Multi-qubit superconducting circuits

Abstract

The invention relates to a superconducting circuit suitable for implementing multiple qubits in a quantum processing unit. The superconducting circuit comprising at least one non-linear inductive element shunted by at least one linear inductive element such that a first impedance along a first path of the non-linear inductive element is different to a second impedance along a second path of the non-linear inductive energy element forming at least two modes of the superconducting circuit that couple to the non-linear inductive element.

Description

[0001]MULTI-QUBIT SUPERCONDUCTING CIRCUITS Technical Field The invention relates to field of superconducting circuits for use in quantum computing, specifically to superconducting circuits that are capable of encoding multiple (i.e. more than one) qubit and allowing multi-qubit operations to be performed natively. Background Superconducting circuits can be used to encode qubits for use in quantum computation. One recent example of a superconducting circuit that can be used to encode a single qubit is the “unimon” qubit, described in Hyyppä, E., Kundu, S., Chan, C.F. et al. Unimon qubit. Nat Commun 13, 6895 (2022). The unimon consists of a single Josephson junction situated in the centre of the centre conductor of a superconducting coplanar-waveguide (CPW) resonator that is grounded at both ends. Due to the non-linearity of the Josephson junction, the normal modes of the resonator with a non-zero current across the junction are converted into anharmonic oscillators that can be used as a qubit. The frequency of each anharmonic mode can be controlled by applying external fluxes through the two superconducting loops of the resonator structure. While it may in principle be possible to use multiple modes to encode multiple qubits, in practice the differences in transition frequencies between the modes that couple to the junction placed in the middle of the structure are too large to allow for effective driving of each single and two-qubit transition using the same drive circuitry and the use of stronger coupling or different drive circuity for different transitions is not practical due to limitation imposed by the qubit environment, the qubit may quickly decay to the drive lines. Summary The invention includes a quantum processing unit that comprises at least one superconducting circuit and at least one phase-biasing element. The superconducting circuit comprises at least one non-linear inductive element asymmetrically shunted by at least one linear inductive element such that a first impedance along a first path of the non- linear inductive element is different to a second impedance along a second path of the non-linear inductive energy element. This forms at least two modes of the superconducting circuit that couple to the non-linear inductive element. The phase-biasing element is configured to bias a superconducting phase difference across the linear inductive element and the non-linear inductive element such that potential energy terms that are quadratic in phase of the linear inductive element and the nonlinear inductive element are at least partly cancelled by one another, and such that the frequencies of the said at least two modes of the superconducting circuit depend on the excitation of at least two modes. The invention also includes a method of performing a multi-qubit gate on the quantum processing unit described above. The method comprises driving the superconducting circuit at a first frequency, which corresponds to a transition frequency of a first qubit when one or more second qubits are in the ground state or first excited state, such that the state of the first qubit is modified based on the state of the one or more second qubits. Further aspects of the invention are set out in the detailed description and in the claims. Brief Description of the Drawings Figure 1 is a schematic of a first superconducting circuit that may be used to encode multiple qubits. Figure 2 is a graph showing three modes of the superconducting circuit of Figure 1. Figure 3A is a schematic of a second superconducting circuit that may be used to encode multiple qubits. Figure 3B is a circuit diagram depicting a lumped-element model corresponding to the superconducting circuit of Figure 3A. Figure 4A is a schematic of a third superconducting circuit that may be used to encode multiple qubits. Figure 4B is a circuit diagram depicting a lumped-element model corresponding to the superconducting circuit of Figure 4A. Figure 5 is a schematic of a fourth superconducting circuit that may be used to encode multiple qubits. Figure 6 is a schematic of a fifth superconducting circuit that may be used to encode multiple qubits. Detailed Description The invention relates to superconducting circuits for use in quantum processing units, in particular for use as superconducting qubits. In the present context, a “superconducting qubit” refers 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. To avoid confusion, where the term “qubit” is used in isolation, it is used to refer to a two-state system in the quantum mechanical sense, while the term “superconducting qubit” is used to refer to a superconducting circuit that is the physical implementation of a qubit. A quantum processing unit (QPU), also referred to as a quantum processor or quantum chip, is a physical (fabricated) chip that contains at least one superconducting qubit or a number of superconducting qubits that are somehow interconnected (e.g., to form quantum logic gates). For example, this