Magic of quantum hypergraph states

Magic, or nonstabilizerness, characterizes the deviation of a quantum state from the set of stabilizer states and plays a fundamental role from quantum state complexity to universal fault-tolerant quantum computing. However, analytical or even numerical characterizations of magic are very challenging, especially in the multi-qubit system, even with a moderate qubit number. Here we systemically and analytically investigate the magic resource of archetypal multipartite quantum states -- quantum hypergraph states, which can be generated by multi-qubit Controlled-phase gates encoded by hypergraphs. We first give the magic formula in terms of the stabilizer R$\mathrm{\acute{e}}$nyi-$\alpha$ entropies for general quantum hypergraph states and prove the magic can not reach the maximal value, if the average degree of the corresponding hypergraph is constant. Then we investigate the statistical behaviors of random hypergraph states and prove the concentration result that typically random hypergraph states can reach the maximal magic. This also suggests an efficient way to generate maximal magic states with random diagonal circuits. Finally, we study some highly symmetric hypergraph states with permutation-symmetry, such as the one whose associated hypergraph is $3$-complete, i.e., any three vertices are connected by a hyperedge. Counterintuitively, such states can only possess constant or even exponentially small magic for $\alpha\geq 2$. Our study advances the understanding of multipartite quantum magic and could lead to applications in quantum computing and quantum many-body physics.

Quantum magic has been extensively studied in the resource framework with various measures proposed [29][30][31][32][33][34][35][36].Most of these measures involve optimization in the stabilizer polytope, which is extremely challenging for multi-qubit systems analytically and numerically [9,13,37].To address this issue, Stabilizer Rényi Entropy (SRE) was recently proposed as a faithful indicator of magic [38,39], which explicitly quantifies magic by the weight distribution of the state projected to all Pauli strings.The simplicity of SRE triggers a series of interesting studies, whose topics include magic measurement protocol [34,40,41], the complexity of wave functions [42,43], quantum information dynamics [44,45], and learning theory [46,47].Even though SREs enable an explicit computation and experimental realization, the evaluation cost scales exponentially with the number of qubits, in general, [40], thereby hindering its application towards large-scale systems.There are some positive progresses using matrix-product-state [48][49][50][51].However, the target states are limited to low-entanglement ones, and the methods are mainly for numerical purposes.Till now, SRE and magic have been unexplored for complex many-qubit states with extensive entanglement.
In this work, we significantly extend the scope of magic quantification to large-scale and highly entangled states.In particular, we systemically and analytically investigate the magic of quantum hypergraph states [52,53], which are generalized from graph states [54,55].Unlike graph states, which are generated by Clifford gates and lack quantum magic, hypergraph states are capable of magic and play an essential role in quantum advantage protocols [56], measurement-based quantum computing (with Pauli measurements) [57,58] and topological order [59][60][61].According to the indices of all Pauli strings, we relate the magic in terms of SRE to a family of induced hypergraphs from the original one.This pictorial expression enables a series of analytical findings as follows.We first show a general upper bound of magic for any hypergraph state with a bounded average degree, for instance, ones whose hypergraphs are defined on lattices like Union-Jack one [61].We further develop general theories that transform the statistical properties of magic into a series of counting problems in the binary domain.Our theories lead to the concentration result that the magic of hypergraph states is typically large and very near the maximal value, showing similar behavior to the unphysical Haar random states [18,38,62].In addition, we analyze the magic of quantum hypergraph states with permutation symmetry.Based on the symmetry simplification and pictorial derivation, we obtain exact analytical results of the stabilizer Rényi-α entropy (SR α E) for different α's, and in particular, find that SR 2 E and SR 1 2 E can be exponentially different for these states.Specifically, SR 1 2 E serves as a lower bound for the robustness of magic, which operationally quantifies the overhead in classical simulation when using ancillary magic states [13,38].Our findings and the developed techniques can advance further investigations of multipartite quantum magic with applications from quantum computing to quantum many-body physics, where especially hypergraph states can serve as an archetypal class of complex states and tractable toy models of other complex quantum systems.

Preliminaries 2.1 Quantum hypergraph states
Graph states are widely recognized for their well-defined structure and robust entanglement capabilities, making them popular in various research fields such as quantum entanglement [63], quantum computing [54,55], and quantum error correction [64].Despite their numerous applications, graph states lack quantum magic and are unsuitable for demonstrating quantum advantage in certain scenarios.To overcome this limitation, the extension from graph states to hypergraph states has been proposed [52].This generalization endows them with the necessary quantum magic while retaining their clear structural advantage, expanding the applicability in quantum computing [56,57].
Here we give a brief introduction to quantum hypergraph states [52,53,65], which is a generalization of graph states.A hypergraph G = (V, E) consists of a vertex set V = {v i |i ∈   11) and (12).Here two 6-bit strings are chosen as ⃗ x = {1, 0, 1, 0, 1, 0} and ⃗ z = {1, 0, 0, 1, 0, 0}.For example, the 2-edge {1, 2} is induced from the 3-edge {1, 2, 3} of G in (a) with x 3 = 1, according to the edge set E We use the label e \ e ′ to denote the minus operation of two edges with e ′ ⊂ e, and specifically e \ {v} is a new edge without the vertex v.The neighbor set of a vertex v i contains all the vertices v j connected to v i by some hyperedge, and the degree ∆(v i ) is the cardinality of this set.The average degree of a hypergraph G is defined as Associate each vertex v i with a qubit, i.e., a local Hilbert space H i ≃ C 2 , and the total Hilbert space is H = ⊗ n i=1 H i with the total dimension d = 2 n .Apply quantum gates on the qubits associated with hyperedges, and one can define the quantum hypergraph state as follows.
Definition 1.Given a hypergraph G = (V, E) with n vertices, the corresponding quantum hypergraph state of n qubits reads where |+⟩ = (2) Note that here S i is not necessarily in the tensor-product of single-qubit Pauli operator {X i , Y i , Z i } and the identity I i , but the eigenvalues of S i are still ±1.Similar to the graph states and stabilizer states [66], hypergraph states can be written as the successive projection to the +1 space of each generator S i as where ⃗ s is a binary vector of dimension n and the summation is over all possible ⃗ s.It is direct to see that any multiplication i S s i i is also a stabilizer of |G⟩.We show the following observation of a very compact form for any stabilizer of |G⟩, i.e., the multiplication of some Pauli X gates and some phase gates determined by the hypergraph structure.For consistency of the following discussion, we let CZ ∅ = −1.
Observation 1.For an n-qubit quantum hypergraph state |G⟩ with the stabilizer generator S i defined in Eq. (2), any stabilizer St(G, ⃗ s) labeled by the vector ⃗ s shows where v(⃗ s) is the set of vertices v i with the corresponding s i = 1.
Observation 1 can be proved by recursively using the commuting relation that , which is left in Appendix A.1.This way, one can finally move the Pauli X gates and phase gates apart.Observation 1 could be of independent interest and applied to other studies of hypergraph states.Eq. ( 4) is very helpful for the following discussions, and for the simplicity of the presentation, hereafter, we always take q ̸ = ∅ and e ∈ E in the product by default.

