High-Fidelity Superconducting Quantum Processors Via Laser-Annealing of Transmon Qubits

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High-ﬁdelity superconducting quantum processors via laser-annealing of transmon qubits Eric J. Zhang, Srikanth Srinivasan, Neereja Sundaresan, Daniela F. Bogorin, Yves Martin, Jared B. Hertzberg, John Timmerwilke, Emily J. Pritchett, Jeng-Bang Yau, Cindy Wang, William Landers, Eric P. Lewandowski, Adinath Narasgond, Sami Rosenblatt, George A. Keefe, Isaac Lauer, Mary Beth Rothwell, Douglas T. McClure, Oliver E. Dial, Jason S. Orcutt, Markus Brink, Jerry M. Chow IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA (Dated: December 15, 2020)

Abstract—Scaling the number of qubits while maintaining high fidelity (> 99.9%) single-qubit gate operations [2], and high-fidelity quantum gates remains a key challenge for quantum two-qubit entangling cross-resonance (CR) gates [19] are computing. Presently, superconducting quantum processors with realized via static qubit-qubit coupling activated by an all- > 50-qubits are actively available. For such systems, fixed- frequency transmons are attractive due to their long coherence microwave drive scheme [2], [20]. A key requirement to enable and noise immunity. However, scaling fixed-frequency archi- high-fidelity CR gates involves the selective addressability of tectures proves challenging due to precise relative frequency fixed-frequency transmons, as well as precisely engineering requirements. Here we employ laser annealing to selectively tune their computational |0i→|1i transitions (f01) for optimal two- transmon qubits into desired frequency patterns. Statistics over qubit interaction [2]. For example, sub-optimal f separation hundreds of annealed qubits demonstrate an empirical tuning 01 precision of 18.5 MHz, with no measurable impact on qubit between neighboring qubits reduces ZX coupling strength coherence. We quantify gate error statistics on a tuned 65-qubit whilst higher-order static ZZ interactions cause accumulation processor, with median two-qubit gate fidelity of 98.7%. Baseline of two-qubit errors as well as ‘spectator’ error propagation tuning statistics yield a frequency-equivalent resistance precision across the lattice [21]. The most likely frequency collisions 3 of 4.7 MHz, sufficient for high-yield scaling beyond 10 qubit and corresponding tolerance bounds resulting from frequency levels. Moving forward, we anticipate selective laser annealing to play a central role in scaling fixed-frequency architectures. crowding of lattice transmons have been quantitatively enumerated in [22]. The principal challenge for scaling fixed-frequency archi- I.INTRODUCTION tectures is mitigating errors arising from lattice frequency col- Recent technological advances have enabled rapid scaling lisions. Typical fabrication tolerances for transmon frequencies of both the physical number of qubits and computational range from 1-2%, with uncertainties dominated by the 2-4% capabilities of quantum computers [1]–[4]. The distinction variation in tunnel junction resistance Rn [22], [23]. Thermal between classical and quantum computing arises from the ex- annealing methods to adjust and stabilize post-fabrication Rn ponentially larger computational subspace available to qubits (and correspondingly, transmon frequencies f01) have been (quantum bits), which may be set to non-classical superposi- explored previously in [24]–[27]. More recently, the LASIQ tions and entangled states. Existing multi-qubit systems built (Laser Annealing of Stochastically Impaired Qubits) tech- on superconducting circuit quantum electrodynamics (cQED) nique was introduced to increase collision-free yield of trans- architectures [5], [6] have been utilized in applications ranging mon lattices by selectively trimming (i.e. tuning) individual from early implementations of Shor’s factoring algorithm [7], qubit frequencies via laser thermal annealing [22]. The LASIQ to quantum chemistry simulations [8]–[10] and accelerated process sets Rn with high-precision, and f01 could be predicted feature mapping in machine-learning [11], [12]. Presently, from Rn according to a power-law relationship resulting from arXiv:2012.08475v1 [quant-ph] 15 Dec 2020 solid-state cQED-based processors have significantly increased the Ambegaokar-Baratoff relations and transmon theory [5], in scale [13], [14], with dozens of physical qubits demon- [28]. An empirical f01(Rn) scatter of σf ' 14 MHz limited the strated on a single quantum chip. As gate fidelities improve ability to predict f01 from Rn, resulting in a post-tune fequency and eventually reach thresholds required for fault-tolerance, precision of equivalently ∼14 MHz [22]. quantum advantage will be exploited to simulate complex Here, we demonstrate LASIQ as a scalable process tool used molecular dynamics and implement quantum algorithms on to reduce two-qubit gate errors by systematically trimming practical scales [15]–[17]. To track the continual progression lattice transmon frequencies to desired patterns. We show of quantum processing power, the quantum volume (QV) laser tuning results on statistical aggregates of > 300 tuned metric is used as an overall measure of the computational qubits, and demonstrate LASIQ baseline frequency-equivalent space available for a given processor [4]. resistance tuning precision of 4.7 MHz from empirical f01(Rn) Amongst the major classes of superconducting qubits, fixed correlations, reaching this precision with 89.5% success rate. frequency transmons [2], [5] operating in the EJ EC Based on cryogenic f01 measurements from our laser-tuned regime are attractive for their low charge dispersion, yielding processors, we empirically find a frequency assignment preci- relatively noise-immune qubits with coherence times (T1, T2) sion of 18.5 MHz, which in addition to the LASIQ tuning pre- exceeding 100 µs [18]. Transmon qubits are amenable to cision includes all deviations arising from pre-cooldown steps 2 including post-tuned chip cleaning and bonding processes. Our X/Y/Z Stage results indicate the precision of trimming f01 is dominated by (a) 4-wire probes the residual f01(Rn) scatter of untuned qubits (σf = 18.1 MHz), and is not limited by the LASIQ trimming process itself. Laser head 532 nm TEC In addition to scaling the number of tuned qubits, we mea- (SHG x-tal) sure functional parameters of multi-qubit chips (coherence and Shutter two-qubit gate-fidelity) to ensure high processor performance. Multi-qubit chip Initial collision risk: We assess the impact of LASIQ tuning on qubit coherence 17 (b) Type 1: f f j,01 k,01 21 23 using a collection of composite (partially tuned) processors, Type 2: fj,01 fk,01 – δ/2 18 24 5.20 Type 3: fj,01 fk,01 – δ showing aggregate mean T1 and T2 times of 79 ± 16 µs 26 Frequency constraints: 25 and 69 ± 26 µs respectively, with no statistically significant 15 5.15 Initial f01 > fpurc (5.2 GHz) variation between the tuned and untuned groups. Our LASIQ [GHz] 22

