DisQ: A Novel Quantum Output State Classification Method on IBM Quantum Computers using OpenPulse
DDisQ: A Novel Quantum Output State Classification Method onIBM Quantum Computers using OpenPulse
Tirthak Patel
Northeastern University
Devesh Tiwari
Northeastern University
ABSTRACT
Superconducting quantum computing technology has ushered ina new era of computational possibilities. While a considerable re-search effort has been geared toward improving the quantum tech-nology and building the software stack to efficiently execute quan-tum algorithms with reduced error rate, effort toward optimizinghow quantum output states are defined and classified for the pur-pose of reducing the error rate is still limited. To this end, this paperproposes DisQ, a quantum output state classification approachwhich reduces error rates of quantum programs on NISQ devices.
Quantum computing is advancing at a rapid pace with the prolifer-ation of different quantum computing technologies and renewedinterest from industry and academia. In addition to the better es-tablished quantum annealing approach which has limited applica-bility [17, 30, 31], different quantum computing technologies arebeing actively explored to build a reliable quantum bit (or qubit),including superconducting qubits, trapped-ion qubits, and photon-based qubits. The most promising of these is the superconductingqubit technology which is primarily pioneered by IBM and Google.In fact, Google recently used their 53-qubit Sycamore quantumcomputing chip to run within a few seconds a task which wouldtake a few days even on large supercomputers [3].State-of-the-art Noisy Intermediate-Scale Quantum (NISQ) com-puters based on superconducting quantum technology are activelybeing used to establish the usefulness of quantum computers withpotential applications ranging from chemistry and physics simula-tions to hard optimization problems [6, 18, 24, 33]. Unfortunately,due to the primitiveness of the technology, current NISQ computershave high error rates. Further, they do not have enough number ofqubits to deploy meaningful error correction. Because of the higherror rate of NISQ machines, the outputs generated by quantumalgorithms run on current NISQ computers are erroneous.A significant amount of research effort has been geared towardunderstanding, debugging, and mitigating the errors of these com-puters. These works have mainly focused on three primary researchaspects: (1) developing quantum error-and-noise-aware simulationframeworks [2, 10, 26], (2) debugging quantum programs to identifycode-level errors [23, 27], and (3) providing a best-effort solutionto the NP-hard problem of mapping a logical quantum algorithmto a physical set of qubits such that the error rate of the producedoutput is minimized [4, 5, 14, 21, 22, 28, 34, 35, 37–39, 41, 43].However, research efforts to address the problem of quantumoutput state classification are still in a nascent stage. This problemrefers to the task of determining whether a qubit’s output energysignal, after a quantum algorithm’s execution, should be classifiedas the | ⟩ state or the | ⟩ state. This is not a straightforward task asthe output state is affected by a variety of error-inducing factors. x y z |0|1 (a) x y z |0|1 (b) x y z |0|1 (c) Figure 1:
The Block Sphere visualizes a qubit’s state. Here, two 𝑅 𝑦 ( 𝜋 / ) gates are applied to a qubit initialized to state | ⟩ . To the best of our knowledge, we propose DisQ, the first method tooptimize the classifier which differentiates between different quantumoutput states using the quantum pulse schedules on an IBM supercon-ducting quantum computer.
DisQ is built on two key insights: (1)The classification methodology, including choice of quantum gatesand classifier shape, can affect the error rate of quantum output; (2)More surprisingly, the quantum output error rate is dependent onthe probability of the output states. Current state-of-art classifieris not aware of these characteristics and hence, suffers from a rel-atively high error rate. DisQ addresses this challenge by buildinga classifier that is trained using multiple micro-benchmarks withdifferent output state probabilities and optimized using a simulatedannealing approach. We proposes two versions of DisQ: circle- andellipse-shaped classifiers. Our evaluation shows that DisQ improvesthe median error rate, the 75 𝑡ℎ percentile error rate and the variabil-ity in error rate. DisQ is practical and suitable for deployment: thetraining data can be collected within 2 minutes during calibration(calibration is the task of determining the optimal parameters todrive the qubit) and the classifier can be optimized within 7 minuteson an Intel Core i7 processor, and can be used for all algorithm runsuntil the next calibration period (approx. every 12 hours). DisQ’s classification toolbox is compatible with IBM’s Python-based Qiskit OpenPulse framework and can be used with any IBMquantum computer which supports OpenPulse. It is available at: http://github.com/GoodwillComputingLab/DISQ . In this section, we provide some background of quantum computing,superconducting quantum computers, and their sources of errors. A qubit is the building block of a quantum computer. A classical bitcan exist in one of two states: 0 or 1. On the other hand, a qubit’s state ( | Ψ ⟩ ), during computation, can be expressed as a superposition ofthe two basis states: | ⟩ and | ⟩ (in bra-ket notation). More formally, | Ψ ⟩ = 𝛼 | ⟩ + 𝛽 | ⟩ , where 𝛼 and 𝛽 are complex numbers such that ∥ 𝛼 ∥ + ∥ 𝛽 ∥ =
