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arXiv:2506.13619

Blocklet concatenation: Low-overhead fault-tolerant protocols for fusion-based quantum computation

Daniel Litinski

Published Jun 16, 2025• 0 citations
arXivHTMLPDF

Abstract

We introduce a construction for protocols for fault-tolerant quantum computing based on code concatenation and transversal gates. These protocols can be interpreted as families of quantum circuits of low-weight stabilizer measurements without strict locality constraints, effectively implementing concatenated codes. However, we primarily study these protocols in the context of photonic fusion-based quantum computing (FBQC), where they yield families of fusion networks with constant-sized resource states. Their high erasure thresholds relative to their resource-state cost establish them as promising candidates to replace surface codes in the context of FBQC. Examples include protocol families using 8-, 10- and 12-qubit resource states, with erasure thresholds of 13.8%, 19.1% and 11.5%, and footprint-per-logical-qubit scaling as $\mathcal{O}(d)$, $\mathcal{O}(d^{1.46})$ and $\mathcal{O}(d^{0.58})$, respectively, where $d$ is the code distance. We also present techniques for performing logical operations, decoding, and implementing the protocols in photonic hardware. Although we focus on photonic FBQC, these ideas may also be of interest in other settings.

DOI
10.48550/arxiv.2506.13619

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We introduce a construction for protocols for fault-tolerant quantum computing based on code concatenation and transversal gates. These protocols can be interpreted as families of quantum circuits of low-weight stabilizer measurements without strict locality constraints, effectively implementing concatenated codes. However, we primarily study these protocols in the context of photonic fusion-based quantum computing (FBQC), where they yield families of fusion networks with constant-sized resource states. Their high erasure thresholds relative to their resource-state cost establish them as promising candidates to replace surface codes in the context of FBQC. Examples include protocol families using 8-, 10- and 12-qubit resource states, with erasure thresholds of 13.8%, 19.1% and 11.5%, and footprint-per-logical-qubit scaling as O(d)\mathcal{O}(d)O(d), O(d1.46)\mathcal{O}(d^{1.46})O(d1.46) and O(d0.58)\mathcal{O}(d^{0.58})O(d0.58), respectively, where ddd is the code distance. We also present techniques for performing logical operations, decoding, and implementing the protocols in photonic hardware. Although we focus on photonic FBQC, these ideas may also be of interest in other settings.
Abstract The essential requirement for fault-tolerant quantum computation (FTQC) is the total protocol design to achieve a fair balance of all the critical factors relevant to its practical realization, such as the space overhead, the threshold, and the modularity. A major obstacle in realizing FTQC with conventional protocols, such as those based on the surface code and the concatenated Steane code, has been the space overhead, i.e., the required number of physical qubits per logical qubit. Protocols based on high-rate quantum low-density parity-check (LDPC) codes gather considerable attention as a way to reduce the space overhead, but problematically, the existing fault-tolerant protocols for such quantum LDPC codes sacrifice other factors. Here, we construct a new fault-tolerant protocol to meet these requirements simultaneously based on more recent progress on the techniques for concatenated codes rather than quantum LDPC codes, achieving a constant space overhead, a high threshold, and flexibility in modular architecture designs. In particular, under a physical error rate of 0.1%, our protocol reduces the space overhead to achieve the logical CNOT error rates 10 −10 and 10 −24 by more than 90% and 96%, respectively, compared to the protocol for the surface code. Furthermore, our protocol achieves the threshold of 2.5% under a conventional circuit-level error model, substantially outperforming that of the surface code. The use of concatenated codes also naturally introduces abstraction layers essential for the modularity of FTQC architectures. These results indicate that the code-concatenation approach opens a way to significantly save qubits in realizing FTQC while fulfilling the other essential requirements for the practical protocol design.
Useful fault-tolerant quantum computers require very large numbers of physical qubits. Quantum computers are often designed as arrays of static qubits executing gates and measurements. Photonic qubits require a different approach. In photonic fusion-based quantum computing (FBQC), the main hardware components are resource-state generators (RSGs) and fusion devices connected via waveguides and switches. RSGs produce small entangled states of a few photonic qubits, whereas fusion devices perform entangling measurements between different resource states, thereby executing computations. In addition, low-loss photonic delays such as optical fiber can be used as fixed-time quantum memories simultaneously storing thousands of photonic qubits. Here, we present a modular architecture for FBQC in which these components are combined to form "interleaving modules" consisting of one RSG with its associated fusion devices and a few fiber delays. Exploiting the multiplicative power of delays, each module can add thousands of physical qubits to the computational Hilbert space. Networks of modules are universal fault-tolerant quantum computers, which we demonstrate using surface codes and lattice surgery as a guiding example. Our numerical analysis shows that in a network of modules containing 1-km-long fiber delays, each RSG can generate four logical distance-35 surface-code qubits while tolerating photon loss rates above 2% in addition to the fiber-delay loss. We illustrate how the combination of interleaving with further uses of non-local fiber connections can reduce the cost of logical operations and facilitate the implementation of unconventional geometries such as periodic boundaries or stellated surface codes. Interleaving applies beyond purely optical architectures, and can also turn many small disconnected matter-qubit devices with transduction to photons into a large-scale quantum computer.
Fault-tolerant complexes describe surface-code fault-tolerant protocols from a single geometric object. We first introduce fusion complexes that define a general family of fusion-based quantum computing (FBQC) fault-tolerant quantum protocols based on surface codes. We show that any 3-dimensional cell complex where each edge has four incident faces gives a valid fusion complex. This construction enables an automated search for fault tolerance schemes, allowing us to identify 627 examples within a moderate search time. We implement this using the open-source software tool Gavrog and present threshold results for a variety of schemes, finding fusion networks with higher erasure and Pauli thresholds than those existing in the literature. We then define more general structures we call fault-tolerant complexes that provide a homological description of fault tolerance from a large family of low-level error models, which include circuit-based computation, floquet-based computation, and FBQC with multi-qubit measurements. This extends the applicability of homological descriptions of fault tolerance, and enables the generation of many new schemes which have not been previously identified. We also define families of fault-tolerant complexes for color codes and 3d single-shot subsystem codes, which enables similar constructive methods, and we present several new examples of each.
As existing approaches to get around the restriction of the no-go theorem generally exhibit high ancillary qubit overhead costs, we propose a scheme for universal fault-tolerant quantum computation by using the pieceable fault-tolerant protocol and code switching techniques. In additional, by utilizing the construction of the pieceable fault-tolerant circuit, we adopt a decoding strategy based on a deep-neural-network algorithm to improve the error threshold of the non-Clifford logical gate circuit. We describe our universal construction in detail with a two-level nonuniform concatenated 25-qubit code and perform numerical simulations to analyze the depolarizing noise threshold of a given universal gate set with this code. The resources required to implement this universal gate set are also estimated to further demonstrate the efficiency of our scheme. We compare these results with the existing universal concatenation methods and conclude that our method outperforms them in terms of the lower bound of the error threshold and qubit resource overhead.
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