Qubit apparatus

Abstract

A circuit quantum electrodynamics, QED, component having a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the component is configured to generate one or more engineered interference effects and / or additional functionality in the circuit QED device.

Description

[0001]QUBIT APPARATUS The present disclosure relates to apparatus and methods for the improved operation of circuit quantum electrodynamics (cQED) devices. In particular, but not exclusively, the present disclosure relates to the use of symmetry breaking perturbations in circuit QED components of circuit QED devices. Circuit quantum electrodynamics (cQED) has emerged as a promising platform for the implementation of quantum computing, enabling the preparation of stable quantum states with good lifetimes and strong interactions. In circuit QED devices, non-linear devices, such as Transmons, are used to create qubit states that can be measured by applying resonant probe pulses to respective readout resonators and detecting the response, such as the phases of reflections. While circuit QED systems have demonstrated significant advances in coherence times and gate fidelities, addressing challenges when scaling such systems remains crucial for fully harnessing the potential of quantum devices. Among the major sources of loss channels in Transmon qubits limiting the coherence time are radiative, dielectric and quasiparticle relaxation mechanisms. One prominent challenge in quantum devices is in isolating a qubit from decohering into its environment while at the same time being able to read out its state in a short time. An important mechanism contributing to decohering of the qubit is radiative relaxation, commonly known as Purcell decay, in which interaction between the qubit and the surrounding electromagnetic environment leads to spontaneously emission of photons from the qubit. Such radiative relaxation ultimately limits the qubit’s lifetime and the ability to retain quantum information. Accordingly, a significant obstacle in the advancement of solid-state quantum devices is achieving high coherence in qubits, such as Transmon superconducting qubits. Whilst suppression of radiative loss channels through Purcell filtering is known, in 2D planar large-scale quantum devices, the number of conventional circuit elements providing Purcell filtering increases significantly with the number of qubits. This poses extra fabrication steps and space limitations for incorporating circuit elements, in addition to potentially introducing unwanted electromagnetic field cross-talk in large planar circuits. Specifically, electromagnetic signals from one circuit element can couple or interfere with neighbouring elements, leading to undesired interactions that could degrade the performance of qubits and filters in large-scale quantum systems. Another challenge in this area concerns control of ZZ couplings between qubits, which can impact gate fidelities by shifting energy levels conditional on qubit states. It can therefore be desirable to control ZZ interactions, for example to minimise them. It can also be useful to deliberately enhance them, for example to implement adiabatic controlled-phase gates and quantum simulations. However, implementing such control may be complex and / or demand extra fabrication steps and / or space requirements. It is an object of the disclosure to at least partly address one or more of the shortcomings in the prior art mentioned above. According to an aspect of the disclosure, there is provided a circuit quantum electrodynamics, QED, component having a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the component is configured to generate one or more engineered interference effects and / or additional functionality in the circuit QED device. Optionally, the one or more engineered interference effects and / or additional functionality comprises a frequency response with a transmission minimum at or near the qubit frequency of the circuit QED device. Optionally, the one or more engineered interference effects and / or additional functionality comprises a frequency adjustment of the qubit frequency of the circuit QED device. Optionally, the one or more engineered interference effects and / or additional functionality comprises a change in the anharmonicity response of the circuit QED device. Optionally, the one or more engineered interference effects and / or additional functionality comprises a change in a resonant frequency of the circuit QED device. Optionally, the one or more engineered interference effects and / or additional functionality are based on a relative increase in the number of pathways between the lowest resonant mode of the circuit QED device and higher order modes of the circuit QED device. Advantageously, the circuit QED components are provided that enable improved operation of circuit QED devices by the control of effects such as the suppression of unwanted decay and error mechanisms through suppressing the Purcell effect and / or ZZ terms in the Hamiltonian for a circuit QED device. Advantageously, such effects may be engineered to provide significant improvements in the relaxation time of a qubit state and / or reduce operational errors in a quantum processor. Beneficially, this approach reduces or avoids the need for additional circuitry, either on-chip or off chip, for the circuit QED device. Optionally, the circuit QED component comprises a first electrode and a second electrode. Optionally, the first and second electrodes are coaxial electrodes. Optionally, the second electrode is an outer electrode formed concentrically around the first electrode, wherein the first and second electrodes are coplanar. Optionally, the first electrode and / or the second electrode is substantially circular, is substantially a regular polygon, is substantially a square or is substantially a triangle. Optionally, the symmetry breaking perturbation comprises a local variation in a dimension of the first and / or second electrodes. Optionally, the second electrode has a major surface that follows a geometry along a closed path, wherein the width of the closed path is substantially uniform for at least the majority of the closed path. Optionally, the symmetry breaking perturbation comprises a local variation in the width of the closed path. Beneficially, circuit QED components, such as transmons and / or resonators, which comprises two electrodes may be modified through the application of a symmetry breaking perturbation to provide enhanced performance in circuit QED devices. Optionally, a surface of the second electrode comprises a planar annulus. Optionally, the symmetry breaking perturbation corresponds to a local variation in the width of the planar annulus. Optionally, the local variation in the width comprises a widening or narrowing of the width of the planar annulus in a radial direction along a portion of the planar annulus. Optionally, the local variation is based on an intersection of a circle or an ellipse with the planar annulus at the outside edge of the planar annulus or the inside edge of the planar annulus. Optionally, a surface of the first electrode comprises a planar circle. Optionally, the symmetry breaking perturbation corresponds to a local variation in the radius of the planar circle. Optionally, the local variation in the radius comprises an increase and / or a decrease of the radius of the planar circle along a portion of the circumference of the planar circle. Optionally, the local variation is based on an intersection of a circle or an ellipse with the outside edge of the planar circle. Advantageously, coaxial configurations of circuit QED components, such as transmons and / or resonators are adapted for improved circuit QED device performance. Optionally, the symmetry breaking perturbation is based on the global geometry of the first electrode and / or the second electrode. Optionally, the global geometry of the second electrode corresponds to a distorted planar annulus, optionally wherein the distorted planar annulus comprises a discontinuity, optionally wherein the distorted planar annulus has a non-circular geometry, optionally wherein the non-circular geometry is an elliptical geometry. Beneficially, circuit QED components are provided in a manner that provides improved fabrication and operation of circuit QED devices. Optionally, the circuit QED component is a transmon, optionally wherein the transmon is a coaxmon or an Xmon. Optionally the circuit QED component comprises a first electrode and a second electrode, wherein the first electrode is a superconducting electrode and the second electrode is superconducting electrode. Optionally, the circuit QED component comprises a Josephson Junction connected between the superconducting electrodes. Optionally, the one or more engineered interference effects and / or additional functionality comprises a transmission minimum based on the relative position of the Josephson Junction and the symmetry breaking perturbation. Optionally, the circuit QED component comprises a first superconducting electrode and / or the second superconducting electrode comprising a planar surface with two or more substantially orthogonal planar portions, optionally wherein the planar geometry is cross- shaped, wherein the symmetry breaking perturbation comprises a local variation in a dimension of the first and / or second superconducting electrodes. Beneficially, symmetry breaking perturbations are applied to transmons thereby to enable improved operation of circuit QED devices. Optionally, the circuit QED component is a resonator. Optionally, the circuit QED component comprises a first resonator electrode and a second resonator electrode. Beneficially, symmetry breaking perturbations are applied to resonators thereby to enable improved operation of circuit QED devices. Optionally, the circuit QED component comprises at least one of a Josephson Junction having a non-linear geometry, a capacitor having a non-linear geometry and an inductor having a non-linear geometry. Advantageously, symmetry breaking perturbations are based on one or more devices having a non-linear geometry, thereby enhancing the operation and functionality of circuit QED devices in which they are used. Optionally, the geometrically symmetrical arrangement has rotational symmetry of order two or higher. Beneficially, a geometrically symmetrical arrangement having rotational symmetry of order two or higher enables the formation of degenerate modes. Accordingly, the application of a symmetry breaking perturbation enables one or more engineered interference effects and / or additional functionality in a circuit QED device comprising the circuit QED component. Optionally, the component is configured to generate transmission minima to suppress transmission at one or more qubit frequencies of a circuit QED device based on at least one of position, shape and / or size of the symmetry breaking perturbation. Advantageously, parameters of the circuit QED component can be manipulated in order to generate one or more engineered interference effects and / or additional functionality. There is also provided a circuit quantum electrodynamics, QED, device comprising: a substrate comprising a first surface and an opposing second surface; one or more circuit QED components on the first surface of the substrate; and one or more circuit QED components on the second surface of the substrate, wherein each of the one or more circuit QED components on the second surface of the substrate is coupled to a respective one of the one or more circuit QED components on the first surface of the substrate. At least one of the one or more circuit QED components on the first surface of the substrate and / or on the second surface of the substrate is the circuit quantum electrodynamics, QED, component having a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the component is configured to generate one or more engineered interference effects and / or additional functionality in the circuit QED device described herein. Optionally, the symmetry breaking perturbation is configured to increase the number of pathways available for photons propagating in the circuit QED device and thereby create a transmission minimum. Optionally, the circuit QED device comprises at least two entangled qubits, wherein at least one of the two entangled qubits has a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in the circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the circuit QED device is configured to generate one or more engineered interference effects in the circuit QED device comprising a reduction in ZZ interactions between the entangled qubits. According to a further aspect of the disclosure, there is provided a method of designing a circuit QED component for a circuit QED device, wherein: the circuit QED component has a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in the circuit QED device, the geometrical arrangement comprising a symmetry breaking perturbation relative to the geometrically symmetric arrangement; and the method comprises designing the symmetry breaking perturbation to generate one or more engineered interference effects and / or additional functionality in the circuit QED device. Further aspects of the disclosure will be apparent from the description and the appended claims. A detailed description of embodiments is described, by way of example only, with reference to the figures in which: Figure 1 shows a perspective view of a circuit QED device (left), a representation of a Josephson Junction (centre) and a circuit diagram for the circuit QED device (right); Figure 2 shows a perspective view of a four-qubit system; Figure 3 shows circuit QED components with symmetry breaking perturbations; Figure 4 shows images of circuit QED components with symmetry breaking perturbations; Figures 5A to 5F show plan views illustrating geometrically symmetric arrangements of circuit QED components; Figures 6A to 6F show plan views of the geometrically symmetric arrangements of circuit QED components of Figures 5A to 5F with additional symmetry breaking perturbations; Figures 7A to 7D show plan views of coaxial circuit QED components; Figures 8A to 8F show plan views illustrating examples of coaxial circuit QED components comprising symmetry breaking perturbations; Figures 9A and 9B show plan views illustrating the creation of symmetry breaking perturbations based on an