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Repeated Quantum Error Detection in a Surface Code
Christian Kraglund Andersen, Ants Remm, Stefania Lazar, Sebastian Krinner, Nathan Lacroix, Graham J. Norris, Mihai Gabureac, Christopher Eichler, Andreas Wallraff
TL;DR
Fault-tolerant quantum computing requires experimentally demonstrated quantum error correction, motivating scalable surface-code implementations. This work repeatedly detects single errors with a seven-qubit superconducting surface code and reports high logical-state fidelity and conditioned lifetimes exceeding those of constituent qubits.
Problem
Fault-tolerant quantum computing requires quantum error correction, but experimentally realizing repeated, high-fidelity ancilla-based stabilizer measurements remains essential for surface-code implementations.
Method
The experiment encodes logical cardinal states in four data qubits and repeatedly measures three stabilizers using three ancilla qubits and low-crosstalk readout.
Results
96.1% average logical fidelity was achieved for the cardinal logical states, while conditioned logical lifetime and coherence time exceeded those of the best physical qubit.
Takeaways & Limitations
The seven-qubit demonstration provides a surface-code error-detection implementation using techniques described as extensible to larger codes and relevant to fault-tolerant quantum computing.
Takeaways & Limitations
At code distance d = 2, the experiment detects only one error per round and cannot unambiguously identify every detected error.
Abstract
from arXiv · showhide
The realization of quantum error correction is an essential ingredient for reaching the full potential of fault-tolerant universal quantum computation. Using a range of different schemes, logical qubits can be redundantly encoded in a set of physical qubits. One such scalable approach is based on the surface code. Here we experimentally implement its smallest viable instance, capable of repeatedly detecting any single error using seven superconducting qubits, four data qubits and three ancilla qubits. Using high-fidelity ancilla-based stabilizer measurements we initialize the cardinal states of the encoded logical qubit with an average logical fidelity of 96.1%. We then repeatedly check for errors using the stabilizer readout and observe that the logical quantum state is preserved with a lifetime and coherence time longer than those of any of the constituent qubits when no errors are detected. Our demonstration of error detection with its resulting enhancement of the conditioned logical qubit coherence times in a 7-qubit surface code is an important step indicating a promising route towards the realization of quantum error correction in the surface code.
INTRODUCTION
The seven-qubit surface code uses repeated ancilla-based stabilizer measurements to detect errors while encoding logical information in a protected code space. This smallest instance can detect one error per measurement round, but cannot identify its type unambiguously.
- Stabilizer codes repeatedly measure commuting multi-qubit operators, projecting qubits into a shared code space.
- Surface-code syndromes arise from changes in ancilla-measured stabilizer outcomes, with a d × d data-qubit grid and d^2−1 ancillas.
- For code distance d = 2, the code detects only one error per stabilizer round and cannot unambiguously distinguish some errors.For example, X errors on D1 and D3 produce the same syndrome.
- The smallest seven-qubit code uses four data qubits and three ancillas to measure two Z-type and one X-type stabilizer.
- The experiment prepares logical cardinal states probabilistically through one stabilizer-measurement cycle conditioned on all ancillas yielding |0⟩, then performs repeated error detection.
IMPLEMENTATION
The experiment implements the seven-qubit surface code on a superconducting device with four data qubits and three ancillas. Its pipelined stabilizer measurements combine entangling gates, basis changes, and multiplexed readout while limiting unwanted interactions and crosstalk.
- Four data qubits D1–D4 carry the logical qubit, while ancillas A1–A3 measure the three surface-code stabilizers.
- The X-type stabilizer is measured with basis-change pulses and ancilla A2; simultaneous Z-type measurements use A1 and A3.
- The seven-qubit device uses transmon qubits, coupling resonators, individual flux and charge lines, readout resonators, and Purcell filters.
- Multiplexed readout probes each feedline with frequency-multiplexed pulses, while Purcell filters suppress readout crosstalk and protect against decay.
