Table Of ContentSubsystem stabilizer codes cannot have a universal set
of transversal gates for even one encoded qudit
Xie Chen,1 Hyeyoun Chung,2 Andrew W. Cross,2,3 Bei Zeng,1 and Isaac L. Chuang1,2
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3IBM Research Division, T. J. Watson Research Center, P. O. Box 218, Yorktown Heights, NY 10598
(Dated: February 2, 2008)
Along-standingopenprobleminfault-tolerantquantumcomputationhasbeentofindauniversal
set of transversal gates. As three of us proved in arXiv: 0706.1382, such a set does not exist for
binary stabilizer codes. Here we generalize our work to show that for subsystem stabilizer codes
in d dimensional Hilbert space, such a universal set of transversal gates cannot exist for even one
8
encodedqudit,foranydimensiond,primeornonprime. Thisresultstronglysupportstheideathat
0
otherprimitives, suchasquantumteleportation, arenecessary for universalfault-tolerant quantum
0
computation, and may bean important factor for fault tolerance noise thresholds.
2
n PACSnumbers: 03.67.Pp,03.67.Lx
a
J
5 I. INTRODUCTION nonprime. However,asbothqubitandquditsystemsoc-
1 curinthenaturalworld,thereisnoreasontoassumethat
] All quantum systems are vulnerable to noise, which a theoretical result should hold solely for 2-dimensional
h canarisefromvarioussourcessuchasuncontrolledinter- systems. If an important negative result were to hold
p onlyinthecasewhend=2,thenthiswouldsuggestthat
actions of the system with the environment, or from im-
- a lot of effort should be directed towards building qudit
t perfectionsintheimplementationofquantumlogicalop-
n systems, as the case when d > 2 would be fundamen-
erations. Moreover,noisecanpropagatethroughaquan-
a tallydifferentfromthecased=2(forinstance,quantum
u tum circuit, affecting qudits throughout the computa-
error-correctingcodesofcertainparametersonlyexistin
q tionalsystem. Thus,ifquantumcomputationistobeim-
the case of d>2 [3]). Therefore, it is important to con-
[ plementedonalargescale,itisessentialtoexploremeth-
sider the case of higher dimensional systems, and in our
ods for protecting quantum information against noise,
1 workweconsiderthecaseofQECCsforarbitraryd,both
and for preventing the spread of errors through a quan-
v prime and nonprime.
0 tum system.
One way of implementing fault-tolerant quantum op-
6 Thetheoryofquantumerror-correctingcodes,coupled
erationsistousetransversalgates[4]. Atransversalgate
3
withfault-tolerantquantumcomputation,offersthehope
2 has a particularly simple form: it is a tensor product of
of resolving both of these problems, and has therefore
. unitaries that each act on only one qudit per encoded
1 greatly improved the long-term prospects for quantum
block [5]. Thus, transversal gates are naturally designed
0 computing technology [1, 2]. Quantum error-correcting
tolimitthe propagationofnoise,asanerroroccuringon
8
codesencodequantuminformationinaformthatismore
0 the kth qudit in a block can only ever propagate to the
resistant to noise. In principle, fault-tolerant quantum
: kth qudit of other blocks of the code, no matter what
v computationthenmakesitpossibletocarryoutarbitrar-
othersequencesofgatesweperformbeforeretrievingthe
Xi ily long quantum computations reliably, provided that encoded information.
the average probability of error per gate is less than a
r Astransversalgatesoffersignificantadvantagesincon-
a certain critical value known as the accuracy threshold
structing fault-tolerantquantum circuits, it is highly de-
[4].
sirable to know exactly which gates can be performed
The precise implementation of fault-tolerant quantum transversally on a given QECC. In the case of certain
computation depends on the quantum error-correcting codes, such as the 7-qudit Steane code for d=2, a num-
code(QECC)used. OnceaQECChasbeenchosen,each ber of different gates can be performed transversally: in
qubit in the original circuit is replaced with an encoded particular, any gate from the Clifford group can be im-
blockofqubits. Protocolsarethenspecifiedforperform- plementedasatransversalgate. Itwouldbewonderfulto
ing fault-tolerant operations on the code, i.e. protocols find a QECC such that universal quantum computation
for each type of logic gates. A fault-tolerant protocol is can be achieved entirely through transversal operations
designed so that if only one component of a procedure on the code. Unfortunately, it is widely believed in the
fails, then the failure causes at most a correctable num- quantum information science community that no such
beroferrorsineachencodedblockofquditsoutputfrom code exists [4].
the procedure. A proof of this belief is of fundamental importance in
Many studies concentrate only on the case of binary the fault-tolerant design of quantum circuits and the es-
QECCs in a d = 2 dimensional Hilbert space, as gener- timationoftheaccuracythreshold,assuchaproofwould
alizations of proofs are often non-trivial when d > 2 is provide valuable information about the fundamental re-
2
sources needed for quantum computation. If there is no a discussion of open problems, in particular the effect
QECCsuchthatauniversalsetofgatescanbeperformed ofcoordinatepermutationonthe possibilityofachieving
transversally on the code, then transversalgates are not universality using transversalgates.
the ultimate primitives for fault-tolerantuniversalquan-
tumcomputation: theymustbesupplementedwithmore
complicated techniques, such as quantum teleportation II. PRELIMINARIES
[6, 7] or state distillation [8].
Severaldifficulties must be overcomein orderto prove A. The generalized Pauli group
that transversality is insufficient for universality. Even
though the gates that can be implemented transversally Inthis sectionwereview the mathematics forthe gen-
on a given QECC depend on the code itself, the result eralized Pauli group d, which will be the main math-
must hold for all error-correcting codes. Furthermore, ematical tool for desPcribing the qudit stabilizer codes.
the logical operation of the gate on the encoded infor- ThegeneralizedPauligroupisgeneratedbytwoelements
mation must be determined from the physical operation X,Z with the commutation relation [12, 13, 14, 15, 16,
of a transversalgate onthe qudits ofa quantum system. 17, 18]
Finally, the important step of generalizing this result for
qudits in a Hilbert space of arbitrary dimension d is not ZX =qXZ, (1)
necessarilystraightforward,particularlyifdisnonprime.
where q is a complex number. Mathematically, we can
Inthispaper,weapproachtheproblemofprovingthat
prove that the associated group generated by Z, X pos-
stabilizercodescannothaveauniversalsetoftransversal
sesses a d dimensional irreducible representation only
gates. Recently,threeofusprovedin[10]thatauniversal
setoftransversalgatesdoesnotexistforbinarystabilizer for qd = 1−[12]. In this article, we take q qd ei2dπ.
