Experimental Demonstration of a Robust and Scalable Flux Qubit

R. Harris,1 J. Johansson,1 A.J. Berkley,1 M.W. Johnson,1 T. Lanting,1 Siyuan Han,2

P. Bunyk,1 E. Ladizinsky,1 T. Oh,1 I. Perminov,1 E. Tolkacheva,1 S. Uchaikin,1 E.Chapple,1 C. Enderud,1 C. Rich,1 M. Thom,1 J. Wang,1 B. Wilson,1 and G. Rose1

1D-Wave Systems, 4401 Still Creek Drive, Burnaby BC, Canada, V5C 6G92Department of Physics and Astronomy, University of Kansas, Lawrence KS, USA, 66045

(Dated: September 24, 2009)

A novel rf-SQUID flux qubit that is robust against fabrication variations in Josephson junctioncritical currents and device inductance has been implemented. Measurements of the persistentcurrent and of the tunneling energy between the two lowest lying states, both in the coherent andincoherent regime, are presented. These experimental results are shown to be in agreement withpredictions of a quantum mechanical Hamiltonian whose parameters were independently calibrated,thus justifying the identification of this device as a flux qubit. In addition, measurements of theflux and critical current noise spectral densities are presented that indicate that these devices withNb wiring are comparable to the best Al wiring rf-SQUIDs reported in the literature thusfar, witha 1/f flux noise spectral density at 1 Hz of 1.3+0.7

−0.5 µΦ0/√

Hz. An explicit formula for converting theobserved flux noise spectral density into a free induction decay time for a flux qubit biased to itsoptimal point and operated in the energy eigenbasis is presented.

I. MOTIVATION

Experimental efforts to develop useful solid state quan-tum information processors have encountered a hostof practical problems that have substantially limitedprogress. While the desire to reduce noise in solid statequbits appears to be the key factor that drives much ofthe recent work in this field, it must be acknowledged thatthere are formidable challenges related to architecture,circuit density, fabrication variation, calibration and con-trol that also deserve attention. For example, a qubitthat is inherently exponentially sensitive to fabricationvariations with no recourse for in-situ correction holdslittle promise in any large scale architecture, even withthe best of modern fabrication facilities. Thus, a qubitdesigned in the absence of information concerning its ul-timate use in a larger scale system may prove to be oflittle utility in the future. In what follows, we present anexperimental demonstration of a novel superconductingflux qubit1 that has been specifically designed to addressseveral issues that pertain to the implementation of alarge scale quantum information processor. While noiseis not the central focus of this article, we nonethelesspresent experimental evidence that, despite its physicalsize and relative complexity, the observed flux noise inthis flux qubit is comparable to the quietest such devicesreported upon in the literature to date.

It has been well established that rf-SQUIDs can beused as qubits given an appropriate choice of device pa-rameters. Such devices can be operated as a flux biasedphase qubit using two intrawell energy levels2 or as aflux qubit using any pair of interwell levels1. This ar-ticle will focus upon an experimental demonstration ofa novel rf-SQUID flux qubit that can be tuned in-situusing solely static flux biases to compensate for fabrica-tion variations in device parameters, both within singlequbits and between multiple qubits. It is stressed that

this latter issue is of critical importance in the develop-ment of useful large scale quantum information proces-sors that could foreseeably involve thousands of qubits3.Note that in this regard, the ion trap approach to build-ing a quantum information processor has a considerableadvantage in that the qubits are intrinsically identical,albeit the challenge is then to characterize and controlthe trapping potential with high fidelity4. While our re-search group’s express interest is in the development of alarge scale superconducting adiabatic quantum optimiza-tion [AQO] processor5,6, it should be noted that manyof the practical problems confronted herein are also ofconcern to those interested in implementing gate modelquantum computation [GMQC] processors7 using super-conducting technologies.

This article is organized as follows: In Section II, atheoretical argument is presented to justify the rf-SQUIDdesign that has been implemented. It is shown that thisdesign is robust against fabrication variations in Joseph-son junction critical current. Second, it is argued why itis necessary to include a tunable inductance in the fluxqubit to account for differences in inductance betweenqubits in a multi-qubit architecture and to compensatefor changes in qubit inductance during operation. There-after, the focus of the article shifts towards an experimen-tal demonstration of the rf-SQUID flux qubit. The archi-tecture of the experimental device and its operation arediscussed in Section III and then a series of experimentsto characterize the rf-SQUID and to highlight its controlare presented in Section IV. Section V contains measure-ments of properties that indicate that this more complexrf-SQUID is indeed a flux qubit. Flux and critical cur-rent noise measurements and a formula for converting themeasured flux noise spectral density into a free induction(Ramsey) decay time are presented in Section VI. A sum-mary of key conclusions is provided in Section VII. De-tailed calculations of rf-SQUID Hamiltonians have beenplaced in the appendices.

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II. RF-SQUID FLUX QUBIT DESIGN

The behavior of most superconducting devices is gov-erned by three types of macroscopic parameters: thecritical currents of any Josephson junctions, the net ca-pacitance across the junctions and the inductance of thesuperconducting wiring. The Hamiltonian for many ofthese devices can generically be written as

H =∑i

[Q2i

2Ci− EJi cos(ϕi)

]+∑n

Un(ϕn − ϕxn)2

2, (1)

where Ci, EJi = IiΦ0/2π and Ii denote the capaci-tance, Josephson energy and critical current of Joseph-son junction i, respectively. The terms in the first sumare readily recognized as being the Hamiltonians of theindividual junctions for which the quantum mechani-cal phase across the junction ϕi and the charge col-lected on the junction Qi obey the commutation relation[Φ0ϕi/2π,Qj ] = i~δij . The index n in the second sum-mation is over closed inductive loops. External fluxesthreading each closed loop, Φxn, have been represented asphases ϕxn ≡ 2πΦxn/Φ0. The quantum mechanical phasedrop experienced by the superconducting condensate cir-culating around any closed loop is denoted as ϕn. Theoverall potential energy scale factor for each closed loopis given by Un ≡ (Φ0/2π)2/Ln. Here, Ln can be eithera geometric inductance from wiring or Josephson induc-tance from large junctions8. Hamiltonian (1) will be usedas the progenitor for all device Hamiltonians that follow.

A. Compound-Compound Josephson JunctionStructure

A sequence of rf-SQUID architectures are depicted inFig. 1. The most primitive version of such a device isdepicted in Fig. 1a, and more complex variants in Figs.1b and 1c. For the single junction rf-SQUID (Fig. 1a), thephase across the junction can be equated to the phasedrop across the body of the rf-SQUID: ϕ1 = ϕq. TheHamiltonian for this device can then be written as

H =Q2q

2Cq+ V (ϕq) ; (2a)

V (ϕq) = Uq

{(ϕq − ϕxq)22

− β cos (ϕq)}

; (2b)

β =2πLqIcq

Φ0, (2c)

with the qubit inductance Lq ≡ Lbody, qubit capacitanceCq ≡ C1 and qubit critical current Icq ≡ I1 in this particu-lar case. If this device has been designed such that β > 1and is flux biased such that ϕxq ≈ π, then the poten-tial energy V (ϕq) will be bistable. With increasing β anappreciable potential energy barrier forms between the

I1j1 jL

I2j2

x

jqx

jqI3j3 jR

I4j4

x

jccjjx

jl jr

I1j1 jcjj I2

j2x

jqx

jq

Lcjj/2

I1j1

jqx

jqLbody

a)

b)

c)

Lbody

Lbody

Lcjj/2

Lccjj/2Lccjj/2

FIG. 1: (color online) a) A single junction rf-SQUID qubit.b) Compound Josephson Junction (CJJ) rf-SQUID qubit. c)Compound-Compound Josephson Junction (CCJJ) rf-SQUIDqubit. Junction critical currents Ii and junction phases ϕi(1 ≤ i ≤ 4) as noted. Net device phases are denoted as ϕα,where α ∈ (`, r, q). External fluxes, Φxn, are represented asphases ϕxn ≡ 2πΦxn/Φ0, where n ∈ (L,R, cjj, ccjj, q). Induc-tance of the rf-SQUID body, CJJ loop and CCJJ loop aredenoted as Lbody, Lcjj and Lccjj, respectively.

two local minima of V (ϕq), through which the two low-est lying states of the rf-SQUID may couple via quantumtunneling. It is these two lowest lying states, which areseparated from all other rf-SQUID states by an energy oforder of the rf-SQUID plasma energy ~ωp ≡ ~/

√LqC1,

that form the basis of a qubit. One can write an effectivelow energy version of Hamiltonian (2a) as9

Hq = −12

[εσz + ∆qσx] , (3)

where ε = 2∣∣Ipq ∣∣ (Φxq − Φ0/2

),∣∣Ipq ∣∣ is the magnitude of

the persistent current that flows about the inductive qloop when the device is biased hard [ε� ∆q] to one sideand ∆q represents the tunneling energy between the oth-erwise degenerate counter-circulating persistent currentstates at Φxq = Φ0/2.

A depiction of the one-dimensional potential energyand the two lowest energy states of an rf-SQUID at de-generacy (Φxq = Φ0/2) for nominal device parameters isshown in Fig. 2. In this diagram, the ground and first ex-cited state are denoted by |g〉 and |e〉, respectively. Thesetwo energy levels constitute the energy eigenbasis of aflux qubit. An alternate representation of these states,which is frequently referred to as either the flux or persis-tent current basis, can be formed by taking the symmet-ric and antisymmetric combinations of the energy eigen-states: |↓〉 = (|g〉+ |e〉) /

√2 and |↑〉 = (|g〉 − |e〉) /

√2,

3

!5 !2.5 0 2.5 5

0

5

10

15

20

25

(!q !!0/2)/Lq (µA)

Ener

gy/h

(GH

z)

|"#|$#

|g#|e#

FIG. 2: (color online) Depiction of the two lowest lying statesof an rf-SQUID at degeneracy (ε = 0) with nomenclature forthe energy basis (|g〉,|e〉) and flux basis (|↓〉,|↑〉) as indicated.

which yield two roughly gaussian shaped wavefunctionsthat are centered about each of the wells shown in Fig. 2.The magnitude of the persistent current used in Eq. (3)is then defined by

∣∣Ipq ∣∣ ≡ |〈↑| (Φq − Φ0/2) /Lq |↑〉|. Thetunneling energy is given by ∆q = 〈e|Hq |e〉 − 〈g|Hq |g〉.

