Optimization of a solid-state electron spin qubit using Gate Set Tomography
Juan P. Dehollain, Juha T. Muhonen, Robin Blume-Kohout, Kenneth M. Rudinger, John King Gamble, Erik Nielsen, Arne Laucht, Stephanie Simmons, Rachpon Kalra, Andrew S. Dzurak, Andrea Morello
OOptimization of a solid-state electron spin qubitusing Gate Set Tomography
Juan P. Dehollain ‡ , Juha T. Muhonen § , RobinBlume-Kohout , , Kenneth M. Rudinger , , John KingGamble , , Erik Nielsen , Arne Laucht , Stephanie Simmons (cid:107) ,Rachpon Kalra , Andrew S. Dzurak and Andrea Morello Centre for Quantum Computation and Communication Technologies, School ofElectrical Engineering and Telecommunications, UNSW Australia, Sydney, NewSouth Wales 2052, Australia Sandia National Laboratories, Albuquerque, New Mexico 87185, USA Center for Computing Research, Sandia National Laboratories, Albuquerque, NewMexico 87185, USAE-mail: [email protected] , [email protected] , [email protected] Abstract.
State of the art qubit systems are reaching the gate fidelities required forscalable quantum computation architectures. Further improvements in the fidelityof quantum gates demands characterization and benchmarking protocols that areefficient, reliable and extremely accurate. Ideally, a benchmarking protocol shouldalso provide information on how to rectify residual errors. Gate Set Tomography(GST) is one such protocol designed to give detailed characterization of as-built qubits.We implemented GST on a high-fidelity electron-spin qubit confined by a single Patom in Si. The results reveal systematic errors that a randomized benchmarkinganalysis could measure but not identify, whereas GST indicated the need for improvedcalibration of the length of the control pulses. After introducing this modification, wemeasured a new benchmark average gate fidelity of 99 . . Keywords : quantum computing, silicon, tomography ‡ Present address: QuTech & Kavli Institute of Nanoscience, TU Delft, 2628 CJ Delft, The Netherlands § Present address: Center for Nanophotonics, FOM Institute AMOLF, 1098 XG, Amsterdam, TheNetherlands (cid:107)
Present address: Department of Physics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 a r X i v : . [ c ond - m a t . m e s - h a ll ] J un ptimization of a solid-state electron spin qubit using Gate Set Tomography
1. Introduction
One of the main challenges in the physical implementation of a universal quantumcomputer lies in designing quantum bits that meet the exquisite operation accuraciesdemanded by fault-tolerant quantum codes. Sophisticated quantum error correctionstrategies [1, 2, 3] have driven required qubit tolerances down into the realm ofexperimental possibility; numerical evidence suggests that gate fidelities as low as 99%might be sufficient for fault-tolerant operation [4, 5]. Gate fidelities above this value havealready been claimed by several qubit systems, including liquid-state NMR [6], atomicions [7, 8, 9], superconducting qubits [10] and single spins in semiconductors [11, 12, 13].However, all of these demonstrations have been achieved in single or few-qubit systemsand it is likely that further optimization will be required in order to maintain the highfidelities above the fault tolerance threshold as the systems scale up. While problemswith low-fidelity qubits can be discerned and addressed easily, improving high-fidelityqubits is more challenging since one must characterize the qubit operation to an ever-increasing degree of accuracy. Quantum Process Tomography (QPT) [14] has been aprimary method for characterizing qubit gates. By preparing a set of input states,applying the gate to be evaluated to each state and measuring the output states viaquantum state tomography, the operator ( G ) corresponding to the applied gate can beextracted. The problem with this method is that it assumes perfect state preparationand measurement (SPAM); therefore, the accuracy in G is limited by the ratio of SPAMto gate errors [15, 16]. Most common quantum error correction codes require muchhigher fidelity on the qubit logic gates than on SPAM [4, 5]. The experimental push toincrease gate fidelities without the need to improve as much in SPAM, is rendering QPTobsolete as a means to characterize qubit gates. Randomized benchmarking (RB) [17, 18]is an alternative protocol for assessing the performance of qubit gates. Random gatesequences are applied to the qubit and the measurement outcome is compared to theexpected result to obtain an average gate fidelity. By observing the survival probabilityas the number of gates in the sequences are increased, we can extract an average gatefidelity which is independent of SPAM. The downside to this protocol is that it outputsa single benchmark for qubit gate performance, without providing further insight intoqubit characteristics and the nature of the errors.Gate Set Tomography (GST) [19] is a tool for characterizing logic operationsin a qubit system. By analysing carefully constructed experiments consisting ofstate preparation, quantum operation sequences, and measurements, it self-consistentlycharacterizes the experimental system. GST operates with minimal assumptions aboutphysical characteristics of the system; it outputs a set of logical gate operators—a gateset—that models the behaviour of the device. Characteristics of the system relevant toquantum information processing can be directly extracted from the gate set, such asrotation angles, relaxation and dephasing rates, and randomized benchmarking decayrates. By computing the goodness of a GST fit (i.e. how well the model fits theexperimental data), one reveals any deviation in the behaviour of the device from an ideal ptimization of a solid-state electron spin qubit using Gate Set Tomography
2. Qubit description and operation
GST is architecture-agnostic , in that it directly characterizes the experimental systemin the language of quantum information processing. Hence, to effectively interpret theGST results to help improve the experiment, it is necessary to understand the underlyingphysics, which we detail below.