≡ ≡
This special case was first introduced by Weyl [19], and
codes. Here we generalize our earlier result by showing
its completeness was first provedby Schwinger [20]. Ob-
the following Main Theorem.
viously, when d=q =1, the generatorsX and Z can be
Main Theorem: For subsystem stabilizer codes in d
regardedas the ordinary coordinates of R2 plane. When
dimensional Hilbert space, a universal set of transversal
d=2, q = 1,the generatorsX and Z can be identified
gates cannot exist for even one encoded qudit, for any
−
with the Pauli matrices σ and σ , and the generalized
dimension d, prime or nonprime. x z
Pauli group 2 is the familiar 1-qubit Pauli group, also
Given that stabilizer codes form the most important P
denoted by .
aconddesw,etlhl-edseivteulaotpioedn ccolnasssideorfedquinanotuurmpreororfoirs-cvoerrryecgteinng- ChoosingPa basis |kidk−=10, we have
eral. Wealsoprovideanalternativeinsightintotheprob-
lem by introducing a different proof technique from the Z k =qk k , (2)
one given in [10], which uses an idea in a recent work | i d| i
by Daniel Gross and Maarten Van den Nest [11]. This where k =X k 0 . This also implies
†
techniqueismoretransparentandaccessiblethantheap- | i | i
proachtaken in [10], and provides more intuition for the X k = k+1 . (3)
| i | i
final result.
In the Z-diagonalrepresentation, the matrices of X and
Our proof has two main stages. Firstly, given a
transversal gate U = n U , we show that there must Z are:
⊗j=1 j
besomerestrictionsonthephysicaloperationsU forev-
j 1 0 0 0 0
ery j. We derive these restrictions by studying the sub- ···
0 q 0 0 0
d
ccoardreisedofouatsbtyabRilaiziners[c9o]d,ew.hTohshisowideedatdhaatteasnbyatcrkantosvwerosrakl Z = ... ... ... ·.·.·. ... ... , (4)
gate on a given stabilizer code must keep some subcodes 0 0 0 ··· qdd−2 0
invariant. This fact allows us to place strong conditions 0 0 0 ··· 0 qdd−1
on the structure of the transversal gate. After describ-
ingtheserestrictionsonthe operationsU ,wethenshow
j
thattheremustbe correspondingrestrictionsonthe log- 0 0 0 0 1
···
ical operation U that prevent universality. 1 0 0 0 0
···
Our paper is organized as follows: we begin by pre- X =0 1 0 0 0. (5)
···
senting some backgroundinformation on the generalized ... ... ... ... ... ...
Pauligroup,stabilizercodes,andtransversalgatesinSec. 0 0 0 1 0
II. We then identify the restrictions on the structure of ···
transversal gates in Sec. III (the binary case d=2) and All the elements of the generalized Pauli group are
Sec. IV (the nonbinary case d > 2). In Sec. V we ana- given by
lyze the effect ofthe restrictionsonthe transversaloper-
ations on the logical gate. We conclude in Sec. VI with qiZjXk i,j,k Z (6)
{ d | ∈ d}
3
Define a basis set of d by
d
B P
= ZjXk j,k Z , (7)
d d
B { | ∈ }
and call the elements in the basis elements of d,
d
B P
thenthegeneralcommutationrelationsforanytwobasis
elements are
ZjXk =qjkXkZj. (8)
d
In addition, we can replace the generators Z and X
with two other elements in the basis. First, let (m,n)
denote the greatest common factor of integers m and n. FIG. 1: Illustration of a transversal gate on r blocks of n
Then if (m1,n1)=1 for m1,n1 Zd, we can define qubitseach. Theblocksarerepresentedbyacollectionofcir-
∈
cles(qubits),groupedintoboxesofn. Therblocksundergoa
X¯ =q−d−21m1n1Zm1Xn1, (9) transversalgatewhoseunitariesUj actonqubitsinthe[blue]
d boxeswith rounded edges.
where the factor before ZmXn is chosen so that X¯ has
the same eigenvalues as X. To maintain Eq. (1), we
define [[n,k+k′,δ′]] code withδ′ δ. The k′ additionallogical
≤
qubits are knownas the gauge qubits, and the original
Z¯ =q−d−21m2n2Zm2Xn2, (10) k logical qudits are known as the protected qubits.
d
A qudit stabilizer code Q is then the vector space
d
where(m ,n )=1form ,n Z ,andm n m n = stabilizedbyasubgroupSofthegeneralizedPauligroup,
2 2 2 2 d 1 2 2 1
1. From another viewpoint, X∈¯ and Z¯ define−a unitary such that qlI / S for l = 0. An [[n,k,δ]] stabilizer code
d ∈ 6
transformation U such that encodes k logical qudits into n physical qudits and can
correct up to δ 1 independent single qudit errors.
X¯ =UXU†, Z¯ =UZU†. (11) Suppose the⌊re−2ar⌋e initially r blocks of k qudits in a d-
dimensionalHilbertspaceandweencode eachblockofk
Bytheabovedefinition,itiseasytocheckthatthesetof
quditsintoQ. Inordertodefineatransversalgateacting
all suchunitary transformationsU forms a group,which ontheser blocks,wemustfirstdefinethelocal unitary
is known as the Clifford group. group. Forthesingleblockcase,thelocalunitarygroup
Finally, we define the n-qudit Pauli group. The famil- is G = U(1) SU(d)n. Each state P has a stabilizer
Q
iar n-qubit Pauligroup n consists of all localoperators subgroup I× G consisting of elements g G that
ofthe formR=αRR(1)P...R(n),where αR ∈{±1,±i}is leave PQ fixeQd⊂under the action gPQg−1 [25].∈For the
an overall phase factor and R(i) is either the identity I
multiblock case with r blocks, the local unitary group is
otlohrceoanlaenoapolefortgahoteuosrPsanou-fqliutmdhieattfrPoircameusliσRgxr,=oσuyαp, oPRrndσ(1za).s..Wt.hRee(cnsa)en,tdwoehffieanrleel sGurbg=roUup(1)IQr×⊂SUG(drr)cno.nsEisaticnhgsotafteelePmQ⊗ernthsasga∈stGarbitlihzaert
α = qk for some k Z is an oveRrall phase factor and leave PQ⊗r fixed under the action gPQ⊗rg−1. The sub-
RR(i) is adn element of∈thedgeneralized Pauli group d. group IQr is known as the local unitary group of PQ⊗r. A
P transversal gate acting on the r blocks is an nr qudit
unitary U that is an element of the local unitary group
B. Stabilizer codes, transversal gates, and encoded IQr ofPQ⊗r. Thegatefactorsintoann-foldtensorproduct
U = n U ofr quditunitariesU . EachU actsonthe
universality ⊗j=1 j j j
jthquditoftherblocks. SeeFIG.1foranillustrationof
atransversalgateappliedtorencodedblocksofnqubits
In this section we introduce definitions and notation
(the case d=2) each.
