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1. Error-correcting codes for fermionic quantum simulation

   https://doi.org/10.21468/SciPostPhys.16.1.033  

   Chen, Yu-An; Gorshkov, Alexey V.; Xu, Yijia (January 2024, SciPost Physics)

   Utilizing the framework of\mathbb{Z}_2 ${\mathbb{Z}}_2$lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any\lfloor \frac{d-1}{2} \rfloor $\lfloor \frac{d-1}{2}\rfloor$-qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically. 

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2. Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes

   https://doi.org/10.1103/PRXQuantum.5.020355  

   Omanakuttan, Sivaprasad; Buchemmavari, Vikas; Gross, Jonathan A; Deutsch, Ivan H; Marvian, Milad (June 2024, PRX Quantum)

   We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous-variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources can exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. A key component is the gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error-correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of 87Sr, and show how to generate the universal gate set, including the rank-preserving gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing. 

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3. Erasure conversion in a high-fidelity Rydberg quantum simulator

   https://doi.org/10.1038/s41586-023-06516-4  

   Scholl, Pascal; Shaw, Adam L.; Tsai, Richard Bing-Shiun; Finkelstein, Ran; Choi, Joonhee; Endres, Manuel (October 2023, Nature)

   Abstract Minimizing and understanding errors is critical for quantum science, both in noisy intermediate scale quantum (NISQ) devices¹and for the quest towards fault-tolerant quantum computation^2,3. Rydberg arrays have emerged as a prominent platform in this context⁴with impressive system sizes^5,6and proposals suggesting how error-correction thresholds could be significantly improved by detecting leakage errors with single-atom resolution^7,8, a form of erasure error conversion^9–12. However, two-qubit entanglement fidelities in Rydberg atom arrays^13,14have lagged behind competitors^15,16and this type of erasure conversion is yet to be realized for matter-based qubits in general. Here we demonstrate both erasure conversion and high-fidelity Bell state generation using a Rydberg quantum simulator^5,6,17,18. When excising data with erasure errors observed via fast imaging of alkaline-earth atoms^19–22, we achieve a Bell state fidelity of$$\ge 0.997{1}_{-13}^{+10}$$ $\ge 0.9971_{-13}^{+10}$, which improves to$$\ge 0.998{5}_{-12}^{+7}$$ $\ge 0.9985_{-12}^{+7}$when correcting for remaining state-preparation errors. We further apply erasure conversion in a quantum simulation experiment for quasi-adiabatic preparation of long-range order across a quantum phase transition, and reveal the otherwise hidden impact of these errors on the simulation outcome. Our work demonstrates the capability for Rydberg-based entanglement to reach fidelities in the 0.999 regime, with higher fidelities a question of technical improvements, and shows how erasure conversion can be utilized in NISQ devices. These techniques could be translated directly to quantum-error-correction codes with the addition of long-lived qubits^7,22–24. 

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4. Logical quantum processor based on reconfigurable atom arrays

   https://doi.org/10.1038/s41586-023-06927-3  

   Bluvstein, Dolev; Evered, Simon J.; Geim, Alexandra A.; Li, Sophie H.; Zhou, Hengyun; Manovitz, Tom; Ebadi, Sepehr; Cain, Madelyn; Kalinowski, Marcin; Hangleiter, Dominik; et al (December 2023, Nature)

   Abstract Suppressing errors is the central challenge for useful quantum computing¹, requiring quantum error correction (QEC)^2–6for large-scale processing. However, the overhead in the realization of error-corrected ‘logical’ qubits, in which information is encoded across many physical qubits for redundancy^2–4, poses substantial challenges to large-scale logical quantum computing. Here we report the realization of a programmable quantum processor based on encoded logical qubits operating with up to 280 physical qubits. Using logical-level control and a zoned architecture in reconfigurable neutral-atom arrays⁷, our system combines high two-qubit gate fidelities⁸, arbitrary connectivity^7,9, as well as fully programmable single-qubit rotations and mid-circuit readout^10–15. Operating this logical processor with various types of encoding, we demonstrate improvement of a two-qubit logic gate by scaling surface-code⁶distance fromd = 3 tod = 7, preparation of colour-code qubits with break-even fidelities⁵, fault-tolerant creation of logical Greenberger–Horne–Zeilinger (GHZ) states and feedforward entanglement teleportation, as well as operation of 40 colour-code qubits. Finally, using 3D [[8,3,2]] code blocks^16,17, we realize computationally complex sampling circuits¹⁸with up to 48 logical qubits entangled with hypercube connectivity¹⁹with 228 logical two-qubit gates and 48 logical CCZ gates²⁰. We find that this logical encoding substantially improves algorithmic performance with error detection, outperforming physical-qubit fidelities at both cross-entropy benchmarking and quantum simulations of fast scrambling^21,22. These results herald the advent of early error-corrected quantum computation and chart a path towards large-scale logical processors. 

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5. Virtualized Logical Qubits: A 2.5D Architecture for Error-Corrected Quantum Computing

   https://doi.org/10.1109/MICRO50266.2020.00026  

   Duckering, Casey; Baker, Jonathan M.; Schuster, David I.; Chong, Frederic T. (October 2020, 53rd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO))

   null (Ed.)

   Current, near-term quantum devices have shown great progress in the last several years culminating recently with a demonstration of quantum supremacy. In the medium-term, however, quantum machines will need to transition to greater reliability through error correction, likely through promising techniques like surface codes which are well suited for near-term devices with limited qubit connectivity. We discover quantum memory, particularly resonant cavities with transmon qubits arranged in a 2.5D architecture, can efficiently implement surface codes with substantial hardware savings and performance/fidelity gains. Specifically, we virtualize logical qubits by storing them in layers of qubit memories connected to each transmon. Surprisingly, distributing each logical qubit across many memories has a minimal impact on fault tolerance and results in substantially more efficient operations. Our design permits fast transversal application of CNOT operations between logical qubits sharing the same physical address (same set of cavities) which are 6x faster than standard lattice surgery CNOTs. We develop a novel embedding which saves approximately 10x in transmons with another 2x savings from an additional optimization for compactness. Although qubit virtualization pays a 10x penalty in serialization, advantages in the transversal CNOT and in area efficiency result in fault-tolerance and performance comparable to conventional 2D transmon-only architectures. Our simulations show our system can achieve fault tolerance comparable to conventional two-dimensional grids while saving substantial hardware. Furthermore, our architecture can produce magic states at 1.22x the baseline rate given a fixed number of transmon qubits. This is a critical benchmark for future fault-tolerant quantum computers as magic states are essential and machines will spend the majority of their resources continuously producing them. This architecture substantially reduces the hardware requirements for fault-tolerant quantum computing and puts within reach a proof-of-concept experimental demonstration of around 10 logical qubits, requiring only 11 transmons and 9 attached cavities in total. 

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