Showing papers on "Quantum error correction published in 2021"

Circuit quantum electrodynamics

Abstract: Quantum-mechanical effects at the macroscopic level were first explored in Josephson-junction-based superconducting circuits in the 1980s. In recent decades, the emergence of quantum information science has intensified research toward using these circuits as qubits in quantum information processors. The realization that superconducting qubits can be made to strongly and controllably interact with microwave photons, the quantized electromagnetic fields stored in superconducting circuits, led to the creation of the field of circuit quantum electrodynamics (QED), the topic of this review. While atomic cavity QED inspired many of the early developments of circuit QED, the latter has now become an independent and thriving field of research in its own right. Circuit QED allows the study and control of light-matter interaction at the quantum level in unprecedented detail. It also plays an essential role in all current approaches to gate-based digital quantum information processing with superconducting circuits. In addition, circuit QED provides a framework for the study of hybrid quantum systems, such as quantum dots, magnons, Rydberg atoms, surface acoustic waves, and mechanical systems interacting with microwave photons. Here the coherent coupling of superconducting qubits to microwave photons in high-quality oscillators focusing on the physics of the Jaynes-Cummings model, its dispersive limit, and the different regimes of light-matter interaction in this system are reviewed. Also discussed is coupling of superconducting circuits to their environment, which is necessary for coherent control and measurements in circuit QED, but which also invariably leads to decoherence. Dispersive qubit readout, a central ingredient in almost all circuit QED experiments, is also described. Following an introduction to these fundamental concepts that are at the heart of circuit QED, important use cases of these ideas in quantum information processing and in quantum optics are discussed. Circuit QED realizes a broad set of concepts that open up new possibilities for the study of quantum physics at the macro scale with superconducting circuits and applications to quantum information science in the widest sense.

A four-qubit germanium quantum processor

Abstract: The prospect of building quantum circuits1,2 using advanced semiconductor manufacturing makes quantum dots an attractive platform for quantum information processing3,4. Extensive studies of various materials have led to demonstrations of two-qubit logic in gallium arsenide5, silicon6–12 and germanium13. However, interconnecting larger numbers of qubits in semiconductor devices has remained a challenge. Here we demonstrate a four-qubit quantum processor based on hole spins in germanium quantum dots. Furthermore, we define the quantum dots in a two-by-two array and obtain controllable coupling along both directions. Qubit logic is implemented all-electrically and the exchange interaction can be pulsed to freely program one-qubit, two-qubit, three-qubit and four-qubit operations, resulting in a compact and highly connected circuit. We execute a quantum logic circuit that generates a four-qubit Greenberger−Horne−Zeilinger state and we obtain coherent evolution by incorporating dynamical decoupling. These results are a step towards quantum error correction and quantum simulation using quantum dots. Using germanium quantum dots, a four-qubit processor capable of single-, two-, three-, and four-qubit gates, demonstrated by the creation of four-qubit Greenberger−Horne−Zeilinger states, is the largest yet realized with solid-state electron spins.

Quantum computer based on color centers in diamond

Abstract: Artificial atoms like the nitrogen vacancy (NV) centers in diamond enable the realization of fully functional qubits in a solid at room temperature. The functionalities of all the parts needed to create a quantum computer, such as quantum error correction, couplings, quantum teleportation, and a quantum repeater, have already been experimentally demonstrated. These achievements are expected to influence the industrial development of quantum information technology as well as quantum sensing. Whereas quantum sensing has been established and a large number of organizations are working on new developments in this area, a quantum computer itself remains elusive due to technical reasons and limitations of the available materials. For example, only in recent months has it become possible to electrically readout the NV spin state at the level of a single center and significantly improve the scalability of NV center production. A number of ideas have been proposed to overcome the above-mentioned limitations. This paper summarizes the status of research in the area, details the most promising concepts for development, and discusses factors limiting progress as well as the most recent developments in the field.

Fault-tolerant control of an error-corrected qubit.

Abstract: Quantum error correction protects fragile quantum information by encoding it into a larger quantum system1,2. These extra degrees of freedom enable the detection and correction of errors, but also increase the control complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while controlling the logical qubit, and are essential for realizing error suppression in practice3–6. Although fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. Here we experimentally demonstrate fault-tolerant circuits for the preparation, measurement, rotation and stabilizer measurement of a Bacon–Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault-tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6 per cent and a Clifford gate error of 0.3 per cent after offline error correction. In addition, we prepare magic states with fidelities that exceed the distillation threshold7, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant control. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved. Fault-tolerant circuits for the control of a logical qubit encoded in 13 trapped ion qubits through a Bacon–Shor quantum error correction code are demonstrated.

