Grajcar, M.

Izmalkov, A.

Love, Peter John

van der Ploeg, S. H. W.

English

Haverford College: Haverford Scholarship

Haverford College Haverford Scholarship Faculty Publications Physics 2006 Four-Qubit Device with Mixed Couplings M. Grajcar A. Izmalkov S. H. W. van der Ploeg Peter John Love Haverford College Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation Four- ux-qubit device with mixed couplings" M. Grajcar, A. Izmalkov, S.H.W. van der Ploeg, S. Linzen, T. Plecenik, Th. Wagner, U. Huebner, E. Il'ichev, H.-G. Meyer, A.Yu. Smirnov, Peter J. Love, Alec Maassen van den Brink, M.H.S. Amin, S. Uchaikin, A.M. Zagoskin, Phys. Rev. Lett. , 96 , 047006 (2006), http://www.arxiv.org/abs/cond-mat/0509557 This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. Four-Qubit Device with Mixed CouplingsM. Grajcar,1,2,3 A. Izmalkov,1 S. H. W. van der Ploeg,1,4 S. Linzen,1 T. Plecenik,1,2 Th. Wagner,1 U. Hübner,1 E. Il’ichev,1,*H.-G. Meyer,1 A. Yu. Smirnov,5 Peter J. Love,5 Alec Maassen van den Brink,5,†M. H. S. Amin,5 S. Uchaikin,5 and A. M. Zagoskin6,‡1Institute for Physical High Technology, P.O. Box 100239, D-07702 Jena, Germany2Department of Solid State Physics, Comenius University, SK-84248 Bratislava, Slovakia3Center of Excellence of the Slovak Academy of Sciences (CENG), Slovakia4MESA+ Research Institute and Faculty of Science and Technology, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands5D-Wave Systems Inc., 100-4401 Still Creek Drive, Burnaby, British Columbia, V5C 6G9 Canada6Physics and Astronomy Dept., The University of British Columbia,6224 Agricultural Road, Vancouver, British Columbia, V6T 1Z1 Canada(Received 21 September 2005; published 2 February 2006)We present the first experimental results on a device with more than two superconducting qubits. Thecircuit consists of four three-junction flux qubits, with simultaneous ferro- and antiferromagnetic couplingimplemented using shared Josephson junctions. Its response, which is dominated by the ground state, ischaracterized using low-frequency impedance measurement with a superconducting tank circuit coupledto the qubits. The results are found to be in excellent agreement with the quantum-mechanical predictions.DOI: 10.1103/PhysRevLett.96.047006 PACS numbers: 85.25.Dq, 03.67.Lx, 85.25.CpThe implementation of one- and two-qubit gates insuperconducting qubit prototypes [1] has confirmed theirutility for quantum computation. Such qubits are readilyfabricated, and highly scalable in principle. A prominentsubcategory consists of superconducting loops interruptedby three Josephson junctions, so-called 3JJ flux qubits [2].If threaded by a near-degenerate magnetic bias flux x 12 0 (0 is the flux quantum), such devices have quantumstates which are superpositions of clockwise (j#i) andcounterclockwise (j"i) circulating supercurrent Ip. Thesmall area of 3JJ qubits reduces their coupling to environ-mental magnetic noise, making comparatively long co-herence times possible [3], but also limits the strengthof their inductive coupling [2,4]. This can be overcomeusing direct galvanic coupling through a shared Josephsonjunction [5]. When reporting its experimental realiza-tion [6], we mentioned that direct coupling can have ad-ditional advantages. First, the coupling strength can bevaried independently of the sample geometry by chang-ing the shared junction’s critical current. Second, whiledirect coupling normally has the antiferromagnetic (AFM)sign just as in the inductive case, a ‘‘twisted’’ design [join-ing two qubit loops in a ‘‘1’’ shape with crossing leads inthe center; see Fig. 1(b)] features ferromagnetic (FM)coupling [7].In this Letter, we pursue this by studying a four-qubitcircuit in which the two types of coupling coexist. This isvery promising from the perspectives of realizing nontri-vial Ising-spin systems [8] and scalable adiabatic quantumcomputing (AQC) [9,10]. However, while our circuit be-haves in full agreement with quantum mechanics andfeatures excellent multiqubit bias control, the measure-ments presented are essentially equilibrium. The system’seffective low-energy (pseudospin) Hamiltonian isH  X4i1iiz  iix  X1	i<j	4Jijiz jz ; (1)where i is the bias on qubit i (implemented through aflux bias x  12 0), i is its tunneling amplitude, Jijis the coupling energy between qubits i and j, and z, xare Pauli matrices in the span of j#i (z  1) and j"i(z  1). Equation (1) adequately