interconnection may be implemented as capacitive and / or inductive couplings, or it may be performed by using any suitable coupling means, such as coupling resonators, tuneable couplers, etc. The QPU is the foundational component of a quantum computing device, also referred to as a quantum computer, which may further include a housing for the QPU, control electronics, and many other components. In general, the quantum computing device may perform different qubit operations by using the superconducting qubit, including reading the state of a qubit, initializing the state of the qubit, and entangling the state of the qubit with the states of other qubits in the quantum computing device, etc. Existing implementation examples 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 photon quantum computers, quantum computers based on defect centres in diamond, etc. Anharmonicity and coherence may be considered as two of the most important propertiesfor single superconducting qubits. The anharmonicity may be defined as ^^ / 2 ^^ =( ^^12 − ^^01) / ℎ, where ^^12 is the energy difference between states 1 and 2 (i.e. the firstexcited state and second excited state), ^^01is the energy difference between states 0 and 1 (i.e. the ground state and first excited state), and ℎ is Planck’s constant. In practice, the anharmonicity affects the shortest possible duration of single-qubit gates, and the anharmonicity should be high enough to perform fast single-qubit gates with small leakage errors to non-computational states. On the other hand, the coherence of qubits may be quantitively described with relaxation time ^^1and coherence time ^^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 may be applied before quantum information has been lost to the environment. The present invention provides a high-coherence high-anharmonicity superconducting qubit design to be used in a QPU. This design is provided by combining phase-biased linear and non-linear inductive elements in a superconducting qubit. The term “phase- biased” refers to biasing a superconducting phase difference across the linear and non- linear inductive elements. The phase difference is biased such that quadratic potential energy terms of the linear and nonlinear inductive-energy elements are cancelled at least partly by one another. In particular, the superconducting phase difference of a circuit element may refer to a physical magnitude defined as where ^^( ^^) is the superconducting phase difference at time ^^, ^^( ^^) is the voltage difference across the circuit element, and Φ0is the flux quantum ℎ / 2 ^^, where ℎ is Planck’s constant and ^^ is the elementary charge. Note that the superconducting phase difference is related to a corresponding branch flux via a scale transformation. The linear inductive elements may be geometric or linear inductors. A geometric or linear inductor refers to a superconducting inductor having a geometric inductance that may be defined as ^^ = Φ / ^^ , where ^^ denotes the electric current through the inductor, and denotes the magnetic flux generated by the current. Geometric inductance depends on the geometry of the inductor. For example, the geometric inductor may be implemented as a wire, coil, or a centre conductor of a distributed-element resonator (in particular, a co- planar waveguide resonator). In the context of the present invention, a non-linear inductive element may be one or more Josephson junctions or kinetic inductors. A kinetic inductor may refer to a nonlinear superconducting inductor whose inductance arises mostly from the inertia of charge carriers in the inductor. In turn, 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.). Assuming that a superconducting qubit is represented as a simple circuit model comprising a linear (geometric) inductor shunting a Josephson junction (or Josephson junctions), the total potential energy ^^ of the circuit model can be expressed as where ^^ is the superconducting phase difference across the linear inductor, ^^^^=1^^is the inductive energy of the linear inductor, ^^^^is the Josephson energy of the Josephson junction, and ^^^^ ^^ ^^is the phase bias of the Josephson junction. Such a phase bias could be achieved, for example, with an external magnetic flux Φ^^ ^^ ^^=Φ0 ^^ ^^ ^^ ^^2^^through a loop formed by the Josephson junction and the linear inductor. In this case, a flux quantization condition would relate the superconducting phase differences across the linear inductor and the Josephson junction as is the superconducting phase difference across the Josephson junction, and ^^ is an integer. If the phase bias ^^^^ ^^ ^^= ± ^^, the quadratic potential energy terms associated with the linear inductor and the Josephson junction have different signs, on account of which they cancel each other at least partly. In other words, the total potential energy may be approximated to the fourth order as where the cancellation of the quadratic potential energy terms is clearly visible. If ^^^^≈ ^^^^, the quartic potential energy term may become large compared with the quadratic potential energy term, thereby resulting in the high anharmonicity of the superconducting qubit corresponding to the above-assumed circuit model. In order to estimate the amount of the cancellation quantitatively for the total potential energy ^^, it should be noted that the potential energy of a phase-biased Josephson junction may be expanded into a Taylor series as where ^^^^, ^^(^^^^ ^^ ^^)denotes the ^^th Taylor series coefficient of the potential energy of a phase-biased Josephson junction. This allows one to