Logical-qubit operations in an error-detecting surface code
J. F. Marques, B. M. Varbanov, M. S. Moreira, H. Ali, N. Muthusubramanian, C. Zachariadis, F. Battistel, M. Beekman, N. Haider, W. Vlothuizen, A. Bruno, B. M. Terhal, L. DiCarlo
LLogical-qubit operations in an error-detecting surface code
J. F. Marques,
1, 2
B. M. Varbanov, M. S. Moreira,
1, 2
H. Ali,
1, 2
N. Muthusubramanian,
1, 2
C. Zachariadis,
1, 2
F. Battistel, M. Beekman,
1, 3
N. Haider,
1, 3
W. Vlothuizen,
1, 3
A. Bruno,
1, 2
B. M. Terhal,
1, 4 and L. DiCarlo
1, 2 QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Kavli Institute of Nanoscience, Delft University of Technology,P.O. Box 5046, 2600 GA Delft, The Netherlands Netherlands Organisation for Applied Scientific Research (TNO),P.O. Box 96864, 2509 JG The Hague, The Netherlands JARA Institute for Quantum Information, Forschungszentrum Juelich, D-52425 Juelich, Germany (Dated: February 26, 2021)We realize a suite of logical operations on a distance-two logical qubit stabilized using repeatederror detection cycles. Logical operations include initialization into arbitrary states, measurementin the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type ofoperation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants,and quantify the difference through detailed characterization. In particular, we demonstrate processtomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration ofhigh-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone onthe road to quantum error correction with higher-distance superconducting surface codes.
INTRODUCTION
Two key capabilities will distinguish an error-corrected quantum computer from present-day noisyintermediate-scale quantum (NISQ) processors [1].First, it will initialize, transform, and measure quan-tum information encoded in logical qubits rather thanphysical qubits. A logical qubit is a highly entangledtwo-dimensional subspace in the larger Hilbert space ofmany more physical qubits. Second, it will use repet-itive quantum parity checks to discretize, signal, and(with aid of a decoder) correct errors occurring in theconstituent physical qubits without destroying the en-coded information [2]. Provided the incidence of phys-ical errors is below a code-specific threshold and thequantum circuits for logical operations and stabiliza-tion are fault-tolerant, the logical error rate can be ex-ponentially suppressed by increasing the distance (re-dundancy) of the quantum error correction (QEC) codeemployed [3]. At present, the exponential suppressionfor specific physical qubit errors (bit-flip or phase-flip)has been experimentaly demonstrated [4, 5] for repeti-tion codes [6–8].Leading experimental quantum platforms have takenkey steps towards implementing QEC codes protect-ing logical qubits from general physical qubit errors.In particular, trapped-ion systems have demonstratedlogical-level initialization, gates and measurements forsingle logical qubits in the Calderbank-Shor-Steane [9]and Bacon-Shor [10] codes. Most recently, entanglingoperations between two logical qubits have been demon-strated in the surface code using lattice surgery [11].However, except for smaller-scale experiments using twoion species [12], trapped-ion experiments in QEC have so far been limited to a single round of stabilization.In parallel, taking advantage of highly-non-demolition measurement in circuit quantum elec-trodynamics [13], superconducting circuits have takenkey strides in repetitive stabilization of two-qubitentanglement [14, 15] and logical qubits. Quan-tum memories based on 3D-cavity logical qubitsin cat [16, 17] and Gottesman-Kitaev-Preskill [18]codes have crossed the memory break-even point.Meanwhile, monolithic architectures have focused onlogical qubit stabilization in a surface code realizedwith a 2D lattice of transmon qubits. Currently, thesurface code [19] is the most attractive QEC codefor solid-state implementation owing to its practicalnearest-neighbor connectivity requirement and higherror threshold. Recent experiments [5, 20] havedemonstrated repetitive stabilization by post-selectionin a surface code which, owing to its small size, iscapable of quantum error detection but not correction.We demonstrate a complete suite of logical-qubit op-erations for this small (distance-2) surface code whilepreserving multi-round stabilization. Our logical oper-ations span initialization anywhere on the logical Blochsphere, measurement in all cardinal bases, and a uni-versal set of single-logical-qubit gates. For each typeof operation, we quantify the increased performance offault-tolerant variants over non-fault-tolerant ones. Weintroduce the notion of a logical Pauli transfer matrixto describe a logical gate, analogous to the procedurecommonly used to describe gates on physical qubits [21].Finally, we compare the performance of two scalable,fault-tolerant stabilizer measurement schemes compat-ible with our quantum hardware architecture [22].The distance-2 surface code (Fig. 1a) uses four data a r X i v : . [ qu a n t - ph ] F e b qubits ( D through D ) to encode one logical qubit,whose two-dimensional codespace is the even-parity(i.e., eigenvalue +1) subspace of the stabilizer set S = { Z D1 Z D3 , X D1 X D2 X D3 X D4 , Z D2 Z D4 } . (1)This codespace has logical Pauli operators Z L = Z D1 Z D2 , Z D3 Z D4 , Z D1 Z D4 , and Z D2 Z D3 , (2) X L = X D1 X D3 and X D2 X D4 , (3)that anti-commute with each other and commute with S , and logical computational basis | L (cid:105) = 1 √ | (cid:105) + | (cid:105) ) , (4) | L (cid:105) = 1 √ | (cid:105) + | (cid:105) ) . (5)Measuring the stabilizers using three ancilla qubits ( A , A and A in Fig. 1a) allows detection of all individualphysical-qubit errors. Such errors change the outcomeof one or more stabilizers to m = − . However, no errorsyndrome combination is unique to a single error. Forinstance, a phase flip in any one data qubit triggers thesame syndrome: m A2 = − . Consequently, this codecannot be used to correct such errors. We thus performstate stabilization by post-selecting runs in which noerror is detected by the stabilizer measurements in anycycle. In this error-detection context, an operation isfault-tolerant if any single-fault produces a non-trivialsyndrome and can therefore be post-selected out [23]. RESULTSStabilizer measurements
Achieving high performance in a code hinges on per-forming projective quantum parity (stabilizer) measure-ments with high assignment fidelity and low additionalbackaction. We implement each of the stabilizers in S using a standard indirect-measurement scheme [24, 25]with a dedicated ancilla. As a fidelity metric, we mea-sure the average probability to correctly assign the par-ity Z D1 Z D3 , Z D1 Z D2 Z D3 Z D4 and Z D1 Z D3 of physicalcomputational states of the data-qubit register, finding . , . and . , respectively (see Fig. S2). Logical state initialization using stabilizermeasurements
A practical means to quantify the backaction of sta-bilizer measurements is using them to initialize logical
Figure 1.
