lambda^{2\beta-4}\left(\mu^{4-2\alpha}-\frac{\rho}{\mathcal{M}^{2\alpha}}\right)}\,\,\,.$
(6)

In an unscreened situation a generalized symmetron has a mass

$\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
4\beta(\beta-\alpha)\frac{\varphi_{0}^{2(\beta-1)}}{\Lambda^{2\beta-4}}\,\,\,,$
(7)

while in case of total screening, this becomes

$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{\rho}{\mathcal{M}^{2}}-\mu^{2}&,\,\,\,\alpha=1\\\
0&,\,\,\,\alpha>1\end{cases}\,\,\,.$ (8)

##  III Lensing

Gravitational lensing became the first experimentally confirmed novel effect
predicted by general relativity when in 1919 the bending of light in the Sun’s
gravitational field was observed [60]. Today gravitational lensing serves as a
valuable tool for indirect observations of DM [61]. If a symmetron fifth force
is supposed to act as an alternative to particle DM, as suggested by the
findings in Refs. [51, 52], then it must be able to explain the observed
roughly $1:5$ ratio between baryonic mass and DM in galaxies [62], i.e.
contribute to the lensing about $5$ times as much as the Newtonian potential
$\Phi$ of a galaxy. In Refs. [53] and [49] it was shown that the standard or
$(1,2)$-symmetron cannot provide such an explanation for the parameter values
required by Ref. [51]. Though, other models of the class of generalized
symmetrons might still be able to serve as sensible alternatives to particle
DM since they show the same qualitative features as the $(1,2)$-symmetron, but
are yet much less constrained.
In order to study the effect of a generalized symmetron on lensing, we now
derive a measure that enables us to compare the contribution from the fifth
force with the one from Newtonian gravity. For this, we follow the discussion
in Ref. [49], which in turn is based on elaborations made in Ref. [63]:
We consider a perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) background

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
a^{2}(\tau)[-(1+2\Psi)d\tau^{2}+(1+2\Phi)\delta_{ij}dx^{i}dx^{j}]$ (9)

with conformal time $d\tau=dt/a(t)$ and Newtonian potentials $\Phi,\Psi\ll 1$,
which fulfill the no-slip condition $\Phi=-\Psi$. Furthermore, we assume that
the thickness of the lens (e.g. a galaxy) is much smaller than the distance
$d_{L}$ between lens and observer (on Earth), and the deflection angles are
very small. This assumption justifies a thin-lens approximation, which
effectively considers the lens to have no thickness and lie within a two-
dimensional plane. This so-called lens plane is perpendicular to the line
between light source and observer. Using this approximation, we introduce the
coordinate system $\\{\tau,r,x^{1}\approx r\theta^{1},x^{2}\approx
r\theta^{2}\\}$ with radial coordinate $r$ and deflection angles
$\theta^{1,2}$. See Fig. 2 for a depiction of the setup.

Figure 2: Schematics of gravitational lensing in two dimensions with thin-lens
approximation: The source sends out light which is distorted by the lens and
then received by an observer. $d_{L}$ denotes the distance between the
observer and the lens plane in which the lens is situated. The radial
coordinate $r$ equals $0$ at the position of the observer and is orthogonal to
the lens plane. $\theta$ denotes the angle between $r$ and the light ray
reaching the observer (here it represents either $\theta_{1}$ or
$\theta_{2}$). The coordinate $x$ is approximated by $r\theta$ since $\theta$
is assumed to be very small. [49]

We introduce the photon momentum $k^{\mu}:=dx^{\mu}/d\kappa$ with some affine
parameter $\kappa$, which gives the geodesic equation

$\displaystyle\frac{dk^{\mu}}{d\kappa}+\Gamma^{\mu}_{\nu\sigma}k^{\nu}k^{\sigma}$
$\displaystyle=$ $\displaystyle 0\,\,\,.$ (10)

The photon momentum can be separated into a background vector $\hat{k}^{\mu}$
and a small perturbation $\delta k^{\mu}$ due to the presence of a lens mass,
such that $k^{\mu}=\hat{k}^{\mu}+\delta k^{\mu}$. Without the lens mass, the
photon trajectory is not subject to deflection, which means, for the
background momentum $\hat{k}^{x^{1,2}}=0$ must hold. Consequently, from the
on-shell condition $k^{\mu}k_{\mu}=0$ follows $\hat{k}^{r}=\hat{k}^{0}$.
Taking all this into account, and considering only terms up to first order in
the Newtonian potentials and the momentum perturbations, for the purely
Newtonian case without any fifth forces, the geodesic Eq. (10) can be used to
derive the lensing force law

$\displaystyle\frac{d^{2}x^{i}}{dr^{2}}$ $\displaystyle=$
$\displaystyle(\Phi-\Psi),_{x^{i}}\,\,\,,\,\,\,i\in\\{1,2\\}\,\,\,.$ (11)

A similar force law can be derived when also including generalized symmetrons.
Though, it must be stressed that solely a conformal coupling of the scalar
field $\varphi$ to the metric tensor is not sufficient to have any effect on
lensing. This was, for example, explicitly shown in Ref. [49], but can also
simply be concluded from the fact that the energy-momentum tensor of a
massless particle like the photon has a vanishing trace. Therefore, the scalar
$\varphi$ must not only couple conformally, but also disformally via

$\displaystyle\bar{g}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
A(\varphi)g_{\mu\nu}+B\varphi,_{\mu}\varphi,_{\nu}\,\,\,,$ (12)
$\displaystyle\bar{g}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
A^{-1}(\varphi)\left(g^{\mu\nu}-\frac{B}{C(\varphi)}\varphi,^{\mu}\varphi,^{\nu}\right)\,\,\,,$
(13)

where $A(\varphi)$ is the conformal factor from Eq. (4), $B$ is the disformal
coupling parameter, which for simplicity is assumed to be a constant, and
$C(\varphi):=A(\varphi)+B(\partial\varphi)^{2}$. $\bar{g}$ denotes the metric
in the Jordan frame and $g$ the metric in the Einstein frame [1].
For the Jordan frame Christoffel symbols we find

$\displaystyle\bar{\Gamma}^{\mu}_{\nu\sigma}$ $\displaystyle=$
$\displaystyle\Gamma^{\mu}_{\nu\sigma}+A^{-1}\left[A,_{(\nu}g^{\mu}_{\sigma)}-\frac{1}{2}A,^{\mu}g_{\nu\sigma}\right]$
(14)
$\displaystyle+\frac{B}{C}\varphi,^{\mu}\left[\varphi,_{\nu\sigma}-\varphi,_{\rho}\Gamma^{\rho}_{\nu\sigma}+\frac{A^{-1}}{2}\varphi,^{\rho}A,_{\rho}g_{\nu\sigma}-A^{-1}A,_{(\nu}\varphi,_{\sigma)}\right]\,\,\,,$

where the second term in the first line represents a contribution which is
arising purely from the conformal coupling and therefore does not affect the
lensing. Substituting Eq. (4) into Eq. (14), following the same procedure as
for the derivation of Eq. (11), and assuming the lens object to be static,
then the force law

$\displaystyle\frac{d^{2}x^{i}}{dr^{2}}$ $\displaystyle=$
$\displaystyle(\Phi-\Psi),_{x^{i}}-\frac{B}{C}\varphi,^{x^{i}}\left[\bigg{(}\varphi,_{yz}-\frac{2\alpha}{\mathcal{M}^{2\alpha}}\varphi^{2\alpha-1}\varphi,_{y}\varphi,_{z}\bigg{)}\frac{dx^{y}}{dr}\frac{dx^{z}}{dr}\right.$
(15)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.-2\mathcal{H}\varphi,_{y}\frac{dx^{y}}{dr}+a^{2}\varphi,_{y}(\Phi-\Psi),^{y}-2\varphi,_{r}\Phi,_{r}\right]\,\,\,,$

where $y,z\in\\{r,x^{1},x^{2}\\}$, and $\mathcal{H}$ denotes the conformal
Hubble parameter, can be obtained. Comparing this force law with the one in
Eq. (11), we find that the term proportional to $B$ in Eq. (15) corresponds to
the contribution of the generalized symmetron fifth force to the lensing
effect.
Next, we assume $a(\text{today})=1$, use the no-slip condition for the
Newtonian potentials, and consider the force law only in the lens plane
$r=d_{L}$, where we expect the lensing to be maximal and $dx/dr$ to vanish:

