Measurement-based quantum computation in a 2D phase of matter

0.43 MB·English

by  Andrew S. Darmawan

Measurement-based quantum computation in a 2D phase of matter
Andrew S. Darmawan,1 Gavin K. Brennen,2 and Stephen D. Bartlett1
1Centre for Engineered Quantum Systems, School of Physics,
The University of Sydney, Sydney, NSW 2006, Australia
2Center for Engineered Quantum Systems, Macquarie University, Sydney, NSW 2109, Australia
Recently it has been shown that the non-local correlations needed for measurement-based quan-
tumcomputation(MBQC)canberevealedinthegroundstateoftheAffleck-Kennedy-Lieb-Tasaki
(AKLT)modelinvolvingnearestneighborspin-3/2interactionsonahoneycomblattice. Thisstate
isnotsingularbutresidesinthedisorderedphaseofgroundstatesofalargefamilyofHamiltonians
characterized by short-range-correlated valence bond solid states. By applying local filtering and
adaptive single-particle measurements we show that most states in the disordered phase can be
reduced to a graph of correlated qubits that is a scalable resource for MBQC. At the transition
betweenthedisorderedandN´eelorderedphaseswefindatransitionfromuniversaltonon-universal
states as witnessed by the scaling of percolation in the reduced graph state.
I. INTRODUCTION Moreover, the tri-cluster state [10] on spin-5/2 parti-
cles is the ground state of a frustration-free, two-body
2
HamiltonianandisauniversalresourceforMBQC.How-
1 Quantum computers use entanglement to efficiently
ever, the Hamiltonians of both of these models lack nat-
0 performtasksthoughttobeintractableonclassicalcom-
2 ural symmetries. Resources with more natural interac-
puters. In one model of quantum computation, called
tionsbasedontheAffleck-Kennedy-Lieb-Tasaki(AKLT)
n measurement-based quantum computation (MBQC) [1,
state (which we define in Sec. II), have also been found.
a 2], the entanglement is prepared in a system of many
J These models are two-body, rotationally symmetric and
particlescalledaresourcestate beforecomputationtakes
5 place. Given this resource state, a quantum algorithm Heisenberg-like. The one-dimensional AKLT state on a
1 chain [3, 11], while not universal, can be used to imple-
proceeds by performing adaptive, single-particle mea-
ment single qubit unitaries. Theoretical constructions
surements,withclassicalprocessingofmeasurementout-
] based on the AKLT state by Cai et al. [12] and Li et
h comes. This approach is convenient for physical imple-
al. [13] were shown to be universal, the latter working
p mentationsbecausesingle-particleoperationsareusually
- atnon-zerotemperaturewithalways-oninteractions. Fi-
less error prone than entangling ones. It is also a fruit-
nt ful theoretical model to investigate computationally use- nally,thetwo-dimensionalAKLTstateonatrivalentlat-
a tice is universal [14, 15].
ful phases of matter which can be studied using well-
u A potential difficulty with these approaches is that re-
established methods from many-body physics.
q quiring an exact Hamiltonian to produce a ground state
[ Ifaresourcestateistoprovideaquantumspeed-up,it
is not robust: a physical Hamiltonian will be perturbed
must have the right kind of entanglement [3]. We call a
2 from the ideal one to some degree. Hence a phase that
resourceuniversal [4]ifwecanefficientlyobtaintheout-
v is universal, rather than a specific ground state, is a
1 putofanarbitraryquantumcomputationbyperforming more realistic computational resource. The computa-
4 single-particle measurements on it. The canonical exam-
tional power of certain cluster state phases have been
7 ple of a universal resource is the cluster state [1].
studied[16–19]. Inaddition,themorenaturalspin-1Hal-
4
Whileitisonethingtoshowthataresourceisuniver- dane phase can be used as a resource to perform single
.
8 sal, for it to be viable we must also be able to prepare it qubitunitaryoperations[20],butnotarbitraryquantum
0 efficientlyandprovidesomeshieldingagainsterrors. Itis computations.
