Adiabatic Cluster State Quantum Computing

0.11 MB·

by  Dave Bacon

Adiabatic Cluster State Quantum Computing
Dave Bacon1,2 and Steven T. Flammia3
1Department of Computer Science & Engineering, University of Washington, Seattle, WA 98195
2Department of Physics, University of Washington, Seattle, WA 98195
3Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada
(Dated: January 15, 2010)
Modelsofquantumcomputationareimportantbecausetheychangethephysicalrequirementsfor
achievinguniversalquantumcomputation(QC).Forexample,one-wayQCrequiresthepreparation
ofanentangled“cluster”statefollowedbyadaptivemeasurementonthisstate,asetofrequirements
which is different from the standard quantum circuit model. Here we introduce a model based on
one-wayQCbutwithoutmeasurements (exceptforthefinalreadout),insteadusingadiabaticdefor-
0 mation of a Hamiltonian whose initial ground state is the cluster state. This opens the possibility
1
touse thecopious results from one-way QC to build more feasible adiabatic schemes.
0
2

n Computers that can exploit the laws of quantum the- can instead simply adiabatically turn on appropriate lo-
a ory can, in principle, outperform today’s classical com- cal fields while turning off portions of the cluster state
J
puters. For example, quantum computers can efficiently in order to perform the QC. Thus we can dispense with

5
factor [1], something classical computers are thought in- measurements in the one-way model (except, of course,
1
capable of doing. Motivated by this fact, a vast amount forthefinalreadout)andinsteaduseadiabaticevolutions

] of ongoing research focuses on figuring out exactly how to enact one-way QC. This model provides many of the
h to build a quantum computer. In addition to differ- advantages of adiabatic control; in particular it retains
p
ent physical mediums for implementing QC, numerous robustnessto deformations ofthe specific adiabatic path
-
t different models for how to achieve QC have been pro- traversedduring the open-loop holonomic evolution [8].
n
posed. While to date each of these models provides Adiabatic dragging. Themaintoolsweuseinthispa-
a
u the same computational power, they differ substantially per are adiabatic changes in a Hamiltonian. Suppose
q on the requirements they put on the physical hard- initially we have a system with Hamiltonian H and the
i
[
ware. The most widely used model of QC is the quan- system is in an energy eigenstate. Then we evolve the

2 tumcircuitmodel,but othermodelsinclude one-way(or system under a time-varying Hamiltonian over a time
v measurement-based) QC [2], holonomic QC [3], univer- period 0 ≤ t ≤ T as H(t) = f(s)H +g(s)H where
i f
8 sal adiabatic QC [4], and topological QC [5]. Here we f(0) = g(1) = 1, f(1) = g(0) = 0 and s = t is a scaled
9 T
proposeanewmodelofcomputingwhichcombinesideas time. If we vary this evolution smoothly and there are
0
from all of these models. In particular we demonstrate no level crossings, then it is always possible to choose
2
. how one can perform one-way QC adiabatically. a T large enough such that at the end of this evolution
2
1 One-way QC [2] is a method for QC in which one cre- we will be in the eigenstate of Hf which is continuously
9 atesaspecific, fixedentangledstateofaquantummany- connected to the initially preparedeigenstate. In partic-
0 bodysystemandthencomputesviaaseriesoflocalmea- ular, if we choose T to be on the order of the minimum
v: surements on the subsystems. The choice of measure- energygapbetweenthe instantaneouseigenstateofH(t)
i ments correspond to unitary gates enacted in the QC and the nearest eigenstates, then with high probability
X
and these measurements are adaptive: that is, the ex- at the end of the above evolution the system will be in
r act measurement being executed depends on the previ- theconnectedeigenstateofthefinalHamiltonian[9]. We
a
ous measurement results. One set of states which can willcallsuchasetupandevolutionanadiabatic dragging.
be used for one-way QC are the class of so-called clus- Recently, adiabatic draggingbetween Hamiltonians with
ter states [6]. Cluster states are defined for any graph, energy eigenstates that are degenerate and are quantum
though not all graphs allow for universal one-way QC. error-correcting codeword states has emerged as a pow-
A cluster state can be defined as a stabilizer code state. erfulprimitiveforbuilding aquantumcomputer[10–12].
