Active resonator reset in the nonlinear dispersive regime of circuit QED
C. C. Bultink, M. A. Rol, T. E. O'Brien, X. Fu, B. C. S. Dikken, C. Dickel, R. F. L. Vermeulen, J. C. de Sterke, A. Bruno, R. N. Schouten, L. DiCarlo
AActive resonator reset in the nonlinear dispersive regime of circuit QED
C. C. Bultink,
1, 2
M. A. Rol,
1, 2
T. E. O’Brien, X. Fu,
1, 4
B. C. S. Dikken,
1, 2
C. Dickel,
1, 2
R. F. L. Vermeulen,
1, 2
J. C. de Sterke,
5, 1
A. Bruno,
1, 2
R. N. Schouten,
1, 2 and L. DiCarlo
1, 2 QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Kavli Institute of Nanoscience, Delft University of Technology,P.O. Box 5046, 2600 GA Delft, The Netherlands Instituut-Lorentz for Theoretical Physics, Leiden University, Leiden, The Netherlands College of Computer, National University of Defense Technology, Changsha, China 410073 Topic Embedded Systems B.V., P.O. Box 440, 5680 AK Best, The Netherlands (Dated: June 28, 2016)We present two pulse schemes for actively depleting measurement photons from a readout res-onator in the nonlinear dispersive regime of circuit QED. One method uses digital feedback condi-tioned on the measurement outcome while the other is unconditional. In the absence of analyticforms and symmetries to exploit in this nonlinear regime, the depletion pulses are numerically op-timized using the Powell method. We shorten the photon depletion time by more than six inverseresonator linewidths compared to passive depletion by waiting. We quantify the benefit by emulat-ing an ancilla qubit performing repeated quantum parity checks in a repetition code. Fast depletionincreases the mean number of cycles to a spurious error detection event from order 1 to 75 at a 1 µ scycle time. Many protocols in quantum information processingrequire interleaving qubit gates and measurements inrapid succession. For example, current experimentalimplementations of quantum error correction (QEC)schemes [1–7] rely on repeated measurements of ancillaqubits to discretize and track errors in the data-carryingpart of the system. Minimizing the QEC cycle time is es-sential to avoid build-up of errors beyond the thresholdfor fault-tolerance.An attractive architecture for QEC codes is circuitquantum electrodynamics (cQED) [8]. Initially imple-mented with superconducting qubits, this scheme hassince grown to include both semiconducting [9] and hy-brid qubit platforms [10, 11]. Readout in cQED in-volves dispersively coupling the qubit to a microwave-frequency resonator causing a qubit-state dependent shiftof the fundamental resonance. This shift can be mea-sured by injecting the resonator with a microwave pho-ton pulse. Inversely however, there is a dual sensitivityof the qubit transition frequency to resonator photons(AC Stark shift [8]), leading to qubit dephasing and de-tuning, as well as gate errors. To ensure photons leavethe resonator before gates recommence, cQED implemen-tations of QEC include a waiting step after measure-ment. During this dead time, which lasts a significantfraction of the QEC cycle, qubits are susceptible to de-coherence. Whilst many prerequisites of measurement incQED devices for QEC have already been demonstrated(including frequency-multiplexed readout via a commonfeedline [12], the use of parametric amplifiers to improvespeed and readout fidelity [13, 14] and null back-actionon untargeted qubits [15]), comparatively little attentionhas been given to the fast depletion of resonator photonspost measurement.Two compatible approaches to accelerate photon de-pletion have been explored. The first increases the res-onator linewidth κ while adding a Purcell filter [2, 16, 17] to avoid enhanced qubit relaxation via the Purcell ef-fect [18]. However, increasing κ enhances the rate ofqubit dephasing due to stray photons [19, 20], introduc-ing a compromise. The second approach is to activelydeplete photons using a counter pulse, as recently demon-strated by McClure et al. [21]. This demonstration ex-ploited useful symmetries available when the resonatorresponse is linear. However, reaching the single-shotreadout fidelity required for QEC often involves drivingthe resonator deep into the nonlinear regime, where nosuch symmetries are available.In this Letter, we propose and demonstrate two meth-ods of active photon depletion in the nonlinear dispersiveregime of cQED. The first uses a homebuilt feedback con-troller to send one of two depletion pulses conditioned onthe declared measurement outcome. The second appliesa universal pulse independent of measurement outcome.We maximize readout fidelity at a measurement powertwo orders of magnitude larger than that inducing thecritical photon number in the resonator [8]. Withoutanalytic