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Alexander Russell

Publications and source records attributed to Alexander Russell.

At least 55 records · Page 3Linked to original sources
arXiv2010

Finding conjugate stabilizer subgroups in PSL(2; q) and related groups

We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an almost 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).

quant-ph↗
arXiv2010

Circuit partitions and #P-complete products of inner products

We present a simple, natural #P-complete problem. Let G be a directed graph, and let k be a positive integer. We define q(G;k) as follows. At each vertex v, we place a k-dimensional complex vector x_v. We take the product, over all edges (u,v), of the inner product . Finally, q(G;k) is the expectation of this product, where the x_v are chosen uniformly and independently from all vectors of norm 1 (or, alternately, from the Gaussian distribution). We show that q(G;k) is proportional to G's cycle partition polynomial, and therefore that it is #P-complete for any k>1.

cs.CC↗
arXiv2009

Efficient Steganography with Provable Security Guarantees

We provide a new provably-secure steganographic encryption protocol that is proven secure in the complexity-theoretic framework of Hopper et al. The fundamental building block of our steganographic encryption protocol is a "one-time stegosystem" that allows two parties to transmit messages of length shorter than the shared key with information-theoretic security guarantees. The employment of a pseudorandom generator (PRG) permits secure transmission of longer messages in the same way that such a generator allows the use of one-time pad encryption for messages longer than the key in symmetric encryption. The advantage of our construction, compared to that of Hopper et al., is that it avoids the use of a pseudorandom function family and instead relies (directly) on a pseudorandom generator in a way that provides linear improvement in the number of applications of the underlying one-way permutation per transmitted bit. This advantageous trade-off is achieved by substituting the pseudorandom function family employed in the previous construction with an appropriate combinatorial construction that has been used extensively in derandomization, namely almost t-wise independent function families.

cs.CR↗
arXiv2009

Approximating the Permanent via Nonabelian Determinants

Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued the possibility of an alternate, purely algebraic, polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. When A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first--and therefore the number of trials we need to obtain a good estimate of the permanent--is (1 + O(1/d))^n when A is the algebra of d by d matrices. These results can be extended to group algebras, and semi-simple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, for constant d--the only case in which an efficient algorithm is known--we show that the critical ratio exceeds 2^{n} / n^{O(d)}. Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are random, in either the Haar measure or the Gaussian measure, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group.

cs.CC↗
arXiv2009

The One-Way Communication Complexity of Group Membership

This paper studies the one-way communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input, a subgroup $H$ of a finite group $G$; Bob receives an element $x \in G$. Alice is permitted to send a single message to Bob, after which he must decide if his input $x$ is an element of $H$. We prove the following upper bounds on the classical communication complexity of this problem in the bounded-error setting: (1) The problem can be solved with $O(\log |G|)$ communication, provided the subgroup $H$ is normal; (2) The problem can be solved with $O(d_{\max} \cdot \log |G|)$ communication, where $d_{\max}$ is the maximum of the dimensions of the irreducible complex representations of $G$; (3) For any prime $p$ not dividing $|G|$, the problem can be solved with $O(d_{\max} \cdot \log p)$ communication, where $d_{\max}$ is the maximum of the dimensions of the irreducible $\F_p$-representations of $G$.

cs.CC↗
arXiv2008

A simple constant-probability RP reduction from NP to Parity P

The proof of Toda's celebrated theorem that the polynomial hierarchy is contained in $¶^{# P}$ relies on the fact that, under mild technical conditions on the complexity class $C$, we have $\exists C \subset BP \cdot \oplus C$. More concretely, there is a randomized reduction which transforms nonempty sets and the empty set, respectively, into sets of odd or even size. The customary method is to invoke Valiant's and Vazirani's randomized reduction from NP to UP, followed by amplification of the resulting success probability from $1/\poly(n)$ to a constant by combining the parities of $\poly(n)$ trials. Here we give a direct algebraic reduction which achieves constant success probability without the need for amplification. Our reduction is very simple, and its analysis relies on well-known properties of the Legendre symbol in finite fields.

cs.CC↗
arXiv2008

Uncertainty Principles for Compact Groups

We establish an operator-theoretic uncertainty principle over arbitrary compact groups, generalizing several previous results. As a consequence, we show that if f is in L^2(G), then the product of the measures of the supports of f and its Fourier transform ^f is at least 1; here, the dual measure is given by the sum, over all irreducible representations V, of d_V rank(^f(V)). For finite groups, our principle implies the following: if P and R are projection operators on the group algebra C[G] such that P commutes with projection onto each group element, and R commutes with left multiplication, then the squared operator norm of PR is at most rank(P)rank(R)/|G|.

math.RT↗
arXiv2008

Quantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs

We study the problem of \emph{local search} on a graph. Given a real-valued black-box function f on the graph's vertices, this is the problem of determining a local minimum of f--a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires $Ω(2^{n/2 - o(1)})$ queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to $Ω(2^{n/2}/n^2)$ and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are $Ω(N^{1/2}/d\log N)$ and $Ω(N^{1/4}/\sqrt{d\log N})$, respectively.

quant-ph↗
arXiv2007

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 Ω(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^{Ω(\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})}.