Quantum magic and stabilizer Rényi entropy
Magic [2,3] quantifies the derivation of a quantum state from the stabilizer states, which is an essential resource for quantum computing complexity and its fault-tolerant realization [4,8].Stabilizer Rényi entropy (SRE) [38,40] was a recently introduced faithful indicator of magic for (pure) multipartite states defined via the probability distribution from the projection onto the Pauli operators as follows.
A quantum state ρ can be decomposed onto the complete Pauli operator basis, i.e., Pauli-Liouville representation, ρ = P ∈Pn 2 −n Tr{P ρ}P. (5) Here we consider n-qubit system and P n is the Pauli group {I i , X i , Y i , Z i } ⊗n ignoring the phase, which can be denoted as Here ⃗ x and ⃗ z are binary vectors of dimension n, X ⃗ x = n i=1 X x i i and similar for Z ⃗ z , and x i z i is some unessential phase.For a pure state ρ = |Ψ⟩ ⟨Ψ|, one can utilize the orthogonality of the Pauli operator basis to express the purity as In this way, the non-negative terms in the summation of the second line can be regarded as a probability distribution, and SR α E of |Ψ⟩ is defined via the Rényi-α entropy of this distribution.
where the offset −n keeps the magic of stabilizer states to be zero.Hereafter, all the log functions are base two otherwise specified.
As an entropy function defined on the domain P n with d 2 elements, a direct upper bound shows M α (|Ψ⟩) ≤ n.Note also that M α monotonically decreases with α by the property of Rényi entropy, and thus M 2 serves as a lower bound for M α≤2 .It is shown that its average on Haar random states, ⟨M 2 (|Ψ⟩)⟩ Ψ∈Haar ≥ log 2 (2 n + 3) − 2 > n − 2 [38], very close to the maximum possible value.We also remark that SR 1 2 E severs as a lower bound of another important measure, robustness of magic, i.e., log(R(|Ψ⟩)) ≥ 1/2M 1 2 (|Ψ⟩) [13], which could quantify the cost of classical simulation.
For ease of the following discussion, we define the closely related quantity α-order Pauli-Liouville(PL) moment as and the corresponding SR α E directly reads

Stabilizer Rényi Entropy of hypergraph states
In this section, we show a general formula of the magic for any hypergraph state by relating the PL-moment and, thus, SRE to a family of induced hypergraphs.This pictorial result enables us to find a general upper bound of magic based on the structure of the corresponding hypergraph, which constrains the magic, especially for the hypergraph states on the lattice.First, let us define a family of hypergraphs G ⃗ x,⃗ z = (V, E ⃗ x,⃗ z ), which are induced from the original hypergraph G.The vertex set V remains the same, and the updated edge set E ⃗ x,⃗ z is determined by two n-bit vectors ⃗ x and ⃗ z shown as follows.Hereafter, all the additions are module 2 on the binary domain otherwise specified.
Here and E (1) ⃗ x,⃗ z denotes the 1-edge set, while E (2) ⃗ x is for the set with 2 or more cardinality edges.Note that E (2) ⃗ x only depends on ⃗ x , and is irrelevant to ⃗ z.Following the definition in Eq. (1), we denote the phase unitary encoded by this hypergraph G ⃗ x,⃗ z as U (G ⃗ x,⃗ z ).
Additionally, we define another induced hypergraph by simplifying the 1-edge set of where E (1) ⃗ z is the simplified 1-edge set, only determined by the index ⃗ z compared to E where U (G ⃗ x,⃗ z ) is the phase unitary determined by the hypergraph G ⃗ x,⃗ z defined in Eq. (11), which is induced from G by the index vectors of P ⃗ x,⃗ z .
The proof is left in Appendix A.2.By applying the above result of Eq. (13) to the definition of Eq. ( 9) and (10) and some further simplification, one has the following general formula for the magic of quantum hypergraph states.

Theorem 1. The α-order PL-moment of a hypergraph state |G⟩ shows
where U (G * ⃗ x,⃗ z ) is the phase unitary determined by the hypergraph G * ⃗ x,⃗ z defined in Eq. (12).The corresponding SR α E reads Compared to Eq. (13) with the phase unitary U (G ⃗ x,⃗ z ), Eq. ( 14) and (15) are only related to simplified one U (G * ⃗ x,⃗ z ).This is based on the fact that the summation over all possible ⃗ z simplifies the summation over all possible E (1) ⃗ x,⃗ z to the one over E (1) ⃗ z .We remark that Ref. [49] also studied the magic of quantum hypergraph states mainly using the measure named min-relative entropy D min (|G⟩), which is related to the maximal fidelity to stabilizer states.In particular, they relate an upper bound of D min (|G⟩) to a minimization procedure of the nonquadraticity of the resulting Boolean function.However, the minimization result can only be obtained for a few example states.On the other hand, M α (|G⟩) shown here can be used as an lower bound of D min (|G⟩) [38,50].
Theorem 1 transforms the PL-moment and also SRE into the calculation of trace of a family of phase unitary.There are totally 4 n such kind of induced U (G * ⃗ x,⃗ z ), which makes the calculation of the magic for a given hypergraph state still challenging.
Specifically, for 3-uniform hypergraph states |G n,3 ⟩, the hyperedges in E (2) ⃗ x are all 2edges so the corresponding gates are CZ gates.Then PL-moment in Eq. ( 14) becomes a summation of Boolean functions, for α = 2, it reads explicitly as where b j,k (⃗ x) = {v i ,v j ,v k }∈E x i represents the summation of all the possible CZ gates for a fix qubit-pair {j, k}.Naively, there are totally 8 n terms to sum the indices of ⃗ x, ⃗ z and also ⃗ a, which looks very sophisticated.Nevertheless, based on the pictural expression in Theorem 1, we can give an upper bound of SRE of general hypergraph states with respect to its average degree.Theorem 2. For any n-qubit hypergraph state |G⟩ whose corresponding graph G has average degree ∆(G), its SR α E with α ≥ 2 is upper bounded by The proof is left in Appendix B. For α = 2, and a large enough ∆(G), the upper bound in Eq. (17) behaves like [1 − log(e) −1 2 −3 ∆(G) ]n.This indicates that for a quantum hypergraph state with ∆(G) bounded by some constant, its magic cannot reach the maximum possible value n − o(n).This bound especially constrains the magic of hypergraph states on the lattice.Moreover, a more direct upper bound for any hypergraph state shows M α (|G⟩) ≤ n α−1 .It is interesting to remark that M α (|G⟩) of α > 2 can not reach the value near n.

