## A quantum time-space lower bound for the counting hierarchy (2007)

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Citations: | 5 - 1 self |

### BibTeX

@TECHREPORT{Melkebeek07aquantum,

author = {Dieter Melkebeek and Thomas Watson},

title = {A quantum time-space lower bound for the counting hierarchy},

institution = {},

year = {2007}

}

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### Abstract

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real ǫ there exists a real c> 1 such that either: • MajMajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n c, or • MajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n d and space n 1−ǫ. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time n 1+o(1) and space n 1−ǫ for any ǫ> 0. The key technical novelty is a time- and space-efficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation. 1

### Citations

1372 |
Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ...sion of the model in Section 2, we derive our results in Section 3, and we conclude with some open problems in Section 4. Throughout we assume basic background in quantum computation; see for example =-=[14, 12]-=-. 2 Models of Quantum Computation In this section we develop the model that we use for the exposition of our arguments. Section 2.1 contains a discussion of the issues that arise in choosing a model o... |

878 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...ter science and physics communities and allow us to accurately measure the resources of time and space. For 3sexample, the model should allow us to express important quantum algorithms such as Shor’s =-=[15]-=- and Grover’s [9] in a way that is natural and faithfully represents their complexities. This forms the overarching issue in choosing a model. Below we discuss eight specific aspects of quantum comput... |

849 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...ysics communities and allow us to accurately measure the resources of time and space. For 3sexample, the model should allow us to express important quantum algorithms such as Shor’s [15] and Grover’s =-=[9]-=- in a way that is natural and faithfully represents their complexities. This forms the overarching issue in choosing a model. Below we discuss eight specific aspects of quantum computation models and ... |

482 | Quantum complexity theory
- Bernstein, Vazirani
- 1997
(Show Context)
Citation Context ... with random access to the input and memory. This makes the lower bounds more meaningful as they do not exploit artifacts due to sequential access. Extending the standard quantum Turing machine model =-=[3]-=- to accommodate random access leads to complications that make the model inconvenient to work with. In Section 2 we discuss these and other issues in detail, and we survey the known models from the li... |

155 |
Quantum computations: algorithms and error correction
- Kitaev
- 1997
(Show Context)
Citation Context ...end on previous measurement outcomes; i.e., we handle more powerful models than uniform quantum circuits. Our simulation makes use of a result on approximating quantum gates due to Solovay and Kitaev =-=[11]-=-. Theorem 1 follows from our simulation and the Allender et al. lower bound. The quantitative strength of our lower bound derives from the latter; our translation does not induce any further weakening... |

109 | Quantum computability
- Adleman, Demarrais, et al.
- 1997
(Show Context)
Citation Context ...e first level. In the quantum setting the situation is worse: we know of no efficient simulations in any level of the polynomial-time hierarchy. The best simulations to date are due to Adleman et al. =-=[1]-=-, who showed that polynomial-time quantum computations with bounded two-sided error can be simulated in PP. Building on this connection, we bring the lower bounds of Allender et al. to bear on bounded... |

109 | Topological quantum computation
- Freedman, Kitaev, et al.
- 2000
(Show Context)
Citation Context ...sion of the model in Section 2, we derive our results in Section 3, and we conclude with some open problems in Section 4. Throughout we assume basic background in quantum computation; see for example =-=[14, 12]-=-. 2 Models of Quantum Computation In this section we develop the model that we use for the exposition of our arguments. Section 2.1 contains a discussion of the issues that arise in choosing a model o... |

94 | Complexity limitations on quantum computation
- FORTNOW, ROGERS
- 1999
(Show Context)
Citation Context ...iently with unbounded-error probabilistic algorithms. Our strategy builds on known simulations of quantum computations without intermediate measurements by probabilistic machines with unbounded error =-=[1, 8]-=-. The basic idea of these simulations is to write the final amplitude of a basis state as a simple linear combination of #P functions, where each #P function counts the number of quantum computation p... |

31 |
PP is as hard as the polynomial time hierarchy
- Toda
- 1991
(Show Context)
Citation Context ...a given Boolean formula ϕ on disjoint variable sets x and y has the property that for at least half of the assignments to x, ϕ is satisfied for at least half of the assignments to y. Recall that Toda =-=[16]-=- proved that the polynomial-time hierarchy reduces to the class PP, which represents polynomialtime probabilistic computations with unbounded two-sided error and forms the first level of the counting ... |

