sign in sign up
<<
Home , Clifford gates

Un-Weyl-ing the Clifford Hierarchy

Tefjol Pllaha1, Narayanan Rengaswamy2, Olav Tirkkonen1, and Robert Calderbank3

1Department of Communications and Networking, Aalto University, Finland 2Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA 3Department of Electrical and Computer Engineering, Duke University, NC, USA

The teleportation model of quantum compu- quantum mechanics that govern our universe. In this tation introduced by Gottesman and Chuang computational model, a consists of (1999) motivated the development of the Clif- a sequence of operations each of which is either a ford hierarchy. Despite its intrinsic value for quantum gate, characterized by a unitary matrix, or , the widespread use of a quantum measurement, characterized by a Hermi- , which is closely related tian matrix (i.e., an observable)[23]. So, a universal to this model, emphasizes the importance of quantum computer must be capable of implement- comprehending the hierarchy. There is cur- ing arbitrary unitary operations and measuring any rently a limited understanding of the structure Hermitian operator on a given set of m . In of this hierarchy, apart from the case of diag- 1999, Gottesman and Chuang demonstrated that such onal unitaries (Cui et al., 2017; Rengaswamy universal quantum computing can be performed just et al. 2019). We explore the structure of the by using the protocol if one second and third levels of the hierarchy, the has access to certain standard resources — Bell-state first level being the ubiquitous Pauli group, via preparation, Bell-basis measurements, and arbitrary the Weyl (i.e., Pauli) expansion of unitaries single- rotations [17]. They defined the Clifford at these levels. In particular, we character- hierarchy as part of their proof, and this has proven ize the support of the standard Clifford opera- to be a useful characterization of a large set of uni- tions on the Pauli group. Since conjugation of tary operations, both in theory and practice. In fact, a Pauli by a third level unitary produces trace- in their teleportation model of computation, the level less Hermitian Cliffords, we characterize their of a unitary in the hierarchy can be interpreted as a Pauli support as well. Semi-Clifford unitaries measure of complexity of implementing it. Further- are known to have ancilla savings in the tele- more, this model is closely related to the currently portation model, and we explore their Pauli widespread scheme of distilling “magic” states and in- support via symplectic transvections. Finally, jecting them via teleportation-like methods in order to we show that, up to multiplication by a Clif- fault-tolerantly execute unitary operations on qubits ford, every third level unitary commutes with encoded in a quantum error-correcting code [3,4]. at least one Pauli matrix. This can be used in- Hence, it is very important to understand the struc- ductively to show that, up to a multiplication ture of this hierarchy since it has important implica- by a Clifford, every third level unitary is sup- tions for fault-tolerant quantum computing. ported on a maximal commutative subgroup The first level of the hierarchy is the Pauli (or of the Pauli group. Additionally, it can be eas- Heisenberg-Weyl) group and the second level is the ily seen that the latter implies the generalized Clifford group, which is defined as the normalizer of semi-Clifford conjecture, proven by Beigi and the Pauli group in the unitary group. Subsequent lev- Shor (2010). We discuss potential applications els C(k) of the hierarchy, for k ≥ 2, are defined recur- in and the design of sively as those unitaries that map to flag gadgets. (k−1) arXiv:2006.14040v4 [quant-ph] 9 Dec 2020 C under conjugation [17] (see (23) for the precise definition). While the first two levels form groups, 1 Introduction it is known that the higher levels are only finite sets of unitary matrices (up to overall phases) and that Quantum computing provides a fundamentally new even when k → ∞, the hierarchy does not encompass approach to computation by exploiting the laws of all unitary matrices (see Example2). Furthermore, each level is closed under left or right multiplication Tefjol Pllaha: tefjol.pllaha@aalto.fi by Cliffords [35]. Narayanan Rengaswamy: [email protected], Most of this work was conducted when N.R. was with the Department of Elec- It is well-known that the Pauli matrices form an trical and Computer Engineering, Duke University, NC, USA orthonormal basis for all square matrices under the Olav Tirkkonen: olav.tirkkonen@aalto.fi Hilbert-Schmidt inner product [16]. Therefore, a nat- Robert Calderbank: [email protected] ural question to consider is to determine the Pauli

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 1 (i.e., Weyl) expansion of all unitaries in the Clifford decomposition of any semi-Clifford operation in terms hierarchy. It is reasonable to expect that the Pauli of diagonal gates and physical permutation operators expansion of elements at a level provides insight into composed of CNOTs and Pauli X’s. Thus, when com- the structure of the hierarchy. Indeed, a subset of bined with [14] and our contribution of characterizing the current authors recently identified a special set the Pauli support of standard Cliffords, this essen- of diagonal unitary matrices in the hierarchy, called tially produces the Weyl expansion of semi-Cliffords. Quadratic Form Diagonal (QFD) gates, and produced Third, Zeng et al. conjectured in the above pa- formulae for their action on Pauli matrices [28]. In per that all unitaries in C(3) are semi-Clifford and all subsequent work, they considered QFD gates con- unitaries in C(k) are generalized semi-Clifford for any structed as tensor products of integer powers of the k. While a semi-Clifford operation maps, by conju- iπ  “T” gate, T := diag 1, exp 4 . There they exam- gation, a maximal commutative subgroup (MCS) of ined the result of conjugating Pauli matrices by such the Pauli group to another MCS of the Paulis, a gen- gates, and fully characterized the Pauli expansion of eralized semi-Clifford operation maps the span (i.e., the (Clifford) result. Then they used this characteri- complex linear combination) of a MCS to the span zation to understand when such a physical operation of another MCS. It is well-known that for m = 1, 2 preserves the code subspace of a stabilizer quantum qubits all unitaries are semi-Clifford, and for m = 3 error-correcting code [27, 29]. This is fundamental qubits the third level is semi-Clifford, so these conjec- because such codes are necessary to make the quan- tures are for k = 3 for all m > 3 and for k ≥ 4 for tum computer tolerate noise, and all operations on all m ≥ 3, respectively. Gottesman and Mochon have the encoded information have to be performed by provided a counterexample for C(3) that disproves the such codespace-preserving physical fault-tolerant op- semi-Clifford conjecture [34]1. Subsequently, Beigi erations. Furthermore, for universal quantum compu- and Shor [1] proved that all unitaries in C(3) are gen- tation we need to implement at least one non-Clifford eralized semi-Clifford operations, thereby settling the gate, and the T gate is one of the easiest non-Clifford conjecture for k = 3. In this paper, we prove the gates to engineer. As a general recipe, one could re- stronger result that for any unitary C from C(3), there place this (tensor product) gate with any high fidelity exists a Clifford G such that GC is supported on a lab operation and attempt to repeat this process to MCS of the Pauli group. Our proof uses a much sim- understand required code structure. pler induction argument based on the fact that any The Heisenberg-Weyl expansion considered in this third level unitary must map (under conjugation) at paper is intimately connected with the Wigner func- least one Pauli to some other Pauli. tions [32] via the Fourier transform. The discrete Finally, the third level of the hierarchy is of par- counterparts are explored in [15, 18, 33]. These in- ticular interest since any third level gate enables uni- sights have played an important role in the simulation versal quantum computation when combined with the (3) and general understanding of magic states, as well as Clifford group [2, 17]. When a C gate acts by con- with non-stabilizer resources [31]. jugation on a Pauli matrix, the result is a Hermitian Clifford, one example being the aforementioned case In this paper, we make contributions towards a few of choosing a C(3) operation that is a tensor product related questions about the hierarchy. First, surpris- of integer powers of T. It is well-known that Clifford ingly, the Pauli support (i.e., the Pauli matrices with transvections (that is, square roots of Hermitian Pauli non-zero coefficients in the Pauli expansion) of even matrices; see (28)) form a different generating set for the well-known Clifford group operations remains un- all Cliffords, compared to the standard Clifford gate known. (Note that conjugating a Pauli matrix by the set mentioned earlier [9, 20, 24]. We prove a neces- transversal T gate produces a Clifford gate, so the sary and sufficient condition for the Paulis involved in aforementioned result already calculates the Pauli ex- the transvection decomposition of an arbitrary Hermi- pansion of certain types of Cliffords.) Hence, we study tian Clifford operator. Since expanding the product the support of the “standard” Clifford operations that of transvections provides the Pauli expansion of these correspond to the standard Clifford gate-set consist- Hermitian Cliffords, this can potentially be applied ing of Hadamard, Phase, Controlled-NOT (CNOT), to extend the aforementioned result on characterizing and Controlled-Z (CZ) gates. stabilizer codes that support transversal T gates to Second, Zeng et al. [35] considered certain oper- other gates from C(3). ations called semi-Clifford unitaries in the hierar- As a different application, flag gadgets have recently chy [19], which have the advantage that they require become popular as a near-term method to detect cor- fewer ancillae than general unitaries in the teleporta- related faults in circuits [10–13, 30]. The idea is to in- tion model of Gottesman and Chuang [36]. They also troduce a multi-qubit Pauli measurement before and showed that any semi-Clifford U can be expressed as after the circuit, using one or more ancilla qubits, such U = G1DG2, where D is diagonal and G1, G2 are that the extra gadget acts trivially in the case of no Clifford operators. Cui et al. [14] have recently char- acterized the diagonal unitaries in the Clifford hierar- 1The authors of [28] were unaware of this result, and they chy. We prove a general result that provides an exact regret reporting that this conjecture remained open.