Surface-7 quantum processor and initial-ization of logical cardinal states. (a) Distance-two sur-face code. (b) Optical image of the quantum hardware withadded false-color to emphasize different circuit elements. (c-f) Estimated physical density matrices, ρ , after targeting thepreparation of the logical cardinal states | L (cid:105) (c), | L (cid:105) (d), | + L (cid:105) (e) and |− L (cid:105) (f). Each state is measured after prepar-ing the data qubits in | (cid:105) , | (cid:105) , | ++++ (cid:105) and | ++ −−(cid:105) ,respectively. The ideal target state density matrix is shownin the shaded wireframe. states. As proposed in Ref. 20, we can prepare arbitrarylogical states by first initializing the data-qubit registerin the product state | ψ (cid:105) = (cid:0) C θ/ | (cid:105) + S θ/ | (cid:105) (cid:1) | (cid:105) (cid:0) C θ/ | (cid:105) + S θ/ e iφ | (cid:105) (cid:1) | (cid:105) (6)using single-qubit rotations R θy on D and R θφ on D acting on | (cid:105) ( C α = cos α and S α = sin α ). A follow-up round of stabilizer measurements ideally projects thefour-qubit state onto the logical state | ψ L (cid:105) = (cid:16) C θ/ | L (cid:105) + S θ/ e iφ | L (cid:105) (cid:17) / (cid:113) C θ/ + S θ/ (7)with probability P = 12 (cid:16) C θ/ + S θ/ (cid:17) . (8)We use this procedure to target initialization of the log-ical cardinal states | L (cid:105) , | L (cid:105) , | + L (cid:105) = (cid:0) | L (cid:105) + | L (cid:105) (cid:1) / √ ,and |− L (cid:105) = (cid:0) | L (cid:105) − | L (cid:105) (cid:1) / √ . For the first two states,the procedure is fault-tolerant according to the defini-tion above. We characterize the produced states us-ing full four-qubit state tomography including readoutcalibration and maximum-likelihood estimation (MLE)(Fig. 1). The fidelity F to the ideal four-qubit tar-get states is . , . , . , and . , re-spectively. For each state, we can extract a logical fi-delity F L by further projecting the obtained four-qubitdensity matrix onto the codespace [20], finding . , . , . , and . , respectively (see Meth-ods). This sharp increase from F to F L demonstratesthat the vast majority of errors introduced by the paritycheck are weight-1 and detectable. A simple modifica-tion makes the initialization of | + L (cid:105) ( |− L (cid:105) ) also fault-tolerant: initialize the data-qubit register in a differentproduct state, namely | ++++ (cid:105) ( | ++ −−(cid:105) ), before per-forming the stabilizer measurements. With this mod-ification, F increases to . ( . ) and F L to . ( . ), matching the performance achievedwhen targetting | L (cid:105) and | L (cid:105) . Logical measurement of arbitrary states
A key feature of a code is the ability to measurelogical operators. In the surface code, we can mea-sure X L ( Z L ) fault-tolerantly, albeit destructively, bysimultaneously measuring all data qubits in the X ( Z )basis to obtain a string of data-qubit outcomes (each +1 or − ). The value assigned to the logical oper-ator is the computed product of data-qubit outcomesas prescribed by Eq. 3 (2). Additionally, the outcomestring is used to compute a value for the stabilizer(s) Figure 2.
Arbitrary logical-state initialization andmeasurement in the logical cardinal bases. (a) As-sembly of data-qubit measurements used to evaluate logicaloperators Z L , X L and Y L with additional error detection.(d) Initialization of logical states using the procedure de-scribed in Eq. 6. (c, e) Z L , X L and Y L logical measurementresults as a function of the gate angles φ (c) and θ (e). Thecolored dashed curves show a fit of the analytical predictionbased on Eqs. 9 and 11 to the data and the dark curve de-notes a bound based on the measured F L of each state. (b,f) Total fraction P of post-selected data as a function of theinput angle for each logical measurement. The dashed curveshows the ideal fraction given by Eq. 8. X D1 X D2 X D3 X D4 ( Z D1 Z D3 and Z D2 Z D4 ), enabling afinal step of error detection (Fig. 2a). Measurementof Y L = + iX L Z L = Y D1 Z D2 X D3 is not fault-tolerant.However, we lower the logical assignment error by alsomeasuring D in the Z basis to compute a value for Z D2 Z D4 and thereby detect bit-flip errors in D and D .We demonstrate Z L , X L and Y L measurements onlogical states prepared on two orthogonal planes of thelogical Bloch sphere. Setting θ = π/ and sweeping φ ,we ideally prepare logical states on the equator (Fig. 2d) | ψ L (cid:105) = (cid:0) | L (cid:105) + e iφ | L (cid:105) (cid:1) / √ . (9)We measure the produced states in the Z L , X L and Y L bases and obtain experimental averages (cid:104) Z L (cid:105) , (cid:104) X L (cid:105) and (cid:104) Y L (cid:105) . As expected, we observe sinusoidal oscillations in (cid:104) X L (cid:105) and (cid:104) Y L (cid:105) and near-zero (cid:104) Z L (cid:105) . We extract logicalassignment fidelities F RL for X L and Y L from the am-plitude of the oscillations and separating the effect ofinitialization error: (2 F RL − F L −
1) = max |(cid:104) O L (cid:105)| , O ∈ { X, Y } . (10)We find F RL = 95 . for X L and . for Y L , whichmanifests the non-fault-tolerant nature of Y L measure-ment. A second manifestation is the higher fraction P of post-selected data in this case (Fig. 2b).Setting φ = 0 and sweeping θ , we then prepare logicalstates on the X L - Z L plane of the logical Bloch sphere(Fig. 2d), ideally | ψ L (cid:105) = (cid:16) C θ/ | L (cid:105) + S θ/ | L (cid:105) (cid:17) / (cid:113) C θ/ + S θ/ . (11)Note that due to the changing overlap of the initialproduct state with the codespace, P is now a functionof θ (Eq. 8). Using the same procedure as above, we ex-tract F RL = 99 . for Z L and . for X L . Althoughboth measurements are fault-tolerant, F RL is higher for Z L . This arises because the Z L measurement is onlyvulnerable to vertical double bit-flip errors while the X L measurement is vulnerable to horizontal and diago-nal double phase-flip errors. Logical gates
Finally, we demonstrate a suite of gates enabling uni-versal logical-qubit control (Fig. 3). Full control ofthe logical qubit requires a gate set comprising Clif-ford and non-Clifford logical gates. Some Clifford gates,like Z L and X L , can be implemented transversally andtherefore fault-tolerantly (Fig. 3d). We perform arbi-trary rotations (generally non-fault-tolerant) about the Z L axis using the standard gate-by-measurement cir-cuit [26] shown in Fig. 3a. In our case, the ancillais physical ( A ), while the qubit transformed is ourlogical qubit. The rotation angle θ is set by the ini-tial ancilla state | A θ (cid:105) = ( | (cid:105) + e iθ | (cid:105) ) / √ . Since wecannot do binary-controlled Z L rotations, we simply Figure 3.