Magic of random hypergraph states
In this section, we study the statistical properties magic of random hypergraph states.First, we define some random hypergraph state ensembles E from the corresponding random hypergraph ensembles.Here, we mainly study random c-uniform hypergraphs, which only own c-edge.A random c-uniform hypergraph ensemble can be determined by the probability p whether there is a c-edge or not among all choices of c vertices.Denote the combination number C c n := n c , and the ensembles are defined formally as follows [68].Definition 3. The (c-uniform) random hypergraph state ensemble E p c of n-qubit system is defined as Here each e i is a distinct c-edge of the n vertices, with totally C c n such edges, U e i acts on the Hilbert space H e i by taking {I e i , CZ e i } from the probability distribution {1 − p, p} respectively, and the initial state |Ψ 0 ⟩ = |+⟩ ⊗N .
Note that the specific gate sequence in Eq. ( 18) is not relevant, as CZ e gates commute with each other.In particular, as p = 1/2, all the elements in E k share an equal probability [68].Hereafter we omit the superscript p of E p c as p = 1/2 for simplicity of notation.See Fig. 2 (a) for an example of a 3-uniform hypergraph state.
The SRE defined in Eq. ( 10) is in the logarithmic function.As such, to analyze the average property of SRE for some state ensembles E, we instead focus on the calculation of the average PL-moment of Eq. ( 9) inside the logarithm, that is, This directly gives a lower bound of the SR α E by the concavity of the logarithmic function.
In particular, for p = 1/2, one has the average PL-moment of the hypergraph ensemble E c defined in Eq. (18) as where Hereafter, we focus on α ≥ 2 ∈ Z and the uniform ensemble E c to calculate the average properties and the fluctuations of the magic, and then extend to non-uniform ensemble E p c later.The adjustment of the probability parameter p can thus change the expected density of the applied gates and also the expected average degree of the corresponding hypergraph, which is discussed in Sec.4.3.We remark that our work is the first to show statistical properties of magic for some realistic ensembles beyond Haar random states [18,38,62].

Average analysis of magic
In this section, our main focus is on the average properties of magic, especially the PLmoment, and then we show quite tight lower bounds of the average SRE.The following theorem transforms the average PL-moment into a counting problem of binary strings.We first show some related definitions of the norm and operations of an n-bit string ⃗ t = {t i }.The 1-norm ⃗ t 1 = i t i , with the addition modulo 2. The Hadamard or Schur product of some bit strings ⃗ t k is the element-wise product, i.e., For any integer α ≥ 2 ∈ Z, the average α-th PL-moment of n-qubit random hypergraph state ensembles E c defined in Eq. (18) shows where N(c, α, n) is the number of 2-tuple (T, ⃗ x), such that the following two constraints are satisfied. where x is an n-bit vector with elements x i , and e c labels all possible c-edges.
The full proof is left in Appendix C.1, and here we give some intuition about the proof.We mainly utilize the replica trick to write the average PL-moment on 2α-copy of the original Hilbert space H ⊗2α d [38,48,68].Consequently, the computational basis of every qubit can be labeled by a 2α-bit string [68], i.e., ⃗ t i of the matrix T for the i-th qubit.The two constraints Eq. ( 23) and Eq.(24), which look a bit complicated at first glance, actually correspond to the induced hypergraph structure of Eq. (12) previously introduced in Sec. 3. In particular, the first constraint Eq. ( 23) accounts for the effect of the expectation from 2α-replica on the edge set ⃗ z , and the second one Eq.( 24) for the edge set ⃗ x , that is, edges introducing multi-qubit controlled gates.See Fig. 2 (b) for an illustration of the constraints when c = 3 and α = 2.Here we explicitly give the first three 4-bit stings ⃗ t 1 , ⃗ t 2 , ⃗ t 3 with respective to the first three qubits, and clearly they satisfy the parity constraint in Eq. (23).The second constraint in Eq. ( 24) is simplified to Eq. (28) as c = 3.For a specific 3-edge e 3 = {1, 2, 3}, its induced constraint reads and ⃗ x should satisfy all such constraint from every 3-edge.(c) A graphic illustration of the 2-tuple (T , X) in Thereom 4. T is a 2α × n × τ 3-order tensor, and X is a n × τ matrix.The first constraint in Eq. ( 32) is just the parity one for each ⃗ t i (s) of T .The second one in Eq. ( 33) is a generalization of Eq. ( 24), by additionally summing the index s ∈ [τ ].For every edge e c , one first calculates Eq. ( 24) on each plane of this cube and then sums all of them in the longitudinal direction.
Before showing refined results later, we first give some preliminary estimation of the counting problem in Theorem 3. Note that Eq. ( 23) is indeed the parity constraint and independent of ⃗ x.Denote the set D = ⃗ t ∥ ⃗ t∥ 1 = 0 of 2α-bit strings as the valid set, one directly has |D| = 2 2α−1 .Each ⃗ t i of T should be taken from D, and consequently the entire matrix T comprises 2 2α−1 n alternatives when Eq. ( 23) satisfied.Furthermore, as for Eq.(24), one needs to find the number of ⃗ x given a specific assignment of T with each ⃗ t i ∈ D. Supposed that ⃗ x = ⃗ 0, it is not hard to check that in this case, Eq. ( 24) induces no constraint on T. If we ignore the constraint of Eq. (24), it is clear that there are totally 2 n distinct ⃗ x.On account of these observations, we have a lower bound and a trivial upper bound of PL-moment as The upper bound is just 1 and is trivial by definition of moments.With a refined analysis of Eq. ( 24) and more exact counting of N(c, α, n), a non-trivial upper bound of the PLmoment is shown as follows for general α and c.
Proposition 2. For any c ≥ 3 ∈ Z and α ≥ 2 ∈ Z, with the qubit number n ≫ α and n ≫ c, the average PL-moment In particular, for c = 3 and α = 2, it shows 11) .
The upper bound in Proposition 2 is a bit loose considering the dependence on the parameters c and α, which is exponential to c and even double-exponential to α.However, suppose one only considers constant c and α and focuses on the relation to the qubit number n, the upper bound then shows ⟨m α ⟩ Ec = O (2 −n ), which matches the lower bound in Eq. (25).We summarize this and its implication to SRE via Eq.(20) in the following corollary.
Corollary 1.For any c ≥ 3 ∈ Z and α ≥ 2 ∈ Z, with n ≫ α and n ≫ c, the average SRE is lower bounded by In particular, for constant c and α, it shows that Note that as α = 2, the lower bound of average SRE shows that the magic of random hypergraph states is nearly the maximal possible value n for any constant c ≥ 3 and sufficiently large qubit number n, which reproduces the result of Haar random states [38].
Furthermore, a more refined analysis can be conducted when focusing on 3-uniform hypergraph states.A direct simplification of Eq. (24) in this case is shown as follows.
Corollary 2. As c = 3, the second constraint in Eq. (24) of Theorem 3 can be simplified to The proof is straightforward.As c = 3, one only needs to sum over all possible q with |q| = 1 and |q| = 2 in Eq. (24).For |q| = 2, the set e 3 \ q only has one element, so the l 1 -norm is just for a single binary vector, which is already constrained by Eq. (23).As a result, the constraint is reduced to that for |q| = 1 as shown in Eq. (28).Based on this simplification, we give the following more accurate counting results.

