Experimental implementation of non-Clifford interleaved randomized benchmarking with a controlled-S gate
Shelly Garion, Naoki Kanazawa, Haggai Landa, David C. McKay, Sarah Sheldon, Andrew W. Cross, Christopher J. Wood
EExperimental implementation of non-Clifford interleaved randomized benchmarkingwith a controlled-S gate
Shelly Garion, ∗ Naoki Kanazawa, † Haggai Landa, David C.McKay, Sarah Sheldon, Andrew W. Cross, and Christopher J. Wood IBM Quantum, IBM Research Haifa, Haifa University Campus, Mount Carmel, Haifa 31905, Israel IBM Quantum, IBM Research Tokyo, 19-21 Nihonbashi Hakozaki-cho, Chuo-ku, Tokyo, 103-8510, Japan IBM Quantum, T.J. Watson Research Center, Yorktown Heights, NY 10598, USA IBM Quantum, Almaden Research Center, San Jose, CA 9512, USA
Hardware efficient transpilation of quantum circuits to a quantum devices native gateset is es-sential for the execution of quantum algorithms on noisy quantum computers. Typical quantumdevices utilize a gateset with a single two-qubit Clifford entangling gate per pair of coupled qubits,however, in some applications access to a non-Clifford two-qubit gate can result in more optimalcircuit decompositions and also allows more flexibility in optimizing over noise. We demonstratecalibration of a low error non-Clifford Controlled- π phase (CS) gate on a cloud based IBM Quantumcomputing using the Qiskit Pulse framework. To measure the gate error of the calibrated CS gatewe perform non-Clifford CNOT-Dihedral interleaved randomized benchmarking. We are able toobtain a gate error of 5 . × − at a gate length 263 ns, which is close to the coherence limit ofthe associated qubits, and lower error than the backends standard calibrated CNOT gate. I. INTRODUCTION
Quantum computation holds great promise for speed-ing up certain classes of problems, however near-termapplications are heavily restricted by the errors that oc-cur on present day noisy quantum devices [1]. To run acomputation on a quantum processor requires first cali-brating a universal gate set – a small set of gates whichcan be used to implement an arbitrary quantum circuit –which has low error rates, and then transpiling the circuitto this set of gates. This transpilation should be done ina hardware-efficient manner to reduce the overall error byminimizing the use of the highest error gates [2]. Two ofthe most significant error sources on current devices areincoherent errors due to interactions with the environ-ment, quantified by the coherence times of device qubits,and calibration errors in the gates used to implement aquantum computation [3, 4].If a gate set could be perfectly calibrated the coher-ence time of the qubits would set the fundamental limiton error rates without active error correction. Thus thegoal of gate calibration is to get as close to the coherencelimit as possible. Current quantum hardware typicallyuse a gate set consisting of arbitrary single-qubit rota-tions and a single entangling two-qubit gate [5]. Stateof the art single-qubit gate error rates in these systemsapproach 2 × − [6], where two-qubit gate errors arearound 10 − [7–9], see also Appendix A. In supercon-ducting qubit systems using fixed-frequency transmonqubits a microwave-only two-qubit entangling gate maybe implemented using the cross-resonance (CR) interac-tion [10]. The CR interaction can be used to implementa high fidelity Controlled-NOT (CNOT) gate [11]. Gate ∗ Corresponding author: [email protected] † Corresponding author: [email protected] sets with a Clifford two-qubit like CNOT are appealingas a variety of averaged errors in a Clifford gateset canbe can be robustly measured using various randomizedbenchmarking (RB) protocols [4, 12–16].In some cases it may be favourable to introduce an ad-ditional two-qubit gate to a gate set if it enables morehardware efficient compilation of relevant circuits, how-ever this adds the overhead of additional calibration andcharacterization of the gate errors. One such gate is theControlled-Phase (CS) gate, which is a non-Clifford two-qubit entangling gate that is universal when combinedwith the Clifford group [17]. The CS gate is particularlyattractive to fixed-frequency transmon qubit systems asit can be implemented using the CR interaction, since itis locally equivalent to √ CNOT. This means it can becalibrated using the same techniques as the CNOT gate,but with a shorter gate duration or lower power, poten-tially leading to a higher fidelity two-qubit gate when cal-ibrated close to the coherence limit. Furthermore the CSgate is a member the CNOT-Dihedral group and can bebenchmarked using CNOT-Dihedral randomized bench-marking [17]. Recently an optimal decomposition algo-rithm for two-qubit circuits into the Clifford + CS gateswas developed [18]. This method minimizes the num-ber of non-Clifford (CS) gates, which is important in thecontext of quantum error correction as non-Clifford gatesrequire additional resources such as magic-state distilla-tion to prepare fault-tolerantly [19]. However, in non-fault tolerant near term devices it is often preferable tominimize the total number of two-qubit gates in a de-composition rather than non-Clifford gates. An opti-mal decomposition for gates generated by the CNOT-Dihedral in terms of the number of CNOT and CS gateshas also recently been developed [20]. Another exam-ple is the Toffoli gate which can be decomposed into 6CNOT gates and single qubit gates, but requires only 5two-qubit gates in its decomposition if the CS and CS − a r X i v : . [ qu a n t - ph ] J u l gates are also available [21].In this work we calibrate CS and CS − gates of varyingdurations on an IBM Quantum system and benchmarkthe gate error rates by performing the first experimentaldemonstration of interleaved CNOT-Dihedral random-ized benchmarking. For specific gate durations we areable to obtain a high-fidelity CS gate approaching thecoherence limit, which due to the shorter CR interactiontime results in a lower error rate than can be obtained fora CNOT gate. In addition to RB we also compute the av-erage gate error of the CS gate using two-qubit quantumprocess tomography (QPT) and compare to the valuesobtained from RB. Pulse-level calibration was done us-ing Qiskit Pulse [22], and the RB and QPT experimentswere implemented using the open source Qiskit comput-ing software stack [23] through the IBM Quantum cloudprovider. II. CNOT-DIHEDRAL RANDOMIZEDBENCHMARKING
We describe the protocol for estimating the averagegate error of the CS gate using interleaved CNOT-Dihedral Randomized Benchmarking, which is a natu-ral generalization of the CNOT-Dihedral RB proceduredescribed in [17] with interleaved RB [13] to estimateindividual gate fidelities for the CS gate CS = i . In the following we let G denote the CNOT-Dihedralgroup on n qubits, g ∈ G denote a unitary element of G ,and I , X , Y , Z denote the single-qubit Pauli matrices. Step 1: Standard CNOT-Dihedral benchmarking.
