Universal Dephasing Noise Injection via Schrodinger Wave Autoregressive Moving Average Models
Andrew Murphy, Jacob Epstein, Gregory Quiroz, Kevin Schultz, Lina Tewala, Kyle McElroy, Colin Trout, Brian Tien-Street, Joan A. Hoffmann, B. D. Clader, Junling Long, David P. Pappas, Timothy M. Sweeney
UUniversal Dephasing Noise Injection via Schrodinger Wave Autoregressive Moving Average Models
Andrew Murphy, Jacob Epstein, Gregory Quiroz, Kevin Schultz, Lina Tewala,
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Kyle McElroy, Colin Trout, Brian Tien-Street, Joan A. Hoffmann, B. D. Clader, Junling Long, David P. Pappas, and Timothy M. Sweeney Johns Hopkins University Applied Physics Laboratory, Laurel, MD, 20723, USA Thomas C. Jenkins Department of Biophysics, Johns Hopkins University, Baltimore, MD, 21218, USA National Institute of Standards and Technology, Boulder, Colorado 80305, USA (Dated: February 12, 2021)
We present and validate a novel method for noise injec-tion of arbitrary spectra in quantum circuits that can beapplied to any system capable of executing arbitrary sin-gle qubit rotations, including cloud-based quantum pro-cessors. As the consequences of temporally-correlatednoise on the performance of quantum algorithms are notwell understood, the capability to engineer and inject suchnoise in quantum systems is paramount. To date, noise in-jection capabilities have been limited and highly platformspecific, requiring low-level access to control hardware.We experimentally validate our universal method by com-paring to a direct hardware-based noise-injection scheme,using a combination of quantum noise spectroscopy andclassical signal analysis to show that the two approachesagree. These results showcase a highly versatile methodfor noise injection that can be utilized by theoretical andexperimental researchers to verify, evaluate, and improvequantum characterization protocols and quantum algo-rithms for sensing and computing.
As qubit technology advances into the Noisy Intermediate-Scale Quantum (NISQ) era, spurred by quantum algorithmsthat promise advantage over classical alternatives, quantumhardware is progressively able to support circuits that are in-creasingly complex in circuit depth and system size. Whilenumerous physical technologies and architectures are beingdeveloped, each system suffers from limitations and sensi-tivities due to environmental and systematic noise sources.Ultimately, these noise sources lead to computational errorsthat destroy the efficacy of a quantum computation. For thisreason, a significant amount of research has focused on char-acterizing [1–8] and addressing (i.e., avoiding [9], suppress-ing [10–13], or correcting [14–18] noise in quantum sys-tems. These efforts have been complemented by evaluationsof quantum algorithmic resilience to specific noise sources[19–23].Engineered noise provides a necessary capability for inves-tigating the effects of noise in a controlled environment. Be-yond providing a means for validation and verification of sys-tem characterization protocols, engineered noise allows oneto examine the susceptibility and robustness of noise charac-terization and mitigation protocols to particular noise types[24]. Furthermore, when sufficiently versatile, engineerednoise can provide key insight into inherent noise resilienceand vulnerabilities of quantum algorithms to targeted noisesources. In these ways, engineered noise enables the devel-opment of robust system characterization techniques, noise mitigation protocols, and error resilient quantum algorithms,all of which can be considered necessary for achieving fault-tolerant quantum computation. To date, engineered noise hasbeen achieved in a few experimental platforms, but a standardmethod for generating well-defined noise of arbitrary spectrahas yet to be identified. Furthermore, methods for injectingengineered noise have been highly platform specific [4, 5],with particular focus on lab-based experimental systems ofsuperconducting transmon qubits [4, 5] or trapped ions [25].These noise injection protocols require either a specific de-vice design [4, 5], or control of specific in-house hardwareplatforms [25].In this article, we present a platform-agnostic techniquefor injecting engineered dephasing noise of arbitrary spectrabased on the Schr¨odinger Wave Autoregressive Moving Aver-age (SchWARMA) model [26]. A generalization of the Au-toregressive Moving Average (ARMA) models used in clas-sical signal processing, SchWARMA models have been pre-viously used to numerically simulate spatio-temporally corre-lated noise processes in multi-qubit systems. Exploiting theframework of SchWARMA-based numerical simulation, weshow that the concept of inserting SchWARMA-designed er-ror gates into an ideal quantum circuit can be leveraged fornoise injection. This is a key advantage of SchWARMA overother noise injection techniques, making it implementable inany situation where access to arbitrary single qubit rotationgates is provided, including cloud devices such as the IBMQuantum Experience (IBMQE) [27]. Focusing in particu-lar on phase noise, we show that SchWARMA can be uti-lized to introduce temporally correlated dephasing errors in real physical systems, and that the SchWARMA approach hasexcellent predictive power as well. While this study centersaround stationary Gaussian phase noise, we emphasize thatthe SchWARMA approach is far-reaching and enables injec-tion of control amplitude noise, as well as non-stationary, non-Gaussian, and spatiotemporally correlated noise.To validate the SchWARMA model as both a statisticalmodel for describing temporally-correlated noise and as agenerative model for mimicking phase-noise on a qubit, wedevelop and implement two SchWARMA-motivated noise in-jection approaches: (1) a gate-based approach – faithful to theoriginal SchWARMA concept – that inserts correlated phaseerrors into an ideal circuit, and (2) a low-cost hardware im-plementation that uses a Software Defined Radio (SDR) toimpart errors directly on the master clock. Approach (2) isless universal and not platform-agnostic, requiring direct ac- a r X i v : . [ qu a n t - ph ] F e b . . . ф SDR (cid:31)(cid:30)(cid:29)(cid:28)(cid:27)(cid:26)(cid:25) (cid:31)(cid:30)(cid:29)(cid:28)(cid:27)(cid:26)(cid:24) (cid:31)(cid:29)(cid:23)(cid:22)
Time (cid:21)(cid:28)(cid:20)(cid:19)(cid:22)(cid:30)(cid:18)(cid:27)(cid:17)(cid:16)(cid:15)(cid:14)(cid:17)(cid:29)(cid:20)(cid:22)(cid:18)(cid:17)(cid:13)(cid:19)(cid:30)(cid:17)(cid:27)(cid:26)(cid:12)(cid:11)(cid:22)(cid:28)(cid:10)(cid:17)(cid:9)(cid:19) (cid:21)(cid:22)(cid:28)(cid:20)(cid:9)(cid:30)(cid:22)(cid:18)(cid:17)(cid:13)(cid:19)(cid:30)(cid:17)(cid:27)(cid:8)(cid:30)(cid:22)(cid:7)(cid:28)(cid:30)(cid:22) (cid:21)(cid:22)(cid:28)(cid:20)(cid:9)(cid:30)(cid:22)(cid:18)(cid:17)(cid:13)(cid:19)(cid:30)(cid:17)(cid:27)(cid:8)(cid:30)(cid:22)(cid:7)(cid:28)(cid:30)(cid:22) G G . . . G N G . . . G N . . . G . . . Time Time τ τ τ τ τ N τ N yx Ψ yx Ψ (a) (b)(c) Noisy QubitNoisy Master Clock
R ( ) Z 1 ф R ( ) Z 2 ф N x x x R ( ) ф Z FIG. 1. Implementation of SchWARMA. (a) Phase noise on a qubit under ideal measurement conditions (upper Bloch sphere) is simulatedby imparting controlled errors on the experimental clock (i.e., the reference frame) with no intentional additional noise on the qubit (lowerBloch sphere). (b) Gate Injection. SchWARMA models an ideal quantum circuit in a noisy environment by interleaving error gates betweeneach control pulse. We implement these error gates as instantaneous rotations of the qubit’s reference frame. (c) SDR-based noise injection.In the SDR setup, the SDR signal is mixed with control pulses to impart errors in the control signal. The phase of the SDR signal is updateddiscretely, every 100 ns, but the signal itself runs continuously over all shots in an experiment. Unlike gate-injection experiments, each shot inan SDR-based noise injection experiment will have a unique phase trajectory of the control signal. cess to hardware for implementation. However, unlike (1),method (2) allows for direct measurement of the injectednoise spectrum. For this reason, we present method (1) asa proof-of-principle for arbitrary phase noise spectra injec-tion using SchWARMA and present method (2) as a validatorfor method (1). The IBMQE serves as a testbed for approach(1), where hardware access is purposefully restricted to presetgates to convey the universality of SchWARMA noise injec-tion on systems with limited access. In addition, an in-housetransmon-based superconducting qubit system, which we willrefer to as the Applied Physics Laboratory (APL) system, isemployed as an evaluation testbed that allows for both gate-based and direct access scenarios to be examined and com-pared.Using the qubit as a probe, the injected noise spectra isreconstructed via Quantum Noise Spectroscopy (QNS) [1–5, 28]. We find strong agreement between the reconstructedand injected noise spectra for both gate- and SDR-basedapproaches. The SDR approach is further validated by directmeasurement of its output with a signal analyzer, displayingagreement with the desired spectrum. Additionally, we showthat forward simulations of the corresponding SchWARMAmodels have strong correlation with the experimentaldata, closing the loop on the validation of the SchWARMAapproach for modeling temporally correlated dephasing noise.
ResultsDephasing Noise Injection.
The dephasing between a qubit and a master clock can be engineered through either dynamiccontrol of a qubit’s phase relative to the master clock orthrough the dynamic control of the phase of the masterclock relative to the qubit. The former case is representedby the upper panel of Fig. 1(a) where the reference framedefined by the local oscillator is stationary, and the qubitstate wanders about the z-axis. Conversely, in the latter case,represented by the lower panel of Fig. 1(a), the qubit stateis stationary while the reference frame wanders [29]. Weartificially induce correlated phase noise using this lattermethod, through dynamical control over the master clockon a single qubit system. The result is an effective qubitresponse that mimics that of a qubit subject to a temporallycorrelated classical noise environment. The phase modulationof the master clock φ ( t ) = φ ctrl ( t ) + φ noise ( t ) is composedof the typical (potentially faulty) control phase φ ctrl ( t ) , withan additional φ noise ( t ) representing the desired injected noiseprocess. Next, we discuss how the SchWARMA formalismcan be used to used to construct φ noise ( t ) . Noise Injection via SchWARMA.
ARMA models are widelyused in the field of time series analysis to model time correla-tions in data. SchWARMA provides a natural path for gen-eralization of these classic techniques from signal process-ing to quantum information. While SchWARMA has beenpreviously utilized as an approach for numerically simulatingclassical temporally correlated noise in quantum circuits, itsutility can be readily extended to noise injection and model S D R F r e q u e n c y N o i s e ( H z ) (b) S e q u e n ce N u m b e r F r e q u e n c y ( M H z ) g ( m s ) . . . (a)0 25 50Sequence Number0 . . . Su r v i v a l P r o b a b ili t y (c) 2 4Frequency (MHz)012 ∆ S ( ω )( H z ) (d) .
396 2 .
400 2 . − L . O . P o w e r( d B m ) FIG. 2. SchWARMA based noise injection and reconstruction. (a)The filter functions of the fixed total-time pulse sequences have sharpand unique dependencies on noise of different frequencies. (b) Noiseof a desired spectrum, in this example bandpass, is generated witheither the gate-based or SDR-based injection method. Inset: A mea-surement of the frequency spectrum of the 2.4 GHz noisy carriersignal out of the SDR. This signal is mixed with an ideal controlsignal and upconverted to the qubit frequency. (c) The survival prob-abilities for each sequence can be analyzed to reconstruct the noiseinfluencing the system (d). prediction.The gate-based noise injection protocol follows a similarrecipe to the numerical approach demonstrated in [26], wherethe noise is injected via error gates interleaved in a quantumcircuit, as is demonstrated in Fig. 1(b). In the case of sin-gle qubit dephasing, a circuit with noise injection can be ex-pressed as U ( φ ) = R z ( φ N ) G N R z ( φ N − ) G N − · · · R z ( φ ) G (1)where G j denotes a single qubit operation in the noise-less circuit and the error gate R z ( φ j ) represents a z -rotationwith time-correlated φ j . Upon specifying properties of thenoise process, e.g., the mean and power spectral density, aSchWARMA model is used to generate a trajectory of tem-porally correlated phases φ = { φ , . . . , φ N } . Averaging anobservable or fidelity metric over many realizations of φ re-sults in the desired dephasing behavior.This form of SchWARMA-based noise injection is equiv-alent to control master clock phase noise injection. In super-conducting qubits, an R z ( φ ) gate is implemented via a virtualframe change that simply updates the control phase (Fig. 1a).In this sense, φ represents a discretized implementation of φ noise ( t ) . When the sampling time of φ noise ( t ) is the funda-mental gate time, t G , φ and φ noise ( t ) are identical. Note thatall gates G j do not need to be equivalent in duration, but ratherjust integer multiples of t G . For example, if gate G j requirestiming nt G , the composition of n error gates are appended tothe circuit following G j .Below, we also introduce an SDR-based SchWARMAnoise injection technique that is distinct from the gate-basedapproach in that it occurs simultaneously and asynchronouslywith the control. Requiring lower level control access, the SDR approach involves injecting phase noise directly into thecontrol lines. As a result, the sampling time of the noise is not necessarily restricted by the gate time t G . However, evenin the case where the sampling time is set by t G , the injectednoise and gates are inherently asynchronous; thus, phaseupdates may frequently occur during pulses. Because theengineered noise in the SDR-based noise injection techniqueis produced independently of the control pulses, this methodallows a more direct comparison between the engineered andmeasured noise via a classical signal analyzer, a method thatcannot be leveraged in the gate-based approach. In additionto these distinct features, we elaborate on other facets of theSDR approach in the discussion section. Noise Injection Validation.
