  We define an ensemble of random Clifford quantum circuits whose output state
undergoes an entanglement phase transition between two volume-law phases as a
function of measurement rate. Our setup maps exactly the output state to the
ground space of a spin glass model. We identify the entanglement phases using
an order parameter that is accessible on a quantum chip. We locate the
transition point and evaluate a critical exponent, revealing spin glass
criticality. Our work establishes an exact statistical mechanics theory of an
entanglement phase transition.
