WO2024189298A1

WO2024189298A1 - Multiplexer for multiplexing signals from a plurality of readout resonators, and a circuit qed apparatus

Info

Publication number: WO2024189298A1
Application number: PCT/GB2023/050569
Authority: WO
Inventors: Mustafa Safaa Ahmed Bakr; Peter James LEEK
Original assignee: Oxford University Innovation Limited
Priority date: 2023-03-10
Filing date: 2023-03-10
Publication date: 2024-09-19

Abstract

A multiplexer is disclosed for multiplexing signals from a plurality of readout resonators of a circuit QED device. In one arrangement, the multiplexer comprises a cavity resonator system comprising one or more inner conductors and an outer conductor. The outer conductor comprises a plurality of openings and surrounds the one or more inner conductors in directions other than towards the openings. A signal port system is coupled to the cavity resonator system and configured to receive multiplexed signals from the readout resonators. The readout resonators are provided in a common plane. One of the inner conductors is configured to capacitively couple to the readout resonators through respective openings in a direction extending perpendicularly or obliquely relative to the plane.

Description

MULTIPLEXER FOR MULTIPLEXING SIGNALS FROM A PLURALITY OF

READOUT RESONATORS, AND A CIRCUIT QED APPARATUS

The present disclosure relates to multiplexing signals from readout resonators of a circuit quantum electrodynamics (QED) device, particularly in the context of qubit control and readout.

To achieve higher computing powers, it is desirable to provide architectures that allow selective control and readout of large numbers of qubits. As the number of qubits increases so do the challenges associated with handling signals for control and readout. Space may be limited, particularly in two-dimensional arrangements, and it is also desirable to minimize heat loads and signal cross-talk.

Three-dimensional connectivity has been proposed to address space restrictions and reduce cross-talk. For example, J. Rahamim et al., “Double-sided coaxial circuit QED with out-of-plane wiring”, Appl. Phys. Lett. 110, 222602 (2017), discloses a double-sided coaxial circuit QED arrangement with out-of-plane wiring.

Multiplexing may be used to reduce the overall number of connections that need to be made for control and/or readout.

In circuit QED devices, qubit states can be measured by applying resonant probe pulses to respective readout resonators and detecting the responses, such as the phases of reflections. A challenge with this approach to manipulating qubits is that the readout resonators can mediate decay of the qubits, thereby undesirably shortening qubit lifetimes and reducing the performance of qubits, such as by degrading the fidelity of operations on the qubit and/or measurements made on the qubit. The problem can become worse if the readout circuitry is configured to provide rapid readout speed, meaning a balance may need to be made between desirably increasing readout speed and undesirably decreasing qubit lifetimes.

Purcell filtering can be used to suppress resonator-mediated qubit decay, but this can be complex to implement, particularly where large numbers of qubits are being addressed and/or where it is desired to use multiplexing.

It is an object of the invention to improve readout from qubits in circuit QED devices.

According to an aspect of the invention, there is provided a multiplexer for multiplexing signals from a plurality of readout resonators of a circuit quantum electrodynamics, QED, device, the multiplexer comprising: a cavity resonator system comprising one or more inner conductors and an outer conductor, the outer conductor comprising a plurality of openings and surrounding the one or more inner conductors in directions other than towards the openings; and a signal port system coupled to the cavity resonator system and configured to receive multiplexed signals from the readout resonators, wherein: the readout resonators are provided in a common plane; and one of the inner conductors is configured to capacitively couple to the readout resonators through respective openings in a direction extending perpendicularly or obliquely relative to the plane.

Using a cavity resonator system configured to provide capacitive couplings to the readout resonators perpendicularly or obliquely in this manner allows the multiplexer to couple efficiently to the qubits in an out of plane direction. The multiplexer can thus implement multiplexing without requiring additional connections or structure to be incorporated into the plane of the readout resonators, thereby avoiding adding complexity within the plane of the readout resonators. There may additionally be more freedom for building structure into the multiplexer in the out of plane direction, thereby allowing properties of the multiplexer to be tuned more freely, thereby facilitating high performance. For example, the inner and/or outer conductors of the cavity resonator system can be configured to provide a wide and/or low attenuation pass band for signal frequencies to be used in multiplexing while at the same time providing effective Purcell filtering of qubit frequencies, all in the same structure.

In an embodiment, the multiplexer comprises a plurality of pillars configured to provide the capacitive couplings to the readout resonators, each pillar connected to one of the inner conductors and configured to extend perpendicularly or obliquely relative to the plane from the inner conductor towards a respective one of the readout resonators and to capacitively couple to the readout resonator through a respective one of the openings. Using pillars facilitates providing a strong capacitive coupling and/or provides a further degree of freedom for varying the capacitive coupling.

In an embodiment, the readout resonators are provided in a regular array defined by repeating instances of a unit cell containing a single readout resonator. A size of a crosssection of the outer conductor in a plane parallel to the common plane of the readout resonators may be substantially equal to or smaller than a total size of the unit cells containing the plurality of readout resonators. Thus, outer conductors corresponding to different pluralities of the readout resonators can be tiled in a plane parallel to the common plane of the readout resonators. This makes efficient use of space and facilitates scaling of the arrangement to large numbers of qubits.

