Tunable Superconducting Qubits with Flux-Independent Coherence
M. D. Hutchings, Jared B. Hertzberg, Yebin Liu, Nicholas T. Bronn, George A. Keefe, Jerry M. Chow, B. L. T. Plourde
TTunable Superconducting Qubits with Flux-Independent Coherence
M. D. Hutchings, J. B. Hertzberg, Y. Liu, N. T. Bronn, G. A. Keefe, J. M. Chow, and B. L. T. Plourde Syracuse University, Department of Physics, Syracuse, NY 13244, USA IBM, TJ Watson Research Center, Yorktown Heights, NY 10598, USA (Dated: February 23, 2017)We have studied the impact of low-frequency magnetic flux noise upon superconducting transmonqubits with various levels of tunability. We find that qubits with weaker tunability exhibit dephasingthat is less sensitive to flux noise. This insight was used to fabricate qubits where dephasing dueto flux noise was suppressed below other dephasing sources, leading to flux-independent dephasingtimes T ∗ ∼ µ s over a tunable range of ∼
340 MHz. Such tunable qubits have the potential tocreate high-fidelity, fault-tolerant qubit gates and fundamentally improve scalability for a quantumprocessor.
Quantum computers have the potential to outperformclassical logic in important technological problems. Apractical quantum processor must be comprised of quan-tum bits (“qubits”) that are isolated from environmentaldecoherence sources yet easily addressable during logicalgate operations. Superconducting qubits are an attrac-tive candidate because of their simple integration withfast control and readout circuitry. In recent years, ad-vances in superconducting qubits have demonstrated howsuch integration may be achieved while maintaining highcoherence [1–3]. Further extensions of qubit coherencewill serve to reduce gate errors, cutting down the num-ber of qubits required for fault-tolerant quantum logic[4, 5].An important aspect of maintaining high qubit coher-ence is the reduction of dephasing. Frequency-tunablequbits are inherently sensitive to dephasing via noise inthe tuning control channel. Tuning via a magnetic fluxthus introduces dephasing via low-frequency flux noise[6–13]. Such noise is ubiquitous in thin-film supercon-ducting devices at low temperatures. Experiments indi-cate a high density of unpaired spins on the thin-filmsurface [14] with fluctuations of these leading to low-frequency flux noise that typically has a 1 /f power spec-trum [13, 15–17]. For any flux-tunable qubit, this fluxnoise leads to significant dephasing whenever the qubitis biased at a point with a large gradient of the qubitenergy with respect to flux.Flux tuning is nonetheless highly advantageous formany quantum circuits, and several classes of quantumlogic gates rely on flux-tunable qubits. In the controlled-phase gate [1, 18], qubit pairs are rapidly tuned into res-onance to create entanglement. Here, both flux noiseand off-resonant coupling to other qubits produce phaseerrors proportional to gate times, with total gate errorscaling as the square of the gate time [19]. Alternatively,fixed-frequency qubits have been employed in schemessuch as the cross resonance (CR) gate [20, 21] to demon-strate aspects of quantum error correction (QEC) [3, 22].Recent efforts with two-qubit devices have extended CRgate fidelities beyond 99% [23]. Larger lattices of fixed-frequency qubits, however, are likely to suffer increas- ingly from frequency crowding. If a qubit’s 0-1 excita-tion frequency overlaps with the 0-1 or 1-2 frequency ofits neighbor, or if the two qubits’ frequencies are veryfar apart, the CR gate between these two qubits will benon-ideal, with the strong possibility of leakage out ofthe computational subspace, or a very weak gate, respec-tively [24]. However, fixed-frequency transmon qubits arechallenging to fabricate to precision better than about200 MHz [25]. Given such imprecision, a hypotheticalseventeen-qubit logic circuit could see up to a quarter ofits gate pairs fail due to frequency crowding (see Sup-plement). Frequency-tunable transmon qubits thereforeappear attractive for use in architectures based on theCR gate.In this Letter, we show how a tunable qubit’s sensitiv-ity to flux noise may be reduced by limiting its extent oftunability. We report results for several different qubitsshowing that the qubit dephasing rate is proportional tothe sensitivity of the qubit frequency to magnetic fluxand to the amplitude of low-frequency flux noise. Fur-thermore, we use the understanding gained through thisstudy to fabricate a qubit whose dephasing due to non-flux dependent sources exceeds its dephasing due to low-frequency flux noise, over a range of more than 300 MHzof tunability. This unique qubit has the potential to re-duce errors in gates employing frequency-tunable qubitsand to evade frequency crowding in qubit lattices em-ploying CR gates. It therefore offers a promising routeto create high-fidelity two-qubit gates that reach fault-tolerant gate operation and to improve the scalability ofsuperconducting qubit devices.Our device adapts a design in which a superconduct-ing quantum interference device (SQUID) serves as theJosephson inductance in a transmon qubit [26]. Here, theJosephson energy, and consequently the qubit 0-1 transi-tion frequency f , may be tuned with a magnetic flux Φwith a period of Φ ≡ h/ e , the magnetic flux quantum,where h is Planck’s constant and e is the electron charge.However, if the two junctions in the SQUID have differentJosephson energies E J and E J , a so-called ‘asymmetrictransmon’ is formed [27]. The greater the difference injunction energies, the smaller the level of tunability. If a r X i v : . [ c ond - m a t . s up r- c on ] F e b E J > E J , we can define the ratio α = E J /E J andthe sum E J Σ = E J + E J . The total flux-dependentJosephson energy E J varies according to the followingexpression from Ref. [26]: E J (Φ) = E J Σ cos (cid:18) π ΦΦ (cid:19)(cid:115) d tan (cid:18) π ΦΦ (cid:19) , (1)where d is given by d = ( α − / (1 + α ).To explore the dephasing behavior of qubits havingsuch tunability, we prepared transmon qubits on twostyles of chip, referred to as sample A and sample B.Both samples employ a multi-qubit planar circuit quan-tum electodynamics (cQED) architecture with eight sep-arate cavity/qubit systems. Qubits on the same chipshould experience the same flux noise level, allowing acomparison of dephasing properties between them. Onsample A, the eight frequency-multiplexed cavities areall coupled to a common feedline for microwave driveand readout (sample details and layout shown in Sup-plement). On this chip, we compare transmons havingjunction ratios α = 7, 4, 1 and a fixed-frequency single-junction qubit. We design for a particular α by varyingthe junction areas in the SQUID, since E J of a junc-tion is directly proportional to its area. For consistency,the single-junction qubit maintained the same SQUIDloop structure with one of the junctions being left open,and all four qubit types were designed to have the same E J Σ . Sample B employs a qubit design similar to thatin [3, 22, 28, 29]. For this device, all qubits have sepa-rate drive and readout microwave lines (layout shown inSupplement). Six qubits were designed to have α = 15while two employed a single junction matching the E J ofthe tunable qubits. The fixed-frequency qubits act as areference for non-flux dependent dephasing on each chip.We used standard photolithographic and etch pro-cesses to pattern the coplanar waveguides, ground plane,and qubit capacitors from Nb sputtered films on Si sub-strates, followed by electron-beam lithography and de-position of conventional Al-AlO x -Al shadow-evaporatedjunctions. While all qubits were similar in design to[3, 22, 23], the transmon capacitor pads and SQUID loopgeometry differed somewhat between samples A and B.Designs are shown in Fig. 1. Each sample was mountedon a dilution refrigerator in its respective lab (sampleA at Syracuse; sample B at IBM) and surrounded byboth room-temperature and cryogenic magnetic shields.Measurements for both samples were performed usingstandard cQED readout techniques [30]. Measurementsignals from both samples were amplified by a low-noiseHEMT amplifier at 4K. In the case of the α = 15 qubiton sample B, additional amplification was provided by aSLUG amplifier [31]. Flux bias was applied to each sam-ple during measurement using a wire coil placed close tothe top of each device. Fabrication details and a discus- Sample A
500 µm 20 µm 500 µm 20 µm
Sample B
FIG. 1. (color online) Optical micrographs of example qubitsfrom samples A and B. Φ / Φ f ( GH z ) Φ / Φ f ( GH z ) FIG. 2. (color online) f vs. flux measured for qubits fromsamples A and B. Solid lines are fits to these tuning curvesbased on Eq. 1. Also included are frequencies of single junc-tion qubits from both samples. Dashed lines for these qubitsto guide the eye. Inset: entire tuning range measured for the α = 1 qubit with the α = 7 qubit included as a comparisonto illustrate the large frequency tunability of an α = 1 qubit. sion of measurement techniques are given in the Supple-ment.Here we present data from four qubits on sample Aand two qubits on sample B, one of each variation fromeach sample. Figure 2 shows the flux dependence of f for each qubit. We have subtracted any fixed flux offsetappearing in the measurement. The α = 15 qubit onsample B had the weakest tunability of these: 337 MHz.Following Eq. (S1) and the expectation that f ∝ √ E J [26], we fit the data in Fig. 2 to find the maximumfrequency f max01 ∝ √ E J Σ and asymmetry parameter d .From the latter we compute α for all tunable qubits andwe find that the measured asymmetry α was within 5%of the designed value. We note that the four sample Aqubits shown in Fig. 2 were designed to have identical E J Σ and therefore identical f max01 , but in fact exhibit a ∼
200 MHz spread, thus illustrating the challenge of fab-ricating qubits to precise frequencies.To assess the effect of flux noise on dephasing, we ob-serve how the latter relates to each qubit’s frequencygradient as a function of flux D Φ = | ∂f /∂ Φ | . Wecharacterize dephasing via measurement of the Ramseydecay time T ∗ , which is sensitive to low-frequency de-phasing noise [7, 9]. We fit these using an exponentialform. Although it has been shown that a dephasing noisesource with a 1 /f power spectrum will result in a Gaus-sian decay envelope [7, 9], flux-independent dephasingsources such as cavity-photon shot-noise [32–34] result inan exponential decay envelope. Ramsey decays for fixed-frequency qubits are therefore well fit with an exponentialdecay envelope. For all of our asymmetric transmons, aswell as a large portion of the dephasing data for the α = 1symmetric device, we find that an exponential decay en-velope is also a good fit. In all of our data, we find thatdifferences between values of T ∗ obtained using an expo-nential or Gaussian fit are systematic but slight. Further-more, assuming a purely exponential decay simply putsan upper bound on the extracted flux noise level. A morecomplete discussion of the nature of our Ramsey decayenvelopes and alternative fitting approaches appears inthe Supplement.Relaxation times T ranged from ∼ − µ s over thesix qubits reported here. In general T increased withdecreasing qubit frequency (Supplement, Fig. 2), con-sistent with dielectric loss and a frequency-independentloss tangent, as observed in other tunable superconduct-ing qubits [35]. For the α = 15 qubit on sample B, a re-duction in T with increasing frequency is also consistentwith Purcell losses to the readout resonator. Qubits onsample A remained sufficiently detuned below the read-out resonators that Purcell loss was not a significant losschannel. T relaxation due to coupling to a flux-biasline, first discussed for inductive coupling in Ref. [26]for a near symmetrical qubit, and for capacitive couplingin Ref. [36], was considered for the qubits studied here.We show in the Supplement that the upper bound on T due to the flux-line coupling for our qubit designs is notsignificantly lower than that reported in [26].To compare dephasing rates among the qubits, we usethe relation Γ φ = 1 /T ∗ − / T [37] to remove the relax-ation contribution. These values are plotted against fluxin Fig. 3. As the curves in Fig. 2 illustrate, the inte-ger and half-integer Φ / Φ points are ‘sweet spots’ where D Φ = 0 and thus the qubit is first-order insensitive toflux noise. All the transmons on sample A clearly ex-hibit a dephasing rate that increases with D Φ and is aminimum at the sweet spots. Second-order sensitivityto flux noise [9, 38] should be negligible in our samplesbecause of the small energy-band curvature. However,the level of Γ φ for the non-tunable qubit on each sam-ple and the tunable qubits at their sweet spots indicatesthe presence of non-flux dependent sources of dephasing.Such background dephasing may arise from other mech-anisms, including cavity-photon shot noise [32], criticalcurrent noise [39], or charge noise affecting the residualcharge dispersion in the transmon design [26]. This back-ground dephasing may be expected to vary from qubit to qubit due to differences in qubit-cavity coupling or cav-ity thermalization, among other effects. Such variationsare commonly observed in multi-qubit devices [3, 22, 28].The Supplement contains dephasing data for additionaldevices similar to those discussed here, illustrating fur-ther variations in background dephasing.For sample A, if we consider only flux-dependent de-phasing, it is evident that Γ φ ∝ D Φ . Furthermore, qubitsof the same geometry on the same chip should experiencesimilar flux noise [14]. The analysis outlined in Ref. [7, 9]may then be used to extract a flux noise level from therelationship between Γ φ and D Φ . We apply a simultane-ous fit of the form mD Φ + b to the α = 1, 4, and 7 qubits,allowing background dephasing b to vary for each qubit,while a single m is common to all. The fit appears as solidlines in Fig. 3. We derive Γ φ = 2 π (cid:112) A Φ | ln (2 πf IR t ) | D Φ following the approach in Ref. [9], where the flux noisepower spectrum is S Φ ( f ) = A Φ / | f | , f IR is the infraredcutoff frequency, taken to be 1 Hz and t is on the orderof 1 / Γ φ , which we take to be 10 µ s in our calculations.Equating mD Φ to Γ φ in the equation above, we may cal-culate the flux noise level on sample A. To determine theuncertainty in the measured flux noise level, we must notonly account for the error in fitting m but also how vari-ations in dephasing time impact the calculation of A Φ values. To account for the latter, we determine the im-pact on extracted A / as t is varied. Adjusting t over arange similar to what we observe experimentally leads toa ∼
10% change A / . The errors we report for all cal-culated A / reflect this added uncertainty. We find that A / = 1 . ± . µ Φ on sample A. This level is compat-ible with previous experimental studies of flux noise insuperconducting flux [6–8, 40, 41] and phase qubits [42].To achieve an even clearer picture of the influence offlux noise on these qubits, we plot Γ φ vs. D Φ for eachqubit in Fig. 4a. Here, D Φ is computed from the fitsto the energy bands of each qubit shown in Fig. 2. Thisyields a linear dependence where the slope can be relatedto the amplitude of the flux noise and the offset corre-sponds to the background dephasing level. In this case,instead of a simultaneous fit we apply a separate fit ofΓ φ = mD Φ + b to each qubit, and we find A / values of1 . ± .
2, 1 . ± . . ± . µ Φ for the α = 7, 4and 1 qubits, respectively. These flux noise levels are allconsistent with past studies of low-frequency flux noisein superconducting devices [6–8, 40–42].In Fig. 4a it can be seen that, for the tunable qubits onsample A, within the range D Φ (cid:46) / Φ , the mea-sured dephasing rate is largely flux-independent withinthe experimental spread. To exploit this insensitivity, wedesigned the tunable transmon on sample B to have D Φ no greater than ∼ / Φ at any point within its tun-ing range, a condition satisfied by having α = 15. Asa result, its sensitivity to 1 /f flux noise appears to besuppressed below the level where background dephasing Φ / Φ Γ φ ( M H z ) FIG. 3. (color online) Γ φ vs. flux measured for qubits fromsamples A and B. Solid lines show a simultaneous fit of theform mD Φ + b to the tunable qubits on sample A. Factor m is common to all three datasets while b is allowed to vary foreach. Γ φ measured for fixed frequency qubits on both samplesincluded with dashed lines to help guide the eye. dominates. Γ φ is essentially flat across the entire tun-ing range, as shown in Fig. 3, with a mean of 58 kHzand experimental scatter of σ = 17 kHz. In comparison,this sample’s fixed-frequency qubit exhibits Γ φ = 72 kHz.Figure 4b shows clearly that T ∗ for the α = 15 qubit onsample B is independent of frequency over the whole tun-ing range.Although no significant flux dependence of the dephas-ing is detectable for sample B, we estimate from our ear-lier expression for Γ φ that the observed scatter is con-sistent with A / of 0.9 µ Φ . Recent progress in under-standing the origins of 1 /f flux noise in SQUIDs [43]has facilitated up to a 5 × reduction in A Φ [44]. Such re-ductions applied to the sample B qubit would reduce itsmaximum flux-noise-driven dephasing below 8 kHz. In a α = 7 qubit tunable over more than 700 MHz, flux noiseof such a level would cause dephasing no greater than17 kHz. Alternatively, in a qubit of 150 MHz tunability,the flux noise seen in sample B would cause dephasingnot exceeding 8 kHz, or only 4 kHz if the flux noise werereduced as in Ref. [44]. We may contrast these valueswith the non-flux-noise-driven dephasing seen in state-of-the-art single-junction transmons used for multi-qubitgate operations: Γ φ = 4 to 8 kHz on 2-qubit samples[23, 45], 10 kHz on 5-qubit samples [29] and 10 to 21 kHzon 7-qubit samples [28].In conclusion, we have shown that by reducing theflux-tunability of a transmon qubit, we can dramaticallylower its sensitivity to 1 /f flux noise. Using this under-standing, we have fabricated a qubit in which the de-phasing rate due to flux noise is suppressed below thelevel set by non-flux dependent sources. This device ex- -1 (GHz/ (a) Γ φ ( M H z ) D Φ (GHz/ Φ ) Frequency (GHz) µ T * ( s ) (b) FIG. 4. (color online) (a) Γ φ vs. D Φ measured for qubitsfrom samples A and B. Solid lines show individual linear fitsto the tunable qubits on sample A, as described in text. Notethe scale is log-log. Γ φ measured for fixed-frequency qubitson both samples, included with dashed lines to help guidethe eye. (b) T ∗ vs. frequency measured for the α = 15 andfixed-frequency qubits on sample B. hibits a flux-independent dephasing rate Γ φ ∼
60 kHzover a tunable range in excess of 300 MHz. This qubit de-sign should be readily adaptable to existing architecturesaimed at the realization of a logically-encoded qubit, inboth frequency-tuned gates and all-microwave gates. Asqubit architectures progress to more complex geometries,this work will enable the implementation of multi-qubitgates without frequency collisions impacting gate perfor-mance. This is a promising route to the creation of high-fidelity two-qubit gates for reaching fault tolerance, thusfundamentally improving the scalability of such systemsfor the creation of a universal quantum computer.We acknowledge support from Intelligence AdvancedResearch Projects Activity (IARPA) under contractW911NF-16-0114. The device fabrication was performedin part at the Cornell NanoScale Facility, a member ofthe National Nanotechnology Coordinated Infrastructure(NNIC) which is supported by the National Science Foun-dation under Grant ECCS-1542081. We thank Y.-K.-K.Fung, J. 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NON-IDEAL FABRICATION IN FIXED FREQUENCY QUBITS
Lattices of coupled qubits are proposed to enable error-correction algorithms such as the ‘surface code’ [S1, S2].Qubits are arranged into a square grid with alternate qubits serving either data or error-checking functions. Bus-couplers provide interaction among adjacent qubits, with up to four qubits attached to each bus. A seven qubit-latticethereby comprises 12 qubit pairs and a seventeen-qubit lattice comprises 34 pairs. However, single junction transmonqubits are challenging to fabricate at precisely set frequencies. Among dozens of identically-fabricated qubits, thefrequencies typically have a spread of σ f ∼
200 MHz [S3]. Such imprecision will inhibit functioning of qubit lattices.Considering a lattice of tansmon qubits of frequency ∼ δ/ π = −
340 MHz, and consideringcross-resonance gate operations, we can estimate the number of undesired interactions among these pairs. Studies ofthe cross-resonance gate [S4] indicate that these gates will be dominated by undesirable interactions if the frequencyseparation | ∆ | between adjacent qubits is equal to zero, a degeneracy between f of the qubits; equal to − δ/ π , adegeneracy between f of one qubit and f of the next; or if | ∆ | > − δ/ π (weak interaction leading to very slowgate operation). In a simple Monte Carlo model, we assign to all points in the lattice a random qubit frequencyfrom a gaussian distribution around 5 GHz, and count the number of degenerate or weak-interaction pairs, taking arange of ± ( δ/ π ) /
20, or ±
17 MHz around each degeneracy. The results appearing in Table I make it evident that thelikelihood of frequency collisions increases as the lattice grows.
Number Mean Numberof QBs σ f of Collisions7 | δ/ π | | δ/ π | | δ/ π | | δ/ π | σ f are considered. DEVICE DESIGN AND FABRICATION
The device for sample A, shown in Fig. S1, has all eight qubit/cavities capacitively coupled to a common feedlinethrough which individual qubit readout was achieved via a single microwave drive and output line. Sample B, shownin Fig. S1, employs a design where all qubits have separate drive and readout microwave lines. As in Ref. [S5] and[S6], this sample is designed as a lattice of coupled qubits for use in multi-qubit gate operations, although no suchoperations are presented in this paper. Coplanar-waveguide buses, half-wave resonant at ∼ . − µ m wideAl traces and Josephson junctions, with the asymmetry in the junctions fabricated by increasing the width of onejunction with respect to the other, while keeping the overlap fixed at 0 . µ m. The sum of the large and small junctionareas was designed to be constant, independent of α . Qubits on sample A had capacitor pads separated by 20 µ mand the Al electrodes separated such that the SQUID loop area was roughly 400 µ m . In sample B, the Nb capacitorpads were separated by 70 µ m. The SQUID comprises a ∼ × µ m Al loop of 2 µ m trace width, placed midwaybetween the capacitor pads and joined to Nb leads extending from the pads. In sample B, the large and small junctiondiffer in both width and overlap. In this sample, all SQUIDs of a given α were fabricated identically but SQUIDs ofdifferent α had different total junction area. MEASUREMENT SETUP
Measurements of sample A were completed in a dilution refrigerator (DR) at Syracuse University (SU), while sampleB was measured in a DR at the IBM TJ Watson Research Center. Both samples were wire-bonded into holdersdesigned to suppress microwave chip modes. Each sample was mounted to the mixing chamber of its respective DRand placed inside a cryoperm magnetic shield, thermally anchored at the mixing chamber. Both SU and IBM DRs hadroom-temperature µ -metal shields. Measurements for both samples were performed using standard cQED readouttechniques [S10].For sample A, room-temperature microwave signals were supplied through attenuated coaxial lines, thermalized ateach stage of the DR and filtered using 10 GHz low pass filters (K&L) thermalized at the mixing chamber. We used atotal of 70 dB of attenuation on the drive-lines: 20 dB at 4 K, 20 dB at 0 . . − .
136 GHz, separated by 20 −
25 MHz. κ/ π linewidths forthese resonators were on the order of a few hundreds of kHz.Figure S1 shows the layout of the sample B chip. The α = 15 asymmetric-SQUID transmon reported in thepaper was located at position Q . It was read out through a coplanar waveguide resonator of frequency 6.559 GHzand linewidth ∼
300 kHz, and was found to have f max = 5 .
387 GHz. The fixed-frequency transmon (5.346 GHz)at position Q was read out through a 6.418 GHz resonator having linewidth ∼
300 kHz. Sample B qubits weremeasured via signal wiring similar to that presented in Refs. [S5, S8, S9, S11]. Drive wiring included 10 dB ofattenuation at 50 K, 10 dB at 4K, 6 dB at 0.7 K, 10 dB at 100 mK, and at the mixing-chamber plate 30 dB ofattenuation plus a homemade ‘Eccosorb’ low-pass filter. Drive signals entered a microwave circulator at the mixingplate. On one set of signal wiring, the 2nd port of the circulator passed directly to qubit Q . In another set of signalwiring, the second port of the circulator passed to several different qubits via a microwave switch. Signals reflected Sample A
Qubit B Q Q Q Q Q Q Q Q Res. Res. Res. Res. Res. Res. Res. Res. Sample B
FIG. S1. (color online) Optical micrographs of samples including higher magnification images of qubits and SQUID loops.Sample B image is a chip of identical design to the ones used for measurements. In sample B image, labels indicate each qubitand its individual readout resonators, while unlabeled resonators are bus resonators. from the device passed back through the circulator to output and amplifier circuitry. Output circuitry compriseda low-pass Cu powder filter, followed by two cryogenic isolators in series, followed by an additional low-pass filter,followed by superconducting NbTi coaxial cable, followed by a low-noise HEMT amplifier at 4K and an additionallow-noise amplifier at room temperature. Low-pass filters were intended to block signals above ∼
10 GHz. In the caseof Q , additional amplification was afforded by a SLUG amplifier [S12] mounted at the mixing stage, biased via twobias-tee networks and isolated from the sample by an additional cryogenic isolator. Output signals were mixed downto 5 MHz before being digitized and averaged. Mixing-plate thermometer indicated a temperature of ∼
15 to 20 mKduring measurements.Magnetic flux was supplied to sample A via a 6-mm inner diameter superconducting wire coil placed 2 mm abovethe sample. A Stanford SRS SIM928 dc voltage source with a room-temperature 2 kΩ resistor in series supplied thebias current to the coil. The flux bias current passed through brass coaxial lines that were thermally anchored ateach stage of the DR, with a 80 MHz π -filter at 4K and a copper powder filter on the mixing chamber. In sample B,a similar wire-wound superconducting coil was mounted about 3 mm above the qubit chip and likewise driven froma SIM928 voltage source through a room-temperature 5 kΩ bias resistor. DC pair wiring (Cu above 4K within thefridge, NbTi below) was used to drive the coil. The coil had a self-inductance of 3.9 mH and mutual inductance tothe SQUID loop of ∼ f as a function of coil current and fit this against Eq. (1) of our paper to enable scaling of Φ and subtract anyoffset flux, as well as to determine f max and asymmetry d . We treat the sign of flux as arbitrary. QUBIT COHERENCE
Coherence data for both samples was collected using an automated measurement algorithm. After applying aprescribed fixed flux, the system determined the qubit frequency from Ramsey fringe fitting, optimized π and π/ T ∗ measurements were completed at a frequency detuned fromthe qubit frequency, with the level of detuning optimized to provide a reasonable number of fringes for fitting. Allraw coherence data was visually checked to confirm that a good quality measurement was achieved. If the automatedtuning routine failed to find the frequency or properly scale the π and π/ T measurements were made at each flux point followed by three T ∗ measurements. Ateach flux point, the reported T and T ∗ values and error bars comprise the mean and standard deviation of the threemeasurements. The corresponding Γ φ value is found from these mean values and its error bar is found by propagatingthe errors in T and T ∗ through via partial derivative and combining these in a quadrature sum. For sample B, at eachflux point first T was measured, then T ∗ , three times in succession. For this device the reported T and T ∗ valuescomprise the mean of the three measurements and the error bars are their standard deviation. Here the reporteddephasing rate Γ φ comprises the mean of the three values of Γ φ = 1 /T ∗ − / T found from the three T , T ∗ pairs,and the error bar is the standard deviation.Figure S2 shows T plotted versus qubit frequency, measured for the qubits discussed in our paper. We observe atrend of increasing T with decreasing qubit frequency. In sample A, each qubit’s quality factor ωT is roughly constant,consistent with dielectric loss and a frequency-independent loss tangent, as observed in other tunable superconductingqubits [S13]. On sample B, T decreases by about 10 µ s from the low to high end of the frequency range, consistentwith Purcell loss to the readout resonator. In addition, fine structure is occasionally observed in Fig. S2 where T drops sharply at specific frequencies. These localized features in the T frequency dependence are observed for alltunable qubits that we have measured. These features, similar to those observed by [S13], are attributed to frequencieswhere a qubit transition is resonant with a two-level system defect on or near the qubit. Additionally, on sample B,at a few frequency points inter-qubit coupling affects relaxation. Where the Q qubit is nearly degenerate to Q (at ∼ Q (at ∼ T ∗ plotted versus flux, measured for the qubits discussed in our paper. For the tunable qubitson sample A, T ∗ is greatest at the qubit sweet-spots and decreases away from these sweet spots as D Φ increases. Inthe α = 15 tunable qubit on sample B, T ∗ is nearly constant over the measured half flux quantum range. The smallfrequency dependence observed in T ∗ in sample B is consistent with the observed variation of T with frequency,leading to the frequency-independent dephasing rate observed for this qubit in Fig. 3 of our paper. T ( µ s ) Frequency (GHz)
FIG. S2. T vs. frequency measured for all qubits discussed in the main paper. Single points included for T values measuredfor the fixed-frequency qubits. Φ / Φ T * 2 ( µ s ) FIG. S3. T ∗ vs. flux measured for the qubits discussed in the main paper. T ∗ measured for the fixed-frequency qubits on bothsamples is included with dashed lines to help guide the eye. −0.5 −0.25 0 0.25 0.510 −2 −1 Φ / Φ T ( s ) FIG. S4. Dependence of T with flux for asymmetric transmons, calculated for the asymmetries discussed in the main paper,due to coupling to an external flux bias following the analysis of Koch et al [S14]. Though in the main paper our symmetricqubit was an α = 1, in this calculation we used α = 1 . T did not diverge at Φ = 0. RELAXATION DUE TO COUPLING TO FLUX BIAS LINE
While using two Josephson junctions to form a dc SQUID for the inductive element of a transmon allows itsfrequency to be tuned via magnetic flux, this opens up an additional channel for energy relaxation via emission intothe dissipative environment across the bias coil that is coupled to the qubit through a mutual inductance. Thiswas first discussed by Koch et al [S14]. regarding a near symmetrical split-junction transmon. We apply the sameanalysis here to study the effect of increasing junction asymmetry on the qubit T through this loss mechanism. Foran asymmetric transmon, Koch et al. show in Eq. (2.17) of Ref. [S14] that the Josephson portion of the qubitHamiltonian can be written in terms of a single phase variable with a shifted minimum that depends upon the qubit’sasymmetry and the applied flux bias. By linearizing this Hamiltonian about the static flux bias point for small noiseamplitudes, Koch et al. compute the relaxation rate for a particular current noise power from the bias impedancecoupled to the SQUID loop through a mutual inductance M . We followed this same analysis for our qubit parameters,assuming harmonic oscillator wavefunctions for the qubit ground and excited state, and obtained the dependence of T due to this mechanism as a function of bias flux. Using our typical device parameters ( E J = 20 GHz, E c = 350 MHz, M = 2 pH, R = 50 Ω) we obtain the intrinsic loss for the asymmetries discussed in our paper, shown in Fig. S4. Thisanalysis agrees with the results described in Ref. [4]. For a 10% junction asymmetry, this contribution results in a T that varies between 25 ms and a few seconds. As the junction asymmetry is increased, the minimum T value, obtainedat odd half-integer multiples of Φ , decreases slightly. However, even for our α = 15 qubit, the calculated value of T due to this mechanism never falls below 10 ms. Therefore, although increasing junction asymmetry does place anupper bound on T of an asymmetric transmon, this level is two orders of magnitude larger than the measured T incurrent state-of-the-art superconducting qubits due to other mechanisms.Also in Ref. [4], Koch et al. described a second loss channel for a transmon related to coupling to the flux-biasline. In this case, the relaxation occurs due to the oscillatory current through the inductive element of the qubit– independent of the presence of a SQUID loop – coupling to the flux-bias line, described by an effective mutualinductance M (cid:48) . This mutual vanishes when the Josephson element of the qubit and the bias line are arrangedsymmetrically. With a moderate coupling asymmetry for an on-chip bias line, Koch et al. estimate that the T corresponding to this loss mechanism would be of the order of 70 ms. Because this mechanism does not directlyinvolve the presence or absence of a SQUID loop for the inductive element, the asymmetry between junctions that weemploy in our asymmetric transmons will not play any role here and this particular limit on T should be no differentfrom that for a conventional transmon. An additional potential relaxation channel may arise due to capacitive couplingto the flux-bias line, as discussed in Ref. [S15]. However, this is expected to be negligible where a bobbin coil is usedas in our experiments. RAMSEY DECAY FITTING
As described in the main paper, our analysis of qubit dephasing rates used a purely exponential fit to all of themeasured Ramsey decays. Here we discuss why this fitting approach is appropriate for all asymmetric qubits and alarge portion of the coherence data measured for the symmetric qubit.Of all the qubits measured in this study, the symmetric α = 1 qubit was most impacted by flux noise away fromthe qubit sweet spot because of its large energy-band gradient. Therefore, to illustrate the impact that flux noisehas upon the Ramsey decay envelope we will consider the Ramsey measurements for this qubit on and off the sweetspot. Example measurements are shown at flux values of 0 and 0.3 Φ in Fig. S5a and b, respectively. At each fluxpoint, we fit the Ramsey decay with both a purely exponential (Fig. S5a I) and purely Gaussian form (Fig S5a II),the residuals of each fit are included to compare the quality of fit in each case. As has been discussed in the mainpaper, at the upper sweet-spot, where D Φ = 0, non-flux dependent background-dephasing should dominate and theRamsey decay should be more readily fit using an exponential. Figure S5a shows that this is indeed the case: thepurely exponential fit provides a more precise fit to the Ramsey decay, with the residuals to this fit being smallerover the entire range compared to those corresponding to the Gaussian fit. The Ramsey decay shown in Fig S5bwas measured at a point where D Φ was the maximum measured for the α = 1 qubit. Here, it is clear that a purelyGaussian form results in a better fit with smaller residuals than an exponential envelope. This indicates that, at thisflux point, the α = 1 qubit is heavily impacted by low-frequency flux noise, as a purely 1/f dephasing source wouldresult in a Gaussian envolope for the decay [S16]. Although a purely Gaussian fit form is useful for illustrating theimpact that flux noise has upon the Ramsey decay form, it is not an optimal quantitative approach for investigatingdephasing in these qubits. This is because tunable transmons dephase not only due to flux noise with a roughly 1 /f power spectrum, but also due to other noise sources with different non-1 /f power spectra [S17–S19]. These othernoise sources generally result in an exponential dephasing envelope. Also, dephasing has an intrinsic loss componentthat is always exponential in nature. Therefore, to accurately fit decay due to dephasing in