intersection of a circuit QED component with another shape; Figures 10A and 10B show plan views illustrating symmetry breaking perturbations based on the varied width of a planar annulus circuit QED component; Figures 11A to 11D show plan views illustrating symmetry breaking perturbations based on the global geometry of a circuit QED component; Figures 12A and 12B show plan views illustrating symmetry breaking perturbations based on the inclusion of components with a non-linear geometry in a circuit QED component; Figure 13 is a simplified schematic of a variation of the device of Figure 1 with a symmetry breaking perturbation (which may also be referred to herein as a symmetry- breaking feature) in the form of a “mouse-bite” on the outer pad ring of the transmon qubit. The equivalent circuit of this device (right) allows for additional cross-coupling capacitors between the qubit, the excited higher modes of the circuits, and the readout control ports, enabling the generation of multiple destructive zeros at different frequencies (which functionality may be referred to as a multi-mode intrinsic Purcell filter); Figure 14 is a graph showing a comparison of single-mode and multi-mode Purcell decay rates as a function of qubit frequency. The dashed curve represents the single-mode Purcell decay rate. The solid curve represents the multi-mode Purcell decay rate. The presence of multiple modes modifies the qubit decay spectrum, leading to a suppressed decay rate at certain frequencies due to interferometric cancellation effects; Figure 15 is a graph depicting finite element (FE) simulation of intrinsic Purcell filtering for the real part of the admittance response of the device of Figure 1 (upper solid line) and a device having a symmetry breaking perturbation (lower solid line); Figure 16 depicts Wideband FE simulation of the real part of the admittance response of the device in Figure 1 (dotted black line) and a device having a symmetry breaking perturbation different to that of Figure 15 (solid black line). Figures 17(a)-(c) are plots of static ZZ shift as a function of detuning (Figure 17(a)), qubit anharmonicity (Figure 17(b)), and drive frequency (Figure 17(c)). Figure 17(d) shows heatmap representations of the static ZZ shift as a function of detuning and drive frequency. The left panel shows the ZZ shift ignoring higher modes, and the right panel incorporates effects of higher-order modes and cross-couplings. The scale on the right represents the magnitude of the static ZZ shift in MHz. Figure 18 depicts simulation results for a four-qubit device showing simulated Purcell-limited relaxation time ^^1−^^^^^^as a function of frequency ^^. Circuit QED devices typically have qubits and resonators that are coupled to form a complex network of interactions. Engineering the interactions to enable effective operation of circuit QED devices without introducing further structures at additional cost of time and materials is a significant problem. Further, the introduction of additional components in a circuit QED device further complicates the interactions and affects the operation of such devices. Advantageously, the devices and methods described herein address at least some of these problems through the controlled engineering of circuit QED components. For example, traditional circuit QED components are formed based on structures with geometrically symmetric arrangements. Such geometrically symmetric arrangements provide significant advantages in terms of design, fabrication and / or operation of circuit QED devices. Counterintuitively, it has been realised that by breaking the symmetry of such geometrically symmetric arrangements, interference effects and / or other functionality can be generated that provides improvements in circuit QED devices. Figure 1 shows a perspective view of a geometrically symmetric arrangement of a circuit QED device, a representation of a Josephson Junction and a circuit diagram for the circuit QED device. The circuit QED device 100 described with reference to Figure 1 is an example of a known device that is subject to improvements through the modification of one or more components of the circuit QED device 100 in order to apply symmetry breaking perturbations, as described in further detail below. As shown in the perspective view of the circuit QED device 100 of Figure 1, the circuit QED device 100 comprises a substrate 106 having a first surface 101A and an opposing second surface 101B. There is a shown a circuit QED component in the form of a transmon qubit 102 provided on the first surface 101A and a further circuit QED component in the form of a readout resonator 104 provided on the second surface 101B. In the example of Figure 1, the substrate 106 is a planar substrate. In further examples, the substrate 106 has any appropriate form for implementing the functionality described herein. Whilst in the example of Figure 1 the qubit is a transmon qubit 102, in further examples any suitable structure is used for providing a qubit. The readout resonator 104 is a lumped element resonator. As shown at Figure 1, the resonator 104 has an inner electrode connected to an outer electrode by a spiral inductor connection. In further examples, the readout resonator 104 takes any appropriate form. In the example of Figure 1, there is also shown a control port 108 and a readout port 110. In the example of Figure 1, the transmon qubit 102 is a coaxmon qubit, comprising a first superconducting electrode 103 and a second superconducting electrode 107. A Josephson Junction is formed between the first superconducting electrode 103 and the second superconducting electrode 107. A representation of a Josephson Junction 109 is shown in Figure 1 (centre). The circuit diagram 112 for the circuit QED device 100 is shown in Figure 1 (right) and illustrates the capacitively coupling between the coaxmon qubit 102 and harmonic LC resonator 104. Figure 2 shows a perspective view of a four-qubit system 200. In a manner analogous to that shown at Figure 1 there is provided a substrate 206. The substrate 206 is a planar substrate having a first surface 201A and an opposing second surface 201B. Four circuit QED components in the form of transmon qubits 202A, 202B, 202C, 202D are shown on the first surface 201A. Each transmon qubit 202A, 202B, 202C, 202D is coupled to a further circuit QED component in the form of a respective readout resonator 204A, 204B, 204C, 204D on the second surface 201B of the substrate 206. The transmon qubits 202A, 202B, 202C, 202D are shown with the same geometry as the transmon qubit 102 of Figure 1, such that an inner electrode is connected to an outer electrode with a connection comprising a Josephson Junction. In further examples, the transmon qubits 202A, 202B, 202C, 202D have any appropriate geometry with additional and / or alternative subcomponents. The resonators 204A, 204B, 204C, 204D are shown with the same geometry as the resonator 104 of Figure 1, such that an inner electrode is connected to an outer electrode with a spiral inductor connection. In further examples, the resonators 2024, 204B, 204C, 204D have any appropriate geometry with additional and / or alternative subcomponents. Each qubit-resonator pair has a respective control port 208A, 208B, 208C, 208D and a respective readout port 210A, 210B, 210C, 210D. Whilst Figure 2 shows a four-qubit system 200, in further examples circuit QED devices and systems can include any appropriate number of qubits to provide required functionality. The modification of circuit QED components through the application of symmetry breaking perturbations, as described herein, is applicable to circuit QED devices and systems of any configuration. In the examples of the single qubit circuit QED device 100 of Figure 1 and the four-qubit circuit device 200 of Figure 2, the circuit QED components, such as the qubits 102, 202 and the resonators 104, 204, have geometrically symmetric arrangements that enable the formation of degenerate modes in the circuit QED devices 100, 200. Advantageously, breaking the symmetry of the geometrically symmetric arrangement enables one or more interference effects and / or additional functionality to be engineered into circuit QED device-based apparatuses to provide improved operation. For example, in circuit QED devices, if a transmon couples to multiple modes simultaneously, the relative phase between these modes can lead to constructive or destructive interference effects. This can impact the effective coupling strength and the dynamics of energy exchange between the transmon and the resonator, and it can be engineered to protect a qubit from radiative decay. In a circuit QED device with a geometrically symmetric arrangement that enables the formation of degenerate modes, the relative phase and magnitude of coupling between a transmon and the degenerate modes can lead to various interference effects. The introduction of symmetry breaking perturbations into the circuit QED device breaks the orthogonality between the degenerate modes and enables multipath coupling between the qubit mode (the lowest resonant mode) of the circuit QED device and the higher order geometric modes of the distributed circuit QED device. Appropriate engineering of the symmetry breaking perturbation enables the generation of one or more engineered interference effects and / or additional functionality. In an example, the use of a symmetry breaking perturbation in a circuit QED device is implemented through the application of a symmetry breaking perturbation to a component of the circuit QED device, such as a transmon and / or a resonator in order to engineer effects such as the suppression of unwanted decay and error mechanisms through suppressing the Purcell effect and or ZZ terms in the Hamiltonian for a circuit QED device. Advantageously, such effects are engineered to provide significant improvements in the relaxation time of a qubit state and / or reduce operational errors in a quantum processor. Beneficially, this approach reduces or avoids the need for additional circuitry, either on-chip or off chip, for the circuit QED device. In an example, the one or more engineered interference effects and / or additional functionality comprises a frequency response with a transmission minimum at or near the qubit frequency of the circuit QED device. In an example, a symmetry breaking perturbation in a multi-mode resonator coupled transmon is used to suppress Purcell decay through quantum interference. In a further example, the one or more engineered interference effects and / or additional functionality comprises a frequency adjustment of the qubit frequency of the circuit QED device. In a further example, the one or more engineered interference effects and / or additional functionality comprises a change in the anharmonicity response of the circuit QED device. In a further example, the one or more engineered interference effects and / or additional functionality comprises a change in a resonant frequency of the circuit QED device. In further examples, the one or more engineered interference effects and / or additional functionality are based on a relative increase in the number of pathways between the lowest resonant mode of the circuit QED device and higher order modes of the circuit QED device. In order to understand how such interference and / or functionality can be engineered for a circuit QED device, the Hamiltonian for a transmon coupled to a transmission line is considered, such as that shown in the circuit QED device 100 of Figure 1. The Hamiltonian of the transmission line with open boundary conditions is described in Equation 1: ^^^^^^^^^^ = ∫ ^^^^ ^^(^^) ^^^+^^^^^(Equation 1) where ^^(^^) is the dispersion relation for the transmission line, giving the frequency of the photons as a function of their wavevector. The total Hamiltonian for a transmon coupled to the transmission line is then shown at Equation 2: ^^ = ^^^^^^^^^^ + ^^^^^^^^^^^^ + ^^^^^^^^ (Equation 2)and can be described as shown at Equation 3: (Equation 3) Since the problem at hand relates to a finite transmission, the allowed values of k (and hence ^^) are quantised to the boundary conditions. For angular ring transmission lines, the boundary condition, and hence periodicity, imposes certain allowed wavevectors k and corresponding frequencies ^^. Assume a ring of circumference L, the boundary condition for the wavefunction is described by Equation 4: ^^(^^ + ^^) = ^^(^^) (Equation 4)The periodic boundary condition leads to allowed wavevectors k of the form shown at Equation 5: (Equation 5) When n is integer (positive, negative, and zero), the corresponding frequencies are given by Equation 6: (Equation 6) where ^^^^is the group velocity of the photons in the ring, where the dispersion relation is known, or can be determined from field solvers. Here the Hamiltonian of the quantised modes of the ring coupled to a transmon simplifies to. (Equation 7)The actual values of ^^^^and the specific modes that couple strongly to the transmon depend on the spatial overlap between the mode patterns in the ring and the electric dipole pattern of the transmon. Consider the Hamiltonian of a transmon coupled to the first ten modes of the angular ring, assuming three pairs are degenerate. The Hamiltonian of the coupling terms is described at Equation 8: (Equation 8) Where φnrepresents the relative phase of the n-th mode at the location of the qubit (the phase associated with the interaction between the qubit and the i-th mode). The phase information can be extracted from the full-wave electromagnetic simulations of the structure. The phases determine the nature of the qubit’s coupling to each mode. For example, one might consider a phase difference of π between two modes at the location of the transmon; their coupling to the qubit would have opposite signs, i.e. −^^^^^^^^^^+and ^^^^^^^^^^. Note that in quantum mechanics, the concept of the absolute or the global phase of a state has no observable consequences. This is because measurements in quantum mechanics yield probabilities, which depends on the magnitude of the inner product of vectors and not their phase. However, relative phases between components of quantum states can have observable effects and are therefore crucial. The value of ∅^^might be set or adjusted based on the physical setup or design of the quantum system. They might arise due to external fields, geometric configurations, or other design