In an embodiment, the outer conductor comprises a conductive shield associated with each of one or more of the pillars, each conductive shield extending adjacent to an opening corresponding to the pillar in a direction parallel to a longitudinal axis of the pillar, the conductive shield being positioned between the pillar and at least one other pillar so as to electrically shield the pillar from the at least one other pillar, the conductive shield optionally surrounding the pillar in all directions perpendicular to the longitudinal axis of the pillar. Providing such conductive shields reduces or avoids cross-talk between signals from neighbouring readout resonators.

In an embodiment, the one or more inner conductors comprises a plurality of the inner conductors. Providing additional inner conductors increases freedom for designing the frequency response of the multiplexer, thereby allowing improvements to Purcell filtering, such as by increasing a number of transmission minima in the frequency response.

In an embodiment, two or more of the inner conductors are spaced apart along a direction perpendicular to the common plane of the readout resonators. Stacking the inner conductors in this manner, along a direction perpendicular to the common plane of the readout resonators, provides extra degrees of freedom for tuning the frequency response of the multiplexer without requiring extra space in directions parallel to the plane of the readout resonators, thus avoiding encroachment on space required for processing signals from other neighbouring groups of qubits. The performance of a multiplexer for a group of qubits can be improved without expanding the in-plane footprint of that multiplexer.

In an embodiment, the inner conductor capacitively coupled to the readout resonators comprises a facing surface comprising surface portions facing respective openings through which the capacitive couplings are provided, wherein the facing surface is free of geometrical discontinuities such as pillars or protrusions and extends continuously and smoothly between and through the surface portions. Providing the capacitive couplings without pillars simplifies construction and/or compactness, as well as facilitating provision of relatively weak capacitive couplings, which is advantageous for some applications.

According to an aspect of the invention, there is provided a circuit quantum electrodynamics, QED, apparatus comprising: a circuit QED device comprising: a substrate comprising a first surface and an opposing second surface; one or more qubits formed on the first surface of the substrate; one or more readout resonators disposed on the second surface of the substrate, wherein each of the one or more readout resonators is coupled to a respective one of the one or more qubits; and the multiplexer of any embodiment of the present disclosure capacitively coupled to the readout resonators, optionally via pillars of the multiplexer being capacitively coupled to respective readout resonators.

Embodiments of the disclosure will now be further described, merely by way of example, with reference to the accompanying drawings.

Figure l is a schematic perspective view of portion of a substrate comprising a qubit and a readout resonator and couplings to a signal port.

Figure 2 is a schematic perspective view of portion of a substrate comprising a plurality of qubits and associated readout resonators coupled to an example multiplexer (outer conductor not shown).

Figure 3 is a schematic side sectional view of an arrangement having a multiplexer of the type shown in Figure 2.

Figure 4 is a magnified schematic perspective view of a region above one of the pillars in the arrangement of Figure 3 to show an example conductive shield for reducing cross-talk.

Figure 5 depicts a simplified model of Purcell filtering in an arrangement of the type depicted in Figure 3.

Figure 6 depicts an example admittance (Y parameter) frequency response for the model of Figure 5.

Figure 7 depicts an example scattering (S parameter) frequency response between two defined ports at the cavity resonator system end for a cavity system resonator of the type depicted in Figure 3.

Figure 8 schematically depicts an array of readout resonators with one unit cell indicated by a broken line box.

Figure 9 depicts the array of readout resonators of Figure 8 and outer conductors of corresponding cavity resonator systems.

Figure 10 is a schematic side sectional view of an arrangement having a multiplexer with a cavity resonator system comprising a single inner conductor attached to a bottom wall of an outer conductor. Figure 11 is a schematic side sectional view of an arrangement having a multiplexer with a cavity resonator system comprising two inner conductors attached to a side wall of an outer conductor.

Figure 12 is a schematic side sectional view of an arrangement having a multiplexer with a cavity resonator system comprising two inner conductors attached to a bottom wall of an outer conductor.

Figure 13 is a schematic perspective view of an arrangement having a multiplexer with a cavity resonator system comprising two inner conductors attached to a bottom wall of an outer conductor, with the two inner conductors coupled capacitively and inductively.

Figure 14 is a schematic side view of an alternative arrangement of the type shown in Figure 13, having a multiplexer with a cavity resonator system comprising two inner conductors attached to a bottom wall of an outer conductor, with the two inner conductors coupled capacitively and inductively.

Figure 15 is a schematic top sectional view of an arrangement having a multiplexer with a cavity resonator system comprising two inner conductors attached to a bottom wall of an outer conductor, with one of the inner conductors radially surrounding the other inner conductor.

Figure 16 is a schematic side sectional view of the arrangement of Figure 15.

Figure 17 is a schematic side sectional view of a variation on the arrangement of Figures 15 and 16 in which the two inner conductors are attached to a side wall of the outer conductor.

Figure 18 is a schematic side sectional view showing a variation on the arrangement of Figure 10 in which capacitive coupling to the readout resonators is implemented without pillars.

The present disclosure relates to a multiplexer for multiplexing signals from a plurality of readout resonators of a circuit quantum electrodynamics QED device.

