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1. Comparing Shor and Steane error correction using the Bacon-Shor code

   https://doi.org/10.1126/sciadv.adp2008  

   Huang, Shilin; Brown, Kenneth R; Cetina, Marko (November 2024, Science Advances)

   Quantum states decohere through interaction with the environment. Quantum error correction can preserve coherence through active feedback wherein quantum information is encoded into a logical state with a high degree of symmetry. Perturbations are detected by measuring the symmetries of the state and corrected by applying gates based on these measurements. To measure the symmetries without perturbing the data, ancillary quantum states are required. Shor error correction uses a separate quantum state for the measurement of each symmetry. Steane error correction maps the perturbations onto a logical ancilla qubit, which is then measured to check several symmetries simultaneously. We experimentally compare Shor and Steane correction of bit flip errors using the Bacon-Shor code implemented in a chain of 23 trapped atomic ions. We find that the Steane method produces fewer errors after a single round of error correction and less disturbance to the data qubits without error correction. 

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2. Between Shor and Steane: A Unifying Construction for Measuring Error Syndromes

   https://doi.org/10.1103/PhysRevLett.127.090505  

   Huang, Shilin; Brown, Kenneth R. (August 2021, Physical Review Letters)
3. Adaptive syndrome measurements for Shor-style error correction

   https://doi.org/10.22331/q-2023-08-08-1075  

   Tansuwannont, Theerapat; Pato, Balint; Brown, Kenneth R. (August 2023, Quantum)

   The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance $d$that can correct up to $t=\lfloor (d-1)/2\rfloor$errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated $t+1$times in a row; in the worst-case scenario, $(t+1{)}^2$rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than $(t+3{)}^2/4-1$rounds and $(t+3{)}^2/4-2$rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance $d$, our FTEC protocols with the adaptive schemes require no more than $d$rounds on average. 

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4. Concatenated Steane code with single-flag syndrome checks

   https://doi.org/10.1103/PhysRevA.110.032411  

   Pato, Balint; Tansuwannont, Theerapat; Brown, Kenneth_R (September 2024, Physical Review A)
5. Subsystem symmetry, spin-glass order, and criticality from random measurements in a two-dimensional Bacon-Shor circuit

   https://doi.org/10.1103/PhysRevB.108.024205  

   Sharma, Vaibhav; Jian, Chao-Ming; Mueller, Erich J (July 2023, Physical Review B)

   We study a 2D measurement-only random circuit motivated by the Bacon-Shor error correcting code. We find a rich phase diagram as one varies the relative probabilities of measuring nearest-neighbor Pauli XX and ZZ check operators. In the Bacon-Shor code, these checks commute with a group of stabilizer and logical operators, which therefore represent conserved quantities. Described as a subsystem symmetry, these conservation laws lead to a continuous phase transition between an X-basis and Z-basis spin-glass order. The two phases are separated by a critical point where the entanglement entropy between two halves of an L × L system scales as L ln L, a logarithmic violation of the area law. We generalize to a model where the check operators break the subsystem symmetries (and the Bacon-Shor code structure). In tension with established heuristics, we find that the phase transition is replaced by a smooth crossover, and the X - and Z -basis spin-glass orders spatially coexist. Additionally, if we approach the line of subsystem symmetries away from the critical point in the phase diagram, some spin-glass order parameters jump discontinuously. 

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