pebble $a^{\prime}$ in $l_{c_{1}}$ has not been reallocated by the end of
$s_{c}$. From this it is easy to see that the tuple $\alpha(\mathbf{a})$ (with
function application defined component-wise on the tuple) is related in
$\mathbb{H}_{n,k}\mathcal{{A}}$. Finally, we verify that $\alpha$ satisfies
the coalgebra laws. The counit law,
$\epsilon\circ\alpha=\text{id}_{\mathcal{{A}}}$ is satisfied by definition.
For comultiplication, it suffices to check that for any $a,b\in A$, if $b$
appears in $s_{c_{a}}=[l_{c_{1}}|\dots|l_{c_{q}}]$ then it appears in exactly
one of the $l_{c_{i}}$ and $\alpha(b)=[[l_{c_{1}}|\dots|l_{c_{i-1}}]|b]$. This
can be seen to hold from the construction above, concluding our proof.

##  5\. Concluding Remarks

The work of Abramsky et al., giving comonadic accounts of pebble games and
their relationship to logic has opened up a number of avenues of research. It
raises the possiblilty of studying logical resources through a categorical
lens and introduces the notion of _coresources_. This view has been applied to
pebble games [3], Ehrenfeucht-Fraïssé games, bisimulation games [4] and also
to quantum resources [1, 2]. In this paper we have extended this approach to
logics with generalised quantifiers.

The construction of the comonad $\mathbb{H}_{n,k}$ introduces interesting new
techniques to this project. The pebbling comonad $\mathbb{P}_{k}$ is graded by
the value of $k$ which we think of as a _coresource_ increasing which
constrains the morphisms. The new parameter $n$ provides a second coresource,
increasing which further constrains the moves of Duplicator. It is interesting
that the resulting comonad can be obtained as a quotient of $\mathbb{P}_{k}$
and the strategy lifting argument developed in Section 3 could prove useful in
other contexts. The morphisms in the Kleisli category correspond to winning
strategies in a new game we introduce which characerises a natural logic: the
positive logic of homomorphism-closed quantifiers. The isomorphisms correspond
to an already established game: Hella’s $n$-bijective game with $k$ pebbles.
This relationship allows for a systematic exploration of variations
characterising a number of natural fragments of the logic with $n$-ary
quantifiers. One natural fragment that is not yet within this framework and
worth investigating is the logic of embedding-closed quantifiers of Haigora
and Luosto [16].

This work opens up a number of perspectives. Logics with generalised
quantifiers have been widely studied in finite model theory. They are less of
interest in themselves and more as tools for proving inexpressibility in
specific extensions of first-order or fixed-point logic. For instance, the
logics with rank operators [8, 14] of great interest in descriptive complexity
have been analysed as fragments of a more general logic with linear-algebraic
quantifiers [7]. It would be interesting to explore whether the comonad
$\mathbb{H}_{n,k}$ could be combined with a vector space construction to
obtain a categorical account of this logic.

More generally, the methods illustrated by our work could provide a way to
deconstruct pebble games into their component parts and find ways of
constructing entirely new forms of games and corresponding logics. The games
we consider and classify are based on Duplicator playing different kinds of
functions (i.e. morphisms on finite sets) and maintaining different kinds of
homomorphisms (i.e. morphisms in the category of $\sigma$-structures). Could
we build reasonable pebble games and logics on other categories? In
particular, can we bring the algebraic pebble games of [10] into this
framework?

## References

  * [1] Abramsky, S., Barbosa, R. S., de Silva, N., and Zapata, O. The Quantum Monad on Relational Structures.  In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) (Dagstuhl, Germany, 2017), K. G. Larsen, H. L. Bodlaender, and J.-F. Raskin, Eds., vol. 83 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 35:1–35:19.
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  * [5] Abramsky, S., and Tzevelekos, N. Introduction to categories and categorical logic.  In New structures for physics. Springer, 2010, pp. 3–94.
  * [6] Dawar, A. Generalized Quantifiers and Logical Reducibilities.  Journal of Logic and Computation 5, 2 (1995), 213–226.
  * [7] Dawar, A., Grädel, E., and Pakusa, W. Approximations of isomorphism and logics with linear-algebraic operators.  In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece (2019), vol. 132 of LIPIcs, pp. 112:1–112:14.
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  * [9] Dawar, A., and Hella, L. The expressive power of finitely many generalized quantifiers.  Information and Computation 123, 2 (1995), 172–184.
  * [10] Dawar, A., and Holm, B. Pebble games with algebraic rules.  Fundamenta Informaticae Vol. 150, nr 3/4 (2017), 281–316.
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  * [12] Ebbinghaus, H.-D. Extended logics: The general framework.  In Model-Theoretic Logics, J. Barwise and S. Feferman, Eds. Springer-Verlag, New York, 1985, pp. 25–76.
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# Multi-Valued Quantum Neurons