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 2 errors but catches catastrophic errors otherwise. A with Im|r being the block matrix with Ir in upper left key requirement to construct flag gadgets for a specific corner and 0 elsewhere, and Im|−r = Im − Im|r. Here application circuit is to determine the best Pauli mea- r = rank (C). The semidirect product S of SD and surement to apply before the circuit and identify the SU corresponds to r = 0, that is, symplectic matrices result of “propagating” the Pauli through the circuit, with C = 0. Let Se be the subgroup of S consisting of i.e., determine the result of conjugating the Pauli by matrices FD(P)FU (S) with P upper triangular. Then 2 the circuit. The simplest case is to use a Pauli opera- Se has size 2m(m−1)/2 · 2m(m+1)/2 = 2m and is a 2- tor that commutes with the circuit. For this purpose, Sylow subgroup of Sp(2m) that contains SU . any Pauli in the centralizer/dual of the support (of Another type of decomposition of Sp(2m) can be the circuit/unitary) would suffice. Hence, our afore- achieved via symplectic transvections Tv := I2m + t 2m 2m said results on characterizing Pauli supports can be Ωv v, v ∈ F2 . Such matrix acts on F2 as x 7−→ 2 applied to determine the Paulis that commute with x + h v | x isv, and thus Tv = I2m. In general, 2 the corresponding circuit. In particular, since flag F ∈ Sp(2m) is said to be an involution if F = I2m gadgets are generally applied only to Clifford circuits, and is said to be hyperbolic if h v | vF is = 0 for all (3) 2m our result that any C element is supported on a v ∈ F2 . It is well-known that symplectic transvec- MCS of the Paulis, up to multiplication by a Clifford, tions generate Sp(2m) [9, 20, 24]. It is shown there provides a way to determine a Pauli that commutes that a non-hyperbolic involution can be written as with a non-Clifford element. Therefore, this insight product of r transvections Tv1 ,..., Tvr , where r = could be used to design flag gadgets beyond Clifford 2m − dim(Fix(F)) = dim(Res(F)) and (subsections of) circuits. 2m Fix(F) := ker(I ⊕ F) := {v ∈ F2 | v = vF}, (7) Res(F) := rs (I ⊕ F) := {v ⊕ vF | v ∈ 2m}. (8) 2 Preliminaries F2 Throughout the paper rs (•) will denote the row space 2.1 The Binary Symplectic Group of a matrix. Note here that, by definition, Fix(F) and Res(F) are dual of each other. The vectors We will denote by GL(n) and Sym(n) the groups of v1,..., vr ∈ Res(F) must be independent, in which n × n invertible and symmetric matrices over the bi- case we say that corresponding transvections are inde- nary field F2, respectively. Addition in F2 will be de- pendent. On the other hand, a hyperbolic involution noted by ⊕. The binary symplectic group Sp(2m) ⊂ can be written as a product of r+1 transvections (r as GL(2m) is the set of 2m × 2m binary matrices that above), r of which are independent and the additional 2m preserve the symplectic inner product in F2 : one is dependent of the others. We will see that the residue space Res(F) of a symplectic F is intimately t t t h (a, b) | (c, d) is = ad ⊕ bc = (a, b)Ω(c, d) , (1) connected with the support (33) of the corresponding Clifford G. On the other hand, the fixed space Fix(F) where being the dual of Res(F) is intimately connected with  0 I  Ω = m m , (2) the Paulis that commute with G. Transvections are I 0 m m the simplest form of involutions in Sp(2m) and will  AB  play a central role throughout the paper for the sim- A matrix F = ∈ Sp(2m) satisfies FΩFt = CD ple reason that they correspond to square roots of Ω, which in turn is equivalent with ABt, CDt ∈ Hermitian Paulis; see (28) and (29). t t Sym(m) and AD ⊕ BC = Im. In Sp(2m) we distinguish two subgroups: 2.2 Quantum Computation     Fix N = 2m. The standard basis vectors of N will P 0m C SD := FD(P) = −t P ∈ GL(m) (3) be indexed by binary vectors and denoted as kets, 0m P that is, e = |vi, v ∈ m, will have 1 in the position =∼ GL(m), v F2 indexed by v and 0 else. The Heisenberg-Weyl group     Im S is defined as SU := FU (S) = S ∈ Sym(m) (4) 0m Im HW := {ikD(a, b) | a, b ∈ m, k ∈ } ⊂ (N), =∼ Sym(m). N F2 Z4 U (9) Then every F ∈ Sp(2m) can be Bruhat-decomposed where t [22, 25] as D(a, b): |vi −→ (−1)bv |v ⊕ ai. (10)

F = FD(P1)FU (S1)FΩ(r)FU (S2)FD(P2), (5) We will denote by PHWN := HWN /{±IN , ±iIN } the projective Heisenberg-Weyl group. Directly by where definition, we have   Im|−r Im|r t FΩ(r) = , (6) bc Im|r Im|−r D(a, b)D(c, d) = (−1) D(a ⊕ c, b ⊕ d). (11)

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 3 t m−k We will also define E(a, b) := iab D(a, b), which con- has dimension 2 . In literature, F+(S) is known stitute the Hermitian matrices in HWN . If follows as the [[m, m − k]] associated to S [23], by (11) that such matrices satisfy which encodes m − k logical qubits to m physical qubits. It follows that a MCS S defines a [[m, 0]] sta- t t E(a, b)E(c, d) = ibc −ad E(a + c, b + d), (12) bilizer code, that is, dim(Fε(S)) = 1. For this reason |ψεi := Fε(S) is called a stabilizer state. Let S = where we view all binary vectors as integer vectors hE1,..., Eki be the stabilizer group generated by the and operations are done modulo 4; see [28, Rem. 1] commuting Hermitian Paulis {E1,..., Ek}. For d ∈ 2m k d1 dk for the meaning of E(a, b) with (a, b) ∈ Z4 . If the F2 , we will denote Sd := h(−1) E1,..., (−1) Eki. arithmetic of the arguments of operators E(a, b) were Then to be done modulo 2 one would have k dn Y IN + (−1) En 1 X Π := = E (21) E(a, b)E(c, d) d 2 2k n=1 E∈S = (−1)h (a, b) | (c, d) is E(c, d)E(a, b) (13) d t t = ibc −ad E(a + c, b + d) (14) is a projection onto F+(Sd), which in turn gives a t t resolution of the identity [23, Sec. 10.5]: = ibc −ad E(a + c, (b ⊕ d) + 2(b ∗ d)) (15) t t t X = ibc −ad (−1)(a+c)(b∗d) E((a ⊕ c) + 2(a ∗ c), b ⊕ d) Πd = IN . (22) k (16) d∈F2 t t t t = ibc −ad (−1)(a⊕c)(b∗d) +(b⊕d)(a∗c) E(a ⊕ c, b ⊕ d). (17) 2.3 The Clifford Hierarchy (k) Above, the asterisk stands for the coordinate-wise The Clifford hierarchy {C , k ≥ 1} is defined re- product. We see that binary arithmetic only ever in- cursively, where the first level is the Heisenberg-Weyl troduces an additional sign. Thus when the sign is not group, and higher levels are defined by relevant (e.g., (25)) we will stick to binary arithmetic. (k) † (k−1) C = {U ∈ U(N) | UHWN U ⊂ C }. (23) Remark 1. From (11) we have that D(a, b) and D(c, d) commute iff h (a, b) | (c, d) is = 0, By definition, the Clifford group CliffN is the second and otherwise they anticommute. Similarly, (12) level of the hierarchy up to overall phases, that is (2) implies E(a, b)E(c, d) = ±E(a + c, b + d) if CliffN := C /U(1). The following example shows h (a, b) | (c, d) is = 0 and E(a, b)E(c, d) = ±iE(a + that the Clifford hierarchy does not exhaust U(N) c, b + d) otherwise. and also motivates the Weyl expansion. A stabilizer is a commutative subgroup of HW N Example 2. Set E1 = E(010, 010), E2 = generated by Hermitian matrices of form ±E(a, b) E(011, 001), E3 = E(001, 111), E4 = E(101, 011). that does not contain −I . Thus either E or −E N Then W = (E1 + E2 + E3 + E4)/2 is easily seen to be belong to a stabilizer, but not both. We will write outside of CliffN . Further, set E = E(100, 000). We S = E(A, B) if the stabilizer S is generated by † † have WEW = E3WE and (E3WE)E(E3WE) = E(a , b ),... E(a , b ), where A and B are k × m 1 1 k k −EE3W. Thus, since multiplication by Paulis (or matrices obtained by stacking ai’s and bi’s. Since S even Cliffords) preserves the level, iterative conjuga- t is abelian, the matrix C = (AB) satisfies CΩC = 0, tion cannot bring E up to the same level as W. and thus the row space of C is a self-orthogonal 2m 2m (isotropic) subspace of F2 , with respect to the sym- Let {e1,..., e2m} be the standard basis of F2 , and 2m plectic inner product. A maximal stabilizer, or a max- consider G ∈ CliffN . Let ci ∈ F2 be such that imal commutative subgroup (MCS) is a stabilizer of m † size 2 . Of particular interest are MCSs GE(ei)G = ±E(ci). (24)

m XN = E(Im, 0m) = {E(a, 0) | a ∈ F2 }, (18) Then the matrix FG whose ith row is ci is a symplec- m tic matrix such that ZN = E(0m, Im) = {E(0, b) | b ∈ F2 }. (19)

† We will refer to their elements as X stabilizers and GE(c)G = ±E(cFG) (25) Z stabilizers, respectively. Naturally, we identify X 2m 2m stabilizers with vectors (a, 0) ∈ F2 and Z stabilizers for all c ∈ F2 . We thus have a group homomorphism 2m with vectors (0, b) ∈ F2 . k Let S be a stabilizer group of size 2 . For ε ∈ Φ : CliffN −→ Sp(2m), G 7−→ FG. (26) {1, −1}, the complex vector space In addition, Φ is surjective with kernel ker Φ = N ∼ Fε(S) := {v ∈ C | Ev = εv for all E ∈ S} (20) PHWN [26], and thus CliffN /PHWN = Sp(2m).

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 4 Given the decomposition (5), one is naturally inter- 3 Support of the Clifford Group ested on preimages of respective symplectic matrices 2m via Φ. Namely, the unitary matrices The set EN = {E(c) | c ∈ F2 } is an orthonormal basis for the vector space MN (C) of N × N complex GD(P) := |vi 7−→ |vPi, matrices with respect to the Hermitian inner product t  vSv mod 4 GU (S) := diag i , (27) 1 † v∈ m h M | N i := Tr(M N). (31) F2 N ⊗r G (r) := (H ) ⊗ I m−r , Ω 2 2 Thus any matrix M ∈ MN (C) is a linear combination where H2 is the 2×2 Hadamard matrix, correspond to X M = α E(c), α = h E(c) | M i ∈ . (32) FD(P), FU (S), and FΩ(r), respectively [6]. Strictly c c C −1 c∈ 2m speaking, a preimage Φ (F) is meant up to HWN F2 (and up to a eighth root of unity which we have dis- We are interested in sums of Pauli matrices that yield regarded throughout [7]). Clifford matrices. The support of M ∈ MN (C) as in (32) with respect to EN is defined as Remark 3. Since Φ is a homomorphism we have that † −1 Φ(G ) = FG . It follows that if G ∈ CliffN is Hermi- supp(M) := {E(c) ∈ HWN | αc 6= 0} (33) tian then FG is a symplectic involution. Conversely, ∼ 2m = {c ∈ F2 | αc 6= 0}. (34) if F is a symplectic involution then G = Φ−1(F) sat- 2 When dealing with the support, we will conveniently isfies G ∈ HWN . switch between the two equivalent definitions. Thus † We will call G ∈ CliffN a Clifford transvection if E(c) ∈ supp(M) iff Tr(M E(c)) 6= 0. We say in this 2m case that M is supported on supp(M). Φ(G) is a symplectic transvection. For v ∈ F2 de- 2 fine Remark 5. (1) Since E(c) differs from D(c) only by a factor ik we see that Tr(M†E(c)) 6= 0 iff IN ± iE(v) † Gv := √ . (28) Tr(M D(c)) 6= 0. Thus, to avoid the additional 2 scaling factor, we will use matrices D(c) when For W ∈ HW we have computing supports/traces. N (2) It follows directly by the definition of support ( and (11) that the support of MD(x) (or D(x)M) † W, if WE(v) = E(v)W, GvWGv = is just the translate {x} + supp(M) for all x ∈ ∓iWE(v), if WE(v) = −E(v)W. 2m F2 . (29) It is clear that HWN is supported on singletons. It follows that a unitary M is a Clifford matrix It follows that Φ(G ) = T , and any Clifford v v iff MEM† is supported on a singleton for all E ∈ transvection is of this form (up to HW ). N HW . On the other hand, for G ∈ Cliff , we have In addition, we mentioned that Sp(2m) is gener- N N |supp(M)| = 2 iff G is a Clifford transvection (up to ated by transvections, thus G ∈ Cliff is a product of N HW ). In general, we have the following immediate Clifford transvections. We have proved the following. N consequence of Proposition4.

