Four-Pole Broadband Purcell Filter

• The four-pole broadband Purcell filter is a fourth-order bandpass network that enables fast, high-fidelity qubit readout while suppressing decay via the Purcell effect.
• Various implementations, including cascaded quarter-wave resonators and spiral CPW networks, use Butterworth and Chebyshev synthesis methods to achieve a flat, wide passband with strong off-band attenuation.
• Experimental results show minimal passband ripple (<0.5 dB), high isolation (>20 to 50 dB), and significant improvements in qubit readout speed and fidelity, validating its practical application in superconducting circuits.

Searching arXiv for papers on four-pole broadband Purcell filters and related superconducting-circuit implementations. A four-pole broadband Purcell filter is a fourth-order band-pass network used in superconducting quantum circuits to reconcile two requirements that are ordinarily in tension: strong coupling between a readout resonator and a feedline for fast, high-fidelity measurement, and suppression of qubit decay through the Purcell effect. In the recent literature, the same functional objective is realized through several closely related microwave architectures, including cascades of four coupled quarter-wave resonators, direct-coupled spiral CPW networks, transmission-line band-pass filters synthesized from prototype coefficients, and tunable filter-plus-readout networks whose effective response is also four-pole. Across these variants, the recurring design target is a flat, wide readout passband together with strong rejection at typical qubit frequencies and compatibility with frequency-multiplexed readout (Luo et al., 17 Mar 2026, Yan et al., 2023, Park et al., 2023, Xiong et al., 15 Sep 2025).

1. Functional role in circuit QED

Fast and high-fidelity qubit readout requires strong coupling between the readout resonator and the feedline, but such coupling unavoidably enhances qubit decay through the Purcell effect. Broadband band-pass Purcell filters are introduced precisely to overcome or relax this trade-off while preserving measurement speed and multiplexing capacity (Luo et al., 17 Mar 2026, Yan et al., 2023). In the planar compact-footprint work, the same motivation is formulated as engineering the admittance of external environments connected to superconducting qubits so that Purcell loss is suppressed without losing the fast measurement speed (Park et al., 2023). In the tunable implementation, the problem is broadened further to include photon-noise-induced dephasing in idle operation, with the filter dynamically reconfigured between read-off and read-on states (Xiong et al., 15 Sep 2025).

In this setting, “four-pole” denotes a fourth-order passband response. A plausible implication is that the defining characteristic is the pole structure of the transfer function rather than a unique physical layout. This is consistent with the fact that one implementation realizes the four poles as four coupled quarter-wave resonators, whereas another describes a four-pole network composed of two filter poles and two load poles associated with readout resonators (Luo et al., 17 Mar 2026, Xiong et al., 15 Sep 2025).

2. Network topologies and pole realizations

In a 3D flip-chip realization, the broadband Purcell filter is implemented by four coupled quarter-wave resonators: two compact spiral resonators at the ends and two unfolded CPW $\lambda/4$ lines in the center, strongly coupled in cascade. The end spirals have $f_{01}\approx 7.68\,\mathrm{GHz}$ and $f_{04}\approx 7.68\,\mathrm{GHz}$, while the central resonators have $f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$. The resulting four-pole cascade produces a $1\,\mathrm{GHz}$ flat passband from $7.20$ to $8.20\,\mathrm{GHz}$ with ripple $<0.5\,\mathrm{dB}$ (Luo et al., 17 Mar 2026).

A different fourth-order realization uses four identical parallel resonators at the center frequency $\omega_0$, coupled by five admittance inverters $J_1\ldots J_5$. In that formulation, the first and last inverters set the input and output coupling to the $f_{01}\approx 7.68\,\mathrm{GHz}$0 feedlines, and the filter is synthesized as a 4th-order maximally-flat Butterworth band-pass. For $f_{01}\approx 7.68\,\mathrm{GHz}$1 and $f_{01}\approx 7.68\,\mathrm{GHz}$2, the reported fractional bandwidth is $f_{01}\approx 7.68\,\mathrm{GHz}$3 (Yan et al., 2023).