1. When the quantum computation is completed, a r X i v : . [ qu a n t - ph ] F e b , Tirthak Patel and Devesh Tiwari Capacitor(Niobium) QubitInductorJosephson Junction(Aluminum) CapacitorReadout ResonatorCoupling Resonator Readout ResonatorCoupling Resonator
Figure 2:
A superconducting qubit is made out of an LC oscillatorcircuit constructed using a capacitor and a Josephson Junction.the qubit’s state is measured or read out . When the qubit is readout, its superposition is destroyed, and it is measured in state | ⟩ (with probability ∥ 𝛼 ∥ ) or in state | ⟩ (with probability ∥ 𝛽 ∥ ).A quantum algorithm is a set of quantum gates sequentiallyapplied to the qubits on a quantum computer. When a quantumalgorithm completes execution, the state of all or some of the qubitsis measured to analyze the output. A qubit’s state can be readout only once during the algorithm execution as it collapses thequbit’s superposition. Therefore, it is only read out at the end of thealgorithm’s execution. Note that the output of a quantum algorithmis probabilistic. Therefore, multiple trials are conducted to get the output probabilities of a qubit state. For example, if a qubit has state | Ψ ⟩ = √ | ⟩ + √ 𝛽 | ⟩ , then the output probability of state | ⟩ is (cid:13)(cid:13)(cid:13) √ (cid:13)(cid:13)(cid:13) = and state | ⟩ is also (cid:13)(cid:13)(cid:13) √ (cid:13)(cid:13)(cid:13) = . Therefore, if 1024 trialsare conducted, then it would be expected that half of the trials (512)would have state | ⟩ and the other half would have state | ⟩ .Quantum gates can be of 1-qubit or 2-qubit variety. A 1-qubit gateoperates on a single qubit and puts it in the desired superposition.A 2-qubit gate entangles two qubits and modifies the superpositionof the target qubit based on the superposition of the control qubit.A general 1-qubit gate has three components: x-, y-, and z- rota-tion components which rotate the qubit about the x-axis, y-axisand z-axis, respectively, on the Bloch Sphere (denoted as ( 𝑅 𝑥 ( 𝜃 ) , 𝑅 𝑦 ( 𝜙 ) , and 𝑅 𝑧 ( 𝛿 ) , respectively, where 𝜃 , 𝜙 , and 𝛿 are the anglesof rotation). The Bloch Sphere is a unit sphere with the | ⟩ staterepresented as a vector pointing toward the positive z-axis andthe | ⟩ state represented on the negative z-axis. While, the qubitstate vector can be pointed in any direction on the Bloch Sphere,when it is measured, it collapses to the | ⟩ or the | ⟩ state, and itsoutput probability is the projection of the vector onto the z-axis.For example, in Fig. 1, the Bloch Sphere shows the state changesafter applying gates to a qubit initialized to state | 𝜓 ⟩ = | ⟩ . First, a 𝑅 𝑦 ( 𝜋 / ) gate is applied. This puts the qubit in equal superposition(0 . | ⟩ and | ⟩ ). Another 𝑅 𝑦 ( 𝜋 / ) gate brings the qubit to state | ⟩ . Arbitrary rotations can be appliedabout any of the three axes to achieve the desired superposition.All 1-qubit rotations have 2-qubit variants ( 𝐶𝑅 𝑥 ( 𝜃 ) , 𝐶𝑅 𝑦 ( 𝜙 ) and 𝐶𝑅 𝑧 ( 𝛿 ) ), where the target qubit is rotated as per the control qubit. IBM uses a circuit-based approach toward developing qubits us-ing superconducting Josephson Junctions. As shown in Fig. 2, asuperconducting niobium linear capacitor and a superconducting
Drive Pulse DurationPulse Amplitude Readout Pulse DurationQubit Drive ChannelQubit Readout Channel
Figure 3:
Microwave pulses are applied to drive the qubit to adesired state and read out the qubit value.aluminium Josephson Junction, which behaves as a non-linear non-dissipative inductor, are developed on a silicon wafer. To build theJosephson Junction, two pieces of weakly coupled superconductingelectrodes are separated by a very thin tunnel barrier which servesas an insulating layer. Overall, this forms a non-linear LC oscillatorwhich behaves like a qubit if the parameters are tuned correctly.The oscillator allows for a two-level quantum system with discretequantum energy levels. The lowest two energy levels are used asthe | ⟩ (ground level) and the | ⟩ (first excited level) states. The superconducting-qubit system can be addressed using externalcontrols. Typically, the superconducting qubit has a frequency inthe range of 4-5 GHz. As shown in Fig. 3, a Microwave tone canbe applied at the qubit frequency to drive the qubit. By applying apulse shape to the Microwave tone, quantum gates can be achieved.Typically, a Microwave pulse with a Gaussian shape is used to drivethe qubit. Frame-of-reference change is used to apply gates alongdifferent axes. The pulse with the maximum amplitude magnitudeis known as the “ 𝜋 Pulse” and it is used to apply the 𝑅 𝑥 ( 𝜋 ) rotation,which transforms the | ⟩ state to the | ⟩ state and vice versa.The qubit state is measured by coupling the qubit to a super-conducting transmission resonator. It is ensured that the readoutresonator frequency is dependent on the state of the qubit. Thus, thequbit state can be determined by probing the resonant frequency.Fig. 3 shows that a long Square Gaussian pulse is applied to measurethe state of the qubit. Lastly, as shown in Fig. 2, 2-qubit entanglinggates are applied using a superconducting coupling bus resonator. There are several sources of error in NISQ technology. Once initial-ized, a qubit can only hold the coherence of its state for a limitedamount of time. There are two types of coherence decays: (1) TheT1 coherence refers to the amplitude damping. (2) The T2 coher-ence refers to the phase damping. NISQ errors also include the gateand readout errors. An erroneous application of the microwavepulses can cause gate errors, i.e., the gate could be put in a slightly-off-the-desired superposition which can lead to incorrect outputprobabilities. The readout resonators are also error-prone and causereadout errors. Refer to Patel et al. [32] for more details about thedifferent sources of error. These errors can be a result of deviationsfrom the optimal pulse parameters such as the frequency at whichthe pulse is applied, its amplitude, and its duration. Moreover, be-cause the technology is still maturing, the qubit’s properties, suchas its frequency, vary. Therefore, the optimal pulse parameters needto be determined on a regular basis. isQ: A Novel Quantum Output State Classification Method on IBM Quantum Computers using OpenPulse , ,