Proposition 4. Any Clifford matrix G ∈ CliffN can Corollary 6. Any Clifford matrix G is supported ei- be written as ther on a group S or on a coset E0S depending on whether G has trace or not. k Y IN + iEn E0 X Proof. Observe that E ∈ HW is traceless unless E = G = E0 √ = p αEE, (30) N n=1 2 |S| E∈S IN . Thus, for G as in (30) we have that Tr(G) 6= 0 iff E0 ∈ S. It also follows that supp(G) = E0S. where E ∈ HW ,S = hE ,..., E i, and α ∈ . 0 N 1 k E C Remark 7. One can easily construct non-Clifford One of the goals of this paper is to determine the matrices supported on a subgroup. For instance, it follows easily from Proposition 13 that T⊗m is sup- αE’s produced by Clifford matrices. In particular we ported on the subspace {0} × m, or alternatively, will see that if E1,..., Ek are independent then αE ∈ F2 on the subgroup ZN of diagonal Paulis. Thus, Corol- {±1, ±i}; see (63)-(64). If G is Hermitian then FG is an involution, and thus k ∈ {r, r + 1} where r = lary6 does not completely characterize CliffN . In fact, for any G ∈ CliffN , we have 2m−dim(Fix(FG)) with at least r transvections being independent. E(c) ∈ supp(G) ⇐⇒ Tr(GE(c)G†)G 6= 0 (35) √  2 λ ⇐⇒ Tr E(cFG)G 6= 0 (36) Note that (IN + i E(v))/ 2 is unitary iff λ = 1, 3, and otherwise it is a projection. ⇐⇒ E(cFG) ∈ supp(G), (37)

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 5 which implies that the support of G is an invariant Proof. (1) By definition we have GD(P) = P subspace of FG. m |vPihv|. Thus, for G = GD(P) we have v∈F2 By definition, in order to understand G ∈ CliffN X X t GD(a, b) = |vPihv| (−1)wb |w ⊕ aihw| it is sufficient to understand its action on HWN , v∈ m w∈ m which thanks to (26) can be understood via the action F2 F2 2m (44) of the corresponding symplectic matrix FG on F2 . X t Consider the fixed space Fix(FG) from (7). Then = (−1)vb |(v ⊕ a)Pihv|. (45) c ∈ Fix(FG) iff m v∈F2 † GE(c)G = ±E(cFG) = ±E(c), (38) m m For a ∈ F2 we will denote Fixa(P) = {v ∈ F2 | that is, iff E(c) either commutes or anticommutes v ⊕ vP = aP}. With this notation we have with G. Let us now consider the Pauli matrices that t  X vb commute with G, that is Tr GD(a, b) = (−1) . (46) v∈Fixa(P) † CG = {c ∈ Fix(FG) | GE(c)G = E(c)}. (39) m Since Fix(P) is a subspace of F2 we have that ⊥ It is then clear that the quotient Fix(FG)/CG cap- D(0, b) ∈ supp(G) iff b ∈ Fix(P) . On the other tures the Pauli matrices that anticommute with G. hand, for x ∈ Fixa(P) we have x ⊕ Fix(P) = ⊥s We will denote by (• ) the dual w.r.t. the symplec- Fixa(P). Indeed, the forward containment is triv- tic inner product (1). With this notation we have the ial and equality follows due to equal cardinalities. following. Thus, if Fixa(P) 6= ∅, we have that D(a, b) ∈ ⊥ ⊥s supp(G) iff b ∈ Fix(P) . Next, recall the subspace Proposition 8. supp(G) ⊆ CG . Res(P) from (8). We have that Fixa(P) 6= ∅ iff −1 Proof. We will show the reverse inclusion of the com- a ∈ Res(P ). We conclude that GD(P) is supported 2m ⊥s −1 ⊥ 2m plements. Indeed, let c ∈ F2 be such that c ∈/ CG . on Res(P ) × Fix(P) ⊂ F2 . Then by definition ⊥ Then, there exists v ∈ CG such that h c | v is = 1. Fix(P) = Res(P). It follows that E(v) commutes with G and anticom- (2) Let G := GU (S). Then mutes with E(c). Thus t t  X (v⊕a)S(v⊕a) +2vb   Tr GD(a, b) = i hv|v ⊕ ai.  † m Tr GE(c) = Tr E(v) E(v)GE(c) (40) v∈F2 (47)  †  = Tr E(v) GE(v)E(c) (41) It follows that D(a, b) ∈ supp(G) only if a = 0. So

 †  from now on we fix a = 0. It is shown in [8, Ap- = −Tr E(v) GE(c)E(v) (42) pendix A] that the sum in (47) is nonzero iff  = −Tr GE(c) , (43) X t t iwSw +2wb 6= 0. (48) which in turn implies Tr(GE(c)) = 0, and hence c ∈/ w∈W supp(G). t Consider the maps χS : w 7−→ wSw and χb : w 7−→ t Next, we completely characterize the supports of +2wb , and put χS,b := χS + χb. For v, w ∈ W we standard Clifford matrices (27) in terms of the invari- have ants (7)-(8) of the defining symplectic matrices. t t χS,b(v ⊕ w) = (v ⊕ w)S(v ⊕ w) + 2(v ⊕ w)b Proposition 9. The support of standard Clifford ma- (49) trices introduced in (27) satisfies the following: t t −1 ⊥ = (v + w)S(v + w) + 2(v + w)b mod 4 (1) supp(GD(P)) = Res(P ) × Fix(P) = −1 (50) Res(P ) × Res(P). t m = χS,b(v) + χS,b(w) + 2wSv (51) (2) Let S ∈ Sym(m) and W = ker(S) = {w ∈ F2 | wS = 0}. If Tr(GU (S)) 6= 0 then supp(GU (S)) = = χS,b(v) + χS,b(w) mod 4, (52) ⊥ {0} × W . Otherwise GU (S) is supported on a coset of {0}×W ⊥. As a consequence, the support where the last equality follows by the fact that wS = χS,b(w) of diagonal Cliffords is completely characterized 0 mod 2. Thus the map w 7−→ i is a character by the row/column space of the associated sym- of W . It follows that D(0, b) ∈ supp(G) iff χS,b(w) = metric S. 0 for all w ∈ W . r 2m By the above argument, it also follows that (3) Let Dr = {(x, 0m−r, x, 0m−r) | x ∈ F2} ⊂ F2 . Tr(G) 6= 0 iff χS(w) = 0 for all w ∈ W . In this Then supp(GΩ(r)) = (1r, 02m−r) ⊕ Dr, where 1r t P wb denotes the all ones vector of size r. As a con- case, (48) reduces to w∈W (−1) 6= 0, which holds ⊥ sequence, partial Hadamard matrices GΩ(r) are iff b ∈ W . Similarly, Tr(G) = 0 iff χS is not the supported on a coset of Res(FΩ(r)). trivial map. Since χS is an even-valued linear map

