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arXiv:cond-mat/9906292 · Physical Review Letters

Fidelity and Leakage of Josephson Qubits

Rosario Fazio, G. Massimo Palma, Jens Siewert

Published Dec 20, 1999• 78 citations
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Abstract

The unit of quantum information is the qubit, a vector in a two-dimensional Hilbert space. On the other hand, quantum hardware often operates in two-dimensional subspaces of vector spaces of higher dimensionality. The presence of higher quantum states may affect the accuracy of quantum information processing. In this Letter we show how to cope with quantum leakage in devices based on small Josephson junctions. While the presence of higher charge states of the junction reduces the fidelity during gate operations we demonstrate that errors can be minimized by appropriately designing and operating the gates.

DOI
10.1103/physrevlett.83.5385

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The unit of quantum information is the qubit, a vector in a two-dimensional Hilbert space. On the other hand, quantum hardware often operates in two-dimensional subspaces of vector spaces of higher dimensionality. The presence of higher quantum states may affect the accuracy of quantum information processing. In this Letter we show how to cope with quantum leakage in devices based on small Josephson junctions. While the presence of higher charge states of the junction reduces the fidelity during gate operations we demonstrate that errors can be minimized by appropriately designing and operating the gates.
In this paper we calculate the loss of fidelity due to quantum leakage for the Josephson charge qubit (JCQ) in virtue of the Mathieu functions. It is shown that for an present typical parameters of JCQ E_{J}/E_{ch}~0.02, the loss of the fidelity per elementary operation is about 10^(-4) which satisfy the DiVincenzo's low decoherence criterion. By appropriately improving the design of the Josephson junction, namely, decreasing E_{J}/E_{ch} to 0.01, the loss of fidelity per elementary operation can decrease to 10^(-6) even smaller.
A quantum bit is a closed two-dimensional Hilbert space, but often experimental systems have three or more energy levels. In a Josephson phase qubit the energy differences between successive levels differ by only a few percent, and hence care must be taken to isolate the two desired levels from the remaining Hilbert space. Here we show via numerical simulations how to restrict operations to the qubit subspace of a three-level Josephson junction system requiring shorter time duration and suffering less error compared with traditional methods. This is achieved by employing amplitude modulated pulses as well as carefully designed sequences of square wave pulses. We also show that tunneling out of higher lying energy levels represents a significant source of decoherence that can be reduced by tuning the system to contain four or more energy levels.
Quantum optimal control theory allows us to design accurate quantum gates. We employ it to design high-fidelity two-bit gates for Josephson charge qubits in the presence of both leakage and noise. Our protocol considerably increases the fidelity of the gate and, more important, it is quite robust in the disruptive presence of 1/f noise. The improvement in the gate performances discussed in this work (errors approximately 10(-3)-10(-4) in realistic cases) allows us to cross the fault tolerance threshold.
In this paper we calculate the loss of fidelity due to quantum leakage for the Josephson charge qubit (JCQ). In this investigation the characteristic values and characteristic states of the Mathieu equation are used. It is shown that for present typical parameters of JCQ, E J /E ch ~ 0.02, the loss of the fidelity per elementary operation is about 10-4 which satisfies DiVincenzo's low decoherence criterion. By appropriately improving the designs of the Josephson junction, namely, decreasing E J /E ch to ~ 0.01, the loss of fidelity per elementary operation can decrease even smaller to 10-5. The first-order nonadiabatic correction is also obtained by using the approach.
Solid-state devices for quantum-bit computation (qubits) are not perfect isolated two-level systems since additional higher energy levels always exist. One example is the Josephson flux qubit, which consists on a mesoscopic superconducting quantum interference device loop with three Josephson junctions operated at or near a magnetic flux of half quantum. We study intrinsic leakage effects, i.e., direct transitions from the allowed qubit states to higher excited states of the system during the application of pulses for quantum computation operations. The system is started in the ground state and rf-magnetic field pulses are applied at the qubit resonant frequency with pulse intensity fp{f}_{p}fp​. A perturbative calculation of the average leakage for small fp{f}_{p}fp​ is performed for this case, obtaining that the leakage is quadratic in fp{f}_{p}fp​, and that it depends mainly on the matrix elements of the supercurrent. Numerical simulations of the time-dependent Schr\"odinger equation corresponding to the full Hamiltonian of this device were also performed. From the simulations we obtain the value of fp{f}_{p}fp​ above which the two-level approximation breaks down, and we estimate the maximum Rabi frequency that can be achieved. We study the leakage as a function of the ratio $\ensuremath{\alpha}$ among the Josephson energies of the junctions of the device, obtaining the best value for minimum leakage $(\ensuremath{\alpha}\ensuremath{\approx}0.85)$. The effects of flux noise on the leakage are also discussed.
Quantum-state engineering, i.e., active control over the coherent dynamics of suitable quantum-mechanical systems, has become a fascinating prospect of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. Low-capacitance Josephson tunneling junctions offer a promising way to realize quantum bits (qubits) for quantum information processing. The article reviews the properties of these devices and the practical and fundamental obstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux) degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltages in one case and by magnetic fields in the other case. Both kinds of device can be fabricated with present technology. In flux qubit devices, an important milestone, the observation of superpositions of different flux states in the system eigenstates, has been achieved. The Josephson charge qubit has even demonstrated coherent superpositions of states readable in the time domain. There are two major problems that must be solved before these devices can be used for quantum information processing. One must have a long phase coherence time, which requires that external sources of dephasing be minimized. The review discusses relevant parameters and provides estimates of the decoherence time. Another problem is in the readout of the final state of the system. This issue is illustrated with a possible realization by a single-electron transistor capacitively coupled to the Josephson device, but general properties of measuring devices are also discussed. Finally, the review describes how the basic physical manipulations on an ideal device can be combined to perform useful operations.