Randomly sample l elements g j , . . . , g j l uniformlyfrom G , and compute the ( l + 1)th element from theinverse of their composition, g j ( l +1) = ( g j l ◦ · · · ◦ g j ) − .Denote by j l the l -tuple ( j , . . . , j l ). For each sequence,we prepare an input state ρ , and apply the compositionof the l + 1 gates that ideally would be S j l := g j ( l +1) ◦ g j l ◦ · · · ◦ g j , and then measure the expecation value of an observable E .Assuming each gate g i has an associated error Λ i ( ρ ),the sequence S j l is implemented as˜ S j l := Λ j ( l +1) ◦ g j ( l +1) ◦ (cid:0) (cid:13) li =1 [Λ j i ◦ g j i ] (cid:1) (1)The expectation value of E is (cid:104) E (cid:105) j l = T r [ E ˜ S j l ( ρ )].Averaging this overlap over K independent sequences of length l gives an estimate of the average sequence fidelity F seq ( l, E, ρ ) := T r [ E ˜ S l ( ρ )] (2)where ˜ S l ( ρ ) := K (cid:80) j l ˜ S j l ( ρ ) is the average quantumchannel.We decompose the input state and this final measure-ment operator in the Pauli basis P (an orthonormal basisof the n -qubit Hermitian operators space, constructed ofsingle-qubit Pauli matrices). This gives ρ = Σ P x P P/ n and E (cid:48) = Σ P e P P . Given that the gate errors are close tothe average of all errors [17], the average sequence fidelityis F seq ( l, E, ρ ) = A Z α lZ + A R α lR + e I where A Z = Σ P ∈Z\{ I } e P x P and A R = Σ P ∈P\Z e P x P ,with Z being tensor products of Z and I gates.Each of the two exponential decays α lZ and α lR canbe observed by choosing appropriate input states. Forexample, if we choose the input state | . . . (cid:105) then F seq = e I + A α lZ where A = Σ P ∈Z\{ I } e P . On the other hand,if we choose | + · · · + (cid:105) then F seq = e I + A + α lR where A + = Σ P ∈X \{ I } e P , with X tensor products of X and I gates.The channel parameters α Z and α R can be extractedby fitting the average sequence fidelity to an exponential.From α Z , α R the average depolarizing channel parameter α for a group element g is given by α = ( α Z + 2 n α R ) / (2 n + 1) (3)and the corresponding average gate error is given by r = (2 n − − α ) / n . (4) Step 2: Interleaved CNOT-Dihedral sequences.
Choose a sequence of unitary gates where the first el-ement g j is chosen uniformly at random from G , thesecond is always chosen to be g , and alternate betweenuniformly random elements from G and fixed g up to the l -th random gate. The ( l + 1) element is chosen to be theinverse of the composition of the first l random gates and l interlaced g gates, g j ( l +1) = ( g ◦ g j l ◦ · · · ◦ g ◦ g j ) − . Weadopt the convention of defining the length of a sequenceby the number of random gates l .For each sequence, we prepare an input state ρ , apply ν j l := g j ( l +1) ◦ g ◦ g j l ◦ · · · ◦ g ◦ g j and measure an operator E .Assuming that the gate g has an associated error Λ g ( ρ )and that each gate g i has an associated error Λ i ( ρ ), thesequence ν j l is implemented as˜ ν j l := Λ j ( l +1) ◦ g j ( l +1) ◦ (cid:0) (cid:13) li =1 [Λ g ◦ g ◦ Λ j i ◦ g j i ] (cid:1) . (5)The overlap with E is T r [ E ˜ ν j l ( ρ )]. Averaging thisoverlap over K independent sequences of length l givesan estimate of the new sequence fidelity F seq ( l, E, ρ ) := T r [ E ˜ ν l ( ρ )]where ˜ ν l ( ρ ) := K (cid:80) j l ˜ ν j l ( ρ ) is the average quantum chan-nel.Similarly to Step 1, we fit F seq ( l, E, ρ ) and obtain thedepolarizing parameter α ¯ g , according to Eq. (3). Usingthe values obtained for α and α ¯ g , the gate error of Λ g ,which is given by r rb g = (2 n − − α ¯ g /α )2 n , (6)and must lie in the range [ r rb g − (cid:15), r rb g + (cid:15) ], where (cid:15) isestimated in [13] Eq. (5). Note that one has to be carefulin interpreting the results of an interleaved experiment,as in some cases (cid:15) might be large compared to r rb g . III. IMPLEMENTING THE CONTROLLED- S GATE
