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Code Transversal gates Gates not transversal operators. Consider the stabilizer = XXXX,ZZZZ S h i [[5, 1, 3]] PH, M3 H, P , CNOT, T where indicates a generating set, so [[7, 1, 3]] H, P , CNOT T h·i [[9, 1, 3]] CNOT H, P , T = IIII,XXXX,ZZZZ,YYYY . [[15, 1, 3]] T , CNOT H S { } m [[2 − 1, 1, 3]] Tm, CNOT H We have supp (XXXX)= 1, 2, 3, 4 and wt (XXXX)= 4, for example. Finally,{ the centralizer} is ( ) = TABLE I C S COLLECTION OF SOME [[n, 1, 3]] CODESANDTHEIRPROPERTIES. ,ZZII,ZIZI,XIXI,XXII . hS i THESECONDCOLUMNLISTSALLOWEDTRANSVERSALGATES, A stabilizer consists of 2m Pauli operators for some nonneg- ANDTHETHIRDCOLUMNGIVESTHEGATESWHICHCANNOT ative integer m n and is generated by m independent Pauli BETRANSVERSALONTHECORRESPONDINGCODES. H IS THE ≤ HADAMARD GATE, P = diag(1,i) IS THE PHASE GATE, T = operators. As the operators in a stabilizer are Hermitian and iπ/4 iπ/2m−2 diag (1,e ) AND Tm = diag (1,e ). FOR THE [[5, 1, 3]] mutually commuting, they can be diagonalized simultaneously. CODE, M3 IS A THREE-QUBIT CLIFFORD OPERATION (SEE PAGE 89 m Definition 2: An n-qubit stabilizer code Q is the joint OF [5]). THE [[2 − 1, 1, 3]] CODE WITH TRANSVERSAL Tm GATE IS A CSS CODECONSTRUCTEDFROMPUNCTUREDBINARY REED- eigenspace of a stabilizer (Q), ∗ MULLER CODE RM (1, m) AND ITS EVEN SUBCODE [15]. S 2 n Q = ψ (C )⊗ R ψ = ψ , R (Q) (1) {| i ∈ | | i | i ∀ ∈ S } where each state vector ψ is assumed to be normalized. Q n m | i additional qubit permutation operations inside code blocks. has dimension 2 − and is called an [[n,k,d]] stabilizer code, Our main result, given in Section III, proves a special case where k = n m is the number of logical qubits and d is − of this “T versus U” incompatibility, where Q is a GF (4)- the minimum distance, which is the weight of the minimum additive code and coordinate permutations are not allowed. weight element in ( ) . The code Q can correct errors of d 1 C S \ S Our proof relies on earlier results by Rains [12] and Van den weight t − . ≤⌊ 2 ⌋ Nest [16], generalized to multiple blocks encoded in additive Example 2: Continuing, we have quantum codes. In Section IV, we prove T vs. U incompati- Q = span 0000 + 1111 , 0011 + 1100 , bility for a single block of qubits encoded in a GF (4)-additive {| i | i | i | i code, by clarifying the effect of coordinate permutations. In 1010 + 0101 , 1001 + 0110 | i | i | i | i} Section V, we consider the allowable transversal gates on so n =4, m =2, dim Q =4, and k =2. From C( ) , we additive codes, using the proof technique we employ. We also see that d =2. Therefore, Q is a [[4, 2, 2]] code. S \ S present a simple construction based on classical divisible codes Each set of mutually commuting independent elements of that yields many quantum codes with non-Clifford transversal n stabilizes a quantum codeword and generates an abelian gates on a single block. We begin in the next section with ( ) CsubgroupS of the centralizer. This leads to the isomorphism some preliminary definitions and terminology. ( )/ = that maps each element X ,Z to a C S S ∼ Gk i i ∈ Gk coset representative X¯i, Z¯i C( )/ [5]. The isomorphism II. PRELIMINARIES ∈ S S associates the k logical qubits to logical Pauli operators X¯i, Z¯i This section reviews definitions and preliminary results for i = 1,...,k, and these operators obey the commutation about additive codes [9], Clifford groups and universality, relations of . Gk automorphism groups, and codes stabilized by minimal el- Example 3: One choice of logical Pauli operators for the ements. Throughout the paper, we only consider GF (4)- [[4, 2, 2]] code is X¯1 = XIXI, Z¯1 = ZZII, X¯2 = XXII, additive codes, i.e. codes on qubits, leaving more general codes and Z¯2 = ZIZI. These satisfy the commutation relations of to future work. We use the stabilizer language to describe 2. GF (4)-additive quantum codes, which are also called binary G stabilizer codes. B. Universality A. Stabilizers and stabilizer codes Stabilizer codes are stabilized by subgroups of the Pauli group, so some unitary operations that map the Pauli group Definition 1: The n-qubit Pauli group n consists of all n G to itself also map the stabilizer to itself, preserving the code 4 4 operators of the form R = αRR1 Rn, where α× 1, i is a phase factor and each⊗···⊗R is either the space. R ∈ {± ± } i 2 2 identity matrix I or one of the Pauli matrices X, Y , or Definition 3: The n-qubit Clifford group n is the group × L Z. A stabilizer is an abelian subgroup of the n-qubit Pauli of unitary operations that map n to itself under conjugation. S G group which does not contain I. A support is a subset One way to specify a gate in n is to give the image of a Gn − L of [n] := 1, 2,...,n . The support supp (R) of an operator generating set of n under that gate. n is generated by the { } G L R n is the set of all i [n] such that Ri differs from the single qubit Hadamard gate, identity,∈ G and the weight wt∈(R) equals the size supp(R) of | | H : (X,Z) (Z,X), (2) the support. The set of elements in n that commute with all → G elements of is the centralizer ( ). the single qubit Phase gate Example 1:S We have the relationC S [XXXX,ZZZZ]=0 4 where XXXX = X⊗ represents a tensor product of Pauli P : (X,Z) ( Y,Z), (3) → − 3 and the two-qubit controlled-not gate Let V(Q) denote the set of logical gates on Q. When Q is understood, we simply say that the gate U is a logical gate. CNOT :(XI,IX,XX,ZI,IZ,ZZ) (4) → The logical gates V(Q) are a group that is homomorphic to (XX,IX,XI,ZI,ZZ,IZ) (5) SU(2k) since it is possible to encode an arbitrary k-qubit state in the code. by the Gottesman-Knill theorem [4], [3]. Example 4: For the [[4, 2, 2]] code, P = 1 (I 4 + X 4 + Definition 4: A set of unitary gates G is (quantum) com- Q 4 ⊗ ⊗ Y 4 + Z 4). Any unitary acting in the code manifold putationally universal if for any n, any unitary operation ⊗ ⊗ U SU(2n) can be approximated to arbitrary accuracy ǫ α( 0000 + 1111 )+ β( 0011 + 1100 )+ in the∈ sup operator norm by a product of gates in G. | i | i | i | i || · || γ( 1010 + 0101 )+ δ( 1001 + 0110 ) In notation, ǫ > 0, V = V1V2 ...Vη(ǫ) where each Vi | i | i | i | i G s.t. V ∀U <∃ ǫ. In this definition, gates in G may∈ is a logical gate. be implicitly|| − mapped|| to isometries on the appropriate 2n- Definition 7: The full automorphism group Aut (Q) of Q dimensional Hilbert space. is the collection of all logical operations on Q of the form The Gottesman-Knill theorem asserts that any set of Clifford PπU where Pπ enacts the coordinate permutation π and group gates can be classically simulated and is therefore not U = U U is a local unitary operation. The product of 1 ⊗···⊗ n (quantum) computationally universal. Quantum teleportation two such operations is a logical operation of the same form, is one technique for circumventing this limit and constructing and operations of this form are clearly invertible, so Aut (Q) computationally universal sets of gates using Clifford group is indeed a group. More formally, the full automorphism gates and measurements of Pauli operators [17], [18]. There group Aut (Q) of Q is sometimes defined as the subgroup is a large set of gates that arise in fault-tolerant quantum of logical operations