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41
Quantum Walks on the Hypercube
 In Proc. of RANDOM 02
, 2002
"... Recently, it has been shown that onedimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the ndimensional hypercube, one in discrete time and one in cont ..."
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Cited by 44 (0 self)
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Recently, it has been shown that onedimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the ndimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the instantaneous mixing time is (p=4)n steps, faster than the Q(nlogn) steps required by the classical walk. In the continuoustime case, the probability distribution is exactly uniform at this time. On the other hand, we show that the average mixing time as defined by Aharonov et al. [AAKV01] is W(n 3=2 ) in the discretetime case, slower than the classical walk, and nonexistent in the continuoustime case. This suggests that the instantaneous mixing time is a more relevant notion than the average mixing time for quantum walks on large, wellconnected graphs. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds predict an exponential average mixing time when applied to the hypercube. 1
Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 42 (7 self)
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
Coins make quantum walks faster
 Proceedings of SODA’05. Also quantph/0402107
"... We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently ..."
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Cited by 31 (5 self)
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We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O ( √ N) and give several extensions of quantum walk search algorithms for general graphs. The coinflip operation needs to be chosen judiciously: we show that another “natural ” choice of coin gives a walk that takes Ω(N) steps. We also show that in 2 dimensions it is sufficient to have a twodimensional coinspace to achieve the time O ( √ N log N). 1
Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 30 (2 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1
Search via quantum walk
 LOGIC PROGRAMMING, PROC. OF THE 1994 INT. SYMP
, 2007
"... We propose a new method for designing quantum search algorithms for finding a “marked ” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimat ..."
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Cited by 25 (7 self)
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We propose a new method for designing quantum search algorithms for finding a “marked ” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [6] and Szegedy [24]. Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.
A hypercontractive inequality for matrixvalued functions with applications to quantum computing and LDCs
"... The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and ..."
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Cited by 21 (4 self)
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The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m<0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak’s quantum random access code bound. It in turn implies strong direct product theorems for the oneway quantum communication complexity of Disjointness and other problems. Second, we prove that errorcorrecting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first “nonquantum” proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.
A lower bound for the bounded round quantum communication complexity of set disjointness
"... We show lower bounds in the multiparty quantum communication complexity model. In this model, there are t parties where the ith party has input Xi ⊆ [n]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combin ..."
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Cited by 14 (5 self)
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We show lower bounds in the multiparty quantum communication complexity model. In this model, there are t parties where the ith party has input Xi ⊆ [n]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combined input (X1,...,Xt). We consider the class of Boolean valued functions whose value depends only on X1 ∩...∩ Xt; that is, for each F in this class there is an fF : 2[n] → {0,1}, such that F(X1,...,Xt) = fF(X1 ∩...∩ Xt). We show that the tparty kround communication complexity of F is Ω(sm(fF)/(k2)), where sm(fF) stands for the monotone sensitivity of fF' and is defined by sm(fF) = ▵ maxS⊆[n] {i : fF(S ∪ {i}) ≠ fF(S)}. For twoparty quantum communication protocols for the set disjointness problem, this implies that the two parties must exchange Ω(n/k2) qubits. An upper bound of O(n/k) can be derived from the O(√n) upper bound due to S. Aaronson and A. Ambainis (2003). For k = 1, our lower bound matches the Ω(n) lower bound observed by H. Buhrman and R. de Wolf (2001) (based on a result of A. Nayak (1999)), and for 2 ≤ k ≪ n14 /, improves the lower bound of Ω(√n) shown by A. Razborov (2002). For protocols with no restrictions on the number of rounds, we can conclude that the two parties must exchange Ω(n13/) qubits. This, however, falls short of the optimal Ω (√n) lower bound shown by A. Razborov (2002). Our result is obtained by adapting to the quantum setting the elegant informationtheoretic arguments of Z. BarYossef et al. (2002). Using this method we can show similar lower bounds for the L∞ function considered in Z. BarYossef et al. (2002).
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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Cited by 13 (1 self)
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
Quantum information processing in continuous time
, 2004
"... Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explo ..."
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Cited by 12 (2 self)
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Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explore two approaches to designing quantum algorithms based on continuoustime Hamiltonian dynamics. In quantum computation by adiabatic evolution, the computer is prepared in the known ground state of a simple Hamiltonian, which is slowly modified so that its ground state encodes the solution to a problem. We argue that this approach should be inherently robust against lowtemperature thermal noise and certain control errors, and we support this claim using simulations. We then show that any adiabatic algorithm can be implemented in a different way, using only a sequence of measurements of the Hamiltonian. We illustrate how this approach can achieve quadratic speedup for the unstructured search problem. We also demonstrate two examples of quantum speedup by quantum walk, a quantum mechanical analog of random walk. First, we consider the problem of searching a region
Quantum communication complexity
 Foundations of Physics
"... Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can ..."
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Cited by 12 (6 self)
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Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the noncommunicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. KEY WORDS: Bell’s theorem; communication complexity; distributed computation; entanglement simulation; pseudotelepathy; spooky communication.