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A leakage-resilient scheme for the measurement of stabilizer operators in superconducting quantum circuits

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A leakage-resilient scheme for the measurement of stabilizer operators in superconducting quantum circuits
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    a  r   X   i  v  :   1   4   0   6 .   2   4   0   4  v   1   [  q  u  a  n   t  -  p   h   ]   1   0   J  u  n   2   0   1   4 A Leakage-Resilient Scheme for the Measurement of Stabilizer Operators inSuperconducting Quantum Circuits Joydip Ghosh 1,  ∗  and Austin G. Fowler 2,3,  † 1 Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada  2 Department of Physics, University of California, Santa Barbara, California 93106, USA 3 Centre for Quantum Computation and Communication Technology,School of Physics, The University of Melbourne, Victoria 3010, Australia  (Dated: June 11, 2014)Superconducting qubits, while promising for scalability and long coherence times, contain morethan two energy levels, and therefore are susceptible to errors generated by the  leakage   of pop-ulation outside of the computational subspace. Such leakage errors are currently considered tobe a prominent roadblock towards fault-tolerant quantum computing with superconducting qubits.Fault-tolerant quantum computing using topological codes is based on sequential measurements of multi-qubit stabilizer operators. Here we propose a leakage-resilient scheme to perform repetitivemeasurements of multi-qubit stabilizer operators in superconducting circuits. Our protocol is basedon SWAP operations between data and ancilla qubits at the end of every cycle, requiring read-outand reset operations on every physical qubit in the system, and thereby preventing persistent leakageerrors from occurring. PACS numbers: 03.67.Lx, 03.67.Pp, 85.25.-j Recent years have witnessed remarkable progress inquantum computing with superconducting components,as far as the scalability and coherence times are con-cerned [1–3]. With major advances in designing scal- able qubits [1, 4, 5] and high-fidelity quantum gates [1, 6–9], a significant fraction of this research endeavor is now directed towards Fault-Tolerant Quantum Comput-ing (FTQC) with superconducting devices [3, 10]. FTQC via topological error-correcting codes, such as the sur-face code [11], requires sequential measurements of multi-qubit stabilizer operators [12]. The fluctuations in the measurement outcomes of such stabilizer operators gen-erate characteristic signatures for various discrete Paulierrors, thereby rendering the error-correcting scheme ro-bust against error models described by a Pauli channel[12].Superconducting qubits comprise more than two en-ergy levels that are often utilized to design optimal two-qubit gates, such as a controlled- σ z (CZ) gate [7–9, 13]. Apart from the decoherence, therefore, superconductingqubits also suffer from errors due to  leakage   of popu-lation outside of the computational subspace, often re-ferred to as leakage errors and currently considered tobe a prominent barrier towards FTQC [1, 14]. While decoherence-induced errors can be approximated by aPauli channel [15], leakage errors lack such a descrip-tion, consequently compromising the fault-tolerance forsuperconducting quantum circuits. It has been shownrecently with numerical simulations, how persistent leak-age errors in superconducting circuits destroy an ancilla-assisted qubit-measurement scheme producing randomfluctuations in the output of the ancilla qubit [14]. In this letter, we propose a multi-qubit stabilizer-measurement scheme, which is resilient to such leakage (a) D  | ψ   /U  cycle  U  cycle ... A  | 0   /  | 0   ... (b) | 0   /  × U  oddcycle × U  evencycle ... | ψ   /  × | 0  ×  ... FIG. 1. (a) Standard scheme for repetitive measurements of stabilizer operators. At the end of each cycle, the ancilla-qubitregister (denoted by ‘A’) gets measured and initialized, whilethe data-qubit register (demoted by ‘D’) never gets measured. U  cycle  denotes the sequence of gate operations required for thecorresponding stabilizer measurement. (b) Leakage-resilientscheme for repetitive measurements of stabilizer operators.SWAP operations are denoted by vertical lines connectingtwo ‘X’ symbols. Since the roles of data- and ancilla-registerget swapped after each cycle, the corresponding sequence of gate operations differ for odd- and even-numbered cycles, asdenoted by  U  oddcycle  and  U  evencycle . errors. A schematic diagram of the standard and ourleakage-resilient scheme for repetitive measurements of stabilizer operators is shown in Fig. 1. In the standard approach, the data qubit register never gets measured,and therefore, any leaked qubit in the data register re-mains leaked for many cycles until it undergoes relax-ation due to decoherence or leaks back to the compu-tational subspace. As shown in Ref. [14], such leaked data qubits generate random noises in the measurementoutcomes of ancilla qubits, effectively spoiling the entirescheme. In our leakage-resilient protocol, we supplementthe standard approach with SWAP operations at the end  2of every measurement-cycle as shown in Fig. 1b. TheSWAP gates exchange the roles of the data-register andthe ancilla-register, requiring us to measure and initial-ize every qubit in alternate cycles, effectively eliminatingthe possibility of leakage errors persisting without com-promising the stabilizer-measurement scheme. ✚✙✛✘✚✙✛✘✚✙✛✘✚✙✛✘ AAD D                 ❅  ❅  ❅  ❅  ❅  ❅  ❅  ❅  1 234ZZXX FIG. 2. (Color online) A Schematic diagram of the architec-ture for measuring two-qubit stabilizer operators. The circlesdenote superconducting qubits (data qubits denoted by ‘D’and ancilla qubits denoted by ‘A’), and lines denote requirednearest-neighbor couplings. The stabilizer operators,  ZZ   and XX  , are measured via qubit-3 and qubit-4 respectively.1  •  H   •  H  ... 2  •  H   •  H  ... 3  | 0   H     • •  H   | 0   ... 4  | 0   H   • •  H   | 0   ... FIG. 3. Standard scheme for repetitive measurements of two-qubit stabilizer operators  XX   and  ZZ  .  H   denotes theHadamard gate, the vertical lines connected by filled circlesdenote CZ gates, and the numbers denote the indices for thequbits in the same order as shown in Fig. 2. The gates inside the dashed rectangle represent  U  cycle  in Fig. 1a. Here, we illustrate the advantage of our protocolassuming a model where both the data- and ancilla-registers consist of two superconducting qubits, and wemeasure two 2-qubit stabilizer operators,  XX   and  ZZ  ,repetitively for many cycles. Fig. 2 shows a schematic di-agram of our architecture, where four nearest-neighbor-coupled superconducting qubits form a 1D circular lat-tice. The standard scheme for stabilizer measurement forthis model is shown in Fig. 3 [15]. As mentioned earlier, in this protocol the data-qubit register (qubits 1 and 2)remains unmeasured throughout the routine, while theancilla-qubit register (qubits 3 and 4) is measured andinitialized every cycle. Note that, if we encode any of the four Bell states in the data register, then under idealgate operations, the final states before reinitialization aregiven by,  | 00  + | 11 √  2  ⊗| 00 −→  | 00  + | 11 √  2  ⊗| 00   | 00 −| 11 √  2  ⊗| 00 −→  | 00 −| 11 √  2  ⊗| 01   | 01  + | 10 √  2  ⊗| 00 −→  | 01  + | 10 √  2  ⊗| 10   | 01 −| 10 √  2  ⊗| 00 −→  | 01 −| 10 √  2  ⊗| 11  , (1)which essentially means that under circuit 3, the four dif- ferent Bell states in the data register are stabilized by theoperators  XX   and  ZZ   as their simultaneous eigenstatescorrespondingto the four different possible combinations.Without any loss of generality, in this work we assumethe Bell state ( | 00  + | 11  ) / √  2 as our encoded initial statein the data register and compare our protocol against thestandard scheme simulating the circuits numerically. 1  | 0  × •  H   •  H   ×  H   • •  H  . . . 2  | 0  × •  H   •  H   ×  H   • •  H  . . . 3  ×  H     • •  H   | 0  × •  H   •  H  . . . 