We calibrate CS gates of varying gate durations usingQiskit Pulse and measure the average gate error usingthe interleaved CNOT-Dihedral RB protocol in II. Weuse the CR pulse sequence as a generator of two-qubitentanglement [10, 24]. The CR pulse is realized by ir-radiating one (control) qubit with a microwave pulse atthe transition frequency of another (target) qubit. Thestimulus drives the quantum state of the target qubitwith the direction of rotation depending on the quantumstate of the control qubit. This controlled rotation isused to create two-qubit entangling gates such as CNOTand CS.The two-qubit system driven by the CR pulse withamplitude A and phase φ can be approximated by aneffective block-diagonal time-independent Hamiltonian[25, 26] H CR ( A, φ ) = (cid:88) P = I,X,Y,Z ω ZP ( A, φ )2 Z ⊗ P (7)+ (cid:88) Q = X,Y,Z ω IQ ( A, φ )2 I ⊗ Q, where the qubit ordering is control ⊗ target, and ω ZP and ω IQ represent the interaction strength of the cor-responding Pauli Hamiltonian terms. In the absence ofnoise, the ideal CR evolution for a constant-amplitudepulse is written as an unitary operator U CR ( A, φ ) = exp (cid:8) − it CR H CR ( A, φ ) (cid:9) , (8)where t CR is the length of the CR pulse. We also definethe unitary operator created by an arbitrary two-qubitgenerator as [ BC ] θ = exp (cid:26) − i θ B ⊗ C ) (cid:27) (9) where B , C are arbitrary single qubit operators, and weuse [ BC ] ≡ [ BC ] π .As can be seen by examining Eq. (7), the CR pulse in-duces three entangling interaction terms ( ZX , ZY , and ZZ ), in addition to potentially many unwanted local ro-tations with different amplitudes. By appropriately cal-ibrating the phase of the CR drive φ , the ZX term isthe dominant term among the interactions and is the keyterm for executing two-qubit gates in this system. Aswith the standard CNOT gate, we can compose a CSgate by isolating the ZX interaction with a refocusingsequence and single qubit pre- and post-rotations: CS = [ IH ] ◦ [ IX ] π ◦ [ ZI ] π ◦ [ ZX ] − π ◦ [ IH ] , (10)where H is the Hadamard operator. As shown inEq. (10), we need to develop the calibration procedureto find an amplitude A and a phase φ where | ω ZX | t CR = π/ ZI term as a result of the off-resonant driving of the control qubit; IX , ZZ and IZ can also be large for transmon qubits [25]. However, thestrengths of ZZ and IZ terms are expected to be negli-gibly weak in our device. We note that both ZI and IX terms commute with the ZX term of interest, while ZI and ZX terms anti-commute with the inversion of thecontrol qubit XI . In addition, the ZI term is the evenfunction and both IX and ZX terms are odd functionsof the drive amplitude A . Accordingly, we can effectivelyeliminate the impact of those unwanted terms with thetwo-pulse echoed CR sequence [27] expressed as U echo ( A, φ ) = [ XI ] ◦ U CR ( − A, φ ) ◦ [ XI ] ◦ U CR ( A, φ ) . (11)This sequence consists of two CR pulses with oppositedrive amplitude, each one followed by a π -rotation refo-cus pulse XI on the control qubit. Here we also assumethe negligible impact of the IY term which is generallyintroduced by the physical crosstalk between the controland the target qubit [11]. A. Gate Calibration and Benchmarks
To experimentally implement the CS and CS † gateswe use the 27 qubit IBM Quantum system ibmq paris with fixed-frequency and dispersively coupled transmonqubits. Qubit 0 and the qubit 1 of this system are as-signed as the control and the target qubit, respectively.The resonance frequency and anharmonicity of the con-trol (target) qubit are 5.072 (5.020) GHz and -336.0 (-321.0) MHz.The pulses realized in practice are not constant-amplitude pulses, rather the amplitude is increased anddecreased smoothly. We implment the CR pulse as aflat top Gaussian, with flat-top length τ sq , and Gaussianrising and falling edges each with length τ edge ( τ CR = τ sq + 2 τ edge ). We use a constant Guassian edge with τ edge = 28 .
16 ns with 14 .
