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This question already has an answer here:

There are different theoretical models for quantum computing like the circuit model or the model of adiabatic quantum computers.

Between which of these models exist polynomial-time reductions?

Note that this question does not aim to cover physical implementations of quantum computers which are already discussed here.

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marked as duplicate by Niel de Beaudrap, bytebuster, u32i64, Discrete lizard, ItamarG3 Apr 2 '18 at 8:55

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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A non-exhaustive list of theoretical models of quantum computation are provided as answers to another question: "What are the methods of quantum computation?".

As to which models are polynomial-time equivalent — the following is an incomplete list of models which are provably universal for polynomial-time quantum computation, assuming perfect control:

  • The unitary circuit model is polynomial-time equivalent to adiabatic quantum computation [arXiv:quant-ph/0405098];
  • The unitary circuit model is polynomial-time equivalent to quantum circuits with intermediate measurements (by the principle of deferred measurement);
  • The one-way measurement-based model is polynomial-time equivalent to unitary circuits.
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