Equivalently,thereisaHamiltonianwithatmost(d+1)- Here we extend these ideas to one-way QC.
qubitinteractions,wheredisthemaximumdegreeofthe 1D degenerate cluster-state model. Begin by consid-
graph, whose ground state is the cluster state (one can eringaline ofnqubitsandadegeneratevariationonthe
replace this Hamiltonian with another involving only 2- one-dimensional cluster state. In particular define the
qubit interactions while retaining the cluster state as an following n−1 commuting operators
approximate ground state [7].) Thus one could imag-
S =[Z] [X] [Z] ,1≤i≤n−2,S =[Z] [X] ,
i i i+1 i+2 n−1 n−1 n
ine engineeringa physicalsystemwith this Hamiltonian,
cooling the system to its ground state, and then doing where X and Z are the corresponding Pauli operators
measurements that enact the cluster state QC. Here we and we use the notation [P] to denote the operator P
i
showthatinsteadofperformingthesemeasurementsone acting on the ith physical qubit. These are n−1 out of
2
the n operatorsusually used to define a cluster state [6]. logical Z is [Z] ) we see that a Hadamard gate has been
i
Definenowthestabilizercodecorrespondingtotheseop- applied to this information. Thus, by turning on a [X]
1
eratorsasthecommon+1eigenstatesofalloftheS ,i.e. term on the first qubit while turning off the term in the
i
|ψi such that S |ψi = |ψi. By standard results in the Hamiltonianwithwhichitanti-commuted,wehaveeffec-
i
theory of stabilizer codes [13], this code space is two di- tivelymovedthisinformationonestepdowntheline,and
mensional (encodes a qubit.) We can define the logical applied a Hadamard gate to the quantum information.
operators for this encoded qubit as Proceedinginductively,ifwefirstadiabaticallyturnon
[X] , then [X] , etc, while turning off the corresponding
X¯ =[X] [Z] and Z¯ =[Z] . (1) 1 2
1 2 1 anticommuting term in the original Hamiltonian we will
end up with the qubit which was originally localized to
Now consider the Hamiltonian
one end of the line moved to the other end of the line,
n−1 alongwitha sequenceofHamadardgatesappliedto this
H0 =−∆XSi. (2) qubit. Throughout this piecewise evolution the energy
i=1 gap will remain constant because each successive adia-
Since the S all commute and have eigenvalues ±1, the batic draggingacts independently. If we proceedto turn
i
groundstatesubspaceofthisHamiltonianisthe+1com- on each [X]i all the way up to the (n−1)st qubit, the
moneigenstateoftheS ’sor,inotherwords,theencoded information originally encoded into the first two qubits
i
qubit defined above. Note that quantum information in will end up exactly on the last qubit. In other words af-
the degenerate ground state can be accessed by measur- terthisevolution,X¯ ismappedto[X]n andZ¯ ismapped
ing or manipulating the encoded Pauli operators which to [Z]n if the chain is odd length and X¯ is mapped to
are themselves localized on the first two qubits. [Z]n and Z¯ is mapped to [X]n otherwise — these dif-
Nowsupposethatweadiabaticallyturnonalocalfield ferences arising from whether an even or odd number of
along the −[X] direction while turning off the S term Hadamards have been applied to the encoded qubit.
1 1
inH ,whichanticommuteswith[X] . Inparticularcon- Single qubit gates. We now show how to modify the
0 1
sider adiabatic dragging from H to H +∆(S −[X] ). abovesetupsuchthatinadditiontopropagatingasingle
0 0 1 1
Notice that while X¯ commutes with [X] , Z¯ does not qubit of information down the one dimensional system,
1
commute with [X] . However because we are in the +1 wealsoapplygatesotherthantheHadamardgatetothe
1
eigenspace of each S , instead of defining the logical Z¯ qubit. This scheme is motivated directly by the one-way
i
as we have done above in Eq. (1) we could also define QCmodelwhereinsteadofmeasuringthequbitalongthe
the encoded Z as Z¯′ = Z¯S = [X] [Z] . If we do this, X direction to propagate the information, we measure
1 2 3
then the encodedqubit commutes with the terms we are along a rotated direction, M(θ) = cos(θ)X + sin(θ)Y.