M. W .AlMasri
CyberSecurity and Systems Research Unit, USIM, Malaysia
Email<EMAIL_ADDRESS>
###### Abstract

The multiple-valued quantum logic is formulated in a systematic way using the
Bargmann representation of quantum basis states. In this approach, the truth
values are represented naturally as unique roots of unity placed on the unit
circle. Consequently, multi-valued quantum neurons are based on the principles
of multiple-valued threshold logic over the field of complex numbers. The
training of MVQN is reduced to the movement along the unit circle. A quantum
neural networks (QNNs) based on multi-valued quantum neurons can be
constructed with complex weights, inputs, and outputs encoded by roots of
unity and an activation function that maps the complex plane into the unit
circle. Such neural networks enjoy fast convergence and higher functionalities
compared with quantum neural networks based on binary input with the same
number of neurons and layers. Possible practical application can be found
using the orbital angular momentum (OAM) of light based QNNs.

##  1 Introduction

Artificial intelligence deals with computational tasks that are difficult to
handle by ordinary computers but easy at the human-level, such as image
recognition, decision-making, translation between languages, speech
recognition etc. Some early attempts to generalize these concepts into quantum
theory can be found in [1, 2]. However, until recently, artificial
intelligence in quantum theory had started to attract reasonable attention in
the physics community due to the massive growing interest in artificial
intelligence and machine learning globally [3, 4, 5, 6]. From a practical
point of view, building quantum neural networks with a large number of neurons
is still challenging. The same difficulty is seen in building any quantum
computational device with large numbers of qubits (currently, we are speaking
about a few hundred qubits in the NISQ era[7]). To understand the problem
better, recall that qubits can be seen as a generalization of the $0$ and $1$
truth values in classical Boolean logic functions, and the main difference
from classical bits stems from the possibility of having superposition states
between $|0\rangle$ and $|1\rangle$ basis states. As we shall argue below,
having ternary possible states composing from $|0\rangle$, $|1\rangle$ and a
superposition state $\alpha|0\rangle+\beta|1\rangle$ where $\alpha$ and
$\beta$ are arbitrary complex numbers not equal to zero can lead to lower
computational cost. This fact may explain why, at the very basic level of the
universe, quantum matter particles (Fermions) have spin-$1/2$ and
consequently, nature at its most fundamental level operates at the minimum
computational cost. It is a widely accepted fact that increasing the number of
possible basis quantum states increases the information capacity a system can
possess. However, due to the decoherence in quantum systems, holding a large
number of qubits for practical computing tasks is still challenging. One
possible cure to this problem is to look for multi-level quantum logic gates
where the truth values can be more than two. This would decrease the number of
fundamental quantum sub-systems in a device with the same capacity of a device
using qubits as the primary information carrier. The higher-dimensional
quantum information carriers are known in the literature as qudits, where $d$
is not limited to 2 [8, 9, 10, 11, 12]. The main advantage of qudits over
qubits is the logarithmic reduction of separate quantum systems needed to span
the quantum memory, and thus provides more computational capabilities compared
with qubits under the same number of subspaces in both cases [8].

From a classical computing perspective, multiple-valued logic has been studied
in detail, and many devices have been proposed based on multiple-valued logic
architectures, such as the full ternary computer ”Setun” and resonant-
tunneling transistor [13, 14, 15, 16, 17, 18]. The main motivation was to
solve the interconnections problem both on-chip and between chips by
increasing the information content per connection [14]. Since all classical
computing processes have an analog at the quantum level, there is no reason
not to expect the possibility of building multiple-valued quantum logic
devices. Certainly, the opposite is not true in general since the space of
possible basis states in any quantum device is always larger than in classical
devices under the same conditions. A general article on multiple-valued
quantum logic could be read from [19]. In this work, we extend the scope of
using multiple-valued logic in the quantum realm in a novel, systematic way
using the Bargmann representation of quantum states. More precisely, we
introduce the multi-valued quantum neurons (MVQNs) and discuss the quantum
neural networks based on them. Finally, we suggest the use of the orbital
angular momentum (OAM) of light in building multi-valued quantum neurons.

##  2 Qubits and complexity

For a given number system, the smaller the radix $r$ the larger the number of
digits required to express a given quantity. The number necessary to represent
a range of $N$ is given by $N=r^{d}$ which is the necessary number of digits
rounded up to the second higher value if necessary. Adopting this terminology,
the cost or complexity of the system hardware is [14]

$C=k(r\times d)=k(r\frac{\log N}{\log r})$ (1)

where $k$ is constant. Differentiating the last equation with respect to $r$
gives the minimum cost, which is when $r=e=2.718$. Since the radix $r$ must be
an integer, the minimum cost happens when $r=3$. This proves that using the
basis states $|0\rangle$ , $|1\rangle$ and one superposition state between
these two basis states leads to a minimum cost. Finally, before we embark on
our main objective of this work, the formulation of multiple-valued quantum
logic in the space of holomorphic functions, it is worthy to mention that our
terminology of the hardware complexity here is of a qualitative signature and
is different from the complexity geometry between quantum states mentioned,
for example in [20].

##  3 Multiple-valued threshold quantum logic

The quantum logic of Birkhoff and Von Neumann refers to the orthocomplemented
lattice of closed linear subspaces of a Hilbert space of quantum states [21].
The conjugation is given by the intersection of two linear subspaces, and the
negation is given by forming an orthogonal complement111Let $V$ be a vector
space over field $F$ equipped with a bilinear form $B$. We define $f$ to be
left-orthogonal to $g$ and $g$ to be right-orthogonal to $f$ if $B(f,g)=0$.
The main difference between classical and quantum logic stems from the failure
of the propositional distributive law in the quantum case, i.e.,
$P\wedge(Q\lor R)\neq(P\wedge Q)\lor(P\wedge R)$ where the symbols $P,Q$ and
$R$ are propositional variables. Thus, disjunction is given by forming the
linear space of two linear subspaces. Consequently, the notion of quantum
logic can be defined in a straightforward manner.

Therefore, disjunction is given by forming the linear space of two linear
subspaces. Consequently, the notion of quantum logic can be defined in a
straightforward manner. In Boolean logic, there are two truth values (false
and true, or 0 and 1). Using more abstract language, these values are elements
of the set $K_{2}=\\{0,1\\}$. In multiple-valued logic , this can be
generalized easily to $K_{k}=\\{0,1,\dots k-1\\}$. However, it is customary in
engineering and especially in the context of neural networks to consider the
Boolean alphabet to be $E_{2}=\\{-1,1\\}$ instead of $K_{2}=\\{0,1\\}$. The
generalization of $E_{2}$ to multiple-valued logic can be achieved using $k$
roots of unity in the complex domain [22], that is,

$\displaystyle
E_{k}=\\{1=\epsilon^{0}_{k},\epsilon_{k},\epsilon_{k}^{2},\dots,\epsilon_{k}^{k-1}\\},$
(2)

where $\epsilon_{k}^{j}=(e^{i2\pi/k})^{j}=e^{i\phi_{j}}$ and $\phi_{j}=2\pi
j/k$ with $j=0,1,\dots k-1$. Therefore, we have $k$ roots of unity positioned
on the unit circle. It is important to emphasize that although the absolute
value of these elements is one, they have different arguments and thus can be
uniquely determined on the unit circle as shown in Figure 1 for
$k=5,j\in\\{0,4\\}$.

Figure 1: The roots of unity for $\epsilon_{5}^{j}$ with $j\in\\{0,4\\}$. As
we will see below, such roots appear in the study of the spin-2 case.

To connect this method with any quantum system, it is more convenient to use
the Bargmann representation of quantum mechanics. In this representation, the
bosonic creation and annihilation operators are given by multiplying or taking
the derivative $(\hat{a}^{\dagger}=z,\hat{a}=\frac{\partial}{\partial z})$
acted on a convergent normalized monomial in the complex plane, i.e.
$\\{\frac{z^{n}}{\sqrt{n!}}\\}$. This was originally proposed by Fock and
developed in a systematic way by Bargmann and Segal[23]. More concretely, we
define the Segal-Bargmann spaces $\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{t})$ as
spaces of holomorphic functions with Gaussian integration measure
$\mu_{t}=(\pi)^{-d}e^{-|z|^{2}/t}$ and inner-product of the form [23, 24]

$\displaystyle\langle f|g\rangle=(\pi
t)^{-d}\int_{\mathbb{C}^{d}}\overline{f}(z)\;g(z)\;e^{-|z|^{2}/t}dz,$ (3)

Where $|z|^{2}=|z_{1}|^{2}+\dots+|z_{d}|^{2}$ and $dz$ is the $2d$-dimensional
Lebesgue measure on $\mathbb{C}^{d}$. On these spaces, we define the
orthonormality condition with $d=1$ for simplicity as

$\displaystyle\frac{1}{\pi}\int_{\mathbb{C}}e^{-|z|^{2}/t}\;\overline{z}^{n}z^{m}dz=t^{n}n!\;\delta_{mn}.$
(4)

The next step is to apply this representation to quantum mechanical systems.
The ideal example to start with is the well-known quantum harmonic oscillator
with one degree of freedom, mass $m$ and frequency $\omega$. In this case, the
creation and annihilation operators are

$\displaystyle\hat{a}=\sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x}+\frac{i}{m\omega}\hat{p}\right)=\frac{\partial}{\partial
z},$ (5)
$\displaystyle\hat{a}^{\dagger}=\sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x}-\frac{i}{m\omega}\hat{p}\right)=z,$
(6)

and the Hamiltonian becomes

$\displaystyle H=\hbar\omega\left(z\frac{d}{dz}+\frac{1}{2}\right),$ (7)

with eigenstates $|n\rangle=\frac{z^{n}}{\sqrt{n!}}$.

In the case of bosonic states such as Fock (number) states,
$|n\rangle=\frac{z^{n}}{\sqrt{n!}}$ or coherent states
$|\alpha\rangle=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{(\alpha
z)^{n}}{n!}$ in Bargmann representation. The multi-valued logic is represented
by the roots of unity.

$\displaystyle E_{N}=\\{\epsilon^{0}_{N},\dots\epsilon^{n}_{N}\\},$ (8)

where $n$ is the energy level index and $N=n+1$. In Figure 2, we show the case
of a four-level system with a zero energy ground state for the harmonic
oscillator and the corresponding roots of unity.

$E(n=0)=0,\rightarrow\epsilon_{5}^{0}=1$$\hbar\omega$$E(n=1)=\hbar\omega$
$\rightarrow$
$\epsilon_{5}$$\hbar\omega$$E(n=2)=2\hbar\omega\rightarrow\epsilon_{5}^{2}$$\hbar\omega$$E(n=3)=3\hbar\omega\rightarrow\epsilon_{5}^{3}$$\hbar\omega$$E(n=4)=4\hbar\omega\rightarrow\epsilon_{5}^{4}$$\mathrm{Energy}$$2\hbar\omega$$3\hbar\omega$
Figure 2: The first four energy levels for a harmonic oscillator with a zero
ground state. In this case, we have the following $k$th roots of unity
$E_{5}=\\{1,\epsilon_{5},\epsilon_{5}^{2},\epsilon_{5}^{3},\epsilon_{5}^{4}\\}$
for multiple-valued logic.

For fermions, due to the anticommutation relations between fermionic creation
and annihilation operators (being Grassmann numbers), it is not possible to
apply the Bargmann representation directly to fermions such as electrons.
Fortunately, this issue can be solved gently using the Bargmann representation
of Jordan-Schwinger mapping, as discussed, for example, in [25]. In this case,
we write the operators $J_{i}$ such as spin operators (in natural units
$\hbar=1$)

