Multi-Mode Purcell Filter Design

• Multi-mode Purcell filters are engineered electromagnetic structures that suppress qubit radiative decay while preserving strong coupling for readout and reset.
• They employ interference among multiple cavity modes, multi-pole networks, and spatial field engineering to shape the environment’s spectral response.
• Key architectures include shared broadband feedline, intrinsic notch, and multi-stage bandpass filters, enabling scalable multiplexed qubit readout and fast reset.

A multi-mode Purcell filter is an engineered electromagnetic environment that suppresses radiative qubit decay while preserving strong coupling where readout or reset must occur. In circuit QED this objective is naturally expressed through the dissipative part of the environment admittance, with $$\Gamma_{\rm P}(\omega_q)\propto \Re\{Y_{\rm env}(\omega_q)\}$$ and $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$; the filter therefore acts by making the environment predominantly reactive at qubit frequencies and transmissive or dissipative at selected resonator, readout, or reset frequencies (Escribano et al., 20 Apr 2026). Taken together, recent work indicates that “multi-mode” is used for several closely related mechanisms: interference among multiple cavity harmonics, explicit multi-pole or multi-stub filter networks, distributed-line notch structures, and spatial field engineering that uses the intrinsic mode structure of the device itself (Patel et al., 14 Mar 2025, Gu et al., 7 Jul 2025).

1. Theoretical basis

In its simplest form, Purcell decay arises when a qubit is dispersively coupled to a lossy readout resonator. The standard single-mode expression is $\gamma_q=\kappa_r(g/\Delta)^2$, with $\kappa_r$ the resonator linewidth, $g$ the qubit–resonator coupling, and $\Delta=\omega_q-\omega_r$ the qubit–resonator detuning (Patel et al., 14 Mar 2025). This formula is already sufficient to expose the core trade-off: increasing $\kappa_r$ accelerates measurement but also increases radiative decay through the readout channel.

The multi-mode generalization replaces a single dissipative pole by a structured environment containing several harmonic or filter modes. A standard description is

$$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$

with corresponding decay

$$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$

up to mode-dependent geometric factors and exact prefactors (Patel et al., 14 Mar 2025). In this form, a multi-mode Purcell filter is any network in which the combined modal structure generates strong suppression of the effective spectral density or admittance at $\omega_q$.

For explicit bandpass Purcell filters, the frequency dependence is often summarized by an effective linewidth. In a readout–filter–line chain, the readout mode acquires

$$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$0

so that the resonator can have a large linewidth $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$1 near the measurement band and a much smaller effective linewidth $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$2 at the qubit frequency. In that language, the suppression factor is $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$3, and the Purcell rate is reduced without changing the underlying dispersive coupling (Sete et al., 2015).

2. Suppression mechanisms

One major mechanism is direct admittance engineering. The shared $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$4-filter proposed for multiplexed superconducting qubits uses two open-ended stubs connected by an in-line transmission line, integrated directly into the feedline after the output coupling capacitor. Its purpose is to suppress $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$5 over a broad qubit band while leaving the readout and reset bands transmissive. In the transmission-line picture, the two stub resonances and the in-line phase delay are chosen so that the standing-wave patterns interfere constructively for suppression across the protected window; the paper identifies $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$6 as the constructive-interference choice and reports a broad stopband of approximately $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$7 GHz with Purcell-limited relaxation times exceeding $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$8 ms over the target region (Escribano et al., 20 Apr 2026).

A second mechanism is the synthesis of higher-order spectral responses below the first filter pole. In the sub-resonant linewidth-plateau approach, a high-pass ladder with alternating series capacitors and shunt inductors is operated so that the readout resonator band lies below the filter’s first resonant mode. The resulting admittances satisfy

$$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$9

which yields an approximately constant readout linewidth across a wide band together with steep low-frequency suppression of qubit decay (Smitham et al., 13 Mar 2025). This makes the filter broadband not because it places the qubit inside a narrow stopband, but because it shapes the sub-resonant admittance landscape into a plateau for readout and a steep roll-off for qubit protection.

A third mechanism is interference among a small number of nearby modes or paths. In one intrinsic three-mode realization, a distributed CPW resonator is engineered so that Mode A serves as a reset channel, Mode B serves as a readout bus, and Mode C acts as a $\gamma_q=\kappa_r(g/\Delta)^2$0 stub mode near the qubit frequency. The notch condition follows from the transfer impedance $\gamma_q=\kappa_r(g/\Delta)^2$1: choosing $\gamma_q=\kappa_r(g/\Delta)^2$2 makes $\gamma_q=\kappa_r(g/\Delta)^2$3, strongly reducing radiative decay at the qubit frequency while leaving the readout mode largely unaffected (Gu et al., 7 Jul 2025). Closely related interference logic appears in coupled readout–filter resonators with an auxiliary notch mode, where the readout mode, filter mode, and effective $\gamma_q=\kappa_r(g/\Delta)^2$4 notch mode together produce a zero in transfer impedance at the qubit frequency (Spring et al., 2024).

