Monitored random-circuit code[1][2][3]


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].


When in the volume-law phase, protects against random projective measurements at a rate \( p < p_c \). While the true code distance is not known, it is conjectured to be proportional to the contiguous distance. As first described in Ref. [6], the contiguous distance for these circuits is defined with respect to all partitions of the system.


Rate can be finite [6], depending on the family of random codes generated by the circuit.


The dynamics of the monitored random circuit can be recast in the language of stabilizer codes [6]. The stabilizer group of the error-correcting code resulting from a monitored Clifford circuit either grows or shrinks with each time step, depending on which projective measurements were performed during the time step.One can construct optimal single-copy encoding operations for strong purification transitions [6]


The recovery operation is the reverse unitary transformation with access to the measurement record (for dynamically generated codes with a strong purification transition) [6]


Above the critical measurement rate \( p_c\), the natural error correction properties of the circuit can no longer protect the information. This can be interpreted as the code threshold.These dynamically generated codes saturate the trade off between density of encoded information and the error rate threshold [6]


Measurement induced quantum phases have been realized in a trapped-ion processor [8].


Connections to information scrambling in black hole physics, as introduced in Section 11 of [5]. In particular, monitored random circuits can be viewed as the Hayden-Preskill recovery problem [9] running backwards in time. In this setting, the volume-law entanglement phase of the monitored circuit describes the phase when information can be recovered from an old black hole (ie, a black hole that is maximally entangled with the early universe).Mapping monitored random circuits to statistical mechanics models can help estimate thresholds and code distances for these systems [10].


  • 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 [11].
  • 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].

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Cite as:
“Monitored random-circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_monitored_random_circuits, title={Monitored random-circuit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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B. Skinner, J. Ruhman, and A. Nahum, “Measurement-Induced Phase Transitions in the Dynamics of Entanglement”, Physical Review X 9, (2019). DOI
Y. Li, X. Chen, and M. P. A. Fisher, “Quantum Zeno effect and the many-body entanglement transition”, Physical Review B 98, (2018). DOI
A. Chan et al., “Unitary-projective entanglement dynamics”, Physical Review B 99, (2019). DOI; 1808.05949
Y. Li, X. Chen, and M. P. A. Fisher, “Measurement-driven entanglement transition in hybrid quantum circuits”, Physical Review B 100, (2019). DOI; 1901.08092
S. Choi et al., “Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition”, Physical Review Letters 125, (2020). DOI; 1903.05124
M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020). DOI; 1905.05195
A. Zabalo et al., “Critical properties of the measurement-induced transition in random quantum circuits”, Physical Review B 101, (2020). DOI; 1911.00008
Crystal Noel et al., “Observation of measurement-induced quantum phases in a trapped-ion quantum computer”. 2106.05881
B. Yoshida, “Soft mode and interior operator in the Hayden-Preskill thought experiment”, Physical Review D 100, (2019). DOI
Y. Li and M. P. A. Fisher, “Statistical mechanics of quantum error correcting codes”, Physical Review B 103, (2021). DOI; 2007.03822
A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits”, Physical Review Letters 127, (2021). DOI; 2011.06595
M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021). DOI; 2107.02194

Cite as:

“Monitored random-circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.