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 6 (mod 4) then there exists c (depending on S) such symplectic FCNOT = FD(P), which is a hyperbolic t that χS(w) = 2wc . But then it is clear that χS,b involution with Fix(FCNOT) = supp(CNOT). Note ⊥ ⊥ is the zero map iff c ⊕ b ∈ W iff b ∈ c ⊕ W . It also that Fix(FCNOT) = CCNOT, and thus equality follows that supp(G) = {0} × (c ⊕ W ⊥) for any such in Proposition8 can be achieved. In addition c as above. FCNOT = T0010T0100T0110. On the complex domain (3) Consider G = GΩ(r) for 0 ≤ r ≤ m, and let we have us first handle r = m, which corresponds to the fully 1 − i (I + iE )(I + iE )(I − iE E ) occupied Hadamard matrix in N = 2m dimensions. CNOT = √ · 1 √ 2 1 2 In this case, we have 2 8 (59) GD(a, b) 1 = (I + E + E − E E ), (60) 1 X t X t 2 1 2 1 2 = √ (−1)vw |vihw| (−1)xb |x ⊕ aihx| N v,w x∈ m F2 where E1 = E(00, 10), E2 = E(01, 00). It follows that (53) CNOT is supported on the MCS generated by E1 and 1 X t t E . = √ (−1)v(x⊕a) +xb |vihx|. (54) 2 N m v,x∈F2 m Example 12. (1) Let b ∈ F2 and consider the t symmetric matrix Sb := b b. In this case The above yields ⊥ [ker(Sb)] = {0√, b}. In addition GU (Sb) = 1 X t (IN ± iE(0, b))/ 2. TrGD(a, b) = √ (−1)v(v⊕a⊕b) . (55) (2) Let S be a diagonal matrix with diagonal dS. Let N m v∈F2 r = wtH(dS) be the number of non-zero elements in d . In this case F (S) is a product of transvec- Then map v 7−→ v(v⊕a⊕b)t is additive, and it is the S U tions T where v = (0, b ) where b is the nth trivial map iff a ⊕ b = 1 . Thus D(a, b) ∈ supp(G) vn n n n m nonzero row of S. Then iff a + b = 1m. It follows that supp(G) = (1m, 0m) ⊕ Dm. Then, for r < m, a similar argument implies that r Y IN + iE(vn) D(a, b) ∈ supp(G) iff a + b = 1 and a = GU (S) = √ , (61) 1:r 1:r r r+1:m 2 br+1:m = 0m−r. The proof is concluded with the n=1 observation that D = Res(F (r)). r Ω from which we may also conclude that Remark 10. Let P ∈ GL(m) and S ∈ Sym(m), Tr(GU (S)) 6= 0. and put G = GD(P)GU (S). Then G = t P vSv We end this section by computing the supports of v∈ m i |vPihv|, which in turn yields, ⊗m F2 the local Clifford group (Cliff2) ⊂ CliffN .   t t Proposition 13 (Support of local Cliffords). Let  X vSv +2vb ⊗m Tr GD(a, b) = Tr  i |(v ⊕ a)Pihv| G = G1⊗· · ·⊗Gm ∈ (Cliff2) , and let Si be the sup- v∈ m 2 F2 port of Gi in F2. Then supp(G) = σ(S1 × · · · × Sm), (56) where σ is the permutation (a1, b1, . . . , am, bm) 7−→ X t t (a , . . . , a , b , . . . , b ). = ivSv +2vb . (57) 1 m 1 m v∈Fixa(P) Proof. The result follows immediately by the fact that the trace function is multiplicative on pure tensors. Now the analysis continues as in Proposition9(2); see also [5, Lem. 6] for further details.

Example 11. We saw from Proposition9 that ma- trices GD(P) are supported on a subspace/subgroup. 4 On Hermitian Clifford matrices This is of course consistent with Corollary6 since these matrices always have trace. Indeed, entry (1,1) Hermitian Clifford matrices, on top of being interest- is always 1 since 0P = 0. The CNOT gate is of form ing on their own right, they also play a prominent role on understanding the third level of the Clifford hier- GD(P) where (3) † archy C . Indeed, by definition, CHWN C ⊂ CliffN  1 1  for all C ∈ C(3). The conjugate action preserves traces P = P−1 = . (58) 0 1 and the Hermitian property. Thus, other than the identity, only traceless Clifford matrices can emerge We have Res(P−1) = {00, 01} and from conjugate action with a third level matrix. With Fix(P)⊥ = {00, 10}. Thus, supp(CNOT) = the same notation as in Proposition4, we see that G † {0000, 0010, 0100, 0110}, as one can directly verify. is traceless iff E0 ∈/ S. Further, CEC must also Here we have m = 2 and CNOT corresponds to the be a Hermitian Clifford matrix for any Hermitian

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 7 Pauli matrix E. The corresponding symplectic ma- For the converse, one could argue similarly to show trix Φ(CEC†) is an involution, and symplectic ma- that the coefficients in (64) are ±1. However, we trices emerging in this way (C fixed, E varies) must point out here that the statement follows immediately commute. This fact, although elementary, is crucial from (62). because any group of commuting involutions is conju- Finally, for Hermitian G as in (62) we argued that 3 gate with some subgroup of SU from (4). This means E0 anticommutes with all En. Thus E0 cannot be that, for instance, the CNOT gate (or any Clifford contained on the commutative group generated by all matrix of form GD(P)) cannot emerge from a third En. As we have mentioned earlier, this implies that level action, despite FCNOT being a symplectic invo- G is traceless. lution; see also Example 11. We have mentioned that any Clifford matrix, is, up Remark 15. Recall Proposition4 where a Clifford to a multiplication by a Pauli, a product of transvec- matrix G is written as a generic sum of Hermitian tions. We have the following structural results if the matrices E. In (64) we have explicitly computed transvections involved are independent. the coefficients αE, and evidently, they are of form Q(v) m i , v ∈ F2 where Q is a quadratic form mod 4. Theorem 14. Let En = E(cn), n = 1, . . . , k, be a set This generalizes a result of [6] (see also [8]) where of k independent Hermitian Pauli matrices. Let also the authors showed that the coefficients of diago- E0 = E(c0) be a Hermitian Pauli matrix. Then, the nal Clifford matrices are determined by a quadratic Clifford matrix form. Indeed, diagonal Clifford matrices are of form k GU (S), S ∈ Sym(m), and we see the aforementioned Y 1 G = E0 √ (I + iEn), (62) quadratic form in (47)-(48). n=1 2 Remark 16. If G ∈ CliffN is Hermitian, we men- is Hermitian iff E anticommutes with all E and all 0 n tioned that the corresponding symplectic matrix FG En commute with each other. As a consequence, if G must be an involution, which we also mentioned can is Hermitian then it is also traceless. be written as a product of k ∈ {r, r+1} transvections, Proof. Let C := (AB) be the k × 2m binary matrix r = 2m − dim(Fix(FG)), where at least r are inde- pendent. Theorem 14 settles the scenario when F is whose nth row is cn = (an, bn). Since all the En are G independent we have that rank (C) = k. Using (12) a product of only independent transvections. When we have that the additional transvection Tr+1 is dependent of the other r transvections, then multiplying G = T1 ··· Tr t E0 X d(I +C)d G = √ i k e E(dC mod 4) with Tr+1 may preserve the support of G (see Exam- 2k d∈ k ple 11) or reduce the support of G. In the latter F2 (63) 1 t instance the supported is reduced by half. Keeping X d(Ik+C)d = √ i e E0E(dC mod 4), track of the support of GT becomes tedious and k r+1 2 k d∈F2 involves sign chasing that depends on the commuta- tivity relation of T1,..., Tr. t where Ce is the k×k matrix whose (i, j) entry is ajbi − t aibj mod 4 if i < j and 0 else. If we write c0 = (a0, b0), then (63) can be further rewritten as 5 On (Generalized) Semi-Clifford Ma- 1 t t t t X d(Ik+C)d dAb0−a0B d trices G = √ i e i E(dC + c0) 2k d∈ k (k) F2 For k ≥ 3 the levels C of the Clifford hierarchy (64) do not form a group, and thus a complete character- It follows immediately that G is Hermitian iff all co- ization becomes challenging. In [35] the authors use efficients in (64) are ±1. the notion of semi-Clifford matrices to achieve par- By looking at the standard basis of k, i.e., d = e , F2 n tial results. A unitary matrix U ∈ U(N) is called and corresponding coefficients, we see that a bt ⊕ n 0 semi-Clifford if there exists a MCS S1 ⊂ HWN such a bt = 1 for n = 1, . . . , k, which in turn means that † 0 n that S2 = US1U is also MCS. Since the Clifford E anticommutes with E . To show that E , E com- 0 n j n group CliffN permutes stabilizers of a given dimen- mute, consider d of weight two with ones in positions sion, a Clifford matrix is trivially semi-Clifford. It is j, n. This corresponds to looking at the coefficient shown in [35] that for m = 1, 2 qubits the Clifford of E0EjEn. Because E0 anticommutes with both hierarchy is comprised of semi-Clifford matrices, and t t t dAb0−a0B d (3) Ej, En, the term i contributes ±1 and for m = 3 qubits the third level C is comprised of t the term idIkd = iwt(d) contributes −1. Thus, the semi-Clifford matrices. Moreover, they show that for m > 2 qubits that there exist non semi-Clifford ma- overall coefficient will be ±1 only if Ej, En commute. trices in each level C(k), k > 3, and conjecture that the 3Two groups S and S0 are called conjugate if there exists g third level C(3) is comprised of semi-Cliffords for any such that S0 = gSg†. number of qubits. The conjecture was disproved by

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 8 Gottesman and Mochon via a counterexample with Remark 19. In [35] it was shown that a semi-Clifford m = 7 qubits; see [1]. On the other hand, the di- (k) C ∈ Cd is of the form C = G1DG2 for some agonal elements of each level, denoted C(k) do form (k) d G1, G2 ∈ CliffN and D ∈ Cd . Theorem 18 further a group [35, Prop. 4], and are completely character- extends this result by characterizing the Clifford ma- ized in [14]. The QFD gates of [28] represent all 1- trices that appear into decomposition of C. Thus, we and 2-local diagonal gates in the hierarchy, and thus obtain a complete characterization of semi-Clifford el- forming a particularly nice subclass of diagonal gates. ements in the Clifford hierarchy. We believe that this Remark 17. Multiplying by Clifford matrix pre- result, along with the notion of the support can be serves the levels of the hierarchy. Thus, without loss used in many applications, e.g., design of flag gad- of generality, we will consider semi-Clifford matrices gets. (and any matrix in the hierarchy) up to multiplica- The argument of Theorem 18 holds in a slightly tion by Clifford matrices. This enables us to ad- more general setting. just any semi-Clifford U matrix so that it fixes any † Remark 20. (1) Let C be any unitary matrix that given given MCS S. Indeed, assume US1U = S2. m † fixes a MCS S = hE1,..., Emi. For d ∈ F2 de- Let G1, G2 ∈ CliffN be such that G1SG = S1 and 1 note S := h(−1)d1 E ,..., (−1)dm E i, and put G S G† = S. Then G UG is a semi-Clifford ma- d 1 m 2 2 2 2 1 F (S ) := |ψ i. For any E ∈ S we have trix that fixes S. As mentioned earlier, by [26, Alg. + d d d 0 1], there exists G3 ∈ CliffN such that G3G2UG1 fixes EC|ψdi = CE |ψdi = ±C|ψdi, (67) S pointwise. 0 m and thus C|ψdi ∈ F+(Sd0 ) for some d ∈ F2 . (k) Theorem 18. Let C ∈ C be a unitary matrix that This means that C|ψdi = λd|ψd0 i for some eigen- fixes the group of diagonal Paulis ZN = E(0m, Im). value λd. In particular, C is a monomial matrix Then C = DE(a, 0)GD(P), for some diagonal D ∈ with respect to the eigenbasis ES := {|ψdi | d ∈ (k) m m}. Cd , P ∈ GL(m), and a ∈ F2 . F2 (2) Let C be a unitary matrix that fixes the span of We will make use of the structure of first order ZN , that is, C maps any diagonal to another di- Reed-Muller codes to prove Theorem 18. Let C = agonal. Then C is a monomial matrix. In partic- t m {(vb mod 2) m | b ∈ }. Then, the first-order v∈F2 F2 ular, any semi-Clifford matrix is a monomial ma- m m−1 Reed-Muller code is the linear [2 , m+1, 2 ]2-code trix up to some Clifford correction; see also [35, Prop. 2]. RM(1, m) = C ∪ {c ⊕ 1 | c ∈ C}. (65) Let C ∈ C(3) be such that it fixes some subgroup Sb The automorphism group of RM(1, m) is the general (of HW ) under conjugation; see also Remark 17. affine group GA(m) of maps v 7−→ vP ⊕ a, P ∈ N m Then, by [26, Alg. 1], there exists G ∈ CliffN , GL(m), a ∈ F2 ; see [21, Chapter 13] for instance. produced as a sequence of transvections, such that Cb := GC fixes Sb point-wise. Let Sb⊥s be all the Pauli Proof of Theorem 18. Assume that C ∈ C(k) fixes matrices that commute with elements of Sb. Proceed- N ZN , and let v ∈ C be a common eigenvector of all ing similarly as in the proof of Proposition8 we have D(0, b) ∈ ZN . Then, by assumption, we have the following.

⊥s E(0, b)Cv = CE(0, b0)v = ±Cv, (66) Proposition 21. supp(Cb ) ⊂ Sb , and thus supp(C) ⊂ S := {EE0 | E ∈ supp(G−1), E0 ∈ Sb⊥s }. which in turn implies that Cv is also a common eigen- Corollary 22. Let C be a unitary matrix and S be vector of Z . Thus C maps the common eigenvector N a MCS. If C fixes S pointwise then supp(C) ⊂ S. |vi to another common eigenvector which is of the The converse is also true. This property characterizes form α |π(v)i for some π(v) ∈ m and α ∈ . In v F2 v C semi-Clifford matrices up to multiplication by Clif- other words, C is a monomial map, that is C = DΠ, ford. In particular, as for Hermitian Clifford matri- where D is the diagonal matrix with entry α in po- v ces, we have that C is supported on a commutative sition π(v) and Π is the permutation v 7−→ π(v). (k) (k) subgroup. The assumption C ∈ C implies D ∈ Cd . By con- struction, the diagonals of Paulis in ZN are of form In the reminder of this section we show that Corol- t (3) vb lary 22 holds for the entire third level C of the hier- ±((−1) )v∈ m , and we point out that the expo- F2 archy (always up to Cliffords). Note that this is not a nents of such diagonals are precisely the elements of trivial step because, as mentioned before, there exist RM(1, m). Thus Π induces an isometry on RM(1, m), elements in C(3) that are not semi-Cliffords [1]. which as we mentioned must be an invertible affine (3) map. That is, Π = E(a, 0)GD(P) for some diago- Lemma 23. For C ∈ C there exists a Pauli Ee such m † nal P ∈ GL(m), and a ∈ F2 . In particular we have that CECe is also a Pauli. As a consequence, there (k) Π ∈ CliffN , which along with the assumption C ∈ C exists a Clifford correction G such that GC fixes (i.e., (k) implies D ∈ Cd . commutes with) some Pauli matrix.

