A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation

TL;DR

This paper demonstrates that a single energy measurement on a translationally invariant qudit chain can replicate any quantum computation, linking energy measurement complexity to quantum computational universality.

Contribution

It introduces a specific Hamiltonian model where a single measurement suffices to perform universal quantum computation, showing the measurement process's computational equivalence.

Findings

Energy measurement complexity scales inverse polynomially with circuit size.

Single-shot energy measurement can implement any quantum circuit.

Energy measurement on qudit chains is as hard as quantum computation.

Abstract

In measurement-based quantum computation, quantum algorithms are implemented via sequences of measurements. We describe a translationally invariant finite-range interaction on a one-dimensional qudit chain and prove that a single-shot measurement of the energy of an appropriate computational basis state with respect to this Hamiltonian provides the output of any quantum circuit. The required measurement accuracy scales inverse polynomially with the size of the simulated quantum circuit. This shows that the implementation of energy measurements on generic qudit chains is as hard as the realization of quantum computation. Here a ''measurement'' is any procedure that samples from the spectral measure induced by the observable and the state under consideration. As opposed to measurement-based quantum computation, the post-measurement state is irrelevant.