measure the cancellation effect present in the total potential energy ^^ using the ratio where ^^^^,2(^^^^ ^^ ^^)denotes the 2nd order Taylor series coefficient of the potential energy of a phase-biased Josephson junction, and ^^ denotes the amount of the cancellation. Acancellation of at least 30% means that ^^ ≥ 0.3. If, for example, the phase bias ^^ ^^ ^^ ^^ =± ^^, then ^^ ^^,2 = − ^^ ^^ implying that In this case the requirement of ^^ ≥ 0.3 implies that the Josephson energy and the inductive energy must satisfy^^ ^^∈ [01^^^^.3,0.3]. In the superconducting circuit of the present invention, the non-linear inductive element is asymmetrically shunted by the linear inductive element such that a first impedance along a first ground path of the non-linear inductive element is different to the second impedance along a second ground path of the non-linear inducive energy element. This forms at least two modes of the superconducting circuit that couple to the non-linear inductive element. Above we have considered just a single mode of the superconducting circuit that arises if the Josephson junction is shunted with an ideal inductor and capacitor. However, in practice, a superconducting circuit made of a coplanar waveguide resonator has many modes. A full derivation of the multimode Hamiltonian can be found in the master’s thesis “Island-free superconducting qubit” by Eric Hyyppä, available at In particular, section 4.2 describes the derivation of the Hamiltonian shown in equation (4.38). Similar multimode Hamiltonians can be obtained in various different ways and optimized such that the cross-Kerr couplings are increased. Since each of these modes may be used to encode a different quantum state, a single superconducting circuit as described above may be used to encode multiple qubits. Encoding multiple qubits in a single superconducting circuit enables the application of native multi-qubit gates, i.e. operations on one or more qubit states that depend on the states of other qubits. It may also reduce the footprint of qubits on the chip, since multiple qubits occupy the same physical volume and also allowing integration of filters and other almost linear modes as well. Figure 1 is a schematic drawing of a superconducting qubit 100 according to a first embodiment of the present invention. The qubit 100 includes a non-linear inductive element 101 and a linear conductive element (formed by first section 102A and second section 102B). In the example depicted in Figure 1, the non-linear inductive element 101 is a Josephson junction and the linear inductive element 102a, 102b is the centre conductor of a grounded coplanar waveguide resonator, which also includes ground plane 103 surrounding the non-linear inductor 101, and where the centre conductor of the co- planar waveguide resonator forms the linear inductive element 102a, 102b. The superconducting circuit 100 may be formed on a suitable substrate, e.g. a silicon wafer. The ground plane and centre conductor are formed of a superconducting material (i.e. a material that exhibits superconductivity under suitable conditions such as low temperature such as aluminum, niobium, titanium, tantalum, and their alloys and nitrides). The centre conductor 102a, 102b is electrically connected to the ground plane 103 at each end of the centre conductor 102a, 102b. At the same time, the superconductor 106 is 10 separated by gaps from the superconducting ground plane on a second pair of opposite sides (i.e. top and bottom sides, as shown in FIG.1). In the published unimon qubit, a Josephson junction is positioned in the centre of the centre conductor of the coplanar waveguide resonator, i.e. such that the length of the centre conductor from the Josephson junction to ground plane is the same on each side of the Josephson junction. In contrast, in the qubit 100 of the embodiment depicted in Figure 1 invention, the non-linear inductive element 100 is displaced from the centre of the centre conductor, i.e. distances D1 and D2 are not equal. As such, the non-linear inductive element (i.e. Josephson junction 101) is asymmetrically shunted by the linear inductive element (i.e. centre conductor 102a, 102b). Therefore, the impedance (first impedance) along a first ground path (i.e. along first portion of the centre conductor 102a) of the non- linear inductive element is different to the impedance (second impedance) along a second ground path (i.e. along second portion of the centre conductor 102b) of the non-linear inducive energy element. Figure 2 shows three modes of the superconducting circuit shown in Figure 1, each of which is coupled to the Josephson junction 101 and therefore also to each other. The first mode is shown with a solid line, the second mode with a short-dashed line and the third mode with a long-dashed line. Note that if the junction was placed in the middle, one of the modes shown in Figure 2 would not couple to the Josephson junction, and hence could not be used as a qubit. The modes of the superconducting circuits described herein are particularly advantageous for encoding multiple qubits in a single circuit, enabling multi-qubit operations (i.e. quantum gates) to be performed natively, i.e. in a single physical operation on the circuit. In particular, the modes of the superconducting circuits exhibit cross-Kerr coupling, which causes changes in the transition frequencies between qubit states of one encoded qubit based on the states of one or more other encoded qubits. Thus, by selecting an appropriate frequency for a driving signal, state transitions for one qubit can be