Quantum information system

Abstract

A quantum information element 10 comprising a first element 18 and a second element 20. The first element comprising a first superconducting non-linear inductor (e.g. Josephson junction) provided betw

Description

TECHNICAL FIELD The present application relates to a method and system for processing of quantum information. BACKGROUND A quantum computer is a computer that stores and processes information in the form of quantum bits (or “qubits”). A quantum computer can take advantage of quantum mechanical phenomena to perform computational processes that are not possible on a classical computer, including performing encoding and processing of information in quantum states to achieve faster processing times than may be possible in a classical computer. To achieve a practical quantum computer, a qubit must be implemented in a physical fashion and information encoded onto the qubit in a manner that is stable. Instabilities in the encoded quantum state can give rise to errors in the encoded information and thus in any process or readout performed on the encoded quantum information. Quantum error detection techniques are thus desirable to identify any errors in encoded information from decoherence or noise. BRIEF DESCRIPTION OF THE DRAWINGS Embodiments will now be described by way of example with reference to the accompanying drawings. Figure 1 schematically illustrates a quantum information element. Figure 2A - 2C illustrates three-wave mixing elements comprising a flux-biased Josephson junction. Figures 3A - 3C illustrate example quantum information elements. Figure 4A illustrates a further example quantum information element. Figure 4B schematically illustrates an equivalent circuit diagram for the quantum information element of Figure 4A. Figures 5 and 6 each illustrate the energy level diagram for a quantum information element. Figure 7 illustrates a quantum information system comprising a quantum information element. Figures 8A and 8B each illustrate a method of quantum information processing for a quantum information element. Figure 8C illustrates an energy level diagram for a coupled quantum information element and resonator. Figure 8D illustrates a method for quantum information processing for a coupled quantum information element and resonator. Figures 9A - 9C each illustrate a first quantum information element coupled to a second quantum information element. Figures 10 and 11 illustrate energy level diagrams for a first quantum information element coupled to a second quantum information element. Figure 12 illustrates a quantum information system comprising a first quantum information element coupled to a second quantum information element. DETAILED DESCRIPTION In a first aspect, there is described a quantum information element comprising: a first element and a second element; the first element comprising a first superconducting non-linear inductor provided between two superconducting islands; and the second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements. A three-wave mixing element is an element that introduces an asymmetry into the Hamiltonian, thereby enabling three-wave mixing of modes within the quantum information element. In a three-wave mixing process, three mode operators may be coupled allowing for transitions between different modes of the quantum information element to be directly addressed. The quantum information element described above provides accessible quantum states with an improved degree of control between the states. When two superconducting islands are coupled together by a superconducting non-linear inductor, or three-wave mixing element, the superconducting islands may be populated with charge carriers, resulting in a quantum state. For example, the superconducting non-linear inductor may be a Josephson junction connected between the superconducting islands (and each element a Josephson element), and the three-wave mixing element may comprise a Josephson junction. Embodiments herein are described in connection with reference to Josephson junctions, but other superconducting non-linear inductors may be used to provide accessible quantum states within the quantum information element. The quantum information element provides accessible quantum states including a pair of states where relaxation between the states has a low probability but a transition between the pair of states is still directly addressable, due to a three-wave mixing term in the Hamiltonian. The three-wave mixing term may be a result of a flux bias or current bias applied to the superconducting non-linear inductor. Such a pair of states within the accessible quantum states may be chosen as a quantum computational subspace, where bit flip errors between states of the subspace are minimised but transitions between states are still accessible. Thus, the quantum information element allows for quantum state encoding and state processing that is simplified while providing increased lifetime of the encoded states. The state energy levels of the quantum information element correspond to excitations above a ground state. Within the accessible energy levels there exist multiple singleexcitation transitions with different transition probabilities between the energy levels. A single-excitation is understood to be a single photon excitation, being the excitation between two states as the result of a single photon interaction with the quantum information element. Due to the shared superconducting island of the quantum information element, the excitation modes of the quantum information element are strongly hybridised, resulting in two different modes of the system, and single excitations of the quantum information element may correspond to single excitations of each mode of the quantum information element. Due to the coupling of two superconducting islands by a Josephson junction that is configured to be biased (for example, by a DC bias current resulting from a flux bias applied to a loop containing the Josephson junction), the symmetry of the Josephson effect between the islands is broken, resulting in a three-wave mixing term in the Hamiltonian. The three-wave mixing term allows for transitions between modes of the quantum information element to be directly addressed in a single photon excitation as sideband transitions of the single mode excitations of the quantum information element. Thus, transitions between state energy levels can be driven to perform single qubit gates in a manner that is quicker, easier and more accurate than in a system of superconducting islands coupled only with Josephson elements in which the phase symmetry is preserved. For example, the accessible energy levels of the quantum information element may comprise a first state, a second state, and a third state, wherein there is a singleexcitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency. The first state and second state may be respectively chosen as a first state and a second state of a computational subspace for the encoding of quantum information. Transitions between the two states of the computational subspace may be directly driven, by a control line, via the three-wave mixing term that couples the two states of the computational subspace. However, the probability of spontaneous decay between the states of the computational subspace via the three-wave mixing term is lower than the probability of spontaneous decay between the first state and the third state or the second state and the third state. Thus, the computational subspace provided by the quantum information elements provided herein allow for quantum information to be encoded and manipulated more accurately with minimum loss of coherence. In some examples, the two states of the computational subspace correspond to the two states having the two lowest energy excitations of the quantum information element from the ground state, which each correspond to a single-excitation from the ground state. However, other computational bases are possible within the energy level diagram. The single-excitations of modes within the quantum information element correspond to a creation of dipole or quadrupole moments across the superconducting islands of the Josephson elements. The first excited state corresponds to in-phase / parallel dipole excitations across the Josephson junctions of the first and second Josephson elements and the second excited state corresponds to antiparallel / out-of-phase excitations across the Josephson junctions of the first and second Josephson elements. This means that any longitudinal relaxation of an excited state within the quantum information element will not manifest as a bit flip error within the computational subspace, but can instead be identified as the ground state being occupied. The qubits determined to be in the ground state can thus be identified as in error, and discarded from any calculation. The possibility of three-wave mixing allows for flux noise to be introduced into the circuit. However, due to the selection of the excited states of the quantum information element as a computational subspace, the flux noise sensitivity of the quantum information elements described is substantially reduced. Thus, the three-wave mixing element provides improvements to the flexibility and simplicity of the quantum information system with minimal effect upon flux noise sensitivity. In an embodiment, the three-wave mixing element comprises a second superconducting non-linear inductor and a superconducting loop, the superconducting loop comprising the second superconducting non-linear inductor, the superconducting loop being configured to receive a bias flux through the superconducting loop. In an embodiment, the first superconducting non-linear inductor is a first Josephson junction and the second superconducting non-linear inductor is a second Josephson junction. In an embodiment, the superconducting loop further comprises at least two third Josephson junctions, wherein the at least two third Josephson junctions are connected in series between the superconducting islands of the second element, and in parallel with the second Josephson junction. In an embodiment, the superconducting loop further comprises an inductor connected between the superconducting islands of the second element, and in parallel with the second Josephson junction. In an embodiment, the three-wave mixing element comprises a second superconducting non-linear inductor and wherein the superconducting junction is configured to be current-biased by a current source. In an embodiment, the three-wave mixing element comprises a conductive loop, the conductive loop comprising the second superconducting non-linear inductor and the current source, wherein the current source is connected in parallel to the second superconducting non-linear inductor and wherein the conductive loop is configured to receive a bias flux through the conductive loop, wherein, optionally, the conductive loop is a superconducting loop. In an embodiment, the quantum information element comprises a split superconducting electrode arranged coaxially with a further superconducting electrode, the split superconducting electrode forming one of the superconducting islands of each of the first element and second element, the further superconducting electrode forming the superconducting island which is common to both the first and second elements. In a further aspect, there is described a quantum information system comprising a quantum information element as described above and a control line, wherein the superconducting islands of the first element are coupled to the