An example of one readout resonator 2 of such a circuit QED device 4 is shown schematically in Figure 1. The circuit QED device 4 comprises a substrate 6. In the example shown the substrate 6 is planar. The substrate 6 comprises a first surface 61 and an opposing second surface 62. In this example, the readout resonator 2 is provided on the second surface 62 and a corresponding qubit 8 is provided on the first surface 61. In this example, the readout resonator 2 is a lumped element resonator and the qubit 8 is a transmon qubit. A control port 10 (e.g., comprising coaxial electrodes) extending perpendicularly to the plane of the substrate 6 provides control signals to the qubit 8. A signal port 12 (e.g., comprising coaxial electrodes) extending perpendicularly to the plane of the substrate 6 on an opposite side of the substrate 6 performs readout of the qubit 8.

In some arrangements, as exemplified in Figure 2, a circuit QED device 4 is provided that comprises a plurality of qubits 8 and associated readout resonators 2. Each of the readout resonators 2 is coupled to a respective one of the qubits 8. Each qubit 8 is also associated with a respective control port 10. In such arrangements having multiple qubits 8, a multiplexer may be provided between the signal port 12 and the readout resonators 2. The multiplexer allows selective readout of multiple qubits via a single signal port 12. The multiplexer may for example interact with the different qubits 8 using signals at different respective frequencies.

In arrangements of the disclosure, the multiplexer comprises a cavity resonator system 15. The cavity resonator system 15 comprises one or more inner conductors 16 and an outer conductor 18. As will be described below, the multiplexer may comprise a plurality of the cavity resonator systems 15 tiled in a plane. The inner and outer conductors will typically be formed from metal. In Figure 2 the outer conductor 18 is not depicted for clarity of presentation but would be present and would surround the inner conductor 16. Figure 3 is a schematic side view of the multiplexer of Figure 2, including depiction of the outer conductor 18.

As exemplified in Figure 3, the outer conductor 18 may comprise a plurality of openings 20. The outer conductor 18 may surround the one or more inner conductors 16 in directions other than towards the openings 20 (e.g., in most directions). The outer conductor 18 thus defines a cavity that contains the inner conductors 16. The readout resonators 2 are outside of the outer conductor 18. One or more of the inner conductors 16 may be configured to capacitively couple to the readout resonators through respective openings 20 in a direction extending perpendicularly or obliquely relative to a common plane of the readout resonators 2.

In some arrangements, the multiplexer further comprises a plurality of pillars 22. The pillars 22 may provide the capacitive couplings to the readout resonators 2. Each pillar 22 may comprise a solid conducting element such as a metal. The pillars 22 shown in the examples are cylindrical but other shapes of pillar may be used. For example, pillars having non-circular cross-sections may be used, such as squares, rectangles, or ellipses or ovals, and/or cross-sections that vary in shape and/or size along a length of the pillar The term “pillar” will be understood to encompass any protrusion positioned and configured to extend in the manner described herein.

Each pillar 22 is connected to one of the inner conductors 16 and configured to extend from the inner conductor 16 towards a respective one of the readout resonators 2 (e.g., in a direction perpendicularly or obliquely to the common plane of the readout resonators 2) and to capacitively couple to the readout resonator 2 through a respective one of the openings 20. The lengths of the pillars 22 may be selected to achieve desired strengths of coupling to the readout resonators 2. In some arrangements, the pillars may even be omitted, for example if a relatively weak coupling is desired and/or for structural simplicity and/or compactness, as exemplified in Figure 18. The pillar 22 may have the same lengths as each other or some or all of the pillars 22 may have different lengths. In some arrangements, as shown particularly in Figure 4, the outer conductor 18 may comprise a conductive shield 25 around each of one or more of the openings 20. Each conductive shield 25 may comprise a conductive wall extending perpendicularly or obliquely to a surface of the outer conductor 18 adjacent to the conductive shield 25. Each conductive shield 25 is associated with one of the pillars 22. Each conductive shield 25 may extend adjacent to an opening 20 corresponding to the pillar 22 in a direction parallel to a longitudinal axis of the pillar 22. The conductive shield 25 may be positioned between the pillar 22 and at least one other pillar 22 so as to electrically shield the pillar 22 from the at least one other pillar 22. The conductive shield 25 may surround the pillar 22 in all directions perpendicular to the longitudinal axis of the pillar 22. The conductive shields 25 may reduce or prevent cross-talk between signals from neighbouring readout resonators 2 (addressed by different respective pillars 22).

The multiplexer comprises a signal port system. The signal port system may comprise a single signal port 12 or a plurality of signal ports 12. The signal port system is coupled to the cavity resonator system 15 and configured to deliver electromagnetic signals (e.g., radio-frequency signals) into and out of the system. The signal port system may receive multiplexed signals from the readout resonators 2. Each signal port 12 may comprise a signal port conductor that capacitively couples to one of the inner conductors 16. In the examples of Figures 2 and 3, the outer conductor 18 surrounds a single inner conductor 16 and a single signal port 12 is provided that is capacitively coupled to the single inner conductor 16. In other arrangements, as exemplified in Figures 13 and 14, the outer conductor 18 surrounds two inner conductors 16 and two signal ports 12 are provided that are capacitively coupled to different respective inner conductors 16. The signal port conductors of each signal port 12 may comprise coaxial conductors, for example a central axial conductor 121 and a peripheral conductor 122 radially surrounding the axial conductor 121. In some arrangements, as depicted in Figure 3, the signal port 12 may couple to the cavity resonator system 15 by arranging for the axial conductor 121 to protrude beyond the peripheral conductor 122 and engage within a through hole 164 in the inner conductor 16. For ease of depiction, the coupling between the signal port system and the cavity resonator system 15 is not shown in all of the examples (e.g., in Figures 10-12 and 15-17) but it is understood that this coupling would be present in the complete configuration.

