Fault-tolerant Computation Research Articles

Modular Quantum Processor with an All-to-All Reconfigurable Router

Superconducting qubits provide a promising approach to large-scale fault-tolerant quantum computing. However, qubit connectivity on a planar surface is typically restricted to only a few neighboring qubits. Achieving longer-range and more flexible connectivity, which is particularly appealing in light of recent developments in error-correcting codes, however, usually involves complex multilayer packaging and external cabling, which is resource intensive and can impose fidelity limitations. Here, we propose and realize a high-speed on-chip quantum processor that supports reconfigurable all-to-all coupling with a large on-off ratio. We implement the design in a four-node quantum processor, built with a modular design comprising a wiring substrate coupled to two separate qubit-bearing substrates, each including two single-qubit nodes. We use this device to demonstrate reconfigurable controlled-Z gates across all qubit pairs, with a benchmarked average fidelity of 96.00%±0.08% and best fidelity of 97.14%±0.07%, limited mainly by dephasing in the qubits. We also generate multiqubit entanglement, distributed across the separate modules, demonstrating GHZ-3 and GHZ-4 states with fidelities of 88.15%±0.24% and 75.18%±0.11%, respectively. This approach promises efficient scaling to larger-scale quantum circuits and offers a pathway for implementing quantum algorithms and error-correction schemes that benefit from enhanced qubit connectivity. Published by the American Physical Society 2024

Decoding quantum color codes with MaxSAT

In classical computing, error-correcting codes are well established and are ubiquitous both in theory and practical applications. For quantum computing, error-correction is essential as well, but harder to realize, coming along with substantial resource overheads and being concomitant with needs for substantial classical computing. Quantum error-correcting codes play a central role on the avenue towards fault-tolerant quantum computation beyond presumed near-term applications. Among those, color codes constitute a particularly important class of quantum codes that have gained interest in recent years due to favourable properties over other codes. As in classical computing, decoding is the problem of inferring an operation to restore an uncorrupted state from a corrupted one and is central in the development of fault-tolerant quantum devices. In this work, we show how the decoding problem for color codes can be reduced to a slight variation of the well-known LightsOut puzzle. We propose a novel decoder for quantum color codes using a formulation as a MaxSAT problem based on this analogy. Furthermore, we optimize the MaxSAT construction and show numerically that the decoding performance of the proposed decoder achieves state-of-the-art decoding performance on color codes. The implementation of the decoder as well as tools to automatically conduct numerical experiments are publicly available as part of the Munich Quantum Toolkit (MQT) on GitHub.

Scalable Architecture for Trapped-Ion Quantum Computing Using rf Traps and Dynamic Optical Potentials

Qubits based on ions trapped in linear radio-frequency traps form a successful platform for quantum computing, due to their high fidelity of operations, all-to-all connectivity, and degree of local control. In principle, there is no fundamental limit to the number of ion-based qubits that can be confined in a single 1D register. However, in practice, there are two main issues associated with long trapped-ion crystals, that stem from the “softening” of their modes of motion, upon scaling up: high heating rates of the ions’ motion and a dense motional spectrum; both impede the performance of high-fidelity qubit operations. Here, we propose a holistic, scalable architecture for quantum computing with large ion crystals that overcomes these issues. Our method relies on dynamically operated optical potentials that instantaneously segment the ion crystal into cells of a manageable size. We show that these cells behave as nearly independent quantum registers, allowing for parallel entangling gates on all cells. The ability to reconfigure the optical potentials guarantees connectivity across the full ion crystal and also enables efficient midcircuit measurements. We study the implementation of large-scale parallel multiqubit entangling gates that operate simultaneously on all cells and present a protocol to compensate for crosstalk errors, enabling full-scale usage of an extensively large register. We illustrate that this architecture is advantageous both for fault-tolerant digital quantum computation and for analog quantum simulations. Published by the American Physical Society 2024