Proposition 3. The average PL-moments of random 3-uniform hypergraph states are
The proofs of Propositions 2 and 3 are left in Appendix C.3 and D.1, respectively.

Variance analysis of magic
In this section, we further investigate the variance of PL-moments, which is helpful in constructing a sharp concentration result of SRE for random hypergraph states in the large n limit.In this way, we find that hypergraph states which are easier to prepare, typically own the maximal magic, similar to the Haar random states.The variance of PL-moment is defined as with the average PL-moment ⟨m α ⟩ E given in Eq. (19).The average PL-moment is investigated in the previous Sec.4.1, and hereafter, we focus on the analysis of the first term, i.e., the average of the 2-th moment of the PL-moment, and then combine them to show the final variance.
As an analog and extension of Theorem 3, for the general τ -th moment of the PLmoment, we transform the calculation of its average on random hypergraph state ensembles into the following counting problem.Theorem 4. For any integer α ≥ 2 ∈ Z, the τ -th moment of the α-th PL-moment of n qubit random hypergraph state ensembles E c shows where N (τ ) (c, α, n) is the number of 2-tuple (T , X), such that the following two constraints are satisfied.
where T is a 2α × n × τ rank-3 binary tensor, with its element 2α × n matrix denoted by , and the element of the i-th binary vector ⃗ t i ) is an n × τ binary matrix with elements x i,s .
Note that Theorem 4 reduces to Theorem 3 as τ = 1, and its proof is similar but more complicated, which is left in Appendix C.2.The counting problem here is more challenging even for τ = 2 of interest, compared to the one in Theorem 3 of Sec.4.1.Instead of showing a general result similar to Proposition 2, we focus on the case where α = 2 and c = 3 here and further give an upper bound for the variance of the PL-moment.Proposition 4. The variance of 2-order PL-moment on the random 3-uniform hypergraph state ensemble E 3 is The proof is by estimating the counting result in Theorem 4 for τ = 2, α = 2 and c = 3, and then combining with ⟨m 2 ⟩ E 3 in Eq. (29), and we leave it in Appendix D.2.Note that the standard variance here ) is exponentially small than its mean value ⟨m α ⟩ E 3 = O(2 −n ) as shown in Eq. (29).Consequently, by utilizing Chebyshev's inequality, there is the concentration of measure effect of magic for the E 3 ensemble.Corollary 3.For a hypergraph state |G n,3 ⟩ of n-qubit chosen randomly from the 3-unifrom hypergraph state ensemble E 3 , the probability that its SR 2 E larger than A similar concentration effect of magic is also observed for Haar random states [18,38,62].However, compared with Haar random states, E 3 are quite easy to prepare, i.e., just by randomly operating three-qubit CCZ gates.

Average magic with general probability p
In the previous two subsections, Sec.4.1 and Sec.4.2, we mainly focus on the magic properties of the hypergraph state ensemble E c .In this section, we extend the ensemble E c to E p c in Definition 3, where each c-edge is randomly selected by the probability parameter p.In particular, we focus on the c = 3 and α = 2 cases and show the following analytical formula for the average PL-moment for any n-qubit system.Theorem 5.The average 2-order PL-moment of the state ensemble E p 3 , i.e., random 3-uniform hypergraph state with each hyperedge selected by probability p, shows Here and ⃗ κ − , with each element κ ± 1≤i≤4 ∈ N being non-negative integer; the summation of the elements of ⃗ κ equals 4 i=1 (κ + i + κ − i ) = n, and the multinomial coefficient n ⃗ κ is short for the combinatorial number The proof is left in Appendix E. We remark that Theorem 5 could be extended to arbitrary constant c cases while the polynomial complexity of calculation holds since one still only needs to sum over some combinatorial number of the vector ⃗ κ.
Note that Eq. (29) of the previous p = 1/2 case is obtained by counting the number of solutions to f (⃗ κ) = 0.For the general p here, we do not have a very compact formula, and the function f (⃗ κ) and also ⟨m 2 ⟩ E p 3 in Eq. (36) look a little tedious.Nevertheless, the calculation of ⟨m 2 ⟩ E p 3 only involves polynomial complexity with respective to the qubitnumber n, that is, O(n 7 log n).This fact enables us to numerically study the relationship between the average magic and the probability p for quite large n, as shown in Fig. 3.
To be specific, Fig. 3 shows the relation between the expected number of hyperedges ⟨n e ⟩ := p • n 3 and the qubit-number n, given the average magic ⟨m 2 ⟩ E p 3 = γn for some constant γ.One can see that for a fixed proportion γ of n, ⟨n e ⟩ is almost linear to n for different γ's, i.e., ⟨n e ⟩ ∼ µn, and thus p ∼ O(n −2 ) far less than 1/2 like before.
For each vertex, the expected number of edges is  and thus the expected average degree ⟨ ∆(G)⟩ is about a constant.This shows the consistency to Theorem 2, where the magic of a bounded-average-degree hypergraph state is also bounded.For γ = 0.999, which is very near the maximal 1, the slope µ ≃ 3.0.It means that a very small p can let the average magic near the maximal value.
The statistical results here may also suggest a dynamical way to generate maximal magic states efficiently.For each step, one operates a CCZ gate on any three-qubit chosen randomly from a 3-edge, and repeats this process for about K = O(n) times.In particular, the numerical result implies that K = 3n may be enough to let SR 2 E reach 0.999n.Moreover, if one parallel applies CCZ gates, constant-depth (depth-3µ) quantum circuit could be sufficient (depth-9 for M 2 (|G⟩) ∼ 0.999n).

