Entangling unitary gates on distant qubits with ancilla feedback
EEntangling unitary gates on distant qubits with ancilla feedback
Kerem Halil Shah and Daniel K. L. Oi SUPA Department Of Physics, University of Strathclyde, Glasgow G4 0NG, UK ∗ By using an ancilla qubit as a mediator, two distant qubits can undergo a non-local entanglingunitary operation. This is desirable for when attempting to scale up or distribute quantum compu-tation by combining fixed static local sets of qubits with ballistic mediators. Using a model drivenby measurements on the ancilla, it is possible to generate a maximally entangling CZ gate whileonly having access to a less entangling gate between the pair qubits and the ancilla. However thisresults in a stochastic process of generating control phase rotation gates where the expected timefor success does not correlate with the entangling power of the connection gate. We explore howone can use feedback into the preparation and measurement parameters of the ancilla to speed upthe expected time to generate a CZ gate between a pair of separated qubits and to leverage strongercoupling strengths for faster times. Surprisingly, by choosing an appropriate strategy, control of abinary discrete parameter achieves comparable speed up to full continuous control of all degrees offreedom of the ancilla.
I. INTRODUCTION
The ability to harness quantum phenomena for infor-mation processing purposes underlies quantum compu-tation (QC). There are several different underlying com-putation models, including gate-based[1], measurement-based [2], adiabatic[3] and topological models [4]. Theirinterest in not only due to their suitability to differentphysical substrates for implementation but also on a morefundamental level as to the sets of resources necessary orsufficient for universal QC.Recently a subset of schemes have arisen based aroundthe use of ancilla systems, such as the ancilla-driven [5, 6],ancilla-control [7] and quantum bus [8] proposals, wherelogical operations are generated by interacting the qubitsof the main register with an ancilla system then perform-ing operations on that ancilla system. The various ancillaschemes are distinguished by their differing requirementsof the interaction between register and ancilla, the op-erations on the ancilla and the number of required in-teractions. For example, the ancilla-control scheme forimplementing a single qubit unitary on a register qubitrequires being able to perform that unitary on the an-cilla [7]; the ancilla-driven scheme requires only arbitraryrotations about a single axis, provided the appropriatemeasurement basis is available for measurements on theancilla [5].Ancilla driven quantum computation is particularlysuited to the use of hybrid physical systems [9, 10] wherethere is a memory register optimised for stability and longcoherence times and a short lived but easily manipulatedancilla system such as NV centre nuclear-electron spins.The model of weaker or arbitrary interaction strength issuited for when the interaction is not tunable such aswhen dealing with scattering between flying and staticqubits [11]. A stable memory register may be the prod-uct of a particularly well chosen physical system but also ∗ Electronic address: [email protected] it could be due to the use of nodes of qubits that areegineered to perform error-correction and fault tolerancecodes locally as in several proposals for networked quan-tum computation [12]. A distributed design may also aidin parallelising circuit design for a time speed up or inaiding scability of a physical implementation. [13, 14]In these cases different local nodes of qubits may haveto be connected by a different physical medium, such asa photon or coherent beam system [15]. Therefore it isuseful to consider ancilla schemes in the context of dis-tributed or networked quantum computation where non-local operations are applied over relatively large separa-tions that inhibit coordination.The problem of entangling a physically separated pairhas been considered before with methods such as theBarrett-Kok double heralding approach [16] where entan-gled states are generated through projecting the systemvia photon pair measurements or Lim, Barrett et al [17]’srepeat-until-success method through Bell basis measure-ments. In contrast, the operation on the qubit pair in An-cilla Driven Quantum Computation remains a reversible,commutable unitary gate that also requires only singlequbit measurements and does not require maximum en-tanglement with the ancilla. This means that the processcan be used in frameworks other than the generation ofcluster states for measurement based quantum compu-tation and can use non-maximal ancilla-register interac-tions. However in the latter case, the process for gener-ated an entangling gate becomes stochastic.This paper examines the use of feedback of ancilla mea-surement results into subsequent generations. The appli-cation of control over the ancilla state is also used tospeed up this stochastic process and to make it behaveaccording to a well defined statistical behaviour. Thisoccurs in the broader theme of how we can trade off therequirement of some resources at a cost of an increasedtime to implement specific operations. a r X i v : . [ qu a n t - ph ] N ov FIG. 1: An circuit diagram example of ancilla driven quan-tum computation using the connection interaction ( H A ⊗ H R ) .CZ [5]. The ancilla is prepared in a fixed state, thechoice of measurement basis is represented by the applicationof the unitary gate J ( β ) to the ancilla in order to draw atten-tion to the symmetry between the action on the ancilla andthe register. II. ANCILLA DRIVEN OPERATIONS