Logical gates and their characterization. (a,b) General gate-by-measurement schemes realizing arbitraryrotations around the Z (a) and X (b) axis of the Blochsphere. (c) Process tomography experiment of the T L gate.Input cardinal logical states are initialized using the methodof Fig. 2. Output states are measured following a secondround of stabilizer measurements. (d) Logical X π/ , Z L and X L gates compiled using the hardware-native gateset.(e) Logical state tomography of input and output states ofthe T L gate. These logical density matrices are obtained byperforming four-qubit tomography of the data qubits andthen projecting onto the codespace. (f) Extracted logicalPauli transfer matrices. Figure 4.
Repetitive error detection using pipelined and parallel stabilizer measurement schemes. (a, b) Gatesequences used to implement the pipelined (a) and parallel (b) stabilizer measurement schemes. Gate duration is
20 ns for single-qubit gates,
60 ns for controlled-Z (CZ) gates and parking [14, 22], and
540 ns for ancilla readout. The orderof CZs in the X D1 X D2 X D3 X D4 stabilizer (blue shaded region) prevents the propagation of ancilla errors into logical qubiterrors [23]. The total cycle duration for the pipelined (parallel) scheme is
840 ns ( ). (c) Estimated Z L expectationvalue, (cid:104) Z L (cid:105) , measured for the | L (cid:105) state versus the duration of the experiment using the pipelined (blue) and the parallel(orange) schemes. We also plot the excited-state probability (right axis) set by the maximum and minimum physical qubit T . (d) Post-selected fraction of data versus the number of error detection cycles n for the pipelined (blue) and parallel(orange) scheme. post-select runs in which the measurement outcome is m A2 = +1 . Choosing θ = π/ implements the non-Clifford T L = Z π/ gate. A similar circuit (Fig. 3b)can be used to perform arbitrary rotations around the X L axis. We compile both circuits using our hardware-native gateset (Figs. 3c,d). To assess logical-gate per-formance, we perform logical process tomography us-ing the procedure illustrated in Fig. 3e for T L . First,we initialize into each of the six logical cardinal states {| L (cid:105) , | L (cid:105) , | + L (cid:105) , |− L (cid:105) , | + i L (cid:105) , |− i L (cid:105)} . We characterizeeach actual input state by four-qubit state tomographyand project to the codespace to obtain a logical densitymatrix. Next, we similarly characterize each outputstate produced by the logical gate and a second roundof stabilizer measurements to detect errors occurred inthe gate. Using this over-complete set of input-outputlogical-state pairs, combined with MLE (see Methods),we extract a logical Pauli transfer matrix (LPTM). Theresulting LPTMs for the non-fault-tolerant T L and X π/ gates as well as the fault-tolerant Z L and X L are shownin Fig. 3e. From the LPTMs, we extract average logicalgate fidelities F GL (Eq. 19) 97.3%, 95.6%, 97.9%, and98.1%, respectively. Pipelined versus parallel stabilizer measurements
A scalable control scheme is fundamental to realizesurface codes with large code distance. To this end,we now compare the performance of two schemes suit- able for the quantum hardware architecture proposedin Ref. 22. These schemes are scalable in the sensethat their cycle duration remains independent of codedistance. The pipelined scheme interleaves the coher-ent operations and ancilla readout steps associated withstabilizer measurements of type X and Z by perform-ing the coherent operations of X ( Z ) type stabilizersduring the readout of Z ( X ) type stabilizers. (Fig. 4a).The parallel scheme performs all ancilla readouts simul-taneously (Fig. 4b). To compare their performance, weinitialize and stabilize | L (cid:105) for up to n = 15 cycles.We separately calibrate the equatorial rotation axis ofrefocusing pulses ( R πϕ i ) in each scheme to extract thebest performance in both schemes. At each n , we takedata back-to-back for the two schemes in order to min-imize the effect of parameter drift, repeating each ex-periment up to × times. Figure 4c shows the Z L measurement outcome averaged over the post-selectedruns. We extract the error-detection rate γ from the n -dependence of the fraction of post-selected data P (Fig. 4d) using the procedure described in Methods.We observe that the error rate is slightly lower for thepipelined scheme ( γ pip ∼ ), most likely due to theshorter duration of the cycle. This superiority is con-sistent across different input logical states (see Fig. S3)with an average ratio γ pip /γ par ∼ . DISCUSSION
We have demonstrated a suite of logical-level initial-ization, gate and measurement operations in a distance-2 superconducting surface code undergoing repetitivestabilizer measurements. For each type of logical oper-ation, we have quantified the increased performance offault-tolerant variants over non-fault-tolerant variants.Table I summarizes all the results. We can initialize thelogical qubit to any point on the logical Bloch sphere,with logical fidelity surpassing Ref. 20. In addition tocharacterizing initialized states using full four-qubit to-mography, we also demonstrate logical measurementsin all logical cardinal bases. Finally, we demonstrate auniversal single-qubit set of logical gates by performinglogical process tomography, introducing the concept ofa logical-level Pauli transfer matrix.With a view towards implementing higher-distancesurface codes using our quantum-hardware architec-ture [22], we have compared the performance of twoscalable stabilization schemes: the pipelined and paral-lel measurement schemes. In this comparison, two mainfactors compete. On one hand, the shorter cycle time fa-vors pipelining. On the other, the pipelining introducesextra dephasing on ancilla qubits of one type duringreadout of the other. The performance of both schemesis comparable, but slightly higher for the pipelinedscheme. From density-matrix simulations discussed indetail in the Supplementary Material, we further un-derstand that conventional qubit errors such as en-ergy relaxation, dephasing and readout assignment er-ror alone do not account for the net error-detection rateobserved in the experiment (Fig. S5). We believe thatthe dominant error source is instead leakage to highertransmon states incurred during CZ gates. Our data(Fig. S4) shows that the error detection scheme suc-cessfully post-selects leakage errors in both the ancillaand data qubits. Learning to identify these non-qubiterrors and to correct them without post-selection is thesubject of ongoing research [27–29] and an outstandingchallenge in the quest for quantum fault-tolerance withhigher-distance superconducting surface codes.
