On the universality of S n -equivariant k -body gates

The importance of symmetries has recently been recognized in quantum machine learning from the simple motto: if a task exhibits a symmetry (given by a group G ), the learning model should respect said symmetry. This can be instantiated via G -equivariant Quantum Neural Networks (QNNs), i.e., parametrized quantum circuits whose gates are generated by operators commuting with a given representation of G . In practice, however, there might be additional restrictions to the types of gates one can use, such as being able to act on at most k qubits. In this work we study how the interplay between symmetry and k -bodyness in the QNN generators aﬀect its expressiveness for the special case of G = S n , the symmetric group. Our results show that if the QNN is generated by one-and two-body S n -equivariant gates, the QNN is semi-universal but not universal. That is, the QNN can generate any arbitrary special unitary matrix in the invariant subspaces, but has no control over the relative phases between them. Then, we show that in order to reach universality one needs to include n -body generators (if n is even) or ( n − 1) -body generators (if n is odd). As such, our results brings us a step closer to better understanding the capabilities and limitations of equivariant QNNs.


I. INTRODUCTION
Quantum Machine Learning (QML) aims to use quantum computers to classify, cluster, and make predictions from classical data encoded in quantum states, or from quantum data produced by some quantum mechanical process [1,2].Despite the tremendous attention QML has received, its true potential and limitations are still unclear [3].Recent results have shown that QML models lacking inductive biases (i.e., models that do not contain information about the specific problem being tackled) can encounter serious trainability and generalization issues [4][5][6][7].This can be understood as a form of "nofree-luch" whereby high-expressiveness, multi-purpose algorithms will have overall poor performance.
Numerous endeavors have been undertaken to create learning models that are tailored specifically to a given task.Among these, Geometric Quantum Machine Learning (GQML) has emerged as one of the most promising approaches [8][9][10][11][12][13][14][15].The fundamental idea behind GQML is to leverage the symmetries present in the task to develop sharp inductive biases for the learning models.For instance, when dealing with the task of classifying nqubit pure states from n-qubit mixed states, GQML suggests using models whose outputs remain invariant under arbitrary unitaries applied to the qubits [8].This is because these unitaries cannot change the purity and are therefore symmetries of the task.
The GQML program consists of several steps.First, one needs to identify the group of transformations preserving some important property of the data (e.g., a symmetry that preserves the labels in supervised learning).
In the case of purity, the group is G = U(2 n ).The next step is to build a model that generates labels that remain invariant under G. Several recent works provide templates for developing such G-invariant models, which can be found in Refs.[9,11].For models based on Quantum Neural Networks (QNNs), one needs to leverage the concept of equivariance [11,16].
An equivariant QNN is a parametrized quantum circuit ansatz that commutes, for all values of parameters, with the representation of G. Figure 1 illustrates this constraint on the set of unitaries that the QNN can express (i.e., to the subspace of G-symmetric operators).Given the practical limitations on implementing manybody gates, it is reasonable to ask whether having access to elementary up-to-k-local equivariant gates suffices for the QNN to express any possible equivariant unitary.This question was first studied in Ref. [17], where the authors found that, as opposed to the wellknown result which states that any unitary can be decomposed in one-and two-qubit gates, certain equivariant unitaries cannot be decomposed in equivariant k-local gates.These result were then extended in Ref. [18][19][20] to rotationally-invariant circuits, tensor product representations of groups acting on qudits, and tensor product representations of Abelian symmetries.
In this work, we will focus on G = S n , the symmetric group of permutations, with its action on n qubits.This group is of special interest as it is the relevant symmetry group for a wide range of learning tasks related to problems defined on sets, graphs and grids, molecular systems, multipartite entanglement, and distributed quantum sensing [15,[21][22][23][24][25][26][27][28][29].Our results can be summarized as follows.First, we show that that S n -equivariant QNNs with 2-body gates are enough to reach semi-universality (and hence subspace controllability), but not universality.This means that the circuit can generate any ar-bitrary special unitary matrix in each of the subspaces but that one has no control over all their relative phases.We demonstrate that this small, albeit important, difference follows from the central projection theorem of Ref. [30].Next, we prove that if one wishes to attain subspace universality, then the QNN must contain n-body S n -equivariant gates (if n is even) or (n − 1)-body gates (if n is odd).Hence, our theorems impose restrictions on the set of unitaries that can be generated when combining symmetry and few-bodyness.We further argue that our results are important for the field of quantum optimal control [31] as they correct a previous result in the literature [32].

