Twisted Interferometry: the topological perspective

4.1 MB·

by  Parsa Bonderson

Twisted Interferometry: the topological perspective
Parsa Bondersona, Lukasz Fidkowskia, Michael Freedmana,b,
Kevin Walkera
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1 aStationQ,Microsoft Research,SantaBarbara, California93106-6105, USA
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2 bDepartmentofMathematics, University ofCalifornia, SantaBarbara, California93106,
USA
c
e
D

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Abstract

]
h
Three manifold topology isused toanalyze theeffect ofanyonic interferometers inwhich
p
- theprobeanyons’pathalonganarmcrossesitself,leadingtoa“twisted”orbraidedspace-
t
n timetrajectoryfortheprobeanyons.InthecaseofIsingnon-Abeliananyons,twistedinter-
a
ferometry isshowntobeabletogenerate atopologically protected π/8-phase gate,which
u
q cannotbegenerated fromquasiparticle braiding.
[

2 Keywords: Interferometry; Anyoniccharge measurement;Topological quantum
v
computation.
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2 PACS: 03.67.Lx,03.65.Vf,03.67.Pp,05.30.Pr
2
8
0
.
1
0 1 Introduction
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1
:
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Anyonicinterferometry[1,2]isapowerfultoolforprocessingtopologicalquantum
i
X
information [3,4,5,6,7,8,9]. Its ability to non-demolitionally measure the collec-
r
a tive anyonic charge of a group of (non-Abelian) anyons, without decohering their
internal state, allows it to generate braid operators [10,11], generate entangling
gates [12,13,14,15], and change between different qubit encodings [14,15]. Any-
onicinterferometryhasbeenthefocusofmyriadexperimentalproposals[16,17,18,19,20,21,22,23,24,25,26
and efforts to physically implement them [30,31,32,33,34,35,36]. As powerful as
anyonic interferometry may be, its potential capabilities have yet to be fully un-
derstood.Inthispaper,weproposeandanalyzeanovelimplementationofanyonic
interferometry that we call “twisted interferometry,” which can significantly aug-
mentitspotentialcapabilities.
Oneoftheprimarypracticalmotivationsforstudyingtwistedinterferometryisthat
it could be used with anyons of the Ising TQFT to generate “magic states,” as we
willdemonstrate. This is significantbecause, if oneonly has theability to perform
PreprintsubmittedtoarXiv 29January2016
braiding operations and untwisted anyonic interferometry measurements for Ising
anyons, then one can only generate the Clifford group operations, which is not
computationally universal and, in fact, can be efficiently simulated on a classical
computer [37]. However, if one supplements these operations with magic states,
then one can also generate π/8-phase gates, which results in a computationally
universalgateset [38].
Theapplicationoftwistedinterferometrytogeneratingtheπ/8-phasegateforIsing
anyons is the latest link in a chain of ideas [39,40,41,42], originating with the un-
publishedworkofBravyiandKitaev,forgeneratingatopologically-protectedcom-
putational universal gate set from the Ising TQFT by utilizing topological opera-
tions. The concept and analysis of twisted interferometry is new, but closely con-
nectedtotheseideas,whichstemfromtheconceptofDehnsurgeryon3-manifolds.
As we will discuss in detail, anyonic interferometry: 1) projectively measures the
