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PHYSICAL REVIEW A 67, 050303͑R͒͑2003͒
Universal quantum computation using exchange interactions and measurements of single- and two-spin observables
Lian-Ao Wu and Daniel A. Lidar Chemical Physics Theory Group, Chemistry Department, University of Toronto, 80 St. George Street, Toronto, Ontario M5S 3H6, Canada ͑Received 17 August 2002; published 28 May 2003͒ We show how to construct a universal set of quantum logic gates using control over exchange interactions and single- and two-spin measurements only. Single-spin unitary operations are teleported between neighbor- ing spins instead of being executed directly, thus potentially eliminating a major difficulty in the construction of several of the most promising proposals for solid-state quantum computation, such as spin-coupled quantum dots, donor-atom nuclear spins in silicon, and electrons on helium. Contrary to previous proposals dealing with this difficulty, our scheme requires no encoding redundancy. We also discuss an application to superconducting phase qubits.
DOI: 10.1103/PhysRevA.67.050303 PACS number͑s͒: 03.67.Lx, 05.30.Ch, 03.65.Fd I. INTRODUCTION the linear optics case, it was shown that photon-photon in- teractions can be induced indirectly via gate teleportation Quantum computers ͑QCs͒ hold great promise for inher- ͓6͔. This idea has its origins in earlier work on fault-tolerant ently faster computation than is possible on their classical constructions for quantum gates ͓7͔ and stochastic program- counterparts, but so far progress in building a large-scale QC mable quantum gates ͓8͔. The same work inspired more re- has been slow. An essential requirement is that a QC should cent results showing that, in fact, measurements and state be capable of performing ‘‘universal quantum computation’’ preparation alone suffice for UQC ͓9–11͔. ͑UQC͒: a set of quantum logic gates ͑unitary transforma- Experimentally, retaining only the absolutely essential in- tions͒ is said to be ‘‘universal’’ if any unitary transformation gredients needed to construct a universal QC may be an im- can be approximated to arbitrary accuracy by a quantum cir- portant simplification. Since read-out is necessary, measure- cuit involving only those gates ͓1͔. One of the chief obstacles ments are inevitable. Here, we propose a minimalistic in constructing large-scale QCs is the seemingly innocuous, approach for universal quantum computation that is particu- but in reality very daunting set of requirements must be met larly well suited to the important class of spin-based QC for universality, according to the standard circuit model ͓1͔: proposals governed by exchange interactions ͓2–4͔, and ͑1͒ preparation of a fiducial initial state ͑initialization͒, ͑2͒ a other proposals governed by effective exchange interactions set of single and two-qubit unitary transformations generat- ͓12͔. In particular, we show that UQC can be performed ing the group of all unitary transformations on the Hilbert using only single- and two-qubit measurements and con- space of the QC ͑computation͒, and ͑3͒ single-qubit mea- trolled exchange interactions, via gate teleportation. We has- surements ͑read out͒. Since initialization can often be per- ten to add that in our case teleportation involves only formed through measurements, requirements ͑1͒ and ͑3͒ do nearest-and next-nearest-neighbor spins, so that no coherent not necessarily imply different experimental procedures and manipulations between macroscopically separated spins are contraints. Until