Topological cluster state quantum computing

a r X i v :0805.3202v 1 [q u a n t -p h ] 21 M a y 2008Topological cluster state quantum computingAustin G.Fowler 1and Kovid Goyal 21Institute for Quantum Computing,University of Waterloo,Waterloo,ON,N2L 3G1,CANADA and 2Institute for Quantum Information,California Institute of Technology,Pasadena,CA 91125,USA(Dated:May 21,2008)The quantum computing scheme described in [1,2],when viewed as a cluster state computation,features a 3-D cluster state,novel adjustable strength error correction capable of correcting general errors through the correction of Z errors only,a threshold error rate approaching 1%and low overhead arbitrarily long-range logical gates.In this work,we review the scheme in detail framing discussion solely in terms of the required 3-D cluster state and its stabilizers.I.INTRODUCTIONClassical computers manipulate bits that can be ex-clusively 0or 1.Quantum computers manipulate quan-tum bits (qubits)that can be placed in arbitrary su-perpositions α|0 +β|1 and entangled with one another (|00 +|11 )/√FIG.1:A3-D18qubit cluster state.This cell is tiled in3-D. mal error chains that begin and end on primal bound-aries are thus undetectable and have the potential to cause logical errors as discussed in Section III.Fig.3 contains examples of primal error chains connected to primal boundaries.Fig.3also contains dual boundaries —lattice boundaries that pass through the centers of primal cells.A primal error chain connected to a dual boundary is always detectable as it changes the parity of the boundary cell containing the endpoint.Fig.4shows eight primal cells and the location of one complete dual cell.Dual cells are centered on the corners of primal cells.Given an unlabeled lattice,the choice of which cells to call primal and which to call dual is completely arbitrary.Dual cells are used in an identical manner to primal cells,meaning they also detect the pres-ence of Z or M X errors on their face qubits.Fig.5shows a dual error chain starting and ending on dual bound-aries.In an analogous manner to primal error chains, the parity of the dual boundary cells containing the chain endpoints remains unchanged.Discussion of when such undetectable errors are dangerous will again be deferred until Section III.Primal and dual error correction occur independent of one another.At this stage it may seem strange that both appear to only focus on Z and M X errors.An X error that occurs just before an M X measurement has no effect on the measurement result or the underlying cluster state after the measurement.An X error that occurs during the preparation of the cluster state,as shown in Fig.6, is equivalent to one or more Z errors on the neighboring qubits as well as an X error just before measurement.As before,we can ignore the X error,and the error correc-tion scheme deals with Z errors.We are now in a position to describe how correction proceeds.Note that only the classical measurement re-sults will be corrected,not any remaining unmeasured qubits.Without loss of generality,let us focus on primal errors.The procedure for correcting dual errors is anal-ogous.Suppose we have a connected lattice of(primal)cells with both primal and dual boundaries.Identify one dimension of the lattice as simulated time.Suppose we measure all qubits in the lattice up to some given sim-ulated time t in the X basis and classically determine which cells in the measured region have odd parity.We need an algorithm to match odd parity cells with each other and with primal boundaries such that there is a high probability the matching corresponds to the errors that caused the odd parity cells.The algorithm we will use is called the minimum weight matching algorithm [15].The minimum weight matching algorithm takes a weighted graph with an even number of vertices and pro-duces a spanning list of disjoint edges with the property that no other list has lower total weight.The cells with odd parity become half the vertices we will feed into the algorithm.For every vertex in this list we add a vertex corresponding to the nearest point on the nearest primal boundary.We make an almost complete graph of these vertices according to the following rules:all boundary vertices are connected to all other boundary vertices with edge weight zero,odd parity cell vertices are connected to all odd parity cell vertices with edge weight equal to the sum of the absolute value of the differences of their three coordinates measured in cells,and odd parity cell vertices are connected to their nearest boundary vertex with edge weight equal to the number of cells that need to be passed through to reach the boundary plus one. When this graph is processed by the minimum