On probabilistic time and space

Jung, Hermann

doi:10.1007/bfb0015756

Cited by 23 publications

(13 citation statements)

References 5 publications

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“…It follows from these simulations that unbounded error, space-bounded PTMs and QTMs are equivalent in power. Furthermore, we have that unbounded error, space-bounded QTMs do not lose power if required to halt absolutely; a result analogous to one proved by Jung [16] for the probabilistic case (see also [1]). …”

Section: Introductionsupporting

confidence: 64%

“…Furthermore, if a given well-behaved probabilistic Turing machine runs in space O(s), the function t in item 2 is computable in space O(s) as well. Finally, note that the classes PrSPACE(s), BP H SPACE(s), and co-C = SPACE(s) remain unchanged if the underlying machine is required to be well-behaved (following from [1,16] in the case of PrSPACE(s)).…”

Section: Quantum Simulations Of Ptmsmentioning

confidence: 99%

“…Furthermore, we may assume that each of these machines, save M (4) , M (6) , M (8) , M (14) , M (16) , and M (18) , performs its respective transformation in a number of steps independent of its argument.…”

Section: ) Xmentioning

confidence: 99%

“…It is proved that any unbounded error QTM running in space s, for s(n)=0(log n) space-constructible, can be simulated by an unbounded error PTM running in space O(s). Our proof of this fact is based on a technique previously used in the probabilistic case (e.g., in [1,16]); the problem of determining if a given quantum machine accepts with probability exceeding 1Â2 is reduced to the problem of comparing determinants of integer matrices. From this fact, we conclude that any QTM running in space s, even in the case of unbounded error and with no restrictions on running time, can be simulated deterministically in NC 2 (2 s ) DSPACE(s 2 ) & DTIME (2 O(s) ) by a result of Borodin, Cook, and Pippenger [7] (see below).…”

Section: Introductionmentioning

confidence: 99%

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Space-Bounded Quantum Complexity

Watrous

1999

Journal of Computer and System Sciences

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This paper investigates the computational power of space-bounded quantum Turing machines. The following facts are proved for space-constructible space bounds s satisfying s(n)=0(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 worst-case 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 one-sided error probabilistic space-bounded classes and prove some simple facts regarding these classes. Academic Press

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Section: Introductionsupporting

confidence: 64%

Section: Quantum Simulations Of Ptmsmentioning

confidence: 99%

Section: ) Xmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Space-Bounded Quantum Complexity

Watrous

1999

Journal of Computer and System Sciences

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“…Savitch's Theorem (Savitch 1970) gives the following relation between deterministic and nondeterministic space-bounded classes: NSPACE(s) ⊆ DSPACE(s 2 ); a quadratic increase in space compensates for any advantage of nondeterministic computation over deterministic computation. A similar relation holds if nondeterministic computation is replaced with probabilistic computation, even if the probabilistic computation has unbounded error and no restrictions on running time (Borodin et al 1983;Jung 1985). Specifically, PrSPACE(s) ⊆ DSPACE(s 2 ).…”

Section: Introductionmentioning

confidence: 89%

On the complexity of simulating space-bounded quantum computations

Watrous

2003

Computational Complexity

104

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This paper studies the space-complexity of predicting the long-term 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 space-bounded 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 ).

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Circuits over PP and PL

Beigel

Fu²

2000

Journal of Computer and System Sciences

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model of oracle gates provides a framework for considering reductions whose strength is intermediate between truth-table and Turing. Improving on a stream of previously published results, we prove that PL and PP are closed under NC 1 reductions. This answers an open problem of Ogihara (1996,``Proc. 37th Ann. IEEE Symp. Found. Computer Sci.''). More generally, we show that NC PP k+1 = AC PP k and NC PL k+1 =AC PL k for all k 0. On the other hand, we construct an oracle A such that NC PP A k {NC PP A k+1 for all integers k 1. Slightly weaker than NC 1 reductions are Boolean formula reductions. We ask whether PL and PP are closed under Boolean formula reductions. This is a nontrivial question despite NC 1 =BF, because that equality is easily seen not to relativize. We prove that P PP log 2 nÂlog log n&T BF PP PrTIME(n O(log n)). Because P PP log 2 nÂlog log n&T 3 PP relative to an oracle, we think it is unlikely that PP is closed under Boolean formula reductions. We also show that PL is unlikely to be closed under BF reductions.

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On probabilistic time and space

Cited by 23 publications

References 5 publications

Space-Bounded Quantum Complexity

Space-Bounded Quantum Complexity

On the complexity of simulating space-bounded quantum computations

Circuits over PP and PL

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