Fault tolerant quantum computation with very high threshold for loss errors

Efficient Concatenated Bosonic Code for Additive Gaussian Noise

Bosonic codes offer noise resilience for quantum information processing. Good performance often comes at a price of complex decoding schemes, limiting their practicality. Here, we propose using a Gottesman-Kitaev-Preskill code to detect and discard error-prone qubits, concatenated with a quantum parity code to handle the residual errors. Our method employs a simple linear-time decoder that nevertheless offers significant performance improvements over the standard decoder. Our Letter may have applications in a wide range of quantum computation and communication scenarios.

Dual-rail encoding with superconducting cavities

The design of quantum hardware that reduces and mitigates errors is essential for practical quantum error correction (QEC) and useful quantum computation. To this end, we introduce the circuit-Quantum Electrodynamics (QED) dual-rail qubit in which our physical qubit is encoded in the single-photon subspace, [Formula: see text], of two superconducting microwave cavities. The dominant photon loss errors can be detected and converted into erasure errors, which are in general much easier to correct. In contrast to linear optics, a circuit-QED implementation of the dual-rail code offers unique capabilities. Using just one additional transmon ancilla per dual-rail qubit, we describe how to perform a gate-based set of universal operations that includes state preparation, logical readout, and parametrizable single and two-qubit gates. Moreover, first-order hardware errors in the cavities and the transmon can be detected and converted to erasure errors in all operations, leaving background Pauli errors that are orders of magnitude smaller. Hence, the dual-rail cavity qubit exhibits a favorable hierarchy of error rates and is expected to perform well below the relevant QEC thresholds with today's coherence times.

High-fidelity gates and mid-circuit erasure conversion in an atomic qubit

The development of scalable, high-fidelity qubits is a key challenge in quantum information science. Neutral atom qubits have progressed rapidly in recent years, demonstrating programmable processors1,2 and quantum simulators with scaling to hundreds of atoms3,4. Exploring new atomic species, such as alkaline earth atoms5-7, or combining multiple species8 can provide new paths to improving coherence, control and scalability. For example, for eventual application in quantum error correction, it is advantageous to realize qubits with structured error models, such as biased Pauli errors9 or conversion of errors into detectable erasures10. Here we demonstrate a new neutral atom qubit using the nuclear spin of a long-lived metastable state in 171Yb. The long coherence time and fast excitation to the Rydberg state allow one- and two-qubit gates with fidelities of 0.9990(1) and 0.980(1), respectively. Importantly, a large fraction of all gate errors result in decays out of the qubit subspace to the ground state. By performing fast, mid-circuit detection of these errors, we convert them into erasure errors; during detection, the induced error probability on qubits remaining in the computational space is less than 10-5. This work establishes metastable 171Yb as a promising platform for realizing fault-tolerant quantum computing.

High-Threshold Quantum Computing by Fusing One-Dimensional Cluster States

We propose a measurement-based model for fault-tolerant quantum computation that can be realized with one-dimensional cluster states and fusion measurements only; basic resources that are readily available with scalable photonic hardware. Our simulations demonstrate high thresholds compared with other measurement-based models realized with basic entangled resources and 2-qubit fusion measurements. Its high tolerance to noise indicates that our practical construction offers a promising route to scalable quantum computing with quantum emitters and linear-optical elements.

All-Microwave Leakage Reduction Units for Quantum Error Correction with Superconducting Transmon Qubits

Minimizing leakage from computational states is a challenge when using many-level systems like superconducting quantum circuits as qubits. We realize and extend the quantum-hardware-efficient, all-microwave leakage reduction unit (LRU) for transmons in a circuit QED architecture proposed by Battistel et al. This LRU effectively reduces leakage in the second- and third-excited transmon states with up to 99% efficacy in 220 ns, with minimum impact on the qubit subspace. As a first application in the context of quantum error correction, we show how multiple simultaneous LRUs can reduce the error detection rate and suppress leakage buildup within 1% in data and ancilla qubits over 50 cycles of a weight-2 stabilizer measurement.