08 ns standard deviation andvary the length of the duration of the square flat-toppulse τ sq . The minimum pulse duration is τ sq = 0ns, yielding a pure Gaussian shape. The overhead ofsingle-qubit gates in the echoed CS sequence in Eq. (10)for the ibmq paris backend is 106.7 ns, giving a to-tal echoed CS gate time of τ CS = 2 τ CR + 106 . optimization level = 1 followed by conversion to apulse schedule [22].We performed calibration to a CR rotation angle ω ZX ( A, φ ) τ CR (cid:39) π/ τ sq . This wasdone by first performing a rough calibration of ( A, φ )by scanning those parameters, followed by the closed-loop fine calibration with standard error amplificationsequences (see Appendix B for details). The calibratedpulse schedule of the CS gate with τ sq = 21 . τ CS =263 . l ∈ (1 , , , , , , , , , l . Each experiment is executed 1024 times forboth input states | (cid:105) and | ++ (cid:105) both with and withoutinterleaving the CS gate. An example of measuredRB decay curves for τ sq = 21.3 ns are shown in Fig.1(b). The exponential fit of the decay curves yields α = 9 . × − and α ¯ g = 9 . × − , givingan estimated average gate error of the CS gate of r rb g = 5 . × − . According to [13], the theoreticalbound of the error is calculated to be [0 , . × − ]. Inaddition to RB we also perform quantum process tomog-raphy (QPT) [29] and compute the average gate fidelityfrom the reconstructed process. QPT was done usingmaximum likelihood estimation with the tomographymodule of Qiskit Ignis [30], using a preparation basis of {| (cid:105) , | (cid:105) , | + (cid:105) , | + i (cid:105)} and measurement basis of { X, Y, Z } for each qubit. We performed 1024 repetations (shots)per basis configuration and correct for measurementerrors using the readout error mitigation technique [31].The average gate error calculated from the tomographicfit for τ sq = 21 . r qpt g = 8 . × − which iscomparable to the value estimated from the interleavedCNOT-Dihedral RB experiment. B. Gate Duration Dependence
We perform the same calibration and benchmarkingprocedures for different flat-top width τ sq from 0 nsto 355.6 ns ( τ CS from 219.3 ns to 930.5 ns) and mea-sure the average gate errors by both the interleavedCNOT-Dihedral RB experiment and QPT. In this ex-periment, we use a reduced set of RB sequence lengths l ∈ (1 , , , , , T and T Time (ns) d0u0d1
VZ( )X( ) X( )VZ( ) VZ( ) VZ( )CR CR + VZ( ) VZ( ) VZ( )Y( ) Y( ) (a) G r o un d s t a t e p o p u l a t i o n (b) RB in |00Interleaved RB in |00RB in |++Interleaved RB in |++
FIG. 1. The CS gate realized with a closed-loop calibra-tion. (a) Pulse schedule with the flat-top width τ sq = 21.3ns. The schedule consists of two CR pulses CR − and CR + on the ControlChannel u0 with echo pulses X ( π ) applied on DriveChannel d0 of the control qubit. Local gates in Eq.(10) are also applied to the
DriveChannel d1 of the targetqubit. Modulation frequencies of d0 and d1 are locked at theresonance frequencies of the control and the target qubits,respectively. The u0 channel is associated with the controlqubit with the frame of the target qubit to drive CR pulses atappropriate phase and frequency. A Circular arrow of VZ ( θ )represents the virtual-Z rotations with rotation angle θ . (b)CNOT-Dihedral interleaved RB. Dotted lines show fit curvesof the ground state population measured by standard RBs in | (cid:105) and | + + (cid:105) basis, while solid lines show fits of interleavedRB. Triangle and cross symbols show raw experiment data of10 different random circuits. with relaxation and Hahn echo sequences [32], respec-tively, to monitor the stability of physical propertiesof qubits. These experiments are inserted immediatelybefore each calibration experiment and yield coherencetimes of T = 59 . ± . . ± . µ s and T =92 . ± . . ± . µ s for the control (target) qubitduring the experiment. Here, the error bars correspondto the standard deviation over the duration of the wholeset of calibration and benchmarking experiments, pre-sented in Fig. 2. The device was accessed via the cloudthrough a fair-share queuing model used in IBM Quan-tum systems. The time in between experiments wasabout 168 minutes on average, thus the experiment couldbe subject to some parameter fluctuations due to noise sq (ns)0.0000.0050.0100.0150.0200.0250.030 A v e r a g e g a t e e rr o r QPTCoherence limitRB220.4 320.4 420.4 520.4 620.4 720.4 