The readout resonators 2 are provided in a common plane, for example on one side of a planar substrate 6 of the circuit QED device. The readout resonators 2 couple to qubits 8 that are also provided in a common plane (parallel to the common plane of the readout resonators 2). The capacitive couplings through the openings 20, perpendicularly or obliquely to the common plane of the readout resonators 2, for example via the pillars 22, thus provide out-of-plane coupling to a plurality of qubits arranged in a common plane.

In some arrangements, the pillars 22, the one or more inner conductors 16 and the outer conductor 18 are configured such that the cavity resonator system 15 acts as a bandpass filter in respect of the multiplexed signals from the readout resonators 2. A multiplexer comprising a cavity resonator system 15 having such a frequency response could thus multiplex using frequencies within or near the pass band of the filter.

In some arrangements, the pillars 22, the one or more inner conductors 16 and the outer conductor 18 are configured such that the cavity resonator system 15 acts to suppress transmission at one or more qubit frequencies of the circuit QED device. The cavity resonator system 15 may thus be configured to perform Purcell filtering. This can be achieved by arranging for the frequency response to have a transmission minimum (which may also be referred to as a notch) at or near qubit frequencies. The pillars 22, the one or more inner conductors 16 and the outer conductor 18 may, for example, be configured to operate in conjunction with each readout resonator 2 to provide multiple propagation paths between the qubit 8 corresponding to the readout resonator 2 and the signal port system. The multiple propagation paths allow cancellation to occur between signals propagating along different paths, which provide the transmission minima. By increasing the number of available paths it is possible to create a larger number of notches. The positions of the notches can be tuned to desired frequencies by appropriate modification of the paths (e.g., by modifying positions, shapes and/or dimensions of the elements involved), for example to control the amplitude and phase between different non-adjacent resonators in the system. Creating a plurality of notches that are near to each other in the frequency domain may create a range of low transmission in the frequency response and thus act to suppress qubit decay over a corresponding range of qubit frequencies.

Figure 5 represents a model of Purcell filtering using a cavity resonator system 15 of the type depicted in Figure 3 with a single inner conductor 16. The calculated frequency response corresponding to the model of Figure 5 is depicted in Figure 6. The vertical axis in Figure 6 shows the Y parameter (admittance response) of the electromagnetic environment of the quantum device looking from the Josephson Junction (transmon qubit). The Josephson Junction is removed, and a lumped port of high impedance is defined instead. The real part of the admittance response is plotted to understand the loss (decay) to the environment.

The four nodes in Figure 5 respectively represent the qubit 8, the readout resonator 2, the cavity resonator system 15 and the signal port 12. The lines connecting the nodes and the respective K values represent couplings between the nodes (the K values refer to impedance or admittance inverters; inverters can be equated to a capacitive coupling or inductive coupling). The coupling between the readout resonator 2 and the qubit 8 provides intrinsic Purcell filtering because it provides two possible paths between the qubit 8 and the signal port 12 and thus provides a possibility for cancellation. Introducing the cavity resonator system 15 between the signal port 12 and the readout resonator 2 provides an additional degree of freedom, making it possible to define two notches in the frequency response. The positions (frequencies) of the notches can be controlled by varying the strength of the respective cross-coupling terms K₄ and K_s, as indicated in Figure 6. Adding further structure to the cavity resonator system 15 could provide further degrees of freedom and further notches, potentially allowing more targeted suppression of qubit frequencies.

Figure 7 depicts the scattering response (S parameter response) 30 of an arrangement using a cavity resonator system 15 of the type depicted in Figure 3 with particular defined ports at the cavity resonator system 15 end and without the circuit QED device. The simulation is performed such that energy propagates from the signal port 12 (which may be referred to as a first signal port 12 in the context of Figure 7) to the quarter wavelength resonator defined by the inner and outer conductors 16 and 18. For the purposes of the plot in Figure 7, the signal port 12 is mirrored along the x-axis to provide a second signal port 12 (not shown) that is coupled to the same quarter wavelength resonator. Figure 7 then shows the frequency response of S12 parameter (transmission from the first signal port 12 to the second signal port 12). Figure 7 show that the cavity resonator system 15 of the type depicted in Figure 3 comprises a pass band 35 where the signal is not attenuated. A multiplexer comprising a cavity resonator system 15 that provides such a frequency response could thus multiplex using frequencies within the pass band 35. The scattering response 30 further comprises two notches 31 and 32 which suppress transmission over a range of frequencies below the pass band 35. Arranging for qubit frequencies to be at or near (e.g., between) the frequencies of the notches will suppress transmission at those qubit frequencies. Suppressing transmission at qubit frequencies increases the coherence times of the qubits and improves performance.