contained in the semidirect product n n computing through quantum teleportation. (S ,SU(2)⊗ ,ν), where ν : S Aut (SU(2)⊗ ) is given n n → Definition 5: The (n) hierarchy is a set of gates that can by Ck be achieved through quantum teleportation and is defined ν(π)(U1 Un)= Uπ(1) Uπ(n) (8) (n) ⊗···⊗ ⊗···⊗ recursively as follows: 1 = n and C G and Sn is the symmetric group of permutations on n items. (n) (n) (n) n n The notation Sn⋉SU(2)⊗ is sometimes used. When Aut (Q) k = U SU(2 ) UgU † k 1 g 1 , (6) C { ∈ | ∈C − ∀ ∈C } is considered as a semidirect product group, an element (n) for k > 1. k is a group only for k = 1 and k = 2 and (π,U1 Un) Aut (Q) acts on codewords as U1 (n) C ⊗···⊗ ∈ ⊗ = . Un and on coordinate labels as π. The product of two C2 Ln ···⊗ The Clifford group generators H,P,CNOT plus any automorphisms in Aut (Q) is other gate outside of the Clifford{ group is computationally} (π ,U)(π , V ) = (π π , (U V ) (U V )), (9) universal [13]. For example, the gates T = diag (1,eiπ/4) 1 2 1 2 π2(1) 1 ⊗···⊗ π2(n) n (1) (1) (3) (3) ∈ 3 2 and TOFFOLI 3 2 are computationally by definition of the semidirect product. universalC \C when taken together∈ C with\C the Clifford group. The full automorphism group contains several interesting subgroups. Consider the logical gates that are local C. Automorphisms of stabilizer codes LU (Q)= U V(Q) U = n U ,U SU(2) (10) { ∈ | ⊗i=1 i i ∈ } An automorphism is a one-to-one, onto map from some and the logical gates that are implemented by permutations domain back to itself that preserves a particular structure of the domain. We are interested in quantum code automorphisms, PAut (Q)= π Sn Pπ V(Q) (11) unitary maps that preserve the code subspace and respect { ∈ | ∈ } where P : S SU(2n) is defined by P ψ ψ ...ψ = a fixed tensor product decomposition of the n-qubit Hilbert π n → π| 1 2 ni space. The weight distribution of an arbitrary operator with ψπ(1)ψπ(2) ...ψπ(n) on the computational basis states. The |semidirect product ofi these groups is contained in the full respect to the Pauli error basis n is invariant under these G automorphism group, i.e. PAut (Q) ⋉ LU (Q) Aut (Q). maps. With respect to the tensor product decomposition, we ⊆ can assign each qubit a coordinate j [n], in which case the In other words, the elements of this subgroup are products quantum code automorphisms are those∈ local operations and of automorphisms for which either Pπ = I or U = I, in coordinate permutations that correspond to logical gates. In the notation of the definition. In general, Aut (Q) may be some cases, these automorphisms correspond to the permuta- strictly larger than PAut (Q) ⋉ LU (Q), as happens with the tion, monomial, and/or field automorphisms of classical codes family of Bacon-Shor codes [8]. The automorphism group of [19]. This section formally defines logical gates and quantum Q as a GF (4)-additive classical code is a subgroup of the full code automorphisms on an encoded block. automorphism group, since classical automorphisms give rise Definition 6: A unitary gate n acting on to quantum automorphisms in the Clifford group. U SU(2 ) n 4 4 ∈ Example 5: For the [[4, 2, 2]], LU (Q) = P ⊗ ,H⊗ = qubits is a logical gate on Q if [U, PQ]=0 where PQ is h i ∼ the orthogonal projector onto Q given by S3 and PAut (Q) = S4. Furthermore, the full automorphism group Aut (Q) = S4 S3 equals the automorphism group 1 × P = R. (7) of the [[4, 2, 2]] as a GF (4)-additive code and PAut (Q) ⋉ Q 2m R (Q) LU (Q) = Aut (Q) [12]. ∈SX 4
D. Fault-tolerance and multiple encoded blocks E. Codes stabilized by minimal elements and the minimal support condition As we alluded to earlier, the reason we find Aut (Q) Definition 9: A minimal support of (Q) is a nonempty interesting is because gates in Aut (Q) are “automatically” set ω [n] such that there exists an elementS in (Q) with fault-tolerant. Fault-tolerant gate failure rates are at least support⊆ω, but no elements exist with support strictlyS contained quadratically suppressed after error-correction. Given some in ω (excluding the identity element, whose support is the positive integer t t, two properties are sufficient (but not ′ empty set). An element in (Q) with minimal support is necessary) for a gate≤ to be fault-tolerant. First, the gate must called a minimal element. ForS each minimal support ω, let take a weight w Pauli operator, 0 w t , to a Pauli operator ′ (Q) denote the stabilizer generated by minimal elements with weight no greater than w under≤ ≤ conjugation. Second, if ω withS support ω and let Q denote the minimal code associated w unitaries in the tensor product decomposition of the gate are ω to ω, stabilized by (Q). Let (Q) denote the minimal replaced by arbitrary quantum operations acting on the same ω support subgroup generatedS by allM minimal elements in (Q). qubits, then the output deviates from the ideal output by the Example 6: Consider the [[5, 1, 3]] code Q whose stabilizerS action of an operator with weight no more than w. Gates in is generated by XZZXI and its cyclic shifts. Every set of Aut (Q) have these properties for any t [n] if we consider ′ 4 contiguous coordinates modulo the boundary is a minimal the permutations to be applied to the qubit∈ labels rather than support: 1, 2, 3, 4 , 2, 3, 4, 5 , 3, 4, 5, 1 , etc. The mini- the quantum state. mal elements{ with} support{ ω }={ 1, 2, 3,}4 are XZZXI, We are also interested in applying logic gates between { } YXXYI, and ZYYZI. Therefore, the minimal code Qω is multiple encoded blocks so that it is possible to simulate a stabilized by ω(Q)= XZZXI,YXXYI . This code is a large logical computation using any stabilizer code we choose. [[4, 2, 2]] [[1S, 1, 1]] code,h since this [[4, 2, 2]]i code is locally In general, each block can be encoded in a different code. equivalent⊗ to the code stabilized by XXXX,ZZZZ by the Logic gates between these blocks could take inputs encoded equivalence I C C I, wherehC : X Y Zi X in one code to outputs encoded in another, as happens with by conjugation.⊗ The⊗[[5,⊗1, 3]] code is the intersection7→ 7→ 7→ of its some logical gates on the polynomial codes [20] or with code minimal codes, meaning Q = ωQω and (Q)= ω ω(Q) teleportation [17]. where the intersection and product∩ runS over the minimalS In this paper, we only consider the simplest situation where supports. Furthermore, (Q)= (Q). Q M S blocks are encoded using the same code and gates do not Given an arbitrary support ω, the projector ρω(Q) obtained map between codes. Our multiblock case with r blocks has by taking the partial trace of PQ over ω¯ := [n] ω is rk qubits encoded in the code Q r for some positive integer \ ⊗ 1 r. The notion of a logical gate is unchanged for the multiblock ρω(Q)= Trω¯ R, (13) r Bω(Q) case: Q is replaced by Q⊗ in the prior definitions. However, R (Q),supp(R) ω the fault-tolerance requirements become: (1) a Pauli operator ∈S X ⊆ where Bω(Q) is the number of elements of S with support with weight wi on input block i and i wi t′ conjugates to ≤ contained in ω including the identity. The projector ρω(Q) Iω¯ a Pauli operator with weight no greater than i wi on each ⊗ P projects onto a subcode Qω of Q, Q Qω, that is stabilized output block and (2) if w t′ unitaries in the tensor product ⊆ ≤ P