QND based cross-Kerr nonlinearity for parity check Mirror +0 Homodyne a bI bI PBS al(bl) al(bl) BBO 2(b2) a2(b2 Pump (aithbuH +ai bir +a2nb2H +a2y bir )o 1HlH ae ae b+3b3)0)ae) ae❖ QND based cross-Kerr nonlinearity for parity check： 1 1 1 1 2 2 2 2 ( ) 0 ( ) 0 i i H H V V a b i i H H V V a b a b a b e e a b a b e e         + + + + + + + +    + + + 1 1 1 1 2 2 2 2 ( ) 0 a b a b a b a b H H V V H H V V + + + + + + + + + + +