Bosonic quantum error correction codes in superconducting quantum circuits

Abstract: Quantum information is vulnerable to environmental noise and experimental imperfections, hindering the reliability of practical quantum information processors. Therefore, quantum error correction (QEC) that can protect quantum information against noise is vital for universal and scalable quantum computation. Among many different experimental platforms, superconducting quantum circuits and bosonic encodings in superconducting microwave modes are appealing for their unprecedented potential in QEC. During the last few years, bosonic QEC is demonstrated to reach the break-even point, i.e. the lifetime of a logical qubit is enhanced to exceed that of any individual components composing the experimental system. Beyond that, universal gate sets and fault-tolerant operations on the bosonic codes are also realized, pushing quantum information processing towards the QEC era. In this article, we review the recent progress of the bosonic codes, including the Gottesman-Kitaev-Preskill codes, cat codes, and binomial codes, and discuss the opportunities of bosonic codes in various quantum applications, ranging from fault-tolerant quantum computation to quantum metrology. We also summarize the challenges associated with the bosonic codes and provide an outlook for the potential research directions in the long terms.

The XZZX surface code.

Abstract: Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation. The surface code is a keystone in quantum error correction, but it does not generally perform well against structured noise and suffers from large overheads. Here, the authors demonstrate that a variant of it has better performance and requires fewer resources, without additional hardware demands.

Correlated charge noise and relaxation errors in superconducting qubits

Abstract: The central challenge in building a quantum computer is error correction. Unlike classical bits, which are susceptible to only one type of error, quantum bits (qubits) are susceptible to two types of error, corresponding to flips of the qubit state about the X and Z directions. Although the Heisenberg uncertainty principle precludes simultaneous monitoring of X- and Z-flips on a single qubit, it is possible to encode quantum information in large arrays of entangled qubits that enable accurate monitoring of all errors in the system, provided that the error rate is low1. Another crucial requirement is that errors cannot be correlated. Here we characterize a superconducting multiqubit circuit and find that charge noise in the chip is highly correlated on a length scale over 600 micrometres; moreover, discrete charge jumps are accompanied by a strong transient reduction of qubit energy relaxation time across the millimetre-scale chip. The resulting correlated errors are explained in terms of the charging event and phonon-mediated quasiparticle generation associated with absorption of γ-rays and cosmic-ray muons in the qubit substrate. Robust quantum error correction will require the development of mitigation strategies to protect multiqubit arrays from correlated errors due to particle impacts. Cosmic-ray particles and γ-rays striking superconducting circuits can generate qubit errors that are spatially correlated across several millimetres, hampering current error-correction approaches.

Quantum information processing with bosonic qubits in circuit QED

Abstract: The unique features of quantum theory offer a powerful new paradigm for information processing. Translating these mathematical abstractions into useful algorithms and applications requires quantum systems with significant complexity and sufficiently low error rates. Such quantum systems must be made from robust hardware that can coherently store, process, and extract the encoded information, as well as possess effective quantum error correction (QEC) protocols to detect and correct errors. Circuit quantum electrodynamics (cQED) provides a promising hardware platform for implementing robust quantum devices. In particular, bosonic encodings in cQED that use multi-photon states of superconducting cavities to encode information have shown success in realizing hardware-efficient QEC. Here, we review recent developments in the theory and implementation of quantum error correction with bosonic codes and report the progress made towards realizing fault-tolerant quantum information processing with cQED devices.

Removing leakage-induced correlated errors in superconducting quantum error correction.

Abstract: Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing.

Realization of real-time fault-tolerant quantum error correction

Abstract: Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the $[[7,1,3]]$ color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of $1.7(6) \times 10^{-3}$, compared to the average physical SPAM error $2.4(8) \times 10^{-3}$ of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.

High-Fidelity Controlled- Z Gate with Maximal Intermediate Leakage Operating at the Speed Limit in a Superconducting Quantum Processor

Abstract: 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.

Two-qubit silicon quantum processor with operation fidelity exceeding 99%.

Abstract: Silicon spin qubits satisfy the necessary criteria for quantum information processing. However, a demonstration of high fidelity state preparation and readout combined with high fidelity single- and two-qubit gates, all of which must be present for quantum error correction, has been lacking. We use a two qubit Si/SiGe quantum processor to demonstrate state preparation and readout with fidelity over 97%, combined with both single- and two-qubit control fidelities exceeding 99%. The operation of the quantum processor is quantitatively characterized using gate set tomography and randomized benchmarking. Our results highlight the potential of silicon spin qubits to become a dominant technology in the development of intermediate-scale quantum processors.

