Clifford Group and Unitary Designs under Symmetry

We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design. Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup. We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries. Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group. This work will open a new perspective into quantum information processing such as randomized benchmarking, and give a deep understanding to many-body systems such as monitored random circuits.


I. INTRODUCTION
Randomness in quantum systems is a ubiquitous concept that underpins the core of quantum information processing and quantum many-body systems [1,2].The uniform randomness plays central role not only in understanding fundamental phenomena such as thermalization [3,4] and information scrambling [5,6], but also realizing efficient quantum communication [7,8] and encryption [9,10].To utilize the beautiful and powerful property of the randomness, there have been significant advancements in engineering of approximations of the Haar unitary ensemble, namely the unitary design.A unitary t-design is an ensemble of unitaries that mimics the Haar random unitaries up to the t-th moment, and it has proven useful in tasks such as data hiding [11], quantum state discrimination [12,13], quantum advantage [14][15][16], quantum gravity [2], to name a few.
One of the most prominent examples of unitary designs is the Clifford group.Initial interest in the Clifford group was primarily in the context of quantum computing, e.g., the classical simulability [17,18].However, currently it is known to be applicable to even wider fields such as the quantum state tomography [19] and hardware verification via randomized benchmarking [20][21][22][23].Although for general qudits, the Clifford group is only a unitary 1-design [24], it elevates to a unitary 2-design if the local Hilbert space dimension is prime [11,25,26].Intriguingly, the multiqubit Clifford group singularly qualifies as a unitary 3-design [27,28].
While the concurrent presence of classical simulability and the pseudorandomness of the multiqubit Clifford group has invoked numerous applications to quantum science [29][30][31][32], we point out that existing studies have focused predominantly on the full ensemble of unitary designs; our comprehension on realistic scenarios with operational constraints/restrictions remains underdeveloped.One of the most outstanding questions pertains to the relationship with symmetry, an essential concept responsible for a wealth of phenomena in the natural sciences.To further explore the physics and quantum information processing under realistic constraints, it is an urgent task to establish how the symmetry impacts the Clifford group.
In this work, we introduce the concept of a symmetric unitary t-design and prove that the multiqubit Clifford group under symmetry forms a symmetric unitary 3-design, if and only if the symmetry constraints are essentially characterized by some Pauli subgroup.We also propose a complete and unique method for constructing symmetric Clifford operators with elementary quantum gates that operate on a maximum of two qubits.We subsequently show that, other classes of symmetry stand in stark contrast, as the Clifford groups under these symmetries are merely symmetric unitary 1-designs.For comprehensive understanding, we have highlighted such a remarkable disparity through practical examples of U (1) and SU(2) symmetries.Finally, we provide numerical evidence for our findings by computing the frame potentials for the Clifford groups under two representative types of symmetries.

II. SETUP
We first overview the conventional Clifford group and unitary designs.The Clifford group on N qubits is defined as the normalizer of the Pauli group in the unitary group U N , i.e., C N := {U ∈ U N |U P N U † = P N }, where P N := {±1, ±i} • {I, X, Y, Z} ⊗N is the group generated by the Pauli operators I, X, Y and Z on each qubit.It is convenient to introduce the t-fold twirling channel to characterize the randomness of a subgroup X of U N as Φ t,X (L) := U ∈X U ⊗t LU †⊗t dµ X (U ), (1) where L is a linear operator acting on tN qubits and µ X denotes the normalized Haar measure on X .We say that the subgroup X is a unitary t-design if Φ t,X = Φ t,U N . ( We note that the definition of unitary designs can be extended for general subsets of the unitary group by considering a distribution on the sets [33], and that our main statement is invariant under the extended definition, as we show in Appendix F. From Eq. ( 1), we see that unitary t-designs with larger t better approximate the Haar random unitaries, which can be regarded as a unitary ∞design.In this regard, it is known that the Clifford group C N is a unitary 3-design but not a 4-design [27,28].Note that unitary t-designs are always t ′ -designs if t > t ′ , but the contrary does not hold in general.
The symmetric Clifford group and symmetric unitary designs are defined as the symmetric generalizations of the conventional ones.In the following, we consider symmetry that can be represented by a subgroup G of U N .We define the G-symmetric Clifford group on N qubits as the group consisting of the Clifford gates commuting with all the elements in G.We can give a rigorous definition as follows: Definition 1. (Symmetric Clifford group.)Let G be a subgroup of U N .The G-symmetric Clifford group C N,G is defined by with the G-symmetric unitary group Now it is natural to define for a subgroup X of U N to be a G-symmetric unitary design if the subgroup approximates the G-symmetric unitary group U N,G .The rigorous definition is as follows: Definition 2. (Symmetric unitary designs.)Let G and X be subgroups of U N .X is a G-symmetric unitary tdesign if the t-fold twirling channel Φ t,X satisfies Note that in these definitions, the symmetry constraint is described by U N,G rather than by G itself, and the conventional definitions are included as the special case when the symmetry is trivial, i.e., G = {I}.

III. MAIN RESULTS
Now we are ready to present our two main results.The first one is the description of the randomness of symmetric Clifford groups in terms of symmetric unitary designs, which we rigorously present in Theorem 1.The second one is the complete and unique construction of symmetric Clifford circuits with elementary gates, which we concisely state in Theorem 2.

