EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6344
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Generalized SWAP and iSWAP and Solutions to the
Yang–Baxter Equation in All Dimensions

Arash Pourkia

College of Engineering and Technology, American University of the Middle East, Kuwait

Abstract. We introduce an infinite family of universal quantum logic gates that includes not only
higher-dimensional versions of the usual SWAP and iSWAP gates, but also their previously known
extensions. This family consists of permutation-like matrices with nonzero entries of the form eiαi ,
where the αi are arbitrary real numbers. Moreover, we show that these gates, which we refer to
as αSWAP, provide unitary solutions to the constant Yang–Baxter equation in all dimensions.

2020 Mathematics Subject Classifications: 81P68, 81R50, 15A30, 81P45

Key Words and Phrases: Qudits, Universal quantum logic gates, Entangling gates, Yang—
Baxter equation

1. Introduction

The d-level qudit-based computing, where d ≥ 3, has been at the center of attention in
recent research. This is partly due to the advantages of d-level qudits (d > 2) over conven-
tional qubits (d = 2) in quantum computing and quantum information. In parallel, there
have been developments in higher-dimensional quantum logic gates, both theoretically and
in terms of implementation [1–12].

In dimension two, the familiar swap gate (SWAP) and iswap gate (iSWAP) are among
the most important quantum gates in quantum computing [9–11, 13, 14]. The use of
SWAP and iSWAP gates is crucial in implementing quantum algorithms, particularly
for moving information to resolve the neighborhood constraints of qubit topology. The
iSWAP gate is also entangling and hence a universal gate. The study of these gates
and their generalizations, from various perspectives and their physical implementations,
remains an active area of research [6, 8–11, 15–17].

On a related topic, the Yang–Baxter equation and its solutions have played a funda-
mental role in several areas of physics and mathematics [18–22]. Unitary solutions to the
constant Yang–Baxter equation for d = 2 are well-known sources of quantum logic gates
[23–25], and are therefore important tools in quantum information theory. They also play
a crucial role in exploring the relationship between quantum entanglement and topological

DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6344

Email address: arash.pourkia@aum.edu.kw (A. Pourkia)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 2 of 14

entanglement [26–30]. The search for new solutions to the Yang–Baxter equation and their
applications is a common endeavor in both mathematics and physics [6, 21, 22, 26, 31–35].

The goal of this paper is to introduce an infinite family of universal quantum logic
gates that not only extends the usual SWAP and iSWAP gates to all dimensions d ≥ 2,
beyond their previously known extensions [15], but also provides an infinite family of
unitary solutions to the constant Yang–Baxter equation. The organization of the paper
is as follows. In Section 2, we recall some preliminary notions and notations. In Section
3, specifically in Definition 1, we give an explicit description of the quantum gate named
αSWAP and present concrete examples. In Subsection 3.1, we prove that αSWAP is a
universal gate for quantum computing. In Subsection 3.2, we prove that αSWAP is a
unitary solution to the Yang–Baxter equation. We conclude with final remarks in Section
4.

2. Preliminaries and Notations

In what follows, to avoid confusion, we use i to denote subscripts, as in αi, and we use
i to denote the imaginary unit

√
−1.

Recall that in quantum computing, quantum logic gates are represented by unitary
matrices acting on a Hilbert space H of dimension d, i.e., H = Cd. A qubit is the
fundamental unit of quantum information in dimension d = 2, while a qudit generalizes
this concept to a higher-dimensional Hilbert space with d ≥ 2. Throughout this paper we
use the standard basis {

|0⟩, |1⟩, |2⟩, · · · , |d− 1⟩
}

for Cd. We also use the computational basis for 2-qudit states. For instance, in dimension
d = 2, the computational basis for 2-qubit states is{

|00⟩, |01⟩, |10⟩, |11⟩
}
.

In dimension d = 3, the computational basis for 2-qutrit states is{
|00⟩, |01⟩, |02⟩, |10⟩, |11⟩, |12⟩, |20⟩, |21⟩, |22⟩

}
.