The aforementioned dual representation of the statesof a flux qubit allows two distinct modes of operation ofthe flux qubit as a binary logical element with a logicalbasis defined by the states |0〉 and |1〉. In the first mode,the logical basis is mapped onto the energy eigenbasis:|0〉 → |g〉 and |1〉 → |e〉. This mode is useful for optimiz-ing the coherence times of flux qubits as the dispersionof Hamiltonian (3) is flat as a function of Φxq to first or-der for ε ≈ 0, thus providing some protection from theeffects of low frequency flux noise10. However, this is nota convenient mode of operation for implementing inter-actions between flux qubits11,12. In the second mode,the logical basis is mapped onto the persistent currentbasis: |0〉 → |↓〉 and |1〉 → |↑〉. This mode of opera-tion facilitates the implementation of inter-qubit interac-tions via inductive couplings, but does so at the expenseof coherence times. GMQC schemes exist that attemptto leverage the benefits of both of the above modes ofoperation13,14,15. On the other hand, those interestedin implementing AQO strictly use the second mode ofoperation cited above. This, very naturally, leads tosome interesting properties: First and foremost, in thecoherent regime at ε = 0, the groundstate maps onto|g〉 = (|0〉+ |1〉) /

√2, which implies that it is a super-

position state with a fixed phase between componentsin the logical basis. Second, the logical basis is not co-incident with the energy eigenbasis, except in the ex-treme limit ε/∆q � 1. As such, the qubit should not beviewed as an otherwise free spin-1/2 in a magnetic field,rather it maps onto an Ising spin subjected to a magneticfield with both a longitudinal (Bz → ε) and a transverse(Bx → ∆q) component16. In this case, it is the competi-

tion between ε and ∆q which dictates the relative ampli-tudes of |↓〉 and |↑〉 in the groundstate wavefunction |g〉,thereby enabling logical operations that make no explicituse of the excited state |e〉. This latter mode of oper-ation of the flux qubit has connections to the fields ofquantum magnetism17 and optimization theory18. Inter-estingly, systems of coupled flux qubits that are operatedin this mode bear considerable resemblance to Feynman’soriginal vision of how to build a quantum computer19.

While much seminal work has been done on single junc-tion and the related 3-Josephson junction rf-SQUID fluxqubit20,21,22,23,24,25,26,27,28,29,30,31, it has been recognizedthat such devices would be impractical in a large scalequantum information processor as their properties areexceptionally sensitive to fabrication variations. In par-ticular, in the regime EJ1 � ~ωp, ∆q ∝ exp(−~ωp/EJ1).Thus, it would be unrealistic to expect a large scale pro-cessor involving a multitude of such devices to yield fromeven the best superconducting fabrication facility. More-over, implementation of AQO requires the ability to ac-tively tune ∆q from being the dominant energy scale inthe qubit to being essentially negligible during the courseof a computation. Thus the single junction rf-SQUID fluxqubit is of limited practical utility and has passed out offavor as a prototype qubit.

The next step in the evolution of the single junctionflux qubit and related variants was the compound Joseph-son junction (CJJ) rf-SQUID, as depicted in Fig. 1b.This device was first reported upon by Han, Lapointeand Lukens32 and was the first type of flux qubit to dis-play signatures of quantum superposition of macroscopicstates33. The CJJ rf-SQUID has been used by other re-search groups13,34,35 and a related 4-Josephson junctiondevice has been proposed20,21. The CJJ rf-SQUID fluxqubit and related variants have reappeared in a gradio-metric configuration in more recent history14,36,37. Here,the single junction of Fig. 1a has been replaced by a fluxbiased dc-SQUID of inductance Lcjj that allows one totune the critical current of the rf-SQUID in-situ. Let theapplied flux threading this structure be denoted by Φxcjj.It is shown in Appendix A that the Hamiltonian for thissystem can be written as

H =∑n

[Q2n

2Cn+ Un

(ϕn − ϕxn)2

2

]−Uqβeff cos

(ϕq − ϕ0

q

),

(4a)where the sum is over n ∈ {q, cjj}, Cq ≡ C1+C2, 1/Ccjj ≡1/C1 +1/C2 and Lq ≡ Lbody +Lcjj/4. The 2-dimensionalpotential energy in Hamiltonian (4a) is characterized by

βeff = β+ cos(ϕcjj

2

)√1 +

[β−β+

tan(ϕcjj/2)]2

; (4b)

ϕ0q ≡ 2π

Φ0q

Φ0= − arctan

(β−β+

tan (ϕcjj/2))

; (4c)

β± ≡ 2πLq (I1 ± I2) /Φ0 . (4d)

4

Note that if cos (ϕcjj/2) < 0, then βeff < 0 in Hamiltonian(4a). This feature provides a natural means of shiftingthe qubit degeneracy point from ϕxq = π, as in the singlejunction rf-SQUID case, to ϕxq ≈ 0. It has been assumedin all that follows that this π-shifted mode of operationof the CCJ rf-SQUID has been invoked.

Hamiltonian (4a) is similar to that of a single junc-tion rf-SQUID modulo the presence of a ϕcjj-dependenttunnel barrier through βeff and an effective critical cur-rent Icq ≡ I1 + I2. For Lcjj/Lq � 1 it is reasonable toassume that ϕcjj ≈ 2πΦxcjj/Φ0. Consequently, the CJJrf-SQUID facilitates in-situ tuning of the tunneling en-ergy through Φxcjj. While this is clearly desirable, onedoes pay for the additional flexibility by adding morecomplexity to the rf-SQUID design and thus more po-tential room for fabrication variations. The minimumachievable barrier height is ultimately limited by any socalled junction asymmetry which leads to finite β−. Inpractice, for β−/β+ = (I1 − I2)/(I1 + I2) . 0.05, thiseffect is of little concern. However, a more insidious ef-fect of junction asymmetry can be seen via the changeof variables ϕq − ϕ0

q → ϕq in Eq. (4a), namely an ap-parent Φxcjj-dependent flux offset: Φxq → Φxq − Φ0

q(Φxcjj).

If the purpose of the CJJ is to simply allow the experi-mentalist to target a particular ∆q, then the presence ofΦ0q(Φ

xcjj) can be readily compensated via the application

of a static flux offset. On the other hand, any mode ofoperation that explicitly requires altering ∆q during thecourse of a quantum computation13,14,15,35,38,39 wouldalso require active compensation for what amounts toa nonlinear crosstalk from Φxcjj to Φxq . While it may bepossible to approximate this effect as a linear crosstalkover a small range of Φxcjj if the junction asymmetry issmall, one would nonetheless need to implement precisetime-dependent flux bias compensation to utilize the CJJrf-SQUID as a flux qubit in any quantum computationscheme. While this may be feasible in laboratory scalesystems, it is by no means desirable nor practical on alarge scale quantum information processor.

A second problem with the CJJ rf-SQUID flux qubitis that one cannot homogenize the qubit parameters

∣∣Ipq ∣∣and ∆q between a multitude of such devices that pos-sess different β± over a broad range of Φxcjj. While onecan accomplish this task to a limited degree in a per-turbative manner about carefully chosen CJJ biases foreach qubit40, the equivalence of

∣∣Ipq ∣∣ and ∆q betweenthose qubits will be approximate at best. Therefore, theCJJ rf-SQUID does not provide a convenient means ofaccommodating fabrication variations between multipleflux qubits in a large scale processor.

Given that the CJJ rf-SQUID provides additional flex-ibility at a cost, it is by no means obvious that one candesign a better rf-SQUID flux qubit by adding even morejunctions. Specifically, it is desirable to have a devicewhose imperfections can be mitigated purely by the ap-plication of time-independent compensation signals. Thenovel rf-SQUID topology shown in Fig. 1c, hereafter re-ferred to as the compound-compound Josephson junction

(CCJJ) rf-SQUID, satisfies this latter constraint. Here,each junction of the CJJ in Fig. 1b has been replaced by adc-SQUID, which will be referred to as left (L) and right(R) minor loops, and will be subjected to external fluxbiases ΦxL and ΦxR, respectively. The role of the CJJ loopin Fig. 1b is now played by the CCJJ loop of inductanceLccjj which will be subjected to an external flux bias Φxccjj.It is shown in Appendix B that if one chooses static val-ues of ΦxL and ΦxR such that the net critical currents ofthe minor loops are equal, then it can be described by aneffective two-dimensional Hamiltonian of the form

H =∑n

[Q2n

2Cn+ Un

(ϕn − ϕxn)2

2

]−Uqβeff cos

(ϕq − ϕ0

q

),

(5a)where the sum is over n ∈ {q, ccjj}, Cq ≡ C1 + C2 +C3 +C4, 1/Cccjj ≡ 1/(C1 +C2) + 1/(C3 +C4) and Lq ≡Lbody + Lccjj/4. The effective 2-dimensional potentialenergy in Hamiltonian (5a) is characterized by

βeff = β+(ΦxL,ΦxR) cos

(ϕccjj − ϕ0

ccjj

2

), (5b)

where β+(ΦxL,ΦxR) = 2πLqIcq (ΦxL,Φ

xR)/Φ0 with

Icq (ΦxL,ΦxR) ≡ (I1+I2) cos

(πΦxLΦ0

)+(I3+I4) cos

(πΦxRΦ0

).

Given an appropriate choice of ΦxL and ΦxR, the q and ccjjloops will possess apparent flux offsets of the form

Φ0q =

Φ0ϕ0q

2π=

Φ0L + Φ0

R

2; (5c)

Φ0ccjj =

Φ0ϕ0ccjj

2π= Φ0

L − Φ0R , (5d)

where Φ0L(R) is given by Eq. (B3c), which is purely a func-

tion of ΦxL(R) and junction critical currents. As such,the apparent flux offsets are independent of Φxccjj. Un-der such conditions, we deem the CCJJ to be balanced.Given that the intended mode of operation is to hold ΦxLand ΦxR constant, then the offset phases ϕ0

L and ϕ0R will

also be constant. The result is that Hamiltonian (5a)for the CCJJ rf-SQUID becomes homologous to that ofan ideal CJJ rf-SQUID [β− = 0 in Eqs. (4b) and (4c)]with apparent static flux offsets. Such static offsets canreadily be calibrated and compensated for in-situ usingeither analog control lines or on-chip programmable fluxsources41. For typical device parameters and junctionvariability on the order of a few percent, these offsets willbe ∼ 1→ 10 mΦ0. Equations 5a-5d with Φ0

q = Φ0ccjj = 0

will be referred to hereafter as the ideal CCJJ rf-SQUIDmodel.

The second advantage of the CCJJ rf-SQUID is thatone can readily accommodate for variations in criti-cal current between multiple flux qubits. Note that inEq. (5b) that the maximum height of the tunnel barrier

5

is governed by β+(ΦxL,ΦxR) ≡ βL(ΦxL) + βR(ΦxR), where

βL(R) is given by Eq. (B3c). One is free to choose anypair of (ΦxL,Φ

xR) such that βL(ΦxL) = βR(ΦxR), as dictated

by Eq. (B6). Consequently, β+ = 2βR(ΦxR) in Eq. (5b).One can then choose ΦxR, which then dictates ΦxL, so asto homogenize β+ between multiple flux qubits. The re-sults is a set of nominally uniform flux qubits where theparticular choice of (ΦxL,Φ

xR) for each qubit merely re-

sults in unique static flux offsets Φ0q and Φ0

ccjj for eachdevice.