The SchWARMA noise injectionprotocol is validated via QNS. We utilize a set of determinis-tically generated Fixed Total Time Pulse Sequences (FTTPS)designed to probe the noise spectrum across a frequency bandof interest. Each sequence S k is comprised of π -pulses thatproduce a unique filter function g k that is well-concentratedin frequency space [Fig. 2(a)]. The FTTPS probe the noisespectrum by altering the number of pulses within a fixed totalsequence time. This protocol is distinct from the Carr-Purcell-Meiboom-Gill (CPMG) based QNS protocol [28] that probesthe noise spectrum by varying the interpulse delay (and to-tal sequence time) using a fixed number of π -pulses, and itis distinct from other fixed total time sequences found from arandom search [4]. Further details on FTTPS are provided inthe supplement.Dephasing noise [Fig. 2(b)] is injected into a single qubitsystem via the SDR or gate-based SchWARMA procedure,thereby corrupting the FTTPS. Survival probabilities are thenmeasured for the FTTPS [Fig. 2(c)], where the survival prob-ability for each probe sequence exhibits a distinct (and sharp)dependence on noise of specific frequencies. The resultingsurvival probabilities are used to construct an estimated noisespectrum [Fig. 2(d)]. Note that we define the survival proba-bility as the probability of the qubit being in a desired state atthe end of a probe FTTPS.Below, we demonstrate that QNS reconstructions agreewith the various SchWARMA generated and injected noisespectra. In addition, we show that SchWARMA can beused to achieve a detailed prediction of the FTTPS survivalprobabilities. In order to separate injected noise from thenative background, the native noise spectrum S nat ( ω ) ismeasured using the procedure described above and subtractedfrom the injected noise spectrum reconstructions S inj ( ω ) toobtain ∆ S ( ω ) = S inj ( ω ) − S nat ( ω ) . Gate-Based Noise Injection.
Gate-based noise injection pro-vides a means for examining quantum algorithms in the pres-ence of engineered noise when hardware access is limited toparticular gate operations. Here, we demonstrate gate-basednoise injection via SchWARMA. The IBMQE serves as arestricted access testbed, where we intentionally utilize thecircuit-level interface to limit circuit operations to the stan-
Frequency (MHz) ∆ S ( ω )( M H z ) (a) ∆ ω = 0 .
18 MHz∆ ω = 0 .
54 MHz∆ ω = 0 .
89 MHz
Frequency (MHz) ∆ S ( ω )( M H z ) (b) ω = 1 .