In dimension d = 2, i.e., for qubit systems, two very important examples of quantum
logic gates are the swap gate (SWAP), denoted here by S, and the iswap gate (iSWAP),
denoted here by iS. They are defined as follows, for all u, v ∈ H :

SWAP : H ⊗ H → H ⊗ H (1)

S(u⊗ v) = v ⊗ u



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 3 of 14

SWAP =


1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1

 (2)

iSWAP : H ⊗ H → H ⊗ H (3)

iS(u⊗ v) = i(v ⊗ u), when v ̸= u,

iS(u⊗ u) = u⊗ u

iSWAP =


1 0 0 0
0 0 i 0
0 i 0 0
0 0 0 1

 (4)

An n-qudit quantum logic gate is a unitary operator on (Cd)⊗n. A quantum logic
gate is said to be entangling if it can produce entangled states from unentangled ones. A
2-qudit gate G (i.e., a gate acting on two d-level qudits) is universal if the set consisting of
G together with all 1-qudit gates is sufficient to generate all qudit gates. It is well known
that a quantum logic gate acting on two d-level qudits is universal for quantum computing
[13] if and only if it is entangling [30].

On a related note, entangling (unitary) solutions of the Yang-Baxter equation [18–20]
have proven useful in quantum information theory, primarily as effective sources of univer-
sal quantum logic gates [23, 25, 30]. They have also been shown to produce non-constant
knot invariants [26], via the Turaev invariant [28]. Consequently, they play an important
role in linking quantum and topological entanglement [23, 26].

Let V be any complex vector space, and let I denote the identity map on V . A linear
operator R : V ⊗V → V ⊗V is said to be a solution to the constant braided Yang–Baxter
equation (BYBE) if [18–20]

R23R12R23 = R12R23R12. (5)

Similarly, a linear map R̂ : V ⊗ V → V ⊗ V is a solution to the constant quantum
Yang–Baxter equation (QYBE) if

R̂12R̂13R̂23 = R̂23R̂13R̂12. (6)

In the above relations, the operators R12, R23, and R13 all belong to End(V ⊗V ⊗V ) and
are defined as follows:

R12 := R⊗ I,



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 4 of 14

R23 := I ⊗R,

R13(u, v) := (I ⊗ S)R12(I ⊗ S),

where the map S : V ⊗ V → V ⊗ V is the usual swap gate (SWAP), sometimes called the
flip map.

Remark 1. Equations (5) and (6) represent two sides of the same coin in the following
sense. For any solution R to equation (5), the transformation R̂ = RS (or SR) yields
a solution to equation (6), and vice versa. Finding all solutions to these equations for
dimensions d > 2 is extremely challenging and remains an open problem [18, 19, 25, 29].

3. αSWAP, a universal quantum logic gate and a solution to the
Yang–Baxter equation

In this section we introduce an infinite family of universal quantum logic gates that
extends the usual SWAP and iSWAP gates to all dimensions d ≥ 2, beyond their pre-
viously found extensions in [15]. We name these gates αSWAP. Here, α represents a
d-tuple, (α1, α2, · · · , αd), in which the αi’s are arbitrary real numbers. Moreover, we show
that αSWAP provides solutions to the constant Yang–Baxter equations (5) and (6) in all
dimensions.

Definition 1. For any d ≥ 2 and for any set of real numbers α1, α2, · · · , αd2, we define
the quantum gate αSWAP (denoted by αS for short) by the following relations:

αSi,j = eiαi , for i = (t− 1)d+ s and j = (s− 1)d+ t, (7)

where 1 ≤ t ≤ d and 1 ≤ s ≤ d. We let αSi,j = 0 for all other values of i, j. Here, αSi,j
denotes the entry of the αSWAP located at row i and column j.

For both d = 2 and d = 3, the gate αSWAP is illustrated below.