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 9 Proof. Let C ∈ C(3) and consider the map Theorem 25. Let C be a unitary matrix from C(3). ( Then there exists a Clifford G such that GC is sup- φC Φ PHWN −−→ CliffN −−→ Sp(2m) ported on a maximal commutative subgroup of HW . ϕC : N E 7−→ CEC† 7−→ Φ(CEC†) Proof. From Lemma 23 we know that there exists where Φ is the map from (26) and PHWN is the some Clifford H such that HC commutes with some projective form of HWN that ignores phases. Then E ∈ HWN . Now consider the group S = hEi and its k ⊥s ker ϕC ⊂ PHWN has size 2 for some k ≥ 0. So, we normalizer in HWN which we denote by S . Then, 2m−k see that G := im ϕC ⊂ Sp(2m) has size 2 . since HC preserves S under conjugation, it is a valid 2m Let G act on F2 \{0}. Since the size of an orbit logical operator for the [[m, m − 1]] stabilizer code de- must divide the size of the group G, each orbit has fined by S, i.e., it maps code states (+1 eigenvectors size a power of 2 as well. However, since the orbits of E) to code states. Denote this logical (m−1)-qubit 2m (3) partition a set of size 2 − 1 (odd), there must exist operation realized by HC as CH. Since HC ∈ C , an orbit of odd size, and that size must be 20 = 1. we know that HC must satisfy the necessary conjuga- 2m This means that there exists 0 6= c ∈ F2 that is fixed tion conditions on logical Paulis as determined by CH. by all symplectic matrices Φ(CEC†). The definition These conditions can only correspond to physical re- of Φ yields that the Hermitian Pauli Ee := E(c) either alizations of logical Clifford gates (since physical Clif- commutes or anticommutes with all CEC†. In other fords cannot realize anything above the second level (2) (3) words CliffN = C ). Hence, we conclude that CH ∈ C † † (in the logical space). ECECe = αECEC Ee, αE = ±1, (68) First, it has already been shown in [35] that op- erations in C(3) are semi-Clifford for 1 and 2 qubits. for all Paulis E. Now put Ce = ECe Ee. Let also σE = Using Corollary 22, this automatically means that up ±1 be such that EEe = σEEEe. We have that to some Clifford correction such gates are supported † † ECe = Ce Ee and ECe = Ce Ee, (69) on a MCS. Therefore, we consider the induction hy- (3) which combined with (68) yields pothesis that for (m − 1) qubits, any C element is supported on a MCS, up to multiplication by some † † αECEC = σECEe Ce (70) Clifford. Applying this hypothesis for CH above, we see that there exists some (m − 1)-qubit logical Clif- for all Paulis E. Now it is easy to see that CCe † is a 0 0 0 † 0 ford G such that G CH is supported on a MCS (of Pauli E , and thus (69) implies CECe = E Ee. size 2m−1). Note that a logical Clifford operation is Remark 24. The proof of Lemma 23 could have defined by its action on logical Paulis. been concluded using the language of stabilizer codes. Let this MCS be generated by logical (m−1)-qubit With the same notation, let Ee be such that it either Paulis E1, E2,..., Em−1, and let E1, E2,..., Em−1 ∈ † S⊥s form their respective physical m-qubit realiza- commutes or anticommutes with all CEC ∈ CliffN . Now consider the stabilizer group S = hEei and the tions in HWN . These realizations are automatically defined once 2(m − 1) m-qubit Pauli operations are corresponding stabilizer codes Fε := Fε(S), for ε = † chosen to be the appropriate physical realizations of ±1 (see (20)). We have that CEC Fε = ±Fε for all Paulis E. In other words, for any v ∈ F , we have the logical X and Z on the (m − 1) logical qubits. As ε 0 that G CH is supported on Ei’s, it clearly commutes with   each one of them. Hence, by taking G0 to be an m- εE · CEC†v = ± CEC† · εEv = ±CEC†v (71) 0 e e qubit Clifford that forms a physical realization of G , 0 P we see that G (HC) commutes with each Ei. Note holds for all E. Next, express C = E αEE and sum both sides in (71) to obtain that, by definition of realizing a logical gate, such a G0 must preserve the stabilizer S, i.e., commute with ! ! X 0 X E, and act on Ei as G acts on Ei. εE αEE v = ε αEE v. (72) e Finally, consider the group hE, E1, E2,..., Em−1i. E E This is clearly a MCS and (G0H)C fixes it point- In other words, εECe v = ε0Cv, and thus C permutes wise. Therefore, by applying Corollary 22 we see that GC := (G0H)C is supported on this MCS. This com- {Fε}. This is equivalent with C mapping Ee to some other Pauli. pletes the induction. Next we prove the main result about the support Lemma 23 constitutes a crucial property of the of gates in C(3), which can then be straightforwardly third level C(3) of the Clifford hierarchy. This prop- used to show that every gate in C(3) is generalized erty is of course exclusive to the third level because semi-Clifford. Recall that a generalized semi-Clifford we highly rely on the fact that im φC is a subgroup of matrix is a unitary matrix that maps under conjuga- CliffN . This in turn enables an induction argument on tion the span of some MCS to the span of some other the number of qubits, rather than the typical induc- MCS. tion arguments on the levels of the Clifford hierarchy.

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 10 It is worth mentioning that the “up to Clifford” is in- multiplication and conjugation (which, surprisingly, deed necessary throughout the paper. For instance, is unknown). This, among other things, would give with regards to Lemma 23, the physical Clifford per- a closed form description of the support of any Clif- mutation ford group element. Next, with the ultimate goal of   1 0 0 0 completely characterizing the third level of the hier-  0 0 0 1  archy, we will consider the converse. That is, find- G =   (73)  0 1 0 0  ing sufficient conditions under which a unitary U, 0 0 1 0 supported on some MCS, belongs to the third level. does not fix (commute with) any Pauli matrix. Simi- We expect the coefficients to be (scaled) eighth roots larly, one can easily produce other instances of exam- of unity that are perhaps determined by third-order ples that require the “Clifford correction”. Reed-Muller codes. Finally, we will use the structural If C ∈ C(3) is supported on a MCS S then it triv- results of this paper to develop flag gadgets for third ially fixes the span of S, and therefore C is a gener- level operators, as well as reduce circuit complexity alized semi-Clifford matrix. The converse is not true for these operators. since, for instance, even the CNOT gate fixes Z4 but obviously is not supported on Z4. In fact, any gate GD(P) fixes ZN by construction but is not supported Acknowledgements on Z (see Proposition9(1)). N The work of TP and OT was supported in part by Corollary 26 ([1, Thm. 1.1]). Every C ∈ C(3) is a the Academy of Finland under the grant 334539. The generalized semi-Clifford matrix. work of NR and RC was supported in part by NSF un- Proof. By Theorem 25, there exists a Clifford correc- der the grant CCF1908730. TP and OT thank Robert tion G such that GC is supported on a MCS S. Then Calderbank for his hospitality during visits to Duke GC fixes the span of S, and C maps the span of S University. to the span of G†SG. Thus C is generalized semi- Clifford. References

6 Conclusions and Future Research [1] Salman Beigi and Peter W. Shor. C3, semi- Clifford and generalized semi-Clifford operations. In this paper we study the Clifford hierarchy via the Quantum Info. Comput., 10(1):41–59, January Pauli/Weyl expansion/support. First, we consider 2010. URL http://arxiv.org/abs/0810.5108. the Clifford group, that is, the second level of the [2] P. Oscar Boykin, Tal Mor, Matthew Pulver, hierarchy. We show that every element of the group Vwani Roychowdhury, and Farrokh Vatan. On is supported on a subgroup of the Pauli group (or a universal and fault-tolerant quantum comput- coset, when traceless). Additionally, we give a closed ing: A novel basis and a new constructive form description of the support of standard group ele- proof of universality for Shor’s basis. page ments, and show that the coefficients are determined 486, USA, 1999. IEEE Computer Society. DOI: by a quadratic form modulo 4. We argue that the https://doi.org/10.1109/SFFCS.1999.814621. Hermitian elements of the group play a prominent [3] Sergey Bravyi and Jeongwan Haah. Magic- role on understanding the third level of the hierar- state distillation with low overhead. chy. For this reason, we treat them separately, and Phys. Rev. A, 86(5):052329, 2012. DOI: among other things, we show that they are supported https://doi.org/10.1103/PhysRevA.86.052329. on a commutative subgroup of the Pauli group (or a [4] Sergey Bravyi and Alexei Kitaev. Uni- coset). Next, we consider the third level of the hier- versal quantum computation with ideal archy. Our treatment is up to Clifford equivalence, Clifford gates and noisy ancillas. Phys. which, at any rate, preservers the levels of the hier- Rev. A, 71:022316, Feb 2005. DOI: archy. We show that, up to such equivalence, every https://doi.org/10.1103/PhysRevA.71.022316. third level matrix commutes with at least one Pauli [5] Robert A. Calderbank, Stephen Howard, matrix. This constitutes the main building block of a Sina Jafarpour, and Jeremy Kent. Sparse powerful induction argument on the number of qubits, approximation and compressed sensing us- which we use to prove that every third level matrix, ing the Reed-Muller sieve. Technical Report, up to Clifford equivalence, is supported on a maxi- Princeton University, TR-888-10, 2010. URL mal commutative subgroup of the Pauli group. We https://www.cs.princeton.edu/research/ believe that such induction argument can be further techreps/TR-888-10. exploited in various aspects of quantum computation [6] Robert A. Calderbank, Peter J. Cameron, and quantum error-correction. William M. Kantor, and Jaap J. Seidel. Z4- In future research, we will consider the behaviour of Kerdock codes, orthogonal spreads, and extremal the support under elementary transformations such as Euclidean line-sets. Proceedings of the London

