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arXiv:quant-ph/0610059 · Physical Review Letters

Robust Controlled-NOT Gates from Almost Any Interaction

Charles D. Hill

Published May 1, 2007• 63 citations
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Abstract

There are many cases where the interaction between two qubits is not precisely known, but single-qubit operations are available. In this Letter we show how, regardless of an incomplete knowledge of the strength or form of the interaction between two qubits, it is often possible to construct a controlled-NOT gate which has arbitrarily high fidelity. In particular, we show that oscillations in the strength of the exchange interaction in solid state Si and Ge structures are correctable.

DOI
10.1103/physrevlett.98.180501

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There are many cases where the interaction between two qubits is not precisely known, but single-qubit operations are available. In this Letter we show how, regardless of an incomplete knowledge of the strength or form of the interaction between two qubits, it is often possible to construct a controlled-NOT gate which has arbitrarily high fidelity. In particular, we show that oscillations in the strength of the exchange interaction in solid state Si and Ge structures are correctable.
We demonstrate how using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterization. Operating the robust CNOT gate in a suitably characterized system means concatenation of the composite pulse is unnecessary, hence reducing operation time, and ensuring the gate operates below the threshold required for fault-tolerant quantum computation.
A self-contained approach to studying the unitary evolution of coupled qubits is introduced, capable of addressing a variety of physical systems described by exchange Hamiltonians containing Rabi terms. The method automatically determines both the Weyl chamber steering trajectory and the accompanying local rotations. Particular attention is paid to the case of anisotropic exchange with tracking controls, which is solved analytically. It is shown that, if computational subspace is well isolated, any exchange interaction can always generate high fidelity, single-step controlled-NOT (CNOT) logic, provided that both qubits can be individually manipulated. The results are then applied to superconducting qubit architectures, for which several CNOT gate implementations are identified. The paper concludes with consideration of two CNOT gate designs having high efficiency and operating with no significant leakage to higher-lying noncomputational states.
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A new approach to single-qubit operations using exchange interactions of single electrons in gate-defined quantum dots has recently been demonstrated: the resonant exchange qubit. We show that two-qubit operations, specifically the controlled phase gate, can be performed between resonant exchange qubits very straightforwardly, using a single exchange pulse. This is in marked contrast to the best known protocols for exchange qubits where such a gate requires many pulses so that leakage processes arising from the exchange interaction can be overcome. For resonant exchange qubits a simple two-qubit gate is possible because in this mode of operation energy conservation suppresses leakage.
There are well-known protocols for performing controlled NOT (CNOT) quantum logic with qubits coupled by particular high-symmetry (Ising or Heisenberg) interactions. However, many architectures being considered for quantum computation involve qubits or qubits and resonators coupled by more complicated and less symmetric interactions. Here we consider a widely applicable model of weakly but otherwise arbitrarily coupled two-level systems, and use quantum gate design techniques to derive a simple and intuitive CNOT construction. Useful variations and extensions of the solution are given for common special cases.