αS =


eiα1 0 0 0
0 0 eiα2 0
0 eiα3 0 0
0 0 0 eiα4



αS =



eiα1 0 0 0 0 0 0 0 0
0 0 0 eiα2 0 0 0 0 0
0 0 0 0 0 0 eiα3 0 0
0 eiα4 0 0 0 0 0 0 0
0 0 0 0 eiα5 0 0 0 0
0 0 0 0 0 0 0 eiα6 0
0 0 eiα7 0 0 0 0 0 0
0 0 0 0 0 eiα8 0 0 0
0 0 0 0 0 0 0 0 eiα9





A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 5 of 14

Remark 2. To give the reader some intuition behind the relations in (7), let us compute
the indices for d = 3. Consider 1 ≤ t ≤ 3 and 1 ≤ s ≤ 3.

If we fix t = 1 and compute the indices for s = 1, 2, 3, we obtain:

αS11 = eiα1 , αS24 = eiα2 , αS37 = eiα3

corresponding to s = 1, s = 2, and s = 3, respectively.
Similarly, fixing t = 2 and computing the indices for s = 1, 2, 3, we get:

αS42 = eiα4 , αS55 = eiα5 , αS68 = eiα6 .

Finally, if we fix t = 3 and compute the indices for s = 1, 2, 3, we have:

αS73 = eiα7 , αS86 = eiα8 , αS99 = eiα9 .

The calculations for d = 2 are similar and even simpler.

Note that the condition t = s corresponds to the diagonal (nonzero) entries αSii, where
i = j = (t − 1)d + t, 1 ≤ t ≤ d. In the case of d = 3, these entries are αS11, αS55, and
αS99, whereas for d = 2, they are only αS11 and αS44.

Remark 3. For clearer illustration, let us write the action of αSWAP (denoted by αS for
short) on the computational basis for 2-qubit and 2-qutrit gates.

For d = 2, the action of αS on the computational basis states is given by:

αS(|00⟩) = eiα1 |00⟩,
αS(|01⟩) = eiα2 |10⟩,
αS(|10⟩) = eiα3 |01⟩,
αS(|11⟩) = eiα4 |11⟩.

Similarly, for d = 3, the action of αS on the computational basis states is:

αS(|00⟩) = eiα1 |00⟩,
αS(|01⟩) = eiα2 |10⟩,
αS(|02⟩) = eiα3 |20⟩,
αS(|10⟩) = eiα4 |01⟩,
αS(|11⟩) = eiα5 |11⟩,
αS(|12⟩) = eiα6 |21⟩,
αS(|20⟩) = eiα7 |02⟩,
αS(|21⟩) = eiα8 |12⟩,
αS(|22⟩) = eiα9 |22⟩.



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 6 of 14

Now let us proceed to some examples. The xSWAP gate introduced in [15] is, in fact,
a special case of the above definition, as illustrated in the following example.

Example 1. Let αi = 0 for the diagonal (nonzero) entries, i.e., for i = j = (t − 1)d + t
with 1 ≤ t ≤ d, and let αi = α otherwise. If we set x = eiα, then this gate coincides with
the xSWAP from [15]. The xSWAP, in turn, includes the usual SWAP for x = 1 (i.e.,
α = 0) and the iSWAP for x = i (i.e., α = π

2 ) in all dimensions. Below, we illustrate the
xSWAP matrices for d = 2 and d = 3.

xSWAP =


1 0 0 0
0 0 x 0
0 x 0 0
0 0 0 1

 , xSWAP =



1 0 0 0 0 0 0 0 0
0 0 0 x 0 0 0 0 0
0 0 0 0 0 0 x 0 0
0 x 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 x 0
0 0 x 0 0 0 0 0 0
0 0 0 0 0 x 0 0 0
0 0 0 0 0 0 0 0 1


.

By choosing different values for the αi’s, the αSWAP generalizes both the SWAP and
iSWAP gates beyond the xSWAP example given above. A few such examples follow.