$\displaystyle J_{x}=\frac{1}{2}\left(z\frac{\partial}{\partial
w}+w\frac{\partial}{\partial z}\right),$ (9) $\displaystyle
J_{y}=\frac{-i}{2}\left(z\frac{\partial}{\partial w}-w\frac{\partial}{\partial
z}\right),$ (10) $\displaystyle
J_{z}=\frac{1}{2}\left(z\frac{\partial}{\partial z}-w\frac{\partial}{\partial
w}\right),$ (11)

with canonical commutation relation $[J_{i},J_{j}]=i\varepsilon_{ijk}J_{k}$ .
In this representation, the normalized states of spin operators are

$\displaystyle|n_{1},n_{2}\rangle=\frac{z^{n_{1}}}{\sqrt{n_{1}!}}\frac{w^{n_{2}}}{\sqrt{n_{2}!}}\equiv
f_{n_{1}n_{2}}$ (12)

with normalization condition

$\displaystyle\langle
m_{1},m_{2}|n_{1},n_{2}\rangle=\delta_{m_{1},n_{1}}\delta_{m_{2},n_{2}}.$ (13)

The occupation numbers $n_{1}$ and $n_{2}$ are connected with the principal
angular momentum quantum numbers $j,m$ defined in the basis $|j,m\rangle$ as
$J_{z}|j,m\rangle=m|j,m\rangle$ and $J^{2}|j,m\rangle=j(j+1)|j,m\rangle$
through the relations $j=\frac{n_{1}+n_{2}}{2},m=\frac{n_{1}-n_{2}}{2},$ where
$m$ runs from $-j$ to $j$ in integer steps.

Since the monomails $\\{\frac{z^{n}}{\sqrt{n!}}\\}$ form an orthonormal basis,
their magnitudes are not associated with measurable quantities, only the
exponent $n$ appears eventually when calculating the expectation values of the
Hamiltonian. Thereby, we may write $f_{n_{1}n_{2}}$ as

$\displaystyle f_{n_{1}n_{2}}=\frac{|z|\;|w|\;e^{2\pi in_{1}/N}e^{2\pi
in_{2}/N}}{\sqrt{n_{1}!n_{2}!}}=\frac{|z|\;|w|\;\epsilon^{n_{1}}_{N}\epsilon^{n_{2}}_{N}}{\sqrt{n_{1}!n_{2}!}}$
(14)

where $N=2j+1$ is the number of possible distinct states for a particle or
multi-level system with spin number $j$. It is very important to impose the
condition $n_{1},n_{2}\in\\{0,N-1\\}$ in 14.

Table 1: The mapping between the roots of unity and the state-functions for $j=1/2,1,$ and $2$ Roots of Unity | State-Function | Quantum Numbers
---|---|---
$(\epsilon_{2}^{1})_{z}(\epsilon_{2}^{0})_{w}=\epsilon^{1}_{2}$ | $f_{10}=z$ | $j=1/2,m=1/2$
$(\epsilon_{2}^{0})_{z}(\epsilon_{2}^{1})_{w}=\epsilon^{1}_{2}$ | $f_{01}=w$ | $j=1/2,m=-1/2$
$(\epsilon^{1}_{3})_{z}(\epsilon^{1}_{3})_{w}=\epsilon^{2}_{3}$ | $f_{11}=zw$ | $j=1,m=0$
$(\epsilon^{2}_{3})_{z}(\epsilon^{0}_{3})_{w}=\epsilon^{2}_{3}$ | $f_{20}=\frac{z^{2}}{\sqrt{2}}$ | $j=1,m=1$
$(\epsilon^{0}_{3})_{z}(\epsilon^{2}_{3})_{w}=\epsilon^{2}_{3}$ | $f_{02}=\frac{w^{2}}{\sqrt{2}}$ | $j=1,m=-1$
$(\epsilon^{4}_{5})_{z}(\epsilon^{0}_{5})_{w}=\epsilon^{4}_{5}$ | $f_{40}=\frac{z^{4}}{\sqrt{24}}$ | $j=2,m=2$
$(\epsilon^{3}_{5})_{z}(\epsilon^{1}_{5})_{w}=\epsilon^{4}_{5}$ | $f_{31}=\frac{z^{3}}{\sqrt{6}}w$ | $j=2,m=1$
$(\epsilon^{2}_{5})_{z}(\epsilon^{2}_{5})_{w}=\epsilon^{4}_{5}$ | $f_{22}=\frac{z^{2}}{\sqrt{2}}\frac{w^{2}}{\sqrt{2}}$ | $j=2.m=0$
$(\epsilon^{1}_{5})_{z}(\epsilon^{3}_{5})_{w}=\epsilon^{4}_{5}$ | $f_{13}=z\frac{w^{3}}{\sqrt{6}}$ | $j=2,m=-1$
$(\epsilon^{0}_{5})_{z}(\epsilon^{4}_{5})_{w}=\epsilon^{4}_{5}$ | $f_{04}=\frac{w^{4}}{\sqrt{24}}$ | $j=2,m=-2$