To summarize up to this point, the CCJJ rf-SQUID isrobust against Josephson junction fabrication variationsboth within an individual rf-SQUID and between a plu-rality of such devices. The variations can be effectivelytuned out purely through the application of static flux bi-ases, which is of considerable advantage when envisioningthe implementation of large scale quantum informationprocessors that use flux qubits.

B. L-tuner

The purpose of the CCJJ structure was to provide ameans of coming to terms with fabrication variations inJosephson junctions both within individual flux qubitsand between sets of such devices. However, junctions arenot the only key parameter that may vary between de-vices, nor are fabrication variations responsible for all ofthe potential variation. In particular, it has been exper-imentally demonstrated that the inductance of a qubitLq that is connected to other qubits via tunable mutualinductances is a function of the coupler configuration42.Let the bare inductance of the qubit in the presence ofno couplers be represented by L0

q and the mutual induc-tance between the qubit and coupler i be represented byMco,i. If the coupler possesses a first order susceptibilityχi, as defined in Ref. 42, then the net inductance of thequbit can be expressed as

Lq = L0q −

∑i

M2co,iχi . (6)

Given that qubit properties such as ∆q can be exponen-tially sensitive to variations in Lq, then it is undesirableto have variations in Lq between multiple flux qubits orto have Lq change during operation. This could havea deleterious impact upon AQO in which it is typicallyassumed that all qubits are identical and they are in-tended to be annealed in unison5. From the perspectiveof GMQC, one could very well attempt to compensate forsuch effects in a CJJ or CCJJ rf-SQUID flux qubit by ad-justing the tunnel barrier height to hold ∆q constant, butdoing so alters

∣∣Ipq ∣∣, which then alters the coupling of thequbit to radiative sources, thus demanding further com-pensation. Consequently, it also makes sense from theperspective of GMQC that one find a means of render-ing Lq uniform between multiple qubits and insensitiveto the settings of inductive coupling elements.

!1

!n

"LTx

Mco,1 Mco,n

FIG. 3: (color online) A CCJJ rf-SQUID with L-tuner con-nected to multiple tunable inductive couplers via transformerswith mutual inductances Mco,i and possessing susceptibilitiesχi. The L-tuner is controlled via the external flux bias ΦxLT

In order to compensate for variations in Lq, we haveinserted a tunable Josephson inductance8 into the CCJJrf-SQUID body, as depicted in Fig. 3. We refer to thiselement as an inductance (L)-tuner. This relatively sim-ple element comprises a dc-SQUID whose critical currentvastly exceeds that of the CCJJ structure, thus ensuringnegligible phase drop across the L-tuner. Assuming thatthe inductance of the L-tuner wiring is negligible, theL-tuner modifies Eq. (6) in the following manner:

Lq = L0q −

∑i

M2co,iχi +

LJ0

cos(πΦxLT /Φ0), (7)

where LJ0 ≡ Φ0/2πIcLT , IcLT is the net critical current ofthe two junctions in the L-tuner and ΦxLT is an externallyapplied flux bias threading the L-tuner loop. For modestflux biases such that IcLT cos(πΦxLT /Φ0)� Icq , Eq. (7) isa reliable model of the physics of the L-tuner.

Given that the L-tuner is only capable of augment-ing Lq, one can only choose to target Lq > L0

q −∑iM

2co,iχ

AFMi +LJ0, where χAFM

i is the maximum anti-ferromagnetic (AFM) susceptibility of inter-qubit coupleri. In practice, we choose to restrict operation of the cou-plers to the range −χAFM

i < χi < χAFMi such that the

maximum qubit inductance that will be encountered isLq > L0

q +∑iM

2co,iχ

AFMi +LJ0. We then choose to pre-

bias ΦxLT for each qubit to match the maximum realizedLq ≡ Lmax

q amongst a set of flux qubits. Thereafter, onecan hold Lq = Lmax

q as couplers are adjusted by invertingEq. (7) to solve for an appropriate value of ΦxLT . Thus,the L-tuner provides a ready means of compensating forsmall variations in Lq between flux qubits and to holdLq constant as inductive inter-qubit coupling elementsare adjusted.

III. DEVICE ARCHITECTURE, FABRICATIONAND READOUT OPERATION

To test the CCJJ rf-SQUID flux qubit, we fabricateda circuit containing 8 such devices with pairwise inter-actions mediated by a network of 16 in-situ tunable CJJrf-SQUID inter-qubit couplers42. Each qubit was also

6

q0 q1 q2 q3

q4

q5

q6

q7RO

CCJJLTCO

a)

500 nmb)

q0 q1 q2 q3

q4

100 µm

c)

FIG. 4: (color online) a) High level schematic of the analogdevices on the device reported upon herein. Qubits are repre-sented as light grey elongated objects and denoted as q0 . . . q7.One representative readout (RO), CCJJ and L-tuner (LT )each have been indicated in dashed boxes. Couplers (CO)are represented as dark objects located at the intersections ofthe qubit bodies. b) SEM of a cross-section of the fabrica-tion profile. Metal layers denoted as BASE, WIRA, WIRBand WIRC. Insulating layers labeled as SiO2. Topmost in-sulator has not been planarized in this test structure, but isplanarized in the full circuit process. An example via (VIA),Josephson junction (JUNC, AlOx/Al) and resistor (RESI) arealso indicated. c) Optical image of a portion of a device com-pleted up to WIRB. Portions of qubits q0 . . . q3 and the en-tirety of q4 are visible.

coupled to its own dedicated quantum flux parametron(QFP)-enabled readout43. A high level schematic of thedevice architecture is shown in Fig. 4a. External flux bi-ases were provided to target devices using a sparse com-bination of analog current bias lines to facilitate devicecalibration and an array of single flux quantum (SFQ)

based on-chip programmable control circuitry (PCC)41.The device was fabricated from an oxidized Si wafer

with Nb/Al/Al2O3/Nb trilayer junctions and four Nbwiring layers separated by planarized plasma enhancedchemical vapor deposited SiO2. A scanning electron mi-crograph of the process cross-section is shown in Fig. 4b.The Nb metal layers have been labeled as BASE, WIRA,WIRB and WIRC. The flux qubit wiring was primarilylocated in WIRB and consisted of 2µm wide leads ar-ranged as an approximately 900µm long differential mi-crostrip located 200 nm above a groundplane in WIRA.CJJ rf-SQUID coupler wiring was primarily located inWIRC, stacked on top of the qubit wiring to provideinductive coupling. PCC flux storage loops were im-plemented as stacked spirals of 13-20 turns of 0.25µmwide wiring with 0.25µm separation in BASE and WIRA(WIRB). Stored flux was picked up by one-turn washersin WIRB (WIRA) and fed into transformers for flux-biasing devices. External control lines were mostly lo-cated in BASE and WIRA. All of these control elementsresided below a groundplane in WIRC. The groundplaneunder the qubits and over the PCC/external control lineswere electrically connected using extended vias in WIRBso as to form a nearly continuous superconducting shieldbetween the analog devices on top and the bias circuitrybelow. To provide biases to target devices with mini-mal parasitic crosstalk, transformers for biasing qubits,couplers, QFPs and dc-SQUIDs using bias lines and/orPCC elements were enclosed in superconducting boxeswith BASE and WIRC forming the top and bottom, re-spectively, and vertical walls formed by extended vias inWIRA and WIRB. Minimal sized openings were placedin the vertical walls through which the bias and targetdevice wiring passed at opposing ends of each box.

An optical image of a portion of a device completedup to WIRB is shown in Fig. 4c. Qubits are visible aselongated objects, WIRB PCC spirals are visible as darkrectangles and WIRB washers are visible as light rect-angles with slits. Note that the extrema of the CCJJrf-SQUID qubits are terminated in unused transformers.These latter elements allow this 8-qubit unit cell to betiled in a larger device with additional inter-qubit CJJcouplers providing the connections between unit cells.

We have studied the properties of all 8 CCJJ rf-SQUIDflux qubits on this chip in detail and report upon one suchdevice herein. To clearly establish the lingua franca ofour work, we have depicted a portion of the multi-qubitcircuit in Fig. 5a. Canonical representations of the exter-nal flux biases needed to operate a qubit, a coupler anda QFP-enabled readout are labeled on the diagram. Thefluxes ΦxL, ΦxR, ΦxLT and Φxco were only ever subjected todc levels in our experiments that were controlled by thePCC. The remaining fluxes and readout current biaseswere driven by a custom-built 128 channel room temper-ature current source. The mutual inductance betweenqubit and QFP (Mq−qfp), between QFP and dc-SQUID(Mqfp-ro), qubit and coupler (Mco,i) and Φxco-dependentinter-qubit mutual inductance (Meff) have also been in-

7

0

0

-!0/2

~!0/3

iro(t)

!ro(t)x

!qfp(t)x

x!latch(t)

t=0 trep

0

b)

-!0 }}

dc-SQUID

QFP

a) Qubit Meff

Coupler

{

QFP

dc-SQUID

!cox

iro

!qfpx

!qx

!latchx

!ccjjx

!rox

!Lx!Rx

Mq-qfp

Mqfp-ro

Mco,i

!LTx

1 0.5 0 0.5 10

0.25

0.5

0.75

1c)

!qfpx (m!0)

Pqfp

FIG. 5: (color online) a) Schematic representation of a por-tion of the circuit reported upon herein. Canonical repre-sentations of all externally controlled flux biases Φxα, readoutcurrent bias iro and key mutual inductances Mα are indi-cated. b) Depiction of latching readout waveform sequence.c) Example QFP state population measurement as a functionof the dc level Φxqfp with no qubit signal. Data have been fitto Eq. (8).

dicated. Further details concerning cryogenics, magneticshielding and signal filtering have been discussed in pre-vious publications41,42,43,44.

Since much of what follows depends upon a clear un-derstanding of our novel QFP-enabled readout mecha-nism, we present a brief review of its operation herein.The flux and readout current waveform sequence in-volved in a single-shot readout is depicted in Fig. 5b.Much like the CJJ qubit44, the QFP can be adiabat-ically annealed from a state with a monostable poten-tial (Φxlatch = −Φ0/2) to a state with a bistable poten-tial (Φxlatch = −Φ0) that supports two counter-circulatingpersistent current states. The matter of which persistentcurrent state prevails at the end of an annealing cycledepends upon the sum of Φxqfp and any signal from thequbit mediated via Mq−qfp. The state of the QFP is thendetermined with high fidelity using a synchronized fluxpulse and current bias ramp applied to the dc-SQUID.The readout process was typically completed within arepetition time trep < 50µs.

