Demonstration of the QCCD trapped-ion quantum computer architecture

J. M. Pino,∗ J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C.

H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson, and B. NeyenhuisHoneywell Quantum Solutions

(Dated: March 13, 2020)

We report on the integration of all necessary ingredients of the trapped-ion QCCD (quantumcharge-coupled device) architecture into a robust, fully-connected, and programmable trapped-ionquantum computer. The system employs 171Yb+ ions for qubits and 138Ba+ ions for sympatheticcooling and is built around a Honeywell cryogenic surface trap capable of arbitrary ion rearrange-ment and parallel gate operations across multiple zones. As a minimal demonstration, we use twospatially-separated interaction zones in parallel to execute arbitrary four-qubit quantum circuits.The architecture is benchmarked at both the component level and the holistic level through a va-riety of means. Individual components including, state preparation and measurement, single-qubitgates, and two-qubit gates, are characterized by randomized benchmarking. Holistic tests includeparallelized randomized benchmarking showing that the cross-talk between different gate regionsis negligible, a teleported CNOT gate utilizing mid-circuit measurement, and a quantum volumemeasurement of 24.

INTRODUCTION

The first demonstration of a universal set of quantumlogic gates was performed with trapped ions [1, 2]. Sincethen, researchers have demonstrated up to 10 minute co-herence times [3], and gate, state preparation, and mea-surement fidelities surpassing those of other competingquantum computing architectures [4–6]. Upon realizingthe basic building blocks of quantum computation, re-searchers began focusing their attention on how to scalesuch a machine to the large number of qubits needed forimplementing complex quantum algorithms. A crucialfeature of trapped-ion based quantum computers is theuse of strong motional coupling amongst ions in a crys-tal to mediate interactions between their internal (qubit)states. While it is possible to make a larger quantumcomputer simply by putting more ions into a single crys-tal, this approach is unlikely to be scalable to very largenumbers of qubits. Efforts toward trapped-ion quan-tum computation have therefore focused on architecturesthat scale the number of ion crystals used and employquantum communication between the crystals. Meth-ods for inter-crystal communication include using pho-tonic links [7, 8] or physically transporting ions betweendifferent crystals. Here we demonstrate the latter ap-proach, which, following the original proposal, we referto as the QCCD (quantum charge-coupled device) archi-tecture [9, 10].

Every proposal for constructing a large-scale quantumcomputer has engineering challenges, and the QCCD ar-chitecture is no different. As described in the originalpaper, difficulties include: (1) constructing a device ca-pable of trapping a large number of small ion crystals,(2) reordering and precise positioning of ion crystals viafast, robust transport operations, (3) clock synchroniza-tion across the inhomogeneous environment of the trap,(4) the likely need for trapping two different ion species,

one for use as a qubit and another for sympathetic cool-ing during the execution of a quantum circuit, and (5)parallelization of both transport and quantum operationsacross the device. Once these engineering challenges areovercome, the result would be a fully connected quan-tum processor that benefits from the high gate fidelitiesdemonstrated for small isolated ion crystals and has apath to scale to a large number of qubits.

Nearly all of these individual difficulties of the QCCDarchitecture have been overcome in various laboratories:cryogenic microfabricated surface traps have been built[11], fast transport operations that induce little heatinghave been demonstrated [12, 13], hyperfine clock-stateshave been shown to provide a robust qubit with high-fidelity qubit operations [14], and multi-species crystalshave been trapped and sympathetically cooled to neartheir ground state [15]. Combining these features into asingle machine creates performance requirements that areseemingly at odds; for example, cross-talk minimizationbenefits from increasing spatial separation at the cost ofmagnetic field homogeneity, inter-zone clock synchroniza-tion, increased transport distances, and detection opticscomplexity. Additionally, the system-level integration ofchallenging and disparate technologies compounds relia-bility requirements for a functioning device, requiring awhole new framework of automation and robustness.

