Quadratic Form Expansions for Unitaries

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by  Niel de Beaudrap

Quadratic Form Expansions for Unitaries
NieldeBeaudrap1,VincentDanos2,ElhamKashefi3,MartinRoetteler4
1 IQC,UniversityofWaterloo
8 2 SchoolofInformatics,UniversityofEdinburgh
0 3 Laboratoired’InformatiquedeGrenoble
0 4 NECLaboratoriesAmerica,Inc.
2

n
Abstract. We introduce techniques to analyze unitary operations in terms of
a
J quadraticformexpansions, aformsimilartoasumoverpathsinthecomputa-
tionalbasiswhenthephasecontributedbyeachpathisdescribedbyaquadratic
6
1 formoverR.Weshowhowtorelatesuchaformtoanentangledresourceakinto
thatoftheone-waymeasurementmodelofquantumcomputing.Usingthis,we

] describevariousconditionsunderwhichitispossibletoefficientlyimplementa
h
unitaryoperationU,eitherwhenprovidedaquadraticformexpansionforU as
p
input,orbyfindingaquadraticformexpansionforU fromotherinputdata.
-
t
n
a 1 Introduction
u
q
[ Intheone-waymeasurementmodel[1,2],quantumstatesaretransformedusing

single-qubit measurements on an entangled state, which is prepared from an
1
v input state by performing controlled-Z operations on pairs ofqubits, including
1 the input system and ancillas prepared in the + state. This model lends itself
6 | i
to ways of analyzing quantum computation which do not naturally arise in the
4
2 circuit model, e.g. with respect to depth complexity [3] and discrete structures
.
1 underlying unitary operations [6,8]. In this article, wepresent another result of
0 thisvariety, byintroducing quadratic formexpansions.
8
0
Definition1. LetV beasetofnelements,andI,O V be(possiblyintersect-
v: ing) subsets. For a binary string x 0,1 V, let x⊆and x be the restriction
I O
i ∈ { }
X ofxtothose bit-positions indexed byelementsofI andO,respectively. Thena
r quadratic formexpansion isamatrix-valued expression oftheform
a
1
U = eiQ(x) x x , (1)
O I
C | ih |
x X0,1 V
∈{ }
U : I O,whereQisareal-valued quadratic formonx,andC C.
H2⊗ → H2⊗ ∈
Quadraticformexpansionsbearaformalsimilaritytoarepresentationofaprop-
agatorofaquantumsystem intermsofasumoverpaths.ForaunitaryU given
as in (1), the outer product x x essentially specifies a particular coeffi-
O I
| ih |
cient, in the row indexed by the substring x and the column indexed by x :
I O
2 NieldeBeaudrap,VincentDanos,ElhamKashefi,MartinRoetteler
the amplitude of the transition between these standard basis states is propor-
tionaltoasumofcomplexunitsspecifiedbyx ,x ,andtheauxiliaryvariables
I O
v V r(I O).
∈ ∪
Representations of unitary transformations as sums over paths is a well-
developed subject in theoretical physics (see e.g. [4,5]); and a representation
of unitaries as a sum over paths was used in [9] to provide a simple proof of
BQP PP.1 However, there are also examples of quadratic form expansions
⊆
which arise without explicitly seeking to represent unitaries in terms of path
integrals: the quantum Fourier Transform over Z can readily be expressed
2n
in such a form, and quadratic form expansions for Clifford group operations
are implicit in the work of Dehaene and de Moor [17], as we will describe in
Section3.3.
Givensuchanexpression foraunitary U,weshowhowtoobtainadecom-
position of U in terms of operations similar to those used in the one-way mea-
surement model. Using this connection, we demonstrate techniques involving
quadraticformexpansionstoefficientlyimplementaunitaryoperator,whenthe
coefficientsofthequadraticformsatisfiescertainconstraintsrelatedto“general-
izedflows”(orgflows)[8]orCliffordgroupoperations.Inparticular,weexhibit
an O(n3/logn) algorithm to obtain a reduced measurement pattern (an algo-
rithmintheone-waymodel)forCliffordgroupoperationsfromadescriptionof
howtheytransform thePauligroup, basedontheresultsof[17].