A. Characterization of pseudorandomness in symmetric Clifford groups
We prove that the G-symmetric Clifford group C N,G is a G-symmetric unitary 3-design if and only if the symmetry constraint by G is essentially described by some Pauli subgroup.This can be rigorously stated as follows: Theorem 1. (Randomness of the Clifford group under symmetry.)Let G be a subgroup of U N .Then, C N,G is a G-symmetric unitary 3-design if and only if U N,G = U N,Q with some subgroup Q of P N .
This theorem provides a guarantee that, under a Pauli symmetry, the symmetric Clifford group maintains its pseudorandomness, which is applicable to various quantum information processing tasks [11][12][13][14][15][16].Moreover, it is remarkable that this theorem also states that the Clifford group maintains the pseudorandomness under symmetry only if the symmetry can be characterized by some Pauli subgroup.We note that symmetry constraints can be captured by U N,G without using G itself, because when two subgroups G and G ′ of U N satisfy U N,G = U N,G ′ , the Gand G ′ -symmetric Clifford groups are identical to each other, and moreover the notions of Gand G ′ -symmetric unitary designs are the same.We do not directly present the condition for G itself, because there are cases when We illustrate how we can use this theorem to know whether symmetric Clifford groups are symmetric unitary 3-designs by taking the following three physically important examples: These groups are isomorphic to Z 2 , U(1) and SU(2), respectively, which appear ubiquitously in quantum systems; first-principles description of electronic structures, atomic, molecular, and optical physics, quantum spin systems, and lattice gauge theory, to name a few.When G is given by Eq. ( 6), G-symmetric Clifford group is a G-symmetric unitary 3-design, because G itself is a Pauli subgroup.In contrast, when G is given by Eq. ( 7) or (8) with N ≥ 2, G-symmetric Clifford group is not a Gsymmetric unitary 3-design, because in these cases U N,G cannot be expressed as U N,Q with any Pauli subgroups Q.We note that when G is given by Eq. ( 7) or (8) with N = 1, G-symmetric Clifford group is again a Gsymmetric unitary 3-design.In fact, G itself is not a Pauli subgroup, but U N,G can be expressed as U N,Q with a Pauli subgroup Q = {I, Z} or P 1 .We can prove that there is no Pauli subgroup Q such that U N,G = U N,Q when G is given by Eq. ( 7) or (8) with N ≥ 2 as follows: First, we suppose that U N,G = U N,Q with some Pauli subgroup Q.Second, we note that we always have Q ⊂ U N,U N,Q .Third, the qubit permutation group S satisfies S ⊂ U N,G , which implies that U N,U N,G ⊂ U N,S .By these three relations, we get , where X (j) is the Pauli X operator on the jth qubit.However, this contradicts with Eq. ( 7) as well as with Eq. (8).
We emphasize that we can completely characterize the randomness of the examples in terms of unitary designs, i.e., we can clarify the maximal t such that C N,G is a G-symmetric unitary t-design.In fact, as expected from the non-symmetric case, we can prove the no-go theorem for G-symmetric unitary 4-designs except for the most constrained case of U N,G = {e iθ I|θ ∈ R}, which we will describe in Theorem 4; generally we have t max = 3 for Pauli symmetry.On the other hand, when G is given by Eq. ( 7) or (8) with N ≥ 2, we get t max = 1, which we will describe in Theorem 3. Note that the single-qubit case is special since we have t max = 3, ∞ for Eq. ( 7) and (8), respectively.This is because the symmetry constraint can be written by Pauli subgroup {I, Z} in the case of Eq. (7), and the G-symmetric Clifford operators are restricted to the identity operator up to phase in the case of Eq. (8).We finally remark that we cannot increase t max by considering a nonuniform mixture in the definition of unitary designs, which we show in Appendix F.
While we guide readers to Appendix B for details on the derivation, it is informative to provide a brief sketch on the proof.In the proof of the "if" part, it is sufficient to show that C N,Q is a Q-symmetric unitary 3design for all Pauli subgroups Q.We explicitly construct a map D with a certain class of symmetric Clifford operators and show that the twirling channels satisfy Φ 3,C N,G = Φ 3,U N,G = D by considering the fixed-points of Φ 3,C N,G and Φ 3,U N,G .We emphasize that the nontrivial and technical contribution of Theorem 1 resides in the "only if" part.Namely, if C N,G is a G-symmetric unitary 3-design, then there exists a Pauli subgroup Q such that U N,G = U N,Q .Concretely, we construct Q as the group generated by the set Q ′ := {Q ∈ {I, X, Y, Z} ⊗N |∃G ∈ G s.t.tr(GQ) ̸ = 0}, where A ⊗n := A ⊗ A ⊗n−1 and A ⊗ B := {A ⊗ B|A ∈ A, B ∈ B} for general operator sets A and B. The inclusion U N,G ⊃ U N,Q directly follows from span(G) ⊂ span(Q), because for any G ∈ G, every Pauli basis in G with a nonzero coefficient is in-cluded in Q.However, the proof of the inverse inclusion U N,G ⊂ U N,Q requires some technical lemmas (see Appendix B 2).We consider the function U → U QU † from U N,G to U N for arbitrary taken Q ∈ Q ′ and show that it is a constant function.