Magic of some symmetric hypergraph states
In this section, we investigate the magic of some hypergraph states with permutation symmetry.We focus on the c-complete hypergraph states with c = 3 and c = n, and our method can be applied to other symmetric hypergraph states.
The motivation for this study is two-fold.First, for any c-complete hypergraph, every vertex is connected to other n − 1 vertices, and thus the average degree being ∆(G c-com ) = n − 1.Consequently, c-complete hypergraph states can give some new insight beyond the one with a bounded average degree whose magic is constrained by Theorem 2. Second, the permutation symmetry significantly simplifies the summation of indices ⃗ x, ⃗ z in Eq. ( 14) from exponential to polynomial.This fact and the pictural magic formula introduced in Theorem 1 enable calculating the spectrum of the PL-components, and thus SR α E for any α.In particular, α needs not to be limited to integers as that in Sec. 4, for instance, here one can take α = 1/2.

Definition 4. An n-partite quantum state |Ψ⟩ owns permutation symmetry, if
Here π is the permutation element in the n-th order symmetric group S n .U {π} is an with {|a⟩} the basis state for a single-party.For example, U {(1,2)} is the swap operator on 2-party.
Due to the permutation symmetry, the PL-component Tr P ⃗ x,⃗ z |Ψ⟩ ⟨Ψ| is not relevant to the specific positions of single-qubit Pauli operators but only depends on the numbers of them, i.e., x i = 1 and z i = 1 and both x i = z i = 1 in P ⃗ x,⃗ z .Denote the corresponding sets of P ⃗ x,⃗ z as  For quantum hypergraph states whose PL-component is proportional to Tr U (G ⃗ x,⃗ z ) as shown in Proposition 1, if they own permutation symmetry, like c-complete hypergraph states, the induced hypergraphs G ⃗ x,⃗ z (and also G * ⃗ x,⃗ z ) are isomorphic to each other if the aforementioned sets share the same cardinality.This observation and the pictural magic formula in Theorem 1 make the calculations intuitive and elegant.
In the following, we first show the results of magic for 3-and n-complete hypergraph states, respectively, and then give a series of comments on them.We leave the detailed proofs in Appendix F. We show their induced hypergraphs G * ⃗ x,⃗ z in Fig. 4. Proposition 5.The 2-order and 1/2-order PL-moment of the quantum 3-complete hypergraph state of n-qubit Consequently, the corresponding Proposition 6.The 2-order and 1/2-order PL-moment of the quantum n-complete hypergraph state of n-qubit |G n-com ⟩ are for n ≥ 2. Consequently, the corresponding SRE M 2 (|G n-com ⟩) shows its maximal value log 32  11 ≈ 1.54 when n = 3, and decreases exponentially with n; M 1/2 (|G n-com ⟩) increases monotonously to 2 log 2 3 as n → ∞.
First, these two examples indicate that hypergraph states with an unbounded average degree could not own maximum magic, opposite to Theorem 2. For |G 3-com ⟩, SR 2 E is about a constant, even though there are extensive CCZ gates to generate it.This phenomenon also appears in entanglement analysis, where over-connected graphs and hypergraph states only own a small amount of entanglement.For instance, the 2-complete graph state is equivalent to the GHZ state, with entanglement entropy being 1 for any bipartition.
For |G n-com ⟩, its SR 2 E even scales like O(2 −n ) for the large n limit.This behavior can be understood via the fidelity to the initial stabilizer state |Ψ 0 ⟩ with zero magic.The n-qubit gate CZ n is nearly an identity operator for a large n, and the fidelity As such, G n-com should have small magic as being close to a stabilizer state.
Second, the SR 2 E and SR 1 2 E are very different for both states.In particular, M 1/2 (|G 3-com ⟩) is near n, however, M 2 (|G 3-com ⟩) is about a constant.This is mainly due to the distribution of the PL-component of |G 3-com ⟩.There are almost half of PL-components are non-zero, but their distribution is very sharp.That is, most of them are extraordinarily small, and on the contrary, only a constant number of them are larger as O(1).This phenomenon may lead to significant consequences in further one-shot manipulations [69,70].

Conclusion and outlook
In this work, we study the magic properties of quantum hypergraph states in terms of SRE.We find that the random hypergraph states typically have near-maximal magic, indicating an efficient way to generate them by dynamically applying few-qubit controlled phase gates.We also show a general upper bound of magic by the degree of the corresponding hypergraph and give a few analytical solutions to states with permutation symmetry.The general pictural formula of magic presented here can advance further explorations of manyqubit magic for large-scale systems.
From the results shown here, there are several directions to investigate further.First, it is direct to generalize the current study to other phase diagonal states [71][72][73], especially those generated by 2-qubit CZ θ gates.CZ θ gate is more feasible in practise than 3-qubit CCZ gate and still can create magic.Second, the permutation symmetry significantly simplifies the calculation of magic, leading to a few analytical results, and it is promising to generalize to other states beyond hypergraph states, for example, the Wstate [43].It is thus interesting to analyze the statistical properties of state ensembles with high symmetry [74].Third, there is a large gap of SR α E for different α's, and this indicates the significance of analyzing the spectrum of the PL-component, which may play an important role similar to the entanglement spectrum [75][76][77].And the implications of this phenomenon to one-shot quantum resource theory [70] and magic distillation of manyqubit states [78] also deserve further studies.Finally, it is known that typically random hypergraph states also have near-maximal entanglement [68].Thus the following important question arises-whether quantum states with near-maximal magic necessarily own a large amount of entanglement?The random states and permutation-symmetric states here give some positive support.Or, more broadly, what are the general relations and constraints between these two key quantum resources [23,79]?The answers to these questions would definitely unveil some ultra-quantum features and benefit quantum information processing.

