# Permutation-invariant quantum circuits

Maximilian Balthasar Mansky<EMAIL_ADDRESS>Department
of Informatics, LMU Munich, 80538 Munich, Germany  Santiago Londoño Castillo
Department of Informatics, LMU Munich, 80538 Munich, Germany  Victor Ramos
Puigvert  Department of Informatics, LMU Munich, 80538 Munich, Germany
Claudia Linnhoff-Popien  Department of Informatics, LMU Munich, 80538 Munich,
Germany

(August 28, 2024)

###### Abstract

The implementation of physical symmetries into problem descriptions allows for
the reduction of parameters and computational complexity. We show the
integration of the permutation symmetry as the most restrictive discrete
symmetry into quantum circuits. The permutation symmetry is the supergroup of
all other discrete groups. We identify the permutation with a
$\operatorname{SWAP}$ operation on the qubits. Based on the extension of the
symmetry into the corresponding Lie algebra, quantum circuit element
construction is shown via exponentiation. This allows for ready integration of
the permutation group symmetry into quantum circuit ansatzes. The scaling of
the number of parameters is found to be $\mathcal{O}(n^{3})$, significantly
lower than the general case and an indication that symmetry restricts the
applicability of quantum computing. We also show how to adapt existing
circuits to be invariant under a permutation symmetry by modification.

##  I Introduction

The mathematical study of group symmetries has proven to be one of the most
successful tools in mathematical physics. From the development of quantum
chromodynamics [1] from $\operatorname{SU}(3)$ symmetries to the discovery of
exact solutions to Einstein's equations from their respective isometry groups
[2] and advancements in string theory from duality symmetries [3], there exist
many instances in physics where the study of the symmetries of a problem led
to its resolution or even to the development of an entirely new theory.

Group theory has developed out of the consideration of physical symmetries and
provides an accurate description of the behaviour of the system under symmetry
operations [4]. In particular, it gives a notion of systems that do not change
under a symmetry operation and thus are symmetry-invariant. Computational
problems from other fields of science can similarly utilize the description of
symmetry operations to simplify the problem and reduce the solution space [5].

The consideration of symmetries is in general independent of the computational
method. For classical computers, an appropriate design of the computational
algorithm allows to capture the underlying symmetry of the solution to be
found. On quantum computers, the discrete symmetry can be integrated into the
quantum circuit [6, 7]. Quantum computers are structurally beneficial for the
study of quantum systems [8] and as such need to be able to capture symmetries
within the system of study.

For a study of a physical system, it is often helpful to look at the extremes
first. Quantum circuit construction in the absence of external symmetries is
possible, both towards a known target [9] and also with a quantum machine
learning ansatz [10]. On the other end, the permutation symmetry acts as the
supergroup of all other discrete symmetries. In the quantum computing case,
this can be realized with the requirement that all qubits are interchangeable
with all other qubits. This assumes that the smallest unit of information
processed in the system is a single qubit. It is also possible to require
larger blocks of qubits representing a block of information to be
interchangeable, or consider the interchangeability of logical qubits [11].

By embedding the physical problem in the description of the quantum computer,
one also imparts additional knowledge on the system. There is always a trade-
off between generalization and specialization of systems [12]. This is also
true for implementing symmetries on a computational problem, as the solution
is then inherently limited to that symmetry-respecting subspace of the total
solution space. In the case where the implemented symmetry does not correspond
to the factual symmetry of the problem, the solution can only be
approximative. On the other hand, if the factual symmetry is correctly
implemented, the lower number of parameters should lead to a faster
convergence. In the context of quantum computing, the symmetry restricts the
size of the available Hilbert space [13].

The structure of the paper is as follows. After an overview of prior and
related work, we first introduce the mathematical fundamentals and show that a
permutation-invariant circuit is well-defined, in terms of its group structure
and corresponding Lie algebra, and has the necessary structure for quantum
computing. A constructive approach for permutation invariant circuits is
introduced in section IV and the scaling of the number of parameters is
analyzed in section V. We close by showing how existing circuits can be
modified to capture a permutation and symmetry and a discussion of our
results.

Figure 1: The main results of this paper. We show a permutation invariance in
a physical problem can be implemented on an quantum computer. The permutation
of inputs translates naturally to a $\operatorname{SWAP}$ invariance on the
quantum circuit. There are circuits that are invariant under the
$\operatorname{SWAP}$ and they form a group $piSU$. The elements of the group
have a natural representation in circuit form, which in turn can be used to
construct permutation invariant quantum circuits.

##  II Related work

The integration of discrete symmetries has been discussed mainly within their
application to quantum machine learning by the parallel independent work of
Larocca et al. [6] and Meyer et al. [7]. They both set the basis for
principally integrating discrete and continuous symmetries as restrictions on
the group of quantum computing, the special unitary group $SU$. Though their
work is thorough in the construction of the theory, neither of them explicitly
constructs quantum circuits.

Discrete symmetries in general and permutation symmetries in general are also
relevant for quantum physics. Here, it is less the computation and rather the
quantum state that is required to exhibit some form of symmetry. Permutation
symmetric quantum states can be used to study entanglement properties [14,
15]. Symmetry requirements of composite systems are also used to reason about
the mutual independence of systems [16].

There is a corresponding effort in classical machine learning to integrate
symmetries in the structure of the model. The most ubiquitous structure is the
use of convolutional neural networks to implement a translation symmetry
across a multidimensional domain [17]. This type of neural network allows one
to identify a signal approximately independent of its position within a larger
block of data. That idea is further extended to implement other symmetries
through group-invariant convolutional neural networks [18]. The overall area
of implementing symmetries into neural networks is called geometric deep
learning [18]. Within the field, graph neural networks are an extreme in this
regard, designed to encapsulate the permutation symmetry inherent to many
graphs [19].

##  III Permutation-invariant quantum circuits and their associated algebras

All operations on a quantum circuit acting on $n$-qubits correspond to an
element of the special unitary group $\operatorname{SU}(2^{n})$, generated by
the Lie algebra $\mathfrak{su}(2^{n})$. In this section, we will provide a
definition of what we mean by permutation invariant circuits at the group
level. Then, by looking at the linearization of the corresponding group at the
identity, i.e. its Lie algebra, the definition of permutation invariance will
be extended to the relevant Lie algebra. The main result of this section will
provide the connection between permutation invariant groups and permutation
invariant algebras.