The physical implementation of the qubit logic states – The qubit used in this studyis the quantum two-level system formed by the spin- states of an electron bound to a P donor, implanted [20] in isotopically purified Si [21]. The fabrication and operationof the device has been described in great detail in references [22, 23, 24, 25, 26]. The spinenergy states are split by an externally applied magnetic field B = 1 .
55 T. The electronspin is coupled to the P spin- nucleus via the hyperfine interaction A = 98 MHz,resulting in a two-spin, four-level system, whose eigenstates are the product states of theelectron and nuclear spins. The relaxation rate of the nuclear spin is orders of magnitudesmaller than the electron relaxation rate, allowing us to operate on a two-level electronspin subsystem with the nuclear spin ‘frozen’ in an energy eigenstate. The qubit logicstates | (cid:105) and | (cid:105) are then the eigenstates of the electron spin |↑(cid:105) and |↓(cid:105) respectively. State preparation and measurement are performed via spin dependent tunnelling ofthe P bound electron to and from a nearby single electron transistor (SET) [22, 23].For this purpose, an aluminium gate stack is fabricated on top of an 8 nm SiO layer,on the surface of the substrate above the donor. The substrate consists of a 1 µ mepilayer of isotopically purified Si with 800 ppm residual Si concentration, grownon a natural silicon wafer [21]. The SET accumulates electrons from n + source-drainregions defined by phosphorus diffusion. The full device structure—as seen in figure 1—contains the SET, a set of gates (DG) used to control the electrochemical potential ofthe donor and an ESR antenna used for qubit state manipulation [27]. The SET is verysensitive to changes in the electrostatic environment, providing high-fidelity detectionof the charge state of the P donor. Its electron island also acts as a reservoir to whichthe donor is tunnel coupled. The device is cooled down in a dilution refrigerator toan electron temperature T e ≈
100 mK. At this temperature, the thermal broadeningof the Fermi sea in the SET island (∆ E F ) is much smaller than the Zeeman splitting ptimization of a solid-state electron spin qubit using Gate Set Tomography P ESR n m SET Si DG Figure 1: Diagram of qubit device and GST model of a qubit. SEM image of the on-chip gate structure of a device identical to the one used here. The aluminium gateshave been false coloured for clarity. Depicted in red are the source-drain n + regionswhich connect the SET to the current measurement electronics. For initialization andmeasurement, the donor gates are pulsed such that µ ↑ > µ SET > µ ↓ , inducing spin-dependent tunnelling between the donor and SET. When applying a gate sequence, theDG are pulsed to higher voltage to prevent the donor electron from tunnelling to theSET. The inset diagram—zoomed from the approximate donor location—represents theBloch sphere of the qubit, consisting on the spin of an electron confined by an implanted P donor, with its nuclear spin frozen in an eigenstate. The GST model treats the qubitas a black box with buttons which allow to initialize ( ρ ), apply each gate in the gateset ( G i , x , y ) and measure ( M ) in the observable basis ( |↑(cid:105) or |↓(cid:105) ).( E Z ) of the donor spin states. By tuning the donor spin electrochemical potentials ( µ ↑ , ↓ )with respect to that of the SET island ( µ SET ), such that µ ↑ > µ SET > µ ↓ , we restrictdonor → island tunnelling to a spin-up electron, and island → donor tunnelling to spin-down electrons [23]. This allows us to perform single-shot readout and initializationwith fidelities > The gate set – Logic gates are applied with electron spin resonance (ESR) pulses.An oscillating magnetic field with amplitude B and frequency ν , matching the qubitESR frequency ν = γ e B + A/ ≈
43 GHz (where γ e = 28 GHz/T is the electrongyromagnetic ratio), will cause the spin qubit state to rotate coherently between |↑(cid:105) and |↓(cid:105) . The frequency of rotation ν can be extracted from the Rabi formula as ν = (cid:115) ( ν − ν ) + (cid:18) B γ e (cid:19) (1)The x axis in the rotating frame of the qubit is defined by the phase of the first microwavepulse applied to it. Subsequent pulses can be phase-shifted by an angle ϕ p to achieve ptimization of a solid-state electron spin qubit using Gate Set Tomography G i G x −