by the subgroup ω(Q) of (Q). decomposition of the gate are replaced by arbitrary quantum S S 1 Example 7: For the [[5, 1, 3]], ρ 1,2,3,4 = (IIII + operations acting on the same qubits, then each output block { } 4 may deviate from the ideal output by no more than a weight XZZX + YXXY + ZYYZ) ∼= P[[4,2,2]]. Definition 10: If , are stabilizer codes, a gate w operator. Q Q′ U = r U1 Un satisfying U ψ = ψ′ Q′ for all ψ Q is Gates in Aut (Q⊗ ) are also fault-tolerant, since the only ⊗···⊗ | i | i ∈ | i ∈ r a local unitary (LU) equivalence from Q to Q′ and Q and Q′ new behavior comes from the fact that PAut (Q)⊗ is not are called locally equivalent codes. If each U then Q generally equal to PAut (Q r). However, Aut (Q r) does not i 1 ⊗ ⊗ and Q are called locally Clifford equivalent codes∈ Land U is contain all of the fault-tolerant gates on r blocks because we ′ a local Clifford (LC) equivalence from Q to Q . In this paper, can interact qubits in different blocks and still satisfy the fault- ′ we sometimes use these terms when referring to the projectors tolerance properties. onto the codes as well. r Definition 8: A transversal r-qubit gate on Q⊗ is a unitary The following results are applied in Section III. r gate U V(Q⊗ ) such that Lemma 1 ([12]): Let Q be a stabilizer code. If U = U ∈ 1 ⊗ Un is a logical gate for Q then [Uω,ρω(Q)] = 0 for n ···⊗ U = j=1Uj, (12) all ω, where Uω = i ωUi. More generally, if Q′ is another ⊗ ⊗ ∈ stabilizer code and U is a local equivalence from Q to Q′ then r where Uj SU(2 ) only acts on the jth qubit of each block. ∈ r Uωρω(Q)Uω† = ρω(Q′) (14) Let Trans (Q⊗ ) denote the r-qubit transversal gates. More generally, we could extend the definition of transver- for all ω. sality to allow coordinate permutations, as in the case of code Proof: U is a local gate, so automorphisms, and still satisfy the fault-tolerance proper- Tr UP U † = U (Tr P )U † = U ρ (Q)U † . (15) ties given above. However, we keep the usual definition of ω¯ Q ω ω¯ Q ω ω ω ω transversality and do not make this extension here. Since U maps from Q to Q′, we obtain the result. 5
By examining subcodes, we can determine if a given gate element N (Q). Fixing any j ω, either (M ) , ∈ Sω 0 ∈ 1 j0 can be a logical gate using Lemma 1. In particular, if U is not (M2)j0 , or (M1M2)j0 equals Nj0 , say (M1)j0 = Nj0 . a logical gate for each minimal code of Q, then U cannot be Then, supp(M1N) is strictly contained in ω, a contradiction. a logical gate for Q. Therefore, if A (Q) 2 then A (Q)=3. The number of ω ≥ ω Definition 11: A stabilizer code is called free of Bell pairs coordinates in the support ω must be even since M1 and M2 if it cannot be written as a tensor product of a stabilizer code commute. | | and a [[2, 0, 2]] code (a Bell pair). A stabilizer code is called The next result shows that any local equivalence between free of trivially encoded qubits if for each j [n] thereS exists two [[2m, 2m 2, 2]] stabilizer codes with the same m 2 an element s such that the jth coordinate∈ of s is not the must be a local− Clifford equivalence. In the m = 1 special≥ identity matrix,∈ Si.e. if S cannot be written as a tensor product case, we have a [[2, 0, 2]] code, i.e. a Bell pair locally Clifford of a stabilizer code and a [[1, 1, 1]] code (a trivially encoded equivalent to the state ( 00 + 11 )/√2, for which the result | i | i qubit). does not hold because V V ∗ is a local equivalence of the Let m(Q) be the union of the minimal supports of a [[2, 0, 2]] for any V SU⊗(2). This special case is the reason stabilizer code Q. The following theorem is a major tool in for introducing the definition∈ of a stabilizer code that is free the solution of our main problem. of Bell pairs. Theorem 1 ([12], [16]): Let Q, Q′ be [[n,k,d]] stabilizer Lemma 3: Fix m 2 and let Q, Q′ be stabilizer codes ≥ 2m codes, not necessarily distinct, that are free of Bell pairs and that are LC equivalent to Q[[2m,2m 2,2]]. If U U(2)⊗ is − ∈ trivially encoded qubits, and let j m(Q). Then any local a local equivalence from Q to Q′ then U 2m. ∈ ∈ L 2m equivalence U from Q to Q′ must have either Uj 1 or Proof: We must show that every U U(2)⊗ satisfying iθR ∈ L ∈ Uj = Le for some L 1, some angle θ, and some R Uρ[[2m,2m 2,2]]U † = ρ[[2m,2m 2,2]] is a local Clifford opera- ∈ L ∈ − − 1. tor. Recall that any 1-qubit unitary operator V U(2) acts on G Proof: For completeness, we include a proof of this the Pauli matrices as ∈ theorem here, though it can be found expressed using slightly σ V σ V † = o X + o Y + o Z, different language in [12], [16], and [21]. The proof requires a 7→ a ax ay az several results about the minimal subcodes of a stabilizer code for every a x,y,z and where (oab) SO(3). In the that we present as Lemmas within the proof body. The first standard basis∈ {0 , 1 ,}2 of R3, the matrix∈ of these results shows that each minimal subcode is either {| i | i | i} 2m m 2m 2m X⊗ + ( 1) Y ⊗ + Z⊗ (17) a quantum error-detecting code or a “classical” code with a − single parity check, neglecting the [[ ω¯ , ω¯ , 1]] part of the is associated to the vector space. | | | | m 3 2m Lemma 2: Let denote the cardinality of the set of v := 00 ... 0 +( 1) 11 ... 1 + 22 ... 2 (R )⊗ (18) Aω(Q) | i − | i | i ∈ elements s with support ω and let Q be a stabilizer code 2m ∈ S acted on by SO(3)⊗ . We must show that every O = O1 with stabilizer . If ω is a minimal support of , then exactly 2m ⊗ O SO(3)⊗ satisfying Ov = v is such that each one of the followingS is true: S ···⊗ 2m ∈ Oi is a monomial matrix (see [19]; a matrix is monomial if it (i) Aω(Q)=1 and ρω(Q) is locally Clifford equivalent to is the product of a permutation matrix and a diagonal matrix). 1 Consider the single qutrit operator ω ω (16) ρ[[ ω , ω 1,1]] := ω (I⊗| | + Z⊗| |), | | | |− 2| | T 0 1 Tr 3,4,...,2m (vv ) 0 1, (19) a projector onto a [[ ω , ω 1, 1]] stabilizer code h | { } | i | | | | − acting on the second qutrit (second copy of R3). The matrix Q[[ ω , ω 1,1]]. | | | |− vvT has 9 nonzero elements, and the partial trace over the last (ii) Aω(Q)=3, ω is even, and ρω(Q) is locally Clifford | | 2m 2 qutrits gives equivalent to − T 1 ω ω Tr 3,4,...,2m (vv )= 00 00 + 11 11 + 22 22 . (20) { } ρ[[2m,2m 2,2]] := ω (I⊗| | + X⊗| | | ih | | ih | | ih | − 2| | ω /2 ω ω Hence the matrix in Eq. 19 equals the rank one projector + ( 1)| | Y ⊗| | + Z⊗| |), 0 0 . Therefore, if Ov = v then the operator − | ih | a projector onto a [[2m, 2m 2, 2]] stabilizer code T T − 0 1 Tr 3,4,...,2m (Ovv O ) 0 1 (21) Q[[2m,2m 2,2]], m = ω /2. h | { } | i − | | Proof: For any minimal support ω, A (Q) 1. If equals 0 0 as well. The operator is given by the matrix ω ≥ | ih | Aω(Q)=1 then ω(Q) is generated by a single element and T T T S O2 0 1(O1 I) Tr 3,4,...,2m (vv )(O1 I) 0 1O2 (22) we are done. If A (Q) 2, let M ,M (Q) I be h | ⊗ { } ⊗ | i ω 1 2 ω (O )2 0 0 distinct elements. These≥ elements must satisfy∈ S I = (\{M )} = 1 00 1 j = O 0 (O )2 0 OT . (23) (M ) = I for all j ω, otherwise supp(6 M M ) 6is 2 1 01 2 2 j 1 2 0 0 (O )2 strictly contained6 in ω, contradicting∈ the fact that ω is a 1 02 minimal support. It follows that supp(M1M2) = ω and where we have factored O2 to the outside. The matrix within (M ) , (M ) , (M M ) equals X,Y,Z up to phase for Eq. 23 equals the rank one projector OT 0 0 O if and only { 1 j 2 j 1 2 j } { } 2 | ih | 2 all j ω. Therefore, I, M1, M2, and M1M2 are the only if exactly one of the elements (O1)00, (O1)01, or (O1)02 elements∈ in (Q). Indeed, suppose there exists a fourth is nonzero. Repeating the argument for every row of O Sω 1 6