Before moving into any constructions or proofs, it is necessary to introduce
some important definitions which will be used throughout the paper and some
notation, which will greatly simplify calculations. It is also important to
note that even though the choice of basis plays a role in the representation
of individual objects such as matrices and vectors, physical conclusions are
independent of the choice of basis. It is possible to construct the same
result in any other basis with the same results. Nonetheless, to carry any
concrete calculations it is necessary to pick a basis, and from now on we will
adhere to the so-called Pauli Basis, which is also the standard across quantum
computing literature [20].

######  Definition III.1 (Pauli Basis).

Given an $n$-qubit system, the Pauli basis for the generators of the Lie group
$\operatorname{SU}(2^{n})$ consists of all $n$-tensor products of the Pauli
matrices $\\{\sigma_{x},\sigma_{y},\sigma_{z},\mathbbm{1}\\}$. There are in
total $4^{n}-1$ such elements, which we will denote using the following
notation for generalized Pauli matrices:

$x_{k}y_{k-1}=\mathbbm{1}\otimes\cdots\otimes\sigma_{y}\otimes\sigma_{x}\otimes\mathbbm{1}\otimes\cdots\otimes\mathbbm{1}$
(1)

where the Pauli matrix $\sigma_{x}$ acts on the $k$th qubit and the
$\sigma_{y}$ matrix acts on the $k-1$th qubit. Matrices for the rotations
around $\sigma_{z}$ are constructed similarly and denoted by $z_{k}$. We will
refer to a vector written in the above form as a Pauli string.

Any element in the $\mathfrak{su}(2^{n})$ algebra can be written as a unique
linear combination of such Pauli strings:

$\mathfrak{su}(2^{n})=\operatorname{span}\left\\{\bigotimes_{i}^{n}\sigma_{i}\right\\}\backslash\mathbbm{1}^{n}.$
(2)

Any element in the $\operatorname{SU}(2^{n})$ group can then be generated from
this basis via the exponential map. The identity matrix $\mathbbm{1}$ is not
part of the algebra since it is not a traceless matrix.

The permutation group that we want to include as a restriction on the special
unitary group $SU$ consists of all possible permutations $\pi_{k}$ between all
available indices. On a quantum computer, the natural choice to reflect
permutation is a permutation of the individual qubits carrying out the
calculation. The order of qubits can be changed with a $\operatorname{SWAP}$
matrix.

The SWAP operation in the Pauli basis is given as the two-qubit unitary
operation represented by the following matrix:

$\operatorname{SWAP}_{1,2}=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{pmatrix}$ (3)

Note that the SWAP gate is a self-inverse operation, such that
$\operatorname{SWAP}_{i,j}\operatorname{SWAP}_{j,i}=\operatorname{SWAP}_{i,j}\operatorname{SWAP}_{i.j}=\mathbbm{1}$.
We refer to the set of all possible $\operatorname{SWAP}$ gates in an
$n$-qubit system as the $\operatorname{SWAP}$-group. This allows us to define
a permutation invariance on the quantum computation.

######  Definition III.2 (Permutation Invariant Circuit).

A circuit $U\in SU(2^{n})$ is said to be permutation invariant if it is fixed
under the adjoint action of any $n$-ary swap:

$\displaystyle\operatorname{SWAP}^{\otimes
n}U\left[\operatorname{SWAP}^{\otimes n}\right]^{\dagger}=U$ (4)

######  Definition III.3 (pi$SU$).

Given an $n$-qubit system, we call the mathematical space related to a
permutation invariant quantum circuit a permutation invariant special unitary
group $piSU$ and its corresponding algebra $pi\mathfrak{su}$.

$\textnormal{pi}SU=\\{SAS=A|\forall A\in SU(2^{n})\wedge\forall
S\in\operatorname{SWAP}\textnormal{-group}\\}$

We suggest a pronunciation with a lax sound, such as "pin" [pI], rather than a
tense sound as in $\pi SU$ [paI].

It is straightforward to show that this subspace of $SU(2^{n})$ is in fact a
subgroup. To show this, we must show that $piSU$ is non-empty and closed under
products and inverse. $piSU$ is trivially non-empty as it contains the
identity element, since $\mathbbm{1}=\operatorname{SWAP}_{i,i}$. It remains to
show closeness under product and inverse. Let $A,B$ be arbitrary elements of
$piSU$, using the definition of $piSU$ and the self-inverse property of
$\operatorname{SWAP}$ operators, we show that $piSU$ is closed under group
multiplication and inverse element:

$\displaystyle\operatorname{SWAP}_{i,j}\left(AB\right)\operatorname{SWAP}_{i,j}$
$\displaystyle=\operatorname{SWAP}_{i,j}\left(\operatorname{SWAP}_{i,j}A\operatorname{SWAP}_{i,j}\right)B\operatorname{SWAP}_{i,j}$
$\displaystyle=A\left(\operatorname{SWAP}_{i,j}B\operatorname{SWAP}_{i,j}\right)=AB,$

and

$\displaystyle\mathbbm{1}=\operatorname{SWAP}_{i,j}\left(AA^{-1}\right)\operatorname{SWAP}_{i,j}$
$\displaystyle=\operatorname{SWAP}_{i,j}\left(\operatorname{SWAP}_{i,j}A\operatorname{SWAP}_{i,j}\right)A^{-1}\operatorname{SWAP}_{i,j}$
$\displaystyle=A\left(\operatorname{SWAP}_{i,j}A^{-1}\operatorname{SWAP}_{i,j}\right)$
$\displaystyle\Rightarrow
A^{-1}=\operatorname{SWAP}_{i,j}A^{-1}\operatorname{SWAP}_{i,j}.$

Thus, we have shown that $piSU$ is a closed subgroup of $SU(2^{n})$. Moreover,
Cartan's closed-subgroup theorem states that if $H$ is a closed subgroup of a
Lie group $G$, then $H$ is an embedded Lie subgroup, see chapter 5 in [21] for
a detailed proof. Moreover, the subgroup-subalgebra correspondence theorem
states that there is a unique Lie sub-algebra generating the subgroup via the
exponential map [21].

Since it has been shown that $piSU$ is a closed subgroup of $SU(2^{n})$, there
must be a unique Lie sub-algebra of $\mathfrak{su}$, which we will refer to as
$pi\mathfrak{su}$, that generates $piSU$.

