Quantum computing is poised to solve practically useful problems which are computationally intractable for classical supercomputers. However, the current generation of quantum computers are limited by errors that may only partially be mitigated by developing higher-quality qubits. Quantum error correction (QEC) will thus be necessary to ensure fault tolerance. QEC protects the logical information by cyclically measuring syndrome information about the errors. An essential part of QEC is the decoder, which uses the syndrome to compute the likely effect of the errors on the logical degrees of freedom and provide a tentative correction. The decoder must be accurate, fast enough to keep pace with the QEC cycle (e.g. on a microsecond timescale for superconducting qubits) and with hard real-time system integration to support logical operations. As such, real-time decoding is essential to realize fault-tolerant quantum computing and to achieve quantum advantage. In this work, we highlight some of the key challenges facing the implementation of real-time decoders while providing a succinct summary of the progress to-date. Furthermore, we lay out our perspective for the future development and provide a possible roadmap for the field of real-time decoding in the next few years. As the quantum hardware is anticipated to scale up, this perspective article will provide a guidance for researchers, focusing on the most pressing issues in real-time decoding and facilitating the development of solutions across quantum, nano and computer science.

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Computers are used all over the place to perform tasks ranging from sending an email to running some complicated numerical simulation. That is brilliant of course, because computers enable us to solve a lot of problems in the world in this way. At the same time, for some of those problems, not even powerful supercomputers are enough to get the result of the computation in any reasonable amount of time. An alternative that might be able to solve some of these problems very quickly are quantum computers. The operations performed by a quantum computer need to be faithful in order to get the right result of the quantum computation. However, nowadays quantum computers are fairly noisy, severely limiting their range of applicability in the near future. Various methods for quantum error correction have been developed, showing that, if error rates are below a certain threshold, one can make the computation as error-free as desired. However, while quantum error correction is starting to be tested in experiments, its performance has been mostly studied with respect to idealized error models. Furthermore, quantum error correction comes at the price of a substantial overhead in number of qubits and number of operations, especially if error rates are just barely below threshold. From a different perspective, error-mitigation techniques that do not need the full machinery of quantum correction have been put forward, fostering hope that noisy near-term devices might run useful applications even without quantum error correction. However, in either case the physical error rates of the fundamental operations are still high. In this thesis we focus on achieving lower error rates for some of the fundamental operations in a quantum computer, specifically for superconducting qubits, and we demonstrate the beneficial impact of these results on quantum error correction in a realistic setting. We develop error models that are physically motivated for superconducting qubits (reviewed in Chapter 2), based on the noise sources to which they are sensitive (reviewed in Chapter 3). The major elements of novelty in our models are the inclusion of leakage, quasi-static flux noise, and distortions of electronic signals. In Chapter 6 we discuss a flux-pulsing technique for controlled-phase gates, named Net Zero. In the first part, we show that the characteristic zero-integral feature protects from long-timescale distortions, echoes out flux noise and uses leakage interference to mitigate leakage, leading to a fast, high-fidelity gate. In the second part, we introduce an updated version of Net Zero, called Sudden Net Zero, that maintains the same advantages and adds easiness of tuneup and straightforward conditional-phase tunability. Diagnosing errors is crucial for correcting them and tuning up gates. In Chapter 7 we introduce Spectral Quantum Tomography, a tomographic method that can provide detailed information about errors in single- and two-qubit gates, in a way that is independent of state-preparation and measurement errors. In particular, we investigate the footprint of relaxation and dephasing, as well as leakage and non-Markovian noise. Leakage outside of the qubit computational subspace is particularly damaging for quantum error correcting codes, in particular stabilizer codes (reviewed in Chapter 4). Leakage-reduction units (reviewed in Chapter 5) can bring a leaked qubit back to the computational subspace, thus restoring part of the loss in performance. Based on the error model developed for two-qubit gates, we study the effect of leakage in quantum error correction using realistic density-matrix simulations. In Chapter 8 we use hidden Markov models to detect leakage in a transmon-qubit-based surface code and improve the logical fidelity by post-selection. The detection is based on recognizing patterns in the stabilizer measurements that can likely be attributed to leakage. In Chapter 9 we introduce a hardware-efficient leakage-reduction scheme to directly remove leakage in a scalable way that does not require extra qubits or time, leading to a reduction of the logical error rate. In particular, we propose two separate leakage-reduction units tailored for data and ancilla qubits, respectively. For data qubits, we apply a microwave pulse that transfers leakage to its dedicated readout resonator, where it quickly decays into the environment. For ancilla qubits, we use a microwave pulse that maps the leaked state to a computational state. These techniques for two-qubit gates, tomography and leakage mitigation contribute to reducing the error rates, benefiting quantum error correction as well as near-term devices. In the Conclusion we give an outlook on the potential challenges in superconducting quantum computing, including tunable couplers, real-time decoding and physical error rates in large devices. @en

