Topologically driven no-superposing theorem with a tight error bound

• URL: http://arxiv.org/abs/2111.02391v3
• Date: Mon, 28 Oct 2024 14:51:07 GMT
• Title: Topologically driven no-superposing theorem with a tight error bound
• Authors: Zuzana Gavorová,
• Abstract summary: 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. Our results exclude the superposition as an elementary gate for quantum computation.
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• 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 exclude the superposition as an elementary gate for quantum computation, and 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.

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