Protecting a bosonic qubit with autonomous quantum error correction

Abstract: To build a universal quantum computer from fragile physical qubits, effective implementation of quantum error correction (QEC)1 is an essential requirement and a central challenge Existing demonstrations of QEC are based on an active schedule of error-syndrome measurements and adaptive recovery operations2,3,4,5,6,7 that are hardware intensive and prone to introducing and propagating errors In principle, QEC can be realized autonomously and continuously by tailoring dissipation within the quantum system1,8,9,10,11,12,13,14, but so far it has remained challenging to achieve the specific form of dissipation required to counter the most prominent errors in a physical platform Here we encode a logical qubit in Schrodinger cat-like multiphoton states15 of a superconducting cavity, and demonstrate a corrective dissipation process that stabilizes an error-syndrome operator: the photon number parity Implemented with continuous-wave control fields only, this passive protocol protects the quantum information by autonomously correcting single-photon-loss errors and boosts the coherence time of the bosonic qubit by over a factor of two Notably, QEC is realized in a modest hardware setup with neither high-fidelity readout nor fast digital feedback, in contrast to the technological sophistication required for prior QEC demonstrations Compatible with additional phase-stabilization and fault-tolerant techniques16,17,18, our experiment suggests quantum dissipation engineering as a resource-efficient alternative or supplement to active QEC in future quantum computing architectures A logical qubit encoded in multi-photon states of a superconducting cavity is protected with autonomous correction of certain quantum errors by tailoring the dissipation it is exposed to

Reducing the impact of radioactivity on quantum circuits in a deep-underground facility.

Abstract: As quantum coherence times of superconducting circuits have increased from nanoseconds to hundreds of microseconds, they are currently one of the leading platforms for quantum information processing. However, coherence needs to further improve by orders of magnitude to reduce the prohibitive hardware overhead of current error correction schemes. Reaching this goal hinges on reducing the density of broken Cooper pairs, so-called quasiparticles. Here, we show that environmental radioactivity is a significant source of nonequilibrium quasiparticles. Moreover, ionizing radiation introduces time-correlated quasiparticle bursts in resonators on the same chip, further complicating quantum error correction. Operating in a deep-underground lead-shielded cryostat decreases the quasiparticle burst rate by a factor thirty and reduces dissipation up to a factor four, showcasing the importance of radiation abatement in future solid-state quantum hardware.

Logical-qubit operations in an error-detecting surface code

Abstract: We realize a suite of logical operations on a distance-two logical qubit 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 through detailed characterization. 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.

Quantum tomography of an entangled three-qubit state in silicon

Abstract: Quantum entanglement is a fundamental property of coherent quantum states and an essential resource for quantum computing1. In large-scale quantum systems, the error accumulation requires concepts for quantum error correction. A first step toward error correction is the creation of genuinely multipartite entanglement, which has served as a performance benchmark for quantum computing platforms such as superconducting circuits2,3, trapped ions4 and nitrogen-vacancy centres in diamond5. Among the candidates for large-scale quantum computing devices, silicon-based spin qubits offer an outstanding nanofabrication capability for scaling-up. Recent studies demonstrated improved coherence times6–8, high-fidelity all-electrical control9–13, high-temperature operation14,15 and quantum entanglement of two spin qubits9,11,12. Here we generated a three-qubit Greenberger–Horne–Zeilinger state using a low-disorder, fully controllable array of three spin qubits in silicon. We performed quantum state tomography16 and obtained a state fidelity of 88.0%. The measurements witness a genuine Greenberger–Horne–Zeilinger class quantum entanglement that cannot be separated into any biseparable state. Our results showcase the potential of silicon-based spin qubit platforms for multiqubit quantum algorithms. Among the candidates for large-scale quantum computing devices, silicon-based spin qubits offer an outstanding nanofabrication capability for scaling-up. In an array of three spin qubits in silicon, high-fidelity state preparation and control enable the creation of a three-qubit Greenberger–Horne–Zeilinger state with 88% state fidelity.

Tunable Coupling Architecture for Fixed-Frequency Transmon Superconducting Qubits.

Abstract: Implementation of high-fidelity 2-qubit operations is a key ingredient for scalable quantum error correction. In superconducting qubit architectures, tunable buses have been explored as a means to higher-fidelity gates. However, these buses introduce new pathways for leakage. Here we present a modified tunable bus architecture appropriate for fixed-frequency qubits in which the adiabaticity restrictions on gate speed are reduced. We characterize this coupler on a range of 2-qubit devices, achieving a maximum gate fidelity of 99.85%. We further show the calibration is stable over one day.

