Simon's problem

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In computational complexity theory and quantum computing, Simon's problem is a computational problem conceived to showcase the efficiency increase a quantum algorithm could have over a classic one. Although the problem itself is of little practical value, it is interesting because it provides an exponential speedup over any classical algorithm (in a black box model).[1]

The problem deals with the model of decision tree complexity or query complexity and was conceived by Daniel Simon in 1994.[2] Simon exhibited a quantum algorithm, usually called Simon's algorithm, that solves the problem exponentially faster than any deterministic or probabilistic classical algorithm, requiring exponentially less computational power than the best classical probabilistic algorithm.

This problem yields an oracle separation between BPP and BQP, unlike the separation provided by the Deutsch-Jozsa algorithm, which separates P and EQP.

Simon's algorithm was also the inspiration for Shor's algorithm. Both problems are special cases of the abelian hidden subgroup problem, which is now known to have efficient quantum algorithms.

Problem description and algorithm[edit]

The input to the problem is a function (implemented by a black box) , promised to satisfy the property that for some we have for all , if and only if or . Note that the case of is allowed, and corresponds to being a permutation. The problem then is to find .

The set of n-bit strings is a vector space under bitwise XOR. Given the promise, the preimage of f is either empty, or forms cosets with n-1 dimensions. Using quantum algorithms, we can, with arbitrarily high probability determine the basis vectors spanning this n-1 subspace since s is a vector orthogonal to all of the basis vectors.

Quantum subroutine in Simon's algorithm

Consider the Hilbert space consisting of the tensor product of the Hilbert space of input strings, and output strings. Using Hadamard operations, we can prepare the initial state

and then call the oracle to transform this state to

Hadamard transforms convert this state to

The following is not a step in the algorithm, but we can understand this quantity better if we factorize out the s out to get

and observe that there are exacly two s, which we shall call and , for which . Since , we can rewrite the above as

where

The coefficient of is only when is . Using the fact that , we see that that's the case precisely when the left hand side

is , which is only the case if is .

The final step in the algorithm is to measure both registers. The only that we can get back is one that satisfies . This determines a subspace which must lie on. Given enough such samples , is constrained to a -dimensional subspace, which means we've determined .

Complexity[edit]

Simon's algorithm requires queries to the black box, whereas a classical algorithm would need at least queries. It is also known that Simon's algorithm is optimal in the sense that any quantum algorithm to solve this problem requires queries.[3][4]

See also[edit]

References[edit]

  1. ^ Arora, Sanjeev and Barak, Boaz. Computational Complexity: A Modern Approach. Cambridge University Press. 
  2. ^ Simon, D.R. (1995), "On the power of quantum computation", Foundations of Computer Science, 1996 Proceedings., 35th Annual Symposium on: 116–123, retrieved 2011-06-06 
  3. ^ Koiran, P.; Nesme, V.; Portier, N. (2007), "The quantum query complexity of the abelian hidden subgroup problem", Theoretical Computer Science, 380 (1-2): 115–126, doi:10.1016/j.tcs.2007.02.057, retrieved 2011-06-06 
  4. ^ Koiran, P.; Nesme, V.; Portier, N. (2005), "A quantum lower bound for the query complexity of Simon's Problem", Proc. ICALP, 3580: 1287–1298, arXiv:quant-ph/0501060Freely accessible, retrieved 2011-06-06