1
hopedthatthesepropertiescanbefoundinnaturalinter- In this paper we investigate the computational power
1
: acting spin systems equipped with an energy gap. Find- of ground states in a spin-3/2 phase of matter originally
v ing universal resources that are natural ground states is studied by Niggemann et al. [21], which includes the 2D
Xi interesting in its own right, because it sheds some light AKLT state. We find that a large portion of the phase
on the intrinsic computational power of natural systems. has ground states that are universal resources, follow-
r
a Unfortunately,theclusterstateisnotanaturalground ing similar methods to [14, 15]. The phase has several
state. Infact,itisimpossibletohaveauniversalresource interesting points including a unique point where only
of spin-1/2 particles that is the unique ground state of projective measurements (as opposed to general POVM
a frustration-free Hamiltonian with only two particle in- measurements)arenecessary,andatransitionincompu-
teractions [5–7]. However, this negative result does not tational power that coincides with the phase boundary.
hold for higher level systems. For example, a gapped The paper is structured as follows. In section II we
Hamiltonian with two-body interactions involving essen- describe the spin-3/2 model defined in [21]. The compu-
tially 8-dimensional systems (on a honeycomb lattice) or tational power of this model is explored in section III
16-dimensional systems (on a square lattice) can pro- by generalising methods used for the 2D AKLT state
duce ground states that are universal resources [8, 9]. [15, 22]. In section IV, we highlight significant features
2
inmodelfromtheperspectiveofMBQC.Wepresentour is a bounded positive operator but our protocol works
conclusions in section V. just as well for a strictly negative with the replacement
D(a) → D(|a|). Importantly, the ground state of H(a)
√
depends only on a and the ground state of H( 3) is
II. MODEL DEFINITIONS |ψ (cid:105). The fully rotationally invariant interaction,
AKLT
H in Eq. (1), corresponds to the choice of param-
AKLT √
Consider a collection of spin-3/2 particles on a honey- eters a = 3 and λm = 1∀m. With periodic boundary
comb lattice interacting via the Hamiltonian conditions or with open boundaries and Heisenberg in-
teractions between spin-1/2 particles and the edges, the
(cid:88)
H = PStot=3, (1) ground state |ψ(a)(cid:105) is unique for a > 0 and can be ob-
AKLT i,j
tainedsimplybyapplyingtheinversedeformationtothe
<i,j>
2D AKLT state
where the sum is over each pair of nearest neighbours
|ψ(a)(cid:105)∝(D(a)−1)⊗N|ψ (cid:105). (5)
and AKLT
UsingMonteCarlosampling,Niggemannet.al[21]found
PS=3 = 243 S(cid:126) ·S(cid:126) + 29 (S(cid:126) ·S(cid:126) )2+ 1 (S(cid:126) ·S(cid:126) )3+ 99 , (2)
i,j 1440 i j 360 i j 90 i j 1152 thegroundstateshadexponentiallydecayingcorrelation
functions below a critical value of a2 = 6.46, while were
projects nearest neighbours i and j onto the seven di-
N´eelorderedabovethisvalue. Thus,Hamiltoniansinthe
mensional subspace of total spin S = 3. We will call
tot 0<a2 <6.46regionareconjecturedtobegapped,while
thismodelthe2DAKLTmodelaftertheauthorsAffleck,
Hamiltonians in the a2 >6.46 region are gapless [25].
Kennedy,LiebandTasakiwhooriginallyproposedit[23].
We will refer to the appearance of N´eel order at a2 =
The AKLT model can be thought of as a deformation of
the Heisenberg model H = (cid:80)S(cid:126)i·S(cid:126)j that preserves full t6h.4e6raesgitohneap2ha<se6t.r4a6nsaitsiotnheinAtKhiLsTmpohdaels.e,Waendwitllhleabree-l
rotational symmetry. Note, however, that unlike the 1D gion a2 >6.46 as the N´eel ordered phase. Note that the
case, the AKLT model and the Heisenberg model are
area law for entanglement holds across this phase transi-
not in the same phase: the Heisenberg model has a N´eel
tion (PEPS dimension is constant), a property that can
ordered ground state, while the AKLT model does not.
only occur in PEPS on graphs of dimension greater than
ThustheAKLTmodelissaidtobeinadisorderedphase.
one [26]. We also note that Schuch et al. [27] have stud-
TheabsenceofN´eelordermakesitamorerealisticmodel
ied classes of PEPS related by this type of symmetry-
for certain systems, e.g. Bi Mn O , which is a spin-3/2
3 4 12 preserving deformation.
antiferromagnetonahoneycomblatticewithoutN´eelor-
der[24]. ThegroundstateoftheAKLTmodel|ψ (cid:105),
AKLT
which we will call the 2D AKLT state, is a valence-bond III. MBQC USING GROUND STATES IN THE
solid, or projected entangled pair state (PEPS). Details AKLT PHASE
of the PEPS construction of ground states are included
in Appendix A.