topological charge inside γ, and 2) decoheres the anyonic entanglement between
thesubsystemsinsideandoutsidetheinterferenceloopγ[43].Bothoperationshave
a3DtopologicalinterpretationinthecontextofChern-Simonstheoryor,moregen-
erally, axiomatic (2+1)D topological quantum field theories (TQFTs). We learned
from Witten [44] that all low energy properties of systems governed by a TQFT
can be calculated in a Euclidean signature diagrammatic formalism called unitary
modular tensor categories (UMTC). This suggests [40,41] that the choice of inter-
ference loopγ shouldnotberestrictedto asimplespace-likeloopinaspatialslice
R2 R2 time,asisthetypicaldesignforaninterferometer,butratherγ mightbe
⊂ ×
a general simple closed curve of space-time. Twisted interferometry explores this
directionbyallowingtheprobeanyons’paththroughthearmsoftheinterferometer
to beself-crossingin R2 (so γ isimmersed in mathematicalterminology).We give
ageneralprocedureforanalyzinginterferometersofthiskind.Intherestrictedcase
oftheIsingTQFT,wedescribeatwistedinterferometerwhichwouldbecapableof
producingmagicstates.
Our strategy is: 1) to start with the UMTC calculation [1,2] which lays bare the
asymptotic behavior of the simplest anyonic Mach-Zehnder interferometer (and
serves as a model for Fabrey-Pe´rot type interferometers in the weak tunneling
limit); 2) describe this behavior in an equivalent topological language; and 3) ex-
ploitthegeneral covarianceinherentinthetopologicaldescription.
TheconcretecalculationusingthemachineryofUMTCsiscarriedoutinacompan-
ionpaper[45],whichalsofocusesonpossiblephysicalimplementationsoftwisted
interferometers. The analysis of the companion paper agrees with the topological
argument presented here and both show how magic state production is achieved
when specializedto theIsingtheory.
2
(cid:1371)
T
2
(cid:3036)(cid:3087)(cid:3141)(cid:3141) (cid:2879)(cid:2869)
A(cid:1857) (cid:1844)(cid:3002)(cid:3003) (cid:1372)
B … B
N 1
(cid:3036)(cid:3087)(cid:3141)
(cid:1857) (cid:1844)(cid:3003)(cid:3002)
T C
1
Fig. 2.1. An idealized Mach-Zehnder interferometer for an anyonic system, where T are
j
beamsplitters.Thetargetanyons(collectivelydenotedA)inthecentralregionshareentan-
glementonlywiththeanyon(s)C outsidethisregion.AbeamofprobeanyonsB ,...,B
1 N
issentthrough theinterferometer anddetected atoneofthetwopossible outputsbyD .
s
2 WhatanAnyonicInterferometer Does inTwo Different Languages
Werecallthebarebonesofanyonicinterferometryinageneralanyoniccontext(as
developedin [1,2]; see[45]fornotationalclarificationand calculationaldetails).
The target anyon A may be a composite of several quasiparticles (anyons), so it is
not necessarily in an eigenstateof charge. In thesimplestcase, which we treat, the
probe quasiparticles B are assumed to be uncorrelated, identical, and simple (not
composites).In fact, tomake thesourcestandard and uncorrelated, the probes will
beindependentlydrawnfromthevacuumtogetherwithanantiparticle(topological
chargeconjugateanyon),whichisthendiscardedandmathematically“tracedout.”
Wewillsimplifythediscussioninthispaperbyalsoassumingtheprobehasdefinite
topologicalchargevaluesB = b,butthegeneralizationisstraightforward.Coming
from the left, probe anyon B encounters first beam splitter T , and then T . The
i 1 2
correspondingtransitionmatricesare:
t r
j j∗
T = . (2.1)
j  
r t
 j − ∗j 
 
