Monitored random-circuit code

Monitored random-circuit code[1–3]

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 nonzero, 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

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

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

Encoding

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]

Decoding

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

Threshold

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]

Realizations

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

Notes

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].

Cousins

Topological code— Topological order can be generated in 2D monitored random circuits [11].
Random stabilizer code— An important sub-family of monitored random-circuit codes are the Clifford monitored random-circuit codes, where unitaries are sampled from the Clifford group [4].
Dynamical code— Both dynamical and monitored random circuit codes can have an instantaneous stabilizer group which evolves through unitary evolution and measurements. However, dynamical codewords are generated via a specific periodic sequence of measurements, while random-circuit codes maintain a stabilizer group after any measurement. Dynamical codes have the additional capability of detecting errors induced during the measurement process; see Appx. A of Ref. [12].
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.

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Block quantum code

Dynamically generated QECC

Random-circuit codeAQECC Quantum

Parents

Random-circuit code

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