B. Construction of symmetric Clifford groups
From the viewpoint of algorithms and experiments, it is crucial to give an explicit construction for the symmetric Clifford operators.In fact, for a Pauli symmetry, we show that the set of symmetric Clifford operators considered in the proof of Theorem 1 actually form a complete and unique expression of the symmetric Clifford operators (see Fig. 1 (a)).We will later discuss the case for non-Pauli symmetry.This is a symmetric extension of the result in Refs.[34,35], where they showed that the standard Clifford operators can be uniquely decomposed by elementary gate sets.
As a preparation for stating the theorem, it is crucial to mention that every Pauli subgroup naturally gives a decomposition into three parts.Concretely, we note that any Pauli subgroup Q can be transformed into the form by some Clifford conjugation action up to phase, i.e., P 0 W QW † = R with some W ∈ C N , where P 0 := {±1, ±i}.We denote the subsystem of N k qubits by A k (k = 1, 2, 3) and the set of indices representing the qubits in A k by Γ k .We can get N 1 , N 2 , N 3 , and W by considering the following two types of induction processes.Let Q be a Pauli subgroup on n qubits.The process is to take a Pauli subgroup to phase with some Clifford operator W ′ .We can conduct the first type of induction process while Q has noncommutative pairs of elements, and the second type of process while Q ̸ = {I} up to phase, as we show in Lemma 14 in Appendix G. N 1 and N 2 are given as the numbers of the first and the second induction processes, respectively, and We can get the Clifford operator W by taking the product of the Clifford operators W ′ in all the induction processes.By using these notations, we can present the following theorem: Theorem 2. (Complete and unique construction of the Clifford group under Pauli symmetry.)Let Q be a subgroup of P N .Then, there exists some W ∈ C N and R in the form of Eq. (9) such that P 0 W QW † = R, and every Q-symmetric Clifford operator U can be uniquely expressed as Fig. 1(a) as with µ j ∈ {0, 1, 2, 3}, ν j,k ∈ {0, 1}, V ∈ C N3 and P j ∈ {I, X, Y, Z} ⊗N3 , where S (j) is the S gate on the jth qubit, CZ (j,k) is the controlled-Z gate on the jth and kth qubit, V acts on the subsystem A 3 , and C(P j ) (j,Γ3) is the controlled-P j gates with the jth qubit as the control qubit and the qubits in the subsystem A 3 as the target qubits, and T means the ordered product, i.e., T The complete and unique expression by Eq. (10) gives an efficient way to generate all the elements of a Qsymmetric Clifford group.In fact, we can understand that it is much more efficient than choosing symmetric elements from the entire Clifford group.Namely, the size of the quotient group of the symmetric Clifford group C N,G divided by the freedom of phase U 0 := {e iθ |θ ∈ R} is given by where we used the fact that there are 4, 2, |C N3 /U 0 | and 4 N3 choices for each µ j , ν j,k , V , and P j , respectively, and . This is much smaller than the size |C N /U 0 | ∼ 2 2N 2 +3N of the entire Clifford group.The reduction rate is exponential with N in a standard setup where N 1 and N 2 are O(1) [37,38], which highlights the significance of the explicit construction of symmetric Clifford operators.We can see that each qubit in A 1 , A 2 and A 3 contributes as 0, 1/2 and 1 qubit in the estimation of the size |C N,G /U 0 |.When we ignore the phase degree of freedom, the size |C N,G /U 0 | of the Gsymmetric Clifford group on N qubits is almost the same as the size |C N2/2+N3 /U 0 | of the entire Clifford group on N 2 /2 + N 3 qubits.
We illustrate the construction of Pauli-symmetric Clifford operators by taking the symmetry Q = {I ⊗4 , X ⊗4 , Y ⊗4 , Z ⊗4 } on four qubits as an example, which appears as the symmetry of the XYZ Hamiltonian with arbitrary connectivity.We know from Theorem 2 that every Q-symmetric Clifford operator U can be uniquely expressed as Fig. 1 (3,4) CNOT (1,2) .We can confirm that the symmetry constraint greatly reduces the size of the Clifford group by seeing that |C 4,Q /U 0 | ∼ 10 8 and |C 4 /U 0 | ∼ 10 13 .Such a striking difference is displayed in more depth in Fig. 2. Here, we indeed find that the existence of Pauli symmetry leads to exponential reduction of |C N,G /U 0 |.As can be seen from Eq. ( 11), we can understand that the entire curve is shifted by N 1 + N 2 /2 in the asymptotic limit, which gives the advantage of using the construction method presented in Theorem 2.
While we leave the detailed proof of this theorem to Appendix C, we here provide the proof sketch.It is sufficient to consider the construction of C N,R with a specific class of Pauli subgroups R given by Eq. ( 9), because there exists some Clifford conjugation action W • W † that gives one-to-one correspondence from C N,Q to C N,R for general Pauli subgroups Q.In the proof of the completeness, the key is to take the Heisenberg picture, i.e., to see how the conjugation action of a unitary operator transforms Pauli operators.For arbitrary U ∈ C N,R , U satisfies U Z (j) U † = Z (j) and U X (j) U † = X (j) for all j ∈ Γ 1 , and U Z (j) U † = Z (j) for all j ∈ Γ 2 .We can inductively construct U ′ such that U ′ U is in the form of Eq. (10), and U ′ Z (j) U ′ † = Z (j) and U ′ X (j) U ′ † = X (j) for all j ∈ Γ 1 and j ∈ Γ 2 .This implies that U ′ is a Clifford operator acting nontrivially only on the subsystem A 3 , and thus U ′ is in the form of Eq. (10).Since U can be written as U = U ′ † (U ′ U ), and both U ′ and U ′ U is in the form of Eq. ( 10), we know that U is in the form of Eq. (10).We prove the uniqueness by the proof by contradiction.Namely, we take arbitrary U ∈ C N,R and suppose that there are two different sets of (µ j , ν j,k , V, P j ) that realize U , and show that they must coincide with each other.

