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An improved method for quantum matrix multiplication

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Abstract

Following the celebrated quantum algorithm for solving linear equations (so-called HHL algorithm), Childs et al. (SIAM J Comput 46:1920–1950, 2017) provided an approach to solve a linear system of equations with exponentially improved dependence on precision. In this note, we aim to complement such a result for applying a matrix to some quantum state, based upon their Chebyshev polynomial approach. A few examples that motivate this application are included, and we further discuss an application of this improved matrix application algorithm explicitly with an efficient quantum procedure.

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Acknowledgements

This work was supported in part by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. We also acknowledge the support from a Seed Grant from Stony Brook University’s Office of the Vice President for Research.

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Appendix A: review of Chebyshev approach

Appendix A: review of Chebyshev approach

Here we make a review of Chebyshev approach that was employed in [2], which is essentially built upon quantum walk technique [6, 8]. What we will describe below is more or less a summary of Sect. 4 in Ref. [2], the result of which was used in our main text.

Let A be a d-sparse (s was used in the main text) \(N \times N\) Hermitian matrix and a procedure, or black box that could query the entries of A. Define the following state on the Hilbert space \(\mathcal {C}^{2N} \otimes \mathcal {C}^{2N}\):

$$\begin{aligned} |\psi _j\rangle = |j\rangle \otimes \frac{1}{\sqrt{d}} \sum _{k \in [N]; A_{ij} \ne 0} \Big ( \sqrt{A_{ij}^*} |k\rangle + \sqrt{1- |A_{ij}|} |k+N\rangle \Big ) \end{aligned}$$
(A.1)

We also define the following isometry T from \(\mathcal {C}^N\) to \(\mathcal {C}^{2N} \otimes \mathcal {C}^{2N}\):

$$\begin{aligned} T = \sum _{j\in [N]} |\psi _j\rangle \langle j|. \end{aligned}$$
(A.2)

We note that T is not unitary, and therefore, what we would need is a unitary version of T, which is \(U_T\) acting as following:

$$\begin{aligned} U_T |0^m\rangle |\phi \rangle = T|\phi \rangle , \end{aligned}$$
(A.3)

for some \(|\phi \rangle \in \mathcal {C}^N\) and \(m = log(2N) +1\).

The so-called walk operator is defined as:

$$\begin{aligned} W = S(2T T^\dagger - I), \end{aligned}$$
(A.4)

where S is the SWAP operator on the space \(\mathcal {C}^{2N} \otimes \mathcal {C}^{2N}\), i.e., \(S |j,k\rangle = |k,j\rangle \). The implementation of W (and \(U_T\)) was explicitly described in Lemma 10 of Ref. [6]. We now summarize the structural property of W that makes it highly beneficial for many quantum algorithms, including the improved quantum linear solver [2] and quantum simulation [6].

Let \(|\lambda \rangle \) and \(\lambda \) be eigenvector and eigenvalue of A/d (note that the scaling by d does not have further systematic problem, as the spectrum remains the same, only eigenvalues got scaled by a factor). Within the subspace spanned by \(T|\lambda \rangle \) and \(ST|\lambda \rangle \), W admits the following block form:

$$\begin{aligned} \begin{bmatrix} \lambda &{} -\sqrt{1-\lambda ^2} \\ \sqrt{1-\lambda ^2} &{} \lambda \end{bmatrix}. \end{aligned}$$
(A.5)

The proof can be found in Lemma 15 of [2]. The above form of W possesses the following remarkable property (Lemma 16 of [2]),

$$\begin{aligned} W^n = \begin{bmatrix} T_n(\lambda ) &{} -\sqrt{1-\lambda ^2} U_{n-1}(\lambda ) \\ \sqrt{1-\lambda ^2} U_{n-1}(\lambda ) &{} T_n(\lambda ) \end{bmatrix}, \end{aligned}$$
(A.6)

where \(T_n\) is the n-th Chebyshev polynomial of the first kind, and \(U_n\) is the n-th Chebyshev polynomial of the second kind. The construction of \(W_n\) is simply product of n times application of W. Therefore, we can construct the unitary \(U = U_T^\dagger W^n U_T\) that acts on \(|0^m\rangle |\phi \rangle \) as follows,

$$\begin{aligned} U|0^m\rangle |\phi \rangle = |0^m\rangle T_n (A) |\phi \rangle + |\Phi _{\perp }\rangle , \end{aligned}$$
(A.7)

where \(|\Phi _{\perp }\rangle \) is orthogonal to \(|0^m\rangle T_n (A) |\phi \rangle \). As we have remarked in the main text, the 1st Chebyshev polynomial of the first kind \(T_1(x)=x\) is sufficient for matrix multiplication purpose, which requires a single execution of W.

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Nghiem, N.A., Wei, TC. An improved method for quantum matrix multiplication. Quantum Inf Process 22, 299 (2023). https://doi.org/10.1007/s11128-023-04054-6

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