expressions and convenient symmetries for thisnonlinear regime, we rely exclusively on numerical opti-mization by Powell’s method [22] to tune up pulses withphysically-motivated shapes, defined by two or four pa-rameters. Both methods shorten the photon depletionby at least 5 /κ compared to depletion by waiting. Weillustrate the benefits of active photon depletion usingan emulation of multi-round quantum error correction.Specifically, we emulate an ancilla qubit performing par-ity checks [15, 23] by subjecting our qubit to repeatedrounds of coherent operations and measurement. Wequantify performance by extracting the mean number ofrounds to a measurement outcome that deviates from theideal result (i.e., an error detection event). With activedepletion, we observe an increase in this mean rounds toevent, RTE, from 15 to 39 due to the reduction of totalcycle time to 1 µ s ∼ /κ . By further fixing the ancilla a r X i v : . [ qu a n t - ph ] J un FIG. 1. (Color online)
Dispersive qubit readout in thenonlinear regime and qubit errors produced by left-over measurement photons. (a) CW feedline transmis-sion spectroscopy as a function of incident power and fre-quency near the low- and high-power fundamentals of thereadout resonator. The qubit is simultaneously driven witha weakly saturating CW tone. The right (left) vertical lineindicates the fundamental f r , | (cid:105) ( f r , | (cid:105) ) for qubit in | (cid:105) ( | (cid:105) )in the linear regime. The dot [also in (b)] indicates the set-tings ( P rf , f rf ) = ( −
93 dBm , . F a as a function of P rf and f rf with the JPA off ( τ r = 1200 ns, τ int = 1500 ns), obtained from histograms with 4000 shots perqubit state. Inset: Turning on the JPA achieves F a = 98 . τ r = 300 ns and an optimized weight function integrating τ int = 400 ns. (c) Illustration of qubits errors induced by left-over photons. At τ d , after an initial measurement pulse ends,AllXY qubit pulse pairs are applied and a final measurementis performed 1000 ns later to measure F . The transient ofthe decaying homodyne signal, P H , fits a single-photon relax-ation time 1 /κ = 250 ns. Insets and (d): F versus pulse pairfor several τ d . The ideal two-step signature is observed onlyat τ d (cid:38) to remain in the ground state, RTE increases to 75. Nu-merical simulations [24] indicate that, when including thesame intrinsic coherence for surrounding data qubits, a 5-qubit repetition code (studied in [2]) would have a logicalerror rate below its pseudo-threshold [25].We employ a 2D cQED chip containing ten trans-mon qubits with dedicated readout resonators, all capac-itively coupled to a common feedline through which allmicrowave control and measurement pulses are applied.We focus on one qubit-resonator pair for all data pre-sented. This qubit is operated at its flux sweetspot, withtransition from ground ( | (cid:105) ) to first-excited ( | (cid:105) ) stateat f q = 6 .
477 GHz, and average relaxation and Hahnecho times T = 25 µ s and T echo2 = 39 µ s. The disper-sively coupled resonator has a low-power fundamental at f r , | (cid:105) = 6 . f r , | (cid:105) = 6 . | (cid:105) ( | (cid:105) ), making the dispersive shift χ/π = − . f r , bare = 6 . P rf (cid:38) −
88 dBm. We calibrate a single-photon power P rf = −
130 dBm using photon-numbersplitting experiments [26, 27] and a critical photon num-ber [8] n crit = (∆ / g ) ≈
33 ( P rf ≈ −
115 dBm) using f r , | (cid:105) − f r , bare = g / π ∆ and ∆ = 2 π ( f q − f r , bare ).Our first objective is to maximize the average assign-ment fidelity of single-shot readout, F a = 1 −
12 ( (cid:15) + (cid:15) ) , where (cid:15) ij is the probability of incorrectly assigning mea-surement result j for input state | i (cid:105) , i, j ∈ { , } . We map F a as a function of the power P rf and frequency f rf of ameasurement pulse of duration τ r = 1200 ns [Fig. 1(b)]. F a is maximized at an intermediate P rf = −
93 dBm,22 dB stronger than the n crit power. The nonlinearity isevidenced by the bending of resonator lineshapes in theaccompanying continuous-wave (CW) transmission spec-troscopy [Fig. 1(a)]. We make two additions to furtherimprove F a . First, we turn on a Josephson paramet-ric amplifier (JPA) as the front-end of our amplificationchain, operating in non-degenerate mode with 14 dB ofgain. The improved signal-to-noise ratio allows shorten-ing τ r to 300 ns. Second, we use an optimized weightfunction (duration τ int = 400 ns) to integrate the de-modulated homodyne signal before thresholding. Thisweight function consists of the difference of the averagedtransients for | (cid:105) and for | (cid:105) [28, 29]. These additionsachieve F a = 98 . (cid:15) = 0 .
