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84
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.
Grassmannian frames with applications to coding and communication
 Appl. Comp. Harmonic Anal
, 2003
"... For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation 〈fk, fl〉  among all frames {fk}k∈I ∈ F. We first analyze finitedimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal sph ..."
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Cited by 227 (13 self)
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For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation 〈fk, fl〉  among all frames {fk}k∈I ∈ F. We first analyze finitedimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with unit norm tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on unit norm tight frames. We then introduce infinitedimensional Grassmannian frames and analyze their connection to unit norm tight frames for frames which are generated by grouplike unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical enginee ..."
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Cited by 103 (5 self)
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Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems.
Highly sparse representations from dictionaries are unique and independent of the sparseness measure
, 2003
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Designing Structured Tight Frames via an Alternating Projection Method
, 2003
"... Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alterna ..."
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Cited by 87 (10 self)
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Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems, which includes the frame design problem. To apply this method, one only needs to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate
Nonbinary quantum codes
 IEEE Trans. Inform. Theory
, 1999
"... Abstract. We present several results on quantum codes over general alphabets (that is, in which the fundamental units may have more than 2 states). In particular, we consider codes derived from finite symplectic geometry assumed to have additional global symmetries. From this standpoint, the analogu ..."
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Cited by 81 (1 self)
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Abstract. We present several results on quantum codes over general alphabets (that is, in which the fundamental units may have more than 2 states). In particular, we consider codes derived from finite symplectic geometry assumed to have additional global symmetries. From this standpoint, the analogues of CalderbankShorSteane codes and of GF(4)linear codes turn out to be special cases of the same construction. This allows us to construct families of quantum codes from certain codes over number fields; in particular, we get analogues of quadratic residue codes, including a singleerror correcting code encoding one letter in five, for any alphabet size. We also consider the problem of faulttolerant computation through such codes, generalizing ideas of Gottesman.
FaultTolerant Error Correction with Efficient Quantum Codes
, 1996
"... We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum errorcorrecting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can fu ..."
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Cited by 51 (4 self)
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We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum errorcorrecting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced grouptheoretic framework for unifying all known quantum codes.
Painless Reconstruction from Magnitudes of Frame Coefficients
 J FOURIER ANAL APPL (2009) 15: 488–501
, 2009
"... The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present line ..."
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Cited by 49 (10 self)
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The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2designs in finitedimensional real or complex Hilbert spaces. Examples of such frames are twouniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a ddimensional Hilbert space.
Equiangular lines, mutually unbiased bases, and spin models
, 2008
"... We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)difference sets, we obtain k 2 −k+1 equiangular lines in C k when k − 1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,λ) we construct sets of n + 1 mutually unbiased ba ..."
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Cited by 43 (1 self)
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We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)difference sets, we obtain k 2 −k+1 equiangular lines in C k when k − 1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,λ) we construct sets of n + 1 mutually unbiased bases in C k. We show how to construct these difference sets from commutative semifields and that all known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.
A GroupTheoretic Framework for the Construction of Packings in Grassmannian Spaces
, 2002
"... By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this ..."
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Cited by 40 (11 self)
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By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with BarnesWall lattices, Kerdock sets and quantumerrorcorrecting codes.