A fourth mechanism is spatial interference. In “waves-in-space Purcell effect” analyses, the relevant object is not merely the modal spectrum but the spatial structure of the qubit and cavity fields. A port placed at a location where the qubit field is weak or null while the cavity field is large can provide intrinsic Purcell protection without an additional filter resonator. The same general logic appears in distributed resonators with couplers placed so that the dressed-qubit mode has a node at the output coupler, thereby suppressing resonator-mediated qubit decay through destructive interference of multiple distributed modes (Patel et al., 14 Mar 2025, Sunada et al., 2022).

3. Principal architectural families

The literature supports several distinct realizations of the multi-mode Purcell-filter concept.

Architecture	Defining mechanism	Representative examples
Shared broadband feedline filter	Two nearby stub modes create a broad stopband in $\gamma_q=\kappa_r(g/\Delta)^2$5	(Escribano et al., 20 Apr 2026)
Multi-stage bandpass filter	Coupled resonant stages synthesize a flat passband and steep stopbands	(Yan et al., 2023, Park et al., 2023)
Intrinsic notch or three-mode filter	Readout, filter, and notch modes cancel transfer at $\gamma_q=\kappa_r(g/\Delta)^2$6	(Gu et al., 7 Jul 2025, Spring et al., 2024)
Distributed intrinsic filter	Coupler placement uses the resonator’s multi-mode structure to create a qubit-frequency node	(Sunada et al., 2022)
Spatial field filter	Port placement exploits weak qubit field and strong cavity field at the same location	(Patel et al., 14 Mar 2025)
Shared 3D cavity filter	A broad 3D cavity passband is combined with intrinsic Purcell filtering and notch engineering	(Bakr et al., 2024)
Mechanical ladder filter	A multi-pole acoustic ladder synthesizes a microwave bandpass environment	(Cleland et al., 2019)

Multi-stage electromagnetic filters are the most direct extension of classical microwave synthesis into circuit QED. A 4-pole Chebyshev bandpass filter implemented with four coupled spiral CPW resonators achieved measured passbands of $\gamma_q=\kappa_r(g/\Delta)^2$7 MHz and $\gamma_q=\kappa_r(g/\Delta)^2$8 MHz around $\gamma_q=\kappa_r(g/\Delta)^2$9 GHz and $\kappa_r$0 GHz, respectively, within a footprint of approximately $\kappa_r$1 mm$\kappa_r$2, and was analyzed as a way to support $\kappa_r$3 to $\kappa_r$4 readout resonators while strongly suppressing Purcell loss outside the passband (Park et al., 2023). Transmission-line implementations of multi-stage bandpass filters proceed from standard low-pass prototypes, convert them into coupled resonant stages, and then realize the couplings with short transmission-line sections; in that framework, adding stages simultaneously broadens the passband and steepens stopband suppression (Yan et al., 2023).

Shared filters emphasize hardware reduction. The shared $\kappa_r$5-filter is placed in the feedline after the output capacitor, so the same structure protects all qubits whose readout resonators dump into that line (Escribano et al., 20 Apr 2026). A related but tunable philosophy appears in broadband tunable filter architectures, where a shared $\kappa_r$6 filter can switch between a read-on regime with $\kappa_r$7 and a read-off regime with $\kappa_r$8, thereby changing the effective resonator linewidths across an entire multiplexed band (Xiong et al., 15 Sep 2025).

Intrinsic filters dispense with explicit extra resonators. Distributed resonators can be positioned so that the output coupler sits at a node of the dressed-qubit mode, leading to more than two orders of magnitude suppression over a $\kappa_r$9 MHz bandwidth (Sunada et al., 2022). Two-point feedline coupling can also produce a bandstop response through destructive interference between capacitive and inductive coupling paths, without dedicated filter elements or impedance mismatch in the feedline (Yen et al., 2024). A further intrinsic extension uses controlled geometric asymmetry in a transmon capacitor to activate mode–mode couplings inside the device itself, creating destructive interference among multiple internal decay pathways (Bakr et al., 13 Jul 2025).