describes a system offlux qubits [2,4,11], although possible limitations [12]remain to be investigated.Our approach follows earlier work using the impedancemeasurement technique (IMT) [13]. This method measuresimpedance shifts in a high-quality superconducting LC(tank) circuit inductively coupled to the system. For onequbit [14], the average loop current I  jbj2  jaj2Ip inan eigenstate aj#i  bj"i changes direction at the anticross-ing point x  12 0 (where jaj  jbj, i.e., the pseudospinground state flips from j#i to j"i and vice versa for theexcited state). This causes a peak in the qubit susceptibility  dI=dx; since dI=dx / d2E=dx2, whereE is theaverage energy and dx is adiabatic (note 15 in Ref. [14]),this also follows from the large energy-band curvature atthe anticrossing. Note that the very presence of such peaksis already a quantum effect, since the classical xcurves feature hysteretic loops [10] in the absence of fluxtunneling. Because of the qubit-tank coupling,  affects thetank’s eigenfrequency !T and hence, under resonant driv-ing, its current-voltage phase angle  / . Thus, the widthand height of such an IMT peak in  provide informationabout the anticrossing, i.e., about  and Ip.For multiple qubits, a more general analysis yields [15]tan  2QTLTX<  E  ER; (2)PRL 96, 047006 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 FEBRUARY 20060031-9007=06=96(4)=047006(4)$23.00 047006-1 © 2006 The American Physical Societywhere LT and QT are the tank’s inductance and qualityfactor, respectively,   eE=T=PeE=T is theBoltzmann factor of the eigenstate ji of H in (1), andwith the matrix elementR X4i;j1ijhjiz jihjjz ji; (3)i are tank-qubit coupling coefficients in Hint IrfPiiiz , and Irf is the oscillating component of thetank current IT  IbT  Irft. The term i  j producesthe IMT peak corresponding to qubit i. Cross terms i j,on the other hand, appear only if ji or ji has nonzeroentanglement between qubits i and j due to their interac-tion. For two weakly coupled qubits [4], the effect ofinteraction can be understood as a correction. The crossterms in R then change the shape of overlapping peaks atcodegeneracies [i.e., when the i in (1) are chosen such thatboth qubits flip simultaneously], which turn out to besmaller (larger) than the sum of the two individual peaksfor AFM (FM) coupling [16].Here, the analysis needs to be extended both beyond twoqubits and to stronger coupling (particularly J >), whichqualitatively changes the locus of the IMT peaks, as canalready be understood classically: neglecting the i in (1),the degeneracy points of the classical multiqubit flux statesare seen to depend on the Jij [6]. One then proceeds byfitting the coefficients of H in (1); see below.Figure 1(a) shows the sample with four Al qubit loopsinside a Nb pancake coil. The areas of the Al loops are10 7:5 m2. Two junctions in each qubit are typically650 150 nm2, while the third is 25% smaller [2]. TheNb coil has LT  105 nH, and together with an externalcapacitance CT  470 pF forms a parallel tank circuit with!T=2  22:703 MHz and QT  !TRTCT  400, whereRT is the effective resistance. The Al loops were fabricatedby e-beam lithography and conventional shadow evapora-tion, and the Nb coil by e-beam lithography and CF4reactive-ion etching.The qubits are coupled to each other both magneticallyand via a shared junction, consisting of parallel junctionsA1–A3 (Fig. 1). For a coupling junction with large criticalcurrent Ic, and assuming identical qubits, J@I2p=2eIc [6].For our system, Ip  250 nA. Because of the relative twist[6] between the qubit loops [Fig. 1(b)], the couplingsq1-q3 and q2-q4 are FM (Jij < 0), while the others areAFM (Jij > 0). The biases i can be varied by adjusting thecurrents Ib1-b4 through the Nb wires, as well as the dccurrent IbT through the tank coil. The qubit-coil mutualACT RTVoutIbT+Idr(t)Nb coilLTq1q2q3q4Ib1Ib2 Ib4Ib3A1A3A2IT(b)FIG. 1 (color online). (a) Electron micrograph of the sample.The central junctions A1-A3 couple the Al qubits q1-q4. Thesurrounding Nb coil is part of the LC tank circuit, used for bothmeasurement and global flux biasing. The Nb lines Ib1-b4 allowasymmetric bias tuning. (b) Schematic diagram. The drivingcurrent Idrt is much smaller than the oscillations in IT andonly needs to compensate for the latter’s losses.