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I want to show that $Q_{\epsilon}(IP) \geq (1-O(\epsilon))n$, where $IP:\{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ is the usual mod 2 inner product.

I have Nayak's lower bound, but I am not sure how to use it in the above.

I have that for some encoding $p:n\rightarrow m$ $m\geq n(1-H(p))$, I feel that I need to use this bound here, but I am not sure how.

Any advice?

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    $\begingroup$ It would be very useful if you mentioned what $Q$ is. $\endgroup$ – Robin Kothari Jul 4 '12 at 6:12
  • $\begingroup$ I guess he is talking about the bound for quantum random access codes. $\endgroup$ – Joe Fitzsimons Jul 4 '12 at 6:20
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    $\begingroup$ I agree with @RobinKothari that it is not clear if you are talking about query-complexity or communication-complexity. You should clarify this in your question and re-tag appropriately. You already found a self-answer for the case of communication complexity. If you are happy with asymptotic statements, then the $\Omega(n)$ lower bound for query-complexity follows trivially from parity. $\endgroup$ – Artem Kaznatcheev Jul 5 '12 at 4:01
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Ok I believe i found what I was looking for.

In Kremer's Thesis "Quantum Communication" he proves that $Q_{\epsilon}(IP)= \Omega(n)$.

here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.6636

page 38.

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