Inthissectionwewilllookathowgroundstatesinthe
Niggemann et al. [21] studied a 5-parameter deforma-
AKLT phase (as defined above) can be used for MBQC.
tion of the 2D AKLT Hamiltonian which is frustration
To do this we generalise the existing method used at the
freeandwhosegroundstateisaone-parameterdeforma-
AKLT point [14, 15], which we will briefly review.
tionoftheAKLTPEPS(seeAppendixA).ThisHamilto-
nian is still two-body nearest neighbour with summands
that preserve two Z2 symmetries, parity and spin flip, A. Protocol at AKLT point
however it breaks full rotational symmetry to a U(1)
symmetry (arbitrary rotations about the z-axis). The
The 2D AKLT state has been shown to be a uni-
deformed even parity Hamiltonian is
versal resource for measurement-based quantum com-
(cid:88) putation [14, 15]. We will summarize the procedure
H(a)= [D(a) ⊗D(a) ]h (a)[D(a) ⊗D(a) ] ,
i j i,j i j for measurement-based quantum computing on the 2D
<i,j> AKLT state by breaking it into two stages: reducing to
(3)
a stochastic graph state, then using this graph state for
where
computation.
3
(cid:88)
h (a)= λ |S =3,m(cid:105)(cid:104)S =3,m|, (4)
i,j |m| tot tot
1. Reduction to a stochastic graph state
m=−3
√ √
Here D(a) = diag( 3/a,1,1, 3/a) in the S basis and Thefirststagereliesontheprincipleofquantumstate
z
the continuous free parameters satisfy: λ ,λ ,λ ,λ >0 reduction [28], where a resource is shown to be universal
0 1 2 3
and a can be positive or negative. In this work we focus byprovingthatitcanbeconvertedintoaknownuniver-
on the regime where a is strictly positive so that D(a) salresourceefficientlybysingle-particlemeasurement. A
3
on a square lattice, which is itself a universal resource
Y Y Z Y Y Y Y Y Y
[28]. Alternatively, in Ref. [15], ‘backbone’ paths are
Z Z Z X X Z X Z X identified through the graph state along which corre-
Z X Y Z Z Z X Z Z X Z lation space qubits can propagate and interact, again
enabling universal quantum computation. Essentially,
(a) X Y Y Y Z (b) X Y (c) X Y
both approaches use a stochastic graph state as a re-
source. Whether this is possible depends on the stochas-
FIG. 1: Illustration of how the graph state is encoded on
tic graph states having certain desirable properties. We
the post-filtering AKLT state. In (a) we illustrate a small
will show how the same approach can be applied to de-
2DAKLTstateonatrivalentlattice,whereeachnodecorre-
spondstoaspin-3/2particle,andislabelledaccordingtothe formed AKLT states.
filter outcome obtained. In (b), we illustrate the graph ob-
tainedbyprocessing(a)suchthatnodesrepresentdomainsof
like outcomes, and edges are bonds between domains. In (c) B. Generalized reduction scheme
we have illustrated the encoded graph state, obtained from
(b) by deleting edges modulo 2. Each node represents an
Here we generalize the above method to show how de-
encoded qubit, and each edge a graph state edge.
