Microwave-activated conditional-phase gate for superconducting qubits
Jerry M. Chow, Jay M. Gambetta, Andrew W. Cross, Seth T. Merkel, Chad Rigetti, M. Steffen
MMicrowave-activated conditional-phase gate for superconducting qubits
Jerry M. Chow, Jay M. Gambetta, Andrew W. Cross, Seth T. Merkel, Chad Rigetti, and M. Steffen
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA (Dated: July 11, 2013)We introduce a new entangling gate between two fixed-frequency qubits statically coupled viaa microwave resonator bus which combines the following desirable qualities: all-microwave control,appreciable qubit separation for reduction of crosstalk and leakage errors, and the ability to functionas a two-qubit conditional-phase gate. A fixed, always-on interaction is explicitly designed betweenhigher energy (non-computational) states of two transmon qubits, and then a conditional-phase gateis ‘activated’ on the otherwise unperturbed qubit subspace via a microwave drive. We implementthis microwave-activated conditional-phase gate with a fidelity from quantum process tomographyof ∼ PACS numbers: 03.67.Ac, 42.50.Pq, 85.25.-j
Superconducting qubits are a prime candidate for scal-ing towards larger quantum processors. Improvementsto coherence times for Josephson junction-based qubits[1–4] have made possible high-fidelity universal gatesas well as the characterization of gate verification andvalidation methods [5–10]. As systems evolve towardsquantum error-correction architectures such as the two-dimensional surface code [11, 12], it is increasingly im-portant to devise and characterize different entanglinggate schemes to determine suitability for large-scale im-plementations.There have been multiple incarnations of univer-sal entangling two-qubit gates for superconductingqubits, whether as i SWAP, controlled-NOT (CNOT), orconditional-Phase (c-Phase), each with their own set ofadvantages and disadvantages. One class of gates includeall of those which rely on the dynamical flux-tunabilityof either the underlying qubits, or some separate sub-circuit. This includes the direct-resonant i SWAP (DRi)[13, 14], the higher-level resonance induced dynamical c-Phase (DP) [15–17], and any variant gates induced viaa dynamic tunable coupling [18, 19]. Another class ofgates contains all those in which the qubits have fixed-frequencies, and only a microwave-modulated passivecoupling in place either directly or via a coupling circuitsuch as a resonator bus [20]. The gates in this class in-clude the resonator sideband induced i SWAP (RSi) [21],the cross-resonance (CR) gate which generates a CNOT[9, 22–24], the Bell-Rabi (BR) single-step entanglementgate [8], the wait c-Phase (WP) gate from an always on ZZ interaction [25], and the driven resonator inducedc-Phase (RIP) [26, 27].The primary advantage of the dynamically tunableclass of gates (DRi and DP) is the ability to operatethe qubits in very different regimes: one in which thequbits are independent with negligible interaction, andone where the two-qubit interaction is maximized. Inthe first regime, single-qubit gates can be applied triv-ially without the need for specialized decoupling schemesas the qubits will not experience significant crosstalk er- rors. In the second regime, the qubits can be tuned tooptimize the two-qubit interaction so as to enable theshortest possible gate times. This means simple single-qubit gates, the possibility of strong two-qubit interac-tions, and low crosstalk errors are enabled by DRi andDP. The main disadvantages of such gates are the re-liance on flux-tunable qubits, which can have reducedcoherence times due to flux-noise [28], the risk of inter-acting with other energy levels in the system during tun-ing, and additional circuit and control complexity due toon-chip tunable flux controls or couplers which supportdynamical tunability.For the case of the microwave two-qubit gates listedabove, the qubits are fixed in frequency, and thereby canbe parked at ‘sweet-spots’ of coherence or made to beun-tunable from the start. Furthermore, the addressinghardware and shaped-microwave controls become analo-gous to those of single-qubit gates. There is additionalcircuit complexity for some of the schemes, specificallythe RSi and CR gates require local microwave address-ability for each qubit. The most significant disadvan-tages for the fixed-frequency gates are tradeoffs to ei-ther coherence or single-qubit control in order to havestronger two-qubit interactions. For RSi and RIP, thetwo-qubit gate interaction is optimized through havingstronger resonator-qubit coupling strengths, g . Yet, in-herent to both schemes is a step in which real photonsexist in the cavity, and with large g , this can lead tosignificant dephasing due to photons during the oper-ation [29]. As for CR, BR, and WP, the qubit-qubitdetunings which would give the strongest two-qubit in-teraction, also happen to result in reduced single-qubitaddressability, although this error can be mitigated viaoptimal-control schemes.In this paper, we introduce an additional entanglinggate to the lexicon of two-qubit gates: the microwave-activated c-Phase (MAP) gate. The MAP gate combinesthe higher-level resonant interactions introduced in theDP gate with the simple microwave controls of fixed-frequency transmons, while also permitting the transition a r X i v : . [ qu a n t - ph ] J u l R ab i m A m p li t ude m[ a . u ] twomqubitmsubspaceinteractionmregion ga(gb( f r equen cy f f f FIG. 1. (color online) Level diagram and two-qubit Rabiamplitude spectroscopy. (a) Representative two-transmon en-ergy ladder up to the three-excitation manifold. An interac-tion of strength √ J between the | (cid:105) and | (cid:105) levels (upperright pink shaded box) gives rise to the MAP interaction, asit causes the energy difference between | (cid:105) and | (cid:105) , denoted E B , to be different from the energy difference between | (cid:105) and | (cid:105) , denoted E A . An off-resonant drive tone from E A will then result in a different phase on the | (cid:105) state relativeto the other states in the computational basis states shadedin grey. (b) Density plot of Rabi amplitude spectroscopy. Byvarying the amplitude of a Rabi drive pulse and sweepingover frequency, it becomes possible to identify the full two-transmon energy landscape in cavity transmission. Startingfrom the right at higher frequencies and sweeping left, firstthe f =5.68 GHz transition of Q2 is encountered, which hasa linear fringing pattern with increasing drive amplitude, sub-sequently followed by multi-photon transitions to the higherlevels of Q2. Around 5.18 GHz, the f of Q1 is observed,and it happens to overlap closely to the f / between separate regimes for single-qubit control andtwo-qubit interaction, turned on or off via microwaves.The key to the MAP gate interaction lies in going beyondthree levels of the transmon, and pre-defining a resonancecondition in the three-excitation manifold. By design-ing transmons such that the | (cid:105) energy transition alignswith the | (cid:105) transition [ | nm (cid:105) refers to n excitations inqubit 1 (Q1) and m excitations in qubit 2 (Q2)], theselevels are split by the inter-qubit interaction ( √ J in thismanifold) and the degeneracy between the | (cid:105) ↔ | (cid:105) and the | (cid:105) ↔ | (cid:105) transition is removed. The netresult is that an applied external drive near resonancewith the | n (cid:105) ↔ | n (cid:105) transition induces a ZZ interac-tion, much like the ac-Stark effect. Here, we implementthe MAP gate on a pair of transmon qubits coupled via aresonator bus designed with a detuning so as to align the | (cid:105) and | (cid:105) transitions. We characterize the interactionand tune-up a c-Phase gate which is verified via quantumprocess tomography (QPT) with a gate fidelity of 87%.The technique is extendable to a general class of two-qubit MAP gates which can arise from other resonanceconditions in the higher manifolds of transmon qubits.Furthermore, although the MAP gate significantly easessingle and two-qubit controls, the burden of scaling thisto larger systems lies in the fabrication of qubits withexplicit resonance conditions in a well-defined window ofenergies.The MAP scheme relies on the presence of higher lev-els in the two transmons, but unlike the DP gate, doesnot require any resonance condition between higher lev-els and computational states (i.e. | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) ).Rather, by careful control over the design of the trans-mons, through capacitance and/or Josephson junctioncritical currents, it is possible to tailor the two differ-ent transmon energy spectra to experience a resonancecondition involving only higher level non-computationalstates.In the case where the energy corresponding to | (cid:105) isaligned with the energy corresponding to | (cid:105) [see leveldiagram of Fig. 1(a)], the interaction between the trans-mons will result in a splitting of these levels by 2 ξ where ξ = 12 (cid:16)(cid:113) J , + ∆ , + ∆ , (cid:17) . (1)Here J , is the matrix element for the interaction be-tween the 12 and 03 levels and ∆ , is the differencebetween the bare energies of the 12 and 03 levels. Fora transmon, J , is approximately equal to √ J and∆ , is approximately equal to ω − ω − δ where ω and ω are the 0-1 transition energies of the two trans-mons and δ is the anharmonicity of the second trans-mon. As illustrated in Fig. 1(a) the transition energiesbetween | (cid:105) and | (cid:105) [labeled E B ] and | (cid:105) and | (cid:105) [la-beled E A ] differ by an amount ξ , i.e. ξ can be viewed asa conditional anharmonicity.The basic principle of the MAP gate is to use thisenergy difference to induce a gate via the ac-Stark effect[30]. The ac-Stark shift of an energy level occurs when anexternal drive with amplitude Ω is nearly resonant witha transition involving that level. The level then shifts byan amount equal to the power of the external drive Ω divided by the difference ∆ d between the transition fre-quency and the drive frequency, provided that the ratioΩ / ∆ d is small. At second order in perturbation theory,we find that the conditional-phase gate has a rate ζ = ( E − E − E ) / (cid:126) ≈ ζ + Ω d ζ (2)where ∆ d = ∆ , − ω d . Here ζ = J , J , (cid:18) , + 1∆ , (cid:19) (3)is the always-on component of the rate, and ζ = J , J , + ∆ d ( ω d − ∆ , ) (4)is the microwave-activated component. While this is areasonable approximation in the small drive limit [seeFig. 2], we find that as the drive strength is increasednumerical simulations predict a saturation of this rate.Understanding this saturation will be a topic of futureresearch. (a)(b) Drive1strength1 Ω /2 π G a t e r a t e ζ / π M H z ] G a t e r a t e ζ / π M H z ] Drive1frequency1 ω d /2 π FIG. 2. (color online) Perturbative theory. (a) At weakdrive strengths (below 5 MHz here), perturbation theory(dashed blue line) agrees with a six level numerical simula-tion (solid black line). Experiments are performed at higherdrive strengths, where perturbation theory does not apply,and the gate rate of the numerical simulation is seen to sat-urate. Transition frequencies are taken from Rabi amplitudespectroscopy measurements [Fig. 1b] and drive frequenciescorrespond to those in Fig. 4. (b) At a drive strength of 5MHz, the perturbation theory (dashed blue line) agrees withnumerics (solid black line) if the drive frequency is sufficientlyfar from a transition.
For our experiment, we design two transmons( ω / π =5.166 GHz and ω / π =5.668 GHz) with a de-tuning close to twice the anharmonicity, δ / π = − (a) 400 450 500 550 600 65000.51 MAPEgateEtimeE Δ tE[ns] Q E | › E s t a t epopu l a t i on Δ tE[ns} Q E | › E s t a t epopu l a t i on oror MeasQ1Q2cav MeasQ1Q2cav (b)(c)(d)
FIG. 3. (color online) MAP interaction tune-up. (a) Pulseprotocol for the simple MAP gate tune-up, involving a stan-dard Ramsey-type experiment, where the phase on Q2 is ob-served via time-separated X π/ pulses (gate length 40 ns), inthe two cases of Q1 in ground or excited state. The MAP in-teraction is tuned via a microwave pulse tuned near f of Q2of varying amplitude and duration ∆ t applied on any driveline. (b) The population of the | (cid:105) state for Q2, show differentRamsey fringes for starting in | (cid:105) (cyan circles) versus | (cid:105) (purple squares), and a [ ZZ ] π gate occurs at 514 ns (dashedline). (c) A modified cross-resonance, refocused MAP gateprotocol which includes a composite pulse of total time ∆ t consisting of three parts: two drive pulses on Q1 at an activa-tion frequency near f of Q2, separated by refocusing pulses X π on both Q1 and Q2. The Ramsey experiment is mod-ified with a Y π/ at the end to observe out-of-phase fringes(d) controlled on the state | (cid:105) (cyan circles) or | (cid:105) (purplesquares). A [ ZZ ] π gate is indicated at the first out-of-phasefringe (dashed line) at a total gate time of 510 ns. Note thatthe upwards trend of the Q2 | (cid:105) state population is a result ofrelaxation, as in the actual experiment protocol an additional X π is applied at the end to both qubits to undo the effect ofthe refocusing X π pulses. MHz. Fig. 1(b) shows a Rabi amplitude spectroscopylandscape, where a strong Rabi-drive pulse of varyingamplitude is applied to the coupling cavity ( ω r / π =8.646GHz) of two transmons. Fringes are observed for eachtransition from the ground state to the n th level oftransmon Q1 or Q2 [labeled f n for each transmon inFig. 1(b)]. The two transmons shown satisfy the MAPcondition from the observation that the f transition ofQ1 aligns with the f / δ / π detuned [31] from the f tran-sition of Q2.The MAP interaction is tuned via a Ramsey experi-ment on Q2 as illustrated in Fig. 3(a). Conditioned onthe state of Q1, different Ramsey fringes [Fig. 3(b)] areobserved when a drive Ω at ω d / π ∼ f of Q2 is ap-plied to the system. A two-qubit c-Phase gate generator[ ZZ ] π = exp( − i π ( Z ⊗ Z )) is realized when the fringesare π out of phase, indicated by the dashed line at a gatetime ∆ t = 514 ns. Although the data in Fig. 3(b) is for Ωapplied directly to Q2, the interaction can also be driventhrough the bus resonator, or in a CR-like scheme viathe Q1 excitation-port, as indicated in Fig. 3(a).The MAP drive Ω can result in additional control er-rors to both qubits, particularly phase errors ZI or IZ due to the ac-Stark effect, and leakage to higher-levelsof Q2 when ω d / π is too close to f of Q2. These canbe mitigated with the modified MAP protocol shown inFig. 3(c), which turns the interaction into a two-qubitClifford generator [9]. In this protocol, the MAP driveis split in half, sandwiched around refocusing X π gateson both qubits, and applied only in the CR-like fashionto Q1. The refocusing pulses remove single-qubit phaseerrors and the CR-like driving of Q1 gives additional pro-tection from leakage to higher levels of Q2. We observethe effect of different Q1 input states on the output of Q2by ending the Ramsey sequence with a Y π/ , which re- − − [ ZZ ] π HH l eng t h H[ μ s ] ZZ H d r i v e H l eng t h H[ μ s ] DriveHfrequencyH[GHz] DriveHpowerH[dBm]aa)ab) ac)ad)
FIG. 4. (color online) MAP interaction versus frequency andpower. (a) Phase difference between modified MAP schemeRamsey experiments starting in | (cid:105) and | (cid:105) swept versusdrive frequency and pulse length, with (b) extracted mini-mum [ ZZ ] π two-qubit gate length. Dashed line at 5.43 GHz,located outside leakage region (5.443 to 5.466 GHz) containingtransitions to | (cid:105) and | (cid:105) . (c) Phase difference for modifiedMAP scheme Ramsey experiments starting in | (cid:105) and | (cid:105) swept versus drive power at 5.43 GHz. (d) Extracted mini-mum [ ZZ ] π two-qubit gate length for (c). Solid black linesare from six-level numerical simulations. O u t pu t P au li O pe r a t o r Experiment -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1Input Pauli Operator Input Pauli Operator
II IX IY IZ XI YI ZI XXXY XZYXYYYZ ZX ZYZZ
IIIXIYIZXIYIZIXXXYXZYXYYYZZXZYZZ
Ideal (a) (b)
IIIXIYIZXIYIZIXXXYXZYXYYYZZXZYZZ
II IX IY IZ XI YI ZI XXXY XZYXYYYZ ZX ZYZZ