A Related Proofs of the general formula of magic for hypergraph states A.1 Proof of Observation 1
We first prove some commuting relation that for any set e ′ ⊆ e by induction.When there is only one element in e ′ , it is just the formula X i CZ e X i = CZ e • CZ e\{v i } .Suppose for some e ′ such that Eq. (42) holds, then for any e ′′ = e ′ ∪ {v i 0 } with e ′ ⊂ e ′′ ⊆ e, one has where the second line is by the induction assumption, and the third line is by the singleelement case.Consequently, Eq. (42) holds for any set e ′ ⊆ e.
It is clear that the stabilizer generators of the initial state |Ψ 0 ⟩ = |+⟩ ⊗n are X i .It is direct to check that S i = U (G)X i U † (G).Then we calculate the product of stabilizer generators as Here the second line we insert many additional i X s i i for each CZ e .Note that in the third line, nontrivial contributions only come from v i ∈ (v(⃗ s) ∩ e), otherwise X i commutes with CZ e .The fourth line is by Eq. (42).

A.2 Proof of Proposition 1
To proceed with the proof, we first show the following simple but useful result about the action of multi-qubit Pauli X gates with any phase unitary U Z = e iθ(⃗ a) |⃗ a⟩ ⟨⃗ a|, which only introduces phase on the computational basis {|⃗ a⟩} for some function θ of the n-bit string ⃗ a.

Lemma 1. Let
Moreover, if U Z 1 and U Z 2 are both phase unitary, Proof.The proof is straightforward by definition.
where U Z only introduces some phase θ(⃗ a) on |⃗ a⟩, and It is clear that CZ e gates used to generate hypergraph states in Eq. ( 1) are phase gates, and we can apply Lemma 1 to calculate its PL-component as follows.By using the stabilizer decomposition of |G⟩ in Eq. ( 2), ( 3) and (4), and the definition of Pauli operator in Eq. ( 6), one has where the third line is by Lemma 1 inducing the constraint δ ⃗ x,⃗ s , and the final line is by the commuting relation indicates that one should only select the stabilizer generator of i S s i i in the summation exactly according to ⃗ x information of the Pauli operator P ⃗ x,⃗ z .
The operator in the trace of the final line of Eq. ( 49) is indeed a phase unitary denoted as which can be represented by a hypergraph G ⃗ x,⃗ z defined in Eq. ( 11) by considering 1hyperedges and other hyperedges separately, similar as in the definition of hypergraph state in Eq. (1).Notice that ν(⃗ x) is the number of e ⊆ v(⃗ x), i.e., it counts the number of q = e.In this case, CZ e\q = CZ ∅ which gives a −1 phase by definition in the second line.Since both Tr P ⃗ x,⃗ z |G⟩ ⟨G| and Tr U (G ⃗ x,⃗ z ) are real numbers, the phases are not essential and by Eq. (49) we finally have B Proof of Theorem 2: upper bound of magic from bounded average degree First, we find the subset Ṽ⃗ x by deleting zero-degree vertices in V .That is, all the vertices now has some neighbour, ⃗ x , v i ∈ e}, which is only determined by E  If there is some vertex v j ∈ V \ Ṽ⃗ x with a 1-hyperedge on it, i.e., there is some Pauli Z operator outside Ṽ⃗ x , one has As a result, we should choose Now by fixing ⃗ x and summarizing all nontrivial vectors ⃗ z This indicates that Let W (⃗ x) = w be the weight of ⃗ x, i.e., the number of ⃗ x , it is bounded by the number of vertices adjacent to those v i with In the third line, we apply the convex property of function f (z) = 1 2 (2α−1)z , and the relation where the degree of each vertex has been summed totally w times.The final line is by the binomial formula.
Finally, the SR α E is bounded by for α ≥ 2.

C Proofs of general formulas for statistical properties of PL-moment C.1 Proof of Theorem 3
To distinguish among different c-uniform hypergraphs, we assign a binary vector n,c .The value of B ec decides whether the hyperedge e c belongs to the edge set of G n,c .The first step is to write m α on 2α-replica [38,48], where the local term Λ (α) i is for the 2α-replica of i-th qubit and where ρ = ρ * for hypergraph states.In this step, the 2α-degree formula is transformed into a 2α-replica tensor, which enables the interchange of the order of the summation of Pauli operators and the trace.Similarly, we can also interchange the order in the calculation of the average PL-moment where the second line is by Eq. (59).The average of the α-fold density matrix in the final of Eq. (62) shows where in the second line we insert the stabilizer decomposition of the hypergraph state.
Here the index r labels all 2α-replica, and the binary vector ⃗ x (r) labels the choice of S i of each qubit for the r-th replica.Without causing any ambiguity, we use St(G n,c , ⃗ x (r) )⊗• • •⊗I for 2α-replica and change the tensor product to the matrix product.Similar simplifications will also be used in the following discussion without further explanations.By using Observation 1, we further get (64) Here in the final line, we put the summation of hypergraph configuration ⃗ B into the phasegate part, since the first Pauli-X part is irrelevant to ⃗ B. This indicates that we can separate the Pauli-X part and the phase-gate part.
As shown in Eq. ( 63), the average α-fold density matrix ρ (α) is a linear combination n,c , ⃗ x (r) ) indexed by {⃗ x (r) } 2α r=1 .By focusing one specific term and inserting the result in Eq. (64), we get Here similar as before we separate the X-part and phase-gate part, and we change the order of the product in the X-part.Due to Lemma 1, the whole trace is nonzero iff It indicates that all the vectors ⃗ x (1) , • • • , ⃗ x (2α) should be identical, which is denoted by ⃗ x in the final line in Eq. (65).The physical meaning is that, to make nonzero contribution, the choice of stabilizers should be the same among 2α replicas.For short, we use C 0 to represent and In the final line, it is in fact an interchange of index: Since C 0 and C 1 are both phase matrices in the Z basis, the trace of their multiplication can be written as where ) is a 2α × n binary matrix denoting the bit string in Z basis whose elements are t j,i .Each row of T represents one replica, and each column represents a corresponding qubit.The first term ⟨T| C 0 |T⟩ is either 0 or 1.It equals to 1 iff ⟨T| On the other hand, the second term for all e c , i.e., q⊆v(⃗ x)∩ec CZ ec\q where v i ∈q x i is by q⊆v(⃗ x) and v k ∈ec\q ⃗ t k 1 is for the phase gate.Here we ignore the case q = e c since in this case CZ ec\q ⊗2α = (−1) 2α = 1.These two conditions are just that of Eq. (23) and Eq.(24) in main text.
Finally, plug Eq. ( 63) and the final line of Eq. (65) into Eq.(62) to get It is clear that now the average is just related to the number of possible ⃗ x and T satisfying conditions (69) and (71), and we finish the proof.

