On the sampling complexity of coherent superpositions

TL;DR

This paper presents an algorithm for efficiently sampling measurement outcomes from superpositions of quantum states, reducing the problem to simpler sampling tasks, and extends previous finite-outcome POVM results to continuous outcomes like Gaussian states.

Contribution

It introduces a sampling algorithm for superpositions that applies to continuous POVMs, linking strong classical simulation to weak simulation in quantum measurement scenarios.

Findings

Algorithm achieves sampling with $O( ext{parameters})$ complexity

Extends results to continuous-outcome POVMs like Gaussian measurements

Reduces classical simulation of quantum measurements to sampling tasks

Abstract

We consider the problem of sampling from the distribution of measurement outcomes when applying a POVM to a superposition $|\Psi\rangle = \sum_{j=0}^{\chi-1} c_j |\psi_j\rangle$ of $\chi$ pure states. We relate this problem to that of drawing samples from the outcome distribution when measuring a single state $|\psi_j\rangle$ in the superposition. Here $j$ is drawn from the distribution $p(j)=|c_j|^2/\|c\|^2_2$ of normalized amplitudes. We give an algorithm which $-$ given $O(\chi \|c\|_2^2 \log1/\delta)$ such samples and calls to oracles evaluating the involved probability density functions $-$ outputs a sample from the target distribution except with probability at most $\delta$. In many cases of interest, the POVM and individual states in the superposition have efficient classical descriptions allowing to evaluate matrix elements of POVM elements and to draw samples from outcome distributions. In such a scenario, our algorithm gives a reduction from strong classical simulation (i.e., the problem of computing outcome probabilities) to weak simulation (i.e., the problem of sampling). In contrast to prior work focusing on finite-outcome POVMs, this reduction also applies to continuous-outcome POVMs. An example is homodyne or heterodyne measurements applied to a superposition of Gaussian states. Here we obtain a sampling algorithm with time complexity $O(N^3 \chi^3 \|c\|_2^2 \log1/\delta)$ for a state of $N$ bosonic modes.