weight matching algorithm,the resulting edge list is highly likely to enable correction of the odd parity cells in a manner that does not introduce logical errors.The classical mea-surement results along an arbitrary path connecting the relevant pairs of cells are bit-flipped resulting in all mea-sured cells having even parity.Note that in a large com-putation such corrective bit-flips would only be applied between pairs of vertices such that at least one vertex of the pair is located at a time earlier than some t−t c where t c depends on the size of the computation.This is to ensure that odd parity cells close to t have a chance to be matched with appropriate partner cell,which may not yet have been measured.III.LOGICAL INITIALIZATION ANDMEASUREMENTLogical qubits can be introduced into the3-D cluster state by making appropriate Z measurements.A con-nected region of the3-D cluster state measured in the Z basis is called a defect.Defects must have a boundary of a single type.Fig.7a shows an example of a minimum volume primal defect.Fig.7b shows a slightly larger example intended to emphasize that all qubits within the boundary of the defect are measured in the Z ba-sis.Note that in the absence of errors the parity of X measurements on all face qubits of a closed boundary is even.This last observation implies that the error correc-FIG.2:A cluster state comprised of64cells of the form shown in Fig.1.Three different Z or M X error chains of length1,2 and3are indicated by thick lines and enlarged qubits.Cells with odd parity are indicated with thick bounding lines.Most of the qubits in the cluster state have not been drawn for clarity.tion scheme can also cope in a natural manner with lostqubits—the parity of any closed boundary around any lost qubits can be checked instead of the individual cells.A primal qubit is a pair of primal defects typically arranged parallel to one another in the direction of sim-ulated time as shown in Fig.8.Note that we define a single time step to correspond to the measurement of asingle layer of the cluster state.We define a primal qubit to be in the|+L state if in a single even time step it is in the simultaneous+1eigenstate of each of the two operators consisting of a ring of single qubit Z operatorsencircling and on the boundary of each defect.Similarly, the simultaneous−1eigenstate of these two boundary operators is defined to be|−L .There is some redundancy in the way we have defined|+L and|−L .It would have been sufficient to focus on a single ring of Z operators around a single defect.Indeed, applying both of these Z rings simultaneously is the log-ical identity operation,with X L being just one of these rings,although it does not matter which ring.For later convenience,when we do not wish to specify which ring, we will use the notation X L.When we need to discuss exactly what operators are being applied,we will write either X1or X2.Furthermore,we choose the logical op-erators to lie on the boundary of the defect purely for ease of definition.While we must choose specific logical oper-ators,not a class of topologically equivalent operators as in general different operators have different eigenvalues since the system does not have any stabilizers consisting solely of Z terms,we could certainly have chosen different specific operators.Primal qubits can be initialized to|+L up to byprod-uct operators via a measurement pattern similar to that shown in Fig.9.Measuring the indicated qubits in the X basis leaves the defects in either the+1or−1eigenstate of X1and X2depending on the parity of the associated X measurements.If we denote the parity(or sum)of the X measurements associated with X1by s1,the state of the logical qubit after initialization will be Z s11Z s22|+L , with Z L=Z1Z2and{X1,Z1}={X2,Z2}=0.The operators Z1and Z2,while not physical unless at least one additional primal boundary is present in the system, are useful for keeping track of byproduct operators af-fecting a single defect.If an additional primal boundary is present,these operators can be represented by chains of Z starting on each defect and ending on this additionalFIG.3:A cluster state with both primal boundaries and dual boundaries,which consist of primal cells cut in half.Examples of the observable parity effects of primal error chains connected to the two types of boundaries are included with odd parity cells indicated by thick boundinglines.FIG.4:Location of a dual cell(shaded)relative to its sur-rounding primal cells.A dualcell contains exactly the ar-rangement of qubits shown in Fig.1.boundary.Note that in the absence of errors all surfaces of X measurements bounded by either X1or X2will have the same parity,as the X stabilizer associated with