Using the definition of the Lie algebra of a Lie group as the linearization of
the group around the identity $\mathbbm{1}$, we will proceed to explicitly
construct the $pi\mathfrak{su}$ sub-algebra. Given the smooth structure of
$piSU$ as an embedded subgroup, pick a smooth path $\gamma(t)$ going through
the origin so that $\gamma(0)=\mathbbm{1}$ and $\gamma(t)\in piSU$ for all
$t\in\mathbbm{R}$. To first order, one can expand around the origin as
follows:

$\gamma(t)=\gamma(0)+t\frac{dA(t)}{dt}|_{t=0}+\mathcal{O}(t^{2})$ (5)

By definition $\frac{dA(t)}{dt}|_{t=0}\in pi\mathfrak{su}$. Let us denote
$\frac{dA(t)}{dt}|_{t=0}=B$, combining it with the above equation gives:

$\gamma(t)=\mathbbm{1}+tB+\mathcal{O}(t^{2})$ (6)

Imposing the condition of invariance under the adjoint action of any element
$S$ of the $\operatorname{SWAP}$-group gives the following constraint of the
elements of the Lie algebra:

$B=SBS,\ \ \forall B\in pi\mathfrak{su}.$ (7)

Therefore, the elements of the $pi\mathfrak{su}$ sub-algebra are skew
Hermitian and traceless matrices invariant under the adjoint action of the
$\operatorname{SWAP}$-group. The relations among the discussed algebras and
groups are depicted in figure 2.

Figure 2: Diagram of the relation between the algebra $\mathfrak{su}$ and its
group $SU$. One can move from the general $SU$ group to the permutation
invariant group $piSU$ with the SWAP invariance and from the algebra
$\mathfrak{su}$ to the algebra $pi\mathfrak{su}$ with a complete
symmetrization. The exponential map connects the algebra $pi\mathfrak{su}$ to
the group $piSU$ exactly.

First, we note that given an arbitrary basis vector of $pi\mathfrak{su}$
written in the form shown in equation 1, the adjoint action of a
$\operatorname{SWAP}$-operator results in a change of order in the Pauli
string, for instance:

$\operatorname{SWAP}_{i,j}(x_{i}y_{j})\operatorname{SWAP}_{i,j}=x_{j}y_{i}.$
(8)

Thus, if we construct a new basis, whose elements are defined as the symmetric
sum of all Pauli strings containing a fixed number of
$\sigma_{x},\sigma_{y},\sigma_{z},\mathbbm{1}$, such a basis will by
construction be $\operatorname{SWAP}$-invariant. Consisting of fully symmetric
linear combinations of $\mathfrak{su}$ vectors, the resulting space meets all
the desired properties for $pi\mathfrak{su}$; skew Hermitian, traceless and
$\operatorname{SWAP}$-invariant. Finally, it remains to show that the
constructed space is in fact a Lie subalgebra, i.e. that is it is closed under
the Lie bracket. Below, we provide an explicit proof for $pi\mathfrak{su}(4)$.
The concept of this proof can be extended to higher $n$, however, it becomes
increasingly difficult to keep track of the indices, so we conjecture that for
a given $n$, it is possible to exhaustively compute all the brackets
explicitly, but we provide no general proof for arbitrary $n$.

######  Proof III.1 (The algebra of $pi\mathfrak{su}(4)$ is closed.).

First of all, notice that all basis elements of $pi\mathfrak{su}(4)$ can be
written, up to constants, either as
$(\sigma_{a}\otimes\sigma_{b}+\sigma_{b}\otimes\sigma_{a})$ or
$(\sigma_{a}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{a})$. Thus, to show
that $pi\mathfrak{su}(4)$ is a subalgebra of $\mathfrak{su}(4)$ one needs to
show that the following commutators are closed under the Lie bracket:

$\displaystyle[(\sigma_{a}\otimes\sigma_{c}+\sigma_{c}\otimes\sigma_{a}),(\sigma_{b}\otimes\sigma_{d}+\sigma_{b}\otimes\sigma_{d})]$
(9)
$\displaystyle[(\sigma_{a}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{a}),(\sigma_{b}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{b})]$
(10)
$\displaystyle[(\sigma_{a}\otimes\sigma_{c}+\sigma_{c}\otimes\sigma_{a}),(\sigma_{b}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{b})].$
(11)

Using the structure constants $\epsilon$ of the Lie algebra $\mathfrak{su}(2)$
and the anti-symmetry of the commutator, it is possible to simplify the
following commutators:

$[\sigma_{i}\otimes\sigma_{k},\sigma_{j}\otimes\sigma_{l}]=2i\epsilon_{ija}\delta_{kl}(\sigma_{a}\otimes\mathbbm{1})+2i\epsilon_{klb}\delta_{ij}(\mathbbm{1}\otimes\sigma_{b}).$

Using the above result, we find the following expressions for the commutators:

$\displaystyle[(\sigma_{a}\otimes\sigma_{c}+\sigma_{c}\otimes\sigma_{a}),$
$\displaystyle(\sigma_{b}\otimes\sigma_{d}+\sigma_{d}\otimes\sigma_{b})]$
$\displaystyle=2i\epsilon_{abe}\delta_{cd}(\sigma_{e}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{e})$
$\displaystyle+2i\epsilon_{cde}\delta_{ab}(\sigma_{e}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{e})$
$\displaystyle+2i\epsilon_{ade}\delta_{cb}(\sigma_{e}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{e})$
$\displaystyle+2i\epsilon_{cbe}\delta_{ad}(\sigma_{e}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{e})$
$\displaystyle[(\sigma_{a}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{a}),$
$\displaystyle(\sigma_{b}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{b})]$
$\displaystyle=2i\epsilon_{abc}(\sigma_{c}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{c})$
$\displaystyle[(\sigma_{a}\otimes\sigma_{c}+\sigma_{c}\otimes\sigma_{a}),$
$\displaystyle(\sigma_{b}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{b})]$
$\displaystyle=2i\epsilon_{abd}(\sigma_{d}\otimes\sigma_{c}+\sigma_{c}\otimes\sigma_{d})$
$\displaystyle+2i\epsilon_{cbd}(\sigma_{d}\otimes\sigma_{a}+\sigma_{a}\otimes\sigma_{d})$

It is straightforward to see that for all $a,b,c,d\in\\{x,y,z\\}$ the above
commutators are elements of $pi\mathfrak{su}(4)$. Thus, we have proved that
$pi\mathfrak{su}(4)$ is closed under the Lie bracket of $\mathfrak{su}(4)$,
and thus, a subalgebra of $\mathfrak{su}(4)$. $\square$

These results demonstrate that the $pi\mathfrak{su}$ Lie subalgebra is spanned
by symmetrized Pauli strings:

$pi\mathfrak{su}(2^{n})=\operatorname{span}\left\\{\sum_{k}\bigotimes_{i}^{n}\pi_{k}\sigma_{i}\right\\}\backslash\mathbbm{1}^{n}.$
(12)

The permutation $\pi_{k}$ and the summation $\sum_{k}$ iterate over all
possible permutations of $[x,y,z,\mathbbm{1}]$ in the Pauli string. Each
element in $pi\mathfrak{su}$ corresponds to the sum of all permutations of the
Pauli matrices contained within its Pauli string. We show this construction
explicitly for the two-qubit case $piSU(4)$ in section IV.1.

