Topologically driven no-superposing theorem with a tight error bound

TL;DR

This paper proves the fundamental impossibility of superposing two unknown quantum states with a tight error bound, using topological arguments to establish limits on quantum state superposition and its sample complexity.

Contribution

It introduces a topological proof of the no-superposing theorem with a tight error bound and sample complexity, advancing understanding of quantum state superposition limits.

Findings

Superposing two unknown states is impossible regardless of sample size.

The paper quantifies the approximation error and sample complexity, showing the bounds are tight.

State tomography's usefulness is limited in superposition tasks unless randomness is tolerated.

Abstract

To better understand quantum computation we can search for its limits or no-gos, especially if analogous limits do not appear in classical computation. Classical computation easily implements and extensively employs the addition of two bit strings, so here we study 'quantum addition': the superposition of two quantum states. We prove the impossibility of superposing two unknown states, no matter how many samples of each state are available. The proof uses topology; a quantum algorithm of any sample complexity corresponds to a continuous function, but the function required by the superposition task cannot be continuous by topological arguments. Our result for the first time quantifies the approximation error and the sample complexity $N$ of the superposition task, and it is tight. We present a trivial algorithm with a large approximation error and $N=1$, and the matching impossibility of any smaller approximation error for any $N$. Consequently, our results limit state tomography as a useful subroutine for the superposition. State tomography is useful only in a model that tolerates randomness in the superposed state. The optimal protocol in this random model remains open.