Theunitaryoperatorrepresentingaprobeanyonpassingthroughtheinterferometer
isgivenby
U = T ΣT (2.2)
2 1
3
0 eiθIIRA−B1
Σ = . (2.3)
 
eiθIR 0
BA
 
 
Thiscan bewrittendiagrammaticallyas
A B
s t r r r r t t t
U = eiθI 1 2∗ 1∗ 2∗ +eiθII 1 2 − ∗1 2 , (2.4)
   
Bs′ A −t1t∗2 −r1∗t∗2s,s′B A r1r2 −t∗1r2s,s′B A
   
where we introduce the notation of writing the directional index s of the probe
quasiparticle as a subscript on its anyonic (topological) charge label, e.g. B . The
s
anyonicstatecomplementarytotheregionbeingprobedwillbedenotedbyC (and
laterbytwo disjointsectorsC and C ).
1 2
ThepassageofasingleprobeB transformsthedensitymatrixρAC forbothsystem
and environmentby
1
ρAC ρAC (s) = Tr Π VU ρB ρAC U V Π , (2.5)
B s † † s
7→ Pr(s) ⊗
h (cid:16) (cid:17) i
f
whereTristhe“quantumtrace,” V representsbraiding,and
f
Pr(s) = Tr[Π VUρU V ] (2.6)
s † †
f
is the probability of measurement outcome s. The effect of this superoperator can
becomputedbyconsideringtheactionontheρAC densitymatrix’sbasiselements,
whichis expresseddiagrammaticallyby
a c
Πs
U
a µ
b(cid:1) f bs (2.7)
a′ µ′
U†
Πs
a′ c′
Fortheoutcomes =(cid:1), thismaybeexpanded as
4
(Faa′cc′)−1 (f,µ,µ′)(e,α,β)
(eX,α,β)h i
a c a c
b
 e
×|t1|2|r2|2 α b β +t1r1∗r2∗t∗2ei(θI−θII) α e β
a c a c
 ′ a′ c ′a ′ c
e
+t∗1r1t2r2e−i(θI−θII)b α β +|r1|2|t2|2 b α e β 
a c a c
′ ′ ′ ′a c
=db (Faa′cc′)−1 (f,µ,µ′)(e,α,β)p(cid:1)aa′e,b[Faa′cc′](e,α,β)(f′,ν,ν′) f′ νν′ (2.8)
(eX,α,β) h i
(f′,ν,ν′)
a c
′ ′
wherewehavedefined
p(cid:1) = t 2 r 2M +t r r t ei(θI θII)M
aa′e,b | 1| | 2| eb 1 1∗ 2∗ ∗2 − ab
+t r t r e i(θI θII)M + r 2 t 2, (2.9)
∗1 1 2 2 − − a∗′b | 1| | 2|
where M is the monodromy matrix M = SabS00 (with S the modular S-matrix),
ab S0aS0b
andθ ,θ arethenon-universalphasesassociatedwithtraversingtheinterferometer
I II
via the two different paths around the interferometry region. A similar calculation
fors =(cid:2)gives
p↑aa′e,b=|t1|2|t2|2Meb −t1r1∗r2∗t∗2ei(θI−θII)Mab
t r t r e i(θI θII)M + r 2 r 2. (2.10)
− ∗1 1 2 2 − − a∗′b | 1| | 2|
Thus,wehavethesingleprobemeasurementprobabilities
Pr(s) = ρAC ps , (2.11)
(a,c;f,µ),(a,c;f,µ) aa0,B
a,c,f,µ
X
and post-measurementstate(foroutcomes)
ρAC
ρAC (s)= (a,c;f,µ),(a′,c′;f,µ′) Fa,c −1
a,a′,cX,c′,f,µ,µ′ (dfdf′)1/2 (cid:20)(cid:16) a′,c′(cid:17) (cid:21)(f,µ,µ′)(e,α,β)
(e,α,β),(f′,ν,ν′)
ps
aa′e,B Fa,c a,c;f ,ν a,c;f ,ν . (2.12)
×Pr(s) a′,c′ (e,α,β)(f′,ν,ν′)| ′ ih ′ ′ ′ ′|
h i
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The next step (which we sketch very lightly here) is to compute probabilities and
theeffect forastreamofN identicalprobeanyonsB, on ρAC. Theresultsare:
Pr(s ,...,s ) = ρAC ps1 ...psN , (2.13)
1 N (a,c;f,µ),(a,c;f,µ) aa0,B aa0,B
a,c,f,µ
X
ρAC
ρAC (s ,...,s ) = (a,c;f,µ),(a′,c′;f,µ′) Fa,c −1
1 N a,a′,cX,c′,f,µ,µ′ (dfdf′)1/2 (cid:20)(cid:16) a′,c′(cid:17) (cid:21)(f,µ,µ′)(e,α,β)
(e,α,β),(f′,ν,ν′)
ps1 ...psN
aa′e,B aa′e,B Fa,c a,c;f ,ν a,c;f ,ν . (2.14)
× Pr(s ,...,s ) a′,c′ (e,α,β)(f′,ν,ν′)| ′ ih ′ ′ ′ ′|
1 N
h i
Itisclearthatthespecificorderofthemeasurementoutcomesisnotimportant,but
only the total number of outcomes of each type matters, and that keeping track of
onlythetotalnumbersleadsto abinomialdistribution.
Forgeneric choices of interferometricparameters: t ,r ,θ , and θ , thesebinomial
j j I II
distributions will concentrate exponentially fast at distinct transmission probabili-
ties associated with the equivalence classes of charge types a where a a if and
′
≡
only if Ma,b = Ma′,b. In the simplest cases, there is a natural choice for the probe
B where every a is distinguished (e.g. for Ising and Fibonacci anyons one selects
b = σ andb = τ, respectively),andhencethe“equivalenceclasses”aresingletons.
In general, theprobabilityofobservingn(out ofN)probes inthe detectoris:
→
N!
Prκ (n)= Pr (κ) pn(1 p )N n, (2.15)
N A n!(N n)! κ − κ −
Xκ −
Pr (κ)= ρAC , (2.16)
A (a,c;f,µ),(a,c;f,µ)
a∈CXκ,c,f,µ
where κ indexes the equivalence classes w.r.t. probe b. The fraction r = n/N
κ
C
of probes measured in the s = detector goes to r = p with probabilityPr (κ),