-21 -20 -19 -18 -17 -16 -15 -140.00.10.20.30.40.50.60.70.8q2q4q1 & q3q4q3q2q1(b)−  tan ΘIbT (µA)(a)FIG. 2 (color online). (a)  tanIbT at Ib1  700 A, Ib2 10 A, and Ib4  250 A (Ib3  0 throughout). The four qubitpeaks do not overlap. (b) The same graph (vertically shifted forclarity) but at Ib1  268 A, where the peaks corresponding toq1 and q3 overlap. The height increase compared to the sum ofthe two nonoverlapping peaks is a signature of FM coupling.Dashed lines are theoretical fits. The fit for q2 is poor because therelatively large tank amplitude washes out the experimentalpeak [13]; a better fit has been obtained for smaller amplitudes(not shown).PRL 96, 047006 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 FEBRUARY 2006047006-2inductances follow from the flux periodicity as Mq1;TMq3;T38:2 pH, Mq2;T  38:6 pH, and Mq4;T  38:9 pH.The sample was thermally anchored to the mixing cham-ber of a dilution refrigerator at a temperature Tmix 10 mK. In general the effective T is affected by noise onexternal leads and so is larger than Tmix. Often, Tmix aloneis given, since T is difficult to determine. One of theadvantages of IMT is that it allows one to determineeffective temperatures [6,14]. From the best theoretical fitwe estimate T  70 mK, well within the expected range.Figure 2 shows peaks in  tan as the overall flux isscanned by changing IbT. For most biases, four peaks canbe distinguished [Fig. 2(a)], each corresponding to degen-eracy in one of the qubits while the others provide asemiclassical static background field. The narrow peakfor q2 indicates a small 2 (12 mK), and incidentallyconfirms that the width of the other traces is not resolutionlimited. When q1 and q3 are codegenerate [Fig. 2(b)], theresulting peak exceeds the sum of their individual peaks,indicating strong FM coupling as mentioned above (this isalso true for q2 and q4).Figure 3(a) plots  tanIbT; Ib1. The four traces areIMT peaks, each corresponding to a single-qubit anticross-ing. Since IbT biases all qubits almost equally, the tracewith the greatest sensitivity to Ib1 can be ascribed to q1 etc.Combining several such cross-sections of the total IMTresponse, one reconstructs the biasing coefficients in xi Mqi;TIbT PjM0qi;bjIbj. The centers of the traces mark theboundaries of the classical stability diagram. Each regionFIG. 3 (color). (a)  tanIbT; Ib1 at Ib2  10 A and Ib4 250 A. The repulsion between the traces (q1; q2), (q2; q3), and(q1; q4) shows AFM coupling between the qubits in those pairs,while the merging of traces (q1; q3) demonstrates FM coupling.The line widths are proportional to the tunneling amplitudes ofthe individual qubits. Since the qubits’ sensitivity to Ib1 de-creases in the order q1, q2, q3, q4 [cf. Figure 1(a)], the variousslopes allow one to identify the trace belonging to each qubit.(b) Theoretical graph for the same parameters.FIG. 4 (color). Same as in Fig. 3, but as a function of IbT andIb2, at Ib1  400 A and Ib4  50 A. The skipped cross-ing at the center corresponds to four-qubit codegeneracy.Because of their FM coupling, the q1 and q3 traces mergeover an extended region. The qubits’ sensitivity to Ib2 decreasesin the order q2, q4, q1;3.PRL 96, 047006 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 FEBRUARY 2006047006-3between the traces corresponds to a different (minimum-potential) flux state of the qubits [17]. The shifts of thetraces after each (skipped) crossing depend on the Jij. Thepeak shapes, on the other hand, carry information aboutthe i. As expected, the traces corresponding to two qubitswith AFM (FM) coupling repel (attract) each other, and inthe FM case merge over a certain distance. Fitting the datato Eq. (1) yields 1  147, 2  12, 3  163,4  165, and J12  J34  163, J14  J23  155,J13  J24  62 (all in mK), andM0 247 71  68143 309  8470 87  18250 195  3550BBB@1CCCA fH; (4)where the missing entries correspond to Ib3 which wasnever used. The theory [Fig. 3(b)] accurately describesthe system’s behavior over the whole parameter space,especially at the codegeneracies.Figure 4(a) shows  tanIbT; Ib2 for a case when allfour qubits become degenerate simultaneously in the fig-ure’s center, where they produce two repelling peaks. FromH in (1) one infers that, between those peaks, the twolowest eigenstates are close to superpositions of j"#"#iand j#"#"i, for which all the coupling energies are negative;at codegeneracy, the superpositions become (anti)symmet-ric [18]. From the viewpoint of AQC, it is encouraging thatthe ground state of H