Alternating coupling elements for crosstalk mitigation

Abstract

There is provided mitigating cross-talk between coupling elements of a quantum circuit. A quantum circuit comprises a configuration of coupling elements arranged between computational elements, wherein the configuration of coupling elements comprises coupling elements of at least two types and the at least two types of the coupling elements are alternated between adjacent coupling elements.

Description

[0001]ALTERNATING COUPLING ELEMENTS FOR CROSSTALK MITIGATION FIELD OF THE INVENTION The invention is generally related to the field of quantum computing. In particular, the invention is related to quantum circuit comprising more than one type of coupling elements. 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"). 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 coupling element which allows states of the superconducting qubits or resonators to interact with each other in a controlled manner. In other words, the coupling element arranged between the superconducting qubits or resonators allows one to implement a quantum gate. Coupling elements are used in superconducting circuitry. Cross-talk between the coupling elements in a superconducting circuit can occur when an adjustment of a coupling element affects a coupling strength of another coupling element. This can cause unwanted interference between different parts of the circuit and potentially degrade its performance. For example, the adjustment, the operation, the idling of one coupling element or the nearest neighboring qubits can affect the circuit properties of the non- nearest neighbour circuitry. For example the characteristic frequencies of the computational elements (i.e. qubits, coupling elements, resonators) and the coupling strengths between those elements are subjected to variations. Examples of some ways in which cross-talk can occur between coupling elements: - Electromagnetic cross-talk: Coupling elements in a superconducting circuitry are often located in close proximity to each other, and the magnetic field from one coupling element can couple to another coupling element, qubit, or resonator. This can cause changes in the magnetic flux and current distribution in the affected component, leading to changes in its coupling strength. - Electrostatic cross-talk: Coupling elements may also couple through capacitive coupling, which occurs when the electric field from one coupling element affects the electric potential of other components. This can occur when the two coupling elements share a common ground or are located close to each other. - Thermal cross-talk: Changes in the temperature of the circuit can affect the characteristics of the coupling elements, including their resonance frequency, quality factor, and coupling strength. This can result in cross-talk between different coupling elements in the circuit. To mitigate cross-talk between coupling elements, several techniques can be used, including physical isolation of the coupling elements, adjusting the layout of the circuit to minimize the proximity of coupling elements, pulse engineering, using shielding to reduce electromagnetic interference, and optimizing the tuning range of the coupling elements to reduce the likelihood of interference. Additionally, simulation and modeling tools can be used to predict, analyze and compensate the potential cross-talk between different coupling elements in a superconducting circuitry and to optimize the circuit design and the pulse driving schemes accordingly. Cross-talk become harder to control when the number of components scales to larger chips. In an example two superconducting qubits can be coupled by a tunable coupling element. The indirect interaction is mediated by the tunable coupling element itself. If the tunable coupling element is implemented as a single-island transmon, the coupling element characteristic frequencies are higher than qubit frequencies. If the tunable coupling element is implemented as a floating transmon, the coupling element characteristic frequencies can be both lower and higher than the qubit frequencies, giving the freedom to choose the circuit configuration. 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 alleviate cross- talk between coupling elements 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 quantum circuit comprising a configuration of coupling elements arranged between computational elements, wherein the configuration of coupling elements comprises coupling elements of at least two types and the at least two types of the coupling elements are alternated between adjacent coupling elements. According to a second aspect there is provided a quantum computing apparatus comprising at least one quantum circuit according to an aspect and a control unit configured to perform quantum computing operations by using coupling elements of the at least one quantum circuit. At least some aspects provide mitigating cross-talk between coupling elements of a quantum circuit. BRIEF DESCRIPTION OF THE DRAWINGS The invention is explained below with reference to the accompanying drawings in which: FIG.1 shows a block diagram of a tunable qubit-qubit coupling circuit in accordance with at least some embodiments; FIG. 2 shows a block diagram of another tunable qubit-qubit coupling circuit in accordance with at least some embodiments; FIG. 3 shows a block diagram of yet another tunable qubit-qubit coupling circuit in accordance with at least some embodiments; FIG. 4 shows a block diagram of a tunable resonator-resonator coupling circuit in accordance with at least some embodiments; FIG. 5 shows a block diagram of a tunable resonator-resonator coupling circuit in accordance with at least some embodiments; FIG. 6 shows a block diagram of a tunable resonator-resonator coupling circuit in accordance with at least some embodiments; and FIG.7 shows a block diagram of a quantum computing apparatus in accordance with at least some embodiments. Fig.8 shows an example of a configuration of coupling elements for a quantum circuit in accordance with at least some embodiments. Fig.9 shows an example of a configuration of coupling elements for a quantum circuit in accordance with at least some embodiments. Fig.10 shows an example of a configuration of coupling elements for a quantum circuit in accordance with at least some embodiments. Fig. 11 shows an example of a multi-layer quantum circuit 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. In a superconducting circuit, neighboring coupling elements refer to two or more coupling elements located in close proximity to each other, typically within a few millimeters. These coupling elements are designed to control the strength of the coupling between different elements of the circuit, such as qubits or resonators, and are typically based on variable inductors or capacitors. Neighboring coupling elements can interact with each other in several ways, which can result in unwanted crosstalk between different parts of the circuit. 