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arXiv:quant-ph/0002016 · Journal of Experimental and Theoretical Physics Letters

Physical implementation of three-qubit gates on a separate quantum particle

A. R. Kessel, Vladimir L. Ermakov

Published Apr 1, 2000• 25 citations
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Abstract

A virtual spin formalism is suggested to demonstrate that a single quantum particle possessing eight suitable discrete energy levels can be used for storing three information qubits and organizing on them a universal set of logical operations that are necessary for constructing an arbitrary quantum algorithm. The formalism can be practically implemented on a nuclear spin 7/2 subjected to resonance rf pulses. A single-pulse realization is found for all quantum gates of a universal set, including a three-qubit gate.

DOI
10.1134/1.568340

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•Mar 1, 2008•0 citations
A virtual spin formalism is suggested to demonstrate that a single quantum particle possessing eight suitable discrete energy levels can be used for storing three information qubits and organizing on them a universal set of logical operations that are necessary for constructing an arbitrary quantum algorithm. The formalism can be practically implemented on a nuclear spin 7/2 subjected to resonance rf pulses. A single-pulse realization is found for all quantum gates of a universal set, including a three-qubit gate.
The possibility of using multilevel quantum systems with a discrete nonequidistant spectrum for the physical implementation of a universal collection of quantum gates, which is required for constructing an arbitrary algorithm for a quantum computer, is studied on the basis of the proposed virtual-spin formalism. It is shown that such a set of gates, including two-qubit gates, can be constructed on a single spin-3/2 nucleus. The assertion that three-qubit gates can also be realized on a single spin-7/2 quantum particle is substantiated.
A conception of virtual quantum information bit - virtual qubit - is introduced. It is shown by means of virtual qubit representation that four states of a single quantum particle is enough for implementation of full set of the gates, which is necessary for creation an arbitrary algorithm for a quantum computer. The physical nature and mutual disposition of four working states is of no significance provided the existence of the suitable selection rules for the particle interaction with the external electromagnetic field pulses.
One qubit is mapped to two level system in the standard model of quantum computation: a typical example is a spin-1/2 system. Further progress requires the use of network of multilevel quantum systems of different physical origin in solids: optical levels, large or strong interacting spins etc. We propose virtual qubit formalism and show how the quantum objects state space can be used as a basis for multiqubit information recording and processing in a quantum computer. It is suggested to map the qubits into solid state NMR spin system collective states instead of the state of the individual spin. Such an approach introduces the stable computational basis without any additional actions and allows to obtain the universal set of quantum gates, which operation time is determined only by the RF pulses.
Quantum walk has been regarded as a primitive to universal quantum computation. In this paper, we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems by using the operations required to describe the single particle discrete-time quantum walk on a position space. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.
Universal quantum computation requires the implementation of arbitrary control operations on the quantum register. In most cases, this is achieved by external control fields acting selectively on each qubit to drive single-qubit operations. In combination with a drift Hamiltonian containing interactions between the qubits, this allows the implementation of arbitrary gate operations. Here, we demonstrate an alternative scheme that does not require local control for all qubits: we implement one- and two-qubit gate operations on a set of target qubits indirectly, through a combination of gates on directly controlled actuator qubits with a drift Hamiltonian that couples actuator and target qubits. Experiments are performed on nuclear spins, using radio-frequency pulses as gate operations and magnetic-dipole couplings for the drift Hamiltonian.
A constructive scheme has been devised to enable mapping of any quantum computation into a spintronic circuit in which the computation is encoded in a basis that is, in principle, immune to quantum decoherence. The scheme is implemented by an algorithm that utilizes multiple physical spins to encode each logical bit in such a way that collective errors affecting all the physical spins do not disturb the logical bit. The scheme is expected to be of use to experimenters working on spintronic implementations of quantum logic. Spintronic computing devices use quantum-mechanical spins (typically, electron spins) to encode logical bits. Bits thus encoded (denoted qubits) are potentially susceptible to errors caused by noise and decoherence. The traditional model of quantum computation is based partly on the assumption that each qubit is implemented by use of a single two-state quantum system, such as an electron or other spin-1.2 particle. It can be surprisingly difficult to achieve certain gate operations . most notably, those of arbitrary 1-qubit gates . in spintronic hardware according to this model. However, ironically, certain 2-qubit interactions (in particular, spin-spin exchange interactions) can be achieved relatively easily in spintronic hardware. Therefore, it would be fortunate if it were possible to implement any 1-qubit gate by use of a spin-spin exchange interaction. While such a direct representation is not possible, it is possible to achieve an arbitrary 1-qubit gate indirectly by means of a sequence of four spin-spin exchange interactions, which could be implemented by use of four exchange gates. Accordingly, the present scheme provides for mapping any 1-qubit gate in the logical basis into an equivalent sequence of at most four spin-spin exchange interactions in the physical (encoded) basis. The complexity of the mathematical derivation of the scheme from basic quantum principles precludes a description within this article; it must suffice to report that the derivation provides explicit constructions for finding the exchange couplings in the physical basis needed to implement any arbitrary 1-qubit gate. These constructions lead to spintronic encodings of quantum logic that are more efficient than those of a previously published scheme that utilizes a universal but fixed set of gates.