An example trace of the population of one of the QFPpersistent current states Pqfp versus Φxqfp, obtained usingthe latching sequence depicted in Fig. 5b, is shown inFig. 5c. This trace was obtained with the qubit potentialheld monostable (Φxccjj = −Φ0/2) such that it presented

minimal flux to the QFP and would therefore not influ-ence Pqfp. The data have been fit to the phenomenolog-ical form

Pqfp =12

[1− tanh

(Φxqfp − Φ0

qfp

w

)](8)

with width w ∼ 0.18 mΦ0 for the trace shown therein.When biased with constant Φxqfp = Φ0

qfp, which we referto as the QFP degeneracy point, this transition in thepopulation statistics can be used as a highly nonlinearflux amplifier for sensing the state of the qubit. Giventhat Mq−qfp = 6.28 ± 0.01 pH for the devices reportedupon herein and that typical qubit persistent currentsin the presence of negligible tunneling

∣∣Ipq ∣∣ & 1µA, thenthe net flux presented by a qubit was 2Mq−qfp

∣∣Ipq ∣∣ &6 mΦ0, which far exceeded w. By this means one canachieve the very high qubit state readout fidelity reportedin Ref. 43. On the other hand, the QFP can be used as alinearized flux sensor by engaging Φxqfp in a feedback loopand actively tracking Φ0

qfp. This latter mode of operationhas been used extensively in obtaining many of the resultspresented herein.

IV. CCJJ RF-SQUID CHARACTERIZATION

The purpose of this section is to present measurementsthat characterize the CCJJ, L-tuner and capacitance ofa CCJJ rf-SQUID. All measurements shown herein havebeen made with a set of standard bias conditions givenby ΦxL = 98.4 mΦ0, ΦxR = −89.3 mΦ0, ΦxLT = 0.344 Φ0

and all inter-qubit couplers tuned to provide Meff = 0,unless indicated otherwise. The logic behind this partic-ular choice of bias conditions will be explained in whatfollows. This section will begin with a description ofthe experimental methods for extracting Lq and Icq frompersistent current measurements. Thereafter, data thatdemonstrate the performance of the CCJJ and L-tunerwill be presented. Finally, this section will conclude withthe determination of Cq from macroscopic resonant tun-neling data.

A. High Precision Persistent CurrentMeasurements

The most direct means of obtaining information re-garding a CCJJ rf-SQUID is to measure the persistentcurrent

∣∣Ipq ∣∣ as a function of Φxccjj. A reasonable firstapproach to measuring this quantity would be to sequen-tially prepare the qubit in one of its persistent currentstates and then the other, and use the QFP in feedbackmode to measure the difference in flux sensed by the QFP,which equals 2Mq−qfp

∣∣Ipq ∣∣. A fundamental problem withthis approach is that it is sensitive to low frequency (LF)flux noise45, which can alter the background flux experi-enced by the QFP between the sequential measurements.

8

0

0

0

-!0/2

-!0/2

~!0/3

+!qi

-!qi

!m

!m+"!m

!m#"!m

iro(t)

!ccjj(t)x

!q(t)x

!ro(t)x

!qfp(t)x

x!latch(t)

t=0 trep

a)

-!0

-!0

Qubit}

} QFP

}dc-SQUID

!ccjjx

b)

11

2

2

3

3

!qfpx

Pqfp

!m !m+"!m!m#"!m

Qubit Init.=+!qiQubit Init.=-!q

i

slope=1/2w 1

0

slope=1/2w

FIG. 6: (color online) a) Low frequency flux noise rejectingqubit persistent current measurement sequence. Waveformsshown are appropriate for measuring

˛Ipq

`Φxccjj

´˛for −Φ0 ≤

Φxccjj ≤ 0. The Φxccjj waveform can be offset by integer Φ0

to measure the periodic behavior of this quantity. Typicalrepetition time is trep ∼ 1 ms. b) Depiction of QFP transitionand correlated changes in QFP population statistics for thetwo different qubit initializations.

For a typical measurement with our apparatus, the act oflocating a single QFP degeneracy point to within 20µΦ0

takes on the order of 1 s, which means that two sequentialmeasurements would only be immune to flux noise below0.5 Hz. We have devised a LF flux noise rejection schemethat takes advantage of the fact that such noise will gen-erate a correlated shift in the apparent degeneracy pointsif the sequential preparations of the qubit can be inter-leaved with single-shot measurements that are performedin rapid succession. If these measurements are performedwith repetition time trep ∼ 1 ms, then the measurementswill be immune to flux noise below ∼ 1 kHz.

A depiction of the LF flux noise rejecting persistentcurrent measurement sequence is shown in Fig. 6a. Thewaveforms comprise two concatenated blocks of sequen-tial annealing of the qubit to a target Φxccjj in the presenceof an alternating polarizing flux bias ±Φiq followed bylatching and single-shot readout of the QFP. The QFP

flux bias is engaged in a differential feedback mode inwhich it is pulsed in alternating directions by an amountδΦm about a mean level Φm. The two single-shot mea-surements yield binary results for the QFP state andthe difference between the two binary results is recorded.Gathering a statistically large number of such differentialmeasurements then yields a differential population mea-surement δPqfp. Conceptually, the measurement worksin the manner depicted in Fig. 6b: the two different ini-tializations of the qubit move the QFP degeneracy pointto some unknown levels Φ0

m± δΦ0m, where Φ0

m representsthe true mean of the degeneracy points at any given in-stant in time and 2δΦ0

m is the true difference in degener-acy points that is independent of time. Focusing on fluxbiases that are close to the degeneracy point, one canlinearize Eq. (8):

Pqfp,± ≈12

+1

2w[Φxqfp −

(Φ0m ± δΦ0

m

)]. (9)

Assuming that the rms LF flux noise Φn � w and thatone has reasonable initial guesses for Φ0

m ± δΦ0m, then

the use of the linear approximation should be justified.Applying Φxqfp = Φm ± δΦm and sufficient repetitions ofthe waveform pattern shown in Fig. 6a, the differentialpopulation will then be of the form

δPqfp = Pqfp,+ − Pqfp,− =1w

[δΦm + δΦ0

m

], (10)

which is independent of Φm and Φ0m. Note that the above

expression contains only two independent variables, wand δΦ0

m, and that δPqfp is purely a linear function ofδΦm. By sampling at three values of δΦm, as depicted bythe pairs of numbered points in Fig. 6b, the independentvariables in Eq. (10) will be overconstrained, thus readilyyielding δΦ0

m. One can then infer the qubit persistentcurrent as follows:∣∣Ipq ∣∣ =

2δΦ0m

2Mq−qfp=

δΦ0m

Mq−qfp. (11)

Example measurements of∣∣Ipq ∣∣ (Φxccjj

)are shown in

Fig. 7. These data, for which 1.5 . |βeff| . 2.5, havebeen fit to the ideal CCJJ rf-SQUID model by findingthe value of ϕq ≡ ϕmin

q for which the potential in Eq. (5a)is minimized: ∣∣Ipq ∣∣ =

Φ0

2π

∣∣ϕminq − ϕxq

∣∣Lq

. (12)

The best fit shown in Fig. 7 was obtained with Lq =265.4 ± 1.0 pH, Lccjj = 26 ± 1 pH and Icq = 3.103 ±0.003µA. For comparison, we had estimated Lq = 273 pHat the standard bias condition for ΦxLT and Lccjj = 20 pHfrom design.

In practice, we have found that the LF flux noise reject-ing method of measuring

∣∣Ipq ∣∣ effectively eliminates anyobservable 1/f component in that measurement’s noisepower spectral density, to within statistical error. Fi-nally, it should be noted that the LF flux noise rejecting

9

!1.5 !1 !0.5 0 0.5 1 1.51

1.5

2

2.5

3

!xccjj /!0

! ! Ip q

! !(µ

A)

DataFit

FIG. 7: (color online) Example measurements of˛Ipq

`Φxccjj

´˛.

method is applicable to any measurement of a differencein flux sensed by a linearized detector. In what followsherein, we have made liberal use of this technique to cali-brate a variety of quantities in-situ using both QFPs andother qubits as flux detectors.

B. CCJJ

In this subsection, the CCJJ has been characterized asa function of ΦxL and ΦxR with all other static flux biasesset to the standard bias condition cited above. Referringto Eq. (B4c), it can be seen that the qubit degeneracypoint Φ0

q is a function of Φxccjj through γ0 if the CCJJhas not been balanced. To accentuate this functionaldependence, one can anneal the CCJJ rf-SQUID withΦxccjj waveforms of opposing polarity about a minimumin |βeff|, as found at Φxccjj = −Φ0/2. The expectation isthat the apparent qubit degeneracy points will be anti-symmetric about the mean given by setting γ0 = 0 inEq. (B4c). The waveform sequence for performing a dif-ferential qubit degeneracy point measurement is depictedin Fig. 8. In this case, the QFP is used as a latchingreadout and the qubit acts as the linearized detector ofits own apparent annealing polarization-dependent fluxoffset. As with the

∣∣Ipq ∣∣ measurement described above,this LF flux noise rejecting procedure returns a differ-ence in apparent flux sensed by the qubit and not theabsolute flux offsets.

To find balanced pairs of (ΦxL,ΦxR) in practice, we set

ΦxR to a constant and used the LF flux noise rejectingprocedure inside a software feedback loop that controlledΦxL to null the difference in apparent degeneracy point toa precision of 20µΦ0. Balanced pairs of (ΦxL,Φ

xR) are

plotted in Fig. 9a. These data have been fit to B6 usingβ−/β+ as a free parameter. The best fit shown in Fig. 9awas obtained with 1 − βR,+/βL,+ = (4.1 ± 0.3) × 10−3,which then indicates an approximately 0.4% asymmetry

0

0

-!0/2

-!0/2

~!0/3

!m!m+"!m

!m#"!m

iro(t)

!ccjj(t)x

!q(t)x

!ro(t)x

!qfp(t)x

x!latch(t)

t=0 trep

0

Qubit}

} QFP

}

dc-SQUID

0

-!0

-!0

FIG. 8: (color online) Schematic of low frequency noise re-jecting differential qubit degeneracy point measurement se-quence. The qubit is annealed with a Φxccjj signal of opposingpolarity in the two frames and the qubit flux bias is controlledvia feedback.

between the pairs of junctions in the L and R loops.A demonstration of how the CCJJ facilitates tuning of

Icq is shown in Fig. 9b. Here, the measurable consequenceof altering Icq that was recorded was a change in

∣∣Ipq ∣∣ atΦxccjj = −Φ0. These data have been fit to the ideal CCJJrf-SQUID model with the substitution

Icq (ΦxR,ΦxL) = I0

c cos(πΦxRΦ0

)(13)

and using the values of Lccjj and Lq obtained from fittingthe data in Fig. 7, but treating I0

c as a free parameter.Here, ΦxL on the left side of Eq. (13) is a function of ΦxRper the CCJJ balancing condition Eq. (B6). The best fitwas obtained with I0

c = 3.25±0.01µA. This latter quan-tity agrees well with the designed critical current of four0.6µm diameter junctions in parallel of 3.56 µA. Thus,it is possible to target a desired Icq by using Eq. (13) toselect ΦxR and then Eq. (B6) to select ΦxL. The stan-dard bias conditions for ΦxL and ΦxR quoted previouslywere chosen so as to homogenize Icq amongst the 8 CCJJrf-SQUIDs on this particular chip.

C. L-Tuner

To characterize the L-tuner, we once again turned tomeasurements of

∣∣Ipq (Φxccjj = −Φ0)∣∣, but this time as a

function of ΦxLT . Persistent current results were then

10

!0.2 !0.1 0 0.1 0.20

0.04

0.08

0.12

0.16

0.2

!xR/!0

!x L/!

0a)

Data

Fit

!0.2 !0.1 0 0.1 0.22.4

2.5

2.6

2.7

2.8

!xR/!0

! ! Ip q

! !(µ

A)

b) DataFit

FIG. 9: (color online) a) Minor lobe balancing data and fitto Eq. (B6). The standard bias conditions for ΦxL and ΦxR areindicated by dashed lines. b)

˛Ipq (Φxccjj = −Φ0)

˛versus ΦxR,

where ΦxL has been chosen using Eq. (B6). The data havebeen fit to the ideal CCJJ rf-SQUID model. The standardbias condition for ΦxR and the resultant

˛Ipq (Φxccjj = −Φ0)

˛are

indicated by dashed lines.

used to infer δLq = Lq(ΦxLT ) − Lq(ΦxLT = 0) using theideal CCJJ rf-SQUID model with Lccjj and Icq held con-stant and treating Lq as a free parameter. The experi-mental results are plotted in Fig. 10a and have been fitto

δLq =LJ0

cos (πΦxLT /Φ0), (14)

and the best fit was obtained with LJ0 = 19.60±0.04 pH.Modeling this latter parameter as Lq0 = Φ0/2πIcLT , weestimate IcLT = 16.79± 0.04µA, which is close to the de-sign value of 16.94µA. The standard bias condition forΦxLT was chosen so as to homogenize Lq amongst the 8CCJJ rf-SQUID flux qubits on this chip and to provideadequate bipolar range to accommodate inter-qubit cou-pler operation.

To demonstrate the use of the L-tuner, we have probeda worst-case scenario in which four CJJ rf-SQUID cou-

!0.4 !0.2 0 0.2 0.40

4

8

12

16

!xLT /!0

!Lq

(pH

)

a)

Data

Fit

0 0.1 0.2 0.3 0.4 0.5 0.62.58

2.6

2.62

2.64

2.66

2.68

!xco/!0

! ! Ip q

! !(µ

A)

b)

Compensation O", Data

Compensation On, Data

Compensation O", Predicted

Compensation On, Predicted

FIG. 10: (color online) a) L-tuner calibration and fit toEq. (14). The standard bias condition for ΦxLT and the resul-tant δLq are indicated by dashed lines. b) Observed changein maximum qubit persistent current with and without activeL-tuner compensation and predictions for both cases.

plers connected to the CCJJ rf-SQUID in question aretuned in unison. Each of the couplers had been in-dependently calibrated per the procedures described inRef. 42, from which we obtained Mco,i ≈ 15.8 pH andχi (Φxco) (i ∈ {1, 2, 3, 4}). Each of these devices pro-vided a maximum AFM inter-qubit mutual inductanceMAFM = M2

co,iχAFM ≈ 1.56 pH, from which one can es-timate χAFM ≈ 6.3 nH−1. Measurements of

∣∣Ipq ∣∣ withand without active L-tuner compensation as a functionof coupler bias Φxco, as applied to all four couplers si-multaneously, are presented in Fig. 10b. The predictionsfrom the ideal CCJJ rf-SQUID model, obtained by us-ing Lq = 265.4 pH (with compensation) and Lq obtainedfrom Eq. (6) (without compensation), are also shown.Note that the two data sets and predictions all agreeto within experimental error at Φxco = 0.5 Φ0, whichcorresponds to the all zero coupling state (Meff = 0).The experimental results obtained without L-tuner com-pensation agree reasonably well with the predicted Φxco-dependence. As compared to the case without compensa-

11

tion, it can be seen that the measured∣∣Ipq ∣∣ show consider-

ably less Φxco-dependence when L-tuner compensation isprovided. However, the data suggest a small systematicdeviation from the inductance models Eqs. (6) and (7).At Φxccjj = −Φ0, for which it is estimated that βeff ≈ 2.43,∣∣Ipq ∣∣ ∝ 1/Lq. Given that the data for the case withoutcompensation are below the model, then it appears thatwe have slightly underestimated the change in Lq. Conse-quently, we have provided insufficient ballast inductancewhen the L-tuner compensation was activated.

D. rf-SQUID Capacitance

Since Icq and Lq directly impact the CCJJ rf-SQUIDpotential in Hamiltonian (5a), it was possible to in-fer CCJJ and L-tuner properties from measurements ofthe groundstate persistent current. In contrast, the rf-SQUID capacitance Cq appears in the kinetic term inHamiltonian (5a). Consequently, one must turn to al-ternate experimental methods that invoke excitations ofthe CCJJ rf-SQUID in order to characterize Cq. Onesuch method is to probe macroscopic resonant tunnel-ing (MRT) between the lowest lying state in one wellinto either the lowest order [LO, n = 0] state or intoa higher order [HO, n > 0] state in the opposing wellof the rf-SQUID double well potential36. The spac-ing of successive HOMRT peaks as a function of rf-SQUID flux bias Φxq will be particularly sensitive toCq. HOMRT has been observed in many different rf-SQUIDs and is a well established quantum mechanicalphenomenon36,46,47. LOMRT proved to be more difficultto observe in practice and was only reported upon rel-atively recently in the literature44. We refer the readerto this latter reference for the experimental method formeasuring MRT rates.

Measurements of the initial decay rate Γ ≡ dP↓/dt|t=0

versus Φxq are shown in Fig. 11a with the order of thetarget level n as indicated. The maximum observable Γwas imposed by the bandwidth of the apparatus, whichwas ∼ 5 MHz. The minimum observable Γ was dictatedby experimental run time constraints. In order to ob-serve many HO resonant peaks within our experimentalbandwidth we have successively raised the tunnel barrierheight in roughly equal intervals by tuning the targetΦxccjj. The result is a cascade of resonant peaks atop amonotonic background.

The authors of Ref. 46 attempted to fit their HOMRTdata to a sum of gaussian broadened lorentzian peaks. Itwas found that they could obtain satisfactory fits withinthe vicinity of the tops of the resonant features but thatthe model was unable to correctly describe the valleysbetween peaks. We had reached the same conclusion withthe very same model as applied to our data. However, itwas empirically observed that we could obtain excellentfits to all of the data by using a model composed of a sumof purely gaussian peaks plus a background that varies

0 5 10 15 2010!5

10!4

10!3

10!2

10!1

100

101

!xq (m!0)

"(µ

s!1)

a)

0

1

2

3

4

5

6

7

8

9

10

11

12

0 2 4 6 8 10 120.1

0.15

0.2

0.25

0.3

n

Wn/2|I

p q|(

m!

0)

b)

0 2 4 6 8 10 120

5

10

15

20

n

!n p/2|I

p q|(

m!

0)

c)

FIG. 11: (color online) a) HOMRT peaks fitted to Eq. (15).Data shown are for Φxccjj/Φ0 = −0.6677, −0.6735, −0.6793,−0.6851, −0.6911 and −0.6970, from left to right, respec-tively. Number of levels in target well n as indicated. b) Bestfit Gaussian width parameter Wn as a function of n. c) Bestfit peak position εnp as a function of n.

exponentially with Φxq :

Γ(Φxq )~

=∑n

√π

8∆2n

Wne−

(ε−εnp )2

2W2n + Γbkgde

Φxq/δΦbkgd , (15)

where ε ≡ 2∣∣Ipq ∣∣Φxq . These fits are shown in Fig. 11a. A

summary of the gaussian width parameter Wn in Fig. 11bis shown solely for informational purposes. We will re-frain from speculating why there is no trace of lorentzianlineshapes or on the origins of the exponential back-ground herein, but rather defer a detailed examinationof HOMRT to a future publication.

For the purposes of this article, the key results totake from the fits shown in Fig. 11a are the positionsof the resonant peaks, as plotted in Fig. 11c. Theseresults indicate that the peak spacing is very uniform:δΦMRT = 1.55±0.01 mΦ0. One can compare δΦMRT with

12

the predictions of the ideal CCJJ rf-SQUID model usingthe previously calibrated Lq = 265.4 pH, Lccjj = 26 pHand Icq = 3.103µA with Cq treated as a free parameter.From such a comparison, we estimate Cq = 190± 2 fF.

The relatively large value of Cq quoted above can bereconciled with the CCJJ rf-SQUID design by notingthat, unlike other rf-SQUID flux qubits reported upon inthe literature, our qubit body resides proximal to a su-perconducting groundplane so as to minimize crosstalk.In this case, the qubit wiring can be viewed as a differ-ential transmission line of length `/2 ∼ 900µm, where` is the total length of qubit wiring, with the effec-tive Josephson junction and a short on opposing ends.The transmission line will present an impedance of theform Z(ω) = −jZ0 tanh(ω`/2ν) to the effective Joseph-son junction, with the phase velocity ν ≡ 1/

√L0C0

defined by the differential inductance per unit lengthL0 ∼ 0.26 pH/µm and capacitance per unit length C0 ∼0.18 fF/µm, as estimated from design. If the separationbetween differential leads is greater than the distance tothe groundplane, then `/2ν ≈

√LbodyCbody/4, where

Cbody ∼ 640 fF is the total capacitance of the qubitwiring to ground. Thus, one can model the high fre-quency behavior of the shorted differential transmissionline as an inductance Lbody and a capacitance Cbody/4connected in parallel with the CCJJ. Taking a reason-able estimated value of 40 fF/µm2 for the capacitanceper unit area of a Josephson junction, one can esti-mate the total capacitance of four 0.6µm diameter junc-tions in parallel to be CJ ∼ 45 fF. Thus we estimateCq = CJ + Cbody/4 ∼ 205 fF, which is in reasonableagreement with the best fit value of Cq quoted above.

With all of the controls of the CCJJ rf-SQUID hav-ing been demonstrated, we reach the first key conclusionof this article: The CCJJ rf-SQUID is a robust devicein that parametric variations, both within an individualdevice and between a multitude of such devices, can beaccounted for using purely static flux biases. These bi-ases have been applied to all 8 CCJJ rf-SQUIDs on thisparticular chip using a truly scalable architecture involv-ing on-chip flux sources that are programmed by only asmall number of address lines41.

