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arXiv:cond-mat/0309491 · Physical Review B

Fast two-bit operations in inductively coupled flux qubits

J. Q. You, Yasunobu Nakamura, Franco Nori

Published Jan 31, 2005• 72 citations
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Abstract

A central problem for implementing efficient quantum computing is how to realize fast operations (both one- and two-bit ones). However, this is difficult to achieve for a collection of qubits, especially for those separated far away, because the interbit coupling is usually much weaker than the intrabit coupling. Here we present an experimentally feasible method to effectively couple two flux qubits via a common inductance and treat both single and coupled flux qubits with more realistic models, which include the loop inductance. The main advantage of our proposal is that a strong interbit coupling can be achieved using a small inductance, so that two-bit as fast as one-bit operations can be easily realized. We also show the flux dependence of the transitions between states for the coupled flux qubits.

DOI
10.1103/physrevb.71.024532

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A central problem for implementing efficient quantum computing is how to realize fast operations (both one- and two-bit ones). However, this is difficult to achieve for a collection of qubits, especially for those separated far away, because the interbit coupling is usually much weaker than the intrabit coupling. Here we present an experimentally feasible method to effectively couple two flux qubits via a common inductance and treat both single and coupled flux qubits with more realistic models, which include the loop inductance. The main advantage of our proposal is that a strong interbit coupling can be achieved using a small inductance, so that two-bit as fast as one-bit operations can be easily realized. We also show the flux dependence of the transitions between states for the coupled flux qubits.
We propose an experimentally realizable method to control the coupling between two flux qubits. In our proposal, the bias fluxes are always fixed for these two inductively coupled qubits. The detuning of these two qubits can be initially chosen to be sufficiently large, so that their initial interbit coupling is almost negligible. When a variable frequency or time-dependent magnetic flux (TDMF) is applied to one of the qubits, a well-chosen frequency of the TDMF can be used to compensate the initial detuning and to couple two qubits. This proposed method avoids fast changes of either qubit frequencies or the amplitudes of the bias magnetic fluxes through the qubit loops, and also offers a remarkable way to implement any logic gate, as well as tomographically measure flux qubit states.
We investigate the galvanic coupling schemes of superconducting flux qubits. From the fundamental boundary conditions, we obtain the effective potential of the coupled system of two or three flux qubits to provide the exact Lagrangian of the system. While usually the two-qubit gate has been investigated approximately, in this study we derive the exact inductive coupling strength between two flux qubits coupled directly and coupled through a connecting central loop. We observe that the inductive coupling strength needs to be included exactly to satisfy the criteria of fault-tolerant quantum computing.
An effective Hamiltonian is derived for two coupled three-Josephson-junction (3JJ) qubits. This is not quite trivial, for the customary ``free'' 3JJ Hamiltonian is written in the limit of zero inductance LLL. Neglecting the self-flux is already dubious for one qubit when it comes to readout, and becomes untenable when discussing inductive coupling. First, inductance effects are analyzed for a single qubit. For small LLL, the self-flux is a ``fast variable,'' which can be eliminated adiabatically. However, the commonly used junction phases are not appropriate ``slow variables,'' and instead one introduces degrees of freedom that are decoupled from the loop current to leading order. In the quantum case, the zero-point fluctuations (LCLCLC oscillations) in the loop current diverge as $L\ensuremath{\rightarrow}0$. While their effect thus formally dominates over the classical self-flux, it merely renormalizes the Josephson couplings of the effective (two-phase) theory. In the coupled case, the strong zero-point fluctuations render the full (six-phase) wave function significantly entangled in leading order. However, in going to the four-phase theory, this uncontrollable entanglement is integrated out completely, leaving a computationally usable mutual-inductance term of the expected form as the effective interaction.
We propose a scheme of realization of fast quantum information transfer and two-qubit entangled state with two different superconducting flux qubits coupled to a single-mode resonator. In this approach, the two lowest levels of each Λ-type flux qubit serve as the logical states and a higher-energy intermediate level is used as the gate manipulation. This proposal does not require equal device parameters for the two flux qubits that is extremely difficult to achieve experimentally. In addition, neither a second-order detuning nor auxiliary qubits are required in this scheme. Moreover, there is no need to change the Rabi frequencies to satisfy the adiabatic passage. Therefore, the operation time can be shortened to 20–30 ns, which is faster by one order of magnitude compared to the schemes employing the second-order detuning technique.
The ability to switch the coupling between quantum bits (qubits) on and off is essential for implementing many quantum-computing algorithms. We demonstrated such control with two flux qubits coupled together through their mutual inductances and through the dc superconducting quantum interference device (SQUID) that reads out their magnetic flux states. A bias current applied to the SQUID in the zero-voltage state induced a change in the dynamic inductance, reducing the coupling energy controllably to zero and reversing its sign.
The faster speed and operational convenience of two-qubit gate with flux bias control makes it an important candidate for future large-scale quantum computers based on high coherence flux qubits. Based on a properly designed two-spin gadget which has small gaps during the evolution of energy levels, we build a CNOT-equivalent gate which can reach a fidelity larger than 99.9% within 40ns. Moreover, we also use the Schrieffer-Wolff Transformation to translate the spin model Ising coefficients schedule to circuit model flux bias schedule for realistic flux qubit circuits coupled by a tunable rf-squid. The two-qubit entangling gate scheme introduced here is suitable for realizing efficient two-qubit gates in the large scale flux qubit systems dominated by inductive couplings. Comparing with the current gate-based quantum computation systems dominated by capacitive couplings, it can resolve the conflict between the speed and a high coherence.