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 . 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 . 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 . 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 .
- Random-circuit code — Monitored random circuits are random circuits where projective measurements are interspersed throughout the circuit and measurement results are recorded.
- Topological code — Topological order can be generated in 2D monitored random circuits .
- 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.
- Hastings-Haah 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. .
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“Monitored random-circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/monitored_random_circuits