Fusion-based quantum computation

The standard primitives of quantum computing include deterministic unitary entangling gates, which are not natural operations in many systems including photonics. Here, we present fusion-based quantum computation, a model for fault tolerant quantum computing constructed from physical primitives readily accessible in photonic systems. These are entangling measurements, called fusions, which are performed on the qubits of small constant sized entangled resource states. Probabilistic photonic gates as well as errors are directly dealt with by the quantum error correction protocol. We show that this computational model can achieve a higher threshold than schemes reported in literature. We present a ballistic scheme which can tolerate a 10.4% probability of suffering photon loss in each fusion, which corresponds to a 2.7% probability of loss of each individual photon. The architecture is also highly modular and has reduced classical processing requirements compared to previous photonic quantum computing architectures.

Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays

Executing quantum algorithms on error-corrected logical qubits is a critical step for scalable quantum computing, but the requisite numbers of qubits and physical error rates are demanding for current experimental hardware. Recently, the development of error correcting codes tailored to particular physical noise models has helped relax these requirements. In this work, we propose a qubit encoding and gate protocol for ¹⁷¹Yb neutral atom qubits that converts the dominant physical errors into erasures, that is, errors in known locations. The key idea is to encode qubits in a metastable electronic level, such that gate errors predominantly result in transitions to disjoint subspaces whose populations can be continuously monitored via fluorescence. We estimate that 98% of errors can be converted into erasures. We quantify the benefit of this approach via circuit-level simulations of the surface code, finding a threshold increase from 0.937% to 4.15%. We also observe a larger code distance near the threshold, leading to a faster decrease in the logical error rate for the same number of physical qubits, which is important for near-term implementations. Erasure conversion should benefit any error correcting code, and may also be applied to design new gates and encodings in other qubit platforms.

In quantum computing, realistic error models can allow tailored correction schemes for specific platforms. Here, while considering the case of qubits encoded in metastable electronic levels of atomic arrays, the authors propose a way to convert a large fraction of occurring errors into detectable leakages, or erasure errors, which are vastly easier to correct.

Efficient generation of entangled multiphoton graph states from a single atom

The central technological appeal of quantum science resides in exploiting quantum effects, such as entanglement, for a variety of applications, including computing, communication and sensing¹. The overarching challenge in these fields is to address, control and protect systems of many qubits against decoherence². Against this backdrop, optical photons, naturally robust and easy to manipulate, represent ideal qubit carriers. However, the most successful technique so far for creating photonic entanglement³ is inherently probabilistic and, therefore, subject to severe scalability limitations. Here we report the implementation of a deterministic protocol^4–6 for the creation of photonic entanglement with a single memory atom in a cavity⁷. We interleave controlled single-photon emissions with tailored atomic qubit rotations to efficiently grow Greenberger–Horne–Zeilinger (GHZ) states⁸ of up to 14 photons and linear cluster states⁹ of up to 12 photons with a fidelity lower bounded by 76(6)% and 56(4)%, respectively. Thanks to a source-to-detection efficiency of 43.18(7)% per photon, we measure these large states about once every minute, which is orders of magnitude faster than in any previous experiment^3,10–13. In the future, this rate could be increased even further, the scheme could be extended to two atoms in a cavity^14,15 or several sources could be quantum mechanically coupled¹⁶, to generate higher-dimensional cluster states¹⁷. Overcoming the limitations encountered by probabilistic schemes for photonic entanglement generation, our results may offer a way towards scalable measurement-based quantum computation^18,19 and communication^20,21.

Using a single memory atom in a cavity, a deterministic protocol is implemented to efficiently grow Greenberger–Horne–Zeilinger and linear cluster states by means of single-photon emissions.