II. RESULTS
A. Background: Dynamical Lie algebra and dynamical Lie group for QNNs Parametrized quantum circuits, or more generally, QNNs, are one of the most widely used computational paradigms to process information in near-term quantum computers due to their high versatility [3,33].In what follows, we will consider QNNs acting on systems of n qubits (with associated d = 2 n -dimensional Hilbert space H) of the form where H m are Hermitian operators taken from a given set of generators G, and θ = (θ 1 , . . .θ M ) ∈ R M are trainable parameters.Here we note that the operators in G constitute a veritable fingerprint for the QNN, as they allow us to differentiate a given architecture from another.
In this work we consider the case where G contains only up-to-k-body gates, i.e., gates acting non-trivially on at most k qubits.Such restriction is usually physically motivated and arises when working with gates that are native to some specific hardware [34].At this point, we find it convenient to recall that in the absence of symmetries, 2-local gates are universal [35,36].That is, they suffice to generate any d-dimensional unitary.However, the same is generally not true when the operators in G are local, but also chosen to respect a certain symmetry group G [17,20].Hence, given these constraints, it is critical to quantify the QNN's expressiveness, i.e., the breadth of unitaries that U (θ) can generate when varying the parameters θ (see Fig. 1(b)).By inspecting Eq. ( 1) one can see that two main factors come into play: the number of parameters M , and the set of generators G.While several measures of expressiveness exist [5,30,37], here we will focus on the so-called Dynamical Lie Algebra (DLA) [38], which captures the potential expressiveness of the QNN (under arbitrary depth, or number of parameters).
Given a set of Hermitian generators G, the DLA is the subspace of operator space u(d) spanned by the repeated in the field of GQML indicate that if the problem has a given relevant symmetry G, then the gates in the QNN should be G-equivariant [11].In this work we consider the case where G = Sn, the permutation group, and study how the expressiveness of the QNN changes when one imposes the additional constraint of few-bodyness on the QNN's gates.In the circuit, gates with the same color share a common parameter.b) The effect of restricting the set of available elementary gates in the QNN is to restrict its expressiveness, i.e., how much of the unitary space it can cover by varying its parameters.Imposing G-equivariance can appropriately reduce the QNN's expressiveness to a region of unitaries respecting the task's symmetry.Imposing additional restrictions, such as few-bodyness, further restricts its expressiveness.
nested commutators of the elements in G.That is, where ⟨ • ⟩ Lie denotes the Lie closure.Notably, the DLA fully determines the ultimate expressiveness of the QNN, as we have Here, G denotes the dynamical Lie group, and is composed of any possible unitary generated by a unitary U (θ) as in Eq. ( 1) for any possible choice of θ and M .When G = SU(d) or G = U(d), the QNN is said to be controllable, or universal, as any unitary can be generated from U (θ) (up to a global phase) [31,39].In what follows, we will study how constraints in G (such as equivariance or few-bodyness) alter the QNN's ability to become universal.For this purpose, we recall the mathematical definition of G-equivariance [9,[11][12][13].Given a symmetry As schematically shown, the associated Lie group contains the set of matrices where an arbitrary unitary can be prepared in each isotypic component.b) Next, we define the special subalgebra of u G (d), which we denote as su G (d). Now the associated Lie group contains all the matrices where an arbitrary unitary can be prepared in each isotypic component, under the additional restriction that the determinant of the overall matrix must be equal to one.c) Finally, we define the centerless subalgebra of u G (d), which we denote as su G cless (d).In this case, the Lie group contains all matrices where an arbitrary special unitary can be prepared in each isotypic component.Note that the difference between su G cless (d) and u G (d) is that the former does not allow us to control the relative phases between the isotypic components.
group G (which we henceforth assume to be compact), we have the following definition.
Definition 1 (Equivariant operators).An operator H is said to be G-equivariant if it commutes with a representation of a given compact group G.That is, if where R is a unitary representation of G on the ddimensional Hilbert space H.
Note that if all the generators in G are equivariant, the QNN can be readily shown to be equivariant itself [9,11].We also highlight the fact that Definition 1 implies where comm(G) denotes the commutant algebra of the (representation R of the) group G, i.e., the associative matrix algebra of linear operators that commute with every element in G: Here, B(H) denotes the space of bounded linear operators in H.
Next, it is fundamental to recall that the representation R admits an isotypic decomposition where r λ is a d λ = dim(r λ )-dimensional irreducible representation (irrep) of G, and 1 1 m λ is an identity of dimension m λ .Hence, m λ denotes the multiplicity associated to each irrep.In this case, the Hilbert space H can be expressed as where H ν λ is a m λ -dimensional space, and ν = 1, • • • , d λ .Moreover, using P ν λ to denote the projector onto the subspace H ν λ , we can focus on the part of the DLA that acts non-trivially on each H ν λ , given by Note that by definition g ν λ = g ν ′ λ , which allows us to simply use the notation g λ := g ν λ .The previous motivates us to define three important Lie subalgebras of u(d).First, we define the maximal where Q is a representation defined by the right-handside of Eq. (10).We then define the maximal special where s[•] denotes keeping the operators with vanishing trace.Finally, we also define the maximal centerless Gsymmetric subalgebra In Fig. 2 we show how the previous three algebras lead to different unitaries in their associated Lie groups.
