Nice video explanation of the SABRE engine by Richard Varvill.
In a sense, it boils down to the problem of changing the hot fast low pressure intake air flow to a cold slow high pressure flow.
In the Sabre engine, techniques somewhat similar to liquid air plants are used: there is a compressor, that is coupled to an expander, the expander runs from the “waste” heat of the compressor. This is efficient, but it is heavy. In Sabre the compressor is the shock cone (the jet engine style compressor only compresses the cooled air).
The heat exchangers are critical, and I wonder how reliable and expensive they will get with the vast temperature envelopes, ice and thin wall problems of the huge number of tubes there are. On the other hand, such heat exchangers could have uses in many other places as well. (Power generation.) The helium in the tubes is especially leak prone.