Doubly Optimal Parallel Wire Cutting without Ancilla Qubits

A restriction in the quality and quantity of available qubits presents a substantial obstacle to the application of near-term and early fault-tolerant quantum computers in practical tasks. To confront this challenge, some techniques for effectively augmenting the system size through classical processing have been proposed; one promising approach is quantum circuit cutting. The main idea of quantum circuit cutting is to decompose an original circuit into smaller subcircuits and combine outputs from these subcircuits to recover the original output. Although this approach enables us to simulate larger quantum circuits beyond physically available circuits, it needs classical overheads quantified by two metrics: the sampling overhead in the number of measurements to reconstruct the original output, and the number of channels in the decomposition. Thus, it is crucial to devise a decomposition method that minimizes both of these metrics, thereby reducing the overall execution time. This paper studies the problem of decomposing the n-qubit identity channel, i.e., n-parallel wire cutting, into a set of local operations and classical communication; then we give an optimal wire-cutting method composed of channels based on mutually unbiased bases that achieves minimal overheads in both the sampling overhead and the number of channels, without ancilla qubits. This is in stark contrast to the existing method that achieves the optimal sampling overhead yet with ancilla qubits. Moreover, we derive a tight lower bound on the number of channels in parallel wire cutting without ancilla systems and show that only our method achieves this lower bound among the existing methods. Notably, our method shows an exponential improvement in the number of channels, compared to the aforementioned ancilla-assisted method that achieves optimal sampling overhead. Our work significantly alleviates the additional overheads for the wire-cutting method and may provide essential components for the early success of quantum computing. Published by the American Physical Society 2024

Optimality of the Howard-Vala T-gate in stabilizer quantum computation

In a remarkable work [Phys. Rev. A 86 022316 (2012)], Howard and Vala introduced a qudit version of the qubit T-gate (i.e., π/8-gate) for any prime dimensional system. This non-Clifford gate is a key ingredient of the paradigm ‘Clifford +T’, which are widely employed in the stabilizer formalism of universal and fault-tolerant quantum computation. Considering the applications and significance of the T-gate, it is desirable to characterize it from various angles. Here we prove that in any prime dimensional system, the Howard-Vala T-gate is optimal, among all diagonal gates, for generating magic resources from stabilizer states when the magic is quantified via the L 1-norm of characteristic functions (Fourier transforms) of quantum states. The quadratic Gaussian sum in number theory plays a key role in establishing this optimality. This highlights an extreme feature of the Howard-Vala T-gate. We further reveal an intrinsic relation between the Howard-Vala T-gate and the Watson-Campbell-Anwar-Browne T-gate [Phys. Rev. A 92 022312 (2015)] for any prime dimensional system.

Weight-Reduced Stabilizer Codes with Lower Overhead

Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024

High-fidelity spin readout via the double latching mechanism

Projective measurement of single-electron spins, or spin readout, is among the most fundamental technologies for spin-based quantum information processing. Implementing spin readout with both high-fidelity and scalability is indispensable for developing fault-tolerant quantum computers in large-scale spin-qubit arrays. To achieve high fidelity, a latching mechanism is useful. However, the fidelity can be decreased by spin relaxation and charge state leakage, and the scalability is currently challenging. Here, we propose and demonstrate a double-latching high-fidelity spin readout scheme, which suppresses errors via an additional latching process. We experimentally show that the double-latching mechanism provides significantly higher fidelity than the conventional latching mechanism and estimate a potential spin readout fidelity of 99.94% using highly spin-dependent tunnel rates. Due to isolation from error-inducing processes, the double-latching mechanism combined with scalable charge readout is expected to be useful for large-scale spin-qubit arrays while maintaining high fidelity.

Information processing at the speed of light.

In recent years, quantum computing has made significant strides, particularly in light-based technology. The introduction of quantum photonic chips has ushered in an era marked by scalability, stability, and cost-effectiveness, paving the way for innovative possibilities within compact footprints. This article provides a comprehensive exploration of photonic quantum computing, covering key aspects such as encoding information in photons, the merits of photonic qubits, and essential photonic device components including light squeezers, quantum light sources, interferometers, photodetectors, and waveguides. The article also examines photonic quantum communication and internet, and its implications for secure systems, detailing implementations such as quantum key distribution and long-distance communication. Emerging trends in quantum communication and essential reconfigurable elements for advancing photonic quantum internet are discussed. The review further navigates the path towards establishing scalable and fault-tolerant photonic quantum computers, highlighting quantum computational advantages achieved using photons. Additionally, the discussion extends to programmable photonic circuits, integrated photonics and transformative applications. Lastly, the review addresses prospects, implications, and challenges in photonic quantum computing, offering valuable insights into current advancements and promising future directions in this technology.