forstudyingstabilizercodesandtransversalgates. LetQ
denote an [[n,k,δ]] binary stabilizer code with stabilizer
S [4,24]. The orthogonalprojectorontoQisdenotedby
P and is given by C. Problem and Strategy
Q
1
P = R. (12) We would like to know if the transversal gates are an
Q 2n encoded quantum computationally universal set for at
R S
X∈ least one of the encoded qudits. If so, then this means
The codemayormaynotbe asubsystemcode [27]. If thatitispossibletoapproximateanysinglequditlogical
the code is asubsystemcode,there arek 0 additional gateononeofthek encodedqudits(wedon’tcarewhich
′
≥
logical qubits, S is generated by n k k independent one) to any accuracy using only transversalgates. Since
′
− −
generators, and the corresponding subspace code is an thetransversalgatesformagroup,wecanformallystate
4
this as follows: Given any encoded single qudit unitary form of transversalgates by studying subcodes of a spe-
gate V on a fixed encoded qudit and an accuracy ǫ> 0, cial form. The following important lemma will be useful
thereisatransversalr-blockgateUǫ suchthat||UǫPQ⊗r− in studying the action of transversal gates on stabilizer
aVPcoQ⊗nrt|r|a<dicǫt.ioWn,ewwhililchasismupmlieesthtahtatthtihseistrtarnusevaernsdaldgearitvees subLceomdems.a 1: Letω [n]beanonemptysubsetofcoor-
⊆
are not computationally universal. dinates,andletω¯ denotetheset[n] ω. Givenatransver-
Our general strategy is to show that the condition of sal gate U = ⊗ni=1Ui, let Uω ≡ \i ωUi. We can then
transversalityplacesrestrictionsontheformofeachU in write ∈
j N
the tensor productexpansionU = n U ofa transver-
sal gate. In Sec. III and IV we d⊗erji=ve1 tjhe exact forms trω¯ UPQ⊗rU† =ρ⊗ωr, (13)
of these restrictions for the cases when d=2 and d>2, h i
respectively. In Sec. V we use these results to show that where ρω is defined as trω¯PQ.
therestrictionsontheU placeenoughconstraintsonthe Toprovethelemma,notethatsinceatransversalgate
j
logical operation U to prevent universality. U is an encoded gate, we can write
trω¯ UPQ⊗rU† =Uωtrω¯ PQ⊗r Uω†
III. THE STRUCTURE OF STABILIZER
h i=U ρ rUh =iρ r, (14)
SUBGROUPS OF STABILIZER CODES: BINARY ω ⊗ω ω† ⊗ω
CASE
which gives the necessary result.
Thislemmatellsusthatanencodedgatealsopreserves
In this section we show that a transversal gate acting
thesubcodesρ r foranyω. Thisresultisusefulbecause
onr blocksofnqubits encodedusingastabilizercodeQ ⊗ω
we can turn it around – if a gate does not preserve sub-
has a severely restricted form. We first introduce some
codes,thenitcannotbe anencodedgate. Notethatitis
definitions that allow us to formally state the restric-
easy to compute the projectorρ onto the subcode from
tions ontransversalgates. An n-qubitunitary operation ω
the projector P onto the original code. We define the
is said to be semi-Cliffordif it sends at least one max- Q
support supp(R) of an element R S to be the set of
imal abelian subgroup of the n-qubit Pauli group Pn to all i [n] such that the ith coordin∈ate R(i) differs from
anothermaximalabeliansubgroupof Pn under conjuga- the i∈dentity. We say that an element R S has full
tion. If T is a semi-Clifford operation, then there exist ∈
support if supp(R)=[n]. We then have
Clifford operations L ,L such that L TL is diagonal
1 2 1 2
[26].
ρ =tr P tr ( R)
An n-qubit unitary operation is said to be general- ω ω¯ Q ∝ ω¯
ized semi-Clifford if it sends the span of one maximal RX∈S
abelian subgroup of to the span of another maximal = tr R= R. (15)
n ω¯
P
abeliansubgroupofPn under conjugation. If T is agen- RX∈S R∈S,sXupp(R)⊆ω
eralizedsemi-Cliffordoperation,thenthereexistClifford
operations L1,L2, and a classical permutation operator The set Sω ={R∈S | supp(R)⊆ω} is the stabilizer
P such that PL TL is diagonal [26]. of the subcode, which is a subgroup of S. The partial
1 2
Our main task in this section is to prove the following trace removes the unencoded qubits at coordinates in ω¯
theorem. from the subcode.
Theorem 1: Given an n-qubit stabilizer code Q free WewillproveTheorem1intwoways,bystudyingtwo
ofBellpairsandtriviallyencodedqubits,letU = n U classes of stabilizer subcodes. In Sec. IIIA we use the
be an element of Ir. Let [n] denote the set of p⊗ojs=it1ivej so-called minimal subcodes of S, and in Sec. IIIB we
Q
integers from 1 to n. Then for each j [n], U is an use subcodes associated with single qubits. For the rest
j
r-qubit generalized semi-Clifford operatio∈n. of this section we will work with an n-qubit stabilizer
code Q with corresponding stabilizer S that satisfies the
Thistheoremplacessevererestrictionsonthephysical
conditions of Theorem 1.
form of a transversalgate U. In Section V, we will show
that these restrictions place corresponding constraints
on the logical gate U, thereby making it impossible to
A. Minimal subcodes and beyond
achieve universality using only transversalgates.