In some arrangements, as exemplified schematically in Figures 8 and 9, the readout resonators 2 are provided in a regular array defined by repeating instances of a unit cell 3 containing a single readout resonator 2. In such embodiments, as shown in Figure 9, a size of a cross-section of the outer conductor 18 in a plane parallel to the common plane of the readout resonators 2 may be substantially equal to or smaller than a total size of the unit cells containing the plurality of readout resonators 2 that are coupled to by pillars 22 that are at least partially within the outer conductor 18. In the example of Figures 8 and 9, the cross-section of each outer conductor 18 is equal to or smaller than four unit cells 3 of the readout resonators 2. The multiplexer may thus comprise a plurality of the cavity resonator systems 15 tiled in a plane parallel to the common plane of the readout resonators 2, each cavity resonator systems 15 containing one of the outer conductors 18. The outer conductors 18 of the cavity resonator systems 15 are aligned with different respective groups of the unit cells of the array of readout resonators 2. In the example shown, each outer conductor 18 is aligned with a respective group of four readout resonators 2.

The plurality of pillars 22 associated with each cavity resonator system 15 may comprise four pillars (as would be appropriate for example of Figures 8 and 9) or more. The plurality of pillars 22 may for example comprise 8 or more, 16 or more, or 32 or more pillars 22.

In some arrangements, as exemplified in Figures 3 and 10-17, each of the one or more inner conductors 16 extends along a respective inner conductor axis 163 from a first axial end 161 of the inner conductor 16 to a second axial end 162 of the inner conductor 16. Each of the one or more inner conductors 16 is furthermore connected to the outer conductor 18 at the first axial end 161 of the inner conductor 16. Each inner conductor 16 may furthermore be spaced apart from the outer conductor 18 at the second axial end 162 of the inner conductor 16. The gap between the second axial end 162 of the inner conductor 16 and a facing surface of the outer conductor 18 may provide a capacitive coupling. The combination of the inner conductor 16 and the capacitive coupling across the gap may be modelled as a resistor in series with a capacitor when analysing the frequency response of the cavity resonator system. The length of each of one or more of the inner conductors 16 may be selected to operate as a quarter wavelength resonator.

In some arrangements, the inner conductor axis 163 of each of one or more of the inner conductors 16 is substantially parallel to the common plane of the readout resonators 2 (e.g., parallel to the plane of the substrate 6). This is the case in the example of Figures 2 and 3 discussed above, where a single inner conductor 16 extends horizontally in the plane of the page along the inner conductor axis 163. It is also the case in the examples of Figures 11 and 17 discussed below. The inner conductors 16 may thus be attached to a side wall of the outer conductor 18 in these arrangements. The side wall is optionally a wall that is angled, for example perpendicularly, relative to the common plane of readout resonators 2 and/or substrate 6. Connecting the inner conductors to a side wall may facilitate providing the coupling between the signal port system and the cavity resonator system 15 as well as allowing the inner conductors to be stacked in an out of plane direction (vertically in the figures).

In some arrangements, the inner conductor axis 163 of each of one or more of the inner conductors 16 is substantially perpendicular to the common plane of the readout resonators 2 (e.g., parallel to the plane of the substrate 6). This is the case in the examples of Figures 10 and 12-16 discussed below. The inner conductors 16 may thus be attached to a bottom wall of the outer conductor 18 in these arrangements.

In some arrangements, as exemplified in Figures 2-4 and 10, the one or more inner conductors 16 comprises only a single inner conductor 16. In other arrangements, as exemplified in Figures 11 to 17, the one or more inner conductors 16 comprises a plurality of the inner conductors 16. Each of the inner conductors 16 may predominantly comprise, consist essentially of, or consist of, a cuboidal body of metal. Providing more inner conductors 16 provides extra degrees of freedom for providing a desirable frequency response. For example, the extra inner conductors 16 may be configured to provide one or more additional notches in the frequency response, which may improve Purcell filtering and thereby increase qubit performance.

As exemplified in Figures 11 and 17, two or more of the inner conductors 16 may be arranged to be spaced apart along a direction perpendicular to the common plane of the readout resonators (i.e., in the z direction in the orientation of the figures). Stacking the inner conductors in the direction perpendicular to the common plane of the readout resonators provides extra degrees of freedom for tuning the frequency response of the multiplexer without requiring extra space in directions parallel to the plane of the readout resonators, thus avoiding encroachment on space required for processing signals from other neighbouring groups of qubits.

Two or more of the inner conductors 16 extend along parallel inner conductor axes 163. The two or more inner conductors 16 may be connected to the outer conductor 18 in a common plane of the outer conductor 18 (e.g., connected to the same planar side of bottom wall of the outer conductor 18). The common plane of the outer conductor 18 may be perpendicular to or parallel to the common plane of the readout resonators. In the examples of Figures 11 to 17, the number of inner conductors 16 in the outer conductor 18 is two and both extend along parallel inner conductor axes 163 (either parallel or perpendicular to the substrate 6). In other arrangements, more than two inner conductors 16 may be provided. In some arrangements, some of the inner conductors 16 may be parallel to each other and one or more others may extend in different directions.

The two or more inner conductors 16 may be capacitively and/or inductively coupled to each other. For example, the uppermost inner conductor 16 and the lowermost inner conductor 16 in Figure 11, or the leftmost inner conductor 16 and the rightmost inner conductor 16 in Figure 12, may be inductively and/or capacitively coupled to each other.

