Quantum processing unit with one or more superconducting qubits based on phase-biased linear and nonlinear inductive energy elements

Abstract

The present invention generally relates to the field of quantum computing, and in particular to a quantum processing unit comprising at least one superconducting qubit based on phase-biased linear and nonlinear inductive energy elements. The superconducting phase difference between the linear inductive energy element and the nonlinear inductive energy element is biased, for example, by an external magnetic field, so that the quadratic potential energy terms of the linear and nonlinear inductive energy elements are at least partially cancelled. In a preferred embodiment, such cancellation is at least 30%. By doing so, it is possible to implement the design of a superconducting qubit with high coherence and high anharmonicity.

Description

[Technical field]

[0001] The present invention relates generally to the field of quantum computing. In particular, the present invention relates to a quantum processing unit comprising at least one superconducting qubit based on phase-biased linear and nonlinear inductive energy elements, and a quantum computer using one or more such quantum processing units. [Background technology]

[0002] Quantum computing devices, also known as quantum computers, use quantum mechanical phenomena such as superposition and entanglement to solve requested computational tasks. Unlike classical computers, which manipulate information in the form of bits (e.g., "1" or "0"), quantum computers manipulate information using quantum bits. Qubit refers to the basic unit of quantum information, as well as quantum devices used to store one or more quantum bits of information (e.g., a superposition of "0" and "1").

[0003] Quantum computers can be implemented based on superconducting circuits with superconducting qubits and resonators. For example, there are several types of superconducting qubits, including charge qubits, transmons, persistent current flux qubits, C-shunt flux qubits, phase qubits, fluxonium, and 0-π qubits. Each of these qubit types has both their advantages and disadvantages. For example, charge qubits have high anharmonicity that is optimal for high-speed single qubit operation, but at the same time, they suffer from very short coherence times due to deleterious dephasing resulting from charge noise. Due to their poor coherence properties, charge qubits, persistent current flux qubits, and phase qubits are not used in current quantum computers.

[0004] The longest measured relaxation and coherence times of superconducting qubits have been achieved with flaxonium, where the Josephson junctions are shunted by a superinductor with large inductance but small capacitance. This inductive shunt makes the flaxonium-based circuitry immune to low-frequency charge noise. The superinductor in a flaxonium qubit is typically implemented by using a Josephson junction array or a superinducting nanowire with high kinetic inductance. Fluxonium is also well protected from magnetic flux noise that couples into the flaxonium-based circuitry primarily through the superinductor.

[0005] However, flaxonium can be difficult to implement and operate. The latter hinders the application of flaxonium in, for example, fast and accurate quantum logic gates. Moreover, so-called heavy flaxonium, implemented by shunting regular flaxonium with large geometry capacitors, may require several photon Raman processes to excite the qubit from its ground state to its excited state. Again, the same drawbacks are inherent to the 0-π qubit. Summary of the Invention [Problem to be solved by the invention]

[0006] This Summary is provided to introduce in a simplified form a selection of concepts that are further described below in the Detailed Description. This Summary is not intended to identify key or essential features of the invention, nor is it intended to be used to limit the scope of the invention.

[0007] It is an object of the present invention to provide a highly coherent and highly anharmonic superconducting qubit design. [Means for solving the problem]

[0008] The above object is achieved by means of the features of the attached independent claims. Further embodiments and examples are evident from the dependent claims, the detailed description and the accompanying drawings.

[0009] According to a first aspect, a quantum processing unit is provided. The quantum processing unit comprises a dielectric substrate and at least one superconducting qubit provided on the dielectric substrate. Each of the at least one superconducting qubit comprises a linear inductive energy element and a nonlinear inductive energy element. The linear inductive energy element is superconducting. Each of the at least one superconducting qubit further comprises a phase bias element configured to bias a superconducting phase difference between the linear inductive energy element and the nonlinear inductive energy element, such that quadratic potential energy terms of the linear inductive energy element and the nonlinear inductive energy element are at least partially cancelled by each other. Such a configuration of the quantum processing unit has the following advantages: - The anharmonicity of a superconducting qubit can be increased by (at least partially) mutual cancellation of the quadratic potential energy terms of the nonlinear and linear inductive energy elements. - Often (but perhaps not always), maximum cancellation can occur in the flux-insensitive sweet spot, where the superconducting qubit is first-order insensitive to flux-noise induced dephasing.

[0010] In one embodiment of the first aspect, the phase bias element is configured to bias the superconducting phase difference such that the quadratic potential energy terms of the linear inductive energy element and the nonlinear inductive energy element are cancelled by at least 30%. Such cancellation can significantly increase the anharmonicity of the superconducting qubit.

[0011] In some embodiments of the first aspect, the linear inductive energy element comprises one or more geometric inductors and the nonlinear inductive energy element comprises one or more Josephson junctions or kinetic inductors, which may make the processing unit according to the first aspect more flexible in use.

[0012] In one embodiment of the first aspect, each of the at least one superconducting qubit further comprises a capacitive energy element. By using the capacitive energy element, it is possible to modify the energy spectrum of the qubit and its susceptibility to different noise sources.

[0013] In an embodiment of the first aspect, the capacitive energy element comprises one or more interdigitated capacitors, gap capacitors, parallel plate capacitors, or junction capacitors, which may make the processing unit according to the first aspect more flexible in use.

[0014] In one embodiment of the first aspect, the phase bias element is configured to bias the superconducting phase difference by generating and passing a magnetic field through at least one superconducting qubit or by applying a predefined voltage to the nonlinear inductive energy element, thereby more effectively biasing the superconducting phase difference.

[0015] In an embodiment of the first aspect, the phase biasing element comprises one or more coils and / or flux bias lines. By using the coils and / or flux lines, flux control can be achieved.

[0016] In one embodiment of the first aspect, the at least one superconducting qubit comprises two or more superconducting qubits capacitively and / or inductively coupled to each other on a dielectric substrate, thereby allowing for the storage and manipulation of multiple qubits, making the quantum processing unit according to the first aspect more flexible in use.

[0017] In one embodiment of the first aspect, the at least one superconducting qubit comprises two or more superconducting qubits. In this embodiment, the quantum processing unit further comprises one or more coupling resonators and / or tunable couplers for coupling the superconducting qubits on the dielectric substrate. By doing so, it is possible to store and manipulate multiple qubits, thereby making the quantum processing unit according to the first aspect more flexible in use.

