Detecting bit-flip errors in a logical qubit using stabilizer measurements

D. Riste∗,1 S. Poletto∗,1 M.-Z. Huang∗,1, 2 A. Bruno,1 V. Vesterinen,1, † O.-P. Saira,1, ‡ and L. DiCarlo1

1QuTech and Kavli Institute of Nanoscience, Delft University of Technology,P.O. Box 5046, 2600 GA Delft, The Netherlands

2Huygens-Kamerlingh Onnes Laboratory, Leiden Institute of Physics,Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

(Dated: November 21, 2014)

Quantum data is susceptible to decoherence in-duced by the environment and to errors in thehardware processing it. A future fault-tolerantquantum computer will use quantum error cor-rection (QEC) to actively protect against both.In the smallest QEC codes [1–5], the informa-tion in one logical qubit is encoded in a two-dimensional subspace of a larger Hilbert spaceof multiple physical qubits. For each code,a set of non-demolition multi-qubit measure-ments, termed stabilizers, can discretize and sig-nal physical qubit errors without collapsing theencoded information. Experimental demonstra-tions of QEC to date, using nuclear magneticresonance [6], trapped ions [7, 8], photons [9],superconducting qubits [10], and NV centers indiamond [11, 12], have circumvented stabilizersat the cost of decoding at the end of a QECcycle. This decoding leaves the quantum infor-mation vulnerable to physical qubit errors un-til re-encoding, violating a basic requirement forfault tolerance. Using a five-qubit superconduct-ing processor, we realize the two parity measure-ments comprising the stabilizers of the three-qubit repetition code [13] protecting one logi-cal qubit from physical bit-flip errors. We con-struct these stabilizers as parallelized indirectmeasurements using ancillary qubits, and evi-dence their non-demolition character by generat-ing three-qubit entanglement from superpositionstates. We demonstrate stabilizer-based quan-tum error detection (QED) by subjecting a logicalqubit to coherent and incoherent bit-flip errors onits constituent physical qubits. While increasedphysical qubit coherence times and shorter QEDblocks are required to actively safeguard quan-tum information, this demonstration is a criticalstep toward larger codes based on multiple paritymeasurements.

A recent roadmap [14] for fault-tolerant quantum com-puting marks a transition from storing quantum data inphysical qubits to QEC-protected logical qubits as the

∗equal contribution.

fourth of seven development stages. Following steadyimprovements in qubit coherence, coherent control, andmeasurement over 15 years, superconducting quantumcircuits are well poised to face this outstanding challengecommon to all quantum computing platforms. Initial ex-periments using superconducting processors include oneround of either bit-flip or phase-flip QEC with decod-ing [10], and the stabilization of one Bell state using dis-sipation engineering [15]. Independent, parallel work [16]demonstrates the detection of general errors on a sin-gle Bell state using stabilizer measurements. We demon-strate stabilizer-based QED on the minimal unit of en-coded quantum information, a logical qubit, restrictingto bit-flip errors.

By analogy to the classical repetition code that mapsbit 0 (1) to 000 (111), the quantum version maps theone-qubit state α |0〉 + β |1〉 to the Greenberger-Horne-Zeilinger-type (GHZ) state α |0t0m0b〉 + β |1t1m1b〉 ofthree data qubits (labelled top, middle, and bottom) [13].The stabilizers of this code consist of two-qubit paritymeasurements described by Hermitian operators ZtZm

and ZmZb. While GHZ-type states are eigenstates ofboth stabilizers with eigenvalue +1, their corruption bya bit-flip error on one data qubit produces eigenstateswith a unique pattern of -1 eigenvalues. Measuring sta-bilizers can thus discretize and signal single bit-flip errorswithout affecting the encoded information (i.e., the prob-ability amplitudes α and β). Depending on the error sig-nalled, the logical qubit is transformed to an orthogonaltwo-dimensional subspace.

This realization of bit-flip QED with stabilizer mea-surements employs a superconducting quantum proces-sor with 12 quantum elements (Fig. 1a) exploiting reso-nant and dispersive regimes of circuit quantum electro-dynamics [17]. Three data transmon qubits (Dt, Dm andDb) encode the logical qubit. Two ancillary transmons(At and Ab), two bus resonators (Bt and Bb), and twodedicated ancilla readout resonators are used for the sta-bilizer measurements. Dedicated readout resonators ondata qubits are used to quantify performance (fidelitymeasures, entanglement witnessing, and state tomogra-phy). All readout resonators couple to one feedline usedfor all qubit control and readout pulses. The feedline out-put couples to a single amplification chain allowing read-out of all qubits by frequency-division multiplexing [18].Ancilla readout fidelity is boosted by a Josephson para-

arX

iv:1

411.

