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International Journal of Quantum Chemistry

Quantum circuits for an effective pure state in NMR quantum computer

Yoshiyuki Mori, Ryuichi Sawae, Minaru Kawamura, Toshio Sakata, Kenichi Takarabe

Published Jan 1, 2005• 6 citations
Publisher

Abstract

Abstract In nuclear magnetic resonance (NMR) quantum computing systems, it is very important to prepare an effective pure state rather than a true pure state. Our main objective is to present a general algorithm that provides quantum circuits for preparing an effective pure state from any mixed state, consisting only of controlled‐not (CNOT) quantum gates for homonuclear spin systems. Our algorithm is based on the product operator approach and could be applicable to NMR quantum computing systems with more than eight qubits. Optimization of the number of controlled‐not quantum gates is discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

DOI
10.1002/qua.20659

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Abstract In nuclear magnetic resonance (NMR) quantum computing systems, it is very important to prepare an effective pure state rather than a true pure state. Our main objective is to present a general algorithm that provides quantum circuits for preparing an effective pure state from any mixed state, consisting only of controlled‐not (CNOT) quantum gates for homonuclear spin systems. Our algorithm is based on the product operator approach and could be applicable to NMR quantum computing systems with more than eight qubits. Optimization of the number of controlled‐not quantum gates is discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
Abstract A fast method to prepare effective pure states for NMR‐quantum computer in homo‐nuclear spin system is proposed. The temporal averaging method has been introduced to prepare an effective pure state in NMR‐quantum computer, but the method requires many experiments, whose number increases exponentially with the number of quantum bits (qubits), and it seems that a huge number of logical gates are required. The proposed method avoids this problem by eliminating a part of elements in the density matrices. This method requires only one ancillary qubit and two different experiments; the steps have been confirmed to be proportional to the number of qubits up to four‐qubit. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Abstract The initialization of states is of importance in nuclear magnetic resonance quantum computing systems. In this article it is shown that the initialization ofn‐qubits states can be realized via a sequence of controlled‐not quantum gates from any mixed state. We present a general algorithm and its optimization to construct a circuit of controlled‐not quantum gates that realizes the initialization using the concept of Gray Codes. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
Abstract The field of quantum computing has grown rapidly in the last years. Most realizations of quantum computers were done on liquid‐state NMR quantum computers so far. We present in a didactic approach how to build small NMR quantum computers with two and three qubits. The two‐qubit quantum computer was implemented on α‐D‐methylglucose using 13C spins, whereas the three‐qubit quantum computer was implemented on 4‐bromo‐1,1,2‐trifluoro‐1‐butene using 19F spins. We describe in detail how to initialize these systems by generating pseudo‐pure states. For initialization, we used the spatial averaging method, which has the advantage that it can be easily understood by product operators. Furthermore, the implementation of quantum logic gate operations will be presented. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40: 25–37, 2012.
Basing on the temporal averaging, schedules to prepare 3-qubit and 4-qubit pesudo-pure states in nuclear magnetic resonance (NMR) quantum computation are presented in this paper. Utilizing Lambda(n) (not) gate, our schedules reduced the experiment number that is needed to prepare pesudo-pure state in a homonuclear spin system: only three experiments are needed to prepare 3-qubit peudo-pure state; and only five experiments are needed to prepare 4-qubit peudo-pure state. Then we implemented the schedule for 3-qubit psudo-pure state in a chain-coupled spin system to confirm the schedule, and gave the simulation result of the schedule for the 4-qubit peudo-pure state.
The premise of nuclear magnetic resonance (NMR) quantum computing is that pseudopure quantum states, for which the populations of all energy levels but one are equal, can be used instead of pure quantum states. However, for systems with a large number of quantum bits (qubits), it would require elaborate pulse sequences with many steps to prepare these pseudopure states. To overcome this problem, a method using pairs of pseudopure states (POPS) instead of individual pseudopure states as the basis for NMR quantum computing is introduced here. The POPS approach has a significant advantage over the existing methods that, regardless of the number of qubits, only a single step is needed to prepare each pair of pseudopure states. A detailed analysis of a 16-level spin system is given, and the results of applying a c2{c}^{2}c2-NOT quantum logic gate (a controlled-NOT gate conditional upon the state of two other qubits) to all the POPS in the system are presented.
Preparation of effective pure states ∣00⋯0⟩\mid 00 \cdots 0 \rangle∣00⋯0⟩ on an NMR quantum computer with any number of spins (Analytical Study of Quantum Information and Related Fields)
A method is proposed to prepare an effective pure state corresponding to the pure state ∣0⟩|0\rangle∣0⟩ ∣0⟩|0\rangle∣0⟩ ⋯∣0⟩\cdots|0\rangle⋯∣0⟩ for any number of spins on a nuclear magnetic resonance quantum computer, where ∣0⟩|0\rangle∣0⟩ stands for the spin-up state of a nuclear spin.The method is based on the spatial labeling proposed by Cory et al.