10 0 1 0 G y − rotations about an axis rotated by ϕ p with respect to x. By controlling B , the pulseduration τ p and ϕ p , we can encode any arbitrary qubit state. The device contains anon-chip broadband (DC-50 GHz) antenna [27] used to transmit ESR pulses to the qubit.The antenna is connected to an Agilent E8267D vector signal generator. The ∼
43 GHzmicrowave signal is modulated by its internal dual arbitrary waveform generator, whichallows precise and simultaneous control of B , τ p and ϕ p . For the experiments presentedhere, we use a fixed B ≈ µ T and calibrate τ p and ϕ p to apply the desired gate. Forthe purpose of GST we will characterize two active gates: G x and G y . G x correspondsto a π/ τ π/ = (4 ν ) − . G y is a π/ G x , but with a relative ϕ p = π/
2. Taken together these twogates are informationally complete, since they generate the single-qubit Clifford group.In addition to the active gates, we include the identity gate G i , where no pulse is appliedfor the same duration τ π/ . This gate characterizes the behaviour of a qubit while it sitsidle, waiting for other operations to finish in the quantum processor. The superoperatorscorresponding to each of these gates are displayed in table 1. The decoherence rates – For the electron spin qubit, the free induction decay andHahn echo decay times have been measured to be T ∗ = 0 .
16 ms and T = 1 msrespectively [26]. Under constant driving, the qubit can maintain its coherence forup to T ρ = 1 . T ≈
3. Gate Set Tomography
GST [19] is a method for characterizing a set of quantum processes (gates), statepreparation, and measurement simultaneously. GST requires no pre-calibration, andas such stands in contrast to state tomography, which requires pre-calibrated gates, andprocess tomography, which requires pre-calibrated state preparation and measurement.Furthermore, GST is able to obtain high-accuracy estimates efficiently, meaning thatthe number of experiments required to obtain a given accuracy, scales optimally with thedesired accuracy. To use GST, one must perform a pre-determined set of experiments.Each experiment consists of 1) state preparation, 2) a sequence of gates, performed oneafter another, and 3) a measurement. Each gate sequence consists of three parts: 1) ashort ‘fiducial’ gate sequence, followed by 2) a ‘germ’ sequence repeated some number of ptimization of a solid-state electron spin qubit using Gate Set Tomography preparation fiducial , germrepeated to max-length , measurement fiducial ) gives the complete list of gate sequencesrequired to run GST. Experiments for each gate sequence are repeated multiple times,and the resulting counts of measurement outcomes serve as input to the GST estimationalgorithms. These algorithms find the best-fit gate set to the experimental data. Becausethe gate set is defined to contain only single-qubit operations, i.e. operations acting ona 2-dimensional Hilbert state space, a gate set cannot capture effects due to additionalHilbert space dimensions. In particular, memory effects due to the environment, whichare an example of what we refer to as ‘non-Markovian noise’, cannot be fit by any as-defined gate set. All physical systems will suffer from some degree of non-Markoviannoise, and GST can detect this by assessing how well the best-fit gate set is able toreproduce the experimental data. The Pearson chi-squared test and the likelihood-ratiotest are used to quantify the ‘goodness-of-fit’.The fiducial gate sequences and germ gate sequences, which are used to constructthe final list of experiments as explained above, depend upon the ideal desired gates. Inour case these gates, given in Table 1, result in the six fiducial sequences { (empty) , G x , G y , G x G x , G x G x G x , G y G y G y } and eleven germ sequences { G x , G y , G i , G x G y , G x G y G i , G x G i G y , G x G i G i ,G y G i G i , G x G x G i G y , G x G y G y G i , G x G x G y G x G y G y } Details of how fiducial and germ sequences are computed can be found in thesupplementary material of reference [29]. We used maximum lengths that wereincreasing powers of two from 1 to 256, which are chosen to include the longest sequencespractical on our particular hardware given signal-to-noise and qubit decoherenceconsiderations. The GST analysis was performed using the open-source pyGSTicode [30].