driven conditionally, based on the state of one or more other qubits. In particular, this coupling may lead to a change in the transition frequency of at least 1%. Changes of several percents and beyond can be possible depending on the chosen parameter values. In other embodiments of the invention, the non-linear inductive element is asymmetrically shunted by the linear inductive element in other ways, but the principle of asymmetric shunting, which results in a first impedance along a first ground path being different to a second impedance along a second ground path remains. For example, in Figure 3A, the circuit includes two non-linear inductive elements (e.g. Josephson junctions) 301a and 301b. Each non-linear inductive element is situated in the centre conductor of a grounded coplanar waveguide resonator 302, and the centre conductor is grounded between the non-linear inductive energy elements. This arrangement could alternatively be seen as two separate grounded coplanar waveguide resonators, each with one non-linear inductive element embedded in the centre conductor, and direct connected to each other at one end, which is also connected to ground. Figure 3B depicts a lumped-element model corresponding to the embodiment depicted in Figure 3A. Considering this model, the classical Hamiltonian of the circuit is based on the kinetic energy arising from the capacitors C1 and C2 is The potential energy arising from the Josephson junction and the linear inductance is The momenta conjugate to for ^^ ∈ {1,2} are respectively, where ℒ = ^^ − ^^ is the Lagrangian of the circuit. By defining the dimensionless charges as ^^^^= ^^^^ / ℏ, the Hamiltonian can be written as where the definition ^^^^ ^^= ^^2 / (2 ^^^^) has been used. By imposing the commutation relations Where ^^^^, ^^′denotes the Kronecker delta function, we obtain the quantum Hamiltonian This Hamiltonian demonstrates that the circuit shown in Figure 3B includes to modes (first row and second row of the equation) inductively coupled to a third mode (third row of the equation). Figure 4A depicts a further alternative arrangement, in which four non-linear inductive elements (e.g. Josephson junctions) are connected to either other in a ring configuration. Figure 4A depicts four non-linear inductive elements, but any number greater than or equal to two may be used. In one option, the non-linear inductive elements are connected to each other by linear inductive elements 404a-d, and each linear inductive element is connected to a ground plane 403 via conductive connections 402a-d. The linear inductive elements may be a coplanar waveguides or other linear inductors. In another option, the non-linear inductive elements are connected to each other by conductive elements 404a-d, and each conductive element is connected to a ground plane 403 by a linear inductive element 402a-d. In another option, the ground plane 403 is replaced by linear inductive elements or conductive elements. A lumped-element model of the configuration depicted in Figure 4A is shown in Figure 4B. Following a similar analysis as used above for Figure 3B, the kinetic energy arising from the capacitors is The potential energy arising from the Josephson junction and the inductors is The momenta conjugate to for ^^ ∈ {1,2,3} are Respectively, where ℒ = ^^ − ^^ is the Lagrangian of the circuit. Equivalently, where ^^∗= ^^1^^2^^3+ ^^2^^3^^4+ ^^1^^2^^4+ ^^1^^3^^4. By defining the dimensionless charges as ^^^^= ^^^^ / ℏ, the Hamiltonian can be written as By imposing the commutation relations The quantum Hamiltonian can be obtained This form demonstrates that the circuit approximately comprises three modes (first, second, and third rows) capacitively (fourth row) and inductively (fifth row) coupled to each other. Above, we considered the linear inductive elements to be either purely inductive elements or distributed elements in the sense that they also bear some capacitance to the ground potential or between other elements. It is also possible to construct several modes that couple to single non-linear inductive element only from linear purely inductive and purely capacitive elements. Note that in a practical realization, of course, any physical element, such as the purely inductive and purely capacitive elements discussed here, has both inductive and capacitive components although one of them may be extremely small. This type of circuit is shown in Figure 5, where the non-linear inductive element 501 is located in parallel with the linear purely inductive element 502a and linear purely capacitive element 502b, and in series with a number of linear purely inductive and capacitive elements such as 503a and 503b. Each series connection of the linear purely inductive and capacitive elements provides a new mode that couples to the non-linear inductive element. By choosing the inductances and capacitances in a way that the different mode frequencies are not very far from each other and that all purely linear inductive elements experience a major cancellation of their linear inductive energies, very strongly coupled modes can be implemented. Several circuits made from non-linear inductive elements and linear purely inductive and capacitive elements may be also coupled together by either non-linear or linear inductive of capacitive elements. Figure 6 illustrates this kind of a coupled ring, where the non-linear inductive element 601 is located in parallel with the linear purely inductive element 602a and linear purely capacitive element 602b, and coupled to adjacent similar system by the coupling element 602c.