$\displaystyle\left.\frac{d^{2}x^{i}}{dr^{2}}\right|_{d_{L}}$ $\displaystyle=$
$\displaystyle\bigg{[}2\Phi,_{x^{i}}\left.-\frac{B}{C}\varphi,^{x^{i}}\bigg{(}\varphi,_{rr}-\frac{2\alpha}{\mathcal{M}^{2\alpha}}\varphi^{2\alpha-1}(\varphi,_{r})^{2}-2\mathcal{H}\varphi,_{r}+2\varphi,_{x^{j}}\Phi,^{x^{j}}\bigg{)}\bigg{]}\right|_{d_{L}}\,\,\,.$
(16)

In order to get a rough numerical estimate of the generalized symmetron
contribution, we assume its source, i.e. a galaxy acting as lens, to be a disk
lying in the lens plane with homogeneous, constant mass density and radius
$R$, leading to a total galaxy mass $M$. Since this approximated lens has a
spherical symmetry, we can restrict our investigation to the case
$\theta^{2}=0$, and consequently only work with $\theta^{1}=:\theta$. Applying
Newton’s shell theorem and considering that the radial coordinate within the
lens plane originating from the disk’s center can be expressed as $r\theta$ in
the observer’s coordinates, the Newtonian potential becomes

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle-\frac{GM}{r\theta}\,\,\,.$
(17)

Some references, including Refs. [49, 55], suggest an approximation for the
symmetron field profile outside a homogeneous, spherically symmetric source of
radius $R$, which is valid for a symmetron mass $m_{\text{out}}$ outside the
source fulfilling $r\theta<m_{\text{out}}^{-1}$. However, in what follows,
this approximation will not always be good. Therefore, we use the symmetron
profile [38]

$\displaystyle\varphi$ $\displaystyle=$ $\displaystyle
v-\frac{D}{r\theta}e^{-m_{\text{out}}r\theta}$ (18)

with

$\displaystyle D$ $\displaystyle:=$
$\displaystyle(v-w)Re^{m_{\text{out}}R}\frac{Rm_{\text{in}}-\mathcal{T}}{Rm_{\text{in}}+Rm_{\text{out}}\mathcal{T}}\,\,\,,$
(19)

where $m_{\text{in}}$ is the field’s mass within the source,
$v:=\varphi_{0,\text{out}}$ and $w:=\varphi_{0,\text{in}}$ are the vev in the
environment surrounding the source and within the source, respectively, and
$\mathcal{T}:=\tanh(m_{\text{in}}R)$. The solution in Eq. (18) requires that
we only consider terms up to first order in $\delta\varphi$, and, at this
order, is even valid for any generalized symmetron model.
Substituting Eqs. (17) and (18) into Eq. (16) leads us to

$\displaystyle\left.\frac{d^{2}x}{dr^{2}}\right|_{d_{L}}$ $\displaystyle=$
$\displaystyle\frac{2GM}{d^{2}_{L}\theta^{2}}\left[1+F\right]$ (20)

with

$\displaystyle F$ $\displaystyle:=$
$\displaystyle\frac{BD^{2}}{Cd_{L}^{4}\theta^{2}}e^{-2m_{\text{out}}d_{L}\theta}(1+m_{\text{out}}d_{L}\theta)\bigg{\\{}\left(1+\frac{d_{L}\theta}{2GM}\right)\bigg{[}1+(1+m_{\text{out}}d_{L}\theta)^{2}$
(21) $\displaystyle+\frac{2\alpha
D}{d_{L}\theta\mathcal{M}^{2\alpha}}\left(v-\frac{D}{d_{L}\theta}e^{-m_{\text{out}}d_{L}\theta}\right)^{2\alpha-1}e^{-m_{\text{out}}d_{L}\theta}(1+m_{\text{out}}d_{L}\theta)^{2}+2\mathcal{H}d_{L}(1+m_{\text{out}}d_{L}\theta)\bigg{]}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1+m_{\text{out}}d_{L}\theta}{\theta^{2}}\bigg{\\}}$

being a term that describes the contribution of the generalized symmetron
fifth force to lensing in comparison to the one originating from Newtonian
gravity. Furthermore, in the lens plane we have:

$\displaystyle C|_{d_{L}}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{\mathcal{M}^{2\alpha}}\left(v-\frac{D}{d_{L}\theta}e^{-m_{\text{out}}d_{L}\theta}\right)^{2\alpha}+(1-2\Phi|_{d_{L}})\frac{BD^{2}}{d_{L}^{4}\theta^{4}}e^{-2m_{\text{out}}d_{L}\theta}(1+m_{\text{out}}d_{L}\theta)^{2}(1+\theta^{2})\,\,\,.$

In Eq. (21) we see that $F\sim D^{2}$, such that from Eq. (19) we can find
$F\sim(v-w)^{2}$. This result implies that if $v$ and $w$ are very similar,
and have the same sign, for example in some situations where the field is
unscreened both in- and outside the source, the contribution of the
generalized fifth force to lensing can be very small. However, it is not
necessarily true that both vev need to have the same sign. Since we are
interested in checking whether it is at all possible to explain the observed
lensing by a generalized symmetron fifth force, we consider the best case
scenario, in which $v=+|v|$ and $w=-|w|$.

##  IV Model parameters

We now want to consider different generalized symmetron models and see for
what points in their parameter spaces the expression in Eq. (21) gives
$F\approx 5$, such that the contribution of a fifth force can be interpreted
as an alternative to particle DM, at least when it comes to lensing.
For this, we consider the same galaxy parameters as in Ref. [49], i.e. we look
at a Milky Way-like galaxy with $M=6\times 10^{11}M_{\odot}\approx 6.67\times
10^{77}\,\text{eV}$ and a scale length $R=5\,\text{kpc}\approx 2.69\times
10^{26}\,\text{eV}^{-1}$ [64]. Ref. [65] reports lensing by galaxies at
redshift $z=1$ under angles of about $1\,\text{arcmin}$. Therefore, we choose
$\theta=\frac{\pi}{10800}$ and $d_{L}\approx 6.60\times
10^{32}\,\text{eV}^{-1}$, where we obtained the latter from $d_{L}\approx
zd_{H}$ [66] using the Hubble length $d_{H}$. For the density around the
galaxy we assume $\rho_{\text{out}}\approx 2.59\times 10^{-11}\,\text{eV}^{4}$
[67]. In addition, we use $G\approx 6.71\times 10^{-57}\,\text{eV}^{-2}$ and
$\mathcal{H}\approx 1.51\times 10^{-33}\,\text{eV}$.
We start the discussion by again looking at the $(1,2)$-symmetron before we
show that moving to larger values of $(\alpha,\beta)$ is beneficial for
complying with the existing constraint $B<5.6\times 10^{-48}\,\text{eV}^{-4}$
[56, 57] on the disformal coupling parameter.

###  IV.1 $(1,2)$-symmetron

Ref. [51] suggested that the $(1,2)$-symmetron could explain the stability and
rotation curves of disk galaxies, and therefore be a possible candidate for an
alternative to particle DM. For this to work, the symmetron parameters were
chosen to be $\mathcal{M}=M_{P}/10$, $v=\mathcal{M}/150$, and $\mu=3\times
10^{-30}\,\text{eV}$, where $M_{P}$ is the reduced Planck mass. In Refs. [53]
and [49] it was shown that those parameter choices require a disformal
coupling parameter $B$ much larger than permitted by the constraints given in
Refs. [56, 57]. This result also held for realistic variations of the galaxy
parameters.
We now study for what values of $\mathcal{M}$ and the self-coupling constant
$\lambda$ the $(1,2)$-symmetron can actually comply with the constraints on
disformal couplings. For the disformal coupling parameter we choose
$B=10^{-49}\,\text{eV}^{-4}$ in order to consider a value that is not yet
excluded by experiments, and for $\mu$ we initially take the same value as in
Ref. [51]. We find that $\lambda\approx 10^{-174.6}$ is the largest and
$\mathcal{M}\approx 10^{58.7}\,\text{eV}$ the smallest possible value in order
to find $F\approx 5$, while simultaneously fulfilling the perturbative
condition $\varphi\ll\mathcal{M}$. Even though there are not necessarily any
restrictions on the permitted values for the coupling constants, it is
certainly peculiar to have a theory with such a small self-coupling and, more
strikingly, a mass scale that is more than $30$ orders of magnitude above the
Planck mass.
Note that for these parameter values the field is not screened within the
galaxy. Furthermore, since in this parameter regime $1/m_{\text{in}}>R$, i.e.
the Compton wavelength of the field is larger than the galaxy scale, the field
adapts to the size of the galaxy and consequently rather takes on the mass
$m_{\text{in}}\approx q/R$ and the vev

$\displaystyle w$ $\displaystyle\approx$
$\displaystyle\pm\sqrt[\beta-1]{\frac{q\Lambda^{\beta-2}}{2\sqrt{\beta(\beta-\alpha)}R}}\,\,\,,$
(23)

where $q$ is a fudge factor that would have to be computed numerically taking
into account more detailed properties of a galaxy, but is assumed to be of
order $1$ (compare with the fudge factor, e.g., in Ref. [68]).
Considering smaller values of the disformal coupling parameter only worsens
the situation, i.e. requires smaller values of $\lambda$ and larger
$\mathcal{M}$. As was also observed in Ref. [49], changing the galaxy
parameters to other realistic values does not significantly improve the
situation.
What remains to be checked is how the result is affected by a change in the
tachyonic mass $\mu$. As it turns out, and can be seen in Fig. 3, the value
for $\mu$ used in Ref. [51] is sitting within a small part of the parameter
space that allows for $F\approx 5$.

Figure 3: Behavior of the function $F(\mu)$ for the $(1,2)$-symmetron with
parameters $\lambda=10^{-174.6}$, $\mathcal{M}=10^{58.7}\,\text{eV}$, and
$B=10^{-49}\,\text{eV}^{-4}$; the black rhombuses with vertical lines depict
the two points where $F=5$.

Looking at Eqs. (18) and (19), and remembering that $F\sim D^{2}$, we see that
significantly increasing or decreasing $\mu$ beyond this small section of the
parameter space, while keeping all other parameters at the same values, leads
to $F$ getting rapidly smaller since the combined exponential function in Eqs.
(18) and (19) becomes smaller due to the enlarged symmetron mass or the vev
reduces, respectively. In order to counteract the declining $F$, such that
$F\approx 5$ can be recovered, we have to consider even smaller values of
$\lambda$ and consequently larger values of $\mathcal{M}$. This is the exact
opposite of what we had hoped to achieve, which is why varying $\mu$ can be
excluded as a possibility for improving the case for the $(1,2)$-symmetron.

###  IV.2 $(1,\beta)$-symmetrons with $\beta\geq 3$

Now we move to the first set of non-standard symmetrons and consider models
with $(1,\beta)$ for $\beta\geq 3$. For those, Eq. (21) has still the same
form as for the $(1,2)$-symmetron, but the vev and the unscreened masses are
different according to Eqs. (6) and (7). Furthermore, instead of using a
dimensionless constant $\lambda$, we now have to work with $\Lambda$ as
another mass scale besides $\mathcal{M}$. Again choosing $\mu=3\times
10^{-30}\,\text{eV}$, we find the results presented in Fig. 4 for
$3\leq\beta\leq 10$.

Figure 4: Values of $\Lambda$ and $\mathcal{M}$ that give $F=5$ for
$(1,\beta\geq 3)$-symmetrons with $\mu=3\times 10^{-30}\,\text{eV}$ up to
$\beta=10$

We observe that going from $\beta=2$ to $\beta=3$ requires a slightly lower
$\mathcal{M}$ for $F\approx 5$ and $\varphi/\mathcal{M}\ll 1$, but the
$\Lambda$ of the $(1,3)$-symmetron must be more than $100$ orders of magnitude
above the Planck mass. This renders this theory to be unrealistic as an
alternative to particle DM.
At $\beta=4$ there is another slight decrease in $\mathcal{M}$, while from
$\beta=5$ on this mass scale increases again until it settles around
$\mathcal{M}\approx 10^{58.3}\,\text{eV}$ even for very large $\beta$. This
means, compared to the standard symmetron, only changing $\beta$ barely
improves the value of the matter coupling mass scale.
The self-coupling constant $\Lambda$ is strictly monotonically decreasing and
approaches $\Lambda\approx 10^{57.2}\,\text{eV}$ for large $\beta$. This
means, the two mass scales in $(1,\beta\geq 3)$-symmetron theories are getting
closer to each other with increasing $\beta$ until they differ only by
approximately $1$ order of magnitude.
Even though there are parts of their parameter spaces for which generalized
symmetrons with $\alpha=1$ can explain the lensing otherwise attributed to
particle DM, they require coupling constants which are at least $30$ order of
magnitude above the Planck mass.

###  IV.3 $(\alpha,\beta)$-symmetrons with $\alpha\geq 2$

Finally, we also consider generalized symmetron models with $\alpha\geq 2$.
Now we encounter a couple of differences to the previously considered models:
Eq. (21) changes with every possible value of $\alpha$, and $\mu$ is either
dimensionless (for $\alpha=2$) or appears in Eq. (5) with negative order (for
$\alpha\geq 3$).
Generally, it can be said that studying these models is numerically more
intricate than it was for the ones we considered before. This is due to the
fact that, within the considered astrophysical setup, $\alpha\geq
2$-symmetrons can lead to extremely small terms that have to be compensated by
extremely large terms in order to find any results of order $1$. For example,
Eq. (21) can be separated into two terms, one outside ($F_{o}$) and one inside
($F_{i}$) the curly brackets. $F_{o}$ tends to become very small since it is
suppressed by at least $B/d_{L}^{3}\theta\sim 10^{-144}\,\text{eV}^{-1}$ (for
$B=10^{-49}\,\text{eV}^{-4}$). While generalized symmetrons with only
$\alpha=1$ seem to not struggle with compensating such smallness by leading to
a sufficiently large $F_{i}$, the models discussed here are more problematic
since they lead to even more extreme values for $F_{o}$ and $F_{i}$.
Beginning with the $(2,3)$-symmetron model, we can find no point in the
parameter space that allows for $F\approx 5$. The closest value we find is
around $F\sim 10^{-139}$. This is caused by $F_{i}$ not reaching sufficiently
large values and whenever it increases for parameter values beyond the maximum
of $F$, $F_{o}$ decreases even faster. Increasing $\beta$ does not lead to
greatly improved results.
Moving to $\alpha=3$, from where on $\mu$ is a mass scale, we again do not
find any point in the parameter space that allows for $F\approx 5$, but can at
least reach $F\sim 10^{-117}$.
The first model we find that allows for $F\approx 5$ is the $(5,7)$-symmetron
around the parameter space point
$(\Lambda,\mathcal{M},\mu)=(10^{37.4},10^{58.7},10^{24.0})\,\text{eV}$.
Interestingly, from $\alpha=5$ on, changing $\beta$ can actually have a
significant impact on the results. However, while for $\alpha=6$ we still need
at least $\beta=\alpha+2$, from $\alpha=7$ on we can obtain $F\approx 5$ even
for $\beta=\alpha+1$.
With increasing $\alpha$ we have to use smaller $\Lambda$ in order to reach
$F\approx 5$, such that we are getting closer to the Planck scale. For
example, in the $(11,12)$-symmetron model we find $F\approx 5$ at the point
$(\Lambda,\mathcal{M},\mu)=(10^{27.3},10^{58.7},10^{24.0})\,\text{eV}$, where
$\Lambda$ is close to $M_{P}$.
As can be seen in the examples presented above, increasing $\alpha$ is useful
for reducing $\Lambda$, but $\mathcal{M}$ remains more than $30$ order of
magnitude above the Planck scale independent of the choice of
$(\alpha,\beta)$.

##  V Conclusions

In this article we discussed generalizations of the symmetron model,
characterized by a pair of positive integers $(\alpha,\beta)$, and
investigated for what parameter values their fifth forces can explain the
difference between baryonic and lens masses of galaxies, which is otherwise
attributed to particle DM. For this, we first reviewed generalized symmetrons
and then derived a measure $F$ that allowed us to compare the lensing
contribution from Newtonian gravity with one from a gravity-like fifth force
induced by a disformally coupling generalized symmetron.
We looked at a Milky Way-like galaxy and checked for a selection of
generalized symmetron models for what parameter space values $F\approx 5$,
which corresponds to the expected ratio between DM and baryonic matter in
galaxies, is fulfilled. With $B=10^{-49}\,\text{eV}^{-4}$ we chose a disformal
coupling parameter close to the maximal value allowed by current experimental
constraints. For the standard symmetron, corresponding to
$(\alpha,\beta)=(1,2)$, we found that a tiny self-coupling constant $\lambda$
and a matter coupling mass scale $\mathcal{M}$ more than $30$ orders of
magnitude larger than the Planck mass $M_{P}$ are required in order to find
$F\approx 5$. Increasing $\beta$, which required us to work with a mass scale
$\Lambda$ instead of a dimensionless $\lambda$, led to both $\Lambda$ and
$\mathcal{M}$ being at least $30$ orders of magnitude above the Planck scale.
Finally, also varying $\alpha$ was most promising since it allowed us to use
generalized symmetron fifth forces as explanations for the observed lensing
excess at mass scales $\Lambda$ and $\mu$ around or even below the Planck
scale. However, in no model it was possible to significantly reduce the value
for $\mathcal{M}$, which means it must always be much larger than $M_{P}$.
From Refs. [53] and [49] we know that the $(1,2)$-symmetron is not able to
successfully act as an alternative to particle DM since it cannot explain
lensing. This is also reflected in the fact that, as we showed in the present
article, the standard symmetron fifth force would require an extremely small
$\lambda$ for $F\approx 5$. In contrast, some generalized symmetron models
with larger $\alpha$ are in so far better in explaining the observed lensing
as they instead require the self-interaction mass parameter $\Lambda$ to be
around the Planck scale, which is a typical value expected for many screened
scalar field models. Though, every model, even for very large $\alpha$ and
$\beta$, required $\mathcal{M}$ to be at least $30$ orders of magnitude above
$M_{P}$ in order to reach $F\approx 5$. There are several possible ways around
this conundrum: either we accept that we have such a large fundamental mass
scale in Nature, we relax the requirement on $F$ and want a generalized
symmetron fifth force to only partially explain DM, we find a way to relax the
requirement $\varphi\ll\mathcal{M}$, or we consider a hybrid model between
modified gravity and particle DM, as was suggested for the standard symmetron
in Ref. [53].
In any case, generalized symmetrons beyond $(\alpha,\beta)=(1,2)$ are
interesting theories to study since, to date, there are no experimental
constraints on their parameter spaces. In addition, redoing the analyses made
in Refs. [51] and [52] for some of the more promising models, for example
theories like $(\alpha,\beta)=(5,7)$ and beyond, might in the future
demonstrate that generalized symmetrons are actually good alternatives to
particle DM.

###### Acknowledgements.

CK is grateful to C. Burrage and M. Pitschmann for useful discussions, and to
C. Voith and B. Burazor Domazet for spotting a sign mistake in the lensing
calculation. This article was supported by the Austrian Science Fund (FWF): P
34240-N, and is based upon work from COST Action COSMIC WISPers CA21106,
supported by COST (European Cooperation in Science and Technology).

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# Nonlinear Addition of Qubit States Using Entangled Quaternionic Powers of
Single-Qubit Gates

Dominic Widdows<EMAIL_ADDRESS>IonQ

###### Abstract

This paper presents a novel way to use the algebra of unit quaternions to
express arbitrary roots or fractional powers of single-qubit gates, and to use
such fractional powers as generators for algebras that combine these
fractional input signals, behaving as a kind of nonlinear addition.

The method works by connecting several well-known equivalences. The group of
all single-qubit gates is $U(2)$, the unitary transformations of
$\mathbf{C}^{2}$. Using an appropriate phase multiplier, every element of
$U(2)$ can be mapped to a corresponding element of $SU(2)$ with unit
determinant, whose quantum mechanical behavior is identical. The group $SU(2)$
is isomorphic to the group of unit quaternions. Powers and roots of unit
quaternions can be constructed by extending de Moivre’s theorem for roots of
complex numbers to the quaternions by selecting a preferred ‘square root of
-1’. Using this chain of equivalences, for any single-qubit gate $A$ and real
exponent $k$, a gate $B$ can be predictably constructed so that $B^{k}=A$.

Different fractions generated in this way can be combined by connecting the
individually rotated qubits to a common ‘sum’ qubit using entangling 2-qubit
CNOT gates. Some of the simplest such algebras are explored — those generated
by roots of the quaternion $\mathbf{k}$ (which corresponds to and $X$-rotation
of the Bloch sphere), the quaternion
$\frac{\sqrt{2}}{2}(\mathbf{i}+\mathbf{k})$ (which corresponds to the Hadamard
gate), and a mixture of these.

One of the goals of this research is to develop quantum versions of classical
components such as the classifier ensembles and activation functions used in
machine learning and artificial intelligence. An example application for text
classification is presented, which uses fractional rotation gates to represent
classifier weights, and classifies new input by using 2-qubit CNOT gates to
collect the appropriate classifier weights in a topic-scoring qubit.

##  I Introduction

Transformations in quantum theory are expected to be continuous and reversible
[9]. For any transformation $A$, we should therefore be able to find some
partial transformation $B$ such that performing this transformation $k$ times
recovers $A$, which is expressed using the equation $B^{k}=A$. The problem of
finding some suitable $B$ can be solved for general matrices $A$ and real
powers $k$, but general solutions rely on detailed machinery such as Schur
decomposition and Padé approximation [10].

For operations on single qubits, the group of possible transformations is much
smaller and simpler than the general case. The qubit state is represented by a
normalized vector $\alpha\ket{0}+\beta\ket{1}$ with
$|\alpha|^{2}+|\beta|^{2}=1$, and the transformations on this state are given
by the unitary group $U(2)$, the subgroup of $GL(2,\mathbf{C})$ that preserves
this invariant. This observation leads to the small set of single-qubit
transformations or ‘gates’ familiar in quantum computing [15, §1.3.1, §4.2].
It is well-known that the space of single-qubit gates corresponds to rotations
of the Bloch sphere $S^{2}$, and that this space is 3-dimensional — for
example, a rotation of $S^{2}$ can be specified by giving a direction for the
rotation axis (2 coordinates, e.g. latitude and longitude), and a rotation
angle (1 more coordinate). A common way to describe this space is to use three
Euler angles, which are the parameters $\theta$, $\phi$ and $\lambda$ in the
‘universal’ or U-gate [2, §1.4]. Euler angles are well-established in many
fields and quite intuitive to understand, but also have well-known drawbacks
including coordinate singularities and gimbal lock [3]. This makes composing
and decomposing sequences of rotations using Euler angles cumbersome, which
has encouraged the adoption of mathematical alternatives including quaternions
for applications in computer science including graphics [13], aerospace, and
virtual reality [11].

This paper demonstrates that the use of quaternions instead of Euler angles
enables fractional powers of single-qubit gates to be constructed easily and
explicitly, and that their behavior when composed into larger circuits can be
predicted simply and effectively. The method relies on connecting the
following observations:

  * •

Every gate represented by a unitary matrix $A$ can also be represented by a
corresponding special unitary matrix (a matrix with unit determinant) by
multiplying the matrix by the phase factor $\det(A)^{-\frac{1}{n}}$, where $n$
is the number of (complex) dimensions which in the case of single-qubit gates
is $n=2$.

  * •

The special unitary group $SU(2)$ is isomorphic to the group of unit
quaternions, and so single-qubit gates can be represented by unit quaternions
[18].

  * •

Fractional real roots of quaternions can be obtained using the quaternionic
version of the de Moivre formula
$(\cos(\theta)+i\sin(\theta))^{n}=\cos(n\theta)+i\sin(n\theta)$ [16].

  * •

The quaternionic root found in this way can be mapped back to the
corresponding special unitary matrix, which can be multiplied by a real root
of the phase factor if we want to recover the original unitary matrix exactly.

  * •

These observations are combined to give an explicit formula for any real power
of a single-qubit gate.

Once these fractional operations are available, they can be combined in simple
‘adder circuits’ that use 2-qubit CNOT gates to connect each of the
fractionally rotated qubits to a common ‘sum’ qubit that collects the various
contributions. The method is applied in a demonstration problem in text
classification. Small fractional rotations are repeated to count the number of
times a word is seen with a particular topic in training. When classifying a
new phrase, these rotation signals are accumulated by connecting the
corresponding word-counting qubits to a scoring qubit using CNOT gates. When
two such fractional rotations through angles $\theta$ and $\varphi$ are
combined into the same scoring qubit, this gives an outcome probability that
is a function $F$ of $\theta$ and $\varphi$. This behavior depends on which
quaternion generators were chosen — for example, the quaternion
$q=\cos(\theta)+\mathbf{k}\sin(\theta)$ corresponding to the unitary gate
$R_{x}(\theta)$ gives rise to a scoring function whose probability of giving
an output $\ket{1}$ from the input angles $\theta$ and $\varphi$ is given by
$\cos^{2}(\theta/2)\sin^{2}(\varphi/2)+\sin^{2}(\theta/2)\cos^{2}(\varphi/2)$.
This application was chosen partly for its relevance to quantum machine
learning, one of whose challenges is to find suitable nonlinear activation
functions [17, §5.4.1].

The rest of this paper is arranged as follows. Section II recalls the basic
properties of quaternions, just enough to explain the passage from single-
qubit unitary operators, to quaternions, to fractional quaternion powers, back
to fractional powers of unitary operators. Section III shows how two of these
fractional rotations can be wired together in a 3-qubit circuit that combines
the contributions from each input fraction, and compares three such functions
arising from different quaternion generators. Section IV demonstrates the
successful application of these components in a simple text classification
problem, where the accurate results also demonstrate that inaccuracies in
gate-level qubit operations do not always lead to application-level errors.
Finally, Section V relates the work in this paper to known alternatives, and
proposes further work expanding on common areas.

##  II Quaternion Powers for Fractional Single-Qubit Gates

It is assumed that the reader is familiar with quantum gates and unitary
matrices (otherwise see [15, §2.1]). Quaternions are less ubiquitous in
quantum theory, so their important properties are recalled here.

Quaternions were discovered in 1843 by William Rowan Hamilton (1805–1865), who
worked closely with Arthur Caley (1821–1895) on the foundations of linear
algebra, introduced the terms ‘vector’, ‘associative’, ‘commutative’, and
‘distributive’ [8], and also developed Hamiltonian mechanics. The quaternions
$\mathbb{H}$ are a 4-dimensional real algebra generated by the identity
element 1 and the symbols $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$,
multiplied according to the quaternion relations

$\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k}\quad\quad\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i}\quad\quad\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}\quad\quad\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=-1$

and the distributive law. (Note the similarity with the vector cross product
in 3-dimensional coordinate geometry, and the algebra of Pauli matrices in
quantum mechanics.) The quaternion algebra is not commutative, though it does
obey the associative law (the investigation of these properties of quaternions
led Hamilton to invent those terms). The quaternions are a division algebra
(an algebra with the property that $ab=0$ implies that $a=0$ or $b=0$).

Each of the generators $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ behaves as a
‘square root of $-1$’, and more generally, so does every imaginary quaternion
$a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ with $a^{2}+b^{2}+c^{2}=1$. Identifying
such an $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ with the imaginary number
$i=\sqrt{-1}$ gives an embedding of the complex numbers
$\mathbb{C}\hookrightarrow\mathbb{H}$. (To avoid confusion in this paper, the
symbol $i$ will be used for the complex square root of $-1$, and the boldface
symbol $\mathbf{i}$ will be used for the imaginary quaternion.) The set of
quaternionic square roots of $-1$ can be identified with the 2-sphere $S^{2}$.
A quaternion $q=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}$ can be
decomposed into its real part $q_{0}$ and its imaginary part
$q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}$, which Hamilton and his
immediate successors referred to as the scalar and vector parts of the
quaternion.

As with complex numbers, the conjugate of a quaternion
$q=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}$ is defined as
$\bar{q}=q_{0}-q_{1}\mathbf{i}-q_{2}\mathbf{j}-q_{3}\mathbf{k}$, the norm of a
quaternion is given by the formula $|q|=q\bar{q}$, and the inverse is given by
$q^{-1}=\bar{q}/|q|$. Quaternions have a polar decomposition
$q=|q|(\cos(\theta)+\gamma\sin(\theta))=|q|e^{\gamma\theta}$, where $\gamma$
is one of the 2-sphere of unit imaginary quaternions. The unit quaternions are
those with unit norm. A unit quaternion $q$ can therefore be written as
$q=\cos(\theta)+\gamma\sin(\theta)$, where $\gamma$ again is an imaginary unit
quaternion $\gamma\in\mathbb{H}$ such that $\gamma^{2}=-1$. The group of unit
quaternions is often called $Sp(1)$, because it is the first in the series of
symplectic groups [6, §7.2], though the relationship between symplectic
geometry and quantum mechanics is not needed in this paper.

See Widdows, [19, Ch 1] for more explanation of quaternion terminology, simple
constructions using quaternions such as 3d and 4d rotations, and isomorphisms
with various real and complex matrix groups and manifolds. The important
relationship between $SU(2)$ and the unit quaternions is explained below.

###  II.1 Real Fractional Roots of Quaternions

Crucially for this work, the de Moivre formula

$(\cos(\theta)+i\sin(\theta))^{n}=\cos(n\theta)+i\sin(n\theta)$

applies unchanged to the quaternionic setting [16]. That is, for a unit
quaternion $q=\cos(\theta)+\gamma\sin(\theta)$ with $\gamma^{2}=-1$,

$q^{n}=\cos(n\theta)+\gamma\sin(n\theta).$ (1)

This adapts immediately to fractional powers: if
$p=q^{n}=\cos(n\theta)+\gamma\sin(n\theta)$ it follows that $q=p^{1/n}$, so
setting $\varphi=n\theta$, it follows that if
$p=\cos(\varphi)+\gamma(\sin\varphi)$, then

$p^{1/n}=\cos(\varphi/n)+\gamma(\sin\varphi/n).$ (2)

In general, quaternion roots, like complex roots, are usually not uniquely
defined. Any multiple of $2\pi$ can be added to the angle $\varphi$ without
changing the quaternion $p$, so typically the equation $q=p^{\frac{1}{n}}$ has
$n$ roots if $n$ is an integer, infinitely many roots if $n$ is irrational,
with special cases for when $p$ is a real number [16].

###  II.2 Single-Qubit Gates and Quaternions

Recall that single-qubit gates are represented by operators in the group
$U(2)$, which can be characterized as those matrices $\\{A\in
GL(2,\mathbf{C}):A\bar{A}^{T}=I\\}$, where $\bar{A}$ is the complex conjugate
matrix, $A^{T}$ the transpose, and $I$ is the identity matrix. All unitary
matrices have determinants in the complex unit circle group $U(1)$, whereas
special unitary matrices are those with determinant equal to 1. The group of
such $2\times 2$ matrices is called $SU(2)$.

The group of unit quaternions under multiplication is isomorphic to the
multiplicative group $SU(2)$ with a correspondence given by

$q=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\longleftrightarrow
A=\begin{pmatrix}a+bi&c+di\\\ -c+di&a-bi\end{pmatrix},\mathrm{where}\
a^{2}+b^{2}+c^{2}+d^{2}=1.$ (3)

It is easy to check that the condition $a^{2}+b^{2}+c^{2}+d^{2}=1$ guarantees
that $\det(A)=1$ and $A\bar{A}^{T}=I$, so $A$ is indeed an element of $SU(2)$.