With the previous algebras in mind, we can introduce three key definitions that will allow us to study controllability and degrees of universality when there are symmetries in play.
Definition 2. The QNN, or its associated DLA, is said to be: • Semi-universal, if • Universal, or completely controllable, if It is clear that complete controllability [31] implies semi-universality [20], which in turn implies subspace controllability [40].The key difference between the unitaries arising from a DLA which is semi-universal but not universal is that one cannot control the relative phases between the unitaries acting within each isotypic component (see Fig. 2).On the other hand, a QNN that is subspace controllable can generate arbitrary unitaries in each subspace, but not necessarily independently as in the semi-universal case.For instance, if ) is subspace controllable but not semi-universal.As we will see below, distinguishing between these three cases will be extremely important to better understand the expressive capacity of QNNs with equivariance and few-bodyness constraints.
To finish this section we recall a fundamental concept: Given a Lie algebra h ⊆ u(d), its center z(h) is composed of all the elements in h that commute with every element in h, i.e., In particular, z(u where e λ ∈ R.Moreover, it also follows that i.e., there are as many elements in the center as irreps in Eq. (10).Finally, let us note that su

B. Main results
In what follows we will consider G to be the symmetric group S n and R the qubit-permuting representation of S n on n qubits, Given a set of equivariant k-local generators, it is natural to ask: can we achieve subspace controllability, or even (semi-)universality?In this section we characterize the expressiveness of permutation-equivariant few-bodynessconstrained ansatze.
First, let us investigate the case of 1-body symmetric operators.For instance, consider the (single element) set where we use X j , Y j , Z j to denote the corresponding Pauli operator acting on the j th qubit.By inspection, n j=1 X j is S n -equivariant, as it is invariant under any qubit permutation.Moreover, we can also see that the corresponding DLA g 1 is a representation of u(1).Next, consider the set for which the DLA g ′ 1 is a representation of su(2).These two cases show that, unsurprisingly, one cannot obtain subspace controllability (let alone semi-universality or universality) using 1-body symmetric operators.
Let us now consider up-to-2-body S n -equivariant generators.In particular, consider the set The expressiveness of G 2 is captured by the following theorem.
Theorem 1.Consider the set G 2 of S n -equivariant generators in Eq. (23).The associated DLA is where ⊞ denotes the Minkowski sum1 and where .
Theorem 1 has several important implications.First, it shows that adding a single S n -equivariant 2-body operator to G ′ 1 significantly increases its expressiveness, as it leads to a DLA g 2 that contains su Sn cless (d) plus the span of an element in the center z(u Sn (d)).
The second implication of Theorem 1 is that the system is semi-universal (and thus subspace controllable) according to Definition 2. However, we note that the previous result also implies that g 2 fails to be universal (see Definition 2).This is in contrast to the claims in Ref. [32] which state that g 2 is universal.To clarify this discrepancy, we show in the Supplemental Material that the proof of universality in Ref. [32] contains an error.
While we leave the full proof of Theorem 1 for the Supplemental Material, here we will explain the intuition and main steps behind it.In particular, we will constructively show that g 2 contains the centerless subalgebra su Sn cless (d) plus the span of a single element in the center z u Sn (d) .The main tool behind deriving the latter result is the central projections condition of [30].
Let us introduce some useful notation.First, we will define the S n -symmetrized Pauli strings.Definition 3 (Symmetrized Pauli strings).Define P (kx,ky,kz) to be the operator corresponding to the sum of all distinct Pauli strings that have k x X symbols, k y Y symbols, and k z Z symbols.
The significance of the C µ operators is explained by the following proposition, proved in the Supplemental Material.(For additional properties of the C µ operators, we also refer the reader to Ref. [18], where they are referred to as C l with l = 2µ.)Proposition 1 (Center of the S n -equivariant Lie algebra).The center of the S n -equivariant Lie algebra u Sn (d) is spanned by the C µ operators: We can see from Eqs. ( 26) and ( 28) that the maximal S n -equivariant centerless subalgebra su Sn cless (d) is given by all linear combinations of the form i 0≤kx+ky+kz≤n c (kx,ky,kz) P (kx,ky,kz) , (30) with c (kx,ky,kz) ∈ R, such that for each integer 0 ≤ µ ≤ ⌊n/2⌋.As previously mentioned (and as seen in Eq. ( 19)), for a DLA to be universal it needs to generate all the elements in su Sn cless (d) plus all the traceless elements in the center z(u Sn (d)).Notably, one can prove the following necessary condition for a DLA to be able to generate elements in the center: Theorem 2 (Central Projections.Result 1 in [30], Restated).Let g be the DLA obtained from a set of generators G.If [H, C] = 0 and Tr[HC] = 0 for every H in G, then Tr[M C] = 0 for every M in g.In particular, for g to contain a nonzero central element C, it is necessary that Tr[HC] ̸ = 0 for some H in G.