Automatic Test Pattern Generation for Robust Quantum Circuit Testing

Quantum circuit testing is essential for detecting potential faults in realistic quantum devices, while the testing process itself also suffers from the inexactness and unreliability of quantum operations. This article alleviates the issue by proposing a novel framework of automatic test pattern generation (ATPG) for robust testing of logical quantum circuits. We introduce the stabilizer projector decomposition (SPD) for representing the quantum test pattern and construct the test application (i.e., state preparation and measurement) using Clifford-only circuits, which are rather robust and efficient as evidenced in the fault-tolerant quantum computation. However, it is generally hard to generate SPDs due to the exponentially growing number of the stabilizer projectors. To circumvent this difficulty, we develop an SPD generation algorithm, as well as several acceleration techniques that can exploit both locality and sparsity in generating SPDs. The effectiveness of our algorithms are validated by (1) theoretical guarantees under reasonable conditions and (2) experimental results on commonly used benchmark circuits, such as Quantum Fourier Transform (QFT), Quantum Volume (QV), and Bernstein-Vazirani (BV) in IBM Qiskit.

Improving Threshold for Fault-Tolerant Color-Code Quantum Computing by Flagged Weight Optimization

Color codes are promising quantum error-correction (QEC) codes because they have an advantage over surface codes in that all Clifford gates can be implemented transversally. However, the thresholds of color codes under circuit-level noise are relatively low, mainly because measurements of their high-weight stabilizer generators cause an increase in the circuit depth and, thus, substantial errors are introduced. This makes color codes not the best candidate for fault-tolerant quantum computing. Here, we propose a method to suppress the impact of such errors by optimizing weights of decoders using conditional error probabilities conditioned on the measurement outcomes of flag qubits. In numerical simulations, we improve the threshold of the (4.8.8) color code under circuit-level noise from 0.14% to around 0.27%, which is calculated by using an integer programming decoder. Furthermore, in the (6.6.6) color code, we achieve a circuit-level threshold of around 0.36%, which is almost the same value as the highest value in the previous studies employing the same noise model. In both cases, the effective code distance is also improved compared to a conventional method that uses a single ancilla qubit for each stabilizer measurement. Thereby, the achieved logical error rates at low physical error rates are almost one order of magnitude lower than those of the conventional method with the same code distance. Even when compared to the single-ancilla method with a higher code distance, considering the increased number of qubits used in our method, we achieve lower logical error rates in most cases. This method can also be applied to other weight-based decoders, making the color codes more promising as candidates for the experimental implementation of QEC. Furthermore, one can utilize this approach to improve a threshold of wider classes of QEC codes, such as high-rate quantum low-density parity-check codes. Published by the American Physical Society 2024

Clifford orbits and stabilizer states

Abstract Stabilizer states serve as “classical objects” in the stabilizer formalism of quantum theory, and play an important role in quantum error correction, fault-tolerant quantum computation, and quantum communication. They provide an efficient description of many basic features of quantum theory and exhibit a rich structure. For prime dimensional systems, they may be defined by two quite different yet equivalent ways: The first is via stabilizer groups (maximal Abelian subgroups of the discrete Heisenberg-Weyl group). The second is via the orbits of the Clifford group acting on any computational basis state. However, in a general dimensional system, this equivalence breaks down, and consequently, it is desirable to classify the difference and relation between the above two approaches to stabilizer states. In this work, we show that these two approaches are equivalent if and only if the system dimension is square-free (i.e., has no square factor). Furthermore, we completely clarify the relation between the Clifford orbits and stabilizer states in an arbitrary dimensional system, derive the explicit expressions of the Clifford orbits and determine their cardinalities.

Research on Quantum State Implementation Based on Topological Insulators

Abstract This paper first introduces the mainstream technology route of quantum computing. The analysis provides the role and advantages of topological insulators in quantum computing, such as providing stable quantum bits, enhancing the coherence and non-locality of quantum bits, and enabling the transmission of quantum information and entanglement allocation. This paper then proposes a quantum state implementation scheme based on Bi2Se3, based on the characteristics and advantages of the quantum Hall effect of topological insulators. It includes calculating the eigenstates of the Hamiltonian of topological insulators, obtaining the spatial distribution of wave functions, and generating a quantum bit platform based on topological insulators. A quantum state implementation system based on topological insulator (Bi2Se3) was constructed, and the system framework and reference configuration were provided. The solution presented in this article is an important research achievement in topological quantum computing, providing more possibilities for the implementation of feasible fault-tolerant quantum computer prototypes.