these qubits, we mustaccount for these exponential decay envelopes in any fitting approach that is not purely exponential.To account for the T contribution to the Ramsey decay envelope in our non-exponential fitting, we take the average T measured at each flux point and separate this from T ∗ in the Ramsey fit function using 1 /T ∗ = 1 /T φ + 1 / T .Therefore, instead of fitting a T ∗ time, we fit T φ directly. To fit the Ramsey using a Gaussian fit form, we square thedephasing exponent within the fitting function [Eq. (S1)]. We can go one step further by not forcing an explicit fitform to the dephasing exponent, but instead adding another fit parameter γ [Eq. (S2)], which would be 1 for a pureexponential and 2 for a pure Gaussian. Although a fit that is not explicitly exponential or Gaussian is not motivateddirectly by a particular theoretical model, by fitting Ramsey decays with this free exponent γ , we gain insight intothe transition from flux-noise dominated dephasing at large D Φ to background dephasing near the sweet-spots. Thetwo separate fit forms described above are given by the following decay functions: f Ramsey ( t ) = A + B { cos ( ωt + δ ) exp ( − Γ t/
2) exp [ − (Γ φ t ) ] } , (S1) f Ramsey ( t ) = A + B { cos ( ωt + δ ) exp ( − Γ t/
2) exp [ − (Γ φ t ) γ ] } , (S2)where A and B are magnitude and offset constants to adjust the arbitrary measured signal, ω is the detuning from thequbit frequency with a phase offset δ , Γ is the intrinsic loss rate (1 /T ) and Γ φ is the dephasing rate. Here, A, B, ω , δ , Γ φ , and γ are fit parameters. All other components are fixed with values determined using the methods discussedabove.This behavior is illustrated in Fig. S6, where we plot γ vs. flux extracted from fits to the Ramsey measurementson the α = 1 qubit using Eq. (S1). In the flux region between +/- 0.1 Φ , γ ≈
1, indicating that the dephasingenvelope is primarily exponential, and thus the dominant dephasing noise affecting the qubits here does not have a1 /f spectrum. At flux bias points further away from the sweet-spot, γ shifts towards 2 as D Φ increases and appearsto level off close to this value at flux biases above ∼ . . Thus, in this bias regime, the dephasing envelope is −
101 T = 2.0 +/ − µ s A m p ( a r b . un it s ) − µ s) R e s i du a l s −
101 T = 2.6 +/ − µ s A m p ( a r b . un it s ) − µ s) R e s i du a l s − Time [ µ s] A m p ( a r b . un it s ) T = 10.0 +/ − µ s0 5 10 15 20 − µ s) R e s i du a l s − A m p ( a r b . un it s ) T = 7.6 +/ − µ s0 5 10 15 20 − µ s) R e s i du a l s Φ = 0 Φ (Upper sweet spot) Φ = 0.3 Φ a) b) I II I II
Exponential fit Exponential fit Gaussian fit Gaussian fit
FIG. S5. Ramsey decay envolopes measured for the α = 1 qubit at a) the sweet-spot Φ = 0 and b) Φ = 0 . where D Φ wasthe largest value measured for this qubit. At each flux point, the Ramsey decay envelopes are fit with both a purely exponential(I) and Gaussian (II) fit form. Functions fitted to the measured data (blue open circles) plotted as solid red lines. primarily Gaussian and the dephasing noise influencing the qubits is predominantly low-frequency in nature with a1 /f -like spectrum [S16, S20].We can also vizualize this variable-exponent fit by plotting γ vs. D Φ rather than Φ, again, for the α = 1 qubit(Fig. S7). In this plot, γ approaches 2 for D Φ values around 6 GHz / Φ . We have also included vertical dashed lineson Fig. S7 indicating the maximum D Φ values reached by the less tunable α = 4 and 7 qubits on sample A. Belowthese D Φ levels, γ is close to 1 implying that the decay envelope is nearly exponential, and thus justifying our use ofan exponential decay for fitting the asymmetrical qubits in the main paper.As yet another approach to fitting the Ramsey decay envelopes, we can employ a function that separates theexponential from background-dephasing from the Gaussian form due to dephasing from noise with a low-frequencytail. For this fit, along with separating out the T contribution to the Ramsey decay envelope, we also determinethe non-flux dependent background-dephasing rate at the sweet-spot, then use this rate as a fixed parameter in thefitting of our Ramsey measurements at any given flux point. We now have a composite Ramsey fit form that hasthree components: a T contribution and background dephasing component that are purely exponential and fixed bythe fitting of separate measurements, plus a Gaussian component to capture the dephasing due to noise with a 1 /f spectrum. This leads to a composite fitting function of the form: f Ramsey ( t ) = A + B { cos ( ωt + δ ) exp ( − Γ t/
2) exp ( − Γ φ,bkg t ) exp [ − (Γ φ t ) ] } , (S3) −0.1 0 0.1 0.2 0.311.21.41.61.82 Φ / Φ γ ( un itl e ss ) FIG. S6. γ vs flux extracted from fits to the Ramsey measurements on the α = 1 qubit using Eq. S2. Φ (GHz/ Φ ) γ ( un itl e ss ) FIG. S7. γ vs D Φ extracted from fits to the Ramsey measurements on the α = 1 qubit using Eq. S2. Dashed lines includedto indicate the maximum D Φ reached by the α = 7 (black dashed line) and α = 4 (blue dot-dashed line) qubits measured onsample A. −1 Φ (GHz/ Φ ) Γ φ ( M H z ) ExponentialGaussian γ exponentComposite FIG. S8. Γ φ vs. D Φ calculated for the α = 1 qubit using the exponential, Gaussian [Eq. (S1)], γ -exponent [Eq. (S2)], andcomposite [Eq. (S3)]fitting forms. where A and B are magnitude and offset constants to adjust the arbitrary measured signal, ω is the detuning fromthe qubit frequency with a phase offset δ , Γ is the intrinsic loss rate (1 /T ), Γ φ,bkg is the background dephasing ratemeasured at D Φ = 0 and Γ φ is the fitted dephasing rate. Here, A, B, ω , δ , and Γ φ are fit parameters. All othercomponents are fixed with values determined using methods discussed above. Though this fit form well separates thedifferent components to dephasing decay, it has one key deficiency: it assumes that the background dephasing rateis frequency independent, which is not necessarily justified, as the background dephasing mechanism may also varywith frequency. To calculate the total dephasing rate using this fit form, we add the constant background dephasingto the fitted Γ φ .To understand how the explicit fitting form impacts the dephasing rate, in Fig. S8 we plot Γ φ vs. D Φ calculatedfor the