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Can anyone help with this problem from, An Introduction to Quantum Computing, Phillip Kaye, Raymond Laflamme, Michele Mosca

Exercise 6.4.2: Show that a probabilistic classical algorithm that makes $O(n)$ queries can with probability at least $1 − \frac{1}{2^n}$ correctly determine whether $f$ is constant or balanced.

Hint: Use the Chernoff bound (Appendix A.1).

I am new to Quantum Computing and deutsch jozsa problem, and Hint is not helping.

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1 Answer 1

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Consider the following algorithm:

1. Choose $n+1$ bitstrings $\{q_1,\dots,q_{n+1}\}\subseteq \{0,1\}^n$ uniformly at random (allowing repetitions).
2. If $f(q_1)=\dots=f(q_{n+1})$ return "CONSTANT"
3. Else return "BALANCED"

Now let's analyze the above algorithm. If $f$ is constant then the algorithm will always be correct. If on the other side $f$ is balanced then the probability that our algorithm will return CONSTANT is bounded by the probability $$\mathbb{P}\left(f(q_1)=f(q_2), f(q_1)=f(q_3), \dots, f(q_1)=f(q_{n+1})\right)$$ The events $f(q_1)=f(q_i)$ are independent and have probability of occurring equal to $\frac{2^{n-1}}{2^n}=\frac{1}{2}$ (since out of the $2^n$ possible bitstring there are $\frac{2^{n}}{2}=2^{n-1}$ that yield the same value of $f$ as $q_1$) and thus the probability of error is at most $\frac{1}{2^n}$, as required. Notice that the Chernoff bound was not used here.

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  • $\begingroup$ Could you elaborate in your answer on how you get the bound? Also how does Chernov ineq come into play? $\endgroup$
    – MonteNero
    Aug 15, 2022 at 4:48
  • $\begingroup$ @MonteNero I have edited my answer to explain a bit more how the probability of error was derived. Is it ok now? $\endgroup$ Aug 15, 2022 at 11:32

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