T T by considering the operators i 1 Tr 3,4,...,2m (Ovv O ) i 1, For coordinates which are not covered by minimal supports h | { } | i i 0, 1, 2 , shows that every row of O1 has exactly one of , the results in Sec. II-E tell us nothing about the allowable ∈ { } S n nonzero entry. O1 is nonsingular therefore O1 is a monomial form of Uj for a transversal gate U = j=1 Uj , so we need matrix. The vector v is symmetric so repeating the analogous another approach for these coordinates. N argument for each operator Oi, i [2m], completes the proof. Let j = R R (Q), j supp(R) and define ∈ the “minimalS { elements”| ∈ of S this set∈ to be ( } ) = R M Sj { ∈ Now we can complete the proof of Theorem 1. Let Q, Q ∄R′ s.t. supp(R′) ( supp(R) . Note that these sets ′ Sj | ∈ Sj } be stabilizer codes, let U be a local equivalence from Q to Q′, do not define codes because they are not necessarily groups. and take a coordinate j m(Q). There is a least one element Lemma 4: If j is not contained in any minimal support of ∈ M (Q) with j ω := supp(M). Either A (Q)=1 or , then for any R, R′ ( j ) such that the jth coordinates ∈ M ∈ ω S ∈M S A (Q)=3 by Lemma 2. satisfy R j = R′ j , we must have supp(R) = supp(R′). ω | 6 | 6 If Aω(Q)=3 then ρω(Q) is LC equivalent to ρ[[ ω , ω 2,2]]. Proof: We prove by contradiction. If there exist R, R′ | | | |− ∈ Moreover, as Q is locally equivalent to Q′, ω is also a minimal ( j ) such that R j = R′ j and supp(R) = supp(R′)= ω, M S | 6 | ω support of (Q′) with Aω(Q′)=3. Therefore, ρω(Q′) is local then up to a local Clifford operation, we have R = X⊗| | and S ω Clifford equivalent to ρ[[ ω , ω 2,2]]. By Lemma 1, Uω maps R′ = Z⊗| |. Without loss of generality, assume j =1. Since ω | | | |− ρ (Q) to ρ (Q ) under conjugation. Note that we must have is minimal in j but not minimal in , there exists an element ω ω ′ S S ω > 2, otherwise Q is not free of Bell pairs. Since ω is F in j whose support supp(F )= ω′ is strictly contained | | | | S \ S even, ω 4. By Lemma 3, Uω ω so Uj 1. in ω, i.e. ω′ ( ω. Since F is not in j , RF , R′F , R′RF | |≥ ∈ L| | ∈ L S ∈ If A (Q)=1 and there are elements R , R , R (Q) ( j ). However, one of RF , R′F , R′RF ( j ) will ω 1 2 3 ∈M M S ∈ M S such that (R1)j = X, (R2)j = Y , and (R3)j = Z, then have support that is strictly contained in ω, contradicting the there exists another minimal element N (Q) such that fact that ω is a minimal support of j . S j µ := supp(N) and M = N . If A∈(Q M)=3 then we Lemma 5: If j is not contained in any minimal support of ∈ j 6 j µ n can apply the previous argument to conclude that U . , then for any transversal gate U = j=1 Uj, one of the j ∈ L1 S Otherwise, Aµ(Q)=1 and following three relations is true: Uj Xj Uj = Xj , UjYj Uj = N ± iθR Yj , Uj Zj Uj = Zj. In other words, Uj = Le for some 1 ω ± ± ρ (Q)= (I⊗| | + M ) (24) L 1, some angle θ, and some R 1. ω 2 ω ω ∈ L ∈ G | | Proof: For any element R (Sj ) with a fixed support 1 µ ∈M ρ (Q)= (I + N ). (25) ω, we have R j = Z up to local Clifford operations by µ µ ⊗| | µ | 2| | Lemma 4. Tracing out all the qubits in ω¯, we get a reduced Since ω and µ are also minimal supports of (Q′) with density matrix ρ with the form S ω Aω(Q′)=1 and Aµ(Q′)=1, there exist unique M ′,N ′ 1 ∈ (31) S(Q′) such that ρω = ω (Ij RI + Zj RZ ), 2| | ⊗ ⊗ 1 ω where RI and RZ are linear operators acting on the other ρω(Q′)= (I⊗| | + M ′ ) (26) ω ω qubits. Since , we have † . 2| | ω j UωρωUω† = ρω UjZj Uj = Zj 1 \{ } ± µ (27) ρµ(Q′)= µ (I⊗| | + Nω′ ). 2| | The following corollary about the elements of the automor- phism group of a stabilizer code is immediate from Lemma 5. Applying Lemma 1 to Uµ and Uν , we have After this work was completed, we learned that the same Uj MjUj† = Mj′ (28) statement was independently obtained by D. Gross and M. Van ± den Nest [23] and that the theorem was first proved in the UjNj U † = N ′ (29) j ± j diploma thesis of D. Gross [24]. from Eqs. 24-27. These identities show that U . Corollary 1: U Aut (Q) for a stabilizer code Q iff j ∈ L1 ∈ Finally, if Aω(Q)=1 and R = (R1)j = (R2)j for any n R , R (Q) then any minimal support µ such that j µ iθj 1 2 U = L diag (1,e ) R†P V(Q) (32) ∈M ∈ π ∈ satisfies Aµ(Q)=1. Applying Lemma 1 to Uµ, we have j=1 O U RU † = R′ (30) for some local Clifford unitaries L= L1 Ln, R = j j ± ⊗···⊗ R1 Rn, product of swap unitaries Pπ enacting the for some R X,Y,Z . Choose L such that LRL = ⊗···⊗ ′ 1 † coordinate permutation π, and angles θ1,...,θn . ∈{ iθR } ∈ L R′. Then Uj = Le and the proof of Theorem 1 is complete. { } III. TRANSVERSALITY VERSUS UNIVERSALITY In this section we prove that there is no universal set of transversal gates for binary stabilizer codes. F. Coordinates not covered by minimal supports n Definition 12: A set A V(Q⊗ ) is encoded computation- It is not always the case the m(Q) = [n], as the following ⊆ n ally universal if, for any n, given U V(Q⊗ ), example shows. ∈ Example 8: Consider a [[6, 2, 2]] code with stabilizer = n n ǫ> 0, V1,...,V A, s.t. UP ⊗ Vi P ⊗ < ǫ. . For S , ∀ ∃ η(ǫ) ∈ || Q − Q || XXXXII,ZZIIZZ,IIIIXX,IIXXZZ j = 1, 2 i ! thereh is no minimal support ω of such thatij ω [22]. Y (33) S ∈ 7
Gates in A may be implicitly mapped to isometries on the where r(i)=0 if R commutes with Mi and r(i)=1 if R appropriate Hilbert space, as in Definition 4. anticommutes with M . R/ ( ) so r(i)=1 for at least one i ∈C S Theorem 2: For any stabilizer code Q that is free of i, and (I Mi)(I + Mi)=0, which gives PQRPQ =0. Bell pairs and trivially encoded qubits, and for all r 1, Lemma− 7: Let Q be a stabilizer code with stabilizer and r ≥ S Trans(Q⊗ ) is not an encoded computationally universal set let α ( ) be a minimum weight element in ( ) . ∈C S \ S C S \ S of gates for even one of the logical qubits of Q. Without loss of generality, α X¯1 (the X¯1 coset of ), Proof: We prove this theorem by contradiction. We first where the subscript indicates what∈ logicalS qubit the logicaS l assume that we can perform universal quantum computation on operator acts on. If the logical Clifford operations H¯1 and P¯1 at least one of the qubits encoded into Q using only transversal on the first encoded qubit are transversal, then there exists gates. Then, we pick an arbitrary minimum weight element β,γ ( ) such that β Z¯1 , γ Y¯1 and supp(α)= α ( ) , and perform appropriate transversal logical supp(∈Cβ) =S supp(\ S γ). ∈ S ∈ S ∈ C S \ S ¯ ¯ ¯ ¯ Clifford operations on α. Finally, we will identify an element Proof: H1 is transversal, so β′′ := H1αH1† Z1 and in ( ) that has support strictly contained in supp(α). ξ := supp(α) = supp(β ). Expand β in the basis∈ ofI Pauli C S \ S ′′ ′′ This contradicts the fact that α is a minimal weight element operators in ( ) , i.e. that the code has the given distance d. WeC S first\ S prove the theorem for A) the single block case and β′′ = bRR + bR′ R′ (35) ′ R C( ),supp(R) ξ R Gn C( ) then generalize it to B) the multiblock case. ∈ S X ⊆ ∈ X\ S where b ,b ′ C. By Lemma 