16 research outputs found
On the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results
It is known that any quantum algorithm for Graph Isomorphism that works
within the framework of the hidden subgroup problem (HSP) must perform highly
entangled measurements across \Omega(n \log n) coset states. One of the only
known models for how such a measurement could be carried out efficiently is
Kuperberg's algorithm for the HSP in the dihedral group, in which quantum
states are adaptively combined and measured according to the decomposition of
tensor products into irreducible representations. This ``quantum sieve'' starts
with coset states, and works its way down towards representations whose
probabilities differ depending on, for example, whether the hidden subgroup is
trivial or nontrivial.
In this paper we show that no such approach can produce a polynomial-time
quantum algorithm for Graph Isomorphism. Specifically, we consider the natural
reduction of Graph Isomorphism to the HSP over the the wreath product S_n\wr
Z_2. Using a recently proved bound on the irreducible characters of S_n, we
show that no algorithm in this family can solve Graph Isomorphism in less than
e^{\Omega(\sqrt{n})} time, no matter what adaptive rule it uses to select and
combine quantum states. In particular, algorithms of this type can offer
essentially no improvement over the best known classical algorithms, which run
in time e^{O(\sqrt{n \log n})}.Comment: An earlier preprint, quant-ph/0609138, gave versions of these results
which were conditional on a group-theoretic conjecture. This version provides
unconditional result
Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula
We study asymptotics of an irreducible representation of the symmetric group Sn corresponding to a balanced Young diagram λ (a Young diagram with at most View the MathML source rows and columns for some fixed constant C) in the limit as n tends to infinity
Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems
The Code Equivalence problem is that of determining whether two given linear
codes are equivalent to each other up to a permutation of the coordinates. This
problem has a direct reduction to a nonabelian hidden subgroup problem (HSP),
suggesting a possible quantum algorithm analogous to Shor's algorithms for
factoring or discrete log. However, we recently showed that in many cases of
interest---including Goppa codes---solving this case of the HSP requires rich,
entangled measurements. Thus, solving these cases of Code Equivalence via
Fourier sampling appears to be out of reach of current families of quantum
algorithms.
Code equivalence is directly related to the security of McEliece-type
cryptosystems in the case where the private code is known to the adversary.
However, for many codes the support splitting algorithm of Sendrier provides a
classical attack in this case. We revisit the claims of our previous article in
the light of these classical attacks, and discuss the particular case of the
Sidelnikov cryptosystem, which is based on Reed-Muller codes
Separation cutoffs for random walk on irreducible representations
Random walk on the irreducible representations of the symmetric and general
linear groups is studied. A separation distance cutoff is proved and the exact
separation distance asymptotics are determined. A key tool is a method for
writing the multiplicities in the Kronecker tensor powers of a fixed
representation as a sum of non-negative terms. Connections are made with the
Lagrange-Sylvester interpolation approach to Markov chains.Comment: 20 page
Asymptotics of characters of symmetric groups related to Stanley character formula
We prove an upper bound for characters of the symmetric groups. Namely, we
show that there exists a constant a>0 with a property that for every Young
diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr
\rho^\lambda(\pi) / Tr \rho^\lambda(e)| < [a max(r(\lambda)/n,
c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors
needed to write \pi\in S_n as a product of transpositions. We also give uniform
estimates for the error term in the Vershik-Kerov's and Biane's character
formulas and give a new formula for free cumulants of the transition measure.Comment: Version 4: Change of title, shortened to 20 pages. Version 3: 24
pages, the title and the list of authors were changed. Version 2: 14 pages,
the title, abstract and the main result were changed. Version 1: 10 pages
(mistake in Lemma 7- which is false
Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler