In the world of circuit quantum electrodynamics, the quest for qubit protection is of paramount importance. Researchers have been exploring various methods to extend the lifetime of qubits, and one promising approach involves the use of broadband bandpass Purcell filters. These filters, when properly implemented, can provide enhanced protection for qubits, allowing for longer coherence times and enabling more robust quantum information processing.

The typical setup for qubit readout circuits involves a coupled-mode configuration, as depicted in Figure 3(a). The strength of the coupling between the qubit and the readout resonator is denoted as *c _{q,r}*, while

*c*represents the coupling strength between the readout resonator and the

_{j,r}*j*-th filter stage. The qubit can decay through the readout resonator and the bandpass filter modes, affecting both the input and output ports.

When no Purcell filter is present, the qubit lifetime (represented as *T _{1,bare}*) is limited. However, by introducing a bandpass filter, the qubit lifetime can be significantly extended. In the original article, the formula for qubit lifetime in the presence of a Purcell filter is presented as:

T_{1} ∝ *Δ _{q,r}*

^{2k+2},

where *Δ _{q,r}* corresponds to the frequency detuning between the qubit and the readout resonator, and

*k*represents the order of the filter.

Further analysis reveals that the use of additional stages and larger qubit-resonator detuning can yield even longer qubit lifetimes. In fact, a higher insertion loss filter, such as one with 20 dB, offers better protection for the qubit compared to a filter with 0 dB insertion loss. This observation holds true for both symmetric and asymmetric filters, with the former exhibiting qubit lifetime scaling as *Δ _{q,r}*

^{2k+2}and the latter following a power-law scaling of

*Δ*

_{q,r}^{2N+2}.

It’s worth noting that as the number of stages increases, the qubit lifetime may deviate from the power-law scaling for the higher insertion loss filter, especially when the qubit-resonator detuning is large. This can be attributed to the finite coupling between the readout resonator and the first stage. However, by further increasing the insertion loss or reducing the coupling strength, the qubit lifetime can approach the expected power-law scaling.

In summary, this study illustrates the significance of additional stages in enhancing qubit protection with broadband bandpass Purcell filters. By carefully optimizing the filter design and considering factors such as insertion loss and qubit-resonator detuning, researchers can pave the way for improved coherence times and more efficient quantum information processing.

### Frequently Asked Questions (FAQ)

#### Q: What is the purpose of a broadband bandpass Purcell filter in circuit quantum electrodynamics?

The purpose of a broadband bandpass Purcell filter is to enhance qubit protection by extending the qubit lifetime, thereby enabling more robust quantum information processing.

#### Q: How does the qubit lifetime change when a Purcell filter is added?

The qubit lifetime increases significantly when a Purcell filter is added, offering better protection against decoherence.

#### Q: What factors affect the qubit lifetime in the presence of a bandpass filter?

The qubit lifetime in the presence of a bandpass filter is influenced by the order of the filter, the qubit-resonator detuning, and the insertion loss of the filter.

#### Q: Can the qubit lifetime be further improved by increasing the number of filter stages?

Yes, adding additional stages to the bandpass filter has been shown to enhance qubit protection and extend the qubit lifetime.

Sources:

– Original Article: [insert link to original article here]

– Supplementary Material: [insert link to supplementary material here]