How does the anharmonicity of transmon qubits aid in selective addressing of energy levels, and what are the typical frequency ranges for (omega_{01}) and (omega_{12})?

by EITCA Academy / Tuesday, 11 June 2024 / Published in Artificial Intelligence, EITC/AI/TFQML TensorFlow Quantum Machine Learning, Quantum supremacy, Control of transmon qubits using a cryogenic CMOS integrated circuit, Examination review

The anharmonicity of transmon qubits plays a pivotal role in the selective addressing of energy levels, which is important for their effective operation in quantum computing systems. To understand this, one must consider the intrinsic properties of transmon qubits, their energy level structure, and how anharmonicity facilitates precise control over quantum states.

Transmon qubits are a type of superconducting qubit that are designed to mitigate charge noise, which is a significant source of decoherence in superconducting circuits. They achieve this by operating in a regime where the ratio of the Josephson energy ( $E_J$ ) to the charging energy ( $E_C$ ) is large, typically $E_J/E_C \gg 1$ . This design choice results in a weakly anharmonic oscillator, meaning that the energy level spacings are not uniform.

Anharmonicity refers to the deviation of a system from being a perfect harmonic oscillator, where the energy levels are equally spaced. In a harmonic oscillator, the energy levels are given by $E_n = \hbar \omega (n + 1/2)$ , where $n$ is an integer, $\hbar$ is the reduced Planck constant, and $\omega$ is the angular frequency. In contrast, for an anharmonic oscillator like a transmon qubit, the energy difference between successive levels decreases as one moves up the energy ladder.

For a transmon qubit, the energy levels $E_n$ can be approximated as:

$E_n \approx -E_J + \sqrt{8 E_J E_C} \left( n + \frac{1}{2} \right) - \frac{E_C}{12} \left(6n^2 + 6n + 3\right).$

The anharmonicity $\alpha$ is defined as the difference in energy between the first and second transitions:

$\alpha = (E_2 - E_1) - (E_1 - E_0).$

This anharmonicity is typically negative, indicating that the energy difference between higher levels is smaller than that between lower levels. The presence of anharmonicity is beneficial for selectively addressing specific transitions between energy levels, which is essential for implementing quantum logic gates and controlling qubit states.

Selective Addressing of Energy Levels

In quantum computing, operations on qubits are performed by applying microwave pulses at specific frequencies corresponding to the energy differences between qubit states. For a transmon qubit, the primary transition of interest is between the ground state $|0\rangle$ and the first excited state $|1\rangle$ . This transition frequency is denoted as $\omega_{01}$ .

The anharmonicity ensures that the transition frequency $\omega_{12}$ between the first excited state $|1\rangle$ and the second excited state $|2\rangle$ is different from $\omega_{01}$ . Specifically, $\omega_{12}$ is lower than $\omega_{01}$ due to the anharmonic nature of the transmon qubit. This separation allows for selective addressing of the $|0\rangle \leftrightarrow |1\rangle$ transition without inadvertently exciting the $|1\rangle \leftrightarrow |2\rangle$ transition.

Mathematically, the transition frequencies can be expressed as:

$\omega_{01} = \frac{E_1 - E_0}{\hbar},$

$\omega_{12} = \frac{E_2 - E_1}{\hbar}.$

Due to the anharmonicity, we have $\omega_{12} < \omega_{01}$ . This difference in frequencies is important for the implementation of high-fidelity quantum gates, as it prevents leakage to higher energy states during qubit manipulation.

Typical Frequency Ranges

The typical frequency ranges for $\omega_{01}$ and $\omega_{12}$ in transmon qubits are determined by the specific design parameters, such as the Josephson energy $E_J$ and the charging energy $E_C$ . Generally, transmon qubits are designed to have $\omega_{01}$ in the range of 4 to 8 GHz. This range is chosen to balance the need for high coherence times and the practical considerations of microwave control and readout.

Given the anharmonicity, the $\omega_{12}$ frequency is slightly lower than $\omega_{01}$ . The anharmonicity $\alpha$ is typically on the order of several hundred MHz, often around -200 to -300 MHz. Therefore, if $\omega_{01}$ is approximately 5 GHz, $\omega_{12}$ would be around 4.7 to 4.8 GHz.

Practical Implications and Examples

To illustrate the practical implications of anharmonicity in transmon qubits, consider the following example:

Suppose we have a transmon qubit with $\omega_{01} = 5 \text{ GHz}$ and an anharmonicity $\alpha = -300 \text{ MHz}$ . The transition frequency $\omega_{12}$ would then be:

$\omega_{12} = \omega_{01} + \alpha = 5 \text{ GHz} - 0.3 \text{ GHz} = 4.7 \text{ GHz}.$

When performing a quantum gate operation, such as a single-qubit rotation, a microwave pulse at 5 GHz would be used to drive the $|0\rangle \leftrightarrow |1\rangle$ transition. The anharmonicity ensures that this pulse does not significantly excite the $|1\rangle \leftrightarrow |2\rangle$ transition, which would require a pulse at 4.7 GHz. This selective addressing is critical for maintaining the integrity of the qubit state and achieving high-fidelity gate operations.

Moreover, the anharmonicity also aids in the implementation of two-qubit gates, such as the controlled-NOT (CNOT) gate or the cross-resonance gate. These gates often rely on the interaction between qubits, mediated by their respective energy levels. The distinct transition frequencies due to anharmonicity help in tuning the interactions and minimizing cross-talk between qubits, which is essential for scalable quantum computing.

Conclusion

The anharmonicity of transmon qubits is a fundamental property that facilitates the selective addressing of energy levels, enabling precise control over qubit states and high-fidelity quantum gate operations. By ensuring that the transition frequencies $\omega_{01}$ and $\omega_{12}$ are distinct, anharmonicity prevents unwanted excitations and leakage to higher energy states, which is important for the reliable operation of quantum computing systems. Typical frequency ranges for $\omega_{01}$ are in the range of 4 to 8 GHz, with $\omega_{12}$ being slightly lower due to the anharmonicity, typically by a few hundred MHz. This separation of frequencies is essential for the effective control and manipulation of transmon qubits in practical quantum computing applications.

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How does the anharmonicity of transmon qubits aid in selective addressing of energy levels, and what are the typical frequency ranges for (omega_{01}) and (omega_{12})?

Selective Addressing of Energy Levels

Typical Frequency Ranges

Practical Implications and Examples

Conclusion

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How does the anharmonicity of transmon qubits aid in selective addressing of energy levels, and what are the typical frequency ranges for (omega_{01}) and (omega_{12})?

Selective Addressing of Energy Levels

Typical Frequency Ranges

Practical Implications and Examples

Conclusion

Other recent questions and answers regarding Control of transmon qubits using a cryogenic CMOS integrated circuit:

More questions and answers: