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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 41 (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 30 (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
Optimal measurements for the dihedral hidden subgroup problem
 Chicago Journal of Theoretical Computer Science
, 2005
"... Abstract. We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the socalled pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp ..."
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Cited by 29 (3 self)
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Abstract. We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the socalled pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density ν = k / log 2 N, where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for ν < 1 the optimal measurement (and hence any measurement) identifies the hidden subgroup with a probability that is exponentially small in log N, while for ν> 1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Ω(log G) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement. 1.
Quantum versus classical proofs and advice
 In preparation
, 2006
"... Abstract: This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA = QCMA. We prove three results about this question. First, we give a “quantum oracle separation ” between QMA and QCMA. More concretely, we show that any quantum algorithm ..."
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Cited by 19 (11 self)
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Abstract: This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA = QCMA. We prove three results about this question. First, we give a “quantum oracle separation ” between QMA and QCMA. More concretely, we show that any quantum algorithm needs Ω queries to find an n� � 2 n m+1 qubit “marked state ” ψ〉, even if given an mbit classical description of ψ 〉 together with a quantum black box that recognizes ψ〉. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previouslyknown case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying nonmembership in finite groups. Under plausible grouptheoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA. ACM Classification: F.1.2, F.1.3
Computational indistinguishability between quantum states and its cryptographic application
 Advances in Cryptology – EUROCRYPT 2005
, 2005
"... We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is “secure ” against any polynomialtime quantum adversary. Our problem QSCDff is to distinguish between two types of random coset s ..."
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Cited by 7 (5 self)
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We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is “secure ” against any polynomialtime quantum adversary. Our problem QSCDff is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonlyused distinction problem between two probability distributions in computational cryptography. As our major contribution, we show three cryptographic properties: (i) QSCDff has the trapdoor property; (ii) the averagecase hardness of QSCDff coincides with its worstcase hardness; and (iii) QSCDff is computationally at least as hard in the worst case as the graph automorphism problem. These cryptographic properties enable us to construct a quantum publickey cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomialtime quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multibit encryption scheme relying on the cryptographic properties of QSCDcyc.
On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems
, 2005
"... ..."
On the Complexity of the Hidden Subgroup Problem
"... Abstract. We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, andOrbit Coset, are equivalent or reducible to Hidden Subgroup. Wealso show that, over permutation groups, the decision version and search version of Hidden Subgroup are polyno ..."
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Cited by 1 (0 self)
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Abstract. We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, andOrbit Coset, are equivalent or reducible to Hidden Subgroup. Wealso show that, over permutation groups, the decision version and search version of Hidden Subgroup are polynomialtime equivalent. For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth. Finally, we give nonadaptive program checkers for Hidden Subgroup and its decision version. 1
Implementation of GroupCovariant POVMs by Orthogonal Measurements
, 2004
"... We consider groupcovariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find an implementation of a given groupcovariant POVM by a ..."
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We consider groupcovariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find an implementation of a given groupcovariant POVM by a quantum circuit using its symmetry. Based on representation theory of the symmetry group we develop a general approach for the implementation of groupcovariant POVMs which consist of rankone operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by WeylHeisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain. 1
Computational Distinguishability between Quantum States: Random Coset States vs. Maximally Mixed States over the Symmetric Groups
, 2004
"... We introduce a new underlying problem for computational cryptographic schemes secure against quantum adversaries. The problem is a distinction problem between quantum states which is a natural generalization of distinction problems between probability distributions, which are commonly used in comput ..."
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We introduce a new underlying problem for computational cryptographic schemes secure against quantum adversaries. The problem is a distinction problem between quantum states which is a natural generalization of distinction problems between probability distributions, which are commonly used in computational cryptography. Specifically speaking, our problem QSCDff is defined as a quantum state computational distinguishability problem between random coset states with a hidden permutation and a maximally mixed state (uniform distribution) over the symmetric group. A similar problem to ours appears in the context of the hidden subgroup problem on the symmetric group in the research of quantum computation and is regarded as a hard problem. In this paper, we show that (i) QSCDff has the trapdoor property; (ii) the averagecase complexity of QSCDff completely coincides with its worstcase complexity; (iii) the computational complexity of QSCDff is lowerbounded by the worstcase hardness of the graph automorphism problem. These properties enable us to construct cryptographic systems. Actually, we show a cryptographic application based on the hardness of QSCDff. Keywords: 1