Patent Description:
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., "<NUM>" or "<NUM>"), 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 "<NUM>" and "<NUM>").

The quantum computer may be implemented based on superconducting circuits comprising superconducting qubits and/or resonators. Tunable interaction between the superconducting qubits or resonators is desirable for most of the quantum computing operations (e.g., large-scale quantum computation and simulation). It may be achieved by inserting an extra circuit element between the superconducting qubits or resonators, namely a tunable coupler which allows states of the superconducting qubits or resonators to interact with each other in a controlled manner. In other words, the tunable coupler arranged between the superconducting qubits or resonators allows one to implement a quantum gate.

A variety of tunable couplers have previously been designed and demonstrated experimentally. For example, according to one existing tunable-coupler setup, two superconducting qubits are capacitively coupled both directly and indirectly. The indirect interaction is mediated by the tunable coupler itself. If the tunable coupler is implemented as a single-island transmon, the sign of the indirect interaction is different than that of the direct interaction at coupler frequencies that are higher than qubit frequencies. Consequently, the direct and indirect coupling terms may cancel each other at a certain coupler frequency (also referred to as an idling frequency). However, in this tunable-coupler setup, the idling frequency is always above the qubit frequencies, which may adversely affect the operation speed and accuracy of two-qubit gates implemented based on such a tunable coupler.

<NPL> describes a floating tunable coupler that does not rely on direct qubit-qubit coupling capacitances to achieve the zero-coupling condition.

<NPL>), describes two fixed-frequency transmon qubits connected by a parametrically driven coupler.

Further embodiments and examples are apparent from the detailed description and the accompanying drawings.

The objective is to provide a technical solution that enables a tunable coupling between linear or nonlinear resonators.

According to a first aspect, a tunable resonator-resonator coupling circuit is provided. The circuit comprises a first resonator with a first frequency and a second resonator with a second frequency. The first resonator is linear or nonlinear, and the second resonator is linear or nonlinear. The first resonator and the second resonator have a direct coupling therebetween. The circuit further comprises a tunable coupling element arranged to provide an indirect coupling between the first resonator and the second resonator. The tunable coupling element has an idling frequency that is below each of the first frequency and the second frequency. The circuit further comprises a first readout resonator configured to readout the first resonator, the first readout resonator having a first operating frequency, and a second readout resonator configured to readout the second resonator, the second readout resonator having a second operating frequency. The tunable coupling element comprises a first superconducting island and a second superconducting island which are both ungrounded. The indirect coupling comprises: (i) a first coupling between the first resonator and the first superconducting island, (ii) a Josephson coupling between the first superconducting island and the second superconducting island, and (iii) a second coupling between the second resonator and the second superconducting island. Each of the first operating frequency and the second operating frequency is above each of the idling frequency of the tunable coupling element, the first frequency of the first resonator, and the second frequency of the second resonator. Since the superconducting islands are floating, i.e. ungrounded, it is possible to provide different signs of coupling frequencies for the resonators and the superconducting islands, which in turn allows the interaction between the first and second resonators to be controlled more efficiently. Moreover, the design, calibration, and operation of the circuit according to the first aspect are significantly easier and simpler compared to the existing analogues, while providing the same or even better performance.

In one embodiment of the first aspect, each of the first resonator and the second resonator is implemented as one of a harmonic oscillator, a coplanar waveguide resonator and a lumped element resonator. These linear resonators may be used as quantum busses, or they may be used to store qubit states. In this embodiment, the tunable coupler may allow on-demand transfer of quantum information between the linear resonators.

In another embodiment of the first aspect, each of the first resonator and the second resonator is implemented as a superconducting qubit. This may allow one to implement a two-qubit gate based on the circuit according to the first aspect.

In one embodiment of the first aspect, the direct coupling is implemented as a non-galvanic (e.g., capacitive or inductive) coupling between the first resonator and the second resonator. Such a capacitive coupling may increase the circuit performance.

In one embodiment of the first aspect, the first coupling is implemented as a capacitive coupling between the first resonator and the first superconducting island, and the second coupling is implemented as a capacitive coupling between the second resonator and the second superconducting island. Such capacitive couplings are easier to implement by using conventional technologies compared, for example, to inductive couplings.

In one embodiment of the first aspect, the tunable coupling element is implemented as a transmon qubit. The transmon qubit is less sensitive to a control noise and a signal noise compared to other standard qubit implementations.

Since the idling frequency of the whole coupling element is below the frequencies of the resonators, it is possible to design the coupling element such that its sweet spot is located at (or close to) the operation point of a two-qubit gate implemented by using superconducting qubits as the nonlinear resonators in the circuit according to the first aspect. This may reduce gate errors arising from the decoherence of the coupling element.

The coupling element will not be resonant with the readout resonators during the circuit operation (or gate operation if the resonators are represented by superconducting qubits), which would otherwise adversely affect the whole circuit operation.

In one embodiment of the first aspect, the indirect coupling further comprises a third coupling between the second resonator and the first superconducting island and a fourth coupling between the first resonator and the second superconducting island. By using these additional couplings, it is possible to increase the applicability and flexibility of the circuit according to the first aspect.

In one embodiment of the first aspect, the third coupling is implemented as a capacitive coupling between the second resonator and the first superconducting island, and the fourth coupling is implemented as a capacitive coupling between the first resonator and the second superconducting island. By using such capacitive couplings, it is possible to make the circuit according to the first aspect more compact.

In one embodiment of the first aspect, at least one of the first superconducting island and the second superconducting island has a capacitive coupling to a ground. This may allow the tunable coupling element to be shielded from other circuit elements, thereby reducing crosstalk.

According to a second aspect, a quantum computing apparatus is provided. The apparatus comprises at least one circuit according to the first aspect and a control unit configured to perform quantum computing operations by using the at least one circuit. By using such one or more tunable resonator-resonator coupling circuits in the quantum computing apparatus, one may increase the computational accuracy and processing speed of the quantum computing apparatus.

Other features and advantages will be apparent upon reading the following detailed description and reviewing the accompanying drawings.

Reference is made to the accompanying drawings in which:.

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 <NUM> 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.

As used in the embodiments disclosed herein, a tunable resonator-resonator coupling circuit may refer to a quantum circuit in which linear or nonlinear resonators are coupled to each other in a controlled manner. One non-restrictive example of the linear resonators may include a harmonic oscillator which is well-known in the art (for this reason, its description is omitted herein). One non-restrictive example of the nonlinear resonators may include a superconducting qubit.

As used in the embodiments disclosed herein, the superconducting 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 superconducting qubit serves as a quantum information storage and processing device. The source of nonlinearity in the superconducting 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.).

According to the embodiments disclosed herein, a quantum computing apparatus, 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 superconducting qubit, initializing the state of the superconducting qubit, and entangling the state of the superconducting qubit with the states of other superconducting qubits in the quantum computing apparatus, etc.) by using the tunable resonator-resonator coupling circuit disclosed herein. Existing implementation examples of such quantum computing apparatuses 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..

It should be noted that there are many tunable qubit-qubit coupling circuits (also referred to as tunable couplers) in the prior art. They are mainly used to implement quantum gates and allow the states of the qubits to interact with each other in a controlled manner. Some of the existing tunable qubit-qubit coupling circuits are discussed below with reference to <FIG>.

<FIG> shows a block diagram of a tunable qubit-qubit coupling circuit <NUM> in accordance with the prior art. The circuit <NUM> comprises a first superconducting qubit <NUM>, a second superconducting qubit <NUM>, and a tunable coupling element <NUM>. The first superconducting qubit <NUM> and the second superconducting qubit <NUM> are grounded and implemented similar to each other, i.e. each of them comprises a parallel connection of a capacitor (C<NUM> or C<NUM>, respectively) and two Josephson junctions (schematically shown as crosses in <FIG>). The first superconducting qubit <NUM> and the second superconducting qubit <NUM> are directly coupled to each other via a capacitor C<NUM>. Additionally, the first superconducting qubit <NUM> and the second superconducting qubit <NUM> are indirectly coupled to each other via the tunable coupling element <NUM>. The tunable coupling element <NUM> comprises a grounded superconducting portion <NUM> and a superconducting island <NUM>. It should be noted that the term "superconducting island" used herein may refer to a circuit component having all its parts galvanically connected to each other with an insignificant self-inductance at all operating frequencies of the circuit. The (upper) grounded superconducting portion <NUM> is provided on one side of a capacitor Cc and the two Josephson junctions, while the (lower) superconducting island <NUM> is provided on another side of the capacitor Cc and the two Josephson junctions. Thus, there is a Josephson coupling between the grounded superconducting portion <NUM> and the superconducting island <NUM>. Furthermore, the superconducting island <NUM> is ungrounded or floating, and is coupled to the first superconducting qubit <NUM> via a capacitor C<NUM>c and to the second superconducting qubit <NUM> via a capacitor C<NUM>c.

