Core Concepts
An efficient procedure is presented for representing a tridiagonal matrix in the Pauli basis, allowing the construction of a Hamiltonian evolution circuit without the use of oracles. The method systematically determines all Pauli strings present in the decomposition and divides them into commuting subsets, with the efficiency in the number of commuting subsets being O(n).
Abstract
The key findings of this work are:
Tridiagonal matrices can be efficiently decomposed into the Pauli basis, with the number of commuting subsets scaling as O(n), where n is the number of qubits.
For an arbitrary tridiagonal matrix, the Pauli strings in the decomposition can only come from a union of (n+1) disjoint sets, each containing 2^n Pauli strings.
For a real tridiagonal matrix, the Pauli strings can only come from a union of 2n+1 disjoint sets, each containing 2^(n-1) or 2^n Pauli strings.
For a real symmetric tridiagonal matrix, the Pauli strings can only come from a union of n+1 disjoint sets, each containing 2^(n-1) or 2^n Pauli strings.
Formulae are provided to efficiently calculate the decomposition weights for the diagonal and off-diagonal elements of the tridiagonal matrix.
The method is demonstrated on the one-dimensional wave equation, showing that the gate complexity as a 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 tridiagonal matrices.
Stats
c1 + c2 + ... + cN-1 = 0
a1^2 + a2^2 + ... + aN-1^2 + b1^2 + b2^2 + ... + bN-1^2 = 2n
Quotes
"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 efficiency is in the number of commuting subsets O(n)."