I am attempting to use Grover's to find elements in an array that satisfy a comparative query (=, <, >, etc). I am using Qiskit, and am using a qRAM to encode my array.

I have experimented with several quantum comparators, all of which seem to be working well offline (i.e. they indeed produce 1 when the query is satisfied). I settled for the IntegerComparator present in Qiskit.

Things start getting weird once I inject my comparator into the oracle of my algorithm. For the array is $[7,6,5,4,3,2,1,0]$, the following happens when testing the query $< n$ for some predetermined $n$:

  • The algorithm produces sensible results for $n \in \{0, 1, 2,3\}$.
  • For $n = 4$, the algorithm doesn't produce any conclusive solutions.
  • For $n \in \{5,6,7\}$, it produces answers the $\geq n$ instead!

Analogous behavior was observed for $> n$. The algorithm works perfectly for the query $==n$.

I have been debugging this for several days, and I can't seem to figure it out. I highly appreciate any kind of help.

This is the high level design of my algorithm: enter image description here


I don't think it's an issue with the algorithm implementation (though I haven't checked the circuit in detail), but rather a typical behavior of Grover's algorithm. The algorithm analysis that tells you the number of iterations that need to be done based on the size of the search space $N$ and the number of solutions to your problem $M$ assumes that $M$ is much smaller than $N$. Once $M$ grows to be large enough, the algorithm starts to misbehave.

In your case, $N = 8$, and when you set $n = 4$ for your query, you have $M = 4 = \frac{N}{2}$ solutions to the search problem. In this case the initial (pre-iteration) probability of obtaining a solution is $\frac{M}{N} = 0.5$, and each Grover's iteration leaves it unchanged, so effectively you're always drawing from a uniform distribution.

When you set $n = 5$ or more, you have $M > \frac{N}{2}$, and your oracle which marks all elements that are less than $5$ differs from an oracle that marks all elements greater than or equal to $5$ only by a global phase of $-1$. Grover's algorithm is not sensitive to this global phase, so it amplifies the probability of the less probable group of outcomes - in this case the probability of getting a non-solution.

You can read more about this in Nielsen and Chuang, section 6.1.4, as quoted in this answer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.