Example 2. Let αi = (−1)
i
2 (
π

2
) for even i, and let αi = π for odd i. Below, we illustrate

this particular αSWAP for d = 2 and d = 3.

αSWAP =


−1 0 0 0
0 0 −i 0
0 −1 0 0
0 0 0 i

 , αSWAP =



−1 0 0 0 0 0 0 0 0
0 0 0 −i 0 0 0 0 0
0 0 0 0 0 0 −1 0 0
0 i 0 0 0 0 0 0 0
0 0 0 0 −1 0 0 0 0
0 0 0 0 0 0 0 −i 0
0 0 −1 0 0 0 0 0 0
0 0 0 0 0 i 0 0 0
0 0 0 0 0 0 0 0 −1


Example 3. Let αi =

π

4
for diagonal (nonzero) entries, i.e., for i = j = (t− 1)d+ t, and

let αi =
π

3
otherwise. We illustrate this αSWAP for d = 2 and d = 3.

αSWAP =


√
2
2 (1 + i) 0 0 0

0 0 1
2(1 +

√
3 i) 0

0 1
2(1 +

√
3 i) 0 0

0 0 0
√
2
2 (1 + i)





A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 7 of 14

αSWAP =



√
2

2
(1 + i) 0 0 0 0 0 0 0 0

0 0 0 1
2
(1 +

√
3 i) 0 0 0 0 0

0 0 0 0 0 0 1
2
(1 +

√
3 i) 0 0

0 1
2
(1 +

√
3 i) 0 0 0 0 0 0 0

0 0 0 0
√

2
2

(1 + i) 0 0 0 0

0 0 0 0 0 0 0 1
2
(1 +

√
3 i) 0

0 0 1
2
(1 +

√
3 i) 0 0 0 0 0 0

0 0 0 0 0 1
2
(1 +

√
3 i) 0 0 0

0 0 0 0 0 0 0 0
√

2
2

(1 + i)


3.1. αSWAP is a universal quantum logic gate

From Definition 1, it is clear that αSWAP is unitary. From [30] it is well known that a
2-qudit gate (i.e., a gate operating on two d-level qudits) is universal (i.e., forms a universal
set together with all 1-qudit gates) if and only if it is entangling (i.e., it can create entan-
gled states from some non-entangled ones). However, the entangling property of αSWAP
is not obvious. This fact is well known and easy to prove only for iSWAP in dimension two.

In this section we will prove that αSWAP is almost always a universal quantum gate.
We use the non-entangling criterion from [26]. To show that any operator R : H ⊗H →
H ⊗H is entangling, it suffices to show that neither R nor RS can be factored as X⊗Y ,
where X and Y are arbitrary operators on H , i.e., X, Y : H → H . Here S is the usual
SWAP gate with the same dimension as R.

Theorem 1. For any d ≥ 2 the αSWAP from Definition 1 is entangling, hence universal
[30], if the following condition is satisfied:

eiα1eiαd2 ̸= eiαdeiαd2−d+1 (8)

Proof. We prove this theorem by showing that the condition (8) is sufficient to ensure
that neither αSWAP nor (αSWAP)S can be factored as X ⊗ Y [26]. In fact, as we will
see, αSWAP can never be factored.

To see this, suppose the contrary: assume that αSWAP can be written as X ⊗ Y for
two d×d matrices X and Y with entries denoted by xij and yij , respectively. By focusing
solely on the diagonal entries at positions (1, 1), (d, d), (d2−d+1, d2−d+1), and (d2, d2)
in both αSWAP and X ⊗ Y , we obtain the following equalities:

On one hand,
(x11y11)(xddydd) = eiα1eiαd2 ,

and on the other hand,
(x11ydd)(xddy11) = (0)(0) = 0.



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 8 of 14

However, the left sides of these expressions must be equal. Thus, assuming that αSWAP
can be factored as X ⊗ Y leads to the contradiction

eiα1eiαd2 = 0.