Theorem 2 indicates that an element in the center cannot be in the DLA if all of the generators have zero projection onto it (in the sense that they are orthogonal with respect to the Frobenius inner product ⟨A, B⟩ = Tr[A † B]).Now we can use the following Theorem, proved in the Supplemental Material.Theorem 3. Consider the set G 2 of S n -equivariant generators in Eq. (23).Then, one has that for all H ∈ G 2 and for all C µ with 0 ≤ µ ≤ ⌊n/2⌋ except µ = 1.
Theorem 3 shows that the set of generators have no projection onto all center elements C µ with 0 ≤ µ ≤ ⌊n/2⌋ except for that with µ = 1.Therefore, according to Theorem 2, no C µ with 0 ≤ µ ≤ ⌊n/2⌋ (except for that with µ = 1) can be generated within the DLA.It is important to remark that the previous does not imply that C 1 actually appears in the DLA, as Theorem 2 only provides a necessary condition.However, one can prove by direct construction that g 2 indeed contains C 1 , as well as any operator in su Sn cless (d).That is, one can prove that the following theorem holds.
Here we note that Theorem 4 is simply a reformulation of Theorem 1 which allows us to directly identify g 2 as containing all elements in su Sn cless (d) plus the span of the center element Q (u(1)) = span R {iC 1 }.In addition, we can infer from Theorem 4 that the dimension of Our previous results proved that sets of generators with S n -equivariant 1-body and 2-body operators are not sufficient to generate su Sn (d).A natural question then is whether this can be fixed by including in the set of k-body S n -equivariant generators (for k ⩾ 3).Defining the set of generators we find that the following result also holds (see the Supplementary Information for a proof).
Theorem 5. Consider the set G k of S n -equivariant generators in Eq. (36).Then, the associated DLA g k is where Theorem 5 shows that one element in the center is generated in the DLA every time one adds a generator with even bodyness.Crucially, this result is in accordance with the central projections condition of Theorem 2. In particular, the central projections condition implies that one cannot generate su Sn (d) unless one has, for every µ from 0 to ⌊n/2⌋, some generator with nonzero projection onto C µ .However, it is easy to see that the following theorem holds.Here we review the main results of our work.In particular, we consider the case where the single-body operators in G are P (1,0,0) and P (0,1,0) and where the k-body operators are of the form P (0,0,k) .Theorem 6.Consider the set G k of S n -equivariant generators in Eq. (36).Then, one has that for all H ∈ G k and for all C µ with µ = 0 and ⌊ k 2 ⌋ < µ ≤ ⌊n/2⌋.
Theorem 6 shows that the operators in G k have no projection onto C µ with µ > ⌊ k 2 ⌋ (or µ = 0).Thus, one finds that the following corollary holds.Corollary 1.Any set consisting of at-most-k-body S nequivariant operators will always be insufficient to generate su Sn (d) unless k = n for n even or k ⩾ n − 1 for n odd.
It is worth noting that Corollary 1 can also follow from the no-go theorem of Ref. [17], along with the fact that the center z u Sn (d) contains n-body operators for n even or (n − 1)-body operators for n odd.
Theorem 5 and Corollary 1 shows that in order for a QNN with S n -equivariant k-body gates to be universal, one needs to include in the set of generators up-to-n-body gates (for n even) or up-to-(n−1)-body gates (for n odd).Hence, as schematically depicted in Fig. 3, this corollary imposes a fundamental limitation of the universality of QNNs with S n -equivariant k-local gates.

III. DISCUSSION
The recent field of GQML has paved the way for studying how adding symmetry information into quantum learning models changes their performance in terms of expressiveness, trainability, and generalization.Recent results has shown that GQML can indeed provide a heuristic advantage for several machine learning tasks [42][43][44][45][46][47][48][49] over their symmetry agnostic counter-parts.In particular, it was shown that the special case of S nequivariant QNNs exhibit the holy grail of desirable properties [12]: absence of barren plateaus, generalization from few training points, and the capacity to be efficiently overparametrized.Still, despite the great promise of GQML there are still many open questions regarding its full capabilities.As an example, the seminal works of Refs.[17][18][19][20], have started to analyze restrictions to universality and their connections to the type of gates used.Moreover, it is has been documented that quantum noise can be quite detrimental for equivariant models [50][51][52] and that reducing a quantum learning model's expressive power too much can potentially make it classically simulable [15,53].As such, we expect that the study of symmetry-enforced models will be a thriving area in the future.