The Josephson junctions used in the circuit <NUM> provide the anharmonicity of the first and second superconducting qubits <NUM>, <NUM> and the tunable coupling element <NUM>. However, if the first and second superconducting qubits <NUM>, <NUM> and the tunable coupling element <NUM> all have a negative anharmonicity (which is usually observed in case of such a transmon regime), this may lead to the following problems:.

<FIG> shows a block diagram of a tunable qubit-qubit coupling circuit <NUM> in accordance with the prior art. Similar to the circuit <NUM>, the circuit <NUM> comprises a first superconducting qubit <NUM>, a second superconducting qubit <NUM>, and a tunable coupling element <NUM>. The first superconducting qubit <NUM> is implemented similar to the first superconducting qubit <NUM> and the second superconducting qubit <NUM>. In particular, the first superconducting qubit <NUM> is grounded and comprises a parallel connection of a capacitor C<NUM> and two Josephson junctions. At the same time, unlike the circuit <NUM>, the second superconducting qubit <NUM> is ungrounded and therefore connected in the circuit <NUM> differently as compared to the connection of the first superconducting qubit <NUM>. The tunable coupling element <NUM> comprises a grounded superconducting portion <NUM> and a (ungrounded) superconducting island <NUM> which are provided in the circuit <NUM> in the same manner as the grounded superconducting portion <NUM> and the superconducting island <NUM>, respectively, in the circuit <NUM>. The first superconducting qubit <NUM> and the second superconducting qubit <NUM> are also directly coupled to each other via a capacitor C<NUM> and indirectly coupled to each other via the tunable coupling element <NUM>. The indirect coupling is provided via capacitors C<NUM>c and C<NUM>c. However, the circuit <NUM> lacks some benefits which may be provided if one replaces the grounded superconducting portion <NUM> with an ungrounded superconducting island.

<FIG> shows a block diagram of a tunable qubit-qubit coupling circuit <NUM> in accordance with the prior art. Similar to the circuit <NUM> and the circuit <NUM>, the circuit <NUM> comprises a first superconducting qubit <NUM>, a second superconducting qubit <NUM>, and a tunable coupling element <NUM>. The first superconducting qubit <NUM> and the second superconducting qubit <NUM> are implemented similar to the first superconducting qubit <NUM> and the second superconducting qubit <NUM>, respectively, in the circuit <NUM>. However, the tunable coupling element <NUM> is connected between the first superconducting qubit <NUM> and the second superconducting qubit <NUM> differently as compared to the tunable coupling elements <NUM>, <NUM>. More specifically, the tunable coupling element <NUM> is provided in the circuit <NUM> such that there is no direct coupling between the first superconducting qubit <NUM> and the second superconducting qubit <NUM>. The first superconducting qubit <NUM> and the second superconducting qubit <NUM> are only indirectly coupled to each other via a first superconducting island <NUM> and a second superconducting island <NUM> which are both included in the tunable coupling element <NUM>. The first superconducting island <NUM> and the second superconducting island <NUM> are both ungrounded, with each of them being coupled to the first superconducting qubit <NUM> and the second superconducting qubit <NUM> via capacitors C1cu, C<NUM>cu, C<NUM>cb, and C<NUM>eb. At the same time, there is a Josephson coupling between the first superconducting island <NUM> and the second superconducting island <NUM>. However, the circuit <NUM> lacks the effective direct qubit-qubit coupling. For example, due to the lack of the capacitor C<NUM> (like in the circuit <NUM> or <NUM>), the two-qubit gates based on the circuit <NUM> may have a limited operation speed compared to a similar circuit with such a capacitor. Moreover, in some practical scalable multi-qubit systems, it is not possible to arrange sufficiently large capacitors C<NUM>cu and C<NUM>cb to provide the fast two-qubit gates.

