PHYSICAL REVIEW X 8, 021047 (2018)
Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates
Tomas Jochym-O’Connor,1,2 Aleksander Kubica,2,3,4 and Theodore J. Yoder5,6 1Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA 2Institute for Quantum Information & Matter, California Institute of Technology, Pasadena, California 91125, USA 3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 4Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 5Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 6IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA
(Received 28 October 2017; revised manuscript received 3 March 2018; published 21 May 2018)
Stabilizer codes are among the most successful quantum error-correcting codes, yet they have important limitations on their ability to fault tolerantly compute. Here, we introduce a new quantity, the disjointness of the stabilizer code, which, roughly speaking, is the number of mostly nonoverlapping representations of any given nontrivial logical Pauli operator. The notion of disjointness proves useful in limiting transversal gates on any error-detecting stabilizer code to a finite level of the Clifford hierarchy. For code families, we can similarly restrict logical operators implemented by constant-depth circuits. For instance, we show that it is impossible, with a constant-depth but possibly geometrically nonlocal circuit, to implement a logical non-Clifford gate on the standard two-dimensional surface code.
DOI: 10.1103/PhysRevX.8.021047 Subject Areas: Quantum Physics, Quantum Information
I. INTRODUCTION fault-tolerant operators, both theoretically and experimen- tally, so it is important to understand exactly what logical Quantum error-correcting codes form the foundation of operators they can implement. scalable quantum computing [1–3]. By construction, quan- Unfortunately, the set of transversal or, more generally, tum codes serve as quantum memories by protecting constant-depth logical operators is inherently limited, with encoded data from a noisy environment and successfully computational universality generally incommensurate with extending the storage time, at least if the noise is sufficiently the error-correction capabilities of the code. In particular, small. However, a quantum computer should do more than there is a no-go theorem from Eastin and Knill, which states just store quantum data; It needs to also apply logical that transversal operators on any nontrivial quantum code operations to the data [4,5]. These operations must therefore belong to a finite group and thus cannot be universal [6,7]. be implemented fault tolerantly upon quantum codes. Similar no-go theorems limiting logical operators to be in a Generally, operators are fault tolerant if they do not finite level of the Clifford hierarchy were derived for couple too many qubits within a particular code block. This transversal single-qubit gates and two-qubit diagonal gates condition is sufficient to limit the spread of errors and also on stabilizer codes [8], as well as for constant-depth, local guarantee that, if parts of the circuitry implementing the circuits on stabilizer and subsystem topological codes [9,10]. operator were to fail, not many qubits would be affected. The latter result has an important implication—one cannot With respect to some partitioning of the code qubits into achieve a universal gate set with constant-depth local circuits small, disjoint subsets Q , a transversal operator acts on i on two-dimensional (2D) topological codes such as those in each subset of qubits Q independently. For a family of i Refs. [11,12]. We also remark that one can consider more codes with increasing size, a constant-depth logical oper- general models beyond stabilizer codes, such as 2D topo- ator is implementable by a constant (independent of the logical quantum field theories, and characterize the set of code size) depth circuit over the subsets Q . Transversal and i gates implementable by locality-preserving unitaries [13,14]. constant-depth circuits are some of the simplest possible Here, we address several related questions regarding transversal and constant-depth logical operators on stabi- lizer codes using a new quantity called the disjointness of Published by the American Physical Society under the terms of the code. The disjointness, roughly speaking, is the number the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of mostly nonoverlapping representatives of any given the author(s) and the published article’s title, journal citation, nontrivial logical Pauli operator. We use the disjointness and DOI. to show that all transversal logical operators on stabilizer
2160-3308=18=8(2)=021047(14) 021047-1 Published by the American Physical Society JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) codes must be in the Clifford hierarchy, as conjectured by suppðAÞ the set of qubits an operator A acts on nontrivially Zeng et al. [7]. Moreover, we find explicit upper bounds on (we later generalize this notion). Since we assume that U is the level attainable. In particular, this result rules out the constant depth, then jsuppð½U; p Þj ¼ OðlÞ. Note that a possibility of transversal implementation of operators that tensor product of Pauli Z operators on qubits along any can be used to bootstrap up the Clifford hierarchy, such as vertical path on the lattice would implement the logical the Toffoli gate [15]. Importantly, our result, when applied Pauli Z¯ [see Fig. 1(a)]. Similarly, Pauli X operators along to families of codes of growing size, restricts constant- any horizontal path implement the logical Pauli X¯ . Thus, depth circuits to the Clifford hierarchy, regardless of their for any operator Q¯ ∈ fX;¯ Z¯ g, we can choose OðlÞ different, geometric locality. For instance, for the 2D surface code on nonoverlapping representatives. Using the pigeonhole a square lattice of size l × OðlÞ, we find that even nonlocal principle, we are guaranteed to find a representative q of constant-depth circuits cannot implement logical non- Q¯ , such that it has constant overlap with ½U; p [see Clifford operators. Asymmetry of logical operators appears Fig. 1(b) for an example]. This, in turn, implies that the in our bounds as a necessary condition for possessing operator ½½U; p ;q is supported on a constant-size region, constant-depth circuits for non-Clifford gates, such as those 1 on 3D color and toric codes [16,17] and on asymmetric 2D jsuppð½½U; p ;q Þj ¼ Oð Þ. Since the distance of the code is Bacon-Shor codes [18]. OðlÞ, the region suppð½½U; p ;q Þ is correctable. We con- clude that ½½U; p ;q can only be a trivial logical operator, and thus U implements a logical Clifford operator. II. INTUITION The property of any stabilizer code that we would like to In this section, we sketch out the proof that constant- abstract from the provided example of the 2D surface code depth circuits, even with gates that are geometrically is the existence of several disjoint representatives of the nonlocal, cannot implement logical non-Clifford operators same logical Pauli operator. In the following sections, we on the 2D surface code of size l × OðlÞ (see Fig. 1). We use introduce a notion of disjointness of a stabilizer code, the following two key ideas: (i) There are many non- which quantitatively captures that property. We remark that overlapping representatives for logical Pauli X¯ and Z¯ the disjointness of the 2D surface code is OðlÞ since we can operators; (ii) logical operators supported on a correctable find a set of OðlÞ nonoverlapping representatives of either ¯ ¯ region are trivial. X or Z. ¯ In order to find out what logical gate a unitary U As a final remark, consider Y. We argued above that ¯ ¯ implements, it is sufficient to characterize the action of ½U; X and ½U; Z are logical Pauli operators, which implies ¯ ¯ ¯ ¯ ¯ U on the logical Pauli operators. Let ½A; B ¼ABA†B† that ½U; Y ¼½U; X X½U; Z X is a logical Pauli as well. Yet, represent the group commutator of two unitaries A and proving this directly by an argument analogous to the one ¯ ¯ B [9]. We know that for any two logical Pauli operators presented for X and Z does not work exactly. This is ¯ P¯ , Q¯ ∈ fX;¯ Z¯ g, if the group commutator ½½U; P¯ ; Q¯ is a because any two different representatives of Y necessarily trivial logical operator, then the unitary U implements a overlap on at least one qubit, and the argument breaks down logical Clifford operator [19]. since it is impossible to find a set of OðlÞ disjoint Let us pick a representative p of the logical operator representatives of Y¯ . Nevertheless, we can find a set of l P¯ ∈ fX;¯ Z¯ g, such that jsuppðpÞj ¼ OðlÞ. We denote by representatives of Y¯ such that any single qubit is in support of at most two representatives from the set. We call such a set of representatives 2-disjoint. Given this 2-disjoint set, the pigeonhole argument can again be used to find a representative q of Y¯ such that ½½U; p ;q ¼Oð1Þ, where p is the previously chosen representative of a logical Pauli P¯ ∈ fX;¯ Z¯ g of weight OðlÞ. As we will see in the rest of the paper, considering overlapping representatives of logical Pauli operators and c-disjoint sets proves very useful in constraining fault-tolerant logical gates.