01 12 f process has been broadly utilized for precise frequency control 5.10 10 6 13 19 of post-fabricated 27-qubit Falcon processors, including the re- 7 20 16 cent QV-128 ibmq montreal system [4], [29]. We demonstrate 5.05 14 LASIQ scaling capabilities by tuning a 65-qubit Humming- 4 5.00 bird processor (cloud-accessible as ibmq manhattan), with a 11

Predicted Qubit Qubit Predicted 1 median two-qubit gate ﬁdelity of 98.7% based on randomized 27 Falcon chip yield: 4.95 2 8 benchmarking [2], [30]. As a scalable frequency trimming tool 3 5 Pre-tune: 3.4 % for ﬁxed frequency transmon architectures, we envision the 9 Post-tune: 51 % (c) LASIQ process to be widely implemented in future genera- 15 tions of superconducting quantum systems.

R [%]R

Δ 5

II.RESULTS 0

A. Tuning a 27-qubit Falcon processor -150150 2 4 6 8 10 12 14 16 18 20 22 24 26

As a practical demonstration of frequency trimming, a 27-

[MHz]

qubit Falcon processor is tuned to predicted frequency targets 01 300 f

Δ -300 LASIQ shift

using the LASIQ setup shown in Fig. 1(a) (described in Target shift Methods). The Falcon chip series are based on a heavy- 450

Initial f01 LASIQtuning hexagonal lattice, which contains the distance-3 hybrid sur- 5.4

face and Bacon-Shor code for error correction [31]. For this [GHz] 5.2

f demonstration, all measurements were performed at ambient conditions, with resulting tuned frequencies estimated from 5.0 Qubit Tuned f01 empirical f01(Rn) correlations. The tuned lattice is depicted in Fig. 1(b), with a color heatmap indicating the post-LASIQ 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627 Lattice qubit number frequency predictions. Nearest-neighbor (NN) qubit frequency spacing (∆fNN) can visually be seen to be separated by Fig. 1. Example of a LASIQ anneal process. (a) Outline of the laser trimming setup [22]. A 532 nm second-harmonic generation laser is sequentially focused 50 MHz ≤ |∆fNN| ≤ 250 MHz, placing qubits comfortably on the junctions of a multi-qubit quantum processor, with thermal annealing within the straddling regime for high-ZX interaction [32]. NN to selectively decrease qubit frequencies (f01) for collision avoidance. (b) Ex- collisions have been avoided with twice the collision tolerance ample of a tuned 27-qubit Falcon lattice. Final predicted f01 are depicted as a heatmap, with initial high-risk NN collision pairs highlighted, and orange (2∆c, see Supplementary Information) from bounds described outlines indicating initial f01 above the bandwidth of Purcell protection. Post- in [22], to protect against two-qubit state hybridization and LASIQ, collision and frequency constraints are resolved. (c) Detail of qubit improve chip yield. Prior to tuning, high-risk pairs within 2∆c anneals. The bottom panel indicates the initial (red) and final (blue) predicted collision bounds have edge borders highlighted as indicated in f01 showing the qubits tuned to distinct frequency setpoints. The middle panel indicates the tuning distance (monotonic negative shifts), along with the lattice graph, and are resolved after the LASIQ process is the desired target shifts (purple diamonds), with an RMS deviation (i.e. complete. Based on Monte Carlo models with a conservative frequency-equivalent resistance tuning precision) of 4.8 MHz, as determined from empirical f (R ) correlations. The top panel depicts the corresponding post-tune spread estimate of σf = 20 MHz (Sec. II-B), we 01 n junction resistance shifts, achieving tuning ranges up to 14.2%. demonstrate substantial increase in the yield rate from 3.4% to 51% for achieving zero NN collisions, corresponding to 15× yield improvement after LASIQ tuning (see Supplementary Information). The target qubit f01 patterns corresponding to protection and avoiding radiative qubit relaxation [33]. Due Fig. 1(b) are shown in the bottom panel of Fig. 1(c), where to the monotonic increase of junction resistance (Rn) intrinsic initial f01 (red) are progressively tuned until they reach pre- to the laser anneal process, qubit frequencies are ‘trimmed’ defined and distinct frequency setpoints (blue). (i.e. reduced) to desired f01 values. The LASIQ process is In addition to NN collision avoidance, all qubits have been engineered to proceed until Rn for all junction reside within tuned to targets at or below fpurc = 5.2 GHz, which in our 0.3% of the target RT (corresponding to ∼10 MHz for typical Falcon design is the cutoff for maintaining good Purcell f01(Rn) correlation coefficients), although a nominal LASIQ 3 approach will typically outperform this upper precision bound (a) untuned (initial) qubit distribution (0.01% bin width) (Sec. II-B). The center panel of Fig. 1(c) shows desired 20 tuned qubit distribution 〈R 〉 = 9.40 kΩ target shifts (purple diamonds) superimposed on final predicted T Gaussian fit (initial distribution) 〈f 〉 = 5.14 GHz frequency tuning amplitudes (green bars), indicating excellent 01 349 qubits successfully tuned agreement with desired target values. The root-mean square 15 (RMS) Rn deviation from RT for this processor is 0.16%, corre- 〈δR 〉 = 6.7% sponding to a frequency-equivalent resistance tuning precision T LASIQ tuning σR = 3.1%