Figures 13 and 14 depict in more detail an example implementation of an arrangement of the type shown in Figure 12 with example features provided to enhance capacitive and inductive coupling.

In the example of Figures 13 and 14, a pair of inner conductors 16 comprises a pair of extensions 164 configured to provide capacitive coupling (e.g., to enhance the capacitive coupling relative to a case where the extensions are not provided) between the inner conductors 16 of the pair. The extensions 164 extend towards each other. In the example shown, each extension comprises an extension arm 165 extending perpendicularly away from the inner conductor and a plate 166 attached or integral with a distal portion of the respective extension arm 165. The two plates 166 face each across a gap that is smaller than a gap between the main (e.g., cuboidal) bodies of the inner conductors 16.

In the example of Figures 13 and 14, the pair of the inner conductors 16 further comprises an inductive extension 167 extending from one inner conductor 16 of the pair (e.g., the one on the left) to the other inner conductor 16 of the pair (e.g., the one on the right).

Figures 15-17 exemplify are variations on arrangements of the type shown in Figures 11-14 (that have two or more inner conductors 16) in which the inner conductors 16 comprise a first inner conductor 16A that surrounds a second inner conductor 16B. The first inner conductor 16A surrounds the second inner conductor 16B in directions perpendicular to the inner conductor axis 163 of the second inner conductor 16B (and/or of the first inner conductor 16A if the inner conductor axes are parallel or coaxial) over at least a portion of the axial length of the second inner conductor 16B. As shown in the examples of Figures 15-17, the first and second inner conductors 16A, 16B may be coaxial with each other. The first inner conductor 16A may have cylindrical symmetry, as exemplified in Figure 15, or any other shape.

In the example of Figures 15 and 16, the first inner conductor 16A comprises an annular shaped surface parallel to the substrate 6. The pillars 22 could in principle be positioned anywhere on the annular surface. In the example of Figure 15, four pillars are provided spaced apart from each other by 90 degrees azimuthally but other arrangements are possible (e.g., with different positioning around the annular surface and/or with different numbers of pillars 22). One or more pillars 22 could alternatively or additionally be positioned on a surface of the second inner conductor 16B.

In the example of Figure 17, pillars are provided on a portion of a radially outer surface of the first inner conductor 16A that is substantially parallel to the substrate 6.

The examples described above with reference to Figures 10-17 all comprise pillars 22. However, as explained earlier pillars 22 are not essential. Any of the arrangements described above with reference to Figures 10-17 may be implemented without pillars 22, for example where a relatively low capacitive coupling strength is desired. As an example, Figure 18 shows a variation on the arrangement shown in Figure 10 in which the pillars 22 are omitted. In such arrangements, the inner conductor 16 capacitively coupled to the readout resonators 2 may comprise a facing surface 168 comprising surface portions 169 facing respective openings 20 through which the capacitive couplings are provided. The facing surface 168 is free of geometrical discontinuities such as pillars or protrusions and extends continuously and smoothly between and through the surface portions 169. The surface portions 169 may for example by coplanar with regions between the surface portions or lie on a same surface of other symmetry, for example on a same cylindrical surface (where the inner conductor 16 comprises a cylindrical surface). Arrangements without pillars 22 may be particularly applicable where the inner conductor axis 163 extends towards the common plane of the readout resonators 2 (rather than being parallel to the common plane), as in the example of Figure 18, because the electric field between the facing surface 168 and the readout resonators 2 will be relatively strong in these cases.

Further embodiments of the disclosure are defined in the following numbered clauses.

1. A multiplexer for multiplexing signals from a plurality of readout resonators of a circuit quantum electrodynamics, QED, device, the multiplexer comprising: a cavity resonator system comprising one or more inner conductors and an outer conductor, the outer conductor comprising a plurality of openings and surrounding the one or more inner conductors in directions other than towards the openings; a plurality of pillars, each pillar connected to one of the inner conductors and configured to extend from the inner conductor towards a respective one of the readout resonators and to capacitively couple to the readout resonator through a respective one of the openings; and a signal port system coupled to the cavity resonator system and configured to receive multiplexed signals from the readout resonators, wherein: the readout resonators are provided in a common plane and the pillars extend perpendicularly or obliquely relative to the plane.

2. The multiplexer of clause 1, wherein the pillars, the one or more inner conductors and the outer conductor are configured such that the cavity resonator system acts as a bandpass filter in respect of the multiplexed signals from the readout resonators.

3. The multiplexer of clause 1 or 2, wherein the pillars, the one or more inner conductors and the outer conductor are configured such that the cavity resonator system acts to suppress transmission at one or more qubit frequencies of the circuit QED device.

4. The multiplexer of clause 3, wherein the pillars, the one or more inner conductors and the outer conductor are configured to operate in conjunction with each readout resonator to provide multiple propagation paths between the qubit corresponding to the readout resonator and the signal port system, the multiple propagation paths providing transmission minima in the frequency response of multiplexer via cancellation between different paths, the transmission minima acting to suppress transmission at one or more qubit frequencies of the circuit QED device. 5. The multiplexer of any preceding clause, wherein the readout resonators are provided in a regular array defined by repeating instances of a unit cell containing a single readout resonator.

