Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

TL;DR

This work introduces an oracle-free method for Hamiltonian simulation by decomposing tridiagonal matrices into Pauli strings and grouping them into internally commuting subsets. For a $2^n\times 2^n$ tridiagonal matrix, the decomposition yields $2n+1$ commuting blocks (or $n+1$ in the Hermitian real-symmetric case), with weights computed explicitly for both diagonal and off-diagonal parts. The approach is validated on the one-dimensional wave equation, showing gate counts scaling as $O(r\,n\,2^n)$ per Trotter step and outperforming oracle-based methods for $n<15$, while using roughly half as many qubits. This framework extends to other tridiagonal-structure Hamiltonians and suggests natural extensions to higher-diagonal and multi-dimensional discretizations; code for the numerical experiments is provided.

Abstract

The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix $2^n \times 2^n$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets $O(n)$. The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as function of the number of qubits is lower than the oracle based approach for $n < 15$ and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.

Figures (3)

• Figure 1: Illustrating the contribution of commuting subsets $S_{m,+}$ to decomposition of the matrix $B$. Same colors contribute to same commuting subset. Colors: $S_{1}$ -- green, $S_{2}$ -- blue, $S_{3}$ -- red, $S_{4}$ -- black.
• Figure 2: Illustrating the contribution of commuting subsets $S_{m}$ to decomposition of the matrix $H$ of size $2^5 \times 2^5$ with $B$ and $B^{\dagger}$ of size $2^4 \times 2^4$. Same colors contribute to same commuting subset. Color code: $S_1$ -- green, $S_2$ -- blue, $S_3$ -- red, $S_4$ -- black. $S_0$ corresponds to the diagonal of each $B$ matrix.
• Figure 3: Number of gates to approximate $e^{-\imath Ht}$ with accuracy $\epsilon = 10^{-5}$ for $1D$ wave equation Hamiltonian \ref{['eq:hamil_wave']}. Points represent number of gates obtained in the simulation. Red solid line shows the theoretical gate scaling given by \ref{['eq:g1d']} for Trotter order $p=2$. The black solid line shows number of gates from reference suau2021practical. Dashed lines are fit to the simulation data with \ref{['eq:approx']}.

Theorems & Definitions (20)

• Proposition 1: Decomposition of an arbitrary tridiagonal matrix
• Proposition 2: Commutativity criterion
• Corollary 1
• Proposition 3: Real tridiagonal matrix
• Corollary 2: Real symmetric tridiagonal matrix
• Corollary 3
• Proposition 4
• proof
• Proposition 4: Commutativity criterion
• proof