The definition of a SWAP-invariant group can be generalized to any other
invariance based on swapping entries of the matrix. The CNOT matrices,

$\displaystyle\operatorname{CNOT}_{1,2}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\\\
0&1&0&0\end{pmatrix},\quad\operatorname{CNOT}_{2,1}=\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{pmatrix}$

individually act as swapping matrices on the index 2 and 4, 3 and 4
respectively. It is possible to then construct a group that is invariant under
each operation – $\operatorname{CNOT}_{1,2}$\- and
$\operatorname{CNOT}_{2,1}$-invariant subgroups of $SU(4)$ and their higher
order analogues. Similarly, one can imagine permutation matrices that act on
any two indices. Effectively these invariant groups are all versions of the
quotient group $SU(2^{n})/S_{n}$, the special unitary group of size $2^{n}$
with $n$ arbitrarily interchangeable indices. The corresponding quantum
circuits may not always be easily constructable nor have a direct application.
They do however follow the same parameter scaling structure of section V and
have the same overall mathematical structure.

##  IV Construction of circuits with permutation symmetry

The elements of $piSU$ can be expressed as quantum circuits, as any element of
$SU$ can [9]. Using the algebra $pi\mathfrak{su}$, it is possible to construct
most of the elements directly and build circuits schematically that are SWAP
invariant. There is a direct construction of a quantum circuit element from a
given element of the algebra [22].

###  IV.1 Full classification for $piSU(4)$

As an example, we provide a complete classification of $piSU(4)$, the
permutation invariant group for two qubits. The structure of the $piSU$ group
can be understood through its accompanying Lie algebra, $pi\mathfrak{su}(4)$.
Its basis are the symmetrized elements of the overalgebra:

$\displaystyle
pi\mathfrak{su}(4)=\sum_{k}\bigotimes^{2}_{i=1}\pi_{k}(\sigma_{i})$ (13)

This means that each combination of Pauli matrices is symmetrized:

$\displaystyle\sigma_{1}$ $\displaystyle=\sigma_{x}\otimes\sigma_{x}$
$\displaystyle\sigma_{2}$
$\displaystyle=\sigma_{x}\otimes\sigma_{y}+\sigma_{y}\otimes\sigma_{x}$
$\displaystyle\sigma_{3}$
$\displaystyle=\sigma_{x}\otimes\sigma_{z}+\sigma_{z}\otimes\sigma_{x}$
$\displaystyle\sigma_{4}$
$\displaystyle=\sigma_{x}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{x}$
$\displaystyle\sigma_{5}$ $\displaystyle=\sigma_{y}\otimes\sigma_{y}$
$\displaystyle\sigma_{6}$
$\displaystyle=\sigma_{y}\otimes\sigma_{z}+\sigma_{z}\otimes\sigma_{y}$
$\displaystyle\sigma_{7}$
$\displaystyle=\sigma_{y}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{y}$
$\displaystyle\sigma_{8}$ $\displaystyle=\sigma_{z}\otimes\sigma_{z}$
$\displaystyle\sigma_{9}$
$\displaystyle=\sigma_{z}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{z}$

For $pi\mathfrak{su}(4)$, it is possible to construct all the circuits
directly from the algebra as there are only two Pauli matrices in each string.
These are shown in table 1. Each string has a central
$\sigma_{x}\otimes\sigma_{x}$ or $\sigma_{z}\otimes\sigma_{z}$ structure as
its backbone. The axis of each Pauli matrix can be changed with $H$ and
$S,S^{\dagger}$ gates [20]. These circuits can be individually and combination
be verified to fulfil equation (4). Their construction can be explained
intuitively as a central entangling structure generated by
$\bigotimes\sigma_{x}$ or $\bigotimes\sigma_{z}$ with a change of basis for
the corresponding qubit.

Table 1: Corresponding circuits for the exponentials of the algebra elements in $pi\mathfrak{su}(4)$ as $\exp(-i\alpha/2\sum_{k}\pi_{k}\bigotimes_{i=1}^{2}\sigma_{i})$. The circuits are constructed on a $\sigma_{x}\otimes\sigma_{x}$ backbone with changes to the qubit axis as necessary. For some circuits, there are optimized visions with fewer gates available. Exponent | Circuit | Optimized circuit
---|---|---
$\sigma_{x}\otimes\sigma_{x}$ |  |
$\sigma_{x}\otimes\sigma_{y}+\sigma_{y}\otimes\sigma_{x}$ |  |
$\sigma_{x}\otimes\sigma_{z}+\sigma_{z}\otimes\sigma_{x}$ |  |
$\sigma_{x}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{x}$ |  |
$\sigma_{y}\otimes\sigma_{y}$ |  |
$\sigma_{y}\otimes\sigma_{z}+\sigma_{z}\otimes\sigma_{y}$ |  |
$\sigma_{y}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{y}$ |  |
$\sigma_{z}\otimes\sigma_{z}$ |  |
$\sigma_{z}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{z}$ |  |

During exponentation, each term of the permutation-invariant algebra has a
complex-valued parameter $\alpha$ that corresponds to the parameter generating
the $X$ rotation in the group. The parameterized gates allow the full $piSU$
group coverage.

The individual circuits can also be optimized to reduce the number of basic
operations within them. It is known that any two-qubit operation can be
expressed with at most three qubits [23, 24]. The four-qubit circuits of table
1 are not optimal in this sense. For symmetry reasons, the circuits should be
expressible with two $\operatorname{CNOT}$s only.

