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58
Exponential Algorithmic Speedup by a Quantum Walk
"... We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fouri ..."
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Cited by 104 (5 self)
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We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk e#ciently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
Quantum Computation and Lattice Problems
 Proc. 43rd Symposium on Foundations of Computer Science
, 2002
"... We present the first explicit connection between quantum computation and lattice problems. Namely, we show a solution to the uniqueSVP under the assumption that there exists... ..."
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Cited by 56 (4 self)
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We present the first explicit connection between quantum computation and lattice problems. Namely, we show a solution to the uniqueSVP under the assumption that there exists...
Adiabatic quantum state generation and statistical zeroknowledge
 in Proc. 35th STOC
, 2003
"... The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem. We systematically study ’quantum state generation’, namely, which superpositions can be efficiently generated. We first show that all problems in Statistical Z ..."
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Cited by 43 (3 self)
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The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem. We systematically study ’quantum state generation’, namely, which superpositions can be efficiently generated. We first show that all problems in Statistical Zero Knowledge (SZK), a class which contains many languages that are natural candidates for BQP, can be reduced to an instance of quantum state generation. This was known before for graph isomorphism, but we give a general recipe for all problems in SZK. We demonstrate the reduction from the problem to its quantum state generation version for three examples: Discrete log, quadratic residuosity and a gap version of closest vector in a lattice. We then develop tools for quantum state generation. For this task, we define the framework of ’adiabatic quantum state generation ’ which uses the language of ground states, spectral gaps and Hamiltonians instead of the standard unitary gate language. This language stems from the recently suggested adiabatic computation model [20] and seems to be especially tailored for the task of quantum state generation. After defining the paradigm, we provide two basic lemmas for adiabatic quantum state generation: • The Sparse Hamiltonian lemma, which gives a general technique for implementing sparse Hamiltonians efficiently, and, • The jagged adiabatic path lemma, which gives conditions for a sequence of Hamiltonians to allow efficient adiabatic state generation. We use our tools to prove that any quantum state which can be generated efficiently in the standard model can also be generated efficiently adiabatically, and vice versa. Finally we show how to apply our techniques to generate superpositions corresponding to limiting distributions of a large class of Markov chains, including the uniform distribution over all perfect
QUANTUM ALGORITHMS FOR SOME HIDDEN SHIFT PROBLEMS
 SIAM J. COMPUT
, 2006
"... Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in th ..."
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Cited by 42 (2 self)
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Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
Limitations of quantum coset states for graph isomorphism
 In Proceedings of the 38th ACM Symposium on Theory of Computing
, 2006
"... It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which enc ..."
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Cited by 31 (10 self)
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It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [MRS05] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n, Fp m) and Gn where G is finite and satisfies a suitable property. 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
Solving the Pell Equation
, 2008
"... We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly a ..."
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Cited by 20 (0 self)
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We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.
The Hidden Subgroup Problem and Quantum Computation Using Group Representations
 SIAM Journal on Computing
, 2003
"... The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonAbelian case is open; an efficient solution would give rise to an ..."
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Cited by 18 (2 self)
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The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonAbelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case algorithm to the nonAbelian case. We show that the algorithm finds the normal core of the hidden subgroup, and that, in particular, normal subgroups can be found. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism. 1
On the impossibility of a quantum sieve algorithm for Graph Isomorphism
, 2006
"... ABSTRACT. 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 efficien ..."
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Cited by 16 (2 self)
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ABSTRACT. 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 polynomialtime quantum algorithm for Graph Isomorphism. Specifically, we consider the natural reduction of Graph Isomorphism to the HSP over the the wreath product Sn ≀ Z2. Using a recently proved bound on the irreducible characters of Sn, we show that no algorithm in this family can solve Graph Isomorphism in less than e Ω( √ 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( √ nlog n) 1.
Efficient quantum algorithms for estimating Gauss sums
, 2002
"... We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we ..."
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Cited by 14 (1 self)
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We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we also give evidence that this problem is hard for classical algorithms. The workings of the quantum algorithm rely on the interaction between the additive characters of the Fourier transform and the multiplicative characters of the Gauss sum.