Real Component − I m ag i n a r y C o m p . (a) Hadamard Output
Real Component − I m ag i n a r y C o m p . | ⟩ | ⟩ (b) Good Discriminant
Real Component − I m ag i n a r y C o m p . | ⟩ | ⟩ (c) Poor Discriminant
Figure 4:
The role of a classifier is to identify the outputs of trialsas | ⟩ or | ⟩ in a manner which minimizes the error. Real Component − I m ag i n a r y C o m p . State | ⟩ (a) Step 1: State | ⟩ Run
Real Component − I m ag i n a r y C o m p . State | ⟩ State | ⟩ (b) Step 2: State | ⟩ Run
Real Component − I m ag i n a r y C o m p . State | ⟩ State | ⟩ (c) Step 3: Discriminant
Figure 5:
In the baseline case, a linear discriminant is used todifferentiate the | ⟩ vs. | ⟩ using a three-step process.For this reason, IBM’s computers are calibrated twice a day, andthe qubit’s error rates change after each calibration. Calibration isthe task of determining qubit properties, such as frequency, andaccordingly, setting the parameters of the microwave pulses for thegate operations. These parameters are then used to perform all theoperations until the next calibration. Also, during the calibrationperiod, the classifier which distinguishes between the | ⟩ and the | ⟩ state is developed (as discussed next in Sec. 3). In this section, we introduce the state classification problem, theexisting state-of-art classification method, and its inefficiencies.
Once a quantum algorithm has completed execution, its final stateis measured. To do this, the readout pulse is run on the readoutchannel (as shown in Fig. 3), and the corresponding signal, whichmeasures the energy state of the qubit, is recorded on the acquirechannel. The signal recorded during the entire acquire durationis then summed up and a single value is returned for each trialfor each qubit. This value is a complex number with a real and animaginary component. It is used to determine if the measured staterepresents the | ⟩ state or the | ⟩ state.As an example, consider that a quantum gate known as theHadamard gate is executed on a qubit initialized to state | ⟩ andits resulting final state is read out. The resulting output valuesof conducting 1024 trials are shown in Fig. 4(a). The Hadamardgate puts a qubit in equal superposition of the | ⟩ and the | ⟩ state(0.5 probability of both states). Therefore, ideally, 512 of these trialoutputs should be classified as state | ⟩ and the other 512 as state | ⟩ .Given the nature of NISQ computers, the ideal classification is notpossible. However, best effort must be made to reduce the outputerror . The output error (in percentage) is defined as the | correctprobability of state | ⟩ − observed probability of state | ⟩ | ×
100 =| correct probability of state | ⟩ − observed probability of state | ⟩ | × | ⟩ is calculated by dividing the numberof trials which resulted in | ⟩ by the total number of trials conducted.Also, by definition, probability of | ⟩ = − probability of | ⟩ .A trivial choice for state classification is a linear classifier. Con-sider the linear classifier used in Fig. 4(b) which classifies 548trial outputs as state | ⟩ . Then, the observed output probabilityof state | ⟩ is 548 / = . | . − . | × = . | ⟩ , resulting in anerror of 18.5%. Thus, developing a classifier which minimizes theoutput error is critical. Note that it is not possible to use differentclassifiers which cater to the output probabilities of different quan-tum algorithms as the output probabilities are not known for realquantum algorithms. Therefore, only one classifier is developed andused for all algorithms. Typically, the classifier is developed duringcalibration, once the qubit’s frequency and pulse parameters, suchas amplitude, are determined. Next, we look at the state-of-art method used for classification [2].
Step I.
First, the qubit is initialized to the | ⟩ state and its value isdirectly readout without running any gate operations. This is donefor multiple trials. The probability of observing state | ⟩ would be 1in this case. Therefore, ideally, the output value of all trials shouldbe classified as belonging to state | ⟩ . Fig. 5(a) shows the outputspattering of 1024 trials of this run on the complex plane. Step II.
Next, the output values for the | ⟩ state need to begenerated. To achieve this, the qubit is initialized to the | ⟩ state.Then, a 𝜋 pulse (i.e., a rotation of 𝜋 about the x-axis – 𝑅 𝑥 ( 𝜋 ) ) isapplied to it to put it in the | ⟩ state (pointing toward the negativez-axis in the Bloch Sphere). It is then measured, as shown in Fig. 3.Ideally, the output value of all trials of this run should be classifiedas belonging to the | ⟩ state. Fig. 5(b) shows the output spatteringof 1024 trials of this run which produces the | ⟩ state. Step III.