In this context, we have shown that quantum circuits with elementary k-body S n -equivariant gates are semiuniversal and subspace controllable (see Definition 2), but not universal.While this result might seem negative (as we cannot reach universality), we are inclined to note that reaching semi-universality is in itself a tall order.The fact that S n -equivariant QNNs with only 2-body gates have enough expressiveness to be semi-universal is a notable result in itself.Crucially, it is not obvious that such high degree of universality will generally hold.Moreover, our results indicate that in order to reach universality, one must include up-to-n-body interactions if n is even, and up-to-(n − 1)-body interactions if n is odd.These results have several implications.First, they highlight the existence of a fundamental limitation to achieving universality from local permutation-invariant gates.Second, they correct a result in the literature which states that one-and two-body S n -equivariant gates are indeed enough for universality.Finally, they portray the power of the central projections condition of Ref. [30].
Here we also note that at a higher level, the central projections condition offers a significant obstruction to the possibility of using only local or few-body gates to achieve universality of gates that respect some symmetry group (see also [18]).Of course, symmetries are necessary for this phenomenon to present itself; for instance, as illustrated by a classical result in quantum computation [35], the collection of all 1-body and 2-body Pauli strings, which do not share any symmetry, is indeed sufficient to generate all Pauli strings.
It is also worth highlighting the fact that our results are intrinsically connected to those in Ref. [18] through the Schur-Weyl duality.Namely, while we study circuits with S n -equivariance, the work in [18] studies SU(2)equivariant circuits.Since the qubit permuting representation of S n and the tensor product representation of SU(2) mutually centralize each other, their centers match.Hence, the conditions imposed by the central projections theorem to the circuit's generators bodyness are exactly the same for SU(2)-or S n -equivariant circuits.That is, Corollary 1 in our work is precisely, and necessarily, the same as the direction of Theorem 1 in [18] that establishes constraints from the central projections condition.This realization then allows us to identify our work as being dual to that in Ref. [18], where each work explores how the central projections conditions affect generators on each "side" of the Schur-Weyl duality.Finally, this connection shows that results obtained for one side of a dual reductive pair will naturally be relevant for the other side.
As a final note, both our work and those in Refs.[17,18] focus primarily on obstructions to universality coming from the central projections condition.However, the characterization of Lie algebras presented in Ref. [30] has three components: linear symmetries, quadratic symmetries, and central projections, with central projections generally having the smallest impact in terms of reducing the dimension of the Lie algebra.As an example, the Lie algebra g 2 studied in this paper has a relatively small dimension deficit of ⌊n/2⌋ compared to the full S n -invariant algebra u Sn (d).The question of whether few-bodyness (or locality) of symmetry-equivariant qubit gates could enforce extra linear or quadratic symmetries has only begun to be studied, for instance in Refs.[19,20], and such possibilities would cause the DLA dimension deficiency to be much larger.We thus believe that a more general investigation into the failure modes for gate sets with additional constraints (beyond symmetry-equivariance) to generate the full symmetry-invariant algebra will be a fruitful area of study.
Note added: After completion of this manuscript, it was brought to our attention that Proposition 1 of our Supplemental Material is also presented in a slightly more general form as Proposition 1 in [18].
Lemma 1 motivates us to define the following set of operators, each of which is a conjugacy-class sum within S n : Definition 6.For each integer 0 ≤ µ ≤ ⌊n/2⌋, define T µ to be the set of all permutations in the symmetric group S n that consist of exactly µ disjoint transpositions.Then define the following operator: (B7) As a quick example, consider the operator obtained from two element transpositions Now, armed with Lemma 1, we can provide a proof for the following lemma (see also [56] for an alternative proof).
Lemma 2. The center of the algebra S n is the span of the L µ operators: Proof.Each conjugacy class of the symmetric group is the set of all permutations of a given cycle type.Hence each T µ is a conjugacy class.And since L µ is a sum over all permutation operators in this conjugacy class, Lemma 1 immediately implies that L µ commutes with all permutation operators.Hence each L µ lies in the center of S n .Now it remains to show that these are all of the central elements of S n .Clearly the T µ are only a small number of the conjugacy classes of S n , so Lemma 1 will not help us with that.However, it is known by an argument involving Young diagrams that the dimension of the center of S n is given by the number of Young diagrams with n cells that have at most 2 rows [57].Such a diagram has µ cells in the second row and n − µ cells in the first row for some integer 0 ≤ µ ≤ ⌊ n 2 ⌋, so there are exactly ⌊ n 2 ⌋ + 1 such diagrams.Hence, this means that there are ⌊ n 2 ⌋ + 1 elements in the center which coincides with the number of L µ operators.Thus, L µ span the center of the algebra S n .