the six faces of a single cell can be used to arbitrarily deform FIG.5:An undetectable dual error chain connecting two dual boundaries.a surface without changing its parity.This implies that the initialization procedure is fault-tolerant when used in conjunction with the error correction described in Sec-tion II.Primal qubits can also be initialized to|0L up to byproduct operators via a measurement pattern similar5+++++FIG.6:Quantum circuit showing how X errors occurring at any point during the preparation of the cluster state are equivalent to potentially multiple Z errors and an X error just before measurement in the X basis,which can be ignored. to that shown in Fig.10.Note that Z L must be cho-sen.It must be a chain of Z in(preferably just one)odd time slice connecting two sections of defect,but there is still considerable freedom within this definition.Further details of the significance of this choice will be given in Section V.The parity s of the X and Z measurements in time slices earlier than the chosen logical operator de-termines the byproduct operator X s L.As drawn,Fig.10is not fault-tolerant.The defect is too narrow to provide any information about errors on the internal qubits measured in the Z basis.Fig.11 shows a larger defect and examples of odd parityfive sided dual cells resulting from X or M Z errors on qubits inside the defect.In addition to demonstrating that primal qubit initial-ization to|0L can be made fault-tolerant,Fig.11shows how appropriate error information is extracted on the surface of a primal defect to permit dual error correction to continue and vice versa.Furthermore,note that Z measurements deeper inside the defect than the outer-most layer are not used in any part of the computation or error correction procedure and as such their results can be discarded.Dual qubit initialization,expressed in terms of dual cells,looks absolutely identical to primal qubit initial-ization.The only difference lies in the interpretation of what the logical operators mean.Because dual cells can be thought of as primal cells one time step offset,a dual measurement pattern of the form shown in Fig.9initial-izes the dual qubit to|0L .Similarly,a dual measurement pattern of the form shown in Fig.10initializes the dual qubit to|+L .The definitions of all X and Z logical and byproduct operators are also reversed.With logical qubits and logical operators now defined, we can now discuss logical errors.In Fig.9,any chain of primal Z and M X errors connecting the two defects is undetectable and changes the state of the logical qubit from|+L to|−L .To make this unlikely,defects must be kept well separated.In Fig.10,any ring of dual Z and M X errors encircling one of the defects is undetectable and changes the state of the logical qubit from|0L to |1L .To make this unlikely,defects must have a suffi-ciently large perimeter.The situation is similar for dual qubits,with the meaning of the two types of logical errors interchanged.Now that we have initialization,logical measurement follows in a straightforward manner.Figs.9–10,reversed in time can be used to measure the logical operators of the qubit.The parity of the measurement results deter-mines the sign of the eigenvalue of the logical operator. Before concluding this section,a few words are in order to justify the choice of logical operators X L and Z L as atfirst glance they appear to commute.Fig.12contains the stabilizer manipulations necessary to show that in fact Z L X L=−X L Z L.Note the importance of both real and simulated time ordering.IV.STATE INJECTIONWe have discussed logical qubit initialization to states |+L and|0L ,measurement in the X L and Z L bases, logical errors and3-D cluster state error correction.We now turn our attention to state injection,specifically the preparation of logical states|0L +e iθ|1L .We assume familiarity with Theorem1of[6]which we state below without proof.Theorem1Let|φ be a cluster state as shown in Fig.13.Sup-pose that after measuring the middle qubits in some set of bases the n input and output qubits,now collectively in state|ψ ,obey the2n eigenvalue equations(X i)I(UX i U†)O|ψ =(−1)λx,i|ψ(Z i)I(UZ i U†)O|ψ =(−1)λz,i|ψ (1) where U is an n qubit unitary operator,(X i)I is a bit-flip of the i th input qubit,(UX i U†)O is an n qubit unitary op-erator acting on the output qubits,λx,i,λz,i∈{0,1}and 1≤i≤n.Then we can perform computation by start-ing with n qubits in an arbitrary state|ψin instead of the usual cluster state input qubits,creating the remainder of the cluster state as usual and then measuring the middle qubits in the set of bases described above and the input qubits in the X basis.If the results of the X measure-ments are{s i},the output qubits|ψout will then be in the state|ψout =UUΣ|ψin (2) where UΣis a byproduct operator given byUΣ=ni=1Xλz,iiZλx,i+s ii.