In this article, we report on the integration of allthese necessary ingredients into a full-stack QCCD quan-tum computer. The device is built around a microfabri-cated cryogenic surface trap (Fig. 1) containing five zonesused for gating operations and ten storage zones. Using171Yb+ and 138Ba+ as the qubit and coolant ions, respec-tively, we demonstrate parallel operation and commu-nication between two adjacent gate zones separated by750µm. Loading the trap with four ions of each species,we show that high fidelity gate operations can be per-formed in parallel on two four-ion mixed-species crystals

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and that fast transport operations allow the qubits tobe reconfigured between those operations in order to ex-ecute arbitrary quantum circuits without the overheadof logical swap gates. Using randomized benchmarking,we demonstrate high-fidelity single-qubit and two-qubitgates in the two zones and show that the cross-talk erroris small. By teleporting a CNOT gate, we demonstratethe ability to perform mid-circuit measurements withoutdamaging the other qubits in the circuit. Finally, wedemonstrate the faithful integration of the different com-ponents by running a quantum-volume test using all fourqubits in these two interaction zones and find a quantumvolume of 24 = 16. The demonstration of these technolo-gies integrated into a single system paves the path forcontinued scaling of trapped-ion quantum computers.

SYSTEM

Hardware

We designed and fabricated a 2D surface trap with seg-mented trap electrodes at Honeywell’s microfabricationproduction facility in Plymouth, MN. The photographand diagram in Fig. 1 shows the RF Paul trap featur-ing a linear geometry for the simultaneous trapping ofmultiple linear chains of ions. The trap consists of fivezones dedicated to optical gates, two zones dedicated tothe storage of ions, one zone with a through-hole for ionloading, and eight auxiliary zones for storing and sort-ing ions. Ions are trapped and transported to differentregions in the trap via dynamic voltage waveforms thatare individually applied to each of the 198 DC electrodes.All five gate zones are capable of the transport required toimplement parallel gate operations and detection acrossthe trap. In this work, we use the storage zone closest tothe loading hole as a gate zone for convenience, as it hasall of the necessary capabilities for the required trans-port. Hereafter, we shall refer to this zone as a gate zonefor ease of reference.

The trap was designed for operation in a cryogenicenvironment and fabricated using gold electrodes withan undercut etch to mitigate stray fields by eliminatingline-of-sight between dielectrics and the ions [16]. Thetrap is cooled to 12.6 K via a cold finger attached to aliquid He flow cryostat with stability better than 2 mK,thereby suppressing gate errors due to anomalous heating[17].

We store quantum information in the 171Yb+ 2S1/2 hy-perfine clock states, ∣0⟩ ≡ ∣F = 0,mF = 0⟩ and ∣1⟩ ≡ ∣F =

1,mF = 0⟩, with a frequency splitting of 12.642821 GHz[18], where F and mF are the quantum numbers for totalangular momentum and its z-projection. We pair eachYb ion with a partner Ba ion for sympathetic cooling,and throughout a circuit, they exist either as a four ioncrystal (Ba-Yb-Yb-Ba or Yb-Ba-Ba-Yb) or as a Yb-Ba

FIG. 1. Illustration of the programmable QCCD quantumcomputing system along with a photograph of the trap. (a)On the right, a picture of the trap. On the left, the infor-mation flow from the user to the trapped qubits. From topto bottom, we illustrate: user, cloud, internal tasking, ma-chine control system, FPGA. The circuits are processed by acompiler as described in the text to generate control signals(purple) sent to both the trap electrodes as well as optoelec-tronic devices that control laser beams. An imaging systemand PMT array collect and count scattered photons, and theresult (green) is sent back to the software stack and user. (b)A schematic of the trap detailing: the loading hole (black),load zone (purple), storage zones (orange), gate zones (blue),as well as auxiliary zones (yellow) for additional qubit storage.The illustration of how a general quantum circuit is carriedout shows that ions already sharing a gate zone are gated,then spatially isolated for single-qubit gates, then the secondand third ions are swapped so that the final two-qubit gatescan be executed. While not shown, readout, two-qubit gates,and single-qubit rotations can all be performed in parallel.

pair.

The qubits are initialized and measured using stan-dard optical pumping and state-dependent fluorescencetechniques [18], and the ions’ fluorescence is imaged ontoa linear PMT array for quantum circuit measurements.Cooling is accomplished via a combination of Dopplercooling and ground-state Raman sideband cooling [19] ofthe Ba ions. The ground-state cooling occurs once af-ter initial Doppler cooling and then again immediatelybefore each two-qubit gate. Gating operations are ac-complished via stimulated Raman transitions driven bytwo different configurations: single-qubit gates are imple-mented using pairs of co-propagating 370.3 nm Ramanbeams with circular polarization, and two-qubit gates areimplemented with additional pairs of beams that couples

3

to an axial mode of motion (along the trapping RF-null).We impose a quantization axis parallel to the trap sur-

face at a 45○ angle with respect to the RF-null. This 5 Gfield is uniform between the two zones to within 0.2 mG,creating non-uniformity in the measured qubit frequen-cies of less than 1 Hz. All laser systems are frequencyand intensity stabilized via closed-loop feedback, and allbeam alignment is handled through automated routines.

Computer Control

We illustrate the software stack from the user down tothe qubits in Fig. 1. The processor is programmed us-ing the quantum circuit model [20]. A quantum circuitis submitted remotely through a cloud-based service andtasked in Honeywell’s internal cloud. The algorithm iscompiled into the various primitives needed to executethe quantum circuit and sent to the machine control sys-tem. This system is responsible for programming thefield-programmable gate array (FPGA) to execute thespecific quantum circuit as well as scheduling and ex-ecuting calibration routines. These automated calibra-tions are either executed on a predetermined time inter-val or triggered when a drift tolerance is exceeded. TheFPGA handles the timing of operational primitives andreal-time decision-making based on mid-circuit measure-ment outcomes. Clock synchronization between qubits ismaintained via a phase-tracking protocol handled by theFPGA, which updates the qubit phases after transportand gate operations to account for phase accumulationgenerated by AC-Stark shifts.

In the rare event of ion-loss or detectable ion-reordering events, the data is discarded at the machinecontrol level, and circuits are repeated as necessary toproduce valid data. Finally, results are reported backthrough the cloud service to the user.

PRIMITIVE OPERATIONS

Trapping and transport

A hole in the trap, shown in Fig. 1, allows a collimatedbeam of neutral atoms from an effusive thermal sourceto enter the trapping region. To load the device, theseatoms are photoionized and Doppler cooled using stan-dard techniques [18] and then transported to one of thegate zones. We initialize the quantum registers with aBa-Yb-Yb-Ba crystal in two of the gate zones in Fig. 1.To enable all-to-all connectivity and entangle any twoqubits, a suite of transport operations, achieved by ap-plying dynamic waveforms to the DC electrodes, allowsthe order of the ions to be arbitrarily rearranged. Thesetransport operations fall into three categories:

=UZZ UMS(ϕ)USQ(ϕ) USQ(−ϕ)

USQ(−ϕ)USQ(ϕ)

FIG. 2. Construction of a phase-insensitive two-qubitgate. The Mølmer-Sørensen interaction generates the uni-tary UMS = exp(−iπ

4(Xsinφ + Y cosφ)⊗2

) whose basis is de-termined by the optical phase φ. Single-qubit operationsdriven by the same laser beams generate the unitary USQ =exp(−iπ

4(Xcosφ + Y sinφ)) and are applied globally to both

qubits. The resulting composite gate is, up to a global phase,equal to Uzz = exp(−iπ

4Z ⊗Z).

Linear transport: A potential well is moved alongthe RF-null from one region of the trap to another [21].During loading, we transport a single ion in a single well;however, during a quantum-circuit, multiple wells at dif-ferent axial positions are moved simultaneously. Notethat in our architecture, these linear transports alwaysinvolve moving a Yb-Ba pair.Split/combine: During a split operation, a single po-

tential well is divided into two potential wells [22], split-ting a 4-ion crystal into two 2-ion crystals. The combineoperation is the time-reversal of the split operation, andboth occur only in the gate zones.Swap: The qubit order of a 4-ion crystal is flipped by

rotating the qubits about an axis perpendicular to theRF-null [23]. These operations only occur in gate zonesand allow us to avoid the quantum circuitry overhead oflogical swaps.

These three transport primitives can be used to rear-range the qubits in a quantum circuit arbitrarily. Onaverage, these transport primitives add less than twoquanta of heating axially. Transport failure events arerare and, as previously stated, are automatically detectedand scrubbed from the data sets so that computations arenot corrupted. A typical duration to reconfigure the ionlocations through transport and re-cool them to near themotional ground state is 3-5 ms. This time scale canbe reduced via faster and more precise transport opera-tions with improved electrode-control electronics as wellas optimized waveforms and electrode geometries.

Quantum Operations

For qubit initialization and measurement, we spatiallyisolate a single Yb-Ba crystal so that resonantly scatteredlight has a minimal effect on the other qubits, allowingmeasurement (and subsequent initialization, if necessary)to be performed at any point in the circuit. Our nativegate set consists of four basic types of gates:

Single-qubit Z rotations: The single-qubit rotationaround Z is done entirely in software by a phase-tracking

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Individual two-qubit RBSimultaneous two-qubit RBS

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(a) (b) (c)

FIG. 3. (a) Single-qubit randomized benchmarking for all four qubits. (b,c) Two-qubit randomized benchmarking results forboth zones. We performed two-qubit randomized benchmarking in each zone, both separately and simultaneously, and observeno statistically significant difference between the two sets of experiments. These measurements indicate that cross-talk errorsdo not play a significant role in our two-qubit gates (see the Appendix for more details).

update as described in the Computer Control section. Weset a resolution limit of π/500 for Z rotations.

Single-qubit X/Y rotations: Stimulated Ramantransitions on an isolated Yb-Ba crystal in a gate zoneare used to apply single-qubit rotations about an arbi-trary axis in the X-Y plane. The transition is driven bythe co-propagating single-qubit beams described in theHardware section. We currently restrict our gate set toπ and π/2 rotations. Arbitrary single-qubit gates sub-mitted by the user are synthesized into at most two X/Yrotations along with an additional Z-rotation.

Two-qubit gates: We operate a Mølmer-Sørenseninteraction [24] in the phase-sensitive configuration [25]and add single-qubit wrapper pulses driven with the samelasers (Fig. 2) to remove the optical phase dependence,resulting in the phase-insensitive entangling gate Uzz =

exp(−iπ4Z ⊗Z).

Global microwave rotations: A microwave antennain the vacuum chamber applies global qubit rotationswith a variable phase. The microwave field amplitude isonly homogeneous over the entire device to within about3% and, therefore, not suitable for logical operations in aquantum circuit. We do, however, use microwaves to sup-press memory errors through dynamical decoupling [26].Since the amplitude is inhomogeneous, we cannot applyglobal microwave π pulses, as is the nominal prescriptionfor dynamical decoupling. To avoid the accumulation ofcoherent errors, we apply pulses in pairs with oppositephases during ground-state cooling without any trans-port operation in between, thereby canceling out smallamplitude errors while mostly preserving the memory er-ror suppression benefits.

SYSTEM PERFORMANCE

To characterize the system, we evaluate the individ-ual components and then compare the results to severalholistic measurements.

Component benchmarking

We benchmark single-qubit gates, our native two-qubitgate, and state preparation and measurement (SPAM)using randomized benchmarking (RB). Results of eachcomponent benchmark are summarized in Table I. Fur-ther details on each procedure are given below:

State preparation and measurement: We extracta SPAM error from the y-intercept of the survival proba-bility in single-qubit RB. Theoretical modeling suggeststhe error from state preparation via optical pumpingis below 10−4, whereas the effective solid angle of ourphoton-detectors currently limits the measurement errorto > 10−3. Thus, we believe that SPAM errors are domi-nated by measurement error.

Single-qubit gates: We simultaneously performsingle-qubit RB with each qubit following the standardClifford-twirl version of RB outlined in Ref. [27], withresults plotted for each qubit in Fig. 3a.

Two-qubit gate: We perform two-qubit RB in eachzone independently and then again simultaneously inboth zones, following the standard Clifford-twirl versionof RB [27]. Results are plotted in Fig. 3(b,c), and we re-port the average infidelity of the two-qubit Clifford gatesscaled by the average number of Uzz gates (1.5).