of 4.8 MHz (using nominal f01(Rn) power-law coefﬁcients for

10 this sample), and is consistent with LASIQ baseline tuning (0.2% bin width) precision as described in Fig. 2. As seen from the center panel, target f01 shifts range from 75 MHz (Qubit 8) to ofqubits Number 375 MHz (Qubit 21) in Fig. 1(c), corresponding to resistance 5 shifts up to ∆Rn = 14.2%. Based on separate calibration measurements over standard junction arrays, resistance tuning peaks at ∼14%, and therefore tuning plans for each processor 0 are designed to be constrained within attainable targets. Upon -12 -10 -8 -6 -4 -2 0 completion of the LASIQ process, a typical quantum processor Distance from target (δRT) [%] is cooled and screened for coherence and single/two-qubit gate ﬁdelity, and assessments of quantum volume are performed (b) 6 tuned (success) qubit distribution [4]. Two-qubit gate error statistics of a tuned and operational log-normal distribution (fit)

65-qubit Hummingbird processor are shown in Sec. II-C. resistance undershoot

anneal success: resistance overshoot B. LASIQ tuning precision |δRT| < 0.3% of RT In prior work, tuning precision estimates were limited by 4 the imprecision in predicting f01 from room-temperature Rn

δRT = 0 → RT target

measurements [22]. Thus, only an upper bound of tuning precision could be determined (σf ' 14 MHz). Presently, we address the limits of LASIQ tuning precision as limited by 2 σRMS = 0.17%

the process itself, rather than our ability to predict cryogenic Density of qubits [%] of qubits Density f01(Rn). We analyze a large sample of 390 tuned qubits, 349 of which successfully tuned to target (defined in Sec. II-A as achieving junction resistance within 0.3% of RT), yielding an aggregate tuning success rate of 89.5% for this experiment. 0 The initial and final distributions of successfully tuned qubits -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 are shown in Fig. 2. We note here that a full range of Distance from target (δRT) [%] tuning (∆R up to ∼14% with respect to initial junction Rn) was performed for this experiment, with anticipated failure Fig. 2. LASIQ tuning outcome statistics. (a) The initial distribution (gray) rates being weighted towards tuning extremes (∆R > 14% of qubits that were successfully tuned to target (orange). The distance from target δRT is the tuning differential normalized to the final target resistance increases undershoot risk, whilst low tuning ∆R < 1% results RT. Orange bars indicate the final distribution (20× reduced bin width for in increased overshoot risk). Such extremes are readily avoided clarity) and shows the 349 qubits tuned to success. (b) Expanded view of the orange distribution shown in (a). Anneal success is defined as a tuned during f01 tuning plan assignment. A detailed analysis of resistance within 0.3% of RT, which was reached by all displayed qubits, and tuning statistics and success rates has been performed for all 89.5% of the 390 tuned qubits (details in Supplementary Information). The qubits (see Supplementary Information). blue/red regions indicate undershoot/overshoot respectively. A log-normal fit is shown by the black curve, which supports the interpretation of LASIQ To understand the consequences of the incremental approach tuning as an incremental resistance growth process. to RT, aggregate tuning statistics of the 349 successfully tuned qubits are depicted in Fig. 2(a). Rn of the initial qubits (gray histogram) are tuned to a tight tolerance about the targets (orange), as displayed on a RT normalized scale. A Gaussian precision shows that the fundamental tuning performance is fit approximates the initial resistance distribution, yielding well below the upper bound of ∼14 MHz determined in [22], a mean fractional tuning distance of 6.7% (with respect to where it was also noted that this bound was dominated by RT). The final distribution is magnified in Fig. 2(b) with a residual scatter in the prediction of cryogenic f01 from room superimposed log-normal curve fit, a characteristic distribution temperature Rn. Based on Monte Carlo simulations, a precision for fractional growth processes consistent with incremental of ≤ 6 MHz is required for high-yield scaling for quantum anneals during the LASIQ process. Averaging over the entire processors beyond 103 qubits [22]. Therefore, our baseline distribution yields a RMS error deviation σR = 0.17% from RT, frequency-equivalent resistance precision of σf < 5 MHz corresponding to σf = ∂f 01/∂Rn ·σR = 4.7 MHz as determined demonstrates LASIQ as a viable post-fabrication trimming by empirical f01(Rn) correlations from our 65-qubit Humming- process for high-yield scaling of fixed frequency transmon bird processor (Fig. 3). Our determination of LASIQ resistance processors. 4

(a) both cases. We note the slight deviation of the power-law 65-Qubit Hummingbird 5.2 10 ﬁt exponent (-0.55) from the nominal one-half expected from the Ambegaokar-Baratoff relations and transmon theory [5], σf = 18.6 MHz 5 [28], which is attributed to non-idealities in predicting f01 from

) ) [GHz] 5.0 room-temperature Rn. Nevertheless, this deviation to ﬁrst-order

Number of qubitsNumber f 0 does not affect the relative frequency spacing between qubits,

-50 -25 0 25 50 and therefore does not impact the precision to which we

Power law fit: f01 residuals [MHz] 4.8 -0.55 assign qubit frequency spacing for collision avoidance. The f01 = 18.2×Rn effective f01 assignment precision may be determined by the residuals of the power-law ﬁt as shown in the inset of Fig. 4.6 Tuned qubits (49) Qubit frequency ( frequency Qubit Untuned qubits (16) 3(a), outlined by a Gaussian distribution with σf = 18.6 MHz Aggregate power-law fit spread, corresponding to 0.38% of the mean qubit frequency (4.84 GHz), and deﬁnes the practical f01 assignment precision 10.0 11.0 12.0 for this chip. Junction resistance (Rn) [kΩ] (b) Fig. 3(b) shows a similar analysis performed over a statis-

tical sampling of 241 qubits from 7 Falcon and 2 Humming-

Tuned chips bird tuned processors. Each processor sample was ﬁt to an

10.010 (241 qubits total) -40 -20 0 20 40 -60 σf = 18.5 MHz individual f01(Rn) curve and residuals are aggregated in the

0.00 upper histogram (dark green), with 1-σf spread of 18.5 MHz,

10 -60 -40 -20 0 20 40 consistent with the single-processor sample observed in Fig. 10 3(a). Results from control (untuned) qubits are shown in the 10.0 Untuned control σf = 18.1 MHz Qubit density [%] Qubitdensity (117 qubits total) bottom panel histogram (gray), which are f01(Rn) residuals

0 extracted from 117 untuned qubits on composite (partially -60 -40 -20 0 20 40 tuned) processors, yielding 1-σf spread of 18.1 MHz. We f residuals (aggregate) [MHz] 01 may therefore conclude that the f01 assignment imprecision resulting from our LASIQ process is a negligible contribu- Fig. 3. Frequency assignment precision based on statistical aggregates of tuned tor to the overall frequency spread of the qubits. We note 27-qubit Falcon and 65-qubit Hummingbird processors. (a) Resistance (Rn) to frequency (f01) correlation for a tuned Hummingbird processor. Cryogenic that although our residual value for both tuned and untuned f01 measurements are plotted against measured junction resistances Rn, with samples is larger than the ∼14 MHz demonstrated in [22], a power-law curve superimposed on the measured data. Both tuned (49 qubits) and untuned (16) qubits are depicted. The inset shows a histogram our Falcon and Hummingbird processors undergo significantly of residuals with a standard deviation of 18.6 MHz, indicating the practical more preparatory steps prior to cooldown and a certain amount precision to which we may assign qubit frequencies. (b) The top panel shows of deviation is anticipated. The prediction imprecision of statistical precision analysis performed for a total of 241 tuned qubits from a f (R ) remains the dominant contributor to overall spread, combination of Falcon and Hummingbird chips, with aggregate f01 residuals 01 n from individual power-law regressions for each chip. The bottom panel shows with relatively smaller contributions from post-tune drifts identical analysis performed for 117 untuned qubits from both processor and pre-cooldown processes. Significant room remains for families. Cryogenic f01 measurements yield 18.5 MHz and 18.1 MHz spread for tuned and untuned qubits respectively, indicating that the LASIQ process improving frequency predictions to reach single-MHz levels does not significantly impact the overall spread of qubit frequencies prior to dictated by the ∼5 MHz LASIQ baseline frequency-equivalent preparatory chip cleaning, bonding and cooldown processes. tuning precision as shown in Fig. 2(b) (details in Sec. III).

C. Qubit coherence and gate fidelity In addition to the residual f01(Rn) prediction scatter, a number of preparatory cleaning, bonding and processor mounting Maintaining high qubit coherence is an essential component steps are undertaken for our quantum processors in the pre- of high-fidelity single- and two-qubit gates, in addition to pre- cooldown stage. A natural question therefore arises as to the cise frequency control. To empirically determine the impact of practical precision achieved for frequency assignment with the laser-tuning on qubit coherence, a composite (partially tuned) inclusion of such processes. Fig. 3(a) shows cryogenic f01 set of 4 Hummingbird processors were cooled and coherence plotted against post-LASIQ measurements of Rn on a 65-qubit was assessed. Out of a total 221 measured qubits from the 4 Hummingbird processor, which comprised 49 tuned qubits composite processors, 162 were LASIQ-tuned and 59 were left and 16 untuned qubits. Post-tuning, the processor underwent untuned (73% fractional tuning rate). The statistical aggregate plasma cleaning and flip-chip bump-bonding to an interposer is shown in Fig. 4 where tuned and untuned relaxation (T1, red) layer prior to mounting in a dilution refrigerator for cooldown and dephasing (T2, blue) times are correlated on a quantile- and screening (see Methods). The entire process occurred quantile plot (left panel). Good correspondence is observed within a 24-hour span to minimize the impact of aging and with respect to linear unity slope, indicating that distributions drift on the junction resistances (and therefore qubit frequen- of coherences remain unaffected by tuning. Visual indicators cies). A power-law fit (dashed curve) conforms well to both of the mean hT1i and hT2i and 1-σ bounds are shown in the tuned and untuned qubits, indicating that no appreciable the shaded ovals, with aggregate (including both tuned and relative shifts due to LASIQ tuning have occurred, and that untuned qubits) relaxation and dephasing times of 79 ± 16 µs the same f01(Rn) prediction may be adequately utilized in and 69 ± 26 µs respectively. A direct comparison between 5

initial detuning post-LASIQ two-qubit detuning

f f T QQ plot Untuned / Untuned / 1010 j k 1 Tuned Tuned T2 QQ plot NN Collisions: Unit slope

Type 1: |Δfj,k| ≅ 0

100 100100 Type 2: |Δfj,k| ≅ δ/2 100 sec] Type 3: |Δf | ≅ δ

Density [%] Density j,k μ [ 03 2 3

10 1000 T 〈ZZ〉 = 69 kHz 〈J〉-200= 2.1 MHz 0 200 σ = 23.2 kHz

& & F

1 〈δ〉 = −332 MHz

T 22

100

50 5050 1010

|ZZ| [kHz] |ZZ|

-1 1 Tuned Tuned -1

1010 0.1 8 Gate error model Gate-error Untuned Qubits: 59 (rotary optimization) density Tuned Qubits: 162 6 0 0 0 4 Hummingbird 〈F〉 = 98.7% (65Q)

0 50 100 A A TB1 B C C DTD2 E E σF = 0.4%

Untuned T1 & T2 [μsec] 2

error Gate

-2

Fig. 4. Impact of LASIQ tuning on qubit relaxation (T1, red) and dephasing 1010-2 0.01 (T2, blue), using composite (partially tuned) Hummingbird processors. The left panel shows a quantile-quantile (QQ) plot of tuned and untuned qubits.

Good linearity with respect to unit slope indicates similar (T1, T2) distributions -200 0 200 1.0 0.0 of both tuned and untuned qubits. The shaded ovals are visual indicators of Two-qubit detuning (Δfj,k) [MHz] PD (norm.) 1-σ spread in relaxation and dephasing times. For tuned (untuned) qubits, hT1i = 80 ± 16 µs (76 ± 15 µs) and hT2i = 68 ± 25 µs (70 ± 26 µs), Fig. 5. Gate errors of a 65-qubit Hummingbird processor after LASIQ tuning. with robust agreement between mean values within statistical error bounds. The bottom panel shows measured CNOT gate errors as a function of two- The right panel box chart (interquartile box range with 10-90% whiskers, 1- qubit detuning (orange points), yielding a median gate fidelity of 98.7% for the 99% outliers indicated by crosses and minima/maxima indicated by horizontal LASIQ-tuned Hummingbird. The shaded (gray) regions indicate approxima- delimiters) demonstrates the negligible effect of the LASIQ process on qubit tive error-rate projections based on CR gate error modeling [34], incorporating coherence. A total of 59 untuned and 162 tuned qubits from 4 Hummingbird typical qubit interaction parameters (frequency and anharmonicity, qubit chips are displayed, corresponding to a total of 221 measured qubits. coupling, gate times), with optional rotary echo pulsing for error minimization. The middle panel depicts the achieved ZZ distribution, indicating well-tailored separation near null-detuning (type-1 NN collision), while maintaining a tight ZZ spread with 69 kHz median. The top panel indicates the distribution of tuned and untuned qubits indicates that for tuned (untuned) tuned two-qubit f01 separation (orange), along with the initial (pre-LASIQ) distribution (blue), indicating high-density of collisions and ZZ amplitudes qubits, hT1i = 80 ± 16 µs (76 ± 15 µs) and hT2i = 68 ± 25 µs prior to LASIQ tuning. (70 ± 26 µs). Corresponding box-plots (interquartile box range, with 10-90% whiskers) are shown in the right panel, demonstrating that within statistical error, no variation in qubit coherence is observed due to the LASIQ process. lower panel. The corresponding ZZ statistics are shown in the As a practical demonstration of LASIQ tuning capabilities, middle panel, indicating good separation near null-detuning a 65-qubit Hummingbird processor is laser-tuned and oper- (type-1 collision), whilst maintaining tight ZZ distribution ationally cloud-accessible as ibmq manhattan. In a similar with median 69 kHz (±23.2 kHz). We note that transmon- fashion to that shown in Fig. 1(a), the LASIQ tuning plan transmon coupling suffers from static-ZZ (i.e. ‘always-on’) is generated by ensuring avoidance of nearest-neighbor (NN) error; however, within the ∆fj,k < δ regime (anharmonicity level degeneracies whilst maintaining level separation within δ ' –330 MHz, see Methods), tailoring the ZZ distribution the straddling regime [32]. Post-LASIQ, the 65-qubit proces- to present magnitudes (∼70 kHz) with collision-free detuning -2 sor was cooled and qubit frequencies were measured, with is sufficient to yield gate errors approaching 10 for typical density of frequency detuning between two-qubit gate pairs two-qubit CR gate durations of ∼400 ns, indicating that our shown in the top panel of Fig. 5 (orange, 10 MHz bin width), two-qubit gate-fidelities are near levels set by coherence [21]. along with the initial two-qubit detuning (blue, 30 MHz bins). Two-qubit gate errors as a function of qubit-pair detuning As-tuned, our processor consists of 72 operational two-qubit are displayed in Fig. 5 (bottom panel), with their associated CR gates, with gate durations ranging from 250 to 600 ns. distribution shown in the adjacent probability density map Collision boundaries (2∆c bounds as derived from [22], [32], (right). Notably, our detuning distribution shows more gates see Supplementary Information) for NN degeneracies are with positive control-target detuning, as expected given the shown in the background, consistent with those utilized in greater ZX interaction in the positive detuning region. A Fig. 1(a) indicating that ∼20% of gates are within ‘high- CR gate error model is also depicted in the figure (see risk’ collision zones. Indeed, Monte Carlo yield modeling Methods), where the shaded background (gray) indicates low- of the untuned (as-fabricated) 65-qubit Hummingbird chip fidelity regions consistent with known frequency collisions indicates an average of 12 NN collisions (assuming nominal described in [22]. As part of our model, an optimizer routine 20 MHz spread with ∆c collision bounds), with an effectively incorporates a standard CR echo sequence with optional target null yield of zero NN collisions. These yield and collision rotary pulsing to determine usable