Quantum State Restoration and Single-Copy Tomography for Ground States of Hamiltonians

Given a single copy of an unknown quantum state, the no-cloning theorem limits the amount of information that can be extracted from it. Given a gapped Hamiltonian, in most situations it is impractical to compute properties of its ground state, even though in principle all the information about the ground state is encoded in the Hamiltonian. We show in this Letter that if you know the Hamiltonian of a system and have a single copy of its ground state, you can use a quantum computer to efficiently compute its local properties. Specifically, in this scenario, we give efficient algorithms that copy small subsystems of the state and estimate the full statistics of any local measurement.

Figure 1

Probability of restoring the state on a given iteration conditioned on all previous iterations failing. Conditioned on failing every time, the first two flat regions are metastable states and the third is stable. In this graph, $|\psi ⟩=\sqrt{1-{10}^{-2}-{10}^{-3}}|0{⟩}_A|0{⟩}_B+\sqrt{{10}^{-2}}|1{⟩}_A|1{⟩}_B+\sqrt{{10}^{-3}}|2{⟩}_A|2{⟩}_B$, $\dim {\mathcal{H}}_B=10$, and the expected number of iterations required is 30.