6. The multiplexer of clause 5, wherein a size of a cross-section of the outer conductor in a plane parallel to the common plane of the readout resonators is substantially equal to or smaller than a total size of the unit cells containing the plurality of readout resonators that are coupled to by pillars that are at least partially within the outer conductor.

7. The multiplexer of clause 5 or 6, wherein the multiplexer comprises a plurality of the cavity resonator systems tiled in a plane parallel to the common plane of the readout resonators, the outer conductors of the cavity resonator systems being aligned with different respective groups of the units cells of the array of readout resonators.

8. The multiplexer of any preceding clause, wherein the plurality of pillars comprises 4 or more, or 8 or more, or 16 or more, or 32 or more pillars.

9. The multiplexer of any preceding clause, wherein the outer conductor comprises a conductive shield associated with each of one or more of the pillars, each conductive shield extending adjacent to an opening corresponding to the pillar in a direction parallel to a longitudinal axis of the pillar, the conductive shield being positioned between the pillar and at least one other pillar so as to electrically shield the pillar from the at least one other pillar, the conductive shield optionally surrounding the pillar in all directions perpendicular to the longitudinal axis of the pillar.

10. The multiplexer of any preceding clause, wherein each of the one or more inner conductors: extends along a respective inner conductor axis from a first axial end of the inner conductor to a second axial end of the inner conductor; and is connected to the outer conductor at the first axial end of the inner conductor.

11. The multiplexer of clause 10, wherein each inner conductor is spaced apart from the outer conductor at the second axial end of the inner conductor.

12. The multiplexer of clause 11, wherein the length of each of one or more of the inner conductors is selected to operate as a quarter wavelength resonator.

13. The multiplexer of any of clauses 10 to 12, wherein the inner conductor axis of each of one or more of the inner conductors is substantially parallel to the common plane of the readout resonators. 14. The multiplexer of any of clauses 10 to 13, wherein the inner conductor axis of each of one or more of the inner conductors is substantially perpendicular to the common plane of the readout resonators.

15. The multiplexer of any of clauses 10 to 14, wherein the one or more inner conductors comprises a plurality of the inner conductors.

16. The multiplexer of clause 15, wherein two or more of the inner conductors are spaced apart along a direction perpendicular to the common plane of the readout resonators.

17. The multiplexer of clause 15 or 16, wherein two or more of the inner conductors extend along parallel inner conductor axes.

18. The multiplexer of any of clauses 15-17, wherein two or more of the inner conductors are connected to the outer conductor in a common plane of the outer conductor, the common plane of the outer conductor being optionally perpendicular to or parallel to the common plane of the readout resonators.

19. The multiplexer of any of clauses 15 to 18, wherein: the two or more inner conductors comprises a first inner conductor and a second inner conductor; and the first inner conductor surrounds the second inner conductor in directions perpendicular to the inner conductor axis of the second inner conductor over at least a portion of the axial length of the second inner conductor.

20. The multiplexer of clause 19, wherein the first and second inner conductors are coaxial.

21. The multiplexer of any of clauses 15 to 20, wherein a pair of the inner conductors comprises a pair of extensions extending towards each other to provide a capacitive coupling between the inner conductors of the pair.

22. The multiplexer of clause 20 or 21, wherein a pair of the inner conductors comprises an inductive extension extending from one inner conductor of the pair to the other inner conductor of the pair.

23. The multiplexer of any preceding clause, wherein the signal port system comprises a signal port, the signal port comprising a signal port conductor capacitively coupled to one of the inner conductors.

24. The multiplexer of clause 23, wherein the signal port system comprises a plurality of signal ports, each signal port comprising a respective signal port conductor capacitively coupled to a respective different one of the inner conductors. 25. A circuit quantum electrodynamics, QED, apparatus comprising: a circuit QED device comprising: a substrate comprising a first surface and an opposing second surface; one or more qubits formed on the first surface of the substrate; one or more readout resonators disposed on the second surface of the substrate, wherein each of the one or more readout resonators is coupled to a respective one of the one or more qubits; and the multiplexer of any preceding clause, wherein the pillars of the multiplexer are capacitively coupled to respective readout resonators.

Claims

1. A multiplexer for multiplexing signals from a plurality of readout resonators of a circuit quantum electrodynamics, QED, device, the multiplexer comprising: a cavity resonator system comprising one or more inner conductors and an outer conductor, the outer conductor comprising a plurality of openings and surrounding the one or more inner conductors in directions other than towards the openings; and a signal port system coupled to the cavity resonator system and configured to receive multiplexed signals from the readout resonators, wherein: the readout resonators are provided in a common plane; and one of the inner conductors is configured to capacitively couple to the readout resonators through respective openings in a direction extending perpendicularly or obliquely relative to the plane.

2. The multiplexer of claim 1, further comprising a plurality of pillars configured to provide the capacitive couplings to the readout resonators, each pillar connected to one of the inner conductors and configured to extend perpendicularly or obliquely relative to the plane from the inner conductor towards a respective one of the readout resonators and to capacitively couple to the readout resonator through a respective one of the openings.

3. The multiplexer of claim 2, wherein the pillars, the one or more inner conductors and the outer conductor are configured such that the cavity resonator system acts as a bandpass filter in respect of the multiplexed signals from the readout resonators.