1% and (cid:15) = 2 . T .The effect of photons leftover from this strong measure-ment is conveniently illustrated with a modified AllXYsequence [30, 31]. AllXY consists of 21 sequences, eachcomprised of one pair of pulses [Fig. 1(d)] applied to thequbit followed by qubit measurement. The qubit pulsesare drawn from the set { I, X, Y, x, y } , where I denotesthe identity, and X and Y ( x and y ) denote π ( π/
2) pulsesaround the x and y axis of the Bloch sphere, respectively.Ideal pulses leave the qubit in | (cid:105) (first 5 pairs), on theequator of the Bloch sphere (next 12), and in | (cid:105) (final4), producing a characteristic two-step signature in thefidelity to | (cid:105) , F [Fig. 1(d)]. The chosen order of pulsepairs reveals clear signatures of errors in many gate pa-rameters [31]. Here, we modify the AllXY sequence byapplying a measurement pulse ending at a time τ d be-fore the start of the qubit pulses. The effect of leftoverphotons on the pulses is clearly visible in Fig. 1(c). At τ d (cid:38) /κ , F displays the expected double step. At τ d ∼ /κ , the characteristic signature of moderate qubitdetuning is observed in the high/low response of pulsepairs x - y and y - x . At τ d ≤ /κ , the detuning is signifi-cant with respect to the Rabi frequency of pulses, whichthus barely excite the qubit.We now focus on the calibration of AllXY as a pho-ton detector, suitable for the optimization of depletion FIG. 2. (Color online)
Two active methods of photon de-pletion compared to passive depletion. (a) Pulse schemefor conditional photon depletion. The controller applies a de-pletion pulse D (at f r , | (cid:105) ) or D (at f r , | (cid:105) ), each with separateamplitude and phase, depending on its declared measurementoutcome. (b) Performance of conditional depletion. Averagephoton number n as a function of τ d for all combinations ofinput qubit state and depletion pulse. Compared to wait-ing, conditional depletion saves ≥ D U , immediately fol-lowing the nominal measurement pulse, has four parameterscorresponding to the amplitude and phase of pulse compo-nents at f r , | (cid:105) and f r , | (cid:105) . (d) Performance of unconditionaldepletion. Unconditional depletion saves ≥ | (cid:105) ( | (cid:105) ). Exponential best fits (curves) to the data in thelinear regime ( n ≤
8) give 1 /κ = 255 ± pulses. Because we miss analytic formulas in the nonlin-ear regime, pulse optimization relies on numerical min-imization of the residual average photon number n us-ing Powell’s method. We choose E AllXY as cost function,defined as the sum of the absolute deviations from theideal-result fit. We find experimentally that E AllXY = αn ( τ d ) + β for n (cid:46)
30. The calibration of coefficients α and β is described in [27]. Measurement noise limits thesensitivity of the detector to δn (cid:38) .
3. These two ordersof magnitude constitute a suitable dynamic range for theoptimizations that follow.Photon depletion by feedback applies one of two de-pletion pulses, D j , conditioned on the declared measure-ment result, j ∈ { , } [Fig. 2(a)]. The pulse D j , a squarepulse of duration τ p = 30 ns, is applied at f r , | j (cid:105) by side-band modulating f rf . The combined delays from round-trip signal propagation (80 ns), the augmented integra-tion window (100 ns), and controller latency (150 ns) make D j arrive 330 ns after the measurement pulse ends.The amplitude and phase of each pulse is separately op-timized using a two-step procedure. Using the modifiedAllXY sequence with the qubit initialized in | i (cid:105) , we firstminimize n at τ d = 1000 ns. This τ d is sufficiently longto avoid saturating the detector and to reach the sensi-tivity limit after a few optimization rounds. Next, weminimize n at τ d = 500 ns. This second optimizationconverges to n ∼ . .