Proving this theorem is not trivial, as we must draw
conclusions about each factor U of the transversal gate 1. Minimal subcodes
j
U, given only information about the action of U on the
entirecodespace. We willprovethetheorembystudying In order to define minimal subcodes, we must first in-
codes thatare stabilizedby subgroupsof S. Sucha code troduce the concept of minimal supports. A support
is known as a stabilizer subcode. We can show that ω is a minimal support of S if there is a nonidentity el-
a transversal gate preserves certain stabilizer subcodes. ement of S with support ω, and there are no elements
This requirement allows us to place restrictions on the with support strictly contained in ω. An element in S
5
with minimal support is called a minimal element [9]. 2. Transversal gates on minimal subcodes
Theconceptofaminimalsupportofastabilizerstatehas
beenextremelyusefulinthestudyoflocalunitaryversus In this section, we place restrictions on the operators
local Clifford equivalence of stabilizer and graph states U ofatransversalgateU = n U whenj iscontained
[21, 28]. Minimal supports have also arisen in classical injsome minimal support of S⊗.j=1 j
coding theory in the context of secret sharing schemes Supposewecanfindminimalelementswhosesupports
[29], coverasubsetofcoordinatesm [n]. Whatcanwelearn
Given a minimal support ω, then all the nonidentity abouttheformofatransversalg⊆ateonthecoordinatesin
elements in Sω have support ω. The following lemma m by studying its action on minimal subcodes? The fol-
of Van den Nest [21] allows us to characterize Sω for a lowing discussion is a generalization of Rains’ approach
minimal ω. [9]. First, recall that Clifford gates are not universal,
Lemma 2: Let A denote the number of nonidentity and if we have a transversal gate constructed from Clif-
ω
elements in S with minimal support ω. Then A = 1 fordgates,then thattransversalgatemustbe somekind
ω ω
or 3. of logical Clifford gate as well. The challenging behav-
Theproofisfairlystraightforward. Bydefinition,there ior comes from non-Clifford gates. Therefore, we will
mustbesomeelementofS withsupportω,soifthereare find it convenient to more or less ignore Clifford gates
no more, A = 1. If there are two elements M,N with altogether. We will move to locally Clifford equivalent
ω
supportω,then their productMN musthavesupportω stabilizer codes freely when studying particular minimal
too, as otherwise ω is not minimal. So A cannot be 2, subcodes. Keepingthisinmind,wecanwritether block
ω
but it can be 3. Suppose there is a fourth element M′ projectors when Aω =1 and Aω =3. If Aω =1, then
with support ω. There are only three nonidentity Pauli
operators, so one of them must appear twice at some ρ⊗ωr ∝(Iω +Zω)⊗r = (Zω)i1 ⊗···⊗(Zω)ir
coordinate in ω. But then we can form another product i∈X{0,1}r
whosesupportis strictly containedin ω,meaningthat ω
= Z(i) ω (18)
is not a minimal support, so A cannot be greater than ⊗| |
ω
3. NoticethatwhenAω =3, ω mustbeeven,otherwise i∈X{0,1}r
| |
the operators will not commute.
wherei denotesthejthbitofi,inthesecondexpression,
j
We can use this result to describe the subcode stabi- and Z(i) = r Zij in the third expression. The Z(i)
lizedby Sω. ByLemma 2, Sω has either2 or4elements. arethePaul⊗i Zj=o1perators,andformamaximalabelian
We denote the coordinates in ω by j 1,2,..., ω ,
subgroup of the r qubit Pauli group. We can define the
∈ { | |}
thoughwewillunderstandthatthisnotationjustindexes Pauli X and Pauli Y operators analogously.
ω – the actual coordinate is the jth element of ω. Com-
It may be helpful to consult FIG. 2 for an illustration
puting the projector ρω onto the subcode stabilized by of one of the summands in Equation (18) as it would
Sω, we find that either look overlayed on FIG. 1. The third expression may be
somewhat confusing because the tensor product “ ω ”
ρω I I+M1 M2 Mω is over the columns of FIG. 2. We do this becaus⊗e|th|e
∝ ⊗···⊗ ⊗ ⊗···⊗ | |
transversalgate,whichwewillapplyshortly,factorsinto
ω times
| | a tensor product over columns too. Similarly, if A =3,
=|I|ω|+{zMω} (16) then ω
or ρ⊗ωr ∝(Iω +Xω+Zω+(−1)|ω|/2Yω)⊗r
wt(ab)
ρω I|ω|+Mω+Nω +(MN)ω, (17) = ( 1)|ω|/2 · Rω(a1,b1)
∝ − ⊗
(a|b)∈X{0,1}2rh i
where Mω and Nω are Pauli operators in S restricted to Rω(a ,b )
r r
ω whose product also has support on ω. It is helpful to ···⊗
wt(ab)
realizethattheseoperatorsareprojectorsonto[[ω , ω = ( 1)ω/2 · R(a,b) ω , (19)
1,1]] and [[ω , ω 2,2]] stabilizer codes, resp|ec|ti|ve|l−y. − | | ⊗| |
We can also| |se|e |th−at there is some Clifford operation (a|b)∈X{0,1}2rh i
that we can apply at each coordinate in ω to transform where R(0,0) = I, R(0,1) = Z, R(1,0) = X, and
thestabilizersofthesesubcodesintohZωiandhXω,Zωi, R(1,1) = Y, (i.e. R(aj,bj) = iaj·bjXajZbj) and also
respectively. These codes are the minimal subcodes R(a,b) = r R(a ,b ). Again, the tensor product in
⊗j=1 j j
associated with the minimal support ω. the third expression is over columns rather than rows.
The extenttowhichastabilizercode canbe described One or both of the projectors we have written are left
by its minimal subcodes depends on the particular sta- unchanged by transversal gates when the gates are re-
bilizer code. For example, the GF(4)-linear codes are strictedtoaminimalsupportω,i.e. U ρ U =ρ . Since
ω ω ω† ω
onefamily ofstabilizercodesthatcanbe describedcom- U IU =I, we can subtract the identity from each side
ω ω†
pletely by their minimal subcodes [9, 21]. from the preceding equation. Rains has shown that it is
6
where α 1 . We can now compute
j
∈{± }
2r 1
−
wwT = ii...i jj...j , (23)
| ih |
i,j=1
X
4r 1
−
vvT = α α ii...i jj...j . (24)
i j
| ih |
i,j=1
X
Following Rains, consider the following operators when
ω 3 (we will come back to ω =2 later),
FIG. 2: Illustration of a single term in the expansion of ρ⊗r | |≥ | |
ω
for the case Aω = 1. Each box is associated to a qubit in 11tr 3,...,ω wwT 1 1 = 1 12, (25)
FIG. 1, and the value of the bit i to the left of the jth row h | { | |} | i | ih |
determines whether that row is Z|ω| or I|ω|. Therefore, the 11tr 3,...,ω vvT 1 1 1 12. (26)
h | { | |} | i ∝| ih |
PauliZoperatoralongeachcolumnisthesameoperatorZ(i),
and it is determined by the bit string i. A factor Uj of a The transversal gate, represented by a rotation O, fixes
transversalgateactsonacolumn(the[blue]boxwithrounded at least one of v or w (Ov =v or Ow =w), so
edges, for example).