820.4 920.4Total gate time CS (ns) FIG. 2. Average gate errors as a function of the flat-topwidth of the CR pulse τ sq estimated by different benchmarktechniques. The corresponding total gate time τ CS is shownin the top axis. Blue circles and red triangles represent r rb g and r qpt g , respectively. The Green dotted line shows the the-oretical lower bound of the average gate error calculated bythe total gate time τ CS and the average T and T values ofthe qubits during the experiment. The filled area representsthe coherence limit with T and T values with variance of1 σ . See text for a detailed discussion. with a long characteristic time [33].Nevertheless, as Fig. 2 shows, our calibration methodprovides highly accurate results and allows to approachthe coherence limit for appropriately chosen gate times.It can also be seen that r rb g and r qpt g show a similartrend as a function of τ sq . This dependence on τ sq agreeswell with the slope predicted by the coherence limit for τ sq (cid:38) . r rb g gives a more robustestimate of the gate error than the r qpt g as it is not as sen-sitive to state prepratation and measurement errors. Theinterleaved CNOT-Dihedral RB experiment also requiresonly 24 circuit executions per single error measurement,while the two-qubit QPT requires 144 circuit executionswith the readout error mitigation. The smaller exper-imental cost to measure r rb g enables us to average theresult over 10 different random circuits, which is empir-ically sufficient to obtain a reproducible outcome, at apractical queuing time with ibmq paris .The nearly stable offset of r rb g from the coherence limitpossibly indicates the presence of coherent errors due toimperfection of calibration. The measured r qpt g obtainedconsistantly smaller error values than r rb g .In the region τ sq (cid:46) . π/ τ CR . The amplitude of crosstalk (cid:112) ω IX + ω IY measured at τ sq = 0 ns is 176.2 kHz, whileone at τ sq = 355 . IX term is refocused and has negligible contribution, the re- mained IY term can still impact on the measured gateerrors. Thus, at τ sq = 0 we calibrate a CS gate with acompensation tone on the target qubit to suppress thephysical crosstalk between qubits (see Appendix C fordetails). The calibrated pulse sequences with and with-out the compensation tone yield r rb g of 2 . × − and2 . × − , respectively. These comparable results in-dicate the physical crosstalk is relatively suppressed inthis quantum device and other noise sources are dom-inant for τ sq (cid:46) . ZZ coupling, CR-induced ZZ inter-action [34–36], and leakage to the higher energy levels[37, 38]. Although a further analysis of the error mecha-nisms in this regime of high-power pulses is beyond thescope of this study, initial results indicate that coherentpopulation transfer out of the two-qubit manifold intothe higher levels, and ZZ interaction terms, are not therelevant mechanisms [39]. At the same time, the coher-ence limit can be further lowered by reducing the timespent on single-qubit gates. At τ sq = 21 . r rb g of 5 . × − , the refocusing pulse andlocal rotations occupy 40% of the total gate time τ CS ,yielding a non-negligible impact on the gate error. IV. CONCLUSION
We have demonstrated calibration of a high fidelitynon-Clifford CS gate on 27 qubit IBM Quantum sys-tem ibmq paris . This gate is not currently included inthe standard basis gates of IBM Quantum systems, andit was calibrated and benchmarked entirely using opensource software available in Qiskit. Since the CS gateis non-Clifford, robust characterization of the averagegate error cannot be done using standard RB. To bench-mark performance of the non-Clifford gate we performedthe first experimental demonstration of two-qubit inter-leaved CNOT-Dihedral RB, which allow efficient and ro-bust characterization of a universal gateset containingthe CS gate.We obtained a minimal gate error of 5 . × − with appropriately shaped echoes and a total gate time of263 . ibmq paris is 1 . × − .Thus the presented CS gate error is comparable with halfthe CNOT error. By performing RB and QPT for a vari-ety of gate lengths we were also able to study the perfor-mance of the CS gate in different regimes and observed abreak down in performance if gate lengths were reducedbelow the best value obtained for 263 . ACKNOWLEDGEMENTS