[0018] In one embodiment of the first aspect, the quantum processing unit further comprises a signal line provided on the dielectric substrate. The signal line is configured to provide a control signal (e.g., from an external control unit) to the superconducting qubit. The signal line may comprise a radio frequency line, and the control signal may include a microwave pulse. The control signal may enable the superconducting qubit to be controlled in a desired manner.

[0019] In an embodiment of the first aspect, the quantum processing unit further comprises a readout line provided on the dielectric substrate. The readout line is configured to measure a state of the superconducting qubit. The readout line may be coupled to the superconducting qubit via a readout resonator. By using the readout line, it is possible to provide a state measurement of the superconducting qubit, thereby making the quantum processing unit according to the first aspect more flexible in use.

[0020] In one embodiment of the first aspect, the at least one quantum bit is configured as a distributed element resonator comprising at least two conductors separated by at least one gap. In this embodiment, at least one of the conductors serves as a linear inductive energy element, and the nonlinear inductive energy element comprises at least one Josephson element incorporated in the distributed element resonator. Furthermore, the phase bias element is configured to bias the superconducting phase difference by generating and passing a magnetic field through at least one gap of the distributed element resonator. By doing so, it is possible to increase the anharmonicity of the superconducting quantum bit.

[0021] In one embodiment of the first aspect, the distributed element resonator is configured as a coplanar waveguide (CPW) resonator. In this embodiment, the at least two conductors are represented by a central superconductor and a superconducting ground plane separated from each other by a gap in the CPW resonator. The central superconductor serves as a linear inductive energy element. Furthermore, in this embodiment, at least one Josephson junction is incorporated in the CPW resonator so that there are no isolated superconducting islands in the quantum processing unit. By using such a CPW resonator, the following advantages can be obtained: The inductance and capacitance of the CPW resonator shunt the Josephson junction, thereby providing protection against low frequency charge noise; - CPW resonator geometry exhibits low dielectric loss; Due to the above-mentioned protection against charge noise, it is possible to avoid severe dephasing of superconducting qubits and thereby achieve long coherence times; - by making the potential energies of the Josephson junction and the central superconductor of the CPW resonator approximately equal to each other, it is possible to use an external magnetic flux to (at least partially) cancel the quadratic energy term in the total potential energy of the superconducting qubit, thereby resulting in high anharmonicity of the superconducting qubit; and Such a structure of the superconducting qubit allows incorporating a relatively small number (e.g. just one, or a small number) of Josephson junctions in the superconducting qubit (compared to prior art superconducting qubits), which makes the manufacturing process of the superconducting qubit, and thus the entire quantum processing unit according to the first aspect, relatively simple and cheap.

[0022] In one embodiment of the first aspect, a central superconductor of a CPW resonator has first and second pairs of opposing sides. A superconducting ground plane is formed on a dielectric substrate such that the central superconductor is galvanically connected to the superconducting ground plane on the first pair of opposing sides and is separated by a gap from the superconducting ground plane on the second pair of opposing sides. Such a configuration of a superconducting qubit has the following advantages: - the grounded central superconductors of the resonators (on opposite sides of the first pair) ensure that there are no isolated superconducting islands in the qubit circuit, which makes the qubit immune to low frequency charge noise; - a grounded central superconductor (on the opposite side of the first pair) of the resonator can turn the superconducting qubit into a gradiometric circuit that is protected against magnetic flux noise, the spatial scale of which exceeds the characteristic width of the resonator; Due to the protection against magnetic flux noise (in addition to the above-mentioned protection against charge noise), it is possible to more effectively avoid severe dephasing of qubit devices and thereby achieve longer coherence times.

[0023] In one embodiment of the first aspect, the ground plane comprises opposing portions physically separated from each other by a central superconductor and a gap. In this embodiment, the opposing portions are connected to each other via an air bridge that extends across the gap and the central superconductor. By doing so, it is possible to suppress parasitic slotline modes of the resonator.

[0024] In one embodiment of the first aspect, a Josephson junction is incorporated into the central superconductor of the CPW resonator, which is interrupted by the Josephson junction, which acts as a nonlinear inductive energy element that increases the anharmonicity of the mode of the superconducting qubit.

[0025] In one embodiment of the first aspect, the central superconductor of the CPW resonator has a parallel connection of two Josephson junctions built into it, which makes the superconducting qubit more flexible in use, for example by doing so it is possible to implement a Superconducting Quantum Interference Device (SQUID) loop in the superconducting qubit.

[0026] In one embodiment of the first aspect, the Josephson junction is centrally located in the central superconductor of the CPW resonator. By locating the Josephson junction in the middle or center of the central superconductor, it is possible to increase the anharmonicity of the mode of the superconducting qubit by at least a factor of two.

[0027] In one embodiment of the first aspect, a superconducting qubit comprises a first Josephson junction embedded in a central superconductor of a CPW resonator and at least one second Josephson junction disposed in one or more of the gaps proximate the first Josephson junction. Each of the at least one second Josephson junction connects the central superconductor to a superconducting ground plane through a corresponding gap. This configuration of the superconducting qubit can allow for a more flexible mode structure and a more flexible energy spectrum for each mode.

[0028] In one embodiment of the first aspect, the at least one second Josephson junction comprises an even number of second Josephson junctions arranged symmetrically with respect to the first Josephson junction, which may enable achieving better operating behavior of the superconducting qubit.

[0029] In one embodiment of the first aspect, the central superconductor of the CPW resonator has a straight or curved shape, which can enable different configurations of the superconducting qubit depending on the specific application.

[0030] In one embodiment of the first aspect, the quantum processing unit further comprises at least one 3D cavity. In this embodiment, the dielectric substrate having at least one superconducting qubit comprises at least one 3D cavity therein. By placing the superconducting qubit inside the at least one 3D cavity, it can be possible to achieve longer relaxation and coherence times due to a reduced surface participation rate.

[0031] According to a second aspect, a quantum computer is provided, comprising at least one qubit device according to the first aspect and a control unit configured to perform computational operations by using at least one quantum processing unit. The use of such a quantum processing unit can improve the efficiency, functionality and processing speed of the quantum computer.

[0032] Other features and advantages of the present invention will become apparent upon reading the following detailed description and reviewing the accompanying drawings.

