High-Fidelity Magic-State Preparation with a Biased-Noise Architecture

Shraddha Singh,1, 2 Andrew S. Darmawan,3, 4 Benjamin J. Brown,5 and Shruti Puri1, 2

1Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA2Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA

3Yukawa Institute of Theoretical Physics (YITP), Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

4JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan5Centre for Engineered Quantum Systems, School of Physics,

University of Sydney, Sydney, New South Wales 2006, Australia(Dated: September 8, 2021)

Magic state distillation is a resource intensive subroutine that consumes noisy input states toproduce high-fidelity resource states that are used to perform logical operations in practical quantum-computing architectures. The resource cost of magic state distillation can be reduced by improvingthe fidelity of the raw input states. To this end, we propose an initialization protocol that offers aquadratic improvement in the error rate of the input magic states in architectures with biased noise.This is achieved by preparing an error-detecting code which detects the dominant errors that occurduring state preparation. We obtain this advantage by exploiting the native gate operations of anunderlying qubit architecture that experiences biases in its noise profile. We perform simulationsto analyze the performance of our protocol with the XZZX surface code. Even at modest physicalparameters with a two-qubit gate error rate of 0.7% and total probability of dominant errors in thegate O(103) larger compared to that of non-dominant errors, we find that our preparation schemedelivers magic states with logical error rate O(10−8) after a single round of the standard 15-to-1distillation protocol; two orders of magnitude lower than using conventional state preparation. Ourapproach therefore promises considerable savings in overheads with near-term technology.

I. INTRODUCTION

The significant resource cost of implementing fault-tolerant logical gates is a major challenge for scalablequantum computation with near-term quantum hard-ware [1–7]. A number of recent studies have shown thatthe structure of noise in the underlying qubit architecturecan be leveraged to improve the performance of quantumerror correction [8–15]. These studies motivate the designof new noise-aware protocols for resource-efficient logicaloperations for fault-tolerant quantum computation.

The planar layout of the surface-code (SC) quantumcomputing architecture [1, 16–18] makes it particularlyappealing for experimental implementation. A practicalway of realizing a non-Clifford gate with the SC is byteleportation where a high-fidelity resource state, calleda magic state, is used by the circuit [19]. High-qualityresource states can be prepared with magic state distilla-tion (MSD) [3, 19–31] where several copies of noisy magicstates are consumed to produce a smaller number of copieswith lower logical error rates. MSD is expected to occupya large fraction of the resources of a SC architecture andit therefore presents a bottleneck in realizing quantumalgorithms [22].

In this work we present a new protocol for preparinghigher-fidelity input states for MSD protocols that istailored for qubit architectures that experience biased-noise such that errors that cause bit-flips are far less likelythan those that lead to phase-flips. In our protocol we usea physical two-qubit diagonal non-Clifford gate to preparea magic state encoded in a two-qubit code capable of

detecting a single dominant error. Therefore, the infidelityof the post-selected states that herald no error scalesquadratically with the physical error probability when thebias is strong and physical error rates are modest. This isa quadratic reduction in the infidelity compared with moreconventional approaches for state preparation [1, 32–35].Detecting more high probability errors results in morestates being discarded, but importantly this only resultsin a minute decrease in the success probability comparedto other approaches based on post-selection [34].

This work follows a bottom-up approach for the designof fault-tolerant protocols. For example, our scheme uti-lizes a recently discovered, bias-preserving controlled-not(CX) gate [36] for detecting errors without affecting thenoise bias of the system. This bias-preserving gate alsoenables us to encode the post-selected state into a high-distance error correcting code required for robust quantumcomputing while maintaining the quadratic improvement.Unlike the CX, single- and two-qubit diagonal gates aretrivially biased [37]. Moreover, in the biased-noise super-conducting Kerr-cat architecture, the two-qubit diagonalgates can be implemented with simple interactions andcan in principle be much faster and higher fidelity thansingle qubit diagonal gates [11, 36, 38]. Consequently,we leverage two-qubit diagonal non-Clifford gates in thisproposal. While, in practice the dominant source of noiseis independent perturbations on physical qubits, theseindependent errors can get correlated due to the actionof the gate. For example, in the bias-preserving CX gatea phase-flip error in the target qubit during the gatepropagates to the control qubit giving rise to correlated

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phase noise [11, 36]. In contrast, the diagonal gates aretransparent to phase-errors in the qubits. Thus, the high-rate independent phase-flip events do not get correlated.Highly precise microwave control in superconducting qubitplatform also ensures that correlated errors due to controlnoise are rare events. The naturally low probability of cor-related errors on diagonal gates ensures that high-fidelitypreparation of magic states in our protocol is possible.