###  IV.2 General construction of elements

The construction of the quantum circuit corresponding to an individual Pauli
string in the algebra is straightforward. The group element is expressed as
$\exp(-i\alpha/2\bigotimes_{i}^{n}\sigma_{i})$. The parameter $\alpha$ is
expressed through a rotation gate on the $X$ axis. The connection to the other
qubits is established through CNOT ladders that flow out on both sides of the
rotation gate. The axis for each Pauli gate corresponding to a qubit can be
changed with $H$ for $Z$ axis and $S,S^{\dagger}$ for the $Y$ axis. Qubits
with a $\mathbbm{1}$ in their corresponding Pauli matrix are not addressed. An
example of this structure is shown in figure 3.

Figure 3: Example of a Pauli string exponential represented as a quantum
circuit. On the backbone of a central $\bigotimes_{i=1}^{5}\sigma_{x}$
structure that creates the central $\operatorname{CNOT}$ ladder, each qubit
gets changed to its appropriate $\sigma_{y}$, $\sigma_{z}$ by a
$S,S^{\dagger}$, $H,H$ pairs on the outside of the ladder, respectively.
Identity elements $\mathbbm{1}$ in the Pauli string are created by skipping
the respective qubit.

The arrangement of $\operatorname{CNOT}$s in the interior is not fixed to a
broadening ladder from one qubit to another. All rearrangements of the qubits
are equivalent, a direct result of the qubit permutation invariance. The
ladder can also be restructured to connect to the rotation matrix directly or
form other shapes, as long as the $\operatorname{CNOT}$ layers are connected
to the central rotation gate increasing from the inside [22, 25].

For a sum of Pauli strings, this approach is not directly extensible. An
exponential of a sum is not trivially the same as a product of exponentials.
However, results from quantum error correction, in particular stabilizer codes
[26] show that if the individual elements within the sum in the algebra
commute, the individual Pauli strings can be exponentiated and combined in the
group. The Pauli strings commute if only two of three Pauli matrices
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ are present [27].

The result that exponents with only two Pauli matrices in the strings commute
also has an intuitive explanation: The vertical $\operatorname{SWAP}$
operations between the qubits are equivalent to horizontal swaps between the
individual elements. This is illustrated in figure 4. This property implies
that the individual elements of the product of exponentials commute. In the
case of Pauli strings consisting of all components, the number of strings is
greater than the number of available $\operatorname{SWAP}$s and it is no
longer possible to arrange the individual strings in an arbitrary order.

Figure 4: The $\operatorname{SWAP}$ operations between qubits are equivalent
to changes in order between the individual elements of the exponential sum of
Pauli string. The underlying $\bigotimes\sigma_{x}$ element between the pairs
of Hadamard gates $H$ is intrinsically invariant under the
$\operatorname{SWAP}$ operation.

In the case where the Pauli strings are directly expressible through
exponentiation, it possible to optimize the resulting CNOT ladders for shorter
circuit depths [25]. The authors achieve a 50% reduction in the number of
CNOTs employed in the test circuits that they use. The naive structure for
circuits requires $2n\sum_{k=1}^{n}(n-k)^{3}=\mathcal{O}(n^{5})$ CNOTs for
construction of the whole group.

The elements of $piSU$ always affect all qubits, but their connectivity varies
depending on how many qubits are addressed at the same time. The Pauli strings
containing only Pauli matrix and identity elements $\mathbbm{1}$ otherwise
have the lowest connectivity, with their corresponding circuits consisting of
equivalent rotation matrices on all qubits only. On the other end, a Pauli
string of only Pauli matrices links all qubits together with
$\operatorname{CNOT}$ gates. In between, for $k$ Pauli matrices present in the
individual Pauli string, $k$ out of $n$ qubits are addressed. This is termed
locality and restrictions of locality can be important on some quantum
computers. While restrictions of locality almost have no effect on the
expressibility of the circuit [28, 29], the same is not true for permutation
symmetric quantum gates with locality restrictions [30, 31]. Here it is
necessary to have all elements of the algebra present in the group to achieve
full expressibility.

From a circuit diagram, it is not visually obvious whether a given quantum
circuit is $\operatorname{SWAP}$ invariant or not. This is a general drawback
of circuit diagrams, in the standard language used in this work as well as its
varieties such as ZX-calculus [32]. Within ZX-calculus, operators that modify
all qubit lines equally and are therefore permutation invariant have been
proposed to be represented by their own special symbol [25]. In the standard
notation, no such symbol is available.

##  V Scaling behaviour

The group $piSU(2^{n})$ captures how unitary transformations on $n$-qubits
behave under permutation symmetries and it provides insights into the inherent
structure and relationships between unitary operators in the presence of
permutation operations. In the following lemma, we establish a fundamental
result that characterizes the dimension of the group $piU(2^{n})$, where we
show that the dimension of $piU(2^{n})$ is given by $(n+3)(n+2)(n+1)/6$.

The dimension of the two-qubit unitary Lie group $U(4)$ can be determined by
counting the Pauli matrices spanning its Lie algebra,
$\sum_{i,j=0}^{3}\sigma_{i}\otimes\sigma_{j}$, giving dim $U(4)=4^{2}$. In
order to determine the dimension of the two-qubit permutation-invariant group
$piU(4)$ we need to account for the symmetry present and account for the
degenerate elements.

The group $piSU(4)$ then has 9 dimensions, due to its unitarity constraint.
The number of elements generalizes via the multinomial theorem [33] to

$\displaystyle pi(\sigma_{i})^{\otimes
n}=\sum_{\begin{subarray}{c}i_{1},i_{2},i_{3},i_{4}\geq 0\\\
i_{1}+i_{2}+i_{3}+i_{4}=n\end{subarray}}\binom{n}{i_{1},i_{2},i_{3},i_{4}}\sigma_{x}^{i_{1}}\sigma_{y}^{i_{2}}\sigma_{z}^{i_{3}}1^{i_{4}}$
(14)

The number of terms in such a multinomial sum is equal to the number of
monomials of $n$th-degree on the variables $x,y,z,1$.

Since we are determining the dimension of the algebra spanned by the
independent terms, each degenerate term needs to be counted as one,

$\\#piSU(2^{n})=\binom{n+4-1}{4-1}-1=\frac{(n+3)(n+2)(n+1)}{6}-1$ (15)

The number of independent terms grows as $n^{3}$ for the permutation invariant
special unitary group.