43 MHz ω = 1 .
07 MHz ω = 0 .
64 MHz − Frequency (MHz) ∆ S ( ω )( H z ) (c) α = 2 α = 1 α = 0 α = − α = − FIG. 3. SchWARMA gate-based noise injection on the IBMQE.Noise reconstructions are shown for (a) bandpass, (b) double band-pass, and (c) /f α noise for different bandwidths, center frequen-cies, and spectral decays. Ideal spectra (dotted lines in panels (a) and(b) and solid lines in panel (c)) are found to be in good agreementwith estimated spectra (solid lines in panels (a) and (b) and symbolsin panel (c)). The data shown here are compiled over five calibrationcycles, with solid lines denoting bootstrapped medians and shadedregions specifying confidence intervals in panels (a) and (b).Note that the confidence intervals are small for a majority of the es-timated spectra. Data has been offset for clarity. dard IBM gate set [30], demonstrating the availability of theapproach to the general user community. Phase noise is in-troduced through U ( λ ) phase gates that are interleaved withthe gates that comprise the sequences that define the probeFTTPS. The time resolution of the noise is set by the X gatetiming, given by the U ( θ, φ, λ ) gate time. The U gate im-plements generic unitary operations using three configurableframe changes interleaved with R x ( π/ and R x ( − π/ ro-tations. The frame changes are used to set the phases θ, φ, λ .Employing the Vigo processor for our experiment, where t G ≈ ns, we prepare a single qubit in the equal superposi-tion state, apply the FTTPS with injected phase noise, apply agate that would return the state to | (cid:105) in the absence of noise,and measure. The experiment is performed over 50 indepen-dent realizations of SchWARMA trajectories for each probesequence, where statistics for each experiment are collectedover 1000 shots. The complete noise injection and noise spec-trum estimation experiment is conducted over five calibrationcycles and bootstrapped to mitigate spectral anomalies solelydue to variations in hardware characteristics across calibra-tion cycles, thereby improving spectral estimates. Refer to thesupplement for further information regarding optimized ex-perimental practices and procedures.Through the SchWARMA gate-based injected noise proto-col, we find good agreement between the desired and recon-structed noise spectra. In Figure 3, we show the results forthree distinct classes of noise spectra: (a) a single bandpassspectrum of varying width, (b) a multi-bandpass spectrumwith varying center frequencies, and (c) /f α noise. We con- sider single bandpass bandwidths of ∆ ω = 0 . , . , . MHz, with a center frequency of ω = 1 . MHz. We ex-plore the versatility of SchWARMA by separating the injectednoise into two bandpass regions centered at ω = 1 . MHzand ω = 1 . MHz, and subsequently, ω = 0 . MHz and ω = 2 . MHz, while keeping the bandwidth of 0.18 MHzconstant for each region. Lastly, we consider injected /f α -type noise with α = − , − , , , . Such frequency depen-dence is commonly found in superconducting [31–35] andsemiconductor [3, 36, 37] qubit systems.The resulting spectral density estimates yield spectral fea-tures, such as peak height, bandwidth, and center frequency,that are found to be in good agreement with the desirednoise profiles. Additional spectral features observed at highfrequency have been determined to be artificial, resultingfrom increased gate error as higher frequency regimes areprobed with sequences containing a greater number of pulses.The manifestation of gate error in dephasing noise spectrahas been previously noted and mitigated using spin-locking(SL) techniques [38]. IBM’s quantum control toolbox [39]provides a means for exploring SL-based methods, as wellas pulse-error compensating sequences for QNS. We focuson the latter and create a set of Robust FTTPS (RFTTPS)composed of calibrated, true R x ( π ) and R x ( − π ) pulses. Ithas long been known in the NMR and dynamical decouplingcommunties that pulse phase adjustments can aid in thesuppression of pulse imperfection errors (e.g., over/under-rotation errors) for pulse-based error mitigation schemes[40–43]. By constructing RFTTPS composed of pulses ofalternating phases (i.e., R x ( π ) and R x ( − π ) ), we effectivelyreduce the effect of pulse imperfections on the estimatedspectra. RFTTPS are employed on the APL system in thesubsequent section, and an additional comparison betweenFTTPS and RFTTPS using on the IBMQE is included inthe supplement. The use of RFTTPS along with additionalsimulations (see supplement) support the conclusion that theobserved high-frequency noise is primarily an artifact of gateerror. Verification with SDR.
