Realizing Repeated Quantum Error Correction in a Distance ...

1 Realizing Repeated Quantum Error Correction in a Distance -Three Surface CodeSebastian Krinner,1, Nathan Lacroix,1, Ants Remm,1 Agustin Di Paolo,2 Elie Genois,2 Catherine Leroux,2 Christoph Hellings,1 Stefania Lazar,1 Francois Swiadek,1 Johannes Herrmann,1 Graham J. Norris,1 ChristianKraglund Andersen,1, Markus M uller,3, 4 Alexandre Blais,2, 5 Christopher Eichler,1and Andreas Wallraff1, 61 Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland2 Institut Quantique and D epartement de Physique,Universit e de Sherbrooke, Sherbrooke J1K2R1 Qu ebec, Canada3 Institute for Quantum Information, RWTH Aachen University, Aachen D-52056, Germany4 Peter Gr unberg Institute, Theoretical Nanoelectronics,Forschungszentrum J ulich, J ulich D-52425, Germany5 Canadian Institute for Advanced Research, Toronto, ON, Canada6 Quantum Center, ETH Zurich, 8093 Zurich, Switzerland(Dated.)

2 December 8, 2021) Quantum computers hold the promise of solving computational problems which are intractableusing conventional methods [1]. For fault-tolerant operation Quantum computers must correct errorsoccurring due to unavoidable decoherence and limited control accuracy [2]. Here, we demonstratequantum Error Correction using the surface code, which is known for its exceptionally high toleranceto errors [3 6]. Using 17 physical qubits in a superconducting circuit we encode Quantum informationin a Distance -three logical qubit building up on recent Distance -two Error detection experiments [7 9]. In an Error Correction cycle taking only s, we demonstrate the preservation of four cardinalstates of the logical qubit. Repeatedly executing the cycle, we measure and decode both bit- andphase-flip Error syndromes using a minimum-weight perfect-matching algorithm in an Error -model-free approach and apply corrections in postprocessing.

3 We find a low Error probability of 3 % percycle when rejecting experimental runs in which leakage is detected. The measured characteristicsof our device agree well with a numerical model. Our demonstration of Repeated , fast and high-performance Quantum Error Correction cycles, together with recent advances in ion traps [10], supportour understanding that fault-tolerant Quantum computation will be practically surface code [4, 11] is a planar realization of Ki-taev s toric code [3] which uses topological features ofa qubit lattice to correct errors in Quantum informationprocessing systems. This code is a prominent contenderto reach fault-tolerant Quantum computation because ofits high Error threshold of about 1 % against Quantum cir-cuit noise [5, 12] and its compatibility with 2D architec-tures.

4 The surface code belongs to the family of stabilizercodes [13, 14] which encode Quantum information into ajoint subspace of definite parities on a set of physical dataqubits to form a logical qubit. Errors are detected usingmeasurements of auxiliary qubits to extract parity infor-mation without collapsing the logical qubit state. Thefault-tolerant operation of a Quantum computer requiresrepeated detection and Correction of both bit- and phase-flip errors on data qubits. With an increasing numberof physical qubits and thus an increasing code distancedthe number of errorsb(d 1)/2cwhich can at leastbe detected and corrected per Error - Correction cycle in-creases, making the code more resilient when Error ratesare sufficiently Correction limited to a single type of Error hasbeen realized with repetition codes in nuclear magneticresonance [15], trapped ions [16], nitrogen-vacancy cen- These authors contributed equally to this work.

5 Current affiliation: QuTech and Kavli Institute for Nanoscience,Delft University of Technology, Delft 2628 CJ, Netherlandsters [17] and superconducting circuits [9, 18]. In single-cycle experiments, fault-tolerant stabilizer measurementsand Correction of both types of errors have been demon-strated with the five-qubit code and the Bacon-Shor code[19 22]. Recently, Error detection in a Distance -two sur-face code has been realized with seven qubits [7 9], andonly very recently, Repeated stabilizer-based Error correc-tion has been demonstrated with a Distance -three colorcode in a trapped ion system [10]. Correction of both bit- and phase-flip errors requiresat least a Distance -three code. In combination with fault-tolerant circuits for Error syndrome measurements, thisguarantees that any single Error on any of the constituentdata and auxiliary qubits or operations can be corrected[14, 23].