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 11 Mathematical Society, 75(2):436–480, 1997. DOI: LU-LC conjecture, diagonal local operations and https://doi.org/10.1112/S0024611597000403. quadratic forms over GF(2). Quantum Inf. Com- [7] Robert A. Calderbank, Eric M. Rains, Peter W. put., 8(3-4):263–281, 2007. URL http://arxiv. Shor, and Neil J. A. Sloane. Quantum error org/abs/0707.4000. correction via codes over GF(4). IEEE Trans. [20] Robert Koenig and John A. Smolin. How to effi- Inform. Theory, 44(4):1369–1387, 1998. DOI: ciently select an arbitrary Clifford group element. https://doi.org/10.1109/18.681315. J. Math. Phys., 55(12):122202, Dec 2014. DOI: [8] Robert A. Calderbank, Stephen Howard, and https://doi.org/10.1063/1.4903507. Sina Jafarpour. Construction of a large class of [21] Florence J. MacWilliams and Neil J. A. Sloane. matrices satisfying a statistical isometry prop- The theory of error-correcting codes, volume 16. erty. In IEEE Journal of Selected Topics in Elsevier, 1977. Signal Processing, Special Issue on Compressive [22] Dmitri Maslov and Martin Roetteler. Shorter Sensing, volume 4, pages 358–374, 2010. DOI: stabilizer circuits via Bruhat decompo- https://doi.org/10.1109/JSTSP.2010.2043161. sition and quantum circuit transforma- [9] David Callan. The generation of Sp(F2) tions. IEEE Transactions on Informa- by transvections. J. Algebra, 42(2):378– tion Theory, 64(7):4729–4738, 2018. DOI: 390, 1976. DOI: https://doi.org/10.1016/0021- https://doi.org/10.1109/TIT.2018.2825602. 8693(76)90105-8. [23] Michael A. Nielsen and Isaac L. Chuang. Quan- [10] Christopher Chamberland and Michael E. Bev- tum computation and . erland. Flag fault-tolerant error correction Cambridge University Press, Cambridge, 2000. with arbitrary distance codes. Quantum, 2:53, [24] Onorato T. O’Meara. Symplectic groups, vol- 2018. DOI: https://doi.org/10.22331/q-2019-05- ume 16 of Mathematical Surveys. American 20-143. Mathematical Society, Providence, R.I., 1978. [11] Rui Chao and Ben W Reichardt. Quantum [25] Tefjol Pllaha, Olav Tirkkonen, and Robert A. Error Correction with Only Two Extra Qubits. Calderbank. Reconstruction of multi-user Phys. Rev. Lett., 121(5):050502, 2018. DOI: binary subspace chirps. In 2020 IEEE https://doi.org/10.1103/PhysRevLett.121.050502. International Symposium on Information [12] Rui Chao and Ben W Reichardt. Fault- Theory (ISIT), pages 531–536, 2020. DOI: tolerant quantum computation with few qubits. https://doi.org/10.1109/ISIT44484.2020.9174412. npj Quantum Inf., 4(1):42, 2018. DOI: [26] Narayanan Rengaswamy, Robert A. Calder- https://doi.org/10.1038/s41534-018-0085-z. bank, Henry D. Pfister, and Swanand Kadhe. [13] Rui Chao and Ben W. Reichardt. Flag fault- Synthesis of logical Clifford operators via tolerant error correction for any stabilizer symplectic geometry. In 2018 IEEE In- code. PRX Quantum, 1:010302, Sep 2020. DOI: ternational Symposium on Information The- https://doi.org/10.1103/PRXQuantum.1.010302. ory (ISIT), pages 791–795, 2018. DOI: [14] Shawn X. Cui, Daniel Gottesman, and Anirudh https://doi.org/10.1109/ISIT.2018.8437652. Krishna. Diagonal gates in the Clifford hierar- [27] Narayanan Rengaswamy, Robert Calderbank, chy. Phys. Rev. A, 95:012329, Jan 2017. DOI: Michael Newman, and Henry D. Pfister. On Op- https://doi.org/10.1103/PhysRevA.95.012329. timality of CSS Codes for Transversal T . arXiv [15] Kathleen S. Gibbons, Matthew J. Hoff- preprint arXiv:1910.09333, 2019. URL http: man, and William K. Wootters. Dis- //arxiv.org/abs/1910.09333. crete phase space based on finite fields. [28] Narayanan Rengaswamy, Robert Calderbank, Phys. Rev. A, 70:062101, Dec 2004. DOI: and Henry D. Pfister. Unifying the Clifford https://doi.org/10.1103/PhysRevA.70.062101. hierarchy via symmetric matrices over rings. [16] Daniel Gottesman. Stabilizer codes and quan- Phys. Rev. A, 100(2):022304, 2019. DOI: tum error correction. PhD thesis, Cali- https://doi.org/10.1103/PhysRevA.100.022304. fornia Institute of Technology, 1997. DOI: [29] Narayanan Rengaswamy, Robert Calderbank, https://doi.org/10.7907/rzr7-dt72. Michael Newman, and Henry D. Pfister. Classical [17] Daniel Gottesman and Isaac L. Chuang. Demon- coding problem from transversal t gates. In 2020 strating the viability of universal quantum com- IEEE International Symposium on Informa- putation using teleportation and single-qubit op- tion Theory (ISIT), pages 1891–1896, 2020. DOI: erations. Nature, 402(6760):390–393, 1999. DOI: https://doi.org/10.1109/ISIT44484.2020.9174408. https://doi.org/10.1038/46503. [30] Theerapat Tansuwannont, Christopher Cham- [18] David Gross. Hudson’s theorem for finite- berland, and Debbie Leung. Flag fault-tolerant dimensional quantum systems. Journal of Math- error correction, measurement, and quantum ematical Physics, 47(12):122107, 2006. DOI: computation for cyclic Calderbank-Shor-Steane https://doi.org/10.1063/1.2393152. codes. Phys. Rev. A, 101(1):012342, 2020. DOI: [19] David Gross and Maarten Van den Nest. The https://doi.org/10.1103/PhysRevA.101.012342.

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 12 [31] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The re- source theory of stabilizer quantum computa- tion. New Journal of Physics, 16(1):013009, jan 2014. DOI: https://doi.org/10.1088/1367- 2630/16/1/013009. [32] Eugene Wigner. On the quantum cor- rection for thermodynamic equilibrium. Phys. Rev., 40:749–759, Jun 1932. DOI: https://doi.org/10.1103/PhysRev.40.749. [33] William K. Wootters. Picturing qubits in phase space. IBM Journal of Research and Development, 48(1):99–110, 2004. DOI: https://doi.org/10.1147/rd.481.0099. [34] Bei Zeng. Quantum operations and codes be- yond the Stabilizer-Clifford framework. PhD the- sis, Massachusetts Institute of Technology, 2009. URL http://hdl.handle.net/1721.1/53235. [35] Bei Zeng, Xie Chen, and Isaac L. Chuang. Semi-Clifford operations, structure of Ck hierarchy, and gate complexity for fault- tolerant quantum computation. Phys. Rev. A, 77:042313, Apr 2008. DOI: https://doi.org/10.1103/PhysRevA.77.042313. [36] Xinlan Zhou, Debbie W. Leung, and Isaac L. Chuang. Methodology for construction. Phys. Rev. A, 62:052316, Oct 2000. DOI: 10.1103/PhysRevA.62.052316.

Accepted in Quantum 2020-11-26, click title to verify. Published under CC-BY 4.0. 13