α = 1 qubit using the four different fitting forms: exponential, Gaussian [Eq. (S1)], γ -exponent [Eq. (S2)],and composite [Eq. (S3)]. We first note that any differences in the rate of dephasing calculated at each point usingthe various fit methods are subtle and the fits are reasonably consistent with one another within the fit error bars andscatter. We do observe, though, that a purely exponential fit results in a dephasing rate that is slightly higher than thevalues from the Guassian fits for all flux points, resulting in the largest slope and thus the highest effective flux-noiselevel. Therefore, we conclude that forcing a purely exponential fit to the Ramsey decay envelopes measured for qubitsthat are strongly influenced by 1 /f flux noise simply puts an upper bound on the absolute flux noise strength. The γ -exponent fitting approach provides a dephasing rate that agrees well with that extracted from the exponential fitform at low D Φ values where background-dephasing processes dominate. However, at higher D Φ values where thequbit is heavily impacted by 1 /f flux noise, the γ -exponent fit provides better agreement with the Gaussian-fitteddephasing rate.The composite fit is rigidly fixed in the Γ φ axis by the value chosen to match the background dephasing rate, inthis case chosen to match the rate observed at the lowest D Φ for the pure exponential fit. For this reason, directcomparisons between this fit and the others at individual flux points is more difficult. Despite all of these potentialissues, the slope of Γ φ vs. D Φ is independent of the chosen background-dephasing rate. Therefore, this composite fitcan be used to calculate a flux-noise level for this α = 1 qubit that takes into account both the exponential nature ofnon-flux dependent dephasing and the Gaussian nature of 1 /f flux-noise decay. Using the same methods outlined inour paper, where we specified Γ φ = 2 π (cid:112) A Φ | ln (2 πf IR t ) | D Φ , following the approach described in Ref. [[S16]], we use0the slope of this composite fit to extract a 1 /f flux noise level of A / = 1 . ± . µ Φ . This ∼
10% reduction in theextracted flux-noise level for the α = 1 qubit compared to the purely exponential fit ( A / = 1 . ± . µ Φ ) bringsit closer to the flux-noise level extracted from the fits to the measurements on the α = 7 and 4 qubits: 1 . ± . µ Φ and 1 . ± . µ Φ , respectively. The Ramsey measurements for these qubits were fit using a purely exponential fitform. It is important to note though, that the ∼
10% reduction in the composite fit extracted flux-noise level for the α = 1 qubit is within the errors associated with our flux-noise calculations.To conclude this fitting study, we have shown that:1. The α = 1 qubit in this study has a Ramsey decay envelope that is more Gaussian in nature at high D Φ valueswhere the dephasing of this qubit is strongly influenced by low-frequency flux noise.2. Though we have discussed different fitting approaches that better model the Ramsey decay envelope of qubitsinfluenced by 1 /f flux-noise, using a purely exponential decay form for the Ramsey decay simply puts an upperbound on the extracted flux noise strength. Also, the value of the flux-noise level and the dephasing rates arecomparable to those we obtained with the various other fitting approaches.3. Using a Ramsey fit function that takes into account both the exponential nature of the T contribution to thedecay envelope and non-flux dependent dephasing, as well as the Gaussian nature of dephasing due to 1 /f fluxnoise, allows us to calculate a flux noise level for the α = 1 qubit that agrees well with the other, asymmetricqubits on the same sample. This is expected, as qubits of the same geometry on the same chip should experiencesimilar flux noise [S21]. DEPHASING RATE DISCUSSION
In Fig. S9 we present dephasing rates for several additional qubits, plotted against D Φ . These qubits were similarto those in our paper, but were prepared on additional chips and measured during additional cools of our cryostats.These data are not included in our paper for reasons of clarity and consistency. However, they are presented here tosupport the observations found in this study across all qubits measured in both of our labs.The first observation we make from Fig. S9 is that a spread in background dephasing rates is measured betweenboth fixed-frequency and tunable qubits. As discussed in our paper, these subtle variations in qubit dephasing rate arenot unexpected and are commonly observed in multi-qubit devices [S5, S8, S9]. While these variations in dephasingrate make the figure somewhat challenging to interpret, we can still draw the same conclusions for this data as thosefrom our main paper. We still observe that the dephasing rate due to flux-noise increases linearly with D Φ for thelower asymmetry qubits. Again at lower D Φ values, below ∼ / Φ , the rate of dephasing is constant within theexperimental spread for all qubits. Here, it is important to note that, for several of the qubits shown here and thosediscussed in our paper, there are specific flux bias points for each qubit where the dephasing rate is anomalously high.These points almost always coincide with places where T drops sharply at specific frequencies, presumably due tolocalized coupling to defects in these qubits. Again, this sharp frequency dependence in T is not unusual for tunablesuperconducting qubits and is consistent with what others have observed [S13].The relatively flux-independent dephasing rate at low D Φ is particularly apparent in the 9:1 qubits we measured.Several of these qubits exhibited the lowest background depahsing rates we observed in our study, between 20 and40 kHz. 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Rev.Lett. , 123602 (2005).[S19] J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,Phys. Rev. A , 042318 (2006).[S20] F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura, and J. S. Tsai, Phys. Rev. Lett. , 167001 (2006).[S21] S. Sendelbach, D. Hover, A. Kittel, M. M¨uck, J. M. Martinis, and R. McDermott, Phys. Rev. Lett. , 227006 (2008). −1 Φ (GHz/ Φ ) Γ φ ( M H z ) FIG. S9. Γ φ vs D Φ for qubits measured during this study that were not included in the main paper. Γ φφ