ref [196] · 2405.09628 · notice #2314 · dispute
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a complementary measure of the dynamics focused on the spread of the operator in the Krylov lattice. The Krylov variance is defined as [187] ∆K(t)2 := DK −1X n=0 n2|φn(t)|2 − DK −1X n=0 n|φn(t)|22 = DK −1X n=0 |φn(t)|2(n − K(t))2 . (27) An alternative definition was considered in [195, 196], which in our notation stands for∆K(t)2/K(t)2. The higher moments of the distribution can be similarly defined [196]. For this, it is interesting to consider the Krylov operator K such that [197] K |On) = n|On) , ⇔ K := DK −1X n=0 n|On)(On| . (28) In other words, the Krylov operator acts as a number operator on the Krylov basis,K = diag(0, 1, 2 · · · , DK− 1). In terms of the definition (28), the Krylov complexity is associated with the expectation value of the Krylov operator K in the time-evolved operator |O(t)), i.
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a complementary measure of the dynamics focused on the spread of the operator in the Krylov lattice. The Krylov variance is defined as [187] ∆K(t)2 := DK −1X n=0 n2|φn(t)|2 − DK −1X n=0 n|φn(t)|22 = DK −1X n=0 |φn(t)|2(n − K(t))2 . (27) An alternative definition was considered in [195, 196], which in our notation stands for∆K(t)2/K(t)2. The higher moments of the distribution can be similarly defined [196]. For this, it is interesting to consider the Krylov operator K such that [197] K |On) = n|On), ⇔ K := DK −1X n=0 n|On)(On| . (28) In other words, the Krylov operator acts as a number operator on the Krylov basis,K = diag(0, 1, 2 · · ·, DK− 1). In terms of the definition (28), the Krylov complexity is associated with the expectation value of the Krylov operator K in the time-evolved operator |O(t)), i