choices. Given three pairs of the ten modes are degenerate, the Hamiltonian can be expanded to: (Equation 9) The total Hamiltonian can be written as: (Equation 10) Given the degeneracy and potential phase differences, the transmon can couple differently to each member of a degenerate pair. The relative phase and magnitude of the coupling determine how the qubit interacts with the degenerate modes and can lead to various interference effects. If the goal is to exploit or control these interference effects, then understanding and potentially tuning the relative phase and coupling strengths becomes crucial in the design and operation of the system. This is where the energy participation ration (EPR) comes into play. EPR in circuit QED provides a way to quantify how much of the energy in a particular mode of a resonator or transmission line is stored in a specific component of the circuit. The concept becomes particularly important when analysing complex superconducting circuits with multiple modes, as in this example. Primarily the coupling coefficients of the transmon to the multiple mode are calculated and the phase information (relative phase difference between the fields at the locations of different components or between different modes) is interpolated, for example through the use of EPR software such as PyEPR, in conjunction with full-wave electromagnetic solves like Ansys HFSS. The relative phase between the coupling to these modes is particularly important for interference effect and engineering mechanisms supressing Purcell decay. Software such as pyEPR straddles the analysis from Maxwell’s to Schrodinger’s equations, and converts the solutions of the distributed microwave (typically eigenmode solutions) to a fully diagonalised spectrum of the energy levels, couplings, and key parameters of a many body quantum Hamiltonian. To determine the non-linear response of the junction ^^^−^^^, the quantum zero-point fluctuations ∅^^and ∅^^are needed, which are calculated from the participation of the junction in the eigenfield solutions. The participation ^^^^of the junction in the mode ^^ ∈{^^, ^^} is defined to the fraction of inductive energy stored in the junction relative to thetotal inductive energy stored in the entire circuit, (Equation 11) where |^^^^> denotes a coherent state of a Fock excitation of mode m. Let the variance of the quantum zero-point fluctuations ∅^^and ∅^^expressed as a function of the classical energy participations ^^^^, ^^ ∅2^^ℏ^^^^^^= 2^^^^(Equation 12) and (Equation 13) Now consider the relationships and constraints in quantum systems involving Josephson junctions. Consider these constraints, which are independent of the specific circuit design or type of Josephson element used: (Equation 14)And (Equation 15)The constraints suggest that a given mode can have at most a total EPR of unity from all dipoles. The mode cannot take more than its full share of energy from all the junctions combined. The EPR sign ^^^^^^represents the relative direction of the current following across the junction, and how the current flow in one mode compares with another. The orthogonality relationship can be written as:(Equation 16)This ensures the contributions from different modes remain distinct and do not overlap. Routinely the independence and distinguishability of different modes in the system are preserved. Where symmetry breaking perturbations are introduced into a circuit QED system, such perturbations can lead to the breaking of orthogonality between its modes. This is routinely accomplished by carefully engineering the amplitude and phase ration of the interaction between the Josephson junction and pairs of higher order degenerate modes. The orthogonality condition above-mentioned related to energy participation, not directly to phase. However, the orthogonality condition can have implications for the relative phase between two components in a quantum system. ^^^^^^and ^^^^^^0str the signs of the EPRs. They can be either +1 or -1 and indicate the relative direction of current or voltage across the component for that mode. The terms represent the energy shared between the components due to their non- orthogonal participation in the modes (Equation 17) If the orthogonality condition is not satisfied, it implies that the two components (indexed by j and j_0) have some degree of coherent energy participation in the system’s modes. In other modes, their contributions to the modes are not entirely orthogonal. This can be engineered to introduce constructive or destructive interference between the components in certain modes, affecting the system’s behaviour such that the system’s resonance frequencies or the strength of interaction between components. In the context of superconducting quantum circuits and the Energy Participation Ratios (EPRs), introducing a symmetry breaking perturbation, such as a perturbative element comprising an additional inductor, capacitor, or junction, can redistribute the energy stored in each mode across the circuit QED components. This can change the EPRs and their signs, potentially violating the orthogonality condition. Some systems, e.g. degenerate mode, have inherent symmetries that enforce orthogonality between certain states or components. A perturbation can break these symmetries, leading to non- orthogonal states. Revisiting the total Hamiltonian to reflect the lack of orthogonality and the importance of the relative phase for describing destructive interference in the system, cross-terms can be added that describe the interaction between different modes: ^^^^^^^^ = ∑^^≠^^ ^^^^^^(^^^^^^^^^^^^+^^ ^^^^ + ^^−^^^^^^^^^^^+^ ^^^^) (Equation 18) where ^^^^^^represents the interaction strength between the mode ^^ and mode ^^ and ^^^^^^is the relative phase between these modes, crucial for understanding the conditions needed for achieving destructive interference. The total Hamiltonian is therefore: ∑^^≠^^ ^^^^^^(^^^^^^^^^^^^+^^ ^^^^ + ^^−^^^^^^^^^^^+^ ^^^^) (Equation 19) By adjusting the and relative phase ^^^^^^values, one can engineer specific interference effects, potentially suppression undesired interactions or enhancing desired ones. For example, relative phase can be indicated (and / or adjusted) by the relative direction of current or voltage across the component for that mode and the Josephson Junction. Accordingly, in circuit QED devices, the application of symmetry breaking perturbations to components having geometrically symmetric arrangements that enable the formation of degenerate modes enables one or more engineered effects and / or additional functionality to be engineered in a circuit QED device. Such symmetry breaking perturbations take any appropriate form that provides the functionality described herein. The following examples show non-exhaustive practical examples of the application of symmetry breaking perturbations to circuit QED components. Figure 3 shows exemplary qubit-resonator pairs which have had symmetry breaking perturbations applied to them in order to generate one or more interference effects and / or functionality in the circuit QED device in which they are implemented. Figure 3 shows qubit-resonator pairs based on the qubit-resonator pairs 202, 204 of Figure 2. There is shown a first qubit 302A paired with a respective first resonator 304A. The first qubit 302A includes a Josephson Junction 309A which has a non-linear geometry that provides a symmetry breaking perturbation. There is shown a second qubit 302B paired with a respective second resonator 304B. The second qubit 302B has symmetry breaking perturbation 314B. There is shown a third qubit 302C paired with a respective third resonator 304C. The third qubit 302C has symmetry breaking perturbation 314C. There is shown a fourth qubit 302D paired with a respective fourth resonator 304D. The fourth qubit 302D includes a Josephson Junction 309D which has a non-linear geometry that provides a symmetry breaking perturbation. In the examples of Figure 3, each of the qubits 304A, 304B, 304C, 304D and each of the resonators 304A, 304B, 304C, 304D have an inner electrode surrounded by an outer electrode. In the case of the qubits 304A, 304B, 304C, 304D, the inner electrode is connected to the outer electrode by a Josephson Junction. In further examples, the transmon qubits 302A, 302B, 302C, 302D have any appropriate geometry with additional and / or alternative subcomponents. In the case of the resonators 304A, 304B, 304C, 304D, the inner electrode is connected to the outer electrode by a spiral inductor connection. In further examples, the resonators 302A, 302B, 302C, 302D have any appropriate geometry with additional and / or alternative subcomponents. Figure 4 shows optical images 400 of the qubits, resonators and Josephson Junctions of Figure 3, showing the practical application of symmetry breaking perturbations to circuit QED components. At Figure 4 there are shown optical images 402A, 402B, 402C, 402D corresponding to the qubits 302A, 302B, 302C, 302D, respectively, described with reference to Figure 3. There are also shown optical images 404A, 404B, 404C, 404D corresponding to the resonators 304A, 304B, 304C, 304D, respectively, described with reference to Figure 3. There are also shown optical images 409A, 409B, 409C, 409D of the Josephson Junctions corresponding to the qubits 302A, 302B, 302C, 320D, respectively, described with reference to Figure 3. Whilst the images of the resonators 404A, 404B, 404C, 404D do not show the application of symmetry breaking perturbations, symmetry breaking perturbations are seen in the images of the qubits 402B, 402C, where portions of the outer electrode ring have been removed with respect to the geometrically symmetric arrangement upon which they are based (such as that of the image of a first qubit 402A). Further, the images of the Josephson Junctions show significant changes in the geometry in the second 409B and fourth 409D images of Josephson Junctions. The distortion of the symmetry of the Josephson Junctions is used to provide symmetry breaking perturbations in some examples. Whilst fabricated examples of symmetry breaking perturbations are described with reference to Figures 3 and 4, in further examples symmetry breaking perturbations are provided by any suitably implemented change to a structure that breaks a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device. Figures 5A to 5F show plan views illustrating a range of common geometrically symmetric arrangements of circuit QED components to which symmetry breaking perturbations may be applied in accordance with the present disclosure. Figure 5A shows a circuit QED component 500A that is substantially circular. Figure 5B shows a circuit QED component 500B that is substantially a square. Figure 5C shows a circuit QED component 500C that is substantially a regular polygon. Whilst the example of Figure 5C shows a circuit QED component 500C having a hexagonal shape, in further examples the circuit QED component is substantially a regular polygon having any number of sides with sufficient symmetry to provide the functionality described herein. Figure 5D shows a circuit QED component 500D that is substantially a triangle. Figure 5E shows a circuit QED component 500E that is substantially cross-shaped with a planar surface having two substantially orthogonal portions. Xmon qubits are commonly formed using circuit QED components having such a substantially cross-shaped geometry, to which symmetry breaking perturbations may be applied in accordance with the present disclosure. Figure 5F shows a circuit QED component 500F that is substantially Y-shaped with a planar surface having three portions with axes rotated substantially 120° with respect to one another around an axis that is substantially perpendicular to the planar surface. Whilst Figures 5A to 5F illustrate the form that circuit QED components may commonly take, in further examples circuit QED components are based on any appropriate geometrically symmetric arrangement that enables the functionality described herein. For example, any geometrically symmetric arrangement with rotational symmetry of order two or higher may be suitable for the formation of degenerate modes in a circuit QED device. Therefore, as described herein, by breaking the degeneracy, an increase in coupling paths enables one or more interference effects and / or added functionality to be engineered into a circuit QED device. In an example, the shapes of the circuit QED components described with reference to Figures 5A to 5F correspond to a shape of a transmon. In an example, the transmon may be a coaxmon or an Xmon. Whilst, in an example, Figures 5A to 5F describe a shape of a transmon, it will be understood that a transmon having such a shape may comprises one or more additional and / or alternative components to provide transmon functionality, with at least part of the transmon having a shape as described with reference to Figures 5A to 5F. In an example, a transmon having such a shape may comprise a first superconducting electrode connected to a second superconducting electrode by a Josephson Junction, wherein the first and / or second superconducting electrodes have a shape as described with reference to Figures 5A to 5F. In further examples, additionally or alternatively, the shapes of the circuit QED components described with reference to Figures 5A to 5F correspond to a shape of a resonator. Whilst, in an example, Figures 5A to 5F describe a shape of a resonator, it will be understood that a resonator having such a shape may comprises additional and / or alternative components to provide resonator functionality. For example, a resonator having such a shape may comprise a first resonator electrode connected to a second resonator electrode by a connective element, which may be a spiral inductor connection. In an example, a resonator having such a shape may comprise a first resonator electrode connected to a second resonator electrode by spiral inductor connection, wherein the first and / or second resonator electrodes have a shape as described with reference to Figures 5A to 5F. In further examples, the first resonator electrode and the second resonator electrode are connected with an inductor connection of a different form. In further examples, additionally or alternatively, the shapes of the circuit QED components described with reference to Figures 5A to 5F correspond to any component of a circuit QED device that can be used to