recently it was thought that computation is required. In our approach, which offers a new perspective on irreducible to measurements, so that requirement ͑2͒ would the requirements for UQC, two important advantages are ob- appear to be an essential component of UQC. However, uni- tained: ͑i͒ we require no encoding redundancy, ͑2͒ the need tary transformations are sometimes very challenging to per- to perform the aforementioned difficult single-spin unitary form. Two important examples are the exceedingly small operations is obviated, and replaced by measurements, which photon-photon interaction that was thought to preclude linear are anyhow necessary. The tradeoff is that the implementa- optics QCs, and the difficult-to-execute single-spin gates in tion of gates becomes probabilistic ͑as in all gate- certain solid-state QC proposals, such as quantum dots ͓2͔ teleportation-based approaches͒, but this probability can be and donor atom nuclear or electron spins in silicon ͓3,4͔. The boosted arbitrarily close to 1, exponentially fast in the num- problem with single-spin unitary gates is that they impose ber of measurements. difficult demands on g-factor engineering of heterostructure materials, and require strong and inhomogeneous magnetic II. SUPERCONDUCTING PHASE QUBITS EXAMPLE fields or microwave manipulations of spins, that are often slow and may cause device heating ͓5͔. For this reason there We begin our discussion with a relatively simple example has recently been a great deal of theoretical activity involv- of the utility of measurement-aided UQC, in the context of ing various qubit encoding schemes, which allow for UQC the proposal to use d-wave grain boundary ͑dGB͒ phase qu- without invoking difficult-to-control single-spin gates. Spe- bits for QC ͓13͔. In this proposal, it is important to reduce cifically, in the case of exchange Hamiltonians, it was shown the constraints on fabrication by removing the need to apply that when qubits are encoded into states of two or more a bias on individual qubits ͓14͔. This bias requires, e.g., the spins, the exchange interaction, possibly supplemented by possibility of applying a local magnetic field on each qubit, static Zeeman splittings, is sufficient to construct a set of and is a major experimental challenge. The effective system ϭ ϩ universal gates; see, e.g., Ref. ͓5͔ and references therein. In Hamiltonian that we consider is then HS HX HZZ , where
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ϭ͚ ⌬ HX i iXi describes phase tunneling, and HZZ ϭ͚ i, jJijZiZ j represents Josephson coupling of qubits; x y z Xi ,Y i ,Zi denote the Pauli matrices , , acting on the ith qubit. We assume continuous control over Jij . In Ref. ͓14͔ it was shown how UQC can be performed given this Hamiltonian, by encoding a logical qubit into two physical qubits, and using sequences of recoupling pulses. Instead, we now show how to implement Zi using measurements, which together with HS is sufficient for UQC. Suppose we start from an unknown state of qubit 1: ͉͘ϭa͉0͘ϩb͉1͘.By ͑ ͒ cooling in the idle state only HX on we can prepare an ancilla qubit 2 in the state (͉0͘ϩ͉1͘)/ͱ2. Then the total state is a͉00͘ϩb͉10͘ϩa͉01͘ϩb͉11͘. Letting the Josephson- Ϫ gate e i Z1Z2/2 act on this state, we obtain a͉00͘ϩei b͉10͘ Ϫ ϩei a͉01͘ϩb͉11͘ϰe i Z1/2͉͉͘0͘ϩei Z1/2͉͉͘1͘. We then ϩ ͑ measure Z2.Ifwefind 1 with probablity 1/2) then the Ϫ i Z1/2͉͉͘ ͘ state has collapsed to e 0 , which is the required FIG. 1. Gate teleportation of single-qubit operation Rz . Initially operation on qubit 1. If we find