superconducting islands of the second element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency, wherein the control line is configured to encode into the quantum information element one of two states of a computational subspace, the first state of the computational subspace corresponding to the first state and the second state of the computational subspace corresponding to the second state; and wherein the control line is configured to address the quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency. wherein, optionally, the control line is arranged out of plane with respect to the quantum information element. In an embodiment, the quantum information system further comprises a readout element configured to measure the quantum state of the quantum information element wherein, optionally, the readout element is arranged out of plane with respect to the quantum information element. In a further aspect, there is described a quantum information system comprising a quantum information element as described above and a readout element, the readout element comprising a resonator and a readout line, wherein the quantum information element is coupled to the resonator such that the coupled resonator element and quantum information element together comprise a plurality of states comprising a first state, a second state, a third state and a fourth state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency, a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency, and a singleexcitation transition of a mode of the resonator between the second state and the fourth state at a third frequency, wherein the first state corresponds to a first state of a computational subspace of the quantum information element, and the second state corresponds to a second state of the computational subspace; and wherein the readout line is configured to perform an initialization operation to encode into the quantum information element one of the first state and second state of the computational subspace, the initialization operation comprising addressing the quantum information element at an initialization frequency, the initialization frequency being the sum of the second frequency and the third frequency. In a further aspect, there is described a quantum information system comprising a first quantum information element and a second quantum information element coupled to the first quantum information element, wherein the first quantum information element is a quantum information element as described above, and wherein the second quantum information element comprises a first element, the first element comprising a first superconducting non-linear inductor provided between two superconducting islands. In an embodiment, the second quantum information element further comprises a second element, the second element comprising a second superconducting non-linear inductor provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements. In an embodiment, the second quantum information element further comprises a second element, the second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements. In an embodiment, the quantum information system further comprises a first control line and a second control line, wherein the first control line is configured to address the first quantum information element and the second control line is configured to address the second quantum information element, wherein, optionally, the quantum information system further comprises a first readout element configured to measure the quantum state of the first quantum information element and a second readout element configured to measure the quantum state of the second quantum information element. In an embodiment, the first control line is configured to address the first quantum information element at a cross resonance frequency on resonance with the second quantum information element. In an embodiment, the first quantum information element and the second quantum information element are coupled together such that the first quantum information element and the second quantum information element together comprise plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the first quantum information element between the third state and the first state at a first frequency, and a single-excitation transition of a second mode of the first quantum information element between the third state and the second state at a second frequency, and wherein the first control line is configured to address the first quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency. In an embodiment, the first quantum information element and the second quantum information element are coupled together such that the first quantum information element and the second quantum information element together comprise plurality of states comprising a first state, a second state, a third state, a fourth state, a fifth state, a sixth state, a seventh state and an eighth state, wherein there is: a single-excitation transition of a first mode of the first quantum information element between the fifth state and the first state and a first frequency; a single-excitation transition of a second mode of the first quantum information element between the fifth state and the third state at a second frequency; a single-excitation transition of a first mode of the second quantum information element between the sixth state and the first state at a third frequency; a single-excitation transition of a second mode of the second quantum information element between the sixth state and the second state at a fourth frequency; a single-excitation transition of the first mode of the first quantum information element between the seventh state and the second state at a fifth frequency; a single-excitation transition of the second mode of the first quantum information element between the seventh state and the fourth state at a sixth frequency; a single-excitation transition of the first mode of the second quantum information element between the eighth state and the third state at a seventh frequency; a single-excitation transition of the second mode of the second quantum information element between the eighth state and the fourth state at an eighth frequency; wherein the first control line and second control line are configured to encode into the quantum information element one of four states of a computational subspace, the first state of the computational subspace corresponding to the first state, the second state of the computational subspace corresponding to the second state, the third state of the computational subspace corresponding to the third state, and the fourth state of the computational subspace corresponding to the fourth state; wherein the control line is configured to address the first quantum information element at a first cross-resonance frequency, the cross-resonance frequency being the difference between the third frequency and the fourth frequency or being the difference between the seventh frequency and the eighth frequency. In a further aspect, there is described a method of quantum information processing in a quantum information element, the quantum information element comprising a first element and a second element; the first element comprising a first superconducting non-linear inductor provided between two superconducting islands; and the second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements; wherein the superconducting islands of the first element are coupled to the superconducting islands of the second element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency, the method comprising: encoding into the quantum information element one of two states of a computational subspace, the first state of the computational subspace corresponding to the first state and the second state of the computational subspace corresponding to the second state; and addressing the quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency. In a further aspect, there is described a computer program, the computer program comprising instructions that, when executed by a processor, cause the processor perform a method of quantum information processing in a quantum information element, the quantum information element comprising a first element and a second element; the first element comprising a first superconducting non-linear inductor provided between two superconducting islands; and the second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements; wherein the superconducting islands of the first element are coupled to the superconducting islands of the second element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency, the method comprising: encoding into the quantum information element one of two states of a computational subspace, the first state of the computational subspace corresponding to the first state and the second state of the computational subspace corresponding to the second state; and addressing the quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency. Figure 1 schematically illustrates a quantum information element 10 configured to store a quantum state. The quantum information element 10 comprises a first element and a second element. The first element comprises a first Josephson junction 18 provided between two superconducting islands 12 and 16, and the second element comprises an element 20 (also called a mixing element or a three-wave mixing element) provided between two superconducting islands 14 and 16. The first element can be considered a Josephson element. Furthermore, each pair of superconducting islands within each element are capacitively coupled together. As shown in Figure 1, the first element comprises first superconducting island (also called a superconducting electrode) 12 and a third superconducting island 16, and the second element comprises a second superconducting island 14 and the third superconducting island 16. Different examples of quantum information element 10 are shown in Figures 3A - 4B, and will be described later. While quantum information element 10 illustrates superconducting junction 18 connecting the first and third superconducting islands, it is understood that alternative superconducting non-linear inductors may be used in the place of Josephson junction 18. The three wave mixing element 20 may comprise a superconducting non-linear inductor, with such a superconducting junction providing a kinetic inductance. Kinetic inductance is an inductance where a change in potential results from a movement of charge carriers (such as Cooper pairs within a superconductor), and is a measure of kinetic energy stored within the charge carriers that move through the junction. A Josephson junction is an example of a superconducting non-linear inductor, where kinetic inductance results from tunnelling of Cooper pairs across the junction boundary. In some embodiments the superconducting junction of the three-wave mixing element is a second Josephson junction 22 (such that the three-wave mixing element comprises a Josephson junction, which may be configured to receive a bias current). When the three-wave mixing element 20 includes a second Josephson junction 22, the second element can be considered a second Josephson element. Another example of a superconducting non-linear inductor may be a superconducting nanowire fabricated such that there exist a limited number of tunnelling channels. The superconducting non-linear inductor of the three-wave mixing element is configured such that the potential of the superconducting non-linear inductor is asymmetric. For example, the superconducting non-linear inductor is configured to receive