2 Connection to the one-way model
2.1 Reviewoftheone-waymodel
We can formulate the one-way measurement model as a way of transforming
quantum states in the following way. Given astate ψ on aset of qubits I (the
| i
input system), weembedI inalarger system V,wherethequbits ofV rI are
preparedinthe + 0 + 1 state.Wethenperformentangling operations on
| i∝ | i | i
the qubits of V, by performing controlled-Z (denoted Z) operations on some
∧
setsofpairsofqubits.(Theseoperationsaresymmetricandcommutewitheach
other, and so we may characterize the entangling stage by a simple graph G
whose vertices are the qubits of V: we call this the entanglement graph of the
procedure.) Wethenmeasure eachofthequbits ofV insomesequence, except
forsomesetofqubitsO V (theoutputsubsystem) whichwillsupport afinal
⊆
quantum state. We may represent the measurement result for each qubit v by a
1Unitarieswereexpressedin[9]intermsofpathswhosephasecontributionsaredescribedby
cubicpolynomialsoverZ ;commentsmadeinSectionVIofthatpaperessentiallyanticipate
2
quadratic form expansions with discretized coefficients. We describe how their techniques
provideameansofconstructingquadraticformexpansionsfromcircuitsinAppendixA.
QuadraticFormExpansionsforUnitaries 3
bits 0,1 whichindexestheorthonormal basisstatesofthemeasurement.
v
∈ { }
The measurement basis for each qubit may depend on the results of previous
measurements, but without loss of generality may be expressed in terms of a
“default” basiswhichisusedwhenallpreceding measurements yieldtheresult
0.Dependingonthemeasurementresults, afinalPaulioperatormaybeapplied
tothequbitsintheoutputsubsystem O.2
In the original formulation of the one-way measurement model, the mea-
surement bases were described by some axis of the Bloch sphere lying on the
XYplane,whichissufficientforuniversalquantum computation. Itisalsoeasy
toprovethatrestrictingthistostateswhichareanangleθ πZfromtheXaxis
∈ 4
issufficient forapproximately universal quantum computation [12]. Whileitis
reasonable toextendbeyondthisforchoicesofmeasurementbases[7],wewill
onlyneedtoconsider measurementbasesfromtheXYplane.
2.2 Phasemapdecompositionsfromquadraticformexpansions
Consider a unitary U given by a quadratic form expansion as in (1), where the
quadratic formQisgivenby
Q(x) = θ x x , (2)
uv u v
u,v V
{ X}⊆
forsomeangles θ ,andwherethesumincludestermsforu = v.Note
{ uv}u,v V
that Q(x) can be expres∈sed as an expectation value x H x , where H is a
h | | i
2-localdiagonal operator:
H = θ 1 1 1 1 + θ 1 1 . (3)
uv | ih |u⊗| ih |v vv| ih |v
u,v V (cid:20) (cid:21) v V
{ uX=}⊆v X∈
6
ThenwemaydecomposeU asfollows:
U x x eiH x x = y y eiH x x
O I O I
∝ | ih | | ih |  | ih |  | ih |
x X0,1 V y X0,1 V x X0,1 V
∈{ } ∈{ } ∈{ }
   
R eiHP , (4)
O I
∝
where P is a unitary embedding which introduces fresh ancillas (indexed by
I
v Ic = V rI) initialized to the + state, and R is a map projecting onto
O
∈ | i
the + stateforallqubitsinOc = V rO(tracingthosequbitsoutafterwards).
| i
2ThereasonforusingthesamevariablesV,I,andOforthesesetsof(labelsfor)qubitsasfor
thesetsinDefinition1willbecomeapparentinthenextsection.
4 NieldeBeaudrap,VincentDanos,ElhamKashefi,MartinRoetteler
Equation (4) is a phase map decomposition [10] for U: that is, it expresses
U intermsofaprocessofpostselecting observables, asfollows.DecomposeH
intotermsH ,H ,andH ,whereH consistsofthe1-localtermsonthequbits
O 1 2 O
ofO,H consists ofthe1-local term ontheremaining qubits, and H contains
1 2
the remaining terms from (3). We then have U RO eiHOeiH1eiH2PI. Note
∝
that eiHO and eiH1 are simply single-qubit Z rotations applied to the elements
of O and Oc respectively, where in each case the qubits v in those sets are ro-
tated by an angle θ . Then, the composite map R˜ = R eiH1 projects each
vv O O
thestateofeachqubit v Oc ontothevector 0 +e iθvv 1 foreachv Oc.
−
∈ | i | i ∈
We then have U = eiHOR˜OeiH2PI, which is a decomposition of U into the
preparation of some number of + states, followed by a diagonal unitary op-
| i
eratorconsisting oftwo-qubit (fractional) controlled-Z operations, followedby
post-selection of states on the Bloch equator for v Oc, and (unconditionally
∈
applied)single-qubitZ rotationsontheremainingqubits.Ifθ 0,π forall
uv
∈ { }
distinctu,v V andforu= v O,theabovedescribespreciselytheactionof
∈ ∈
ameasurement-based computation inwhichthequbits v Oc aremeasured in
∈
theeigenbasesofobservablesoftheformM( θ ) = cos(θ )X sin(θ )Y,
vv vv vv
− −
in the special case where all measurements result in the +1 eigenstate (which
wemaylabelwiththebits = 0).
v
If weare able toextend the above into acomplete measurement algorithm,
withdefined behavior when not allmeasurements yield aspecific outcome, we
obtain a measurement-based algorithm for U: we discuss this problem in the
next section. Conversely, from every measurement based algorithm, we may
obtainaquadratic formexpansion:
Theorem1. Everyunitaryoperatoronnqubitsmaybeexpressedbyaquadratic
form expansion with I = O = n, and where the quadratic form has coeffi-
| | | |
cients θ 0,π for all cross-terms x x and π < θ 6 π for all terms
uv u v vv
∈ { } −
x2. Furthermore, any unitary can be approximated to arbitrary precision by
v
suchanexpansion wherewefurther requireθ πZ.
vv ∈ 4
Proof. From[11](andusingthenotation ofthatarticle), themeasurement pat-
ternXsuM αE N performs theunitarytransformation J(α) = 1 1 eiα
v u− uv v √2 1 eiα
for α R, from the state space of a qubit u to that of a “fresh” qubit v. T−hese
∈ (cid:2) (cid:3)
operations generate SU(2), and generate a group dense in SU(2) if we restrict
toα πZ,by[12].
∈ 4
Foranynqubit unitary U,thereexists ameasurement pattern composed of
suchpatternstogetherwithtwo-qubitcontrolled-Z operations(whichwedenote
Z)whichimplementsU.LetGbetheentanglement graphofthispattern, and
∧
I andObethequbitsdefiningtheinputspaceandoutputspace(respectively)of
themeasurementpattern.By[6],inthismeasurementpattern,theprobabilityof
QuadraticFormExpansionsforUnitaries 5
every measurement resulting in the +1 eigenvalue (i.e. s = 0 for all v Oc)
v
∈
isnon-zero. Then,U R eiHP ,where
O I
∝
H = π 1 1 1 1 α 1 1 . (5)
| ih |u ⊗| ih |v − v| ih |v
uv E(G) (cid:20) (cid:21) v Oc
∈X X∈
By(4),thisyieldsaquadratic formexpansion forU,with
Q(x) = πx x α x2 . (6)
u v − v v
uv E(G) v Oc
∈X X∈
For a quadratic form expansion approximating U, it is sufficient to consider
measurementpatterns approximating U usinganglesα πZ. 2
v ∈ 4
2.3 MeasurementPatternInterpolation
Asweremarkedabove,theconnectionfromquadraticformexpansionstophase
mapdecompositions mayallowustoobtainanimplementation forU,provided
we can determine how to adapt measurements in case the measurements for
qubitsv Oc donotallyieldtheresults = 0.
