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arXiv:0901.0001 · Physical Review A

Controlled-NOT logic with nonresonant Josephson phase qubits

Andrei Galiautdinov

Published Apr 13, 2009• 8 citations
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Abstract

We establish theoretical bounds on qubit detuning for some of the previously proposed controlled-NOT (CNOT) logic gate implementations with weakly coupled Josephson phase qubits. It is found that in the two-step, $\sqrt{i\text{SWAP}}$-based case the value of the detuning during the entangling operations must not exceed $2g$, where $g$ is the characteristic coupling constant. In the single-step case we consider two practical, physically distinct implementations, in which one of the qubits is driven by a concurrent rf pulse of fixed frequency. We find that when the local drive is applied to the ``reference'' qubit (with which it is in resonance), the detuning should not exceed $g$. If the drive is applied to the ``detuned'' qubit, generation of the perfect CNOT gate is possible at any value of detuning provided that the amplitude of the pulse can be made arbitrarily large.

DOI
10.1103/physreva.79.042316

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We establish theoretical bounds on qubit detuning for some of the previously proposed controlled-NOT (CNOT) logic gate implementations with weakly coupled Josephson phase qubits. It is found that in the two-step, iSWAP\sqrt{i\text{SWAP}}iSWAP​-based case the value of the detuning during the entangling operations must not exceed 2g2g2g, where ggg is the characteristic coupling constant. In the single-step case we consider two practical, physically distinct implementations, in which one of the qubits is driven by a concurrent rf pulse of fixed frequency. We find that when the local drive is applied to the reference'' qubit (with which it is in resonance), the detuning should not exceed $g$. If the drive is applied to the detuned'' qubit, generation of the perfect CNOT gate is possible at any value of detuning provided that the amplitude of the pulse can be made arbitrarily large.
Building on the previous results of the Weyl chamber steering method, we demonstrate how to generate high-fidelity controlled-NOT (CNOT) gates by direct application of certain physically relevant Hamiltonians with fixed coupling constants containing Rabi terms. Such Hamiltonians are often used to describe two superconducting qubits driven by local rf pulses. It is found that in order to achieve 100% fidelity in a system with capacitive coupling of strength ggg, one Rabi term suffices. We give the exact values of the physical parameters needed to implement such CNOT gates. The gate time and all possible Rabi frequencies are found to be $t=\ensuremath{\pi}∕(2g)$ and ${\ensuremath{\Omega}}_{1}∕g=\sqrt{64{n}^{2}\ensuremath{-}1},n=1,2,3,\dots{}$. Generation of a perfect CNOT gate in a system with inductive coupling, characterized by additional constant kkk, requires the presence of both Rabi terms. The gate time is again $t=\ensuremath{\pi}∕(2g)$, but now there is an infinite number of solutions, each of which is valid in a certain range of kkk and is characterized by a pair of integers (n,m)(n,m)(n,m), $\frac{{\ensuremath{\Omega}}_{1,2}}{g}=\sqrt{16{n}^{2}\ensuremath{-}{(\frac{k\ensuremath{-}1}{2})}^{2}}\ifmmode\pm\else\textpm\fi{}\sqrt{16{m}^{2}\ensuremath{-}{(\frac{k+1}{2})}^{2}}$. We distinguish two cases, depending on the sign of the coupling constant: (i) the antiferromagnetic case $(k\ensuremath{\geqslant}0)$ with $n\ensuremath{\geqslant}m=0,1,2,\dots{}$ and (ii) the ferromagnetic case $(k\ensuremath{\leqslant}0)$ with n>m=0,1,2,…n>m=0,1,2,\dots{}n>m=0,1,2,…. We conclude with consideration of fidelity degradation by switching to resonance. Simulation of time evolution based on the fourth-order Magnus expansion reveals characteristics of the gate similar to those found in the exact case, with slightly shorter gate time and shifted values of the Rabi frequencies.
Superconducting qubits with tunable coupling are ideally suited for fast and accurate implementation of quantum logic. Here we present a simple approach, based on Weyl chamber steering, to CNOT gate design for inductively coupled phase qubits with tunable coupling strength g. In the presence of simultaneous rf pulses on the individual qubits that appropriately track the coupling strength as it is varied, we show that an infinite family of switching sequences preserving the time integral or "area" of g can be used to generate CNOT logic. We demonstrate our approach by considering time-dependencies most likely to be used in actual implementations: trapezoidal, sine, and soft quartic (also known as Landau's hat).
We study the coupled-qubit oscillation driven by an oscillating field. When the period of the nonresonant mode is commensurate with that of the resonant mode of the Rabi oscillation, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate oscillations, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of oscillating field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and qubit energy gap in experiments.
We study the coupled-qubit oscillation driven by an oscillating field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi oscillation, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate oscillations, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of oscillating field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and qubit energy gap in experiments.
We present a theoretical analysis of the selective darkening method for implementing quantum controlled-NOT (CNOT) gates. This method, which we have recently proposed and demonstrated, consists of driving two transversely coupled quantum bits (qubits) with a driving field that is resonant with one of the two qubits. For specific relative amplitudes and phases of the driving field felt by the two qubits, one of the two transitions in the degenerate pair is darkened or, in other words, becomes forbidden by effective selection rules. In these driving conditions, the evolution of the two-qubit state realizes a CNOT gate. The gate speed is found to be limited only by the coupling energy J , which is the fundamental speed limit for any entangling gate. Numerical simulations show that at gate speeds corresponding to 0.48 J and 0.07 J , the gate fidelity is 99% and 99.99%, respectively, and increases further for lower gate speeds. In addition, the effect of higher-lying energy levels and weak anharmonicity is studied, as well as the scalability of the method to systems of multiple qubits. We conclude that in all these respects this method is competitive with existing schemes for creating entanglement, with the added advantages of being applicable for qubits operating at fixed frequencies (either by design or for the exploitation of coherence sweet-spots) and having the simplicity of microwave-only operation.
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.