provide the functionality described herein. Figures 6A to 6F show plan views of the geometrically symmetric arrangements of circuit QED components of Figures 5A to 5F with additional symmetry breaking perturbations. As described above with reference to Figures 5A to 5F, the circuit QED components of Figures 6A to 6F may be any appropriate circuit QED component, including transmons and / or resonators. Figure 6A shows the circuit QED component 500A of Figure 5A that is substantially circular with an additional symmetry breaking perturbation 602A. Figure 6B shows a circuit QED component 500B of Figure 5B that is substantially a square with an additional symmetry breaking perturbation 602B. Figure 5C shows the circuit QED component 500C of Figure 5C that is substantially a regular polygon with an additional symmetry breaking perturbation 602C. Figure 6D shows the circuit QED component 500D of Figure 5D that is substantially a triangle with an additional symmetry breaking perturbation 602D. Figure 6E shows the circuit QED component 500E of Figure 5E that is substantially cross-shaped with a planar surface having two substantially orthogonal portions with an additional symmetry breaking perturbation 602E. Figure 6F shows the circuit QED component 500F of Figure 5F that is substantially Y-shaped with a planar surface having three portions with axes rotated substantially 120° with respect to one another around an axis that is substantially perpendicular to the planar surface with an additional symmetry breaking perturbation 602F. Whilst Figures 6A to 6F show the application of symmetry breaking perturbations to provide particular forms of symmetry breaking perturbations, in further examples the symmetry breaking perturbations have different and / or additional forms. Whilst Figures 5A to 5F and Figures 6A to 6F illustrate the general principle of applying symmetry breaking perturbations to geometrically symmetric arrangements that enable the formation of degenerate modes, Figures 7A to 12B show more detailed implementations of the application of symmetry breaking perturbations to circuit QED components formed with two primary electrodes. Advantageously, coaxial resonator and coaxial qubit components may both be provided with electrodes to which symmetry breaking perturbations are applied. In further examples, advantageously, symmetry breaking perturbations are applied to different formations of circuit QED components with two primary electrodes, for example transmons where the first superconducting electrode and / or the second superconducting electrode comprise a planar surface with two or more substantially orthogonal planar portions, optionally wherein the planar geometry is cross-shaped, wherein the symmetry breaking perturbation comprises a local variation in a dimension of the first and / or second superconducting electrodes. Figures 7A to 7D show exemplary plan views of coaxial circuit QED components to which symmetry breaking perturbations are applicable in order to break the geometrically symmetric arrangement that enables the formation of degenerate modes. In an example, the circuit QED components of Figures 7A to 7D are transmons or resonators. Accordingly, in an example, the first and second electrodes of transmon circuit QED components described herein are superconducting electrodes. When the circuit QED components are transmons, the first superconducting electrode may be in communication with the second superconducting electrode by a Josephson Junction connected between the first and second superconducting electrodes, as shown by the transmons 102, 202, 302, 402 described with reference to Figures 1 to 4. In further examples, transmons are provided with different and / or alternative configurations of Josephson Junctions to connect the first and second superconducting electrodes. In a further example, the first and second electrodes of resonators described herein are resonator electrodes. When the circuit QED components are resonators, the first resonator electrode may be in communication with the second resonator electrode via a connection between the first resonator electrode and the second resonator electrode, such as the spiral inductor connection illustrated by the resonators 104, 204, 304, 404 described with reference to Figures 1 to 4. In further examples, resonators are provided with different and / or alternative configurations of connections between the first and second resonator electrodes. Figure 7A shows a circuit QED component 700A having a first electrode 703A and a second electrode 707A. The first electrode 703A and the second electrode 707A are coaxial electrodes such that the second electrode 707A is an outer electrode formed concentrically around the first electrode 703A, which is an inner electrode. The first electrode 703A and the second electrode 707A are coplanar. The first electrode 703A and the second electrode 707A are substantially circular electrodes. In the example of Figure 7A, the surface of the second electrode 707A is a planar annulus. Figure 7B shows a circuit QED component 700B having a first electrode 703B and a second electrode 707B arranged in a similar manner to the electrodes 703A, 707A of the circuit QED component 700A of Figure 7A. In contrast to the arrangement of Figure 7A, the circuit QED component 700B of Figure 7B has electrodes 703B, 707B that are substantially a square. Similarly, Figure 7C shows a circuit QED component 700C having a first electrode 703C and a second electrode 707C arranged in a similar to the circuit QED components 700A, 700B of Figures 7A and 7B, respectively, but with electrodes 703C, 707C that are substantially a regular polygon. Similarly, Figure 7D shows a circuit QED component 700D having a first electrode 703D and a second electrode 707D arranged in a similar to the circuit QED components 700A, 700B, 700C of Figures 7A,7B and 7C, respectively, but with electrodes 703D, 707D that are substantially a triangle. The examples of Figures 7A to 7D have an outer electrode 707A, 707B, 707C, 707D that has a major surface that follows a geometry along a closed path, wherein the width of the closed path is substantially uniform for at least the majority of the closed path. Whilst the coaxial configurations of Figures 7A to 7D provide benefits with respect to operation and fabrication of circuit QED components, therefore providing appealing forms to which symmetry breaking perturbations are applied, in further examples circuit QED components with symmetry breaking perturbations are based on any geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device, such as any geometrically symmetric coaxial arrangement of a first and second electrode. Figures 8A to 8F show plan views illustrating examples of coaxial circuit QED components comprising symmetry breaking perturbations. The examples of Figures 8A to 8F are based on the geometrically symmetric arrangement of the coaxial circuit QED component described with reference to Figure 7A, where the second electrode 707A has a surface comprising a planar annulus. In an example, the circuit QED components of Figures 8A to 8F are transmons or resonators. In an example, when the circuit QED components are transmons, the first electrodes 803 described at Figures 8A to 8F may be connected to the second electrodes 807 by a component such as a Josephson Junction. In such examples, the first electrodes 803 and the second electrodes 807 are superconducting electrodes. In further examples, transmons are provided with different and / or alternative configurations of Josephson Junctions to connect the first and second superconducting electrodes. In an example, when the circuit QED components are resonators, the first electrodes 803 described at Figures 8A to 8F may be connected to the second electrodes 807 by a connective member, which may be a spiral inductor connection. In such examples, the first electrodes 803 and the second electrodes 807 are resonator electrodes. In further examples, resonators are provided with different and / or alternative configurations of connections between the first and second resonator electrodes. In further examples, symmetry breaking perturbations are applied analogously to any geometrically symmetric arrangement of a coaxial circuit QED component including, but not limited to, those shown at Figures 7B to 7D. Each of the examples of Figures 8A to 8F shows symmetry breaking perturbations comprising a local variation in a dimension of the first and / or second electrode. Figure 8A shows a circuit QED component 800A comprising a first electrode 803A surrounded by a second electrode 807A. The second electrode 807A is a planar annulus with a portion removed from the outside edge of the planar annulus to provide a symmetry breaking perturbation 814A. The removal of a portion of the planar annulus results in a local variation in the width of the planar annulus in a radial direction along a portion of the planar annulus. In the example of Figure 8A, the local variation in the width of the planar annulus is a narrowing of the width of the planar annulus in a radial direction along a portion of the planar annulus. Symmetry breaking perturbations of this type may be referred to as “mouse bite” perturbations. Figure 8B shows a circuit QED component 800B comprising a first electrode 803B surrounded by a second electrode 807B. The second electrode 807B is a planar annulus with a portion removed from the inside edge of the planar annulus to provide a symmetry breaking perturbation 814B. The removal of a portion of the planar annulus results in a local variation in the width of the planar annulus in a radial direction along a portion of the planar annulus. In the example of Figure 8B, the local variation in the width of the planar annulus is a narrowing of the width of the planar annulus in a radial direction along a portion of the planar annulus. Figure 8C shows a circuit QED component 800C comprising a first electrode 803C surrounded by a second electrode 807C. The second electrode 807C is a planar annulus with a portion added to the outside edge of the planar annulus to provide a symmetry breaking perturbation 814C. The addition of a portion to the planar annulus results in a local variation in the width of the planar annulus in a radial direction along a portion of the planar annulus. In the example of Figure 8C, the local variation in the width of the planar annulus is a widening of the width of the planar annulus in a radial direction along a portion of the planar annulus. Figure 8D shows a circuit QED component 800D comprising a first electrode 803D surrounded by a second electrode 807D. The second electrode 807D is a planar annulus with a portion added to the inside edge of the planar annulus to provide a symmetry breaking perturbation 814D. The addition of a portion to the planar annulus results in a local variation in the width of the planar annulus in a radial direction along a portion of the planar annulus. In the example of Figure 8D, the local variation in the width of the planar annulus is a widening of the width of the planar annulus in a radial direction along a portion of the planar annulus. The examples of Figures 8A to 8D have an outer electrode 807A, 807B, 807C, 807D that has a major surface that follows a geometry along a closed path, wherein the width of the closed path is substantially uniform for at least the majority of the closed path. In each case, the symmetry breaking perturbation 814A, 814B, 814C, 814D associated with the respective outer electrode 807A, 807B, 807C, 807D comprises a local variation in the width of the closed path. Whilst Figures 8A to 8D show local variations in the width of the closed path of the planar annulus of the second electrode 807A, 807B, 807C, 807D corresponding to either a narrowing or widening of the width of the planar annulus of the second electrode807A, 807B, 807C, 807D, in further examples the symmetry breaking perturbation comprises any combination of widening and narrowing to provide the functionality described herein. Figure 8E shows a circuit QED component 800E comprising a first electrode 803E surrounded by a second electrode 807E. The surface of the first electrode 803E comprises a planar circle with a portion added to provide a symmetry breaking perturbation 814E. The symmetry breaking perturbation 814E corresponds to a local variation in the radius of the planar circular of the first electrode 803E. In the example of Figure 8E, the local variation in the radius of the planar circle comprises an increase in the radius of the planar circle along a portion of the circumference of the planar circle. Figure 8F shows a circuit QED component 800F comprising a first electrode 803F surrounded by a second electrode 807F. The surface of the first electrode 803F comprises a planar circle with a portion removed to provide a symmetry breaking perturbation 814F. The symmetry breaking perturbation 814F corresponds to a local variation in the radius of the planar circular of the first electrode 803F. In the example of Figure 8F, the local variation in the radius of the planar circle comprises a decrease in the radius of the planar circle along a portion of the circumference of the planar circle. Whilst Figures 8A to 8F each show a single symmetry breaking perturbation associated with one of the inner or outer electrodes, in further examples any circuit QED component may comprise any number and combination of symmetry breaking perturbations, for example on the inside and / or outside of the out electrode and / or on the inner electrode. The symmetry breaking perturbations described herein are fabricated using any suitable technique. In an example, in order to engineer interference and / or additional functionality in a circuit QED device, the fabrication of symmetry breaking perturbations based on predefined shapes facilitates control and reproducibility. Figures 9A and 9B show plan views illustrating the creation of symmetry breaking perturbations based on an intersection of a circuit QED component with another shape. In an example, the circuit QED components described with reference to Figures 9A and 9B may be resonators or transmons. In an example, when the circuit QED components are transmons, the first electrodes 903 described at Figures 9A and 9B may be connected to the second electrodes 907 by a component such as a Josephson Junction. In such examples, the first electrodes 903 and the second