[0033] The invention is described below with reference to the accompanying drawings. [Brief description of the drawings]

[0034] [Figure 1] FIG. 1 is a schematic top view of a quantum processing unit (QPU) according to a first exemplary embodiment. [Diagram 2] FIG. 2 shows a circuit model used to derive the Hamiltonian of a superconducting qubit included in the QPU shown in FIG. 1 when the superconducting qubit is exposed to an external magnetic flux. [Diagram 3] FIG. 1 shows the DC Josephson phase as a function of external magnetic flux. [Figure 4]FIG. 1 shows the frequencies of the four lowest frequency normal modes as a function of external magnetic flux. [Diagram 5] FIG. 13 shows the anharmonicity of the four lowest frequency normal modes as a function of external magnetic flux. [Figure 6] FIG. 1 shows the potential energy landscape and phase-based wave functions for the four lowest energy states of the lowest frequency normal mode. [Figure 7] FIG. 11 is a close-up view of the anharmonicity and qubit frequency associated with the lowest frequency normal mode around the sweet spot Φdiff / Φ0=0.5. [Figure 8A] FIG. 2 illustrates a schematic capacitive coupling between superconducting qubits according to a first illustrative embodiment. [Figure 8B] FIG. 2 illustrates a schematic inductive coupling between superconducting qubits according to a first illustrative embodiment. [Figure 9] FIG. 1 is a schematic top view illustrating a QPU according to a second exemplary embodiment. [Figure 10] FIG. 13 is a schematic top view illustrating a QPU according to a third exemplary embodiment. [Figure 11] FIG. 13 is a schematic top view illustrating a QPU according to a fourth exemplary embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0035] Various embodiments of the present invention will be further described in more detail with reference to the accompanying drawings. However, the present 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. On the contrary, these embodiments are provided to provide a thorough and complete description of the present invention.

[0036] According to the detailed description, it will be clear to those skilled in the art that the scope of the present invention encompasses any embodiment disclosed herein, regardless of whether the embodiment is implemented independently or in cooperation with any other embodiment of the present invention. For example, the device disclosed in the present invention can be actually implemented by using any number of the embodiments provided herein. Furthermore, it should be understood that any embodiment of the present invention can be implemented using one or more of the elements presented in the appended claims.

[0037] The word "exemplary" is used herein to mean "used as an illustration." Unless otherwise specified, any embodiment described herein as "exemplary" should not be construed as preferred or advantageous over other embodiments.

[0038] Any location designation terms such as "left", "right", "up", "down", "upper", "lower", "top", "lower", etc. may be used herein for convenience to describe the relationship of one element or feature to one or more other elements or features according to the drawings. It should be clear that the location designation terms are intended to encompass different orientations of the devices disclosed herein in addition to the orientation depicted in the figures. As an example, if the device in the figures is imaginarily rotated 90 degrees clockwise, the elements or features described as "left" and "right" relative to the other elements or features would be oriented "above" and "below" the other elements or features, respectively. Thus, the location designation terms used herein should not be considered as any limitation of the present invention.

[0039] Although numerical classifier terms such as "first", "second", etc. may be used herein to describe various embodiments, it should be understood that these embodiments should not be limited by these numerical classifier terms. These numerical classifier terms are used herein only to distinguish one embodiment from another. Thus, the first embodiment discussed below could be referred to as the second embodiment without departing from the teachings of the present invention.

[0040] As used in the embodiments disclosed herein, a superconducting qubit can refer to a superconducting quantum device that is configured to store one or more quantum bits of information (or qubits for short). In this sense, a superconducting qubit acts as a quantum information storage and processing device.

[0041] According to embodiments disclosed herein, a quantum processing unit (QPU), also referred to as a quantum processor or quantum chip, can refer to a physical (fabricated) chip that includes at least one superconducting qubit, or several superconducting qubits that are interconnected in some way (e.g., to form a quantum logic gate). For example, this interconnection can be implemented as capacitive and / or inductive coupling, or by using any suitable coupling means, such as coupled resonators, tunable couplers, etc. A QPU is a basic building block of a quantum computing device, also referred to as a quantum computer, which can further include an enclosure for the QPU, control electronics, and many other components. In general, a quantum computing device can use superconducting qubits to perform different qubit operations, including reading out the state of the qubit, initializing the state of the qubit, and entangling the state of the qubit with the states of other qubits in the quantum computing device. Examples of existing implementations of such quantum computing devices include superconducting quantum computers, trapped ion quantum computers, quantum computers based on spins in semiconductors, quantum computers based on cavity quantum electrodynamics, optical photonic quantum computers, quantum computers based on defect centers in diamond, etc.

[0042] It should be noted that anharmonicity and coherence can be considered as two of the most important properties of a single superconducting qubit. Anharmonicity is the function of α / (2π)=(E 12 -E 01 ) / h, where E 12 is the energy difference between states 1 and 2, and E 01is the energy difference between states 0 and 1, and h is the Planck constant. In practice, the anharmonicity influences the shortest possible duration of a single qubit gate, and the anharmonicity should be high enough to implement fast single qubit gates with small leakage errors for non-computing states. On the other hand, the coherence of a qubit depends on the relaxation time T 1 and the coherence time T 2 In general, a large ratio between the coherence / relaxation time and the gate duration is desirable, since this determines the number of quantum gates that can be applied before quantum information is lost to the environment.

[0043] Exemplary embodiments disclosed herein provide a high-coherence, highly anharmonic superconducting qubit design for use in QPUs. The design is realized by combining phase-biased linear and nonlinear inductive energy elements in a superconducting qubit. The term "phase bias" as used herein refers to biasing the superconducting phase difference across the linear and nonlinear inductive energy elements. To the knowledge of the authors herein, such a combination of phase-biased linear and nonlinear inductive energy elements has not yet been used in superconducting qubits known from the prior art. It is important to note that the superconducting phase difference is biased such that the quadratic potential energy terms of the linear and nonlinear inductive energy elements are at least partially cancelled by each other. More quantitative metrics for measuring the cancellation are discussed below. In preferred embodiments, such cancellation is at least 30%.

[0044] In the exemplary embodiments disclosed herein, the superconducting phase difference of a circuit element can be referred to as a physical magnitude defined as follows:

[0045]

number

[0046] Where:

[0047]

number

[0048] is the superconducting phase difference at time t, V(t) is the corresponding voltage difference across the circuit elements, and Φ 0 = h / (2e) is the magnetic flux quantum and e is the charge of an electron. Note that the superconducting phase difference is related to the corresponding branch magnetic flux via a scale transformation.