The Pauli matrices [15, §2.1.3] are obtained from this construction as
follows:

$\displaystyle 1\in\mathbf{H}\longleftrightarrow\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\equiv\sigma_{0}\equiv I$
$\displaystyle\mathbf{i}\in\mathbf{H}\longleftrightarrow\begin{bmatrix}i&0\\\
0&-i\end{bmatrix}\equiv i\sigma_{z}\equiv iZ$
$\displaystyle\mathbf{j}\in\mathbf{H}\longleftrightarrow\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\equiv-i\sigma_{y}\equiv-iY$
$\displaystyle\mathbf{k}\in\mathbf{H}\longleftrightarrow\begin{bmatrix}0&i\\\
i&0\end{bmatrix}\equiv i\sigma_{z}\equiv iX$

Note the multiplication by $\pm i$ to recover the Pauli matrices $\sigma_{x}$,
$\sigma_{y}$, $\sigma_{z}$ in their most standard form. This is allowed
because multiplying by any global phase factor $e^{i\theta}$ makes no physical
difference, and is necessary to be able to find a quaternion corresponding to
every unitary transformation. This strategy generalizes to other unitary
matrices: since the group $U(2)$ is more general than the normal subgroup
$SU(2)$, not all unitary matrices representing single-qubit quantum gates have
corresponding quaternions given by equation 3, but all unitary matrices are
phase-equivalent to a special unitary matrix for which the quaternion
correspondence works. An appropriate special unitary matrix can be found as
follows. For $A\in U(n)$, suppose that $\det(A)=e^{i\theta}$. It follows that
$\det(e^{-\frac{i\theta}{n}}A)=1$, and that this operator produces the same
physical state as $A$. Thus we define a mapping
$A\rightarrow\det(A)^{-\frac{1}{n}}A$, which assigns to each unitary matrix a
special unitary matrix with the same physical effect. This mapping is in fact
a fibration mapping $U(n)\mapsto SU(n)$.

The path for finding fractional real powers of a single-qubit quantum logic
gate $A$ now takes shape. To find a gate $B$ such that $B^{k}=A$ for some
power $k$, we follow the following steps:

  * •

Take any single-qubit gate / unitary matrix $A\in U(2)$.

  * •

Map it to a representative special unitary matrix in $A^{\prime}\in SU(2)$
using the formula $A^{\prime}=\det(A)^{-\frac{1}{2}}A$.

  * •

Map $A^{\prime}\in SU(2)$ to a unit quaternion $q$ using the correspondence in
equation 3.

  * •

Take the $k^{th}$ root $q^{\frac{1}{k}}$ using equation 2.

  * •

Map this back to a special unitary matrix $B^{\prime}$, again using equation
3. From the isomorphism between unit quaternions and $SU(2)$, it follows that
$(B^{\prime})^{k}=A^{\prime}$.

  * •

(Optional) To recover the matrix $A$ as well as the phase-equivalent
$A^{\prime}$, multiply $B^{\prime}$ by a phase factor
$\det(A)^{\frac{1}{2k}}$, and verify that if
$B=\det(A)^{\frac{1}{2k}}B^{\prime}$, then $B^{k}=A$.

For dimension $n=2$, note that each unitary matrix $A$ has two possible
special unitary images under the fibration mapping
$A\rightarrow\det(A)^{-\frac{1}{2}}A$, because if $\det(A)=1$ then
$\det(-A)=1$. This corresponds to the multiplicity of roots in the expression
$\det(A)^{-\frac{1}{2}}$. In algebraic terms, $U(2)$ is not a direct product
$SU(2)\oplus U(1)$, though there is a short exact sequence $1\rightarrow
SU(2)\rightarrow U(2)\rightarrow U(1)\rightarrow 1$. In terms of the
correspondence with unit quaternions, this amounts to the observation that the
unit quaternions $q$ and $-q$ produce the same gate operation.

##  III Component Application — Weighted Addition

The original motivation for this work was to create a kind of quantum addition
for fractional states, for use as a component in machine learning operations
such as building classification / decision processes, and for activation
functions in neural networks. This section demonstrates this compositional
behavior.

The basic idea is that fractional rotations in various qubits can be combined
by entangling each rotation qubit with a common target or scoring qubit using
CNOT gates. An example circuit implementation is shown in Figure 1. There are
two ‘summand’ qubits $q_{0}$ and $q_{1}$, each of these has been put through a
single-qubit fractional gate transformation, and these qubits are each
connected to a third ‘sum’ qubit $q_{2}$. The result in the sum qubit is then
measured. In this example, the rotation $\mathrm{R_{X}}(\theta)$ was used,
which corresponds to the quaternion $\cos(\theta)+\mathbf{k}\sin(\theta)$.
This example is particularly well-behaved, both algebraically and for using
standard gate notation, though other quaternion generators can be used.

Figure 1: An Example Addition Circuit

In the rest of this section, the following generators were used:

$\mathbf{k}^{\alpha}\longleftrightarrow\begin{bmatrix}\cos(\frac{\pi\alpha}{2})&i\sin(\frac{\pi\alpha}{2})\\\
i\sin(\frac{\pi\alpha}{2})&\cos(\frac{\pi\alpha}{2})\end{bmatrix}\longleftrightarrow
X^{\alpha}$ (4)

$\left(\frac{1}{\sqrt{2}}(\mathbf{i}+\mathbf{k})\right)^{\alpha}\longleftrightarrow\begin{bmatrix}\cos(\frac{\pi\alpha}{2})+\frac{i}{\sqrt{2}}\sin(\frac{\pi\alpha}{2})&\frac{i}{\sqrt{2}}\sin(\frac{\pi\alpha}{2})\\\
\frac{i}{\sqrt{2}}\sin(\frac{\pi\alpha}{2})&\cos(\frac{\pi\alpha}{2})-\frac{i}{\sqrt{2}}\sin(\frac{\pi\alpha}{2})\end{bmatrix}\longleftrightarrow
H^{\alpha}$ (5)

Note that when $\alpha=1$, these matrices become $\begin{bmatrix}0&i\\\
i&0\end{bmatrix}$ and $\frac{1}{\sqrt{2}}\begin{bmatrix}i&i\\\
i&-i\end{bmatrix}$, which are the standard $X$ and $H$ (Hadamard) rotation
gates [15, §4.2], modulo a phase factor of $i$ to make the determinants equal
to 1. This demonstrates the convenience of the quaternion formulation of
Section II — while the fractional $X$-rotations of equation 4 are standard in
the literature, the fractional Hadamard gate of equation 5 is uncommon, but
the quaternion recipe makes writing down an appropriate expression for
fractional powers of $H$ straightforward. (This point is revisited in Section
V.)

Figure 2: Results of Addition Circuits Fractional Powers of X Rotation and Hadamard Gate as Generators |  |
---|---|---
Fractional X rotations | Fractional Hadamards | Mixture of X and Hadamard

Example outcomes using combinations of these generators in parametrized
circuits like those of Figure 1 are given in in Figure 2. Here the horizontal
axes represent the angles $\theta$ and $\varphi$ through which the two summand
qubits are rotated in Figure 1, and the vertical axis represents the
probability of measuring a $\ket{1}$ outcome in the sum qubit. Around the
origin, each combination is monotonically increasing as both the input angles
increase, so for small angles, the result can be regarded as ‘nonlinear
addition’. Larger angles exhibit periodic behavior as expected. The outcomes
are different depending on the fractional generators used. With the fractional
Hadamard generators, if either summand is around the value $\pi$, the output
is close to a maximum of $\frac{1}{2}$, which can be thought of as a
continuous analogy to logical disjunction if the common maximum value is
rescaled to $1$. (Such a rescaling makes sense with this Hadamard algebra
because it only takes values in $[0,\frac{1}{2}]$.) The X rotation generators
produce undulating waveforms with probabilities throughout the range $[0,1]$,
and less obviously, if either input is $\pi/2$, the outcome probabilities are
always $50:50$. This is an intriguing alternative — if the Hadamard
disjunction shape can be summarized as “whatever you say, I still say that one
option is true”, the X rotation logic can be summarized as “whatever you say,
I still say that each option is equally likely”. With a mixture of X rotation
and Hadamard generators, we observe a combination of these behaviors.