(a) (b) III. PRELIMINARIES FIG. 1. (a) The 2D surface code has X-vertex and Z-plaquette Let us consider systems composed of m-dimensional stabilizer generators and many nonoverlapping representatives of qudits, m ≥ 2. The Pauli group on a set of n qudits, denoted logical Pauli X¯ (green) and Z¯ (blue) operators. (b) For a given Pn, is generated by the X- and Z-type operators [20] representative of Z¯ (blue) and any constant-depth circuit U with possibly geometrically nonlocal gates (depicted in yellow), one Xm−1 Xm−1 j can always find a representative of X¯ (green), such that UZU¯ † X ¼ jj ⊕ 1ihjj;Z¼ ωmjjihjj; ð1Þ and X¯ overlap only on a constant number of qubits. j¼0 j¼0
021047-2 DISJOINTNESS OF STABILIZER CODES AND … PHYS. REV. X 8, 021047 (2018) where addition ⊕ inside bra-kets is modulo m and transversal, it means that U is a tensor product of ωm ¼ expð2πi=mÞ. Letting Un denote the group of n-qudit single-qudit unitaries. However, we consider a more unitaries, we note that Pn is a subgroup of Un because general definition of a transversal gate [23] Partition the X, Z ∈ Un. set of n physical qudits, labeled by integers from Any Pauli group P can be used to define a hierarchy of ½n ¼f1; 2; …;ng, into N disjoint, nonempty subsets n-qudit unitaries called the Clifford hierarchy [21]. The Qi ⊆ ½n , namely, Mth level of this hierarchy is a finite set of unitaries (if the global phases are ignored) recursively defined as ½n ¼Q1 ∪ Q2 ∪ … ∪ QN: ð7Þ
C1ðPÞ¼P; ð2Þ Then, we say that an n-qudit unitary U is transversal if it N ⊗ P ∈ U ∶ ∈ P ∀ ∈ P can be decomposed as U ¼ Ui, where each unitary Ui CMð Þ¼fU n ½U; p CM−1ð Þ; p g: ð3Þ i¼1 acts only on qudits in the subset Qi. The support of U, The first and second levels of the hierarchy correspond to denoted by suppðUÞ ⊆ ½N , is the index set of all subsets the Pauli and Clifford groups, respectively. Qi, on which U acts nontrivially. The typical notion of In this article, we focus our attention on a particularly transversal gate now simply corresponds to the partition popular class of quantum codes—stabilizer codes [22]. into single qudits, Qi ¼fig. A stabilizer code is defined by the stabilizer group We emphasize that for a given code, the set of transversal S … ⊆ P − ¼hs1;s2; ;sn−ki n, which is generated by n k logical operators can depend on the choice of the qudit mutually commuting Pauli operators. The codespace C is a partition. In particular, if the partition is not fixed, then one m ⊗n subspace of the Hilbert space H ≃ ðC Þ on n qudits, can achieve a universal gate set of transversal operators, as which is the simultaneous ðþ1Þ-eigenspace of all stabilizer in the following example [24]. ⟦ ⟧ generators si. We denote by n; k a qudit stabilizer code, Example 1. Consider the [[105,1]] code, which is a which uses n physical qudits to encode k logical ones. concatenation of the Steane 7-qubit code with the 15-qubit For any stabilizer code, a logical operator is a unitary on Reed-Muller code. We illustrate this code in Fig. 2 as a the Hilbert space H that maps states in C to states in C.In 7 × 15 array of qubits. We consider two qubit partitions: particular, logical Pauli operators can be found as elements (a) Each Qi is a subset of 7 qubits from the ith column; of the normalizer N ðSÞ of the stabilizer group S in the (b) each Q is a subset of 15 qubits from the ith row. With P 2 ¯ ¯ ∈ P i Pauli group n. We choose k generators Xi, Zi n of respect to the first and second partitions, the [[105,1]] code P¯ 2π 8 the logical Pauli group k that commute with all stabilizer has, correspondingly, transversal logical T ¼ diagð1;e i= Þ generators and also satisfy and Hadamard gates. For more details, see Ref. [24].