The compact planar spiral-CPW implementation is described as a 4-pole Chebyshev response with $f_{01}\approx 7.68\,\mathrm{GHz}$4 ripple, realized by four $f_{01}\approx 7.68\,\mathrm{GHz}$5 spiral CPW resonators of characteristic impedance $f_{01}\approx 7.68\,\mathrm{GHz}$6. For a design around $f_{01}\approx 7.68\,\mathrm{GHz}$7 and $f_{01}\approx 7.68\,\mathrm{GHz}$8, the text gives $f_{01}\approx 7.68\,\mathrm{GHz}$9 and lists the corresponding prototype-derived couplings and external quality factor (Park et al., 2023).

The tunable broadband filter maps onto a four-pole band-pass network in which two poles are realized by a $f_{04}\approx 7.68\,\mathrm{GHz}$0 filter section with bare inductance $f_{04}\approx 7.68\,\mathrm{GHz}$1 and capacitance $f_{04}\approx 7.68\,\mathrm{GHz}$2, and two additional poles are realized by $f_{04}\approx 7.68\,\mathrm{GHz}$3 readout resonators coupled to the filter via small capacitors $f_{04}\approx 7.68\,\mathrm{GHz}$4. A SQUID inductance $f_{04}\approx 7.68\,\mathrm{GHz}$5 terminates one end of the filter and tunes both the center frequency and filter $f_{04}\approx 7.68\,\mathrm{GHz}$6 (Xiong et al., 15 Sep 2025).

3. Synthesis, coupling-matrix methods, and transfer functions

One analytical route employs a coupling-matrix description. In the 3D flip-chip filter, a $f_{04}\approx 7.68\,\mathrm{GHz}$7 matrix $f_{04}\approx 7.68\,\mathrm{GHz}$8 is assembled, with $f_{04}\approx 7.68\,\mathrm{GHz}$9 filter resonators plus source and load, and entries

$f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$0

together with

$f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$1

Using the normalized low-pass frequency

$f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$2

the network matrix is

$f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$3

and the scattering parameters are

$f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$4

In that synthesis, the four poles are the eigenvalues of $f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$5, and they are placed by choosing the mutual couplings $f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$6, $f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$7, $f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$8 and external couplings $f_{02}\approx f_{03}\approx 7.70\,\mathrm{GHz}$9 according to standard Chebyshev-prototype or maximally flat synthesis for a $1\,\mathrm{GHz}$0 ($1\,\mathrm{GHz}$1 FBW) bandpass (Luo et al., 17 Mar 2026).

The same work gives a geometry-driven synthesis procedure. The design chooses $1\,\mathrm{GHz}$2 and $1\,\mathrm{GHz}$3, determines resonator lengths from

$1\,\mathrm{GHz}$4

uses conformal mapping to compute $1\,\mathrm{GHz}$5 and $1\,\mathrm{GHz}$6 from $1\,\mathrm{GHz}$7, $1\,\mathrm{GHz}$8, and $1\,\mathrm{GHz}$9, extracts mutual coupling $7.20$0 from two-resonator eigenmode analysis via $7.20$1, and sets $7.20$2 from the tapped-line position $7.20$3 using

$7.20$4

The reported kinetic-inductance ratio is $7.20$5 for $7.20$6 Nb on Si (Luo et al., 17 Mar 2026).

Prototype-coefficient synthesis appears in both Butterworth and Chebyshev variants. For the $7.20$7 Butterworth filter, the couplings and port decay rates are

$7.20$8

with quoted products $7.20$9, $8.20\,\mathrm{GHz}$0, $8.20\,\mathrm{GHz}$1, $8.20\,\mathrm{GHz}$2, and $8.20\,\mathrm{GHz}$3 (Yan et al., 2023). For the 4-pole Chebyshev filter with $8.20\,\mathrm{GHz}$4 ripple, the normalized prototype values are listed as $8.20\,\mathrm{GHz}$5, $8.20\,\mathrm{GHz}$6, $8.20\,\mathrm{GHz}$7, $8.20\,\mathrm{GHz}$8, $8.20\,\mathrm{GHz}$9, and $<0.5\,\mathrm{dB}$0, which lead to

$<0.5\,\mathrm{dB}$1

with the corresponding symmetric end couplings (Park et al., 2023).

