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99
Faulttolerant quantum computation
 In Proc. 37th FOCS
, 1996
"... It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information i ..."
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Cited by 264 (5 self)
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It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum errorcorrecting codes without decoding this data. 1.
An Introduction to Quantum Computing for NonPhysicists
 Los Alamos Physics Preprint Archive http://xxx.lanl.gov/abs/quantph/9809016
, 2000
"... ..."
Synthesis of Reversible Logic Circuits
, 2003
"... Reversible or informationlossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minim ..."
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Cited by 93 (6 self)
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Reversible or informationlossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant inputoutput linepairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting the distribution of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.
Quantum search of spatial regions
 THEORY OF COMPUTING
, 2005
"... Can Grover’s algorithm speed up search of a physical region—for example a 2D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Beniof ..."
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Cited by 84 (7 self)
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Can Grover’s algorithm speed up search of a physical region—for example a 2D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a ddimensional hypercube in time O ( √ n) for d ≥ 3, or O ( √ nlog 5/2 n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almosttight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of ‘locality’ for unitary matrices acting on graphs. As an application of our results, we give an O (√ n)qubit communication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov.
The quantum query complexity of approximating the median and related statistics
 STOC'99
, 1999
"... Let X = (z,, , z,,) be a sequence of n numbers. For 6> 0, we say that 5; is an eapproximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):. We consider the quantum query complexity of computing an capprox ..."
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Cited by 74 (1 self)
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Let X = (z,, , z,,) be a sequence of n numbers. For 6> 0, we say that 5; is an eapproximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):. We consider the quantum query complexity of computing an capproximate median, given the sequence X as an oracle. We prove a lower bound of n(min{t,n}) queries for any quantum algorithm that computes an rapproximate median with any constant probability greater than l/2. We also show how an capproximate median may be computed with 0 ( $ log(t) log log ( $)) oracle queries, which rep resents an improvement over an earlier algorithm due to Grover [ll, 121. Thus, the lower bound we obtain is essentially optimal. The upper and the lower bound both hold in the comparison tree model as well. Our lower bound result is an application of the polynomial paradigm recently introduced to quantum complexity theory by Be & et ol. [l]. The main ingredient in the proof is a polynomial degree lower bound far real multilinear polynomials that “approximate” symmetric partial boolean functions. The degree bound extends a result of Patti [15] and also immediately yields lower bounds for the problems of approximating the kthsmallest element, approximating the mean of a sequence of numbers, and approximately counting the number of ones of a boolean function. All bounds obtained come within a polylogarithmic factor of the optimal (as we show by presenting algorithms where no such optimal or near optimal algorithms were known), thus demonstrating the power of the polynomial method.
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 56 (8 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.
Reversible Logic Circuit Synthesis
 In International Conference on Computer Aided Design
, 2002
"... Reversible, or informationlossless, circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement for quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates ..."
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Cited by 42 (2 self)
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Reversible, or informationlossless, circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement for quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant inputoutput linepairs (temporary storage channels). We propose new constructions for reversible circuits composed of NOT, ControlledNOT, and TOFFOLI gates (the CNT gate library) based on permutation theory. A new algorithm is given to synthesize optimal reversible circuits using an arbitrary gate library, and we describe much faster heuristic algorithms. We also pursue applications of the proposed techniques to the synthesis of quantum circuits.
Introduction to Grassmann Manifolds and Quantum
 Computation, J. Applied Math
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Quantum complexity of integration
 J. COMPLEXITY
, 2001
"... It is known that quantum computers yield a speedup for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on [0, 1] d and define γ by γ = (k + ..."
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Cited by 37 (4 self)
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It is known that quantum computers yield a speedup for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on [0, 1] d and define γ by γ = (k + α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are and comp(F k,α
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 29 (15 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