V. QUBIT PROPERTIES

The purpose of the CCJJ rf-SQUID is to provide anas ideal as possible flux qubit1. By this statement, it ismeant that the physics of the two lowest lying states ofthe device can be described by an effective Hamiltonianof the form Eq. (3) with ε = 2

∣∣Ipq ∣∣ (Φxq − Φ0q

),∣∣Ipq ∣∣ being

the magnitude of the persistent current that flows aboutthe inductive loop when the device is biased hard to oneside, Φ0

q being a static flux offset and ∆q representing thetunneling energy between the lowest lying states when bi-ased at its degeneracy point Φxq = Φ0

q. Thus,∣∣Ipq ∣∣ and ∆q

are the defining properties of a flux qubit, regardless of itstopology9. Given the complexity of a six junction device

with five closed superconducting loops, it is quite justifi-able to question whether the CCJJ rf-SQUID constitutesa qubit. These concerns will be directly addressed hereinby demonstrating that measured

∣∣Ipq ∣∣ and ∆q agree withthe predictions of the quantum mechanical Hamiltonian(5a) given independently calibrated values of Lq, Lccjj,Icq and Cq.

Before proceeding, it is worth providing some con-text in regards to the choice of experimental methodsthat have been described below. For those researchersattempting to implement GMQC using resonant elec-tromagnetic fields to prepare states and mediate inter-actions between qubits, experiments that involve highfrequency pulse sequences to drive excitations in thequbit (such as Rabi oscillations22, Ramsey fringes22,30

and spin-echo22,30,31) are the natural modality for study-ing quantum effects. Such experiments are convenient inthis case as the methods can be viewed as basic gate oper-ations within this intended mode of operation. However,such methods are not the exclusive means of characteriz-ing quantum resources. For those who wish to use precisedc pulses to implement GMQC or whose interests lie indeveloping hardware for AQO, it is far more convenientto have a set of tools for characterizing quantum mechan-ical properties that require only low bandwidth bias con-trols. Such methods, some appropriate in the coherentregime48,49 and others in the incoherent regime36,44,50,have been reported in the literature. We have made useof such low frequency methods as our apparatuses typ-ically possess 128 low bandwidth bias lines to facilitatethe adiabatic manipulation of a large number of devices.

One possible means of probing quantum mechanicaltunneling between the two lowest lying states of a CCJJrf-SQUID is via MRT44. Example LOMRT decay ratedata are shown in Fig. 12a. We show results for bothinitializations, |↓〉 and |↑〉, and fits to gaussian peaks, asdetailed in Ref. 44:

Γ(Φxq )~

=√π

8∆2q

We−

(ε−εp)2

2W2 . (16)

A summary of the fit parameters εp and W versus Φxccjj

is shown in Fig. 12b. We also provide estimates of thedevice temperature using the formula

kBTMRT =W 2

2εp. (17)

As expected, TMRT shows no discernible Φxccjj-dependence and is scattered about a mean value of53± 2 mK. A summary of ∆q versus Φxccjj will be shownin conjunction with more experimental results at the endof this section. For further details concerning LOMRT,the reader is directed to Ref. 44.

A second possible means of probing ∆q is via a Landau-Zener experiment50. In principle, this method should beapplicable in both the coherent and incoherent regime. Inpractice, we have found it only possible to probe the de-vice to modestly larger ∆q than we can reach via LOMRT

13

!1 !0.5 0 0.5 110!5

10!4

10!3

10!2

10!1

100

!xq (m!0)

"(µ

s!1)

a)

!0.668 !0.666 !0.664 !0.662 !0.6630

40

50

60

70

80

!xccjj /!0

Tem

pera

ture

(mK

)

b) W/kB

!p/kB

TMRT

FIG. 12: (color online) a) Example LOMRT peaks fitted toEq. (16). Data shown are for Φxccjj/Φ0 = −0.6621, −0.6642and −0.6663, from top to bottom, respectively. Data fromthe qubit initialized in |↓〉 (|↑〉) are indicated by solid (hol-low) points. b) Energy scales obtained from fitting multipleLOMRT traces.

purely due to the low bandwidth of our bias lines. Resultsfrom such experiments on the CCJJ rf-SQUID flux qubitwill be summarized at the end of this section. We see nofundamental limitation that would prevent others withhigher bandwidth apparatuses to explore the physics ofthe CJJ or CCJJ flux qubit at the crossover between thecoherent and incoherent regimes using the Landau-Zenermethod.

In order to probe the qubit tunnel splitting in the co-herent regime using low bandwidth bias lines, we havedeveloped a new experimental procedure for sensing theexpectation value of the qubit persistent current, simi-lar in spirit to other techniques already reported in theliterature48. An unfortunate consequence of the choiceof design parameters for our high fidelity QFP-enabledreadout scheme is that the QFP is relatively strongly cou-pled to the qubit, thus limiting its utility as a detectorwhen the qubit tunnel barrier is suppressed. One can cir-cumvent this problem within our device architecture by

tuning an inter-qubit coupler to a finite inductance andusing a second qubit as a latching sensor, in much thesame manner as a QFP. Consider two flux qubits coupledvia a mutual inductance Meff. The system Hamiltoniancan then be modeled as

H = −∑

i∈{q,d}

12

[εiσ

(i)z + ∆iσ

(i)x

]+ Jσ(q)

z σ(d)z , (18)

where J ≡ Meff|Ipq ||Ipd |. Let qubit q be the flux source

and qubit d serve the role of the detector whose tun-nel barrier is adiabatically raised during the course of ameasurement, just as in a QFP single shot measurementdepicted in Fig. 5. In the limit ∆d → 0 one can writeanalytic expressions for the dispersion of the four lowestenergies of Hamiltonian (18):

E1± = ± 12

√(εq − 2J)2 + ∆2

1 − 12εd ;

E2± = ± 12

√(εq + 2J)2 + ∆2

1 + 12εd .

(19)

As with the QFP, let the flux bias of the detector qubit beengaged in a feedback loop to track its degeneracy pointwhere Pd,↓ = 1/2. Assuming Boltzmann statistics for thethermal occupation of the four levels given by Eq. (19),this condition is met when

Pd,↓ =12

=e−E2−/kBT + e−E2+/kBT∑

α∈{1±,2±} e−Eα/kBT

. (20)

Setting Pd,↓ = 1/2 in Eq. (20) and solving for ε2 thenyields an analytic formula for the balancing condition:

εd =F (+)− F (−)

2+ kBT ln

(1 + e−F (+)/kBT

1 + e−F (−)/kBT

); (21)

F (±) ≡√

(εq ± 2J)2 + ∆21 .

While Eq. (21) may look unfamiliar, it readily reducesto an intuitive result in the limit of small coupling J �∆1 and T → 0:

εd ≈Meff|Ipq |εq√

ε2q + ∆2q

= Meff 〈g| Ipq |g〉 , (22)

where |g〉 denotes the groundstate of the source qubitand Ipq ≡

∣∣Ipq ∣∣σ(q)z is the source qubit persistent current

operator. Thus Eq. (21) is an expression for the expecta-tion value of the source qubit’s groundstate persistentcurrent in the presence of backaction from the detec-tor and finite temperature. Setting εi = 2|Ipi |Φxi andrearranging then gives an expression for the flux biasof the detector qubit as a function of flux bias appliedto the source qubit. Given independent calibrations ofMeff = 1.56 ± 0.01 pH for a particular coupler set toΦxco = 0 on this chip, T = 54 ± 3 mK from LOMRTfits and |Ipd | = 1.25 ± 0.02µA at the CCJJ bias wherethe LOMRT rate approaches the bandwidth of our bias

14

!4 !2 0 2 4!1

!0.5

0

0.5

1

!xq (m!0)

!x d

(m!

0)

1+

1-

2+

2-

3+

3-

slope ! 1"(2J)2+!2

q

slope # !

2Me" |Ipq |

FIG. 13: (color online) Example coupled flux trace taken atΦxccjj = −0.6513 Φ0 used to extract large ∆ in the coherentregime.

lines, one can then envision tracing out Φxd versus Φxq andfitting to Eq. (21) to extract the source qubit parameters|Ipq | and ∆q .

An example Φxd versus Φxq data set for source CCJJ fluxbias Φxccjj = −0.6513 Φ0 is shown in Fig. 13. The solidcurve in this plot corresponds to a fit to Eq. (21) witha small background slope that we denote as χ. We haveconfirmed from the ideal CCJJ rf-SQUID model that χ isdue to the diamagnetic response of the source rf-SQUIDto changing Φxq . This feature becomes more pronouncedwith increasing Cq and is peaked at the value of Φxccjj forwhich the source qubit potential becomes monostable,βeff = 1. Nonetheless, the model also indicates that χ inno way modifies the dynamics of the rf-SQUID, thus thequbit model still applies. From fitting these particulardata, we obtained |Ipq | = 0.72 ± 0.04µA and ∆q/h =2.64± 0.24 GHz.

In practice we have found it inefficient to take detailedtraces of Φxd versus Φxq as this procedure is susceptibleto corruption by LF flux noise in the detector qubit. Asan alternative approach, we have adapted the LF fluxnoise rejecting procedures introduced in the last sectionof this article to measure a series of three differentialflux levels in the detector qubit. The waveforms neededto accomplish this task are depicted in Fig. 14. Here,the dc-SQUID and QFP connected to the detector qubitare used in latching readout mode while the detectorqubit is annealed in the presence of a differential flux biasΦm ± δΦm which is controlled via feedback. Meanwhile,the source qubit’s CCJJ bias is pulsed to an intermediatelevel −Φ0 < Φxccjj < −Φ0/2 in the presence of an initial-ization flux bias ±Φiq. By choosing two appropriate pairsof levels ±Φiq, as indicated by the solid points 1± and 2±in Fig. 13, one can extract

∣∣Ipq ∣∣ and χ from the two dif-ferential flux measurements. In order to extract ∆q, wethen choose a pair of ±Φiq in the centre of the trace, asindicated by the solid points 3±, from which we obtain

0

0

0

-!0/2

-!0/2

~!0/3

!m!m+"!m

!m#"!m

iro(t)

!ccjj(t)x

!q(t)x

!ro(t)x

!qfp(t)x

x!latch(t)

t=0 trep

0

-!0/2

+!qi

-!qi

!ccjj(t)x

!q(t)x

Sourc

e Q

ubit

}

Det

ecto

r Q

ubit

}} Q

FP

}

dc-

SQ

UID

-!0

-!0

-!0

FIG. 14: (color online) Depiction of large ∆q measurementwaveforms. The waveform sequence is similar to that of Fig. 6,albeit the source qubit’s tunnel barrier is partially suppressed(−Φ0/2 < Φxccjj < −Φ0) and a second qubit (as opposed to aQFP) serves as the flux detector.

the central slope dΦxd/dΦxq . Taking the first derivative ofEq. (21) and evaluating at Φxq = 0 yields

dΦxddΦxq

− χ =2Meff

∣∣Ipq ∣∣2√(2J)2 + ∆2

q

tanh

√

(2J)2 + ∆2q

2kbT

. (23)

Given independent estimates of all other parameters, onecan then extract ∆q from this final differential flux mea-surement.