In an ancilla driven model, a qubit in a memory reg-ister interacts with an ancilla qubit in a prepared state,then the ancilla is measured and the resulting back-actionon the register is unitary depending on the parameters ofthe preparation, interaction and measurement. If the in-teraction is locally equivalent to e − i π σ z ⊗ σ z (CZ type) or e − i π ( σ x ⊗ σ x + σ y ⊗ σ y ) (CZ.SWAP type) then one can gen-erate an arbitrary rotation angle β on the register qubitby performing a rotation by β n the ancilla before mea-surement. E.g. Using a CZ gate and an ancilla preparedin the | + (cid:105) state, performing a rotation about the ˆ x axis, R ˆ x ( β ), on the ancilla then measuring in the 0/1 basis;this enacts Z j R ˆ z ( β ), where j = 0 , H a ⊗ H r ) . CZ, where we have in-cluded Hadamard local gates as part of the interaction,then by accounting for the extra local effects on the an-cilla by applying J ( β ) = H.e − iβσ z instead before themeasurement, X j .J ( β ) acts on the register. The class J ( β ) allows one to feed forward the measurement resultsto commute through the Pauli correction into a singlepost-correction and to apply any single qubit unitary(up to global phase), U ≡ J (0) J ( α ) J ( β ) J ( γ ) for some α, β, γ [18].Crucially, by applying this same interaction betweenthe ancilla and two subsequent register qubits, a CZ gatecan be generated on the register, up to local gate correc-tions, thus providing the resources for universal quantumcomputation.However if the interaction is instead equivalent to e − iασ z ⊗ σ z or e − i ( α x σ x ⊗ σ x + α y σ y ⊗ σ y ) for 0 < α < π thenthe rotation angle β becomes random in a way withoutsimple post-corrections [19]. This applies also for thetwo qubit entangling gate: a gate locally equivalent toa Control γ rotation, C ( γ ), is generated with random γ depending on the measurement result. FIG. 2: A two qubit gate in the ADQC scheme, using thesame connection interaction twice, enacts a deterministic en-tangling gate with probabilistic local unitary effects. Here,the choice of measurement basis is fixed.FIG. 3: A circuit for generating a two qubit control-unitarygate with an interaction parametrised by an arbitrary cou-pling strength α : ∆ α = e − iασ z ⊗ σ z . Local gate differences onthe ancilla can be accounted for by the setting of U i , U a and U g . Local gate effects on the register can by accounted by ameasurement dependent post-correction V / . V / will com-mute with ∆ α thus several applications can be treated witha single post-correction. It may however be possible to generate a chosen C ( γ )gate if a probabilistic achievement time is allowed. Anygate generated by the use of a connection interaction inthe local equivalence class of e − iασ z ⊗ σ z will also be ableto be diagonalised in the computational basis by localunitary gate operations. If the local operations can bedirectly created, or created by the use of well engineeredqubits and interactions in a local node, then the diago-nalised products of each generation will all commute andthe random behaviour will map to a random walk on acircle. The gates generated would be of the general form diag ( e iφ , e iφ , e iφ , e iφ ) which is locally equivalent to aControl- R ˆ z (Φ) gate where Φ = φ − φ − φ + φ . Theangles are mapped to a point on a circle. So at each timeinterval, an angle is randomly gated to represent the gategeneration and is added to the sum of all previous gategenerations; if the new sum lies with a target region,given by π ± (cid:15) ), for some chosen error (cid:15) , the operationsare halted. FIG. 4: The distribution of the target region hitting timesgiven by the simulation of the gate generation with an in-teraction with α = π . The mean hitting time is 74.1, thestandard deviation is 74.5. III. UNGUIDED BEHAVIOUR OF RANDOMGENERATION
The probabilistic nature of the generation of gates inthis scheme leaves us with the problem of understandingthe statistics of the time it takes to reach any arbitrarygate.We simulated the creation of a CZ equivalent gate byuse of the circuit in figure 3 with the choice of U i = I , U a = R ˆ x ( π ) and U g = H i.e. preparation and mea-surement in the X eigenstate basis. These settings arenot unique to the gates generated. This was performed10,000 times, with an error bound of π/