(3)a.)b.)FIG.7:a.)Primal defect of minimum size created via a single Z measurement. b.)All qubits inside the boundary of larger defects are measured in the Z basis.FIG.8:A primal qubit consisting of two primal defects with X L=X1=X2and Z L indicated.Consider Fig.14.This is very similar to Fig.10but with a single M Z measurement removed(enlarged qubit). We define this single qubit to be our input cluster using the language of Theorem1from[6].Using the language of[2],this qubit is called an S-qubit.Fig.14also shows an output cluster.Qubits inside the two defects,which are measured in the Z basis,and all other qubits,which are measured in the X basis,form the middle cluster. After measuring the middle cluster,we are left with two classes of stabilizers linking either single qubit X on the input cluster X(I)with one of the rings of X operatorson the output cluster X(O)1,X(O)2or single qubit Z on the input cluster Z(I)with the chain of Z operators onthe output cluster Z(O)L.Explicitly,after measuring all middle cluster qubits we are left with the eigenvalue equa-tionsX(I)X(O)1|ψ =(−1)λ1|ψX(I)X(O)2|ψ =(−1)λ2|ψ2FIG.9:Initializing a primal qubit to the|+L state.After the indicated X measurements,the two defects are left in known eigenstates of their associated X L operators,X1and X2,which are both rings of single-qubit Zoperators.FIG.10:Initializing a primal qubit to the|0L state.After the indicated Z and X measurements,the U-shaped defect is left in a known eigenstate of the Z L operator,which is a specific chosen chain of single-qubit Z operators.Z(I)Z(O)L|ψ =(−1)λZ|ψ (4) whereλ1,λ2andλZ denote the appropriate sums of middle qubit measurements.Note that these sums are independent of the precise surface to use to connect the input and output clusters in the absence of errors. Note that,in contrast to Section III,the output X L operators are rings of X on an odd time step not rings of Z on an even time step.When organizing a cluster so that Theorem1applies,the X L operators must be such that measuring them gives information about the under-lying logical state at that point in time.While measuring a ring of Z on an even time step would certainly give in-formation about the underlying logical state,such a ring of measurements would not be fault-tolerant and would furthermore not be consistent with the logical gates de-scribed in Section V.With knowledge of the parity of all X measurements in the surface attached to the ring of X operators indicated in Fig.14,measurement of thefinal ring of X operators gives the parity of the X L operator as defined in Fig.9and does so in a manner that is lo-cally fault-tolerant and consistent with the logical gates we will describe.Keeping all of the above in mind,and denoting the result of measuring the S-qubit in the X basis by s,we can use Theorem1from[6]to infer that the measurement pattern of Fig.14maps the state of the S-qubit to a logical state encoded in two defects via the relation|ψ →XλZ L Zλ1+s1Zλ2+s2|ψL .(5) Note that given the dependence on s,and the close prox-imity of the two defects around the S-qubit,Fig.14is not fault-tolerant.If the S-qubit is prepared in an arbitrary stateα|0 +β|1 before being entangled with its neighboring qubits, an arbitrary logical stateα|0L +β|1L can be obtained. In practice,it is likely that the cluster state will be pre-paredfirst,implying that the S-qubit can only be ro-tated in the Z basis as such rotations commute with the controlled-Z operators used to construct the clus-FIG.11:Three examples of X or M Z errors on qubits measured in the Z basis inside a defect.The leftmost examples are errors on thefirst layer of qubits inside the defect which can be both detected and corrected by determining the parity offive sided dual cells touching the error.The rightmost example is sufficiently deep inside the defect that no nontrivial stabilizers intersect it and therefore the error can be ignored.ter state.Rotation before measurement could be re-placed with measurement in a rotated basis.Either way,this would limit the class of injectable states to (|0 +e iθ|1 )/√2via an S-qubit,up to byproduct operators,measuring the rings of X operators shown in Fig.15a will perform an X L measurement,but measuring the chain of X opera-tors and two Z operators shown in Fig.15b will provide no information about the logical state.Conversely,if the primal qubit was initialized to|0L ,measuring the chain of X operators and two Z operators shown in Fig.15will perform a Z L measurement with the rings of X measure-ments again providing no information about the logical state.This fact cannot be