METHODSDevice
We use a seven-transmon superconduting processor(Fig. 1b) featuring the quantum-hardware architectureproposed in Ref. 22. We employ flux-tunable transmonsarranged in three frequency groups: a high-frequencygroup for D and D ; a middle-frequency group for A , Logical operation Characteristic Logical fidelity metric value (%) I n i t . | L (cid:105) FT F L | L (cid:105) FT 99.97 | + L (cid:105) Non-FT/FT 97.02/99.78 |− L (cid:105) Non-FT/FT 95.54/99.64 M e a s . Z L FT F RL X L FT . ∗ Y L Non-FT 87.5 G a t e Z L FT F GL X L FT 97.9 X π/ Non-FT 95.6 T L Non-FT 97.3
Table I.
Summary of logical initialization, measure-ment, and gate operations and their performance.
Fault-tolerant operations are labelled FT and non-fault tol-erant ones Non-FT. ∗ Weighted average of values extractedfrom Figs. 2c,d. A and A ; and a low-frequency group for D and D .Each transmon is transversely coupled to its nearestneighbor using a dedicated coupling bus resonator andfeatures an individual microwave drive line for single-qubit gates, a flux line for two-qubit gates, and a disper-sively coupled readout resonator with Purcell filter forreadout [15, 30]. All transmons are flux biased to theirmaximal frequency (i.e., flux sweetspot [31]). Qubit re-laxation ( T ) and dephasing ( T ) times lie in the range27—102 µ s and 55—117 µ s , respectively. Detailed in-formation on the implementation and performance ofsingle- and two-qubit gates can be found in Ref. 32.Device characteristics are also summarized in Table S1. State tomography
To perform state tomography on the prepared logicalstates, we measure the − expectation values of data-qubit Pauli observables, p i = (cid:104) σ i (cid:105) , σ i ∈ { I, X, Y, Z } ⊗ (except I ⊗ ). These are used to construct the densitymatrix ρ = − (cid:88) i =0 p i σ i , (12)with p = 1 corresponding to σ = I ⊗ . Due to statis-tical uncertainty in the measurement, the constructedstate, ρ , might lack the physicality characteristic of adensity matrix, that is, Tr( ρ ) = 1 and ρ ≥ . Specif-ically, ρ might not satisfy the latter constraint, whilethe former is automatically satisfied by p = 1 . To en-force these constraints, we use a maximum-likelihoodmethod [21] to find the physical density matrix, ρ ph ,that is closest to the measured state, where close-ness is defined in terms of best matching the mea-surement results. We thus minimize the cost function (cid:80) − i =0 | p i − Tr( ρ ph σ i ) | , subject to Tr( ρ ph ) = 1 and ρ ph ≥ . We find the optimal ρ optph using the convex-optimization package cvxpy via cvx-fit in Qiskit [33].The fidelity to a target pure state, | ψ (cid:105) , is then com-puted as F = (cid:104) ψ | ρ optph | ψ (cid:105) . (13)One can further project ρ ph onto the codespace to ob-tain a logical state ρ L using ρ L = 12 (cid:88) i Tr( ρ ph σ L i )Tr( ρ ph I L ) σ L i , σ L i ∈ { I L , X L , Y L , Z L } , (14)where I L is the projector onto the codespace. Here, wecan compute the logical fidelity F L using Eq. 13. Process tomography in the codespace
A general single-qubit gate can be described [21] bya Pauli transfer matrix (PTM) R that maps an in-put state described by p i = (cid:104) σ i (cid:105) , σ i ∈ { I, X, Y, Z } , with p = 1 , to an output state p (cid:48) : p (cid:48) j = (cid:88) i R ij p i . (15)To construct R in the codespace, weuse an overcomplete set of input states, {| L (cid:105) , | L (cid:105) , | + L (cid:105) , |− L (cid:105) , | + i L (cid:105) , |− i L (cid:105)} , and their corre-sponding output states and perform linear inversion.The input and output logical states are characterizedusing state tomography of the data qubits to findthe four-qubit state ρ , which is then projected to thecodespace using: p L i = Tr( ρσ L i )Tr( ρI L ) , σ L i ∈ { I L , X L , Y L , Z L } , (16)We find that all the measured logical states already sat-isfy the constraints of a physical density matrix. Thisis likely to happen as one-qubit states that are not verypure usually lie within the Bloch sphere even withinthe uncertainty in the measurement. The constructedLPTM, however, might not satisfy the constraints of aphysical quantum channel, that is, trace preservationand complete positivity (TPCP). These are better ex-pressed by switching from the PTM representation tothe Choi representation. The Choi state ρ R can becomputed as ρ R = 14 (cid:88) i,j R ij σ Tj ⊗ σ i , (17) where the first tensor-product factor corresponds toan auxiliary subsystem. The TPCP constraints are Tr( ρ R ph ) = 1 , ρ R ph ≥ and Tr ( ρ R ph ) = 1 / , where Tr isthe partial trace over the auxiliary subsystem. In otherwords, ρ R ph is a density matrix satisfying an extra con-straint. We then find the optimal ρ R , optph using the sameconvex-optimization methods as for state tomographyand adding this extra constraint [21, 34]. We computethe corresponding LPTM via ( R optph ) ij = Tr( ρ R , optph σ Tj ⊗ σ i ) (18)and the average logical gate fidelity using F GL = Tr( R † ideal R optph ) + 26 , (19)where R ideal is the LPTM of the ideal target gate. Extraction of error-detection rate
