## Quantum computation tree logic – model checking and complete calculus

### Cached

### Download Links

Venue: | International Journal of Quantum Information |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Baltazar_quantumcomputation,

author = {P. Baltazar and R. Chadha and P. Mateus},

title = {Quantum computation tree logic – model checking and complete calculus},

journal = {International Journal of Quantum Information},

year = {},

pages = {2008}

}

### OpenURL

### Abstract

Logics for reasoning about quantum states and their evolution have been given in the literature. In this paper we consider Quantum Computation Tree Logic (QCTL), which adds temporal modalities to exogenous quantum propositional logic. We give a sound and complete axiomatization of QCTL and combine the standard CTL model-checking algorithm with the dEQPL model-checking algorithm to obtain a model-checking algorithm for QCTL. Finally we illustrate the use of the logic by reasoning about the BB84 key distribution protocol.

### Citations

844 | Design and synthesis of synchronization skeletons using branchingtime temporal logic - Clarke, Emerson - 1982 |

646 |
Quantum Cryptography: Public-key Distribution and Coin Tossing
- Bennett, Brassard
- 1984
(Show Context)
Citation Context ...putations to be O(1) by using floating point representation for the real numbers. ⋄ 4 Example: BB84 Protocol In this section we reason about a simplified version of the BB84 key distribution protocol =-=[6]-=- to illustrate the power of QCTL. We assume the reader is conversant with this protocol since it will not be presented here. For the sake of simplicity, we consider that the protocol distributes a key... |

381 | Formal methods : state of the art and future directions
- Clarke
- 1996
(Show Context)
Citation Context ...tes [22, 21, 16, 9] and quantum programs [15, 19, 1, 13, 2, 20, 3, 8]. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols =-=[11, 17]-=-. Herein, we present a temporal logic for reasoning about evolution of quantum systems composed of a fixed finite set of qubits. Our starting point is the logic dEQPL for reasoning about quantum state... |

288 |
Algorithms in Real Algebraic Geometry
- Basu, Pollack, et al.
- 2003
(Show Context)
Citation Context ...nough to interpret analytical formulas. We say that an analytical formula κ is a valid analytical formula if it holds for any assignment. It is a well-known fact from the theory of real closed fields =-=[5]-=- that the set of valid analytical formulas so defined is decidable. However, we shall not go into details of this result and will focus our attention on reasoning about quantum aspects only. The axiom... |

223 | Megiddo N., A logic for reasoning about probability
- Fagin, Halpern
- 1990
(Show Context)
Citation Context ...te. The models of this logic are basically the quantum states of the finite qubit system. We give a sound and complete axiomatization of this state logic. The completeness proof, which is inspired by =-=[9, 14]-=-, also suggests a decision procedure for the theorem-hood problem and we compute the complexity of the decision procedure assuming that all basic integer operations (addition, subtraction, multiplicat... |

172 |
Revised report on the algorithmic language ALGOL 60
- Naur
- 1960
(Show Context)
Citation Context ...tum formulas, are constructed from comparison formulas (formulas that compare terms) using propositional connectives. We present language of dEQPL in Table 1 using an abstract version of BNF notation =-=[18]-=- for a compact presentation of inductive definitions and discuss the language in detail below. Classical formulas α := ⊥ ⫾ qb ⫾ (α ⇒ α) Table 1: Language of efficient EQPL Term language (with the prov... |

160 | A categorical semantics of quantum protocols
- Abramsky, Coecke
- 2004
(Show Context)
Citation Context ...ions such as information processing, security, distributed systems and randomized algorithms. This has attracted research in formal reasoning about quantum states [22, 21, 16, 9] and quantum programs =-=[15, 19, 1, 13, 2, 20, 3, 8]-=-. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols [11, 17]. Herein, we present a temporal logic for reasoning about evo... |

65 | Formal methods for cryptographic protocol analysis: emerging issues and trends
- Meadows
(Show Context)
Citation Context ...tes [22, 21, 16, 9] and quantum programs [15, 19, 1, 13, 2, 20, 3, 8]. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols =-=[11, 17]-=-. Herein, we present a temporal logic for reasoning about evolution of quantum systems composed of a fixed finite set of qubits. Our starting point is the logic dEQPL for reasoning about quantum state... |

63 | Non-deterministic quantum programming
- Zuliani
- 2004
(Show Context)
Citation Context ...ions such as information processing, security, distributed systems and randomized algorithms. This has attracted research in formal reasoning about quantum states [22, 21, 16, 9] and quantum programs =-=[15, 19, 1, 13, 2, 20, 3, 8]-=-. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols [11, 17]. Herein, we present a temporal logic for reasoning about evo... |

48 | Conventions for quantum pseudocode
- Knill
- 1996
(Show Context)
Citation Context ...ions such as information processing, security, distributed systems and randomized algorithms. This has attracted research in formal reasoning about quantum states [22, 21, 16, 9] and quantum programs =-=[15, 19, 1, 13, 2, 20, 3, 8]-=-. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols [11, 17]. Herein, we present a temporal logic for reasoning about evo... |

47 | A compiler for a functional quantum programming language
- Grattage, Altenkirch
- 2005
(Show Context)
Citation Context |

32 | Weakly complete axiomatization of exogenous quantum propositional logic
- Mateus, Sernadas
- 2006
(Show Context)
Citation Context ...ce due to a big potential in applications such as information processing, security, distributed systems and randomized algorithms. This has attracted research in formal reasoning about quantum states =-=[22, 21, 16, 9]-=- and quantum programs [15, 19, 1, 13, 2, 20, 3, 8]. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols [11, 17]. Herein, w... |

24 | Quantum weakest preconditions
- D’Hondt, Panangaden
- 2006
(Show Context)
Citation Context |

12 |
Extending classical logic for reasoning about quantum systems
- Chadha, Mateus, et al.
- 2009
(Show Context)
Citation Context ...ce due to a big potential in applications such as information processing, security, distributed systems and randomized algorithms. This has attracted research in formal reasoning about quantum states =-=[22, 21, 16, 9]-=- and quantum programs [15, 19, 1, 13, 2, 20, 3, 8]. Formal methods have proved to be successful in design and verification of 1classical distributed systems and security protocols [11, 17]. Herein, w... |

9 |
LQP: The Dynamic Logic of Quantum Information
- Baltag, Smets
- 2006
(Show Context)
Citation Context |

7 |
and Bernd-Holger Schlingloff. Model checking
- Clarke
- 2001
(Show Context)
Citation Context ...ff there is a path s1s2 . . . with s = s1 such that for some i ≥ 1 K, s1 ⊩CTL η2 and K, sjρ ⊩CTL η1 for 1 ≤ j < i. 10Axiomatization. The temporal logic CTL enjoys a sound and complete axiomatization =-=[12]-=-. In order to give the axiomatization, we need to introduce some useful abbreviations• (AXθ) for ⊟ EX(⊟ θ); • (EFθ) for ⊟(E[(⊟ ⊥)Uθ]); • (AGθ) for ⊟(EF(⊟ θ)); • (EGθ) for ⊟(AF(⊟ θ)); • A[θ1Uθ2] for ⊟(... |

6 |
Reasoning about quantum imperative programs
- Chadha, Mateus, et al.
(Show Context)
Citation Context |

2 | P.Mateus. Verifying probabilistic systems with EpCTL. Awaiting publication
- Baltazar
- 2008
(Show Context)
Citation Context ...nitary transformations. We give a sound and complete axiomatization of QCTL capitalizing on the complete axiomatization of dEQPL and CTL. The proof of completeness follow the techniques introduced in =-=[7, 4]-=-. Finally, combine the standard CTL model-checking algorithm with the dEQPL model-checking algorithm to obtain a model-checking algorithm for QCTL. Finally, we note that we do not explicitly deal with... |