As a brief aside, it is worth mentioning that the theorem that we cited above regarding Young diagrams depends on the fact that we are working with qubits.For example, if we were working with qutrits instead of qubits, then one would instead need to look at Young diagrams with at most 3 rows [57].
Up to this point, we have shown that the L µ operators span z(S n ), so all that remains is to show that the C µ operators actually have the same span as the L µ operators.We demonstrate this now via the following lemma.Lemma 3.For each integer 0 ≤ µ ≤ ⌊n/2⌋, As a result, we can also write Corollary 2. For each integer 0 ≤ µ ≤ ⌊n/2⌋, the L µ operators and the C µ ′ operators for 0 ≤ µ ′ ≤ µ have the same span.In particular, the full set of (⌊n/2⌋ + 1) L µ operators has the same span as the full set of (⌊n/2⌋ + 1) C µ operators.
Proof.The transposition , and an element π ∈ T µ takes the form π = (α which after expansion implies that R(π) consists solely of Pauli strings with an even number of X symbols, an even number of Y symbols, and an even number of Z symbols.Hence the same is true of L µ , and since L µ is S n -invariant, we know that L µ is a linear combination of symmetrized Pauli strings P (2a,2b,2c) .The coefficient of P (2a,2b,2c) in L µ equals 2 −µ times the number of permutations π ∈ T µ for which any individual Pauli string within P (2a,2b,2c) appears in the expansion of R(π).So, consider an arbitrary Pauli string with 2a X's, 2b Y 's, 2b Z's, and n − 2(a + b + c) I's.The 2a qubits with X's must be paired up into a transpositions within π.The number of ways to choose how they are paired up is The same is true for the 2b qubits with Y 's, and the same is also true for the 2c qubits with Z's.Hence we additionally get factors of (2b)! 2 b b! and (2c)! 2 c c! .The remaining freedom in defining π comes from the number of ways to pair up some of the n − 2(µ − f ) qubits with I's into the remaining d transpositions, where we have defined f = µ − (a + b + c) for convenience.The number of ways to do this is Putting this all together, we obtain that the coefficient of the Pauli string in question within L µ is We conclude that which completes the first part of this proof.Now, for the second part of this proof, simply let µ ′ = a + b + c = µ − f and regroup the summation as follows: To summarize, we now know from Lemma 2 that the L µ operators span z(S n ), and we also know from Corollary 2 (an immediate consequence of Lemma 3) that the L µ operators and the C µ operators have the same span.We conclude that the C µ operators span z(S n ), which is exactly the statement of Proposition 2. In addition, since z(u Sn (d)) = z(S n ), we can also conclude that the C µ operators span z(u Sn (d)), as stated in Proposition 1.
After completion of this manuscript, it was brought to our attention that Proposition 1 in this manuscript is also presented in a slightly more general form as Proposition 1 in [18].Some of the lemmas used to prove this proposition, namely Lemma 2, Lemma 3, and Corollary 2, are also presented in [18].The L µ and C µ operators in this manuscript correspond respectively to the B m and C l operators in [18], with m = 2µ and l = 2µ.
We now present two lemmas that will help us greatly in systematically constructing the operators we claim to be in a given g.The basic idea is that, under certain conditions, one can "hop" by one or two spaces within the barycentric lattice of level-k symmetrized Pauli strings (see Sup. Fig. 2).We first present the lemma that allows one to hop by one space.
Proof.If k y , k z ≥ 1, then simply use the commutator [P (kx,ky,kz) , P (1,0,0) ] ∝ (k y + 1)P (kx,ky+1,kz−1) − (k z + 1)P (kx,ky−1,kz+1) , (D13) where we have used the fact that P (1,0,0) is in g.Then, since P (kx,ky+1,kz−1) ∈ g, we can see that P (kx,ky−1,kz+1) must also be in g.If k z = 0 but k y ≥ 1, then just use the commutator Note that, although the lemma focuses on hopping one space only in certain directions, the lemma stills work regardless of which direction one hops, so long as one stays on level k.For instance, as schematically shown in Sup.Fig. 3, taking the commutator with P (1,0,0) corresponds to hopping in the direction where one keeps the x-coordinate constant, but one increases (decreases) the y-coordinate by 1 and respectively decreases (increases) the z-coordinate by 1.Similarly, taking the commutator with P (0,1,0) corresponds to hopping in the direction where one keeps the y-coordinate constant, and taking the commutator with P (0,0,1) corresponds to hopping in the direction where one keeps the z-coordinate constant (see Sup. Fig. 3).
In the case where k z ≤ 1, the method is exactly the same.Simply take the commutator with P (1,0,0) twice.The result is a linear combination of P (kx,ky,kz) and P (kx,ky−2,kz+2) , where the coefficient of the latter is nonzero.(The only difference from the case above is that P (kx,ky+2,kz−2) no longer exists, since k z ≤ 1.) Since we are given that P (kx,ky,kz) ∈ g, we conclude that P (kx,ky−2,kz+2) ∈ g, as desired.