C.2 Proof of Theorem 4
Similar to that of Theorem 3 in Sec.C.1, the first step is to write the moment of the PL-moment in the following replica form.
In this way, we only need to replace α with τ α in Eq. ( 63) and Eq. ( 64), and replace in Eq. (65) to get (ignoring the 2 −2τ αn prefactor) where and where With a similar argument as that in Eq. (68), one can transform these phase matrices to the conditions Eq. (32) and Eq.(33) according to the expression of C 0,s and C 1 (X) respectively.

C.3 Proof of Proposition 2
We prove the result mainly by recursion.First define a series of more general conditions C ξ with parameter ξ ∈ [c] describing the cardinality of the hyperedge.
Notice that as in main text here q ̸ = ∅ unless otherwise specified.It is clear the condition Eq. ( 24) is just C ξ=c .
We further add the case q = ∅ and define another similar series of conditions C * ξ for ξ ∈ [c] as Eq. ( 79) results in the following powerful restriction on the elements in C * ξ .For any ξ−1 , suppose their corresponding ⃗ t are the same, i.e., ⃗ t i 1 = ⃗ t i 2 , then consider two sets e (1) ξ−1 and e (2) (80) In the first line, we use the precondition that C * ξ is satisfied.In the third line, we use the fact that ⃗ t i 1 = ⃗ t i 2 so those two summations in differ only in x i 1 and x i 2 .The final line is by Eq. (79).As a result, Eq. (80) shows that given T, all the vertices except those in e (0) ξ−1 share the same valid vector ⃗ t, should have the same value of x i .
Similarly, for any element (T, ⃗ x) in ∆C ξ , one can use the same strategy to find the same result starting from the second line of Eq. (80).That is, given T, all the vertices except those in e (0) ξ−1 which share the same valid vector ⃗ t, should take the same value of x i .Now fix T, consider ⃗ x such that (T, ⃗ x) ∈ ∆C * ξ .There are totally 2 2α−1 possible ⃗ t satisfying Eq. ( 23), the number of choices of ⃗ x is no more than 2 ξ−1 • 2 2 2α−1 .Here 2 ξ−1 denotes the number of choices of x i whose v i ∈ e (0) ξ−1 in Eq. (79). 2 2 2α−1 denotes the number of choices of other x i outside e (0) ξ−1 , which should be the same for the same-valued ⃗ t i .Therefore, by counting all 2 (2α−1)n possible T, the size of ∆C * ξ is One can similarly get For the initial condition when ξ = 2, C * 2 becomes where we use the fact that ⃗ t i 1 = 0.Although it gives no limitation on ⃗ x, it limits the valid T. Indeed, this is a matching problem, so there are no more than Then by recursion, one has and for α ≥ 2.
We remark that in the final step, we use (2α − 1)!! ≤ 4 • 2 2 2α−1 as α ≥ 2 for simplicity.But such inequality is quite loose, so one can use the original value to give a relatively tighter bound.

D Proofs of the concentration of magic for random 3-uniform hypergraph states
All the proofs in this section are based on the simplified constraint shown in Corollary 2.

D.1 Proof of Proposition 3
When c = 3, we inherit the spirit of the proof of Proposition 2 in Sec C.3 by considering C * 2 in Eq. (83) and C 3 in Corollary 2. Suppose one finds two vectors satisfying ⃗ t i 1 ⊙ ⃗ t i 2 1 = 1, then besides i 1 , i 2 , the x i whose corresponding ⃗ t i share the same-valued of vector, should also be the same.
In fact, we go one step further to show that these result also applies to i 1 , i 2 as follows.To be specific, suppose one finds two vectors satisfying ⃗ t i 1 ⊙ ⃗ t i 2 1 = 1, and also ⃗ t i 3 = ⃗ t i 1 , then focus on the edge e 3 = {v i 1 , v i 2 , v i 3 }, Eq. (28) shows On the other hand, if there is no such pair with ⃗ t i 1 ⊙ ⃗ t i 2 1 = 1, there is no constraint on ⃗ x at all.
We finally have the upper bound (90)

D.2 Proof of Proposition 4
From Theorem 4, one can directly derive the corresponding counting problem when α = 2 and c = 3: ⃗ t Following the proof of Proposition 2 in Sec.C.3 and Proposition 3 in Sec.D.1, similar conditions can be derived in the same way.a) Suppose we can find v k 1 and v k 2 such that ⃗ t (1) Then for any other two v i 1 and v i 2 with the same ( ⃗ t (1) , ⃗ t (2) ), consider 3-hyperedges (92) This indicates that x i,1 corresponding to the same ( ⃗ t (1) i , ⃗ t (2) i ) should be the same except for v k 1 and v k 2 .With a similar analysis as that in Appendix D.1, one can remove the exception by considering e One can derive a similar conclusion: x i,2 corresponding to the same ( ⃗ t (1) i , ⃗ t (2) i ) should be the same.c) Suppose we can find v k 1 and v k 2 such that ⃗ t (1) One can derive a similar conclusion: x i,1 + x i,2 corresponding to the same ( ⃗ t (1) i , ⃗ t (2) i ) should be the same.
Any two of the conditions in cases a), b) and c) could lead to the conclusion that the (x i,1 , x i 2 ) whose corresponding ⃗ t i (1) , ⃗ t i (2) share the same-valued of vector, should also be the same.In this way, the number of choices of X is no more than 4 2 according to the analysis in Sec.D.1.Also, if only one of the conditions in cases a), b) and c) is satisfied, the number of choices of X should be no more than ≡ 0 so the number of choices of T (2) is 3 • 2 n − 2. In this case, since the number of choices of T (1) is no more than 4 n − 3 • 2 n + 2, the total number of (T , X) is upper bounded by 4 . The same result holds for that in case b).When only the assumption in case c) is satisfied, the total number of T is no more than 15 • 4 n , where 15 comes from all possible choices of 4 different ( ⃗ d (1) , ⃗ d (2) ) such that only c) is satisfied.When any two of the conditions in cases a), b) and c) are satisfied, the number of possible T is no more than (4 n − 3 • 2 n + 2) 2 .With the above conditions, by considering the equivalence between ⃗ d and ⃗ 1 − ⃗ d one can obtain in total Then one gets with the final line by Chebyshev's inequality.