We also perform the simultaneous RB method, as de-scribed in Ref. [28], generalized to two-qubit RB. Thismethod has two parts: (1) identify differences in RB de-cay rates between applying gates individually and simul-taneously, and (2) identify correlated errors when ap-plying gates simultaneously (also similar to Ref. [29]).From both analyses, we find no statistically significantevidence of cross-talk errors (see the Appendix). Whilethis method does not capture all possible cross-talk er-rors, we believe it is a strong indication that these typesof errors are minimal in this system.

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Component Zone Avg. Zone 1 Zone 2

SPAM (simultaneous SQ RB) 3(1) × 10−3 [3(1) × 10−3, 3(1) × 10−3] [2(1) × 10−3, 3(1) × 10−3]

Single-qubit gates (simultaneous RB) 1.1(3) × 10−4 [1.4(3) × 10−4, 1.2(2) × 10−4] [9(2) × 10−5, 1.0(3) × 10−4]

Two-qubit gates (individual RB) 7.9(4) × 10−3 6.7(5) × 10−3 9.0(4) × 10−3

Two-qubit gates (simultaneous RB) 8.0(4) × 10−3 7.2(4) × 10−3 8.8(4) × 10−3

TABLE I. Summary of component benchmarking results. Numbers are the average error per operation. For SPAM andsingle-qubit gates, the two numbers represent the measured values for each qubit in the corresponding zone.

Mid-circuit measurement

Splitting and spatially isolating ion-crystals enablemeasurement of a single qubit in the middle of a cir-cuit without damaging the quantum information storedin the other qubits. This capability, along with the abil-ity to reinitialize the qubit after it is measured, enablesseveral new algorithmic capabilities, including quantumerror correction [20]. Recently, the NIST group demon-strated mid-circuit measurements through a teleportedCNOT gate circuit [30], which we use as our algorithmicbenchmark for this primitive.

Quantum gate teleportation is a protocol in which apair of maximally entangled qubits are used as a re-source for applying a gate between a pair of remotedata qubits [31]. The protocol requires local entanglingoperations, mid-circuit measurements, and classically-conditioned quantum gates. The circuit for teleporta-tion of the CNOT gate is shown in Fig. 4a. Here,qubits q1 and q2 are initially prepared in the Bell state∣ψB⟩ =

1√2( ∣00⟩+ ∣11⟩ ). Two rounds of CNOT gates, each

followed by a measurement and conditional gate, result ina circuit that is logically equivalent to a CNOT controlledon q0 and targeting q3 (see Fig. 4). To realize this cir-cuit, transport operations are required to distribute theBell state between two zones for remote gating. EachCNOT in the circuit is compiled into a native Uzz gateand single-qubit rotations.

To efficiently benchmark the teleported CNOT gate,we use the method of Ref. [32] for bounding the fidelityof a process from its action on two mutually-unbiasedbases. This method amounts to verifying the followingquantum truth table:

CNOT ∶

∣00⟩↦ ∣00⟩ , ∣++⟩↦ ∣++⟩ ,

∣01⟩↦ ∣11⟩ , ∣+−⟩↦ ∣+−⟩ ,

∣10⟩↦ ∣10⟩ , ∣−+⟩↦ ∣−−⟩ ,

∣11⟩↦ ∣01⟩ , ∣−−⟩↦ ∣−+⟩ ,

where the states are labeled ∣q3 q0⟩. We prepare q0 and q3in each state of the {∣0⟩ , ∣1⟩} and {∣+⟩ , ∣−⟩} bases, applythe circuit in Fig. 4a and measure in the appropriatebasis. We repeat the circuit 500 times for each input stateand randomize the order in which the eight variations arerun. The data is shown in Fig. 4b.

(a) q0q1q2q3

H

H

Z

X=

(b)

FIG. 4. (a) Circuit implementing a teleported CNOT gate,with q0 and q3 the control and target qubits, respectively.(b) Bar plots showing the distribution of measurement out-comes when qubits q0 and q3 are prepared and measured inthe {∣0⟩ , ∣1⟩} and {∣+⟩ , ∣−⟩} bases.