We now use Lemmas 5 and 6 to efficiently hop around the barycentric lattice and construct the operators in g 2 .This idea is captured in the following lemma and in Sup.Fig. 4: Lemma 7. If P (k−1,1,0) ∈ g 2 , P (0,k−1,1) ∈ g 2 , and P (1,0,k−1) ∈ g 2 , then P (kx,ky,kz) ∈ g 2 for any ordered triple (k x , k y , k z ) at level k where at least one of k x , k y , k z is odd.In particular, if k = k x + k y + k z is odd, then any ordered triple at level k will satisfy this, which means that every operator at level k is in g 2 .
Proof.In this proof we will first show that if then P (k−1,1,0) ∈ g 2 , then P (kx,ky,kz) ∈ g 2 for any ordered triple (k x , k y , k z ) at level k where k y is odd.Once this result is established, we can use Lemma 4 to obtain the remaining cases.
given P (0,0,2) as a generator.Sure enough, we already generate all of these operators in the proof of Lemma 4. Other than those exceptions, we will ultimately be unable to generate P (kx,ky,kz) operators in isolation if k x , k y , k z are all even.However, as we will see, we can still generate all combinations of these operators that satisfy the condition of Theorem 4. This is what we show now, and it is the last step of the proof: Proof.For µ = 0, note that C 0 = I, and the singular constraint is c (0,0,0) = 0, so we do not need to generate anything here.From now on, assume that µ ≥ 2.
First, we show that, if (2a, 2b, 2c) and (2a ′ , 2b ′ , 2c ′ ) are two ordered triples on the same level that are separated by a distance of just 2 on the barycentric lattice, then we can produce a linear combination using just P (2a,2b,2c) and P (2a ′ ,2b ′ ,2c ′ ) that lies in g 2 .We will call these operators two-coordinate operators for convenience.For example, suppose that a ′ = a, b ′ = b + 1, and c ′ = c − 1.Then we can take the following commutator: Notice that the two coefficients, −2c and 2(b + 1), indeed satisfy the required condition: If one wishes to do repeat the same calculation as above, but for two coordinates separated in the direction that keeps the y-coordinate or z-coordinate constant, then one needs to take a commutator with P (0,1,0) or P (0,0,1) , respectively.
We will now use these two-coordinate operators to produce an arbitrary linear combination of the form given in the statement of the lemma.In particular we will start by drawing an analogy to a very simple linear algebra exercise: In R n , with e j for 1 ≤ j ≤ n as the standard basis vectors, it is easy to show that the n − 1 vectors v j = e j − e j+1 span the space of all vectors whose components add up to 0. In particular, suppose one wants to produce the vector Then one just take the appropriate amount of v 1 to match the first component, then the appropriate amount of v 2 to match the second component, and so on, until one takes the appropriate amount of v n−1 to match the (n − 1) th component.Since c 1 + • • • + c n = 0, the n th component will automatically equal c n .As we can see, following the previous procedure, one successfully produces the vector (c 1 , • • • , c n ).Supplementary Figure 5. Ordering symmetrized Pauli strings in the barycentric lattice.In the figure we schematically show how a "snaking" pattern can be created in the barycentric lattice that lines up all operators P (2a,2b,2c) such that a+b+c = k for some fixed value of k.Notice that the operators have a distance of 2 from their neighbours in the line-up (here we have shown k = 4).The previous means that we can create a two-coordinate operator as in for each pair of neighbours in the line-up.
To complete our proof, we will imitate the linear algebra exercise we just presented.Take the N µ = µ+2 2 operators P (2a,2b,2c) such that a + b + c = k for a fixed value of µ, and line them up in an order such that each coordinate has distance 2 from its nearest neighbors in the line.For each j from 1 to N µ , label the coordinate at position j in the line as (2a j , 2b j , 2c j ).There are many ways to do this, but one simple way is a "snaking" pattern that divides the operators into rows based on decreasing values of a.As schematically shown in Sup.Fig. 5, one starts with the first row, which is just P (2µ,0,0) .Then goes to the next row, which will have P (2(µ−1),2,0) and P (2(µ−1),0,2) in that order.Then one transitions to the next row and traverse it backward, so that one crosses P (2(µ−2),0,4) , P (2(µ−2),2,2) , and P (2(µ−2),4,0) , in that order.If one follows this pattern of alternating traversing each successive row forward or backward, one will eventually reached the end of the lattice (see Sup. Fig. 5).Now, for each j from 1 to N µ − 1, we can construct a two-coordinate operator T j out of P (2aj ,2bj ,2cj ) and P (2aj+1,2bj+1,2cj+1) .Now, suppose we wish to generate the operator M = a+b+c=µ c (2a,2b,2c) P (2a,2b,2c) for any set of coefficients c (2a,2b,2c) such that a+b+c=µ c (2a,2b,2c) a!b!c! = 0. Then start with the right amount of T 1 so that the coefficient of P (2a1,2b1,2c1) matches that in M .Then add the right amount of T 2 so that the coefficient of P (2a2,2b2,2c2) matches that in M .Keep doing this, until you finally add the right amount of T Nµ−1 so that the coefficient of P (2a N k −1 ,2b Nµ−1 ,2c Nµ−1 ) matches that in M .Due to the central projections condition, the coefficient of P (2a Nµ ,2b Nµ ,2c Nµ ) will now automatically match that in M .Therefore, we have constructed our desired operator, so we are done.