turning on and off (S and [X] .) Thus the quantum in- Importantly,however,ourschemeproceedswithoutadap-
1 1
formationinthisencodedsubspaceisnottouched. How- tive operations. Consider mimicking the above scheme,
ever since S1 anticommutes with [X]1, the information butinsteadofturningonsuccessive−∆[X]iswhileturn-
in S1 is changed. To see how this evolutionproceeds, we ing off the appropriate anticommuting terms in H0 (the
can consider a code in which we promote S1 into an en- −∆[Z]i[X]i+1[Z]i+2 terms)weinsteadturnonsuccessive
codedPauliZ operatorand[X]1 isitsconjugateencoded −∆Mi terms where Mi = [M(−θi)]i is a set of rotated
X operator. The adiabatic evolution is then simply be- localfields,1≤i≤n−1. Weclaimthatthiswilltakethe
tween these the two encoded Pauli operators (i.e. from qubitlocalizedto one endofthe line andpropagateitto
anencoded−∆Z¯ toanencoded−∆X¯ whereadenotes theotherendofthelinewhileapplyingagatedependent
a a
thisnewlydefinedencodedqubit.) Suchanevolutionhas on the choice of θi.
no level crossing and an energy gap for reasonable adia- To analyze this scheme it is easier to work in a frame
batic interpolations which is proportionalto ∆. Thus at of reference in which the ith qubit has been rotated by
the end of this evolution we will be in the +1 eigenstate U(θi)=exp(−iθi[Z]i/2). It is convenientto take θ1 =0,
of [X] along with all the remaining S . In other word whichwe will now assume. Consideragainn qubits ona
1 i
we are in the stabilizer code with stabilizer generators line anddefine nowthe rotated stabilizer code operators:
[X] ,S ,S ,...,S . The information in the degener-
1 2 3 n−1
atesubspace,whichoriginallywasrepresentedviatheen- Ti = [Z]i[XUi+1]i+1[Z]i+2, 1≤i≤n−2,
coded operators X¯ =[X]1[Z]2 and Z¯ =[Z]1 is now rep- Tn−1 = [Z]n−1[X]n, (3)
resented by X¯′ =[X] [Z] and Z¯′ =[X] [Z] . However,
1 2 2 3
sinceweareinthe+1eigenstateof[X] thisisequivalent where we use the superscript to denote conjugation,
1
to the encoded operator X¯′′ = [Z] and Z¯′′ = [X] [Z] . PU = UPU†, and U = U(θ ). Note that this con-
2 2 2 i i
In other words the information which was originally en- jugation does not change the fact that these operators
coded in the first two qubits, after the above adiabatic commute and square to identity, and therefore we can
dragging,willbeinthesecondandthirdqubit. Usingthe again define a codespace as the joint +1 eigenspace of
same logical Pauli encoding (logical X is [X] [Z] and these operators. Let H be the initial Hamiltonian for
i i+1 0
3
our system as in Eq. (2), but now with the rotated the procedure we have described for adiabatically drag-
stabilizer operators T substituted for S . Again, ini- ging the initial Hamiltonian, we are always turning off a
i i
tially we can define the information in the degener- −∆T while turning on a −∆[X] . Then not only does
i i
ate subspace as localized to the first two qubits with [α] commute with these terms (because the Pα,β is
i+1 i+1
X¯ = [X]1[Z]2 and Z¯ = [Z]1. Now imagine adiabati- made up entirely of a product of [X]j’s with j < i+1),
cally dragging H0 to H0+∆(T1−[X]1), then dragging and thus is untouched by the evolution, but by an ar-
to H0+∆(T1+T2−[X]1−[X]2), etc. We claim that at gument identical to the untwisted Hamiltonian case, we
the end of this scheme we will end up with the quantum end each such dragging in the +1 eigenvalue of −∆[X] .
i
informationinX¯ andZ¯propagatedtothelastqubitwith Thus we end up exactly in the subspace where the gate
a gate dependent on θi applied to this information. Ui+1H has been applied and the quantum information
To see this, we proceed in three steps. First we shifted one site down the chain for each such adiabatic
will show that using the rotated stabilizer operators it dragging. Thefinaleffectfortheturningonalln−1[X]
i
is possible to write the logical qubit in a form where in order is that the sequence of gates H 2 (U H)
Qi=n−2 i+1
each Xi (except i = n) commutes with this informa- is applied to the quantum information.
tion. Define the followingoperatorsforα,β ∈{X,Y,Z}:
Inrecap,wehaveshownthatbystartingwithaHamil-
α¯i = Pβ(Piα,β[β]i)Ci,i+1 where Ci,i+1 is the controlled tonian which is a negative sum of twisted stabilizer op-
phase gate between the i and (i + 1)st qubits except
erators T and then turning off the T ’s while turning on
i i
when i = n in which case we define C = I. We
n,n+1 the [X] ’s sequentially,we haveenacteda gate whichde-
i
claim that these new Pauli operators are, under the ro-
pends on the angles θ . This is equivalent to using the
i
tated stabilizer code generated by the T ’s, equivalent to
i standardclusterstateHamiltonianfromEq.(2)withthe
the logical operators X¯ = [X] [Z] , Y¯ = [Y] [Z] , and
1 2 1 2 unrotated S stabilizer operators as the initial Hamilto-
Z¯ =[Z] ,withtheconditionthatthePα,β’sareasumof i
1 nian and using rotated magentic fields [M(−θ )] for the
i i
products of [X] operatorsfor j <i. This can be proven
j piecewise final Hamiltonians. Note that we did not work
inductively. The base case corresponds to Pα,β = δ I
α,β in a rotating frame for the final qubit and therefore the
where X¯ =X¯ and Z¯ =Z¯. Now assume the hypothesis
1 1 informationendsupexactlyonthelastqubitofthisevo-
is true for the ith operators. Examine, for example, X¯
i lution. Throughout this piecewise evolution the energy
and expand the controlled-phase:
gapis constant(independent of the length of the chain.)
The gates enacted are universal for single qubit gates.
X¯ =PX,X[X] [Z] +PX,Y[Y] [Z] +PX,Z[Z] . (4)
i i i i+1 i i i+1 i i
State preparation. Inthe previoussectionweenacted
Recall that the T operators act as identity on the gates on the degenerate ground state of a Hamiltonian.
i
codespace and thus can be inserted into this sum in any Wenowshowhowitispossibletopreparequantuminfor-
manner to yieldany equivalentoperator(overthe code.) mation in a particular state, with the Hamiltonian non-
Left multiplying X¯ by T for the last two terms yields degenerate, and then propagate the information down
i i
the line while turning the Hamiltonian into one with a
X¯i = PiX,X[X]i[Z]i+1+PiX,Z[XUi+1]i+1[Z]i+2 degenerate ground state where this encoded information
−iPiX,Y[X]i[XUi+1]i+1[Z]i+1[Z]i+2 (5) lives. Consider, for example, our original Hamiltonian
in Eq. (2) but now with the full cluster state Hamilto-
Expanding out XUi+1, we find that nian H0′ = H0−∆S0 where S0 = [X]1[Z]2. The ground
state of H′ is now not degenerate and corresponds, in
0
PiX+,1X = cos(θi+1)PiX,Z +sin(θi+1)[X]iPiX,Y our previous picture of H0 to being in the +1 eigen-
PX,Y = sin(θ )PX,Z −cos(θ )[X] PX,Y state of X¯. Consider first adiabatically dragging H0′ to
i+1 i+1 i i+1 i i H′+∆(S −[X] ). Attheendofthisevolutionwewillbe
0 1 1
PiX+,1Z = [X]iPiX,X (6) in the +1 eigenspace of [X]1 as before. Since we started
inthe+1eigenspaceofX¯ wewillbeinthe+1eigenspace
Similar relations hold for Y¯i+1 and Z¯i+1 with the impor- ofX¯′ =[Z]2. NextadiabaticallydragtheHamiltonianto
tant property that the new Piα+,1βs are functions of the H0′+∆(S0+S1+S2−[X]1−[X]2). Notice thatwehave
previous Pα,βs and [X] s. This proves our statement. to turn off two stabilizer generators while turning on a
i i
Buttheseexpressionsalsoprovemuchmore. Inpartic- singlefield. Thisimpliesthatwemustincreasethedegen-
ular if we restrict the above equivalence to the +1 sub- eracy of the ground state. We will see that this second
space of [X] , then we see (when we calculate out all dragging, despite increasing the degeneracy, ends with
i
nine new Piα+,1β’s) that the relationship between the α¯i the system in the +1 eigenstate of the X¯′′ =[X]3[Z]4.