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In this paper all experiments are performed via cloudaccess to IBM Quantum system ibmq paris . The back-end provider calibrates single-qubit and two-qubit basisgates on a regular basis and provides pulse schedules andgate errors to users. The gate error distribution at thetime of experiment (2020-05-20 05:48 UTC) is shown inFig. 3. The averaged single-qubit gate error is 5 . × − ,while that of two-qubit gates is 1 . × − . The single-qubit gate error of the qubit 0 and 1, which are use inthe CS gate, are 4 . × − and 3 . × − , respectivly.The two-qubit CNOT gate error between these qubits is1 . × − . H error rate (%)
CNOT error rate (%)
FIG. 3. Distribution of single-qubit and two-qubit gate er-rors of ibmq paris at the time of experiment. Single-qubitgate errors measured by the Hadamard operation are shownin nodes of the qubit coupling map, while two-qubit gate er-rors measured by CNOT operation are shown in graph edges.Error values are represented by color maps shown in the bot-tom.
Appendix B: Calibrating CS Gate
The single qubit gates used for the echo sequence andlocal rotations are provided by ibmq paris . We calibratethe CR pulse amplitude A and its phase φ by the roughparameter scan followed by the closed-loop calibration.These parameters are determined based on the two-pulseechoed CR sequence U echo shown in Eq. (11). This ap-proach simplifies the calibration, namely, we don’t needto take non-negligible ZI and IX terms into accountwhen we fit the experimental results for calibration pa-rameters. Calibrated sequence U echo ∼ [ ZX ] π is used torealize the CS with local rotations shown in Eq. (10).
1. Rough Parameter Scan
We initialized both qubits in the ground state and per-form a rough scan of the CR pulse amplitude with thepulse schedule: S scan A ( A ) ≡ U echo ( A, . The schedule is follwed by the measurement of the targetqubit in the Z -basis. The sinusoidal fit for the measuredpopulation of the target qubit with S scan A with different A gives an estimate of the CR amplitude A where the angleof controlled rotation is approximately π/
4. A typicalexperimental result for τ sq = 21 . A , we scan the CR phase with two pulseschedules S scan φg and S scan φe : S scan φg ( φ ) ≡ [ IZ ] π ◦ [ IX ] π ◦ U echo ( A , φ ) , S scan φe ( φ ) ≡ [ IZ ] π ◦ [ IX ] π ◦ U echo ( A , φ ) ◦ [ XI ] . z (a) y (b) |00|01 N z (c) Initial experimentFinal experiment N z (d) CRCR+Compensation
FIG. 4. Typical experimental results for calibration experi-ments. Measured population is converted into the expectationvalue of Pauli operators. (a) Rough amplitude calibration.The blue and black line show the cosinusoidal fit for the ex-perimental results and the optimal amplitude A . (b) Roughphase calibration. The blue and red line show the cosinu-soidal fit for the experimental result of S scan φg ( φ ) and S scan φe ( φ ),respectivly. The black line show the optimal phase φ . (c)Rough amplitude calibration. The solid and dotted line showthe fit for the result of initial ( A = 0 . A = 0 . N -dependent decay andbaseline F ( N ) = e − αN cos(4( π/ δ A ) N + π/ aN + b . Here α, a and b are additional fit parameters introduced empiri-cally. The residual error per gate after the final experimentis − . × − rad., which is lower than the threshold of10 − π . (d) Compensation tone calibration. The blue and redline show the result of S xy4 without and with the calibratedcompensation tone, respectivly. The cosinusoidal fit with de-cay for those curves yields crosstalk amplitude of 176.2 kHzand 6.7 kHz. All data in (a)–(c) are measured with τ sq = 21 . τ sq = 0 ns. The schedule S scan φg ( S scan φe ) drives the echo sequence U echo ( A , φ ) twice with the control qubit of the ground(excited) state. Note that the last two operations cor-respond to the projection into Y -basis for the followingmeasurement. The flip of the state of the control qubitleads the controlled rotation of the target qubit state withopposite direction as illustrated in Fig. 4(b). This oppo-site rotation of π/ θ = θ − φ of the target qubit Bloch sphere yields measured outcomeof ∓ S scan φg and S scan φe , respectivly, at the optimalphase φ = φ where θ = 0. Here θ is the phase offsetfrom the unknown transfer function of the coaxial cableassembly [42]. The phase φ gives a rough estimate of theCR phase where the ZX term of interest is maximizedwhile the unwanted ZY term is eliminated.