two-qubit detuning regions outcomes are typical of untuned processor samples of this within the straddling regime. For a gate-error of 1%, a total size. Post-LASIQ, the collisions are resolved, as evidenced by usable frequency range of 380 MHz is available (optimized the high median two-qubit gate fidelity (98.7%) shown in the using J = 1.75 MHz in the depicted model), which reduces 6 to 350 MHz (130 MHz) for error targets of 0.5% (0.1%). are ∼100 nm, which yield as-fabricated frequency devia- Finally, we note that our depicted model does not incorporate tions near 2%, corresponding to ∼100 MHz of qubit fre- the impacts of classical crosstalk and coherence, the latter quency spread. Transmon qubit designs follow those described of which limits gate error in the desirable detuning regions. in [4], [22], with coupled qubits in a fixed-frequency lattice We note that although our collisions bounds and unitary architecture. Nominal qubit anharmonicities are engineered gate model have similar qualitative outcomes, further work is near δ ' –330 MHz. Following junction evaporation, pre- required to determine exact collision constraints and identify screening of viable processor candidates are performed via high-fidelity detuning regimes as lattice sizes are progressively room-temperature resistance measurements of all Josephson increased. junctions, with transmon qubit frequencies estimated through wafer-level resistance-to-frequency power-law fits from prior III.DISCUSSION control experiments. Candidates are ranked based on suit- ability for LASIQ tuning into collision-free tuning plans and Significant yield improvement and high two-qubit gate within the Purcell filter bandwidth, all under the constraint fidelities for both Falcon and Hummingbird processors demon- of tuning range limits (typical maximum resistance tuning of strate LASIQ as an effective post-fabrication frequency trim- ∼14%). Following this pre-selection process, a frequency plan ming technique for multi-qubit processors based on fixed- is generated and tested through Monte Carlo collision and frequency transmon architectures. Selective laser annealing yield modeling [22]. Automated LASIQ tuning is performed offers a compelling and scalable solution to the problem of fre- as defined by the tuning plan for each transmon qubit to quency crowding, with LASIQ being readily adaptable to the a junction resistance precision band of 0.3%, which yields scaling of qubits on progressively larger quantum processors. an as-tuned RMS frequency-equivalent resistance precision Based on Monte Carlo models [22] the 4.7 MHz frequency- ∼5 MHz (Sec. II-B). Post-LASIQ, the multi-qubit processors equivalent resistance tuning precision of LASIQ allows high- are plasma-cleaned and flip-chip bonded to an interposer layer yield scaling up to and beyond the 1000-qubit level. At present, for Purcell filtering and readout. The bonded processors are cryogenic f measurements indicate a practical precision 01 mounted into a dilution refrigerator for cryogenic screening, of 18.5 MHz; however, statistical analysis of cryogenic-to- including coherence measurements and single- and two-qubit ambient thermal cycling in a dilution refrigerator yields a re- gate-error analysis. The post-LASIQ cleaning, bonding and cool stability of 5.7 MHz for our multi-qubit processors, which mounting steps are coordinated to occur within a 24-hour span may be leveraged to obtain f (R ) predictions with similar 01 n to minimize junction aging and drifts, thereby maintaining accuracy and approach the baseline frequency assignment the qubits faithfully outside collision bounds of the intended precision allowed by LASIQ. Finally, we note that despite tuning frequency plans. our present emphasis on nearest-neighbor (NN) collisions, errors arising from next-NN qubits (e.g. spectator collisions) are a rising concern with lattice scaling, and future work will C. Gate-error model incorporate tuning plans to minimize such errors, as well as lattice frequency pattern optimization for yield maximization. Our gate error estimates are the result of time-domain simulations of the Schrodinger equation within the full Duffing IV. METHODS model of two coupled transmon qubits [32]. CR pulses are modeled as “Gaussian square” in shape. For each set of A. Laser annealing for transmon frequency allocation parameters, CR gate amplitudes are tuned up in a two-pulse The LASIQ frequency tuning system is depicted in Fig. 1(a), echo for a fixed gate time by minimizing the Bloch vector [4]. whereby a diode-pumped solid-state laser (532 nm) is actively A rotary pulse is additionally applied to the target qubit aligned and focused on a multi-qubit quantum processor to se- during the CR pulses, the amplitude of which is optimized to lectively anneal individual transmon qubits [22]. A timed shut- minimize gate error, as described in [21]. 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J.T. and E.J.P. developed the numerical models wafer-scale Josephson junction resistance variation in superconducting for gate error estimation. W.L., E.P.L. and A.N. performed pre- quantum coherent circuits,” Supercond. Sci. Tech., vol. 33, no. 6, p. cooldown preparations for the multi-qubit chips. E.J.Z., J.S.O., 06LT02, 2020. [24] N. Muthusubramanian, A. Bruno, B. Tarasinski, A. Fognini, R. Hagen, S.S., N.S., J.T., E.P. and J.M.C. prepared the manuscript with L. Dicarlo, and Nandini Muthusubramanian Team, “Local trimming of input from all authors. 8