6 While the work we discuss here focuses on digi-tal encoding of Quantum information, continuous variableencoding, for example in harmonic oscillator states, con-stitutes an alternative approach to Quantum Error cor-rection (QEC), see for example Refs. 24 Distance -THREE SURFACE CODE INSUPERCONDUCTING CIRCUITSE xperimentally Realizing a Distance -three surface coderequires nine data qubits and eight auxiliary qubits, alsoreferred to in the literature as ancilla or measurementqubits [23, 28, 29]. The qubits are arranged in a di-agonal, planar square lattice, the edges of which [quant-ph] 7 Dec 20212 CouplingElementsQubitsReadoutResonatorsP urcellFiltersFeedLinesFluxLinesDriveLine sc1 mmD7D4D1D8D5D2D9D6D3X4X3X2X1Z1Z2Z3Z42-st ate3-stateabZ1Z3Z2Z4X1X2D2X3D3X4D4D5D6D7 D8D9D1Z^LX^LS^Z1S^Z2S^X1S^X2S^X3S^X4S^Z3 S^Z4dFIG.

7 1. Device concept, architecture and representation of the Distance -three surface code consistingof data qubits (red circles), Z-type (green circles) and X-type auxiliary qubits (blue circles), with their connectivity indicatedby gray lines. The data qubits participating in the weight-three logical operators ZLand XLare indicated by solid blacklines. Green (blue) plaquettes indicate X-type (Z-type) stabilizer micrograph of the device realizingthe concept inawith 17 transmon qubits, see legend for circuit elements and text for details. The qubit lattice is rotatedby 45 degrees with respect arrangement in three distinct bands for idling data qubits (red circles), idlingZ/X-type auxiliary qubits (green/blue circles), and readout resonators (violet open circles).

8 The qubit-frequency tuning rangesare indicated by vertical distributions (integrated histograms) of single-qubit gate (pink), simultaneoustwo-qubit gate (cyan), two-state (red) and three-state readout errors (light red).shown in gray in the schematic Fig. 1a. The data qubitsDj,j= , (red dots) form a 3 3 array and are in-terlaced with auxiliary qubitsAi, labeled Xi(blue) andZi,i= (green). We realized this arrangementin a superconducting circuit using 17 transmon qubits[30] (yellow) capacitively coupled to each other along theedges of the square array with 1 mm long coplanarwaveguide segments (turquoise), see Fig. 1b. We discussthe fabrication of this device in Appendix the auxiliary qubits Xiand Ziwe measure theparity of the neighboring two or four data qubits Dj,which are located at the vertices of the blue and greenplaquettes, in the X or Z basis (Fig.)

9 1a,b). In a Z-basismeasurement, if an odd number of the data qubits in-volved in the parity operator under consideration is inthe|1 -state the auxiliary qubit state is flipped. Onthe other hand, if an even number of data qubits is in|1 the auxiliary qubit state remains unchanged. Theequivalent is true in the X basis for an even or oddnumber of data qubits in the| -state. Here,|0 ,|1 are the transmon qubit ground and first excited states,and| = (|0 |1 )/ 2 are their superpositions. Tomap the parity of the data qubits Djonto the corre-sponding auxiliary qubit, we effectively use a sequenceof controlled-not gates with the data qubits as controland the auxiliary qubit as target, and subsequently mea-3sure the state of the auxiliary qubit in the Z basis usingsingle-shot readout.

10 In the Z basis, a single bit-flip errorof any individual data qubit leads to a change of parity,as does a single phase-flip in the X basis. Hence, mea-surements of changes of data-qubit parities allow us todetect and identify phase-flip or bit-flip errors as longas they occur sufficiently rarely [31]. These parity mea-surements are also referred to as stabilizer measurements[13, 14]. The corresponding mutually-commuting weight-two (or weight-four) stabilizer operators SXi= 2(4)j=1 Xjand SZi= 2(4)j=1 Zjof the surface code are products oftwo (or four) Pauli- Xor - Zoperators of the data qubitsjlocated at the vertices of a given data-qubit outcomessAi= 1 of individual stabilizers SAiare extracted from the observed change of auxiliaryqubit state from one cycle to the next and indicate evenor odd parity, our experiments, the stabilizer gate sequence is re-alized as two or four controlled-phase (CZ) gates [32 34] (see Methods Section A) between data and auxiliaryqubits, operated in a high and a low frequency band (seeFig.)