Self-stabilizing multivalued consensus in asynchronous crash-prone systems

The multivalued consensus problem is a fundamental issue in fault-tolerant distributed computing. It encompasses a wide range of agreement problems where processes must unanimously decide on a specific value v∈V, with |V|≥2. Existing solutions that handle process crash failures simplify the multivalued consensus problem by reducing it to the binary consensus problem. Examples of such solutions include Mostéfaoui-Raynal-Tronel [IPL 2000] and Zhang-Chen [IPL 2009].In this work, we aim to design an even more reliable solution by leveraging the concept of self-stabilization, which provides a strong form of fault tolerance. Self-stabilizing algorithms can recover from transient faults, which represent any deviation from the system's intended behavior (as long as the algorithm code remains intact) in addition to process and communication failures.To the best of our knowledge, this work presents the first self-stabilizing algorithm for multivalued consensus in asynchronous message-passing systems susceptible to process failures and transient faults. Our solution uses, at most, n concurrent invocations of binary consensus. This is another way we advance state-of-the-art solutions compared to previous non-self-stabilizing ones. For example, Mostéfaoui-Raynal-Tronel's solution requires an unbounded number of sequential invocations of binary consensus.

Overcoming the coherence time barrier in quantum machine learning on temporal data

The practical implementation of many quantum algorithms known today is limited by the coherence time of the executing quantum hardware and quantum sampling noise. Here we present a machine learning algorithm, NISQRC, for qubit-based quantum systems that enables inference on temporal data over durations unconstrained by decoherence. NISQRC leverages mid-circuit measurements and deterministic reset operations to reduce circuit executions, while still maintaining an appropriate length persistent temporal memory in the quantum system, confirmed through the proposed Volterra Series analysis. This enables NISQRC to overcome not only limitations imposed by finite coherence, but also information scrambling in monitored circuits and sampling noise, problems that persist even in hypothetical fault-tolerant quantum computers that have yet to be realized. To validate our approach, we consider the channel equalization task to recover test signal symbols that are subject to a distorting channel. Through simulations and experiments on a 7-qubit quantum processor we demonstrate that NISQRC can recover arbitrarily long test signals, not limited by coherence time.

Gate operations for superconducting qubits and non-Markovianity

While the accuracy of qubit operations has been greatly improved in the last decade, further development is demanded to achieve the ultimate goal: a fault-tolerant quantum computer that can solve real-world problems more efficiently than classical computers. With growing fidelities even subtle effects of environmental noise such as qubit–reservoir correlations and non-Markovian dynamics turn into the focus for both circuit design and control. To guide progress, we disclose, in a numerically rigorous manner, a comprehensive picture of the single-qubit dynamics in presence of a broad class of noise sources and for entire sequences of gate operations. Thermal reservoirs ranging from Ohmic to deep 1/fɛ-like sub-Ohmic behavior are considered to imitate realistic scenarios for superconducting qubits. Apart from dynamical features, fidelities of the qubit performance over entire sequences are analyzed as a figure of merit. The relevance of retarded feedback and long-range qubit–reservoir correlations is demonstrated on a quantitative level, thus, providing a deeper understanding of the limitations of performances for current devices and guiding the design of future ones. Published by the American Physical Society 2024

Machine learning for efficient generation of universal photonic quantum computing resources

We present numerical results from simulations using deep reinforcement learning to control a measurement-based quantum processor—a time-multiplexed optical circuit sampled by photon-number-resolving detection—and find it generates squeezed cat states quasi-deterministically, with an average success rate of 98%, far outperforming all other proposals. Since squeezed cat states are deterministic precursors to the Gottesman–Kitaev–Preskill (GKP) bosonic error code, this is a key result for enabling fault tolerant photonic quantum computing. Informed by these simulations, we also discovered a one-step quantum circuit of constant parameters that can generate GKP states with high probability, though not deterministically.

Empowering a qudit-based quantum processor by traversing the dual bosonic ladder

High-dimensional quantum information processing has emerged as a promising avenue to transcend hardware limitations and advance the frontiers of quantum technologies. Harnessing the untapped potential of the so-called qudits necessitates the development of quantum protocols beyond the established qubit methodologies. Here, we present a robust, hardware-efficient, and scalable approach for operating multidimensional solid-state systems using Raman-assisted two-photon interactions. We then utilize them to construct extensible multi-qubit operations, realize highly entangled multidimensional states including atomic squeezed states and Schrödinger cat states, and implement programmable entanglement distribution along a qudit array. Our work illuminates the quantum electrodynamics of strongly driven multi-qudit systems and provides the experimental foundation for the future development of high-dimensional quantum applications such as quantum sensing and fault-tolerant quantum computing.