Therefore, αSWAP cannot be factored. Below, we illustrate this argument by highlighting
the relevant entries with boxes:

αSWAP =



eiα1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ eiαd2


=

 x11 ∗ ∗
∗ ∗ ∗
∗ ∗ xdd

⊗

 y11 ∗ ∗
∗ ∗ ∗
∗ ∗ ydd



=



x11y11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ x11ydd ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ xddy11 ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xddydd


For (αSWAP)S, a similar argument (illustrated below with matrices and boxed entries)

shows that assuming (αSWAP)S = X ⊗ Y leads to the equality

eiα1eiαd2 = eiαdeiαd2−d+1 .

Therefore, if
eiα1eiαd2 ̸= eiαdeiαd2−d+1 ,

then (αSWAP)S cannot be factored.



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 9 of 14

(αSWAP)S =



eiα1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ eiαd ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ eiαd2−d+1 ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ eiαd2


=

 x11 ∗ ∗
∗ ∗ ∗
∗ ∗ xdd

⊗

 y11 ∗ ∗
∗ ∗ ∗
∗ ∗ ydd



=



x11y11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ x11ydd ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ xddy11 ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xddydd


This completes the proof.

Remark 4. The criterion in Theorem 1 shows that αSWAP is almost always entangling,
hence universal. This includes the gates from Example 1 (except for x = 1) and Example
2.

Remark 5. To directly prove a 2-qudit gate G is universal, one must show that the set
containing G and all 1-qudit gates can generate all other qudit gates. Proving this directly
is not always easy, even in dimension d = 2. This is why, in the proof of Theorem 1, we
instead used the powerful Brylinskis’ criterion [30] and the non-entangling criterion from
[26].

However, below we illustrate a worked example showing that αSWAP explicitly gener-
ates entanglement, i.e., turns an unentangled state representation into an entangled one.

Example 4. Consider the following gate for d = 2 from Example 2:

αSWAP =


−1 0 0 0
0 0 −i 0
0 −1 0 0
0 0 0 i





A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 10 of 14

If we apply this gate to the unentangled state

|ψ⟩ = 1

2
(|00⟩+ |01⟩+ |10⟩+ |11⟩),

the outcome is the entangled state

αSWAP|ψ⟩ = 1

2
(−|00⟩ − |01⟩ − i|10⟩+ i|11⟩).

3.2. αSWAP is a solution to the constant Yang–Baxter equation

Theorem 2. For any d ≥ 2, the gate R = αSWAP from Definition 1 is a solution to the
Yang–Baxter equation (5). In other words, the gate R̂ = (αSWAP)S, where S denotes the
usual SWAP gate, is a solution to the Yang–Baxter equation (6) (refer to Remark 1).

Proof. For notational simplicity, we define ai = eiαi . Given the straightforward struc-
ture of αSWAP from Definition 1, specifically that the only nonzero entries of αSWAP
are ai located at row i = (t − 1)d + s and column j = (s − 1)d + t with 1 ≤ t ≤ d and
1 ≤ s ≤ d, we observe that the only nonzero blocks in R12 = αSWAP ⊗ I are of the
form diag(ai, ai, . . . , ai), located at the block positions indexed by i = (t − 1)d + s and
j = (s− 1)d+ t, where 1 ≤ t ≤ d and 1 ≤ s ≤ d.