[0049] A linear inductive energy element may be represented by a geometric or linear inductor, which in the exemplary embodiments disclosed herein may refer to a superconducting inductor having a geometric inductance that may be defined as follows: L=Φ / I where I denotes the current through the inductor and Φ denotes the magnetic flux generated by the current. The geometric inductance depends on the geometry of the inductor. For example, a geometric inductor can be implemented as a wire, a coil, or the central conductor of a distributed element resonator (particularly a CPW resonator) depending on the specific application.

[0050] The nonlinear inductive energy element may be represented by one or more Josephson junctions or kinetic inductors. In the exemplary embodiments disclosed herein, a kinetic inductor may refer to a nonlinear superconducting inductor whose inductance arises mostly from the inertia of charge carriers in the inductor. The term "Josephson junction" is then used herein in its ordinary sense and may refer to a quantum mechanical device made of two superconducting electrodes separated by a barrier (e.g., a thin insulating tunnel barrier, an ordinary metal, a semiconductor, a ferromagnetic material, etc.).

[0051] We now explain how the above-mentioned mutual cancellation of the quadratic potential energy terms of the linear and nonlinear inductive energy components affects the anharmonicity of a superconducting qubit. Assuming that the superconducting qubit is represented as a simple circuit model with a linear (geometric) inductor shunting a Josephson junction (or junctions), the total potential energy of the circuit model is given by

[0052]

number

[0053] Where:

[0054]

number

[0055] denotes the superconducting phase difference between both ends of the linear inductor,

[0056]

number

[0057] is the dielectric energy of a linear inductor, and E J is the Josephson energy of the Josephson junction,

[0058]

number

[0059] is the phase bias of the Josephson junction. Such a phase bias can be, for example, the external magnetic flux passing through the loop formed by the Josephson junction and a linear inductor.

[0060]

number

[0061] Note that this can be achieved by: In this case, the flux quantization condition relates the superconducting phase difference across the linear inductor and the Josephson junction as:

[0062]

number

[0063] Where:

[0064]

number

[0065] is the superconducting phase difference across the Josephson junction, where m is an integer.

[0066] The phase bias is

[0067]

number

[0068] When θ is equal to θ, the quadratic potential energy terms associated with the linear inductor and the Josephson junction have different signs, and therefore they at least partially cancel each other. In other words, the total potential energy can be approximated by a fourth-order expression as follows:

[0069]

number

[0070] Here the cancellation of the quadratic potential energy terms can be clearly seen.

[0071]

number

[0072] , the fourth-order potential energy term may be large compared to the second-order potential energy term, which leads to high anharmonicity of the superconducting qubit corresponding to the circuit model assumed above.

[0073] To quantitatively estimate the amount of cancellation in terms of the total potential energy U, it is noted that the potential energy of a phase-biased Josephson junction can be expanded into a Taylor series as follows:

[0074]

number

[0075] Where:

[0076]

number

[0077] denotes the coefficient of the k-th Taylor series of the potential energy of a phase-biased Josephson junction. This makes it possible to measure the cancellation efficiency present in the total potential energy U by using the ratio

[0078]

number

[0079] Where:

[0080]

number

[0081] denotes the coefficient of the second-order Taylor series of the potential energy of the phase-biased Josephson junction, and β denotes the amount of cancellation. At least 30% cancellation means that β≧0.3. For example, if the phase bias is

[0082]

number

[0083] If it is equal to E J,2 =-E J means the following formula:

[0084]

number

[0085] In this case, the requirement of β≧0.3 is satisfied when the Josephson energy and the dielectric energy are

[0086]

number

[0087] This means that the following conditions must be met:

[0088] In some embodiments, the circuit model assumed above may be supplemented with a capacitive energy element also positioned to shunt the Josephson junction. Such a capacitive energy element may be implemented as one or more interdigitated capacitors, gap capacitors, parallel plate capacitors, or junction capacitors.

[0089] In some embodiments, one or more superconducting qubits, each represented by a combination of phase-biased linear and nonlinear inductive energy elements, may be provided on a dielectric substrate, hi some embodiments, the superconducting qubits (together with the dielectric substrate) may be further disposed inside one or more 3D cavities.

[0090] FIG. 1 shows a schematic top view of a QPU 100 according to a first exemplary embodiment. As shown in FIG. 1, the QPU 100 comprises a dielectric substrate 102 and a superconducting qubit 104 disposed on the dielectric substrate 102. In the first embodiment, the superconducting qubit 104 is configured as a CPW resonator comprising a central superconductor 106 and a superconducting ground plane 108. The superconductor 106 is electrically connected to the superconducting ground plane 108 on opposing sides of a first pair (i.e., left and right sides as shown in FIG. 1). At the same time, the superconductor 106 is separated from the superconducting ground plane 108 on opposing sides of a second pair (i.e., top and bottom sides as shown in FIG. 1) by equal gaps 110 and 112. In this case, the superconductor 106 serves as a linear inductive energy element of the superconducting qubit 104. In the case of a nonlinear inductive energy element, it is represented by a single Josephson junction 114 built into the superconductor 106, such that there is no superconducting island in the QPU 100. Here, the superconducting island may be referred to as a Cooper pair box connected via a tunnel junction with the central superconductor 106. As another example, a superconducting island is formed between two Josephson junctions built in series within the central conductor. Note that all building elements of the QPU 100 in FIG. 1 are not drawn to scale for convenience. Furthermore, the shapes of the central superconductor 106 and the superconducting ground plane 108 are also exemplary and may be modified depending on the specific application.

[0091] Superconducting qubit 104 is configured as a CPW resonator, although this should not be considered any limitation of the present invention. In other embodiments, superconducting qubit 104 may be configured as any type of distributed element resonator (one example of which is a CPW resonator), or superconducting qubit 104 may be configured as any other combination of linear and nonlinear inductive energy elements that are configured to be phase biased such that their quadratic potential energy terms at least partially cancel each other.

[0092] In the case of the Josephson junction 114, it may interrupt the central superconductor 106, as shown in Figure 1. In one embodiment, the Josephson junction 114 may be incorporated into the central superconductor 106 such that the current flowing through the central superconductor 106 is equal on both sides of the Josephson junction 114. In another embodiment, the Josephson junction 114 is centrally located within the central superconductor 106.

[0093] To achieve the above-mentioned cancellation, the QPU 100 should also include a phase bias element (not shown in FIG. 1). In a first embodiment, the phase bias element biases the magnetic flux Φ ext,1 and Φ ext,2 It is intended to generate and pass a phase bias in a suitable manner, thereby realizing the phase bias. Thanks to the two parallel loops, the superconducting qubit 104 is gradient, meaning that it is protected against magnetic flux noise whose spatial scale exceeds the width of the superconducting qubit 104. This phase bias leads to at least partial mutual cancellation of the quadratic potential energy terms of the superconductor 106 and the Josephson junction 114, thereby improving the anharmonicity of the superconducting qubit 104. The phase bias element may comprise one or more coils and / or one or more flux bias lines to realize the flux control. The flux bias lines may be implemented as superconducting wires on the dielectric substrate 102, and the magnetic field may be generated by turning a current flowing through the wire. In some other embodiments, such a phase bias element may be configured to realize the phase bias by applying a suitable voltage to the Josephson junction 114 instead of or in addition to passing a magnetic field through the gaps 110 and 112.

[0094] Because the Josephson junctions 114 are incorporated into the CPW resonator such that isolated superconducting islands are not formed, the inductance and capacitance of the CPW resonator shunt the Josephson junctions 114, providing protection against dephasing resulting from low frequency charge noise. Due to the inductive shunt, the superconducting qubit 104 should be completely immune to low frequency charge noise due to its topology, unlike commonly employed transmon qubits, where only a few lowest energy levels are well protected against charge noise.

[0095] 1, the superconducting ground plane 108 comprises opposing upper 108-1 and lower 108-2 portions that are physically separated from one another by the central superconductor 106 and gaps 110, 112. In one embodiment, these opposing portions may be connected to one another via air bridges that extend across the central superconductor 106 and gaps 110, 112 to suppress parasitic slotline modes of the CPW resonator.

[0096] 2 shows a circuit model 200 used to derive the Hamiltonian of the superconducting qubit 104 included in the QPU 100 when the qubit is exposed to an external magnetic flux. According to the circuit model 200, a CPW resonator of length 2l is modeled by using N lumped element inductors and capacitors. In addition, the Josephson junction 114 is assumed to be located between the capacitors 202 and 204 with subscripts J and J+1. Due to the gradient nature of the two loops, the external magnetic flux in the following equations is considered as the (scaled) difference of the external magnetic flux on the two sides of the central superconductor 106, i.e., Φ diff =(Φ ext,2 -Φ ext,1 ) / 2. Using the circuit model 200, it is possible to write the classical kinetic energy term T and potential energy term U for the circuit as:

[0097]

number

[0098] Where:

[0099]

number

[0100] is the voltage V i is the nodal flux across the i-th capacitor at Φ diff,i is the external magnetic flux through the ith loop, Δx=2l / N is the length scale for the discretization, and c tot is the total capacitance per unit length of the CPW resonator, and l tot is the total inductance per unit length of the CPW resonator, and E J is the Josephson energy, and C J is the capacitance of the Josephson junction 114, and Φ 0 is the magnetic flux quantum as above. In addition, the dot above the symbol denotes the time derivative.

[0101] Using the Lagrangian formalism, we can then derive the classical equations of motion for the nodal fluxes in the CPW resonator. In the continuum limit Δx→0, we obtain

[0102]

number

[0103] Here, Ψ i →ψ(x i ) is the position x i corresponds to the continuum limit of the nodal flux in Φ diff,i / (sΔx) → B diff denotes the effective magnetic field difference, and s is the distance between the central superconductor 106 and the superconducting ground plane 108. Using the Lagrangian formalism, the position corresponding to the left electrode of the Josephson junction 114

[0104]

number

[0105] Boundary conditions for the nodal flux at can also be derived.

[0106]

number

[0107] where Δψ=Ψ J+1 -Ψ J is the branch flux across the Josephson junction 114, and I c =2πE J / Φ 0 is the critical current of the Josephson junction 114, and Φ diff =Σ i Φ diff,i is the total external magnetic flux difference. In the above equation, Φ diff,J / Δx→Φ diff The uniform magnetic field assumption is utilized to write / (2l), where 2l is the length of the central superconductor 106. Note that similar boundary conditions can be derived for the right electrode of the Josephson junction 114. Further boundary conditions ψ(-l)=0 and ψ(l)=0 arise from the grounding of the central superconductor 106.

[0108] Based on the classical equations of motion and boundary conditions, it follows that the (classical) product magnetic flux can be described as a linear combination of a dc supercurrent and an infinite number of oscillatory normal modes, i.e.

[0109]

number

[0110] Here, φ 0 is the time-independent coefficient of the "dc mode", and u 0(x) is the corresponding envelope function. In intuitive terms, the dc supercurrent biases the Josephson junction 114, thereby changing the effective Josephson inductance seen by the oscillatory (ac) normal mode. Here, {u n (x)} is the envelope function of the oscillatory ac mode, and {ψ n (t)} are the corresponding time-dependent coefficients. Importantly, the envelope functions and the corresponding modal frequencies can be derived using the above equations of motion and the above boundary conditions.

[0111] Since we use a CPW resonator with Josephson junctions 114 implemented as the superconducting qubit 104, it will be observed that the nonlinearity of the Josephson junctions 114 converts some of the normal modes into anharmonic oscillators. In what follows, we focus on the mth mode and assume that we wish to operate it as a qubit. With this in mind, it is possible to derive a single-mode approximation for the quantum Hamiltonian given by

[0112]

number

[0113] Where:

[0114]

number

[0115] is the effective charging energy associated with the mth mode,

[0116]

number

[0117] is the charge operator of the mth mode,

[0118]

number

[0119] is the effective dielectric energy of the mth mode,

[0120]

number

[0121] is the phase operator corresponding to the mth mode,

[0122]

number

[0123] is the dielectric energy associated with the total linear inductance of the CPW resonator,

[0124]

number

[0125] is the phase bias corresponding to the dc current,

[0126]

number

[0127] denotes the phase associated with the external magnetic flux. The phase operator and the charge operator are given by the commutation relation

[0128]

number

[0129] Note that the conjugate operator satisfies: where i is the imaginary unit.

[0130] The mth mode of the superconducting qubit 104 is treated quantum mechanically in the above Hamiltonian, but in the dc Josephson phase

[0131]

number

[0132] Note that is treated as a static variable that is calculated based on the semiclassical theory. According to the semiclassical theory, the dc Josephson phase is given by the flux quantization condition

[0133]

number

[0134] Where:

[0135]

number

[0136] is the branch flux associated with the dc Josephson phase.