IV. U(1) AND SU(2)-SYMMETRIC CLIFFORD GROUPS
As prominent examples of non-Pauli symmetries, we clarify the property of the G-symmetric Clifford group when G is given by Eq. ( 7) or (8) on multiple qubits.Concretely, in these cases, the G-symmetric Clifford group N < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 P M x b e Q p j 4 B m 4 u L H A = = < / l a t e x i t > (0, 0, N) < l a t e x i t s h a 1 _ b a s e 6 4 = " C 6 E C Z 4 s u 7 5 m e 9 + + V Size of symmetric Clifford groups |CN,G/U0| under various Pauli symmetries.Here, we show the scaling for symmetries such that the numbers of qubits in A1, A2, A3 are given as (N1, N2, N3) = (2, 0, N − 2), (0, 4, N − 4), and (0, 6, N − 6).C N,G is a G-symmetric unitary 1-design, but not a 2design.This property characterizes the randomness of the symmetric Clifford group under U(1) and SU(2) symmetries given by Eqs. ( 7) and (8).We rigorously present this statement as a theorem.
Let us remark that, as for 2-designs, we can actually show no-go theorems in a more general class of symmetries, which are described as a tensor product of representations of a nontrivial connected Lie subgroup of a unitary group.This type of symmetry represents the conservation of the total observables on the system.In the proof of 1-designs and the disproof of 2-designs, we use the proof idea of the "if" part and the "only if" part in the proof of Theorem 1, respectively.
By using the result and the proof idea of Theorem 2, we have also found a complete and unique expression for the G-symmetric Clifford operators when the symmetry is given by Eq. ( 7) or (8).Concretely, in the case of Eq. ( 7), every G-symmetric Clifford operator U is uniquely expressed as Fig. 3 (a) as with µ j ∈ {0, 1, 2, 3}, ν j,k ∈ {0, 1}, σ ∈ S N and c ∈ U 0 , where K σ is the permutation operator that brings the jth qubit to the σ(j)th qubit.It follows that the size of the quotient group of the symmetric Clifford group C N,G divided by the freedom of phase is In the case of Eq. ( 8), the G-symmetric Clifford operators are restricted to cK σ with c ∈ U 0 and σ ∈ S N as expressed in Fig. 3 (b).The size of U N,G /U 0 is N !. See Theorem 8 in Appendix C for details.

V. UNITARY 4-DESIGNS
We can show that the G-symmetric Clifford group C N,G is not a G-symmetric unitary 4-design except for the trivial case when the G-symmetric unitary subgroup U N,G has only scalar multiples of I.This theorem and Theorem 1 imply that under a nontrivial Pauli symmetry, the symmetric Clifford group is a symmetric unitary 3-design but not a 4-design.We can prove this theorem by using the proof idea used in the "only if" part of Theorem 1 as follows: Proof.Since the "if" part is trivial, it is sufficient to prove the "only if" part.Suppose that C N,G is a Gsymmetric unitary 4-design.We define L ∈ L(H ⊗4 ) by L := P ∈P + N P ⊗4 , where P + N := {I, X, Y, Z} ⊗N .We take arbitrary U ∈ C N,G .By Lemma 17 in Appendix G, we can take a function s U : P + N → {±1} and a bijection h U on P + N such that U P U † = s U (P )h U (P ) for all P ∈ P + N .We note that s U and h U are dependent on U .By using the definitions of L, s U and h U , we get < l a t e x i t s h a 1 _ b a s e 6 4 = " B l d   for the case of U(1) symmetry on a 4-qubit system.Here the G-symmetric Clifford group is only a G-symmetric unitary 1-design, and not a 2-design.The numerical simulation is performed using the library Qulacs [39].
Since this holds for all U ∈ C N,G and C N,G is a G-symmetric unitary 4-design, by Lemma 10 in Appendix B, we have U ⊗4 LU †⊗4 = L for all U ∈ U N,G .
We therefore get U P U † = P for all U ∈ U N,G and P ∈ P + N by Lemma 11 in Appendix B. This implies that any U ∈ U N,G satisfies P U P = U for all P ∈ P + N , which is equivalent to U = e iθ I with some θ ∈ R.This means that U N,G ⊂ U 0 I. Since U N,G ⊃ U 0 I always holds, we get U N,G = U 0 I.

