Superconducting circuit comprising multi-resonator structures and quantum computing system

Abstract

There is provided reduced effect of effect of variations in kinetic inductance to a relative frequency. A quantum circuit comprises multi-resonator structures configured to couple qubits to a common feed line, wherein each of the multi-resonator structures comprises a readout resonator and a Purcell-filter resonator, and the readout resonator and the Purcell-filter resonator are configured to have matching inductive contributions to a resonator impedance.

Description

[0001]SUPERCONDUCTING CIRCUIT COMPRISING MULTI-RESONATOR STRUCTURES AND QUANTUM COMPUTING SYSTEM FIELD OF THE INVENTION The invention is generally related to the field of quantum computing. In particular, the invention is related to a supreconducting circuit comprising multi-resonator structures. BACKGROUND OF THE INVENTION A quantum computing device, also referred to as a quantum computer, uses quantum mechanical phenomena, such as superposition and entanglement, to perform required quantum computing operations. Unlike a conventional computer that manipulates information in the form of bits (e.g., "1" or "0"), the quantum computer manipulates information using qubits. A qubit may refer not only to a basic unit of quantum information but also to a quantum device that is used to store one or more qubits of information (e.g., the superposition of "0" and "1"). Quantum computing devices can be implemented by superconducting circuits that typically use frequency multiplexed qubit readout using one or more resonators per qubit, where multiple qubits are read out by a single shared probe line. The qubits may be coupled to the single shared probe line by a readout resonator and one or more Purcell-filter resonators. The resonators have to be tuned relative to each to each other for accurate and reliable readout of the qubit states. Typically these resonators are implemented as transmission line resonators, for example made from coplanar waveguides (CPW) or microstrip lines, which are widely used in superconducting circuits due to their compatibility with fabrication techniques and low loss characteristics at cryogenic temperatures. A transmission line resonator is a one- dimensional structure that consists of a transmission line with a specific length and termination conditions to create a resonant mode at a desired frequency. In a transmission line resonator, the transmission line serves as a waveguide for propagating electromagnetic waves. The resonant behavior of a transmission line resonator is determined by its impedance, propagation constant, length and termination conditions. By carefully designing the length, impedance and propagation constant of the transmission line, a standing wave pattern can be established, creating resonance at a specific frequency. Transmission line resonators can be coupled to other elements in a superconducting quantum circuit, such as qubits or other resonators, to enable interactions and perform desired quantum operations. The coupling strength can be controlled by adjusting the geometry and placement of the coupling elements. In transmission line resonators, the impedance and propagation constant, and thus the resonant frequency of the resonator, is determined by a capacitive and an inductive contribution of the transmission line resonator to electrical properties of the circuit elements, depending on the detailed geometry of the transmission line. The capacitive contribution in a resonator arises from the presence of capacitance in the circuit. The capacitive contribution affects the behavior of the resonator by storing energy in an electric field. The inductive contribution in a resonator is related to the presence of inductance in the circuit. The inductive contribution affects the resonator's behavior by storing energy in a magnetic field. The combined capacitive and inductive contributions in a resonator determine its resonant frequency. In some superconducting materials, the inductive part of the resonator frequency is significantly affected by a so-called kinetic inductance. This may be desirable to enable smaller physical size of the resonator; or it may be undesired but an effect of the material details that are chosen for other reasons. Either way, the kinetic inductance component only contributes to the inductive part of the resonance frequency. The kinetic inductance effect is strongest in low-dimensional and thin-film superconductors. Examples of materials commonly used in superconducting resonators, where kinetic inductance is significant, include: Aluminum (Al): Aluminum is a widely used superconductor in superconducting quantum circuits. It has a relatively high critical temperature and exhibits strong kinetic inductance effects. Niobium (Nb): Niobium is another commonly used superconductor in the fabrication of superconducting resonators and qubits. It has a higher critical temperature compared to aluminum and exhibits significant kinetic inductance. Titanium Nitride (TiN): Titanium nitride is a superconducting material with a relatively high critical temperature. It is often used in high-frequency resonators due to its low microwave losses and substantial kinetic inductance. Other Superconductors: Other superconducting materials, such as tantalum (Ta), niobium nitride (NbN), and certain alloy combinations, have also been used in superconducting resonators and exhibit kinetic inductance effects. SUMMARY This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary is not intended to identify key features of the invention, nor is it intended to be used to limit the scope of the invention. The objective of the invention is to provide a technical solution to reduce effect of variations of kinetic inductance to relative frequency of a resonator in a superconducting circuit. The objective above is achieved by the features of the independent claims in the appended claims. Further embodiments and examples are apparent from the dependent claims, the detailed description and the accompanying drawings. According to a first aspect, there is provided a superconducting circuit comprising multi-resonator structures configured to couple qubits to a common feed line, wherein each of the multi-resonator structures comprises a readout resonator and a Purcell-filter resonator, and the readout resonator and the Purcell-filter resonator are configured to have matching inductive contributions to a resonator impedance. According to a second aspect there is provided a quantum computing system comprising at least one superconducting circuit according to an aspect and a control unit configured to perform quantum computing operations based on frequency-multiplexed readout of qubits connected to the common feedline by the multi-resonator structures. At least some aspects provide reducing effect of variations of kinetic inductance to relative frequency of a resonator in a superconducting circuit. BRIEF DESCRIPTION OF THE DRAWINGS The invention is explained below with reference to the accompanying drawings in which: Fig.1 illustrates an example of a superconducting circuit in accordance with at least some embodiments; Fig. 2 illustrates an example of a multi-resonator structure for coupling to a qubit of a superconducting circuit in accordance with at least some embodiments; and FIG.3 shows a block diagram of a quantum computing system in accordance with at least some embodiments. DETAILED DESCRIPTION Various embodiments of the invention are further described in more detail with reference to the accompanying drawings. However, the invention may be embodied in many other forms and should not be construed as limited to any certain structure or function discussed in the following description. In contrast, these embodiments are provided to make the description of the invention detailed and complete. According to the detailed description, it will be apparent to the ones skilled in the art that the scope of the invention encompasses any embodiment thereof, which is disclosed herein, irrespective of whether this embodiment is implemented independently or in concert with any other embodiment of the invention. For example, the circuits and apparatus disclosed herein may be implemented in practice by using any numbers of the embodiments provided herein. Furthermore, it should be understood that any embodiment of the invention may be implemented using one or more of the elements presented in the appended claims. The word "exemplary" is used herein in the meaning of "used as an illustration". Unless otherwise stated, any embodiment described herein as "exemplary" should not be construed as preferable or having an advantage over other embodiments. Any positioning terminology, such as "left", "right", "upper", "lower", etc., may be used herein for convenience to describe one element's or feature's relationship to one or more other elements or features in accordance with the figures. It should be apparent that the positioning terminology is intended to encompass different orientations of the circuit disclosed herein, in addition to the orientation (s) depicted in the figures. As an example, if one imaginatively rotates the circuit in the figures 90 degrees clockwise, elements or features described as "left" and "right" relative to other elements or features would then be oriented, respectively, "above" and "below" the other elements or features. Therefore, the positioning terminology used herein should not be construed as any limitation of the invention. Although the numerative terminology, such as "first", "second", etc., may be used herein to describe various embodiments and the features thereof, it should be understood that the embodiments and the features thereof should not be limited by this numerative terminology. This numerative terminology is used herein only to distinguish one embodiment or feature from another embodiment or feature. Thus, a first embodiment discussed below could be called a second embodiment and vice versa, without departing from the teachings of the invention. The following embodiments are exemplary. Although the specification may refer to "an", "one", or "some" embodiment(s) in several locations, this does not necessarily mean that each such reference is to the same embodiment(s), or that the feature only applies to a single embodiment. Single features of different embodiments may also be combined to provide other embodiments. As used herein, “at least one of the following: ” and “at least one of ” and similar wording, where the list of two or more elements are joined by “and” or “or”, mean at least any one of the elements, or at least any two or more of the elements, or