Our SDR-based approach to noiseinjection relies on direct hardware access. This approach isnot platform agnostic and is presented as a means to validateSchWARMA gate-based noise injection by directly compar-ing results against a control line signal with the synthesizednoise spectrum. In order to meet the requirements of lower-level access, we turn to the APL system to perform this com-parison.An SDR is configured to implement a SchWARMA modeland produce φ noise ( t ) . The control signal sums the desiredcontrol phase with the phase-noise spectrum output of theSDR, acting as a faulty control signal with a drifting masterclock. The SDR outputs a continuous 2.4 GHz carrier sig-nal with pseudo-random phase changes that are updated every100 ns. While the phase changes may be random, one ad-vantage of SchWARMA is that temporal correlations are per-mitted, meaning that the pseudo-random phase changes are Frequency (MHz) ∆ S ( ω )( M H z ) (a) ∆ ω = 0.2 MHz∆ ω = 1 MHz Frequency (MHz) ∆ S ( ω )( M H z ) (b) ω = 0.5 MHz ω = 1 MHz∆ ω = 0.5 MHz − Frequency (MHz) ∆ S ( ω )( H z ) (c) α = 2 α = 1 α = 0 α = -1 α = -2 FIG. 4. Verification of SDR-based SchWARMA noise injection.Noise reconstructions are shown for (a) bandpass, (b) double band-pass, and (c) /f α noise for different bandwidths, center frequenciesand spectral decays. The data shown are reconstructed from the av-erage survival probability of 10,000 unique shots per sequence, eachwith a unique phase trajectory. Points represent noise reconstruc-tions, and solid lines are measurements performed on the generatednoise by a signal analyzer. In figures (a) and (c) plots are offset forclarity. biased by previously applied phase updates that mimic the de-phasing error gates in the gate-based approach. In software,these fluctuations are shaped from white-noise into an engi-neered, SchWARMA-generated noise spectrum. We measurethe SDR output on a signal analyzer with a 90 kHz band-width as a ground-truth comparison for spectral reconstruc-tions (Fig. 2(b) inset).After confirmation via signal analyzer, we perform a se-ries of RFTTPS qubit experiments in the APL system withthe SDR as the master clock and the clock phase controlledby various SchWARMA models. The APL system includesa fixed-frequency transmon qubit held at 20 mK at the mix-ing chamber stage of a dilution refrigerator. The qubit ( ≈ . GHz) is coupled to a readout resonator ( ≈ GHz) and thestate of the qubit is determined by dispersive measurementtechniques. See the supplement for further detail about SDRinjection and the APL experimental setup. Experiments usingthe SDR approach are performed over 10,000 unique shots,and the R x ( ± π ) pulses are calibrated to have t G = 100 nsfor both SDR injection and gate-based injection in the APLexperimental setup. The fixed total time of noise injectionexperiments in the APL setup is 12.8 µ s, well within the mea-sured coherence times of the qubit ( T ≈ µ s and T ≈ µ s). Before measurement, we apply a R x ( π/ gate whichwould excite an ideally behaved qubit in the absence of noiseto the | (cid:105) state. Using the same QNS analysis applied in thegate-based approach we reconstruct the injected noise spectra.As is the case with the gate-based approach, the SDR injec-tion approach can be used to produce a variety of phase-noisespectra. In Fig. 4, we show the results for (a) a single band-pass spectrum of varying width and noise power, (b) a multi- Sequence . . . . . Su r v i v a l P r o b a b ili t y Frequency (MHz) ∆ S ( ω )( M H z ) ∆ ω = 0 .