VI. VERIFICATION VIA FRAME POTENTIALS
We can give a numerical evidence to Theorems 1 and 2 by computing the frame potentials, which are defined for a subgroup X of U N,G as [2] F t (X ) := (15) where the integral is replaced by a summation if X is a finite set up to phase.Similarly to the conventional case [2], we can measure the distance between the tfold twirling channels Φ t,X and Φ t,U N,G by the frame potentials.It is therefore straightforward to show that F t (X ) ≥ F t (U N,G ), where the equality holds if and only if X is a G-symmetric unitary t-design.
Figure 4 (a)(b) clearly shows that the Clifford group C N,G under Pauli symmetry is a G-symmetric unitary 3design but not a 4-design.In sharp contrast, Fig. 4 (c)(d) shows that, when the Clifford group is constrained by G = {(e iθZ ) ⊗N |θ ∈ R}, which is isomorphic to U(1), then the G-symmetric Clifford group C N,G is only a Gsymmetric unitary 1-design and not a 2-design.
We remark that, in order to compute frame potentials numerically, we have sampled symmetric Clifford operators uniformly from the entire C N,G .For instance, in the case of Pauli-symmetric case, we have utilized the complete and unique construction provided in Theorem 2 (or Fig. 1); we have randomly chosen the parameters µ j ∈ {0, 1, 2, 3}, ν j,k ∈ {0, 1}, P j ∈ {I, X, Y, Z} ⊗N3 with a uniform probability, and have also employed the method in Ref. [35] in order to uniformly choose Clifford operators V ∈ C N3 acting on the subsystem A 3 .This can also be done in a similar way for U(1) and SU(2)-symmetric cases as well based on Fig. 3 and Eq.(12).
It is beneficial to mention that the frame potentials of the symmetric unitary groups can be written with those of the unitary groups of several dimensions.In order to state the result, we explain the irreducible decomposition of group representations [40].We consider the regular representation ρ of the group G, i.e., ρ(G) := G for all G ∈ G. Since ρ is a unitary representation, ρ is completely reducible, and thus there exist some set of pairs {(I λ , J λ )} of spaces such that the Hilbert space H of the N qubits is decomposed into and with inequivalent irreducible representations ρ λ of G on I λ and the identity operator I on J λ .This implies that λ is the index for inequivalent irreducible representations, I λ is the representation space, and J λ is the multiplicity space.
Theorem 5. (Formula for frame potentials of symmetric unitary groups.)Let G be a subgroup of U N and the regular representation ρ of G be irreducibly decomposed in the form of Eq. (17).Then, the frame potential F t (U N,G ) of the G-symmetric unitary group U N,G is given by with S t := {(t λ )| λ t λ = t, t λ ∈ Z, t λ ≥ 0} and the unitary group U(J λ ) on J λ .
We illustrate the result of Theorem 5 in the case of the Pauli symmetries Q, which are unitarily equivalent to the Pauli subgroup R given by Eq. ( 9), as we have explained in Sec.III B. In this case, λ is the index for specifying the sequence of the eigenvalues of Z j for j ∈ Γ 2 .We take I λ as the Hilbert space of the subsystems A 1 and A 2 which is the simultaneous eigenspace of (Z j ) j∈Γ2 with the eigenvalues specified by λ, and also take J λ as the Hilbert space of the subsystem A 3 .This means that there are 2 N2 choices for λ, and for each of them, the dimensions of I λ and J λ are 2 N1 and 2 N3 , respectively.When t ≤ 2 N3 , Eq. ( 18) gives the following simple form: where we used F t λ (U(J λ )) = t λ ! in the first line, and the multinomial theorem in the third line.
Proof.We are going to prove Eq. ( 18) by considering the generating function of F t (U N,G ).For a unitary subgroup X , we define f X by where we used in the second to last line, and we note that the definition of the frame potential (Eq. ( 15)) is equivalent to F t (X ) = U ∈X |tr(U )| 2t dµ X (U ) by the left invariance of the Haar measure.By Eq. (2.26) of Ref. [40], every U ∈ U N,G can be written as with the identity operator I on I λ and some U λ ∈ U(J λ ), and we have Since Eq. ( 23) gives one-to-one correspondence between U N,G and the set of U(J λ ), by Eqs. ( 20) and ( 24), we get By using Eq. ( 21) and comparing the coefficients of z 2t of both sides of Eq. ( 25), we get Eq.( 18).

VII. DISCUSSION
In this paper, we have generalized the unitary 3-design property of the multiqubit Clifford group into symmetric cases.We have rigorously shown that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry is given by some Pauli subgroup (Theorem 1), and also have provided a way to generate all the elements without redundancy (Theorem 2).Furthermore, we have also proven that two physically important class of U(1) and SU(2) symmetries, which cannot be reduced to Pauli subgroups, only yields symmetric unitary 1-designs.Finally, for numerical validation, we have computed the frame potentials by randomly sampling symmetric unitaries, and have confirmed that our findings indeed hold.
We can derive another property of the symmetric Clifford group with respect to locality, from the results about the construction method of the symmetric Clifford operators.As we have shown in Theorem 2 and after Theorem 3, under the Pauli, U(1), and SU(2) symmetries, all the symmetric Clifford operators can be constructed with 2-qubit local symmetric Clifford operators.It can be seen as a symmetric generalization of the fact that all the Clifford operators can be constructed with 2-qubit local Clifford operators [34,35].This stands in contrast to the result in Ref. [41], which shows that while all the unitary operators can be constructed with local unitary operators, some symmetric unitary operators cannot be constructed with local symmetric unitary operators.
We envision two important future directions.First, it is theoretically crucial to reveal the requirement to achieve symmetric unitary designs in the approximate sense.While it is known for the non-symmetric case that the approximate t-designs require only polynomial gate depth with respect to both the qubit count N and order t [42,43], it is far from trivial whether this is also true for the symmetric case.Second, it is practically intriguing to develop a constructive way to generate symmetric Clifford circuits under general symmetry G that cannot be described by some Pauli subgroup.While we provide such an example for both U(1) and SU(2) symmetry, it is important to construct circuits in an automated way for general situations, in particular when one is interested in performing Clifford gate simulation for the purpose of quantum many-body simulation [44].