The fraction of post-selected data P in the repetitiveerror detection experiment (Fig. 4b) decays exponen-tially with the number of cycles n . This is consistentwith a constant error-detection rate per cycle γ . Weextract this rate by fitting the function P ( n ) = A (1 − γ ) n . (20) [1] J. Preskill, Quantum , 79 (2018).[2] B. M. Terhal, Rev. Mod. Phys. , 307 (2015).[3] J. M. Martinis, npj Quantum Inf. , 15005 (2015).[4] J. Kelly, R. Barends, A. G. Fowler, A. Megrant,E. Jeffrey, T. White, D. Sank, J. Mutus, B. Camp-bell, Y. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi,C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan,A. Vainsencher, A. N. Cleland, J. Wenner, and J. M.Martinis, Nature , 66 (2015).[5] Z. Chen, K. J. Satzinger, J. Atalaya, A. N. Korotkov,A. Dunsworth, D. Sank, C. Quintana, M. McEwen,R. Barends, P. V. Klimov, S. Hong, C. Jones,A. Petukhov, D. Kafri, S. Demura, B. Burkett, C. Gid-ney, A. G. Fowler, H. Putterman, I. Aleiner, F. Arute,K. Arya, R. Babbush, J. C. Bardin, A. Bengtsson,A. Bourassa, M. Broughton, B. B. Buckley, D. A.Buell, N. Bushnell, B. Chiaro, R. Collins, W. Court-ney, A. R. Derk, D. Eppens, C. Erickson, E. Farhi,B. Foxen, M. Giustina, J. A. Gross, M. P. Harri-gan, S. D. Harrington, J. Hilton, A. Ho, T. Huang,W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey,Z. Jiang, K. Kechedzhi, S. Kim, F. Kostritsa, D. Land-huis, P. Laptev, E. Lucero, O. Martin, J. R. Mc-Clean, T. McCourt, X. Mi, K. C. Miao, M. Mohseni, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley,C. Neill, M. Newman, M. Y. Niu, T. E. O’Brien,A. Opremcak, E. Ostby, B. Pató, N. Redd, P. Roushan,N. C. Rubin, V. Shvarts, D. Strain, M. Szalay,M. D. Trevithick, B. Villalonga, T. White, Z. J. Yao,P. Yeh, A. Zalcman, H. Neven, S. Boixo, V. Smelyan-skiy, Y. Chen, A. Megrant, and J. Kelly, (2021),arXiv:2102.06132 [quant-ph].[6] D. Ristè, S. Poletto, M. Z. Huang, A. Bruno, V. Vesteri-nen, O. P. Saira, and L. DiCarlo, Nat. Commun. , 6983(2015).[7] J. Cramer, N. Kalb, M. A. Rol, B. Hensen,M. Markham, D. J. Twitchen, R. Hanson, and T. H.Taminiau, Nat. Commun. , 11526 (2016).[8] D. Ristè, L. C. G. Govia, B. Donovan, S. D. Fallek,W. D. Kalfus, M. Brink, N. T. Bronn, and T. A. Ohki,npj Quantum Information , 71 (2020).[9] D. Nigg, M. Müller, E. A. Martinez, P. Schindler,M. Hennrich, T. Monz, M. A. Martin-Delgado, andR. Blatt, Science , 302 (2014).[10] L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu,D. Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina,and C. Monroe, (2021), arXiv:2009.11482 [quant-ph].[11] A. Erhard, H. P. Nautrup, M. Meth, L. Postler,R. Stricker, M. Stadler, V. Negnevitsky, M. Ringbauer,P. Schindler, H. J. Briegel, et al. , Nature , 220(2021).[12] V. Negnevitsky, M. Marinelli, K. K. Mehta, H.-Y. Lo,C. Flühmann, and J. P. Home, Nature , 527 (2018).[13] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A , 062320 (2004).[14] C. K. Andersen, A. Remm, S. Lazar, S. Krinner,J. Heinsoo, J.-C. Besse, M. Gabureac, A. Wallraff, andC. Eichler, npj Quantum Information , 1 (2019).[15] C. C. Bultink, T. E. O’Brien, R. Vollmer, N. Muthusub-ramanian, M. W. Beekman, M. A. Rol, X. Fu,B. Tarasinski, V. Ostroukh, B. Varbanov, A. Bruno,and L. DiCarlo, Science Advances , 10.1126/sci-adv.aay3050 (2020).[16] N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Legh-tas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin,L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J.Schoelkopf, Nature , 441 (2016).[17] L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu,H. Wang, Y. P. Song, C.-L. Zou, S. M. Girvin, L.-M.Duan, and L. Sun, Nat. Phys. 10.1038/s41567-018-0414-3 (2019).[18] P. Campagne-Ibarcq, A. Eickbusch, S. Touzard,E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Rein-hold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio,M. Mirrahimi, and M. H. Devoret, Nature , 368(2020).[19] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.Cleland, Phys. Rev. A , 032324 (2012).[20] C. K. Andersen, A. Remm, S. Lazar, S. Krinner,N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, andA. Wallraff, Nat. Phys. , 875.[21] J. M. Chow, J. M. Gambetta, A. D. Córcoles, S. T.Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A.Keefe, M. B. Rothwell, J. R. Rozen, M. B. Ketchen,and M. Steffen, Phys. Rev. Lett. , 060501 (2012). [22] R. Versluis, S. Poletto, N. Khammassi, B. M. Tarasin-ski, N. Haider, D. J. Michalak, A. Bruno, K. Bertels,and L. DiCarlo, Phys. Rev. App. , 034021 (2017).[23] Y. Tomita and K. M. Svore, Phys. Rev. A , 062320(2014).[24] O.-P. Saira, J. P. Groen, J. Cramer, M. Meretska,G. de Lange, and L. DiCarlo, Phys. Rev. Lett. ,070502 (2014).[25] M. Takita, A. D. Córcoles, E. Magesan, B. Abdo,M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta,Phys. Rev. Lett. , 210505 (2016).[26] P. Aliferis, D. Gottesman, and J. Preskill, Quantum Inf.Comput. , 97 (2005).[27] B. M. Varbanov, F. Battistel, B. M. Tarasinski, V. P.Ostroukh, T. E. O’Brien, L. DiCarlo, and B. M. Terhal,npj Quantum Information , 102 (2020).[28] M. McEwen, D. Kafri, Z. Chen, J. Atalaya, K. J.Satzinger, C. Quintana, P. V. Klimov, D. Sank, C. Gid-ney, A. G. Fowler, F. Arute, K. Arya, B. Buck-ley, B. Burkett, N. Bushnell, B. Chiaro, R. Collins,S. Demura, A. Dunsworth, C. Erickson, B. Foxen,M. Giustina, T. Huang, S. Hong, E. Jeffrey, S. Kim,K. Kechedzhi, F. Kostritsa, P. Laptev, A. Megrant,X. Mi, J. Mutus, O. Naaman, M. Neeley, C. Neill,M. Niu, A. Paler, N. Redd, P. Roushan, T. C. White,J. Yao, P. Yeh, A. Zalcman, Y. Chen, V. N. Smelyan-skiy, J. M. Martinis, H. Neven, J. Kelly, A. N. Ko-rotkov, A. G. Petukhov, and R. Barends, (2021),arXiv:2102.06131 [quant-ph].[29] F. Battistel, B. M. Varbanov, and B. M. Terhal, (2021),arXiv:2102.08336 [quant-ph].[30] J. Heinsoo, C. K. Andersen, A. Remm, S. Krinner,T. Walter, Y. Salathé, S. Gasparinetti, J.-C. Besse,A. Potočnik, A. Wallraff, and C. Eichler, Phys. Rev.App. , 034040 (2018).[31] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster,B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer,L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J.Schoelkopf, Phys. Rev. B , 180502(R) (2008).[32] V. Negîrneac, H. Ali, N. Muthusubramanian, F. Bat-tistel, R. Sagastizabal, M. S. Moreira, J. F. Marques,W. Vlothuizen, M. Beekman, N. Haider, A. Bruno, andL. DiCarlo, (2020), arXiv:2008.07411 [quant-ph].[33] Qiskit: An open-source framework for quantum com-puting (2019).[34] J. de Jong, Implementation of a fault-tolerant SWAPoperation on the IBM 5-qubit device , Master’s thesis,Delft University of Technology (2019).
ACKNOWLEDGEMENTS
We thank R. Sagastizabal, M. Sarsby andT. Stavenga for experimental assistance, and G. Calu-sine and W. Oliver for providing the traveling-waveparametric amplifiers used in the readout amplificationchain. This research is supported by the Office of theDirector of National Intelligence (ODNI), IntelligenceAdvanced Research Projects Activity (IARPA), viathe U.S. Army Research Office Grant No. W911NF-16-1-0071, and by Intel Corporation. The views andconclusions contained herein are those of the authorsand should not be interpreted as necessarily repre-senting the official policies or endorsements, eitherexpressed or implied, of the ODNI, IARPA, or theU.S. Government. B. M. V., F. B. and B. M. T. aresupported by ERC Grant EQEC No. 682726.
AUTHOR CONTRIBUTIONS
J. F. M. performed the experiment and data analysis.M. B., N. H. and L. D. C. designed the device. N. M., C. Z. and A. B. fabricated the device. J. F. M. and H. A.calibrated the device. M. S. M. and W. V. designed thecontrol electronics. B. M. V. performed the numericalsimulations and F. B. implemented the MLE method.B. M. T. supervised the theory work. J. F. M. andL. D. C. wrote the manuscript with contributions fromB. M. V., F. B. and B. M. T., and feedback from allcoauthors. L. D. C. supervised the project.
COMPETING INTERESTS
The authors declare no competing interests.0
SUPPLEMENTAL MATERIAL FOR ’LOGICAL-QUBIT OPERATIONS IN AN ERROR-DETECTINGSURFACE CODE’
This supplement provides additional information in support of statements and claims made in the main text.
DEVICE CHARACTERISTICS
Qubit D D D D A A A Qubit transition frequency at sweetspot, ω q / π (GHz) 6.433 6.253 4.535 4.561 5.770 5.881 5.785Transmon anharmonicity, α/ π (MHz) -280 — -320 — -290 -285 —Readout frequency, ω r / π (GHz) 7.493 7.384 6.913 6.645 7.226 7.058 7.101Relaxation time, T ( µ s ) 27 44 32 102 38 58 43Ramsey dephasing time, T ∗ ( µ s ) 44 55 51 103 55 60 52Echo dephasing time, T ( µ s ) 59 70 55 117 69 79 73Best multiplexed readout fidelity, F RO , (%) 98.6 98.9 96.0 96.5 98.6 94.2 98.9Table S1. Summary of frequency, coherence and readout parameters of the seven transmons. Coherence times are obtainedusing standard time-domain measurements [S1]. Note that temporal fluctuations of several µ s are typical for these values.The multiplexed readout fidelity, F RO , is the average assignment fidelity [S2] extracted from single-shot readout histogramsafter mitigating residual excitation using initialization by measurement and post-selection [S3, S4].Figure S1. Residual ZZ -coupling matrix. Measured residual ZZ coupling between all transmon pairs at the bias point(their simultaneous flux sweetspot [S5]). Each matrix element denotes the frequency shift that the target qubit experiencesdue to the spectator qubit being in the excited state, | (cid:105) . The procedure used for this measurement is similar to the onedescribed in Ref. S6. PARITY-CHECK PERFORMANCE
Figure S2. Characterization of the assignment fidelity of Z -type parity checks (a) Z D1 Z D3 , (b) Z D1 Z D2 Z D3 Z D4 , and (c) Z D2 Z D4 implemented using A , A , and A , respectively. Each parity check is benchmarked by preparing the relevantdata qubits in a computational state and then measuring the probability of ancilla outcome m Ai = − . Measured (ideal)probabilities are shown as solid blue bars (black wireframe). From the measured probabilities we extract average assignmentfidelities . , . and . , respectively. STATE STABILIZATION
Figure S3. Stabilization of logical cardinal states by repetitive error detection using the pipelined and parallel schemes.From left to right, the stabilized logical states are | L (cid:105) , | L (cid:105) , | + L (cid:105) and |− L (cid:105) . For each logical state, the top panel showsthe evolution of the relevant logical operator as a function of number of cycles, n , plotted versus wall-clock time. Errorbars are estimated based on the statistical uncertainty given by P ( n ) . The shaded area indicates the range of physicalqubit T values (a and b) and T values (c and d) plotted on the right-axis. Each bottom panel shows the correspondingpost-selected fraction of data, P ( n ) . NUMERICAL ANALYSISLeakage in experiment
Figure S4. Single-shot readout histograms obtained at cycle n over all shots (red) and the post-selected shots based ondetecting no error in any cycles up to n (blue) for D (left), D (middle) and A (right) and at cycle n = 1 (top row), n = 8 (middle row) and n = 15 (bottom row). The dashed black lines indicate the thresholds used to discriminate | (cid:105) from | (cid:105) . We observe a clear signature of leakage accumulation with the increasing number of error-detection cycles inthe single-shot readout histograms obtained at the end of each experiment. In Fig. S4 we show examples of thisaccumulation for D , D and A at cycles n = 1 , n = 8 and n = 15 . For dispersive readout, a transmon in state | (cid:105) induces a different frequency shift in the readout resonator compared to state | (cid:105) or | (cid:105) . The increased number ofdata points at n = 8 and n = 15 shown in Fig. S4, following a Gaussian distribution with a mean and standarddeviation different from those observed at n = 1 is thus a clear manifestation of leakage to the higher-excitedstates (mostly to | (cid:105) ). We believe that the dominant source of leakage in our processor are the CZ gates [S7, S8].However, the leakage rate L for each gate has not been experimentally characterized, e.g., by performing leakage-modified randomized benchmarking experiments [S9, S10]. This is because our CZ tune-up procedure is performedin a parity-check block unit. This maximizes the performance of the parity-check but makes the gate unfit forrandomized benchmarking protocols. We can estimate the population p L ( n ) in the leakage subspace L at cycle n from the single-shot readout histograms. We perform a fit of a triple Gaussian model to the histograms fromwhich we extract the voltage that allows for the best discrimination of | (cid:105) from | (cid:105) and | (cid:105) . The leaked population p L ( n ) is then given by the fraction of shots declared as | (cid:105) over the total number of shots. Assuming that leakageis only induced by the CZ gates (on the transmon being fluxed to perform the gate) and that each CZ gate hasthe same leakage rate L , we can use the Markovian model presented in Ref. S11 to estimate the L value leadingto the observed population p L ( n ) . This analysis gives a L estimate in the approximate range − for mosttransmons. However, we do not consider these estimates to be accurate due to the low fidelity with which | (cid:105) canbe distinguished from | (cid:105) and instead treat L as a free parameter in our simulations (see below).The histograms of the post-selected shots in Fig. S4 demonstrate that post-selection rejects runs where leak-age on those transmons occurred. Thus, while leakage may considerably impact the error-detection rate in theexperiment [S11], we do not expect it to significantly affect the fidelity of the logical initialization, and gates.3 Density-matrix simulations