v
∈
In a measurement pattern performing N measurements, the computation
mayfollowanyof2N branches, corresponding tothedifferentcombinations of
measurement results. Let us call the branch in which every measurement pro-
duces theresult s = 0thepositive branch ofthemeasurement pattern.3 With-
v
out loss of generality, we may restrict our attention to patterns where no clas-
sical feed-forward is required in the positive branch: then, the positive branch
ofameasurementpattern ischaracterized bythegeometry(G,I,O)ofthepat-
tern (where G is the entanglement graph of the measurement algorithm, and
I,O V(G) are the sets of qubits defining the input/output space of the pat-
⊆
tern),andtheanglesa = α definingthemeasurementstobeperformed.
{ v}v Oc
Toextendthedescriptionof∈thepositivebranchofameasurementalgorithm
into a complete measurement algorithm performing a unitary is the subject of
thefollowingproblem:
MeasurementPatternInterpolation (MPI). For input data (G,I,O,a), de-
scribingaunitaryembeddingU asthepositivebranchofameasurementpattern
with geometry (G,I,O) and performing measurements a, determine if there a
measurement pattern P with geometry (G,I,O) which performs the transfor-
mationU.
3Thischoice of terminology referstoall measurements yielding the+1eigenvalues of their
respectiveobservablesM( θ ).
vv
−
6 NieldeBeaudrap,VincentDanos,ElhamKashefi,MartinRoetteler
This problem is open, and seems to be difficult in general. We may attempt to
maketheproblemeasierbyconsidering amorerestricted problem:
GenericMeasurementPatternInterpolation (GMPI). For an input geome-
try (G,I,O), determine if there exist measurement patterns P(a) parameter-
ized by a choice a of measurement angles, each with geometry (G,I,O), such
thatthepatternP(a)performsaunitaryembedding foralla.
GMPIaddresses,inanangle-independent manner,thesubjectofthestructureof
measurementpatternswhichperformunitarytransformations. Aspecialcaseof
the GMPIwhich has been solved are those geometries (G,I,O) which have a
“generalized flow”(orgflow), whicharethe“yes” instances ofGMPIsuch that
the patterns P(a) yield maximally random outcomes on all of their measure-
ments [8]. The following is the definition of gflows in [13], for measurements
restricted totheXYplane:4
Definition2. Given a geometry (G,I,O) for a measurement pattern, a gflow
isapair(g,4),wheregisafunctionfromOctosubsetsofIcand4isapartial
order,suchthatthefollowingconditions holdforalluandv inthegraphG:
v g(u) = u v, (7a)
∈ ⇒ ≺
v odd(g(u)) = u4 v, (7b)
∈ ⇒
u odd(g(u)), (7c)
∈
whereodd(S)isthesetofverticesadjacenttoanoddnumberofelementsofS.
Here, u 4 v essentially represents, for two qubits u and v, that v is measured
no earlier than u; a gflow then specifies an ordering in which the qubits are to
be measured (with the function g providing a description of how to adapt later
measurements).MhallaandPerdrix[13]presentanalgorithmwhichdetermines
ifageometry hasagflowinthissense inpolynomial time,whichinturnyields
apolynomial time solution tothe GMPIfor that case. Asa result, any instance
oftheMPIwherethegeometry(G,I,O)hasagflowcanbeefficientlysolved.
