Dynamical code[1,2]

Cousins

• Honeycomb (6.6.6) color code— The parent topological phase of the 2D DA color code is realized by two copies of the 6.6.6 color code [2].
• Cubic honeycomb color code— The parent topological phase of the 3D DA color code is realized by three copies of the cubic honeycomb color code [2].
• Kitaev surface code— One of the instantaneous stabilizer codes of the 2D DA color code are stacks of surface codes
• Abelian topological code— Useful measurement sequences of dynamical codes can be extracted from topological quantum field theory [2].
• Approximate quantum error-correcting code (AQECC)— Approximate versions of dynamical codes have been formulated [5].
• Subsystem qubit stabilizer code— A dynamical code can be viewed as a subsystem stabilizer code, albeit one with less logical qubits.
• Monitored random-circuit 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. [1].
• Majorana stabilizer code— Dynamical codes are viable candidates for storage in Majorana-qubit devices [6].
• Qubit stabilizer code— Using ZX calculus, an \([[n,k,d]]\) qubit stabilizer code admitting stabilizer generators of weight no more than \(m\) can be Floquetified into an \([[n+\lceil m/2 \rceil+\ell,k,d^{\prime}]]\) dynamical code with single- and two-qubit operations, where \(\ell \leq \log_{2} m\) and \(d^{\prime} \geq d\) [7] (see also Ref. [8]). A more general locality-preserving spacetime concatenation procedure yields a dynamical code out of any qubit stabilizer code by structuring measurement gadgets using low-weight measurements while ensuring the preservation of logical information [9]. In particular, spacetime concatenation reformulates the notion of a dynamical code associated with a stabilizer code in terms of code concatenation for every qubit (spatial concatenation) and measurements between these codes (temporal concatenation), leading to a temporal evolution of the stabilizer state [9]. A matrix rank condition on the bond operators connecting the gadgets, called the Bond-Kernel-Rank Condition, and a strict locality preservation condition (SLPC), along with preservation of the operator algebra of the stabilizer code under the gadget action, preserves fault-tolerance and the spacetime distance of the code [9].