In the tunable implementation, the filter susceptibility is written as

$<0.5\,\mathrm{dB}$2

and, to leading order in weak coupling,

$<0.5\,\mathrm{dB}$3

The SQUID sets

$<0.5\,\mathrm{dB}$4

and the paper also writes

$<0.5\,\mathrm{dB}$5

This formalism connects the four-pole response directly to dynamic tuning between narrow and broad filter states (Xiong et al., 15 Sep 2025).

4. Physical implementations and fabrication strategies

The reported physical realizations span flip-chip, planar CPW, and distributed transmission-line formats. The 3D flip-chip design uses a two-chip stack of $<0.5\,\mathrm{dB}$6 Si chips separated by SU-8 spacers with $<0.5\,\mathrm{dB}$7 and bonded with indium bump bonds. The bottom chip carries the feedline and spirals; the top chip carries CPW lines and readout resonators. Metallization is $<0.5\,\mathrm{dB}$8 Nb patterned by standard lithography, and the total footprint is reported as $<0.5\,\mathrm{dB}$9 (Luo et al., 17 Mar 2026).

The compact planar design uses a spiral-CPW realization in a single Nb layer and standard CPW processing. Its footprint is reported as $\omega_0$0, and the text states that the filter integrates seamlessly into typical qubit-readout chips without new fabrication steps (Park et al., 2023). The transmission-line Butterworth implementation realizes each inverter $\omega_0$1 as a short-circuited stub of electrical length $\omega_0$2 and each resonator element as a $\omega_0$3 line of length $\omega_0$4, with $\omega_0$5 and synthesis formulas given for $\omega_0$6, $\omega_0$7, and $\omega_0$8 (Yan et al., 2023). The tunable design uses a distributed-element $\omega_0$9 geometry with a SQUID-terminated filter section and parallel-coupled readout resonators (Xiong et al., 15 Sep 2025).

Work	Realization	Reported fabrication or layout feature
(Luo et al., 17 Mar 2026)	3D flip-chip, four coupled $J_1\ldots J_5$0 resonators	Two $J_1\ldots J_5$1 Si chips, SU-8 spacers, indium bump bonds, $J_1\ldots J_5$2 Nb
(Park et al., 2023)	Spiral-CPW 4-pole Chebyshev	$J_1\ldots J_5$3 footprint, one Nb layer, standard CPW processing
(Yan et al., 2023)	Transmission-line Butterworth	Short-circuited stubs plus CPW $J_1\ldots J_5$4 lines
(Xiong et al., 15 Sep 2025)	Tunable $J_1\ldots J_5$5 filter plus $J_1\ldots J_5$6 readout poles	SQUID-terminated distributed filter

Taken together, these implementations show that broadband four-pole filtering is not tied to a single packaging technology. A plausible implication is that the choice among flip-chip, planar compact, and SQUID-tunable formats is governed mainly by whether the priority is footprint, analytical tractability from geometry, or dynamic control of the readout environment.