- The central ancilla A2 connects to four neighboring qubits, enabling the connectivity required by the seven-qubit surface-code geometry.
- The gate sequence is implemented on a cryogenic device characterized with time-domain and randomized-benchmarking methods.
RESULTS
The experiment prepares logical states in a seven-qubit surface code and repeatedly detects errors through ancilla-based stabilizer measurements. Conditioned on detecting no errors, the encoded state retains logical lifetimes and coherence times exceeding those of the best physical qubits.
- Stabilizer measurements: 95.0%, 83.5% and 91.8% are the success probabilities for measuring ZD1ZD3, ZD1ZD2ZD3ZD4 and ZD2ZD4, respectively.The probabilities are calculated from overlap with the ideal parity-measurement outcomes.
- Logical-state preparation: A probabilistic encoding scheme projects initialized product states onto target logical states when all syndrome results are |0⟩.The scheme prepares |0⟩L, |1⟩L, |+⟩L and |−⟩L using one stabilizer-measurement cycle.
- Logical-state preparation: 25.1% is the measured probability of obtaining |0⟩ for all ancilla qubits, versus 50% ideally, while the physical-state fidelity is Fphys = 70.3%.The reduced physical fidelity is dominated by qubit decoherence, with a 5-degree phase error accumulated over the 1.92 µs cycle.
- Logical-state preparation: 98.2%, 94.2%, 94.8% and 97.3% are the logical fidelities for |0⟩L, |+⟩L, |−⟩L and |1⟩L, respectively.These fidelities are obtained after projecting the physical density matrix onto the logical subspace.
- Repeated error detection: 62.7 ± 9.4 µs and 72.5 ± 32.9 µs are the measured logical lifetime and coherence time, exceeding the best physical-qubit values of 16.8 µs and 21.5 µs.The measurements are conditioned on detecting no error and on the final data-qubit stabilizer conditions being satisfied.
- Repeated error detection: 3.1% ± 0.45% and 2.6 ± 1.3% are the logical XL and ZL error probabilities per stabilizer-measurement cycle.Numerical simulations reproduce the observed logical expectation-value decays within experimental error bars, with simulated times of 44.2 µs and 59.6 µs.
DISCUSSION
The experiment implemented a seven-qubit surface code for repeated quantum error detection and prepared logical cardinal states with high fidelity. Conditioned on detecting no errors, the logical qubit showed extended lifetime and coherence time, supporting scalability toward fault-tolerant quantum computing.
- DISCUSSION: 96.1% average fidelity was achieved for preparing the logical states |0⟩L, |1⟩L, |+⟩L and |−⟩L.The probability of remaining within the logical subspace was around 70% because of accumulated errors during stabilizer measurement cycles.
- DISCUSSION: Repeated error detection extended the logical qubit’s lifetime and coherence time when no errors were detected.The data were postselected on ancilla outcomes and final data-qubit measurements satisfying the stabilizer conditions.
- DISCUSSION: The gate sequence is extensible to larger surface codes and supports superconducting devices as a route toward fault-tolerant quantum computing.The implementation uses repeated ancilla measurements with minimal detrimental effects on the data qubits.
Appendix A: Pulse sequence
The pulse sequence implements repeated stabilizer measurements with dynamical decoupling and net-zero flux pulses. These waveform choices support controlled qubit interactions while limiting memory effects.
- Appendix A: Pulse sequence: Dynamical decoupling pulses are applied to ancilla qubits during stabilizer measurements and to data qubits between stabilizer cycles.All qubits are parked at their upper sweetspot during the sequence.
- Appendix A: Pulse sequence: The net-zero flux pulse has zero integrated area, limiting memory effects on the qubits.The pulse shape is used while the qubits remain parked at their upper sweetspot.
Appendix B: Device Fabrication and Characterization
The device is a seven-qubit superconducting circuit, and its parameters and control waveforms were characterized experimentally. The appendix also identifies the pulse components used during stabilizer measurement.
