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Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates
 Quantum Information and Computation
, 2013
"... In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z [ 1√ 2, i], in the singlequbit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Had ..."
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Cited by 19 (4 self)
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In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z [ 1√ 2, i], in the singlequbit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard and T gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring Z [ 1√ 2, i] holds in the nqubit case. 1
A family of norms with applications in quantum information theory
"... Abstract. We consider the problem of computing the family of operator norms recently introduced in [1]. We develop a family of semidefinite programs that can be used to exactly compute them in small dimensions and bound them in general. Some theoretical consequences follow from the duality theory o ..."
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Cited by 9 (5 self)
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Abstract. We consider the problem of computing the family of operator norms recently introduced in [1]. We develop a family of semidefinite programs that can be used to exactly compute them in small dimensions and bound them in general. Some theoretical consequences follow from the duality theory of semidefinite programming, including a new constructive proof that there are nonpositive partial transpose Werner states that are rundistillable for arbitrary r. Several examples are considered via a MATLAB implementation of the semidefinite program, including the case of Werner states and randomly generated states via the Bures measure, and approximate distributions of the norms are provided. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states. 1.
Parallelized Solution to Semidefinite Programmings in Quantum Complexity Theory
, 2010
"... In this paper we present an equilibrium value based framework for solving SDPs via the multiplicative weight update method which is different from the one in Kale’s thesis [Kal07]. One of the main advantages of the new framework is that we can guarantee the convertibility from approximate to exact ..."
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Cited by 4 (4 self)
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In this paper we present an equilibrium value based framework for solving SDPs via the multiplicative weight update method which is different from the one in Kale’s thesis [Kal07]. One of the main advantages of the new framework is that we can guarantee the convertibility from approximate to exact feasibility in a much more general class of SDPs than previous result. Another advantage is the design of the oracle which is necessary for applying the multiplicative weight update method is much simplified in general cases. This leads to an alternative and easier solutions to the SDPs used in the previous results QIP(2)⊆PSPACE [JUW09] and QMAM=PSPACE [JJUW09]. Furthermore, we provide a generic form of SDPs which can be solved in the similar way. By parallelizing every step in our solution, we are able to solve a class of SDPs in NC. Although our motivation is from quantum computing, our result will also apply directly to any SDP which satisfies our conditions. In addition to the new framework for solving SDPs, we also provide a novel framework which improves the range of equilibrium value problems that can be solved via the multiplicative weight update method. Before this work we are only able to calculate the equilibrium value where one of the two convex sets needs to be the set of density operators. Our work demonstrates that in the case when one set is the set of density operators with further linear constraints, we are still able to approximate the equilibrium value to high precision via the multiplicative weight update method.
Computational Distinguishability of Quantum Channels
, 2009
"... The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that h ..."
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The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixedunitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixedunitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.