5. Measured response, suppression, and multiplexed readout

The 3D flip-chip device was characterized at $J_1\ldots J_5$7. HFSS simulation gave a flat $J_1\ldots J_5$8 $J_1\ldots J_5$9 from $f_{01}\approx 7.68\,\mathrm{GHz}$00 to $f_{01}\approx 7.68\,\mathrm{GHz}$01 and $f_{01}\approx 7.68\,\mathrm{GHz}$02 suppression at $f_{01}\approx 7.68\,\mathrm{GHz}$03. The measured raw insertion was $f_{01}\approx 7.68\,\mathrm{GHz}$04 due to wiring loss; after correction, the passband was flat to $f_{01}\approx 7.68\,\mathrm{GHz}$05 with $f_{01}\approx 7.68\,\mathrm{GHz}$06 width and center at $f_{01}\approx 7.68\,\mathrm{GHz}$07, corresponding to a $f_{01}\approx 7.68\,\mathrm{GHz}$08 downshift from kinetic inductance. At qubit frequency near $f_{01}\approx 7.68\,\mathrm{GHz}$09, the stopband showed $f_{01}\approx 7.68\,\mathrm{GHz}$10 suppression. The same chip integrated six floating readout resonators at $f_{01}\approx 7.68\,\mathrm{GHz}$11–$f_{01}\approx 7.68\,\mathrm{GHz}$12 with $f_{01}\approx 7.68\,\mathrm{GHz}$13 spacing, three coupled to CPW#2 and three to CPW#3, and the reported fit errors were $f_{01}\approx 7.68\,\mathrm{GHz}$14 versus simulation and $f_{01}\approx 7.68\,\mathrm{GHz}$15 except one (Luo et al., 17 Mar 2026).

The tunable broadband filter reported a broad on-state response of $f_{01}\approx 7.68\,\mathrm{GHz}$16, insertion loss in the passband below $f_{01}\approx 7.68\,\mathrm{GHz}$17, and stopband attenuation exceeding $f_{01}\approx 7.68\,\mathrm{GHz}$18 within $f_{01}\approx 7.68\,\mathrm{GHz}$19 of $f_{01}\approx 7.68\,\mathrm{GHz}$20. Dynamic tuning suppressed photon-noise-induced dephasing by a factor of $f_{01}\approx 7.68\,\mathrm{GHz}$21 in idle status, with $f_{01}\approx 7.68\,\mathrm{GHz}$22 increasing from $f_{01}\approx 7.68\,\mathrm{GHz}$23 to $f_{01}\approx 7.68\,\mathrm{GHz}$24. In measurement status, the filter enabled $f_{01}\approx 7.68\,\mathrm{GHz}$25 single-shot readout fidelity with a $f_{01}\approx 7.68\,\mathrm{GHz}$26 pulse and $f_{01}\approx 7.68\,\mathrm{GHz}$27 fidelity in $f_{01}\approx 7.68\,\mathrm{GHz}$28 using a multilevel readout protocol. Simultaneous readout of three qubits using $f_{01}\approx 7.68\,\mathrm{GHz}$29 pulses achieved an average fidelity of $f_{01}\approx 7.68\,\mathrm{GHz}$30 with $f_{01}\approx 7.68\,\mathrm{GHz}$31 crosstalk, while repeated measurements gave $f_{01}\approx 7.68\,\mathrm{GHz}$32 QND fidelity and leakage below $f_{01}\approx 7.68\,\mathrm{GHz}$33 (Xiong et al., 15 Sep 2025).

The compact planar 4-pole filter was characterized at $f_{01}\approx 7.68\,\mathrm{GHz}$34 with a calibrated VNA. For PF-C, the reported $f_{01}\approx 7.68\,\mathrm{GHz}$35 band extends from $f_{01}\approx 7.68\,\mathrm{GHz}$36 to $f_{01}\approx 7.68\,\mathrm{GHz}$37, giving $f_{01}\approx 7.68\,\mathrm{GHz}$38 and $f_{01}\approx 7.68\,\mathrm{GHz}$39, with pass-band ripple $f_{01}\approx 7.68\,\mathrm{GHz}$40 and attenuation $f_{01}\approx 7.68\,\mathrm{GHz}$41 just outside the band. FEM-based Purcell estimates gave, for two-port readout with two filters, $f_{01}\approx 7.68\,\mathrm{GHz}$42 at $f_{01}\approx 7.68\,\mathrm{GHz}$43 and $f_{01}\approx 7.68\,\mathrm{GHz}$44 (Park et al., 2023).