thus is globally entangled [19]; for aquantification, see Ref. [20]. With all parameters alreadydetermined from Fig. 3 and a few other cross-sections likeit, the theoretical comparison in Fig. 4(b) (and many othernontrivial cross-sections) can be regarded as having no freeparameters, making the agreement remarkable.Complete agreement between experiment andtheory was also obtained for a second sample, with1  2  60, 3  130, 4  110, J12  J23  300,J14  J34  330, and J13  J24  90 (all in mK).However, the lowest effective temperature attained wasonly 300 mK.In conclusion, ferromagnetic and antiferromagnetic cou-plings between four 3JJ flux qubits have been realizedsimultaneously with shared Josephson junctions, with acoupling strength significantly exceeding the inductiveone. (Incidentally, the results also show that direct galvaniccoupling can be used for more general circuits than thelinear arrays considered in Ref. [21].) The data fully agreewith a quantum-mechanical description to the experimen-tal accuracy. Currently, quasiequilibrium impedance mea-surement already provides valuable information comple-mentary to state readout, notably the determination of theHamiltonian. With a faster and possibly qubit-selectivedetection method, the circuit used looks very promisingfor studying adiabatic bias manipulations.M. G. was supported by Grants No. VEGA 1/2011/05and No. APVT-51-016604, E. I. by the RSFQubit project,and A. Z. by the NSERC Discovery Grants Program. Wethank A. J. Berkley, D. A. Lidar, and M. F. H. Steininger fordiscussions.*Electronic address: ilichev@ipht-jena.de†Electronic address: alec@dwavesys.com‡Electronic address: zagoskin@physics.ubc.ca[1] Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Nature(London) 398, 786 (1999); D. Vion et al., Science 296,886 (2002); T. Yamamoto, Yu. A. Pashkin, O. Astafiev,Y. Nakamura, and J. S. Tsai, Nature (London) 425, 941(2003).[2] J. E. Mooij et al., Science 285, 1036 (1999).[3] E. Il’ichev et al., Phys. Rev. Lett. 91, 097906 (2003);P. Bertet et al., Phys. Rev. Lett. 95, 257002 (2005).[4] A. Izmalkov et al., Phys. Rev. Lett. 93, 037003 (2004).[5] L. S. Levitov, T. P. Orlando, J. B. Majer, and J. E. Mooij,cond-mat/0108266.[6] M. Grajcar et al., Phys. Rev. B 72, 020503(R) (2005).[7] For other designs with FM coupled flux qubits, see J. Q.You, Y. Nakamura, and F. Nori, Phys. Rev. B 71, 024532(2005) and Ref. [6].[8] F. Barahona, J. Phys. A 15, 3241 (1982).[9] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser,quant-ph/0001106; W. M. Kaminsky and S. Lloyd,quant-ph/0211152;W. M. Kaminsky, S. Lloyd, and T. P.Orlando, quant-ph/0403090.[10] M. Grajcar, A. Izmalkov, and E. Il’ichev, Phys. Rev. B 71,144501 (2005).[11] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E.Mooij, Science 299, 1869 (2003); T. P. Orlando et al.,Phys. Rev. B 60, 15 398 (1999); A. Lupaşcu, C. J. M.Verwijs, R. N. Schouten, C. J. P. M. Harmans, and J. E.Mooij, Phys. Rev. Lett. 93, 177006 (2004).[12] A. Maassen van den Brink, Phys. Rev. B 71, 064503(2005).[13] Ya. S. Greenberg et al., Phys. Rev. B 66, 214525 (2002).[14] M. Grajcar et al., Phys. Rev. B 69, 060501(R) (2004).[15] A. Yu. Smirnov, cond-mat/0312635.[16] This is plausible: for stronger AFM coupling the codege-neracy develops into an j"#i $ j#"i anticrossing line withno net flux change, hence no IMT signature at all in thesymmetric case [6]. On the other hand, for strong FMcoupling, an j""i $ j##i line develops, i.e., a sharp second-order anticrossing with a large IMT response.[17] The stability diagram is completed by connecting theopposing corners where traces corresponding to skippedcrossings meet. These invisible lines correspond to thej"#i $ j#"i anticrossings.[18] However, the minimum energy gap E1  E0  8 mKbetween j1i and j0i (due to fourth-order tunneling inperturbative language) was less than the thermal energy,partly because of the small 2. Therefore, we do not claimselectivity between these GHZ-type states.[19] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).[20] P. J. Love et al. (to be published).[21] J. R. Butcher, M.Sc. thesis, Delft University of Tech-nology, 2002.PRL 96, 047006 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending3 FEBRUARY 2006047006-4

Four-Qubit Device with Mixed Couplings

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Four-Qubit Device with Mixed Couplings

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