κ A
→
andthetargetanyondensitymatrixwillgenericallycollapseontothecorresponding
“fixed states.”
The asymptotic operation N of a generically tuned anyonic interferometer
→ ∞
convergestoafixedstateofchargesectorκwithprobabilityPr (κ)and:1)projects
A
the anyonic state onto the subspace where the A anyons have collective anyonic
chargein ,and2)decoheresallanyonicentanglementbetweensubsystemAand
κ
C
C that the probes can detect. The sector κ may be a single charge or a collection
ofcharges withidenticalmonodromyelementswith theprobes,i.e. Ma,B = Ma′,B
fora,a .TheanyonicentanglementbetweenAandC isdescribedintheform
′ κ
∈ C
of anyonic charge lines connecting these subsystems, i.e. the charge lines labeled
by charge e in the preceding analysis, where the contribution of a diagram to the
density matrix will be removed if M = 1. Convergence to such a fixed state is
e,B
6
6
C (cid:1371)
T
2
2
A (cid:1372)
B … B
N 1
T C
1
1
Fig. 2.2. An idealized Mach-Zehnder interferometer where the anyons C entangled with
thetargetanyonsAareseparated intotworegionsC andC .
1 2
based on Gaussian statistics, therefore exponentially precise as a function of the
numberN ofprobeparticles.
In thesimplestcase, Ma,b = Ma′,b a = a′ and theindistinguishableequivalence
⇒
classesC = a aresingletons,i.e.alltopologicalchargesaredistinguished.The
κa { }
correspondingfixedstatedensitymatrixis:
Pr (c a) Pr (c a)
ρAC = A | I = A | a,c;f ,ν a,c;f ,ν , (2.17)
κa d d ac d d | ′ ih ′ |
c a c c,f′,ν a c
X X
where
ρA
(a,c;f,µ)(a,c;f,µ)
f,µ
Pr (c a) = . (2.18)
A | P ρA
(a,c;f,µ)(a,c;f,µ)
c,f,µ
P
(The formulae for the general case can be found in [1,2].) From this point on, we
focus onlyonthesecases wheretheprobedistinguishesalltopologicalcharges.
This is a convenient place to note a modest generalization, where the complemen-
tarychargeC isdividedintotworegionsseparatedbytheinterferometer,whichwe
similarly denote as C and C , respectively. In some experimental setups — e.g.
1 2
a Fabrey-Pe´rot interferometer on a quantum Hall bar — each arm of the interfer-
ometer individually will separate the region with charge A from a complementary
region with respective charges C and C , which could both be nontrivial. This
1 2
situation is depicted for the idealized Mach-Zehnder interferometer in Fig. 2.2. In
this circumstance, all charge lines from A to C and from A to C are (separately)
1 2
decohered iftheycan bedetected bytheprobes B.
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a c c a c
1 2
α
2
α α
3 2
α α α α
4 1 4 1
e
e e
2 1
α
3
a′ c′ c′1 a′ c′2
(a) (b)
Fig. 2.3. (a) For a single region of complementary anyons C, we show the four positions
for the probe loops corresponding to the four terms of Eq. (2.8). (b) For two regions of
complementarychargeC andC ,thefourpositionsofprobeloopsareshownonthemore
1 2
complicated the target system (with complementary anyons) density matrix components.
(The 4-valent vertex is understood to be resolved into appropriate trivalent vertices.) α
j
denotestheweightwithwhichthecorresponding probeloopconfiguration entersthemea-
surementsuperoperator.
In Fig. 2.3, we compare the diagrammatic terms that arise for a single C region
formulation to when there are two regions C and C . For probe b and measure-
1 2
ment outcome s =(cid:1), the four probe loop configurations enter the measurement
superoperatorwithweights
α(cid:1)= t 2 r 2, (2.19)
1 | 1| | 2|
α(cid:1)=t r r t ei(θI θII), (2.20)
2 1 1∗ 2∗ ∗2 −
α(cid:1)=t r t r e i(θI θII), (2.21)
3 ∗1 1 2 2 − −
α(cid:1)= r 2 t 2, (2.22)
4 | 1| | 2|
as inEq. (2.8). Fors =(cid:2),theseare
α(cid:2)= t 2 t 2, (2.23)
1 | 1| | 2|
α(cid:2)= t r r t ei(θI θII), (2.24)
2 − 1 1∗ 2∗ ∗2 −
α(cid:2)= t r t r e i(θI θII), (2.25)
3 − ∗1 1 2 2 − −
α(cid:2)= r 2 r 2. (2.26)
4 | 1| | 2|
GivenN uncorrelatedidenticalprobeanyons,thereare4N configurationsofprobe
loops, each probe choosing from the four positions, with the single probe weights
(depending on a given probe’s measurement outcome) being multiplied together
for the overall superoperator. For the two probe loop positions which cross in
Fig. 2.3(b), repeated copies will nest according to the pattern of later probe loops
having larger radius. We will see shortly that the detail of the nesting patterns are
irrelevantinthelargeN limit.
8
Accordingtothecalculationjustsummarized,theneteffectofrunningtheinterfer-
ometer on the target system with density matrix ρAC, up to corrections that decay
exponentiallyinN, isthat thesuperpositionofthese4N configurations resultsina
measurement of the collectivecharge of anyons A onto charge valuea, with prob-
ability
Pr (a) = Tr ρACΠA , (2.27)
AC a
h i
and post-measurementdensitymatrix f
C A C
2 1
1
ρAaC = Pr (a) ω0 ωa ρAC ωa ω0 . (2.28)
AC
C A C
2′ ′ 1′
(All topological charge lines drawn here have zero framing, i.e. there are no twists
intheframe.)Theω -loops
a
= = S S (2.29)
0a a∗x
ωa ωa¯ Xx x
havetheeffectofprojectingallchargelinespassingthroughtheloopontocollective
charge a. Thus, the ω -loops effectively cut charge lines. This allows the ω -loops
0 a
to be moved to encircle only the A and A lines, i.e. one can perform a handle
′
slide of the loop around the ω -loops (see Section 3.1). Thus, the ω -loops effect
0 a
projection of anyons A into collective charge sector a. When there is only one
region of complementary anyons C, e.g. if there are no C anyons, then the action
2
of the ω -loop between A and C is trivial. Notice that the ω-loops here occur in
0 2
preciselythesamepositionsas thefourpossibleprobeloopconfigurations.
Having depicted the effect of interferometry in terms of ω-loops, we make a ge-
ometric observation for later use: the effects of interferometry are localized to a
certain quasi-1D region of space-time surrounding the ω-loops called a “handle
body.”TheseareindicatedinFig.2.4astheregionsH andH forthesingleregion
′
C and two regionC and C configuration of complementary anyons.The handle-
1 2
bodies H and H model the complementary regions surrounding the ρAC density
′
matrix operator. This enables us to make calculations for twisted interferometry
simplyby computingoperatorswithintransformedcoordinates.
9
H′ ωa
H
ω
a β γ
ω
a γ
ω0 ω0 ω0
γ¯
β¯ γ¯
ω
a
(a) (b)
Fig. 2.4. (a) Genus 2 handle body H and (b) genus 3 handle body H , within which the
′
effectofinterferometry islocalized inEq.(2.28).Somecurvesin∂H and∂H arelabeled
′
forlaterreference.
3 TopologicalExplanations
The goal of this section is to explain the topological nature of interferometry. In
Section3.1,wefirstreviewsomepuretopologybackgroundon3-manifoldsurgery
and the handle slide property. In Section 3.2, we apply this machinery to interfer-
ometry, with the basic idea being that in the limit of large N, the exact partition
function,givenby 4N terms withprobeanyonWilsonloops,can effectivelybede-
scribed by a small number of Dehn surgeries. Although this abstract topological
approachmayatfirstseemlikeoverkill,itprovesitsutilitywhenwetrytogeneral-
izetothecaseoftwistedinterferometry,whichisintroducedinSection3.3.Indeed,
as shownin Section 3.4, twistinghas anatural descriptionintheeffectivetopolog-
ical language: to compute the partition function in the twisted case, all we have
to do is modify the gluing of a certain handle body by some twists. Section 3.5,
although not necessary in the logical flow of the paper, develops a stand-alone,
purely topological perspective on interferometry. Finally, in Section 3.6, we apply
all this machinery to the case of the Ising UMTC, and describe the simplifications
thatarise.
3.1 SurgeryandtheHandleSlideProperty
“Handles”areacombinatorialtoolforassemblingsmoothd-manifoldswithbound-
ary out of littlepieces, which are individuallycopies of d-balls. Our main focus is
d = 4, since we will manipulate within a (2 + 1)D TQFT using a representation
where the 3D space-time is the boundary of a 4D bulk. Note, however, that the
handlebodiesdrawn inFig. 2.4are 3D,being subsetsofthespace-timeitself.
Let Bd denote the unit ball in Rd. There are d + 1 types of d-dimensional han-
10