100 and the re-sulting distribution is displayed in figure 4.The distribution of hitting times, when put into suf-ficiently broad bins, can be described by an exponentialtail. One of the results is an irregular angle that dependson the coupling strength while the other is π for all cou-plings. Aside from the initial probability of success inthe first step by generating the angle π , we expect thatone of the results could be used to reach the target withinthe bound if applied a large number of times scaling withthe error according to the dimensions of the space, in thiscase, linearly. Indeed, we found a linear scaling.A result would be expected at around the time wherethis large number matches of the expected number ofsuccesses of that gate and the number of π results gen-erated is even. The probabilistic nature of this causes adistribution around this point so there is a length of timeassociated with an opportunity for success. This then re-peats in a decaying tails. This probability of success ina fixed time interval ends up accruing the properties of asimilarly described probability distribution, the geomet-ric distribution. This is not an exact description but wewill later find that it allows a comparison to a case withan exact solution.When employing a strategy for feedback, it is neces-sary to have a picture in which the control parameters relate to the ultimate property desired for optimisation:the number of gate generations required. Due to the de-caying tail distribution of hitting times in the case with-out feedback, we expect that an important feature of thehitting time statistic is the minimum number of stepsrequired to create a finite probability of hitting the tar-get. In the following section, we discuss strategies basedaround the principle of optimising the probability of suc-cess in a minimal number of steps. Because there is al-ways a probability of generating CZ with any coupling bypreparation and measurement in the X eigenstate basis,it is possible to perform a “one step” strategy: maximisethe probability to hit the target in the next step. IV. STRATEGIES FOR GUIDED GATEGENERATION
In the ancilla-driven scheme, an ancilla is prepared ina specific basis, undergoes an interaction with one reg-ister qubit using a specific connection interaction, theancilla is then interacted with the second register qubitand then measured in an appropriate basis. The ancillais controlled by the choice of preparation state and mea-surement basis provided that any local unitary actions onthe ancilla in the intermediate time between interactionsaccount for the conditions for a unitary, entangling gener-ated gate on the register. This restriction results in onlytwo degrees of freedom. In the context of a spatially sep-arated pair in a distributed network, the two parameterscan be seen as one requiring the preparation of the an-cilla performed by Alice and the other the measurementperformed by Bob. If you have only 1 degree of freedomthen the task of setting up the strategy can be placedon only one partner. Alice could prepare a sequence ofqubits and then send them to Bob. Bob then only has tomeasure them in a fixed basis with minimal instructionsabout what to do after a certain measurement result. Wealso consider how a strategy might affect the complexityof Bob’s instructions. Since Bob only has two measure-ment results, he must at some point receive a string ofbinary with the instruction to stop at the point whenthe instructions don’t match the string; perhaps thoseso interested could examine all possible strings and theentropy of instructions for particular strategies, we willjust be focusing on a question surrounding a simple casewhere the stopping condition is always the same mea-surement result for Bob. If Bob was looking for signalscoming out of two ports, Bob would just wait until heone of those ports thus we call this a ”one port” strat-egy. This might also be important in an experimentalcontext if the measurement process is prone to a partic-ular measurement bias.So we consider • What if we have control over only 1 degree of free-dom instead of 2? • What if we use only 1 port instead of 2?Since the process is probabilistic, we are interested notjust in the expected number of ancilla one side may haveto send to another but more the total number that haveto be prepared to ensure a certain probability of success.This also can be seen as a division of tasks- Alice pre-pares a number of ancilla qubits N s.t. P ( n < N ) ≥ . n steps, the probability needs to be increased by n .The value of N can also be approximated by a linearmultiple of the expectation value so they enforce a re-striction on when a “multi step” strategy is viable- a twostep strategy should double the probabilities and so on. A. The one step strategy
At each step set the conditions so that one measure-ment result generates a gate which corresponds to theangle difference between the present point on the circleand the point π . If this measurement does not occur, findthe distance between the present point and the point π and attempt to generate that gate. Upon every failure,find the new distance between the target and the currentpoint and attempt to generate that gate. FIG. 5: The probability tree of the “one step” strategy. Thefirst step will always require generating a CZ equivalent gate,the other result will be dependent on the coupling and willthen dictate all future conditions. The conditions are reset ateach step with each new ancilla.