A summary of experimental values of the qubit pa-rameters

∣∣Ipq ∣∣ and ∆q versus Φxccjj is shown in Fig. 15.Here, we have taken ∆q from LOMRT and Landau-Zener experiments in the incoherent regime and fromthe LF flux noise rejecting persistent current procedurediscussed above in the coherent regime. The large gapbetween the three sets of measurements is due to two rea-sons: First, the relatively low bandwidth of our bias linesdoes not allow us to perform MRT or Landau-Zener mea-surements at higher ∆q where the dynamics are faster.Second, while the coherent regime method worked for∆q > kBT , it proved difficult to reliably extract ∆q inthe opposite limit. As such, we cannot make any pre-cise statements regarding the value of Φxccjj which serves

15

!0.67 !0.665 !0.66 !0.655 !0.65 !0.6450.4

0.6

0.8

1

1.2

1.4

!xccjj/!0

|Ip q|(

µA

)

a) DataTheory

!0.67 !0.665 !0.66 !0.655 !0.65 !0.645105

106

107

108

109

1010

!xccjj /!0

"q/h

(Hz)

b)kbT/h

MRT Data

LZ Data

!g| Ipq |g" Data

Theory

FIG. 15: (color online) a) Magnitude of the persistent cur-rent

˛Ipq

˛as a function of Φxccjj. b) Tunneling energy ∆q be-

tween two lowest lying states of the CCJJ rf-SQUID as afunction of Φxccjj, as characterized by macroscopic resonanttunneling [MRT] and Landau-Zener [LZ] in the incoherent

regime and coupled groundstate persistent current (〈g| Ipq |g〉)in the coherent regime. Solid curves are the predictions of theideal CCJJ rf-SQUID model using independently calibratedLq, Lccjj, I

cq and Cq with no free parameters.

as the delineation between the coherent and incoherentregimes based upon the data shown in Fig. 15b. Reg-ulating the device at lower temperature would assist inextending the utility of the coherent regime method tolower ∆q. On the other hand, given that Eq. (16) predictsthat Γ ∝ ∆2

q, one would have to augment the experimen-tal bandwidth by at least two orders of magnitude togain one order of magnitude in ∆q via either MRT or LZexperiments.

The solid curves in Fig. 15 were generated with theideal CCJJ rf-SQUID model using the independently cal-ibrated Lq = 265.4 pH, Lccjj = 26 pH, Icq = 3.103µA andCq = 190 fF. Note that there are no free parameters. Itcan be seen that the agreement between theory and ex-periment is quite reasonable. Thus we reach the secondkey conclusion of this article: The CCJJ rf-SQUID can

10!4 10!3 10!2 10!1 10010!11

10!10

10!9

10!8

Frequency (Hz)

S!(f

)(!

2 0/H

z)

FIG. 16: (color online) Low frequency flux noise in the CCJJrf-SQUID flux qubit. Data [points] have been fit to Eq. (24)[solid curve].

be identified as a flux qubit as the measured∣∣Ipq ∣∣ and

∆q agree with the predictions of a quantum mechanicalHamiltonian whose parameters were independently cali-brated.

VI. NOISE

With the identification of the CCJJ rf-SQUID as aflux qubit firmly established, we now turn to assessingthe relative quality of this device in comparison to otherflux qubits reported upon in the literature. In this sec-tion, we present measurements of the low frequency fluxand critical current spectral noise densities, SΦ(f) andSI(f), respectively. Finally, we provide explicit links be-tween SΦ(f) and the free induction (Ramsey) decay timeT ∗2 that would be relevant were this flux qubit to be usedas an element in a gate model quantum information pro-cessor.

A. Flux Noise

Low frequency (1/f) flux noise is ubiquitous in super-conducting devices and is considered a serious impedi-ment to the development of large scale solid state quan-tum information processors45. We have performed sys-tematic studies of this property using a large number offlux qubits of varying geometry51 and, more recently, as afunction of materials and fabrication parameters. Theselatter studies have aided in the reduction of the ampli-tude of 1/f flux noise in our devices and will be thesubject of a forthcoming publication. Using the methodsdescribed in Ref. 51, we have generated the one-sidedflux noise power spectral density SΦ(f) shown in Fig. 16.

16

These data have been fit to the generic form

S(f) =A2

fα+ wn , (24)

with best fit parameters α = 0.95 ± 0.05,√wn = 9.7 ±

0.5µΦ0/√

Hz and amplitude A such that√SΦ(1 Hz) =

1.3+0.7−0.5 µΦ0/

√Hz. Thus we reach the third key conclu-

sion of this article: We have demonstrated that it ispossible to achieve 1/f flux noise levels with Nb wiringthat are as low as the best Al wire qubits reported inthe literature30,31,45. Moreover, we have measured simi-lar spectra from a large number of identical flux qubits,both on the same and different chips, and can state withconfidence that the 1/f amplitude reported herein is re-producible. Given the experimentally observed geometricscaling of SΦ(1 Hz) in Ref. 51 and the relatively large sizeof our flux qubit bodies, we consider the prospects of ob-serving even lower 1/f noise in smaller flux qubits fromour fabrication facility to be very promising.

B. Critical Current Noise

A second noise metric of note is the critical currentnoise spectral density SI(f). This quantity has beenstudied extensively and a detailed comparison of exper-imental results is presented in Ref. 52. A recent studyof the temperature and geometric dependence of criti-cal current noise has been published in Ref. 53. Basedupon Eq. (18) of Ref. 52, we estimate that the 1/f crit-ical current noise from a single 0.6µm diameter junc-tion, as found in the CCJJ rf-SQUID flux qubit, willhave an amplitude such that

√SI(1 Hz) ∼ 0.2 pA/

√Hz.

Unfortunately, we were unable to directly measure criti-cal current noise in the flux qubit. While the QFP-enablereadout provided high fidelity qubit state discriminationwhen qubits are fully annealed to Φxccjj = −Φ0, this read-out mechanism simply lacked the sensitivity required forperforming high resolution critical current noise measure-ments. In lieu of a measurement of SI(f) from a qubit,we have characterized this quantity for the dc-SQUIDconnected to the qubit in question. The dc-SQUID hadtwo 0.6µm junctions connected in parallel. A time traceof the calibrated switching current Isw ≈ Ic was obtainedby repeating the waveform sequence depicted in Fig. 5bexcept with Φxlatch = −Φ0/2 at all time (QFP disabled,minimum persistent current) and Φxro = 0 to provideminimum sensitivity to flux noise. Assuming that thecritical current noise from each junction is uncorrelated,the best that we could establish was an upper boundof√SI(1 Hz) . 7 pA/

√Hz for a single 0.6µm diameter

junction.Given the upper bound cited above for critical current

noise from a single junction, we now turn to assessing therelative impact of this quantity upon the CCJJ rf-SQUIDflux qubit. It is shown in Appendix B that fluctuations inthe critical currents of the individual junctions of a CCJJ

generate apparent flux noise in the flux qubit by modu-lating Φ0

q. Inserting critical current fluctuations of mag-nitude δIc . 7 pA/

√Hz and a mean junction critical cur-

rent Ic = Icq/4 ∼ 0.8µA into Eq. (B10) yields qubit de-generacy point fluctuations

∣∣δΦ0q

∣∣ . 0.1µΦ0/√

Hz. Thisfinal result is at least one order of magnitude smaller thanthe amplitude of 1/f flux noise inferred from the data inFig. 16. As such, we consider the effects of critical currentnoise in the CCJJ rf-SQUID to be tolerable.

C. Estimation of T ∗2

While measurements of noise power spectral densitiesare the most direct way of reporting upon and comparingbetween different qubits, our research group is frequentlyasked what is the dephasing time for our flux qubits.The answer presumably depends very strongly upon biassettings, for recall that we have measured properties ofthe CCJJ rf-SQUID flux qubit in both the coherent andincoherent regime. Given that our apparatuses containonly low bandwidth bias lines for enabling AQO, we areunable to measure dephasing within our own laboratory.Collaborative efforts to measure dephasing for our fluxqubits are in progress. In the meantime, we provide arough estimate below for our flux qubits if they werebiased to the optimal point, Φxq = Φ0

q based upon themeasured SΦ(f) and subjected to a free induction decay,or Ramsey fringe, experiment. Referring to Eq. (33a) ofRef. 54 and key results from Ref. 55, the mean squaredphase noise for a flux qubit at the optimal point will begiven by

⟨φ2n(t)

⟩=

1~2

(2∣∣Ipq ∣∣)42∆2

∫ ∆/h

fm

dfSΦ2(f)sin2(πft)

(πf)2, (25)

where SΦ2(ω) represents the quadratic flux noise spec-tral density and fm is the measurement cutoff frequency.Assuming that the first order spectral density SΦ(ω) =2πA2/ω, then SΦ2(ω) can be written as

SΦ2(ω) =1

2π

∫dte−iωt

⟨Φ2n(t)Φ2

n(0)⟩

=1

2π

∫dte−iωt

∫dω′

2πA2

ω′

∫dω′′

2πA2

ω′′

= 8π2A4 ln (ω/ωir)ω

, (26)

where ωir ≡ 2πfir denotes an infrared cutoff of the 1/fnoise spectral density. Inserting Eq. (26) into Eq. (25)and rendering the integral dimensionless then yields:

⟨φ2n(t)

⟩=t2

~2

(2∣∣Ipq ∣∣A)4π∆2

∫ ∆t/h

fmint

dxln (x/firt) sin2(πx)

x3,

(27)where fmin = max

[fm fir

]. We have numerically stud-

ied the behavior of the integral in Eq. (27). In the very

17

long measurement time limit the integral is cut off by fir

and the integral varies as 1/t2, which then cancels thefactor of t2 in the numerator of Eq. (27). This meansthat the mean squared phase noise eventually reaches afinite limit. However, the more experimentally relevantlimit is fm � fir , for which we found empirically thatthe integral varies roughly as 5× [ln (fm/fir)]

2 over manyorders of magnitude in the argument of the logarithm. Inthis latter limit the result is independent of t, so Eq. (27)can be rewritten as

⟨φ2n(t)

⟩= t2/(T ∗2 )2, which then yields

the following formula for T ∗2 :

T ∗2 ≈

[1~2

(2∣∣Ipq ∣∣A)4π∆2

5 ln (fm/fir)

]−1/2

. (28)

Since flux noise spectra seem to obey the 1/f formdown to at least 0.1 mHz and researchers are generallyconcerned with dephasing over times of order 1µs, thenit is fair to consider fm/fir ∼ 1010. For a nominal valueof Φxccjj such that the flux qubit is in the coherent regime,say −0.652 Φ0, the qubit parameters are ∆q/h ≈ 2 GHzand

∣∣Ipq ∣∣ ≈ 0.7µA. Substituting these quantities intoEq. (27) then yields T ∗2 ∼ 150 ns. This estimate of thedephasing time is comparable to that observed in consid-erably smaller flux qubits with comparable 1/f flux noiselevels30,31.