−δ In contrast, we fix a partition and prove limitations on ¯ ¯ ij ½Xi; Zj ¼ωm I; ð4Þ logical operators with respect to that partition. For instance, in this fixed-partition scenario, Ref. [6] implies that the ¯ ¯ ¯ ¯ ½Xi; Xj ¼½Zi; Zj ¼I: ð5Þ group of transversal operators is finite and therefore not universal. We define L to be the set of all nontrivial logical Pauli Transversal unitaries are a special case of what we call operators as follows: q-local operators of depth h (with respect to the partition fQig). A unitary U is q-local of depth one, if it is Yk L S ¯ ai ¯ aiþk ∶ ∈ 0 1 … − 1 2k 0 2k ¼ Xi Zi a f ; ; ;m g nf g : ð6Þ i¼1
We remark that each element G ∈ L is a coset of S in N ðSÞ, although in examples we abuse notation and equate G with the logical Pauli that it corresponds to (e.g., X¯ ¼ SX¯ ∈ L). Also, G contains jSj¼mn−k representa- (a)(a) (b)(b) tives of the same nontrivial logical operator. FIG. 2. Two different partitions of the [[105,1]] qubit stabilizer code, which is a concatenation of the 7-qubit and 15-qubit codes. IV. TRANSVERSAL GATES Each row of the 7 × 15 array is an instance of the 15-qubit code that encodes one of the qubits of the 7-qubit code. Depending on All manipulations of a quantum code should be fault the partition of the qubits, the code can have either a transversal tolerant, in the sense that they do not spread errors logical (a) T ¼ diagð1; exp 2πi=8Þ or (b) Hadamard gate. Here, a throughout the system in an uncontrollable way. The transversal unitary with respect to a given partition is a tensor simplest example of such a manipulation is a transversal product of unitaries, each of which is supported only on qubits logical operator U. Typically, when one says U is within one subset of the partition (red or green box).
021047-3 JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) transversal with respect to a second, “coarse-grained” Now, we are ready to define the disjointness Δ of a code. partition fRjg, where each Rj is the union of at most q Definition 1 (disjointness). For any n-qudit stabilizer L of the Qi. Accordingly, a q-local unitary of depth h is a code with the set of nontrivial logical operators and a product of hq-local unitaries of depth one. We note that qudit partition ½n ¼Q1 ∪ Q2 ∪ … ∪ QN, the disjointness transversal operators are 1-local unitaries of depth one. is defined as
Δ ¼ max min ΔcðGÞ: ð12Þ V. DISTANCE AND DISJOINTNESS c≥1 G∈L A fundamental property of stabilizer codes is the dis- We illustrate disjointness with the following example of the tance. Typically, one says that the code has distance d if it 2D surface code. can detect any error that affects at most d − 1 qudits. Here, Example 3. Consider the 2D surface code of size however, we consider distance with respect to the qudit l × l encoding one logical qubit [25] and the single- partition fQig. First, we define the distance dðGÞ of the ¯ 2 − 1 ∈ L qubit partition. We have d↑ ¼ dðYÞ¼ l and d↓ ¼ nontrivial logical operator G to be the size of the ¯ ¯ smallest support of any of its representatives: dðXÞ¼dðZÞ¼l. Moreover, there are exactly l represent- atives of X¯ with weight l, and they are all disjoint. Thus, Δ ¯ Δ ¯ dðGÞ¼minjsuppðgÞj: ð8Þ 1ðXÞ¼l. Similarly, 1ðZÞ¼l. In contrast, different g∈G representatives of Y¯ necessarily overlap, but we can nevertheless find l representatives of minimal weight Then, we introduce two notions of the distance d↓ and d↑, 2l − 1, such that each qubit is in the support of at most the min- and max-distance of the code, as follows: two of them. Those representatives of Y¯ form a 2-disjoint ¯ ¯ set. Thus, Δ1ðYÞ¼1,butΔ2ðYÞ¼l=2.Now,byEq.(12), d↓ ¼ min dðGÞ;d↑ ¼ max dðGÞ: ð9Þ Δ ≥ Δ G∈L G∈L minG∈L cðGÞðGÞ, where cðGÞ may depend on G. ¯ ¯ ¯ Thus, even by just evaluating Δ1ðXÞ, Δ2ðYÞ, and Δ1ðZÞ, We call a code “error detecting” iff its min-distance d↓ is we are able to conclude that the disjointness Δ of the greater than 1. Note that the min-distance d↓ is never surface code satisfies Δ ≥ l=2. greater than the (standard) distance d of the code. Also, if The disjointness Δ turns out to be an important quantity we choose a single-qudit partition, then those two quan- characterizing stabilizer codes. In particular, we use it to tities coincide, d↓ ¼ d. find bounds on the level of the logical Clifford hierarchy In this article, we propose a new quantity for quantum achievable with transversal (see Theorem 5 in Sec. VI)or stabilizer codes, the disjointness, which proves remarkably constant-depth (see Theorem 9 in Sec. VII) logical uni- useful for establishing limitations on logical gates. First, for taries. To facilitate further discussion, we present key any nontrivial logical operator G ∈ L and a positive integer properties of the disjointness. ⟦ ⟧ c ≥ 1, we define c-disjointness Δ ðGÞ to be the maximal Lemma 2 [properties of disjointness]. For any n; k c ∪ ∪ ∪ number (divided by c) of representatives of G chosen in stabilizer code and any partition ½n ¼Q1 Q2 QN, such a way that at most c representatives have support on the disjointness satisfies (i) 1 ≤ Δ ≤ minðd↓;N=d↑Þ. any Qi, a subset of the qudit partition: (ii) Δ > 1 iff the stabilizer code is error detecting, 1 i.e., d↓ > 1. Δ ∶ ∈ cðGÞ¼ maxfjAj at most c elements a A — c A⊆G Proof. We begin by proving four bounds on c-disjointness that together imply case (i). In particular, have support on any