The Butterworth bandpass implementation reported measured transmission with $f_{01}\approx 7.68\,\mathrm{GHz}$45 points at $f_{01}\approx 7.68\,\mathrm{GHz}$46 and $f_{01}\approx 7.68\,\mathrm{GHz}$47, i.e. a $f_{01}\approx 7.68\,\mathrm{GHz}$48 bandwidth centered at $f_{01}\approx 7.68\,\mathrm{GHz}$49, passband ripple $f_{01}\approx 7.68\,\mathrm{GHz}$50, and off-band isolation below $f_{01}\approx 7.68\,\mathrm{GHz}$51 beyond $f_{01}\approx 7.68\,\mathrm{GHz}$52 from $f_{01}\approx 7.68\,\mathrm{GHz}$53. With the filter, qubit $f_{01}\approx 7.68\,\mathrm{GHz}$54 reached $f_{01}\approx 7.68\,\mathrm{GHz}$55, whereas the Purcell limit without the filter would be $f_{01}\approx 7.68\,\mathrm{GHz}$56 at $f_{01}\approx 7.68\,\mathrm{GHz}$57. Four qubits with readout resonators at $f_{01}\approx 7.68\,\mathrm{GHz}$58, $f_{01}\approx 7.68\,\mathrm{GHz}$59, $f_{01}\approx 7.68\,\mathrm{GHz}$60, and $f_{01}\approx 7.68\,\mathrm{GHz}$61 were read out simultaneously across the $f_{01}\approx 7.68\,\mathrm{GHz}$62 band (Yan et al., 2023).

6. Trade-offs, tolerances, and broader interpretation

Fabrication tolerance is treated as a central design parameter rather than a secondary consideration. In the 3D flip-chip device, the measured chip spacing was $f_{01}\approx 7.68\,\mathrm{GHz}$63, leading to $f_{01}\approx 7.68\,\mathrm{GHz}$64 variations of approximately $f_{01}\approx 7.68\,\mathrm{GHz}$65, while over-the-air capacitive coupling between spiral-to-line patches was described as robust against lateral misalignments. The same analytical model includes $f_{01}\approx 7.68\,\mathrm{GHz}$66 through conformal mapping, allowing geometry-to-response prediction to include packaging variation explicitly (Luo et al., 17 Mar 2026).

The tunable filter introduces a different set of trade-offs. Resonator $f_{01}\approx 7.68\,\mathrm{GHz}$67 is ultimately limited by the minimum SQUID inductance, producing an ON/OFF ratio of about $f_{01}\approx 7.68\,\mathrm{GHz}$68. Larger $f_{01}\approx 7.68\,\mathrm{GHz}$69 speeds readout but increases back-action and leakage; the reported leakage is $f_{01}\approx 7.68\,\mathrm{GHz}$70. The off-state linewidth $f_{01}\approx 7.68\,\mathrm{GHz}$71 must remain $f_{01}\approx 7.68\,\mathrm{GHz}$72 to protect data qubits against parasitic measurement-induced dephasing, and cross-talk is suppressed by ensuring $f_{01}\approx 7.68\,\mathrm{GHz}$73 detuning among resonators together with a flat passband (Xiong et al., 15 Sep 2025).

The filter order changes the asymptotic suppression of Purcell decay. In the bare case, the Butterworth paper writes

$f_{01}\approx 7.68\,\mathrm{GHz}$74

whereas with an $f_{01}\approx 7.68\,\mathrm{GHz}$75 bandpass filter the far off-resonant limit gives $f_{01}\approx 7.68\,\mathrm{GHz}$76 for the symmetric design (Yan et al., 2023). This suggests that the practical value of four-pole filtering lies not only in widening the usable readout band, but also in reshaping the qubit’s electromagnetic environment much more steeply away from the passband.

A common oversimplification is to treat Purcell protection and broadband multiplexing as mutually exclusive. The cited works do not present the trade-off as eliminated in a universal sense, but they do show several strategies for relaxing it: adding filter poles, using maximally-flat or Chebyshev synthesis, reducing footprint so that more filtering can be deployed, and introducing tunability so that idle and measurement conditions are not forced to share the same linewidth (Luo et al., 17 Mar 2026, Yan et al., 2023, Park et al., 2023, Xiong et al., 15 Sep 2025).