7) for | (cid:105) ( | (cid:105) ), reducing τ d byat least 5 /κ compared to passive depletion [Fig. 2(b)].An incorrect assignment by the feedback controller leadsto less effective depletion but still outperforms passivedepletion. We have also explored conditional depletionfor various pulse lengths while fixing τ d = 500 ns. Weobserve a systematic evolution of the optimal depletionpulse parameters but no further reduction of n [27].Unconditional depletion uses a universal depletionpulse D U starting immediately after the measurementpulse (there is no latency cost) [Fig. 2(c)]. This pulseis composed by summing two square pulses of duration τ p = 330 ns with independent amplitude and phase, gen-erated by sideband modulating f rf at f r , | (cid:105) and f r , | (cid:105) .These four parameters are numerically optimized usingthe sum of n for | (cid:105) and | (cid:105) as cost function and a similartwo-step procedure (with τ d = 400 ns in the second step)as for the conditional pulses. The minimization achieves n ∼ . .
4) for | (cid:105) ( | (cid:105) ) and reduces τ d by more than6 /κ compared to passive depletion [Fig. 2(d)]. We havealso explored unconditional depletion for various pulselengths while fixing τ d = 400 ns [27]. We find a smoothvariation of optimal pulse parameters, and a small im-provement in residual n with τ p = 270 ns. However,the overlap of the depletion pulse with the measurementintegration window reduces the readout fidelity at thissetting.We quantify the merits of these active photon depletionschemes with an experiment motivated by current effortsin multi-round quantum error correction (QEC). Specif-ically, we emulate an ancilla qubit undergoing the rapidsuccession of interleaved coherent interaction and mea-surement steps when performing repetitive parity checkson data-carrying qubits in a repetition code. We replaceeach conditional-phase (c-phase) gate in the interactionstep with idling for an equivalent time (40 ns), reducingthe coherent step to a 200 ns echo sequence that ideallyflips the ancilla each round [Fig. 3(a)]. As performancemetric, we measure the average number of rounds to anerror detection event, RTE. An error event is marked bythe first deviation of qubit measurement results from theideal alternating sequence. Imperfections reducing RTEinclude qubit relaxation, dephasing and detuning duringthe interaction step, and measurement errors due to read-out discrimination infidelity, 1 − F d (defined as the over-lap fraction of gaussian best fits to the single-shot readouthistograms [32]). To differentiate these sources of ancillahardware errors, we keep track of two types of detectionevents, determined by the measurement outcome in theround following the first deviation (Fig. 3(b), similar to R y /2 R y - /2 R y DepletionRepeat
Q1Q2A cycle cycle (ns) {0,1}
FIG. 3. (Color online)
Emulated multi-round QEC: flip-ping ancilla qubit. (a) Block diagram for repeating paritymeasurements in a repetition code. The ancilla A performs anindirect measurement of the parity of data qubits Q and Q by a coherent 200 ns interaction step followed by measure-ment. This emulation replaces the c-phase gates by idling,reducing the coherent step to a simple echo sequence thatideally flips the ancilla each round. The measurement stepis followed by a depletion step of duration τ d , after which anew cycle begins. (b) Single trace of digitized measurementoutcomes. An event is detected whenever the measurementoutcome first deviates from the ideal alternating sequence.Two types of event, s and d , are distinguished by the mea-surement outcome on the next round. (c) Average rounds toevent as a function of τ d . The unconditional method improvesRTE by a factor 2 . τ d by a factor > .
7. (d) Per-round probability of type- s event versus τ d .Added curves are obtained from the two models described inthe text. Ref. [33]). Events of type s can result, for example, froma single ancilla bit flip or from measurement errors in twoconsecutive rounds. In turn, events of type d can resultfrom one measurement error or from ancilla bit flips intwo consecutive rounds. Because photon-induced errorsprimarily lead to single ancilla bit flips, we also extractthe probability of encountering an event of type s percycle, p s , and investigate its τ d dependence.Decreasing τ d trades off T -induced errors for photon-induced errors. For passive depletion, RTE is maximizedto 14 . τ d = 2200 ns [Fig. 3(c)]. At this optimal point, depletion occupies most of the total QEC cycletime τ cycle = 2700 ns. The active depletion methodsreach a higher RTE by balancing the trade-off at lower τ d . As in the optimization, we find that unconditionaldepletion performs best, improving the maximal RTE to39 . τ cycle to 1200 ns.The essential features of RTE for the three deple-tion schemes are well captured by two theory models(see [27]). The simple model includes only qubit re-laxation and non-photon-induced dephasing (calibratedusing standard T and T echo2 measurements). The exten-sive model also includes photon-induced qubit dephasingand detuning during the idling steps (modeled followingRef. [34] with photon dynamics of Fig. 2), and a mea-sured 1 − F d = 0 .
1% for readout. As we do not modelphoton-induced pulse errors, we restrict the extensivemodel to n <
8. The good agreement observed betweenthe extensive model and experiment demonstrates thenon-demolition character of the measurement and con-firms the n calibration.The multi-round QEC emulation can be made moresensitive to leftover photons by harnessing the asymme-try of the qubit decay channel. Specifically, changing thepolarity of the final π/ | (cid:105) during measurement and depletion. This change extendsthe sensitivity of RTE to n by extending its ceiling to 168[Fig. 4], which is τ d independent and set by intrinsic deco-herence in the coherent step and readout discriminationinfidelity. Clearly, unconditional depletion outperformsconditional and passive depletion, but the reduction ofRTE to 50 at short τ d evidences the performance limitreached by our choice of pulses. In a QEC context, thekey benefit of active depletion in this non-flipping variantwill be an increase in RTE due to lower per-cycle proba-bility of data qubit errors, afforded by reducing τ cycle by6 /κ . Evidently, this effect is not captured by our emula-tion, which is only sensitive to ancilla hardware errors.These RTE experiments motivate two points for dis-cussion and outlook. First, they highlight the impor-tance of using digital feedback [35] in QEC to keep an-cillas in the ground state as much as possible (as usedin a cat code [7]). Conveniently, this feedback has re-laxed latency requirements ( τ d + 160 ns in our example),because the conditional action can be chosen to be thepolarity of the final π/ | (cid:105) followingevery measurement [36] and the ability to tune withoutinterrupting ongoing error correction [37].In summary, we have investigated two active meth-ods for fast photon depletion in the nonlinear regime ofcQED, relying on numerical optimization to successfullyoutperform passive depletion by more than 6 /κ . Activephoton depletion will find application in quantum com-puting scenarios interleaving qubit measurements with cycle (ns) FIG. 4. (Color online)
Emulated multi-round QEC: non-flipping ancilla qubit in | (cid:105) . This variant uses the sequenceof Fig. 3(a) but with opposite polarity on the final π/ τ d , for ancilla starting in | (cid:105) . RTE is no longer sensitive to qubit relaxation during τ d ,increasing the sensitivity to n . The ceiling of ∼
168 reached atlong τ d is set by intrinsic decoherence in the coherent step andreadout discrimination infidelity. (b) Per-round probability ofencountering event of type s as a function of τ d . The simpleand extensive models include the same calibrated errors as inFig. 3. coherent qubit operations. Here, we have focused on theexample of quantum error correction, emulating an an-cilla qubit performing repetitive parity checks in a repe-tition code. Future experiments will focus on combiningactive depletion with Purcell filtering to further reduceQEC cycle time from the achieved 1 µ s to ∼
500 ns, suffi-cient to cross the error pseudo-threshold in small surfacecodes at state-of-the-art transmon relaxation times [25].
ACKNOWLEDGMENTS
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This supplement provides additional figures and a de-scription of the theoretical models used to model theQEC emulation experiments.