at least all the elements. Superconducting computational elements are the building blocks of superconducting digital and quantum computers. These elements are based on superconducting electronic devices that can operate at very low temperatures (typically below 4 Kelvin) and exhibit several unique properties, including zero resistance, high speed, and low power consumption. Here are some of the most common types of superconducting computational elements: - Josephson junction: A Josephson junction is a superconducting device that consists of two superconductors separated by a thin insulating layer. It exhibits a non-linear current-voltage characteristic, and can be used as a switch, memory element, or amplifier in superconducting circuits. - Superconducting quantum interference device (SQUID): A SQUID is a superconducting loop with two Josephson junctions that exhibits a highly sensitive response to magnetic fields. It can be used as a sensor for magnetic fields or as a readout element for qubits in superconducting quantum computing. - Flux qubit: A flux qubit is a superconducting loop with a Josephson junction that exhibits a quantized magnetic flux, which can be used as a qubit in quantum computing. The quantum states of the qubit are controlled by changing the external magnetic flux through the loop. - Transmon qubit: A transmon qubit is a superconducting circuit that consists of a capacitively-shunted Josephson junction. It exhibits a low sensitivity to charge noise, which makes it a popular choice for quantum computing applications. - Superconducting microwave resonator: A superconducting microwave resonator is a device that consists of a superconducting loop or cavity that can store and manipulate electromagnetic waves at specific frequencies. It allows for the exchange of quantum information between qubits and facilitates coherent interactions. It can be used as a quantum bus or as a filter element in superconducting circuits. These superconducting computational elements can be combined to build complex digital and quantum computing systems. By leveraging the unique properties of superconducting electronics, these systems can achieve high computational power and energy efficiency, which makes them attractive for a wide range of applications, including cryptography, optimization, and simulation. As used in the embodiments disclosed herein, a qubit may refer to a superconducting quantum device configured to store one or more quantum bits of information (or qubits for short). In this sense, the qubit serves as a quantum information storage and processing device. The source of nonlinearity in the qubit may be represented by one or more Josephson junctions. The term "Josephson junction" is used herein in its ordinary meaning and may refer to a quantum mechanical device made of two superconducting electrodes which are separated by a barrier (e.g., a thin insulating tunnel barrier, normal metal, semiconductor, ferromagnet, etc.). Kinetic inductance arises due to the motion of charge carriers, typically electrons, in a material. It is a phenomenon that is particularly relevant in certain types of superconductors, where electrons form Cooper pairs and exhibit macroscopic quantum mechanical behavior. In a superconductor, the motion of Cooper pairs gives rise to the phenomenon of supercurrent, which is the flow of electric current without any resistance. As the Cooper pairs move through the material, they experience an effective inertia known as the kinetic inductance. The kinetic inductance is a result of the conservation of momentum in the superconducting state. When an external force is applied to the Cooper pairs, they resist changes in their motion due to their effective mass and the quantum mechanical nature of their behavior. This resistance to changes in current flow is analogous to the behavior of an inductor in a conventional circuit, hence the term "kinetic inductance." The kinetic inductance is proportional to the effective mass of the Cooper pairs and their velocity. It depends on the properties of the superconducting material, such as its critical temperature, coherence length, and penetration depth. These material properties determine the strength of the Cooper pair coupling and their behavior in response to external forces. Variations of kinetic inductance affect relative frequency of a resonator in a superconducting circuit. The relative frequency refers to the change in frequency compared to a reference frequency. It can be expressed as a ratio or percentage change. The ratio may be obtained by dividing a frequency difference between the resonant frequency of the resonator and the reference frequency with the reference frequency itself. In the context of the superconducting circuit, the relative frequency could be the change in resonant frequency due to variations in the kinetic inductance. According to the embodiments disclosed herein, a quantum computing system, also referred to as a quantum computer, may refer to an apparatus that is configured to perform different quantum computing operations (e.g., qubit operations, such as reading the state of a qubit, initializing the state of the qubit, and entangling the state of the qubit with the states of other qubits in the quantum computing apparatus, etc.) based on frequency-multiplexed readout of qubits connected to a common