The last step is to construct a discriminator. Fig. 5(b)shows that there could be some overlap between the values gen-erated by the | ⟩ state and the values generated by the | ⟩ state.This is due to the various gate and coherence errors mentioned inSec. 2 which contribute to erroneous output. Therefore, a perfectdistinction between the | ⟩ and the | ⟩ state, which would resultin an output error of 0%, is not possible to achieve. However, besteffort must be made to reduce the output error. Currently, a lineardiscriminator is used to minimize this output error by generating adiscriminator which maximizes the distance between the means ofthe classified samples, as shown in Fig. 5(c). Points that fall abovethe line are classified as | ⟩ and ones that fall below the line areclassified as | ⟩ . For all the runs conducted until the next calibration,this linear discriminator is used to classify the qubit state. The existing state classification method or the baseline methoduses only two runs – one with output state | ⟩ with probability 1and the other with output state | ⟩ with probability 1 – to developthe linear discriminant. As such, applying this discriminant toruns with other output probabilities can lead to several undesiredconsequences. To demonstrate this, we ran over one thousand , Tirthak Patel and Devesh Tiwari Output Probability of State | ⟩ E rr o r Figure 6:
Different error rates are observed for runs with differentoutput probabilities of state | ⟩ . The bars show the median in eachbin; the range indicators show the spread from the 25 𝑡ℎ percentileerror to the 75 𝑡ℎ percentile error. Run the Simulated Annealing Engine and Generate the D IS Q Classifier
Generate Training Micro-Benchmarks to Execute on a Quantum Device Execute the Corresponding Pulse Schedules and Collect Data
Use the D IS Q Classifier until the Next Calibration
Cycle
Figure 7:
Overview of the steps when classifying states using DisQ.micro-benchmark runs for a period of 10 days on IBM’s 1-qubitArmonk quantum computer. For each run, a 𝑈 ( 𝑅 𝑥 ( 𝜃 ) , 𝑅 𝑦 ( 𝜙 ) , 𝑅 𝑧 ( 𝛿 )) gate with three random rotation angles about each of thethree axes, selected uniformly between − 𝜋 and 𝜋 , is applied to aqubit initialized to state | ⟩ . This results in an arbitrary outputprobability of state | ⟩ between 0 and 1. Fig. 6 shows the errorrates of runs when they are binned in increments of 0.1 of theircorrect output probabilities. We ensured that all bins have thesame number of runs. Fig. 6 shows several interesting results. (1) The output error is highly correlated with the output probabilityof state | ⟩ . For example, the median error when the outputprobability of state | ⟩ is 0.9-1 is 3.5% but the median error whenthe output probability of state | ⟩ is 0-0.1 is 12.5%, which is 3.5 × worse compared to the former’s error rate. Some of this differenceis due to the fact that when the output probability of state | ⟩ is 0-0.1, it means that state | ⟩ is expected for most of the trials.But, due to the coherence properties of qubits, state | ⟩ is morelikely to lose its coherence and drop from the first excited state tothe ground state. However, a degradation in error rate of 3.5 × isunacceptable. Quantum algorithms with lower output probabilityof state | ⟩ should not observe a much higher error rate simplydue to the makeup of their output probabilities. An algorithm’soutput probability of states cannot be controlled or modified asit is an intrinsic property of the algorithm. Therefore, the outputstate classifier should be developed in accordance with theseunintentional outcomes, such that similar error rates are observedregardless of the output probability of state | ⟩ . (2) The error rate is high even for a single U3 gate used for themicro-benchmarks; a typical quantum algorithm consists of manygate operations and these errors get compounded. The medianerror across all the runs is about 5%, while the 75 𝑡ℎ percentileerror is 9%. While the error rate is high in isolation, note thatthe error rate compounds when more gates are applied aseach gate operation introduces its own error rate while also Run 1024 TrialsMeasureApply to Qubit Generate Unitary Matrix Correct Output Probability D IS Q Classifier
Observed Output Probability Calculate ErrorGenerate U3
Figure 8:
Processing and calculating error of a micro-benchmark.increasing the probability of state losing coherence as more timeis spent before measuring the qubit state. Therefore, effort isneeded to decrease the median error rate and especially, the 75 𝑡ℎ percentile error rate as 25% of the runs have more than 9% error rate. (3) The spread or variability in error rate is high even among runswith similar output probabilities of state | ⟩ . For example, in Fig. 6,consider the 0.2-0.3 bin. It has a median error of 8.75%. But it hasa significant error spread, i.e., the 75 𝑡ℎ percentile error - the 25 𝑡ℎ percentile error, of 6.5%. Similarly, other bins also have a high spread.The fact that the error rate varies so much even among runs withsimilar output probability of state | ⟩ indicates that the error ratesare not stable and the results are non-reproducible. Therefore, itneeds to be ensured the spread of the error rates is minimized. In this section, we describe DisQ, an approach to mitigate the afore-mentioned issues in Sec. 3. Fig. 7 provides a high-level overviewof