20 MHz∆ ω = 0 .
50 MHz∆ ω = 1 .
00 MHz
FIG. 5. Comparison between gate-based (solid lines) and SDR-based(dashed lines) SchWARMA noise injection for bandpass dephasingnoise with three different bandwidths. Good agreement betweenthe approaches is found for all spectra considered; thus, validatingSchWARMA gate-based noise injection. Data for ∆ ω = 0 . , . MHz are vertically offset by 0.2 for survival probabilities (left) and1 MHz for spectrum reconstructions (right). bandpass spectrum, and (c) /f α noise. We consider singlebandpass bandwidths of ∆ ω = 0 . , MHz with a center fre-quency ω = 1 MHz. The double bandpass noise spectrumis engineered to have center frequencies ω = 0 . MHz and ω = 1 MHz, with noise bandwidths ∆ ω = 0 . MHz. Lastly,we consider /f α -type noise with α = − , − , , , , as wedid with gate-based experiments.Reconstructed spectra are compared against the measuredfrequency-noise spectrum of the SDR output signal (mea-sured on a signal analyzer) for validation. To achieve fits,the measured frequency spectra of the SDR output signal areeach multiplied by a scaling factor, constant over experimentalsetup, to account for attenuation of the signal between roomtemperature and the qubit. Overall, the reconstructed noisespectra match the injected spectra well. The measured band-widths, peak heights, and center frequencies of the injectedbandpass and double bandpass spectra are in good agreementwith the injected noise spectra. The behavior of /f α spectramatch the expected power dependencies best within approx-imately . MHz ≤ ω ≤ MHz (Fig. 4 (c)). At frequen-cies above 2 MHz, the innate roll off in the SDR output dis-torts the injected noise, leading to inaccuracies in the recon-structed noise spectra. Despite these distortions, the overallshape of the reconstructed spectra remains in good agreementwith the expected noise spectra measured at the output of theSDR (solid lines).The combination of the signal analyzer output and thereconstructed spectra provides confidence that the SDR-basedapproach injects phase noise in the desired manner. Wecan transfer this confidence to the gate-based approach bycomparing reconstructions of gate-based noise injection ofidentical spectra on the same device. Fig. 5 shows recon-structions of gate-based SchWARMA injected noise plottedon top of nominally identical noise injected using the SDRexperimental approach. Gate-based injection experimentsperformed on the APL experimental setup are measured over200 unique SchWARMA trajectories, with 1000 shots pertrajectory. We reiterate that both measurements in Fig. 5are performed on the same APL experimental setup. These . . Pred. SDRAct. SDR Pred. GateAct. Gate . . Pred.Act.
Probe Sequence Number Su r v i v a l P r o b a b ili t y FIG. 6. Comparison of SchWARMA predicted to actual experimen-tally measured survival probabilities using FTTPS.
Top:
SDR- andgate-based noise injection of a wide bandpass spectrum on the APLexperimental platform.
Bottom:
Gate-based injection of a doublebandpass spectrum on the IBMQE. results show excellent agreement between the two methods,with the primary discrepancy between the approaches due tofinite sampling effects on the measurements and number ofSchWARMA trajectories used in the gate-based approach.Further comparisons between the gate-based and SDRapproaches are included in the supplement.
SchWARMA Model Prediction.
Much like classicalARMA, a SchWARMA model’s associated power spectrumcan be used to model real physical systems and predict inclosed form their average response to given stimuli (here pulsesequences). This effectively closes the loop between the gen-eration and injection processes. Here, using the known in-jected SchWARMA model, we fit a few ancillary parametersto experimentally generated FTTPS data. Fig. 6 shows the re-sults of this fitting process for sample experiments performedvia SDR and gate-based noise injection (details of this processare found in the following paragraphs). The top panel showsSDR and gate-based data for the 1 MHz bandpass experimentswhose reconstructions are shown in Figs. 4 and 5, and the bot-tom panel displays gate-based data from the double bandpassexperiment with ω =1.07 MHz shown in Fig. 3(a). The accu-racy of these fits show that SchWARMA models can be usedto effectively predict the expected survival probabilities. Thiscapability conveys that SchWARMA can be a powerful toolfor parametric modeling of temporally correlated noise envi-ronments and therefore, improved quantum system dynamicssimulation informed by experimental data.Given a FTTPS probe sequence k (with n k R x ( π ) -pulses)and filter function g k , we predict the survival probability p k using the model p k = 12 + 12 exp[ − g k ( S nat ( α ) + S inj ) − c n k − c n k ] (2) where S nat ( α ) is a parametric model of the native dephasingnoise power spectrum (see below), S inj is the (known) injectedspectrum, and c and c capture the effects of stochastic (i.e., X − dephasing) and coherent errors on the R x ( π ) -pulses, re-spectively. The pulse error terms are meant to capture the ar-tificial high frequency features observed in the spectral esti-mates shown above, most notably in the case of the IBMQEresults where