| Project/Area Number |
11440028
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Tohoku University (2001)
Nagoya University (1999-2000) |
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Principal Investigator |
OZAWA Masanao Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (40126313)
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| Co-Investigator(Kenkyū-buntansha) |
IHARA Shunsuke Nagoya University, School of Informatics, Professor, 情報文化学部, 教授 (00023200)
TSUKAJI Tasuie Nagoya University, Graduate School of Human Informatics, Research Associate, 大学院・人間情報学研究科, 助手 (70291961)
YASUMOTO Masahiro Nagoya University, Graduate School of Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (10144114)
MATSUBARA Yo Nagoya University, Graduate School of Human Informatics, Associate Professor, 大学院・人間情報学研究科, 助教授 (30242788)
MATSUMOTO Hiroyuki Nagoya University, Graduate School of Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (00190538)
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| Project Period (FY) |
1999 – 2001
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| Keywords | quantum computing / quantum Turing machines / quantum circuits / quantum complexity theory / quantum gates / quantum operation / fault-tolerant quantum computing / controlled NOT gate |
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| Research Abstract |
The following results have been obtained on mathematical foundations on quantum Turing machine and quantumcircuits: (1) Local transition functions of quantum Turing machines (QTM) are generally characterized including multitape cases. (2) The notion of uniform quantum circuit families (UQCF) was introduced for the first time and developed their complexity theory nd proved the computational equivalence between QTMs and UQCF in Monte Caro type computations. (3) In order to solve the halting problem for QTMs, it has been proved that under a refined halting protocol measurements of halting flag do not disturb the probability distribution of the output of computations.
The following results have been obtained on physical implementations of quantum logicgates: (1) Conservation laws limit theaccuracy of physical implementations of elementary quantum logic gates. (2) Although the SWAP gate has no conflict with the conservation law, the controlled-NOT gate, which is one of the universal quantum
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logic gates, cannot be implemented by any 2-qubit rotationally invariant unitary operation within error probability 1/16.. (3) If the computational basis is represented by a component of spin and physical implementations obey the angular momentum conservation law, any physically realizable quantum logicgates with n qubit ancilla cannot implement the controlled-NOT gate within the error probability 1/(4n^2). (4) An analogous relation holds for bosonic ancillae with the size defined through the average number of photons. Any set of universal gates inevitably obeys a related limitation with error probability O(n^<-2>). (5) The current theory demands the threshold error probability 10^5 10^6 for each quantum gate. Thus, a single controlled-NOT gate would not be in reality a unitary operation on a 2-qubitsystem but would be a unitary operation on a system with at least 100 qubits. (6) The present investigation suggests that the current choice of the computational basis should be modified so that the computational basis commutes with the conserved quantity. Less
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