4. Optimizing the qubit operation with GST
Each cycle of initialization, gate sequence and measurement was repeated 100 timesfor each of the 2737 sequences constructed for GST. The number of |↑(cid:105) measurementoutcomes was recorded for each sequence and the results were fed back to pyGSTi foranalysis. Figure 2(a) shows a plot of the spin-up fraction P ↑ for all the pulse sequencesapplied. For an ideal qubit, a sequence can have one of three possible P ↑ outcomes: 0,0 .
5, 1 (since the gates in our gate set consist of π/ P ↑ values in the experimentaldataset. Figure 2(b) shows a table with the estimated gates extracted from GST,highlighting on a separate column the rotation angle implicit in these gate operators. ptimization of a solid-state electron spin qubit using Gate Set Tomography W it h τ π / ca li b r a ti on W it hou t τ π / ca li b r a ti on Sequence number S p i n - up p r opo r ti on S p i n - up p r opo r ti on Average sequence length ( μ s)(a) (b)(d)(c) Gate GST estimate Rot. angleRot. axis
Figure 2: GST results. (a) Raw data points obtained after implementing each of thedesigned gate sequences and repeating them 100 times to extract the spin-up proportion P ↑ for each sequence. We number the sequences from 0 to 2736 as shown in the bottomaxis labels, and they increase in length as shown in the top axis labels. Dashed linesshow target outcomes for an ideal qubit. (b) Post-processed GST results includingthe gate operators extracted from the data, and the rotation axis and angle implied bythese operators. (c,d) GST data and results after optimizing the pulse length calibrationprotocol to improve the τ π/ accuracy.Both G x and G y show rotation angles of 0 . π , which corresponds to a 4 .
4% under-rotation from the optimal 0 . π . Prior to the development of GST, we performed a qubitoptimization using the randomized benchmarking protocol [13]. RB returns a value forgate fidelity but does not provide any characterization of the gates. Therefore, qubitoptimization is achieved by performing sweeps of intuitively chosen qubit operationparameters and searching for the parameter combination which yields the highest gatefidelity. In the RB study, we analysed gate fidelities for different pulse shapes, ESRsignal amplitudes and rise times of the pulses. We found a maximum Clifford gatefidelity F G = 99 . B = 12 µ T(corresponding to τ π = 3 µ s). However, in that study we did not correctly accountfor the fact that the fixed rise times imply that the area under the time-dependentpulse amplitude—which determines the rotation—is not linear with pulse length. Thiseffect is insignificant for long pulse lengths, but becomes more noticeable as τ p becomescomparable to the rise time. This calibration protocol was designed to only calibrate τ π and, for the rise time and pulse lengths used in our experiment, τ π / .
4% shorter in ptimization of a solid-state electron spin qubit using Gate Set Tomography Number of Clifford gates N R ec ov e r y p r ob a b ilit y ( b l ac k ) S p i n - up p r opo r ti on ( b l u e / r e d ) Figure 3: Randomized benchmarking with optimized pulse length calibration. Each ofthe small dots correspond to the P ↑ extracted from 200 repetitions of a sequence; here,red(blue) dots correspond to sequences where the final state was chosen to be |↑(cid:105) ( |↓(cid:105) ).Large black dots correspond to the overall correct recovery probability P as describedin the main text. The solid line is a fit to the data using (2), yielding C = 0 . p = 0 . F G = 99 . N . The dashed line uses p = 0 . F G = 99 .
