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86
Quantum algorithms for the triangle problem
 PROCEEDINGS OF SODA’05
, 2005
"... We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is b ..."
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Cited by 60 (11 self)
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We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.
The twoeigenvalue problem and density of Jones representation of braid groups
 Commun. Math. Phys.
, 2002
"... ..."
Hidden translation and orbit coset in quantum computing
 In Proc. 35th ACM STOC
, 2003
"... We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently ..."
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Cited by 38 (6 self)
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We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z n p, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful selfreducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G. Our selfreducibility framework combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group. 1
Limits on the Power of Quantum Statistical ZeroKnowledge
, 2003
"... In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK ..."
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Cited by 30 (4 self)
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In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
"... We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general famil ..."
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Cited by 24 (5 self)
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We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. This family encompasses the wellknown plat and trace closures, generalizing results recently obtained by Aharonov, Jones and Landau. We base our algorithms on a local qubit implementation of the unitary JonesWenzl representations of the braid group which makes the underlying representation theory apparent, allowing us to provide an algorithm for approximating the HOMFLYPT twovariable polynomial of the trace closure of a braid at certain pairs of values as well. Next, we provide a selfcontained proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. This theorem was originally proved by Freedman, Larsen and Wang in the context of topological quantum computation, and the necessary notion of approximation was later provided by Bordewich et al. Our proof is simpler as it uses a more natural encoding of twoqubit unitaries into the rectangular representation of the eightstrand braid group. We then give QCMAcomplete and PSPACEcomplete problems which are based on braids. Finally, we conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #Phard problem, while learning its most significant bit is PPhard, without taking the usual route through the Tutte polynomial and graph coloring. 1
3Local Hamiltonian is QMAcomplete
 Quantum Information and Computation
"... It has been shown by Kitaev that the 5local Hamiltonian problem is QMAcomplete. Here we reduce the locality of the problem by showing that 3local Hamiltonian is already QMAcomplete. 1 ..."
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Cited by 19 (4 self)
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It has been shown by Kitaev that the 5local Hamiltonian problem is QMAcomplete. Here we reduce the locality of the problem by showing that 3local Hamiltonian is already QMAcomplete. 1
Sophisticated Quantum Search Without Entanglement
, 2000
"... Although entanglement is widely considered to be necessary for quantum algorithms to improve on classical ones, Lloyd has observed recently that Grover's quantum search algorithm can be implemented without entanglement, by replacing multiple particles with a single particle having exponentially many ..."
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Cited by 18 (6 self)
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Although entanglement is widely considered to be necessary for quantum algorithms to improve on classical ones, Lloyd has observed recently that Grover's quantum search algorithm can be implemented without entanglement, by replacing multiple particles with a single particle having exponentially many states. We explain that this maneuver removes entanglement from any quantum algorithm. But all physical resources must be accounted for to quantify algorithm complexity, and this scheme typically incurs exponential costs in some other resource(s). In particular, we demonstrate that a recent experimental realization requires exponentially increasing precision. There is, however, a quantum algorithm which searches a `sophisticated' database (not unlike a Web search engine) with a single query, but which we show does not require entanglement even for multiparticle implementations. 1999 Physics and Astronomy Classification Scheme: 03.67.Lx, 32.80.Rm. 2000 American Mathematical Society Subject ...
Classical simulation of noninteractingfermion quantum circuits
 Phys. Rev. A
"... We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [1] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend ..."
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Cited by 14 (1 self)
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We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [1] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend the result to noninteracting fermions with arbitrary pairwise interactions, where gates can be conditioned on outcomes of complete von Neumann measurements in the computational basis on other fermionic modes in the circuit. This last result is in remarkable contrast with the case of noninteracting bosons where universal quantum computation can be achieved by allowing gates to be conditioned on classical bits [2].
A lattice problem in quantum NP
 In Proc. 44th IEEE Symposium on Foundations of Computer Science
, 2003
"... We consider coGapSV P √ n, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM∩coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first nontrivial upper bound on the quantum complexity of a l ..."
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Cited by 14 (2 self)
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We consider coGapSV P √ n, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM∩coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first nontrivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+. Working with the QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover’s possibility to cheat. We hope that these ideas will lead to further developments in the field. 1
On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 13 (7 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.