Resource-Efficient Topological Fault-Tolerant Quantum Computation with Hybrid Entanglement of Light

We propose an all-linear-optical scheme to ballistically generate a cluster state for measurement-based topological fault-tolerant quantum computation using hybrid photonic qubits entangled in a continuous-discrete domain. Availability of near-deterministic Bell-state measurements on hybrid qubits is exploited for this purpose. In the presence of photon losses, we show that our scheme leads to a significant enhancement in both tolerable photon-loss rate and resource overheads. More specifically, we report a photon-loss threshold of ∼3.3×10^{-3}, which is higher than those of known optical schemes under a reasonable error model. Furthermore, resource overheads to achieve logical error rate of 10^{-6}(10^{-15}) is estimated to be ∼8.5×10^{5}(1.7×10^{7}), which is significantly less by multiple orders of magnitude compared to other reported values in the literature.

A fault-tolerant non-Clifford gate for the surface code in two dimensions

Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive development. This alleviates the need for distillation or higher-dimensional components to complete a universal gate set. The operation uses both local transversal gates and code deformations over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. These decoding algorithms allow us to draw upon the advantages of three-dimensional models using only a two-dimensional array of live qubits. Our gate is completed using parity checks of weight no greater than four. We therefore expect it to be amenable with near-future technology. As the gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation considerably.

Drive-noise tolerant optical switching inspired by composite pulses

Electro-optic modulators within Mach-Zehnder interferometers are a common construction for optical switches in integrated photonics. A challenge faced when operating at high switching speeds is that noise from the electronic drive signals will effect switching performance. Inspired by the Mach-Zehnder lattice switching devices of Van Campenhout et al. [Opt. Express17(26), 23793 (2009).] and techniques from the field of Nuclear Magnetic Resonance known as composite pulses, we present switches which offer protection against drive-noise in both the on and off state of the switch for both the phase and intensity information encoded in the switched optical mode.

Extendibility Limits the Performance of Quantum Processors

Resource theories in quantum information science are helpful for the study and quantification of the performance of information-processing tasks that involve quantum systems. These resource theories also find applications in other areas of study; e.g., the resource theories of entanglement and coherence have found use and implications in the study of quantum thermodynamics and memory effects in quantum dynamics. In this paper, we introduce the resource theory of unextendibility, which is associated with the inability of extending quantum entanglement in a given quantum state to multiple parties. The free states in this resource theory are the k-extendible states, and the free channels are k-extendible channels, which preserve the class of k-extendible states. We make use of this resource theory to derive nonasymptotic, upper bounds on the rate at which quantum communication or entanglement preservation is possible by utilizing an arbitrary quantum channel a finite number of times, along with the assistance of k-extendible channels at no cost. We then show that the bounds obtained are significantly tighter than previously known bounds for quantum communication over both the depolarizing and erasure channels.

Cluster State Generation with Spin-Orbit Coupled Fermionic Atoms in Optical Lattices

Measurement-based quantum computation, an alternative paradigm for quantum information processing, uses simple measurements on qubits prepared in cluster states, a class of multiparty entangled states with useful properties. Here we propose and analyze a scheme that takes advantage of the interplay between spin-orbit coupling and superexchange interactions, in the presence of a coherent drive, to deterministically generate macroscopic arrays of cluster states in fermionic alkaline earth atoms trapped in three dimensional (3D) optical lattices. The scheme dynamically generates cluster states without the need of engineered transport, and is robust in the presence of holes, a typical imperfection in cold atom Mott insulators. The protocol is of particular relevance for the new generation of 3D optical lattice clocks with coherence times >10  s, 2 orders of magnitude larger than the cluster state generation time. We propose the use of collective measurements and time reversal of the Hamiltonian to benchmark the underlying Ising model dynamics and the generated many-body correlations.

Percolation thresholds for photonic quantum computing

Despite linear-optical fusion (Bell measurement) being probabilistic, photonic cluster states for universal quantum computation can be prepared without feed-forward by fusing small n-photon entangled clusters, if the success probability of each fusion attempt is above a threshold, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\lambda }}_{\mathrm{c}}^{(n)}$$\end{document}λc(n). We prove a general bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\lambda }}_{\mathrm{c}}^{(n)} \ge 1/(n - 1)$$\end{document}λc(n)≥1∕(n-1), and develop a conceptual method to construct long-range-connected clusters where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\lambda }}_{\mathrm{c}}^{(n)}$$\end{document}λc(n) becomes the bond percolation threshold of a logical graph. This mapping lets us find constructions that require lower fusion success probabilities than currently known, and settle a heretofore open question by showing that a universal cluster state can be created by fusing 3-photon clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integrated-photonic realizations.

Universal cluster states for quantum computing can be assembled without feed-forward by fusing n-photon clusters with linear optics if the fusion success probability is above a threshold p. The authors bound p in terms of n and provide protocols for n = 3 clusters requiring lower fusion probability than before.

Fan-out Estimation in Spin-based Quantum Computer Scale-up

Solid-state spin-based qubits offer good prospects for scaling based on their long coherence times and nexus to large-scale electronic scale-up technologies. However, high-threshold quantum error correction requires a two-dimensional qubit array operating in parallel, posing significant challenges in fabrication and control. While architectures incorporating distributed quantum control meet this challenge head-on, most designs rely on individual control and readout of all qubits with high gate densities. We analysed the fan-out routing overhead of a dedicated control line architecture, basing the analysis on a generalised solid-state spin qubit platform parameterised to encompass Coulomb confined (e.g. donor based spin qubits) or electrostatically confined (e.g. quantum dot based spin qubits) implementations. The spatial scalability under this model is estimated using standard electronic routing methods and present-day fabrication constraints. Based on reasonable assumptions for qubit control and readout we estimate 10²–10⁵ physical qubits, depending on the quantum interconnect implementation, can be integrated and fanned-out independently. Assuming relatively long control-free interconnects the scalability can be extended. Ultimately, the universal quantum computation may necessitate a much higher number of integrated qubits, indicating that higher dimensional electronics fabrication and/or multiplexed distributed control and readout schemes may be the preferredstrategy for large-scale implementation.

Cat Codes with Optimal Decoherence Suppression for a Lossy Bosonic Channel

We investigate cat codes that can correct multiple excitation losses and identify two types of logical errors: bit-flip errors due to excessive excitation loss and dephasing errors due to quantum backaction from the environment. We show that selected choices of logical subspace and coherent amplitude significantly reduce dephasing errors. The trade-off between the two major errors enables optimized performance of cat codes in terms of minimized decoherence. With high coupling efficiency, we show that one-way quantum repeaters with cat codes feature a boosted secure communication rate per mode when compared to conventional encoding schemes, showcasing the promising potential of quantum information processing with continuous variable quantum codes.

Noise Threshold and Resource Cost of Fault-Tolerant Quantum Computing with Majorana Fermions in Hybrid Systems

Fault-tolerant quantum computing in systems composed of both Majorana fermions and topologically unprotected quantum systems, e.g., superconducting circuits or quantum dots, is studied in this Letter. Errors caused by topologically unprotected quantum systems need to be corrected with error-correction schemes, for instance, the surface code. We find that the error-correction performance of such a hybrid topological quantum computer is not superior to a normal quantum computer unless the topological charge of Majorana fermions is insusceptible to noise. If errors changing the topological charge are rare, the fault-tolerance threshold is much higher than the threshold of a normal quantum computer and a surface-code logical qubit could be encoded in only tens of topological qubits instead of about 1,000 normal qubits.

Foliated Quantum Error-Correcting Codes

We show how to construct a large class of quantum error-correcting codes, known as Calderbank-Steane-Shor codes, from highly entangled cluster states. This becomes a primitive in a protocol that foliates a series of such cluster states into a much larger cluster state, implementing foliated quantum error correction. We exemplify this construction with several familiar quantum error-correction codes and propose a generic method for decoding foliated codes. We numerically evaluate the error-correction performance of a family of finite-rate Calderbank-Steane-Shor codes known as turbo codes, finding that they perform well over moderate depth foliations. Foliated codes have applications for quantum repeaters and fault-tolerant measurement-based quantum computation.