We incorporate our initialization protocol into aquantum-computing architecture based on the XZZXcode [10, 11]; a surface code that is tailored to correctbiased noise. With this setup we find improvements inthe fidelity of the injected magic state, leading to moreeffective MSD. For example, even with a modest CX gateinfidelity of ∼ 0.7%, and average bias O(103), we find thata raw XZZX magic state of size 5× 25 (equivalent to 441data and ancilla qubits) can be prepared, with ∼ 94% suc-cess rate, at an error rate of ∼ 0.1%. The average bias isdefined as the total probability of phase-flip errors relativeto that of other errors in the gate. After consuming theseraw states in one round of 15-to-1 distillation protocol [19],a single copy of magic state can be produced at an errorrate of O(10−8). This error rate is, for example, sufficientfor realizing quantum simulations with quantum advan-tage without further rounds of distillation [39–41]. On theother hand, the error rate after one round of distillationwith raw magic states prepared using the standard schemeis two orders of magnitude larger. These numerical resultscorrespond to the case when noise in the CX gates is anorder of magnitude larger than other operations in thesyndrome extraction circuit, as is typically the case withbiased-noise cat qubits [11]. When the CX gates are asnoisy as other components in the circuit, the protocolproposed here gives a greater advantage over the standardapproach. Other approaches have been studied for imple-menting non-Clifford gates with codes tailored to biasednoise. In [42] for example, a magic state is initialized inthe repetition code with success rate that decreases expo-nentially with the code size even in the absence of errors.This is in contrast to our proposal which prepares themagic state deterministically in the absence of errors andheralding errors only costs a small decrease in the successrate. Moreover, our scheme only requires two-qubit gateswhich are experimentally easy to realize and is effectiveeven with modest amounts of bias achievable in near-termexperiments. Proposals in Refs. [12, 15] on the otherhand use three-qubit entangling gates.

This paper is structured as follows. Sections II and IIIdescribe our protocol and the effect of noise, respectively.Results from simulations are presented in Section IV.We offer concluding remarks in Section V. Appendicesprovide some supporting material and describes possibleimprovements to our protocol with practical three-qubitdiagonal non-Clifford gates.

(a) (b)

FIG. 1. (a) Illustration of the rectangular XZZX code withdata qubits on the vertices of a rotated grid. The stabilizersare the product of two Pauli X and two Pauli Z operators onqubits arranged on the vertices around each face. The distanceto X and Z errors is dx and dz respectively. The logical qubitPauli XL(ZL) are the product of Pauli X(Z) on the qubitsalong the blue and red edges respectively. The order in whichqubits are coupled to the ancilla at the center of each face(not shown) is indicated by the red arrow. (b) Circuit forstabilizer measurements. The ancilla is prepared in state |+〉,then coupled to data qubits with CX and CZ gates and finallyread out in the X basis.

II. THE PROTOCOL

We demonstrate our protocol with the XZZX code [10]defined on a rectangular lattice of size dx × dz shownin Fig 1(a). Data qubits are placed on the vertices ofthe lattice, and dx and dz respectively denote the codedistance with respect to pure X and Z errors. The sta-bilizers of the code are of the form X ⊗ Z ⊗ Z ⊗X onthe qubits around each face, as shown in Fig 1(a). Thelogical operator XL is the product of Pauli X operatorsof the qubits along a vertical edge and ZL is the prod-uct of Pauli Z operators of the qubits along a horizontaledge. The stabilizer measurement circuit is illustrated inFigure 1(b). An ancilla qubit, placed at the center ofeach face, is initialized in |+〉. Next, a sequence of CXand CZ gates is applied in the order shown in Fig. 1(a),and finally the ancilla is measured in the X basis.

The injection protocol proceeds in two stages similarto that presented in [34]. In stage I, a small XZZX codeof size dx,1 × dz,1 is prepared in the magic state. Someerrors are detected, but not corrected, at this stage. Stateswhere no errors are detected proceed to stage II wherethe code is grown to a larger distance; dx,2 × dz,2. Ourprotocol goes beyond the preparation protocol in [34] inthat, as an intermediate step in stage I, we prepare a two-qubit error detecting code that detects a single dominanterror acting on the raw magic state before it is injectedinto the stage I code. This gives a quadratic improvementto fidelity of the input state. The detailed steps in ourprotocol are given below.

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I II

FIG. 2. Illustration of the protocol for magic state preparation.In stage I the qubits in region I are initialized as shown, aZZ(θ) gate is applied to the two grey qubits, and the stabilizersare measured twice. The faces shaded in grey mark the fixedstabilizers for stage I. After stage I is successful and a dx,1×dz,1magic state is prepared, qubits in region II are initialized asshown. Stage II is then implemented and the dx,1 × dz,1 stateis grown to a dx,2 × dz,2 state, where stabilizers are measuredfor dm = dz,2 rounds.

Stage I

Stage I proceeds over three separate steps.

• Step 1: Physical qubits in region I are initialized asshown in Fig 2. The qubits marked in green and blueare initialized in state |0〉 and |+〉 respectively. Thetwo qubits on the top left corner, marked in grey, areinitialized in |+〉. In the following, the stabilizerson the faces shaded in grey will be referred to asfixed stabilizers.

• Step 2: A two-qubit ZZ(θ) = e−iθZ⊗Z gate is ap-plied on the two qubits at the top left which arehighlighted in grey in Fig 2.

• Step 3: All the stabilizers are measured twice andstabilizer measurement outcomes or syndromes arerecorded. If the outcome of measuring any fixedstabilizer is −1 or if the measurement outcomesfrom the two rounds are not identical, then an errorhas been detected. In this case the state is discardedand stage I is restarted. Otherwise, the code is sentto stage II.

Let us give some motivation for these steps. In the absenceof errors, the initial product state in step 1 is the +1

eigenstate of the fixed stabilizers.In step 2, the ZZ(θ) gate entangles the two grey qubits,

while the rest of the qubits remain un-entangled. For ageneral angle θ, which is not an integral multiple of π/4,this is a non-Clifford gate. We can think of the grey qubitsas forming a two-qubit repetition code with Z ′L = Z ⊗ Zand X ′L = X⊗I. In this picture, the effect of the physical

ZZ(θ) gate is to non-transversally apply a logical e−iθZ′L

gate to the two-qubit repetition code. After this step, thestate of the physical qubits on the XL and ZL edge is the+1 eigenstate of cos(2θ)XL + sin(2θ)YL. Observe that inthe absence of errors, the physical qubits remain in the+1 eigenstate of the fixed stabilizers.

The first measurement round of step 3 projects the sys-tem into an eigenspace of the stabilizers and the logicalqubit is realized. In the absence of errors, the syndromescorresponding to the fixed stabilizers will be +1, whilethose corresponding to the unmarked stabilizers can beeither +1 or −1. Moreover, in the absence of errors, mea-surement outcomes from the two measurement rounds instep 3 will be identical. Because the stabilizers commutewith the logical operators, the resulting logical qubit stateis the +1 eigenstate of cos(2θ)XL+sin(2θ)YL. Thus whenθ = π/8, the dx,1 × dz,1 code is initialized in the logicalmagic state |m〉L = |0〉L + eiπ/4 |1〉L. If the target state is|+Y 〉L, then θ = π/4 is used. Thus, by tuning θ, arbitrarystates in the X − Y plane of the Bloch sphere can beprepared.

Stage II

Stage II proceeds to encode the magic state into a largersurface code, pending an appropriate heralded outcome atstage I [34]. Physical qubits in region II are initialized asshown in Fig. 2. All the stabilizers of the dx,2 × dz,2 codeare measured for dm rounds and error correction is per-formed using standard decoding algorithms like minimumweight perfect matching [10, 17, 43, 44]. Subsequentlythe state may be sent for MSD.

Let us remark that there is some freedom in choosingthe initial state of qubits in regions I and II. The initialstate pattern shown in Fig 2 works well for the rangeof parameters used in section IV. Appendix C gives anexample of an alternative pattern.

III. NOISE

Here we argue that our scheme is tolerant to a singledephasing error on a data qubit or an ancilla qubit dur-ing preparation, idling, or any of the gates, to a singlemeasurement error, or to a single correlated dephasingerror that occurs during CX and CZ gates. As a conse-quence, when bit-flip errors are absent, the preparationerror rate is O(p2), with p the probability of a dominant

4

error. This improvement remains significant for realisticnoise models with high but finite bias η, where 1/η (η � 1) is the factor by which the probability of a non-Z erroris suppressed compared to that of the dominant Z error.In this case, undetectable preparation errors can occurat rate O(p/η). It follows that if η is large relative top−1, we obtain a quadratic improvement in the fidelity ofinjected magic states at finite bias compared to standardinjection protocols. At very small p we obtain an improve-ment by a factor of 1/η in preparation fidelity; O(p/η).The competition between the contribution of infidelitydue to high rate and low rate errors can be determinedby numerical experiments such as those we describe inSection IV. For the following qualitative discussion weconcentrate on errors at stage I because this will be thedominant source of infidelity given sufficiently large dx,2and dz,2 at stage II.

We assume a Pauli approximation to a biased cir-cuit noise model. Each single-qubit operation, includ-ing preparation and idling, is followed by a Pauli errorQ = {I,X, Y, Z} that occurs with probability pQ. Faultymeasurements are modelled by flipping a given measure-ment outcome with probability pM . Errors in two-qubitgates are modelled by applying a Pauli error Q = QC⊗QTwith QC , QT ∈ {I,X, Y, Z} with probability PQ beforethe gate where QC(QT ) denotes the error acting on thecontrol(target) qubit of the gate. Our protocol is de-signed to be highly effective against Z-biased noise wherepZ , pZI , pIZ , pZZ , and pM are significantly larger thanthe probabilities of other non-trivial, i.e., non-identity,error events and we take pZZ to be small in the ZZ(θ)gate following experimentally well motivated argumentsgiven below.

We now demonstrate that our protocol is robust againsta single high-rate error event in a biased-noise architecture.Over steps 1-3, a Z error on any of the qubits highlightedin grey and blue will cause the syndromes correspondingto the fixed stabilizers to change to −1. Thus, these errorsare detected in step 3. A Z error on the qubits markedin green before the first measurement round of step 3will not cause a logical error. A Z error on these qubitsin the second measurement round of step 3 will resultin a mismatch of the syndromes, corresponding to theunshaded stabilizers in region I, in the two measurementrounds. Hence, this error is also detected in step 3. A Zerror on an ancilla or a measurement error will also bedetected as it will either cause the outcome of measuring afixed stabilizer to be −1 or cause a mismatch of stabilizermeasurement outcomes from the first and second rounds.

So far we have ignored correlated errors introduced bythe two-qubit gates. During a correlated error, two qubitssimultaneously suffer from phase-flips with a probabilitythat can be greater than the probability of independentphase-flips on the two qubits. In case of pure-dephasingnoise, the CX or CZ gates acting between data and ancillaqubits do not lead to correlated errors on the data qubits.