ignored if we wish to calculate the byproduct operators.With the above discussion in mind,consider Fig.16a. This shows appropriate input,middle and output clus-ters for logical qubits initialized to|+L or(|0L + e iθ|1L )/√FIG.12:Anticommutation of primal qubit logical operators X L (ring)and Z L (chain).Multiplying Z L by a ring of face stabilizers yields the second stabilizer configuration.Multi-plying by a ring of X operators completes the demonstration of anticommutation.input qubits middle qubits output qubitsFIG.13:Schematic of three disjoint sets of qubits coupled via an arbitrary network of controlled-Z gates.All middle qubits are initialized to |+ .the parity of the two Z measurements and chain of X measurements.Reversing the definitions of X L and Z L once more,we obtain a logical identity gate of the form|ψL →X λZ +s L Z λ11Z λ22|ψL .Note that the only difference between this mapping and that associated with Fig.16a is the calculation of byproduct operators associatedwitha.)b.)FIG.14:a.)Stabilizer connecting single-qubit Z on the input cluster (enlarged qubit)with Z L on the output cluster. b.)An example of a stabilizer connecting single-qubit X on the input cluster with a ring of X operators on the output cluster.a.)b.)FIG.15:a.)Odd time slice through a primal qubit.b.)Even time slice through a primal qubit.Only one of the indicated measurement patterns will ever give information about the state of a given logical qubit.The state the logical qubit was initialized to determines which pattern.Initial states |+L and (|0L +e iθ|1L )/√2FIG.16: a.)Appropriate input and output clusters and logical operator definitions for a logical qubit initialized to|+L or (|0L +e iθ|1L )/√2.The definitions required to discuss logical CNOT further are shown in Fig.17a.After selecting appropriate input and output clusters based on the state the primal qubit was initialized to and measuring the middle qubits,we are left withZ1d Z1d|ψL =(−1)λZ1d|ψLZ2d Z2d|ψL =(−1)λZ2d|ψLX1p X1p|ψL =(−1)λX1p|ψLX2p X2p|ψL =(−1)λX2p|ψLZ p(Z p⊗Z2d)|ψL =(−1)λZp|ψLX d(X d⊗X1p)|ψL =(−1)λXd|ψL (8) This corresponds to logical CNOT,which we denote by Λdp.Restricting our attention,for the moment,to pri-mal qubits initialized to(|0L +e iθ|1L )/√11s s a.)b.)c.)FIG.17: a.)Logical operator definitions and measurement parities used to discuss logical CNOT.b.)Stabilizer linking X d with X d X 1p that commutes with the X and Z measure-ments on its surface. c.)Similar stabilizer linking Z p with Z 2d Z p .(Z 2d Z p )λX 1p Z λX 1p+λX 2p2p X λZp +s Zp1pZ λXd +s Xd 2dΛdp |ψL(10)Note that calculating byproduct operators depends cru-cially on associating logical operators with specific ringsand chains of physical operators,not topologically equiv-alent classes of rings and chains.Note also that these rings and chains must be chosen such that the output cluster and output logical operators associated with one logical gate become the input cluster and input logical operators of the next logical gate.We do not yet quite have what we need —a logical0L ancilla +Lcontrol intarget inancilla dual a.)b.)M Z M Xcontrol out target outc.)control in control out target intarget out0L ancilla +Lcontrol intarget inancilla dual FIG.18: a.)Circuit comprised of logical gates described in the text that simulates logical CNOT between two primal qubits. b.)Equivalent braiding of defects. c.)Equivalent simplified braiding of defects.CNOT between two primal qubits.Consider Fig.18a [2].This shows how an additional primal and dual an-cilla qubit can be used to simulate logical CNOT between two primal qubits.Essentially,the first CNOT and as-sociated measurement converts the control primal qubit into a dual qubit,the second CNOT performs the neces-sary logical operation and the third CNOT converts the dual qubit back into a primal qubit.Fig.18b shows a braiding of defects equivalent to Fig.18a and a simplified braiding is shown in Fig.18c [2].Note that byproduct operators arising from all three CNOTs as well as the inherent randomness of both logical measurements must be tracked.Note that the discussion of this section does not depend on the direction of simulated time.Provided appropri-ate input and output clusters are tracked along with all measurements,all diagrams in this section can be arbi-trarily deformed,provided the topology is preserved,and arbitrarily oriented with respect to simulated time.VI.CONCLUSIONWe have presented a thorough review of [1,2],dis-cussing how a specific 3-D cluster state can be used to12 perform general error correction despite only detecting Zand M X errors directly and detailing fault-tolerant ini-tialization of|0L and|+L ,Z L and X L measurement,non fault-tolerant preparation of(|0L +e iθ|1L )/√。