Evaluating a weighted sum by shared measurements
Assume factor separation holds at a fixed parameter value. Enumerate tuples with one integer from -D through D per measurement; with no measurements, try the single empty tuple. For each tuple: (1) Scan each group's Boolean assignments to find a representative with those measurements. Skip the tuple if any group has none. (2) Scan each group again, keeping assignments with the required measurements. Evaluate A_g using that local assignment and the other groups' representatives, then sum these weights into B_g. (3) Multiply all B_g and add the product to a total initialized to zero. Return the total. Separation makes outside representatives interchangeable. Negative or fractional weights are handled by exact arithmetic.
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Math
Learn After
Why the shared-measurement algorithm includes each contribution once
A worked weighted sum using shared measurements
When the shared-measurement enumeration can be small
Restricting the measurement sum to assignments with selected bits equal to one
Keeping polynomial corrections fixed while summing assignments
Splitting the constructed measurements into independent group contributions
Learn After
Why the shared-measurement algorithm includes each contribution once
A worked weighted sum using shared measurements
When the shared-measurement enumeration can be small
Restricting the measurement sum to assignments with selected bits equal to one
Keeping polynomial corrections fixed while summing assignments
Splitting the constructed measurements into independent group contributions