Understanding the setting of the conditions of the gategeneration can be understood with a minor review ofthe Bloch sphere picture of the entanglement condition(see figure 6). Since the non-local part of the Cartandecomposition of the connection interaction is diagonal inthe computational basis, the ancilla before measurement
FIG. 6: The final state of the ancilla can be seen as a mixtureof four points on the Bloch sphere. In order for the post-measurement action to be unitary the four points must lieon a circle and the measurement basis axis go through thecentre. The cap sizes dictate the probability of the results,the distribution of the points around the circle affects theentangling power of the gate generated. The result within theminor cap is the more likely but will generate a less entanglinggate. By adapting the parameters β and θ these two aspectscan be controlled. has evolved as | a (cid:105) (cid:80) ij c ij | i (cid:105)| j (cid:105) → (cid:80) ij c (cid:48) ij | a ij (cid:105)| i (cid:105)| j (cid:105) . Thefinal states | a ij (cid:105) will becos (cid:18) θ − ( − i β (cid:19) | (cid:105) + e i ( − j α sin (cid:18) θ − ( − i β (cid:19) (1)These will map to four points on the Bloch sphere. Theangle β must be set by operations before and after thefirst interaction, the angle θ is set before the second in-teraction and must be known so that a measurement canbe applied which is mutually unbiased to all four points.Given any coupling strength, at the start of the strat-egy, the first attempt to generate CZ is performed thesame way: the ancilla is prepared in the + state and thenmeasured in the |±(cid:105) basis with the “ − ” (port 1) resultgenerating a gate equivalent to CZ ( C ( π )). The “+”(port 0) result would generate a blow back gate C (Φ ).There is a sense of direction with the gate generation; oneport gives C ( −| Φ | ), the other C (+ | Φ | ), clockwise oranticlockwise around the circle that represents the C ( γ )group. One can simply switch the direction associationof the ports by performing a bit flip either immediatelybefore or after transmission from Alice to Bob, so we willignore the exact sign requirements in the notation fromhere on and simply note the need to flip. Having trav-elled “clockwise”, the best next step is to continue in thatdirection and generate C ( π − | Φ | ). If Φ is small then π − Φ will be large enough that it can only be generatedfrom port 1, the port with larger Φ but smaller proba-bilities upper-bounded by . Another feature of port 1is that the probability increases as the preparation andmeasurement variables ( β, θ ) are increased and for a fixedΦ , θ increases with β . Therefore the for optimal proba-bility, it is only needed to fix one of these parameters tothe maximum and vary the other. So an 1 port strategyis effectively also a 1 degree of freedom strategy wherethe only task is finding the gate and the parameters forthe next step.At every step n , there is one gate that matches success C (Φ ,n ) and a failure gate C (Φ ,n , therefore to be at step n , the current action on the register system is the productof previous failures C ( −| Φ , | + Φ , + ... + Φ ,n − ). Thenext gate to be generated for success must be C ( π − ( | Φ , | − Φ , − ... − Φ ,n − ).The magnitude of the angle Φ of both ports increaseswith the probability of success of Φ . So because thelargest angle to be generated is π in the first step, the firststep has the highest probability of success and also thehighest value of the failure gate Φ . Therefore π − | Φ , | is the smallest value and has the smallest probability ofsuccess. These two first values provide a bound on thebehaviour of the strategy. The cumulative distributionfunction based measure, P ( n < N ), can be comparedto the CDF of constant probability for each step usingthe extreme probabilities: 1 − (1 − p ) n < P ( n < N ) < − (1 − p ) n .This also means that the first step provides the thresh-old for when a two port strategy is viable: when is π − | Φ , | small enough that it can be generated fromport 0? Since Φ , is also the maximum Φ , it must bewhen Φ , = π . Port 0 has a different ( β, θ ) for fixedΦ relationship and it’s probabilities are optimised awayfrom the fixed measurement conditions so this is also thethreshold for when a 2 degree of freedom strategy can beinvolved. B. The “flip-undo” strategy
Up until now we have discussed the ability to manip-ulate the ancilla with the assumption that we can ex-ercise any arbitrary single qubit unitary gate. This isin line with the requirements of the ancilla-driven andancilla-control schemes. However we have also developeda scheme for exploring what can be done with as simplean action on the ancilla as possible: we have only avail-able to us a fixed preparation state, a fixed measurementbasis and the choice of whether or not to implement a bitflip gate, X - specifically the bit flip required to changethe sense of direction of the ports. What this providesis the ability to attempt to undo a previous action hencethe designation the “flip-undo” strategy.After attempting to generate a C ( π ) gate in a single step, if the result failed, attempt to go back to the origin.Whether you have arrived at the origin or not, attemptto generate a C ( π ) gate with the product of the nextgeneration. If one fails again, repeat the process fromthe second step. Repeat until success. FIG. 7: The probability tree of the “flip-undo” strategy re-ceives all possible points in the strategy after 2 steps. Afterthe first step, it can be seen as a repeat-until-success strategywhere the time to repeat is 2 gate generations.