a bias current to bias the superconducting non-linear inductor. In some embodiments described herein, the superconducting non-linear inductor is provided in the path of a circuit loop arranged between the superconducting islands, and the bias current is in the form of a DC-current bias that is a persistent current loop that results from a flux bias provided through the loop. Different examples of three-wave mixing element 20 are shown in Figures 2A - 2C, wherein in each of the three-wave mixing elements 20 of Figures 2A - 2C, the three-wave mixing element comprises a loop 26. Each three-wave mixing element is shown as connected between two terminals, where each terminal is configured to be a connected to a respective superconductive island of the superconducting islands 14 and 16 above. In each of the three-wave mixing elements 20 of Figures 2A - 2C, the three-wave mixing element comprises a conductive loop 26. The second Josephson junction 22 is located in the path of the loop 26, and the loop 26 configured to receive a bias flux 0b threaded through the loop to induce a current loop Ib in the loop 26. The current loop Ib thus a current bias provided to the Josephson junction 22. The bias flux 0b may be provided to the circuit by an inductive element, such as a coil, arranged proximate (e.g. adjacent) to the superconducting loop. For example, the inductive element may be formed on the same substrate as the quantum information element 10, or may be separate to the quantum information element and arranged out of plane of the quantum information element. The inductive element may be tuned by a control system (e.g. a control system comprising processor 50 of Figure 7, to be described later) to adjust the value of the bias flux 0b (e.g. by adjusting the current supplied to the inductive element). Thus, the control system can tune the extent of the three-wave mixing and thus the properties of the quantum information element 10. While the presence of a loop may increase the sensitivity of the quantum information element to flux variations (e.g. from the environment), the quantum information elements described herein have an inherently low sensitivity to flux noise due to the selection of the computational subspace in which bit flip errors have low probability of occurring. In some embodiments, and as shown in Figure 2A, the loop 26 further comprises an inductor 27 connected between the superconducting islands of the second Josephson element, and in parallel with the second Josephson junction 22. The inductor 27 is of a different type to the superconducting non-linear inductor of the three-wave mixing element. For example, the inductor may provide a geometric inductance, which is an inductance resulting from the geometry of the inductor (e.g. a coil inductor, where energy is stored in the magnetic field). For example, the inductor may be a linear inductor. In some embodiments, and as shown in Figure 2B, the loop 26 further comprises at least two third Josephson junctions 28i - 28n, wherein the at least two third Josephson junctions 28i - 28n are connected in series between the terminals of the three wave mixing element (and thus between the superconducting islands of the second element), and the series-connected third Josephson junctions 28i - 28n are connected in parallel with the second Josephson junction 22. In general, as discussed above, the three-wave mixing elements provide an asymmetric effective potential for phase between the two superconducting islands. When using a superconducting loop between two terminals as shown in Figure 2A -2C, the second Josephson junction 22 is provided in a first branch of the superconducting loop and will provide a potential proportional to cos^. The total effective potential for phase between the superconducting islands is asymmetric when the potential for phase across the second branch is different from the potential for phase across the first branch, such as in the embodiments of Figures 2A - 2C. For example, Figure 2A provides a potential proportional to ip2 and Figure 2B provides a potential proportional to cos(i(j / N), where N is at least two. When there is no element in second branch, a different potential may still result (e.g. from intrinsic inductance in the wire of the branch) thereby providing the asymmetry). In alternative embodiments to Figures 2A and 2B (not shown), the loop may comprise at least one third Josephson junction connected in series with an inductor 27, wherein the series-connected third Josephson junction and inductor 27 are connected in parallel with the second Josephson junction 22. In some embodiments, as shown in Figure 2C, the loop comprises a current source 24 such that the current source provides a bias current lA across the second Josephson junction 22. As shown in Figure 2C, the current source 24 may be coupled across the second Josephson junction 22 (e.g. coupled to the terminals of the Josephson junction or coupled across the two superconducting islands of the second Josephson element). The bias current lA and bias current Ib combine to bias the Josephson junction 22. In the embodiments of Figures 2A, 2B and 2C, the conductive loop 26 may be a superconducting loop to minimise resistance and provide a stable biasing current in response to the flux bias. The embodiments of Figures 2A - 2C represent alternative configurations of mixing element comprising a Josephson junction 22 configured such that the Josephson junction is biased via a bias flux. However, it is to be understood that alternative superconducting non-linear inductors may instead be deployed in each case. Furthermore, an alternative superconducting non-linear inductor may also be deployed in place of each of Josephson junctions 28i to 28n. In an alternative embodiment, the superconducting non-linear inductor (e.g. second Josephson junction) may be configured to be current biased only by receiving a current from a current source. In these embodiments, the second Josephson junction 22 of the three-wave mixing element 20 is coupled to a current source such that the current source provides a bias current Ib across the second Josephson junction 22. The current source may be coupled across the second Josephson junction 22 (e.g. coupled to the terminals of the Josephson junction or coupled across the two superconducting islands of the second Josephson element). The current source 24 may form part of the three-wave mixing element 20 (and therefore the quantum information element 10). For example, the current source 24 may be fabricated onto the same surface substrate as the quantum information element 10. Alternatively, the current source 24 may be an external current source coupled to the second Josephson junction 22 of the quantum information element 10. The current source, being coupled to the Josephson junction 22, may form a conductive loop in a manner similar to Figure 2C. Thus, examples of this alternative embodiment may be considered topologically equivalent to Figure 2C where no flux bias is applied. While the biasing of the second Josephson junction 22 by a current source may increase the sensitivity of the quantum information element to current variations (e.g. from the environment), the quantum information elements 10 described herein have an inherently low sensitivity to current noise due to the selection of the computational subspace in which bit flip errors have low probability of occurring In each case described above, the mixing element 20 is configured to provide a bias current across the Josephson junction (either by means of a flux bias to a loop or by a current bias from a coupled current source). Thus, in each case, the three-wave mixing element is an element configured to provide a three-wave mixing term in the Hamiltonian of the quantum information system. This results from the effect of a current bias to the Josephson junction, which modifies the Josephson equation from a function in Josephson phase that is symmetric (i.e. an even function) to a function in phase that is asymmetric (i.e. an odd function). This effect can be understood from considering a Josephson junction for which there is no bias current applied. In this case, the superconducting current across the junction follows Equation 1: I = Ic sin(^) (Equation 1) For all currents I <Ic. Ic is the critical current of the Josephson junction, and xp is the Josephson phase (being the phase difference of the Ginzburg-Landau order parameters of each superconductors connected by the junction). The corresponding energy of the Josephson junction is E(^) = -E;cos(^), where the Josephson Energy Ej is E, = 1^(0)1 = and is a parameter of the Josephson junction. The energy of the Josephson junction is thus symmetric. In one example of a Josephson junction being provided with a bias current, the superconducting current across the junction follows Equation 2: I = Ic sin(^) + IB (Equation 2) Where IB is the bias current. The corresponding Energy across the current-biased Josephson junction is E(xp) = -E;cos(^) += -^^cosG / 0 - Thus, the energy stored in the Josephson junction is an odd function of the Josephson phase (and thus asymmetric in Josephson phase), meaning the symmetry of the system has been broken by the current bias. The above example is provided for explanatory purposes to demonstrate the breaking of symmetry upon application of a general current bias to a Josephson junction. In general, in a physical system deploying a Josephson junction configured to have an asymmetric potential (including an asymmetric potential introduced by providing a current bias), for small deviations 8 from the Josephson phase around a phase ip0 at the minimum energy of the Josephson Junction (i.e. 8 = ip - V>o). the Josephson energy of the current-biased Josephson junction can be expanded as: E(^) = D282 + D383 + D484 + 0(55) Where the cubic term 53 results from the asymmetric phase relation in the Josephson energy. The cubic term, when rewritten in terms of bosonic creation and annihilation operators for a single mode of across a pair of superconducting islands 3+ and a, has the form V'3 = + «)3- As will be explained below, when coupled to a second pair of superconducting islands, such as in the quantum information element 10 of Figure 1, and addressed by a drive field of electromagnetic radiation (e.g. a microwave field), the three-wave mixing term results in transitions between single-mode energy levels across two pairs of superconducting islands (e.g. sideband transitions). The same principle applies to other superconducting non-linear inductors; the current biasing of the junction results in a three-wave mixing term and enables sideband transitions to single mode excitations of the quantum information element 10. The above described three-wave mixing elements of Figures 2A, 2B and 2C are example implementations, and it is to be understood that the quantum information elements 10 described herein may include alternative three-wave mixing elements. When a Josephson junction is deployed as the superconducting non-linear inductor in the three-wave mixing element 20, the three-wave mixing element 20 is understood to be an element that is configured to coupled two superconducting islands together in a manner that provides a Josephson effect that is asymmetric with respect to the Josephson phase (i.e. is an asymmetric function of Josephson phase). Figures 3A - 3C illustrate different configurations of the quantum information element 10 of Figure 1, in accordance with embodiments as described herein. Each of the quantum information elements 10 of Figures 3A - 3C may include any one of the mixing elements