feedline by multi- resonator structures. Existing implementation examples of such quantum computing systems may include superconducting quantum computers, trapped ion quantum computers, quantum computers based on spins in semiconductors, quantum computers based on cavity quantum electrodynamics, optical photon quantum computers, quantum computers based on defect centers in diamond, etc. According to the embodiments disclosed herein, a multi-resonator structure comprises a readout resonator and a Purcell-filter resonator which are configured to have matching inductive contributions to a resonator impedance. In this way effect of variations in kinetic inductance to a relative frequency of the readout resonator and to a relative frequency of the Purcell-filter resonator may be reduced. The exemplary embodiments disclosed herein provide a technical solution that allows mitigating or even eliminating the above-sounded drawbacks of the prior art. Fig.1 illustrates an example of a superconducting circuit 100 in accordance with at least some embodiments. The superconducting circuit 100 comprises a common feed line 102 and qubits, ‘T’, coupled to the common feed line by multi-resonator structures 104, 106, 108 each of which comprises a Purcell-filter resonator ‘P’ and a readout resonator ‘R’. The readout resonator and the Purcell-filter resonator are configured to have matching inductive contributions to a resonator impedance for reducing effect of variations in kinetic inductance to a relative frequency of the readout resonator and to a relative frequency of the Purcell-filter resonator. An inductive contribution to a resonator impedance refers to a portion of the resonator's impedance or reactance that arises from inductance. The inductive contribution is related to the presence of inductors or elements that exhibit inductance within the resonator structure. When a resonator is operating near its resonant frequency, the inductive contribution becomes significant and affects the behavior of the circuit. It manifests as a reactive component that contributes to the impedance of the resonator. In a resonator, the inductive contribution can interact with the capacitive contribution, which arises from the presence of capacitors or elements exhibiting capacitance. Together, the inductive and capacitive contributions determine the overall impedance or reactance of the resonator, and the propagation constant of the resonator, which affect its resonant frequency and response to signals. In an example, the qubits ‘T’ are transmons. A qubit with qubit transition frequency fQ may be coupled with strength g to a dedicated λ / 4 readout resonator ‘R’ with frequency fR. The ‘P’ and ‘R’ associated with each qubit have a large detuning ∆RQ = 2π ×(fR− fQ) >> g / whereby R is shifted 2χ when ‘T’ is excited from its ground to a first-excited state. R is also coupled with strength J to a dedicated λ / 4 Purcell-filter resonator ‘P’ at fP. ‘P’ is coupled with rate κ to a 50 Ω feedline 102 whose output is connected to an amplification chain with a traveling-wave parametric amplifier (TWPA) at its front end. In an example, ∆PR / 2π = fP– fR= 0, so that ‘R’ and ‘P’ hybridize into two readout modes, a readout resonator mode and a Purcell filter mode, that frequency-split by 2J, each with effective linewidth κeff = κ / 2 and dispersive shift 2χeff = χ. It should be noted that high κeff provides fast readout as it sets the rate at which the readout mode builds / depletes internal photon population when the pulse is turned on / off. It should be noted that in practice, the readout resonator mode and the Purcell filter mode can have a larger line width than the readout resonator mode, and a dispersive shift of the Purcell filter mode may be negligible. Independent readout of multiple transmons ‘T’ using the common feedline 102 is possible by frequency-division multiplexing (FDM) provided that the hybridized modes for different transmons do not overlap in frequency. For each transmon, a readout pulse of duration τp is applied to the feedline input, at a frequency f where transmission is dependent on qubit state. Frequency-division multiplexed (FDM) readout of qubits is a technique used in quantum computing to simultaneously read out multiple qubits by assigning a distinct frequency to each qubit. It is a multiplexing technique that allows for parallelized readout of qubits in a quantum system. In an example, in FDM readout of qubits ‘T’ of the superconducting circuit 100, each qubit is coupled to a readout resonator ‘R’ at specific frequency fR. The resonator serves as a mediator between the qubit and the measurement apparatus. By assigning different frequencies to each qubit-resonator pair, the qubits can be individually addressed and their states can be read out independently. The readout process involves applying a microwave pulse to the readout resonators via the common feed line 102 and measuring the response of the readout resonators. The response of each readout resonator corresponds to the state of the associated qubit. By analyzing the frequencies and amplitudes of the response signals, the state of each qubit can be determined. In an example, the readout resonator ‘R’, or a measurement resonator, is a component for a superconducting quantum circuit for the measurement and readout of qubits ‘T’. The readout resonator is configured to detect and extract information about the state of a qubit by interacting with the qubit's electromagnetic field. In an example, the readout resonator operates as a high-quality factor (Q-factor) microwave resonator, which means it has a narrow bandwidth and can store energy for a longer period of time. The resonator is typically coupled to the qubit such that their electromagnetic fields interact, allowing for the transfer of information. When the qubit is in a particular state, such as a ground state or an excited state, it produces a specific microwave frequency signal. This signal is coupled to the readout resonator, causing a change in the resonant frequency or quality factor of the resonator. By measuring a change in the resonator's properties, the state of the qubit can be inferred. The readout resonator is typically designed to have a frequency that is distinct from the frequencies used for qubit control and manipulation. This separation allows for the independent measurement of the qubit state without interference from other control signals. To extract information from the readout resonator, various measurement techniques can be employed. For example, the reflection or transmission of a microwave probe signal can be analyzed to determine the resonator's properties and infer the qubit state. This measurement can be performed using sensitive microwave detectors, such as cryogenic amplifiers, which can detect the small changes in the resonator's response. In an example, the Purcell-filter resonator ‘P’ is a component for a superconducting quantum circuit which is configured to modify transmission of specific frequencies on a transmission line based on the principles of the Purcell effect. The Purcell effect is a phenomenon in quantum optics that describes the modification of the spontaneous emission rate of an excited quantum system in the presence of a resonant electromagnetic environment. When an excited quantum system, such as an atom, quantum dot, or qubit, interacts with its surrounding electromagnetic environment, the Purcell effect comes into play. The emission of a photon from the excited state of the quantum system can be enhanced or suppressed depending on the properties of the environment. The Purcell effect is typically described in terms of the Purcell factor (F), which quantifies the modification of the spontaneous emission rate. The Purcell factor is given by: F = (3 / 4π^2) * (λ / n)^3 * Q, where λ is the wavelength of the emitted photon, n is the refractive index of the environment, and Q is the quality factor of the resonator or cavity in which the quantum system is embedded. The Purcell effect arises due to the modification of the density of electromagnetic states in the environment. The density of states affects the probability of the quantum system to emit a photon at a particular frequency. In a resonant environment, the density of states is modified, leading to changes in the spontaneous emission rate. In an example the readout resonator ‘R’ and the Purcell-filter resonator ‘P’ may be transmission line resonators. It should be noted that the readout resonator and the Purcell-filter resonator are possible such as at least one of the following: lumped element resonators; or distributed resonator elements. A lumped element resonator is a type of resonant circuit that consists of discrete or "lumped" components rather than distributed elements. It is designed to exhibit resonant behavior at a specific frequency or a narrow range of frequencies. In a lumped element resonator, the key components typically include capacitors and inductors. These discrete components are connected together in a circuit configuration that forms a resonant structure. The resonant frequency of the circuit is determined by the values of the capacitors and inductors and the overall circuit topology. A distributed resonator comprises a transmission line that is designed in a specific configuration that allows for the formation of standing waves at a particular frequency or a narrow range of frequencies. The distributed nature of the resonator means that the resonance occurs along the length of the transmission line, and the components are distributed rather than concentrated at a specific point. The resonant frequency of a distributed resonator is determined by the physical properties of the transmission line, such as its length, characteristic impedance, and the speed of propagation of the signals along the line. The resonant behavior arises due to the constructive interference of the forward and reflected waves along the transmission line. In an example in accordance with at least some embodiments, the readout resonator ‘R’ and the Purcell-filter resonator ‘P’ are transmission line resonators, and line / gap widths of the readout resonator and the Purcell-filter resonator are adjusted such that the total sensitivity to kinetic inductance is the same for each of the resonators. In an example, a transmission line comprises at least two conductors and the line / gap widths refer to the distance between the two conductors of the resonator. The line / gap width affects the capacitance of the resonator, which in turn affects its resonant frequency and quality factor. In an example in accordance with at least some embodiments, relative resonant frequencies of the readout resonators are controlled for spacing the qubits on the common feed line at readout for frequency-division multiplexed (FDM) readout of the qubits. At each of the multi-resonator structures, resonant frequency of a Purcell-filter resonator is tuned relative to a resonant frequency of a readout resonator. In this way frequency spacing of the qubits on the common feed line at readout is controlled for frequency-division multiplexed (FDM) readout of the qubits, while variations in kinetic inductance to a relative frequency of the readout resonator and to a relative frequency of the Purcell-filter resonator can be reduced. Fig.2 illustrates an example of a multi-resonator structure for coupling to a qubit of a superconducting circuit in accordance with at least some embodiments. The superconducting circuit may be the superconducting circuit of Fig.1. The multi-resonator structure provides reducing effect of variations in kinetic inductance to a relative frequency of resonators forming the multi-resonator structure. The multi-resonator structure comprises a readout resonator 206 coupled to a Purcell-filter resonator 208. The readout resonator and the Purcell-filter resonator are configured to have matching inductive contributions to a resonator impedance for reducing effect of variations in kinetic inductance to a relative frequency of the readout resonator and to a relative frequency of the Purcell-filter resonator. The readout resonator and the Purcell filter may comprise transmission lines 205, 215 that couple the multi-resonator structure to the qubit 202 and to a common feed line 204. One of the transmission lines, also referred to a readout transmission line 205, may form at least a part, e.g. a section, of the readout resonator. The readout transmission line may extend between the qubit and a capacitor element 214 configured to couple the readout transmission line to the Purcell-filter, or a Purcell-filter transmission line 215 that forms at least a part, e.g. a section, of the Purcell-filter. The Purcell-filter transmission line 215 is connected to the capacitor element 214 for coupling with the readout transmission line 205 and to a capacitor element 216 for coupling with the common feed line 204. In this way the readout transmission line and the Purcell-filter transmission line provide propagation of signals from the qubit 202 to the common feedline. A further capacitor element 210 that is connected to the ground may be tapped to the readout transmission line 205 for tuning coupling of the qubit to the common feed line and frequency difference between the readout resonator and the Purcell-filter resonator, whereby effect of variations in kinetic inductance to a relative frequency may be reduced. It should be noted that the inductive contributions of the readout resonator and the Purcell-filter resonator may matched in various ways. When the inductive contributions of the readout resonator and the Purcell-filter resonator are matched the inductive contributions may be the substantially the same. The matching of the inductive contributions may be provided for example based on at least one of the following: - matching distributions of capacitive and inductive contribution to frequency of the readout resonator and the Purcell-filter resonator; - the readout resonator and the Purcell-filter resonator are transmission line resonators, where each of the resonators has the same transmission line dimensions (line widths and length), and the difference in coupling and / or required frequency differences is tuned by adding extra capacitive loading to ground; or - the readout resonator and the Purcell-filter resonator are transmission line resonators where the line / gap widths of the transmission line are adjusted in each resonator such that the total sensitivity to kinetic inductance is the same for each resonator; or - the readout resonator and the Purcell-filter resonator are lumped elements or distributed elements, where the sensitivity to kinetic inductance is tuned by simulation of the total structure and adding capacitive or inductive compensation elements to minimize sensitivity. In an example in accordance with at least some embodiments, the readout resonator 206 comprises a readout transmission line 205 coupled to a qubit 202, and a capacitor element 210 is tapped to the readout transmission line. In this way distributions of capacitive and inductive contribution to frequency of the readout resonator and the Purcell-filter resonator may be controlled, whereby the readout resonator and the Purcell- filter resonator may be tuned for controlling their frequency difference. In an example in accordance with at least some embodiments, the Purcell-filter resonator 208 is a transmission line resonator coupled to the readout transmission line 205 and the common feed line 204. Matching inductive contributions of the readout resonator and Purcell-filter resonator provides controlling frequency difference between the readout resonator and the Purcell-filter resonator and readout speed of the qubit via the common feedline. In an example in accordance with at least some embodiments, the readout resonator 206 and the Purcell-filter resonator 208 are configured for inductive coupling with the readout transmission line 205. In an example in accordance with at least some embodiments, the readout resonator 206 and the Purcell-filter resonator 208 comprise loops that are arranged on opposite sides of the readout transmission line 205 coupled to a qubit 202. In this way a readout resonator loop and a Purcell-filter resonator loop may be positioned laterally around the readout transmission line, which facilitates inductive coupling of the readout resonator loop readout resonator and the Purcell-filter resonator loop with the readout transmission line 205. In an example in accordance with at least some embodiments, the readout resonator 206 and the Purcell-filter resonator 208 are transmission line resonators, and a length of the readout resonator and a length of the Purcell-filter resonator are the same. In this way distributions of capacitive and inductive contribution to frequency of the readout resonator and the Purcell-filter resonator may be controlled, whereby the readout resonator and the Purcell-filter resonator may be tuned for controlling their frequency difference. FIG. 3 shows a block diagram of a quantum computing system 300 in accordance with at least some embodiments. The system 300 comprises at least one superconducting circuit 302 and a control unit 304. The circuit 302 may be implemented based on one or more circuits described with Fig 2. The control unit 304 is configured to perform quantum computing operations by using the at least one circuit 302. The system 300 may further comprise a memory 306 storing executable instructions 308 which, when executed by the control unit 304, may cause the control unit 304 to perform the quantum computing operations. The control unit 304 may also store the result(s) of the quantum computing operations to the memory 306. It should be noted that the number, arrangement and interconnection of the constructive elements constituting the system 300, which are shown in FIG. 3, are not intended to be any limitation of the present disclosure, but merely used to provide a general idea of how the constructive elements may be implemented within the system 300. For example, the system 300 may comprise two or more circuits 302 each implemented based on one or more circuits 300 Fig 3. The control unit 304 may refer a central processing unit (CPU), general-purpose processor, single-purpose processor, microcontroller, microprocessor, application specific integrated circuit (ASIC), field programmable gate array (FPGA), digital signal processor (DSP), complex programmable logic device, etc. It should be also noted that the control unit 304 may be implemented as any combination of one or more of the aforesaid. As an example, the control unit 304 may be a combination of two or more microprocessors. The memory 306 may be implemented as a classical nonvolatile or volatile memory used in the modern electronic computing machines. As an example, the nonvolatile memory may include Read-Only Memory (ROM), ferroelectric Random-Access Memory (RAM), Programmable ROM (PROM), Electrically Erasable PROM (EEPROM), solid state drive (SSD), flash memory, magnetic disk storage (such as hard drives and magnetic tapes), optical disc storage (such as CD, DVD and Blu-ray discs), etc. As for the volatile memory, examples thereof include Dynamic RAM, Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDR SDRAM), Static RAM, etc. The executable instructions 308 stored in the memory 306 may be configured as a computer executable code which causes the control unit 304 to perform the quantum computing operations by using the circuit 302. The computer executable code for carrying out the quantum computing operations may be written in any combination of one or more programming languages, such as Java, C++, or the like. In some examples, the computer executable code may be in the form of a high-level language or in a pre-compiled form, and be generated by an interpreter (also pre-stored in the memory 306) on the fly. Electromagnetic couplings between components may be, for example, capacitive, inductive, or both capacitive and inductive. In some embodiments, couplings between components are couplings through a space between two chips on which two components are respectively positioned. For example, in some implementations, a qubit positioned on a first chip may be electromagnetically coupled to a readout resonator positioned on a second separate chip that is bump bonded to and facing the first chip. In some embodiments, couplings are in-plane, with a direction of a coupling electric field and / or magnetic field being substantially in a plane of a side of a chip on which both of two components are positioned. For example, in some implementations, a qubit on a first surface of a substrate may be electromagnetically coupled to a readout resonator on the first surface of the substrate, or the qubit on the first surface of the substrate may be electromagnetically coupled to a readout resonator on a second surface of the substrate. The first surface and the second surface may be on opposite sides of the substrate. The substrate may be a silicon substrate or a sapphire substrate, for example. Superconducting quantum circuits comprise circuit components for performing computational operations based on quantum operations. Components of the superconducting quantum circuits, also referred to quantum circuit components, or quantum computing circuit components, disclosed herein include circuit