Continuous measurement-based holonomic quantum computation

We propose a scheme to generate holonomies using the Quantum Zeno effect, enabling logical unitary operations on quantum stabilizer codes purely through measurements. The quantum error-correcting code space is adiabatically rotated by successively measuring the rotated stabilizer generators. When the rotation is sufficiently slow, the state remains confined to the instantaneous code space by the Zeno effect; otherwise, measurement-induced jumps can occur into a rotated orthogonal subspace. If the rotation completes a closed loop, the code state acquires a holonomy. We analytically derive the sequence of rotated stabilizer generators that produce a desired holonomy, and find the total time required to implement this procedure with a given success probability. We also present a method to perform error correction simultaneously during the holonomic evolution. By decoding the measurement outcomes, we identify the rotated syndrome space into which the system is projected; then the holonomic path can be steered mid-flight to achieve a desired emulated holonomy. This method poses an advantage at suppressing non-Markovian noise. Finally, we establish conditions on the code that preserve the correctability of a given error set. When a code fails to meet the error-correcting conditions, our protocol remains applicable by augmenting the code with at most two ancilla qubits.