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 coupling elements. The coupling elements may be at least one of the following: tunable coupling elements; or fixed coupling elements; or bus resonators; tunable resonator- resonator coupling circuits; or tunable qubit- qubit coupling circuits. 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. The coupling elements may be used to implement quantum gates and quantum buses to allow the states of the qubits to interact with each other in a controlled manner. 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 mitigating cross-talk between coupling elements of a superconducting circuit. At least some embodiments disclosed herein provide both direct and indirect couplings between linear or nonlinear resonators in a quantum circuit. The indirect coupling is provided by using a coupling element, e.g. 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. 1 shows a block diagram of a tunable qubit-qubit coupling circuit 100 in accordance with at least some embodiments. The circuit 100 comprises a first superconducting qubit 102, a second superconducting qubit 104, and a tunable coupling element 106. The first superconducting qubit 102 and the second superconducting qubit 104 are grounded and implemented similar to each other, i.e. each of them comprises a parallel connection of a capacitor (C1 or C2, respectively) and two Josephson junctions (schematically shown as crosses in FIG.1). The first superconducting qubit 102 and the second superconducting qubit 104 are directly coupled to each other via a capacitor C12. Additionally, the first superconducting qubit 102 and the second superconducting qubit 104 are indirectly coupled to each other via the tunable coupling element 106. The tunable coupling element 106 comprises a grounded superconducting portion 108 and a superconducting island 110. 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 108 is provided on one side of a capacitor Cc and the two Josephson junctions, while the (lower) superconducting island 110 is provided on another side of the capacitor Ccand the two Josephson junctions. Thus, there is a Josephson coupling between the grounded superconducting portion 108 and the superconducting island 110. Furthermore, the superconducting island 110 is ungrounded or floating, and is coupled to the first superconducting qubit 102 via a capacitor C1c and to the second superconducting qubit 104 via a capacitor C2c. The Josephson junctions used in the circuit 100 provide the anharmonicity of the first and second superconducting qubits 102, 104 and the tunable coupling element 106. However, if the first and second superconducting qubits 102, 104 and the tunable coupling element 106 all have a negative anharmonicity (which is usually observed in case of such a transmon regime), this may lead to the following problems: - the idling frequency of the tunable coupling element 106 is above the qubit frequencies of the superconducting qubits 102, 104; - the tunable coupling element 106 may become resonant with readout resonators (which may be additionally connected to the circuit 100) during the operation of the circuit 100 if the readout resonators have frequencies that are above the qubit frequencies; - during the operation of the circuit 100, the decoherence of the tunable coupling element 106 may also lead to a situation that the operating frequency of the tunable coupling element 106 is far away from its flux sweet spot (It should be noted that the sweet spot is a frequency of the tunable coupling element 106, which is first-order insensitive to a tuning parameter. The tuning parameter may be represented by a magnetic field flux through a Superconducting Quantum Interference Device (SQUID) loop for transmon qubits, and by a charge bias for Cooper-Pair Box (CPB) qubits. For example, in case of using transmons with a symmetric SQUID, the sweet spot is at a maximum qubit frequency); and - coupling with higher excited states of the tunable coupling element 106 may reduce the operation speed of the circuit 100. FIG. 2 shows a block diagram of a tunable qubit-qubit coupling circuit 200 in accordance with at least some embodiments. Similar to the circuit 100, the circuit 200 comprises a first superconducting qubit 202, a second superconducting qubit 204, and a tunable coupling element 206. The first superconducting qubit 202 is implemented similar to the first superconducting qubit 102 and the second superconducting qubit 104. In particular, the first superconducting qubit 202 is grounded and comprises a parallel connection of a capacitor C1and two Josephson junctions. At the same time, unlike the circuit 100, the second superconducting qubit 204 is ungrounded and therefore connected in the circuit 200 differently as compared to the connection of the first superconducting qubit 202. The tunable coupling element 206 comprises a grounded superconducting portion 208 and a (ungrounded) superconducting island 210 which are provided in the circuit 200 in the same manner as the grounded superconducting portion 108 and the superconducting island 110, respectively, in the circuit 100. The first superconducting qubit 202 and the second superconducting qubit 204 are also directly coupled to each other via a capacitor C12 and indirectly coupled to each