quant-ph↗
arXiv2007

Quantum Algorithms for Simon's Problem Over General Groups

Daniel Simon's 1994 discovery of an efficient quantum algorithm for solving the hidden subgroup problem (HSP) over Z_2^n provided one of the first algebraic problems for which quantum computers are exponentially faster than their classical counterparts. In this paper, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G, this is the problem of recovering an involution m = (m_1, ..., m_n) in G^n from an oracle f with the property that f(x) = f(xy) iff y equals m or the identity. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G^n, where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G^n have a simple product structure, they share important representation-theoretic properties with the symmetric groups S_n, where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called ``standard method'' requires highly entangled measurements on the tensor product of many coset states. Here we give quantum algorithms with time complexity 2^O(sqrt(n log n)) that recover hidden involutions m = (m_1, ..., m_n) in G^n where, as in Simon's problem, each m_i is either the identity or the conjugate of a known element k, and there is a character X of G for which X(k) = -X(1)$. Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the ``missing harmonic'' approach of Moore and Russell. These are the first nontrivial hidden subgroup algorithms for group families that require highly entangled multiregister Fourier sampling.

quant-ph↗
arXiv2007

A classical one-way function to confound quantum adversaries

The promise of quantum computation and its consequences for complexity-theoretic cryptography motivates an immediate search for cryptosystems which can be implemented with current technology, but which remain secure even in the presence of quantum computers. Inspired by recent negative results pertaining to the nonabelian hidden subgroup problem, we present here a classical algebraic function $f_V(M)$ of a matrix $M$ which we believe is a one-way function secure against quantum attacks. Specifically, inverting $f_V$ reduces naturally to solving a hidden subgroup problem over the general linear group (which is at least as hard as the hidden subgroup problem over the symmetric group). We also demonstrate a reduction from Graph Isomorphism to the problem of inverting $f_V$; unlike Graph Isomorphism, however, the function $f_V$ is random self-reducible and therefore uniformly hard. These results suggest that, unlike Shor's algorithm for the discrete logarithm--which is, so far, the only successful quantum attack on a classical one-way function--quantum attacks based on the hidden subgroup problem are unlikely to work. We also show that reconstructing any entry of $M$, or the trace of $M$, with nonnegligible advantage is essentially as hard as inverting $f_V$. Finally, $f_V$ can be efficiently computed and the number of output bits is less than $1+ε$ times the number of input bits for any $ε> 0$.

quant-ph↗
arXiv2006

On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

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 give strong evidence that no such approach can succeed 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. We show, modulo a group-theoretic conjecture regarding the asymptotic characters of the symmetric group, that no matter what rule we use to adaptively combine quantum states, there is a constant b > 0 such that no algorithm in this family can solve Graph Isomorphism in e^{b sqrt{n}} time. In particular, such algorithms are essentially no better than the best known classical algorithms, whose running time is e^{O(sqrt{n \log n})}.

quant-ph↗
arXiv2006

Explicit Multiregister Measurements for Hidden Subgroup Problems

We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses $k=\log_2 |G|$ registers, and each of the $2^k$ subsets of the registers contributes some information. While this does not, in general, yield an efficient algorithm, it generalizes the relationship between Subset Sum and the HSP in the dihedral group, and sheds some light on how quantum algorithms for Graph Isomorphism might work.

quant-ph↗
arXiv2006

Decoherence in Quantum Walks on the Hypercube

We study a natural notion of decoherence on quantum random walks over the hypercube. We prove that in this model there is a decoherence threshold beneath which the essential properties of the hypercubic quantum walk, such as linear mixing times, are preserved. Beyond the threshold, we prove that the walks behave like their classical counterparts.

quant-ph↗
arXiv2005

Tight Results on Multiregister Fourier Sampling: Quantum Measurements for Graph Isomorphism Require Entanglement

We establish a general method for proving bounds on the information that can be extracted via arbitrary entangled measurements on tensor products of hidden subgroup coset states. When applied to the symmetric group, the method yields an Omega(n log n) lower bound on the number of coset states over which we must perform an entangled measurement in order to obtain non-negligible information about a hidden involution. These results are tight to within a multiplicative constant and apply, in particular, to the case relevant for the Graph Isomorphism problem. Part of our proof was obtained after learning from Hallgren, Roetteler, and Sen that they had obtained similar results.

quant-ph↗
arXiv2005

Strong Fourier Sampling Fails over $G^n$

We present a negative result regarding the hidden subgroup problem on the powers $G^n$ of a fixed group $G$. Under a condition on the base group $G$, we prove that strong Fourier sampling cannot distinguish some subgroups of $G^n$. Since strong sampling is in fact the optimal measurement on a coset state, this shows that we have no hope of efficiently solving the hidden subgroup problem over these groups with separable measurements on coset states (that is, using any polynomial number of single-register coset state experiments). Base groups satisfying our condition include all nonabelian simple groups. We apply our results to show that there exist uniform families of nilpotent groups whose normal series factors have constant size and yet are immune to strong Fourier sampling.

quant-ph↗
arXiv2005

The Symmetric Group Defies Strong Fourier Sampling: Part I

We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem. Specifically, we show that the hidden subgroup problem over the symmetric group cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. Our results apply to the special case relevant to the Graph Isomorphism problem.

quant-ph↗