###  III.1 Worked Example for Fractional X-Rotations

This section works through the linear algebra of the circuit in Figure 1 with
the generators set to be X rotations which correspond to the quaternion
$\mathbf{k}$. For convenience, we can set $\theta=\pi\alpha$, so that
$\mathbf{k}^{\alpha}$ and $\mathrm{R_{X}}(\theta)$ are basically the same, and
the equivalence in equation 4 leads to the well-known form

$R_{x}(\theta)=\begin{bmatrix}\cos(\theta/2)&-i\sin(\theta/2)\\\
-i\sin(\theta/2)&\cos(\theta/2)\end{bmatrix}$

Consider the behavior of the quantum circuit in Figure 1 where operations on
the top two qubits are $R_{x}(\theta)$ and $R_{x}(\varphi)$. For ease of
notation, write $a=\cos(\theta/2)$, $b=-i\sin(\theta/2)$, $c=\cos(\varphi/2)$,
$d=-i\sin(\varphi/2)$, so that rotation matrices are just

$R_{x}(\theta)=\begin{bmatrix}a&b\\\ b&a\end{bmatrix}\quad\mathrm{and}\quad
R_{x}(\varphi)=\begin{bmatrix}c&d\\\ d&c\end{bmatrix}$ (6)

Now consider the combination of a single such rotation and a CNOT gate:

The action of this circuit on the 2-qubit basis states $\ket{00}$, $\ket{01}$,
$\ket{10}$, $\ket{11}$ is given by the matrix

$\mathrm{CNOT}\circ(R_{x}(\theta)\otimes I_{2\times
2})=\begin{bmatrix}a&0&b&0\\\ 0&a&0&b\\\ 0&b&0&a\\\ b&0&a&0\\\ \end{bmatrix}$

and the corresponding matrices for the 3-qubit versions of this component are
shown in Table 1. (The rows and columns of these matrices act upon the tensor
product states in the big-endian order
$\ket{000},\ket{001},\ldots,\ket{111}$.)

Table 1: Circuits and unitary operator matrices for the given circuit components (zeros omitted) |  | $\begin{bmatrix}a&&&&b\\\ &a&&&&b\\\ &&a&&&&b\\\ &&&a&&&&b\\\ &b&&&&a\\\ b&&&&a\\\ &&&b&&&&a\\\ &&b&&&&a\\\ \end{bmatrix}$
---|---|---
|  | $\begin{bmatrix}c&&d\\\ &c&&d\\\ &d&&c\\\ d&&c\\\ &&&&c&&d\\\ &&&&&c&&d\\\ &&&&&d&&c\\\ &&&&d&&c\\\ \end{bmatrix}$

The combined operation of the adder circuit of Figure 1 is given by the
product of these two matrices. To skip tedious calculation, note that we are
only concerned with the action on the initial zero state $\ket{000}$, which
corresponds to the vector $(1,0,\ldots,0)$. The first matrix in Table 1 maps
this vector to $(a,0,0,0,0,b,0,0)$, which the second matrix then maps to
$(ac,0,0,ad,0,bc,bd,0)$. The nonzero entries correspond to the states
$\ket{000}$, $\ket{011}$, $\ket{101}$, $\ket{110}$, and the measurement is
performed on the last qubit, so the probability of measuring a $\ket{1}$ state
for this qubit is given by the sum of the squares of the $\ket{011}$ and
$\ket{101}$ components. It follows that

$P(q_{2}=\ket{1})=|ad|^{2}+|bc|^{2}=\cos^{2}(\theta/2)\sin^{2}(\varphi/2)+\sin^{2}(\theta/2)\cos^{2}(\varphi/2).$
(7)

This shows that the probability of observing a $\ket{1}$ in the target qubit
is given by a nonlinear combination of the inputs $\theta$ and $\varphi$,
which near the origin is monotonically increasing in both $\theta$ and
$\varphi$, and exhibits periodic behavior over wider ranges. Less obviously,
if either angle is equal to $\pi/2$, the outcome is a $50:50$ split. For
example, if $\theta=\pi/2$, then
$\cos^{2}(\theta)=\sin^{2}(\theta)=\frac{1}{2}$, so

$\cos^{2}(\theta/2)\sin^{2}(\varphi/2)+\sin^{2}(\theta/2)\cos^{2}(\varphi/2)=\frac{1}{2}(\sin^{2}(\varphi/2)+\cos^{2}(\varphi/2))=\frac{1}{2}.$

###  III.2 Generalization to Other Quaternion Generators

While the construction above is particularly straightforward for X-rotations,
it generalizes to all unit quaternions relatively simply. Recall from equation
3 that the quaternion
$q=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}$ corresponds to the
matrix

$\begin{bmatrix}q_{0}+q_{1}i&q_{2}+q_{3}i\\\ -q_{2}+q_{3}i&q_{0}-q_{1}i\\\
\end{bmatrix}$

so if we write $\alpha=q_{0}+q_{1}i$ and $\beta=q_{2}+q_{3}i$, then equation 3
leads to the correspondence

$q=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}=\alpha+\beta\mathbf{j}\longleftrightarrow
A=\begin{bmatrix}\alpha&\beta\\\
-\overline{\beta}&\overline{\alpha}\end{bmatrix}.$ (8)

Note that in this process, the quaternion $\mathbf{i}$ is singled out and
identified with the complex number $i$, which amounts to choosing a particular
complex structure that identifies the quaternions $\mathbb{H}$ with the
complex vector space $\mathbb{C}^{2}$ [19, §1.1.2]. The matrix $A$ is only
slightly more complicated than those representing X-rotations in equation 6.
Relaxing the restriction that $A$ and $B$ should be fractional X-rotations,
equation 6 becomes

$A=\begin{bmatrix}\alpha&\beta\\\
-\overline{\beta}&\overline{\alpha}\end{bmatrix}\quad\mathrm{and}\quad
B=\begin{bmatrix}\gamma&\delta\\\
-\overline{\delta}&\overline{\gamma}\end{bmatrix}.$ (9)

The matrix multiplication argument from the previous section can now be
repeated, this time taking care the distinguish the top and bottom rows of the
rotation matrices. This leads to a generalization of the result
$P(q_{2}=\ket{1})=|ad|^{2}+|bc|^{2}$ in equation 7, which is that

$P(q_{2}=\ket{1})=|\alpha\overline{\delta}|^{2}+|\overline{\beta}\gamma|^{2}.$

This process also illustrates some of the useful flexibility of quaternion
representations. Earlier in this paper, the calculation of powers and roots
for a particular quaternion could be done most simply by selecting a
privileged square root of $-1$ for that quaternion, and using the
corresponding polar representation (equation 1). Now instead we want to reason
about the combined effect of two quaternions, which is done easily by using a
common square root of $-1$, and the corresponding matrix representation.

The circuits and methodologies here extend to summing more than two inputs:
this is done by adding more summand qubits and connecting them to the same sum
qubit using more CNOT gates. Initial computational experiments show that at
least some of the properties found with two qubits persist with more qubits,
including monotonic increasing behavior in each variable near the origin, and
sometimes if any input puts the output in a $50:50$ state, the result cannot
be dislodged from there. It is possible that an explicit expression like
equation 9 can be found that generalizes to more input quaternions: this has
not been done yet. However, the next section demonstrates the practical
effectiveness of such circuits in a machine learning task.

##  IV System Application: Bag of Words Classification

A ‘bag of words’ classifier in natural language processing (NLP) is one that
classifies texts or documents based purely on the words occurring in the
document, irrespective of the order they come in. For example, given the
phrases “horse chestnut” and “chestnut horse”, it has no way of knowing that
the first is a kind of tree and the second is a kind of animal — all it knows
is that horse makes the topic animal more likely and chestnut makes the topic
tree more likely, and it would score the phrases “horse chestnut” and
“chestnut horse” identically. Though primitive, bag of words techniques were
remarkably successful for decades (especially in information retrieval [20]).

The process for using the adder components of Section III to build a
classifier was as follows:

  * •

We are given a training corpus of (topic, sentence) pairs, with $T$ topics and
a vocabulary of $W$ different words.

  * •

A circuit is created using $T(W+1)$ qubits. (This only works for very small
training and test sets, hence this example would not yet scale to large
datasets.)

  * •

The first $TW$ qubits store the (word, topic) scores, and the last $T$ qubits
are the ‘topic counters’ that accumulate a score for each topic at
classification time.

  * •

During training, if the training phrase $(t,[w_{1},\dots,w_{n}])$ is
encountered, each of the qubits corresponding to the pair $(t,w_{i})$ is
incremented by a small angle.