formed 2D AKLT states can be reduced to stochastic
graph states using a modified version of the {F ,F ,F }
x y z
measurement. For a≥1 (we will consider the a<1 case
three-outcomefilteringmeasurementisperformedonev-
in section IV 3) we define three measurement operators
eryparticle. Define|m(cid:105) tobethespin-3/2statesatisfy-
b as
ing S |m(cid:105) =m|m(cid:105) where S is the spin-3/2 component
b b b b
along the b axis, b ∈ {x,y,z}, and m ∈ {32,12,−12,−32}. (cid:115)4(cid:18) a2 (cid:19)
The measurement operators for the initial filtering are F (a)= D(a)†F D(a),
x 3 1+a2 x
chosen to be {F ,F ,F } where
x y z
(cid:115)
F =(cid:114)2(cid:0)|3(cid:105) (cid:104)3|+|−3(cid:105) (cid:104)−3|(cid:1) . (6) Fy(a)= 43(cid:18)1+a2a2(cid:19)D(a)†FyD(a),
b 3 2 b 2 2 b 2
(cid:114)
(a2−1)
These operators satisfy the completion relation Fz(a)=a D(a)†FzD(a). (7)
6
F†F + F†F + F†F = I and thus form a valid
x x y y z z
set of measurement operators, i.e., {E ,E ,E } := Numerical prefactors are included to ensure that
x y z
{F†F ,F†F ,F†F }isaPOVM[29]. Themeasurement, F (a)†F (a)+F (a)†F (a)+F (a)†F (a)=I. Themea-
x x y y z z x x y y z z
applied globally, projects each spin-3/2 system onto a surement operators F (a), F (a), and F (a), like F , F ,
x y z x y
two dimensional, or qubit, subspace. We label each par- F ,areprojectionsontoqubitsubspaces,uptoaconstant
z
ticle either X, Y, or Z according to the outcome of this factor.
measurement. The resulting collection of spin-3/2 par- Thereductionprocedureinvolvesperformingthismea-
ticles encodes a graph state, which can be proven using surement on every particle of the deformed ground state
thestabilizerformalism[14]orbyusingatensornetwork (D(a)−1)⊗N|ψ (cid:105). The resulting state after subject-
AKLT
description [15]. ingeveryparticleofthedeformedAKLTstate|ψ(a)(cid:105)toa
The graph state is encoded as follows (we have illus- POVM measurement {F (a),F (a),F (a)} is equivalent
x y z
trated the encoding in Fig. 1). A domain is defined as tothestateobtainedbymeasuringtheundeformedstate
a connected set of particles with the same label. Each |ψ (cid:105) using the POVM {F ,F ,F } and getting the
AKLT x y z
domain encodes a single qubit in the graph state. An same outcomes, up to local unitaries.
edge exists between two encoded qubits if an odd num- Thus we can apply the existing methods in section
ber of bonds (in the original honeycomb lattice) connect IIIA2toresultingstochasticgraphstatesawayfromthe
the corresponding domains. AKLT point. However the success of these methods de-
We remark that the reduction of 2D AKLT state via pendsonthestochasticgraphstateshavingcertainprop-
a three-outcome POVM to a stochastic graph state is erties. The statistics that determine these properties are
similartothereductionofthetri-clusterstatetoagraph dependent on the value of a, as we will explain in the
state described in [28], however the graphs produced in following section.
the latter are deterministic.
C. Statistical model
2. Using stochastic graphs state as resources
Because each particle is measured with a three-
In the second stage of this method, the stochastic outcome POVM, the total number of possible outcomes
graph state is used for MBQC. Following Ref. [14], arbi- is3N whereN isthenumberofspin-3/2particles. Some
trary quantum computations may be performed by first of these outcomes correspond to computationally useful
converting the post-filter graph state into a cluster state graph states (e.g. if every domain had size one), while
4
D. Identifying computationally powerful ground
states
He√re we will show that, beyond the AKLT point at
a = 3, there is a range of a values that have univer-
sal ground states. The reduction process in section IIIB
produces stochastic graph states with statistics given by
Eq. (9). Forsomefilteroutcomesitispossibletoconvert
the stochastic graph state to a cluster state on a honey-
(a) (b)
comb lattice, which is itself a universal resource [33]. A
FIG. 2: A typical reduction outcome at the AKLT point. In ground state at a given value of a is universal if we can
(a)eachnodecorrespondstoaspin-3/2particle,andedgesare reduce it to a honeycomb cluster state efficiently, i.e., if
drawn between nearest neighbours. The nodes are coloured we can produce honeycomb cluster states of size N from
according to which outcome was obtained: Z outcomes are a ground state with poly(N) particles in poly(N) time.
cyan, X outcomes are magenta and Y outcomes are yellow. There are two conditions that will ensure this is possible
In (b) the resulting graph state is drawn. Each node corre- [14, 22]:
sponds to a qubit (a domain of like measurement outcomes),
and edges correspond to graph state edges. Periodic bound- 1. The maximum domain size scales no faster than
aryconditionsareimposedand,forclarity,someverticesand logarithmically with the lattice size;
edgeshavenotbeendisplayed. Thegraphhasmanycrossings,
making it useful for MBQC. 2. The probability of the stochastic graph state hav-
ing a crossing (a path of edges connecting opposite
boundaries of the graph) tends to one in the limit
some will not (e.g. if every measurement outcome was
of large N.