$\displaystyle\dim(piSU(2^{n}))=\mathcal{O}(n^{3})$ (16)

This is significantly less than the number of parameters in the full, non-
symmetric group $\dim(SU(2^{n}))=\mathcal{O}(4^{n})$. It is a consequence of
the fact that calculations executed on a quantum computer that exhibit this
symmetry group cannot make use of the non-symmetric part of the Hilbert space.
These circuits are constrained to the symmetric part and their scaling
behaviour with respect to the number of qubits is severely limited. This is
true for both quantum machine learning approaches utilizing an ansatz as well
as schematic approaches that prescribe a construction of gates on the qubits.
It has been shown that the restrictiveness of the symmetry prevents a quantum
speed-up relative to classical computers [34].

##  VI Circuit modification

In this section we show how the permutation symmetry can be implemented into a
quantum circuit ansatz. On the basis of a commonly used variational circuit
element, we show how to extend a circuit to have permutable blocks of
information and how to modify an existing circuit to capture permutation
symmetry across all circuits. Adopting a known circuit to exhibit permutation
symmetry allows us to capture the present knowledge about the behaviour of the
quantum circuit and extend it to recognize the underlying symmetry. The
starting circuit in both cases is shown in figure 5. It consists of two layers
of parametrized rotation matrices and a strongly entangling layer with
$\operatorname{CNOT}$s [10].

Figure 5: An example of a variational circuit of three qubits that is
subsequently modified to be permutation invariant with a copy of itself and
intrinsically permutation invariant between all qubits. The circuit is
constructed of two layers of rotation matrices with parametrized angles and a
layer of $\operatorname{CNOT}$ elements that provide entanglement. Several of
these elements can be concatenated to form a variational quantum circuit [10].

It is also possible to construct a quantum circuit ansatz by combining the
elements that are presented above. Due to the group properties of $piSU$, the
combination of elements in any order is still a permutation symmetric circuit.

###  VI.1 Symmetry by extending a circuit

In many quantum algorithms, the mapping of information to qubits is not one-
to-one. Rather, input is expressed across several qubits, but the input itself
may exhibit a permutation symmetry. We illustrate the procedure by considering
an extension of the starting variational circuit of figure 5. The circuit is
modified by doubling the circuit to mimic two permutation invariant blocks.
Now each of the two blocks of three qubits corresponds to some information,
but the two blocks are interchangeable. An illustration of that starting
circuit is shown in figure 6. The symmetrization requirement is then
$\\{\operatorname{SWAP}_{1,4},\operatorname{SWAP}_{2,5},\operatorname{SWAP}_{3,6}\\}$.

Figure 6: The starting point for a permutation symmetric quantum circuit by
extension. The base circuit of figure 5 is doubled up into two separate
quantum circuits. The two blocks need to be connected to be permutation
invariant.

The first rotations in the $X$ and $Y$ axis can be symmetrized either as
$\sigma_{x}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{x}$ or as
$\sigma_{x}\otimes\sigma_{x}$ and
$\sigma_{y}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{y}$ or
$\sigma_{y}\otimes\sigma_{y}$ respectively. The first option corresponds to
setting the corresponding parameters in each block to the same value, such
that $\alpha_{i}=\beta_{i}$, the second one introduces additional entanglement
between the two blocks via $XX$ ($YY$) gates. The difference is illustrated in
figure 7, where the $X$ gates are symmetrized as $\sigma_{x}\otimes\sigma_{x}$
and the $Y$ gates as
$\sigma_{y}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{y}$. The two
approaches are not equivalent. Depending on the application, one or the other
may be preferable.

The second part of the variational circuit, the dense entanglement layer, also
requires symmetrization. In contrast to the strongly entangling layer itself
above, two layers or more layers can be made permutation invariant. Similar to
the horizontal and vertical permutation of elements shown in section IV.2, a
vertical permutation of elements through $\operatorname{SWAP}$ operations
implies a horizontal exchange of elements. The circuit needs to be expanded to
also contain the elements that would be generated under an exchange of qubits.
All $\operatorname{CNOT}$ gates that would appear under a
$\operatorname{SWAP}$ operation are present in the circuit. The green
$\operatorname{CNOT}$ gates added to the pairs of $\operatorname{CNOT}$ gates
are abelian with the original ones, therefore the order of the
$\operatorname{CNOT}$ gates does not matter.

Figure 7: The modified variational circuit for two permutation symmetric
blocks of information. The parametrized $X$ and $Y$ rotations can be made
permutation invariany by coupling them to $\sigma_{x}\otimes\sigma_{x}$, shown
for the $X$ rotation, or by equalizing their rotation parameters,
corresponding to $\sigma_{y}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{y}$
for the $Y$ rotation. The entangling layer of $\operatorname{CNOT}$s
incorporates the additional coonections that would appear under a
$\operatorname{SWAP}$. These additional $\operatorname{CNOT}$ gates are drawn
in green.

The final circuit in figure 7 is invariant under the $\operatorname{SWAP}$
group
$\\{\operatorname{SWAP}_{1,4},\operatorname{SWAP}_{2,5},\operatorname{SWAP}_{3,6}\\}$.
It is not invariant under $\operatorname{SWAP}$ operations between qubits
within a block. The permutation-invariant circuits have the same number of
parameters as the starting circuit shown in figure 5, independent of the
number of additional blocks added to the overall circuit. This a consequence
of the restricted choices for parameterized gates in the direct translation.
More parameters can be added to circuit by concatenating some of the
permutation invariant quantum circuit elements presented above.

###  VI.2 Symmetry by changing a circuit

In the case where one has an existing circuit and now wants to impose a
permutation symmetry on all inputs and outputs of the circuits, the circuit
ansatz needs to be symmetrized. Rather than constructing a new circuit from
scratch, the knowledge obtained so far can be used in the modified circuit. We
illustrate the methods on a simple yet common variational quantum circuit with
three qubits, consisting of two layers of rotation matrices and a dense
entanglement layer [35]. The base circuit is illustrated in figure 5. These
first two layers of single-qubit rotations contain the parameters of the
circuit that are optimized during a training loop. The symmetrization
requirement is that all qubits can be permuted, such that the circuit is
invariant under any combination of
$\\{\operatorname{SWAP}_{1,2},\operatorname{SWAP}_{2,3},\operatorname{SWAP}_{1,3}\\}$.