4  ×  H   • •  H   | 0  × •  H   •  H  . . . FIG. 4. The SWAP-based leakage-resilient scheme for repeti-tive measurements of two-qubit  XX   and  ZZ   stabilizer oper-ators. In order to render this repetitive stabilizer measure-ment scheme leakage-resilient, we introduce SWAP op-erations between the data and ancilla registers as shownin Fig. 4. SWAP operations between two quantum reg-isters essentially mean sequential SWAP gates betweenthe  k th qubits in both the registers, for all  k . In order tonot transfer or propagate the leakage errors, we expressSWAP operations (between  | 0   and an arbitrary state | ψ  ) as, | ψ  × •  H   •  H  ≡| 0  ×  H     •  H   • (2)We note that substituting SWAP gates with two CZ gatesalong with some single-qubit Hadamard operations, infact, introduces a negligible overhead in circuit depthbecause of internal cancellations, and the reduced cir-cuit for our SWAP-based stabilizer measurement schemeis shown in Fig. 5. First we discuss how to construct the model for leakage errors for CZ gates in superconduct-ing circuits and then compare our scheme (5) with thestandard scheme (3) via numerical simulations.  3 1  | 0   H   • •  H   •  H   •  H  2  | 0   H   •  H   • •  H   •  H   •  H   • • •  H  3    •  H   •  H   | 0   H   • •  H  ... 4  •  H   • • •  H   | 0   H   •  H   • •  H   •  H  ... FIG. 5. The leakage-resilient scheme for stabilizer measurement where SWAP gates are replaced by CZ and Hadamard gatesas shown in Eq.(2). The gates inside the left dashed rectangle are repeated for odd cycles, while the gates inside the rightrectangle are repeated for even cycles. In order to estimate the dominant contribution for theleakage errors, we model the superconducting qubits asthree-level systems (or qutrits) and parametrize the cor-responding single- and two-qutrit quantum gates, as dis-cussed in Ref. [14]. The Hadamard gate for a qutrit (inthe basis  {| 0  , | 1  , | 2 } ) is given by, H   =  1 √  21 √  2  0 1 √  2  −  1 √  2  00 0 1  ,  (3)which is equivalent to the standard Hadamard gate act-ing on the computational subspace and an Identity act-ing on the  | 2  state. Single-qubit gates can be done withfidelities an order of magnitude higher than that of two-qubit gates [6, 16], and will therefore be assumed to be ideal throughout this work.The CZ gate considered here is performed betweentwo superconducting qubits by tuning and detuning thequbit-frequencies so as to mix the population in theavoided crossing of   | 11   and  | 20   eigenstates, therebyacquiring the phase ( π  for CZ) across the  | 11   state[1, 7, 8, 13]. The dominant error-mechanism for this avoided-crossing-basedCZ gate is generated by the resid-ual non-adiabatic population-transfer in single-excitation( {| 01  , | 10 } ) and double-excitation ( {| 02  , | 20  , | 11 } )subspaces. As shown in Ref. [14], such a CZ gate can beparametrized with two generators,  S   (generates the idealpart of the CZ gate) and  S  ′  (generates the first-ordererror terms), where the parameters are chosen from afull-scale Hamiltonian simulation. The generators  S   and S  ′  can be thought of as Hermitian matrices generatingthe non-ideal CZ gate, also obtainable from a full-scalesimulation of the time-dependent control-Hamiltonian, U  CZ  =  e i ( S  + S  ′ ) .  (4)In the two-qutrit tensor-product basis  | 00  , | 01 | 02  , | 10  , | 11  , | 12  , | 20  , | 21  , | 22   , the generators  S   and  S  ′  are given by [14], S   =  0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0  ξ  1  0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0  π  0 0 0 00 0 0 0 0  ξ  2  0 0 00 0 0 0 0 0  π  0 00 0 0 0 0 0 0  ξ  3  00 0 0 0 0 0 0 0  ξ  4  , S  ′  =  0 0 0 0 0 0 0 0 00  ζ  1  0  iχ 1 e iφ 1 0 0 0 0 00 0 0 0  iχ 2 e iφ 2 0 0 0 00  − iχ 1 e − iφ 1 0  ζ  2  0 0 0 0 00 0  − iχ 2 e − iφ 2 0  ζ  3  0  iχ 3 e iφ 3 0 00 0 0 0 0 0 0  iχ 4 e iφ 4 00 0 0 0  − iχ 3 e − iφ 3 0  ζ  4  0 00 0 0 0 0  − iχ 4 e − iφ 4 0 0 00 0 0 0 0 0 0 0 0  . (5)Since the CZ gate is based on mixing of population inthe avoided level-crossing between  | 11   and  | 20   states,so we have   11 | S  | 11   =   20 | S  | 20   =  π . The remainingparameters  ξ  1 − 4  are the dynamical phases acquired bythe corresponding basis states and are assumed to berandom numbers between 0 and 2 π  for our simulation.The parameters in  S  ′ ,  χ 1 − 4  and  ζ  1 − 4 , are assumed to besmall ( ∼ 10 − 2 ), while the angles  φ 1 − 4  take arbitrary val-ues between 0 and 2 π . The parameters  χ 1  and  χ 4  causepopulation-transfer in the single- and triple-excitationsubspaces respectively, while  χ 2 , 3  are responsible formixing of population in the double-excitation subspace.Since the population transfer probabilities in {| 01  , | 10 } (single-excitation subspace),  {| 02  , | 11  , | 20 }  (double-excitation subspace), and  {| 12  , | 21 }  (triple-excitationsubspace) subspaces scalewith | χ i | 2 [14], our choiceof pa-rameters bounds the intrinsic gate errors to about 10 − 4 ,which is consistent with what has been obtained from a  4full-scale simulation of the control-Hamiltonian for cur-rent gate-design schemes [7, 8]. 