From Table 1, we find that for each spin $j$ the products of the roots of
unity are always $\epsilon_{N}^{N-1}$. To distinguish each state with the same
spin number $j$ but a different magnetic number $m$, we must take the roots of
unity $\epsilon_{N}^{n_{1}}$ in the $z$-plane and $\epsilon_{N}^{n_{2}}$ in
the $w$-plane separately. The product of $\epsilon_{N}^{n_{1}}$ and
$\epsilon_{N}^{n_{2}}$ is always fixed and assumes the value

$\epsilon_{N}^{n_{1}}\epsilon_{N}^{n_{2}}=\epsilon_{N}^{N-1}$ (15)

##  4 Complex-Valued Quantum Neural Networks

Following the quantum perceptron presented in [2], we assume the existence of
quibts $\\{|x_{i}\rangle\\}$

$\displaystyle|x_{j}\rangle=\alpha_{j}|0\rangle_{j}+\beta_{j}|1\rangle_{j}$
(16)

where $\alpha_{j},\beta_{j}$ are complex coefficients satisfying
$|\alpha_{j}|^{2}+|\beta_{j}|^{2}=1$. The states $|0\rangle_{j}$ and
$|1\rangle_{j}$ are different polarization states of light. For a given
quantum system with $n$ inputs of the form
$|x_{1}\rangle,|x_{2}\rangle\dots|x_{j}\rangle$. The output $|y\rangle$ at
discrete time $t$ reads

$\displaystyle|y(t)\rangle=\hat{F}\sum_{j=1}^{n}\hat{\omega}_{j}|x_{j}\rangle$
(17)

where the weights $\hat{\omega}_{j}$ are $2\times 2$ matrices acting on the
basis states $|0\rangle_{j}$ and $|1\rangle_{j}$. The operator $\hat{F}$ is
unknown and can be determined by applying a network of quantum gates. For
simplicity, let’s assume $\hat{F}=1$ to be the identity operator; the output
reduces to

$\displaystyle|y(t)\rangle=\sum_{j=1}^{n}\hat{\omega}_{j}(t)|x_{j}\rangle$
(18)

In analogy to the classical case, the learning rule is defined by [2]

$\displaystyle\hat{\omega_{j}}(t+1)=\hat{\omega}_{j}(t)+\eta\left(|d\rangle-|y(t)\rangle\right)\langle
x_{j}|$ (19)

Here $|d\rangle$ is the desired output. The weights 19 directs the quantum
perceptron into the desired state $|d\rangle$. To prove this, we take the
square-module of the difference between the desired and real inputs ,

$\displaystyle||\;|d\rangle-|y(t+1)\rangle||^{2}=||\;|d\rangle-\sum_{j=1}^{n}\hat{\omega}_{j}(t+1)|x_{j}\rangle\;||^{2}$
(20) $\displaystyle=(1-\eta n)^{2}||\;|d\rangle-|y(t)\rangle||^{2}$ (21)

For small step sizes $\eta$ ($0<\eta<\frac{1}{n}$) and for normalized inputs
$\langle x_{i}|x_{j}\rangle=\delta_{ij}$, the result of iteration converges
towards the desired state $|d\rangle$ [2].

Using the Bargmann representation, the inputs can be written as

$|x_{j}\rangle=\alpha_{j}z_{j}+\beta_{j}w_{j}$ (22)

Plugging this into 18, we find

$\displaystyle|y(t)\rangle=\sum_{j=1}^{n}\omega_{j}\left(\alpha_{j}z_{j}+\beta_{j}w_{j}\right)$
(23)

and the learning rule 19 becomes

$\displaystyle\omega_{j}(t+1)=\omega_{j}(t)+\eta\left(|d\rangle-|y(t)\rangle\right)\langle
x_{j}|$ (24)

In this relation, $\omega_{j}$ is complex-valued root of unity.

