Hamiltonian Eigenvalue Transformation by Tridiagonal Gadgets

2026-10-02

AI Summary

This study addresses the limitation that analog devices struggle to implement non-local polynomial Hamiltonian operations, which constrains quantum algorithm efficiency. We propose a tridiagonal gadget method that constructs a triangle family using a short ancillary qubit chain. By combining Chebyshev duplication identities with adiabatic optimization techniques, this approach transforms an arbitrary polynomial P(H) into the equivalent action of a single time-independent local Hamiltonian on a specific subspace. Replacing high-depth circuits with constant locality overhead, the method significantly reduces ancillary resource requirements and optimizes energy scaling. It achieves eigenstate filtering with geometrically decaying error and enables polynomial-scaling local driving, while simulations perfectly reproduce the spectral properties of the ideal algorithm.

Abstract

An analog device implements a local Hamiltonian H, but a polynomial P (H) is in general not local, so the device cannot implement it. We carry P (H), to any prescribed accuracy, as the action of a single time-independent local Hamiltonian on an explicitly described invariant subspace. Short chains of ancilla qubits are attached to H. A chain of 2m sites has a unique isolated eigenvalue that is an analytic function of the input vanishing to order exactly 2m, because the input must cross the chain and come back before it can shift the energy at the far end. Chains of different lengths therefore form a triangular family, and a weighted sum of them reproduces a prescribed polynomial term by term. Even chains give the even part and odd chains the odd part, so any polynomial is reached. We prove uniqueness, analyticity and coefficient bounds uniform in the chain length for inputs H of norm below any r < 1. Where the circuit model pays degree 2l in 2l sequential oracle calls, the result here is one Hamiltonian of locality two more than that of H, which pays the degree in O(l^2 + log2^(1/eps)) ancillas and in energy scale. As an application we filter a marked eigenstate of a local Hamiltonian. Composing a synthesised square along the Chebyshev doubling identity works, but its energy scale grows quasi-polynomially with the degree. Iterating instead the exact eigenvalue branch of the two-site chain k times gives a filter on k + 1 ancilla qubits whose error decays geometrically in k, at an energy scale polynomial in the inverse passband width and independent of k, the locality growing by one per stage. In adiabatic optimisation with a rank-one driver, the construction replaces that non-local driver, by a Hamiltonian with O(log n) ancillas and O(log n)- body terms, at an energy scale polynomial in n. Simulations including every ancilla reproduce the spectrum of the ideal algorithm.

Problem

• Hamiltonian Eigenvalue Transformation
• Local Hamiltonian
• Polynomial Transformation
• Analog Quantum Device
• Ancilla Qubits

Innovation

• Hamiltonian Eigenvalue Transformation
• Tridiagonal Gadgets
• Ancilla Qubits
• Polynomial Filtering
• Adiabatic Optimization