## Quantum branching programs and space-bounded nonuniform quantum complexity (2005)

Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Sauerhoff05quantumbranching,

author = {Martin Sauerhoff and Detlef Sieling},

title = {Quantum branching programs and space-bounded nonuniform quantum complexity},

journal = {THEORETICAL COMPUTER SCIENCE},

year = {2005},

pages = {177--225}

}

### OpenURL

### Abstract

In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs (QBPs), which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between QBPs and nonuniform quantum Turing machines are presented, which allow to transfer lower and upper bound results between the two models. Using additional insights about the connection between running time and the precision of amplitudes, it is shown that nonuniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Quantum ordered binary decision diagrams (QOBDDs) are a restricted variant of QBPs, which can be considered as nonuniform analog of one-way quantum finite automata. In the second part of the paper, lower and upper bound results for QOBDDs are presented in order to compare variants of QOBDDs with their deterministic and randomized counterparts. In the third part QBPs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.

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Citation Context ...w by a 1-edge and v and w are labeled by the same variable. Reversible BPs are obviously special QBPs. Furthermore, as proved by ˇSpalek [37] using a similar construction of Lange, McKenzie, and Tapp =-=[24]-=- for Turing machines, any (possibly non-reversible) BP of size s(n) = Ω(n) can be efficiently simulated by a reversible one of size poly(s(n)). This implies: Proposition 2.7 ([37]): If the function f ... |

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Citation Context ...ctly one entry 1. It is well-known that PERM = (PERMn)n∈ does not have polynomial size read-once branching programs (Krause, Meinel and Waack [22]) and, therefore, no polynomial size OBDDs either. In =-=[32]-=- (see also [41]), a polynomial size randomized OBDD with one-sided error for PERM has been designed using the so-called fingerprinting technique. We show here how this construction can be modified to ... |

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Citation Context ...ontains labels of the form iw,iv p(α)/m, the same holds for the entries of M. Hence, part (ii) of Lemma 4.14 yields the claimed result. □ Proof of Lemma 4.9. The main idea is similar to that of Simon =-=[36]-=- for limiting the running time of probabilistic Turing machines. We simulate G step-by-step. Before each simulation step, we stop and reject the input with fixed, small probability. This realizes a “p... |

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Citation Context ...atrices, c some positive constant, and the representation can even be obtained by an algorithm with running time polynomial in d and log(1/ε). We use a similar result due to Harrow, Recht, and Chuang =-=[15]-=- that does not provide an efficient algorithm but allows a representation using even fewer matrices. Define the unitary matrices ( √ 1 2 −1 2 √ −1 1 V1 = 1 √ 5 ) , V2 = 1 √ 5 ( ) 1 2 , and V3 = −2 1 1... |

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Citation Context ...eterministic π-OBDD representing INDn requires size 2 n , while the same function can be computed by zero error π-QOBDDs with failure probability ε of size 2 (1−ε)n+O(log n) . Hromkovič and Schnitger =-=[18]-=- have used a similar function to prove an analogous result for classical Las Vegas and deterministic one-way communication complexity and the special case of failure probability ε = 1/2. Proof. The lo... |

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Citation Context ...bles. This yields a randomized OBDD G ′ for Rϑ,k,b,n that is at most as large as G. We prove the desired lower bound for G ′ using the standard lower bound technique for randomized OBDDs (see, e. g., =-=[34]-=-). Observe that the variable order π ′ consists of three parts belonging to the different matrices A, B, C in some arbitrary order. Let C1 be the set of nodes which are reached by all paths on which e... |

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Citation Context ...s of branching programs have been proposed and used for studying space-bounded nonuniform quantum complexity. First, Ablayev, Gainutdinova, and Karpinski [1] and Nakanishi, Hamaguchi, and Kashiwabara =-=[25]-=- have introduced quantum variants of OBDDs (ordered binary decision diagrams) which are branching programs where the input variables may only be read once in a fixed order during each computation. Abl... |

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Citation Context ...plexity. Recently, a couple of quantum variants of branching programs have been proposed and used for studying space-bounded nonuniform quantum complexity. First, Ablayev, Gainutdinova, and Karpinski =-=[1]-=- and Nakanishi, Hamaguchi, and Kashiwabara [25] have introduced quantum variants of OBDDs (ordered binary decision diagrams) which are branching programs where the input variables may only be read onc... |

8 |
Separating the eraser Turing machine classes Le, NLe, co-NLe and Pe
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Citation Context ...akes the value 1 iff each row and each column contains exactly one entry 1. It is well-known that PERM = (PERMn)n∈ does not have polynomial size read-once branching programs (Krause, Meinel and Waack =-=[22]-=-) and, therefore, no polynomial size OBDDs either. In [32] (see also [41]), a polynomial size randomized OBDD with one-sided error for PERM has been designed using the so-called fingerprinting techniq... |

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Citation Context ... bounded error, and unbounded error are defined as usual. It can be shown that cycles in randomized branching programs with unbounded error can be removed at the cost of a polynomial increase in size =-=[33]-=-. Randomized variants of the restricted models of BPs from Definition 2.2 are obtained by applying the respective restriction to the nodes labeled by variables on each computation path. Next, we defin... |

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(Show Context)
Citation Context ...garithmic width for quantum OBDDs. Nakanishi, Hamaguchi, and Kashiwabara have obtained a similar gap, but their lower bound even holds for randomized OBDDs. More recently, Ablayev, Moore, and Pollett =-=[2]-=- have proved that the class of functions that can be exactly computed by oblivious width-2 quantum branching programs of polynomial size coincides with the class NC 1 . On the other hand, they have pr... |

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3 |
Space Complexity of Quantum Computation
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(Show Context)
Citation Context ...ful application of the ideas of this simulation also yields that width-2 randomized branching program with small bounded error for the majority function require superpolynomial size. Finally, ˇSpalek =-=[37]-=- has studied a general model 2of quantum branching programs and has independently come up with a definition similar to that used here. Furthermore, he has also presented exact simulations between qua... |

3 | On quantum and classical space-bounded processes with algebraic transition amplitudes - Watrous - 1999 |

1 |
On quantum and classical space-bounded processes with algebraic transition amplitudes
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(Show Context)
Citation Context ...ctically motivated to work with bounded running time, but it is not clear what kind of bounds can be chosen without restricting the computational power of the space-bounded models considered here. In =-=[38]-=- and implicitly also in [40], Watrous has investigated this question for unidirectional uniform QTMs and has obtained answers analogous to the situation for probabilistic TMs. He has shown that unidir... |