Ϫ1 then the state is Alice has ͉͘ and ͉0͘. Bob has ͉1͘. Time proceeds from left to iZ /2 1 e 1 ͉͉͘1͘, which is an erred state. To correct it we apply right. Starting from the three-qubit state ͉͉͘01͘, the task is to Ϫ i Z1Z2 ͉ the pulse e , which takes the erred state to the correct obtain Rz ͘. The protocol shown succeeds with probability 1/2. state When it fails the operation R† is applied instead. Fractions give the ϪiZ /2 z Ϫe 1 ͉͉͘1͘. We then reinitialize the ancilla qubit. This probability of a branch; 0 and 1 in a gray box are possible measure- method for implementing Zi succeeds with certainty after ment outcomes of the observable in the preceding gray box. See one measurement, possibly requiring ͑with probability 1/2) text for full details. one correction step. ͑we use units where បϭ1), where ␣ϭ͐tdtЈJ␣(tЈ), and we III. EXCHANGE BASED PROPOSALS have suppressed the qubit indices for clarity. In preparation We now turn to the QC proposals based on exchange of our main result, we first prove the following: Gϭ Ќ z ϵ interactions, e.g., Refs. ͓2–4,12͔. In these systems, that are Proposition. The set ͕Uij( , ), R j exp(i /  ͖ ϭ some of the more promising candidates for scalable QC, the 4 j ) ( x,z) is universal for quantum computation. qubit-qubit interaction can be written as an axially symmet- Proof. A set of continuous one-qubit unitary gates and any ric exchange interaction of the form two-body Hamiltonian entangling qubits are universal for ͓ ͔ ex͑ ͒ϭ Ќ͑ ͒͑ ϩ ͒ϩ z ͑ ͒ ͑ ͒ quantum computation 15 . The exchange Hamiltonian Hij t Jij t XiX j Y iY j Jij t ZiZ j , 1 ex Hij clearly can generate entanglement, so it suffices to ␣ ␣ϭЌ where Jij(t)( ,z) are controllable coupling constants. show that we can generate all single-qubit transfor- z ϭ Þ G The XY (XXZ) model is the case when Jij 0( 0). The mations using . Two of the Pauli matrices are z ϭ Ќ ϭϪ 2 ؠ Heisenberg interaction is the case when Jij(t) Jij(t). See given simply by j iRj . Now, let CA exp(i B) Ref. ͓5͔ for a classification of various QC models by the type ϵexp(ϪiA)exp(iB)exp(ϩiA); two useful identities for of exchange interaction. In agreement with the QC proposals anticommuting A,B with A2ϭI ͑the identity͒ are ͓ ͔ Ќ 2–4,12 , we assume here that Jij(t) is competely control- lable and allow that Jz (t) may not be controllable. The /2ؠ ϪiBϭ iB /4ؠ ϪiBϭ AB ij CA e e , CA e e . method we present here works equally well for all three types of exchange interactions, thus unifying all exchange- Ϫ i X1X2 based proposals under a single universality framework. Since Using this, we first generate e ϭ z /2ؠ z U12( /2, )C U12( /2, ), which takes six elemen- all terms in Hex(t) commute, it is simple to show that, in the X1 ordered basis ͕͉00͘,͉01͘,͉10͘,͉11͖͘ it generates a unitary two- tary steps ͓where an elementary step is defined as one of the Ќ z qubit evolution operator of the form operations Uij( , ),R j]. Second, as we show below, our gate-teleportation procedure can prepare R† just as effi- U ͑Ќ,z͒ j ij ͓ † ϭϪ 3 ciently as R j also note that R j (R j) ], so that with t Ϫ Ϫ Ϫ i Y 1X2ϭ /4ؠ i X1X2 ϭ ͫϪ ͵ Ј ex͑ Ј͒ͬ two additional steps we have e CZ e . Fi- exp i dt Hij t 1 nally, with a total of 8ϩ6ϩ8ϭ22 elementary steps we have ϪiZ /4 ϪiX X Ϫiz e 1ϭC ؠe 1 2, where is arbitrary. Similarly, we e Y 1X2 Ϫ Ϫ i Y 1 /4 /4 z Ќ z Ќ can generate e in 22 steps using C instead of C . ei cos 2 Ϫiei sin 2 X1 Z1 ϭ Using a standard Euler angle construction we can generate z Ќ z Ќ Ϫ ͩ Ϫ i i ͪ i Z1 ie sin 2 e cos 2 arbitrary single-qubit operations by composing e and ϪiY z e 1 ͓1͔. eϪi It is important to note that optimization of the number of ͑2͒ steps given in the proof above may be possible. We now