1 1 = 1 tr OwwTOT 1
2 1 3,...,ω 1
| ih | h | { | |} | i
2r 1
−
convenienttoviewtheprojectorsasvectorsinEuclidean =O 1 (O I)ii ii(OT I)1 OT
2h |1 1⊗ | ih | 1 ⊗ | i1 2
space acted on by rotations. This association will let us i=1
X
show that rotations fixing these vectors have a special 2r 1
−
form. The r qubit gateU acts by conjugationona non- =O (O )2 i i OT (27)
identity r qubit Pauli majtrix R (s indexes the 4r 1 2" 1 1,i| ih |2# 2
s − Xi=1
nonidentity Paulis) as
or
UjRsUj† = αtsRt. (20) |1ih1|2 ∝h1|1tr{3,...,|ω|}OvvTOT|1i1
Rt∈XPr−{I} 4r−1
=O 1 α 2(O I)ii ii(OT I)1 OT
Here denotes the r qubit Pauli group. The identity 2h |1 | i| 1⊗ | ih | 1 ⊗ | i1 2
matrixPrdoes not appear on the right hand side because Xi=1
4r 1
Uj is unitary and Rs is traceless, so the image must be =O − (O )2 i i OT. (28)
traceless. The coefficients αts must be real because Rs 2" 1 1,i| ih |2# 2
i=1
is Hermitian. Furthermore, α α =δ X
because R is unitary. So, wRet∈cParn−{rIe}pretss1entts2U bsy1sa2 In the case where O acts on v, “case v”, we can con-
s j
P
matrix Oj in SO(4r 1) whose real entries are αts, cludethattheentirefirstrowofO1hasonenonzeroentry,
s,t [4r 1], and who−se columns are orthonormal. The and the square of this real entry must be 1. Considering
∈ −
inverse unitary Uj† is represented by the transpose OjT analogous operators, and understanding that Oj is non-
and its columns are orthonormal, so both the rows and singular, we conclude that Oj is a monomial matrix for
columns are orthonormal. We can represent the non- “case v”, so the corresponding unitary must normalize
identity r qubit Pauli matrices by the canonical basis the Pauli group, i.e. it must be Clifford.
vectors 1 , 2 ,..., 4r 1 ofR4r 1. Forconcreteness, In the case where O acts on w, “case w”, the opera-
−
we can{a|ssioc|iaite the| la−beli}i of i to the binary repre- tor only has rank 1 if one of (O1)1,i is nonzero and the
sentation (ab) 0,1 2r or to t|hie Pauli representation rest are zero for i [2r 1]. However, the equation is
iwt(a·b)X(a)Z| (b)∈. {Cont}inuing,we cannow write the sub- only satisfied if the∈nonz−ero entry is ±1 since O2 is an
code projectors as vectors in (R4r 1) ω , using “ ” to orthogonalmatrix. Therefore,consideringanalogousop-
− ⊗| |
denote this mapping. For Aω =1, 7→ erators,Oj hasamonomialsubblockfor“casew”,where
j ω and ω is a minimal support, and the south and
∈
2r 1 east subblocks are zero, i.e.
−
ρ⊗ωr−I 7→ |ii...ii=:w (21) M 0
i=1 O = , (29)
X ω times 1 0 M
| | ′
(cid:18) (cid:19)
| {z }
and for Aω =3, where M is a monomial matrix whose nonzero entries
are 1 and M is in SO(4r 2r). Therefore, the cor-
′
4r−1 resp±onding unitary matrix m−ust normalize the Pauli Z
ρ r I α ii...i =:v, (22)
⊗ω − 7→ i| i operators that correspond to the rows and columns of
Xi=1 ω times the M matrix.
| |
| {z }
7
Therefore,wehavethe followingresults. Ifω isamin- different Paulis at coordinate j, then they have differ-
imal support, ω 4, and A =3,then U is anr qubit ent supports. Indeed, suppose there are two elements
ω j
| |≥
Clifford gate for j ω. If A = 1, and ω 3, on the R ,R M that differ on the jth coordinate and sat-
ω a b j
∈ | | ≥ ∈
other hand, then up to local Clifford gates U is an r isfy ω := supp(R ) = supp(R ). Then R R M and
j a b a b j
∈
qubit unitary thatnormalizesPauliZ operatorsbut acts R R , R , and R exhaust the Paulis on the jth coor-
a b a b
arbitrarily on Pauli X operators. In both cases, U is a dinate. So, up to local Clifford operations R = X ω
j a ⊗| |
semi-Clifford operation. and R = Z ω . Since there was no minimal support
b ⊗| |
The case A = 3 and ω = 2 is a special case. In containing j, there exists some R S S such that
ω | | ∈ \ j
this case, the minimal subcode is a [[2,0,2]], which we supp(R) ω. Furthermore, R R,R R,R R R S be-
a b a b j
⊂ ∈
know to be a Bell pair. The Bell pair is preserved by a cause R / S . However, one of these three elements has
j
∈
continuumoflocalrotationsU U ,soitisanedgecase support strictly contained in ω, contradicting the defini-
∗
⊗
that we must discard. Since the possible Paulioperators tion of M .
j
are exhaused on ω, the stabilizer code must be of the Indeed,supposethecoordinatej isnotinanyminimal
formS =S ρ . Therefore,theBellpairisactually support. Take any R M and let ω = supp(R). With-
′⊗ [[2,0,2]] ∈ j
appendedtothe code anddoes notimproveits ability to outlossofgenerality,supposeR(j) =Z. Byourprevious
detecterrorsonanyencodedqubit. Ifabinarystabilizer argument, ρ contains elements from M that only have
ω j
code cannot be decomposed as Q = Q′ [[2,0,2]], then Pauli Z at coordinate j and are supported entirely on
⊗
the code is free of Bell pairs. ω. It also contains elements from S that have support
The cases A = 1 and ω = 1 or ω = 2 are special strictly containedinω but haveidentity atcoordinatej.