Z). Let σ =σ ,...,σ be a sequence of filter outcomes
1 N
where σ is the filter outcome on spin i and is either X,
i Condition1ensuresthatproducinggraphstateswithan
Y or Z. At the AKLT point it was shown in [14] that
arbitrary number of qubits is possible. It also rules out
the probability of obtaining a particular σ is
the possibility of an macroscopic domain, which would
1 produce star-shaped graphs states (see Fig. 3c for an ex-
p(σ)= 2|V(σ)|−|E(σ)|, (8)
Z ample that is not universal for MBQC). If condition 1
is satisfied then condition 2 will imply the existence of a
where |V(σ)| is the number of domains for a given out-
extensivenumberofcrossingsinbothlatticedimensions,
come, |E(σ)| is the number of inter-domain bonds be-
whichguaranteestheexistenceofahoneycombsubgraph
fore deleting edges in the reduction to a graph state,
and Z = (cid:80) 2|V(σ(cid:48))|−|E(σ(cid:48))| is a normalisation factor. [16], and hence the universality of the state.
σ(cid:48) We performed Monte Carlo sampling over the distri-
A typical filter outcome, sampled from this distribution,
bution (9) to determine which values of a correspond
is shown in Fig. 2.
to ground states satisfying these two conditions. We
InAppendixBweexplainhowtouseEq.(7)tocompare
performed simulations on lattices of varying size up to
thenorms(henceprobabilities)ofthepost-filterstatesat
√ √ 120×120 spins. Details of the numerical methods used
a(cid:54)= 3tothoseata= 3. Theprobabilityofobtaining
are provided in Appendix C. Samples of resulting graph
a particular filter outcome σ with deformation a is
states are displayed in Fig. 3. We found that maxi-
1 (cid:18)a2−1(cid:19)Nz(σ) mum domain sizes scale logarithmically in the region
p(σ,a)= 2|V(σ)|−|E(σ)|, (9) 1 ≤ a2 < 6.46 while a macroscopic (of extensive size)
Z(a) 2 domain appears at a2 ≥ 6.46. The scaling of maximum
where |V(σ)| and |E(σ)| are as above, N (σ) is the total domain size at selected values of a is presented in Fig. 4
z
numberofZ filteroutcomes. Thesestatisticsareequiva- and the probability of obtaining a macroscopic domain
lent to a Potts-like spin model in the canonical ensemble asafunctionofaispresentedinFig.5forvariouslattice
sizes.
To assess condition 2, we directly checked for the exis-
1
p(σ,a)= e−β(E(σ)−V(σ)−B(a)Nz(σ)), (10) tenceofacrossingoftheresultingstochasticgraphstate.
Z(a)
We found numerically that the resulting graphs have
wheretheE(σ)termisthePottsHamiltonian[30],V(σ) crossings with probability one for lattices of size 20×20
isanon-localclustercountingtermsimilartotherandom andlarger(upto120×120,thelimitofoursimulations),
clustermodel[31,32],B(a)N (σ)isanexternalfieldterm forallvaluesofa. Ourconclusionisthatthereiscompu-
z
with strength B(a) = log (a2−1)−1, and the inverse tationally powerful region that ends at a2 =6.46±0.05,
2
temperature β = log 2 is constant. This shows that the upper limit corresponding to the boundary between
e
varying a to deform the AKLT model is like varying an the AKLT phase and the N´eel phase.
external magnetic field in terms of the statistics of the Due to the presence of a macroscopic domain of fil-
filter outcomes. ter outcomes, our method for MBQC fails for a2 >6.46,
5
1
n
ai0.8
m
o
d
g
nin0.6
n
a
p
s
(a) (b) bility of 0.4 4600xx4600
a
b 80x80
o0.2
Pr 100x100
120x120
AKLT−Neel
0
6 6.2 6.4 6.6 6.8
a2
FIG.5: Probabilityofadomainthatspansthelatticevsafor
variouslatticesizesclosetothecriticalpoint. Theprobability
tends to a step function as N → ∞ with a discontinuity at
(c)
a = 6.46. This shows there is an macroscopic domain above
this point, thus graph states produced above this point are
FIG. 3: The largest connected component of graphs sampled not expected to be computationally useful.
fromEq. (9)at(a)a2 =1,(b)a2 =5.70and(c)a2 =6.96on
a20×20lattice. WeusethesamecoloringasFig. 2and,for
clarity, some edges have not been displayed. Graphs (a) and
howeverwehaven’truledoutthepossibilitythatground
(b)areinthecomputationallyusefulregion, while(c)isnot.