other via the tunable coupling element 206. The indirect coupling is provided via capacitors C1cand C2c. However, the circuit 200 lacks some benefits which may be provided if one replaces the grounded superconducting portion 208 with an ungrounded superconducting island. FIG. 3 shows a block diagram of a tunable qubit-qubit coupling circuit 300 in accordance with at least some embodiments. Similar to the circuit 100 and the circuit 200, the circuit 300 comprises a first superconducting qubit 302, a second superconducting qubit 304, and a tunable coupling element 306. The first superconducting qubit 302 and the second superconducting qubit 304 are implemented similar to the first superconducting qubit 102 and the second superconducting qubit 104, respectively, in the circuit 100. However, the tunable coupling element 306 is connected between the first superconducting qubit 302 and the second superconducting qubit 304 differently as compared to the tunable coupling elements 106, 206. More specifically, the tunable coupling element 306 is provided in the circuit 300 such that there is no direct coupling between the first superconducting qubit 302 and the second superconducting qubit 304. The first superconducting qubit 302 and the second superconducting qubit 304 are only indirectly coupled to each other via a first superconducting island 308 and a second superconducting island 310 which are both included in the tunable coupling element 306. The first superconducting island 308 and the second superconducting island 310 are both ungrounded, with each of them being coupled to the first superconducting qubit 302 and the second superconducting qubit 304 via capacitors C1cu, C2cu, C1cband C2cb. At the same time, there is a Josephson coupling between the first superconducting island 308 and the second superconducting island 310. However, the circuit 300 lacks the effective direct qubit-qubit coupling. For example, due to the lack of the capacitor C12 (like in the circuit 100 or 200), the two-qubit gates based on the circuit 300 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 C1cuand C1cbto provide the fast two- qubit gates. FIG.4 shows a block diagram of a tunable resonator-resonator coupling circuit 400 in accordance with at least some embodiments. The circuit 400 comprises a first nonlinear resonator 402, a second nonlinear resonator 404, and a tunable coupling element 406. In this exemplary embodiment, each of the first and second nonlinear resonators 402 and 404 is implemented as a superconducting qubit (similar to those shown in FIG.1). For this reason, the circuit 400 may be used to implement a two-qubit gate. However, the circuit 400 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.5), or the circuit 400 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. 4, the first superconducting qubit 402 and the second superconducting qubit 404 are directly coupled to each other via a capacitor C12 and indirectly coupled to each other via the tunable coupling element 406. The tunable coupling element 406 comprises a first superconducting island 408 and a second superconducting island 410 which are both ungrounded. Moreover, the first superconducting island 408 and the second superconducting island 410 are arranged in the circuit 400 such that each of them is directly coupled only to one of the first superconducting qubit 402 and the second superconducting qubit 404. More specifically, there are a first non- galvanic coupling between the first superconducting qubit 402 and the first superconducting island 408 and a second non-galvanic coupling between the second superconducting qubit 404 and the second superconducting island 410. The first and second non-galvanic couplings are both capacitive, i.e. implemented via capacitors C1cand C2c, 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 402 and 404 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 408 and the second superconducting island 410 are coupled to each other via Josephson junctions (see the crosses in FIG. 4), i.e. have a Josephson coupling therebetween. Thus, the indirect coupling between the first superconducting qubit 402 and the second superconducting qubit 404 comprises the above-mentioned first non- galvanic coupling, Josephson coupling, and second non- galvanic coupling. As can be seen in FIG. 4, the arrangement area of the first superconducting island 408 is confined by one plate of the capacitor C1c, one plate of a capacitor Cc, and the two Josephson junction. As for the second superconducting island 410, its arrangement area is confined by one plate of the capacitor C2c, another plate of the capacitor Cc, and the two Josephson junctions. Such arrangements of the first superconducting island 408 and the second superconducting island 410 are given by way of example only. In some other embodiments, the first superconducting island 408 and the second superconducting island 410 may be arranged such that the indirect coupling between the first and second superconducting qubits 402 and 404 comprises an additional third (galvanic or non-galvanic) coupling between the second superconducting qubit 404 and the first superconducting island 408 and an additional fourth (galvanic or non-galvanic) coupling between the first superconducting qubit 402 and the second superconducting island 410. 