A correlated Z ⊗ Z error in any one of these gates inthe first round of step 3, will either cause the outcome ofmeasuring a fixed stabilizer to be −1 or cause a mismatchof stabilizer measurement outcomes and hence will bedetected. Moreover, a Z ⊗ Z error in the second roundwill be corrected by subsequent rounds of error correctionin stage II. A correlated Z ⊗ Z error in the ZZ(θ) gatewill cause a logical error which will not be detected ineither stage I or II. However, these are expected to be low-rate errors in superconducting biased-noise architecturesince independent phase-noise in the qubits don’t getcorrelated and control and crosstalk errors can be easilymitigated (see further discussion in section V). Thus, a Z⊗Z error in the ZZ(θ) gate will not limit the performanceof the scheme in practice. There are several instances ofindependent errors occurring simultaneously on two ormore qubits which will also not be detected. For example,simultaneous phase-flip errors during initialization of thetwo grey qubits will go undetected.

In summary, we find that the proposed scheme is robustagainst a single Z error during preparation, idling, or anyof the gates, or a correlated Z ⊗ Z error in the CX andCZ gates, or a single measurement error. These errors aredetected and discarded in stage I or corrected in stageII. Thus, our protocol has a finite success rate whichdecreases with increase in the number of locations atwhich a fault can occur. Hence, for a high enough successrate, the distance of the code in stage I should not be toolarge.

In order to determine the scaling of the logical errorrate as a function of the probability of high-rate errors,we consider a physically realistic noise model where eachqubit is subject to independent phase-flip errors withidentical probability p. In this case, pZ = p for thesingle-qubit operations, pZI = p, pIZ = pZZ = p/2 forthe CX gates, and pZI = p, pIZ = p, pZZ = p2 for thediagonal gates. Errors in the measurement can also beassumed to be pM = O(p). Thus in the absence of non-Znoise, the logical error rate of the injected magic state ispL = O(p2). The error-channel used to obtain this scalingis justified because in the bias-preserving CX gates a Zerror on the target qubit propagates as a combinationof a Z error on the target and a Z ⊗ Z error on thetarget and control qubits, giving pIZ , pZZ = p/2 [11, 36].Such error-correlations cannot be trivially introducedin the diagonal gates since they can be implementedin an error-transparent manner using interactions thatcommute with physical Z errors in qubits [36]. Hence, theprobability of two qubit Z ⊗ Z errors is the same as theprobability of two independent Z errors for the diagonalgates, pZZ = pIZ · pZI = p2.

5

A. Noise modelling in simulations

We now describe the circuit noise model used to obtainthe numerical results presented in the next section. Inbiased-noise qubits the CX gate is the slowest operationand total noise in the CX gate can be much greater thanthat in the diagonal two-qubit gates. In particular in theKerr-cat qubit architecture, the probability of phase-fliperrors during the CX gate can be an order of magnitudegreater than that of the CZ gate [11] unless sophisticatedcontrol techniques are applied [45]. So we show numericalresults for two noise models: (A) CX slower than CZ,and (B) CX as fast as CZ. In both these cases, for thediagonal CZ, ZZ(θ) gates we use pIZ , pZI and pZZ asdescribed before, and the probability of other non-trivialtwo qubit errors = p/η. For the single-qubit preparationerrors, idling errors on data qubits while the ancillas arebeing measured, and errors on some of the qubits whichidle during CZ gates, we use pZ = p and pX = pY = p/η.Measurement errors are applied with probability p+ p/η.To model the fast CX gate in (B) we use, pZI , pIZ , pZZas described before and the probability of other non-trivial two qubit errors = p/η. In this case, the errorchannel applied to qubits which idle during the CX gateis identical to that applied to qubits which idle duringthe CZ gate. In (A), for the CX and single-qubit idlingerrors during this gate we use the same channel as (B)but with p replaced by 10p.

For numerical results we use two biases η = 104 andη = 103, for which the average gate bias in the CX gateis ∼ 1667 and ∼ 167 respectively. The average gate biasis defined as the ratio of the sum of the probabilitiesof I ⊗ Z,Z ⊗ I, and Z ⊗ Z error and the sum of theprobabilities of all other non-trivial errors. We start witha dx,1×dz,1 = 1×3 code in stage I and grow it to a largerdx,2 × dz,2 code with dm = dz,2.

For comparison we also present the logical error rateand success rate obtained when the standard scheme basedon using a single-qubit Z(θ) = e−iθZ gate, as describedin Appendix B, is used. For the error model of this gatewe use pZ = p and the probability of other non-trivialsingle-qubit errors = p/η. We keep the probability ofphase-flip error per qubit in the ZZ(θ) and Z(θ) gate tobe the same even though in practice the former can besmaller.

IV. RESULTS

Finally, we present numerical results that demonstratethe advantage of our scheme for logical magic state prepa-ration, and subsequently for distillation with practicalsystem parameters. Figure 3 shows the total logical errorrate εrawL of the output XZZX magic state and success rateas a function of the total error rate of the physical CXgate (pCX) for the noise model (A) and for three different

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FIG. 3. Logical error rate (εrawL ) and success rate after dmrounds of error correction in stage II with noise model A (CXslower than CZ) so that pCX = 20p + 120p/η. The bias isη = 104 in (a,c) and η = 103 in (b,d). The code size in stage Iis dx,1× dz,1 = 1× 3. Stage II code sizes dx,2× dz,2 are shownin the legend, with dm = dz,2. The results for our scheme areshown using solid lines and that for the standard approach areshown using dotted lines. Error bars indicate standard error ofthe mean. Each data point is generated with 105 Monte-Carlosamples.

dx,2 × dz,2.