andα¯ is α¯ =α¯Ui+1H. Inother words,with this re- To see this note that while both S and S do not
i+1 i+1 i 1 2
striction,theeffectonthe encodedquantuminformation commute with [X] , S S does. Thus the eigenvalue
2 1 2
in this new form is as if the gate U H has been ap- of S S is preserved while turning on [X] . If we then
i+1 1 2 2
plied to the quantum information. Further note that in rewrite S +S as S (I +S S ), then if we are in the
1 2 1 1 2
4
−1 eigenspace of S S then this term vanishes, but if fields (we note that this initial state can also be piece-
1 2
we are in the +1 eigenspace then in this space the op- wise adiabatically prepared [15].) Consider a quantum
erator effectively acts as 2S (or equivalently 2S ). We circuit made up of gates from a universal gate set such
1 2
can then consider the code where we promote S to an as {HU(π),H,(H ⊗H)C } (other sets are also possi-
1 4 i,j
encoded Z operator and [X] to an encoded X opera- ble) along with the preparation in the +1 eigenstate of
2
tor, and then at the end of the evolution we will be in the Pauli X operator. Then one can map the graph of
the +1 eigenstate of [X] , and we are also in the +1 this circuit onto a cluster state graph using the above
2
eigenstateofS S (due to this operatorcommutingwith elements in such a way that one can also prescribe local
1 2
[X] .) Translating this into the coding language, we are fields which, when turned on piecewise, enact the quan-
2
in the +1 eigenstate of a stabilizer code with genera- tum circuit (or equivalently one can us a twisted cluster
tors [X] ,[X] ,[X] [Z] ,S ,...,S , which is equivalent state Hamiltonian and local fields all along X.)
1 2 3 4 3 n
to saying that we are in the +1 eigenstate of the 1D Conclusion. We haveshownhowtoperformone-way
cluster state with n−2 qubits but prepared in the +1 QConaclusterstateusingonlypiecewiseadiabaticevo-
eigenstate of the encoded X¯ at one end of this chain. If lutions. This scheme shares many of the traits of the
we wish to apply gates to this information, we can pro- recently introduced primitive of adiabatic gate telepor-
ceedasaboveby applyingrotatedlocalfields orrotating tation [12]: it has a robustness to the adiabatic path,
thestabilizerHamiltonian. Itisimportanttorealizethat for example. Further, as in [12] we can use perturbation
the above evolution has gone from a non-degenerate to theory gadgets [7] to implement this entire scheme us-
a degenerate ground state, so that the energy gap van- ing only two-qubit interactions instead of the four-qubit
ishes. However over the subspaces defined by the con- interactions we have presented; it would be interesting
servedquantityS1S2 theenergygapisconstantandthus to make this calculation explicit. Our model shows the
the adiabatic theorem holds. In fact, the same situation novelty of starting with a global entangled ground state
occurs in the creation of anyons in topological QC [14]. and then piecewise turning on local fields to do QC. We
Two-qubit gates. Let us now show how to apply two- have also shown how it is possible to use cluster states
qubit gates. The idea, just as in one-way QC, is to use and their parent Hamiltonians to perform QC without
a Hamiltonian which has a coupling between two chains resortingtoadaptivemeasurements. ACSQCthusopens
which support single qubits. To see how this works let up a new way to adapt the numerous results of one-way
usanalyzeaclusterstateHamiltonianwithadegenerate QC to viable adiabatic architectures.
ground subspace and a single coupling between two en- While preparing this manuscript we learned of similar
coded qubits. Consider the six-qubit initial Hamiltonian resultsforsinglequbitcircuitsusingtheAKLTstate[16].