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 400. Additionally, one or both of the superconducting islands 408 and 410 may also have a coupling to the ground via additional capacitors. Furthermore, the tunable coupling element 406 may be configured such that its idling frequency is below qubit frequencies of the first and second superconducting qubits 402 and 404. In this case, the tunable coupling element 406 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 400). This may reduce gate errors arising from the decoherence of the coupling element 406. In one embodiment, the tunable coupling element 406 may be implemented as a transmon qubit. By using the transmon qubit as a "mediator" between the first and second superconducting qubits 402 and 404, it is possible to reduce sensitivity to a charge noise. In one embodiment, the circuit 400 may further comprise 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 may be set a value higher than the value of the operating frequency of the tunable coupling element 406. In this case, during the operation of the circuit 400, there is no resonance between the tunable coupling element 406 and the readout resonators, which would otherwise adversely affect the operation of the circuit 400. FIG.5 shows a block diagram of a tunable resonator-resonator coupling circuit 500 in accordance with at least some embodiments. The circuit 500 comprises a first linear resonator 502, a second linear resonator 504, and a tunable coupling element 506. The tunable coupling element 506 comprises a first superconducting island 508 and a second superconducting island 510, which are implemented and arranged in the same manner as the first superconducting island 408 and the second superconducting island 410, respectively, in the circuit 400. At the same time, the circuit 500 differs from the circuit 400 by the presence of the linear resonators (but not the nonlinear resonators). In particular, the first and second linear resonators 502 and 504 are schematically shown as harmonic oscillators in FIG.5. In some other embodiments, at least one of the first and second linear resonators 502 and 504 may be implemented as a coplanar waveguide resonator or a lumped element resonator. The circuit 500 may have similar advantages as the circuit 400, and the embodiments discussed above in respect of the circuit 400 may be equally related to the circuit 500. FIG.6 shows a block diagram of a tunable resonator-resonator coupling circuit 600 in accordance with at least some embodiments. The circuit 600 comprises a first resonator 602, a second resonator 604, and a tunable coupling element 606. The tunable coupling element 606 comprises a first superconducting island 608 and a second superconducting island 610, which are implemented and arranged in the same manner as the superconducting islands in the circuit 400 or the circuit 500. At the same time, the circuit 600 differs from the circuit 400 and the circuit 500 in that the first resonator 602 and the second resonator 604 are of different types. More specifically, the first resonator 602 is nonlinear and implemented as a superconducting qubit, while the second resonator 604 is linear and implemented as a harmonic oscillator. It should be apparent that, if required, the first resonator 602 may be of linear type, while the second resonator 604 may be of nonlinear type. In the circuit 600, the second resonator 604 (i.e. the harmonic oscillator) may be used as a quantum bus for the first resonator 602 (i.e. the superconducting qubit), which may be very valuable in some applications. FIG. 7 shows a block diagram of a quantum computing apparatus 700 in accordance with at least some embodiments. The apparatus 700 comprises at least one quantum circuit 702 and a control unit 704. The circuit 702 may be implemented as one of the circuits 800, 900, 1000 and 1100 in Figs.8 to 11. The control unit 704 is configured to perform quantum computing operations by using coupling elements of the at least one circuit 702. The apparatus 700 may further comprise a memory 706 storing executable instructions 708 which, when executed by the control unit 704, may cause the control unit 704 to perform the quantum computing operations. The control unit 704 may also store the result(s) of the quantum computing operations to the memory 706. It should be noted that the number, arrangement and interconnection of the constructive elements constituting the apparatus 700, which are shown in FIG. 7, 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 700. For example, the apparatus 700 may comprise two or more circuits 702 each implemented as the circuit 800, 900, 1000 or 1100, or may comprise any combination of the circuits 800, 900, 1000 and 1100, depending on the quantum computing operations to be performed. The control unit 704 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 704 may be implemented as any combination of one or more of the aforesaid. As an example, the control unit 704 may be a combination of two or more microprocessors . The memory 706 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 708 stored in the memory 706 may be configured as a computer executable code which causes the control unit 704 to perform the quantum computing operations by using the circuit 702. 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 706) on the fly. Figs. 8, 9 and 10 shows examples of configurations of coupling elements for quantum circuits in accordance with at least some embodiments. The quantum circuits 800, 900, 1000 comprise superconducting computational elements, or computational elements, 806, 906, 1006 and coupling elements 802, 804, 902, 904, 1002, 1004 arranged between the computational elements. The configurations are illustrated by two- dimensional configurations with respect to X- and Y-axis that have been arranged to extend along dimensions, e.g. in width and length dimensions, of planar surfaces of the substrates. Examples of the coupling elements are described with Figs.1 to 6. Legend 808, 908, 1008 of components of each of the quantum circuits is illustrated next to the quantum circuits 800, 900, 1000. The configurations comprise coupling elements arranged between computational elements. The two-dimensional configurations may comprise at least one of: rows; or columns; or rays; formed by the coupling elements. In this way the configurations may assume geometrical shapes comprising at least one of: a lattice; or a fan; or a star. Each of the configurations may comprise coupling elements of at least two types, e.g. coupling elements of Type 1804, 902, 1002 and coupling elements of Type 2 804, 904, 1004. Types of the coupling elements are alternated between adjacent coupling elements, whereby cross-talk between coupling elements of the same type may be mitigated. It should be noted that adjacent coupling elements may be regarded at least those coupling elements that are at planar positions, e.g. at X-Y - positions, that are next to each other without