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arXiv:quant-ph/0210157 · Physical Review Letters

Quantum Computing with an Always-On Heisenberg Interaction

Simon C. Benjamin, Sougato Bose

Published Jun 19, 2003• 190 citations
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Abstract

Many promising schemes for quantum computing (QC) involve switching "on" and "off" a physical coupling between qubits. This may prove extremely difficult to achieve experimentally. Here we show that systems with a constant Heisenberg coupling can be employed for QC if we actively "tune" the transition energies of individual qubits. Moreover, we can collectively tune the qubits to obtain an exceptionally simple scheme: computations are controlled via a single "switch" of only six settings. Our schemes are applicable to a wide range of physical implementations, from excitons and spins in quantum dots through to bulk magnets.

DOI
10.1103/physrevlett.90.247901

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Many promising schemes for quantum computing (QC) involve switching "on" and "off" a physical coupling between qubits. This may prove extremely difficult to achieve experimentally. Here we show that systems with a constant Heisenberg coupling can be employed for QC if we actively "tune" the transition energies of individual qubits. Moreover, we can collectively tune the qubits to obtain an exceptionally simple scheme: computations are controlled via a single "switch" of only six settings. Our schemes are applicable to a wide range of physical implementations, from excitons and spins in quantum dots through to bulk magnets.
Recently, there has been interest in the idea of quantum computing without control of the physical interactions between component qubits. This is highly appealing since the 'switching' of such interactions is a principal difficulty in creating real devices. It has been established that one can employ 'always-on' interactions in a one-dimensional Heisenberg chain, provided that one can tune the Zeeman energies of the individual (pseudo-)spins. It is important to generalize this scheme to higher-dimensional networks, since a real device would probably be of that kind. Such generalizations have been proposed, but only at the severe cost that the efficiency of qubit storage must fall . Here we propose the use of multi-qubit gates within such higher-dimensional arrays, finding a novel three-qubit gate that can in fact increase the efficiency beyond the linear model. Thus, we are able to propose higher-dimensional networks that can constitute a better embodiment of the 'always-on' concept—a substantial step towards bringing this novel idea to full fruition.
Recently there has been interest in the idea of quantum computing without control of the physical interactions between component qubits. This is highly appealing since the switching of such interactions is a principle difficulty in creating real devices. It has been established that one can employ 'always on' interactions in a one-dimensional Heisenberg chain, provided that one can tune the Zeeman energies of the individual (pseudo-)spins. It is important to generalize this scheme to higher dimensional networks, since a real device would probably be of that kind. Such generalisations have been proposed, but only at the serve cost that the efficiency of qubit storage must *fall*. Here we propose the use of multi-qubit gates within such higher-dimensional arrays, finding a novel three-qubit gate that can in fact increase the efficiency beyond the linear model. Thus we are able to propose higher dimensional networks that can constitute a better embodiment of the 'always on' concept - a substantial step toward bringing this novel concept to full fruition.
If the interaction between qubits is not diagonal in the computational basis (e.g., the Heisenberg interaction), then one must be able to "switch it off" in order to prevent uncontrolled propagation of states. Therefore, schemes for quantum computing typically demand local control of the interaction strength between each pair of neighboring qubits. Here we demonstrate that this degree of control is not necessary: it suffices to switch the interaction collectively, something that can in principle be achieved by global fields rather than with local manipulations. This observation may offer a significant simplification for various implementations, including solid state and optical lattice schemes.
It has recently been shown that one can perform quantum computation in a Heisenberg chain in which the interactions are always on, provided that one can abruptly tune the Zeeman energies of the individual (pseudo)spins. Here we provide a more complete analysis of this scheme, including several generalizations. We generalize the interaction to an anisotropic form (incorporating the XYXYXY interaction as a limit), providing a proof that a chain coupled in this fashion tends to an effective Ising chain in the limit of far off-resonant spins. We derive the primitive two-qubit gate that results from exploiting abrupt Zeeman tuning with such an interaction. We also demonstrate, via numerical simulation, that the same basic scheme functions in the case of smoothly shifted Zeeman energies. We conclude with some remarks regarding generalizations to two- and three-dimensional arrays.
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.