UPPLEMENTARY NFORMATION 0 V. S I 10 In this supplementary section we present and expand upon 91% results of (A) Monte Carlo yield modeling of the tuned (and 51% comparison against the untuned) 27-qubit Falcon processor

presented in Sec II-A, and (B) Statistical analysis of tuning 10-1

= 4.8 MHz 4.8 =

= 20 MHz 20 =

free yield free

success rates as described in Sec. II-B. 3.4% σ 2.5% A. 27-qubit Falcon yield

Collision -2 In Sec II-A, an example 27-qubit Falcon processor was 10 Zero-collision yield (tuned) LASIQ-tuned for nearest-neighbor (NN) type 1-4 collision Zero-collision yield (untuned) removal. To ascertain the impact of tuning on the NN collision- 3 6 10 20 30 40 free yield rates, a Monte Carlo simulation [22] has been Frequency deviation (σ ) [MHz] performed for the initial (pre-tuned) and final (post-tuned) f transmon qubit frequencies, as predicted by typical power-law Supplementary S1. Monte Carlo yield modeling for nearest-neighbor (NN) f01(Rn) curves for our sample. As described in [22], our model collision-free operation of a 27-qubit Falcon processor, based on pre- and incorporates a random deviation (i.e. normal spread) from post-LASIQ frequency predictions determined in Sec II-A(Fig. 1). Pre-tuning (red), the yield curve is poorly conditioned and indicates < 5% NN collision- the target qubit frequencies, which may arise from empirical free yield for frequency deviations up to 40 MHz. Low yields for small spreads qubit spread from cleaning, bonding and cooldown processes are due to the presence of initial type 1-4 collisions that exist on the as- (18.5 MHz), tuning imprecision (4.7 MHz), or other factors fabricated chip. Post-tuning, the final yield curve (blue) indicates ≥ 10× improved yield rates. For nominal post-tuned spreading (∼20 MHz, Sec. II- resulting in deviations from the desired target (Sec. II-B). We C), our models indicate 51% NN collision free yield. At the limit of LASIQ model the yield by adding to each target qubit frequency a tuning precision predicted for this chip (4.8 MHz), this yield improves to random variation consistent with this frequency spread, and above 90%. count the proportion of NN collision-free outcomes. Fig. S1 shows the result of our yield models, indicating both Control f f Target the untuned (red) and tuned yield rates (blue) as a function Qubit j j k Qubit k of target frequency deviation. As-fabricated, the chip yields a significant number of type 1-4 NN collisions, as evidenced Type Condition Bounds (ΔC) Bounds (2ΔC) by the poorly conditioned untuned yield curve (red). The 1 |Δfj,k| = 0 ± 17 MHz ± 34 MHz low yields for small σf results from the presence of existing collisions immediately after fabrication and prior to tuning. 2 |Δfj,k| = δ/2 ± 4 MHz ± 8 MHz However, even accounting for large random spreads, the initial untuned lattice frequency pattern is highly collision prone, 3 |Δfj,k| = δ ± 30 MHz ± 60 MHz with yields remaining under 5% for σf < 40 MHz. The poorly 4 |Δfj,k| > δ (slow gate) (slow gate) conditioned untuned yield curve is a general characteristic of fixed-frequency architectures that exhibit frequency crowding. Supplementary S2. Nearest-neighbor (NN) frequency collision bounds uti- The primary goal of our LASIQ tuning method to arrange our lized in Monte Carlo yield modeling and the generation of NN collision-free lattice frequency into well conditioned yield curves with high tuning plans. Four collision types are identified, which correspond to level probability of achieving a NN collision-free configuration. hybridization (type-1), excitation of the control or target qubit into the non- computational |2i state (type-2 and type-3), as well as ‘slow gates’ (type- Fig. S2 tabulates the various NN collision types, as well as 4) where excess two-qubit detuning results in diminished ZX interaction. tolerance bounds around which collision avoidance is success- Qubit anharmonicities are engineered near δ ' –330 MHz (see Methods). fully achieved. As described in the main text and elaborated The quantitative collision bounds (∆c) corresponding to gate errors below 1% are displayed in the third column, as determined in [22], [32]. The final in [22], four main collision types are enumerated. Type-1 cor- column (2∆c) indicates double bounds utilized in generating NN collision- responds to two-qubit hybridization, with degenerate |0i→|1i free tuning plans, serving as a ‘best practice’ protocol for successfully yielding control (j) and target (k) level spacings. Type-2 results from NN collision-free candidates. two-photon excitation of j into the non-computational |2i state, and type-3 poses similar risk when fj,01 = fk,12 and vice-versa. Type-4 corresponds to a ‘slow gate,’ whereby reduced ZX drifts, and therefore yields NN collision-free tuned candidates interaction strength occurs due to two-qubit detuning beyond with increased confidence. the straddling regime, resulting in longer gate operating times. The tuning plan for this particular Falcon processor is The ∆c collision bounds are displayed in the third column outlined in Fig. 1(a), with the aim of tuning out NN collisions of Fig. S2, numerically determined to yield ≤ 1% two-qubit and ensuring frequencies reside within the Purcell filter band- gate errors [22], [32], and are consequently utilized in the width, while maintaining resistance tuning < 14%. All qubits Monte Carlo collision and yield models depicted in Fig. S1. are successfully tuned to frequency targets with a predicted In the process of generating tuning plans for LASIQ, we 4.8 MHz precision, as outlined in Sec II-A. The resulting generally adopt the ‘best practice’ protocol of doubling these Monte Carlo yield model of the tuned lattice pattern is shown to 2∆c bounds shown in the final column of Fig. S2, which by the well-conditioned (blue) curve in Fig. S1, indicating provide tuning plans with greater robustness to post-tuned a typical yield roll-off as the frequency deviation increases. 9

For nominal practical spreads (∼20 MHz, Sec. II-B), we 100 calculate a yield improvement of 15×, from 3.4% to 51%,

indicating the efﬁcacy of the LASIQ tuning method. A similar

80 comparison is performed for the baseline precision achievable aggregate success rate = 89.5% (tuned within 0.3% desired target) by LASIQ, which is predicted at 4.8 MHz for this chip (based [%] Success 400 on 0.16% post-LASIQ resistance precision), and indicates NN 0 QUARTILE 21 4 QUARTILE6 2 QUARTILE8 3 10 QUARTILE12 4 14 collision-free yields beyond 90%. As lattice sizes are scaled 300 3

to 10 qubit levels, practical tuning precisions approaching 200 cumulative targets

. qubits . these single-MHz levels will be required to ensure appre- cumulative success 100 tuned (success) = 349 ciable yields. Presently, a major contributor to our practical total targets = 390 trimming precision results from the imprecision of predicting Cumul 0 0 successfully2 tuned4 qubits6 8 10 12 14 f01 from Rn [22], and signiﬁcant gains may be achieved by 40 desired target distribution leveraging processor recool cycles to tailor f01(Rn) predictions. Gaussian fit (success) 〈RT〉 = 7.2% σ = 3.5% In particular, statistics on qubit frequency deviations during R cryogenic cycling yields a recool stability of 5.7 MHz (see 30

Sec. III), which provides a practical pathway to approach the

baseline precision of ∼5 MHz set by the frequency-equivalent 20 resistance precision of our LASIQ process.

Number of qubits of Number 10 B. Tuning success statistics 0 In Sec. II-B, a tuning sample of 349 Falcon qubits out of 0 2 4 6 8 10 12 14 390 total qubits were tuned to resistance targets (RT), spanning Qubit tuning distance (ΔR) [%] a tuning range up to 14.5% with 89.5% success rate. In this section we break down the tuning success statistics of the Supplementary S3. Aggregate statistics of 390 qubits from trial tuning iterations of Falcon processors. The bottom panel indicates the desired tuning tuned qubits, and outline failure mechanisms which induce resistance targets and actual tuning distance (∆R, shown by the striped red deviations from target resistances extending beyond the desired and shaded gray histograms respectively). A Gaussian distribution over the target band. successfully tuned qubits (349 out of 390) is shown, with a mean tuning range of 7.2 ± 3.5%. The cumulative number of tuned qubits is shown in Fig. S3 shows the tuning statistics of all 390 qubits used in the middle panel, with total tuned qubits (red) and successfully tuned qubits our precision benchmarking experiments. The bottom panel in- (black) displayed for comparison. The deviations demonstrate that at higher dicates the desired tuning histogram distribution (red, striped) desired tuning distances, a greater failure rate occurs due to difficulty reaching the desired ∆R. This is further corroborated in the top panel (green squares) with respect to initial junction resistance, superimposed upon indicating the tuning success rate for each ∆R class of qubits. Near the the successfully tuned qubits (gray). Tuning success is defined wings of the curve, we observe dips in success rate, with the junctions with as tuning to within 0.3% band around R , and the LASIQ sys- lower target ∆R suffering generally from overshoot, whereas higher target T ∆R values suffer from undershoot. The total aggregate success rate (defined tem will proceed incrementally towards this goal. A Gaussian as tuning to within 0.3% target resistance band) is 89.5%. distribution of the successfully tuned qubits is also depicted, with a mean tuned distance of 7.2% (σ = 3.5%). We note here that each of the Falcon qubits was tuned as part of a even for significant overshoot, NN collisions are unlikely to tuning plan designed for NN collision avoidance, and hence be induced unless the deviation between tuned resistances and our tuning distances are representative of realistic processor targets approach the frequency spacing between NN qubits. samples. It is notable that deviations between the desired (red, In the latter case, undershoot effects appear for junction striped) and actual (gray) tuning distributions increasingly targets beyond ∆R > 10%, as evidenced by the slow tapering depart at both lower and higher tuning ranges, indicating that down of success rates in the top panel of Fig. S3. This is missed targets occur at the tuning distribution extremities. a consequence of junctions reaching their maximum tuning The central panel of Fig. S3 indicates the cumulative plot limit, which based on calibration measurements on junction of LASIQ tuned qubits, with both tuned total (red) and suc- arrays, is near 14%, with a spread of σ ∼ 2%. In almost all cessfully tuned (black) qubits shown. Corresponding success cases, tuning plans are engineered to reside under 14% tuning, rates for each tuning distance (assessed at 1% intervals) are to ensure that almost all qubits on a given processor have high displayed in the top panel, which indicates that failure rates probability of reaching desired targets. Nevertheless, that sta- are most likely (< 90% success) for tuning targets under 1% tistical nature of the tuning range causes outlier qubits which and above 10%. Below, we consider each case in turn. saturate at lower ∆R, resulting in monotonically decreasing In the former case, small desired tuning distances are success rates as observed in Fig. S3. Progress in scaling lattices exposed to risk of overshooting (i.e. ∆R > RT − Rn). This of transmon qubits will benefit greatly from determining the is a result of the statistical spread in resistance progression, origin of this resistance limit, and the best-practice fabrication and small desired resistance shifts suffer increased risk of processes through which this may be increased. accidental increase beyond desired targets. Despite this risk, we note that typical optimal frequency separation between qubits in our tuning plans are on the order ∼100 MHz, and