A different special case of the Measurement Pattern Interpolation problem
which has been solved is that where the measurement angles are restricted to
multiples of π (orslightly moregenerally, wherethemeasurement observables
2
are Pauli operations). In this case, as noted in[7], no measurement adaptations
are necessary, and the corrections can be determined via the stabilizer formal-
ism[16].
In the following section, we apply these solutions to special cases of the
MPIto describe how to synthesize implementations for a unitary U given by a
quadratic formexpansion.
4Theoriginaldefinitionofgflowsin[8]alsoallowsforYZplaneandXZplanemeasurements,
whichdonotplayaroleeitherinouranalysisorin[13].
QuadraticFormExpansionsforUnitaries 7
3 Synthesis viameasurement pattern interpolation
In order to apply the partial solutions to the MPI described above, it will be
usefultodefinethefollowing:
Definition3. Foraquadratic formexpansion
1
eiQ(x) x x where Q(x) = θ x x , (8)
O I uv u v
C | ih |
x∈X{0,1}V {uX,v}⊆V
the geometry induced by the quadratic form is a triple (G,I,O), where G is
a weighted graph with vertex-set V, edge-set uv u= v andθ = 0 , and
uv
{ | 6 6 }
edge-weights W (uv) = θ /π.
G uv
Because we can require π < θ 6 π for all u,v V , we may without
uv
− ∈
loss ofgenerality restrict Gtohave edge-weights 1 < W (uv) 6 1.Wewill
G
−
assumethatthisholdsfortheremainderofthearticle,andspeakofedgesbeing
eitherofunitweight orfractional weight.
In this section, weconsider the problem of synthesizing an efficient imple-
mentation of unitaries U interms ofthe geometry induced byaquadratic form
expansion for U by reduction to the solved cases of the Measurement Pattern
Interpolation problem discussed intheprevioussection.
3.1 Measurementpatternsynthesisviagflows
Consider a geometry (G,I,O) induced by a quadratic form expansion for a
unitaryembedding U,whereGhasonlyedgesofunitweight:then(G,I,O)is
alsoageometryforameasurementpattern.Toobtainameasurementpatternfor
U,itsufficestofindagflowfor(G,I,O):inthatcase,byTheorem2of[8],for
anychoiceofmeasurement anglesa = α ,wemayconsiderthepattern
{ v}v Oc
∈
<
 Zv Xv Muαu Euv Nu (9)
! ! " #" #
u Oc v odd(g(u)) v g(u) u v u Ic
 Y∈ ∈ O ∈O  Y∼ Y∈
 v=u 
6
 
wheretheleft-handproductmaybeorderedright-to-left inanylinearextension
of the order 4, and denotes the adjacency relation of G. This pattern thus
∼
steersthereducedstateaftereverymeasurementtothestatewhichwouldoccur
if the result had been the +1 eigenvalue. Every branch of the pattern then per-
formsthesameoperation asthepositive branch, andsothepattern implements
a unitary operation U. To obtain a pattern in standard form (with corrections
8 NieldeBeaudrap,VincentDanos,ElhamKashefi,MartinRoetteler
only on output qubits), it is sufficient to propagate the corrections to the left,
absorbing themintothemeasurementbases.
In [13], an O(n4) algorithm is provided to determine whether or not a ge-
ometry (G,I,O) has a gflow where every qubit is to be measured in the XY
plane(andobtainoneinthecasethatoneexists),wheren = V(G) .Themea-
| |
surement pattern of (9) can be constructed in time O(n2) by first producing a
pattern where corrections undo byproduct operations after each measurement,
commuting these corrections to the end, and simplifying; the resulting pattern
willhaveO(n)operations eachwithcomplexity O(n).Thus:
Theorem2. For a unitary embedding U given as a quadratic form expansion
with geometry (G,I,O) with unit edge-weights, there is an O(n4) algorithm
which either determines that (G,I,O) has no gflow, or constructs a measure-
mentpattern consisting ofO(n2)operations5 implementing U (using measure-
mentangles ofarbitrary precision), wheren = V(G) .