. Connectedness of symmetric unitary groups 39
Appendix A: General Remarks for Detailed Proofs In the following, we present the detailed proofs of the theorems in the main text.In Appendix B, we prove Theorem 1 in the main text, which states that the symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint can be written as the commutativity with a Pauli subgroup.In Appendix C, we prove Theorem 2 in the main text, which gives a one-to-one correspondence from the sets of elementary gates to the symmetric Clifford gates.In Appendix D, we prove the former half of Theorem 3 in the main text.We show that the symmetric Clifford group is a symmetric unitary 1-design under the U(1) and SU(2) symmetries, which are not Pauli symmetries.In Appendix E, we prove the latter half of Theorem 3 in the main text in a more general form.We take a larger class of symmetries than the one in Appendix D, and show that the symmetric Clifford group is not a symmetric unitary 2-design for those symmetries.In Appendix G, we present the technical lemmas that we use in the proofs of the statements above.
Before going into the details, we introduce the notations in the following appendices.For general Hilbert spaces K and K ′ , we denote the set of all linear operators from K to K ′ , all linear operators on K, and all unitary operators on K by L(K → K ′ ), L(K) and U(K), respectively.We denote the Hilbert space for N qubits by H.We denote the unitary group on N qubits by U N .We denote the Pauli group on N qubits by P N .We define the Clifford group on N qubits as the normalizer of P N and denote it by C N .As for Clifford operators on a single qubit, we denote the Pauli-X, Y and Z, Hadamard and S operators on the jth qubit by X j , Y j , Z j , H j and S j , respectively.As for Clifford operators on two qubits, we denote the controlled-NOT, the controlled-Z and the SWAP operators on the jth and kth qubits by CNOT j,k , CZ j,k and SWAP j,k , respectively, where the jth qubit is the control qubit and the kth qubit is the target qubit of CNOT j,k .We denote the set of Pauli operators without phase by P + N := {I, X, Y, Z} ⊗N , where A ⊗n := A ⊗ A ⊗n−1 and A ⊗ B := {A ⊗ B|A ∈ A, B ∈ B} for general operator sets A and B. For convenience, we formally define U 0 := {e iθ | θ ∈ R} and P 0 := {±1, ±i}.We denote the symmetric group of degree M by S M .We denote by ⟨O⟩ the group generated by the operators in a set O. We denote a ≡ b (mod r) when a − b is an integer multiple of r.
if and only if the symmetry constraint is described by some Pauli subgroup.
First, we define the symmetric Clifford group as the symmetric subgroup of the conventional Clifford group. where This definition includes the conventional Clifford group C N as the special case when G = {I}.
Next, we define the notion of symmetric unitary design.For a subgroup G of U N and t ∈ N, we define G-symmetric unitary t-designs as the group that approximate U N,G .
where Φ t,X is the t-fold twirling channel defined by with the normalized Haar measure on X and E t,U is the t-fold unitary conjugation map on L(H ⊗t ) defined by This definition includes the standard unitary designs as the special case when G = {I}.This type of unitary designs are sometimes called unweighted unitary designs in comparison with weighted unitary designs, where non-uniform mixtures of E t,U are considered (see Definition 5).As we show in Theorem 11, the conditions for a symmetric Clifford group being unweighted and weighted unitary designs are equivalent to each other.It is therefore sufficient to focus only on unweighted unitary designs, and we express them simply as unitary designs in the following.
We are going to prove that C N,G is a G-symmetric unitary 3-design if and only if the symmetry condition can be described by the commutativity with some Pauli subgroup.This can be rigorously stated as follows: Theorem 6. (Restatement of Theorem 1.) Let G be a subgroup of U N .Then, C N,G is a G-symmetric unitary 3-design if and only if U N,G = U N,Q with some subgroup Q of the Pauli group P N .
We present the overall structure of the proof of this theorem in Fig. 5.We prove the "if" part in Proposition 1, and the "only if" part in Proposition 2.

Proof of the "if " part of Theorem 1 (Theorem 6)
The "if" part of Theorem 6 is equivalent to the statement that C N,Q is a Q-symmetric unitary 3-design for all Pauli subgroups Q, which we show in the following proposition.This is because if U N,G = U N,Q , then C N,G = C N,Q and G-symmetric unitary 3-designs are the same as Q-symmetric unitary 3-designs.
By the definition of unitary t-designs, the goal is to prove Φ C N,Q = Φ U N,Q .In the following, we introduce two useful properties of Φ t,X for X = C N,Q and U N,Q .
As the first property, we cannot distinguish whether there is a symmetric unitary conjugation action before the action of Φ t,X .This can be directly proven by the right invariance of µ X .We also use this lemma in the proofs of Theorems 9 and 11.

Proposition 2.
If  !, is a -symmetric unitary 3-design,  !, is a -symmetric unitary 3-design for  ⊂  ! .then ∀ ∈  !, and  ∈  !-satisfying ′ ) ≠ 0,  , = .Lemma 1.Let N ∈ N, t ∈ N, G be a subgroup of U N , X = C N,G or U N,G , Φ t,X be defined by Eq. (B4), and C t,G be the set of all t-fold G-symmetric Clifford conjugation mixture maps defined by

Lemma 14
Then, for any t-fold G-symmetric Clifford mixture map Proof.Since D ∈ C t,G , D can be written as D = n j=1 λ j E t,Uj with some λ 1 , λ 2 , ..., λ n ∈ R and U 1 , U 2 , ..., U n ∈ C N,G satisfying n j=1 λ j = 1.For any j ∈ {1, 2, ..., n}, we get where we used the right invariance of µ X .We note that the Haar measure on a compact Lie group X is right-invariant by Corollary 8.31 of Ref. [45].We therefore get As the second property, we introduce trivial fixed-points of Φ t,X for X = C N,Q and U N,Q in an explicit form.
,X be defined by Eq. (B4), and M G be the linear subspace of L(H ⊗3 ) defined by where the span is taken over the field C and V σ ∈ U(H ⊗3 ) is the permutation operator that brings the jth copy of qubits to the σ(j)th qubits, i.e., As for Eq.(B11), we note that the state of the jth copy of qubits after the action of V σ is the same as that of the σ −1 (j)th copy of qubits before the action, because V σ brings σ −1 (j)th copy of qubits to the jth copy of qubits.
Proof.Since Φ 3,X is a linear map and M G is a linear subspace spanned by Although we only require the fact that all the points in M G are fixed-points of Φ 3,U N,G in our proof, we can prove that the set of all the fixed-points of Φ 3,U N,G corresponds with M U N,U N,G by using the result of Ref. [46].
In order to connect Lemmas 1 and 2 to the proof of Proposition 1, it is sufficient to find a map We present the existence of such a map D as a lemma.Lemma 3. Let N ∈ N, Q be a subgroup of P N , C 3,Q be the set all t-fold Q-symmetric Clifford conjugation mixture maps defined by Eq. (B6) and M Q be defined by Eq. (B10).Then, there exists a map In order to simplify the proof of this lemma, we show that the statements of this lemma for two symmetry groups are equivalent if the two groups can be transformed into each other by some Clifford conjugation action up to phase.Lemma 4. Let N ∈ N, G and G ′ be subgroups of U N satisfying U 0 G ′ = U 0 W GW † with some W ∈ C N , and C t,G and C t,G ′ be the sets of all t-fold Gand G ′ -symmetric Clifford conjugation mixture maps defined by Eq. (B6).Then, there exists a map is sufficient only to prove the "if" part.We suppose that there exists a map D ∈ C 3,G such that D(L) ∈ M G for all L ∈ L(H ⊗3 ).By the definition of C 3,G , D can be written as with some n ∈ N, U 1 , U 2 , ..., U n ∈ C N,G , and λ 1 , λ 2 , ..., λ n ∈ R satisfying n j=1 λ j = 1.We define Then, D ′ ∈ C 3,G by noting that W U j W † ∈ C N,G ′ for all j ∈ {1, 2, ..., n}.By this definition, we also know that for any L ∈ L(H ⊗t ), Now we note that any Pauli subgroup Q can generally be transformed into a Pauli subgroup R in the form of with some N 1 , N 2 , N 3 ≥ 0 up to phase via some Clifford conjugation action, which we prove in Lemma 14 in Appendix G.By combining this property and Lemma 4, we know that it is sufficient only to prove Lemma 3 when Q is given as R in the form of Eq. (B16).Since we are going to deal with three copies of the system each of which is decomposed into three subsystems, we define the notations for explicit presentation of the Hilbert space on which an operator acts or in which a vector exists.When we explicitly show that an operator O acts on a Hilbert space K and a vector |Ψ⟩ exists in K, we denote O (K)  and |Ψ⟩ (K) , respectively.The notations for Hilbert spaces are as follows: In order to distinguish the Hilbert spaces H associated with the 3 copies of the N qubits that we consider in the context of unitary 3-designs, we denote the 3 Hilbert spaces by H 1 , H 2 and H 3 (see Fig. 6 (a)).The symmetry R induces a natural decomposition of each Hilbert space H j into three parts H j 1 , H j 2 and H j 3 of N 1 N 2 and N 3 qubits, correspondingly to the representation of R. We also denote the Hilbert space of the lth qubit in H j k by H j k,l .(see Fig. 6 (b)).We denote the tensor product of the three spaces of H 1 k , H 2 k and H 3 k by H tot k (see Fig. 6 (c)).We may refer to H j k simply as H k when we need not specify j.
In the proof of Lemma 3, we focus on the following four types of R-symmetric Clifford operators; the S gates on a qubit in H 2 , the controlled-Z gates on two qubits in H 2 , the Clifford gates on qubits in H 3 , and the controlled-Pauli gates with a control qubit in H 2 and target qubits in H 3 (see Fig. 6 (d)).We are going to see their properties one by one in the four lemmas below.
Lemma 5. Let N ∈ N, R be defined by Eq. (B16), C 3,R be the set of all 3-fold R-symmetric Clifford conjugation mixture maps defined by Eq. (B6), m ∈ {1, 2, ..., N 2 }, D 1,m be defined by and K ∈ L(H ⊗3 ) be in the form of Since the Clifford group C N3 is a unitary 3-design [27,28], O ′ can also be written as For any U ∈ U N3 , by the left invariance of µ U N 3 , we get This implies that O ′ commutes with U ⊗3 for all U ∈ U N3 .By Theorem 7.15 of Ref. [47], O ′ can be written as Finally, we prove the property of the mixture of the controlled-Pauli gates on H 2 and H 3 .Here we fix the control qubit as the mth qubit in H 2 .
Proof.Since the controlled-Pauli operators can be expressed as products of the controlled-X, Y, and Z operators, we can confirm that C(Q) (H j 2,m ,H j 3 ) ∈ C N,R .D 4,m is an affine combination of the conjugation actions of C(Q) ⊗3 for Q ∈ P + N3 , and thus we can also confirm that D 4,m ∈ C 3,R .By noting that we know that We note that By Eqs.(B38) and (B39), we get First, we consider the case when x σ(j),m = y j,m for all j ∈ {1, 2, 3}.In this case, we have By Eqs.(B41) and (B41), we get where c := 1 and σ ′ := σ.Since σ ′ = σ, we get x σ ′ (j),m = x σ(j),m = y j,m for all j ∈ {1, 2, 3} and m ′ ∈ {1, 2, ..., m}.
By combining the five lemmas above, we prove Lemma 3.