02004006008001000012(a)    R  e  a   d  o  u   t  v  a   l  u  e   0200400600800100000.51Cycle    P  r  o   b  a   b   i   l   i   t  y (b)   success of predictionleakagequbit 3qubit 4 FIG. 6. (Color online) Results for the standard approachto repetitive measurements of two-qubit stabilizer operators, XX   and  ZZ  . (a) The readouts of two ancilla qubits (qubits3 and 4) are shown for various consecutive cycles. The tworegions having random rapid fluctuations indicate that eitherof the two data qubits or both is leaked during those measure-ment cycles. (b) The solid black curve shows the probabilitythat either of the two data qubits is in the  | 2   state at the endof every measurement cycle. The regions having almost unitleakage probability exactly coincide with the randomly fluc-tuating ancilla outcomes in (a). The solid gray (green) curveshows the overlap between the state encoded in the data reg-ister and the prediction of it from the corresponding ancillaoutputs at the end of each cycle. 02004006008001000012(a)    R  e  a   d  o  u   t  v  a   l  u  e   0200400600800100001Cycle    P  r  o   b  a   b   i   l   i   t  y (b)   qubit 3qubit 4success of predictionleakage FIG. 7. (Color online) Results for our SWAP-based approachto repetitive measurements of two-qubit stabilizer operators, XX   and  ZZ  . (a) The readouts of two ancilla qubits (qubits3 and 4) are shown for various consecutive cycles. Unlike theresults obtained for the standard scheme, no rapidly fluctu-ating signals are observed in this case. (b) The solid blackcurve shows the probability that either of two data qubitsis in  | 2   state at the end of every measurement cycle. Nounit probability for leakage is observed for consecutive cycles.Peaks indicate probabilities for isolated leakage errors. Thesolid gray (green) curve shows the overlap between the stateencoded in the data register and the prediction of it from thecorresponding ancilla outputs at the end of each cycle. Having parametrized the required quantum gates, wenow simulate the quantum circuits shown in Fig. 3 andin Fig. 5. In order to compare our scheme against thestandard approach, we here assume the no-decoherencelimit (i.e.,  T  1 , 2  → ∞ ). We emphasize that the intro-duction of decoherence in our calculation only amountsto some more randomly occurred ‘steps’ in the readoutvalues that are neither relevant for, nor influence the con-clusions of this work.The simulation of the standard approach (circuitshown in Fig. 3) is shown in Fig. 6. The measurement outcomes from the ancilla qubits (qubits 3 and 4) areshown in Fig. 6a for many consecutive cycles. In the pres-ence of leakage errors, we observe regions having randomand rapid fluctuations in the ancilla outcomes, a charac-teristic signature for a leakage error on either of the dataqubits. Such a signature has been observed in Ref. [14] in the context of a repetitive ancilla-assisted measurementof a single data qubit, and with a rigorous analysis, itssrcin has been conclusively associated with the leakageerror in the data qubit. In order to show the connectionbetween this noise and the data-qubit leakage error moreexplicitly, in Fig. 6b we plot the probability (black curve)that either of the data qubits is leaked at the end of ev-ery cycle. For cycles with rapid fluctuations in the out-comes of the ancilla qubits, we also observe a near-unitprobability for leakage errors, clearly signifying the de-structive consequences of data-qubit-leakage for the stan-dard stabilizer-measurement scheme. Using Eq.