The inspiration for this scheme comes from question-ing why the two qubit gate is equivalent to C ( γ ) and not C ( − γ ). The answer comes from the Bloch sphere pictureof the four possible states of the ancilla after interactionand their orientation. In the middle of the procedure, itis only two states dependent on the first register qubit: (cid:80) i | a i (cid:105)| i (cid:105) (cid:80) j c (cid:48)(cid:48) ij | j (cid:105) . On the Bloch sphere, these will betwo points, one above the other on the same verticalplane, in order for the conditions for the gate to be uni-tary and entangling to be fulfilled. Which is above whichdetermines the sign of the rotation angle. So if one wasable to flip the orientation the sign would change. Thiscan be done by introducing an X gate on the ancilla inbetween the two connection interactions.It should seem obvious that in a case where either C ( π )or C ( γ ) is generated and the target is CZ = C ( π ) that itis preferable to label a result C ( γ ) a failure and attemptto undo it in order to try again to generate CZ directly.Yet that then creates a possible result where the sequenceproduct is C † ( π ) C ( γ ) = C ( γ + π ). Again the apparentbest decision is to attempt to undo C ( γ ) as this will nowimmediately lead to the target gate. In the following step,the only two possible product sequences must result in C ( π ) or ( C ( γ ) which makes employing this strategy forma closed loop.Now that there is a description and probability treefor a finite number of points on the circle, we can find anexact description of the time statistics using the recursiverelationships between the expectation times at differentpoints, if we take the probability of generating C ( π ) inthe first step to be p :¯ n = p + (1 − p )(¯ n + 1)¯ n = p (¯ n + 1) + (1 − p )(¯ n + 1)¯ n = p (¯ n + 1) + (1 − p ) ⇒ ¯ n = 1 + 1 p Another way to look at it is that after the probabilityof success in the first step, there is a 2 step time intervalwhich results in a probability of success of 2 p (1 − p ) whichcan be repeated. So an exact geometric distribution tailis formed where the expected time is 2 . p (1 − p ) and so¯ n = p. − p ) . ( p (1 − p ) + 1) = 1 + p . The cumulativedensity function is given by 1 − (1 − p )(1 − p (1 − p ) k where the number of transmitted ancilla qubits is 2 k + 1. V. NUMERICAL RESULTS
We found the parameters and resulting probabilitiesfor continuing with the one-step strategy for 500 stepsfor a range of coupling strengths of the connection inter-action. The full range for 0 < α < π was covered for the1 port 1 degree of freedom strategy where the degree offreedom was represented by the preparation parameter β . We then found more values for the range of couplingstrengths that starts just before the threshold for the twoport strategy. In this range we then found the values fora two port strategy where one could only vary β and per-form no optimisation of port 0 and then found them againfor when optimisation over β & θ is allowed. Finally wechecked for just the one port strategy, the probabilitiesfor each step when the measurement parameter is the al-lowed of degree of freedom rather than the preparationand this did turn out to give the exact same results.The order of improvement between different one-stepstrategies is less than a single step in the expectationtime. The 1 port strategy tends towards an expectationnumber of 2 while the 2 port/ 2 degree of freedom strat-egy tends toward 1.5 ; the 2 port/ 1 d.o.f. approachhas a peak in improvement near the middle of the rangebut at very close to the maximum coupling returns backto the 1 port strategy. This scale of improvement canbe expected from the behaviour of the probability of ei-ther port. As α → π , Φ , → − π , π − | Φ , | becomesvery small and thus the probability out of port 0 in thesecond step tends to 1. Any consideration of multi-stepstrategies in this range is therefore of limited advantage-the behaviour where failure in the first step improves theprobability of success in the second step which we wouldexpect to be a feature of any two step strategy is alreadya feature of the one step strategy with two ports andaccess to both parameters.The viability of a multi-step strategy at the lower cou-pling step range can be examined using the lower bound FIG. 8: A comparison of the unguided (dotted black) gener-ation against the one step (solid red) and flip-undo (dashedblue) strategy with their expected hitting times against cou-pling strength. Relative to the unguided approach, the flip-undo strategy is nearly as completely effective as the one stepstrategy while required less ancilla control. The unguidedexpectation times are not well correlated with the couplingstrength: while