The exemplary embodiments disclosed herein provide a technical solution that allows mitigating or even eliminating the above-sounded drawbacks of the prior art. In particular, the technical solution disclosed herein involves providing both direct and indirect couplings between linear or nonlinear resonators in a quantum circuit. The indirect coupling is provided by using a tunable coupling element that comprises two ungrounded superconducting islands. Since the superconducting islands are ungrounded, it is possible to provide different signs of coupling frequencies for the resonators and the superconducting islands, which in turn allows the interaction between the first and second resonators to be controlled more efficiently. Moreover, the design, calibration, and operation of the quantum circuit with such a tunable coupling element are easier compared to the existing analogues, while providing the same or even better performance.

<FIG> shows a block diagram of a tunable resonator-resonator coupling circuit <NUM> in accordance with a first exemplary embodiment of the invention. The circuit <NUM> comprises a first nonlinear resonator <NUM>, a second nonlinear resonator <NUM>, and a tunable coupling element <NUM>. In this exemplary embodiment, each of the first and second nonlinear resonators <NUM> and <NUM> is implemented as a superconducting qubit (similar to those shown in <FIG>). For this reason, the circuit <NUM> may be used to implement a two-qubit gate. However, the circuit <NUM> is not limited to this application and, for example, may be used to implement gates based on two or more linear resonators (e.g., harmonic oscillators, as will be described below with reference to <FIG>), or the circuit <NUM> may be used to implement quantum gates between qutrits (in a three-level quantum system) or qudits (in a d-level quantum system). The qudits and qutrits are supported by the nonlinear resonators since they have an infinite amount of energy levels. It should be also noted that the qubits may be formed if one restricts the operation of the nonlinear resonator to two lowest energy eigenstates.

Referring back to <FIG>, the first superconducting qubit <NUM> and the second superconducting qubit <NUM> are directly coupled to each other via a capacitor C<NUM> and indirectly coupled to each other via the tunable coupling element <NUM>. The tunable coupling element <NUM> comprises a first superconducting island <NUM> and a second superconducting island <NUM> which are both ungrounded. Moreover, the first superconducting island <NUM> and the second superconducting island <NUM> are arranged in the circuit <NUM> such that each of them is directly coupled only to one of the first superconducting qubit <NUM> and the second superconducting qubit <NUM>. More specifically, there are a first non-galvanic coupling between the first superconducting qubit <NUM> and the first superconducting island <NUM> and a second non-galvanic coupling between the second superconducting qubit <NUM> and the second superconducting island <NUM>. The first and second non-galvanic couplings are both capacitive, i.e. implemented via capacitors C<NUM>c and C<NUM>c, respectively. However, in some other embodiments, at least one of the first non-galvanic coupling, the second non-galvanic coupling, and the direct coupling between the first and second superconducting qubits <NUM> and <NUM> may be inductive, if required and depending on particular applications. Moreover, in some other embodiments, the direct coupling itself may be implemented as a galvanic coupling, and at least one of the first non-galvanic coupling and the second non-galvanic coupling may be replaced with a galvanic coupling. The first superconducting island <NUM> and the second superconducting island <NUM> are coupled to each other via Josephson junctions (see the crosses in <FIG>), i.e. have a Josephson coupling therebetween. Thus, the indirect coupling between the first superconducting qubit <NUM> and the second superconducting qubit <NUM> comprises the above-mentioned first non-galvanic coupling, Josephson coupling, and second non-galvanic coupling.

