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29
Extracting randomness from samplable distributions
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, ..."
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Cited by 55 (8 self)
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The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, we consider the problem of deterministically converting a weak source of randomness into an almost uniform distribution. Previously, deterministic extraction procedures were known only for sources satisfying strong independence requirements. In this paper, we look at sources which are samplable, i.e. can be generated by an efficient sampling algorithm. We seek an efficient deterministic procedure that, given a sample from any samplable distribution of sufficiently large minentropy, gives an almost uniformly distributed output. We explore the conditions under which such deterministic extractors exist. We observe that no deterministic extractor exists if the sampler is allowed to use more computational resources than the extractor. On the other hand, if the extractor is allowed (polynomially) more resources than the sampler, we show that deterministic extraction becomes possible. This is true unconditionally in the nonuniform setting (i.e., when the extractor can be computed by a small circuit), and (necessarily) relies on complexity assumptions in the uniform setting. One of our uniform constructions is as follows: assuming that there are problems in���ÌÁÅ�ÇÒthat are not solvable by subexponentialsize circuits with¦� gates, there is an efficient extractor that transforms any samplable distribution of lengthÒand minentropy Ò into an output distribution of length ÇÒ, whereis any sufficiently small constant. The running time of the extractor is polynomial inÒand the circuit complexity of the sampler. These extractors are based on a connection be
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
SpaceBounded Quantum Complexity
 Journal of Computer and System Sciences
, 1999
"... This paper investigates the computational power of spacebounded quantum Turing machines. The following facts are proved for spaceconstructible space bounds s satisfying s(n) = Ω(log n). 1. Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Tur ..."
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Cited by 22 (5 self)
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This paper investigates the computational power of spacebounded quantum Turing machines. The following facts are proved for spaceconstructible space bounds s satisfying s(n) = Ω(log n). 1. Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Turing machine (PTM) running in space O(s). No assumptions on the probability of error or running time for the QTM are required, although it is assumed that all transition amplitudes of the QTM are rational. 2. Any PTM that runs in space s and halts absolutely (i.e., has finite worstcase running time) can be simulated by a QTM running in space O(s). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the case of unbounded error, the QTM may be taken to halt absolutely. We therefore have that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power, and furthermore that any QTM running in space s can be simulated deterministically in NC 2 (2 s) ⊆ DSPACE(s 2) ∩ DTIME ( 2 O(s)). We also consider quantum analogues of nondeterministic and onesided error probabilistic spacebounded classes, and prove some simple facts regarding these classes. 1 1
Quantum Circuits: Fanout, Parity, and Counting
 In Los Alamos Preprint archives
, 1999
"... Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constantdepth circuits with AND and OR gates of arbitrary fanin, and QACC 0 [q], where nary MODq gates are also allowed. We show that it is possible to make a ‘cat ’ state on n qubits in constant depth if ..."
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Cited by 17 (1 self)
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Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constantdepth circuits with AND and OR gates of arbitrary fanin, and QACC 0 [q], where nary MODq gates are also allowed. We show that it is possible to make a ‘cat ’ state on n qubits in constant depth if and only if we can construct a parity or MOD2 gate in constant depth; therefore, any circuit class that can fan out a qubit to n copies in constant depth also includes QACC 0 [2]. In addition, we prove the somewhat surprising result that parity or fanout allows us to construct MODq gates in constant depth for any q, so QACC 0 [2] = QACC 0. Since ACC 0 [p] ̸ = ACC 0 [q] whenever p and q are mutually prime, QACC 0 [2] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. 1
Computational analogues of entropy
 In 11th International Conference on Random Structures and Algorithms
, 2003
"... Abstract. Minentropy is a statistical measure of the amount of randomness that a particular distribution contains. In this paper we investigate the notion of computational minentropy which is the computational analog of statistical minentropy. We consider three possible definitions for this notio ..."
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Cited by 16 (1 self)
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Abstract. Minentropy is a statistical measure of the amount of randomness that a particular distribution contains. In this paper we investigate the notion of computational minentropy which is the computational analog of statistical minentropy. We consider three possible definitions for this notion, and show equivalence and separation results for these definitions in various computational models. We also study whether or not certain properties of statistical minentropy have a computational analog. In particular, we consider the following questions: 1. Let X be a distribution with high computational minentropy. Does one get a pseudorandom distribution when applying a “randomness extractor ” on X? 2. Let X and Y be (possibly dependent) random variables. Is the computational minentropy of (X, Y) at least as large as the computational minentropy of X? 3. Let X be a distribution over {0, 1} n that is “weakly unpredictable” in the sense that it is hard to predict a constant fraction of the coordinates of X with a constant bias. Does X have computational minentropy Ω(n)? We show that the answers to these questions depend on the computational model considered. In some natural models the answer is false and in others the answer is true. Our positive results for the third question exhibit models in which the “hybrid argument bottleneck ” in “moving from a distinguisher to a predictor ” can be avoided. 1
Counting, Fanout, And The Complexity Of Quantum Acc
, 2002
"... q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upp ..."
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Cited by 15 (2 self)
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q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We dene classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomialsize circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth. Keywords: quantum computation, quantum & circuit complexity, threshold circuit Communicated by : R Cleve & J Watrous 1. Introduction Advances in quantum computation
On the Complexity of Simulating SpaceBounded Quantum Computations
 Computational Complexity
, 2003
"... This paper studies the spacecomplexity of predicting the longterm behavior of a class of stochastic processes based on evolutions and measurements of quantum mechanical systems. These processes generalize a wide range of both quantum and classical spacebounded computations, including unbounded er ..."
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Cited by 12 (0 self)
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This paper studies the spacecomplexity of predicting the longterm behavior of a class of stochastic processes based on evolutions and measurements of quantum mechanical systems. These processes generalize a wide range of both quantum and classical spacebounded computations, including unbounded error computations given by machines having algebraic number transition amplitudes or probabilities. It is proved that any space s quantum stochastic process from this class can be simulated probabilistically with unbounded error in space O(s), and therefore deterministically in space O(s 2). 1
Relationships Between Quantum and Classical SpaceBounded Complexity Classes
 In Proceedings of the 13th IEEE Conference on Computational Complexity
, 1998
"... This paper investigates the relative power of spacebounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number o ..."
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Cited by 11 (1 self)
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This paper investigates the relative power of spacebounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number of steps) can be simulated in space O(s) by a quantum Turing machine (QTM). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the unbounded error case, the QTM may be taken to halt absolutely. 2. Any QTM running in space s can be simulated by an unbounded error PTM running in space O(s). No assumptions on the probability of error or running time for the QTM are required, but it is assumed that all transition amplitudes of the quantum machine are rational. It follows that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in...
RandomnessEfficient Sampling within NC1
 In Proc. 10th Intern. Workshop on Randomization and Computation (RANDOM
, 2006
"... We construct a randomnessefficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC 0 [⊕]). Our sampler matches the parameters achieved by random walks on constantdegree expander graphs, allowing us to apply a variety expanderbased tec ..."
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Cited by 9 (0 self)
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We construct a randomnessefficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC 0 [⊕]). Our sampler matches the parameters achieved by random walks on constantdegree expander graphs, allowing us to apply a variety expanderbased techniques within NC 1. For example, we obtain the following results: