Simon's algorithm
Simon's algorithm, conceived by Daniel Simon in 1994,[1] is a quantum algorithm which solves a black-box problem exponentially faster than any classical algorithm, including probabilistic algorithms. The algorithm uses 
queries to the black box, whereas the best classical randomized algorithm necessarily needs 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.[2][3]
We have a random oracle relative to which BPP and BQP are separated. This is an improvement over the Deutsch-Jozsa algorithm separating P from EQP. The oracle needs to be quantum and nondecohering.
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 s.
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.
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
We perform a simultaneous measurement of both registers. If 
, we have destructive interference. So, only the subspace 
is picked out. Given enough samples of y, we can figure out the n-1 basis vectors, and compute s.
Although the problem itself is of little practical value it is interesting because it provides an exponential speedup over any classical algorithm. Moreover, it 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.
See also
References
- ^ Simon, D.R. (1994), "On the power of quantum computation", Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on: 116–123, http://citeseer.ist.psu.edu/viewdoc/download;jsessionid=0D01FC6F66C81F349B409BAE3F515CAC?doi=10.1.1.51.5477&rep=rep1&type=pdf, retrieved 2011-06-06
- ^ 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, http://perso.ens-lyon.fr/pascal.koiran/Publis/lip.05-17.ps, retrieved 2011-06-06
- ^ 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/0501060, http://perso.ens-lyon.fr/pascal.koiran/Publis/icalp05.ps, retrieved 2011-06-06
|
|||||||||||||||||||||||||||||||||||||||