U(1) and SU(2) symmetries
Next, we take U(1) and SU(2) symmetries given by Eqs.(7) and (8) in the main text as examples of non-Pauli symmetries, and present complete and unique constructions for the symmetric Clifford groups.In both cases, every symmetric Clifford operator can be written as the product of a permutation operator and a Pauli-symmetric Clifford operator as shown in Fig. 3 in the main text.Theorem 8. (Construction of the U(1) and SU(2)-symmetric Clifford groups.)Let N ∈ N, and G 1 and G 2 be given by Then, for any U ∈ C N,G1 , there uniquely exist {µ j } N j=1 ∈ {0, 1, 2, 3} N , {ν j,k } 1≤j<k≤N ∈ {0, 1} N (N −1)/2 , σ ∈ S N and c ∈ U 0 such that and for any U ∈ C N,G2 , there uniquely exist σ ∈ S N and c ∈ U 0 such that This means that {(λ j , U j )} n j=1 is a weighted G-symmetric unitary t-design.Next, we prove the "only if" part.We suppose that there exists n ∈ N, λ 1 , λ 2 , ..., λ n ∈ R and U 1 , U 2 , ..., U n ∈ C N,G such that {(λ j , U j )} n j=1 is a weighted unitary t-design.We define a map D on L(H ⊗t ) by D :=  This means that C N,G is an unweighted G-symmetric unitary t-design.

Theorem 4 .
(No-go theorem for symmetric unitary 4designs.)Let G be a subgroup of U N .Then, C N,G is a G-symmetric unitary 4-design if and only if U N,G = U 0 I.

2 <
g 6 / o x S S m M M P v P Y c l r G I N O T 6 3 g h O c 4 i z 0 p E S U M W W 8 n a q E A s 0 o v o Q S + w A b P 4 q A < / l a t e x i t > t = l a t e x i t s h a 1 _ b a s e 6 4 = " x 6 j x a 8 P b H 1 P w A E 9 n d R l t a d 5 f d U S j p D w R d 6 9 C p I C L 6 G V 3 6 A x 3 8 B 0 V H g y 4 d e l 0 X o q R 6 h 5 l 5 5 p n 3 e e e Z G c X S N U c w 1 v Z J A 4 N D w y P + 0 c D Y + E Q w F J 6 c y j t m w 1 Z 5 T j V 1 0 y 4 q s s N 1 z e A 5 o Q m d F y 2 b y 3

FIG. 4 .
FIG.4.Frame potentials Ft(X ) of G-symmetric groups X = CN,G, UN,G computed by numerically taking the average over randomly generated unitaries.Here we compare the results for the Pauli symmetry R given by Eq. (9) with N1 = 1, N2 = 2, N3 = 3 and the U(1) symmetry given by Eq. (7) with N = 4. (a) Frame potentials Ft(X ) for the case of the Pauli symmetry computed by taking the average over 10 6 samples.(b) Size scaling of the relative error of Ft(CN,G) against the theoretical lower bound Ft(UN,G).Here, we independently generate M samples for U and U ′ respectively, and compute the mean value of |tr(U U ′ † )| 2t .As we increase the total data size M 2 , the errors become smaller for t ≤ 3, while they remain finite for t = 4. Panels (c) and (d) show the results for the case of U(1) symmetry on a 4-qubit system.Here the G-symmetric Clifford group is only a G-symmetric unitary 1-design, and not a 2-design.The numerical simulation is performed using the library Qulacs[39].

CONTENTSA.
General Remarks for Detailed Proofs 10 B. Proof of Theorem 1 (Unitary 3-designs) 10 1. Proof of the "if" part of Theorem 1 (Theorem 6) 11 2. Proof of the "only if" part of Theorem 1 (Theorem 6) 21 C. Proof of Theorem 2 (Construction of symmetric Clifford groups) of Pauli subgroups into the standard form 35 2. Property of 3-bit sequences 37 3. Property of t-fold mixture maps 38 4. Bijections induced by Clifford operators 38 5

FIG. 6 .
FIG.6.Setup of Proposition 1 and the notations of the Hilbert spaces in the proof.(a) In the proof of unitary 3-designs, we consider unitary operations U on 3 copies of a Hilbert space, which we denote by H 1 , H 2 and H 3 .When we explicitly show that a unitary operator U acts on H j , we denote U (H j ) .(b) The symmetry R decomposes each Hilbert space H j into three parts; H j 1 , H j 2 and H j 3 for the N1, N2 and N3 qubits, correspondingly to the representation of R. The figure is for the case when N1 = 1, N2 = 2 and N3 = 2.We denote the Hilbert space for the lth qubit in H j k by H j k,l .(c) We defineH tot k := H 1 k ⊗ H 2 k ⊗ H 3 k for k = 1, 2, 3.We note that the total Hilbert space H ⊗3 can be expressed in two ways as H 1 ⊗ H 2 ⊗ H 3 and as H tot 1 ⊗ H tot 2 ⊗ H tot 3 .(d) As R-symmetric Clifford operations, we focus on four types of gates; the S gates on qubits in H2, the controlled-Z gates on two qubits in H2, the Clifford gates on qubits in H3, and the controlled-Pauli gates with a control qubit in H2 and target qubits in H3.

D
= Φ t,U N,G .(F4) Since D ∈ C t,G , by Lemma 1, we get Φ t,C N,G • D = Φ t,C N,G .(F5) Since µ U N,G is left-invariant and µ C N,G is normalized, we get