Hamiltonian Simulation and Linear Combination of Unitary Decomposition of Structured Matrices

Abstract

To treat a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operations, or gates: this is the quantum description of the problem. These operations are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BE). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hermitian (LCH) or a Linear Combination of Unitary (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. Qubitization constructs a specific query that respects single-qubit behavior when expressed in the appropriate basis. In this work, we extend the notion of qubitization to Hamiltonian matrices used to map the problem of interest. These methods concern almost all the problems implemented on QPUs: from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), or sparse matrices. This work underlines interesting properties associated with the qubitized Hamiltonian basic gate decomposition. It includes the ability to switch from LCH to LCU, to map non-Hermitian problems, and to construct the different quantum circuit primitives (queries) needed for the quantum description of the problem. We also provide a list of qubitized Hamiltonians that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.

Figures (16)

• Figure 1: Application of the one-qubit gate $\widehat{U}$ in the qubitized subspace $\{ \ket{\lambda}, \ket{\lambda_{\perp}} \}$. Here, $\widehat{V_{i}} = (\ket{0}_{5} \bra{\lambda_{i}} + \ket{1}_{5} \bra{\lambda_{\perp}} + \ket{0}^{\otimes 2}_{0, 2} \bra{\perp}) \otimes \ket{\chi}_{1, 3, 4}$.
• Figure 2: Application of the one qubit gate $\widehat{U}$ in the qubitized subspace $\{ \ket{\lambda_{\Sigma}}, \ket{\lambda_{\perp, \Sigma}} \}$. The new degenerated-states reductor $\widehat{V_{\Sigma}}^{(\dag)}$ (and its conjugate) are the gates in the dashed box. Here, $\widehat{V_{1}} = (\ket{0}_{3} \bra{\lambda_{1}} + \ket{1}_{3} \bra{\lambda_{\perp, 1}} + \ket{0}_{0} \bra{\perp_{1}}) \otimes \ket{\chi_{1}}_{1, 2}$ and $\widehat{V_{2}} = (\ket{0}_{4} \bra{\lambda_{2}} + \ket{1}_{4} \bra{\lambda_{\perp, 2}} + \ket{0}_{5} \bra{\perp}) \otimes \ket{\chi}_{6}$.
• Figure 3: being of \ref{['fig_qc_basic']}'s $\widehat{H_{i}}$ when the first qubit reads $\ket{0}_{\mathcal{B}}$.
• Figure 4: The two proposed measurement circuits allow to evaluate the expectation value: $\bra{\psi} \widehat{H_{\Sigma}} \ket{\psi}$ from \ref{['fig_qc_tensor_product']}. Using the left circuit: $\bra{\psi} \widehat{H_{\Sigma}} \ket{\psi} = |\bra{\psi} \ket{0}_{3} \otimes \ket{11}_{0, 5}|^{2} - |\bra{\psi} \ket{1}_{3} \otimes \ket{11}_{0, 5}|^{2}$. The gate $\widehat{M}$ is generally equal to $\widehat{I}$ but can be changed to perform a qubitized space tomography. Using the right circuit: $\bra{\psi} \widehat{H_{\Sigma}} \ket{\psi} = |\bra{\psi} \ket{00}_{3, 4} \otimes \ket{11}_{0, 5}|^{2} + |\bra{\psi} \ket{1}_{3, 4} \otimes \ket{11}_{0, 5}|^{2} - |\bra{\psi} \ket{01}_{3, 4} \otimes \ket{11}_{0, 5}|^{2} - |\bra{\psi} \ket{10}_{3, 4} \otimes \ket{11}_{0, 5}|^{2}$. This second circuit needs slightly more complex classical post-processing (i.e. checking the hamming weight parity) and does not allow a qubitized space tomography. However, only a shallower change of basis is required.
• Figure 5: Symmetric Toeplitz matrix: $\widehat{H}_{m-1, m}^{toep}$ in black, Toeplitz matrix: $\widehat{M}(m-1)^{\dag}$ in blue and directed circulant matrix: $\widehat{M}(m-2)^{\dag}$ in red.