Using our scheme, we find that when bias is largeη = 104, εrawL is approximately independent of the codesize and the curvature of εrawL (pCX) indicates a non-lineardependence of εrawL on the physical error rate. This followsfrom the discussion in section III, according to which thedominant source of uncorrectable errors is two phase-flipevents, or two faulty-measurement outcomes, or a combi-nation of these in the initial 1× 3 code. The deviationsbetween εrawL for different code sizes in Fig. 3 is mainlydue to small but non-zero bit-flip noise. By numerical fit-ting of the component of ZL error in εrawL for η = 104, wefind that this component scales as ((4.48± 0.07)× 103)p2

or (11.2±0.2)p2CX. In contrast, with the standard scheme,the curvature for εrawL (pCX) indicates a linear dependenceon the physical error rate even if the bias is large. In thiscase, with numerical fitting we find that ZL componentof error in εrawL scales as (11.6± 0.5)p or (0.58± 0.02)pCX.

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FIG. 4. Logical error rate (εrawL ) and success rate after dmrounds of error correction in stage II with noise model B (CXas fast as CZ) so that pCX = 2p+ 12p/η. The bias is η = 104

and the code size in stage I is dx,1 × dz,1 = 1 × 3. Stage IIcode sizes dx,2 × dz,2 are shown in the legend, with dm = dz,2.The results for our scheme are shown using solid lines andthat for the standard approach are shown using dotted lines.Error bars indicate standard error of the mean. Each datapoint is generated with 105 Monte-Carlo samples.

Details for the fitting and different components of thetotal logical error rate are given in Appendix A.

Results in Fig. 3(a) show that εrawL can be about anorder of magnitude lower than the physical error rateof the noisiest gate in the system. For example, whenpCX = 0.67% and η = 104, the infidelity of the injectedmagic state in the 3×15 code is = 0.07%. The probabilityof success is high = 94.4%. For an order of magnitudelower bias η = 103, εrawL increases and is still somewhatindependent of the code size in the given range of pCX.Moreover, due to greater contribution from the non-Zerrors, the curve εrawL (pCX) starts to flatten out. Nonethe-less, the scheme introduced here prepares a XZZX magicstate with a significantly lower error rate than the stan-dard approach for both η = 104 and η = 103. The abilityto detect more errors with our scheme leads to a smalldecrease in the success rate compared to the standardapproach.

In Fig. 4(a,b) we present εrawL and success rate as afunction of pCX for the noise model (B). We use η = 104

and again we find that the scheme based on ZZ(π/8)gate outperforms the standard approach. For example,even when the physical error rate in the two-qubit gatesis as high as 0.45%, the infidelity of the injected 3× 15magic state is five-fold lower ∼ 0.11%, while that withthe standard scheme is higher ∼ 0.66%.

The impact of our protocol becomes evident from thesubsequent reduction in cost for MSD. If the infidelityof the raw injected state is εrawL , then after a round of15-to-1 distillation protocol the logical error rate can bemade arbitrarily close to 35(εrawL )

3, if sufficiently large

code dx,2 × dz,2 is used so that errors in the distillation

circuit are negligible [19]. Consider Fig. 3 and note that

εrawL = 0.11% or 35(εrawL )3 ∼ 4.7 × 10−8 when pCX =

0.67%, η = 104, and dx,2× dz,2× dm = 5× 25× 25. Fromnumerical simulations we have confirmed that for the samenoise channel the logical error rate for dm = 25 roundsof error correction with 5× 25 code is � 10−8. Thus, wefind that after one round of distillation a magic state witherror rate O(10−8) can be realized with a 5× 25 XZZXcode. In contrast, with the standard approach, for thesame sized code and physical gate errors, εrawL = 0.33%,so that only an error rate of O(10−6) will be possible afterone round of distillation.

V. SUMMARY AND DISCUSSION

To summarize, we have introduced a protocol to prepareraw encoded states with low error rate by exploitingfeatures of biased-noise hardware. This in turn reducesthe overhead cost of MSD for such systems.

The protocol is robust against the typical errors of abiased circuit noise model. To gain an advantage over thestandard protocol, the probability of two-qubit correlatedphase-flip errors in the ZZ(θ) gate must be low relative tothe probability of two independent single-qubit phase-fliperrors. We expect this to be the case with Kerr-cat qubits.

While correlated phase-flip errors may be induced dueto virtual transitions to the excited states caused bythe microwave drive that realizes the ZZ(θ) gate, suchnoise can be mitigated by pulse shaping or by addingcounter-diabatic drives [45]. Another source of correlatederrors is crosstalk which can be mitigated by appropriatefrequency arrangement of qubits [46]. Thus, while we donot believe correlated errors will be a significant issue,further investigation in mitigating such errors is calledfor, which will be made possible by rapid advances inbiased-noise qubit technology.

We expect that the simple protocol we have proposedcan be widely generalized and adapted to other magicstate preparation schemes. For example, it might beinteresting to determine if further improvements can beachieved by combining our ideas with recent developmentsusing flag qubits [47, 48]. We could also consider usingthe state-preparation protocol with other codes, and weexpect that there may be some room for optimization ofthe initialization strategy we have presented. We discussthese suggestions in Appendix C.

Our work shows the value of carefully analysing thecircuit operations that are available with the underly-ing platform to ease the requirements of fault-tolerantquantum logical operations. To begin with, with the ar-chitecture we have considered here, we might expect toobtain an additional order of magnitude reduction in thepreparation error by using a three-qubit ZZZ(θ) entan-gling gate. We discuss this gate in Appendix D. Movingforward, the discovery of better multi-qubit entangling

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FIG. 5. XL and ZL error rate in the magic state for η =104 (a,b) and η = 103 (c,d) for noise model (A). The blackdashed lines in (b,d) is found by fitting ZL error rate in themagic state prepared using our scheme, at low p and largedistances, to Ap2. In (b) we use the solid lines correspondingto dx,2 × dz,2 = 3× 15 and dx,2 × dz,2 = 5× 25 for the fit andfind A = (4.48 ± 0.07) × 103. In (d) we use the solid linescorresponding to dx,2×dz,2 = 11×11 and dx,2×dz,2 = 15×15for the fit and find A = (4.34± 0.09)× 103.

gates that can be built using near-term technology, couldgive us better error-corrected devices that are essentialfor practical quantum computing.

ACKNOWLEDGEMENTS

SS and SP are supported by the Army Research Office(ARO) under grant number W911NF-18-1-0212. ASD wassupported by JST, PRESTO Grant No. JPMJPR1917,Japan. BJB is supported by the Australian ResearchCouncil via the Centre of Excellence in Engineered Quan-tum Systems (EQUS) project number CE170100009.

Appendix A: Logical error decomposition

Figure 5 shows the component of XL and ZL errors inthe total error rate presented in Fig. 3 of the main text.For small p, we find a quadratic dependence of ZL errorson p (Ap2) when the scheme introduced in this work isused. On the other hand, the dependence of ZL errors onp is linear when the standard protocol is used. In Fig. 5(b)

Standard

0.25 0.50 0.750.00 1.00

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(a) (b)

FIG. 6. XL and ZL error rate in the magic state for η = 104

for noise model (B). The black dashed lines in (b) is foundby fitting ZL error rate in the magic state prepared usingour scheme to Ap2. We use the solid lines corresponding todx,2 × dz,2 = 3 × 15 and dx,2 × dz,2 = 5 × 25 for the fit andfind A = (1.78± 0.06)× 102.

we fit ZL for dx,2× dz,2 = 3× 15 and dx,2× dz,2 = 5× 25to Ap2 and find A = (4.48± 0.07)× 103. In Fig. 5(d) wefit ZL for dx,2 × dz,2 = 11× 11 and dx,2 × dz,2 = 15× 15to Ap2 and find A = (4.34± 0.09)× 103. This confirmsthe analysis in section III, according to which, ZL errorrate, or equivalently A, should be independent of thecode size in stage II if dz,2 is large enough. Because ofthe initialization pattern chosen in stage II, the XL errorrate is expected to grow with the distance dz,2. This canbe understood from the fact that bit-flip errors on anyone of the dz,2 qubits in the top row of block II will beun-correctable. However, since the bias is large, failuredue to such error events is not too large. It is possible toprevent such errors from accumulating, especially whenthe bias is small, by using a larger dx,1 in stage I or byusing an alternative initialization strategy in stage II, likediscussed in the Appendix C.

Figures 6 shows the component of XL and ZL errors inthe total error rate presented in Fig. 4 of the the main text.We fit ZL for dx,2× dz,2 = 3× 15 and dx,2× dz,2 = 5× 25to Ap2 and find A = (1.78± 0.06)× 102.

Appendix B: Standard protocol based on thesingle-qubit Z(θ) gate

The numerical results corresponding to the standardscheme used in Figs. 3,4 were produced by modifying thesteps in Stage I of the protocol described in the main textas follows:

• Step 1: Physical qubits in region I are initialized asshown in Fig 7.

• Step 2: A Z(θ) = e−iθZ gate is applied on the qubiton the top left, highlighted in grey in Fig 7. The

8

FIG. 7. Qubit arrangement in stage I of the standard schemeused for comparison in this paper. The faces shaded in greymark the fixed stabilizers for stage I. Stage II is identical toFig. 2

fixed stabilizers are shown in grey.

• Step 3: All the stabilizers are measured twice andstabilizer measurement outcomes or syndromes arerecorded. If the outcome of measuring any fixedstabilizers is −1 or if the measurement outcomesfrom the two rounds are not identical, then an errorhas been detected. In this case the state is discardedand stage I is started afresh. Otherwise, the code issent to stage II.

Appendix C: Possibilities for further optimization inthe XZZX Code and other surface codes

Our protocol can be understood as preparing a 1× 2surface code magic state directly by using a physical two-qubit operation ZZ(θ). Next, the 1×2 code is grown intoa dx,1× dz,1 code in stage I in a standard way and all thestabilizers are measured twice. Only when no errors aredetected, the dx,1×dz,1 code is grown into dx,2×dz,2 codeand subsequent rounds of error correction are performed.In both the growing steps, the initial state of the qubits(apart from the qubits forming the original 1× 2 code) ischosen so that the logical operators grow correctly andto maximise the number of errors that can be detected orcorrected. For example, an alternate initialization patternis shown in Fig 8 which would be more beneficial whennoise is not too strongly biased. While we mainly focusedon the XZZX code, this basic procedure outlined abovecan also be applied to other surface code families, like thetailored surface code. The main common component is tostart with two qubits in |+〉⊗ |+〉 state and place them inthe magic state of a 1× 2 SC using the two-qubit ZZ(θ)gate. To illustrate, a possible arrangement of qubit statesfor the tailored surface code is shown in Fig. 9.

I II

FIG. 8. Illustration of the protocol for preparing the magicstate in the XZZX code with alternate stage II initializationpattern. The faces shaded in grey mark the fixed stabilizersfor stage I.

Appendix D: Protocol with ZZZ(θ) gate

In biased-noise cat qubits it is possible to realize a three-qubit ZZZ(θ) = e−iθZ⊗Z⊗Z gate. It can be activatedparametrically via four-wave mixing and can be easilyimplemented with the current circuit-QED toolbox [36].In fact, operations requiring similar interactions havealready been realized in several experiments [49–52]. Withsuch a gate, it is possible to directly prepare a 1× 3 codein the magic state. Following the procedure in section II,the 1× 3 code can be first grown to a dx,1 × dz,1 code bymeasuring the stabilizers thrice in stage I and the statepost-selected on no error-detection can be grown to adx,2×dz,2 code in stage II. When the bias is large and theprobability of three-qubit phase-flip error in the ZZZ(θ)gate is small, the probability of a logical error scales asO(p3phy). Alternatively, error detection in stage I can beskipped and the 1× 3 code can be directly grown into adx,2 × dz,2 code. In this case, the logical error probabilityis dominated by the failure rate of the 1 × 3 code andscales as O(p2phy). In general, the protocol can be adapted

to use a k-qubit Zk(θ) gate.

[1] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.Cleland, Surface codes: Towards practical large-scalequantum computation, Physical Review A 86, 032324(2012).

[2] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, andM. Troyer, Elucidating reaction mechanisms on quan-

9

I II

FIG. 9. Arrangement of qubits for preparing the magic statecos(π/8) |+i〉L − i sin(π/8) |−i〉L in the tailored surface code.This code has two types of stabilizers: product of PauliY, Y, Y, Y on the qubits around the white squares and productof Pauli X,X,X,X on the qubits around the grey squares. Atthe boundaries the stabilizers are product of X,X and Y, Yon two qubits. The fixed stabilizers for stage I are markedusing black lines. The ZZ(θ) gate is applied to the two greyqubits on the top left.

tum computers, Proceedings of the National Academy ofSciences 114, 7555 (2017).

[3] J. O’Gorman and E. T. Campbell, Quantum computationwith realistic magic-state factories, Physical Review A95, 032338 (2017).

[4] E. Campbell, A. Khurana, and A. Montanaro, Applyingquantum algorithms to constraint satisfaction problems,Quantum 3, 167 (2019).

[5] Y. R. Sanders, D. W. Berry, P. C. Costa, L. W. Tessler,N. Wiebe, C. Gidney, H. Neven, and R. Babbush, Compila-tion of fault-tolerant quantum heuristics for combinatorialoptimization, PRX Quantum 1, 020312 (2020).

[6] R. Babbush, J. R. McClean, M. Newman, C. Gidney,S. Boixo, and H. Neven, Focus beyond quadratic speedupsfor error-corrected quantum advantage, PRX Quantum2, 010103 (2021).

[7] C. Gidney and M. Eker̊a, How to factor 2048 bit rsaintegers in 8 hours using 20 million noisy qubits, Quantum5, 433 (2021).

[8] D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultra-high error threshold for surface codes with biased noise,Physical review letters 120, 050505 (2018).

[9] D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi,S. D. Bartlett, and S. T. Flammia, Tailoring surface codesfor highly biased noise, Physical Review X 9, 041031(2019).

[10] J. P. B. Ataides, D. K. Tuckett, S. D. Bartlett, S. T.Flammia, and B. J. Brown, The xzzx surface code, Naturecommunications 12, 1 (2021).

[11] A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K.Tuckett, and S. Puri, Practical quantum error correction

with the xzzx code and kerr-cat qubits, arXiv preprintarXiv:2104.09539 (2021).

[12] C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T.Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C.Bohdanowicz, S. T. Flammia, A. Keller, et al., Buildinga fault-tolerant quantum computer using concatenatedcat codes, arXiv preprint arXiv:2012.04108 (2020).

[13] O. Higgott and N. P. Breuckmann, Subsystem codes withhigh thresholds by gauge fixing and reduced qubit over-head, arXiv preprint arXiv:2010.09626 (2020).

[14] S. Huang and K. R. Brown, Fault-tolerant compass codes,Physical Review A 101, 042312 (2020).

[15] J. Guillaud and M. Mirrahimi, Error rates and resourceoverheads of repetition cat qubits, Physical Review A103, 042413 (2021).

[16] A. Y. Kitaev, Fault-tolerant quantum computation byanyons, Annals of Physics 303, 2 (2003).

[17] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topolog-ical quantum memory, Journal of Mathematical Physics43, 4452 (2002).

[18] S. B. Bravyi and A. Y. Kitaev, Quantum codes on alattice with boundary, arXiv preprint quant-ph/9811052(1998).

[19] S. Bravyi and A. Kitaev, Universal quantum computa-tion with ideal clifford gates and noisy ancillas, PhysicalReview A 71, 022316 (2005).

[20] B. W. Reichardt, Quantum universality from magic statesdistillation applied to css codes, Quantum InformationProcessing 4, 251 (2005).

[21] S. Bravyi and J. Haah, Magic-state distillation with lowoverhead, Physical Review A 86, 052329 (2012).

[22] A. G. Fowler, S. J. Devitt, and C. Jones, Surface codeimplementation of block code state distillation, Scientificreports 3, 1 (2013).

[23] A. M. Meier, B. Eastin, and E. Knill, Magic-statedistillation with the four-qubit code, arXiv preprintarXiv:1204.4221 (2012).

[24] C. Jones, Multilevel distillation of magic states for quan-tum computing, Physical Review A 87, 042305 (2013).

[25] G. Duclos-Cianci and K. M. Svore, Distillation of nonsta-bilizer states for universal quantum computation, PhysicalReview A 88, 042325 (2013).

[26] G. Duclos-Cianci and D. Poulin, Reducing the quantum-computing overhead with complex gate distillation, Phys-ical Review A 91, 042315 (2015).

[27] E. T. Campbell and M. Howard, Unified framework formagic state distillation and multiqubit gate synthesiswith reduced resource cost, Physical Review A 95, 022316(2017).

[28] J. Haah and M. B. Hastings, Codes and protocols fordistilling t, controlled-s, and toffoli gates, Quantum 2, 71(2018).

[29] E. T. Campbell and M. Howard, Magic state parity-checker with pre-distilled components, Quantum 2, 56(2018).

[30] C. Gidney and A. G. Fowler, Efficient magic state factorieswith a catalyzed |ccz〉 to 2|t〉 transformation, Quantum3, 135 (2019).

[31] D. Litinski, Magic state distillation: Not as costly as youthink, Quantum 3, 205 (2019).

[32] C. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter,Surface code quantum computing by lattice surgery, NewJournal of Physics 14, 123011 (2012).

10

[33] A. J. Landahl and C. Ryan-Anderson, Quantum com-puting by color-code lattice surgery, arXiv preprintarXiv:1407.5103 (2014).

[34] Y. Li, A magic state’s fidelity can be superior to theoperations that created it, New Journal of Physics 17,023037 (2015).

[35] Y.-H. Luo, M.-C. Chen, M. Erhard, H.-S. Zhong, D. Wu,H.-Y. Tang, Q. Zhao, X.-L. Wang, K. Fujii, L. Li, N.-L.Liu, K. Nemoto, W. J. Munro, C.-Y. Lu, A. Zeilinger, andJ.-W. Pan, Quantum teleportation of physical qubits intological code spaces, Proceedings of the National Academyof Sciences 118, e2026250118 (2021).

[36] S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini,P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais,S. T. Flammia, and S. M. Girvin, Bias-preserving gateswith stabilized cat qubits, Science advances 6, eaay5901(2020).

[37] P. Aliferis and J. Preskill, Fault-tolerant quantum compu-tation against biased noise, Physical Review A 78, 052331(2008).

[38] S. Puri, S. Boutin, and A. Blais, Engineering the quantumstates of light in a kerr-nonlinear resonator by two-photondriving, npj Quantum Information 3, 1 (2017).

[39] R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. Mc-Clean, A. Paler, A. Fowler, and H. Neven, Encodingelectronic spectra in quantum circuits with linear t com-plexity, Physical Review X 8, 041015 (2018).

[40] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, andY. Su, Toward the first quantum simulation with quan-tum speedup, Proceedings of the National Academy ofSciences 115, 9456 (2018).

[41] Y. Nam and D. Maslov, Low-cost quantum circuits forclassically intractable instances of the hamiltonian dy-namics simulation problem, npj Quantum Information 5,1 (2019).

[42] P. Webster, S. D. Bartlett, and D. Poulin, Reducing theoverhead for quantum computation when noise is biased,

Physical Review A 92, 062309 (2015).[43] J. Edmonds, Paths, trees, and flowers, Canadian Journal

of mathematics 17, 449 (1965).[44] V. Kolmogorov, Blossom v: a new implementation of a

minimum cost perfect matching algorithm, MathematicalProgramming Computation 1, 43 (2009).

[45] Q. Xu, J. K. Iverson, F. G. Brandao, and L. Jiang, Engi-neering fast bias-preserving gates on stabilized cat qubits,arXiv preprint arXiv:2105.13908 (2021).

[46] J. M. Gambetta, J. M. Chow, and M. Steffen, Buildinglogical qubits in a superconducting quantum computingsystem, npj Quantum Information 3, 1 (2017).

[47] C. Chamberland and A. W. Cross, Fault-tolerant magicstate preparation with flag qubits, Quantum 3, 143 (2019).

[48] C. Chamberland and K. Noh, Very low overhead fault-tolerant magic state preparation using redundant ancillaencoding and flag qubits, npj Quantum Information 6, 1(2020).

[49] Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis,A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J.Hatridge, et al., Confining the state of light to a quantummanifold by engineered two-photon loss, Science 347, 853(2015).

[50] S. Touzard, A. Grimm, Z. Leghtas, S. O. Mundhada,P. Reinhold, C. Axline, M. Reagor, K. Chou, J. Blumoff,K. M. Sliwa, et al., Coherent oscillations inside a quantummanifold stabilized by dissipation, Physical Review X 8,

021005 (2018).[51] A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada,

S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, andM. H. Devoret, Stabilization and operation of a kerr-catqubit, Nature 584, 205 (2020).

[52] R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Del-becq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas,Exponential suppression of bit-flips in a qubit encoded inan oscillator, Nature Physics 16, 509 (2020).