Scaling fault-tolerant quantum computing beyond a single processor demands logical—logical entanglement. While surface codes excel in local error correction, creating robust, distributed entanglement remains challenging. In this work, we introduce Entanglement Ejection—a protocol for generating physical—logical entanglement from a single logical qubit. This subroutine establishes logical—logical entanglement via entanglement swapping with a physical Bell pair provided by a quantum network. Unlike common injection-based methods, our protocol is deterministic, requiring no post-selection and thereby has the potential of reducing spatio-temporal resource overhead. We show via numerical simulations under a superconducting-specific noise model that this potential resource efficiency comes at the cost of a slightly higher, yet comparable, logical error rate.
Scaling fault-tolerant quantum computing beyond a single processor demands logical—logical entanglement. While surface codes excel in local error correction, creating robust, distributed entanglement remains challenging. In this work, we introduce Entanglement Ejection—a protocol for generating physical—logical entanglement from a single logical qubit. This subroutine establishes logical—logical entanglement via entanglement swapping with a physical Bell pair provided by a quantum network. Unlike common injection-based methods, our protocol is deterministic, requiring no post-selection and thereby has the potential of reducing spatio-temporal resource overhead. We show via numerical simulations under a superconducting-specific noise model that this potential resource efficiency comes at the cost of a slightly higher, yet comparable, logical error rate.
The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{\ensuremath{-}3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.
Fault-tolerant logical operations are fundamental for scalable quantum computation. Here, we report the experimental realization of lattice-surgery operations between a pair of distance-three surface-code logical qubits on a planar superconducting processor. During repeated syndrome extraction cycles, the logical qubits exhibit per-cycle error rates of 0.0365(2) and 0.0282(1), respectively, after leakage events are rejected. By leveraging joint initialization and lattice splitting, we deterministically prepare a logical Bell state, confirming genuine bipartite entanglement via the error-corrected logical state fidelity. We further execute a two-qubit Deutsch-Jozsa algorithm at the logical level to demonstrate algorithmic utility in a fault-tolerant framework. Finally, to achieve universal control, we implement magic-state injection and gate teleportation to realize continuous non-Clifford rotations about the logical X axis. For the logical RX(π/4) gate, we achieve a logical gate fidelity of 0.943−9+10 conditioned on the absence of detected errors. These results establish lattice surgery as a practical and versatile paradigm for logical computation in near-term surface-code architectures, representing a critical milestone toward scalable fault-tolerant quantum advantage in superconducting circuits.
Fault-tolerant quantum computing based on the surface code requires magic states to achieve universality. The initially prepared magic states have low fidelity and need to be purified by a costly procedure called magic state distillation. The high spatial-temporal cost for distillation can be reduced by improving the fidelity of initial magic states. Prior works have investigated state injection approaches on the regular surface code. In this work, we propose a magic state injection method for the rotated surface code that requires fewer physical qubits to encode one logical qubit. Analytical results show that our method could achieve lower logical error rates than the state-of-the-art approach for the regular surface code in [1]. When both the two-qubit gate and single-qubit operation error rates are p, the new method results in a logical error rate to 34p/15 compared to 46p/15 in [1]. This fidelity improvement and fewer qubit encoding requirement would help reduce the overhead of fault-tolerant quantum computation, which is crucial for practical implementation in the near term.
We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side.
We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.
Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol's performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.
Quantum computing has the potential to transform information technology by offering algorithms for certain tasks, such as quantum simulation, that are vastly more efficient than what is possible with any classical device. But experimentally implementing practical quantum information processing is a very difficult task. Here we study two important, and closely related, aspects of this challenge: architectures for quantum computing, and quantum error correction Exquisite quantum control has now been achieved in small ion traps, in nitrogen-vacancy centres and in superconducting qubit clusters, but the challenge remains of how to scale these systems to build practical quantum devices. In Part I of this thesis we analyse one approach to building a scalable quantum computer by networking together many simple processor cells, thus avoiding the need to create a single complex structure. The difficulty is that realistic quantum links are very error prone. Here we describe a method by which even these error-prone cells can perform quantum error correction. Groups of cells generate and purify shared resource states, which then enable stabilization of topologically encoded data. Given a realistically noisy network (10% error rate) we find that our protocol can succeed provided that all intra-cell error rates are below 0.8%. Furthermore, we show that with some adjustments, the protocols we employ can be made robust also against high levels of loss in the network interconnects. We go on to analyse the potential running speed of such a device. Using levels of fidelity that are either already achievable in experimental systems, or will be in the near-future, we find that employing a surface code approach in a highly noisy and lossy network architecture can result in kilohertz computer clock speeds. In Part II we consider the question of quantum error correction beyond the surface code. We consider several families of topological codes, and determine the minimum requirements to demonstrate proof-of-principle error suppression in each type of code. A particularly promising code is the gauge color code, which admits a universal transversal gate set. Furthermore, a recent result of Bombin shows the gauge color code supports an error-correction protocol that achieves tolerance to noisy measurements without the need for repeated measurements, so called single-shot error correction. Here, we demonstrate the promise of single-shot error correction by designing a decoder and investigating its performance. We simulate fault-tolerant error correction with the gauge color code, and estimate a sustainable error rate, i.e. the threshold for the long time limit, of ~0.31% for a phenomenological noise model using a simple decoding algorithm.