4. Design methodologies and quantitative regimes

The most explicit broadband shared design rules are given by the shared $g$0-filter. One first chooses the qubit band to be protected, then selects stub lengths $g$1 so that their quarter-wave resonances $g$2 lie near the lower and upper band edges, and finally chooses the in-line length $g$3 to maximize constructive interference of the standing-wave patterns. For realistic parameters with $g$4 mm, $g$5 mm, and $g$6 mm, simulations show a broad window of enhanced $g$7 from approximately $g$8 to $g$9 GHz, with $\Delta=\omega_q-\omega_r$0 ms in the target band $\Delta=\omega_q-\omega_r$1–$\Delta=\omega_q-\omega_r$2 GHz and values reaching $\Delta=\omega_q-\omega_r$3 ms near the center; a double $\Delta=\omega_q-\omega_r$4-filter, placed at both ends of the feedline, can exceed $\Delta=\omega_q-\omega_r$5 s in the protected band (Escribano et al., 20 Apr 2026).

The intrinsic three-mode CPW architecture follows a different design logic. Mode A is placed below the qubit band and used as a dissipative reset channel, Mode B is placed in the readout band and used as a passband bus, and the geometry between the qubit coupling point and the output capacitor is chosen so that $\Delta=\omega_q-\omega_r$6, producing $\Delta=\omega_q-\omega_r$7. In the demonstrated device, Mode A and Mode B were extracted at $\Delta=\omega_q-\omega_r$8 GHz and $\Delta=\omega_q-\omega_r$9 GHz, while the intrinsic notch yielded $\kappa_r$0 ms over $\kappa_r$1–$\kappa_r$2 GHz and $\kappa_r$3 s at exact notch alignment (Gu et al., 7 Jul 2025).

Compact multi-pole filters are typically designed from classical prototypes. The 4-pole Chebyshev filters PF-C and PF-M targeted $\kappa_r$4 MHz and $\kappa_r$5 MHz bandwidths at $\kappa_r$6 GHz and $\kappa_r$7 GHz, and measured $\kappa_r$8 MHz and $\kappa_r$9 MHz around $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$0 GHz and $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$1 GHz, respectively. In finite-element Purcell analysis, one filter already produced a large figure of merit, while two filters in a two-port geometry gave $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$2 ms and $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$3 at $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$4 GHz (Park et al., 2023). The transmission-line multi-stage bandpass formulation reaches similar conclusions from a coupled-mode viewpoint: for asymmetric $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$5-stage filters, the paper finds $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$6, so higher order steepens the qubit-band suppression while preserving the readout passband (Yan et al., 2023).

Sub-resonant wideband filters replace passband placement near a filter pole by a linewidth plateau below the first pole. A 4th-order implementation coupled to four readout resonators at $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$7, $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$8, $$H=\frac{1}{2}\hbar\omega_q \sigma_z + \sum_n \hbar\omega_n a_n^\dagger a_n + \sum_n \hbar g_n (a_n + a_n^\dagger)(\sigma_+ + \sigma_-),$$9, and $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$0 GHz produced measured linewidths of $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$1, $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$2, $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$3, and $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$4 MHz, respectively, while preserving Purcell protection for a tunable qubit operated below the plateau (Smitham et al., 13 Mar 2025).

5. Readout, reset, multiplexing, and scalability

A defining advantage of multi-mode Purcell filters is that they can separate the qubit band from the readout or reset bands without sacrificing hardware efficiency. In the shared $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$5-filter architecture, a single feedline element protects all qubits in the engineered band, maintains the readout resonator frequency in a high-transmission region, and preserves a separate reset mode. Simulations of a four-qubit multiplexed layout show Purcell-limited $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$6 ms for all qubits parked within the approximately $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$7 GHz protected window (Escribano et al., 20 Apr 2026).

The same separation can be exploited for reset. In the intrinsic three-mode CPW filter, Mode A functions as a dissipative reset channel below the qubit band, Mode B acts as the readout bus, and Mode C supplies the Purcell notch. In that device, unconditional $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$8 reset reached residual excitation below $$\Gamma_{\text{Purcell}} \sim \sum_n \kappa_n \left|\frac{g_n}{\omega_q-\omega_n}\right|^2,$$9 in $\omega_q$0 ns, selective $\omega_q$1 leakage reduction reached $\omega_q$2 population in $\omega_q$3 ns, and cascaded $\omega_q$4 reset required $\omega_q$5 ns (Gu et al., 7 Jul 2025). Distributed intrinsic filters can reach similar operating points by combining a low-$\omega_q$6 resonator with an intrinsic notch: a resonator-mediated intrinsic filter demonstrated $\omega_q$7-ns readout with $\omega_q$8 fidelity and a $\omega_q$9-ns reset with residual excitation of less than $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$00 (Sunada et al., 2022).