Experimental realization of non-adiabatic universal quantum gates using geometric Landau-Zener-Stückelberg interferometry

High fidelity universal gates for quantum bits form an essential ingredient of quantum information processing. In particular, geometric gates have attracted attention because they have a higher intrinsic resistance to certain errors. However, their realization remains a challenge because of the need for complicated quantum control on a multi-level structure as well as meeting the adiabatic condition within a short decoherence time. Here, we demonstrate non-adiabatic quantum operations for a two-level system by applying a well-controlled geometric Landau-Zener-Stückelberg interferometry. By characterizing the gate quality, we also investigate the operation in the presence of realistic dephasing. Furthermore, the result provides an essential model suitable for understanding an interplay of geometric phase and Landau-Zener-Stückelberg process which are well explored separately.

A surface code quantum computer in silicon

The exceptionally long quantum coherence times of phosphorus donor nuclear spin qubits in silicon, coupled with the proven scalability of silicon-based nano-electronics, make them attractive candidates for large-scale quantum computing. However, the high threshold of topological quantum error correction can only be captured in a two-dimensional array of qubits operating synchronously and in parallel-posing formidable fabrication and control challenges. We present an architecture that addresses these problems through a novel shared-control paradigm that is particularly suited to the natural uniformity of the phosphorus donor nuclear spin qubit states and electronic confinement. The architecture comprises a two-dimensional lattice of donor qubits sandwiched between two vertically separated control layers forming a mutually perpendicular crisscross gate array. Shared-control lines facilitate loading/unloading of single electrons to specific donors, thereby activating multiple qubits in parallel across the array on which the required operations for surface code quantum error correction are carried out by global spin control. The complexities of independent qubit control, wave function engineering, and ad hoc quantum interconnects are explicitly avoided. With many of the basic elements of fabrication and control based on demonstrated techniques and with simulated quantum operation below the surface code error threshold, the architecture represents a new pathway for large-scale quantum information processing in silicon and potentially in other qubit systems where uniformity can be exploited.

Nearly deterministic bell measurement for multiphoton qubits and its application to quantum information processing

We propose a Bell-measurement scheme by employing a logical qubit in Greenberger-Horne-Zeilinger entanglement with an arbitrary number of photons. Remarkably, the success probability of the Bell measurement as well as teleportation of the Greenberger-Horne-Zeilinger entanglement can be made arbitrarily high using only linear optics elements and photon on-off measurements as the number of photons increases. Our scheme outperforms previous proposals using single-photon qubits when comparing the success probabilities in terms of the average photon usages. It has another important advantage for experimental feasibility in that it does not require photon-number-resolving measurements. Our proposal provides an alternative candidate for all-optical quantum information processing.

Ultrafast and fault-tolerant quantum communication across long distances

Quantum repeaters (QRs) provide a way of enabling long distance quantum communication by establishing entangled qubits between remote locations. In this Letter, we investigate a new approach to QRs in which quantum information can be faithfully transmitted via a noisy channel without the use of long distance teleportation, thus eliminating the need to establish remote entangled links. Our approach makes use of small encoding blocks to fault-tolerantly correct both operational and photon loss errors. We describe a way to optimize the resource requirement for these QRs with the aim of the generation of a secure key. Numerical calculations indicate that the number of quantum memory bits at each repeater station required for the generation of one secure key has favorable polylogarithmic scaling with the distance across which the communication is desired.

High-fidelity cluster state generation for ultracold atoms in an optical lattice

We propose a method for generating high-fidelity multipartite spin entanglement of ultracold atoms in an optical lattice in a short operation time with a scalable manner, which is suitable for measurement-based quantum computation. To perform the desired operations based on the perturbative spin-spin interactions, we propose to actively utilize the extra degrees of freedom (DOFs) usually neglected in the perturbative treatment but included in the Hubbard Hamiltonian of atoms, such as, (pseudo-)charge and orbital DOFs. Our method simultaneously achieves high fidelity, short operation time, and scalability by overcoming the following fundamental problem: enhancing the interaction strength for shortening the operation time breaks the perturbative condition of the interaction and inevitably induces unwanted correlations among the spin and extra DOFs.