electrodes 907 are superconducting electrodes. In further examples, transmons are provided with different and / or alternative configurations of Josephson Junctions to connect the first and second superconducting electrodes. In an example, when the circuit QED components are resonators, the first electrodes 903 described at Figures 9A and 9B may be connected to the second electrodes 907 by a connective member, which may be a spiral inductor connection. In such examples, the first electrodes 903 and the second electrodes 907 are resonator electrodes. In further examples, resonators are provided with different and / or alternative configurations of connections between the first and second resonator electrodes. Figure 9A shows a circuit QED component 900A comprising a first electrode 903A surrounded by a second electrode 907A. The second electrode 907A is a planar annulus with a portion removed from the outside edge of the planar annulus to provide a symmetry breaking perturbation 914A. In the example of Figure 9A, the symmetry breaking perturbation 914A is based on an intersection of a circle 916A with the second electrode 907A. Figure 9B shows a circuit QED component 900B comprising a first electrode 903B surrounded by a second electrode 907B. The second electrode 907B is a planar annulus with a portion removed from the outside edge of the planar annulus to provide a symmetry breaking perturbation 914B. In the example of Figure 9B, the symmetry breaking perturbation 914B is based on an intersection of an ellipse 916B with the second electrode 907B. Whilst the examples of Figures 9A and 9B show the removal of a portion of the second electrodes 907A, 907B based on the intersection with a circle 916A and an ellipse 916B respectively, in further examples the intersection with a circle 916A and / or an ellipse is used to form an addition to the width of the planar annulus of a circuit QED component to provide a symmetry breaking perturbation. Whilst the symmetry breaking perturbations 914A, 914B described with reference to Figures 9A and 9B respectively are based intersections with a circle 916A and an ellipse 916B, in further examples any shape is used additionally or alternatively to provide one or more symmetry breaking perturbation. For example, Figures 10A and 10B show plan views illustrating symmetry breaking perturbations based on the varied width of a planar annulus circuit QED component, where the local variations in width of the second electrodes 1014A, 1014B are based on the intersection of regular polygons with the second electrodes 1007A, 1007B. In an example, the circuit QED components described with reference to Figures 10A and 10B may be resonators or transmons. In an example, when the circuit QED components are transmons, the first electrodes 1003 described at Figures 10A and 10B may be connected to the second electrodes 1007 by a component such as a Josephson Junction. In such examples, the first electrodes 1003 and the second electrodes 1007 are superconducting electrodes. In further examples, transmons are provided with different and / or alternative configurations of Josephson Junctions to connect the first and second superconducting electrodes. In an example, when the circuit QED components are resonators, the first electrodes 1003 described at Figures 10A and 10B may be connected to the second electrodes 1007 by a connective member, which may be a spiral inductor connection. In such examples, the first electrodes 1003 and the second electrodes 1007 are resonator electrodes. In further examples, resonators are provided with different and / or alternative configurations of connections between the first and second resonator electrodes. In the example of Figure 10A there is shown a circuit QED component 1000A comprising a first electrode 1003A surrounded by a second electrode 1007A. The second electrode 1007A is a planar annulus with a portion removed from both the inside edge and the outside edge of the planar annulus to provide a symmetry breaking perturbation 1014A. The symmetry breaking perturbation 1014A comprises a local variation in the width of the planar annulus in a radial direction along a portion of the planar annulus and is based on the intersection of regular polygons with the second electrodes 1007A. Figure 10B shows a circuit QED component 1000B comprising a first electrode 1003B surrounded by a second electrode 1007B. The second electrode 1007B is a planar annulus with a portion added to both the inside edge and the outside edge of the planar annulus to provide a symmetry breaking perturbation 1014B. The symmetry breaking perturbation 1014A comprises a local variation in the width of the planar annulus in a radial direction along a portion of the planar annulus and is based on the intersection of regular polygons with the second electrodes 1007B. Whilst the geometry of circuit QED components can be altered to provide symmetry breaking perturbations based on local variations, in further examples, additionally or alternatively, the global geometry of the circuit QED components can be varied in order to provide symmetry breaking perturbations. Figures 11A to 11D show plan views illustrating symmetry breaking perturbations based on the global geometry of a circuit QED component. In an example, the circuit QED components described with reference to Figures 11A to 11D may be resonators or transmons. In an example, when the circuit QED components are transmons, the first electrodes 1103 described at Figures 11A to 11D may be connected to the second electrodes 1107 by a component such as a Josephson Junction. In such examples, the first electrodes 1103 and the second electrodes 1107 are superconducting electrodes. In further examples, transmons are provided with different and / or alternative configurations of Josephson Junctions to connect the first and second superconducting electrodes In an example, when the circuit QED components are resonators, the first electrodes 1103 described at Figures 11A and 11B may be connected to the second electrodes 1107 by a connective member, which may be a spiral inductor connection. In such examples, the first electrodes 1103 and the second electrodes 1107 are resonator electrodes. In further examples, resonators are provided with different and / or alternative configurations of connections between the first and second resonator electrodes. Figure 11A shows a circuit QED component 1100A that has a symmetry breaking perturbation based on a global distortion of a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device. The example of Figure 11A has a circuit QED component 1100A that has a geometry that is distorted with respect to the geometrically symmetric arrangement described with reference to the coaxial, coplanar circuit QED component 700A of Figure 7A. The circuit QED component 1100A has a second electrode 1107A that corresponds to a distorted planar annulus. The variation from a circular planar annulus to distorted planar annulus breaks the symmetry in a circuit QED component. Figure 11B shows a circuit QED component 1100B that has a symmetry breaking perturbation based on a global distortion of a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device. The example of Figure 11B has a circuit QED component 1100B that has a geometry that is distorted with respect to the geometrically symmetric arrangement described with reference to the coaxial, coplanar circuit QED component 700A of Figure 7A. The circuit QED component 1100B has a first electrode 1107B that corresponds to a distorted circle. The variation from a planar circle surface to distorted planar circle surface breaks the symmetry in a circuit QED component. Whilst Figures 11A and 11B illustrate the application of a symmetry breaking perturbation based on the global distortion of one electrode, in further examples, any combination of distortion of the first and / or second electrode is used to provide the engineered interference and / or functionality described herein. Figure 11C shows a circuit QED component 1100C that has a symmetry breaking perturbation based on a global distortion of a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device. The example of Figure 11C has a circuit QED component 1100A that has a geometry that is distorted with respect to the geometrically symmetric arrangement described with reference to the coaxial, coplanar circuit QED component 700A of Figure 7A. The circuit QED component 1100C has a second electrode 1107C that corresponds to a distorted planar annulus. In the example of Figure 11C, the second electrode comprises a discontinuity 1114C, thereby to introduce a symmetry breaking perturbation. The variation from a circular planar annulus with a geometry that follows along a closed path, to distorted planar annulus with a discontinuity in the closed path that otherwise follows along a closed path breaks the symmetry in a circuit QED component. Figure 11D shows a circuit QED component 1100D that has a symmetry breaking perturbation based on a global distortion of a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device. The example of Figure 11D has a circuit QED component 1100D that has a geometry that is distorted with respect to the geometrically symmetric arrangement described with reference to the coaxial, coplanar circuit QED component 700A of Figure 7A. The circuit QED component 1100D has a second electrode 1107D that corresponds to a distorted planar annulus. In the example of Figure 11D, the outer edge of the second electrode 1107D follows a path of linear intersections such that the overall geometry of the outer edge of the second electrode 1107D corresponds to a hexagon. The inner edge of second electrode 1107D corresponds to a circle. The variation from a circular planar annulus to distorted planar annulus breaks the symmetry in a circuit QED component. Whilst symmetry breaking perturbations have been described for a number of exemplary situations, it is understood that in further examples, any appropriate combination of features is provided in order to provide a symmetry breaking perturbation. For example, a symmetry breaking perturbation may be based on the combination of a global distortion and / or one or more location variations that break the symmetry of a geometrically symmetric arrangement that enables the formation of degenerate modes. In an example, the arrangement described with reference to Figure 11C may be further adapted by the inclusion of a geometrically non-linear component, such as a Josephson Junction, formed within the discontinuous gap of the second electrode 1107C. Whilst symmetry breaking perturbations can be introduced into circuit QED components based on local and / or global variations with respect to geometrically symmetric arrangements, in further examples the inclusion of one or more components with a non-linear geometry can provide a symmetry breaking perturbation. Figures 12A and 12B show plan views illustrating symmetry breaking perturbations based on the inclusion of components with a non-linear geometry in a circuit QED component. The examples of Figures 12A and 12B are based on the geometrically symmetric arrangement of the coaxial circuit QED component described with reference to Figure 7A, where the second electrode 707A has a surface comprising a planar annulus. Both Figures 12A and 12B show symmetry breaking perturbations comprising additional components with non-linear geometry. In an example, the circuit QED components described with reference to Figures 12A and 12B may be resonators or transmons. In an example, when the circuit QED components are transmons, the first electrodes 1203 described at Figures 12A and 12B may be connected to the second electrodes 1207 by a component such as a Josephson Junction, which may be an additional or alternative Josephson Junction to the components providing the symmetry breaking perturbations 1214 described with reference to Figures 12A and 12B. In such examples, the first electrodes 1203 and the second electrodes 1207 are superconducting electrodes. In further examples, transmons are provided with different and / or alternative configurations of Josephson Junctions to connect the first and second superconducting electrodes. In an example, when the circuit QED components are resonators, the first electrodes 1203 described at Figures 12A and 12B may be connected to the second electrodes 1207 by a connective member, which may be a spiral inductor connection. In such examples, the first electrodes 1203 and the second electrodes 207 are resonator electrodes. In further examples, resonators are provided with different and / or alternative configurations of connections between the first and second resonator electrodes. Figure 12A shows a circuit QED component 1200A comprising a first electrode 1203A surrounded by a second electrode 1207A. A symmetry breaking perturbation 1214A corresponds to two components arranged between the first electrode 123A and the second electrode 1207A. In an example, the additional components correspond to Josephson Junctions. For example, where the circuit QED component 1200A is a transmon which includes a Josephson Junction between a first superconducting electrode 1203A and a second superconducting electrode 1207A, a further Josephson Junction is used to break the symmetry of the structure in order to engineer one or more interference effects and / or provide extra functionality. Figure 12B shows a circuit QED component 1200B comprising a first electrode 1203B surrounded by a second electrode 1207B. A symmetry breaking perturbation 1214B corresponds to the form of a further structure on the circuit QED component 1200B. For example, where the circuit QED component 1200B is a transmon, the form of the Josephson Junction between a first superconducting electrode 1203B and a second superconducting electrode 1207B is manipulated in order to engineer one or more interference effects and / or provide extra functionality. In an example, the one or more engineered interference effects and / or additional functionality comprises a transmission minimum based on the relative position of a Josephson Junction and the symmetry breaking perturbation. In further examples, symmetry breaking perturbations include alternative and / or additional perturbative elements, such as a capacitor having a non-linear geometry, inductor having a non-linear geometry and / or any other perturbative element having a non- linear