E Proof of Theorem 5: average magic for non-uniform ensembles
We can directly follow through the proof of Theorem 3 to Eq. (65), Eq. (66) and Eq.(67).
The only difference is to replace the two coefficients 1 2 with (1 − p) and p in the expression of C 1 (⃗ x) to get where the term 2 n comes from the ⃗ t ⇔ ⃗ 1 − ⃗ t equivalence.
F Proofs of the magic of hypergraph states with permutation symmetry F.1 Proof of Proposition 5 Our proof is mainly based on the magic formula in Eq. (16).With the permutation symmetry, the structure of the induced hypergraph G * ⃗ x,⃗ z is only determined by m = |A x |, m 1 = |A z,x | and m 0 = |A z,−x |.There are two cases determined by the parity of m, as shown in Fig. 4 (a).a) m is odd.In this case, if both v j , v k ∈ A x or v j , v k ∈ A −x , the number of v i such that {v i ,v j ,v k }∈E x i = 0. Thus, the edge set E You Zhou : you_zhou@fudan.edu.cnAccepted in Quantum 2024-05-09, click title to verify.Published under CC-BY 4.0.
[n]} and a hyperedge set E = {e j |j ∈ [m], e j ⊆ V, e j ̸ = ∅}.Here [n] = {1, 2, • • • , n} and [m] = {1, 2, • • • , m} are the index sets of vertices and hyperedges, respectively.A hyperedge e can connect c ≥ 1 vertices, denoted as |e| = c, and we call it a c-edge.There are totally n c=1 n c = 2 n − 1 possible hyperedges.If ∀e ∈ E, |e| = c, the hypergraph is called c-uniform hypergraph.And a c-complete hypergraph contains all such c-edges.See Fig. 1 (a) for an instance of a hypergraph with six vertices and four hyperedges.

( 1 )Proposition 1 .
⃗ x,⃗ z in Eq. (11).See Fig. 1 (b) for an illustration of an induced hypergraph.The corresponding phase unitary is denoted as U (G * ⃗ x,⃗ z ).The following result relates the PL-component of any hypergraph state to the induced phase unitary.Given a hypergraph state |G⟩, the square of the PL-component respective to the Pauli operator P ⃗ x,⃗ z shows

Observation 2 .
one has the following general observation.For an n-partite quantum state |Ψ⟩ owning permutation symmetry defined in Def. 4, its PL-component Tr P ⃗ x,⃗ z |Ψ⟩ ⟨Ψ| is directly related to the cardinality of the sets m = |A x |, m 1 = |A z,x | and m 0 = |A z,−x |, but not the specific positions.That is, two components take the same value if these sets share the same cardinality.

Figure 4 :
Figure 4: Illustration of induced hypergraphs for c-complete hypergraph states.Here the shaded areas label the corresponding sets.Due to the permutation symmetry, the induced hypergraph G * ⃗ x,⃗ z defined in Eq. (11) and (12) is determined by |A x |, |A z,x | and |A z,−x |.Since E (1) ⃗ z in Eq. (12) only introduces 1-edges, for simplicity, we take ⃗ z = 0 and thus |A z,x | = |A z,−x | = 0.In this way, the structure of G * ⃗ x, ⃗ 0 only depends on m = |A x |.(a) 3-complete hypergraph with 8 vertices: G * ⃗ x, ⃗ 0 is related to even/odd of m, and it is either two complete-graphs of A x and A −x or complete bipartite graph between them.(b) n-complete hypergraph with 6 vertices with m = 2: G * ⃗ x, ⃗ 0 in this case contains all the hyperedges A −x ⊆ e ′ ⊂ V .See Appendix F for more detailed discussions.

( 2 )
⃗ x and thus ⃗ x.In this way, for each induced hypergraph G * ⃗ x,⃗ z , we define the reduced hypergraph G * ⃗ x,⃗ z = { Ṽ⃗ x , Ẽ * ⃗ x,⃗ z }, and here Ẽ * ⃗ x,⃗ z is just maintained by reducing E * ⃗ x,⃗ z on the subset Ṽ⃗ x .Actually, this step reduces all the vertices that are not connected to other vertices and delete all the related 1-hyperedges in E (1) ⃗ z .

)
For convenience, we also use C ξ (or C * ξ ) to denote all the 2-tuples (T, ⃗ x) satisfying C ξ (or C * ξ ) and the parity constraint in Eq. (23) in main text.Denote N ξ = |C ξ | (and N * ξ = |C * ξ |) as the number of them, so N(c, α, n) = N c in Theorem 3. Furthermore, define ∆C * ξ as the set of all the (T, ⃗ x) satisfying condition C * ξ but not satisfying condition C * ξ−1 and denote ∆N * ξ = |∆C * ξ |.Similarly, define ∆C ξ as the set of all the (T, ⃗ x) satisfying condition C ξ but not satisfying condition C * ξ−1 and denote ∆N ξ = |∆C ξ |.Notice that in the definition of ∆C ξ , the set is C * ξ−1 but not C ξ−1 .Then we directly have the inequalities of the set cardinality N * ξ ≤ N * ξ−1 + ∆N * ξ and N ξ ≤ N * ξ−1 + ∆N ξ , and we give the estimation of both ∆N * ξ and ∆N ξ as follows.By definition, for any element (T, ⃗ x) in ∆C * ξ , one can find a set e (0)
|1⟩), the phase unitary U (G) is completely determined by the hypergraph G, and CZ e = v i ∈e I i − 2 v i ∈e |1⟩ i ⟨1| the generalized Controlled-Z gate acting non-trivially on the support of edge e.Quantum hypergraph states are generally not traditional stabilizer states since CZ e are not Clifford gates as |e| > 2. Fortunately, one can still apply the generalized stabilizer formalism as follows.
) ) is nonzero, ⃗ x can take all the 2 n values.On the other hand, there exists vector-pair whose Hadamard product is 1, and the x i should be the same if the corresponding t i are the same.We thus usex ⃗ d 0 , x ⃗ d 1 , x ⃗ d 2 , x ⃗ d 3 to denote the possible values, if their corresponding n( ⃗ d) is nonzero.Actually, we find that there are only 4 possible ⃗ x in any of these cases.First, x ⃗ d 0 must keep zero, so the existence of ⃗ d 0 in When α ≥ 3, there are totally 2 2α−2 nonequivalent ⃗ d.On the one hand, if all⃗ t i 1 ⊙ ⃗ t i 2 1 = 0 with ⃗ t i 1 , ⃗ t i 2 ∈T, there is no constraint on ⃗ x, which is actually C * 2 in Eq. (83) with couting result in Eq. (84).On the one hand, if ⃗ ̸ = 0, it is not hard to see that as long as x ⃗ d and x ⃗ d ′ are fixed, all the other x i are also determined.To be specific, considering e 3 4 • 2 n .Otherwise, one should count all the 4 n different X.