Fast multiplexed readout is another central application. In a compact intrinsic three-mode notch architecture, effective readout-mode linewidths of $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$01–$$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$02 MHz enabled $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$03-ns simultaneous readout of four qubits with average assignment fidelity $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$04, and one channel exceeded $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$05 (Spring et al., 2024). In a tunable broadband shared-filter architecture, dynamic control of the filter reduced photon-noise-induced dephasing by a factor of $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$06 in idle status while allowing $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$07 single-shot fidelity with a $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$08 ns readout pulse, $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$09 in $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$10 ns using a multilevel protocol, $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$11 average fidelity for simultaneous three-qubit readout, and $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$12 QND fidelity over repeated measurements (Xiong et al., 15 Sep 2025).

Shared 3D implementations provide a complementary scaling route. A re-entrant cavity filter operating as a large-linewidth bandpass around $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$13 GHz with $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$14 dB bandwidth approximately $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$15 GHz coupled to four readout resonators at $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$16, $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$17, $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$18, and $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$19 GHz. In that device, single-qubit readout fidelity was $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$20–$$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$21, simultaneous four-qubit assignment was $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$22, and measurement-induced dephasing crosstalk remained below $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$23 kHz (Bakr et al., 2024). This suggests that out-of-plane shared filters can function as broadband, multi-channel Purcell filters without occupying on-chip area.

6. Distinctions, misconceptions, and limitations

A recurrent conceptual ambiguity concerns the phrase “multi-mode Purcell effect.” One meaning is the conventional interference between multiple cavity harmonics, which produces lifetime sweet spots between resonances and is most relevant when the qubit frequency lies between cavity modes, typically above the fundamental. Another meaning is the broader engineering of a structured environment with several filter, resonator, or spatial modes. The “waves-in-space Purcell effect” work emphasizes that, for qubits below the fundamental, a spatial field-null mechanism can dominate and is “quite distinct from the multi-mode Purcell effect” in the usual harmonic-interference sense (Patel et al., 14 Mar 2025). The distinction matters because below-fundamental protection may come from port placement rather than from cancellation between adjacent harmonics.

A second misconception is that multi-mode Purcell filters necessarily require extra filter resonators. Several recent results contradict that narrow interpretation. Interferometric two-point feedline coupling creates a notch in the effective admittance without dedicated filter components (Yen et al., 2024). Distributed resonators can realize intrinsic Purcell filters purely by coupler placement (Sunada et al., 2022). Controlled symmetry breaking in a transmon capacitor can activate internal mode–mode couplings and produce destructive interference among multiple decay paths, with one measured qubit showing average $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$24s while similar qubits on the same device showed $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$25, $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$26, and $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$27s (Bakr et al., 13 Jul 2025). Taken together, these results indicate that “multi-mode” may describe the modal structure of the whole electromagnetic device, not only added external filter stages.

The principal limitations are architecture dependent. Broadband shared filters can lose protection if resonances do not overlap or if the phase condition is mistuned; in the shared $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$28-filter, the paper notes that if the in-line length is chosen incorrectly the protected band shrinks or fragments (Escribano et al., 20 Apr 2026). Compact multi-pole filters are sensitive to packaging, kinetic inductance, and parasitic resonances; measured passbands of spiral Chebyshev filters were only reproduced quantitatively after including low-temperature dielectric constant and thin-film kinetic inductance, and two-port layouts showed unintended resonances that produced dips in $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$29 at specific detunings (Park et al., 2023). Sub-resonant plateau filters are also sensitive to package mismatch, which distorted the flatness of measured linewidths relative to the ideal ladder response (Smitham et al., 13 Mar 2025). Spatial filters can demand precise alignment of port position and qubit frequency; in WISPE-type geometries, the sweet spot is narrow, and high Purcell $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$30 requires precise placement (Patel et al., 14 Mar 2025).

The aggregate literature suggests a unifying view. A multi-mode Purcell filter is best understood not as a single circuit topology but as a design strategy: engineer the full structured environment so that $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$31 is strongly suppressed across the relevant qubit band while $$T_{\rm P}(\omega_q)=C_{\Sigma,q}/\Re\{Y_{\rm env}(\omega_q)\}$$32 remains large where readout, reset, or multiplexed transport is needed. Whether that strategy is realized by explicit multi-pole filters, shared broadband feedline structures, intrinsic notch modes, distributed resonator placement, or symmetry-broken internal modes is a matter of implementation rather than principle (Escribano et al., 20 Apr 2026, Patel et al., 14 Mar 2025, Bakr et al., 13 Jul 2025).