- Appendix B: Device Fabrication and Characterization: Seven superconducting qubits are coupled in the geometry required for the surface-code experiment.The device is fabricated on high-resistivity intrinsic silicon using a 150 nm niobium film, photolithography, reactive ion etching, and airbridges.
- Appendix B: Device Fabrication and Characterization: Qubit parameters were extracted using standard spectroscopy and time-domain methods.Residual ZZ coupling was measured with Ramsey experiments, and randomized benchmarking characterized single-qubit Clifford performance.
- Appendix B: Device Fabrication and Characterization: AWG waveforms combine microwave gate pulses, readout pulses, and zero-area flux pulses during stabilizer measurement cycles.Shaded areas indicate the interacting qubit pairs during flux pulses.
Appendix C: Readout Characterization
Multiplexed readout characterizes all seven qubits while allowing selective addressing of subsets. Ancilla readout introduces less than 0.3% phase error on any data qubit.
- Appendix C: Readout Characterization: Multiplexed readout selectively addresses any subset of the seven qubits using qubit-dependent pulse and integration durations.D1, D2, A1, A2 and A3 use 200 ns pulses with 300 ns integration, while D3 and D4 use 300 ns pulses with 400 ns integration.
- Appendix C: Readout Characterization: Measurement-induced dephasing is quantified by interleaving data-qubit Ramsey pulses with ancilla readout.Additional dephasing rates Γij are obtained for data qubits during readout pulses.
- Appendix C: Readout Characterization: Less than 0.3% phase error is induced on any data qubit by measuring the ancilla qubits.The phase-error probability is calculated from the additional dephasing rate and the readout time.
Appendix D: Experimental setup
The setup characterizes qubit connectivity, gate performance, and readout behavior in a seven-qubit superconducting device using Ramsey and randomized-benchmarking measurements.
- Cryogenic control and readout: The seven-qubit device uses microwave and flux control through filtered, attenuated cryogenic wiring and arbitrary waveform generators.The control AWGs provide eight channels and a 2.4 GSa/s sampling rate.
- Device characterization: Residual ZZ-coupling is measured between all qubit pairs using Ramsey experiments with the pulsed qubit prepared in either ground or excited state.Pairs labeled gray have no direct coupling.
- Device characterization: Single-qubit errors per Clifford and average controlled-phase-gate errors are characterized using randomized benchmarking.Single-qubit errors are measured for each qubit, while controlled-phase errors use interleaved randomized benchmarking.
- Cryogenic control and readout: Output signals are filtered, amplified by cryogenic and room-temperature amplifiers, downconverted, and processed using weighted integration units.The setup includes HEMT and WAMP amplification stages before signal processing.
Appendix E: Numerical Simulations
The simulations model seven-qubit dynamics with a time-dependent Hamiltonian, incoherent processes, residual ZZ coupling, and ancilla-readout disturbance.
- System dynamics: The seven-qubit dynamics are modeled numerically with a master equation whose time-dependent Hamiltonian represents the applied gate sequence.The Hamiltonian is treated as piece-wise constant to simplify the system’s time evolution.
- Error modeling: Residual ZZ coupling is included in the Hamiltonian, while incoherent errors are represented by Lindblad terms.The residual coupling parameter is taken from measurements shown in Fig. 7.
- Error modeling: Qubit lifetime and Ramsey decoherence time are used to parameterize the modeled incoherent processes.These parameters are denoted T1,i and T2,i for qubit i.
- Ancilla measurement: Ancilla measurements are simulated with POVM operators based on experimentally determined readout assignment probabilities.Minimal-disturbance POVMs are used, and outcome probabilities are computed with the trace rule.
- Ancilla measurement: Simulated readout effects include simultaneous seven-qubit assignment errors and measurement-induced dephasing during readout.The corresponding quantities are represented in the assignment-error and dephasing-rate measurements.