The proof is now complete, so let us summarize what we have done.As shown in Theorem 2 of the main text, we first used the central projections condition to show that being orthogonal to every C µ except µ = 1 is a necessary condition for an operator to be in g 2 .After that, we showed that g 2 indeed contains all operators that satisfy that condition.As shown in Lemma 8, we successfully generated P (kx,ky,kz) where at least one of k x , k y , k z is odd.The form of the C µ operators makes it clear that all such P (kx,ky,kz) are already orthogonal to all of the C µ operators.Finally, as shown in Lemma 9, we successfully generated all linear combinations of the P (2a,2b,2c) operators on level 2µ that are orthogonal to C µ .Note that we could ignore µ = 1 because we already showed in Lemma 4 that g 2 contains all 6 level-2 operators.
The only operators in u Sn (d) that are "missing" from g 2 are the C µ operators for 0 ≤ µ ≤ ⌊n/2⌋ except µ = 1.This includes C 0 = I, but it also includes the operators C µ for 2 ≤ µ ≤ ⌊n/2⌋.As a result, we obtain dim(L) = As a final observation for this section, note that the requirement of orthogonality with C 0 = I is what prevents any operator with a nonzero P (0,0,0) = I component from being in g 2 .Of course, the identity will be in the center of the commutant of any set of generators, so the central projections condition of Ref. [30] will always exclude operators with nonzero trace from being in the algebra if the generators are all traceless.Of course, the fact that the identity cannot be in such an algebra is by itself a pretty trivial observation, since it is such a common fact that the commutator of two finite-dimensional operators is traceless.But this work already serves to demonstrate that the central projections condition can be used to prove a highly non-obvious result.It is interesting to note that the central projection generalizes such a seemingly mundane statement as the fact that the commutator of two finite dimensional operators is traceless, and yet it is powerful enough to crop up in unexpected ways.

Figure 1 .
Figure 1.Summary of our main results.a) Recent resultsin the field of GQML indicate that if the problem has a given relevant symmetry G, then the gates in the QNN should be G-equivariant[11].In this work we consider the case where G = Sn, the permutation group, and study how the expressiveness of the QNN changes when one imposes the additional constraint of few-bodyness on the QNN's gates.In the circuit, gates with the same color share a common parameter.b) The effect of restricting the set of available elementary gates in the QNN is to restrict its expressiveness, i.e., how much of the unitary space it can cover by varying its parameters.Imposing G-equivariance can appropriately reduce the QNN's expressiveness to a region of unitaries respecting the task's symmetry.Imposing additional restrictions, such as few-bodyness, further restricts its expressiveness.

Figure 2 .
Figure2.Important Lie algebras and the irrep structure of the elements in the associated Lie groups.In the main text we have defined three important subalgebras.a) The first is the maximal G-symmetric subalgebra u G (d).As schematically shown, the associated Lie group contains the set of matrices where an arbitrary unitary can be prepared in each isotypic component.b) Next, we define the special subalgebra of u G (d), which we denote as su G (d). Now the associated Lie group contains all the matrices where an arbitrary unitary can be prepared in each isotypic component, under the additional restriction that the determinant of the overall matrix must be equal to one.c) Finally, we define the centerless subalgebra of u G (d), which we denote as su G cless (d).In this case, the Lie group contains all matrices where an arbitrary special unitary can be prepared in each isotypic component.Note that the difference between su G cless (d) and u G (d) is that the former does not allow us to control the relative phases between the isotypic components.

Theorem 4 .
Consider the set G 2 of S n -equivariant generators in Eq.(23).The associated DLA g 2 contains all linear combinations of the form i 0≤kx+ky+kz≤n c (kx,ky,kz) P (kx,ky,kz) ,

Figure 3 .
Figure 3. Sn-equivariance, few-bodyness, and DLA.Here we review the main results of our work.In particular, we consider the case where the single-body operators in G are P (1,0,0) and P (0,1,0) and where the k-body operators are of the form P (0,0,k) .