(1), it is possible to predict the quantum state of the data-qubitregister, based on the outcomes obtained in the ancillaqubits [17]. In Fig. 6b, we also plot the probability of  success for such predictions (green curve) at the end of each cycle. It is observed that this success-probabilityis enormously compromised for many consecutive cycleswhere the data qubits remain leaked, essentially indicat-ing a catastrophic failure of the standard scheme underleakage errors.We also simulate our SWAP-based scheme (circuitshown in Fig. 5) and the results are shown in Fig. 7. Fig. 7a shows the outputs of the ancilla qubits for manyconsecutive cycles and, unlike standard protocol, norapid random fluctuations are observed in the ancilla out-comes for this case. The probability of a leakage error inthe data register is shown (black curve) in Fig. 7b. Incontrast with the standard scheme, we only observe iso-lated peaks, which means even if there is a data-qubitleakage-event in one cycle, it gets completely removed insubsequent measurement cycles, because in this approachwe are measuring and initializing every physical qubit inthe system in alternate cycles. Fig. 7b also shows theprobability of successful prediction (green curve) of thetwo-qubit state encoded in the data-qubit register. No-tice that the predictions only get compromised wheneverthere is a leakage error either in the data-register or inthe ancilla-register, and since the leakage errors are iso-lated, so are the failure probabilities. The discrete well-separated peaks in the leakage error plot (or the down-ward peaks in the curve showing the probability of suc-cessful predictions) explicitly signify the resilience of ourSWAP-based scheme against leakage errors. The densityof these peaks is determined by the probability of leakage  5errors occurring in the physical qubits, and with betteroptimization techniques it is possible to suppress such er-rors even further [9]. In the standard scheme, however, since the data-register never gets measured, even a singledata-qubit leakage-eventruins the entire scheme, becausethe leaked qubit remains leaked either until it undergoesrelaxationto the ground state due to decoherence or leaksback to the computational subspace.In summary, we have devised a leakage-resilient pro-tocol for repetitive measurements of multi-qubit stabi-lizer operators in superconducting circuits. In the stan-dard approach, the data-qubit register never gets mea-sured, and therefore a leaked data qubit remains leakedfor many consecutive cycles, producing random fluctu-ations in the measurement of the ancilla register. Ourscheme relies on SWAP operations between the data-and the ancilla-register, thereby changing the role of these registers each cycle. This requires us to performthe readout and reset operations on every qubit in thequantum circuit, which essentially eliminates the possi-bility of long-lived leakage errors occurring. Sequentialmeasurements of such multi-qubit stabilizer operatorsarekey steps to performing FTQC using Topological error-correcting codes, such as the surface code. Our analy-sis motivates the use of SWAP-based gate sequences forstabilizer-based error-correctingschemes, rendering themfault-tolerant against the leakage errors.We thank Michael Geller for helpful discussions. Thisresearch was funded by the US Office of the Director of National Intelligence (ODNI), Intelligence Advanced Re-search Projects Activity (IARPA), through the US ArmyResearch Office grant No. W911NF-10-1-0334. All state-ments of fact, opinion or conclusions contained herein arethose of the authors and should not be construed as repre-senting the official views or policies of IARPA, the ODNI,or the US Government. J.G. gratefully acknowledges thefinancial support from NSERC, AITF and University of Calgary’s Eyes High Fellowship Program. ∗ ghoshj@ucalgary.ca † austingfowler@gmail.com[1] R. Barends, J. Kelly, A. 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Wellstood,Phys. Rev. Lett.  91 , 167005 (2003).[14] J. Ghosh, A. G. Fowler, J. M. Martinis, and M. R. Geller,Phys. Rev. A  88 , 062329 (2013).[15] M. R. Geller and Z. Zhou,Phys. Rev. A  88 , 012314 (2013).[16] J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen,B. Chiaro, A. Dunsworth, A. G. Fowler, I.-C. Hoi, E. Jef-frey, A. Megrant, J. Mutus, C. Neill, P. J. J. O‘Malley,C. Quintana, P. Roushan, D. Sank, A. Vainsencher,J. Wenner, T. C. White, A. N. Cleland, and J. M. Marti-nis, ArXiv e-prints (2014), arXiv:1403.0035 [quant-ph].[17] If the ancilla is in  | 2   state, we need to map it to somecomputational state for the purpose of such a prediction,and we map the  | 2   state in ancilla qubit to the  | 1   statein this work. While the choice is arbitrary, it does notchange the conclusions of this work.
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