4. The multiplexer of claim 2 or 3, wherein the pillars, the one or more inner conductors and the outer conductor are configured such that the cavity resonator system acts to suppress transmission at one or more qubit frequencies of the circuit QED device.

5. The multiplexer of claim 4, wherein the pillars, the one or more inner conductors and the outer conductor are configured to operate in conjunction with each readout resonator to provide multiple propagation paths between the qubit corresponding to the readout resonator and the signal port system, the multiple propagation paths providing transmission minima in the frequency response of multiplexer via cancellation between different paths, the transmission minima acting to suppress transmission at one or more qubit frequencies of the circuit QED device.

6. The multiplexer of any of claims 2 to 5, wherein the readout resonators are provided in a regular array defined by repeating instances of a unit cell containing a single readout resonator.

7. The multiplexer of claim 6, wherein a size of a cross-section of the outer conductor in a plane parallel to the common plane of the readout resonators is substantially equal to or smaller than a total size of the unit cells containing the plurality of readout resonators that are coupled to by pillars that are at least partially within the outer conductor.

8. The multiplexer of claim 6 or 7, wherein the multiplexer comprises a plurality of the cavity resonator systems tiled in a plane parallel to the common plane of the readout resonators, the outer conductors of the cavity resonator systems being aligned with different respective groups of the units cells of the array of readout resonators.

9. The multiplexer of any of claims 2 to 8, wherein the plurality of pillars comprises 4 or more, or 8 or more, or 16 or more, or 32 or more pillars.

10. The multiplexer of any of claims 2 to 9, wherein the outer conductor comprises a conductive shield associated with each of one or more of the pillars, each conductive shield extending adjacent to an opening corresponding to the pillar in a direction parallel to a longitudinal axis of the pillar, the conductive shield being positioned between the pillar and at least one other pillar so as to electrically shield the pillar from the at least one other pillar, the conductive shield optionally surrounding the pillar in all directions perpendicular to the longitudinal axis of the pillar.

11. The multiplexer of claim 1, wherein the inner conductor capacitively coupled to the readout resonators comprises a facing surface comprising surface portions facing respective openings through which the capacitive couplings are provided, wherein the facing surface is free of geometrical discontinuities such as pillars or protrusions and extends continuously and smoothly between and through the surface portions.

12. The multiplexer of any preceding claim, wherein each of the one or more inner conductors: extends along a respective inner conductor axis from a first axial end of the inner conductor to a second axial end of the inner conductor; and is connected to the outer conductor at the first axial end of the inner conductor.

13. The multiplexer of claim 12, wherein each inner conductor is spaced apart from the outer conductor at the second axial end of the inner conductor.

14. The multiplexer of claim 13, wherein the length of each of one or more of the inner conductors is selected to operate as a quarter wavelength resonator.

15. The multiplexer of any of claims 12 to 14, wherein the inner conductor axis of each of one or more of the inner conductors is substantially parallel to the common plane of the readout resonators.

16. The multiplexer of any of claims 12 to 15, wherein the inner conductor axis of each of one or more of the inner conductors is substantially perpendicular to the common plane of the readout resonators.

17. The multiplexer of any of claims 12 to 16, wherein the one or more inner conductors comprises a plurality of the inner conductors.

18. The multiplexer of claim 17, wherein two or more of the inner conductors are spaced apart along a direction perpendicular to the common plane of the readout resonators.

19. The multiplexer of claim 17 or 18, wherein two or more of the inner conductors extend along parallel inner conductor axes.

20. The multiplexer of any of claims 17-19, wherein two or more of the inner conductors are connected to the outer conductor in a common plane of the outer conductor, the common plane of the outer conductor being optionally perpendicular to or parallel to the common plane of the readout resonators.

21. The multiplexer of any of claims 17 to 20, wherein: the two or more inner conductors comprises a first inner conductor and a second inner conductor; and the first inner conductor surrounds the second inner conductor in directions perpendicular to the inner conductor axis of the second inner conductor over at least a portion of the axial length of the second inner conductor.

22. The multiplexer of claim 21, wherein the first and second inner conductors are coaxial.

23. The multiplexer of any of claims 17 to 22, wherein a pair of the inner conductors comprises a pair of extensions extending towards each other to provide a capacitive coupling between the inner conductors of the pair.

24. The multiplexer of claim 22 or 23, wherein a pair of the inner conductors comprises an inductive extension extending from one inner conductor of the pair to the other inner conductor of the pair.

25. The multiplexer of any preceding claim, wherein the signal port system comprises a signal port, the signal port comprising a signal port conductor capacitively coupled to one of the inner conductors.

26. The multiplexer of claim 25, wherein the signal port system comprises a plurality of signal ports, each signal port comprising a respective signal port conductor capacitively coupled to a respective different one of the inner conductors.

27. A circuit quantum electrodynamics, QED, apparatus comprising: a circuit QED device comprising: a substrate comprising a first surface and an opposing second surface; one or more qubits formed on the first surface of the substrate; one or more readout resonators disposed on the second surface of the substrate, wherein each of the one or more readout resonators is coupled to a respective one of the one or more qubits; and the multiplexer of any preceding claim capacitively coupled to the readout resonators.