[0137] In general, the anharmonicity α of a given mode m / (2π) can be calculated numerically by performing the following steps: - First, determine the dc Josephson phase using the flux quantization condition given above - Then solve for the (classical) normal mode frequencies using the following equations, derived from the above equations of motion and boundary conditions:

[0138]

number

[0139] Where:

[0140]

number

[0141] is the wave number of the mth mode, and L J =Φ 0 / (2πI c ) is the effective Josephson inductance, and - Finally, to obtain the quantized energy spectrum of the mode of interest for a given external magnetic flux, we use the single-mode Hamiltonian

[0142]

number

[0143] Numerically diagonalize the mth mode energy spectrum to find the qubit frequency ω q / (2π) and the corresponding anharmonicity α m It is easy to understand by evaluating / (2π).

[0144] Due to the large capacitance per unit length of the CPW resonator, the anharmonicity of the superconducting qubit 104 is only small unless the parameters of the circuit model 200 are appropriately selected and a suitable external magnetic flux is applied. However, it is possible for the external magnetic flux to be equal to half the magnetic flux quantum, i.e., Φ diff / Φ 0 = ±0.5, assuming the Josephson inductance is larger than the total inductance of the CPW resonator, the dc Josephson phase is

[0145]

number

[0146] If the linear inductance of the CPW resonator is only slightly smaller than the Josephson inductance, the inductive energy

[0147]

number

[0148] and Josephson energy E J Using experimentally achievable values ​​of the parameters of the circuit model 200, the authors show that the external magnetic flux is Φ diff / Φ 0 We find that when the θ is tuned to =±0.5, the anharmonicity of the lowest frequency mode exceeds (well) 500 MHz for a qubit frequency of about 5 GHz. It is important to note that this also corresponds to the flux-insensitive sweet spot that protects the superconducting qubit 104 against flux-noise induced dephasing. Some numerical results are illustrated in Figs. 3-7.

[0149] More specifically, FIG. 3 shows one possible inductance ratio, 2ll. tot / L J =L CPW / L J For =0.77, the external magnetic flux difference Φ diff DC Josephson phase as a function of

[0150]

number

[0151] Figure 4 shows the external magnetic flux Φ diff The frequencies of the four lowest frequency normal modes, f m =ω m / (2π). Figure 5 shows the external magnetic flux Φ diff The anharmonicity α of the four lowest frequency normal modes as a function of m / (2π). Φ diff / Φ 0 Note the large anharmonicity of the lowest frequency mode at Φ = 0.5 (> 500 MHz). diff / Φ 0 Potential energy landscape for the four lowest energy states of the lowest frequency normal modes at =0.5

[0152]

number

[0153] and the phase-based wave function.

[0154]

number

[0155] The anharmonicity α associated with the lowest frequency normal mode around 1 / (2π) and the qubit frequency ω q An enlarged view of / (2π) is shown. diff / Φ 0 Note that =0.5 corresponds to the flux insensitive sweet spot with these parameter values. The parameters used to obtain the numerical results shown in Figures 3-7 correspond to the parameter set given in Table 1 below. Table 1. Exemplary parameters used to estimate the frequencies and anharmonicities of normal modes in a superconducting qubit 104 exposed to an external magnetic flux.

[0156] [Table 1]

[0157] In Table 1, x J / l∈[-1,1] is the (relative) position of the Josephson junction 114 in the central superconductor 106 (x J / l=0 corresponds to the Josephson junction 114 located at the center of the central superconductor 106), k 0 = w / (w + 2s) is a ratio describing the geometry of the CPW resonator, where w is the width of the central superconductor 106, s is the gap between the central superconductor 106 and the superconducting ground plane (i.e., gap 110 or 112), and ε effis the effective dielectric constant of the CPW resonator, and l k is the motional inductance per unit length of the resonator, and l g is the geometric inductance per unit length of the resonator.

[0158]

number

[0159] is the characteristic impedance of the CPW resonator.

[0160] To further improve the anharmonicity, the central superconductor 106 of the CPW resonator can be fabricated from a superconducting material with high kinetic inductance, such as a superconducting thin film. This will increase the inductance of the CPW resonator relative to the capacitance. As a result, the total capacitance of the CPW resonator can be reduced, which will improve the anharmonicity of the superconducting qubit 104. In such a circuit model, the anharmonicity can exceed 200 MHz even without an external magnetic flux, and well above 1 GHz with an external magnetic flux. However, superconducting thin films tend to be relatively lossy, and therefore an increase in anharmonicity may be accompanied by a significant decrease in relaxation and coherence times. For this reason, an approach based on an external magnetic flux without a superconducting thin film appears to be the most promising route towards a high-coherence, highly anharmonic superconducting qubit.

[0161] 8A and 8B show schematic capacitive and inductive couplings between superconducting qubits according to a first exemplary embodiment. More specifically, FIG. 8A shows a schematic top view of a QPU 800 with a combination of two superconducting qubits 104 capacitively coupled to each other. FIG. 8B shows a schematic top view of a QPU 802 with a combination of three superconducting qubits 104 inductively coupled to each other. In FIG. 8A and 8B, white indicates the central superconductor 106, the superconducting ground plane 108, and the Josephson junctions 114 in each superconducting qubit 104, while black indicates the gaps 110 and 112 in each superconducting qubit 104. It will be clear to one skilled in the art that the number of superconducting qubits 104 shown in FIG. 8A and 8B is merely for illustration and should not be considered as any limitation of the present invention. Furthermore, it should be noted again that the sizes of the QPUs 800 and 802 and their building blocks are not to scale for convenience.

[0162] FIG. 9 shows a schematic top view of a QPU 900 according to a second exemplary embodiment. Similar to the QPU 100 in the first exemplary embodiment, the QPU 900 comprises a dielectric substrate 902 and a superconducting qubit 904 disposed on the dielectric substrate 902. In the second embodiment, the superconducting qubit 904 is also configured as a CPW resonator comprising a central superconductor 906 and a superconducting ground plane 908. The superconductor 906 is electrically connected to the superconducting ground plane 908 on opposing sides of a first pair (i.e., left and right sides as shown in FIG. 9). At the same time, the superconductor 906 is separated from the superconducting ground plane 908 on opposing sides of a second pair (i.e., top and bottom sides as shown in FIG. 9) by equal gaps 910 and 912. The superconductor 906 serves as a linear inductive energy element for the superconducting qubit 904. In contrast to the first embodiment, the nonlinear inductive energy element in the second embodiment is represented by a combination of two parallel Josephson junctions 914 and 916 incorporated in the superconductor 906, such that there are no superconducting islands in the QPU 900. Such an arrangement of Josephson junctions 914 and 916 forms a SQUID loop. Here, phase bias can be realized by passing a magnetic field through the gaps 910 and 912 and the SQUID loop. It should also be noted that all the building blocks of the QPU 900 are shown in FIG. 9 not to scale for convenience. Moreover, the shapes of the central superconductor 906 and the superconducting ground plane 908 are also exemplary and can be changed depending on the specific application.