5542

v1 [

quan

t-ph

] 2

0 N

ov 2

014

2

Dt

Dm

Db

BtBb

Ab

Ata

1

6

2 3

45

7

1 mm

Dt

Db

Dm

ZtZm

Pt

Pb

e,o e,o

U

Encoding Errors

ZmZb

Detection

|ym⟩

|0⟩

|0⟩

At

AbBb

BtRj'

π/2

Rj”

π/2

e,oe,o

|0⟩

|0⟩

|0⟩

|0⟩

Rxπ/2

Rxπ

Rxπ/2

b c Detection

RxJ

RxJ

RxJ

FIG. 1: Quantum processor and gate sequence for im-plementing and characterizing bit-flip QED by sta-bilizer measurements. a, Photograph of the processorshowing the position and interconnections of data qubits (Dt,Dm, Db), ancilla qubits (At, Ab), buses (Bt, Bb), and ded-icated readout resonators. These resonators couple to onecommon feedline to which all readout and microwave controlpulses are applied [18]. Flux-bias lines (ports 2-5, 7) allowcontrol of the qubit transition frequencies on ns timescale(Extended Data Fig. 1). Details of the processor, includ-ing fabrication, parameters and performance benchmarks, areprovided in Methods and Extended Data Table 1. b, Blockdiagram for characterizing bit-flip QED by parallelized par-ity measurements of pairs (Dt, Dm) and (Dm, Db). The Dm

state |ψm〉 = α |0m〉 + β |1m〉 is first encoded into the logi-cal qubit state |ψL〉 = α |1t1m1b〉 + β |0t0m0b〉. Coherent orincoherent bit-flip errors are then introduced on data qubitswith independent single-bit-flip probability perr. ParallelizedZtZm and ZmZb stabilizer measurements discretize these er-rors, and the two-bit measurement result PtPb is interpretedas signalling either no error or error on one qubit. c, Gatesequence implementing the stabilizer measurements by par-allelized interaction with ancilla qubits and projective ancillameasurements. Each ancilla is prepared in a superpositionstate that is transferred to the respective bus with an iSWAPgate (diagonal lines). Consecutive CPHASE gates betweeneach bus and the coupled data qubits (vertical lines) encodethe data-qubit parity in the quantum phase of the bus super-position state. The final iSWAP transfers this state to theancilla, and the latter is then projectively measured in the|±〉 basis. Halfway through the interaction step, a refocusingπ pulse is applied to Dm to reduce inhomogeneous dephasing.

metric amplifier (JPA) [19] with bandwidth covering bothancilla readout frequencies (9 MHz apart).

Building on recent developments [20, 21], we constructquantum non-demolition stabilizer measurements in atwo-step process combining entanglement with ancillaqubits and their projective measurement. Measuring thestabilizer ZtZm involves an iSWAP gate between At andBt, two CPHASE gates between Bt and each of Dt and

a b

c

Top parity Bottom parity

eo eo

Cou

nt fr

actio

n (%

)A

ssig

nmen

t pro

babi

lity

Vt (mV) Vb (mV)

15

10

5

0-5 0 5 10 -5 0 5 10

000001010

101110111

011100

|DtDmDb⟩

|DtDmDb⟩

1.0

0.8

0.6

0.4

0.2

0.0000 001 010 011 100 101 110 111

P(ee)P(eo)P(oe)P(oo)

FIG. 2: Characterization of stabilizer measurements.Single-shot histograms for top (a) and bottom (b) ancillareadout signals Vt and Vb at the end of the sequence in Fig. 1c,with data-qubit computational states as input. The chosenthresholds for discretization of Vt and Vb (dashed verticallines) maximize the parity assignment fidelities. c, Double-parity assignment probabilities for each computational stateinput. The dashed horizontal line at 0.91 marks the loss ofaverage assignment fidelity exclusively from ancilla readouterrors.

Dm, and a final iSWAP transferring the Bt state ontoAt. These interactions correlate joint states of Dt andDm with even/odd (e/o) number of excitations with or-thogonal states of At. Subsequently, At is measured byinterrogating its dispersively coupled resonator. Conve-niently, the interaction and measurement steps neededfor both stabilizers can be partially parallelized (Fig. 1c).(Note that a refocusing π pulse is applied to Dm after itsinteractions to minimize its inhomogeneous dephasing.)

We begin characterizing these stabilizer measurementsby testing their ability to detect the parities of the com-putational states |itjmkb〉, i, j, k ∈ 0, 1. Because all ofthese states are eigenstates of ZtZm and ZmZb, a fixedtwo-bit measurement outcome PtPb ∈ ee, eo, oe, oo isexpected for each one. Histograms of declared doubleparities clearly reveal the correlation (Fig. 2). The aver-age assignment fidelity of 71%, defined as the probabil-ity of correct double-parity assignment averaged over theeight states, is limited by errors in the interaction step(separate calibrations of ancilla readout errors set a 91%upper bound).