components for performing the quantum operations. That is, the superconducting quantum circuit components are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain superconducting quantum circuit components, such as qubits, may be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit components include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), readout resonators, readout transmission lines, amplifiers, and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUID or DC SQUID), among others. In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements may be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operations on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements may be used to transmit data to and / or receive data from the quantum circuit components through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors. Any or all of the components mentioned in this disclosure, including the qubits, the readout resonators, the readout transmission lines, and the amplifiers, may be made of a superconductor material, such as aluminum, niobium, or titanium nitride, among other superconductor materials. The components may include both superconductor and non- superconductor material. Fabrication of the circuit elements disclosed herein may entail the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein may entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process may include, e.g., wet etching techniques, dry etching techniques, or lift- off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography). During operation of a quantum computational system that uses circuit elements formed, in part, from superconductors, such as the circuit elements described herein, the circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (also referred to as superconducting) material may be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature. In certain implementations, control signals for the quantum circuit components (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit components. The control signals may be provided in digital and / or analog form. While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation may also be implemented in multiple implementations separately or in any suitable sub-combination. Although the exemplary embodiments of the invention are described herein, it should be noted that various changes and modifications could be made in the embodiments of the invention, without departing from the scope of legal protection which is defined by the appended claims. In the appended claims, the word "comprising" does not exclude other elements or operations, and the indefinite article "a" or "an" does not exclude a plurality. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination may in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. Particular embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims may be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. LIST OF SYMBOLS AND ABBREVIATIONS

Claims

CLAIMS 1. A superconducting circuit comprising multi-resonator structures configured to couple qubits to a common feed line, wherein each of the multi-resonator structures comprises a readout resonator and a Purcell-filter resonator, and the readout resonator and the Purcell-filter resonator are configured to have matching inductive contributions to a resonator impedance.

2. The superconducting circuit of claim 1, wherein the readout resonator comprises a readout transmission line coupled to a qubit, and a capacitor element is tapped to the readout transmission line for tuning the readout resonator with the Purcell- filter resonator.

3. The superconducting circuit of claim 1 or 2, wherein the Purcell-filter resonator comprises a transmission line coupled to the readout transmission line and the common feed line.

4. The superconducting circuit of claim 1, 2 or 3, wherein the readout resonator and the Purcell-filter resonator are configured for inductive coupling with the readout transmission line.

5. The superconducting circuit of claim 2, 3 or 4, wherein the readout resonator and the Purcell-filter resonator are arranged on opposite sides of the readout transmission line coupled to a qubit.

6. The superconducting circuit of any of claims 1 to 5, wherein the Purcell-filter resonator and the readout resonator are transmission line resonators and line / gap widths of the readout resonator and the Purcell-filter resonator are adjusted such that a sensitivity to kinetic inductance of the Purcell-filter resonator is the same with a sensitivity to kinetic inductance of the readout resonator.

7. The superconducting circuit of any of claims 1 to 6, wherein relative resonant frequencies of the readout resonators are controlled for spacing the qubits on the common feed line at readout for frequency-division multiplexed, FDM, readout of the qubits, and at each of the multi-resonator structures, a frequency of a Purcell- filter resonator is tuned relative to a frequency of a readout resonator.

8. A quantum computing system comprising at least one superconducting circuit according to any of claims 1 to 7 and a control unit configured to perform quantumcomputing operations based on frequency-multiplexed readout of qubits connected to the common feedline by the multi-resonator structures.

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