I. ADDITIONAL FIGURESA. Experimental setup
Figure S1 shows the device and experimental setup,including a full wiring diagram. The chip containsten transmon qubit-resonator pairs. All experimentspresented target pair 2. The experimental setup issimilar to that of previous experiments [1], but withan important addition labeled QuTech Control Box.This homebuilt controller, comprised of 4 intercon-nected field-programmable gate arrays (Altera CycloneIV), has digitizing and waveform generation capabilities.The 2-channel digitizer samples with 8-bit resolution at200 MSamples / s. The 6-channel waveform generator pro-duces qubit and resonator pulse envelopes with 14-bitresolution at 200 MSamples / s. B. Photon number calibration
Figure S2 contains the calibration of the photon num-ber using AllXY error ( E AllXY ) as a detector. E AllXY isdefined as the average absolute deviation from the ideal2-step result in an AllXY experiment. To calibrate thedetector the resonator is populated using a long (1800 ns)readout pulse with a varying pulse amplitude before mea-suring the AllXY. This pulse amplitude is converted toan average photon number using the single-photon powerthat is extracted from a photon number splitting exper-iment. We fit the form E AllXY = αn + β to the datafor each input state separately, with α and β as free pa-rameters. The best-fit functions are used throughout theexperiment to convert E AllXY to n . C. Constant excited state QEC emulation
Figure S3 shows the emulated multi-round QEC for anon-flipping ancilla when the qubit is initialized in the ex-cited state. This variant of the emulation uses the samesequence as Fig. 4 but with the qubit initialized in | (cid:105) .Varying τ d , we find the optimum tradeoff between errorsinduced by leftover photons and by relaxation for thethree methods. Unconditional depletions performs best,increasing RTE by a factor 2 . τ d . The high photon number detunes the qubit so much that qubit pulses areinoperative, causing the qubit to remain in the same stateand yielding long strings of identical, expected measure-ment outcomes. D. Optimal depletion pulse characterization
Figures S4 and S5 summarize our further investigationof depletion-pulse optimizations for conditional and un-conditional depletion, respectively. For a variety of pulselengths τ p , the optimized pulse amplitudes and phaseparameters are shown, along with residual photon levelsand results for multi-round QEC emulation.For conditional depletion, the optimal amplitude A ( A ) of D ( D ) decreases smoothly as τ p increases,whereas the optimal phase φ ( φ ) remains constant. Thediscrimination infidelity 1 − F d is inferred from single-shot readout histogram experiments and is defined as thefraction of overlap of the best-fit gaussians. The residualphoton number and readout discrimination infidelity donot show any dependence on τ p . As expected, there isno dependence of the fidelity on τ p as there is no over-lap between the depletion pulse and integration window.The average rounds to event and per-round probabilityof type- s event for emulated QEC in the flipping config-uration do not show any dependence on τ p either.For unconditional depletion, the optimal values of thefour parameters defining the universal depletion pulse D U evolve smoothly as τ p is varied. The optimized n firstdecreases weakly with decreasing τ p but increases sharplyfor τ p <
250 ns. A smooth decrease in F d is observed fordecreasing τ p . We attribute this effect to the overlapbetween D U and the measurement integration window.RTE is unchanged for τ p >
270 ns, suggesting a trade-offbetween errors due to n and F d . This trade-off is reflectedin the corresponding increase of per-round probability oftype- s event. MITEQ AFS335-ULN, +30 dBMITEQ AFS3 10-ULN, +30 dB300 K I Q3 K20 mK SRSSR445A20 dB 20 dB10 dBI Q
Readout tones
Flux bias
R&SSGS100A S Data acquisition
JPA pump Qubit drives
FIG. S1.
Experimental setup.
Photograph of the cQED chip and complete wiring diagram of electronic components insideand outside the He/ He dilution refrigerator (Leiden Cryogenics CF-450). The chip contains ten transmon qubits individuallycoupled to dedicated readout resonators. All resonators couple capacitively to the common feedline traversing the chip. Alldata shown correspond to qubit-resonator pair 2. Dark features traversing the coplanar waveguide transmission lines are NbTiNbridges which interconnect ground planes and suppress slot-line mode propagation. qubit drive
FIG. S2.
Calibration of photon number using AllXYerror. E AllXY measured directly after a readout pulse of1800 ns duration drives the resonator into a steady-state pho-ton population, n , for input states | (cid:105) and | (cid:105) . The lines showa bilinear fit to the form E AllXY = αn + β . Inset: photon-number splitting experiment [2] used to calibrate the single-photon power level, P rf ∼ −
130 dBm. cycle (ns)
FIG. S3.
Emulated multi-round QEC: non-flipping an-cilla in | (cid:105) . This variant of the emulation uses the samesequence as Fig. 4 but with the qubit initialized in | (cid:105) . (a)Mean rounds to error detection event, RTE, as a function of τ d . (b) Per-round probability of encountering event of type s as a function of τ d . Added curves correspond to the simpleand extensive models described in Sec. II. FIG. S4.
