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Interactive and Noninteractive Zero Knowledge are Equivalent in the Help Model
, 2007
"... We show that interactive and noninteractive zeroknowledge are equivalent in the ‘help model’ of BenOr and Gutfreund (J. Cryptology, 2003). In this model, the shared reference string is generated by a probabilistic polynomialtime dealer who is given access to the statement to be proven. Our result ..."
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Cited by 6 (1 self)
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We show that interactive and noninteractive zeroknowledge are equivalent in the ‘help model’ of BenOr and Gutfreund (J. Cryptology, 2003). In this model, the shared reference string is generated by a probabilistic polynomialtime dealer who is given access to the statement to be proven. Our results do not rely on any unproven complexity assumptions and hold for statistical zero knowledge, for computational zero knowledge restricted to AM, and for quantum zero knowledge when the help is a pure quantum state.
Efficient quantum tensor product expanders and kdesigns
, 2008
"... We give an efficient construction of constantdegree, constantgap quantum ktensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k = O(n / log n), where n is the number of qubits. An immed ..."
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We give an efficient construction of constantdegree, constantgap quantum ktensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k = O(n / log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of approximate unitary kdesigns on n qubits for any k = O(n / log n). 1
Quantum Expanders: Motivation and Constructions
 THEORY OF COMPUTING
, 2009
"... We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Philips and Sarna ..."
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Cited by 3 (0 self)
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We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Philips and Sarnak [29]. The second construction is combinatorial, and is based on a quantum variant of the ZigZag product introduced by Reingold, Vadhan and Wigderson [37]. Both constructions are of constant degree, and the second one is explicit. Using another construction of quantum expanders by Ambainis and Smith [6], we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK–complete (where QSZK is the class of languages having a zeroknowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof follows the classical proof structure that the entropy difference problem is SZK–complete, but crucially depends on the use of quantum expanders.
An explicit construction of quantum expanders
, 2007
"... Quantum expanders are a natural generalization of classical expanders. These objects were introduced and studied by [1, 3, 4]. In this note we show how to construct explicit, constantdegree quantum expanders. The construction is essentially the classical ZigZag expander construction of [5], applie ..."
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Cited by 2 (0 self)
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Quantum expanders are a natural generalization of classical expanders. These objects were introduced and studied by [1, 3, 4]. In this note we show how to construct explicit, constantdegree quantum expanders. The construction is essentially the classical ZigZag expander construction of [5], applied to quantum expanders.
Testing quantum expanders is coQMAcomplete
, 2012
"... A quantum expander is a unital quantum channel that is rapidly mixing, has only a few Kraus operators, and can be implemented efficiently on a quantum computer. We consider the problem of estimating the mixing time (i.e., the spectral gap) of a quantum expander. We show that this problem is coQMAc ..."
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A quantum expander is a unital quantum channel that is rapidly mixing, has only a few Kraus operators, and can be implemented efficiently on a quantum computer. We consider the problem of estimating the mixing time (i.e., the spectral gap) of a quantum expander. We show that this problem is coQMAcomplete. This has applications to testing randomized constructions of quantum expanders, and studying thermalization of open quantum systems. 1
arXiv:0909.5347v2 [quantph] 28 May 2010 1 A quantum version of Wielandt's inequality
"... AbstractIn this paper, Wielandt's inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zer ..."
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AbstractIn this paper, Wielandt's inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zeroerror capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustrationfree Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interactionrange of Hamiltonians with unique MPS ground state.