ω
| | | |
casesaswell. Inthefirstcase,thequbitatthecoordinate In notation, ρ = Z R + I R . Now, we
ω i j ⊗ i k j ⊗ k
j ω is in a product state with the rest of the code. We canapplyasimilarargumenttothe caseweencountered
∈ P P
can discardthis case by insisting that Q=Q [[1,0,1]] earlier for A =1 and ω =2. The form of the subcode
′ ω
6 ⊗ | |
is free of single qubit states, but this isn’t necessary projector is too weak for us to take a trace over other
becauseitiscoveredbythe statements ofTheorem1. In coordinates, but, like before, we observe that U must
j
the second case, we do not have enough qubits to “lock keep the span of Pauli Z operators invariant, i.e. U is
j
the state to the diagonal” by projecting onto the first a generalizedsemi-Clifford operation. We have therefore
qubit. Therefore, we can only say that proved Theorem 1 using minimal subcodes.
2r 1
−
O1" |iihi|#O1T =tr2Ow B. Subcodes associated with single qubits
i=1
X
=tr2w In this section we introduce the single qubit sub-
2r 1 codes,andusethesesubcodestoproveTheorem1. This
−
= i i, (30) approach provides a more intuitive, accessible proof, as
| ih |
i=1 the single qubit subcodes are easier to visualize and un-
X
derstand than the minimal subcodes introduced in Sec-
i.e. that Uj maps linear combinations of Pauli Z opera- tion IIIA.
tors to linear combinations of Pauli Z operators. There-
fore, in this case, U is a generalized semi-Clifford oper-
j
ation. 1. Single qubit subgroups and subcodes
The single qubit subcode associated with a coordi-
3. Coordinates not covered by minimal subcodes
nate i [n] is the subcode with projector ρ = tr P ,
ω ω¯ Q
∈
whereω = i . Wedenotethe projectorforthissubcode
{ }
Ingeneral,however,astabilizercodeneednotbecom- byρ . Thesingle qubit subgroupS i associatedwith
i
h i
pletely describedby its minimalelements, i.e. wecannot i is the set R S R(i) = I . We define the support
{ ∈ | }
always find a minimal support containing a coordinate of a subgroup S i to be the set supp(R). The
R S i
j [n]. In this section, we place restrictions on the op- single qubit subchoides and subgrou∪ps∈hahvie been used by
er∈ators U in a transversal gate U = n U when j is Gross and Van den Nest to study the local unitary and
j ⊗j=1 j
not contained in a minimal support. local Clifford equivalence of stabilizer and graph states
Suppose we cannot find a minimal support containing [11]. We will generalize some of their methods to prove
coordinate j. Take the set S := R R S(Q),j Theorem 1.
j
{ | ∈ ∈
supp(R) of stabilizerelements with supportonj. Since WebeginbyreviewingtwolemmasbyGrossetal.[11].
}
we assume that the code does not have trivially encoded For every subgroup G of S, we let [S : G] denote the
qubits, S is nonempty. Of those elements in S , we index of G in S.
j j
can single out the set of restricted minimal elements Lemma 3: Let S be a stabilizer on n qubits, and
M := R S ∄R S ,supp(R) supp(R) . letS i denotethesinglequbitsubgroupassociatedwith
j j ′ j ′
{ ∈ | ∈ ⊂ } h i
Now we can show that if two elements in M have i [n]. Then [S :S i ]=1,2, or 4 for every i [n].
j
∈ h i ∈
8
Lemma 4: Let Π be the smallest subgroup of S con- U on such a code is a local Clifford operation. Thus U
j
taining all the single qubit subgroups S i . We then ob- is a Clifford operation, and therefore generalized semi-
h i
tain one of three cases. Either S =Π, or [S :Π]= 2, or Clifford.
[S : Π] = 4. If Π has index 4 in S, then the stabilizer If [S :Π]=2, then the stabilizer S can be partitioned
code associated with S must be a [2m,2m 2,2] code. intotwocosetsofΠasS =Π hΠ, whereh S Π. We
− ∪ ∈ \
Note that we can write Π as the set R R R R can see from the definition of Π that h has full support.
1 2 n i
{ ··· | ∈
S i , i [n] . Togetherwithourassumptionthatj supp(Π),thisim-
h i ∈ } 6∈
pliesthatforeveryR S,wemusthaveR(j) I,h(j) .
∈ ∈{ }
It follows that [S : S j ] = 2. We can then partition
h i
2. Transversal gates on single qubit subcodes S into two cosets of S j as S = S j gS j , where
h i h i ∪ h i
g S S j .
∈ \ h i
Following a similar approach to Sec. IIIA2, we show Defining ρ j 1 R, it follows fromthe def-
h i≡ 2n R S j
thatifa coordinatej [n]is containedinthe supportof inition of Q that ∈ h i
∈ P
some single qubit subgroupS i , then the corresponding
operator U in a transversal ghaite U = n U is gener- P =(I I + g)ρ j . (33)
j ⊗j=1 j Q ⊗···⊗ h i
alized semi-Clifford.
n times
We provethe result by induction. If n=2,then up to
local Clifford equivalence plus permutations of the two We now compute the| pro{zjecto}r ρj for the single qubit
qubits the only stabilizer code Q satisfying the require- subcode associated with j. We find that
ments of Theorem 1 has the projector
ρ = R
j
1
PQ = 2(I ⊗I +Z⊗Z). (31) R∈S,suXpp(R)⊆{j}
=I+g(j), (34)
It is straightforward to verify that the result holds for
this code. (See p. 9 in [10]. The relevant case is ω =2 where the second equality follows from the form of P
Q
| |
and Aω =1.) giveninEquation(33). Wecanseethatg(j) X,Y,Z .
∈{ }
In the induction step of the proof, let n ≥ 3 and sup- As we have Ujρ⊗jrUj† =ρ⊗jr by Lemma 1, it follows that
pose that the result has been verified for all n <n. Let
′ U maps linear combinations of Pauli g(j) operators to
Q be a stabilizer code on n qubits satisfying the require- j
linearcombinationsofPaulig(j) operators. ThereforeU
mentsofTheorem1andletU = n U beatransversal j
⊗j=1 j is a generalized semi-Clifford operation. We have thus
gateonQ. Foreveryi [n],define the setω =[n] i .
∈ i \{ } proved Theorem 1 using single qubit subcodes.
Using Lemma 1, we find that
Uωiρ⊗ωirUω†i =ρ⊗ωir, (32) IV. THE STRUCTURE OF STABILIZER
where U is the restrictionof U to ω and ρ is defined SUBGROUPS OF STABILIZER CODES:
as tr Pωi. Since ρ is the projectior forωai stabilizer NONBINARY CASE
ω¯i Q ωi
code on n 1 qubits, and satisfies the requirements of
−
Theorem1,wecanapplytheinductionhypothesistothe Inmanyquantumcomputationalproblems,thedimen-
code corresponding to ρ for every i [n]. This proves sion of the computational unit plays an important role.
ωi ∈
thatU is generalizedsemi-Cliffordforeveryj [n]that Here, we would like to understand its effect on the set
j
∈
is contained in the support of some S i . of possible transversal gates. That is, we want to find
h i
out, in the qudit settings, whether transversal gates can
form a universal set of gates for one of the encoded logi-
3. Coordinates not covered by single qubit subcodes calquditsandifnot,whatoperationscanbetransversal.
We will follow a line similar to that in the qubit case
It could be the case that there is a coordinate j [n] but with emphasis on parts that are different and need
thatisnotcontainedinthesupportofanyS i . How∈ever, special notice. First, we study the physical restrictions
itisstillpossibletoshowthatthecorrespondhiingoperator on transversal gates by analyzing the transformation of
U inatransversalgateU = n U isgeneralizedsemi- stabilizer subcodes under such transversaloperations.
Cjlifford. ⊗j=1 j Our main task in this section is to prove the following
Suppose that the coordinate j is not contained in the theorem.
support of any S i . From the form of Π defined in Sec. Theorem 2: Givenad-dimensionaln-quditstabilizer
h i
IIIB1, we can see that j supp(Π). It follows that Π codeQfreeofBellpairsandtriviallyencodedqudits,let
is strictly contained in S. 6∈By Lemma 4, Π therefore has U Ir. Then for each j [n] either
∈ Q ∈
index 2 or 4 in S. If [S : Π] = 4, then we know that (1) U is an r qudit Clifford gate, or
j
the code Q associated with S is a [2m,2m 2,2] code. (2) U keeps a subgroup of the r-qudit Pauli group in-
j
−
By Lemma 3 in [10], we find that the transversal gate variant under conjugation, or
9
(3) U keeps the span of a subgroup of the r-qudit Pauli We can further simplify the form of the minimal sub-
j
group invariant under conjugation. code. Note that while the operators R must commute,
i
Hereby“Bellpairs”wemeanthetwo-quditmaximally the samedoesnotholdforthe R(1). However,nomatter
i
entangled states, which are states locally equivalent to whatthecommutationfactorsareforthesingle-quditop-
the state 1 d−1 ii . erators, the subcode weight ω is such that they vanish
√d i=0 | i | |
For the rest of this section we will work with a d- for the Ri. Thus we need not concernourselveswith the
dimensionalnP-quditstabilizercodeQwithcorresponding commutation relations of the Pauli operators R(1) and
i
stabilizer S that satisfies the conditions of Theorem 2. simply treat them as commutative. In this way, we are
dealingwiththequotientgroup d = d/C ,where d
A. Minimal subcodes and beyond iisstthheecoenneteqruodfitPda.uTlihgergoruopuapndPdC1P∗is=thP{enI1,aqdfiIPn,i.t.e.aqbdd−el1iPaI1n}
P1 P1∗
groupformedbythedirectproductoftwocyclic-dgroups
1. Minimal subcodes thataregeneratedbyXandZrespectively. Itssubgroups
are of the form Zm or Xm1,Zm2 , where m, m and
1
h i h i
In this section we again make use of the technique of m arefactorsofd. The minimalsubcodesarethe codes
2
minimal subcodes in order to place restrictions on the stabilized by these subgroups.
form of a transversal gate. The generalization of the Wecannowexplicitlywriteouttheprojectorsformin-
binary case is mostly straightforward. We continue to imal subcodes. Denote the number of generators for a
use Rains’ technique of viewing the projectors onto the subcode by N . WhenN =1, the r block projectorcan
g g
codespace as vectors, and the transversal gates as rota- be written as
tions acting on these vectors. However, when d > 2 the
ρ r (I ω +(Zm)ω + +(Z(p 1)m)ω ) r
non-zero entries of the rotation matrices are not neces- ⊗ω ∝ | | | | ··· − | | ⊗
sarily 1, but can be any complex number of modulus = ((Zm)ω)i1 ((Zm)ω)ir
± ⊗···⊗
1tr.anAssvearsraelsgualtt,etUhe=restnricUtioninsTplhaecoerdemon2tdhieffefrorsmlighoftlya i∈{0X...p−1}r
from those of Theore⊗mj=11, stjating that U preserves the = Z(i)⊗|ω|. (36)
j
span of a subgroup of the generalized Pauli group under i∈{0X...p−1}r
conjugation, rather than a maximal abelian subgroup of This differs from the qubit expression only in that each
the Pauli group. component of i can take p different values, rather than
Asinthe binarycase,webeginbytryingtodetermine two (p not necessarily prime). Similarly, the projector
the structure of the projector onto a minimal subcode. ρ r when N =2 is given by
⊗ω g
Given a minimal support ω, we again use S to denote
ω r
the subgroup of S generated by the elements of S with ρ r ((Zm1)c(Xm2)d) ⊗
⊗ω ∝
support ω. The minimal subcode corresponding to ω is
thecodestabilizedbySω. WecanlisttheelementsofSω (cid:16)dc∈∈{{00X......pp12−−11}} (cid:17)
as I,R1,...,Rm, where = Rω(a ,b ) Rω(a ,b )
1 1 r r
⊗···⊗
R1 =R1(1)R1(2)...R1(|ω|) ab∈{00..X....pp21−11}rr
∈{ − }
R2 =R2(1)R2(2)...R2(|ω|) = R(a,b)⊗|ω|, (37)
... ab∈{00..X....pp21−11}rr
∈{ − }
Rm =Rm(1)Rm(2)...Rm(|ω|). (35) where R(ai,bi) = (Zm1)ai(Xm2)bi and R(a,b) =
r R(a ,b ).