Q g: ð10Þ i let G, G0 ∈ L be two noncommuting, nontrivial logical operators. In other words, ½g; g0 ≠ I for all g ∈ G and We call the set A in Eq. (10) c-disjoint. To build intuition 0 ∈ 0 1 ≤ ≤ n−k about the c-disjointness, consider a small example. g G . Then, for any c m (recall m is the qudit Example 2. Consider the ⟦4; 2⟧ qubit code with the dimension), stabilizer group S ¼hX⊗4;Z⊗4i and the single-qubit par- 1 ≤ Δ ≤ n−k tition. There are four equivalent logical operators imple- cðGÞ m =c; ð13Þ ¯ menting a logical X1 ¼ X1X2, which form a set 0 ΔcðGÞ ≤ dðG Þ; ð14Þ G ¼fX1X2;X3X4;Y1Y2Z3Z4;Z1Z2Y3Y4g: ð11Þ ΔcðGÞdðGÞ ≤ N: ð15Þ The set fX1X2;X3X4g is a maximal 1-disjoint set, fX1X2;X3X4;Y1Y2Z3Z4g is a maximal 2-disjoint set, Moreover, each upper bound holds for all c ≥ 1. n−k and G itself is a maximal 3-disjoint set. Thus, Δ1ðGÞ¼2, The lower bound in Eq. (13) is true because any c ≤ m Δ2ðGÞ¼3=2, Δ3ðGÞ¼4=3. elements of G form a c-disjoint set of size c. The upper
021047-4 DISJOINTNESS OF STABILIZER CODES AND … PHYS. REV. X 8, 021047 (2018) bound in Eq. (13) results because any c-disjoint set A ⊆ G then i ∈ suppðgR Þ for the j such that i ∈ Rj, whereas if − j satisfies jAj ≤ jGj¼mn k. As a result of the upper bound, i ∈ H, then i ∈ suppðgÞ. n−k for any c>m , ΔcðGÞ < 1. Along with the lower bound, Thus, the set A can serve as an example of a c-disjoint Δ − Δ this implies minc≥1 cðGÞ¼min1≤c≤mn k cðGÞ for all subset of G, and we obtain a lower bound ΔcðGÞ ≥ G ∈ L, which simplifies the definition of disjointness jAj=c ¼ 1 þ 1=c > 1 on the c-disjointness of G. This, in in Eq. (12). turn, implies that the disjointness Δ of the code is greater For Eq. (14), choose a maximal c-disjoint set A ⊆ G and than 1, Δ ≥ 1 þ 1=⌈d↑=ðd↓ − 1Þ⌉ > 1, finishing the proof a representative g0 ∈ G0 of minimal support. In other words, of case (ii). □ 0 0 jAj¼cΔcðGÞ and jsuppðg Þj ¼ dðG Þ. By definition, every Certain codes even have disjointness saturating the upper g ∈ A does not commute with g0. Thus, g and g0 must have bound in Lemma 2(i), as in the following example. nontrivial overlap, jsuppðgÞ ∩ suppðg0Þj ≥ 1. Consider any Example 4. Consider the family of Reed-Muller codes ⊆ Dþ1 collection H ½N of some qudit subsets Qi. Since at most ⟦n ¼ 2 − 1;k¼ 1⟧ for D ≥ 2, which coincides with a c elements of A intersect at any subset of qudits Qi,wehave family of color codes of distance three in D spatial the inequality dimensions [28–30]. We consider the single-qubit partition. X X X The two smallest codes in this family correspond to the jsuppðgÞ ∩ Hj¼ jsuppðgÞ ∩ figj ð16Þ 7-qubit Steane and the 15-qubit Reed-Muller codes. The ¯ ¯ ¯ ¯ g∈A g∈A i∈H distance of logical X, Y, Z operators satisfies dðXÞ¼ ¯ D ¯ X dðYÞ¼2 − 1 and dðZÞ¼3. Thus, d↓ ¼ 3 and D Dþ1 ¯ ≤ c · 1 ¼ cjHj: ð17Þ d↑ ¼ 2 − 1. There are 2 representatives of X, and i∈H 2Dþ1 − 1 of them have minimal support. The set of minimal P P ¯ representatives of X is, in fact, d↑-disjoint, and therefore Therefore, cΔ ðGÞ¼jAj¼ ∈ 1 ≤ ∈ jsuppðgÞ ∩ c g A g A Δ ¯ ¯ 0 ≤ 0 d↑ ðXÞ¼n=d↑. Moreover, for each representative g of X, suppðg Þj cdðG Þ, proving Eq. (14). ¯ Similarly, Eq. (15) follows from Eqs. (16) and (17) one can always find at least one representative of Z (and ¯ Δ ¯ by setting H ¼½N and using jsuppðgÞj ≥ minp∈Gj thus of Y) supported on suppðgÞ. We obtain that d↑ ðZÞ, Δ ¯ ≥ Δ ¯ suppðpÞj ¼ dðGÞ. d↑ ðYÞ d↑ ðXÞ¼n=d↑, which results in a bound – To get case (i) from Eqs. (13) (15), note that they each on the disjointness Δ ≥ n=d↑. However, Δ ≤ n=d↑ from hold for all c, so we can replace Δ G with max ≥1Δ G cð Þ c cð Þ Lemma 2(i), implying Δ ¼ n=d↑. in all three equations. Since Eq. (13) also holds for all c Δ G ∈ L 1 ≤ Δ The -disjointness cð Þ of a nontrivial logical operator G , minimizing it over G immediately implies G ∈ L quantifies how well G can be “cleaned” (in the sense 0 ∈ L 0 as well. In Eq. (14), take G such that dðG Þ¼d↓ of Ref. [26]) from an arbitrary subset of qudits. We (and G to be any anticommuting logical Pauli), and in conclude this section with a useful lemma that is needed ∈ L Δ ≤ Eq. (15), take G so that dðGÞ¼d↑ to conclude to prove the main results of our work. Δ ≤ Δ ≤ Δ ≤ maxc≥1 cðGÞ d↓ and maxc≥1 cðGÞ N=d↑, Lemma 4 (scrubbing lemma). Consider a nontrivial respectively. logical operator G ∈ L and a collection H ⊆ ½N of qudit We now prove case (ii). First, note that the implication n−k subsets Qi. For any 1 ≤ c ≤ m , there exists a represen- d↓ ¼ 1 ⇒ Δ ¼ 1 follows from case (i). To show tative g ∈ G such that d↓ > 1 ⇒ Δ > 1, we establish a stronger fact: For all G ∈ L,ifd↓ > 1, then for c ⌈d G = d↓ − 1 ⌉,wehave ¼ ð Þ ð Þ ΔcðGÞjsuppðgÞ ∩ Hj ≤ jHj: ð18Þ ΔcðGÞ ≥ 1 þ 1=⌈dðGÞ=ðd↓ − 1Þ⌉. We make use of the following version of the cleaning lemma. Proof.—Let A ⊆ G be a maximal c-disjoint set, jAj¼ 0 Lemma 3 (cleaning lemma [26,27]). For any nontrivial cΔ ðGÞ. Then, choosing g ¼ argmin 0∈ jsuppðg Þ ∩ Hj, ∈ L ⊆ c g A logical operator G and any collection R ½N of qudit we have subsets Qi such that jRj 021047-5 JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) whenever ΔcðGÞ > 1, which is exactly the situation for Clifford hierarchy, we recursively obtain that KM−j is a error-detecting stabilizer codes; see Lemma 2(ii). logical operator from the jth level. In particular, K0 must be ¯ in the Mth level CM. □ VI. LIMITATIONS ON TRANSVERSAL GATES We remark that Theorem 5 implies that transversal operators on a single code block of any error-detecting In this section, we use the disjointness to bound the code must be in a finite level of the Clifford hierarchy. transversal logical gates on any error-detecting stabilizer Namely, from Lemma 2(ii), we get Δ > 1, and thus we can code to the Clifford hierarchy of the logical Pauli group always find an integer M ¼ ⌈ logΔðd↑=d↓Þ⌉ satisfying ¯ P¯ CM ¼ CMð Þ. We start with a theorem for transversal Eq. (21). We illustrate Theorem 5 with the following operators on a single code block, which we later generalize examples. to operators between r code blocks. Example 5. The non-CSS 5-qubit stabilizer code Theorem 5. Consider a stabilizer code with min-dis- [31,32] has the stabilizer group S ¼hZ1Z2X3X5; tance d↓, max-distance d↑, and disjointness Δ.IfM is an X1Z2Z3X4;X2Z3Z4X5;X1X3Z4Z5i and logical Pauli rep- integer satisfying ¯ ⊗5 ¯ ⊗5 resentatives X ¼ X , and Z ¼ Z has d↑ ¼ d↓ ¼ 3 and M−1 Δ ¼ 5=3 with respect to the single-qubit partition. Thus, d↑ 021047-6 DISJOINTNESS OF STABILIZER CODES AND … PHYS. REV. X 8, 021047 (2018) the bound on the level is not conveniently stated in terms of Proof.Q —First, we express the transversal operator Δ d↓, d↑, of the base code but rather in terms of the A ¼ i∈suppðAÞAi as a product of operators Ai, each of effective multiblock code. We address both of these issues which is supported only on one of the qudit subsets, i.e., in Appendix B, with more detailed arguments for the 1 † ⊆ † jsuppðAiÞj ¼ . Then, suppðAi Þ suppðUAiU Þ and multicode block case. We summarize the results with the † ≤ h I ∩ jsuppðUAiU Þj Qq . Let Q¼ suppðUÞ suppðAÞ, and following version of Theorem 5 for stabilizer codes with † † then ½U; A ¼ð ∈I UA U Þ ∈I A . Note that for any multiple code blocks. i i i i two operators V and W, we have suppðVWÞ ⊆ suppðVÞ ∪ Theorem 7 (multi code block case). Consider an ⟦n; k⟧ suppðWÞ. Using this fact, we get stabilizer code constructed from m-dimensional qudits. With respect to a partition of the qudits into N subsets † suppð½U; A Þ ⊆⋃suppðUA U†Þ ∪⋃suppðA Þð29Þ Q , let the code’s parameters be d↓, d↑, and Δ.Now, i i i i∈I i∈I consider r code blocks of this code, and let r0 ¼ − minðr; N!mn kÞ.IfM is an integer satisfying † ¼ ⋃suppðUAiU Þ; ð30Þ i∈I 0 r0 M−1 r d↑ð1 − ð1 − 1=ΔÞ Þ 021047-7 JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) therefore sufficient to consider the limit of Eq. (34).We gates implemented via constant-depth (possibly geometri- arrive at the following corollary. cally nonlocal) circuits are limited to the Mth level of the Corollary 10. If for a family of stabilizer codes Clifford hierarchy, where M ¼bðD − sÞ=scþ1. Note that ⟦nðlÞ;kðlÞ⟧ with parameters d↓ðlÞ, d↑ðlÞ, and ΔðlÞ there as in Example 9, the greater the asymmetry of the support exists an integer M satisfying of the logical operators (or, in other words, the difference in the dimensionality of those operators), the higher the level d↑ l of the Clifford hierarchy that is accessible. It is unclear ð Þ 0 lim M−1 ¼ ; ð35Þ l→∞ d↓ðlÞΔðlÞ though how to bound disjointness on more exotic topo- logical codes with fractal-like logical operators, such as ’ then for any constants q and h, all q-local logical gates of Haah s cubic code [34]. ¯ depth h are in the Mth level of the hierarchy CM. We require that the limit vanish with l (rather than, say, VIII. DISCUSSION just being less than 1) so that we can ignore the factors of We have provided explicit upper bounds on the level of constant locality and depth that appear in Eq. (34). We conclude this section with a few examples illustrating the Clifford hierarchy that is accessible for logical operators the usefulness of Corollary 10. on any stabilizer code, which are implemented by trans- Example 8. Consider the family of surface codes on versal and constant-depth circuits. We expect our tech- square lattices of size l × l. As shown in Example 3, the niques to apply similarly to stabilizer codes composed of qudits, which differ in local dimension. As long as code parameters are d↓ðlÞ¼l, d↑ðlÞ¼2l − 1, and stabilizers and Pauli logical operators are tensor products ΔðlÞ ≥ l=2. Since for M>1 we have of Pauli operators on physical qudits, the results and proofs presented should carry through. For example, an extension d↑ðlÞ 2l − 1 0 ≤ ≤ 2M−1 → 0; ð36Þ of our theorems to the dressed logical operators in stabilizer Δ M−1 M →∞ d↓ðlÞ ðlÞ l ½l subsystem codes is provided in Appendix A. At the same time, an alternative hierarchy bounding then constant-depth, constant-locality circuits on surface theorem exists without making use of the disjointness. We codes can only implement logical Clifford gates. notice that in the proof of Theorem 5 instead of Lemma 4, Surprisingly, asymmetric 2D codes can have transversal we could use the following simple corollary of the Cleaning logical non-Clifford gates. For instance, asymmetric Lemma 3: For any nontrivial logical operator G ∈ L Bacon-Shor codes have the transversal logical CCZ gate and a collection H ⊆ ½N of qudit subsets Qi satisfying [18]. We emphasize that the asymmetry in the weight of jHj ≤ d↓ − 1, one can find a representative g ∈ G such different logical Pauli operators affects the ability to bound that jH ∩ suppðgÞj ≤ jHj − ðd↓ − 1Þ. We follow the same logical gates. recursive reduction of support of K as in Theorem 5 