Construction of Antisymmetric Variational Quantum States with Real Space Representation.

Electronic state calculations using quantum computers are mostly based on the second quantized formulation, which is suitable for qubit representation. Another way to describe electronic states on a quantum computer is based on the first quantized formulation, which is expected to achieve smaller scaling with respect to the number of basis functions than the second quantized formulation. Among basis functions, a real space basis is an attractive option for quantum dynamics simulations in the fault-tolerant quantum computation (FTQC) era. A major difficulty in the first quantized algorithm with a real space basis is state preparation for many-body electronic systems. This difficulty stems from the antisymmetry of electrons, and it is not straightforward to construct antisymmetric quantum states on a quantum circuit. In this study, we provide a design principle for constructing variational quantum circuits to prepare an antisymmetric quantum state. The proposed circuit generates the superposition of exponentially many Slater determinants, that is, multiconfiguration state, which provides a systematic approach to approximating the exact ground state. We performed the variational quantum eigensolver (VQE) to obtain the ground state of a one-dimensional hydrogen molecular system. As a result, the proposed circuit well reproduced the exact antisymmetric ground state and its energy, whereas the conventional variational circuit yielded neither the antisymmetric nor the symmetric state. Furthermore, we analyzed the many-body wave functions based on the quantum information theory, which illustrated the relation between the electron correlation and the quantum entanglement.

Quantum resource estimation for large scale quantum algorithms

Quantum algorithms are often represented in terms of quantum circuits operating on ideal (logical) qubits. However, the practical implementation of these algorithms poses significant challenges. Many quantum algorithms require a substantial number of logical qubits, and the inherent susceptibility to errors of quantum computers require quantum error correction. The integration of error correction introduces overhead in terms of both space (physical qubits required) and runtime (how long the algorithm needs to be run for). This paper addresses the complexity of comparing classical and quantum algorithms, primarily stemming from the additional quantum error correction overhead. We propose a comprehensive framework that facilitates a direct and meaningful comparison between classical and quantum algorithms. By acknowledging and addressing the challenges introduced by quantum error correction, our framework aims to provide a clearer understanding of the comparative performance of classical and quantum computing approaches. This work contributes to understanding the practical viability and potential advantages of quantum algorithms in real-world applications.We apply our framework to quantum cryptanalysis, since it is well known that quantum algorithms can break factoring and discrete logarithm based cryptography and weaken symmetric cryptography and hash functions. In order to estimate the real-world impact of these attacks, apart from tracking the development of fault-tolerant quantum computers it is important to have an estimate of the resources needed to implement these quantum attacks. This analysis provides state-of-the art snap-shot estimates of the realistic costs of implementing quantum attacks on these important cryptographic algorithms, assuming quantum fault-tolerance is achieved using surface code methods, and spanning a range of potential error rates. These estimates serve as a guide for gauging the realistic impact of these algorithms and for benchmarking the impact of future advances in quantum algorithms, circuit synthesis and optimization, fault-tolerance methods and physical error rates.

Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons

Hybridizing different degrees of freedom or physical platforms potentially offers various advantages in building scalable quantum architectures. Here, we introduce a fault-tolerant hybrid quantum computation by building on the advantages of both discrete-variable (DV) and continuous-variable (CV) systems. In particular, we define a CV-DV hybrid qubit with a bosonic cat code and a single photon, which is implementable in current photonic platforms. Due to the cat code encoded in the CV part, the predominant loss errors are readily correctable without multiqubit encoding, while the logical basis is inherently orthogonal due to the DV part. We design fault-tolerant architectures by concatenating hybrid qubits and an outer DV quantum error-correction code such as a topological code, exploring their potential merit in developing scalable quantum computation. We demonstrate by numerical simulations that our scheme is at least an order of magnitude more resource efficient compared to all previous proposals in photonic platforms, allowing us to achieve a record-high loss threshold among existing CV and hybrid approaches. We discuss the realization of our approach not only in all-photonic platforms but also in other hybrid platforms including superconducting and trapped-ion systems, which allows us to find various efficient routes toward fault-tolerant quantum computing. Published by the American Physical Society 2024