  * •

During classification, for the test phrase $[w_{1},\dots,w_{n}]$, each of the
qubits $(t_{j},w_{i})$ is connected to the $t_{j}$ classification qubit using
a CNOT gate, following the template component circuit in Figure 1.

  * •

In this way, each of the topic classification qubits becomes an adder circuit
scoring qubit for the word scores for that topic.

  * •

Each of the topic qubits is measured and (after an appropriate number of
shots), the topic qubit with the highest probability of giving a $\ket{1}$
output is chosen.

Figure 3: Classification Circuit for Two Words and Two Topics

A minimal example circuit is shown in Figure 3. This circuit would arise from
a training corpus with two training phrases, such as:

I played football | $\rightarrow$ | sport
---|---|---
I played guitar | $\rightarrow$ | music

The implementation requires a classical preprocessing step which marks
football as $\mathit{word}_{1}$, guitar as $\mathit{word}_{2}$, sport as $T1$
and music as $T2$ (and ignores the other words since they are common to both
topics). Then the quantum part of the training process prepares the circuit in
Figure 3, without the CNOT gates. For the quantum classification process,
consider the phrase “I kicked the football”. The classical register is used to
recognize the word football as $\mathit{word}_{1}$, and so CNOT gates are
added from the $word_{1},T1$ and $word_{1},T2$ qubits to the scoring qubits
for each topic. Each of the topic qubits is measured (completing the circuit
outline in Figure 3), and the topic that gets the most $\ket{1}$ outcomes over
a statistically significant number of shots is chosen as the appropriate topic
label.

This setup requires parametrization and tuning, and is still too basic to
compete at scale with classifiers built using large language models [7, Ch
16]. However, it performs well on a small dataset when compared to existing
quantum alternatives. In particular, using a rotation angle of $\pi/24$ and a
vocabulary reduced to 9 most salient words, the classifier achieved 100%
accuracy on the test set from the lambeq datasets used in [12], which contains
70 training phrases and 30 test phrases. This compares favorably with the
accuracy of 83% reported in [12], which was achieved using just a 6 qubit
register, but a much more linguistically sophisticated semantic modelling
framework which requires a full grammatical parse tree as a classical
preprocessing step. It should be noted that the dataset in question is
artificially simple and was created specifically to enable experiments of this
nature: this more than anything else explains why it was possible to obtain
100% accuracy. Our result here does not demonstrate that one method is better
than the other in general — rather, it showed that a simpler model performed
more effectively at this specific task at the cost of using more qubits, and
that the components developed in this paper can be used to build circuits that
produce successful results on real quantum hardware.

An obvious weakness of the classification circuit in Figure 3 is that the CNOT
gate is its own inverse, so repeated operations cancel one another out rather
than reinforcing one another. Given the test sentence “I like football,
football is fun!”, the repetition of the topical word football would cause the
association with the correct topic to be forgotten rather than accentuated.
Another issue that remains to be investigated is automating the choice of a
fractional power or rotation angle, which should vary between different words
depending on their salience. There are also some positive observations to be
made: in particular, the accuracy of the eventual classification decisions
demonstrates that even with noisy intermediate scale quantum (NISQ) hardware,
accurate results can be obtained in cases where the application-level outcomes
are suitably stable. In this demonstration case, the circuit only needs to
show that one topic is significantly more relevant than another, and this
high-level decision was unaffected by small inaccuracies.

##  V Related and Further Work

This paper has demonstrated a quaternionic method for finding arbitrary real
powers of single-qubit gates, the use of these fractional powers in a simple
addition component, and the use of several such components in a classifier
circuit. This final section compares these approaches to available
alternatives.

The method of combining fractional rotations is different from the traditional
“quantum adder” circuit proposed in [5], which is for integer addition, and
relies on using $n$-qubits to discretely represent $2^{n}$ bits. Optimizations
obtained using the Fast Fourier Transform [4] speed up the computation, but
still use the same discrete representation rather than using the continuous
nature of vector coordinates or angles to represent a continuum of real
numbers. A natural question that arises is whether it is possible just to add
two state vectors in a quantum computer, but this does not work directly: the
physical impossibility of a unitary transformation that takes $\ket{\Psi_{1}}$
and $\ket{\Psi_{2}}$ as inputs and produces an output
$\ket{\Psi_{1}}+\ket{\Psi_{2}}$ is discussed by [1]. (It is related to the no-
cloning rule: if $\ket{\Psi_{1}}=0$, in many circumstances this would be
equivalent to copying $\ket{\Psi_{2}}$ into the output state, which is
impossible.)

The collection and combination of inputs from several components and
outputting a score is ubiquitous in neural networks — different typesetting
would make the circuit in Figure 1 look exactly like a network unit such as a
perceptron [7, Ch 10]. The power of neural networks arises partly because the
use of nonlinear activation functions enables them to approximate nonlinear as
well as linear functions, and this is an acknowledged problem for quantum
machine learning, because unitary transformations are by definition linear
[17, §5.4.1]. This article has shown that, though matrices only represent
linear operators, when these are used to describe angles of unit quaternions,
the algebraic operations on those angles are not at all linear. This motivates
further work on testing these components on different machine learning
problems, using a variety of quaternion generators, to investigate which lead
to the most successful nonlinear activation functions.

The equivalence between unit quaternions and special unitary matrices is well-
known, and quaternions have been used to model operations on the Bloch sphere
[18]. This work uses the product of quaternions to model gate composition, but
does not use quaternions to form powers and roots of gate operations.
Generators and roots of quantum gates are explored by [14]. Their approach
works for multi-qubit gates as well as single-qubit gates, but only for gates
$A$ such that $A^{2}=I$. An approximate method for calculating arbitrary real
powers of square matrices is presented by Higham and Lin, [10], who also
review several other algorithms for this. This is a general solution to the
matrix power problem, but relies on methods such as Schur decomposition and
Padé approximation, which are computationally detailed and less intuitively
direct. By contrast, the method developed here is particularly simple for
taking powers of the $2\times 2$ unitary matrices that represent single-qubit
quantum gates. Due to its direct mathematical construction, the quaternionic
method has particular geodesic properties, in the sense that it takes a smooth
shortest-path through the quaternion group, and hence the gate space $U(2)$ or
the phase-free subgroup $SU(2)$, which may lend itself to exact and smooth
implementation.

The geodesic property also helps to explain why the notion of ‘fractional
rotations’ does not work well with Euler angles. For example, it is well-known
that a Hadamard gate can be decomposed as a $90^{\circ}$ rotation around the
Y-axis, followed by a $180^{\circ}$ rotation around the X-axis [15, §1.3.1].
However, since rotations do not commute, it does not follow that performing a
$9^{\circ}$ Y-rotation followed by an $18^{\circ}$ X-rotation 10 times in
succession recovers the operation of a Hadamard gate — a calculation shows
that this maps the state $\ket{0}$ to
$(0.445-0.05i)\ket{0}+(0.632-0.632j)\ket{1}$ rather than the correct state
$\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$. The method of using quaternions instead
of Euler angles could be described as finding a coordinate system in which
dividing up rotations in such a way does work correctly. The drawbacks of
Euler angles (such as coordinate singularities, gimbal lock, and cumbersome
composition rules) are well-known [3], and such frustration has motivated the
use of quaternions in areas of computer science including graphics [13], 3d
simulation, and augmented reality [11] — from this point of view, this paper
extends some of the established benefits of quaternions over Euler angles to
applications in quantum computing.

##  VI Conclusion

This paper demonstrated three main innovations:

  1. 1.

The use of quaternion algrebra to represent fractional powers of any single-
qubit gate.

  2. 2.

The use of two CNOT gates to combine such fractions into a function of both
inputs that can be chosen to be monotonically increasing over known intervals.

  3. 3.

The arrangement of several such CNOT combinations to build a language topic
classifier that performed perfectly accurately in a small experiment on
quantum hardware.

The formulations developed here all have some novel aspects, though each of
these advances can be derived quite easily from well-established mathematics,
physics, and computer science. The small but successful classification example
shows that the techniques can already be applied in a real application on
quantum hardware.

##  VII Acknowledgements

Thanks for helpful discussions go particularly to Lazaro Calderin, Vandiver
Chaplin, Jon Donovan, Chris Monroe, Daiwei Song, and Chase Zimmerman.

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