the procedure that DisQ uses to develop a discriminating stateclassifier. When a qubit is being calibrated, DisQ should be executedto obtain the state classifier. First, DisQ generates different micro-benchmarks with a diverse range of output probabilities of state | ⟩ to cover the entire spectrum of possible output state probabilities.Next, it executes these micro-benchmarks on the qubit which needsto be calibrated and obtains the raw complex data. Post this, DisQruns a black-box simulated annealing engine which optimizes theparameters of the discriminating classifier in a manner which mini-mizes the median error and the spread of the error of the trainingdata. This classifier can then be used for all quantum algorithmsexecuted on that qubit until the next calibration is performed. The first order of business is to construct the micro-benchmarkswhose output is fed to the state-classifier for discriminating the | ⟩ and the | ⟩ states. The purpose of these micro-benchmarks is tocover the full spectrum of output probabilities of state | ⟩ from 0 to1. As discussed earlier, the reason for this purpose is to ensure thatthe developed classifier minimizes and achieves equal error rate forquantum algorithms with all types of output probabilities.We achieve this on IBM’s Qiskit framework by applying the 𝑈 ( 𝑅 𝑥 ( 𝜃 ) , 𝑅 𝑦 ( 𝜙 ) , 𝑅 𝑧 ( 𝛿 )) gate with three random rotation anglesabout each of the three axes, selected uniformly between − 𝜋 and 𝜋 , to a qubit initialized to state | ⟩ . This results in an arbitrarycorrect output probability of state | ⟩ between 0 and 1 which canbe calculated from the corresponding unitary matrix of the 𝑈 | ⟩ bin in incrementsof 0.1 (i.e., 0 − . , . − . , . . . , . −
1) contains equal number of isQ: A Novel Quantum Output State Classification Method on IBM Quantum Computers using OpenPulse , ,
Real Component − − I m ag i n a r y C o m p o n e n t | ⟩ | ⟩ (a) Linear Classifier
HighOverlapZones (b)
High-Overlap Zones
Figure 9:
A linear classifier does not take into account high-overlapzones which are error-prone because they have a similar density ofpoints belonging to the | ⟩ and | ⟩ states. Real Component − − I m ag i n a r y C o m p o n e n t | ⟩ | ⟩ (a) C-DisQ: Circle Classifier
Real Component − − I m ag i n a r y C o m p o n e n t | ⟩ | ⟩ (b) E-DisQ: Ellipse Classifier
Figure 10:
Visual representation of C-DisQ (circle classifier) andE-DisQ (ellipse classifier), which focus on low-overlap zones.
Algorithm 1
DisQ’s simulated annealing engine. Input:
Training data 𝑇 data , number of iteration 𝑁 iter , initial temperature 𝑇 , and cooling coefficient 𝛼 Best configuration 𝐶 𝑏𝑒𝑠𝑡 (sampled randomly) Best objective 𝑃 best ⇐ Objective ( 𝑇 data ,𝐶 best ) for 𝑖 = , . . . , 𝑁 iters do Random neighbor configuration 𝐶 new . Corresponding 𝑃 new ⇐ Objective ( 𝑇 data ,𝐶 new ) . Energy 𝐸 ← 𝑃 best − 𝑃 new . if 𝐸 > or 𝐸 < and random () < 𝑒 𝐸 / 𝑇 then 𝐶 best ⇐ 𝐶 new 𝑃 best ⇐ 𝑃 new end if 𝑇 ⇐ 𝛼 × 𝑇 end for Output
Best discriminator configuration 𝐶 best micro-benchmarks. These micro-benchmarks are then executed ona quantum computer with 1024 trials each, their observed outputprobability is calculated using DisQ classifier, and their output erroris calculated, as shown in Fig. 8. The next step for DisQ is to classify. The primary consideration when developing a discriminating classi-fier is to determine the shape of the classifier. The existing approachis to use a linear discriminant as shown in Fig. 9(a), which showsan example density plot of the output of multiple training micro-benchmarks. The points above the line are classified as state | ⟩ , andthe ones below are classified as state | ⟩ . Thus, a linear discriminantclassifies points generated by all the trials that are conducted. However, during the design of DisQ, our experimental results re-vealed that it might not be suitable to use all the trial outputs . AsFig. 9(b) shows, there are certain zones on the complex plane whereboth, points belonging to state | ⟩ and points belonging to state | ⟩ , exist with an equal density. For example, 45% of the pointsfalling in these regions could represent state | ⟩ and the remaining65% could represent state | ⟩ . Thus, these zones are not ideal forclassification purposes as points in these regions are highly likelyto be | ⟩ or | ⟩ . Informed by these observations, we designed DisQto ignore such high-overlap zones and instead, focus on low-overlapzones.
For example, such low-overlap zones could be where pointdensity of state | ⟩ is 99% and that of state | ⟩ is 1%.A linear classifier cannot achieve this. In fact, any line-based clas-sifier (quadratic, cubic, etc.) accounts for all trials by design. To solvethis problem, we propose two classifiers which can identify low-overlapzones and not consider high-overlap zones: a circle discriminant classi-fier (C-DisQ) and an ellipse discriminant classifier (E-DisQ ). Examplesof both of these classifiers are given in Fig. 10. As shown in thefigure, both of these classifiers can focus on low-overlap zones andavoid high-overlap zones. For example, consider an algorithm thatis run with 1024 trials, and 400 fall in the | ⟩ region, 500 fall in the | ⟩ region, and 124 fall outside both regions. Then, those 124 trialscan be ignored, and instead the observed output probability of state | ⟩ can be calculated as 400 /( + ) = .