VII. CONCLUSIONS

One can draw three key conclusions from the work pre-sented herein: First, the CCJJ rf-SQUID is a robust andscalable device in that it allows for in-situ correction forparametric variations in Josephson junction critical cur-rents and device inductance, both within and betweenflux qubits using only static flux biases. Second, the mea-sured flux qubit properties, namely the persistent current∣∣Ipq ∣∣ and tunneling energy ∆q, agree with the predictionsof a quantum mechanical Hamiltonian whose parametershave been independently calibrated, thus justifying theidentification of this device as a flux qubit. Third, ithas been experimentally demonstrated that the low fre-quency flux noise in this all Nb wiring flux qubit is com-parable to the best all Al wiring devices reported uponin the literature. Taken in summation, these three con-clusions represent a significant step forward in the devel-opment of useful large scale superconducting quantuminformation processors.

We thank J. Hilton, P. Spear, A. Tcaciuc, F. Cioata,M. Amin, F. Brito, D. Averin, A. Kleinsasser and G. Ker-ber for useful discussions. Siyuan Han was supported inpart by NSF Grant No. DMR-0325551.

APPENDIX A: CJJ RF-SQUID

Let the qubit and cjj loop phases be defined as

ϕq ≡ (ϕ1 + ϕ2) /2 , (A1a)

ϕcjj ≡ ϕ1 − ϕ2 , (A1b)

respectively. Furthermore, assume that the CJJ loop hasan inductance Lcjj that is divided symmetrically betweenthe two paths. Using trigonometric relations, one canwrite a Hamiltonian for this system in terms of modes inthe q and cjj loops that has the following form:

H =∑n

[Q2n

2Cn+ Un

(ϕn − ϕxn)2

2

]−Uqβ+ cos

(ϕcjj

2

)cos (ϕq)

+Uqβ− sin(ϕcjj

2

)sin (ϕq) ; (A2a)

β± =2πLq (I1 ± I2)

Φ0, (A2b)

where the sum is over n ∈ {q, cjj}, Cq ≡ C1+C2, 1/Ccjj ≡1/C1+1/C2, Un ≡ (Φ0/2π)2/Ln, Lq ≡ Lbody+Lcjj/4 and[Φ0ϕn/2π,Qn] = i~. The Josephson potential energyof Hamiltonian (A2a) can be rearranged by defining anangle θ such that tan θ = (β−/β+) tan (ϕcjj/2). Furthertrigonometric manipulation then yields Eqs. (4a)-(4d).

APPENDIX B: CCJJ RF-SQUID

Following the same logic as for the CJJ rf-SQUID, onecan define four orthogonal quantum mechanical degreesof freedom as follows:

ϕL ≡ ϕ1 − ϕ2 ; (B1a)ϕR ≡ ϕ3 − ϕ4 ; (B1b)

ϕccjj ≡ ϕ` − ϕr =ϕ1 + ϕ2

2− ϕ3 + ϕ4

2; (B1c)

ϕq ≡ϕ` + ϕr

2=ϕ1 + ϕ2 + ϕ3 + ϕ4

4. (B1d)

Using the same strategy as in Appendix A, one can usetrigonometric identities to first express the Josephson po-tential in terms of the L and R loop modes:

H =∑n

Q2n

2Cn+∑m

Um(ϕm − ϕxm)2

2

−UqβL+ cos(ϕL

2

)cos (ϕ`)

+UqβL− sin(ϕL

2

)sin (ϕ`)

−UqβR+ cos(ϕR

2

)cos (ϕr)

+UqβR− sin(ϕR

2

)sin (ϕr) ; (B2a)

βL(R),± ≡2πLq

(I1(3) ± I2(4)

)Φ0

, (B2b)

where the first sum is over n ∈ {L,R, `, r} and the secondsum is over closed inductive loops m ∈ {L,R, ccjj, q}. As

18

before, each of the modes obey the commutation relation[Φ0ϕn/2π,Qn] = i~. Here, 1/CL(R) = 1/C1(3) + 1/C2(4),C`(r) = C1(3) + C2(4) and Um = (Φ0/2π)2/Lm.

We have found it adequate for our work to assumethat LL,R/Lq � 1, which then allows one to reducethe four dimensional system given in Hamiltonian (B2a)to two dimensions. Consequently, we will substituteϕL(R) = ϕxL(R) and ignore the L and R kinetic termshenceforth. Assuming that the inductance of the ccjjloop is divided equally between the two branches one canthen write Lq = Lbody +Lccjj/4. With these approxima-tions and the θ strategy presented in Appendix A, onecan rearrange the Josephson potential terms to yield thefollowing:

H =∑n

Q2n

2Cn+∑m

Um(ϕm − ϕxm)2

2

−UqβL cos(ϕ` − ϕ0

L

)−UqβR cos

(ϕr − ϕ0

R

); (B3a)

βL(R) = βL(R),+ cos(ϕxL(R)

2

)(B3b)

×

√1 +

[βL(R),−

βL(R),+tan

(ϕxL(R)

2

)]2

;

ϕ0L(R) = − arctan

(βL(R),−

βL(R),+tan(ϕxL(R)/2)

), (B3c)

where the first sum is over n ∈ {`, r} and the secondsum is over m ∈ {ccjj, q}. The Josephson potential isgiven by a sum of two cosines, as encountered in the CJJrf-SQUID derivation of Hamiltonian (A2a) from Hamil-tonian (1). These two terms can be rewritten in the samemanner by defining β± = βL ± βR. The result, similarto Hamiltonian (A2a), can then be subjected to the θstrategy to yield

H =∑n

[Q2n

2Cn+ Un

(ϕn − ϕxn)2

2

]−Uqβeff cos

(ϕq − ϕ0

q

), (B4a)

where the sum is over n ∈ {q, ccjj} and the capacitancesare defined as Cq = C1 + C2 + C3 + C4 and 1/Cccjj =1/(C1 + C2) + 1/(C3 + C4). The other parameters aredefined as

βeff = β+ cos(γ

2

)√1 +

[β−β+

tan(γ

2

)]2

; (B4b)

ϕ0q =

ϕ0L + ϕ0

R

2+ γ0 ; (B4c)

γ ≡ ϕccjj −(ϕ0L − ϕ0

R

); (B4d)

γ0 ≡ − arctan(β−β+

tan(γ/2))

; (B4e)

β± ≡ βL ± βR . (B4f)

Hamiltonian (B4a) inherits much of its complexityfrom junction asymmetry both within the minor loops,which gives rise to ϕ0

L(R), and effective junction asymme-try between the minor loops, which gives rise to γ0. Forarbitrary external flux biases and nominal spread in junc-tion critical current, the CCJJ rf-SQUID offers no obvi-ous advantage over the CJJ rf-SQUID. However, uponchoosing biases ΦxL and ΦxR such that

βL = βR , (B5)

then β− = 0, and consequently γ0 = 0. With these sub-stitutions, Hamiltonian (B4a) yields Hamiltonian (5a).Note that for βL(R),−/βL(R),+ � 1 and modest ΦxL(R)

that the so-called CCJJ balancing condition given byEqs. (B3b) and (B5) can be written approximately as

βL,+ cos(ϕxL2

)≈ βR,+ cos

(ϕxR2

),

which, upon solving for ϕxL yields

ΦxL =2πΦ0

arccos[βR,+βL,+

cos(πΦxRΦ0

)]. (B6)

It is possible for critical current noise to couple intothe ϕq degree of freedom in any compound junction rf-SQUID qubit via modulation of the junction asymmetry-dependent apparent qubit flux offset Φ0

q. In the case ofthe CCJJ rf-SQUID, all three quantities on the right sideof Eq. (B4c) are ultimately related to the critical cur-rents of the individual junctions. Given typical junctionparameter spreads from our fabrication facility,∣∣∣∣βL(R),−

βL(R),+

∣∣∣∣ =∣∣∣∣I1(3) − I2(4)

I1(3) + I2(4)

∣∣∣∣ ∼ O(0.01) ,

so one can write an approximate expression for ϕ0L(R)

using Eq. (B3c):

ϕ0L(R) ≈ −

I1(3) − I2(4)

I1(3) + I2(4)tan

(ϕxL(R)

2

)≈ −

I1(3) − I2(4)

2Ictan

(ϕxL(R)

2

), (B7)

and for γ0 using Eqs. (B3b), (B4d) and (B4e):

γ0 ≈(I3 + I4) cos

(ϕxR2

)− (I1 + I2) cos

(ϕxL2

)(I1 + I2) cos

(ϕxL2

)+ (I3 + I4) cos

(ϕxR2

) tan(γ

2

)

≈(I3 + I4) cos

(ϕxR2

)− (I1 + I2) cos

(ϕxL2

)2Ic[cos(ϕxL2

)+ cos

(ϕxR2

)] tan(γ

2

),

(B8)

19

where Ic represents the mean critical current of a singlejunction. The CCJJ rf-SQUID is intended to be oper-ated with only small flux biases in the minor loops, thuscos(ϕxL2

)≈ cos

(ϕxR2

)≈ 1. It is also reasonable to as-

sume that γ ≈ ϕxccjj as the corrections to tan(γ/2) fromϕ0L(R) and from the effective two-dimensionality of the

rf-SQUID potential will be very small. Inserting Eqs. B7and B8 into Eq. (B4c) then yields

ϕ0q ≈ −

I12Ic

[tan

(ϕxL2

)+

12

tan(ϕxccjj

2

)]− I2

2Ic

[− tan

(ϕxL2

)+

12

tan(ϕxccjj

2

)]− I3

2Ic

[tan

(ϕxR2

)− 1

2tan

(ϕxccjj

2

)]− I4

2Ic

[− tan

(ϕxR2

)− 1

2tan

(ϕxccjj

2

)]. (B9)

For the typical operating parameters described in this

article, ΦxL(R)/Φ0 ∼ 0.1 and the device acts as a qubit forΦxccjj/Φ0 ∼ 0.65. For these flux biases, the magnitude ofthe terms within the square braces in Eq. (B9) are all oforder 1. Therefore, for general flux bias conditions, theapparent qubit flux offset is roughly given by

Φ0q ≈ −

Φ0

4π(I1 + I2)− (I3 + I4)

Ic.

Assume that each junction experiences critical currentfluctuations of magnitude δIc. If each junction’s fluctua-tions are independent, then the root mean square varia-tion of the qubit degeneracy point

∣∣δΦ0q

∣∣ will be

∣∣δΦ0q

∣∣ ≈ Φ0

2πδIcIc

. (B10)

Thus, critical current fluctuations generate apparent fluxnoise in the CCJJ rf-SQUID flux qubit.

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