Hamiltonian Decoded Quantum Interferometry

Understand

We introduce Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm that utilizes coherent Bell measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian optimization to classical decoding.

• For a signed Pauli Hamiltonian $H$ and any degree-$\ell$ polynomial ${P}$, HDQI prepares a purification of the density matrix $\rho_{P}(H) \propto {P}^2(H)$ by solving a combination of two tasks: decoding $\ell$ errors on a classical code defined by $H$, and preparing a pilot state that encodes the anti-commutation structure of $H$.
• Choosing $P(x)$ to approximate $\exp(-\beta x/2)$ yields Gibbs states at inverse temperature $\beta$; other choices prepare approximate ground states, microcanonical ensembles, and other spectral filters.
• For local Hamiltonians, the corresponding decoding problem is that of LDPC codes.