IQP computations with intermediate measurements

TL;DR

This paper explores the computational power of IQP circuits with intermediate measurements, showing how different measurement strategies affect universality and stability under errors.

Contribution

It demonstrates that certain measurement schemes preserve IQP's computational power, and introduces a universal model using only CZ gates and adaptive X measurements with specific input states.

Findings

Non-adaptive and adaptive X measurements do not increase IQP power.

Adaptive Z measurements make IQP quantum universal.

A universal model with CZ gates and specific input states is proposed.

Abstract

We consider the computational model of IQP circuits (in which all computational steps are $X$ basis diagonal gates), supplemented by intermediate $X$ or $Z$ basis measurements. We show that if we allow non-adaptive or adaptive $X$ basis measurements, or allow non-adaptive $Z$ basis measurements, then the computational power remains the same as that of the original IQP model; and with adaptive $Z$ basis measurements the model becomes quantum universal. Furthermore we show that the computational model having circuits of only $CZ$ gates and adaptive $X$ basis measurements, with input states that are tensor products of 1-qubit states from the set $\{ |+\rangle, |1\rangle,\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle), \frac{1}{\sqrt{2}}(|0\rangle+e^{i\pi/4}|1\rangle) \} $, is quantum universal. In contrast to the relation of IQP to PH collapse, all our results here are manifestly stable under small additive implementational errors.