2. Closed-loop Fine Calibration
We use the roughly estimated parameters ( A , φ ) asan initial guess of closed-loop calibrations. We first opti-mize the CR pulse amplitude with following experiment: S fine A ( A ) ≡ U echo ( A, φ ) N ◦ [ IX ] π , where N is number of repeated sequences. This scheduleprepares the target qubit in the superposition state andrepat the echo sequence 4 N times to apply a controlledrotation of N π . Because the initial guess of A is esti-mated by the parameter scan in the coarse precision witha finite error δ A , repeating S fine A for different N can ac-cumulate δ A and this error appears as over rotation fromthe superposition state, as shown in Fig. 4(c). The fitfor the over rotation as a function of N yields precise es-timate of δ A , and we iteratively update the initial guessto optimize the CR pulse amplitude to A where δ A ∼ N = 0 , , ..., − π rad.With the optimized amplitude A , we tune the CRphase with following experiment: S fine φ ( φ ) ≡ [ IY ] π ◦ ( U echo ( A , φ ) ◦ [ IY ]) N ◦ [ IX ] π . This sequence also accumulates the small phase error δ φ as function of N . We iteratively update the CR phaseuntil the same threshold value with the amplitude cali-bration to obrain the optimal phase φ where δ φ ∼ Appendix C: Crosstalk Estimation
The unwanted local rotation terms IX and IY canbe simultaneously amplified with the following sequencecombined with the XY-4 dynamical decoupling [43] onthe control qubit: S xy4 ≡ ([ Y I ] ◦ U CR ◦ [ XI ] ◦ U CR ) N , where U CR = U CR ( A , φ ). Here, the CR pulse withthe same sign is repeatedly applied while changing thestate of control qubit. This pulse sequence refocuses (andhence eliminates) controlled rotation terms such as ZX and ZY , allowing us to precisely estimate the strengthof weak local rotation terms (cid:112) ω IX + ω IY , amplified inthe absence of strong two-qubit interactions.This technique can be used to calibrate a compensationtone that eliminates the IY term caused by the physicalcrosstalk between qubits [11]. The compensation tone isapplied to the drive channel of the target qubit d1 , inparallel with U CR . This single-qubit pulse is shaped as a flat-top pulse with Gaussian edges of identical durationas the U CR pulse, with its own calibrated amplitude andphase ( A (cid:48) , φ (cid:48) ). First, we repeat S xy4 for N = 0 , , , ..., Z expectation value of the target qubit. The fit for theoscillation over the total CR gate time 8 τ CR N yields thestrength of the total unwanted local rotation terms. At τ sq = 0 ns, the unwanted local rotation strength of 176.2kHz was observed. This strength was reduced to 6.7 kHzwith the calibrated compensation tone with A (cid:48) = 0 . φ (cid:48) = − ..