9% [13], scaled with the same C forcomparison.rotation than τ π/ , as identified by GST.We corrected the issue by including a separate τ π/ calibration step in the protocol.The data plot in figure 2(c)—taken after implementing the optimized calibrationprotocol—shows significantly less scatter in the data, a first indication that the gatesare closer to the target gates. This is confirmed by the GST results in figure 2(d), nowindicating G x and G y rotations within 0 .
7% of the target.The ancillary files contain the full GST reports generated by pyGSTi. Additionally,we have supplied the data files constructed from the experiments, along with the Pythonnotebook used to generate the report. Instructions on how to use these files to generatethe reports can be found in the pyGSTi project website [30].To confirm the improvement in the gate calibration, we perform randomizedbenchmarking using the optimized calibration protocol. The randomized benchmarkingprotocol was implemented using the same Clifford gate set as in reference [13]. Theprotocol tests sequences with increasing number of Clifford gates N . To construct thesequences, a set of N Clifford gates is selected at random; a final state ( |↑(cid:105) or |↓(cid:105) )is also chosen at random and a final gate is added to the random gate sequence suchthat the spin is flipped to this final state. This sequence is repeated 200 times tocompute P ↑ . For each N , 20 different random sequences are measured. From the data ptimization of a solid-state electron spin qubit using Gate Set Tomography N , we can extract the overall probability of recovering thecorrect state P = 0 .
5( ¯ P ( ↑ ) ↑ +(1 − ¯ P ( ↓ ) ↑ )), where ¯ P ( ↑ ) ↑ is the mean value of P ↑ from sequenceswhere the final state was chosen to be |↑(cid:105) ( |↓(cid:105) for ¯ P ( ↓ ) ↑ ). P ( N ) can then fitted [31] to: P = C p N + 0 . C is a constant determined by SPAM errors and p determines the gate fidelity F G = (1 + p ) /
2. From the results shown in figure 3, we extract F G = 99 . P electron-spin qubit.
5. Non-Markovian noise
The accuracy of GST relies greatly on the stability of the qubit over the timescale ofthe experiment. Essentially, GST assumes that the qubit is ‘the same qubit’ wheneach sequence is being applied. Any slow drift in the environment will reduce GST’sability to fit the data using a Markovian model, and thereby reduce the reliability of itsestimates. While GST is able to detect and crudely quantify such non-Markovian noise(e.g. slow drift results in decreasing goodness-of-fit with increasing sequence length), itis as yet unable to assign meaningful error bars to account for this noise. An analysis ofthe goodness-of-fit from GST reveals that the experimental dataset violates the fittedMarkovian model by up to 250 times the standard deviation returned by the fit (seesupplementary GST reports for more details). This is a strong indicator that there arehigh levels of non-Markovian noise present in the system.We attribute the majority of the non-Markovian noise to jumps on the order of10 kHz in the qubit resonance frequency, which happen on timescales on the order of10 minutes (figure 4). These jumps likely arise from single nuclear spin flips from either Si or other ionized P in the vicinity of the qubit. Recalling (1), a shift in the ESRfrequency will modify the Rabi oscillation frequency, which in turn will cause an errorin the pulse rotation. With the B used in our experiments, a 10 kHz detuning willcause a ∼ .