Let f1 and f2 be the average success probabilities overthe {∣0⟩ , ∣1⟩} and {∣+⟩ , ∣−⟩} bases, respectively. The aver-age fidelity Favg[33] of the teleported CNOT gate is lowerbounded according to

Favg ≥4

5(f1 + f2) −

3

5. (1)

This bound does not account for errors in the statepreparations and measurements required for benchmark-ing. However, since these errors are much smaller thanthe full circuit error, Eq. (1) serves as a useful proxyfor the circuit performance. We find f1 = 0.933(6) andf2 = 0.941(5), yielding Favg ≥ 0.899(6).

Quantum Volume

Quantum volume (QV) is a full-circuit benchmark thatis dependent on both qubit number and gate fidelity [34].By weighing both metrics, QV attempts to estimate theeffective power of the quantum computer. In practice,a QV test consists of running random O(n) depth cir-cuits with n qubits, as shown in Fig. 5a for n = 4. The

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ExperimentNoisy simulationIdeal simulation

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FIG. 5. (a) Quantum volume circuits for n = 4. As de-scribed in Ref. [34], the test is run by interleaving randomSU(4) gates, which each require three entangling gates, withrandom permutations of qubits (denoted Π). While somearchitectures will need to use quantum logic gates for thepermutations, the QCCD architecture can use physical swapoperations that have a negligible effect on the stored quan-tum information. (b) Results for QV test for n = 2,3 and 4.For each test, we performed 100 randomly generated circuitsfrom Qiskit repeated 500 times each with distribution overcircuits plotted in blue. We also ran a noisy simulation inQiskit with depolarizing errors with rates estimated from RBexperiments, which are plotted in red. Green crosses give theaverage of the ideal outcomes for the 100 circuits run. Thegreen dashed line is the average over all ideal circuits whilethe black dashed line gives the passing threshold.

structure of these random circuits makes QV sensitive toerrors that may be missed by some component bench-marks and could degrade performance in generic com-putations. The circuits are then classically simulated,and the measured outputs are tested against the heavy-outcome criteria. If the circuits pass this test 2/3’s of thetime with two sigma confidence, then the system is saidto have QV = 2n.

We performed the QV test for n = 2,3, and 4, each with100 random circuits generated by Qiskit [35]. We passedQV for n = 2 with 77.58% of the circuits and 99.56% con-fidence, n = 3 with 83.28% of the circuits with 99.9996%confidence, and n = 4 with 76.77% of the circuits and99.16% confidence (results are plotted in Fig. 5b). Allconfidence intervals are calculated from the method out-lined in Ref. [34]. Each QV circuit was run without anyoptimization or approximations. For example, in then = 4 test, each circuit contained exactly 24 two-qubitgates.

Fig. 5 also shows theoretical simulations of the circuitoutcomes assuming a depolarizing noise channel for thetwo-qubit gates, with a noise magnitude extracted fromtwo-qubit RB. The excellent agreement provides furtherevidence that the QCCD architecture fulfills its most am-bitious goal: maintaining the gate fidelities achievable insmall ion crystals while executing arbitrary circuits onmultiple qubits.

DISCUSSION

The goal of this manuscript is to present and validate arealization of the QCCD architecture, and to benchmarkthe full set of required operations. Two gate zones wereoperated in parallel to execute arbitrary quantum circuitson four qubits, and we have shown evidence that coher-ence times, transport reliability, and robust sympatheticcooling are sufficient to execute large depth circuits.

Together with the verification of extremely low cross-talk between neighboring gate zones, these results sug-gest a path forward in which the primary near-term ob-stacles to scalability are technical in nature and generallyless severe than those already overcome in the work pre-sented here. We are currently pursuing a scaling of theoptical delivery to encompass all five gates zones, andeven without further improvements to gate fidelities, weexpect to increase the accessible quantum volume. Fi-nally, while we recognize the challenges ahead, we believethe successful integration of QCCD primitives demon-strated in this manuscript paves the way toward near-term intermediate-scale devices with fidelities approach-ing state-of-the-art demonstrations in small ion crystals.

ACKNOWLEDGEMENTS

This work was made possible by a large group of peo-ple, and the authors would like to thank the entire Hon-eywell Quantum Solutions team for their many contribu-tions.