[0163] FIG. 10 shows a schematic top view of a QPU 1000 according to a third exemplary embodiment. Similar to the QPU 100 in the first exemplary embodiment and the QPU 900 in the second embodiment, the QPU 1000 comprises a dielectric substrate 1002 and a superconducting qubit 1004 disposed on the dielectric substrate 1002. In the third embodiment, the superconducting qubit 1004 is also configured as a CPW resonator comprising a central superconductor 1006 and a superconducting ground plane 1008. The superconductor 1006 is electrically connected to the superconducting ground plane 1008 on opposite sides of a first pair (i.e., on the left and right sides as shown in FIG. 10). At the same time, the superconductor 1006 is separated from the superconducting ground plane 1008 on opposite sides of a second pair (i.e., on the top and bottom sides as shown in FIG. 10) by equal gaps 1010 and 1012. The superconductor 1006 serves as a linear inductive energy element of the superconducting qubit 1004. In contrast to the first and second embodiments, the nonlinear inductive energy element in the third embodiment is represented by a combination of three Josephson junctions 1014, 1016, and 1018. The Josephson junction 1014 is incorporated in the central superconductor 1006, while the Josephson junctions 1016 and 1018 are disposed in the upper gap 1010 near the Josephson junction 1014, such that the Josephson junctions 1016 and 1018 connect the central superconductor 1006 to the ground plane 1008. The illustrated arrangement of the Josephson junctions 1014, 1016, and 1018 is not limiting and can be changed depending on the specific application. For example, one of the Josephson junctions 1016 and 1018 can be omitted or disposed in the other lower gap 1012 near the Josephson junction 1014. Importantly, Josephson junctions 1014, 1016, and 1018 are still incorporated into superconducting qubit 1004, such that there are no superconducting islands in QPU 1000. Meanwhile, phase bias can be achieved in the same manner as in the first embodiment, namely, by passing a magnetic field through gaps 1010 and 1012. Note again that all building blocks of QPU 1000 in FIG. 10 are not to scale for convenience.Additionally, the shapes of the central superconductor 1006 and the superconducting ground plane 1008 are also exemplary and may be varied depending on the particular application.

[0164] FIG. 11 shows a schematic top view of a QPU 1100 according to a fourth exemplary embodiment. Similar to the QPU 100 in the first embodiment, the QPU 900 in the second embodiment, and the QPU 1000 in the third embodiment, the QPU 1100 comprises a dielectric substrate 1102 and a superconducting qubit 1104 disposed on the dielectric substrate 1102. In the fourth embodiment, the superconducting qubit 1104 is also configured as a CPW resonator comprising a central superconductor 1106 and a superconducting ground plane 1108. The superconductor 1106 is electrically connected to the superconducting ground plane 1108 on opposite sides of the first pair (i.e., on the left and right sides as shown in FIG. 11). At the same time, the superconductor 1106 is separated from the superconducting ground plane 1108 on opposite sides of the second pair (i.e., on the top and bottom sides as shown in FIG. 11) by equal gaps 1110 and 1112. The superconductor 1106 serves as a linear inductive energy element for the superconducting qubit 1104. In contrast to the first, second, and third embodiments, the nonlinear inductive energy element in the fourth embodiment is represented by a combination of five Josephson junctions 1114, 1116, 1118, 1120, and 1122. While the Josephson junction 1114 is incorporated in the central superconductor 1106, the Josephson junctions 1116-1122 are disposed in the gaps 1110 and 1112 adjacent to the Josephson junction 1114, such that the Josephson junctions 1116 and 1118 connect the central superconductor 1106 to the ground plane 1108 via the upper gap 1110, and the Josephson junctions 1120 and 1122 connect the central superconductor 1106 to the ground plane 1108 via the lower gap 1112. The illustrated arrangement of Josephson junctions 1114-1122 is not limiting and can be modified depending on the specific application. For example, one or more of Josephson junctions 1116-1122 can be omitted. Importantly, Josephson junctions 1114-1122 are still incorporated into superconducting qubit 1104 such that there are no superconducting islands in QPU 1100. Meanwhile, phase bias can be achieved in the same manner as in the first and third embodiments, namely by passing a magnetic field through gaps 1110 and 1112.It should also be noted that all of the building blocks of QPU 1100 are shown out of scale in Figure 11 for convenience. Furthermore, the shapes of central superconductor 1106 and superconducting ground plane 1108 are also exemplary and may be varied depending on the particular application.

[0165] In the third and fourth embodiments, when there is an even number of Josephson junctions in the gap between the central superconductor and the superconducting ground plane, these Josephson junctions may be positioned symmetrically or asymmetrically with respect to the Josephson junction incorporated in the central superconductor, depending on the particular application.

[0166] In some other embodiments, the QPU (e.g., any of QPUs 100, 900-1100) further comprises a signal line disposed on the dielectric substrate. The signal line can be used to provide control signals to the superconducting qubits (e.g., from an external control unit, or from control electronics if the QPU is used in a quantum computer). The signal line can comprise a radio frequency line, and the control signal can include microwave pulses. The control signal can enable the superconducting qubits to be controlled in a desired manner.

[0167] In some other embodiments, the QPU (e.g., any of QPUs 100, 900-1100) further comprises a readout line disposed on the dielectric substrate. The readout line may be included in the QPU in combination with the signal line. The readout line may be coupled to the superconducting qubit via a readout resonator. The readout line may be used to take state measurements of the superconducting qubit, if desired.