Claims

Claims 1. A quantum processing unit comprising a superconducting circuit (100, 300, 400, 500, 600) and a phase-biasing element, the superconducting circuit comprising at least one non-linear inductive element (101, 301a-b, 401a-d, 501, 601) shunted by at least one linear inductive element (102a-b, 302, 402a-d, 502a-b, 602a-c), wherein: the non-linear inductive element is asymmetrically shunted by the linear inductive element such that a first impedance along a first path of the non-linear inductive element is different to a second impedance along a second path of the non-linear inductive energy element forming at least two modes of the superconducting circuit that couple to the non-linear inductive element; and the phase-biasing element is configured to bias a superconducting phase difference across the linear inductive element and the non-linear inductive element such that potential energy terms that are quadratic in phase of the linear inductive element and the nonlinear inductive element are at least partly cancelled by one another, and such that the frequencies of the said at least two modes of the superconducting circuit depend on the excitation of at least two modes.

2. The quantum processing unit of claim 1, wherein at least two modes of the superconducting circuit are suitable for use as qubit states.

3. The quantum processing unit of claim 1 or 2, wherein the quantum processing unit is configured to use at least two modes of the superconducting circuit as qubit states.

4. The quantum processing unit of claim 2 or 3, wherein the frequencies of each of the at least two modes of the superconducting circuit used as qubit states are within 50% of each other, or within 30%, or within 10%.

5. The quantum processing unit of any of claims 2 to 4, wherein the at least two modes of the superconducting circuit exhibit cross-Kerr coupling.