geometry. Advantageously, the application of a symmetry breaking perturbation to a circuit QED component that has an arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device is controllable to exploit one or more interference effects and / or additional functionality. In an example, at least one or the position, shape and / or size of the symmetry breaking perturbation is controlled in order to generate one or more engineered interference effects and / or engineered functionality in a circuit QED device. In an example, circuit QED components in accordance with the disclosure are implemented in a circuit QED device, such as the circuit QED devices described with reference to Figures 1 to 3. In examples there is a circuit quantum electrodynamics, QED, device comprising: a substrate comprising a first surface and an opposing second surface; one or more circuit QED components on the first surface of the substrate; and one or more circuit QED components on the second surface of the substrate, wherein each of the one or more circuit QED components on the second surface of the substrate is coupled to a respective one of the one or more circuit QED components on the first surface of the substrate, wherein at least one of the one or more circuit QED components on the first surface of the substrate and / or on the second surface of the substrate is a circuit QED component in accordance with the disclosure, such as a transmon and / or resonator. Beneficially, in such devices, the symmetry breaking perturbation may be configured to increase the number of pathways available for photons propagating in the circuit QED device and thereby create a transmission minimum. In an example, the circuit QED device comprises at least two entangled qubits, wherein at least one of the two entangled qubits has a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in the circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the circuit QED device is configured to generate one or more engineered interference effects in the circuit QED device comprising a reduction in ZZ interactions between the entangled qubits. Beneficially, in an example of a circuit QED device comprising entangled qubits, a symmetry breaking perturbation is engineered in order to generate an engineered interference effect comprising a reduction in ZZ interactions between entangled qubits. Further, advantageously, the application of a symmetry breaking perturbation in accordance with the disclosure may be engineered in order to reduce gate errors in superconducting qubits by the suppression of ZZ terms. In further examples, the application of a symmetry breaking perturbation in accordance with the disclosure may be engineered in order to reduce errors associated with the electromagnetic environment of a transmon in a circuit QED device. Advantageously, in an example, a circuit QED component is configured to generate one or more engineered interference effects and or functionality of a circuit QED device based on at least one of position, shape and / or size of the symmetry breaking perturbation. For example, the component may be configured to generate transmission minima to suppress transmission at one or more qubit frequencies. In accordance with the disclosure, methods of designing a circuit QED component for a circuit QED device are thus provided. The circuit QED component may take any of the forms described herein, such as for example with reference to the figures. The circuit QED component has a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in the circuit QED device. The geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement. The symmetry breaking perturbation may take any of the forms described above. The method comprises designing the symmetry breaking perturbation. The designing of the symmetry breaking perturbation may be a computer- implemented method. The method may be performed on any suitable combination of hardware, firmware and / or software that is capable of performing the required data processing. The method may comprise iteratively varying a symmetry breaking perturbation until desired properties are achieved. Computer simulations may be performed to evaluate the effect of different designs of symmetry breaking perturbation. The symmetry breaking perturbation is designed in such a way as to generate one or more engineered interference effects and / or additional functionality in the circuit QED device. The one or more engineered interference effects and / or additional functionality in the circuit QED device may take any of the forms described elsewhere herein. As mentioned elsewhere in the disclosure, the designing of the symmetry breaking perturbation may comprises selecting a size, location and / or shape of the symmetry breaking perturbation. The size, location and / or shape of the symmetry breaking perturbation may be selected to define the strength and / or sign of coupling between different resonant modes in the circuit QED device, such as a qubit mode, a resonator mode, and / or higher-order geometric modes, to provide one or more desired engineered interference effects and / or additional functionality. The design parameters may be tuned for example to generate poles at designated frequencies. More generally, the one or more engineered interference effects and / or additional functionality may comprise any one or more of the desired interference effects and / or additional functionalities disclosed elsewhere herein, such as: a frequency response with a transmission minimum at or near the qubit frequency of the circuit QED device; a frequency adjustment of the qubit frequency of the circuit QED device; a change in the anharmonicity response of the circuit QED device; a change in a resonant frequency of the circuit QED device; and / or a change in a variation of ZZ interactions as a function of one or more of the following: qubit detuning, qubit anharmonicity, and qubit drive frequency. The change in a variation of ZZ interactions as a function of one or more of qubit detuning, qubit anharmonicity, and qubit drive frequency is discussed in further detail below with reference to Figures 17(a)-(d), where it is noted that both boosting and suppressing ZZ may be useful in practice and that the approach of the present disclosure effectively provides a valuable new degree of freedom for controlling ZZ coupling as a function of various relevant parameters (e.g., qubit detuning, qubit anharmonicity, and qubit drive frequency). As mentioned above, in some configurations the circuit QED device comprises a substrate comprising a first surface and an opposing second surface and the circuit QED component is on the first surface of the substrate. In such a case, the symmetry breaking perturbation may be designed to generate the one or more engineered interference effects by introducing interference between multiple resonant modes associated with respective pathways between or within circuit QED components supported by the substrate. Examples of such higher modes are depicted in Figure 13 (see, for example, “Qubit higher modes” and “Resonator higher modes”) and discussed in further detail below. Desirably, the symmetry breaking perturbation is designed to generate the one or more engineered interference effects by introducing interference between multiple resonant modes involving higher-order resonant modes of circuit QED components that are either qubits or readout resonators. Thus, the one or more engineered interference effects can be achieved entirely via resonances within “on-chip” circuitry that is directly associated with the fundamental circuit QED functionality, thus removing or avoiding the need to provide additional filters, on-chip or off-chip, to suppress losses due to the Purcell effect. Once the circuit QED component, or multiple circuit QED components, have been designed as described above, the circuit QED component and / or circuit QED components can be manufactured using any suitable known technique and assembled to form a circuit QED device. FURTHER ANALYSIS AND EXPLANATIONS In circuit QED, the qubit state is typically inferred by measuring the state‐ dependent frequency shift of a resonator via homodyne detection. However, this readout pathway also introduces a decay channel: energy can leak from the qubit through the resonator to the transmission line, a phenomenon known as the Purcell effect. In principle, the Purcell rate can be lowered by increasing the qubit–resonator detuning, decreasing the qubit–resonator coupling, or reducing the resonator bandwidth through damping. Unfortunately, these measures prolong the required measurement time and thus trade qubit protection for slower readout. Ideally, the Purcell rate should be reduced without compromising measurement fidelity. A Purcell filter accomplishes this by impeding photon propagation at the qubit frequency while allowing microwave fields at the resonator’s readout frequency to pass. Conventional filters often involve adding another resonator (on-chip or off-chip), which alters the system’s spectral response and mitigates the leakage channel but increases fabrication complexity. By contrast, multi-path interference in the qubit–resonator–transmission line network can achieve the same suppression through destructive interference, eliminating the need for added filter resonators. Existing techniques also include dedicated filter modes, such as low-quality- factor bandpass filters, multi-stage bandpass filters, or “intrinsic” Purcell filters that rely on standing waves in the feedline. Although these methods can be effective, they often increase chip footprint or demand precise placement and calibration. In the embodiments described here, a simpler interference-based approach obviates extra resonators or filter components, remains compatible with various readout schemes, and offers a more scalable, integrated solution to suppress the Purcell effect while maintaining high-fidelity qubit readout. The present disclosure proposes an interferometric approach to Purcell suppression that eliminates the need for additional filter components, making it well-suited for scalable circuit QED architectures. Rather than relying on external resonators, we consider all intrinsic resonant modes present in the circuit, including those arising from the shunt capacitance geometry. By analyzing the spatial symmetry of the circuit and optimizing the multipath coupling matrix, we demonstrate that destructive interference can be engineered within the readout resonator-feedline system. This method enables precise control over interference effects, leading to significant suppression of resonator-mediated qubit decay, even in fixed-frequency transmons, without increasing circuit complexity. The transmon qubit is a weakly anharmonic superconducting oscillator, effectively modelled as a Cooper-Pair Box (CPB) shunted by a large capacitance. This large shunt capacitance, which suppresses charge noise, is typically implemented using interdigitated capacitors, ring resonators, or cross-shaped structures. The transmon's operating frequency, generally ranging from a few GHz to 10 GHz, is determined by the Josephson junction (JJ) inductance and capacitance, as well as the shunt capacitor's capacitance. In practical designs, the shunt capacitance is fixed during fabrication to achieve a target qubit frequency and anharmonicity. Typical qubit capacitances are on the order of 100 fF, with device lengths exceeding 1 mm. A simplified circuit QED qubit-readout setup is shown in Figure 1. In this configuration, the qubit is dispersively coupled to a single-mode readout resonator, allowing for state-dependent frequency shifts to be measured via homodyne detection. This interaction can be described using the Jaynes-Cummings Hamiltonian, where ^^^^and ^^^^denote the qubit and resonator frequencies, respectively. The operators ^^^^, ^^+, and ^^−correspond to the Pauli matrices acting on the qubit states, while ^^ and represent the annihilation and creation operators for the resonator mode. The parameter ^^ defines the qubit-resonator coupling strength. The terms and ^^^^represent the drive amplitudes acting on the qubit and resonator, respectively, and the frequencies correspond to the external drive applied to each component. While this single-mode approximation provides an intuitive framework for describing qubit-resonator interactions, it does not fully account for the distributed nature of the qubit’s shunt capacitance, nor does it capture the presence of higher-order resonant modes within the system. The transmon’s and readout resonator large capacitors (inductor) introduce additional electromagnetic modes, which interact with the qubit and modify its relaxation pathways. The large shunt capacitor and surrounding circuit components support multiple higher- frequency modes in addition to the fundamental resonant mode. To accurately model these effects, we perform eigenmode simulations using HFSS, revealing the existence of additional self-resonances within the system. For example, a spiral resonator designed with a fundamental frequency ^^0= 9.1 GHz exhibits additional resonances at ^^1^^= 21.9 GHz and ^^2^^= 35.7 GHz. Similarly, eigenmode simulations of the transmon qubit’s capacitive pads in isolation reveal fundamental frequencies at ^^0^^^^= 55.2 GHz and ^^0ℎ^^= 55.2 GHz, corresponding to charge oscillations along the vertical and horizontal axes, respectively. In symmetric configurations, these higher-order modes remain decoupled from the qubit. However, introducing symmetry-breaking perturbations, such as a mouse-bite defect, allows for mode hybridization, leading to enhanced interactions between the qubit and higher-frequency modes. TABLE I: Real part of the frequency for each eigenmode: To account for these effects, we extend the Jaynes- Cummings Hamiltonian to incorporate multiple interacting modes, where ^^^^^^defines the coupling strength between modes ^^ and ^^, while ^^^^^^represents the relative phase between these coupled modes. The parameter ^^^^specifies the phase of each mode at the qubit’s location. By engineering the phase relationships between these modes, destructive interference conditions can be introduced, leading to an intrinsic Purcell filter that suppresses energy relaxation at specific frequencies. The relative phase and magnitude of these couplings depend on the qubit’s placement and circuit geometry, necessitating full-wave electromagnetic