and Example 9. Consider the stabilizer code family of j obtain that if M is an integer satisfying asymmetric Bacon-Shor codes in the Z gauge on square a ≥ 1 lattices l × l , a . The code parameters d↓ðlÞ¼l and − 1 − 1 a d↑ 021047-8 DISJOINTNESS OF STABILIZER CODES AND … PHYS. REV. X 8, 021047 (2018) work could be done to compare bounds obtained from National Defense Science and Engineering Graduate Eqs. (38) and (21). (NDSEG) Fellowship program and also an IBM Ph.D. While our main results are derived without assumptions Fellowship award. The authors acknowledge the MIT of geometric locality, we can derive even stronger bounds Open Access Article Publication Subvention Fund as well by assuming geometric locality of the circuits. For instance, as IQIM for support in making this work available for open the D-dimensional surface code encoding one logical qubit access publication. cannot implement non-Clifford logical operators with geometrically local, constant-depth circuits. The argument follows exactly the same lines as that in Sec. II, relying APPENDIX A: SUBSYSTEM CODES essentially on the ability to choose representatives g1, g2 Subsystem stabilizer codes are subspace stabilizer codes of any two logical Pauli operators such that ½½n; k; d for which we designate kl qubits as logical and kg jsuppðg1Þ ∩ suppðg2Þj ¼ Oð1Þ. Since a geometrically local qubits as gauge, kl þ kg ¼ k. Only the logical qubits circuit U cannot greatly distort the support of these contain protected information; the state of the gauge qubits 1 representatives, jsuppð½½U;g1 ;g2 Þj¼Oð Þ as well. The can change arbitrarily during a computation [37]. same argument holds if one superimposes D copies of Therefore, a logical operator U (a unitary on n qubits), the D-dimensional surface code, each encoding one logical by definition, acts on the logical þ gauge qubits as a tensor qubit; geometrically local, constant-depth circuits still product A ⊗ B (a unitary on k qubits), with A acting on the implement at most logical Cliffords. However, a gate from logical qubits and B on the gauge qubits. It cannot be D−1 the Dth level (namely C Z) is accessible if one rotates otherwise—if U were not a tensor product over logical and the D copies relative to one another [17]. gauge, then the change to the logical state would depend on The notion of disjointness for stabilizer codes, which we the state of the gauge qubits, contradicting the principle that introduced, appears to be difficult to calculate exactly. If the state of the gauge qubits is irrelevant. stabilizer codes have some underlying structure, such as Now, we consider a transversal logical operator U on the Reed-Muller codes in Example 4 or topological codes in subsystem code (such that it decomposes like A ⊗ B as Examples 8 and 9, then we can find bounds on the above) and place A somewhere in the Clifford hierarchy. As disjointness, and this usually suffices to establish limits before, we can use the fact that A ∈ C if and only if, for all on the accessible level of the Clifford hierarchy. We believe m p ∈ C1, that it is a challenging open problem to find efficient j methods to compute (or approximate) the disjointness for ½…½½A; p1 ;p2 …;pm ¼I: ðA1Þ an arbitrary stabilizer code. This problem, however, might be substantially simpler for topological codes, where one However, because ½a ⊗ b; c ⊗ d ¼½a; c ⊗ ½b; d , we can could exploit code and lattice symmetries. Also, it would be instead show that for all pj ∈ C1, there are unitaries βj and interesting to find possible relations to other new stabilizer B on the gauge qubits such that code quantities, such as the price [36]. ½…½½A ⊗ B; p1 ⊗ β1 ;p2 ⊗ β2 ; …;pm ⊗ βm ¼I ⊗ B: ACKNOWLEDGMENTS ðA2Þ The authors would like to thank Ben Brown, Steve If we take β to also be Pauli operators, then p ⊗ β is just Flammia, and Daniel Gottesman for helpful discussions. j j j a so-called “dressed” logical Pauli operator of the sub- In particular, we would like to thank Michael Beverland for system code, which can be implemented transversally. comments on the manuscript and for also showing us that the Consider a subsystem code with gauge group T and a transversal gates of all stabilizer and subsystem codes are logical Pauli operator p¯. The whole set of logically restricted to the Clifford hierarchy in unpublished work with equivalent dressed operators is G ¼ pT¯ , a coset of the John Preskill [35]. T. J. acknowledges the support from the gauge group, which we call a dressed coset. Note that in the Walter Burke Institute for Theoretical Physics in the form of case of subspace stabilizer codes (discussed in the main the Sherman Fairchild Fellowship. A. K. acknowledges text) with the stabilizer group S, we considered G ¼ pS¯ . funding provided by the Simons Foundation through the The dressed distance of G is the size of the support of the “It from Qubit” Collaboration, as well as by the Institute for smallest element of G, Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565), with dðGÞ¼minjsuppðgÞj: ðA3Þ support from the Gordon and Betty Moore Foundation g∈G (GBMF-12500028). Research at Perimeter Institute is sup- ported by the Government of Canada through Industry If L is the set of dressed cosets for all nontrivial logical Canada and by the Province of Ontario through the Pauli operators (i.e., all 4kl − 1 of them), then the sub- Ministry of