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arXiv:2404.15989 · PRX Quantum

Creating Entangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes

Bence Hetényi, James R. Wootton

Published Dec 3, 2024• 29 citations
arXivHTMLPDF

Abstract

Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid-state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hexagonal (“heavy-hex”) architecture of current IBM Quantum devices. In such cases, it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error-correcting codes, many qubits remain effectively unused. In this work, we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133-qubit IBM Quantum device. Using transversal controlled- X () gates and lattice surgery, we demonstrate entanglement between these logical qubits with code distance up to d = 4 and five rounds of stabilizer-measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical X X , Y Y , and Z Z observables. With this, we verify the violation of Bell’s inequality for both the d = 2 case with postselection featuring a fidelity of 94% and the d = 3 instance using only quantum error correction. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.040334

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Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid-state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hexagonal (“heavy-hex”) architecture of current IBM Quantum devices. In such cases, it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error-correcting codes, many qubits remain effectively unused. In this work, we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133-qubit IBM Quantum device. Using transversal controlled- X () gates and lattice surgery, we demonstrate entanglement between these logical qubits with code distance up to d = 4 and five rounds of stabilizer-measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical X X , Y Y , and Z Z observables. With this, we verify the violation of Bell’s inequality for both the d = 2 case with postselection featuring a fidelity of 94% and the d = 3 instance using only quantum error correction. Published by the American Physical Society 2024
Low-overhead quantum error-correction schemes are essential for enabling quantum computation on registers containing multiple logical qubits. For planar architectures with limited nearest-neighbor qubit connectivity, the surface code has emerged as the leading paradigm. Recent theoretical and experimental work has shown that a physical-qubit connectivity of degree three is sufficient to implement fault-tolerant quantum error correction. In this work, we study lattice surgery in the context of such trivalent architectures and introduce scalable circuit constructions to implement it. Compared with the four-valent measurement scheme, the trivalent lattice-surgery protocol reduces the required resources by O(d)\mathcal{O}(d)O(d) qubits out of a total qubit count of O(d2)\mathcal{O}(d^2)O(d2) and by O(d)\mathcal{O}(d)O(d) two-qubit gates out of a total two-qubit gate count of O(d3)\mathcal{O}(d^3)O(d3). We benchmark the logical fidelity of both lattice-surgery schemes in terms of experimentally realistic simulations targeting an implementation with a fluxonium qubit based architecture and find a potential improvement of up to ≈25%\approx25\%≈25% for distance-three. These results open a way for scalable planar trivalent qubit architectures to host a surface-code-based logical quantum processor.
Quantum error correction is needed for quantum computers to be capable of executing algorithms using hundreds of logical qubits in a fault-tolerant manner. Recent experiments have progressed towards this by demonstrating sufficiently low error rates for state preservation of a single logical qubit. However, quantum computation algorithms also require that these logical qubits can be entangled and that gate operations can be performed on them. Lattice surgery is a technique that offers a practical approach for implementing such gates, particularly in planar quantum processor layouts. Here we demonstrate lattice surgery between two distance-three repetition-code qubits by splitting a single distance-three surface-code qubit. Using a quantum circuit that is fault-tolerant for bit-flip errors, we achieve an improvement in the value of the decoded ZZ logical two-qubit observable compared with a similar non-encoded circuit. We therefore demonstrate the functional building blocks needed for lattice-surgery operations on larger-distance codes based on superconducting circuits.
Abstract Quantum error correction is needed for quantum computers to be capable of executing algorithms using hundreds of logical qubits in a fault-tolerant manner. Recent experiments have progressed towards this by demonstrating sufficiently low error rates for state preservation of a single logical qubit. However, quantum computation algorithms also require that these logical qubits can be entangled and that gate operations can be performed on them. Lattice surgery is a technique that offers a practical approach for implementing such gates, particularly in planar quantum processor layouts. Here we demonstrate lattice surgery between two distance-three repetition-code qubits by splitting a single distance-three surface-code qubit. Using a quantum circuit that is fault-tolerant for bit-flip errors, we achieve an improvement in the value of the decoded Z Z logical two-qubit observable compared with a similar non-encoded circuit. We therefore demonstrate the functional building blocks needed for lattice-surgery operations on larger-distance codes based on superconducting circuits.
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)0.0365(2)0.0365(2) and 0.0282(1)0.0282(1)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 XXX axis. For the logical RX(π/4)R_{X}(π/4)RX​(π/4) gate, we achieve a logical gate fidelity of 0.943−9+100.943_{-9}^{+10}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.
Abstract The development of quantum computing architectures from early designs and current noisy devices to fully fledged quantum computers hinges on achieving fault tolerance using quantum error correction1–4. However, these correction capabilities come with an overhead for performing the necessary fault-tolerant logical operations on logical qubits (qubits that are encoded in ensembles of physical qubits and protected by error-correction codes)5–8. One of the most resource-efficient ways to implement logical operations is lattice surgery9–11, where groups of physical qubits, arranged on lattices, can be merged and split to realize entangling gates and teleport logical information. Here we report the experimental realization of lattice surgery between two qubits protected via a topological error-correction code in a ten-qubit ion-trap quantum information processor. In this system, we can carry out the necessary quantum non-demolition measurements through a series of local and entangling gates, as well as measurements on auxiliary qubits. In particular, we demonstrate entanglement between two logical qubits and we implement logical state teleportation between them. The demonstration of these operations—fundamental building blocks for quantum computation—through lattice surgery represents a step towards the efficient realization of fault-tolerant quantum computation.
Topological quantum error correction is a milestone in the scaling roadmap of quantum computers, which targets circuits with trillions of gates that would allow running quantum algorithms for real-world problems. The square-lattice surface code has become the workhorse to address this challenge, as it poses milder requirements on current devices both in terms of required error rates and small local connectivities. In some platforms, however, the connectivities are kept even lower in order to minimise gate errors at the hardware level, which limits the error correcting codes that can be directly implemented on them. In this work, we make a comparative study of possible strategies to overcome this limitation for the heavy-hexagonal lattice, the architecture of current IBM superconducting quantum computers. We explore two complementary strategies: the search for an efficient embedding of the surface code into the heavy-hexagonal lattice, as well as the use of codes whose connectivity requirements are naturally tailored to this architecture, such as subsystem-type and Floquet codes. Using noise models of increased complexity, we assess the performance of these strategies for IBM devices in terms of their error thresholds and qubit footprints. An optimized SWAP-based embedding of the surface code is found to be the most promising strategy towards a near-term demonstration of quantum error correction advantage.