any intermediary coupling elements between them. A series of the planar positions, thus the coupling elements, may form a geometrical shape. The geometrical shape may formed by sub-shapes such as at least one of: rows; or columns; or rays. In an example, each coupling element 802, 804, 902, 904, 1002, 1004 is arranged between a pair of computational elements 806, 906, 1006 for electromagnetic coupling of the computational elements, which allows tunable interaction between the computational elements. The interaction between the computational elements facilitates the computational elements to perform computational operations based on quantum operations. In addition to the coupling of the computational elements for facilitating their interaction each coupling element may induce unwanted coupling, or cross-talk, to one or more other coupling elements, or neighboring coupling elements, of the quantum circuit provided the other coupling elements are within a range of electromagnetic coupling. The neighboring coupling elements for a given coupling element comprise at least coupling elements that are adjacent to the given coupling element. Alternating types of the coupling elements within a configuration of coupling elements provides that distances between coupling elements of the same type are increased, whereby cross-talk may be mitigated. In an example in accordance with at least some embodiments, the quantum circuit 900, 1000 comprises pairs 926, 1026 of computational elements 906, 1006 coupled by a coupling element of a first type 904, 1002 of at least two types of coupling elements, and the pairs of the computational elements are coupled with each other by coupling elements of a second type 902, 1004 of the at least two types of coupling elements. In this way the same type of coupling elements are linked by the computational elements and the different type of coupling element, whereby cross-talk between the coupling elements of the same type is mitigated. In an example in accordance with at least some embodiments, the types of the coupling elements 802, 804, 902, 904, 1002, 1004 are alternated between at least one of: adjacent rows of coupling elements; or adjacent columns of coupling elements; or adjacent row positions of rows of coupling elements; or coupling elements at adjacent column positions of columns of coupling elements; or adjacent rays of coupling elements. Examples of the configurations of coupling elements 902, 904, 1002, 1004 for the quantum circuits 800, 900, 1000 comprise a lattice-shaped configuration comprising rows and columns of coupling elements; or a fan-shaped configuration of coupling elements; or a circular configuration of coupling elements. The fan-shaped configuration of coupling elements comprises rays of coupling elements from a central computational element over a limited central angle. The circular configuration of coupling elements comprises rays of coupling elements over an omnidirectional central angle around a central computational element. In an example in accordance with at least some embodiments, the at least two type of coupling elements of the quantum circuit 800, 900, 1000 comprise at least a coupling element of a first type that has a first set of characteristic frequencies, and a tunable coupling element of a second type has a first set of characteristic frequencies and first set of frequencies and the second set of frequencies are at least partially different. A set of frequencies, e.g. the first / second set of frequencies, may comprise an operating frequency, an idling frequency and a design frequency. At the operating frequency, the coupling element is configured to perform coupling, thus to transfer energy between its ports. At the idling frequency, coupling, thus transfer of energy between ports of the coupling element is not enabled. At the design frequency, coupling of the coupling element is optimized. In an example in accordance with at least some embodiments, the computational elements are qubits or resonators coupled to qubits, and the coupling elements 902, 904, 1002, 1004 comprise at least two of the following types: grounded qubits with resonance frequency above the qubit frequencies; or symmetric floating qubits with resonance frequency below the qubit frequencies; or asymmetric floating qubits with resonance frequency above the qubit frequencies. The different types of coupling elements enable mitigating cross-talk due to differences between their resonance frequencies with respect to qubit frequencies, whereby operating and idling frequencies of different types of coupling element may be separated. In an example of grounded qubits with resonance frequency above the qubit frequencies, a coupling element comprises a tunable qubit, e.g. a transmon, where the qubit superconducting island is connected to ground through a tunable non-linear element. The tunable non-linear element may be a SQUID. In an example of floating qubits, a coupling element comprises a tunable qubit, e.g. a transmon, where two superconducting islands shunt a tunable non- linear element, e.g. SQUID. Referring to Fig. 8 the quantum circuit 800 comprises a circular, or a star- shaped, configuration of coupling elements. The tunable 802, 804 coupling elements are arranged in rays between a central computational element and computational elements at edges of the configuration, or edge computational elements. The star-shaped configuration is illustrated by eight edge computational elements 806 arranged around the central computational element. However, it should be noted that the star-shaped configuration may comprise a smaller or larger number of edge computational elements. Types of the coupling elements are arranged to alternate between adjacent rays of coupling elements around the central computational element, whereby each coupling element may be of a different type than its adjacent coupling element on either side of the coupling element. The coupling elements may be arranged omnidirectionally around the central computational element. The coupling elements may be spaced evenly around the central computational element for further mitigating cross-talk between