For instance, when d = 2, R12 takes the form:

R12 =



a1 0 0 0 0 0 0 0
0 a1 0 0 0 0 0 0

0 0 0 0 a2 0 0 0
0 0 0 0 0 a2 0 0

0 0 a3 0 0 0 0 0
0 0 0 a3 0 0 0 0

0 0 0 0 0 0 a4 0
0 0 0 0 0 0 0 a4


Similarly, the only nonzero blocks in R23 = I ⊗ αSWAP are its diagonal blocks, each

being a copy of αSWAP. For example, when d = 2, R23 is given by:

R23 =



a1 0 0 0 0 0 0 0
0 0 a2 0 0 0 0 0
0 a3 0 0 0 0 0 0
0 0 0 a4 0 0 0 0

0 0 0 0 a1 0 0 0
0 0 0 0 0 0 a2 0
0 0 0 0 0 a3 0 0
0 0 0 0 0 0 0 a4


Recall the general identity for any three matrices X, Y , and Z, where the matrix

product is defined: the (i, j)-entry of their product is given by;

(XY Z)i,j =
∑
k,l

(X)i,k(Y )k,l(Z)l,j (9)



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 11 of 14

Using equation (9), it is not very difficult to verify that for R = αSWAP, both sides
of equation (5) are equal, i.e., R23R12R23 = R12R23R12. For example, when d = 2, both
sides of equation (5) evaluate to:

a31 0 0 0 0 0 0 0
0 0 0 0 a1a

2
2 0 0 0

0 0 a1a2a3 0 0 0 0 0
0 0 0 0 0 0 a22a4 0
0 a1a

2
3 0 0 0 0 0 0

0 0 0 0 0 a2a3a4 0 0
0 0 0 a23a4 0 0 0 0
0 0 0 0 0 0 0 a34


This completes the proof.

Remark 6. It is clear from the above proof that the result of Theorem 2 holds for any
arbitrary set of complex numbers a1, a2, · · · , ad.

Examples of such solutions include (but are not limited to) the gates from Examples
1, 2, and 3. In what follows, we present another simple but interesting example.

Example 5. For this example, we let αi = 0 when i = (t − 1)d + t for 1 ≤ t ≤ d, and
otherwise, if i is even then αi = 0, and if i is odd then αi = π. Below we illustrate this
example for dimensions two and three.

αSWAP =


1 0 0 0
0 0 1 0
0 −1 0 0
0 0 0 1



αSWAP =



1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 −1 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 −1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1


4. Concluding remarks

Due to the advantages of qudit-based computing (d > 2) compared to qubit-based
computing (d = 2), there has been an ever-increasing interest in the applications of d-level



A. Pourkia / Eur. J. Pure Appl. Math, 18 (3) (2025), 6344 12 of 14

qudits in quantum computing and quantum information in recent decades. Unitary solu-
tions to the Yang–Baxter equation (5) are well-known candidates for (universal) quantum
logic gates. On the other hand, the search for new solutions to the Yang–Baxter equation
or its applications is a common endeavor in both mathematics and physics.

In the present paper, we introduce an infinite family of universal quantum logic gates,
αSWAP which, on one hand, includes the usual SWAP and iSWAP gates extended to
all dimensions d ≥ 2, along with previously found generalizations. On the other hand,
it provides unitary solutions to the constant Yang–Baxter equation. The α in αSWAP
represents a d-tuple (α1, α2, · · · , αd) for arbitrary real numbers αi.

Let us conclude by mentioning some future research directions related to the present
work. One possible direction is to generalize αSWAP to more general (and not necessarily
unitary) solutions of the Yang–Baxter equation. It would also be interesting to investigate
how αSWAP can be expressed as a linear combination of tensor products of generalized
(higher-dimensional) Pauli matrices. These research directions will be pursued in a sequel
to this paper.

Acknowledgements

First and foremost, I am deeply grateful to the esteemed editor and reviewers for their
invaluable comments and suggestions, which have undoubtedly enhanced the quality of
this paper. I am also thankful to Dr. Sinan Kapcak [36] for introducing me to SageMath,
which proved invaluable in generating large matrices and testing the validity of several
formulas. Last but not least, I extend my thanks to Mrs. Elena Ryzhova for her assistance
with proofreading the manuscript and for the insightful conversations.

This work was supported by ongoing institutional funding. No additional grants to
carry out or direct this particular research were obtained. There are no conflicts of interest
regarding this work.

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