FTTPS without pulse-error compensation areused. We use Eq. (2) to set up a nonlinear least-squares prob-lem to estimate the parameters α and c i using a set of FTTPS.We further expand upon the motivation for the model andpresent additional results in the supplement.The native noise S nat is, in general, qubit and systemdependent. For the APL system (Fig. 6 top), we found thata combination of a Lorentzian spectrum A/ (1 + ω /ω c ) ,with unknown amplitude A and cutoff frequency ω c , and awhite noise floor with power σ captured the observed nativedephasing noise (i.e., α = { A, ω c , σ } ), which exhibitedlow-frequency energy that was not well modeled by thewhite noise floor nor the control noise terms c i . For the IBMsystem (Fig. 6 bottom), we found that there were isolatedresonances at the 0th and 8th sequences, and beyond thatthe noise was well approximated by a white noise floor.Thus, instead of introducing additional model terms in S nat to account for these two sequences (i.e., the native noise inthose frequency bands), we instead ignored these sequencesin the estimation (as the introduction of two more ancillaryterms would perfectly match these points without changingthe estimates for the unknown white noise floor power σ and c i ). Again, we emphasize that Fig. 6 shows strong agreementof the least-squares fits to the measured experimental data,and that we are only fitting over the ancillary backgroundand control noise terms, not on any terms associated with theinjected spectrum. Discussion
Although quantum systems are commonly subject to tem-porally correlated noise, the impact of such noise processeson the performance of quantum algorithms is not wellunderstood. Here, we present a novel method for engineeringand injecting temporally correlated dephasing noise thatcan be used to elucidate key features of noise resilienceand vulnerability in quantum algorithms for noise charac-terization, sensing, and computing. The approach enablesnoise injection of arbitrary noise spectra via SchWARMA,a statistical tool for modeling, estimating, and generatingsemiclassical noise in quantum systems. SchWARMA isused to synthesize engineered phase noise spectra that areintroduced by adjusting the control master clock to emulateclassical temporally correlated dephasing noise processes.We show that the experimental system dynamics generatedby engineered noise can be predicted by SchWARMA-basedmodels; therefore, closing the loop between generation andinjection processes. Our results indicate that SchWARMAis a highly flexible tool for injecting engineered noise anddeveloping experimental data-informed parametric noisemodels for simulating quantum system dynamics. The tech-niques presented here can be readily extended to multi-axisnoise, non-Gaussian and nonstationary noise, and multi-qubitscenarios. This points towards the generalizability andapplicability of SchWARMA as a tool for understanding,evaluating, and improving quantum algorithms subjected to awide range of temporally correlated noise processes relevantto current and future hardware platforms.
Acknowledgements
The authors acknowledge JHU/APL for the infrastructure in-vestments required to perform these experiments. In addition,the authors acknowledge Ben Palmer, Neda Foroozani, andKevin Osborn from the Laboratory for Physical Sciences fortechnical support and guidance. The authors acknowledgeRussell Lake, Xian Wu, and Hsiang-Sheng Ku for chip designand fabrication. G.Q. and L.T. acknowledge funding from theDOE Office of Science, Office of Advanced Scientific Com-puting Research (ASCR) QCATS program, under fieldworkproposal number ERKJ347. G.Q., K.S., and L.T. acknowl-edge support from ARO MURI grant W911NF-18-1-0218.B.D.C. acknowledges support from DOE Office of ScienceARQC Grant DE-SC0020316. J.L and D.P acknowledgesupport from National Science Foundation (Award No.1839136) and the DOE through FNAL. A.M., J.E., G.Q.,K.S., K.M., C.T., B.T.S., J.A.H., B.D.C., and T. S. acknowl-edge support from the United States Department of Defense.The views and conclusions contained in this document arethose of the authors and should not be interpreted as repre-senting the official policies, either expressly or implied, of theUnited States Department of Defense or the U.S. Government.
Author Contributions
K.S., B.D.C., J.A.H., and T.S. conceived of the experimentsperformed at APL. A.M. and J.E. performed the in-house ex-periments. J.L designed the device for APL experiments.B.T.S., K.M., and T.S. provided experimental assistance.G.Q., K.S., and B.D.C. conceived the IBMQE cloud-based ex-periments. G.Q. and L.T. performed the IBMQE experiments.K.S. designed and implemented SchWARMA prediction anal-ysis and numerical simulations. K.S. designed the noise spec-troscopy sequences with feedback from J.E. and G.Q.. A.M.,J.E., G.Q., K.S., and L.T. analyzed the data. G.Q. coordi-nated manuscript writing with substantial contributions fromA.M., J.E., K.S., K.M., C.T., and T.S.. Various aspects of theproject were managed by G.Q., J.A.H., D.P., and T.S. Finally,all authors interpreted data and contributed to editing of themanuscript. [1] Romach, Y. et al.
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