For any Pauli operator g, define its order p to be the ⊗j=1 j j
minimal positive integer that satisfies gp = I. It is easy
to see that for each Ri ∈ Sω, the operators Ri(j) must 2. Transversal gates on minimal subcodes
be of the same order. Otherwise there wouldexist a cer-
tain power m of R such that Rm had a support strictly
i i We can now use the techniques of Sec. IIIA2 to place
contained in ω, contradicting the assumption that ω is
restrictionsontheoperatorsU ofatransversalgateU =
minimal. It can be checked that each Pauli subgroup j
n U suchthatj iscontainedinsomeminimalsupport
I,R(j),...,R(j) at a particular coordinate j has the ⊗j=1 j
{ 1 m } ω. ThePauligroupformsabasisforanyoperatoronthe
same structure, i.e. they have the same multiplication
d dimensional Hilbert space. Therefore, conjugation of
table. This set of subgroups have the same order and
a Pauli operator by tranversal gates can be seen as a
their elements correspond. Therefore, up to local Clif-
unitary transform in the operator space given by
ford operations, R = (R(1)) ω . Each minimal sub-
i i ⊗| |
codeisthenrepresentedbyasingle-quditPaulisubgroup UjRsUj† = αtsRt, (38)
{I,R1(1),...,Rm(1)}. Rt∈XBrd−{I}
10
where d denotesthebasisset(definedinEquation7)of invariant under conjugation. We have therefore proved
Br
ther-quditPauligroup. Theunitarityofthetransforma- Theorem 2 using minimal subcodes.
tion can be easily proved as in the qubit case. However,
unlike the qubit case, α is in general a complex num-
ts
ber as the Pauli operators R are not necessarily Hermi- B. Subcodes associated with single qudits
t
tian. Thus we can represent each transversalgate U on
j
the code space by a matrix Vj SU(d2r 1). We as- In this section we introduce the single qudit sub-
sociate the basis elements XaZ∈b a,b =−0,...d 1 codes,andusethesesubcodes toproveTheorem2. The
{ | − }
of the generalized Pauli group with the basis vectors definitions and results are similar to those of Sec. IIIB,
i i = 0,...d2r 1 . Then the subcode projectors but have been adapted for the case when d > 2. The
c{a|ni a|gain be mappe−d in}to vectors in (Cd2r 1) ω . generalization is mostly straightforward, but requires a
− ⊗| |
When N =1, we find that few adjustments when d is nonprime. The most signifi-
g
cantdifferenceliesinthequditversionsofLemmas3and
ρ r I ii...i =:w (39) 4, which no longer give specific values for the indices of
⊗ω − 7→ | i S i and Π in S, but give bounds instead. This slight
Xi |ω| times rehlaixation still allows us to prove the necessary result.
| {z }
The summation is over all vectors i that correspond to
| i
Pauli matrices (Zm)i1 (Zm)ir in Equation (36).
⊗···⊗ 1. Single qudit subgroups and subcodes
When N = 2, the mapping takes the same form ex-
g
cept that the summation is over all vectors that corre-
spond to Pauli matrices R(a,b) = r (Zm1)ai(Xm2)bi The single qudit subcode associated with a coordi-
⊗i=1 nate i [n] is the subcode with projector ρ = tr P ,
in Equation (37). ω ω¯ Q
∈
whereω = i . Wedenotethe projectorforthissubcode
Rains’ technique still works here to ensure that when
{ }
byρ . Thesingle qudit subgroupS i associatedwith
ω 3, the matrix V is either monomial itself or has a i
| |≥ j i is the set R S R(i) = I . As ihnithe case d = 2,
monomial subblock as in Equation (29). As mentioned
{ ∈ | }
we define the support of a subgroup S i to be the set
at the beginning of this section, the only difference is
h i
supp(R).
that the non-zero entries in the monomial subblock are R S i
∪ W∈ehwi ill now generalize the two lemmas of Gross et
notnecessarily 1,butcanbeanycomplexnumberwith
± al.[11] that we introduced in Sec. IIIB1.
modulus 1. Therefore we find that the transversal gate
Lemma 5: LetS beastabilizeronnqudits,andS i
U iseitherCliffordornormalizesasubgroupofthePauli
j h i
the single qudit subcode associated with i [n]. Then
group.
[S :S i ] d2 for every i [n]. ∈
Now we deal with the case when ω 2. As the
h i ≤ ∈
operatorsX ω andZ ω donotcomm| u|te≤foranyd 3 To prove this, note that since S i is a subgroup of S,
⊗| | ⊗| | h i
≥ we can partition S into N cosets of S i where N =[S :
when ω 2, we are only concerned with the case when
h i
| |≤ S i ]. We can therefore write
thePaulioperatorsatcoordinatejareapropersubgroup
h i
of all the Pauli operators. When ω = 2, we can prove
S =S i g S i g S i (40)
| | 1 N 1
asbeforethatatransversalgateU preservesthespanof h i∪ h i∪···∪ − h i
j
acertainsubgroupofthePauligroupunderconjugation. for N 1 elements g1,...,gN 1 S. Two elements
When ω = 1, if we require that the physical qudit and ga,gb −S belong to different c−oset∈s of S i if and only
logical|qu|dit must have the same dimension, we are left if their∈jth coordinates g(j) and g(j) diffehr.iThus, there
a b
only with a trivially encoded qudit–a case that can be can be at most d2 cosets of S i , as any basis element
h i
discarded. g of the generalized Pauli group can be written in the
formZk1Xk2 fork ,k 0,1,...,d 1 . Itfollowsthat
1 2
[S :S i ] d2, and the∈le{mma is pro−ved}.
3. Coordinates not covered by minimal subcodes Lemh mi ≤a 6: Let Π be the smallest subgroup of S
containing all the single qudit subgroups S i . Then
h i
Now that we have dealt with the coordinates that are [S :Π] d2. Let X n,Z n be the groupgeneratedby
⊗ ⊗
contained in some minimal support, we can go back to X n an≤d Z n. Ifh[S :Π]=di2, then the stabilizer S can
⊗ ⊗
seewhathappenswhenajthcoordinateofthestabilizer bewrittenuptolocalCliffordoperationsas X n,Z n ,
⊗ ⊗
h i
code is not covered by any minimal support. As in the where X and Z are the generators of the generalized r-
qubitcase,weremovealltherestrictedminimalelements qudit Pauli group.
M := R S ∄R S ,supp(R) supp(R) from To prove the first part of the lemma, we use the fact
j j ′ j ′
{ ∈ | ∈ ⊂ }
the set S of stabilizer elements covering the coordinate that S = G[S :G] for any subgroup G of S. As every
j
| | | |
j. We can again prove, as in the qubit case, that for single qudit subgroup S i is contained in Π, it follows
h i
a fixed support (containing j) the Pauli operators at j that S i Π for every i [n]. Thus, we find that
in the minimal elements form a proper subgroup of the [S :Π|] h i[|S≤:S| i|] d2. ∈
≤ h i ≤
1-qudit Pauli group. In this way, we can deduce that To prove the second part of the lemma, assume that
U must keep the span of a subgroup of Pauli operators [S :Π]=d2. As in the case d=2,we can write Π as the
j