Figure S5. Post-selected fraction P as a function of the number n of error-detection cycles. The experimental P (bluedots) is compared to numerical simulation under various models (solid curves). (a) Simulated P obtained by incrementaladdition of error sources starting from the no-error (Model 0, gray); qubit relaxation and dephasing (Model 1, yellow);extra dephasing due to flux noise away from the sweetspot (Model 2, amber); state preparation and measurement errors(Model 3, orange); and crosstalk due to residual ZZ interactions (Model 4, red). (b) Simulated P for Model 5 adding CZgate leakage with 4 different values of L , the leakage per CZ gate, assumed equal for all CZ gates. We perform numerical density-matrix simulations using the quantumsim package [S12] to study the impactof the expected error sources on the performance of the code. We focus on repetitive error detection using thepipelined scheme and with the logical qubit initialized in | L (cid:105) . In Fig. S5a, we show the post-selected fraction P ( n ) as a function of the number n of error-detection cycles for a series of models. Model 0 is a no-error model,which we take as the starting point of the comparison. Model 1 adds amplitude and phase damping experiencedby the transmon. Model 2 adds the increased dephasing away from the sweetspot arising from flux noise. Model3 adds residual qubit excitation and readout (SPAM) errors. Finally, Model 4 adds crosstalk due to the residual ZZ coupling during the coherent operations of the stabilizer measurement circuits. The details of each modeland their input parameters drawn from experiment are detailed below. We find that the dominant contributorsto the error-detection rate are SPAM errors and decoherence. However, we also observe that the noise sourcesincluded through Model 4 clearly fail to quantitatively capture the decay of the post-selected fraction observed inexperiment.We believe that an important factor behind the observed discrepancy is the presence of leakage, as suggestedby the single-shot readout histograms in Fig. S4. We consider the leakage per CZ gate L as a free parameterand assume the same value for all CZ gates. We simulate the post-selected fraction for a range of L values,shown in Fig. S5b. We observe that L ≈ produces a good match with experiment, suggesting that leakagesignificantly impacts the error-detection rate observed. This value of L is significantly higher than achieved inRef. S8, which used the same device. However, note that in this earlier experiment CZ gates were characterizedwhile keeping all other qubits in | (cid:105) . Spectator transmons with residual ZZ coupling to either of the transmonsinvolved in a CZ gate can increase L when not in | (cid:105) (which is certainly the case in the present experiment).Note that leakage may also be further induced by the measurement [S13], an effect that we do not consider in oursimulation. However, the assumption that all CZ gates have the same L , the approximations used in our models,and other error sources that we have not considered here may lead to an overestimation of the true L .Leakage is an important error source to consider in quantum error correction experiments of larger distance4codes, requiring either post-selection based on detection [S11] or the use of leakage reduction units [S14]. We leavethe detailed investigation of the exact leakage rates in our experiment and the mechanisms leading to them tofuture work. Error models
Lastly, we detail the error models used in the numerical simulations in Fig. S5.
Model 1
We take into account transmons decoherence by including an amplitude-damping channel parameterized by therelaxation time T and a phase-damping channel parameterized by the pure-dephasing time at the sweetspot T max φ = 1 T − T , where T is the echo dephasing time (see Table S1). The qutrit Kraus operators defining these channels aredetailed in Ref. S11 and we similarly introduce these channels during idling periods and symmetrically aroundeach single-qubit or two-qubit gate (each period lasting half the duration of the gate). Model 2
We consider the pure-dephasing rate /T φ = 2 π √ ln 2 AD φ + 1 /T max φ away from the sweetspot due to the fast-frequency components of the /f flux noise, where D φ is the flux sensitivity at a given qubit frequency and A isthe scaling parameter for the flux-noise spectral density. We use a √ A ≈ µ Φ , the average of the extracted √ A values for D , A and A obtained by fitting the measured decrease of T as a function of the applied flux bias,following the model described above. This allows us to estimate the dephasing time at the CZ interaction andparking frequencies, which then parameterize the applied amplitude-phase damping channel inserted during thoseoperations [S11]. We neglect the slow-frequency components of the flux noise due to the use of sudden Net Zeropulses, which echo out this noise to first order [S7, S8]. Model 3
We further include state-preparation and measurement errors. We consider residual qubit excitations, whereinstead of the transmon being initialized in | (cid:105) at the start of the experiment it is instead excited to | (cid:105) witha probability p e . We extract p e for each transmon from a double-Gaussian fit to the histogram of the single-shot readout voltages with the transmon nominally initialized in | (cid:105) [S4]. We model measurement errors viathe POVM operators M i = (cid:80) j =0 (cid:112) P ( i | j ) | j (cid:105) (cid:104) j | for i ∈ , , being the measurement outcome, while P ( i | j ) isthe probability of measuring the qubit in state | i (cid:105) when having prepared state | j (cid:105) . We extract the probability P (Q = | i (cid:105) ) = Tr (cid:16) M † i M i ρ (cid:17) of measuring qubit Q in state | i (cid:105) from simulation, where ρ is the density matrix,while application of the POVM transforms ρ → M i ρM † i /P (Q = | i (cid:105) ) . In our simulations we condition on thedetection of no error and thus we calculate P (Q = | (cid:105) ) and then apply M to the state ρ . We obtain P (0 | j ) for j ∈ , from the experimental assignment fidelity matrix [S15] (where a heralded initialization protocol was usedto prepare the qubits in | (cid:105) [S3]) and we assume P (0 |
2) = 0 , consistent with the observed histograms in Fig. S4.At the end of each experiment with n error-detection cycles we calculate the probability P fn of obtaining trivialsyndromes from the final measurements of the data qubits (see Results). From this and from the probability P n (A i = | (cid:105) ) of measuring ancilla A i in | (cid:105) at cycle n , we calculate the post-selected fraction of experimentsdefined as P ( n ) = P fn (cid:81) n (cid:81) i =1 P n ( A i = | (cid:105) ) .5 Model 4
We consider the crosstalk due to residual ZZ interactions between transmons. The CZ gates involved ina parity check are jointly calibrated to minimize phase errors for the whole check as one block (see Fig. S2).Instead of modeling this crosstalk as an always-on interaction and taking into account the details of the checkcalibration, we instead capture the net effect of this noise by including it as single-qubit and two-qubit phaseerrors in each CZ gate. This assumes that the crosstalk only occurs between transmons that are directly coupled,which the measured frequency shifts observed in Fig. S1 validate. We characterize the phases picked up duringthe CZ gates using k × k − Ramsey experiments for a check involving a total of k transmons (including theancilla). In each experiment, we perform a Ramsey experiment on one transmon labelled Q k . Q k is initialized ina maximal superposition using a R − π/ x pulse, while the remaining k − transmons are prepared in each of the k − computational states | l (cid:105) . Following this initialization, the parity check is performed, followed by a rotationof R − π/ φ (while the other transmons are rotated back to | (cid:105) ) and by a measurement of Q k . By varying theaxis of rotation φ , we extract the phase φ k Ram ( l ) picked up by Q k with the remaining transmons in state | l (cid:105) .We perform this procedure for each of the k transmons of the check, resulting in a total of k × k − measuredphases, which are arranged in a column vector (cid:126)φ Ram . We parameterize each CZ gate used in the parity check bya matrix diag (cid:0) , e iφ , e iφ , e iφ (cid:1) . The column vector (cid:126)φ CZ then contains all of the phases parameterizing eachof the k − CZ gates involved in the parity checks, with k = 3 for the Z D1 Z D3 and Z D2 Z D4 checks and k = 5 for the Z D1 Z D2 Z D3 Z D4 check. We can express each of the measured phases in the Ramsey experiment as a linearcombination of the acquired phases as a result of the CZ interactions between transmons, i.e., (cid:126)φ Ram = A(cid:126)φ CZ ,where the matrix A encodes the linear dependence. Given the measured (cid:126)φ Ram we perform an optimization to findthe closest (cid:126)φ CZ as given bymin (cid:126)φ CZ (cid:88) i (cid:88) j A ij (cid:126)φ CZ j − (cid:126)φ Ram i , subject to ≤ (cid:126)φ CZ j < π. The optimal (cid:126)φ CZ then captures the net effect of the ZZ crosstalk during the parity checks, which we include in thesimulation. We do not model phase errors accrued during the ancilla readout, since in our simulation we conditionon each ancilla being measured in | (cid:105) . Model 5
We model leakage due to CZ gates following the model and numerical implementation presented in Ref. S11.Here, we do not consider the phases picked up when non-leaked transmons interact with leaked ones (the leakage-conditional phases [S11]) and we set them to their ideal values. We also neglect higher-order leakage effects, such asexcitation to higher-excited states or leakage mobility. Thus, we only consider the exchange of population between | (cid:105) and | (cid:105) given by L , except for the CZ between A and D , where the population is instead exchanged with | (cid:105) as we use the | (cid:105) - | (cid:105) avoided crossing for this gate [S8].There remain several relevant error sources beyond those included in our numerical simulation. For example, wedo not include dephasing of data or other ancilla qubits induced by ancilla measurement, which we expect to be arelevant error source for comparing the performance of the pipelined and parallel schemes. Also, we only considerthe net effect of crosstalk due to residual ZZ interactions during coherent operations of the parity-check circuits,which we include via errors in the single-qubit and two-qubit phases in the CZ gates. Thus, we do not capturethe crosstalk present whenever an ancilla is projected to state | (cid:105) by the readout but declared to be in | (cid:105) instead.Furthermore, as ZZ crosstalk does not commute with the amplitude damping included during the execution ofthe circuit, we are not capturing the increased phase error rate that this leads to. [S1] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, App. Phys. Rev. , 021318 (2019). [S2] C. C. Bultink, B. Tarasinski, N. Haandbaek, S. Poletto, N. Haider, D. J. Michalak, A. Bruno, and L. DiCarlo, App.Phys. Lett. , 092601 (2018).[S3] D. Ristè, J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo, Phys. Rev. Lett. , 050507 (2012).[S4] T. Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potočnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger,C. Eichler, and A. Wallraff, Phys. Rev. App. , 054020 (2017).[S5] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio,M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. B , 180502(R) (2008).[S6] R. Sagastizabal, S. P. Premaratne, B. A. Klaver, M. A. Rol, V. Negîrneac, M. Moreira, X. Zou, S. Johri, N. Muthusub-ramanian, M. Beekman, C. Zachariadis, V. P. Ostroukh, N. Haider, A. Bruno, A. Y. Matsuura, and L. DiCarlo, (2020),arXiv:2012.03895 [quant-ph].[S7] M. A. Rol, F. Battistel, F. K. Malinowski, C. C. Bultink, B. M. Tarasinski, R. Vollmer, N. Haider, N. Muthusubra-manian, A. Bruno, B. M. Terhal, and L. DiCarlo, Phys. Rev. Lett. , 120502 (2019).[S8] V. Negîrneac, H. Ali, N. Muthusubramanian, F. Battistel, R. Sagastizabal, M. S. Moreira, J. F. Marques,W. Vlothuizen, M. Beekman, N. Haider, A. Bruno, and L. DiCarlo, (2020), arXiv:2008.07411 [quant-ph].[S9] C. J. Wood and J. M. Gambetta, Phys. Rev. A , 032306 (2018).[S10] S. Asaad, C. Dickel, S. Poletto, A. Bruno, N. K. Langford, M. A. Rol, D. Deurloo, and L. DiCarlo, npj Quantum Inf. , 16029 (2016).[S11] B. M. Varbanov, F. Battistel, B. M. Tarasinski, V. P. Ostroukh, T. E. O’Brien, L. DiCarlo, and B. M. Terhal, npjQuantum Information , 102 (2020).[S12] T. E. O’Brien, B. M. Tarasinski, and L. DiCarlo, npj Quantum Information (2017).[S13] D. Sank, Z. Chen, M. Khezri, J. Kelly, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. Fowler,E. Jeffrey, E. Lucero, A. Megrant, J. Mutus, M. Neeley, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan,A. Vainsencher, T. White, J. Wenner, A. N. Korotkov, and J. M. Martinis, Phys. Rev. Lett. , 190503 (2016).[S14] F. Battistel, B. M. Varbanov, and B. M. Terhal, (2021), arXiv:2102.08336 [quant-ph].[S15] J. Heinsoo, C. K. Andersen, A. Remm, S. Krinner, T. Walter, Y. Salathé, S. Gasparinetti, J.-C. Besse, A. Potočnik,A. Wallraff, and C. Eichler, Phys. Rev. App.10