Demonstration of fault-tolerant universal quantum gate operations

Abstract: Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error correcting codes. When manipulating the logical quantum states, it is imperative that errors caused by imperfect operations do not spread uncontrollably through the quantum register. This requires that all operations on the quantum register obey a fault-tolerant circuit design which, in general, increases the complexity of the implementation. Here, we demonstrate a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. In particular, we make use of the recently introduced paradigm of flag fault tolerance, where the absence or presence of dangerous errors is heralded by usage of few ancillary 'flag' qubits. We perform a logical two-qubit CNOT-gate between two instances of the seven qubit color code, and we also fault-tolerantly prepare a logical magic state. We then realize a fault-tolerant logical T-gate by injecting the magic state via teleportation from one logical qubit onto the other. We observe the hallmark feature of fault tolerance, a superior performance compared to a non-fault-tolerant implementation. In combination with recently demonstrated repeated quantum error correction cycles these results open the door to error-corrected universal quantum computation.

Virtual Distillation for Quantum Error Mitigation

Abstract: Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring $M$ copies of a noisy state $\rho$. This enables us to estimate expectation values with respect to a state with dramatically reduced error, $\rho^M/ \mathrm{Tr}(\rho^M)$, without explicitly preparing it, hence the name "virtual distillation". As $M$ increases, this state approaches the closest pure state to $\rho$, exponentially quickly. We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant eigenvector of $\rho$). We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows. Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.

Fault-tolerant operation of a logical qubit in a diamond quantum processor

Abstract: Solid-state spin qubits are a promising platform for quantum computation and quantum networks. Recent experiments have demonstrated high-quality control over multi-qubit systems, elementary quantum algorithms and non-fault-tolerant error correction. Large-scale systems will require using error-corrected logical qubits that are operated fault-tolerantly, so that reliable computation is possible despite noisy operations. Overcoming imperfections in this way remains a major outstanding challenge for quantum science. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the 5-qubit code with a recently discovered flag protocol that enables fault-tolerance using a total of seven qubits. We encode the logical qubit using a novel protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. While future improvements in fidelity and the number of qubits will be required, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards large-scale quantum information processing based on solid-state spins.

Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem

Abstract: We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols to characterize the QECC performance measured in terms of the worst-case entanglement fidelity. The theorem is applicable to a large class of decoherence models, including erasure and depolarizing noise. Our approach is unorthodox, as instead of following the established path of utilizing QECCs to mitigate noise in quantum metrological protocols, we apply methods of quantum metrology to explore the limitations of QECCs.

Holographic tensor network models and quantum error correction: A topical review

Abstract: Recent progress in studies of holographic dualities, originally motivated by insights from string theory, has led to a confluence with concepts and techniques from quantum information theory. A particularly successful approach has involved capturing holographic properties by means of tensor networks which not only give rise to physically meaningful correlations of holographic boundary states, but also reproduce and refine features of quantum error correction in holography. This topical review provides an overview over recent successful realizations of such models. It does so by building on an introduction of the theoretical foundations of AdS/CFT and necessary quantum information concepts, many of which have themselves developed into independent, rapidly evolving research fields.

Quantum Error Correction with the Gottesman-Kitaev-Preskill Code

Abstract: A dive on the GKP code: Going from the visionary work that showed how to realize error correction based on continuous variables to the technological advances that allow extracting its power.

Limitations of optimization algorithms on noisy quantum devices

Abstract: Recent successes in producing intermediate-scale quantum devices have focused interest on establishing whether near-term devices could outperform classical computers for practical applications. A central question is whether noise can be overcome in the absence of quantum error correction or if it fundamentally restricts any potential quantum advantage. We present a transparent way of comparing classical and quantum algorithms running on noisy devices for a large family of tasks that includes optimization and variational eigenstate solving. Our approach is based on entropic inequalities that determine how fast the quantum state converges to the fixed point of the noise model, together with established classical methods of Gibbs state simulation. Our techniques are extremely versatile and so may be applied to a large variety of algorithms, noise models and quantum computing architectures. We use our result to provide estimates for problems within reach of current experiments, such as quantum annealers or variational quantum algorithms. The bounds we obtain indicate that substantial quantum advantages are unlikely for classical optimization unless noise rates are decreased by orders of magnitude or the topology of the problem matches that of the device. This is the case even if the number of available qubits increases substantially. Current quantum computers do not have error correction, which means noise may prevent them outperforming classical devices in useful tasks. An analysis of quantum optimization shows that current noise levels are too high to produce a quantum advantage.

Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits

Abstract: Measurement-induced entanglement transitions in quantum dynamics represent a new class of nonequilibrium transitions, akin to thresholds in quantum error correction. In this work, the authors explore the universal properties of entanglement transitions in one- and two-dimensional monitored Clifford circuits, using a graph-state-based algorithm to unravel geometric properties of entanglement. A study of entanglement clusters in the steady state reveals that, despite similarities in the bulk, the surface critical exponents show strong deviations from classical percolation.

Error Correction of Quantum Reference Frame Information

Abstract: The existence of quantum error correcting codes is one of the most counterintuitive and potentially technologically important discoveries of quantum information theory. However, standard error correction refers to abstract quantum information, i.e., information that is independent of the physical incarnation of the systems used for storing the information. There are, however, other forms of information that are physical - one of the most ubiquitous being reference frame information. Here we analyze the problem of error correcting physical information. The basic question we seek to answer is whether or not such error correction is possible and, if so, what limitations govern the process. The main challenge is that the systems used for transmitting physical information, in addition to any actions applied to them, must necessarily obey these limitations. Encoding and decoding operations that obey a restrictive set of limitations need not exist a priori. We focus on the case of erasure errors, and we first show that the problem is equivalent to quantum error correction using group-covariant encodings. We prove a no-go theorem showing that that no finite dimensional, group-covariant quantum codes exist for Lie groups with an infinitesimal generator (e.g., U(1), SU(2), and SO(3)). We then explain how one can circumvent this no-go theorem using infinite dimensional codes, and we give an explicit example of a covariant quantum error correcting code using continuous variables for the group U(1). Finally, we demonstrate that all finite groups have finite dimensional codes, giving both an explicit construction and a randomized approximate construction with exponentially better parameters.

Practical Quantum Error Correction with the XZZX Code and Kerr-Cat Qubits

Abstract: An architecture for scalable fault-tolerant quantum computing based on Kerr-cat qubits and the XZZX code is shown to be viable with parameters that can be realized in current experiments.

Experimental Pauli-frame randomization on a superconducting qubit

Abstract: The promise of quantum computing with imperfect qubits relies on the ability of a quantum computing system to scale cheaply through error correction and fault tolerance. While fault tolerance requires relatively mild assumptions about the nature of qubit errors, the overhead associated with coherent and non-Markovian errors can be orders of magnitude larger than the overhead associated with purely stochastic Markovian errors. One proposal to address this challenge is to randomize the circuits of interest, shaping the errors to be stochastic Pauli errors but leaving the aggregate computation unaffected. The randomization technique can also suppress couplings to slow degrees of freedom associated with non-Markovian evolution. Here, we demonstrate the implementation of Pauli-frame randomization in a superconducting circuit system, exploiting a flexible programming and control infrastructure to achieve this with low effort. We use high-accuracy gate-set tomography to characterize in detail the properties of the circuit error, with and without the randomization procedure, which allows us to make rigorous statements about Markovianity as well as the nature of the observed errors. We demonstrate that randomization suppresses signatures of non-Markovian evolution to statistically insignificant levels, from a Markovian model violation ranging from $43\ensuremath{\sigma}$ to $1987\ensuremath{\sigma}$, down to violations between $0.3\ensuremath{\sigma}$ and $2.7\ensuremath{\sigma}$ under randomization. Moreover, we demonstrate that, under randomization, the experimental errors are well described by a Pauli error model, with model violations that are similarly insignificant (between $0.8\ensuremath{\sigma}$ and $2.7\ensuremath{\sigma}$). Importantly, all these improvements in the model accuracy were obtained without degradation to fidelity, and with some improvements to error rates as quantified by the diamond norm. This demonstrates the ability of Pauli-frame randomization to shape noise into forms that are more benign for quantum error correction and fault tolerance.

New perspectives on covariant quantum error correction

Abstract: Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.

QECOOL: On-Line Quantum Error Correction with a Superconducting Decoder for Surface Code

Abstract: Due to the low error tolerance of a qubit, detecting and correcting errors on it is essential for fault-tolerant quantum computing. Surface code (SC) associated with its decoding algorithm is one of the most promising quantum error correction (QEC) methods. QEC needs to be very power-efficient since the power budget is limited inside of a dilution refrigerator for superconducting qubits by which one of the most successful quantum computers (QCs) is built. In this paper, we propose an online-QEC algorithm and its hardware implementation with SFQ based superconducting digital circuits. We design a key building block of the proposed hardware with an SFQ cell library and evaluate it by the SPICE-level simulation. Each logic element is composed of about 3000 Josephson junctions and power consumption is about $2.78 \mu \mathrm{W}$ when operating with 2 GHz clock frequency which meets the required decoding speed. Our decoder is simulated on a quantum error simulator for code distances 5 to 13 and achieves a 1.0% accuracy threshold.