  Multi-qubit parity measurements are essential to quantum error correction.
Current realizations of these measurements often rely on ancilla qubits, a
method that is sensitive to faulty two-qubit gates and which requires
significant experimental overhead. We propose a hardware-efficient multi-qubit
parity measurement exploiting the bifurcation dynamics of a parametrically
driven nonlinear oscillator. This approach takes advantage of the resonator's
parametric oscillation threshold which is a function of the joint parity of
dispersively coupled qubits, leading to high-amplitude oscillations for one
parity subspace and no oscillation for the other. We present analytical and
numerical results for two- and four-qubit parity measurements with
high-fidelity readout preserving the parity eigenpaces. Moreover, we discuss a
possible realization which can be readily implemented with the current circuit
QED experimental toolbox. These results could lead to significant
simplifications in the experimental implementation of quantum error correction,
and notably of the surface code.