Characterization of conditional depletion asa function of depletion pulse length τ p . The dashed lineindicates τ p = 30 ns, used in Figs. 2 to 4 and Fig. S3. Alldata were taken at a fixed τ d = 500 ns. (a) Optimal pulseparameters. (b) Residual photon number for both qubit statesand discrimination fidelity F d . (c) Average rounds to eventand per-round probability of type- s event for emulated QECin the flipping configuration. II. THEORETICAL MODELS
We use two models to compare to data in Figs. 3, 4,and S3 labelled simple and extensive. The simple modelincludes ancilla relaxation and intrinsic dephasing, pro-viding an upper bound for the performance of the emu-lated multi-round QEC circuit. The extensive model fur-ther includes ancilla readout error and detuning and de-phasing from the photon-induced AC Stark shift. Thesemodels use separately calibrated parameters.The ancilla sans photon field is modeled consideringamplitude and phase damping as in [3]. Single-qubitgates are approximated as 40 ns decay windows with per-fect instantaneous pulses in the middle. This leads to thefollowing scheme: τ d + 20 ns of T decay, followed by a π/ T echo2 decay (with a π pulse inthe middle), another π/ T decay.Measurement is modeled as a perfect state update S ,followed by a τ r = 300 ns decay window, and a secondstate update S . The measurement signal is conditionedboth on the state post- S ( | ψ i ) and post- S ( | ψ o ). If | ψ i = FIG. S5.
Characterization of unconditional depletionas a function of depletion pulse length τ p . The dashedline indicates τ p = 330 ns, used in Figs. 2 to 4 and Fig. S3.All data were taken at a fixed depletion time of τ d = 400 ns(a) Optimal pulse parameters. (b) Residual photon numberfor both qubit states and discrimination fidelity F d . (c) Aver-age rounds to event and per-round probability of s -event foremulated QEC in the flipping configuration. | ψ o no decay occurred, and the incorrect measurement isreturned with probability 1 −F d = 0 .
1% [Fig. S4(b)]. Theonly other possibility is for a single decay event (as we donot allow excitations). To zeroth order in τ r /T ≈ / dρ qb dt = − i ¯ ω a + B σ z , ρ qb ]+ γ D [ σ − ] ρ qb + γ φ + Γ d D [ σ z ] ρ qb . (S1)Here, D [ X ] is the Lindblad operator D [ X ] ρ = XρX † − X † Xρ − ρX † X , γ = 1 /T and γ φ the pure dephas-ing rate [ γ φ = ( T echo2 ) − − T − = (177 µ s) − ]. ¯ ω a is aconstant rotation around the z axis of the Bloch sphere,and so is canceled by the π pulse in the coherent phase.Γ d = 2 χ Im( α α ∗ ) is the measurement-induced dephas-ing, with α , the qubit-state-dependent photon field am-plitude and 2 χ the dispersive shift per photon. This con-tributes a decay to the off-diagonal element of the density0matrix during the coherent phase, multiplying it byexp (cid:20) − (cid:90) Γ d ( t ) (cid:21) , (S2)where the integral is taken over the coherent time win-dow. B = 2 χ Re( α α ∗ ) is the AC Stark shift, which de-tunes the ancilla by an amount equal to the differencein the average photon number over the two parts of thecoherent phase. This multiplies the off-diagonal terms bya complex phase φ Stark = (cid:90) t A B ( t ) − (cid:90) t B B ( t ) . (S3)Here, t A and t B are the time windows in the coherentphase on either side of the π pulse. The magnitude of the photon fields post-depletion is taken from Fig. 2, andexperiences an exponential decay at a rate that is ob-tained by fitting curves to the same figure. The phasedifference between the fields associated with the groundand excited state grows at a rate 2 χ , as extracted fromFig. 1. As we do not model photon-induced pulse errors,we restrict our modeling to n <
8, where these effects arenegligible.The experiment is simulated by storing the error-freeancilla population as a unnormalized density matrix andapplying repeated cycles of the circuit. At each measure-ment step, the fraction of the density matrix that corre-sponded to an event is removed and the correspondingprobability stored. The removed fraction of the densitymatrix in evolved for one more cycle in order to extractthe event type probabilities. This is repeated until theremaining population is less than 10 − . [1] D. Rist`e, S. Poletto, M. Z. Huang, A. Bruno, V. Vesteri-nen, O. P. Saira, and L. DiCarlo, Nat. Commun. (2015).[2] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff,J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, M. H.Devoret, S. M. Givin, and R. J. Schoelkopf, Nature ,515 (2007). [3] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[4] A. Frisk Kockum, L. Tornberg, and G. Johansson, Phys.Rev. A85