EAOA qubit stabilizer code[1]

Description

Entanglement-assisted operator-algebra (EAOA) qubit code that becomes an OA qubit stabilizer code on \(n+e\) qubits once the receiver’s halves of its \(e\) noiseless ebits are counted as physical qubits. Each anticommuting pair of generators of a possibly non-Abelian subgroup \(H\) of the \(n\)-qubit Pauli group is extended by \(X\) and \(Z\) on one ebit so that the extended group \(S\) is Abelian. The stabilizer subspace of \(S\) then carries gauge qubits, logical qubits, and orthogonal copies of itself that encode classical strings [1].

The isotropic part of \(H\) has \(s\) commuting generators and its symplectic part has \(e\) anticommuting pairs, so the stabilizer subspace of \(S\) encodes \(n-e-s\) qubits. These split into \(r\) gauge qubits acted on by gauge operators \(G_0\) and \(k\) logical qubits acted on by logical operators \(L_0\), all of which act trivially on the ebits. A set \(T_0\) of \(c_b\) representatives of distinct cosets of the normalizer \(N(S)\), containing the identity and called transversal operators, maps the code space to \(c_b\) mutually orthogonal sectors, one per encoded classical string. The transversal operators can also be chosen to act trivially on the ebits. The sender encodes the \(k\) logical qubits, the \(s\) isotropic qubits, and her halves of the ebits into the \(n\) transmitted qubits. The receiver measures the generators of \(S\) on all \(n+e\) qubits. Such a code is denoted by \([[n,k;r,e,c_b]]\) or \([[n,k,d;r,e,c_b]]\), with \(c_b=2^c\) when the hybrid component encodes \(c\) classical bits [1].

EA qubit stabilizer codes are the case \(G_0=\varnothing\) and \(T_0=\{I\}\), EA subsystem stabilizer codes the case \(T_0=\{I\}\), and EA hybrid subspace codes the case \(G_0=\varnothing\). The family thus encompasses ordinary entanglement-assisted qubit stabilizer codes, entanglement-assisted subsystem stabilizer codes, entanglement-assisted hybrid stabilizer codes, and operator-algebra generalizations described within that stabilizer formalism. It contains EA hybrid subspace and EA subsystem stabilizer codes that lie outside the earlier EACQ and EAOQECC formalisms [1].

There exist four constructions of EAOA qubit stabilizer codes from subsystem codes with distance lower bounds [1; Table 1]. Gauge fixing turns \(y\) gauge pairs into stabilizer and transversal operators, \([[n,k,d;r,e,c_b]] \to [[n,k,d';r-y,e,\leq 2^y c_b]]\). The clean-qubits construction replaces \(e\) isotropic stabilizers by ebit-extended symplectic pairs, \([[n,k,d;r,0,c_b]] \to [[n-e,k,d';r,e,c_b]]\). Entanglement-assisted gauge fixing turns \(e\) anticommuting gauge pairs into ebit-extended stabilizers, \([[n,k,d;r,0,c_b]] \to [[n,k,d';r-e,e,c_b]]\). General gauge fixing combines the first and third, \([[n,k,d;r,e,c_b]] \to [[n,k,d';r-y_I-y_S,e+y_S,\leq 2^{y_I}c_b]]\) [1]. For example, the clean-qubits construction applied to a \([[15,1,2;6,0,3]]\) hybrid subsystem code derived from the \([[15,1,3;6]]\) subsystem color code yields a \([[13,1,3;6,2,3]]\) code [1; Sec. 7.2]. Its distance rises from two to three because two of its qubits become noiseless ebits.