I Block encoding
When discussing the number of queries to an oracle we do not distinguish between and its controlled version. The asymptotic notations , are used for the limit and . We use to mean
multiplied by a poly-logarithmic part. Sometimes we do not distinguish between the different ways of measuring error, e.g. in terms of fidelity or 2-norm distance of density matrices, since the query complexity is logarithmic in the error defined in both ways. Floating-point arithmetic is assumed to be exact for conciseness. If floating-point error is taken into account this will only lead to a logarithmic multiplicative overhead in the number of primitive gates, and a logarithmic additive overhead in the number of qubits needed.
The technique of block-encoding has been recently discussed extensively GilyenSuLowEtAl2019; LowChuang2019. Here we discuss how to construct block-encoding for which is used in eigenstate filtering, and , , and which are used in QLSP and in particular the Hamiltonian simulation of AQC. We first introduce a simple technique we need to use repeatedly.
Given , an -block-encoding of where , we want to construct a block encoding of for some . This is in fact a special case of the linear combination of unitaries (LCU) technique introduced in ChildsKothariSomma2017. Let
and . Since , we have
where . Therefore Fig. S1 gives an -block-encoding of .
Therefore we may construct an -block-encoding of . We remark that here we do not need the phase shift gate since . This is at the same time a -block-encoding of .
Now we construct a block-encoding of with . Let
be the reflection operator about the hyperplane orthogonal to. Then is the reflection about the hyperplane orthogonal to . Note that . Therefore we can use the technique illustrated in Fig. S1 to construct a -block-encoding of . Here . Since , we naturally obtain a -block-encoding of . We denote the block-encoding as
For the block-encoding of , first note that
From the block-encoding of , we can construct the block-encoding of controlled- by replacing all gates with their controlled counterparts. The block matrix in the middle is . For a -sparse matrix , we have a -block-encoding of , and therefore we obtain a block-encoding of . Then we can use the result for the product of block-encoded matrix (GilyenSuLowEtAl2019, Lemma 30) to obtain a -block-encoding of , denoted as .
Ii Gate-based implementation of time-optimal adiabatic quantum computing
Consider the adiabatic evolution
Where for and defined in (2). It is proved in AnLin2019 that the gap between and the rest of the eigenvalues of is lower bounded by . With this bound it is proved that in order to get an -approximate solution of the QLSP for a positive definite we need to run for time using the optimal scheduling (AnLin2019, Theorem 1).
In order to carry out AQC efficiently using a gate-based implementation, we use the recently developed time-dependent Hamiltonian simulation method based on truncated Dyson series introduced in LowWiebe2018. In Hamiltonian simulation, several types of input models for the Hamiltonian are in use. Hamiltonians can be input as a linear combination of unitaries BerryChildsCleveEtAl2015, using its sparsity structure AharonovTaShma2003; LowChuang2017, or using its block-encoding LowChuang2019; LowWiebe2018. For a time-dependent Hamiltonian Low and Wiebe designed an input model based on block-encoding named HAM-T (LowWiebe2018, Definition 2), as a block-encoding of where is a time step and is the Hamiltonian at this time step.
In the gate-based implementation of the time-optimal AQC, we construct HAM-T in Fig. S2. We need to use the block-encodings and introduced in the previous section. We denote and as the number of ancilla qubits used in the two block-encodings. We know that and . Our construction of HAM-T satisfies
for any .
In this unitary HAM-T we also need the unitary
to compute the scheduling function needed in the time-optimal AQC, and the unitaries
where . Here is used for preparing the linear combination . Without the circuit would be a -block-encoding of , but with it becomes a -block-encoding, so that the normalizing factor is time-independent, as is required for the input model in LowWiebe2018.
For the AQC with positive definite we have and . For indefinite case we have and .
Following Corollary 4 of LowWiebe2018, we may analyze the different components of costs in the Hamiltonian simulation of AQC. For time evolution from to , HAM-T is a -block-encoding of . With the scheduling function given in AnLin2019 we have and . We choose and by Theorem 1 of AnLin2019 we have . We only need to simulate up to constant precision, and therefore we can set . The costs are then
Queries to HAM-T: ,
Primitive gates: .
Iii The matrix dilation method
In order to extend the time-optimal AQC method to Hermitian indefinite matrices, we follow (AnLin2019, Theorem 2), where and are given by
Here and . The dimension of the dilated matrices is . The lower bound for the gap of then becomes SubasiSommaOrsucci2019. The initial state is and the goal is to obtain . After running the AQC we can remove the second qubit by measuring it with respect to the basis and accepting the result corresponding to . The resulting query complexity remains unchanged. We remark that the matrix dilation here is only needed for AQC. The eigenstate filtering procedure can still be applied to the original matrix of dimension .
For a general matrix, we may first consider the extended linear system. Define adjoint QLSP as , and consider an extended QLSP in dimension where
Here is a Hermitian matrix of dimension , with condition number and , and solves the extended QLSP. Therefore the time-optimal AQC can be applied to the Hermitian matrix to prepare an -approximation of and simultaneously. The dimension of the corresponding matrices is . Again the matrix dilation method used in Eq. (S4) is not needed for the eigenstate filtering step.
Iv Optimal Chebyshev filtering polynomial
In this section we prove Lemma 2. We define
then . We need to use the following lemma:
For any satisfying for all , for all .
We prove by contradiction. If there exists such that for all and there exists such that , then letting , we want to show has at least distinct zeros.
First note that there exist such that , and . Therefore there exist such that , and . In other words, maps each and to , and the mapping is bijective for each interval. Because , there exists for each such that . Therefore and give us distinct zeros. Another zero can be found at as . Therefore there are distinct zeros.
However is of degree at most . This shows . This is clearly impossible since . ∎
Therefore any , solves the minimax problem
This implies (i) of Lemma 2. To prove (ii), we need to use the following lemma:
Let be the -th Chebyshev polynomial, then
The Chebyshev polynomial can be rewritten as for . Let , then . The choice of does not change the value of , so we choose . Since for , we have . Thus . ∎
We use this lemma to prove (ii). Since , when , we have . Thus by the above lemma we have . Since for , we have the inequality in (ii). (iii) follows straightforwardly from the monotonicity of Chebyshev polynomials outside of .