The quantum perceptron is based on two truth values:
$E_{2}=\\{1,\epsilon_{2}\\}$. In the following, we formulate the multi-valued
quantum neural networks in a way similar to the complex-valued neural networks
done in [22],[26], [27], [28]. This was made possible by virtue of the mapping
of quantum states in vector Hilbert space into the Hilbert space of analytic
functions with Gaussian measure mentioned in the previous section, which can
be read in detail from [23, 24, 25].

The multi-valued quantum neuron (MVQN) performs a mapping between $n$ inputs
and a single output described by a multiple-valued function $f$ of $n$
variables $f(x_{1},x_{2},\dots x_{n})$ with $n+1$ complex-valued weights

$\displaystyle f(x_{1},x_{2},\dots
x_{n})=P(\omega_{0}+\omega_{1}x_{1}+\dots+\omega_{n}x_{n})$ (25)

where $P$ is the activation function of MVQN :

$\displaystyle P(z)=\mathrm{csign}(z)=\epsilon^{j}_{k},\;\;2\pi
j/k\leq\mathrm{arg}\;z<2\pi(j+1)/k.$ (26)

For small step sizes $\eta$ ($0<\eta<\frac{1}{n}$) and for normalized inputs
$\langle x_{i}|x_{j}\rangle=\delta_{ij}$, the result of iteration converges
Multi-valued quantum neuron training is equivalent to movement along the unit
circle. Since both the output values and desired values are positioned on the
unit circle. The movement of the output value towards the desired value will
always be in the correct direction. The shortest path for this movement is
determined in such a way that the error between the output and desired values
is minimized.

Let us have $d$ $n$-dimensional learning samples. This means that our MVQN has
$n$ inputs $(x_{1},\dots x_{n})$. Let ${\bf f}$ be an $d$-dimensional vector
column of desired output values. Finally, let $x_{1},\dots x_{n}$ to be the
$d$-dimensional vector columns of all possible values of inputs
$x^{i}_{1},\dots,x^{i}_{n}$ where $i=1,\dots d$. The Hebbian learning rule
states that the weights $\omega_{1},\omega_{2},\dots,\omega_{n}$ must be
calculated as dot products of ${\bf f}$ and vectors $x_{1},x_{2},\dots x_{n}$
respectively,

$\displaystyle\omega_{j}=\langle{\bf
f}|x_{j}\rangle=f_{1}x_{j}^{1}+\dots+f_{d}x^{d}_{j}$ (27)

where $j=0,\dots,n$. The normalized Hebbian rule is given by division over
$d$,

$\displaystyle\omega_{j}=\frac{1}{d}\langle{\bf
f}|x_{j}\rangle,\;\;j=0,\dots,n$ (28)

Typically, learning algorithms are iterative processes that check whether the
inputs and outputs are closer to the desired values with a reasonable error or
not. This in turn suggests the need for using an error-correction learning
rule that determines how the weights must be adjusted to guarantee that the
output is very close to the desired result. The error-correction rule for MVQN
with weight vectors $W=(\omega_{0},\omega_{1},\dots,\omega_{n})$ and
$X=(x_{1},\dots,x_{n})$ is

$\displaystyle W_{r+1}=W_{r}+\frac{\alpha_{r}}{n+1}\delta\overline{X}$ (29)

where $\delta$ is the difference between the desired and actual values, i.e.
$\delta=\epsilon_{N}^{d}-\epsilon^{a}_{N}$, $\alpha_{r}$ is the learning rate,
and $\overline{X}$ is the complex conjugate of $X$.

The next step is to realize multi-valued quantum neurons experimentally. This
would have advantages in building neuromorphic hardware devices that operate
with complex inputs, outputs, and weights represented precisely by roots of
unity, as we mentioned above. One possible experimental application could be
found in the orbital angular momentum of light [29]. This could be done
unambiguously by associating the unique root of unity $\epsilon^{n}_{N}$ for
each beam with a fixed orbital quantum number $\ell$ in analogy to table 1
presented in section 3.

## Acknowledgment

I am grateful to USIM for the postdoctoral fellowship and to Qatar University
for its kind hospitality under the visiting lecturer scheme.

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