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UNIVERSAL QUANTUM COMPUTATION USING . . . PHYSICAL REVIEW A 67, 050303͑R͒͑2003͒
ϭ ͉ ͉ show that the single-qubit gates R j can be implemented S 1 then the outcome is R1z ͘1 T0͘23 , equiprobably. In using cooling, weak spin measurements, and evolution under the latter case Alice ends up with the desired operation ͓Fig. exchange Hamiltonians of the Heisenberg, XY,orXXZ type. 1͑e͔͒. ͓ ͔ † Our method is inspired by the gate-teleportation idea 6–10 . In a similar manner one can generate Rx or Rx acting on We proceed in two cycles. In cycle ͑i͒, consider a spin ͑our an arbitrary qubit state ͉͘. Let ͉Ϯ͘ denote the Ϯ1 eigen- ͒ ͉͘ϭ ͉ ͘ϩ ͉ ͘ x ‘‘data qubit’’ in an unknown state a 0 b 1 , and states of the Pauli operator .AsintheRz case above, first two additional ͑‘‘ancilla’’͒ spins, as shown in Fig. 1. Our task prepare a singlet state ͉S͘ϭ1/ͱ2(͉Ϫϩ͘Ϫ͉ϩϪ͘) on the is to apply the one-qubit operation R to the data qubit. As in ancilla spins 2,3 by cooling. Then perform a single-spin gate teleporation, we require an entangled pair of ancilla x measurement of the observable j on each ancilla, which spins. However, it turns out that rather than one of the will yield either ͉ϩϪ͘ or ͉Ϫϩ͘. For definiteness assume Bell states we need an entangled state that has a phase of ͉ϩϪ that the outcome was ͘23 . Observing that in the i between its components. To obtain this state, we first ͕͉ϩϪ͘ ͉Ϫϩ͖͘ exϭϪ Ќ ϩ Ќϩ z ˜ ex , subspace, Hij JijI (Jij Jij)X, where turn on the exchange interaction H23 between the ancilla Ќ ˜ ͉ϩϪ͘↔͉Ϫϩ͘ Ϫz z ͉ϩϪ͘ X: , it follows that U( /4 0 , 0) spins such that J Ͼ0. The eigenvalues ͑eigenstates͒ are Ќ Ϫ ЌϪ z ЌϪ z z z ͉ ͉ ͉ ͉ ϭ Ϫi ͱ ͉ϩϪ͘Ϫ ͉Ϫϩ͘ ͕ 2J J ,2J J ,J ,J ͖ ( S͘, T0͘, 00͘, 11͘), where e / 2( i ), so that we have a means of ͉ ϭ ͱ ͉ Ϫ͉ ͉ ϭ ͱ ͉ ϩ͉ generating an entangled initial state. The unknown state ͉͘ S͘ 1/ 2( 01͘ 10͘), T0͘ 1/ 2( 01͘ 10͘) are the sin- 1 Ќ z ͉͘ϭ ͉ϩ͘ϩ ͉Ϫ͘ glet and one of the triplet states. Provided J ϾϪJ ͓which is of the data qubit can be expressed as ax bx , ϭ ϩ ͱ ϭ Ϫ ͱ ͑ the case for all QC proposals of interest, in which either where ax (a b)/ 2 and bx (a b)/ 2. Then neglect- Ќ Ќ sgn(J )ϭsgn(Jz), or Jzϭ0] and we cool the system signifi- ing the overall phase eϪi ), cantly below Ϫ2JЌϪJz, the resulting ground state is ͉S͘. ͉͘ ͑ Ϫz z ͉͒ϩϪ͘ We then perform a single-spin measurement of the observ- 1U23 /4 0 , 0 23 able z on one or both of the ancillas, which will yield either j ϭ 1 ͉ ͘ ͉͘ ϩ 1 ͉ x ͘ † ͉͘ ͉01͘ or ͉10͘. For definiteness, assume that the outcome was 2 r* S 12R3x 3 2 r T0 12R3x 3 ͉01͘. We then immediately apply an exchange pulse to the ϩ͑1/ͱ2͒͑a ͉ϩϩϪ͘Ϫib ͉ϪϪϩ͘), ͓ ͑ ͔͒ z ͉ ͘ϭ iz ͱ ͉ ͘ x x ancilla spins Fig. 1 a : U( /8, 0) 10 e / 2( 01 Ϫ ͉ ͓ ͑ ͔͒ x i 10͘) as follows from Eq. 2 . The total state of the three where ͉T ͘ϭ1/ͱ2(͉ϩϪ͘ϩ͉Ϫϩ͘) is a triplet state, a zero z 0 ͑ i x ϩx spins then reads neglecting an overall phase e ): eigenstate of the observable 1 2 . The gate-teleportation procedure is now repeated to yield R or R† . First, Alice ͉͘ ͑ z ͉͒ ͘ x x 1U23 /8, 0 10 23 ជ 2 ϭ ͑ measures the total spin S12 . If she find S 0 with probabil- ϭ͑ ͱ ͒͑ ͉ ͘Ϫ ͉ ͘ ϩ 1 ͉ ͘ † ͉͘ ity 1/4) Bob has spin 3 in the desired state R ͉͘ . If she 1/ 2 a 001 ib 110 ) 2 r T0 12R3z 3 3x 3 finds Sϭ1 then she proceeds to measure the total length Ϫ 1 r*͉S͘ R ͉͘ , ͑3͒ 2ϭ 1 x ϩx 2 2 12 3z 3 of the x component Sx 4 ( 1 