The circuit can be adapted to reflect a permutation symmetry by working layer
by layer. The one-qubit rotations express the algebra element
$\alpha_{1}\sigma_{x}\otimes\mathbbm{1}\otimes\mathbbm{1}+\alpha_{2}\mathbbm{1}\otimes\sigma_{x}\otimes\mathbbm{1}+\alpha_{3}\mathbbm{1}\otimes\mathbbm{1}\otimes\sigma_{x}$.
The obvious approach to symmetrization is then to set the values of the
parameters $\alpha_{1}=\alpha_{2}=\alpha_{3}$. This corresponds directly to
the structure present in the original ansatz. An alternative is to implement
$\alpha\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{x}$ or
$\alpha(\sigma_{x}\otimes\sigma_{x}\otimes\mathbbm{1}+\sigma_{x}\otimes\mathbbm{1}\otimes\sigma_{x}+\mathbbm{1}\otimes\sigma_{x}\otimes\sigma_{x}$.
These structures provide a different path through the available Hilbert space
and can be more suitable to some problems. These alternatives are illustrated
in figure 8.

Figure 8: The circuits corresponding to
$\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{x}$ and
$\sigma_{x}\otimes\sigma_{x}\otimes\mathbbm{1}+\sigma_{x}\otimes\mathbbm{1}\otimes\sigma_{x}+\mathbbm{1}\otimes\sigma_{x}\otimes\sigma_{x}$.

For the second part of the starting variational circuit of figure 5, the same
consideration of symmetrizing across all qubits applies. This
$\operatorname{CNOT}$ structure, often referred to as a "strongly entangling
layer" [36], acts to interchange superpositions between the different qubits.
The construction can also be understood as a permutation matrix in
$SU(2^{n})/S_{2^{n}}$, with order $2^{n}-1$. This structure is not part of
$piSU$ and cannot readily be made to fit into it. This is in contrast with the
previous example of symmetrizing by extending a circuit. If the circuit layer
is modified in the same way as above, by adding the $\operatorname{CNOT}$
elements that would appear under a permutation, the resulting structure is not
abelian and therefore not permutation invariant.

The element can then be replaced with one from $piSU$, for example one that
introduces a rotation around the central permutation invariant axis, such as
$\sum_{k}\pi_{k}(\sigma_{x}\otimes\sigma_{x}\otimes\mathbbm{1})$. In cases
where the number of $\operatorname{CNOT}$s is restricted or important, a good
replacement would be $\sigma_{x}\otimes\sigma_{x}\otimes\sigma_{x}$. Both
circuit element alternatives add entangling structures to the circuit. They
also provide additional training parameters to the circuit, which were reduced
in the prior rotation gate block.

The presented approach can be readily adopted for higher numbers of qubits.
With increasing qubit number comes the consideration of how many $k$-local
structures are necessary and reasonable to implement on the system of choice.
Compared to the non-symmetrized ansatz, the fraction of parameters to non-
parametrized two-qubit gates is low, but the size of the available Hilbert
space is also only growing moderately.

##  VII Discussion and future work

In this work, we show the direct representation of permutational symmetries in
quantum circuits. Implementing a physical symmetry in a computation is
important because it allows to reduce the number of parameters in the quantum
circuit while maintaining the necessary expressibility of the circuit. A
general circuit that is able to cover the whole Hilbert space can achieve the
same solution, but by restricting to the necessary space the quantum circuit
is able to reflect the same calculation with less computational depth.

The mathematical basis, in particular closure of the permutation invariant
subgroup of $SU$, enables the construction of quantum circuits. They follow a
clear structure and are generally directly expressible as quantum circuit
elements. As a result, they can be directly implemented into the quantum
circuit construction toolkits used for quantum machine learning and related
ansatz-based approaches.

The approach of constructing permutation invariant blocks of qubits also
facilitates higher order structures on the physical qubits. The presented
approach can be extended to structures of logical qubits or to incorporate
quantum error correction structures. It shows that including discrete group
symmetries into quantum systems is not only theoretically possible but
concrete quantum circuits can be realized, at least for the permutation
symmetry.

The present work deals with permutation symmetries, which act as the
supergroup for any other discrete symmetry group. This choice is deliberate,
as it makes the mathematics of invariants easy to understand. However, it also
limits the scope of applicability to select problems. An exploration of how to
implement any discrete symmetry as a restriction on the special unitary group
is possible expansion of our work here.

###### Acknowledgements.

The authors acknowledge funding from the German Federal Ministry of Education
and Research (BMBF) under the funding program "Förderprogramm
Quantentechnologien – von den Grundlagen zum Markt" (funding program quantum
technologies – from basic research to market), project BAIQO, 13N16089.