44. This is in contrastto a linear classifier where all trails have to be considered, evenones which fall in high-overlap error-prone zones.However, it is non-trivial to determine the parameters which de-fine the C-DisQ and E-DisQ classifiers. C-DisQ has six parameters,three for each circle (x-coordinate, y-coordinate, and radius). E-DisQ has ten parameters, five for each of the ellipses (x-coordinate,y-coordinate, width, height, and angle of rotation). While, C-DisQhas fewer parameters to optimize, E-DisQ has better precision overthe classification region which can potentially lead to a lower er-ror rate (both are evaluated in Sec. 5). Nonetheless, the problemof determining their parameters, while minimizing the error rateand error spread calculated based on the micro-benchmark mea-sured data, is NP-hard. Therefore, next, we discuss the black-boxsimulated annealing engine used to optimize the two classifiers.
The goal of DisQ’s simulated annealing engine is to obtain the parameter configuration for C-DisQ and E-DisQ such that the givenobjective is nearly-minimized (for the micro-benchmark measureddata). A parameter configuration is one permutation of six parame-ters for C-DisQ and ten parameters for E-DisQ, which define thecharacteristics of the circles or ellipses, respectively.The first step is to design the optimization objective. One optionis to minimize the median error of the training micro-benchmarks.However, just minimizing the median might create a scenario asshown in Fig. 6, where runs with a high output probability of state | ⟩ have a much lower error rate than runs with a low outputprobability. Therefore, to avoid this scenario the spread of the erroramong the runs (i.e., spread is defined as the 75 𝑡ℎ percentile error − the 25 𝑡ℎ percentile error) is also minimized. However, if just thespread is chosen as the objective, then the median error might notbe optimized (for example, spread can be minimized by choosing , Tirthak Patel and Devesh Tiwari parameters such that all runs have an equally high error). Therefore,to create a balance between the median and the spread, the objectivethat DisQ if to minimize the median error + the spread of the error (we evaluate all three options in Sec. 5).Next, Algorithm 1 shows how DisQ’s simulated annealing en-gine iteratively steers toward a parameter configuration whichminimizes the objective. Initially, the annealing environment hasa high temperature, which means that it has a high likelihood ofexploring different configuration neighborhoods. As the algorithmprogresses, the temperature reduces and the algorithm narrows inon a near-optimal configuration neighborhood. At each iteration,a random configuration in the neighborhood of the current bestconfiguration is sampled. A neighborhood is defined as all pointswithin one unit for all of the parameters (six for C-DisQ and tenfor E-DisQ). The objective of this sampled configuration is calcu-lated for the training dataset. Depending on the current energy andtemperature (as defined in Algorithm 1), it is determined if the bestconfiguration should be updated to the newly sampled configura-tion. The best parameter configuration is obtained at the end of theexecution, which can be used to classify the output of all succeedingquantum algorithm executions until the next calibration.
Scaling to multiple qubits.
All qubits on a quantum computerneed to be calibrated and separate classifiers need to be developedfor all of them. The classification time overhead of DisQ does notincrease with the number of qubits as every step of DisQ can runin parallel for all the qubits. This includes generating and executingthe micro-benchmarks and independently running the simulatedannealing engine concurrently.
We use IBM’s Armonk quantum computing machine (specificationsare provided in Table 1) to conduct our experiments. While IBMhas other machines available, Armonk is the only one which allowsobtaining the pre-classification raw complex values (OpenPulse).For demonstrating robustness, DisQ is evaluated and comparedagainst IBM’s existing state-of-art classifier over multiple days.The classifier’s performance is evaluated using a validation dataset.The training and validation dataset are collected using the samemethodology: 100 randomized U3-based micro-benchmarks, eachwith a different output probability of state | ⟩ , 10 belonging to eachprobability bin in increments of 0.1. In particular, DisQ is evaluatedacross ten days, with a classifier developed for each day (eachcalibration). The two variants of DisQ, C-DisQ (circle classifier) andE-DisQ (ellipse classifier), are compared against the default Qiskitclassifier, referred to as the baseline classifier. The metrics usedinclude the median error (of the 100 validation micro-benchmarksacross the 10 days), the 75 𝑡ℎ percentile error, and the error spread(the 75 𝑡ℎ percentile error − the 25 𝑡ℎ percentile error). DisQ achieves a lower median error, 75 𝑡ℎ percentile error,and error spread than the baseline classifier. Fig. 11 shows thatboth C-DisQ and E-DisQ reduce the median error of the single-gateruns from 5% to 4%. Such a reduction can significantly improve theperformance of quantum algorithms and programs with multiple
Table 1:
Specifications of Armonk Quantum Computer.