Research and Innovation. T. Y. is grateful for system code also has a dressed min-distance and a dressed support from the Department of Defense (DoD) through the max-distance, 021047-9 JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) d↓ ¼ mindðGÞ; ðA4Þ Gauge fixing refers to the process of measuring the kg G∈L gauge qubits of a subsystem stabilizer code and placing ⊗ them in some fixed stabilizer state (e.g., 0 kg ). The result d↑ ¼ maxdðGÞ: ðA5Þ j i G∈L is a subspace stabilizer code (since there are effectively no more gauge operators), which we call a gauge-fixed code. Likewise, we define the c-disjointness of a dressed coset This process is not transversal by our definitions, but it G to be allows one to switch between using transversal logical 1 operators on the subsystem code and transversal logical ΔcðGÞ¼ maxfjAj∶ at most c elements a ∈ A operators on the gauge-fixed code. Logical operators can be c A⊆G transversal on one of these two codes without being have support on any Qig; ðA6Þ transversal on the other, and thus, together, they can get around the no-go theorems to form a universal set. i.e., exactly the same as for the nondressed cosets of a The following example exemplifies how gauge fixing subspace code. Note that, if the stabilizer of the code is works. The 15-qubit Reed-Muller subsystem code has S ¼ ZðTÞ (i.e., the center of the gauge group), then kl ¼ 1 logical qubit and kg ¼ 6 gauge qubits [38]. The subsystem code has a transversal H gate that acts as H on ⊗6 ΔcðpS¯ Þ ≤ ΔcðpT¯ Þ; ðA7Þ the logical qubit and H on the gauge qubits. The gauge- fixed code in which the gauge qubits are in the state j0⊗6i simply because S ≤ T. The disjointness of the code is has a transversal T gate. Gauge fixing can be used after defined as before as well: applying the transversal H to return the gauge qubits to j0⊗6i so that T can be applied. Δ ¼ max min ΔcðGÞ: ðA8Þ c≥1 G∈L In addition, fault-tolerant operators would be transversal operators that, when applied to one gauge-fixed code, Now, the key to proving a Clifford hierarchy bound is a transform to another gauge-fixed code (of the same under- scrubbing theorem. But since all these definitions for lying subsystem code) while applying a logical operator to subsystem codes are the same as for the subspace versions, the logical qubits. More generally, Bravyi and König [9] the scrubbing lemma is also the same as stated in Lemma 4. define transversal morphisms, which are transversal oper- In other words, for any c ≥ 1, G ∈ L, and H ⊆ ½N ,it ators that take a codespace of one subspace stabilizer code allows us to find a dressed Pauli operator g ∈ G such that CA to another CB. Both codes involve the same number of physical and logical qubits. Following arguments like in the main text, the disjointness bounds such logical operators to ΔcðGÞjsuppðgÞ ∩ Hj ≤ jHj: ðA9Þ the Mth level of the hierarchy if Using this scrubbing lemma to reduce the support of the ðAÞ ðBÞ ΔðBÞ M−1 nested commutators K0 ¼ U, Kj ¼½Kj−1;gj below the d↑ 021047-10 DISJOINTNESS OF STABILIZER CODES AND … PHYS. REV. X 8, 021047 (2018) so the number of vertices in the graph is finite. Taking any where GðbÞ ∈ L ∪ fSg are logical cosets of the base code code in the graph, say, C0, we can, by assumption, apply a and at least one is nontrivial (i.e., in L). ∈ L ⊆ universal and thus infinite set of logical gates to it via Lemma 11. Let G eff and H ½N . Then, for any transversal morphisms, so C0 has an infinite number of c1;c2; …;cr, there exists a representative g ∈ G such that edges. Choose some neighbor C1 of C0. There must be an infinite number of different transversal morphisms (i.e., Y jsuppðgÞ ∩ Hj ≤ 1 − ð1 − Δ ðGðbÞÞ−1Þ jHj; ðB3Þ edges) that take C0 to C1. Yet, Eq. (A11) implies that there cb should only be a finite number (bounded within some level b of the hierarchy) of such logical operators. It also does not where the product ranges only over nontrivial cosets in the seem possible to get around this by adding or subtracting decomposition of G [Eq. (B2)]. qubits so that the codes change size, unless one were Proof.—Without loss of generality, we say that only the willing to add an infinite number of qubits over the course first r0 ≤ r cosets in Eq. (B2) are nontrivial. We decompose of the computation. r g ∈ G as g ¼ ⊗ gðbÞ with gðbÞ ∈ GðbÞ. Our task is to find b¼1 APPENDIX B: LOGICAL GATES ON MULTIPLE gðbÞ such that Eq. (B3) holds. Start by noting CODE BLOCKS In this section, we describe how to restrict gates on r0 suppðgÞ¼ ⋃ suppðgðbÞÞ; ðB4Þ multiple code blocks to the Clifford hierarchy. We flesh out b¼1 the argument in the main text by giving formulas for d↓, d↑, Δ of the multicode block code in terms of those for the r0 ∩ ⋃ ( ∩ ðbÞ ) single code block. Then, we argue how the bound on the H suppðgÞ¼ H suppðg Þ : ðB5Þ b¼1 level of the Clifford hierarchy obtainable by transversal gates can be made independent of the number of code Say that we have already chosen gð1Þ;gð2Þ; …;gðj−1Þ. Then, blocks. we need only minimize the intersection of gðjÞ with Consider r code blocks of the same [39] ⟦n; k⟧ base stabilizer code, each with identical [40] qudit partitions, j−1 ðbÞ ≔ −⋃( ∩ ðbÞ ) which we write as fQi g, where the superscript b ¼ Hj−1 H H suppðg Þ ; ðB6Þ 1; 2; …;rrepresents the code block. Like in the main text, b¼1 ðbÞ we say there are N subsets Qi for each b, and we use d↓, the