simulations to optimize interference conditions. The interaction between the qubit and these higher-order modes can be quantified using the Energy Participation Ratio (EPR) method, which provides insight into how much of a mode’s energy is stored in specific circuit components. Table II presents the calculated energy participation factors for the Josephson junction (JJ) in different higher- order modes, comparing the symmetric and symmetry-broken configurations. TABLE II: Energy participation of the junction in the qubit pad geometric modes without (EPR sym) and with (EPR usym) the presence of a symmetry breaking perturbation: A key observation from Table II is that introducing a symmetry-breaking perturbation (such as a mouse-bite defect in the outer pad ring of the qubit) leads to a two- order-of-magnitude increase in the participation of the JJ in higher-order modes. This increase suggests that, in the unperturbed symmetric configuration, the qubit remains largely decoupled from these modes. However, once symmetry is broken, previously orthogonal modes hybridize, leading to enhanced coupling between the qubit and higher- frequency circuit modes. This enhanced participation indicates that higher-order resonances, which were negligible in the symmetric case, now play a significant role in determining the qubit’s relaxation dynamics. To quantitatively analyze this effect, we derive the qubit’s Purcell decay rate in a multi-mode system, considering multipath interference. The linearized Heisenberg-Langevin equation of motion for the qubit state, coupled to ^^ resonator where ^^−represents the qubit lowering operator and denotes the coupling strength between the qubit and the i-th mode. Similarly, the equation of motion for the resonator modes takes the form where ^^^^is the coupling strength between the qubit and the i-th mode, ^^^^^^represents the interaction strength between mode i and mode j, is the damping rate of mode i, and ^^^^is the frequency of mode i. The operators ^^−, and ^^^^represent the qubit lowering operator and resonator field amplitudes. Since we are interested in how the resonator interacts with the qubit, it is convenient to rewrite the equation in a frame rotating at the qubit frequency ^^^^. This transformation is commonly used in quantum optics and superconducting circuits where detuning is Now, rewrite the resonator mode equation in this rotating frame: which simplifies to: At steady state, where ^̇^^^ = 0, solving for ^^^^ yields Substituting this expression into the equation of motion for the qubit and summing over all modes, we obtain For total destructive interference, the interaction terms must cancel at specific frequencies, leading to the formation of notch filters in the qubit’s decay spectrum. The total Purcell decay rate, which represents the energy relaxation rate of the qubit in the presence of multiple resonator modes, is given by If the phase conditions satisfy The destructive interference condition cancels contributions at specific frequencies, resulting in notch filters in the decay spectrum. These notch filters selectively suppress the Purcell effect, preventing unwanted energy leakage from the qubit. Figure 14 presents a comparison between the single-mode and multi-mode Purcell decay rates as a function of qubit frequency, where the readout frequency is 9GHz. The solid curve represents the multi-mode Purcell decay scenario, where the qubit interacts not only with the readout resonator but also with additional higher-order resonator modes. The introduction of these additional modes causes a reduced Purcell effect over a wide range of frequencies relative to the standard Purcell decay formula represented by the dashed curve, and includes a minimum representing complete cancellation of the Purcell effect between 6 and 7 GHz. The decay rate is suppressed due to destructive interference conditions between multiple resonator pathways. This suppression arises due to the multi-path interference mechanism, which introduces transmission zeros in the spectral response, effectively functioning as a Purcell-protecting notch filter. The presence of these engineered notch frequencies confirms that interferometric coupling of multiple resonator modes can provide an effective means of controlling the qubit relaxation rate without requiring additional on-chip Purcell filters. This approach is particularly advantageous for scalable superconducting quantum processors, where minimizing design complexity while maintaining high coherence times is crucial. This technique, known as Purcell filtering, enables fast readout without compromising qubit relaxation times. The radiative lifetime, ^^^^^^^^, is determined by the total qubit capacitance and the admittance seen by the qubit, which quantifies spontaneous decay due to the Purcell effect. While dielectric losses, quasiparticles, and other relaxation mechanisms are also present in practical devices, these effects lie outside the scope of this discussion. By treating ^^^^^^^^as a frequency-dependent quantity, we can design filters with tailored stopbands and passbands to suppress qubit decay. In a general linear electromagnetic environment, ^^^^^^^^can be accurately computed using the following semiclassical relation: where ^^(^^) is the admittance seen by the qubit, and represents the effective resistance due to external loss through control ports. As illustrated in Figures 1 and 13, the transmon qubit and its environment can be modeled as a classical LC oscillator coupled to an impedance ^^(^^). The admittance response of the transmon, with and without symmetry-breaking geometry (e.g., a mouse-bite perturbation), is shown in Figure 15. By modifying the transmon geometry, we effectively engineer the admittance ^^(^^) and introduce three real-frequency transmission zeros—two below the readout resonator frequency and one above. These cancellation zeros suppress unwanted radiative decay by decoupling the qubit from certain control and readout lines. The underlying physical mechanism responsible for these interference-induced transmission zeros will be detailed in the following section. The ability to provide transmission zeros both above and below the readout resonator frequency provides valuable additional design flexibility. In a practical application, for example, a readout resonator could be engineered to resonate below the qubit frequency, for example at 3GHz with the qubit at 5GHz. Notches can then be engineered notches to be above 3 GHz for suppression of Purcell decay. In practical systems with 100s or 1000s of qubits, it is possible to engineer some of the readout resonators to be above their qubit frequencies and others below giving the quantum engineer another degree of freedom for control, which could assist with reducing or avoiding frequency crowding. Furthermore, control electronics and microwave components operating at lower RF frequences, say 1-6GHz, are typically cheaper than electronics and components configured to operate above 6GHz. We now demonstrate the application of the methods developed earlier to implement Purcell filters in example circuit designs. We begin by analyzing the device depicted in Figure 1, which consists of a single transmon qubit coupled to a single-mode shunt LC resonator. The qubit and resonator frequencies are set to 5 GHz and 9 GHz, respectively, by choosing capacitance values of ^^^^= 200 pF and ^^^^= 117 pF and tuning the inductances ^^^^and ^^^^. The resonator linewidth is set to ^^^^= 5 MHz. The capacitance matrix for this system is computed using the Ansys Maxwell solver. TABLE III: Capacitance Matrix (pF) : Next, we introduce a symmetry-breaking geometry on the outer ring of the qubit pad, as illustrated in Figure 13, while keeping the qubit and resonator frequencies unchanged. The size, location, and shape of these geometries serve as design parameters that dictate the strength and sign of the coupling between the qubit mode, resonator mode, and higher-order geometric modes, thereby generating poles at designated frequencies. The number of poles is determined by the detuning between the qubit and resonator modes, the coupling strength, and the relative phases of the higher modes. The capacitance matrix is computed for a mouse-bite geometry with a radius of 0.35mm, positioned 90 degrees from the Josephson junctions, using the Ansys Maxwell solver. This modification introduces significant changes in the capacitive network between the ground plane, the control lines, and the qubit and resonator pads. Notably, we observe a two-order-of- magnitude increase in the coupling between the control lines and between the qubit control line and the resonator, indicating a substantial increase in the strength of cross-coupling interactions. TABLE IV: Capacitance Matrix (pF) : We then analyze the admittance response of both the original and modified devices using a driven-terminal simulation, where an excitation port is defined at the qubit junction to probe its microwave environment. The results are illustrated in Figure 16 and reveal two cancellation zeros at 5.83 GHz and 6.3 GHz in the modified device, in contrast to the absence of cancellation zeros in the original device. Additionally, new higher-order modes emerge across the broader frequency spectrum. Figures 15 and 16 represent results from devices having different symmetry breaking perturbations. The following text provides further detail about enhancement of ZZ interactions.We consider four higher-order modes labeled ^^ ∈ {1,2,3,4} with frequencies ^^^^. Eachmode couples to qubit q with a strength ^^^^,^^. They may have mode-mode cross-couplings ^^^^,^^. All couplings, frequencies, and drives are assumed well off resonance so that a second-order Schrieffer-Wolff-type expansion is valid. We first consider the static ZZ interaction. Wei et al. (“Quantum crosstalk cancellation for fast entangling gates and improved multi-qubit performance”, Phys. Rev. Lett.129, 060501 (2022)), hereby incorporated by reference in its entirety, shows that for two Duffing transmons with bare coupling J, the “static” crosstalk is given by: where Δ01 = ^^0 − This is an approximation but is often accurate if Δ01, ^^ ≫ ^^. Foreach higher-order mode m with frequency ^^^^, we define the qubit-mode detunings as: and the qubit-drive detunings as: We also define ^^^^,^^as the qubit-mode coupling and ^^^^as the qubit anharmonicity. In the siZZle approximation (defined in Wei et al.), each mode contributes an added second- order term. A common shorthand is to define an “effective” ^^^^as: which lumps in the relevant denominators. Generalizing Equation (6) from Wei et al. to each mode m, we obtain: where we typically take − ^^^^) with small corrections from ^^^^. This formulaincorporates standard approximations, including off-resonant expansions, rotating-wave approximation, and small ^^^^,^^. If mode m and mode n have a cross-coupling ^^^^,^^, additional second-order interference paths arise. Symbolically, these terms take the form: where additional factors of ^^^^and relevant drive detunings appear. The precise denominators may become complicated, but the underlying principle remains the same: each cross-coupling opens up an extra channel for a virtual photon to pass from qubit 0 toqubit 1 via the mode transition ^^ → ^^. Thus, the total (approximate) ZZ shift is given by: In practice, we typically define functions for and Δ(^^,^^)and sum them numerically. Figure 17(a) is a plot showing the static ZZ shift as a function of the detuningbetween qubit frequencies ^^^^0 − ^^^^1. The dashed line represents the ZZ shift withouthigher modes, and the solid line includes contributions from four higher-order modes and cross-couplings. Figure 17(b) is a plot of the static ZZ shift as a function of the qubit anharmonicity^^^^0 = ^^^^1. The dashed line shows the ZZ shift ignoring higher modes, while the solid lineincludes the additional contributions from higher-order modes and cross-couplings. Figure 17(c) is a plot of the static ZZ shift as a function of the drive frequency ^^^^. The dashed line shows the contribution without higher modes, and the solid line includes the effects of the four higher-order modes and cross-couplings. Figure 17(d) shows heatmap representations of the static ZZ shift as a function ofdetuning ^^^^0 − ^^^^1 (horizontal axis) and drive frequency (vertical axis). The leftpanel shows the ZZ shift ignoring higher modes, and the right panel incorporates effects of higher-order modes and cross-couplings. The scale on the right represents the magnitude of the static ZZ shift in MHz. Figures 17(a)-(d) show a clear ZZ boost over a specific range of parameters important for the siZZle gate, namely qubits' detuning, anharmonicity and drive frequency. We can also see the boost happens with + / - signs of ZZ. Also, we see some very low values of ZZ as you can see in the 2D sweep are much lower than the standard case, which basically a cancellation of ZZ. This can also be seen by the solid lines crossing through zero at various points in the curves shown in Figures 17(a)-(c). Thus, the approach of the present disclosure provides both ZZ enhancement and ZZ suppression at selected frequencies. Unlike the solutions in the literature where we need to add a tunable coupler, or lamda / 4 tranmission line, etc to achieve similar functionality, the approach of the present disclosure provides this behaviour based solely on consideration of the additional higher order modes of the circuit. Boosting ZZ and suppressing ZZ may both be useful in practice. For example, suppression of ZZ coupling may be desirable in a particular set of gates, e.g. cross- resonance gate, to improve gate fidelity. On the other hand, in some implementations, it may be desirable to enhance (boost) ZZ instead and run gates based on ZZ, named native z-gates. The approach of the present disclosure, using higher order interference effects, thus effectively provides a valuable new degree of freedom for controlling ZZ coupling as a function of various relevant parameters (e.g., qubit detuning, qubit anharmonicity, and qubit drive frequency). Figure 18 depicts simulation results for a four-qubit device showing simulated Purcell-limited relaxation time ^^1−^^^^^^as a function of frequency ^^ for the four qubits, with the respective curves labelled Q1-Q4. In this example, qubit Q3 is provided with a symmetry breaking perturbation, which can be seen to provide extra structure in ^^1−^^^^^^.

Claims

CLAIMS 1. A circuit quantum electrodynamics, QED, component having a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in a circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the component is configured to generate one or more engineered interference effects and / or additional functionality in the circuit QED device.

2. The circuit QED component according to claim 1, wherein the one or more engineered interference effects and / or additional functionality comprises a frequency response with a transmission minimum at or near the qubit frequency of the circuit QED device.

3. The circuit QED component according to any preceding claim, wherein the one or more engineered interference effects and / or additional functionality comprises a frequency adjustment of the qubit frequency of the circuit QED device.

4. The circuit QED component according to any preceding claim, wherein the one or more engineered interference effects and / or additional functionality comprises a change in the anharmonicity response of the circuit QED device.

5. The circuit QED component according to any preceding claim, wherein the one or more engineered interference effects and / or additional functionality comprises a change in a resonant frequency of the circuit QED device.

6. The circuit QED component according to any preceding claim, wherein the one or more engineered interference effects and / or additional functionality comprises a change in a variation of ZZ interactions as a function of one or more of the following: qubit detuning, qubit anharmonicity, and qubit drive frequency.

7. The circuit QED component according to any preceding claim, wherein the one or more engineered interference effects and / or additional functionality are based on a relativeincrease in the number of pathways between the lowest resonant mode of the circuit QED device and higher order modes of the circuit QED device.

8. The circuit QED component according to any preceding claim, comprising a first electrode and a second electrode.

9. The circuit QED component according to claim 8, wherein the first and second electrodes are coaxial electrodes.

10. The circuit QED component according to claim 9, wherein the second electrode is an outer electrode formed concentrically around the first electrode, wherein the first and second electrodes are coplanar.

11. The circuit QED component according to any of claims 8 to 10, wherein the first electrode and / or the second electrode is substantially circular, is substantially a regular polygon, is substantially a square or is substantially a triangle.

12. The circuit QED component according to any of claims 8 to 11, wherein the symmetry breaking perturbation comprises a local variation in a dimension of the first and / or second electrodes.

13. The circuit QED component according to any of claims 8 to 12, wherein the second electrode has a major surface that follows a geometry along a closed path, wherein the width of the closed path is substantially uniform for at least the majority of the closed path.

14. The circuit QED component according to claim 13, wherein the symmetry breaking perturbation comprises a local variation in the width of the closed path.

15. The circuit QED component according to claim 10, wherein a surface of the second electrode comprises a planar annulus.

16. The circuit QED component according to claim 15, wherein the symmetry breaking perturbation corresponds to a local variation in the width of the planar annulus.

17. The circuit QED component according to claim 16, wherein the local variation in the width comprises a widening or narrowing of the width of the planar annulus in a radial direction along a portion of the planar annulus.

18. The circuit QED component according to claim 16, wherein the local variation is based on an intersection of a circle or an ellipse with the planar annulus at the outside edge of the planar annulus or the inside edge of the planar annulus.

19. The circuit QED component according to any of claim 8 to 18, wherein the symmetry breaking perturbation is based on the global geometry of the first electrode and / or the second electrode.

20. The circuit QED component according to claim 19, wherein the global geometry of the second electrode corresponds to a distorted planar annulus, optionally wherein the distorted planar annulus comprises a discontinuity, optionally wherein the distorted planar annulus has a non-circular geometry, optionally wherein the non-circular geometry is an elliptical geometry.

21. The circuit QED component according to any of claims 8 to 20, wherein a surface of the first electrode comprises a planar circle.

22. The circuit QED component according to claim 21, wherein the symmetry breaking perturbation corresponds to a local variation in the radius of the planar circle.

23. The circuit QED component according to claim 22, wherein the local variation in the radius comprises an increase and / or a decrease of the radius of the planar circle along a portion of the circumference of the planar circle.

24. The circuit QED component according to claim 23, wherein the local variation is based on an intersection of a circle or an ellipse with the outside edge of the planar circle.

25. The circuit QED component according to any preceding claim, wherein the circuit QED component is a transmon, optionally wherein the transmon is a coaxmon or an Xmon.

26. The circuit QED component according to any of claims 8 to 25, wherein the first electrode is a superconducting electrode and the second electrode is superconducting electrode.

27. The circuit QED component according to claim 26, wherein the circuit QED component comprises a Josephson Junction connected between the superconducting electrodes.

28. The circuit QED component according to claim 27, wherein the one or more engineered interference effects and / or additional functionality comprises a transmission minimum based on the relative position of the Josephson Junction and the symmetry breaking perturbation.

29. The circuit QED component according to any of claims 26 to 28, wherein the first superconducting electrode and / or the second superconducting electrode comprise a planar surface with two or more substantially orthogonal planar portions, optionally wherein the planar geometry is cross-shaped, wherein the symmetry breaking perturbation comprises a local variation in a dimension of the first and / or second superconducting electrodes.

30. The circuit QED component according to any of claims 1 to 24, wherein the circuit QED component is a resonator.

31. The circuit QED component according to claim 30 when dependent on any of claims 8 to 24, wherein the first electrode is a first resonator electrode and the second electrode is a second resonator electrode.

32. The circuit QED component according to any preceding claim, comprising at least one of a Josephson Junction having a non-linear geometry, a capacitor having a non-linear geometry and an inductor having a non-linear geometry.

33. The circuit QED component according to any preceding claim, wherein the geometrically symmetrical arrangement has rotational symmetry of order two or higher.

34. The circuit QED component according to any preceding claim wherein the component is configured to generate transmission minima to suppress transmission at oneor more qubit frequencies of a circuit QED device based on at least one of position, shape and / or size of the symmetry breaking perturbation.

35. A circuit quantum electrodynamics, QED, device comprising: a substrate comprising a first surface and an opposing second surface; one or more circuit QED components on the first surface of the substrate; and one or more circuit QED components on the second surface of the substrate, wherein each of the one or more circuit QED components on the second surface of the substrate is coupled to a respective one of the one or more circuit QED components on the first surface of the substrate, wherein at least one of the one or more circuit QED components on the first surface of the substrate and / or on the second surface of the substrate is the circuit QED component according to any of claims 1 to 34.

36. The circuit QED device according to claim 35, wherein the symmetry breaking perturbation is configured to increase the number of pathways available for photons propagating in the circuit QED device and thereby create a transmission minimum.

37. The circuit quantum electrodynamics, QED, device according to claim 35 or claim 36 comprising at least two entangled qubits, wherein at least one of the two entangled qubits has a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes in the circuit QED device, wherein the geometrical arrangement comprises a symmetry breaking perturbation relative to the geometrically symmetric arrangement such that the circuit QED device is configured to generate one or more engineered interference effects in the circuit QED device comprising a reduction in ZZ interactions between the entangled qubits.

38. A method of designing a circuit QED component for a circuit QED device, wherein: the circuit QED component has a geometrical arrangement based on a geometrically symmetric arrangement that enables the formation of degenerate modes inthe circuit QED device, the geometrical arrangement comprising a symmetry breaking perturbation relative to the geometrically symmetric arrangement; and the method comprises designing the symmetry breaking perturbation to generate one or more engineered interference effects and / or additional functionality in the circuit QED device.

39. The method of claim 38, wherein the circuit QED component is the circuit QED component of any of claims 1 to 34.

40. The method of claim 38 or 39, wherein the designing the symmetry breaking perturbation comprises selecting a size, location and / or shape of the symmetry breaking perturbation.

41. The method of claim 40, wherein the size, location and / or shape of the symmetry breaking perturbation is or are selected to define the strength and / or sign of coupling between different resonant modes in the circuit QED device to provide the one or more engineered interference effects and / or additional functionality.

42. The method of any of claims 38 to 41, wherein the one or more engineered interference effects and / or additional functionality comprise: a frequency response with a transmission minimum at or near the qubit frequency of the circuit QED device; a frequency adjustment of the qubit frequency of the circuit QED device; a change in the anharmonicity response of the circuit QED device; a change in a resonant frequency of the circuit QED device; and / or a change in a variation of ZZ interactions as a function of one or more of the following: qubit detuning, qubit anharmonicity, and qubit drive frequency.

43. The method of any of claims 38 to 42, wherein: the circuit QED device comprises a substrate comprising a first surface and an opposing second surface and the circuit QED component is on the first surface of the substrate; andthe symmetry breaking perturbation is designed to generate the one or more engineered interference effects and / or additional functionality by introducing interference between multiple resonant modes associated with respective pathways between or within circuit QED components supported by the substrate.

44. The method of any of claims 38 to 43, wherein the symmetry breaking perturbation is designed to generate the one or more engineered interference effects and / or additional functionality by introducing interference between multiple resonant modes involving higher-order resonant modes of circuit QED components that are either qubits or readout resonators.

45. A method of manufacturing a circuit QED component, comprising designing a circuit QED component using the method of any of claims 38 to 44 and manufacturing a circuit QED component according to the design.

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