In(a)therearenoZ outcomesandcrossingsaremoresparse states in this region can be used as computational re-
than at the AKLT point. In (b) there are large Z domains, sourcesusinganothermethod. However,theuniversality
which appear as vertices of high degree. In (c) we are in of such ground states is unlikely as ground states above
thesupercriticalregionandthereisaZ domainspanningthe the critical point of a2 = 6.46 are N´eel ordered. While
lattice. ThisresultsinagraphthatisnotuniversalforMBQC states with long range order are usually expected to not
(it has a tree structure and cannot be efficiently reduced to be universal for MBQC, we warn that exceptions have
graph state on a honeycomb lattice).
been found [34].
IV. EXPLORING THE PHASE
7000 In this section we will highlight significant features of
1.00
3.00 the model characterised by particular values of a.
6000 6.21
6.31
6.46
e5000
z y=0.423x+108
n si 1. a2 =3, a2 =∞
mai4000
o
m d At a2 = 3 we have the AKLT state. This point is
u3000
m optimal in the sense that it produces graph states with
Maxi2000 the most qubits. In contrast, as a2 → ∞ the inverse
deformation D(a)−1 tends towards a projection onto the
1000 spa√cespannedby|±23(cid:105)z resultinginaGHZgroundstate
1/ 2(|↑↓↑↓...(cid:105)+(−1)N/2|↓↑↓↑...(cid:105))where|↑(cid:105)=|3(cid:105) and
2 z
00 5000 10000 15000 |↓(cid:105) = |−23(cid:105)z. Any measurement sequence on this state
Number of spins can be simulated efficiently on a classical computer.
FIG. 4: Maximum domain size vs. number of spins in the
ground state for selected values of a. A straight line is fitted
to the a2 =6.46 data points. For values of a2 <6.46 domain 2. a2 =1
sizes scale logarithmically.
Note that F (1) = 0 and therefore the filtering mea-
z
surement in Eq. (7) at a2 = 1 has only two outcomes,
6
F (1) and F (1). We define the orthonormal basis
x y
|0(cid:105):= 1(cid:0)|3(cid:105) +|−1(cid:105) +|1(cid:105) +|−3(cid:105) (cid:1) ,
2 2 z 2 z 2 z 2 z
|1(cid:105):= 1(cid:0)|3(cid:105) −|−1(cid:105) +|1(cid:105) −|−3(cid:105) (cid:1) ,
2 2 z 2 z 2 z 2 z
|2(cid:105):= 1(cid:0)|3(cid:105) +i|−1(cid:105) −|1(cid:105) −i|−3(cid:105) (cid:1) ,
2 2 z 2 z 2 z 2 z
FIG. 6: Illustration of PEPS ground states. Black dots rep-
|3(cid:105):= 1(cid:0)|3(cid:105) −i|−1(cid:105) −|1(cid:105) +i|−3(cid:105) (cid:1) . (11) resentvirtualspin-1/2particles,linesconnectingthemrepre-
2 2 z 2 z 2 z 2 z sent singlet bonds and large circles are locations of physical
particles.
Then we can write F (1) = |0(cid:105)(cid:104)0|+|1(cid:105)(cid:104)1| and F (1) =
x y
|2(cid:105)(cid:104)2| + |3(cid:105)(cid:104)3|, which are projections onto orthogonal
spaces. Hence the a2 =1 ground state is special in that
it is an open question whether or not this quantum com-
it requires only projective measurements to be universal
putational phase is gapped, it is known that the ground
for MBQC.
state for 0 < a < ∞ with periodic boundaries (or open
boundaries with Heisenberg interactions with qubits on
the boundaries) is unique [21]. Any size dependent gap
3. a2 <1
in the disordered phase would be expected to scale at
worst as an inverse polynomial in system size N.
