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 intermittent 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].

In the volume-law or mixed phase (\( p < p_c \) for some critical probability \(p_c\)), the channel-capacity density remains nonzero on polynomial timescales and the purification time grows exponentially with system size [6]. The monitored dynamics projects the system into a random error-correcting code, and for strong purification transitions this code can be capacity-achieving for the future unraveled evolution of the channel [6]. In the area-law or pure phase (\( p > p_c \)), the channel-capacity density vanishes and the system purifies rapidly [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 \( p < p_c \), protects against monitored projective measurements by dynamically encoding information into a late-time code space. For one-dimensional stabilizer circuits, the average contiguous code length is efficiently computable and upper bounds the code distance; it is subextensive deep in the mixed phase and appears extensive near \( p_c \) [6].

Rate

Rate can be finite for \( p < p_c \) and vanishes for \( p > p_c \); in the 1+1-dimensional random Clifford model, the residual entropy density equals the channel-capacity density in the mixed phase [6].

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.For strong purification transitions, the monitored dynamics itself implements a single-copy capacity-achieving encoding for the future unraveled evolution of the channel [6].

Decoding

With access to the measurement record, recovery operations can reverse the future unraveled evolution with high fidelity; on the code space, the induced dynamics becomes effectively unitary in the thermodynamic limit [6].

Threshold

At the purification threshold \( p_c \), the channel-capacity density changes from finite to zero; above \( p_c \), the natural error-correction properties of the circuit can no longer protect an extensive amount of information [6].These dynamically generated codes saturate the trade-off between the 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 [5; Sec. 11]. 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.