Here's another option. I was thinking more of creating it for an iTVF (but leaving that part to you).

DECLARE @String varchar(20) = '00001001001';

WITH e1(N) AS(

SELECT N FROM (VALUES(0),(0),(0),(0),(0),(0),(0),(0),(0),(0))e(N)

),

cteTally(N) AS(

SELECT TOP(LEN(@String)) ROW_NUMBER() OVER(ORDER BY(SELECT NULL)) - 1 FROM e1 a, e1 b

)

SELECT STUFF((SELECT ',' + CAST( POWER(2,N) AS varchar(5))

FROM cteTally

WHERE SUBSTRING( @String, LEN(@String) - N, 1) = 1

ORDER BY N DESC

FOR XML PATH('')), 1, 1, '')

By the way, it was a surprise that ^ is not power in SQL and I had to use the POWER function instead. I haven't used it in a long, long time (if I ever used it before).