As can be seen in <FIG>, the arrangement area of the first superconducting island <NUM> is confined by one plate of the capacitor C<NUM>c, one plate of a capacitor Cc, and the two Josephson junction. As for the second superconducting island <NUM>, its arrangement area is confined by one plate of the capacitor C<NUM>c, another plate of the capacitor Cc, and the two Josephson junctions. Such arrangements of the first superconducting island <NUM> and the second superconducting island <NUM> are given by way of example only. In some other embodiments, the first superconducting island <NUM> and the second superconducting island <NUM> may be arranged such that the indirect coupling between the first and second superconducting qubits <NUM> and <NUM> comprises an additional third (galvanic or non-galvanic) coupling between the second superconducting qubit <NUM> and the first superconducting island <NUM> and an additional fourth (galvanic or non-galvanic) coupling between the first superconducting qubit <NUM> and the second superconducting island <NUM>. Again, the third and fourth couplings may be capacitive or inductive, if required and depending on particular applications. By using these additional couplings, it is possible to increase the applicability and flexibility of the circuit <NUM>. Additionally, one or both of the superconducting islands <NUM> and <NUM> may also have a coupling to the ground via additional capacitors.

Furthermore, the tunable coupling element <NUM> is configured such that its idling frequency is below qubit frequencies of the first and second superconducting qubits <NUM> and <NUM>. In this case, the tunable coupling element <NUM> may have a sweet spot located at (or close to) the operation point of a two-qubit gate (which is implemented based on the circuit <NUM>). This may reduce gate errors arising from the decoherence of the coupling element <NUM>.

In one embodiment, the tunable coupling element <NUM> may be implemented as a transmon qubit. By using the transmon qubit as a "mediator" between the first and second superconducting qubits <NUM> and <NUM>, it is possible to reduce sensitivity to a charge noise.

The circuit <NUM> further comprises a first readout resonator and a second readout resonator (which are not shown in the figures). The first readout resonator is configured to readout the first resonator and has a first operating frequency. The second readout resonator is configured to readout the second resonator and has a second operating frequency. Each of the first operating frequency and the second operating frequency is set to a value higher than the value of the operating frequency of the tunable coupling element <NUM>. In this case, during the operation of the circuit <NUM>, there is no resonance between the tunable coupling element <NUM> and the readout resonators, which would otherwise adversely affect the operation of the circuit <NUM>.

<FIG> shows a block diagram of a tunable resonator-resonator coupling circuit <NUM> in accordance with a second exemplary embodiment of the invention. The circuit <NUM> comprises a first linear resonator <NUM>, a second linear resonator <NUM>, and a tunable coupling element <NUM>. The tunable coupling element <NUM> comprises a first superconducting island <NUM> and a second superconducting island <NUM>, which are implemented and arranged in the same manner as the first superconducting island <NUM> and the second superconducting island <NUM>, respectively, in the circuit <NUM>. At the same time, the circuit <NUM> differs from the circuit <NUM> by the presence of the linear resonators (but not the nonlinear resonators). In particular, the first and second linear resonators <NUM> and <NUM> are schematically shown as harmonic oscillators in <FIG>. In some other embodiments, at least one of the first and second linear resonators <NUM> and <NUM> may be implemented as a coplanar waveguide resonator or a lumped element resonator. The circuit <NUM> may have similar advantages as the circuit <NUM>, and the embodiments discussed above in respect of the circuit <NUM> may be equally related to the circuit <NUM>.

<FIG> shows a block diagram of a tunable resonator-resonator coupling circuit <NUM> in accordance with a third exemplary embodiment of the invention. The circuit <NUM> comprises a first resonator <NUM>, a second resonator <NUM>, and a tunable coupling element <NUM>. The tunable coupling element <NUM> comprises a first superconducting island <NUM> and a second superconducting island <NUM>, which are implemented and arranged in the same manner as the superconducting islands in the circuit <NUM> or the circuit <NUM>. At the same time, the circuit <NUM> differs from the circuit <NUM> and the circuit <NUM> in that the first resonator <NUM> and the second resonator <NUM> are of different types. More specifically, the first resonator <NUM> is nonlinear and implemented as a superconducting qubit, while the second resonator <NUM> is linear and implemented as a harmonic oscillator. It should be apparent that, if required, the first resonator <NUM> may be of linear type, while the second resonator <NUM> may be of nonlinear type. In the circuit <NUM>, the second resonator <NUM> (i.e. the harmonic oscillator) may be used as a quantum bus for the first resonator <NUM> (i.e. the superconducting qubit), which may be very valuable in some applications.