DB was supported by the NSF under grants 0803478,
H =−∆([Z] [X] [Z] [Z] +[Z] [X] )+(a↔b)
2 1,a 2,a 3,a 2,b 2,a 3,a 0829937, and 0916400 and by DARPA under QuEST
grant FA-9550-09-1-0044. STF and research at Perime-
where the encoded qubits will be associated with a and
teraresupportedbytheGovernmentofCanadathrough
b and (a ↔ b) denotes same term with the a and b la-
IndustryCanadaandbytheProvinceofOntariothrough
bels reversed. This Hamiltonian is degenerate, but now
the Ministry of Research & Innovation.
therearetwoqubitsofdegeneracy,correspondingto log-
ical operators X¯ = [X] [Z] and Z¯ = [Z] with
γ 1,γ 2,γ γ 1,γ
γ ∈ {a,b}. Now suppose that we turn on −∆([X] +
1,a [1] P.W.Shor,inProceedingsofthe35thAnnualSymposium
[X] +[X] +[X] )whileturningoffH (wecouldpro-
1,b 2,a 2,b 2 on the Foundations of Computer Science, pp. 124–134
ceed by turning each of these on separately and achieve
(IEEE Computer Society, Los Alamitos, CA, 1994).
similar results.) Using the four stabilizer terms in the [2] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,
Hamiltonian above we can rewrite the encoded opera- 5188 (2001).
tors as X¯′ = [X] [X] and Z¯′ = [X] [Z] [X] [3] P.ZanardiandM.Rasetti,Phys.Lett.A264,94(1999).
γ 1,γ 3,γ γ 2,γ 3,γ 3,¬γ
where ¬a = b and ¬b = a. Using an argument simi- [4] D. Aharonov et al., in 45th Annual Symposium on the
Foundations of Computer Science, pp. 42–51 (IEEE
lar to the single-qubit gates, we will end up in the +1
Computer Society, Los Alamitos, CA, USA,2004).
eigenstate of the Xi,γ operators, i ∈ {1,2}. Over this [5] A. Kitaev, Ann.of Phys. 303, 2 (2003).
eigenspace, the logical operators become X¯f = [X]3,γ [6] M. A.Nielsen, Rep.on Math. Phys.57, 147 (2006).
and Z¯f = [Z]3,γ[X]3,γ¯. This is equivalent to performing [7] S.D.BartlettandT.Rudolph,Phys.Rev.A74,040302
a Hadamard on each encoded qubit, a controlled phase (2006).
gatebetweenthesequbits,andthenaHadamardoneach [8] D. Kult, J. ˚Aberg, and E. Sjo¨qvist, Phys. Rev. A 74,
qubit again. 022106 (2006).
[9] G. Schaller, S.Mostame, and R.Schu¨tzhold, Phys. Rev.
Adiabatic Cluster State QC (ACSQC). We now see
A 73, 062307 (2006).
how to build a quantum computer using piecewise adia-
[10] O. Oreshkov, T. A. Brun, and D. A. Lidar, Phys. Rev.
batic evolutions from a Hamiltonian whose ground state Lett. 102, 070502 (2009).
is a cluster state to a Hamiltonian consisting of local [11] O. Oreshkov,Phys. Rev.Lett. 103, 090502 (2009).
5
[12] D. Bacon and S. T. Flammia, Phys. Rev. Lett. 103, preparation (unpublished).
120504 (2009). [15] G. Schaller, Phys. Rev.A 78, 032328 (2008).
[13] D.Gottesman, Ph.D. thesis, California Insituteof Tech- [16] J. Renes, privatecommunication (unpublished).
nology, Pasadena, CA, 1997.
[14] D. Bacon, S. T. Flammia, A. Landahl, and A. Neels, in