6. The quantum processing unit of claim 5, wherein the cross-Kerr coupling between the at least two modes of the superconducting circuit used as qubit states is at least 1% of the frequency of one of the modes.

7. The quantum processing unit of any preceding claim, wherein the phase-biasing element is configured to bias the superconducting phase difference such that the quadratic potential energy terms of the linear inductive element and the non-linear inductive element are cancelled by one another by at least 30%.

8. The quantum processing unit of any preceding claim, wherein the at least one non- linear inductive element comprises a non-linear inductive element situated in the centre conductor of a grounded coplanar waveguide resonator and offset from the centre of the centre conductor such that the first impedance measured along a first section of the centre conductor connected to a first terminal of the non-linear inductive element is different to the second impedance measured along a second section of the centre conductor connected to a second terminal of the non-linear inductive element.

9. The quantum processing unit of any preceding claim, wherein the non-linear inductive element is a Josephson junction.

10. The quantum processing unit of any of claims 1 to 6, wherein the at least one non- linear inductive element comprises two non-linear inductive elements, wherein each non-linear inductive element is situated in the centre conductor of a grounded coplanar waveguide resonator, and wherein the centre conductor is grounded between the non-linear inductive energy elements.

11. The quantum processing unit of any preceding claim, wherein the at least one non- linear inductive element comprises at least two non-linear inductive elements (401a-d), wherein the at least two non-linear inductive elements are connected in a ring, and wherein the at least two non-linear inductive elements are connected to each other by the same number of linear inductive elements (404a-d).

12. The quantum processing unit of claim 11, wherein each linear inductive element is connected to a ground plane (403).

13. The quantum processing unit of any of claims 1 to 10, wherein the at least one non-linear inductive element comprises at least two non-linear inductive elements (401a-d), wherein the at least two non-linear inductive elements are connected in a ring, and wherein the at least three non-linear inductive elements are connected to each other by the same number of conductor elements (404a-d).

14. The quantum processing unit of claim 13, wherein each conductor element is connected to a ground plane (403) by a linear inductive element (402a-d).

15. The quantum processing unit of any of claims 1 to 10, wherein the at least one non-linear inductive element comprises at least two non-linear inductive elements (401a-d), wherein the at least two non-linear inductive elements are connected in a ring, and wherein each non-linear inductive element is connected in parallel with a linear inductive element (602a) and linear capacitive element (602b).

16. The quantum processing unit of claim 13, wherein the ring is connected to ground.

17. The quantum processing unit of any of claims 11 to 16, wherein the linear inductive elements are coplanar waveguides.

18. The quantum processing unit of any of claims 1 to 10, wherein the at least one non-linear inductive element (501) is connected in parallel with a linear inductive element (502a) and linear capacitive element (502b), and in series with a number of linear inductive and capacitive elements (503a, 503b), such that each series connection of a linear inductive element and linear capacitive element provides a new mode that couples to the non-linear inductive element.

19. The quantum processing unit of any preceding claim, further comprising and second superconducting circuit and one or more coupling resonators and / or tuneable couplers for coupling the superconducting circuits.

20. A method of performing a multi-qubit gate on the quantum processing unit of any preceding claim, the method comprising driving the superconducting circuit at a first frequency, wherein the first frequency corresponds to a transition frequency of a first qubit when one or more second qubits are in the ground state or first excited state, such that the state of the first qubit is modified based on the state of the one or more second qubits.

21. The method of claim 20, wherein several driving frequencies, each chosen to drive a first qubit based on different values of one or more second qubits, are at least partially simultaneously applied.

22. The method of claims 20 and 21, wherein the frequency of at least one driving tone is tuned in time to optimize the fidelity of the desired quantum operation arising from the drive.

23. The quantum processing unit of any preceding claim, wherein different parts of the quantum processor are placed in different physical substrates.

24. The method of any of the preceding claim, wherein native multiqubit operations applied on the qubits is used to speed up the execution of quantum algorithms.

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