Online Date 10-16-2019Number of Qubits 1Drive Frequency ≈ 𝜋 Pulse Shape Gaussian 𝜋 Pulse Amplitude ≈ 𝑗𝜋 Pulse Duration ≈ 𝜇 sReadout Frequency ≈ ≈ 𝑗 Readout Pulse Duration ≈ 𝜇 s Baseline Classifier
C-DisQ E-DisQ M e d i a n E rr o r t h P e r ce n t il e E rr o r E rr o r Sp r e a d Figure 11:
Both variants of DisQ achieve a lower median error,75 𝑡ℎ percentile error, and error spread than the baseline classifier. Output Probability of State | ⟩ M e d i a n E rr o r Baseline Classifier
C-DisQ E-DisQ
Figure 12:
C-DisQ and E-DisQ achieve a more equal error rateacross runs with output probabilities of state | ⟩ from 0 to 1.gates. For instance, if five 5%-error gates are applied, the overallerror rate would be 1 − ( − . ) = − ( − . ) = 𝑡ℎ percentile error is reducedfrom 9.5% to 6.5%. With the baseline classifier, 25% of runs have anerror rate of more than 9.5%, while with DisQ, 75% of runs have anerror rate of less than 6.5% and 50% of runs have an error of lessthan 4%. Note that E-DisQ performs slightly better than C-DisQfor the median error as well as the 75 𝑡ℎ percentile error because itenables a higher level of precision over the classification region.Next, the error spread drops from 7% to ≈ DisQ achieves a better equalization of the error rate com-pared to the baseline classifier.
Fig. 12 shows the median errorsfor runs with different output probability of state | ⟩ for the base-line classifier, C-DisQ, and E-DisQ. Both C-DisQ and E-DisQ havesimilar performance, achieve a more equal distribution of error forall output probabilities, especially reducing the error rate of runswith output probability of state | ⟩ less than 0.5. For instance, themedian error rate of the 0-0.1 bin is reduced from more than 12%to less than 6%. This is a reduction of more than 50% in the errorrate which has a significant impact on the stability of output of the isQ: A Novel Quantum Output State Classification Method on IBM Quantum Computers using OpenPulse , , Objective = Median Spread Median + Spread M e d i a n E rr o r t h P e r ce n t il e E rr o r E rr o r Sp r e a d (a) C-DisQ: Circle Classifier
Objective = Median Spread Median + Spread M e d i a n E rr o r t h P e r ce n t il e E rr o r E rr o r Sp r e a d (b) E-DisQ: Ellipse Classifier
Figure 13:
Optimizing just the median achieves similar median er-ror and error spread as optimizing the median + the spread. Settingthe objective to just the error spread performs worse.
Baseline Classifier
C-DisQ E-DisQ . . . . . . D a il y V a r i a b ili t y i n M e d i a n E rr o r D a il y V a r i a b ili t y i n t h P e r ce n t il e E rr o r . . . . . . D a il y V a r i a b ili t y i n E rr o r Sp r e a d (a) Objective = Median
Baseline Classifier
C-DisQ E-DisQ . . . . . . D a il y V a r i a b ili t y i n M e d i a n E rr o r . . . . . . D a il y V a r i a b ili t y i n t h P e r ce n t il e E rr o r . . . . D a il y V a r i a b ili t y i n E rr o r Sp r e a d (b) Objective = Median + Spread
Figure 14:
The day-to-day variability is lower with DisQ when theobjective is to minimize the median + the spread. When just themedian is minimized, day-to-day variability is much higher.runs with output probability of state | ⟩ of less than 0.1. On theother hand, as a result of the distribution equalization, the errorrates of runs with output probability of state | ⟩ of more than 0.6have increased. However, this increase is comparably smaller thanthe decrease in error rate for other runs. Overall, DisQ achievesthe goal of reducing the median error, while also equalizing theerror rates across different outputs by also optimizing the errorspread. Next, we look at the impact of optimizing different objectivefunctions, including ones where the spread is not optimized. Optimizing the median achieves similar low median errorand error spread as optimizing the median + the spread.However, when the objective function is set to the median,
Fixed Classifier AlwaysNew Classifier Daily M e d i a n E rr o r t h P e r ce n t il e E rr o r E rr o r Sp r e a d (a) C-DisQ: Circle Classifier
Fixed Classifier AlwaysNew Classifier Daily M e d i a n E rr o r t h P e r ce n t il e E rr o r E rr o r Sp r e a d (b) E-DisQ: Ellipse Classifier
Figure 15:
Developing a new classifier every calibration achieveslower error rate than using the same classifier across all calibrations. day-to-day variability is much worse.
Fig. 13(a) and (b) showthe median error, the 75 𝑡ℎ percentile error, and the error spread forchoosing different objectives for the simulated annealing engineto optimize C-DisQ and E-DisQ, respectively. Evidently, in bothcases, optimizing just the spread gives worse performance thanoptimizing the other two metrics. In fact, even the error spread ishigher by over 1% for C-DisQ and 0.5% for E-DisQ when the errorspread is optimized. The reason for this is that because optimizingthe error spread does not focus on optimizing the median error,the configuration neighborhoods where the median is optimizedare not explored. But those neighborhoods have the potential toreduce the spread just by reducing the median, as we observe in thecase when the median is optimized. Optimizing the median givessimilar results as optimizing the median + the spread, with thelatter performing slightly worse for C-DisQ and better for E-DisQ.However, this does not indicate that it is advisable to optimizejust the median. Fig. 14(a) and (b) show the daily variability inerror metrics when just the median is optimized and when themedian + the spread is optimized, respectively. The daily variabilityis calculated as the spread of a given metric across the 10 days.For example, if the median error is considered, the median error iscalculated for the validation dataset for each of the 10 days, andthen, the spread of those 10 samples is indicated in the figure. Thebaseline case has a daily variability of 2% error even in the case ofthe median error, which is a high variability. Interestingly, usingthe E-DisQ classifier with the median + the spread as the objectivereduces that variability to less than 1%, which is a 50% reduction inerror variability. This makes the error rates more stable from oneday to another, just improving the reproducibility of the results.However, using the E-DisQ classifier with just the median as theobjective has similar daily variability of median error as the baselinecase. In fact, it has higher daily variability for the 75 𝑡ℎ percentileerror and the error spread. This shows that while optimizing justthe median error has good overall results, its daily variability isvery high, making the results less reliable. On the other hand, DisQwith the median + the spread as the objective, can reduce the dailyvariability and improve the stability considerably. Developing a new classifier every calibration achieves lowererror rate and spread than using a fixed classifier across allcalibrations, for both C-DisQ and E-DisQ.