<FIG> shows a block diagram of a quantum computing apparatus <NUM> in accordance with one exemplary embodiment of the invention. The apparatus <NUM> comprises a tunable resonator-resonator coupling circuit <NUM> and a control unit <NUM>. The circuit <NUM> may be implemented as one of the circuits <NUM>, <NUM> and <NUM>. The control unit <NUM> is configured to perform quantum computing operations by using the circuit <NUM>. The apparatus <NUM> may further comprise a memory <NUM> storing executable instructions <NUM> which, when executed by the control unit <NUM>, may cause the control unit <NUM> to perform the quantum computing operations. The control unit <NUM> may also store the result(s) of the quantum computing operations to the memory <NUM>. It should be noted that the number, arrangement and interconnection of the constructive elements constituting the apparatus <NUM>, which are shown in <FIG>, 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 apparatus <NUM>. For example, the apparatus <NUM> may comprise two or more circuits <NUM> each implemented as the circuit <NUM>, <NUM> or <NUM>, or may comprise any combination of the circuits <NUM>, <NUM> and <NUM>, depending on the quantum computing operations to be performed.

The control unit <NUM> 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 <NUM> may be implemented as any combination of one or more of the aforesaid. As an example, the control unit <NUM> may be a combination of two or more microprocessors.

The memory <NUM> 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 <NUM> stored in the memory <NUM> may be configured as a computer executable code which causes the control unit <NUM> to perform the quantum computing operations by using the circuit <NUM>. 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 <NUM>) on the fly.

Claim 1:
A tunable resonator-resonator coupling circuit (<NUM>, <NUM>, <NUM>) comprising:
- a first resonator (<NUM>, <NUM>, <NUM>), the first resonator (<NUM>, <NUM>, <NUM>) being linear or nonlinear and having a first frequency;
- a second resonator (<NUM>, <NUM>, <NUM>), the second resonator (<NUM>, <NUM>, <NUM>) being linear or nonlinear and having a second frequency, the first resonator (<NUM>, <NUM>, <NUM>) and the second resonator (<NUM>, <NUM>, <NUM>) having a direct coupling therebetween;
- a tunable coupling element (<NUM>, <NUM>, <NUM>) arranged to provide an indirect coupling between the first resonator (<NUM>, <NUM>, <NUM>) and the second resonator (<NUM>, <NUM>, <NUM>), the tunable coupling element (<NUM>, <NUM>, <NUM>) having an idling frequency that is below each of the first frequency and the second frequency;
- a first readout resonator configured to readout the first resonator (<NUM>, <NUM>, <NUM>), the first readout resonator having a first operating frequency; and
- a second readout resonator configured to readout the second resonator (<NUM>, <NUM>, <NUM>), the second readout resonator having a second operating frequency;
wherein the tunable coupling element (<NUM>, <NUM>, <NUM>) comprises a first superconducting island (<NUM>, <NUM>, <NUM>) and a second superconducting island (<NUM>, <NUM>, <NUM>) which are both ungrounded;
wherein the indirect coupling comprises: (i) a first coupling between the first resonator (<NUM>, <NUM>, <NUM>) and the first superconducting island (<NUM>, <NUM>, <NUM>), (ii) a Josephson coupling between the first superconducting island (<NUM>, <NUM>, <NUM>) and the second superconducting island (<NUM>, <NUM>, <NUM>), and (iii) a second coupling between the second resonator (<NUM>, <NUM>, <NUM>) and the second superconducting island (<NUM>, <NUM>, <NUM>); and wherein each of the first operating frequency and the second operating frequency is above each of the idling frequency of the tunable coupling element (<NUM>, <NUM>, <NUM>), the first frequency of the first resonator (<NUM>, <NUM>, <NUM>) and the second frequency of the second resonator (<NUM>, <NUM>, <NUM>).