ThefilteringmeasurementinEq.(7)isnotwell-defined
Apracticallimitationofthemethodisthatitdepends
for a2 <1. Here we will provide a casual analysis of how
onpreciseknowledgeoftheparametera. Performingthe
states within region may be useful. For a2 <1 we define
procedure with an assumed value of a that differs from
a new measurement with the operators
thatoftheactualgroundstatewillyieldaresourcestate
that differs from the cluster state. The effect will be
{aF (a),aF (a),E(a)} (12)
x y that X and Y outcomes cause errors in the correlation
√ √
spaceinwhichthecomputationtakesplace(Z outcomes,
where E(a) := diag(0, 1−a2, 1−a2,0). The F (a)
x
however,areerrorfree). Itisnotevenclearthattheseer-
and F (a) outcomes produce graph state qubits as be-
y
rors can be corrected using standard techniques, as they
fore, howeverE(a)outcomesmustbetreatedseparately.
Whena2 isveryslightlylessthan1,thestatewillbelike may not correspond to linear completely-positive trace-
the a2 =1 state except for a few isolated E(a) sites. At preserving maps on the correlation space [35]. Whether
there exists a method that is independent of the exact
anE(a)sitewecanmeasuresurroundingX andY qubits
value of the deformation, analogous to [20], remains to
in a disentangling basis (corresponding to a Z cluster
be seen. Another question is if other deformations to
state measurement), effectively disentangling E(a) sites
the 2D AKLT model (e.g. ones that preserve full rota-
fromtheothers. However,aswedecreaseatowardszero,
tionalsymmetry)yieldcomputationallypowerfulground
the number of E(a) outcomes increases, and eventually
states.
we cannot cut them out of the lattice without adversely
affectingtheconnectivityofthegraph. Hencewepredict
a critical value of 0 < a < 1 below which this measure-
VI. ACKNOWLEDGEMENTS
ment produces states that are not universal for MBQC.
We leave a detailed analysis of the 0 < a < 1 region to
future investigation. We thank Akimasa Miyake for helpful comments and
Andrew Darmawan thanks Tzu-Chieh Wei for helpful
discussions. ThisresearchwassupportedbytheARCvia
V. CONCLUSION theCentreofExcellenceinEngineeredQuantumSystems
(EQuS), project number CE110001013.
Insummary,wehavestudiedthecomputationalpower
of a spin-3/2 AKLT phase that preserves U(1) and Z
2
Appendix A: Ground states as PEPS
symmetries [21]. By mapping measurement outcomes
to a classical spin model we identified three regions: a
region with ground states that are universal resources The ground states in Eq. (5) can b√e written as PEPS.
(1≤a2 <6.46), a region that is unlikely to be computa- We place a singlet state |ψ−(cid:105) = 1/ 2(|01(cid:105) − |10(cid:105)) on
tionally powerful (a2 ≥ 6.46), and a region that we can- eachedgeofthehoneycomblattice,where|0(cid:105)and|1(cid:105)are
not say much about (0 < a2 < 1). Significant points in- virtual spin-1/2 states. This places three virtual spin-
cludethe2DAKLTstate(a2 =3),astatewhichrequires 1/2 particles at each vertex, where a vertex corresponds
only projective measurements (a2 = 1), a GHZ state to the location of a single physical spin-3/2 particle, as
(a2 =∞)andthephasetransition(a2 =6.46)whichcor- illustrated in Fig. 6. We obtain the physical ground
responds to a transition in computational power. While state by applying the map D(a)Υ to the three spin-1/2
7
state label state on A sites state on B sites where the second term is the probability ra-
tio at the AKLT point, shown in [22] to be
|z↑(cid:105) |m=3/2(cid:105) |m=−3/2(cid:105)
2|V(σ)|−|E(σ)|−|V(σ(cid:48))|+|E(σ(cid:48))|. Thus we have
|z∧(cid:105) −|m=1/2(cid:105) |m=−1/2(cid:105)
|z∨(cid:105) |m=−1/2(cid:105) |m=1/2(cid:105)
|z↓(cid:105) −|m=−3/2(cid:105) |m=3/2(cid:105)
p(σ,a) (cid:18)a2−1(cid:19)Nz(σ)−Nz(σ(cid:48))
TABLE I: New basis state labels for convenience. The hon- =
p(σ(cid:48),a) 2
eycomb lattices is bipartitioned into A and B sites, where A
sites have a bond pointing down, and B sites have a bond ×2|V(σ)|−|E(σ)|−|V(σ(cid:48))|+|E(σ(cid:48))|,
pointing up.
particlesateachsitewhereΥistheprojectionontospin- which is equivalent to Eq. (9).
3/2. Hence the ground state can be written as
(cid:79) (cid:79)
|ψ(a)(cid:105)∝ (D(a)Υ) |ψ−(cid:105) , (A1)
v e
v∈V e∈E
whichmeansthatsingletsareplacedoneveryedgeofthe
honeycomblatticeE andtheprojectionsD(a)Υmapthe
Appendix C: Monte Carlo sampling
threevirtualspin-1/2particlesateachvertextophysical
spin-3/2 particles.
TosimplifythePEPStensors,wedefineanewspin-3/2
We sampled the distribution in Eq. (9) using the
basisinTableI.Thisgivesthegroundstatesthedefining
Metropolis-Hastings algorithm with single-spin flip dy-
three-index tensors
namics, as was done by Wei et al. [14]. We used essen-
tially the same procedure as [14], however some changes
A[z↑]=|0(cid:105) |0(cid:105) |0(cid:105) , (A2)
u/d l r were made to work with values of a2 (cid:54)= 3. For one, we
(cid:0)
A[z∧]=1/a |1(cid:105)u/d|0(cid:105)l|0(cid:105)r used (9) to obtain a generalised a-dependent Metropolis
+|0(cid:105) |1(cid:105) |0(cid:105) +|1(cid:105) |0(cid:105) |0(cid:105) (cid:1), (A3) ratio,
u/d l r u/d l r
(cid:0)
A[z∨]=1/a |1(cid:105) |1(cid:105) |0(cid:105)
u/d l r
(cid:1)
+|0(cid:105) |1(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) |1(cid:105) , (A4)
u/d l r u/d l r (cid:18)a2−1(cid:19)Nz(σ(cid:48))−Nz(σ)
A[z↓]=|1(cid:105)u/d|1(cid:105)l|1(cid:105)r. (A5) r =
2
×2|V(σ(cid:48))|−|E(σ(cid:48))|−|V(σ)|+|E(σ)|
Appendix B: Distribution of measurement outcomes
We obtain the probability distribution in Eq. (9) by
where σ is a filter configuration in the Markov chain,
calculating the ratio
andσ(cid:48) istheproposednextfilterconfiguration(obtained
p(σ,a) (cid:104)ψ(a)|{F (a)}|ψ(a)(cid:105) by flipping a single spin in σ). We also generalised the
p(σ(cid:48),a) = (cid:104)ψ(a)|{Fσσ(cid:48)(a)}|ψ(a)(cid:105), (B1) starting filter configuration to depend on a, to reduce
burn-in time. This initial configuration was obtained by
where σ and σ(cid:48) are two filter outcomes, and {F (a)} = assigning a label (X,Y or Z) independently to each spin
σ
F† (a)F (a)⊗···⊗F† (a)F (a). Thea-dependenceof with probabilities
σ1 σ1 σN σN
the probability ratio is contained in the numerical pref-
actorsofEq.(7),andthenormsofD(a)|±3(cid:105) . Using
2 x,y,z
this we can rewrite Eq. (B1), with the a-dependence as (cid:12) (cid:12)
(cid:12)a2 − 1(cid:12)
a separate factor (cid:12) 4 4(cid:12)
p = , (C1)
z 1+(cid:12)(cid:12)a2 − 1(cid:12)(cid:12)
p(σ,a) (cid:18)a2−1(cid:19)Nz(σ)−Nz(σ(cid:48)) 4 4
= px =py =(1−pz)/2, (C2)
p(σ(cid:48),a) 2
√ √ √
(cid:104)ψ( 3)|{F ( 3)}|ψ( 3)(cid:105)
× √ σ √ √ , (B2)
(cid:104)ψ( 3)|{Fσ(cid:48)( 3)}|ψ( 3)(cid:105) where pb is the probability of assigning the label b. This
√
(cid:18)a2−1(cid:19)Nz(σ)−Nz(σ(cid:48)) p(σ, 3) istheprobabilitydistributionobtainedbyneglectingcor-
= √ , (B3) relationsbetweenfilteroutcomes(the2|V(σ)|+|E(σ)| term
2 p(σ(cid:48), 3)
in Eq. (9)).
8
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