One might questionthe significance of developing a new classifier every calibration,after all, the output of the same qubit is being classified. However,Fig. 15 shows the importance of developing a new classifier after , Tirthak Patel and Devesh Tiwari
Real Component − − I m ag i n a r y C o m p o n e n t | ⟩ | ⟩ Real Component − − I m ag i n a r y C o m p o n e n t | ⟩ | ⟩ Figure 16:
Difference in classifiers when a fixed classifier is usedalways (broken blue lines) vs. when a new classifier is developedevery calibration (red solid lines).
C-DisQ E-DisQ M e d i a n R un t i m e ( m i n s ) t h P e r ce n t il e R un t i m e ( m i n s ) . . . R un t i m e Sp r e a d ( m i n s ) Figure 17:
Both C-DisQ and E-DisQ classifiers have similar lowruntimes which are very stable across different executions.each calibration. For both C-DisQ and E-DisQ, using a fixed classi-fier (the one which performs the best among all the ones generatedover the 10 days) for all 10 calibrations performs worse than gen-erating a new classifier after every calibration. For example, forC-DisQ using a fixed classifier has an error spread of 6%, whileusing a new classifier daily has an error spread of 4.3%.The reason for this is that due to the instability of the qubits, themeasurement points generated daily might have different densities,requiring different classifiers as shown in Fig. 16. Here, the brokenblue line shows the fixed best classifier across the 10 days, whilethe red lines indicate the classifiers which were the best for thetwo days shown. Evidently, the regions considered by the fixedclassifier are quite different than the regions considered by theday-specific classifiers, even though they have some intersectingportions. However, the disjoint portions are the ones which actuallycontribute to the higher error rate of the fixed classifier.
Lastly, both C-DisQ and E-DisQ classifiers have similar lowruntimes for executing the simulated annealing engine.
Theoverhead of running the micro-benchmarks on the quantum com-puter is very small as it can execute the entire batch of 100 runs,each with 1024 trials (100k trials in total), within 2 minutes, whichis comparable to the baseline case as most of the time is consumedin setting up and initializing the qubits. Once, the training datasetis generated, then the simulated annealing engine needs to be exe-cuted. Fig. 17 shows that the execution time of DisQ’s simulatedannealing engine is less than 7 minutes on our local 4.20 GHz In-tel Core i7-7700K machine for both C-DisQ and E-DisQ and thespread of these runtimes is ≈ Previous works have proposed qubit Quantum Error Correction(QEC) codes, which have an overhead of more than 10 physicalqubits per logical qubit [7, 20, 36, 40]. These methods are thusuntenable for current NISQ devices, which require a low-overhead(in terms of the number of ancillary qubits required) solution toreduce the error so that they can execute quantum algorithms.On the other hand, a large amount of focus has been dedicatedtoward optimizing the execution of quantum algorithms. This in-cludes developing frameworks and compilers to optimize the map-ping of a quantum algorithm to a quantum computer such as IBM’sQiskit compiler and Google’s Cirq framework [1, 2, 11, 16, 19]. Inconjunction with these industry-led efforts, academic research hasalso focused on reducing the error rates of quantum algorithmsby proposing debugging methods, simulation strategies, and noise-and-error-aware algorithm mapping approaches [4, 5, 8, 12, 14, 18,21, 22, 25, 27–29, 32, 34, 35, 35, 38, 39, 41–43]. For example, Muraliet al. [28] propose a method to reduce error rate by performingcross-talk-aware algorithm mapping to the qubits.While these works focus on the higher-level problems of analgorithm’s execution stack, IBM’s OpenPulse, which is the frame-work to apply the pulses for quantum gates, has also been previ-ously leveraged to solve problems including optimizing compila-tion [2, 9, 13, 15]. However, none of the above works have optimizedthe state classification problem by using raw output data of applyingpulses, and thus, these works can be used in a compatible mannerwith DisQ to minimize the error rates of NISQ devices.
This paper presented DisQ, the first work to optimize state classi-fiers which differentiate between different quantum states. DisQdemonstrates that the output error and its variability on NISQdevices can be reduced just by optimizing the classification method-ology, without any hardware or compiler modifications. DisQ isavailable at: http://github.com/GoodwillComputingLab/DISQ . Acknowledgment.
We are thankful to anonymous reviewers for the con-structive feedback and Northeastern University for supporting this work.
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