| |
3.2 Circuitsynthesisviaflows
Ageometry(G,I,O)whichhasfractional edgeslies,atfirstglance, outside of
thedomainoftheMeasurement Interpolation Problemsdescribed above. How-
ever, given a quadratic form expansion with such a geometry, we may still be
able to synthesize a circuit for a unitary U represented by that expansion by
considering flows, which correspond togflowswhere thefunction g mapseach
vertex v Oc to a singleton set: we may say (f,4) is a flow if and only if
∈
(g ,4)isagflow,whereg (v) = f(v) .
f f { }
GeometrieswhichhaveflowsareasolvablespecialcaseoftheGMPI,where
theresultingmeasurementpatternsarevery“circuit-like”. Specifically,thepos-
itivebranchofameasurement patternwhosegeometry hasaflowcanberepre-
sentedbyacircuitwiththefollowingcharacteristics [6]:
– edges of the form vf(v) for v Oc correspond to J( α )gates on some
v
∈ −
wire,separating twowiresegmentswhichwelabelv andf(v);
– edgesuv E(G)foru = f(v)andv = f(u)correspond to Z operations
∈ 6 6 ∧
actingonthewiresegmentslabelledbyuandv;
– wires whose initial segments are labelled by vertices of I accept arbitrary
inputstates,whilethoselabelledbyverticesIcrimg(f)takeinput + .
| i
Intheaboveformulation, theedgesoftheformvf(v)canbeinterpreted as
implementingsingle-qubitteleportation,inwhichcaseafullyentanglingunitary
5 Theseoperationsmayinvolvemeasurementanglesofarbitraryprecision.Acorrespondingap-
proximatemeasurementpatternmayuseO(n2+npolylog(n/ε))operationsbytheSolovay-
KitaevTheorem[14],whereεistheprecisionofthecoefficientsofQ.
QuadraticFormExpansionsforUnitaries 9
is important in order to transfer the information of the “source” qubit to the
“target” qubit upon measurement. However, considering the analysis of [6], it
is not important that the edges of the second kind above be fully entangling
operations:usingsuchedgestorepresentfractionalpowersof Z willalsoyield
∧
unitarycircuits. Thismotivatesthefollowingdefinition:
Definition4. Suppose (G,I,O) is a geometry of a quadratic form expansion
for a unitary transformation U. We may say that (f,4) is a fractional-edge
flow for (G,I,O) if it is a flow for that geometry, and for all ab E(G) with
∈
W (ab) < 1,wehavef(a)= bandf(b)= a.
G
6 6
If (G,I,O) has a fractional-edge flow, we may synthesize a circuit from a
quadratic form expansion for U using the description above, where edges ab
of fractional weight correspond to ZWG(ab) gates on the wire segments la-
∧
belled by aand b rather than simple Z gates. Wewill make use the following
∧
easily verified Lemma to consider how to compose/decompose quadratic form
expansions:
Lemma1. LetU ,U bematricesgivenbyquadratic formexpansions
1 2
1
U = eiQj(x) x x . (10)
j C Oj Ij
j
x∈{X0,1}Vj (cid:12) (cid:11)(cid:10) (cid:12)
(cid:12) (cid:12)
Inthefollowing, C = C1C2,andsumsareover 0,1 V1∪V2.
{ }
(i) IfV V = I = O ,thenU U = 1 eiQ1(x)+iQ2(x) x x .
1 ∩ 2 2 1 2 1 C | O2ih I1|
x
(ii) IfV andV aredisjoint,thenU U =P1 eiQ1(x)+iQ2(x) x x ,
1 2 1⊗ 2 C | Oih I|
whereI = I I andO = O O . x
1 2 1 2
∪ ∪ P
Weprovethecircuitconstructiongivenbyinductingonthenumberofedges
of fractional weight. For the base case, if (G,I,O) has no fractional-weight
edges at all, we may synthesize a circuit for U as above, as it corresponds to a
normal measurement pattern with aflow,and so falls under the analysis of [6].
Wemaytheninductforgeometrieswithfractionaledge-weightsifwecanshow
wecandecomposethegeometryintooneswithfewerfractional edge-weights.
Foranyarbitrary fractional edgeab E(G)andeacheachz O,wemay
∈ ∈
definem(ab,z)tobethemaximalvertexv V(G)intheordering4subjectto
∈
z being in the orbit of v under f (that is, z = fℓ(v) for some ℓ > 0), such that
atleastoneofv 4 aorv 4bholds.ForasetS V(G),letG[S]representthe
⊆
subgraph of G induced by S (i.e. by deleting all vertices in G not in S). Then,
definethefollowingsubgraphs ofG,andcorresponding geometries:
10 NieldeBeaudrap,VincentDanos,ElhamKashefi,MartinRoetteler
(G,I,O)
(G ,I,V ) (G ,V ,V ) (G ,V ,O)
1 2 2 2 2 3 2
V
2
=
a a ◦ a ◦ a
b b b b
I V1 I V2 V2 V2
V3 O O
Fig.1. Illustration of the decomposion of a quadratic form expansion about an
edgeab,expressedintermsofgeometries.V isasetofmaximalverticesunder
2
theconstraint ofbeingbounded fromabove,bytheverticesaandb,inapartial
order 4 associated with a fractional-edge flow. Arrows represent the action of
thecorresponding fractional-edge flowfunction, f.
– Let V be the set of vertices m(ab,z) for each z Oc: it is easy to show
2
∈
thata,b V .LetG = G[V ],andlet = (G ,V ,V ).
2 2 2 2 2 2 2
∈ G
– LetV bethesetofverticesu V(G)suchthatu 4v forsomev V ;let
1 2
∈ ∈
G = G[V ]r uv u,v V ;andlet = (G ,I,V ).
1 1 2 1 1 2
∈ G
– LetV bethesetofverticesu V(G)suchthatu <v forsomev V ;let
3 (cid:8) (cid:12) ∈(cid:9) ∈ 2
G = G[V ]r uv(cid:12)u,v V ;andlet = (G ,V ,O).
3 3 2 3 3 2
∈ G
Thisdecomposes th(cid:8)e ge(cid:12)ometry (G(cid:9),I,O) into three geometries withfractional-
(cid:12)
edgeflows,asillustrated inFigure1.
LetQ beaquadraticformon 0,1 V1 consistingofthetermsx x ofQfor
1 u v
u V or v V , but not both; Q{ be}a quadratic form on 0,1 V2 consisting
1 1 2
∈ ∈ { }
ofthetermsx x ofQfordistinctu,v V ;andsimilarlyletQ bedefinedon
u v 2 3
0,1 V3,andconsistoftheremaining te∈rmsofQ.ThenQ ,Q ,andQ define
1 2 3
{ }
quadratic form expansions for some operations U , U , and U (respectively)
1 2 3
withgeometries , ,and (respectively).
1 2 3
G G G
– U2 inparticularwillbeaproductofoperations∧ZWG(uv) fordistinctu,v ∈
V , as it is a quadratic form expansion whose input and output indices co-
2
incide.ThenU canberepresented asacircuitwithawireforeachu V ,
2 2
∈
withfractional controlled-Z gates ZWG(uv) foreachedgeuv E(G).
∧ ∈
– Both and have fractional-edge flows, but fewer fractional edges than
1 3
G G
(G,I,O). By induction, U and U are also unitary embeddings, and have
1 3
circuitswithwire-segmentsconnected byJ(θ )gates(whereθ aretheco-
v v
efficientsofthetermsx2 ineachquadraticform)andpossiblyfractional Z
v ∧
gates(asinthecaseforU ).
2
– In the circuits described above, the terminal wire-segments for U and (a
1
subsetof)theinitialwire-segmentsforU havethesamelabelsasthewires
3