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I would like to construct a spot curve from supplied bond prices. I know that the curve has to be constructed from dirty prices (i.e. the ones that include accrued interest). However, from FittedBondCurve.cpp example posted on quantlib.org, it appears that FixedRateBondHelper class is initialized with clean prices.
So, my question is: does it mean that FixedRateBondHelper takes care of computing accrued interest and converting clean price to dirty price? Or is it something that a user should do? I believe it's the former but wanted to make sure.

The helper doesn't, but the fitting algorithm does. If you look at the FittedBondDiscountCurve::FittingMethod::FittingCost::value method, you'll cringe a bit at the nested inner classes, but then you'll see that the model price is calculated by adding the discounted future cash flows and subtracting the accrued amount.
A further note: in recent releases, the bond helpers have been given the possibility to work with quoted dirty prices when bootstrapping a curve (see the last parameter of their constructors, useCleanPrice, which defaults to true but can be set to false to use dirty prices. However, the FittedBondDiscountCurve class is not yet aware of this change, and thus setting useCleanPrice to false would break the algorithm. I'll try to fix this in a future release.