6, b = 0 for at least one R R ∈ R 6 A. The single block case (r =1) R C( ) in the first term of Eq. 35. The operator β′ := ∈ S The first problem we encounter is that general transversal PQβ′′PQ Z¯1 is a linear combination of elements of C( ), ∈ I S gates, even those that implement logical Clifford gates, might β = b R + b ′ R , (36) not map logical Pauli operators back into the Pauli group. This ′ R R ′ R C( ) ,supp(R)=ξ R′ ,supp(R′) ξ behavior potentially takes us outside the stabilizer formalism. ∈ S \SX ∈S X ⊆ Definition 13: The generalized stabilizer (Q) of a quan- where the terms R C( ) must have support ξ since α ∈ S \ S tum code Q is the group of all unitary operatorsI that fix the has minimum weight in C( ) . Considering the action of S \ S code space, i.e. β′ on a basis of Q, it is clear that there is a term bββ where b =0, β Z¯ , and supp(β)= ξ. (Q)= U SU(2n) U ψ = ψ , ψ Q . β 6 ∈ 1S I { ∈ | | i | i ∀| i ∈ } Similarly, since P¯1 is transversal, there must exist γ Z¯1 , and supp(γ)= ξ. ∈ S The transversal T gate on the 15-qubit Reed-Muller code Remark 1: Note in the proof of the above lemma, we as- ¯ 15 is one example of this problem since it maps X = X⊗ to sume that H¯1 is exactly transversal, i.e. ǫ =0 in Definition 12. an element ( 1 (X Y )) 15. This element is a representative However, the proof is also valid for an arbitrarily small . √2 ⊗ ǫ> 0 1 − of (X¯ + Y¯ ) but has many more terms in its expansion in Indeed, in this case β′′ / Z¯1 , but β′′ must have a non- √2 ∈ I the Pauli basis. These terms result from an operator in the negligible component in Z¯1 to approximate H¯1. Hence, when I generalized stabilizer . expanding β′′ in the Pauli basis, there must exist a β Z¯1 ∈ S The 9-qubit Shor codeI gives another example. A basis for such that supp(β)= ξ, i.e. the same argument holds even for this code is an arbitrarily small ǫ> 0. ¯ 3 3 Remark 2: The choice of α X1 is made without loss 0/1 ( 000 + 111 )⊗ ( 000 111 )⊗ , (34) ∈ S | i∝ | i | i ± | i − | i of generality, since for a given stabilizer code, we have the iθZ iθZ from which it is clear that e 1 e− 2 (QShor) S. This freedom to define logical Pauli operators, and this freedom 9 ∈ I \ gate does not map X¯ = X⊗ back to the Clifford group, even can be viewed as a “choice of basis”. What is more, since though it is both transversal and logically an identity gate in we assume universal quantum computation can be performed the logical Clifford group. transversally on the code, then no matter what basis (of In spite of these possibilities, we will now see that we the logical Pauli operators) we choose, H¯1 and P¯1 must be can avoid further complication and stay within the powerful transversal. On the other hand, sometimes we would like to stabilizer formalism. fix our choice of basis, as in the case of a subsystem code, to First, we review a well-known fact about stabilizer codes. clearly distinguish some logical qubits (protected qubits) from Lemma 6: Let = M1,...,Mn k be the stabilizer of an other logical qubits (gauge qubits). In this case, we can choose S h − i [[n,k,d]] code Q. For any n-qubit Pauli operator R/ ( ), α as a minimum weight element in X¯ , Y¯ , Z¯ , where ∈ C S { sS sS sS} we have PQRPQ =0 where PQ is the projector onto the code s is a distinguished logical qubit. Starting from this choice of subspace. α, one can see that the arguments hold for subsystem codes as Proof: We have well as subspace codes, because the distance of the subsystem n k code is defined with respect to this subgroup. − P (I + M ) and Remark 3: The procedure of identifying β Z¯1 from Q ∝ i ∈ S i=1 β′′ Z¯1 in the proof of Lemma 7 is general in the following Yn k ∈ I − sense. We can begin with a minimum weight element of r(i) PQR R (I + ( 1) Mi), α X¯ C( ) and apply any transversal logical ∝ − ∈ 1S ⊂ S \ S i=1 Clifford gate to generate a representative β C( ) of Y ∈ S \ S 8 the corresponding logical Pauli operator such that supp(α)= B. The multiblock case (r> 1) supp(β). This procedure is used a few times in our proof, so Now we consider the case with r blocks. A superscript (i), we name this procedure the “ procedure”. (i) I → S i [r], denotes a particular block. For example, U acts on Now we can begin the proof of Theorem 2. Assume the∈ ith block. First, we generalize Theorem 1 and Lemma 5 that Trans (Q) is encoded computationally universal. Let to the multiblock case. ¯ α X1 C( ) be a minimum weight element Lemma 9: Let Q be an [[n,k,d]] stabilizer code free of Bell ∈ S ⊂ S \ S in C( ) . Applying the “ procedure” to both pairs and trivially encoded qubits, and let U be a transversal ¯ S ¯\ S ¯I → S ¯ r H1 and P1, we obtain β Z1 and γ Y1 such that gate on Q⊗ . Then for each j [n] either Uj r or supp(α) = supp(β) = supp(∈ γS) =: ξ and∈ ξS = d. The ∈r ∈ L Uj = L1VL2 where L1,L2 1⊗ are local Clifford gates next lemma puts these logical operators into a| simple| form ∈ L (i) and V either normalizes the group Zj ,i [r] , of Pauli for convenience. Z operators or keeps the linear spanh± of its∈ groupi elements ¯ ¯ ¯ Lemma 8: If α X1 , β Z1 , and γ Y1 , all have invariant. ∈ S ∈ S ∈ S the same support ξ, and ξ = d is the minimum distance Proof: Lemma 1 and Lemma 2 can be generalized to the | | of the code, then there exists a local Clifford operation that the multiblock case with almost the same proof, which we ξ ξ /2 ξ ξ transforms α, γ, and β to X⊗| |, ( 1)| | Y ⊗| |, and Z⊗| |, do not repeat here. In the multiblock case, the corresponding − respectively. results of Lemma 2 read Proof: Let ξ = i1,i2,...,i ξ and write α = r ω ω r { | |} Aω(Q) = 1 : Sω⊗ (Q)= I⊗ ,Z⊗ ⊗ αi1 αi2 ...αi|ξ| , β = βi1 βi2 ...βi|ξ| , where each αik and βik , { } A (Q) = 3 : k [ ξ ], are one of the three Pauli matrices Xik ,Yik , or Zik , ω ∈ | | r ω ω ( ω /2) ω ω r neglecting phase factors i or 1. S⊗ (Q) = I⊗ , X⊗ , ( 1) | | Y ⊗ ,Z⊗ ⊗ , ± − ω { − } Apart from a phase factor, αik = βik for all k [ ξ ]. 6 ¯ ∈ | | and the corresponding equation of Eq. 14 in Lemma 1 is Indeed, if for some k, αik = βik , then supp(Y ) = ξ, a 6 r r contradiction. Uωρω⊗ (Q′)Uω† = ρω⊗ (Q). (38) Therefore, for each k [ ξ ] there exists a single qubit When Aω(Q)= 3, we need to generalize the result of ∈ | | r Clifford operation Lik 1 such that Lik αik Li† = Xik and r 2m ∈ L k Lemma 3. In particular, if U = i=1 Uj U(2 )⊗ Li βi L† = Zi . The local Clifford operation r r ∈ k k ik k satisfies Uρ[[2⊗ m,2m 2,2]]U † = ρ[[2⊗ m,2m 2,2]], then for each − N − j ω, U U(2r) is a Clifford operator. Indeed, any r- j j qubit∈ unitary∈ operator V U(2r) acts on a Pauli operator Lξ = Lik (37) ∈ σa1 σa2 ...σar as Ok=1 σ σ ...σ V σ σ ...σ V † applies the desired transformation. a1 a2 ar 7→ a1 a2 ar 3 Applying Lemma 8, we obtain a local Clifford operation = o σ σ ...σ , that we apply to Q. Now we have a locally Clifford equivalent a1...ar i1...ir i1 i2 ir i1,...ir=0 code Q′ for which supp α = supp β = supp γ = ξ and α = X ξ ξ /2 ξ ξ for every nonidentity Pauli string a, where (o ) X⊗| | X¯1 , γ = ( 1)| | Y ⊗| | Y¯1 and β = Z⊗| | a,i1,...ir ∈ S − ∈ S ∈ SO(4r 1) and o =0. We can rearrange the numbering∈ Z¯1 . a,0,...,0 S of the− coordinates in r such that the coordinate Note that if ξ = d is even, then the validity of Lemma 8 ρ[[2⊗ m,2m 2,2]] − already leads| to| a contradiction since α,β,γ must anti- r(a 1) + b denotes the ath qubit of the bth block. In the − r 4r 1 r commute with each other. However, if ξ = d is odd, we standard basis 0 , ..., 4 2 of R − , ρ[[2⊗ m,2m 2,2]] is {| i | − i} − need to continue the proof. | | associated to the vector n 4r 2 By Theorem 1 and Lemma 5, if U = U is a − r j=1 j 4 1 2m v := bj jj...j (R − )⊗ (39) transversal gate on one block, then either Uj 1 for all | i ∈ iθR N∈ L j=0 j ξ or Uj = Le for one or more j ξ, where L 1, X ∈ ∈ ∈ L θ R, and R 1. If Uj 1 for all j ξ then, for the where bj 1 . The vector is acted on by orthogonal ∈ ∈ G ∈ L ∈ ∈ {± r} 2m first encoded qubit of Q′, the only transversal operations are matrices in SO(4 1)⊗ . For a code free of Bell pairs, we logical Clifford operations. have ω 4. By reasoning− similar to the proof of Lemma 3, if | |≥ r 2m Therefore, there must exist a coordinate j ξ, such that O = O1 O2m SO(4 1)⊗ satisfies Ov = v, then iθZ ∈ ⊗···⊗ ∈ − r U = e up to a local Clifford operation. Since H¯ is each Oi is a monomial matrix. This implies that Uj U(2 ) j 1 ∈ transversal, when expanding H¯ βH¯ † X¯ in the basis of is a Clifford operator for each j ω. 1 1 ∈ 1I ∈ Pauli operators, using the “ procedure”, we know that When Aω(Q)=1, it is possible to follow reasoning similar I → S there exists α′ X¯1 and supp(α′)= ξ. Furthermore, since to the proof of Theorem 1. Now the equations analogous to ¯ ¯ ∈ S Eqs. 24 are (H1)j Z(H1)j† = Z, we have (α′)j = Zj, i.e. α′ restricted ± ¯ r to the jth qubit is Zj. We know γ′ = iα′β Y1 , and 1 ⊗ ∈ S r ω (40) (γ′)j = Ij . Therefore, supp(γ′) is strictly contained in ξ. ρω(Q)⊗ = ω (I⊗| | + Mω) 2| | However, this contradicts the fact that α is a minimal weight r r 1 µ ⊗ element in ( ) . This concludes the proof of Theorem 2 ρ (Q)⊗ = (I⊗| | + N ) . (41) C S \ S µ 2 µ µ for the single block case. | | 9
ω Up to local Clifford operations, we can choose Mω = X⊗| | ω =2, we conclude that Uj must keep one of the three spaces mu | | (i) r (i) r and Nµ = Z⊗| | of linear operators span( X ), span( Y ), or h± j ii=1 h± j ii=1 We again rearrange the numbering of the coordinates of span( Z(i) r ) invariant. r j i=1 ρω(Q)⊗ such that the coordinate r(a 1)+b denotes the ath Givenh± thesei generalizations of Theorem 1 and Lemma 5 to − r qubit of the bth block. In the standard basis 0 , 1 , ..., 2 the multiblock case, we now show that all the arguments in the 2r 1 r {| i | i | − 2 of R − , the matrix ρ (Q)⊗ is associated to the vector i} ω proof of the single block case can be naturally carried to the 2r 2 multiblock case. Most importantly, we show that the “ ” − 2r 1 ω I → S v := jj...j (R − )⊗| | (42) procedure is still valid. To specify the “ ” procedure for | i ∈ I → S j=0 the multiblock case, we first need to generalize the concept X of the generalized stabilizer defined in Definition 13 to the r ω acted on by SO(2 1)⊗| |. − multiblock case. Note for any coordinate j [n], if there are elements r ∈ Definition 14: The generalized stabilizer (Q⊗ ) of an r- R1, R2, R3 (Q) such that (R1)j = X, (R2)j = Y , r I ∈ M block quantum code Q⊗ is the group of all unitary operators and (R3)j = Z, then we have both ω 2 and µ 2 that fix the code space, i.e. [15]. Following similar reasoning to the| | proof ≥ of Lemma| | ≥ 3, r ω r nr r if O = O1 O ω SO(2 1)⊗| | satisfies (Q⊗ )= U SU(2 ) U ψ = ψ , ψ Q⊗ . ⊗ ··· ⊗ | | ∈ − I { ∈ | | i | i ∀| i ∈ } Ov = v, then each Oi has a monomial subblock. Therefore, (i) r (i) r Similar to the single block case, we start by assuming Uj ( Xj i=1) for each j ω, where ( Xj i=1) ∈ N h± i (∈i) N h± i that universal quantum computation can be performed using is normalizer of the group X r of Pauli X operators (1) j i=1 transversal gates. Then H¯ , the logical Hadamard operator acting at the th coordinateh± of thei th block. Similarly for 1 j i acting on the first logical qubit of the first block, is transversal. r, (i) r for each , where ρµ(Q)⊗ Uj ( Zj i=1) j µ Let α(1) be a minimal weight element of ( ) . Without (i) ∈ N h± i (i)∈ ( Z r ) is normalizer of the group Z r of Pauli (1) ¯ C S \ S ¯ (1) N h± j ii=1 h± j ii=1 loss of generality, we assume α X1 . Then H1 will operators. (i) r (i) r is a subgroup (1) ¯(1) ∈ r S Z ( Xj i=1) ( Zj i=1) transform α to some β′ Z1 (Q⊗ ). This is to say, β′ N h± i ∩ N h± i r r ∈ I of the Clifford group, therefore Uj U(2 ) is a Clifford acting on P ⊗ is a logical Z operation on the first logical ∈ Q operator for all j ω µ. If instead (Ra)j = (Rb)j for all qubit of the first block, and identity on the other r 1 blocks. ∈ ∩ (i) r ′′ r −′ Ra, Rb (Q) and if ω 3, then Uj ( Z ) (1) (i) ∈ M | | ≥ ∈ N h± j ii=1 However, this does not mean that β′ = β δ , where up to local Clifford operations, but Uj is not necessarily a i=2 (i) Clifford operator. β′′ Z¯1 and δ′ (Q) for all i =2,...,rN, because ∈ S ∈ I If ω = 2 and Aω = 1, then the form of the vector in n | | ¯ (1) Eq. 42 leads to different behavior when r > 1. When r =1, H1 = Uj , (44) there is only one term in the summation, so Uj ( Xj ). j=1 ∈ N h±(i) r i O However, when r > 1, generally Uj / ( X ). j i=1 where each Uj acts on r qubits. (i) r∈ N h± i Nevertheless Uj must keep span( Xj i=1) invariant, where Expanding β′ in the basis of nr qubit Pauli operators. For (i) r h± i span( Xj i=1 ) is the space of linear operators spanned the same reason shown in the proof of Lemma 7, there must h± i i (i) r be at least one term in the expansion which has the form by the group Xj i=1 with coefficients in C. Indeed, consider Eq. 42h± wheni ω = 2. For convenience, let ω = r |T | r β(1) δ(i), (45) 1, 2 . We have Tr2(vv ) = j=1 j j . If Ov = v, then { } r | ih | Tr (OvvT OT )= j j . However, i=2 2 j=1 | ih | P O where β Z¯ , and δ(i) for all i = 2,...,r. So, the Tr (OvvT OPT )= Tr (O vvT OT ) 1 2 2 1 1 generalization∈ ofS the “ ∈ S procedure” to the multiblock r case is clear: Pauli operatorsI → acting S on a code are either logical = O j j OT 1| ih | 1 Pauli operators (on any number of qubits and any number of j=1 r X r blocks) or they map the code PQ⊗ to an orthogonal subspace. Nevertheless, this is an important observation. = ( (O ) (O ) ′ ) k k′ . 1 jk 1 jk | ih | k,k′ j=1 Now β ( ) is a logical Z operation acting on X X the first logical∈ C S qubit\ S of a single block of the code. Due r ′ ′ (1) (1) (1) Therefore j=1(O1)jk(O1)jk = δkk for all k, k′ r and to Eq. 12, supp(β ) supp(α ). However α is a r ≤ ⊆ j=1(O1)jk(O1)jk′ = 0 for any one of k > r or k′ > r, minimal weight element of ( ) , therefore we have P C S \ S which means (O1)jk =0 for all k > r. supp(β(1)) = supp(α(1)) := ξ. For convenience, we now drop P Finally, we need to generalize Lemma 5 to the multiblock the superscript (1) when referring to these logical operators. case. Recalling Eq. 31, we now have Since P¯(1) is transversal, then there exists a γ Y¯ which 1 ∈ 1S r has the same support as α. Now we have α X¯ , β Z¯ 1 ⊗ ∈ 1S ∈ 1S ρω = (Ij RI + Zj RZ ) . (43) and γ Y¯ such that supp(γ) = supp(β) = supp(α). 2 ω ⊗ ⊗ 1 | | Like∈ theS single block case, by Lemma 8, there is a local r ξ ¯ This projector can be associated with a vector v = j=1 jj . Clifford operation such that α = X⊗| | X1 , β = | i ξ /2 ξ ξ ∈ S Using the same technique as for the case where Aω =1 and ( 1)| | Y ⊗| | Y¯1 and γ = Z⊗| | Z¯1 . If ξ = d is P − ∈ S ∈ S | | 10