The next test probes the ability of each stabilizer todiscern two-qubit parity subspaces while preserving co-

3

a b

c d

W(F+)W(F-)

W(Y+)W(Y-)

kj

data fit

0.50

0.25

0.00

-0.25

-0.50

Preparation angle j of Dm (deg)-180 -90 0 90 180 -180 -90 0 90 180

Top Bottom

WY

]

data fit

Preparation angle j of Dm (deg)-180 -90 0 90 180

MY

]

4

2

0

-2

-4

by msm’tunitary

0.5

0.25

111110101100011010001000

111

000001

010011

100101

110

|rj,k|

data fit

FIG. 3: Generation of two- and three-qubit entangle-ment by stabilizer measurements. Starting with the dataqubits in the state |+t〉

(|0m〉+ eiϕ |1m〉

)|+b〉 /

√2, we selec-

tively perform stabilizer measurements by activating the cor-responding ancilla (applying initial π/2 rotation in Fig. 1c).a, b, Performing one parity measurement generates entangle-ment between the paired data qubits. Measured average 〈W〉of the four witnesses operators W(Φ±) and W(Ψ±) involvingthe data qubits paired by activating the top (a) or bottom(b) ancilla only and postselection on Pt = o and Pb = o,respectively. Entanglement is witnessed whenever 〈W〉 < 0.The weak oscillations in 〈W(Ψ±)〉 result from false positives,which we have partially reduced here by postselecting morestrongly than the threshold maximizing the average parity as-signment fidelity. A dual witnessing by 〈W(Ψ±)〉 is observedby postselection on e. c, Measured average of the Merminoperator M with both ancillas activated and data stronglypostselected on PtPb = oo (black circles). Three-qubit en-tanglement is witnessed whenever |M| > 2. A stronger vi-olation of the Mermin inequality is observed when target-ing the GHZ state |GHZ〉 = (|0t0m0b〉+ |1t1m1b〉) /

√2 using

unitary gates only (white circles). d, Tomography (absolutevalue of the density matrix elements) of the |M|-maximizingstate generated by double-parity measurement. The fidelityF = 〈GHZ| ρ |GHZ〉 is 73%. For comparison, targeting thisstate with gates achieves F = 82%.

herence within each. Specifically, we target the gener-ation of two- and three-qubit entanglement (2QE and3QE) via single and double stabilizer measurements ona maximal superposition state. The gate sequence inFig. 1c is executed with Dt and Db both prepared in|+〉 = (|0〉 + |1〉)/

√2 and Dm in

(|0〉+ eiϕ |1〉

)/√

2.First, we activate one stabilizer by performing the

initial π/2 rotation only on the corresponding an-cilla, and measure the data-qubit-pair witness opera-tors W(Φ±) = (II ∓XX ± Y Y − ZZ) /4, W(Ψ±) =(II ∓XX ∓ Y Y + ZZ) /4 [22] based on fidelity to even-and odd-parity Bell states, respectively. Each of theseoperators witnesses 2QE whenever the expectation value〈W〉 < 0. With postselection on result o, 〈W(Φ+)〉 and〈W(Φ−)〉 jointly witness 2QE at almost all values of ϕ(Figs. 3a and 3b).

We continue building multi-qubit entanglement by ac-tivating both parity measurements and postselecting onthe two-bit result (Figs. 3c, 3d, and Extended DataFig. 2). Ideally, PtPb = oo collapses the maximalsuperposition onto the GHZ-type state |GHZ(ϕ)〉 =(|0t0m0b〉+ e−iϕ |1t1m1b〉

)/√

2. Genuine 3QE is wit-nessed whenever |〈M〉| > 2, where M is the Merminoperator XtXmXb − YtYmXb − YtXmYb −XtYmYb [23].With postselection on PtPb = oo, 〈M〉 versus ϕ reaches2.5 (best fit, Fig. 3c). Full state tomography at the op-timal ϕ reveals a fidelity 〈GHZ(0)| ρ |GHZ(0)〉 = 73% tothe ideal GHZ state (Fig. 3d).

This 3QE-by-measurement protocol can also beused to perform the encoding step of bit-flip QEC.Ideally, the state |+t〉 (α |0m〉+ β |1m〉) |+b〉 is mappedonto α |1t1m1b〉 + β |0t0m0b〉 up to the transforma-tion XtXb, Xt, Xb, I signalled by PtPb = ee, eo,oe, oo, respectively. Postselection on PtPb = oo(Extended Data Fig. 3) encodes with 73% fidelity, aver-aged over the six cardinal input states of Dm,

∣∣ψjm

⟩∈

|0〉 , |1〉 , |±〉 = (|0〉 ± |1〉) /√

2, |±i〉 = (|0〉 ± i |1〉) /√

2

).For comparison, implementing the standard unitaryencoding [10, 24, 25] using our gate toolbox (ExtendedData Fig. 4) achieves 82% average fidelity.

Finally, we use this encoding by gates to demon-strate bit-flip QED by parallelized stabilizer measure-ments (Fig. 4a). Bit-flip errors are coherently added viaX rotations by an angle θ, yielding a single-qubit bit-flipprobability perr = sin2 (θ/2) (adding incoherent errors atthis stage yields very similar results, see Methods andExtended Data Fig. 5). We consider two scenarios: er-rors added on only one data qubit (1), and errors addedon all three (3). We first quantify QED performanceusing the average fidelity F3Q to the ideal three-qubit

state accounting for the subspace transformation Cpq =Xm, XmXb, XtXm, I signalled by PtPb = ee, eo, oe, oo (inorder):

F3Q =1

6

∑j

∑pq

ppq

⟨ψjL

∣∣∣ Cpqρ(j, pq)C†pq

∣∣∣ψjL

⟩(QED).

Here,∣∣∣ψj

L

⟩is the ideal encoded cardinal state, ppq is the

measured probability of PtPb = pq, and ρ(j, pq) is theexperimental pq-conditioned density matrix. The nearconstancy of F3Q(perr) in scenario (1) and the second-order dependence in (3) (Fig. 4b) reflect the ability of

4

Dt

Db

Dm |ym⟩

|0⟩

|0⟩

Rxπ

aEncoding

RxJ

RxJ

Errors QED/Idle Errors Ideal decoder

F3Q

FL

F3Q

EncodingEncoding

Fide

lity

1.0

0.8

0.6

0.4

0.2

0.01.00.80.60.40.20.01.00.80.60.40.20.0

b c

Single bit-flip error probability, perr

no QEDScenario: (1) (3)

QED

no QED, (3)QED, (3)

Logi

cal f

idel

ity F

L

1.0

0.8

0.6

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dno QED, (3)QED, (3)

Error combination0/0 0/1 1/0 1/1a 1/1b

RxJ

RxJ

RxJ

RxJ Rx

π

FL

FIG. 4: Detection of bit-flip errors. a, Sequence usedto assess performance of bit-flip QED. After encoding bygates, either coherent (θ ∈ [0, π]) or incoherent (θ = 0 orπ) errors are introduced with single-qubit bit-flip probabil-ity perr. Next, parallelized stabilizer measurements are eitherperformed or replaced by an equivalent idling period. Partialtomography at this point is used to obtain the three-qubitfidelity F3Q and the logical fidelity FL. The calculation of FL

assumes incoherent second-round errors with the same perrand a perfect decoding (dashed boxes). b, Three-qubit fi-delity F3Q as a function of perr with and without QED undertwo scenarios: coherent errors applied on Dm (1) and on alldata qubits (3). The dashed line indicates the fidelity ceil-ing imposed by encoding errors. c, FL as function of perr,obtained from the same data as in b. The individual con-tributions of the six cardinal states

∣∣ψjm

⟩to F3Q and FL are

shown in Extended Data Fig. 7. d, FL for all combinationsof one and zero incoherent errors before and after QED oridling. Error combinations are labelled m/n, with m (n) thenumber of errors before (after) QED or idling. The case 1/1is divided in two: errors on the same data qubit (1/1a) or ondifferent qubits (1/1b).

the stabilizers to discretize and signal single-qubit bit-flip errors without decoding.

To assess the ability of QED to detect added errorswithout penalizing for intrinsic decoherence and encodingerrors, we compare to F3Q with the stabilizer interactionsreplaced by idling for equal duration (with a refocusing

Dm pulse):

F3Q =1

6

∑j

⟨ψjL

∣∣∣Xmρ(j)Xm

∣∣∣ψjL

⟩(no QED).

Without QED, one expects a linear decrease in F3Q in (1)as one bit flip orthogonally transforms the encoded state.The slight curvature observed reflects residual coherenterrors in encoding. The non-monotonicity of F3Q in (3)reflects that triple errors perform a logical bit flip, whichleaves |+L〉 and |−L〉 unchanged. Comparing curves sug-gests that QED provides net gains for perr & 15% in (1)and for perr & 10% in (3) (Fig. 4b).

However, the true merit of QED hinges on the abilityto suppress the accumulation of errors. We believe that abetter comparison is the logical state fidelity FL followingtwo rounds of errors with QED or idling in between. FL

is defined as the fidelity to the initial unencoded Dm statefollowing an ideal decoder D (Fig. 4a) that is resilient toa bit-flip error remaining in any one qubit. For example,with QED and a second-round error E,

FL =1

6

∑jpq

ppq⟨ψjm

∣∣Trt,b

[DECpqρ(j, pq)C†pqE

†D†]∣∣ψj

m

⟩.

Here we consider scenario (3) and only incoherent second-round errors. We expect QED to win over idling in selectcases, such as single errors on both rounds but on differ-ent qubits, all of which we observe (Fig. 4d and also Ex-tended Data Fig. 6). Weighing in all possible cases (from0 to 3 errors in each round) according to their proba-bility, we find that the current fidelity of the stabilizermeasurements precludes boosting FL by QED at any perr(Fig. 4c). This stricter comparison sets the benchmarkfor gauging future improvements in QED.

In summary, we have realized parallel stabilizer mea-surements with ancillary qubits and used them toperform bit-flip QED in a superconducting circuit.Stabilizer-based QED can detect bit-flip errors on dataqubits while maintaining the encoding at the logical level,thus meeting a necessary condition for fault-tolerantquantum computing. Evidently, it remains a priority toextend qubit coherence times and shorten the QED stepin order to boost logical fidelity by QED. Future workwill also target the completion of several QEC cycles,using digital feedback control [26] to correct inferred er-rors or adapting logical operations in accordance to thesubspace transformations signalled by the stabilizer mea-surements. In the longer term, parallelized ancilla-basedparity measurements as demonstrated here may be usedto protect a logical qubit against general errors with aSteane [5, 27] or small surface code [28].

5

METHODS

Processor fabrication. The integrated circuit is fab-ricated on a c-plane sapphire substrate. A NbTiN film(80 nm) is reactively sputtered at 3 mTorr in a 5% N2

in Ar atmosphere, resulting in a superconducting crit-ical temperature of 15.5 K and normal-state resistiv-ity of 110 µΩcm. This film is e-beam patterned usingSAL601 resist and etched by SF6/O2 RIE to define allcoplanar waveguide structures: feedline, resonators, andflux-bias lines. The transmon Josephson junctions andshunting interdigitated capacitors are patterned usingPMGI/PMMA e-beam lithographed resist and double-angle shadow evaporation of Al with intermediate ox-idization. Air bridges are added to suppress slot-linepropagation modes, to connect ground planes, and to al-low the crossing of transmission lines (Extended DataFig. 8). Bridge fabrication starts with a 6 µm thickPMGI layer which is patterned and then reflowed at220C for 5 min, producing a gently arched profile. Asecond MAA/PMMA resist layer is spun and e-beam pat-terned to define the bridge geometry. Finally, Ti (5 nm)and Al (450 nm) are e-beam evaporated. The 2 mm by7 mm chip is diced and cleaned in 88C NMP for 30 min.

Experimental setup. The quantum processor is an-chored to the mixing chamber plate of a dilution refrig-erator with 15 − 20 mK base temperature. A detailedschematic of the experimental setup at all temperaturestages is shown in Extended Data Fig. 8. The singlecoaxial line for readout and microwave control has in-line attenuators and absorptive low-pass filters providingthermalization, noise reduction, and infrared radiationshielding. Coaxial lines for flux control are broadbandattenuated and bandwidth limited (1 GHz) with reactiveand absorptive low-pass filters.

Qubit control. Most microwave pulses for X andY qubit rotations have a Gaussian envelope in the mainquadrature (5 ns sigma and 20 ns total duration), anda derivative-of-Gaussian envelope in the other (DRAGpulses [29]). Wah-Wah pulses [30] combining DRAG withsideband modulation are used for Dt and Ab to avoidleakage in Dm and Db, respectively. Taking advantageof the proximity in frequency between Dt and At, andbetween Dm and Ab, we coherently control the five qubitsby sideband modulation of three carriers (Extended DataFig. 8).

Flux pulses for iSWAPs are sudden (12 ns duration),while those for CPHASEs are mostly fast adiabatic [31](40 ns). The pulse for CPHASE between Dm and Bt iskept sudden (19 ns) to avoid leakage during the crossingof Dm through Bb. Pulse distortion resulting from theflux control bandwidth is minimized by manual optimiza-tion of convolution kernels.

Qubit readout. The five qubits are readout by fre-quency division multiplexing [18]. The readout pulses

for data and ancilla qubits are separately generated bysideband modulation of two carriers.

The amplitude and duration of readout pulses are cho-sen to maximize assignment fidelity. Dt, Dm, and Db

readout pulses have 1200, 1000, and 700 ns duration,respectively. The signal-to-noise boost provided by theJPA allows shorter ancilla qubit readouts, 600 ns (550 ns)for At (Ab). The amplified feedline output is split anddown-converted with two local oscillators. The two sig-nals are amplified, digitized, demodulated, and inte-grated to yield one voltage for each qubit measured. Thelow crosstalk between the qubit readouts is evidenced bysimultaneous measurement immediately following prepa-ration of the 32 combinations of the five qubits in either|0〉 or |1〉 (Extended Data Fig. 9).

Using the method of Ref. 20 based on Hahn echo se-quences, we have bound the dephasing of each data qubitinduced by the ancilla measurements to less than 1%(data not shown). Since data-qubit fidelity loss duringancilla measurements is dominated by intrinsic decoher-ence and our main interest is to quantify the ability of sta-bilizers to detect the intentionally added errors, we haveopted to advance the data qubit measurements, makingthem simultaneous to those of ancillas (Extended DataFig. 4).

Initialization. The four qubits Dt, Db, At, Ab andtwo buses Bt, Bb are initialized to their ground stateby postselection on six measurements performed beforeany encoding or manipulation protocol. The buses areinitialized by swapping states with their coupled ancillaimmediately after initialization of the latter. Dm is ini-tialized by swapping its excitation (∼ 10%) with that ofBb (∼ 1%). The postselected fraction of runs (50−60%)have a residual excitation of 1 − 2% in every quantumelement.

Gate sequence. Gates are parallelized as much aspossible. We note two important exceptions. Becauseof frequency crowding and the common feedline, pulsestargeting one qubit induce ac Stark shifts on untargetedqubits. We serialize single-qubit control to restrict theeffect of these shifts to residual phase rotations on unad-dressed qubits. Also, the first iSWAP between Bt and At

and CPHASE between Bt and Dm (Fig. 1c) are appliedbefore populating Bb to avoid a strong dispersive shiftof Dm. All others iSWAPS, CPHASE gates and ancillameasurements are simultaneous.

Incoherent errors. We have also tested stabilizer-based QED with incoherent first-round errors generatedusing π rotations (Extended Data Fig. 5). Following en-coding of a Dm cardinal input state

∣∣ψjm

⟩, we apply the

eight combinations of error/no error on the three dataqubits. We calculate F3Q and FL for each combinationand weigh by the corresponding probability.

6

ACKNOWLEDGMENTS

We thank K.W. Lehnert for providing the parametricamplifier, D.J. Thoen and T.M. Klapwijk for sputteringof NbTiN films, K.M. Svore, T.H. Taminiau, D.P. DiVin-cenzo, E. Magesan, and J.M. Gambetta for fruitful dis-cussions, and N.K. Langford, G. de Lange, L.M.K. Van-dersypen, and R. Hanson for helpful comments on themanuscript. We acknowledge funding from the Nether-lands Organization for Scientific Research (NWO), theDutch Organization for Fundamental Research on Mat-ter (FOM), and the EU FP7 project ScaleQIT.

AUTHOR CONTRIBUTIONS

A.B. fabricated the processor, with design input fromO.-P.S. and L.D.C. O.-P.S. and V.V. performed the ini-tial tune-up. D.R., M.-Z.H., and S.P. performed mea-surements and data analysis. S.P., D.R. and L.D.C. pre-pared the manuscript with feedback from all other au-thors. L.D.C. supervised the project.Correspondence and requests for materials should be ad-dressed to L.D.C. (l.dicarlo@tudelft.nl).

† Present address: VTT Technical Research Centre of Fin-land, P.O. Box 1000, 02044 VTT, Finland.

‡ Present address: Low Temperature Laboratory (OVLL),Aalto University, P.O. Box 15100, FI-00076 Aalto, Fin-land.

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[15] Shankar, S. et al. Autonomously stabilized entanglementbetween two superconducting quantum bits. Nature 504,419–22 (2013).

[16] Corcoles, A. D. et al. Detecting arbitrary quantum errorsvia stabilizer measurements on a sublattice of the surfacecode. arXiv:quant-ph/1410.6419 (2014).

[17] Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. &Schoelkopf, R. J. Cavity quantum electrodynamics forsuperconducting electrical circuits: An architecture forquantum computation. Phys. Rev. A 69, 062320 (2004).

[18] Jerger, M. et al. Frequency division multiplexing readoutand simultaneous manipulation of an array of flux qubits.Appl. Phys. Lett. 101, 042604 (2012).

[19] Castellanos-Beltran, M. A., Irwin, K. D., Hilton, G. C.,Vale, L. R. & Lehnert, K. W. Amplification and squeez-ing of quantum noise with a tunable Josephson metama-terial. Nature Phys. 4, 929 (2008).

[20] Saira, O.-P. et al. Entanglement genesis by ancilla-basedparity measurement in 2D circuit QED. Phys. Rev. Lett.112, 070502 (2014).

[21] Chow, J. M. et al. Implementing a strand of a scalablefault-tolerant quantum computing fabric. Nature Comm.5, 4015 (2014).

[22] Horodecki, R., Horodecki, P., Horodecki, M. &Horodecki, K. Quantum entanglement. Rev. Mod. Phys.81, 865 (2009).

[23] Mermin, N. D. Extreme quantum entanglement in a su-perposition of macroscopically distinct states. Phys. Rev.Lett. 65, 1838 (1990).

[24] DiCarlo, L. et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature467, 574 (2010).

[25] Neeley, M. et al. Generation of three-qubit entangledstates using superconducting phase qubits. Nature 467,570 (2010).

[26] Riste, D., Bultink, C. C., Lehnert, K. W. & DiCarlo, L.Feedback control of a solid-state qubit using high-fidelityprojective measurement. Phys. Rev. Lett. 109, 240502(2012).

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7

max f01, GHzoperation point f01, GHzT1, µsT2, µs*

T2 , µsecho

g/2π to Bt, MHzg/2π to Bb, MHzreadout resonator fr, GHz

χ/π, MHzκ/π, MHzaverage assignment fidelity

5.755 6.181 6.788 6.002 6.452 4.80 5.525.755 6.065 6.748 5.985 6.452 4.80 5.52

7 13 6 9 10 7 63 2 4 3 0.77 13 5 4 378 48 - 51 -

1.7 2.1 1.5 0.9 0.989% 82% 95% 94% 95%

13 11

Dt Dm Db At Ab Bt Bb

- 57 58 - 48

-0.6 -0.3 -1.0 -1.6 -2.07.599 7.787 7.998 7.095 7.086

Extended Data Table 1: Summary of the main device parameters.

a

b

c

100-001 100-010

110-020110-200110-011

101-011101-002101-200

Dm flux-pulse amplitude (V)

Dm fl

ux-p

ulse

leng

th (n

s)

0.0 0.5 1.5 1.5 2.0 2.5 3.0

0

10

20

30

400

10

20

30

400

10

20

30

40

Extended Data Figure 1: Vacuum Rabi oscillations between Dm and the buses. Coherent oscillations between Dm

(initially in |1〉) and both buses, as a function of flux pulse amplitude and duration. Buses are prepared in |BtBb〉 = |00〉 (a),|BtBb〉 = |10〉 (b), and |BtBb〉 = |01〉 (c). Labels indicate the corresponding transition with notation |DmBtBb〉.

8

0.5

0.25

0

ano conditioning

e0.5

0.25

0

d0.5

0.25

0

c0.5

0.25

0

b0.5

0.25

0

kj

|rj,k|

111

000001

010011

100101

110

111110101100011010001000

oe oo

eoee

Extended Data Figure 2: Three-qubit entanglement by parallelized stabilizer measurements on a maximal su-perposition state. Density-matrix elements (absolute values) of the states obtained by postselection on different stabi-lizer measurement results: (a) No postselection; (b) PtPb = ee, fidelity 〈GHZ|XbXtρXbXt |GHZ〉 = 67%; (c) PtPb = eo,〈GHZ|XtρXt |GHZ〉 = 67%; (d) PtPb = oe, 〈GHZ|XbρXb |GHZ〉 = 65%; (e) PtPb = oo, 〈GHZ| ρ |GHZ〉 = 68%. Note that theparities of the final state differ from the detected ones due to the refocusing π pulse on Dm. In contrast to Fig. 3, conditioninghere is performed using the Vt (Vb) threshold maximizing the top (bottom) parity assignment fidelity.

9

a

b1

0.5

-0.5

1

0.5

-0.5

c1

0.5

-0.5

1

0.5

-0.5

0

1

0.5

-0.5

0

1

0.5

-0.5

0

ReIrj,kM ImIrj,kM

kj

111110

101100

011010

001000

111

000001010011100101110

|0⟩

|1⟩

|0⟩+Â|1⟩◊2

111110

101100

011010

001000

111

000001010011100101110

111110

101100

011010

001000

111

000001010011100101110

Extended Data Figure 3: Encoding by measurement. Density-matrix elements (real and imaginary parts) of the stateobtained by stabilizer measurements on the state |+t〉 |ψm〉 |+b〉 and with strong postselection on PtPb = oo (as in Fig. 3), with|ψm〉 = |0〉 (a), |ψm〉 = (|0〉 + i |1〉)/

√2 (b); |ψm〉 = |1〉 (c). Due to the refocusing pulse on Dm, the state |0〉 (|1〉) is encoded

in |1t1m1b〉 (|0t0m0b〉).

At

AbBb

Bt |0⟩|0⟩

|0⟩

|0⟩

320 ns

Dt

Db

Dm

|0⟩

|0⟩

|0⟩

Encoding DetectionTomographicprerotations Measurements

600 ns

550 ns

1200 ns

1000 ns

700 ns

e,o

e,o

350 nsErrors

RxJ

RxJ

RxJ

RXπ/2

RXπ/2

Rxπ

Rj'

π/2

Rj''

π/2

Rf'

π/2

Rf''

π/2R-Y

π/2

R-Yπ/2

Prep Rxπ

Tomo

Tomo

Tomo

90 ns 90 ns

Extended Data Figure 4: Quantum circuit for QED characterization. The quantum circuit for QED characterizationhas six steps: initialization (not shown), encoding, addition of bit-flip errors, detection, tomographic pre-rotation pulses, andmeasurements.

10

perr

1.00.80.60.40.20.0perr

1.00.80.60.40.20.0

a b

Thre

e-qu

bit f

idel

ity F

3Q

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

Logi

cal f

idel

ity F

L

coherent incoherent

First-round errorscoherent incoherent

First-round errors

QED, scenario (3) QED, scenario (3)

Extended Data Figure 5: Comparison between coherent and incoherent added errors. Comparison of fidelities F3Q

(a) and FL (b) for coherent (circles, same data as in Fig. 4b) and incoherent (triangles) errors applied on the first round andfor scenario (3) with QED. As expected, the curves closely overlap.

Logi

cal f

idel

ity F

L

1.0

0.8

0.6

0.4

0.2

0.0

no QED, scenario (3)QED, scenario (3)

Error combination0/0 0/1 1/0 1/1a 3/1 2/3 3/2 3/31/1b 0/2 2/0 1/2a 1/2b 2/1a 2/1b 2/2a 2/2b 0/3 3/0 1/3

Extended Data Figure 6: Comparison of logical fidelities FL for all combinations of first- and second-round errorswith and without QED. Same notation for error combinations as in Fig. 4d. Labels 1/1a and 2/2a indicate first- andsecond-round errors on the same qubits. Labels 1/2a and 2/1a indicate that one qubit undergoes errors in both rounds. Greenregions indicate the error combinations for which QED is expected to win over idling. Grey regions indicate the opposite. Forall other combinations, QED and idling would ideally tie.

11

a b

c d

perr

1.00.60.40.20.0perr

1.00.80.60.40.20.0

QED,scenario (1)

QED,scenario (3)

no QED,scenario (1)

no QED,scenario (3)

Thre

e-qu

bit f

idel

ity F

3Q

1.0

0.8

0.6

0.4

0.2

0.01.0

0.8

0.6

0.4

0.2

0.0

|+⟩|-⟩|+Â⟩|-Â⟩|0⟩|1⟩

0.8perr

1.00.80.60.40.20.0

e

f

Logi

cal f

idel

ity F

L no QED,scenario (3)

QED,scenario (3)

1.0

0.8

0.6

0.4

0.2

0.01.0

0.8

0.6

0.4

0.2

0.0

Extended Data Figure 7: Three-qubit and logical state fidelities for the six cardinal input states of Dm undercoherent bit-flip errors. a,b, F3Q for scenario (1) without and with QED, respectively. c,d, F3Q for scenario (3) withoutand with QED, respectively. e,f, FL for scenario (3) without and with QED, respectively.

12

MITEQ AFS335-ULN, +30 dB

MITEQ AFS3 10-ULN, +30 dB

300 K

I Q

3 K

20 mK

M-CZFBT-6GW

SRSSR445A

20 dB 20 dB

10 dB

3 dB

I Q

AgilentE8257D

Flux bias Qubit drives Readouttones

JPApump

Dataacquisition

1 2S

1 2S

DC block DC block

monitoring

monitoring

Alazar ATS9870

M-CBLP-50

1 2S

M-CVLFX-1050

Homemadeeccosorb filters

3

7

456

1 2

20 dB

10 dB

LNF LNC4_8A +40 dB

TektronixAWG5014

JPA

TektronixAWG5014

TektronixAWG5014

I Q

2 3S

1

isolatorscirculator

3 dB

Homemadecurrent

sources

2S

1

TektronixAWG5014

I Q I Q

32S

1

Dt At DmAbDb RAt RAbRDt RDbRDm

20 dB

Homemade eccosorb filtersupercond. coax

Extended Data Figure 8: Experimental setup and device details. Complete wiring of electronic components outsideand inside the 3He/4He dilution refrigerator (Leiden Cryogenics CF-450). Inset: False-color scanning electron micrographshowing processor details. Coplanar waveguide structures (resonators, feedline, and flux bias lines) are patterned on a NbTiNthin film (gold) on sapphire (gray). Al/Ti air bridges (blue) allow cross-overs between coplanar waveguide transmission lines,interconnections of ground planes, and suppression of slot-line mode propagation.

Nor

mal

ized

read

out s

igna

l 1

-11

-1

Dt AtDm

DbAb

|AtAbDtDmDb⟩ (decimal)|0⟩ |8⟩ |16⟩ |24⟩

a

b

Extended Data Figure 9: Low-crosstalk simultaneous qubit readouts. Averaged and normalized readouts of the data (a)and ancilla (b) qubits immediately after preparing the five qubits in the 32 combinations of |0〉 and |1〉.