set of partitions in H that are still unaffected. By Lemma Δ d↑, to denote the quantities of the base code. 4, we can find gðjÞ ∈ GðjÞ such that The effective stabilizer code is formed by treating all r code blocks as a ⟦rn; rk⟧ stabilizer code, and the qudits of ðjÞ ∩ ≤ Δ ðjÞ jsuppðg Þ Hj−1j jHj−1j= cj ðG Þ: ðB7Þ the effective code can be partitioned into subsets fQig, each consisting of one subset from each of the r code blocks, Note the relations r ðbÞ Q ¼ ⋃ Q : ðB1Þ H0 H; B8 i σbðiÞ ¼ ð Þ b¼1 ðjÞ H ¼ H −1 − (suppðg Þ ∩ H −1); ðB9Þ Here, σb∶½N → ½N is an (arbitrary) permutation of the j j j partitions of code block b. This completes the partitioning ’ − ∩ of the effective code in such a way that the effective code s H Hr0 ¼ H suppðgÞ: ðB10Þ min-distance equals that of the base code, d↓;eff ¼ d↓.We also note the simple bound on the effective code’s max- Thus, Eq. (B7) implies ≤ distance d↑;eff rd↑. ðjÞ The final quantity to address is the disjointness of the jHjj¼jHj−1 − (suppðg Þ ∩ Hj−1)j effective code Δ . To do this, we prove a more general ðjÞ −1 eff ≥ (1 − Δ ðG Þ )jH −1j: ðB11Þ version of Lemma 4 for multiple code blocks, and we let cj j Δ this inform the definition of eff. Start by establishing some L Repetitive use of Eq. (B11) gives us the bound notation. Let eff denote the set of nontrivial logical cosets of the effective code. Note that G ∈ L means, by eff Yr0 definition, that ≥ (1 − Δ ðbÞ −1) jHr0 j cb ðG Þ jHj; ðB12Þ b¼1 r G ¼ ⊗ GðbÞ; ðB2Þ b¼1 from which we conclude 021047-11 JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) Yr0 − ∩ ≤ 1− (1−Δ ðbÞ −1) jH Hr0 j¼jH suppðgÞj cb ðG Þ jHj: b¼1 ðB13Þ This completes the proof. □ We can simplify Eq. (B3) by choosing specific cb such ðbÞ that Δ ≤ Δc ðG Þ and find a g ∈ G such that b 4 FIG. 3. Making a C X gate with controls c1, c2, c3, c4 and target t from five Toffoli gates and two ancillas. jH ∩ suppðgÞj ≤ (1 − ð1 − 1=ΔÞr0 )jHjðB14Þ ≤ (1 − 1 − 1 Δ r) ð = Þ jHj: ðB15Þ H ∩ suppðgðb1ÞÞ¼H ∩ suppðgðb2ÞÞ: ðB19Þ The latter form of the right-hand side implies that defining Moreover, we are guaranteed to start repeating cosets in the decomposition in Eq. (B2) when r>mn−k, so effectively, 1 n−k Δ ≔ ðB16Þ we can replace r in Eq. (B17) with minðr; m Þ, thus eff 1 − ð1 − 1=ΔÞr achieving an r-independent bound. When σb is arbitrary, we can make the same argument ðb1Þ ðb2Þ σ σ will result in a theorem analogous to Theorem 5 but for when G ¼ G and b1 ¼ b2 . Since there are finitely multiple code blocks. many permutations as well, we can replace r in Eq. (B17) M−1 Theorem 12. If d↑ 021047-12 DISJOINTNESS OF STABILIZER CODES AND … PHYS. REV. X 8, 021047 (2018) g1 ∈ G1 ∈ L to have minimal support jsuppðg1Þj ¼ dðG1Þ [12] H. Bombin and M. A. Martin-Delgado, Topological Quan- so that tum Distillation, Phys. Rev. Lett. 97, 180501 (2006). [13] M. E. Beverland, O. Buerschaper, R. Koenig, F. Pastawski, J. Preskill, and S. Sijher, Protected Gates for Topological jsuppðK1Þj ≤ 2dðG1Þ; ðC1Þ Quantum Field Theories, J. Math. Phys. (N.Y.) 57, 022201 † (2016). simply because Pg1P may have disjoint support from g1. [14] P. Webster and S. D. Bartlett, Locality-Preserving Logical Bounding the supports of Kj can then be done exactly as in Operators in Topological Stabiliser Codes, Phys. Rev. A the proof of Theorem 5. More generally, the argument for 97, 012330 (2018). Theorem 7 found in Appendix B can incorporate the [15] M. Newman and Y. Shi, Limitations on Transversal observation in Eq. (C1) to show the following. Computation through Quantum Homomorphic Encryption, Theorem 14. Consider a stabilizer code with quantities arXiv:1704.07798. 0 n−k [16] H. Bombín, Gauge Color Codes: Optimal Transversal d↓, d↑, Δ. Let r ¼ minðr; N!m Þ.If Gates and Gauge Fixing in Topological Stabilizer Codes, New J. Phys. 17, 083002 (2015). 0 r0 M−1 2r d↑(1 − ð1 − 1=ΔÞ ) 021047-13 JOCHYM-O’CONNOR, KUBICA, and YODER PHYS. REV. X 8, 021047 (2018) † † [33] Since Kj ¼ Kj−1gjKj−1gj , one immediately obtains [38] A. Paetznick and B. W. Reichardt, Universal Fault- hj ≤ 2hj−1 þ 2, where h0 ¼ h. However, the transversal Tolerant Quantum Computation with Only Transversal operator gj can be absorbed into neighboring gates in the Gates and Error Correction, Phys. Rev. Lett. 111, circuit and, as a result, does not increase the circuit depth. 090505 (2013). Thus, we can remove the additive constant from the [39] Taking the blocks to be the same is for simplification of the recursion. argument only. For instance, we do not have to deal with [34] J. Haah, Local Stabilizer Codes in Three Dimensions different quantities d↓, d↑, Δ for each code. Running without String Logical Operators, Phys. Rev. A 83, through a more general argument where the stabilizer codes 042330 (2011). are allowed to be different is possible, and similarly, this [35] M. E. Beverland and J. Preskill (private communication). restricts transversal gates to the Clifford hierarchy. [36] F. Pastawski and J. Preskill, Code Properties from [40] Again, the sameness of the partitions can be relaxed at the Holographic Geometries, Phys. Rev. X 7, 021022 cost of notational encumbrance. (2017). [41] M. A. Nielsen and I. L. Chuang, Quantum Computation [37] D. Poulin, Stabilizer Formalism for Operator Quantum and Quantum Information (Cambridge University Press, Error Correction, Phys. Rev. Lett. 95, 230504 (2005). Cambridge, England, 2010). 021047-14