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New Journal of Physics

Multiqubit logic gates in NMR quantum computing

Mark D. Price, Timothy F. Havel, David G. Cory

Published May 31, 2000• 26 citations
Publisher

Abstract

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).

DOI
10.1088/1367-2630/2/1/10

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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 (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present-day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR-based QIP capable of reaching 10—30 qubits.
Nuclear magnetic resonance (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present-day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR-based QIP capable of reaching 10-30 qubits.
Decoherence-free subspaces protect quantum information from the effects of noise that is correlated across the physical qubits used to implement them. Given the ability to impose suitable Hamiltonians upon such a multi-qubit system, one can also implement a set of logical gates which enables universal computation on this information without compromising this protection. Real physical systems, however, seldom come with the correct Hamiltonians built-in, let alone the ability to turn them off and on at will. In the course of our development of quantum information processing devices based on liquid-state NMR, we have found the task of operating on quantum information encoded in decoherence-free subspaces rather more challenging than is commonly assumed. This contribution presents an overview of these challenges and the methods we have developed for overcoming them in practice. These methods promise to be broadly applicable to many of the physical systems proposed for the implementation of quantum information processing devices.
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.
Quantum computing and quantum information science are two recently discovered and rapidly growing fields of physics that show substantial promise in providing new and valuable technologies in the foreseeable future. Large-scale quantum computers, if ever realized experimentally, are likely to outperform their classical counterparts in a number of important computational tasks, the most important of which may be the accurate simulation of many-body quantum systems such as the ones encountered in physics, chemistry and life sciences. This thesis investigates the problem of controlling quantum systems for the purpose of performing quantum information processing tasks. The problem is approached from a theoretical and simulational viewpoint. The work contained here encompasses a range of levels of abstraction. Firstly, we discuss the decomposition of abstract multiqubit logic gates into sequences of simple elementary gates. Secondly, we study the local commutational properties of two-qubit gates using local gate invariants. Thirdly, we develop methods for the physical implementation of the elementary gates through the control of specific quantum systems, possibly in the presence of noise and decoherence. We present a new, almost optimal n-qubit gate decomposition based on the cosine-sine decomposition, which utilizes a likewise new intermediate quantum circuit structure we call a uniformly controlled gate. We then show how they can be used in constructing a general state transformation circuit. Both of the resulting circuits can be efficiently implemented using nearest-neighbor gates which makes their physical realization simpler. A local gate invariant is introduced which can be used to assess the suitability of two-qubit gates for serving as the entangling gate in elementary gate libraries. Finally, we develop numerical optimization methods for finding near-optimal control sequences for generating oneand two-qubit gates, both in closed quantum systems and in the presence of Markovian noise.
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, 13C{}^{13}\mathrm{C}13C alanine.