FIG. 5. (color online) Quantum process tomography for[ ZZ ] π with gate time ∆ t = 510 ns. (a) Experimentally ex-tracted Pauli transfer matrix with gate fidelity F mle = 0 . ZZ ] π (with form ( X ⊗ X )exp( − i π ( Z ⊗ Z ))). sults in the oscillations shown in Fig 3(d). Here, a [ ZZ ] π is indicated by the dashed line at a total gate time (twoMAP drives of 235 ns and single-qubit gate of 40 ns) of510 ns. The contrast reduction and upward drift in signalis likely due to relaxation, decoherence and higher-orderleakage effects.We perform the refocused MAP gate scheme at vary-ing ω d / π and powers Ω to characterize the interaction.Fig. 4(a) shows the phase difference between the Ramseyfringes when starting in | (cid:105) and | (cid:105) , scanning over thedrive frequency. We plot the extracted optimal gate timefor a [ ZZ ] π versus ω d / π in Fig. 4(b), which diverges inthe region between 5.443 to 5.466 GHz, due to directleakage channels to | (cid:105) and | (cid:105) . Nonetheless, the MAPgate is clearly observable at frequencies detuned fromthis leakage regime. By parking outside this region witha drive frequency of 5.43 GHz, we scan the MAP interac-tion versus Ω [(Figs. 4(c-d)], saturating to a minimumtotal gate length of 510 ns.A six-level numerical simulation is performed, withenergy-level frequencies extracted from Rabi amplitudespectroscopy experiments for both qubits. The simula-tions are shown as solid black lines in Fig. 4. We find thatthese numerical simulations reproduce key features of theexperiment. However, the gate time is very dependent onthe exact values for the frequencies and only small varia-tions in these values lead to very different predicted gatetimes. While perturbation theory predicts the behaviorat low drive powers, an exact model that explains thegate duration at high powers will be the topic of futureinvestigations. It should also be appreciated that froma control implementation standpoint, the MAP gate isconsiderably simpler than CR [24] or BR [8], as there isnot a stringent requirement that the phase of the drivebe particularly locked to other controls in the system.Finally, we perform QPT (Fig. 5) by prepar-ing an overcomplete set of 36 states generated by { I, X π , X ± π/ , Y ± π/ } , applying the 510 ns [ ZZ ] π gate,and performing full state tomography using the bus res-onator as a joint readout [32]. We use a semidefinite post-processing algorithm and the Pauli transfer matrix repre-sentation [5] to represent the process matrix, from whichwe obtain a gate fidelity of F mle = 0 . F raw = 0 . η = − . T , T ) of (6 , µ s for both qubitsand the total gate time of 510 ns, likely with some contri-bution from state preparation and measurement (SPAM)errors [5]. SPAM errors can to lead to non-physical ef-fects that make determination of error bars difficult [10].This experiment represents a proof-of-principle of theMAP gate. Further improvements in gate speed will bepossible with more accurate frequency placement of thequbits as well as pulse shaping including amplitude andfrequency modulations. Preliminary simulations suggestthat gate speeds on the order of 1 /J should be possi-ble, and clearly further theoretical work is necessary tooptimize gate performance.In conclusion, we have demonstrated a microwave-activated c-Phase gate, utilizing a fixed resonance con-dition in the higher-energy manifolds of two transmonqubits. With a refocused implementation of the MAPgate, we achieve an optimal [ ZZ ] π in 510 ns and extractgate fidelity of ≈
87% from QPT. The refocused MAPgate is easily extendable to randomized benchmarkingmethods [9, 33] and will be explored in future work. TheMAP scheme can be further generalized to any pair ofmulti-level quantum systems, defining static resonanceconditions in non-computational energy states which canbe driven to change relative phases on computationalstates. For superconducting qubits, the MAP scheme isa gate for consideration in larger fixed-frequency quan-tum processors, but places more stringent boundaries onfabrication.We acknowledge contributions from A. D. C´orcoles,D. DiVincenzo, G. A. Keefe, J. Rohrs, M. B. Rothwell,and J. R. Rozen. We acknowledge support from IARPAunder contract W911NF-10-1-0324. All statements offact, opinion or conclusions contained herein are thoseof the authors and should not be construed as represent-ing the official views or policies of the U.S. Government. [1] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair,G. Catelani, A. P. Sears, B. R. 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