2% error in pulse rotation. This is well within the accuracy capabilities ofGST.While GST and RB are expected to agree to within their respective error barson gates with Markovian errors, they respond very differently to the slow drift thatcauses non-Markovian behaviour in the system. Drift in the qubit resonance frequencyproduces coherent (unitary) errors in the gates, but ones that vary in time. RBis largely insensitive to coherent errors of any kind [32, 33]. Large non-Markoviandrifts in detuning frequency can cause the RB decay curve to become noticeably non-exponential [12, 31]; however, in the results presented here this effect is too subtle toobserve. GST, on the other hand, is very sensitive to non-Markovian noise—but hasno mechanism for it. GST misclassifies this kind of non-Markovian noise (caused byslow drift) as stochastic noise. Therefore, while RB underestimates the total noise,GST overestimates the stochastic noise. For this reason, simulated RB using the GSTestimated gate set from the optimized system (figure 2d), predicts an average Clifford ptimization of a solid-state electron spin qubit using Gate Set Tomography Amplitude of detuning jumps (kHz) C oun t s (a) (b) C oun t s Time between detuning jumps (min)
Figure 4: Statistical characterization of random jumps in the qubit resonance frequency.(a) Histogram of the amplitude of the observed frequency shifts; (b) Histogram ofthe time interval between frequency jumps. This data is obtained from repeatedresonance frequency calibrations over a period of ∼
40 hours. The calibration procedureis described in the main text. To obtain this dataset, a total of 791 calibrationswere performed with 3 minute intervals, and a total of 34 frequency jumps abovethe the threshold were recorded. The sampling rate and total length of the Ramseymeasurement is set such that the frequency resolution of the calibration is 1 kHz andthe maximum detuning detection is 100 kHz. The mean values of each dataset are: (a)10 kHz and (b) 28 minutes. The Pearson correlation coefficient using the two datasetsis − . − F G = 0 . −F G = 0 . ptimization of a solid-state electron spin qubit using Gate Set Tomography
11a threshold of 5 kHz, the output frequency of the MW signal generator is adjustedand the Ramsey fringe measurement is performed again; the process is repeated untilthe detuning frequency is found to be within the threshold. The calibration takes onaverage ∼ ∼
20 minutes. When performinglong experiments (such as those required by GST), the experiment needs to be pausedevery time a resonance calibration is performed. Therefore, increasing the frequencywith which the calibration is performed will unmanageably extend the total experimentduration. A different approach to minimize (but not eliminate) the impact of driftand/or non-Markovian noise is to interleave the ‘shots’ of each GST sequence [35].Currently, we take 100 single-shot measurements per sequence consecutively, and runthrough each sequence in a single ‘sweep’. By performing interleaving, the measurementsare taken in 100 sequence sweeps with 1 single-shot per sequence (or, more feasibly,repeating 100 /N sweeps and taking N shots for each sequence during each sweep).Interleaving would ensure that the data for each sequence are sampled from the fullspan of time for which the experiment runs. It does not eliminate non-Markovianbehaviour (drift still has a significant impact on long sequences even with interleaving),but would result in a more reliable and meaningful estimate. However, this method isimpractical with our current experimental setup, because the most time-consuming stepin the experiment is loading a new sequence onto the arbitrary waveform generator, whilerepeating a measurement once a sequence is loaded is relatively much faster. Therefore,attempting to perform an adequate amount of interleaving would unmanageably increasethe total duration of the experiment. Furthermore, this would not address the root ofthe problem: qubit drift over time that would become problematic when running realquantum circuits. Moving forward, an approach to correct this non-Markovian noise isto use dynamically corrected gates [36, 37, 38], where the gate sequence is interleavedwith a dynamical decoupling sequence in order to suppress gate errors and decoherenceeffects from low-frequency noise sources. This approach has been successfully appliedand verified to correct non-Markovian noise using GST for a trapped-ion qubit [29],which leads us to believe that it would also be successful here. Another possible solutionis to implement a Hamiltonian estimation protocol [39], which could potentially allowus to increase the speed and frequency of the detuning frequency calibration.
6. Conclusion
Gate Set Tomography is a protocol designed to characterize and optimize qubit systems.By applying GST to the P electron spin qubit in Si, we were able to identify a 4 . . . ptimization of a solid-state electron spin qubit using Gate Set Tomography Acknowledgments
We thank F. Hudson for assistance in device fabrication and design, K. Itoh forsupplying the Si epilayer, and J. McCallum and D. Jamieson for the P donorimplantation. This research was funded by the Australian Research Council Centreof Excellence for Quantum Computation and Communication Technology (projectno. CE110001027) and the US Army Research Office (W911NF-13-1-0024). Theauthors acknowledge support from the Australian National Fabrication Facility. SandiaNational Laboratories is a multi-program laboratory managed and operated by SandiaCorporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S.Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000. JKG gratefully acknowledges support from the Sandia NationalLaboratories Truman Fellowship Program, which is funded by the Laboratory DirectedResearch and Development (LDRD) program.
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