2) , yielding, provided she finds S2ϭ0, the state ͉Tx ͘ R† ͉͘ with probability where rϭexp(Ϫi/4) and the subscripts denote the spin in- x 0 12 3x 3 1/3. If, on the other hand, she finds S2ϭ1, i.e., the state is dex. x ͉ϩϩϪ͘Ϫ ͉ϪϪϩ͘ ជ 2 At this point Alice makes a weak measurement of her ax ibx , then by letting Bob measure S23 , † ͉͘ ͉ ͘ ͉͘ ͉ x ͘ spins ͓Fig. 1͑b͔͒. Let Sជ ϭ 1 (ជ ϩជ ) be the total spin of the states R1x 1 S 23 or R1x 1 T0 23 are obtained, with ij 2 i j equal probabilities. qubits i, j; Alice measures Sជ 2 with eigenvalues S(Sϩ1). It 12 Figure 1 summarizes the protocol we have described thus follows that if the measurement yields 0, then the state has ͉͘ ͉ ͉ far. The overall effect is to transform the input state to collapsed to S͘12R3z ͘3. In this case, which occurs with † either the output state R͉͘ or R ͉͘, equiprobably. probability 1/4, Bob has R ͉͘ , and we are done ͓Fig.  3z 3 We have now arrived at cycle ͑ii͒, in which we must fix 1͑c͒, bottom͔. If, on the other hand, Alice finds Sϭ1, then the erred state R† ͉͘ ( jϭ1 or 3). To do so we essentially the normalized postmeasurement state is j j repeat the procedure shown in Fig. 1. We explicitly discuss ͑1/ͱ3͓͒r͉T ͘ R† ͉͘ ϩaͱ2͉͑001͘Ϫib͉110͘)]. ͑4͒ one example; all other cases are similar. Suppose we obtain 0 12 3z 3 † ͉͘ ͉ ͘ ͓ ͑ ͔͒ the erred state R1z 1 S 23 Fig. 1 e . It can be rewritten as Similar to the gate-teleportation protocol ͓9,10͔, Alice and † ͉͘ ͉ ͘ ϭϪ͑ ͱ ͒͑ ͉ ͘Ϫ ͉ ͘ Bob now need to engage in a series of correction steps. In the rR1z 1 S 23 i/ 2 a 001 ib 110 ) 2 1 z z 2 1 z z next step, Alice measures S ϭ ( ϩ ) ϭ (Iϩ ) 1 1 z 4 1 2 2 1 2 Ϫ r͉S͘ R† ͉͘ ϩ r*͉T ͘ R ͉͘ , ͓ ͑ ͒ ͔ 2ϭ 2 12 3z 3 2 0 12 3z 3 Fig. 1 c , top . If Alice finds Sz 0 then with probability 1/3 ͉ ͘ † ͉͘ the state collapses to T0 12R3z 3 and Bob ends up with which up to unimportant phases is identical to Eq. ͑3͒, except †͉͘ ͓ the opposite of the desired operation, namely, Rz Fig. that the position of R† and R has flipped. Correspondingly, ͑ ͒ ͔ 3z 3z 1 d , bottom . We describe the required corrective flipping the decision pathway in Fig. 1 will therefore lead to ͑ ͒ 2ϭ action below, in Cycle ii . If Alice finds Sz 1, the correct action R͉͘ with probability 1/2, while the over- ͉ ͘Ϫ ͉ ͘ϭ ͱ † ͉͘ ͉ ͘ † then the state is a 001 ib 110 1/ 2(r*R1z 1 S 23 all probability of obtaining the faulty outcome R͉͘ after ϩ ͉͘ ͉ ͘ ជ 2 rR1z 1 T0 23). Bob now measures S23 . If he finds S the second cycle of measurements is 1/4. Clearly, after n ϭ † ͉͘ ͉ ͘ 0 then the state has collapsed to R1z 1 S 23 , while if measurement cycles as shown in Fig. 1, the probability for
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L.-A. WU AND D. A. LIDAR PHYSICAL REVIEW A 67, 050303͑R͒͑2003͒ the correct outcome is 1 –2Ϫn. The expected number of mea- suring spin qubits ͓17͔. The requirements for single-spin surements per cycle is 1 1 ϩ3 3 2 1 ϭ1, and the expected measurements have already been met using an rf-SET 4 4 3 2 ͓ ͔ ͚ϱ Ϫnϭ coupled to a Si:Te donor 3 . Another possibility, adapted to number of measurement cycles needed is nϭ1n2 2. † ͉͘ ϭ quantum dots, is to use a dot attached to current leads. This We note that in the case of the erred state R jz j ( j 1or ͒ results in Coulomb blockade peaks and spin-polarized cur- 3 , similarly to the case discussed in Sec. II there is an alter- rents uniquely associated with the spin state on the dot ͓2͑a͔͒. native that is potentially simpler than repeating the measure- The same methods are easily adaptable to distinguishing a ment scheme of Fig. 