## Description

Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators. An important sub-family consists of Clifford monitored random circuits, where unitaries are sampled from the Clifford group [4]. When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy. The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities [5,6].

Monitored random circuits have a finite information capacity that decays exponentially with respect to system size [6]. When \( p = 0 \), the random circuit achieves channel capacity, meaning that it stores the most amount of information possible. This notion quantifies the recoverability of information and the reversability of the system under the monitored random dynamics. In the volume-law phase (\( p < p_c \) for some critical probability \(p_c\)), the channel capacity remains non-zero, and the monitored channel projects an initial state into a random error-correcting code [6]. With appropriately chosen evolution operators and measurements, the code is a stabilizer code whose parameters depend on time, \( [[n,k(t),d(t)]] \). A similar notion applies to Haar random circuits with measurements [7].

## Protection

## Rate

## Encoding

## Decoding

## Threshold

## Realizations

## Notes

## Parent

- Random-circuit code — Monitored random circuits are random circuits where projective measurements are interspersed throughout the circuit and measurement results are recorded.

## Cousins

- Topological code — Topological order can be generated in 2D monitored random circuits [11].
- Crystalline-circuit qubit code — Projective measurements can be included in crystalline-circuit codes in a spacetime translation-invariant fashion, turning such codes into monitored crystalline-circuit codes. However, the unit cell of measurements must be large enough to avoid purification.
- Floquet code — Both Floquet and monitored random circuit codes can have an instantaneous stabilizer group which evolves through unitary evolution and measurements. However, Floquet codewords are generated via a specific sequence of measurements, while random-circuit codes maintain a stabilizer group after any measurement. Floquet codes have the additional capability of detecting errors induced during the measurement process; see Appx. A of Ref. [12].

## References

- [1]
- B. Skinner, J. Ruhman, and A. Nahum, “Measurement-Induced Phase Transitions in the Dynamics of Entanglement”, Physical Review X 9, (2019) DOI
- [2]
- Y. Li, X. Chen, and M. P. A. Fisher, “Quantum Zeno effect and the many-body entanglement transition”, Physical Review B 98, (2018) DOI
- [3]
- A. Chan et al., “Unitary-projective entanglement dynamics”, Physical Review B 99, (2019) arXiv:1808.05949 DOI
- [4]
- Y. Li, X. Chen, and M. P. A. Fisher, “Measurement-driven entanglement transition in hybrid quantum circuits”, Physical Review B 100, (2019) arXiv:1901.08092 DOI
- [5]
- S. Choi et al., “Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition”, Physical Review Letters 125, (2020) arXiv:1903.05124 DOI
- [6]
- M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020) arXiv:1905.05195 DOI
- [7]
- A. Zabalo et al., “Critical properties of the measurement-induced transition in random quantum circuits”, Physical Review B 101, (2020) arXiv:1911.00008 DOI
- [8]
- C. Noel et al., “Measurement-induced quantum phases realized in a trapped-ion quantum computer”, Nature Physics 18, 760 (2022) arXiv:2106.05881 DOI
- [9]
- B. Yoshida, “Soft mode and interior operator in the Hayden-Preskill thought experiment”, Physical Review D 100, (2019) DOI
- [10]
- Y. Li and M. P. A. Fisher, “Statistical mechanics of quantum error correcting codes”, Physical Review B 103, (2021) arXiv:2007.03822 DOI
- [11]
- A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits”, Physical Review Letters 127, (2021) arXiv:2011.06595 DOI
- [12]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI

## Page edit log

- Victor V. Albert (2021-12-29) — most recent
- Elizabeth R. Bennewitz (2021-12-15)

## Cite as:

“Monitored random-circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/monitored_random_circuits