the coupling elements. Referring to Figs 9 and 10 the quantum circuits comprise lattice-shaped configurations of coupling elements arranged between computational elements 906, 1006. The lattice-shaped configuration comprises rows 914, 1014 and columns 916, 1016 of coupling elements. The computational elements are arranged in arrays. The arrays of computational elements comprise computational elements in rows, from 1 to 8, and in columns, from A to E. In an example, each of the rows 914, 1014 of coupling elements 902, 904, 1002, 1002 are arranged between two adjacent rows of computational elements for coupling computational elements of the adjacent rows, e.g. rows 1, 2 and 3. The rows of computational elements and the rows of coupling elements may extend in directions that are substantially parallel with each other. Adjacent rows of computational elements may be offset 922, 1022 with respect to one another in their longitudinal directions, whereby distances between neighboring computational elements at the adjacent rows may be increased. Similarly, columns 916, 1016 of coupling elements may be arranged between adjacent columns of computational elements, e.g. cols. A, B and C, for coupling computational elements of the adjacent columns. The columns of computational elements and the columns of coupling elements may extend in directions that are substantially parallel with each other. In example in accordance with at least some embodiments, the quantum circuit 900 comprises a lattice-shaped configuration, where types of coupling elements 902, 904 are alternated between rows 914 of the configuration. Since the types of coupling elements are alternated between rows of the configuration, adjacent coupling elements, for a given coupling element are of different types at adjacent rows, e.g. rows 1 and 2, whereby cross-talk between the coupling elements may be mitigated. Accordingly, each row has coupling elements of a single type, adjacent rows have coupling elements of different types and types of the coupling elements are alternated within each column. For example, if the first row of the lattice-shaped configuration has coupling elements of Type 1902, then the second row that is adjacent row to the first row has coupling elements of Type 2904 and the third row that is adjacent row to the second row has coupling elements of Type 1902. In this way the types of coupling elements are alternated between rows 914 of the configuration. Consequently, types of the coupling elements are alternated within each column 916 as follows, starting from the first row 914: Type 1; Type 2; Type 1; etc. It should be noted that types of coupling elements may be alternated between columns of the configuration in a similar manner as described with rows above. Referring to Fig. 10, the quantum circuit 1000 comprises a lattice-shaped configuration of coupling elements 1002, 1004, where types of the coupling elements are alternated within rows 1014 of the configuration and orders at which types of the coupling elements are alternated within each row are alternated between rows of the configuration. In this way adjacent coupling elements, for a given coupling element, at adjacent column positions and at adjacent row positions, thus in longitudinal directions of the columns and rows, are of different types, whereby cross-talk between coupling elements in the direction of the rows and in the direction of the columns may be mitigated. For example, each row may have at least two types of coupling elements that are alternated according to a first order and adjacent rows have different orders, e.g. a second order, at which the coupling elements are alternated within the rows. In this way types of the coupling elements are alternated within each column and row, which supports mitigating cross-talk between coupling elements of the same type. In an example, the first order for alternating the coupling elements may comprise that a first coupling element of a row is of Type 1 1002. Then, types of the coupling elements of the row may be arranged to alternate according to the first order as follows: Type 1; Type 2; Type 1; etc. Then, the second order for alternating the coupling elements may comprise that a first coupling element of a row is of Type 21004. Then, types of the coupling elements the row may be arranged to alternate according to the second order as follows: Type 2; Type 1; Type 2; etc. It should be noted that the orders at which types of the coupling elements are alternated within each column may be alternated between columns in a similar manner as described with rows above. It should be appreciated that even if Figs.8, 9 and 10 have been described with reference to quantum circuits 800, 900, 1000 that have two types of coupling elements, the quantum circuits may comprise more than two types of coupling elements, whereby cross-talk between the same types of coupling elements may be further mitigated. Having more than two types of coupling elements provides that a number of different types of coupling elements may be increased between same type of coupling elements of the quantum circuit. For example, whereas alternating based on two types of coupling elements may be based on the same types of coupling elements in a row, column and / or ray being separated by one different type of coupling element at a row position, column position and / or ray that is adjacent to the coupling elements, increasing the number of types of coupling elements used for the quantum circuit by a number N that is an integer value, increases also the number of different types of coupling elements between the same type of coupling elements by N. Similarly, having more than two types of coupling elements provides that a number of orders at which types of the coupling elements are alternated within each row or column may be increased. For example, increasing the number of types of coupling elements used for the quantum circuit by a number N that is an integer value, increases also the number of orders at which types of the coupling elements are alternated within each row or column. Fig.11 shows an example of a multi-layer quantum circuit in accordance with at least some embodiments. The multi-layer quantum circuit 1100 comprises quantum circuits on a plurality of layers 1102, 1104 that are stacked on top of each other whereby coupling elements at one layer 1104 may have adjacent coupling elements on an adjacent layer 1102, e.g. a layer that is stacked on top of the layer. The multi-layer quantum circuit is illustrated with respect to X-, Y-, and Z-axis, where the X- and Y-axis extend along planar surfaces of the layers of the multi-layer quantum circuit and the Z- axis extends through the layers of the multi-layer quantum circuit in a direction that is substantially transverse to the planar surfaces of the layers. Coupling elements 1112, 1114 of each of the quantum circuits may be alternated according to types of the coupling elements within each of the layers in accordance to described with Figs. 8 to 10. Additionally, the types of the coupling elements may be alternated between adjacent layers, e.g. layers that are arranged on top of each other. Alternating types of the coupling elements between adjacent coupling elements at adjacent layers of the multi-layer quantum circuit facilitates mitigating cross-talk between the quantum circuits at the adjacent layers. It should be note that, although the multi-layer quantum circuit is illustrated to comprise two types of coupling elements, the multi-layer quantum circuit may comprise further types of coupling elements on one or more of the layers. In an example in accordance with at least some embodiments, the types of the coupling elements of the multi-layer quantum circuit are alternated between at least one of: adjacent rows of coupling elements; or adjacent columns of coupling elements; or adjacent row positions of rows of coupling elements; or coupling elements at adjacent column positions of columns of coupling elements; or adjacent rays of coupling elements. It should be noted that the types may be alternated between coupling elements at adjacent layers of the multi-layer quantum circuit and / or between coupling elements within single layer of the multi-layer quantum circuit. In an example, the multi-layer quantum circuit may be a flip-chip, where quantum circuits, or chips, are bonded with each other by solder bumps 1106 or metallic balls for electrically interconnecting the chips. The bonding may be performed by the solder bumps at active sides of the chips, whereby one of the chips is flipped with respect the other chip. In an example, the types of the coupling elements 1112, 1114 may be alternated between the layers 1102, 1104 of the multi-layer quantum circuit by rotating configurations of coupling elements of the layers with respect to each other. In an example, the configurations of coupling elements may be rotated about the Z-axis, whereby same type of coupling elements at the layers may be misaligned. In an example, each of the interconnected layers may comprise a configuration of coupling elements in accordance to described with Fig.8. Then, if the configurations at the layers would not be rotated with respect to each other, coupling elements of the same type could be on top of each other, thus aligned, and cross-talk between the tunable could be caused. However, the configurations may be rotated with respect to each other for misaligning the coupling elements of the same type at the adjacent layers with each other, positions of the coupling elements are offset in a planar direction, e.g. in a direction of the X and / or Y-axis, whereby the coupling elements of the same type are not on top of each other. Accordingly, at a misaligned position, at a coupling element of one of the interconnected layers is aligned with another type of coupling elements on the other layer. Accordingly, alternating types of the coupling elements between the layers of the multi-layer quantum circuit facilitates mitigating cross-talk between the coupling elements of the interconnected layers. An example of suitable rotation between two configurations described with Fig.8 on interconnected layers is 45 degrees for an even distribution of the eight coupling elements around the central computational element. It should be noted that the configurations described in Figs.9 and 10 may be rotated in a similar manner about the Z-axis to misalign same type of coupling elements between interconnected layers. An example of suitable rotation between the configurations described with Figs.9 and 10 on interconnected layers is 90 degrees. In an example, the types of the coupling elements may be alternated between the adjacent layers of the multi-layer quantum circuit by defining coupling elements for the adjacent layers at a planar position X, Y to comprise different types of coupling elements. Accordingly, for two layers that are on top of each other, the multi-layer quantum circuit comprises at a single planar position X, Y two coupling elements that are on different adjacent layers. Then at the single planar position X, Y, a type of a coupling element at a lower layer is different than a type of a coupling element at a higher layer. It should be noted that the coupling elements may be arranged in rows, columns and / or rays that may form configurations such as described with Figs.8 to 10. Therefore, a given row, column or ray at a given planar position on one layer may be adjacent to a row, column or ray at the same, or substantially, the same planar position on another layer. Types of the coupling elements at the given planar position of the different layers may be alternated, whereby they are of different type which mitigates cross-talk between the layers. 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 silicon substrate may be electromagnetically coupled to a readout resonator on the first surface of the silicon substrate. Quantum circuits comprise circuit components for performing computational operations based on quantum operations. Components of the 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 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 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 coupling elements) 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.