## References

  * [1] M. Gell-Mann, The eightfold way: A theory of strong interaction symmetry 10.2172/4008239.
  * Stephani  _et al._ [2003] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt,  _Exact solutions of Einstein's field equations_, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, 2003).
  * de la Ossa and Quevedo [1993] X. C. de la Ossa and F. Quevedo, Duality symmetries from nonAbelian isometries in string theory, Nucl. Phys. B 403, 377 (1993), arXiv:hep-th/9210021 .
  * Sternberg and Sternberg [1995] S. Sternberg and S. Sternberg,  _Group theory and physics_ (Cambridge university press, 1995).
  * Hamermesh [2012] M. Hamermesh,  _Group theory and its application to physical problems_ (Courier Corporation, 2012).
  * Larocca  _et al._ [2022] M. Larocca, F. Sauvage, F. M. Sbahi, G. Verdon, P. J. Coles, and M. Cerezo, Group-Invariant Quantum Machine Learning, PRX Quantum 3, 030341 (2022), publisher: American Physical Society.
  * Meyer  _et al._ [2023] J. J. Meyer, M. Mularski, E. Gil-Fuster, A. A. Mele, F. Arzani, A. Wilms, and J. Eisert, Exploiting Symmetry in Variational Quantum Machine Learning, PRX Quantum 4, 010328 (2023), publisher: American Physical Society.
  * Georgescu  _et al._ [2014] I. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Reviews of Modern Physics 86, 153 (2014), publisher: American Physical Society.
  * Mansky  _et al._ [2023a] M. B. Mansky, S. L. Castillo, V. R. Puigvert, and C. Linnhoff-Popien, Near-optimal quantum circuit construction via Cartan decomposition, Physical Review A 108, 052607 (2023a), publisher: American Physical Society.
  * Schuld  _et al._ [2015] M. Schuld, I. Sinayskiy, and F. Petruccione, An introduction to quantum machine learning, Contemporary Physics 56, 172 (2015).
  * Nigg  _et al._ [2014] D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, Quantum computations on a topologically encoded qubit, Science 345, 302 (2014), publisher: American Association for the Advancement of Science.
  * Giles and Maxwell [1987] C. L. Giles and T. Maxwell, Learning, invariance, and generalization in high-order neural networks, Applied Optics 26, 4972 (1987), publisher: Optica Publishing Group.
  * Ragone  _et al._ [2023] M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. O. Marrero, M. Larocca, and M. Cerezo, A Unified Theory of Barren Plateaus for Deep Parametrized Quantum Circuits (2023), arXiv:2309.09342 [quant-ph].
  * Dür  _et al._ [2000] W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Physical Review A 62, 062314 (2000), publisher: American Physical Society.
  * Markham [2011] D. J. H. Markham, Entanglement and symmetry in permutation-symmetric states, Physical Review A 83, 042332 (2011).
  * Renner [2007] R. Renner, Symmetry of large physical systems implies independence of subsystems, Nature Physics 3, 645 (2007), number: 9 Publisher: Nature Publishing Group.
  * Wiatowski and Bölcskei [2018] T. Wiatowski and H. Bölcskei, A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction, IEEE Transactions on Information Theory 64, 1845 (2018), conference Name: IEEE Transactions on Information Theory.
  * Bronstein  _et al._ [2021] M. M. Bronstein, J. Bruna, T. Cohen, and P. Veličković, Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021), arXiv:2104.13478 [cs, stat].
  * Zhou  _et al._ [2020] J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun, Graph neural networks: A review of methods and applications, AI Open 1, 57 (2020).
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum computation and quantum information_ , 10th ed. (Cambridge University Press, Cambridge ; New York, 2010).
  * Lee [2012] J. M. Lee,  _Introduction to Smooth Manifolds_, Graduate Texts in Mathematics, Vol. 218 (Springer New York, New York, NY, 2012).
  * Mansky  _et al._ [2023b] M. B. Mansky, V. R. Puigvert, S. L. Castillo, and C. Linnhoff-Popien, Decomposition Algorithm of an Arbitrary Pauli Exponential Through a Quantum Circuit, in  _2023 IEEE International Conference on Quantum Computing and Engineering (QCE)_, Vol. 01 (2023) pp. 434–442.
  * Vatan and Williams [2004] F. Vatan and C. Williams, Optimal quantum circuits for general two-qubit gates, Physical Review A 69, 032315 (2004), publisher: American Physical Society.
  * Shende  _et al._ [2004] V. V. Shende, I. L. Markov, and S. S. Bullock, Minimal universal two-qubit controlled-NOT-based circuits, Physical Review A 69, 062321 (2004), publisher: American Physical Society.
  * Cowtan  _et al._ [2020] A. Cowtan, S. Dilkes, R. Duncan, W. Simmons, and S. Sivarajah, Phase Gadget Synthesis for Shallow Circuits, in  _Proceedings 16th International Conference on Quantum Physics and Logic_, Vol. 318 (Chapman University, Orange, California, US, 2020) pp. 213–228, arXiv:1906.01734 [quant-ph].
  * Gottesman [1997] D. Gottesman, Stabilizer Codes and Quantum Error Correction (1997), arXiv:quant-ph/9705052.
  * Djordjevic [2022] I. Djordjevic,  _Quantum Communication, Quantum Networks, and Quantum Sensing_ (Elsevier, 2022).
  * DiVincenzo [1995] D. P. DiVincenzo, Two-bit gates are universal for quantum computation, Physical Review A 51, 1015 (1995), publisher: American Physical Society.
  * Lloyd [1995] S. Lloyd, Almost Any Quantum Logic Gate is Universal, Physical Review Letters 75, 346 (1995), publisher: American Physical Society.
  * Marvian [2022] I. Marvian, Restrictions on realizable unitary operations imposed by symmetry and locality, Nature Physics 18, 283 (2022), number: 3 Publisher: Nature Publishing Group.
  * Kazi  _et al._ [2023] S. Kazi, M. Larocca, and M. Cerezo, On the universality of $S_n$-equivariant $k$-body gates (2023), arXiv:2303.00728 [quant-ph, stat].
  * van de Wetering [2020] J. van de Wetering, ZX-calculus for the working quantum computer scientist (2020), arXiv:2012.13966 [quant-ph].
  * Brualdi [2008] R. A. Brualdi,  _Introductory combinatorics_ , 5th ed. (Pearson/Prentice Hall, Upper Saddle River, N.J, 2008) open Library ID: OL22514657M.
  * Ben-David  _et al._ [2020] S. Ben-David, A. M. Childs, A. Gilyén, W. Kretschmer, S. Podder, and D. Wang, Symmetries, Graph Properties, and Quantum Speedups, in  _2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)_ (2020) pp. 649–660, iSSN: 2575-8454.
  * Schuld and Petruccione [2021] M. Schuld and F. Petruccione, Variational Circuits as Machine Learning Models, in  _Machine Learning with Quantum Computers_, Quantum Science and Technology, edited by M. Schuld and F. Petruccione (Springer International Publishing, Cham, 2021) pp. 177–215.
  * Bergholm  _et al._ [2022] V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V. Ajith, M. S. Alam, G. Alonso-Linaje, B. AkashNarayanan, A. Asadi, J. M. Arrazola, U. Azad, S. Banning, C. Blank, T. R. Bromley, B. A. Cordier, J. Ceroni, A. Delgado, O. Di Matteo, A. Dusko, T. Garg, D. Guala, A. Hayes, R. Hill, A. Ijaz, T. Isacsson, D. Ittah, S. Jahangiri, P. Jain, E. Jiang, A. Khandelwal, K. Kottmann, R. A. Lang, C. Lee, T. Loke, A. Lowe, K. McKiernan, J. J. Meyer, J. A. Montañez-Barrera, R. Moyard, Z. Niu, L. J. O'Riordan, S. Oud, A. Panigrahi, C.-Y. Park, D. Polatajko, N. Quesada, C. Roberts, N. Sá, I. Schoch, B. Shi, S. Shu, S. Sim, A. Singh, I. Strandberg, J. Soni, A. Száva, S. Thabet, R. A. Vargas-Hernández, T. Vincent, N. Vitucci, M. Weber, D. Wierichs, R. Wiersema, M. Willmann, V. Wong, S. Zhang, and N. Killoran, PennyLane: Automatic differentiation of hybrid quantum-classical computations (2022), arXiv:1811.04968 [physics, physics:quant-ph].