[0168] Although exemplary embodiments of the present invention are described herein, it should be noted that various modifications and changes can be made to the embodiments of the present invention without departing from the scope of legal protection defined by the appended claims. In the appended claims, the word "comprising" does not exclude other elements or operations, and the indefinite articles "a" or "an" do not exclude a plural form. The mere fact that certain actions are recited in mutually different dependent claims does not indicate that a combination of these actions cannot be used to advantage. [Explanation of symbols]

[0169] 100 QPUs 102 Dielectric Substrate 104 Superconducting qubits 106 Central Superconductor 108 Superconducting Ground Plane 108-1 Upper part 108-2 Lower part 110 Gap 112 Gap 114 Josephson Junction 200 Circuit Models 202 Capacitor 204 Capacitor 800 QPUs 802 QPUs 900 QPUs 902 Dielectric Substrate 904 Superconducting qubit 906 Central Superconductor 908 Superconducting Ground Plane 910 Gap 912 Gap 914 Josephson Junction 916 Josephson Junction 1000 QPUs 1002 Dielectric substrate 1004 Superconducting qubit 1006 Central Superconductor 1008 Superconducting Ground Plane 1010 Upper gap 1012 Lower gap 1014 Josephson Junction 1016 Josephson Junction 1018 Josephson Junction 1100 QPUs 1102 Dielectric substrate 1104 Superconducting qubit 1106 Central Superconductor 1108 Superconducting Ground Plane 1110 Upper gap 1112 Lower gap 1114 Josephson Junction 1116 Josephson Junction 1118 Josephson Junction 1120 Josephson Junction 1122 Josephson Junction

Claims

1. A dielectric substrate; At least one superconducting qubit provided on the dielectric substrate, each of the at least one superconducting qubit comprising: a linear inductive energy element that is superconducting; and Nonlinear Inductive Energy Elements At least one superconducting qubit comprising: Equipped with A quantum processing unit, wherein the linear inductive energy element and the nonlinear inductive energy element are configured to bias a superconducting phase difference between the linear inductive energy element and the nonlinear inductive energy element by a phase bias element such that quadratic potential energy terms associated with the linear inductive energy element and the nonlinear inductive energy element are at least partially canceled by each other.

2. The unit described in claim 1, wherein the linear inductive energy element and the nonlinear inductive energy element are configured such that by biasing the superconducting phase difference by the phase bias element, the quadratic potential energy terms associated with the linear inductive energy element and the nonlinear inductive energy element are canceled out by each other by at least 30%.

3. The unit of claim 1 or 2, wherein the linear inductive energy element comprises one or more geometric inductors.

4. 4. A unit according to claim 1, wherein the nonlinear inductive energy element comprises one or more Josephson junctions or motional inductors.

5. 5. The unit of claim 1, wherein each of the at least one superconducting qubit further comprises a capacitive energy element.

6. The unit of claim 5 , wherein the capacitive energy element comprises one or more interdigitated capacitors, gap capacitors, parallel plate capacitors, or junction capacitors.

7. A unit described in any one of claims 1 to 6, wherein the linear inductive energy element and the nonlinear inductive energy element are configured such that the superconducting phase difference is biased by the phase bias element by generating and passing a magnetic field through the at least one superconducting quantum bit or by applying a predetermined voltage to the nonlinear inductive energy element.

8. 8. The unit of claim 1 , wherein the at least one superconducting qubit comprises two or more superconducting qubits on the dielectric substrate that are capacitively and / or inductively coupled to each other.

9. 9. The unit of claim 1, wherein the at least one superconducting qubit comprises two or more superconducting qubits, the unit further comprising one or more coupled resonators and / or tunable couplers on the dielectric substrate for coupling the superconducting qubits.

10. 10. The unit of claim 1, further comprising a signal line disposed on the dielectric substrate, the signal line configured to provide a control signal to the at least one superconducting qubit.

11. The unit of claim 10 , wherein the signal line comprises a radio frequency line and the control signal comprises a microwave pulse.

12. 12. The unit of claim 1, further comprising a readout line provided on the dielectric substrate, the readout line configured to measure a state of the at least one superconducting qubit.

13. 13. The unit of claim 12, further comprising a readout resonator disposed on the dielectric substrate, the readout line being coupled to the at least one superconducting qubit through the readout resonator.

14. at least one quantum bit is configured as a distributed element resonator comprising at least two conductors separated by at least one gap; at least one of the at least two conductors serves as the linear inductive energy element, and the nonlinear inductive energy element comprises at least one Josephson junction integrated into the distributed element resonator; 14. The unit of claim 1, wherein the linear inductive energy element and the nonlinear inductive energy element are configured such that the superconducting phase difference is biased by the phase bias element by generating and passing a magnetic field through the at least one gap of the distributed element resonator.

15. 15. The unit of claim 14, wherein the distributed element resonator is configured as a coplanar waveguide (CPW) resonator, the at least two conductors comprising a central superconductor and a superconducting ground plane, the central superconductor serves as the linear inductive energy element, and at least one Josephson junction is incorporated into the CPW resonator such that the quantum processing unit does not have isolated superconducting islands.

16. 16. The unit of claim 15, wherein the superconducting ground plane is formed on the dielectric substrate such that the central superconductor of the CPW resonator has a first pair of opposing sides and a second pair of opposing sides, the central superconductor being electrically connected to the superconducting ground plane on the first pair of opposing sides and separated from the superconducting ground plane by the gap on the second pair of opposing sides.

17. 17. The unit of claim 15 or 16, wherein the superconducting ground plane comprises opposing portions physically separated from one another by the central superconductor and the gap, the opposing portions being connected to one another via an air bridge extending across the gap and the central superconductor.

18. 18. A unit according to any one of claims 15 to 17, wherein the at least one Josephson junction is embedded in the central superconductor.

19. 20. The unit of claim 18, wherein the at least one Josephson junction comprises a parallel connection of two Josephson junctions.

20. 20. A unit as claimed in claim 18 or 19, wherein the at least one Josephson junction is centrally located within the central superconductor.

21. The at least one Josephson junction is a first Josephson junction embedded in the central superconductor; at least one second Josephson junction disposed in one or more of the gaps proximate the first Josephson junction, each of the at least one second Josephson junction connecting the central superconductor to the superconducting ground plane through a corresponding one of the gaps; A unit according to any one of claims 15 to 17, comprising:

22. 22. The unit of claim 21, wherein the at least one second Josephson junction comprises an even number of second Josephson junctions symmetrically disposed about the first Josephson junction.

23. 23. A unit as claimed in claim 21 or 22, wherein the first Josephson junction is centrally located within the central superconductor.

24. 24. A unit according to any one of claims 15 to 23, wherein the central superconductor has a straight or curved shape.

25. 25. The unit of any one of claims 1 to 24, further comprising at least one 3D cavity, the dielectric substrate having the at least one superconducting qubit being provided within the at least one 3D cavity.

26. 26. A quantum computer comprising at least one quantum processing unit according to any one of claims 1 to 25 and a control unit configured to perform computational operations by using said at least one quantum processing unit.