29 | Time-space tradeoffs for satisfiability
- Fortnow
(Show Context)
Citation Context ...ar for time and logarithmic for space. However, since the late 1990’s we have seen a number of results that rule out certain nontrivial combinations of time and space complexity. One line of research =-=[6, 7, 19, 5, 20]-=-, initiated by Fortnow, focuses on proving stronger and stronger time lower bounds for deterministic algorithms that solve satisfiability in small space. For subpolynomial (i.e., n o(1) ) space bounds... |

25 | Time-space lower bounds for satisfiability
- Fortnow, Lipton, et al.
(Show Context)
Citation Context ...ar for time and logarithmic for space. However, since the late 1990’s we have seen a number of results that rule out certain nontrivial combinations of time and space complexity. One line of research =-=[6, 7, 19, 5, 20]-=-, initiated by Fortnow, focuses on proving stronger and stronger time lower bounds for deterministic algorithms that solve satisfiability in small space. For subpolynomial (i.e., n o(1) ) space bounds... |

21 | Space-bounded quantum complexity
- Watrous
- 1999
(Show Context)
Citation Context ...hat the machine always halts, meaning that it reaches a superposition in which all nonhalting basis configurations have zero amplitude. Bernstein and Vazirani detail how to design such mechanisms. In =-=[18]-=-, Watrous considers a model similar to Bernstein and Vazirani’s, but with one readwrite work tape and a read-only input tape not counting toward the space bound. The model naturally allows for subline... |

19 |
Division is in uniform TC 0
- Hesse
- 2001
(Show Context)
Citation Context ...er et al. does yield this; however, the constant c is very close to 1, and determining it would require a complicated analysis involving constant-depth threshold circuitry for iterated multiplication =-=[10]-=-. Perhaps there is a way to remove the need for this circuitry in the quantum setting. A major goal is to prove quantum time-space lower bounds for problems that are simpler than MajMajSAT. Ideally we... |

18 | Time-space tradeoffs in the counting hierarchy
- ALLENDER, KOUCK´Y, et al.
- 2001
(Show Context)
Citation Context ...ltime hierarchy; Σ1SAT corresponds to satisfiability. Proving nontrivial time-space lower bounds for satisfiability on randomized algorithms with bounded two-sided error remains open. Allender et al. =-=[2]-=- considered the even more powerful (but physically unrealistic) model of probabilistic algorithms with unbounded error 1 . They settled for problems that are even harder than ΣℓSAT for any fixed ℓ, na... |

18 |
NIELSEN: The Solovay-Kitaev algorithm
- DAWSON, A
(Show Context)
Citation Context ...U and the gates in S are computable in time poly(k). The proof by Solovay and Kitaev gives an algorithm for computing an approximation in time polylog(1/ǫ), ignoring the complexity of arithmetic (see =-=[4]-=- and Section 8.3 of [12]). We cannot do exact arithmetic since the entries of our gates may require infinitely many bits to specify, but in each step of the algorithm it suffices to work with poly(ǫ)-... |

16 | Time-space lower bounds for the polynomial-time hierarchy on randomized machines
- Diehl, Melkebeek
(Show Context)
Citation Context ...ar for time and logarithmic for space. However, since the late 1990’s we have seen a number of results that rule out certain nontrivial combinations of time and space complexity. One line of research =-=[6, 7, 19, 5, 20]-=-, initiated by Fortnow, focuses on proving stronger and stronger time lower bounds for deterministic algorithms that solve satisfiability in small space. For subpolynomial (i.e., n o(1) ) space bounds... |

13 | Inductive time-space lower bounds for SAT and related problems
- Williams
(Show Context)
Citation Context |

12 |
Melkebeek. A survey of lower bounds for satisfiability and related problems
- van
(Show Context)
Citation Context ...ds by Allender et al. is also somewhat weaker. In particular, they showed that no probabilistic algorithm can solve MajMajSAT in time n 1+o(1) and space n 1−ǫ for any positive constant ǫ. We refer to =-=[13]-=- for a detailed survey of the past work on time-space lower bounds for satisfiability and related problems, including a presentation of the Allender et al. lower bound that is slightly different from ... |

12 | On the complexity of simulating space-bounded quantum computations
- Watrous
(Show Context)
Citation Context ... to the quantum computation path. Since this behavior does not appear to be physically realizable in the foreseeable future anyway, the complications arising from it are in some sense unjustified. In =-=[17]-=-, Watrous considers a different model of space-bounded quantum computation. This model is essentially a classical Turing machine with an additional quantum work tape and a fixedsize quantum register. ... |

12 | Time-Space Tradeoffs for Counting NP Solutions Modulo Integers
- Williams
- 2007
(Show Context)
Citation Context |