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Physical Review A

Generalized methods for the development of quantum logic gates for an NMR quantum information processor

Mark D. Price, Shyamal S. Somaroo, A. E. Dunlop, Timothy F. Havel, David G. Cory

Published Oct 1, 1999• 65 citations
Publisher

Abstract

Logic gates such as the controlled-NOT (c-NOT) and Toffoli gates play a key role in quantum information processing (QIP) and quantum computing. A natural extension of such gates would necessarily operate on one quantum bit (qubit) conditional on the state of the remaining qubits in the system. We show that such selective gates, termed $(\mathrm{controlled}{)}^{n}\ensuremath{-}\mathrm{NOT}$ gates, or ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates, are convenient in nuclear magnetic resonance (NMR) implementations of QIP and are straightforward to implement. NMR pulse sequences for these gates can be built using classical methods as well as insights from geometric algebra. These methods yield equivalent NMR pulse sequences for the generation of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for any number of control spins. In this work, a catalog of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for systems of as many as 16 spins is provided along with an experimental implementation of a ${\mathrm{c}}^{3}\ensuremath{-}\mathrm{N}\mathrm{O}\mathrm{T}$ gate on a four spin system, ${}^{13}\mathrm{C}$ alanine.

DOI
10.1103/physreva.60.2777

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Logic gates such as the controlled-NOT (c-NOT) and Toffoli gates play a key role in quantum information processing (QIP) and quantum computing. A natural extension of such gates would necessarily operate on one quantum bit (qubit) conditional on the state of the remaining qubits in the system. We show that such selective gates, termed (controlled)n\ensuremath−NOT(\mathrm{controlled}{)}^{n}\ensuremath{-}\mathrm{NOT}(controlled)n\ensuremath−NOT gates, or cn\ensuremath−NOT{\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}cn\ensuremath−NOT gates, are convenient in nuclear magnetic resonance (NMR) implementations of QIP and are straightforward to implement. NMR pulse sequences for these gates can be built using classical methods as well as insights from geometric algebra. These methods yield equivalent NMR pulse sequences for the generation of cn\ensuremath−NOT{\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}cn\ensuremath−NOT gates for any number of control spins. In this work, a catalog of cn\ensuremath−NOT{\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}cn\ensuremath−NOT gates for systems of as many as 16 spins is provided along with an experimental implementation of a c3\ensuremath−NOT{\mathrm{c}}^{3}\ensuremath{-}\mathrm{N}\mathrm{O}\mathrm{T}c3\ensuremath−NOT gate on a four spin system, 13C{}^{13}\mathrm{C}13C alanine.
Much of the current understanding of logic gates for quantum computation relies on the application of one- and two-qubit gates in order to implement a universal set of logic gates, as well as other gates. Such ideas stem from the notion that the Hamiltonians available for quantum computation are one or two-body processes. However, present day NMR implementations of quantum information processing (QIP) rely on the internal Hamiltonian of a liquid state system in which there are many two-body processes that occur simultaneously. Such use of the internal Hamiltonian allows for the creation of `multiqubit' logic gates, i.e. logic gates that operate on many qubits simultaneously and are more efficient than a sequence of one- and two-qubit rotations that effect the same operation. Such larger qubit operations offer a universal set of gates (even when not all couplings within the internal Hamiltonian are readily accessible) as well as more convenient and efficient implementations of the Hadamard transform, the controlled-NOT gate, and a quantum Fourier transform that scales linearly (assuming all couplings can be turned on and off at will).
Nuclear magnetic resonance spectroscopy has demonstrated significant experimental progress toward the development of quantum computations. The developments so far have taken place mainly through the use of spin 12 nuclei. In this paper we describe the use of a spin 32 nucleus, oriented in a liquid crystal matrix for the creation of pseudopure states and the implementation of a complete set of two-qubit reversible logic gates using single-quantum transition-selective pulses, extending the range of practice of NMR toward quantum computation.
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.
Composite pulses, originally developed in nuclear magnetic resonance (NMR), have found widespread use in experimental quantum information processing (QIP) to reduce the effects of systematic errors. Most pulses used so far have simply been adapted from existing NMR designs, and while techniques have been developed for designing composite pulses with arbitrary precision, the results have been quite complicated and have found little application. Here, I describe techniques for designing short but effective composite pulses to implement robust not gates, bringing together existing insights from NMR and QIP, and present some composite pulses.
In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.
The three -qubit Toffoli gate is equivalent to the combination of two two -qubit controlled NOT gates and three two -qubit controlled phase -shift gates.The controlled NOT and phase -shift gates respectively comprise a combination of nuclear -magnetic -resonance pulse sequences and a free development operator of two nuclear spins with time.According to the order in which the controlled NOT and phase -shift gates act on the nuclear spin system, the nuclear -magnetic - resonance -based physical realization of a quantum Toffoli gate is achieved.The time -dependent Schrödinger equation is numerically solved by the Suzuki formula to confirm the correctness and feasibility of the realization of the quantum Toffoli gate based on nuclear magnetic resonance.