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Simultaneous two-qubit randomized benchmarking

Simultaneous RB was originally proposed in Ref. [28]to measure addressability in single-qubit gates, and herewe describe the extension to two-qubit gates. Two-qubitsimultaneous RB is performed by three experiments: (1)RB in zone one while doing no gates in zone two, (2) RBin zone two while doing no gates in zone one, and (3) RBin both zones simultaneously.

For reference, we fit the RB data to the zeroth-orderRB decay equation

p(`) = Aα` +B, (2)

where p(`) is the average survival frequency for a length` RB sequence, A is the SPAM parameter, α is the RBdecay rate, and B is the asymptote, which is typicallyfixed by randomization of the final measurement [36, 37].We then calculate the uncertainty by finding the stan-dard deviation of a semi-parametric bootstrap resampleoutlined in Ref. [38] and also used in Ref. [36].

The data is then analyzed in two steps:Addressability analysis: Addressability errors are

induced in one zone from operations in the other zone.For example, if laser light from zone one is leaking intozone two, then that will cause an error in zone two butnot zone one. Addressability errors are quantified as thedifference between the decay rates from experiments oneand two, αz for z = 1, 2, and the decay rates from ex-periment three, αboth

z , [28],

γz = ∣αz − αbothz ∣. (3)

We compare the decay rates instead of the fidelity, asdone in Ref. [28]. As summarized in Table II, we seethat the measured values of γz are within one standarddeviation of zero.Correlation analysis: Correlation errors are those

that are shared between the zones when running RB si-multaneously. For example, if the laser power changes

8

Zone αz αbothz γz

1 0.9866(9) 0.9856(8) 9(10) × 10−4

2 0.9820(8) 0.9824(8) 4(12) × 10−4

TABLE II. Addressability error estimates for simultaneousRB.

𝛼𝛼1,1

𝛼𝛼1,2

𝛼𝛼1,3

𝛼𝛼2,1 𝛼𝛼2,2 𝛼𝛼2,3

FIG. 6. Correlation parameters for simultaneous two-qubitRB. Each square in the plot represents a value of δi,j withnumbers scaled by 10−4. Each parameter is within one stan-dard deviation of zero.

when applying gates to both zones, this will cause an er-ror in both zones. In Ref. [28], it was shown that thesetypes of errors could be identified by studying differentoutcomes in simultaneous RB experiments. They iden-tified the group structure of the two-subsystem Cliffordgroup C⊗21 and showed that it leads to three distinct de-cay rates α1, α2, and β (with ‘both’ superscript droppedsince we only consider simultaneous RB experiments forthis analysis). The first two decays correspond to errorson each qubit, but the third decay corresponds to corre-lated errors. If the simultaneous gates do not cause any

additional correlated errors, then α1α2 = β. However, ifadditional correlated errors exist, then α1α2 ≤ β. Themagnitude of the error is then defined as

δ = α1α2 − β. (4)

For parallel two-qubit gates, things are more compli-cated due to the group structure of C⊗22 . There are 15different decay rates, including six that correspond to theindividual two-qubit subsystems αz,i for i = 1,2,3, and 9that correspond to correlated errors βi,j for i, j = 1,2,3.Physically, these correspond to different types of corre-lated errors that affect the different combinations of thefour total qubits. To quantify the correlation errors wecalculate

δi,j = ∣α1,iα2,j − βi,j ∣. (5)

The results are summarized in Fig. 6, which plots δi,jfor all 9 values. The minimum standard deviationover all correlation parameters estimated from the semi-parametric bootstrap is 4.1×10−4. Therefore, we see thatall correlation parameters are within one standard devi-ation of zero.

Simultaneous two-qubit RB indicates that addressabil-ity and correlation errors are consistent with zero withhigh-confidence. However, this does not mean all cross-talk errors are zero in the system. Cross-talk is a broadterm and may refer to any context-dependence of the sys-tem when parallel gates are applied, which could includeerrors that are missed with these experiments. For exam-ple, there may be errors present that do not change theaverage fidelity of an RB experiment but still cause dif-ferent effects in an arbitrary quantum circuit. However,from this analysis, we see that the effect of cross-talk withrandom circuits is small and likely has a minimal effect intypical circuits. This conclusion is further verified withother example algorithms like QV.