even there is a contradiction, since α, β, γ must anti-commute Therefore, there exists j′ ξ′ such that either (a) only one ∈ with each other. of X,Y,Z appears in (Q) at coordinate j′ or (b) there { } M When ξ = d is odd, we need the following arguments. is no minimal element with support at j′. Without loss of | | If for all coordinates j ξ, there are elements R , R , R generality, we assume that X appears at coordinate j′ in case ∈ 1 2 3 ∈ (Q) such that (R1)j = X, (R2)j = Y and (R3)j = Z, (a). Since F¯1 can be performed via some transversal gate plus thenM U U(2r) is a Clifford operator for all j ξ by permutation, we have j ∈ ∈ Lemma 9. Therefore, all the possible logical operations that 1 are transversal on the first encoded qubit must be Clifford F¯ α′F¯† = η′ X¯ + Y¯ + Z¯ (Q). (48) 1 1 ∈ √ 1 1 1 I operations, contradicting the assumption that universality can 3 be achieved by transversal gates. Again applying the procedure to η′ we know there I → S Therefore, there exists a coordinate j ξ such that exist α′′ X¯1 , β′′ Y¯1 , and γ′′ Z¯1 such that ∈ ∈ S ∈ S ∈ S either (a) there is no minimal support containing j or (b) supp α′′ = supp β′′ = supp γ′′ = ξ′′. And ξ′′ = ξ = d. | | | | (R1)j = (R2)j = I for all R1, R2 (Q). Use the “ The permutation maps j′ to j′′. However, we know that 6 (1) (1)∈M I → S procedure” to expand H¯ β(H¯ ) in the basis of Pauli η′ j′′ = X, and this is also true in case (b) by similar reasoning 1 1 † | operators and extract α′ X¯1 with supp(α′)= ξ. Since H¯1 to Lemma 5, hence α′′ j′′ = β′′ j′′ = γ′′ j′′ = X. Then ∈ S (i) ¯ | | | is transversal, U must keep span( Z r ) invariant, up to γ′′′ = iα′′β′′ Z1 such that iα′′β′′ j′′ = I. Therefore, j h± j ii=1 ∈ S | a local Clifford operation. This means that the jth coordinate supp(γ′′′) is strictly contained in ξ, which contradicts the fact that is a minimal weight element in . of α′ is either Ij or Zj. The former is not possible since α ( ) r C S \ S supp(α′) = ξ. For the later, γ′′ = iα′β Y¯1 , and the If Aut(Q) is replaced by Aut(Q⊗ ), the theorem still holds ∈ S r jth coordinate of γ′′ is Ij . Therefore, supp(γ′′) is strictly because we can view Q⊗ as another stabilizer code. However, contained in ξ. However, this contradicts the fact that α is a it is not a simple generalization to allow permutations between minimal weight element in ( ) . transversal gates acting on r > 1 blocks. This is because C S \ S permutations are permitted to be different on each block and may also be performed between blocks. IV. THE EFFECT OF COORDINATE PERMUTATIONS In this section we discuss the effect of coordinate permuta- V. APPLICATIONS AND EXAMPLES tions. Theorem 3: For any stabilizer code Q free of trivially In this section, we apply the proof techniques we have used encoded qubits, Aut(Q) is not an encoded computationally in previous sections to reveal more facts about the form of universal set of gates for any logical qubit. transversal non-Clifford gates. First, we describe the form of Proof: Choose a minimum weight element α ( ) . transversal non-Clifford gates on stabilizer codes. We explore Without loss of generality, assume α X¯ and supp(∈CαS)=\ Sξ. further properties of allowable transversal gates in the single ∈ 1S Define a single qubit non-Clifford gate F by block case and discuss how the allowable transversal gates relate to the theory of classical divisible codes. Finally, we 1 F : X X′ = (X + Y + Z); Z Z′ (46) review a CSS code construction based on Reed-Muller codes → √3 → that yields quantum codes with various minimum distances where Z′ is any operator that is unitary, Hermitian and and transversal non-Clifford gates. anticommuting with X′. We cannot use the idea of applying Corollary 1 gives a form for an arbitrary stabilizer code H¯1 and P¯1 from within Aut(Q) since they might involve automorphism. Similarly, in the multiblock case, Lemma 9 different permutations. We instead assume the logical gate provides possible forms of Uj for any transversal gate U = n F¯1 can be approximated to an arbitrary accuracy by gates in j=1 Uj . These forms prevent certain kinds logical gates from Aut (Q). Then we have being transversal on a stabilizer code. N Corollary 2: An r-qubit logical gate U such that Uj / r ¯ ¯ 1 ¯ ¯ ¯ ∈ L F1αF1† = η X1 + Y1 + Z1 (Q) (47) for all j is transversal on a stabilizer code only if U keeps the ∈ √3 I ¯ r operator space span( Zi i=1) invariant up to a local Clifford ¯ h± i Applying the procedure to η, we find α′ X¯ , operation. Here, Zi denotes the logical Pauli Z operator on the I → S ∈ 1S β′ Y¯ , and γ′ Z¯ such that supp α′ = supp β′ = ith encoded qubit. ∈ 1S ∈ 1S supp γ′ = ξ′ and ξ′ = ξ = d. By Lemma 8, we can find a Remark 4: This is a direct corollary from Theorem 2 in | | | | ξ′ locally Clifford equivalent code such that α′ = X⊗| | X¯1 , Sec. III and Theorem 3 in Sec. IV. ξ′ /2 ξ′ ξ ∈ S β′ = ( 1)| | Y ⊗| | Y¯ and γ′ = Z⊗| | Z¯ . Again, Example 9: Consider the three-qubit bit-flip code with sta- − ∈ 1S ∈ 1S d must be odd. bilizer = Z1Z2,Z2Z3 , and choose 0 L = 000 , 1 L = S { } | i | i | i If for all coordinates j ξ, there are elements R , R , R 111 . The Toffoli gate is transversal on this code and is given ∈ 1 2 3 ∈ | i 3 (Q) such that (R1)j = X, (R2)j = Y , and (R3)j = Z, by Toffoli = j=1 Toffolij . The Toffoli gate up to a local M 3 then for U Aut(Q), Uj U(2) is a Clifford operator for Clifford is not in ( Zi i=1); however, the Toffoli gate up ∈ ∈ N N h± i 3 all j ξ by Theorem 1. Permutations are Clifford operations to a local Clifford does keep span( Zi i=1) invariant. ∈ h± i as well, so all possible transversal logical operations on the Remark 5: If Uj up to a local Clifford keeps first encoded qubit must be Clifford operations, contradicting span( Z(i) r ) invariant, i.e. U transforms any diagonal h± j ii=1 j the assumption that the transversal gates are a universal set. matrix to a diagonal matrix, then Uj is a monomial 11 matrix. Similarly, if U keeps span( X(i) r ) (or where wt c denotes the Hamming weight of a classical code- h± j ii=1 span( Y (i) r )) invariant, then U is a monomial matrix word. h± j ii=1 j in the Xj (or Yj ) representation. This does not necessarily The corollary’s condition can be satisfied if and only if the n mean that U = j=1 Uj is a monomial matrix (in one of weight of all the codewords in C1 are divisible by a common the X,Y,Z representations) in the 2nr dimensional Hilbert divisor. N space, since in general some of the Uj might be Clifford Definition 15: A classical linear code is said to be divisible operations. by ∆ if ∆ divides the weight of each codeword. A classical Remark 6: Corollary 2 also applies to a set of gates. A set linear code is divisible if it has a divisor larger than 1. An of gates Vi, i = 1,...,k, (Vi)j / r for all j [n], is [n, k] classical code can be viewed as a pair (V, Λ) where V ∈ L ∈ transversal on a stabilizer code only if all of the Vi up to the is a k-dimensional binary vector space and Λ= λ1,...,λn r { } same local Clifford keep the operator space span( Zi ) is a multiset of n members of the dual space V that serve h± ii=1 ∗ invariant. to encode v V as c = (λ1(v),...,λn(v)) and the image of Example 10: The set of gates Hadamard, Toffoli can- V in 0, 1 n∈is k-dimensional. The b-fold replication of C is { } not both be transversal on any stabilizer code, since (V, rΛ){ where} rΛ is the multiset in which each member of Λ r Hadamard keeps span( Yi i=1) invariant and Toffoli keeps appears r times. span( Z r ) invariant.h± Thesei observations imply that all h± iii=1 The following theorem, which is less general than that transversal gates are Clifford, but Toffoli is not Clifford. Note proven in [26], gives evidence (though not a proof) that Hadamard, Toffoli is “universal” for quantum computation π the allowable value θ might only be (k+2) , which implies in{ a sense that all} the real gates can be approximated to an 2 U (1) (see Definition 5). It would be interesting if all arbitrary accuracy [25]. k of the∈ C transversal gates for stabilizer codes lie within the Now we restrict ourselves to the single block case. Up to k hierarchy. C local Clifford equivalence, Corollary 1 and Corollary 2 say that the unitary part of a code automorphism is a diagonal Theorem 4 ([26]): Let C be an [n, k] classical binary code k 1 gate. Therefore, we may restrict our discussion of the essential that is divisible by ∆, and let b = ∆/gcd(∆, 2 − ). Then C non-Clifford elements of Aut (Q) to diagonal gates, because is equivalent to a b-fold replicated code, possibly with some we can imagine considering the diagonal automorphisms for added 0-coordinates. all locally Clifford equivalent codes and their permutation The Reed-Muller codes are well-known examples of divisi- equivalent codes to find all of the non-Clifford automorphisms. ble codes. Furthermore, they are nested in a suitable way and We further restrict ourselves to the case where the stabilizer their dual codes are also Reed-Muller codes, which makes code is CSS code. them amenable to the CSS construction. In particular: Lemma 10: Let Q be a CSS code CSS(C1, C2) constructed Theorem 5 (1.10.1, [19]): Let RM(r, m) be the rth order from classical binary codes C < C . Then Reed-Muller code with block size n = 2m and 0 r m. 2⊥ 1 ≤ ≤ n Then V = diag (1,eiθℓ ) Aut (Q) (49) ∈ (i) RM(i,m) RM(j, m), 0 i j m ℓ=1 ⊆ r m≤ ≤ ≤ O (ii) dim RM(r, m)= i=0 i m r iff c,c C and a C /C , (iii) d =2 − ′ 2⊥ 1 2⊥ P ∀ ∈ ∀ ∈ (iv) RM(m,m)⊥ = 0 and if 0 r < m then { } ≤ θℓ = θℓ mod 2π. (50) RM(r, m)⊥ = RM(m r 1,m). ′ − − ℓ supp (a+c) ℓ supp (a+c ) m/r 1 ∈ X ∈ X Lemma 11: RM(r, m) is divisible by ∆=2⌊ ⌋− . Proof: The states Corollary 4: Let even(RM ∗(r, m)) = C2⊥ < C1 = RM ∗(r, m) where 0 < r m/2 . Then CSS(C1, C2) is a˜ a + c ,a C /C⊥, (51) 1 2 an [[n = 2m 1, 1, d = ≤min ⌊(2m ⌋r 1, 2r+1 1)]] code | i∝ ⊥ | i ∈ − c C2 − n − i2π/∆− X∈ with a transversal gate G = j=1 diag (1,e ) enacting i2π/∆ (1) ⊗ m/r 1 are a basis for Q. V is diagonal, so V c = v(c) c for c C1 G¯ = diag (1,e− ) where ∆=2⌊ ⌋− . | i | i ∈ ∈Clog2 ∆ and a factor v(c) C that is a sum of angles. V is a logical For instance, the [[2m 1, 1, 3]] CSS codes constructed from operation so V a˜ ∈ Q, which is possible for a diagonal gate − | i ∈ the first-order punctured Reed-Muller code R∗(1,m) and its iff v(a+c)= v(a+c′) for all a C /C⊥ and all c,c′ C⊥. ∈ 1 2 ∈ 2 even subcode even(R∗(1,m)) support the transversal gate π ¯ exp( i 2m−1 Z) [15], [11]. The smallest of these, a [[15, 1, 3]] We now restrict to the case where the angles θℓ = θ are all mentioned− in the introduction, has found application in magic equal. state distillation schemes [27] and measurement-based fault- Corollary 3: Let Q be a CSS code constructed from clas- tolerance schemes [28]. If we choose parameters m = 8 and sical binary codes C2⊥ < C1. A gate V Aut (Q) is a r =2 then we have a [[255, 1, 7]] code with transversal T , but ∈ iθ tensor product of n diagonal unitaries Vθ = diag (1,e ) iff this is not competitive with the concatenated [[15, 1, 3]] code. c,c′ C⊥ and a C /C⊥, ∀ ∈ 2 ∀ ∈ 1 2 We leave open the possibility that other families of classical θ divisible codes give better CSS codes with d > 3 or k > 1 (wt (a + c) wt (a + c′)) Z, (52) 2π − ∈ and transversal non-Clifford gates. 12
VI. CONCLUSION [7] M. Oskin, F. Chong, and I. Chuang, “A practical architecture for reliable quantum computers,” IEEE Computer, vol. 35, pp. 79–87, 2002. We have proven that a binary stabilizer code with a quantum [8] P. Aliferis and A. Cross, “Subsystem fault-tolerance with the bacon-shor computationally universal set of transversal gates for even one code,” Phys. Rev. Lett., vol. 98, no. 220502, 2007, quant-ph/0610063. of its encoded logical qubits cannot exist, even when those [9] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction via codes over gf(4),” IEEE Trans. Inf. transversal gates act between any number of encoded blocks. Theory, vol. 44, pp. 1369–1387, 1998, quant-ph/9608006. Also proven is that even when coordinate permutations are [10] A. M. Steane, “Multiple particle interference and quantum error cor- allowed, universality cannot be achieved for any single block rection,” Proc. Roy. Soc. Lond., vol. 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Zurek, “Threshold accuracy for quantum in the automorphism group of the code is essentially limited computation,” 1996, quant-ph/9610011. to diagonal gates conjugated by Clifford operations, together [15] B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary with coordinate permutations. This observation suggests a versus local clifford equivalence of stabilizer and graph states,” Phys. Rev. A, vol. 75, no. 3, 2007, quant-ph/0611214. broad family of quantum CSS codes that can be derived from [16] M. V. den Nest, J. Dehaene, and B. D. Moor, “On local unitary versus classical divisible codes and that exhibit the attainable non- local clifford equivalence of stabilizer states,” Phys. Rev. A, vol. 71, no. Clifford single-block transversal gates. In general, it is not 062323, 2005, quant-ph/0411115. [17] D. Gottesman and I. 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We could further http://gross.qipc.org/. relax the definitions of transversality and conditions for fault- [25] Y. Shi, “Both toffoli and controlled-not need little help to do universal quantum computation,” Quantum Information and Computation, vol. 3, tolerance so that each Ui acts on a small number of qubits in no. 1, pp. 84–92, 2003, quant-ph/0205115. each block. This latter method is fault-tolerant provided that [26] H. N. Ward, “An introduction to divisible codes,” Designs, Codes, and each Ui acts on fewer than t qubits. Finally, the generalization Cryptography, vol. 17, pp. 73–79, 1999. to nonbinary stabilizer codes, and further to arbitrary quantum [27] S. Bravyi and A. Kitaev, “Universal quantum computation with ideal clifford gates and noisy ancillas,” Phys. Rev. A, vol. 71, no. 022316, codes, remain open possibilities. 2005, quant-ph/0403025. [28] R. Raussendorf, J. Harrington, and K. Goyal, “A fault-tolerant one-way ACKNOWLEDGMENTS quantum computer,” Ann. Phys., vol. 321, no. 9, pp. 2242–2270, 2006, quant-ph/0510135. We thank Panos Aliferis, Sergey Bravyi, David DiVincenzo, Ben Reichardt, Graeme Smith, John Smolin, and Barbara Ter- hal for comments, criticisms, and corrections. AC is partially supported by a research internship at IBM.
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