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I The gap of for the Hermitian positive definite
The Hamiltonian can be written in the block matrix form as
Let be an eigenvalue of , then
where the second equality holds because the bottom two blocks are commutable. Thus is an eigenvalue of , and equals the smallest non-zero eigenvalue of . Applying a proposition of matrices that and have the same non-zero eigenvalues, also equals the smallest non-zero eigenvalue of .
Now we focus on the matrix . Note that is the unique eigenstate corresponding to the eigenvalue 0, all eigenstates corresponding to non-zero eigenvalues must be orthogonal with . Therefore
Ii Relations among Different Measurements of Accuracy
The quantum adiabatic theorem (Jansen et al., 2007, Theorem 3) states that for any ,
We will show that also serves as an error bound for the density distance and bounds the fidelity from below.
Note that is the eigenstate for both and corresponding the 0 eigenvalue, we have , and thus . Together with the initial condition , the overlap of and remains to be 0 for the whole time period, i.e. Since is a rank-2 projector, we have . Therefore the error used in the adiabatic theorem becomes
Since is exactly the fidelity , the fidelity can be bounded from below by .
Furthermore, by using , the distance between and can be bounded by the error of the fidelity as
The proof of Theorem 1 and Theorem 2 can be completed by carefully analyzing the -dependence of each term in given in Eq. (3). Note that in both cases , and we introduce a constant with for the proof of Theorem 1 and for the proof of Theorem 2 due to the different scaling parameter of . We first compute the derivatives of
by chain rule as
Then the first two terms of can be rewritten as
Here stands for a general positive constant independent of . To compute the remaining two terms of , we use the following change of variable
and the last two terms of become
Summarize all terms above, an upper bound of is
Finally, since for
The leading term of the bound is when .
Iv Details of the numerical examples
For concreteness, for the Hermitian positive definite example, we choose . Here
is an orthogonal matrix obtained by Gram-Schmidt orthogonalization (implemented via a QR factorization) of the discretized periodic Laplacian operator given by
is chosen to be a diagonal matrix with diagonals uniformly distributed in. More precisely, with . Such construction ensures to be a Hermitian positive definite matrix which satisfies and the condition number of is . We choose where is the set of the column vectors of . Here .
For the non-Hermitian positive definite example, we choose . Here and are the same as those in the Hermitian positive definite case, except that the dimension is reduced to . is an orthogonal matrix obtained by Gram-Schmidt orthogonalization of the matrix
Such construction ensures to be non-Hermitian, satisfying and the condition number of is . We choose the same as that in the Hermitian positive definite example.