Two-photon quantum interference in integrated multi-mode interference devices

Multi-mode interference (MMI) devices fabricated in silicon oxynitride (SiON) with a refractive index contrast of 2.4% provide a highly compact and stable platform for multi-photon non-classical interference. MMI devices can introduce which-path information for photons propagating in the multi-mode section which can result in degradation of this non-classical interference. We theoretically derive the visibility of quantum interference of two photons injected in a MMI device and predict near unity visibility for compact SiON devices. We complement the theoretical results by experimentally demonstrating visibilities of up to 97.7% in 2×2 MMI devices without the requirement of narrow-band photons.

Blind topological measurement-based quantum computation

Blind quantum computation is a novel secure quantum-computing protocol that enables Alice, who does not have sufficient quantum technology at her disposal, to delegate her quantum computation to Bob, who has a fully fledged quantum computer, in such a way that Bob cannot learn anything about Alice's input, output and algorithm. A recent proof-of-principle experiment demonstrating blind quantum computation in an optical system has raised new challenges regarding the scalability of blind quantum computation in realistic noisy conditions. Here we show that fault-tolerant blind quantum computation is possible in a topologically protected manner using the Raussendorf–Harrington–Goyal scheme. The error threshold of our scheme is 4.3×10⁻³, which is comparable to that (7.5×10⁻³) of non-blind topological quantum computation. As the error per gate of the order 10⁻³ was already achieved in some experimental systems, our result implies that secure cloud quantum computation is within reach.

Blind quantum computation is a protocol that permits an algorithm, its input and output to be kept secret from the owner of the computational resource doing the calculation. Morimae and Fujii propose a strategy for topologically protected fault-tolerant blind quantum computation that is robust to environmental noise.

Towards practical classical processing for the surface code

The surface code is unarguably the leading quantum error correction code for 2D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an n×n lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in O(n2) average time per round of error correction. We also describe how to parallelize the algorithm to achieve O(1) average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.

Experimental demonstration of topological error correction

Scalable quantum computing can be achieved only if quantum bits are manipulated in a fault-tolerant fashion. Topological error correction--a method that combines topological quantum computation with quantum error correction--has the highest known tolerable error rate for a local architecture. The technique makes use of cluster states with topological properties and requires only nearest-neighbour interactions. Here we report the experimental demonstration of topological error correction with an eight-photon cluster state. We show that a correlation can be protected against a single error on any quantum bit. Also, when all quantum bits are simultaneously subjected to errors with equal probability, the effective error rate can be significantly reduced. Our work demonstrates the viability of topological error correction for fault-tolerant quantum information processing.

Fault-tolerant topological one-way quantum computation with probabilistic two-qubit gates

We propose a scalable way to construct a 3D cluster state for fault-tolerant topological one-way computation (TOWC) even if the entangling two-qubit gates succeed with a small probability. It is shown that fault-tolerant TOWC can be performed with the success probability of the two-qubit gate such as 0.5 (0.1) provided that the unheralded error probability of the two-qubit gate is less than 0.040% (0.016%). Furthermore, the resource usage is considerably suppressed compared to the conventional fault-tolerant schemes with probabilistic two-qubit gates.

Fault tolerant quantum computation with nondeterministic gates

In certain approaches to quantum computing the operations between qubits are nondeterministic and likely to fail. For example, a distributed quantum processor would achieve scalability by networking together many small components; operations between components should be assumed to be failure prone. In the ultimate limit of this architecture each component contains only one qubit. Here we derive thresholds for fault-tolerant quantum computation under this extreme paradigm. We find that computation is supported for remarkably high failure rates (exceeding 90%) providing that failures are heralded; meanwhile the rate of unknown errors should not exceed 2 in 10(4) operations.