1. Provided the exchange Hamiltonian ͑ ជ 2 z spin singlet from a triplet measurement of S12), as discussed is of the XY type, or of the XXZ type with a tunable J ͓ ͔ exchange parameter, one can simply apply the correction op- in Refs. 2,3 . The idea is to create an energy difference ϭ † ͉͘ ͉͘ between the two states, and then to observe conductance. erator U j2( /2,0) Z jZ2 to R jz j , yielding R jz j as re- ␣␣ quired. Finally, we note that Nielsen ͓9͔ has discussed the Measurement of the observable 1 2 is possible, e.g., given ␣ ␣ conditions for making a gate-teleportation procedure of the devices that measure 1 and 2 and have a nonlinear re- type we have proposed here, fault tolerant. sponse: the devices must be coupled and tuned into a regime where the linear response coefficient vanishes, leaving only ␣␣ IV. SPIN MEASUREMENTS the second order contribution due to 1 2 . A concrete ex- ample is provided by quadratic detection using magnetome- Our proposal requires measurement of the following ob- ters in the context of Josephson-junction qabits, as discussed z x ͑ ͒ ជ 2 in Ref. ͓18͔. An alternative method is optical measurement: servables: , single-spin projective measurements , S12 ␣ ␣ ͑distinguishing a singlet from a triplet state͒, ͓distin- in the singlet state (Sϭ0) there is no Faraday rotation, while 1 2 ϭ ͓ ͑ ͔͒ guishing whether spins are parallel or antiparallel in the in the triplet state (S 1) there is 2 a . Finally, we note that ͉0͘,͉1͘ (␣ϭz)or͉0͘Ϯ͉1͘ (␣ϭx) basis͔. Underlying our quantum error correction relies on measurements of multi- ␣␣ ͓ ͔ proposal is the assumption that these measurements are ͑or spin observables such as 1 2 1 , so the development of will be͒ easier to perform than the joint requirement of these techniques is as inevitable as that of single-spin mea- single-qubit gates and single-spin measurements. This as- surements. sumption is partly motivated by the observation that mea- surements are necessary, and hence the technology to per- V. CONCLUSION form them will be perfected. Let us now briefly survey the subject of spin measurements ͑for a detailed discussion see, We have proposed a gate-teleportation method for univer- e.g., Refs. ͓2,3͔͒. While direct measurement of spin is diffi- sal quantum computation that is uniformly applicable to cult because of the tiny magnetic moment ͑although possible, Heisenberg, XY, and XXZ-type exchange interaction-based in principle, e.g., optically via the Faraday rotation ͓16͔͒, QC proposals, and that replaces single-spin unitary gates by spin measurement can be converted into charge detection via measurements of single- and two-spin observables. We hope the Pauli exclusion principle. For example, a donor defect in that the flexibility offered by this approach will provide a Si can bind a second electron by 1 meV, provided the second useful alternative route towards the realization of universal electron has opposite spin to the first electron. Thus, spin quantum computers. measurement becomes electrical charge detection. This is the essential idea behind spin-resonance transistors. Vrijen et al. ACKNOWLEDGMENTS ͓4͑a͔͒ have proposed a field-effect transistor ͑FET͒ operating at low temperatures based on this idea. Alternatively, mea- The present study was sponsored by the DARPA-QuIST suring the charge of single electrons is routine using single- program ͑managed by AFOSR under Agreement No. electron transistors ͑SET͒, and has been proposed for mea- F49620-01-1-0468͒ and by DWave Systems, Inc.
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