Leakage outside of the qubit computational subspace poses a threatening challenge to quantum error correction (QEC). We propose a scheme using two leakage-reduction units (LRUs) that mitigate these issues for a transmon-based surface code, without requiring an overhead in terms of hardware or QEC-cycle time as in previous proposals. For data qubits, we consider a microwave drive to transfer leakage to the readout resonator, where it quickly decays, ensuring that this negligibly disturbs the computational states for realistic system parameters. For ancilla qubits, we apply a |1↔|2π pulse conditioned on the measurement outcome. Using density-matrix simulations of the distance-3 surface code, we show that the average leakage lifetime is reduced to almost one QEC cycle, even when the LRUs are implemented with limited fidelity. Furthermore, we show that this leads to a significant reduction of the logical error rate. This LRU scheme opens the prospect for near-term scalable QEC demonstrations.

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Simple tuneup of fast two-qubit gates is essential for the scaling of quantum processors. We introduce the sudden variant (SNZ) of the net zero scheme realizing controlled-Z (CZ) gates by flux control of transmon frequency. SNZ CZ gates realized in a multitransmon processor operate at the speed limit of transverse coupling between computational and noncomputational states by maximizing intermediate leakage. Beyond speed, the key advantage of SNZ is tuneup simplicity, owing to the regular structure of conditional phase and leakage as a function of two control parameters. SNZ is compatible with scalable schemes for quantum error correction and adaptable to generalized conditional-phase gates useful in intermediate-scale applications.

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Future fault-tolerant quantum computers will require storing and processing quantum data in logical qubits. Here we realize a suite of logical operations on a distance-2 surface code qubit built from seven physical qubits and stabilized using repeated error-detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.

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Future fault-tolerant quantum computers will require storing and processing quantum data in logical qubits. Here we realize a suite of logical operations on a distance-2 surface code qubit built from seven physical qubits and stabilized using repeated error-detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.

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Future fault-tolerant quantum computers will require storing and processing quantum data in logical qubits. Here we realize a suite of logical operations on a distance-2 surface code qubit built from seven physical qubits and stabilized using repeated error-detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.

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Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding less than half of the data for the given noise parameters. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.@en

Conditional-phase (cz) gates in transmons can be realized by flux pulsing computational states towards resonance with noncomputational ones. We present a 40 ns cz gate based on a bipolar flux pulse suppressing leakage (0.1%) by interference and approaching the speed limit set by exchange coupling. This pulse harnesses a built-in echo to enhance fidelity (99.1%) and is robust to long-timescale distortion in the flux-control line, ensuring repeatability. Numerical simulations matching experiment show that fidelity is limited by high-frequency dephasing and leakage by short-timescale distortion.

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We introduce spectral quantum tomography, a simple method to extract the eigenvalues of a noisy few-qubit gate, represented by a trace-preserving superoperator, in a SPAM-resistant fashion, using low resources in terms of gate sequence length. The eigenvalues provide detailed gate information, supplementary to known gate-quality measures such as the gate fidelity, and can be used as a gate diagnostic tool. We apply our method to one- and two-qubit gates on two different superconducting systems available in the cloud, namely the QuTech Quantum Infinity and the IBM Quantum Experience. We discuss how cross-talk, leakage and non-Markovian errors affect the eigenvalue data.

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