Claims

CLAIMS 1. A quantum circuit comprising a configuration of coupling elements arranged between computational elements, wherein the configuration of coupling elements comprises coupling elements of at least two types and the at least two types of the coupling elements are alternated between adjacent coupling elements.

2. The quantum circuit of claim 1, wherein a coupling element of a first type has a first set of characteristic frequencies, and a coupling element of a second type has a first set of characteristic frequencies and first set of frequencies and the second set of frequencies are at least partially different.

3. The quantum circuit of claim 1, comprising pairs of computational elements coupled by a coupling element of a first type of at least two types of coupling elements, and the pairs of the computational elements are coupled with each other by coupling elements of a second type of the at least two types of coupling elements.

4. The quantum circuit of claim 1 or 2, wherein the at least two types of coupling elements are alternated between rows of the configuration.

5. The quantum circuit according to any of claims 1 to 3, wherein the at least two types of the coupling elements are alternated within rows, or columns, of the configuration and orders at which the at least two types of the coupling elements are alternated within each row are alternated between rows, or columns, of the configuration.

6. The quantum circuit according to any of claims 1 to 5, comprising a plurality of layers stacked on top of each other, and the at least two types of the coupling elements are alternated between adjacent layers of the plurality of layers.

7. The quantum circuit according to any of claims 1 to 6, wherein the at least two types of the coupling elements are alternated between at least one of: adjacent rows of coupling elements; or adjacent columns of coupling elements; or adjacent row positions of rows of coupling elements; or coupling elements at adjacent column positions of columns of coupling elements; or adjacent rays of coupling elements.

8. The quantum circuit according to any of claims 1 to 7, wherein the computational elements are qubits or resonators coupled to qubits, and the at least two types of the coupling elements comprise at least two of the following types: grounded transmons with resonance frequency above the qubit frequencies; or symmetric floating transmons with resonance frequency below the qubit frequencies; or asymmetric floating transmon with resonance frequency above the qubit frequencies.

9. A quantum computing apparatus comprising at least one quantum circuit according to any of claims 1 to 8 and a control unit configured to perform quantum computing operations by using coupling elements of the at least one quantum circuit.

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