significant differences in step size from largecoupling differences do impact on the number of steps, theability to approach arbitrarily close to the target relates to thedifference of the step size with rational divisions of π makingfor small scale chaotic behaviour.FIG. 9: A comparison of the one step (solid red) strategyagainst the flip-undo (dashed blue) strategy: the number ofancilla that need to be prepared to guarantee a 99.9% chanceof success. on the probabilities of success in each step found fromthe probability in the second step. This value describesthe behaviour of a geometric distribution that boundsthe behaviour of the one step strategy; for a multi-stepstrategy to be effective it must at least improve upon thisand since a multi-step strategy takes place over n steps,it must improve the probability by at least a factor of n yet this will be limited by the maximum value of 1. Infigure 11 we can see what the maximum possible num-ber of steps for a multi-step strategy could be for anyimprovement to be possible.The most striking result is that the “flip-undo” strat-egy has very little cost in the expectation time com- FIG. 10: A comparison of strategies for different numbersof degrees of freedom at high coupling strengths. The solidred curve represents the 1 port/1 d.o.f. approach, the dottedblack curve is the 2 port/ 1 d.o.f. strategy and the dashed bluecurve is the 2 port/2 d.o.f. strategy. The threshold occurs atapproximately 0 . π .FIG. 11: The maximum number of steps in a strategy asallowed by the hard limit of p for a given coupling, displayedover the top half of the range of coupling strengths. pared to the one-step strategy. The gap between it andthe 1 port approach only approaches a maximum of onestep however the effect is more significant when consid-ering the minimum number of ancilla required to secure P ( n ≤ N ) ≥ .
999 but the relative effect is diluted asthe coupling strength gets weaker.
VI. CONCLUSION
In summary, we have analysed a implementation of amaximally entangling gate between distant qubits medi-ated by interaction with flying ancilla with an arbitrarybut fixed coupling strength. Due to the stochastic na-ture of the measurements and the non-determinism ofthe induced gate sequence, the time required for success is random [19]. By use of feedback on the ancilla prepa-ration or the measurement basis, some improvements canbe made over a stationary random walk strategy.We have examined how the addition of local unitarygate control on the ancilla qubit can speed up the ex-pected time for implementation and reduce the totalnumber of ancilla qubit required for a given fidelity.What has been found is that the improvement from con-trol over additional degrees of freedom is small which maybe important in the context of distributed or networkedquantum computation.If the task is distributed between two separated de-vices, co-ordination between the devices only allows forsome speed up past a threshold coupling strength. Theeventual speed is small and indeed the benefits of apply-ing any control and feedback can be mostly realised bythe inclusion of only one single extra operation on theancilla: the ability to choose to apply a bit flip. Thedominant factor appears to be the group structure of thegates that are generated during the process and the abil-ity to apply a bit flip to the ancilla ensures that only fourpossible gates can be generated which leads to a speedup over the generation of a continuous group.Yet to be investigated are using interactions of the class e − i ( α x σ x ⊗ σ x + α y σ y ⊗ σ y ) to generate gates in its own class.However the abelian structure and single parameter ofthe C ( γ ) gates has been a large part of the simplificationof the analysis and possible speed up and one would ex-pect that by having a two-parameter target, one wouldsquare the order of the expectation times. A two param-eter interaction would be better created by applying twosingle parameter interactions with local unitary gates be-tween them.The strategies employed seek to minimise the com-munication between two parties attempting to generatea shared entangling gate. The strategy can be workedout by A and instructions transmitted to B before send-ing any ancilla; the local post-corrections can be com-muted through so B’s instructions can be transmittedafter all have been sent. This allows us to envisagea scenario in which A sends B a packet of N ancillaewhere P ( n < N ) ≥ . [1] D. Deutsch. Quantum computational networks. Proceed-ings of the Royal Society of London. A. Mathematical andPhysical Sciences , 425(1868):73–90, 1989.[2] Robert Raussendorf and Hans J. Briegel. A one-wayquantum computer.
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