described herein. Figure 3A illustrates one example configuration of quantum information element, in which the first superconducting island 12 includes an electrode and multiple subelectrodes. The multiple sub-electrodes extend from one side of the electrode. The second superconducting island 14 includes an electrode and multiple sub-electrodes, where the multiple sub-electrodes extend from one side of the electrode. The configuration of the first superconducting island 12 may mirror the configuration of the second superconducting island 14 such that the sub electrodes of each of the first and second superconducting islands face one another. The third superconducting island 16 includes an electrode, a first plurality of sub-electrodes and a second plurality of subelectrodes. The third superconducting island 16 is located between the first and second superconducting islands, with the first plurality of sub-electrodes extending toward the first superconducting island 12 and located inbetween the sub-electrodes of the first superconducting island, and with the second plurality of sub-electrodes extending toward the second superconducting island 14 and located inbetween the subelectrodes of the second superconducting island 14. For example, the sub-electrodes of the third superconducting island 16 may interlock with the sub-electrodes of the first and second superconducting islands 12 and 14. The quantum information element 10 of Figure 1A is thus symmetric along its length, with identical distances between the first superconducting island 12 and third superconducting island 16 and between the second superconducting island 14 and third superconducting island 16. Figures 3B and 3C illustrate alternative configurations of superconducting islands to form quantum information element 10. In these examples, the quantum information element comprises two superconducting electrode patterns formed coaxially, where one superconducting electrode pattern forms the third superconducting island and the second electrode pattern is split to form the first and second superconducting islands. In the embodiment of Figure 3B, the split electrode pattern is formed in the centre of the element with the third superconducting island being a continuous ring surrounding the first and second superconducting islands. Each of the split electrodes is illustrated with a hemispherical cross-section, for example a half circle split by a distance, but other cross-sections are possible. In the embodiment of Figure 3C, the third superconducting pattern is placed at the centre of the coaxial arrangement with the first and second electrode patterns being formed from a split ring of superconducting material surrounding the third superconducting electrode, to form two symmetric arcs of identical radius centred on the third superconducting island. In the embodiment of Figure 3B the diameter of the outer ring is about 1mm, and the distance between the inner islands is about 250pm. In Figures 3A - 3C, the Josephson junction inductance is about 10nH, corresponding to a Josephson energy of about 15-20 GHz. Alternative configurations to 3B and 3C are possible. For example, the Josephson junction 18 may be located between the first superconducting island 12 and second superconducting island 14 instead of between the first superconducting island 12 and third superconducting island 16. Alternatively, the three-wave mixing element may be located between the first superconducting island 12 and second superconducting island 14 instead of between the second superconducting island 14 and third superconducting island 16. Figure 4A shows one example of a quantum information element 10 utilising a three-wave mixing element 20 (being the quantum information element 10 of Figure 3B that includes a three-wave mixing element of Figure 2B, in which N = 2). An equivalent circuit diagram is shown in Figure 4B. The use of the inner split-electrode configuration within the quantum information element reduces cross-talk between the quantum information element and any coupled neighbouring information elements (such as in embodiments described below). In each of the above examples of quantum information element 10, the capacitive coupling between each pair of islands is due to a spatial separation between the islands of each element, and the capacitance of the elements coupling the superconducting islands together (i.e. the first Josephson junction 18 and the three-wave mixing element 20 including the second Josephson junction). For example, referring to Figures 4A and 4B, the first Josephson element includes a first Josephson junction J1 connecting the first superconducting island 12 to the third superconducting island 16, and the first superconducting island 12 and the third superconducting island 16 are capacitively coupled by a capacitance C1. The second Josephson element includes a second Josephson junction J2, a first third Josephson Junction J3-1 and a second third Josephson junction J3-2 connecting the second superconducting island 14 to the third superconducting island 16, and the second superconducting island 14 and the third superconducting island 16 are capacitively coupled by a capacitance C2. The first Josephson junction J1 has a Josephson energy Eji, the second Josephson junction J2 has a Josephson energy Ej2, the first third Josephson junction J3-1 has a Josephson energy Ej3-i and the second third Josephson junction J3-2 has a Josephson energy Ej3-2. Furthermore, the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson element via a coupling between the first superconducting island 12 and the second superconducting island 14. As shown in Figure 4B, the coupling between the first and second Josephson elements is a capacitive coupling of capacitance Cm between the first superconducting island 12 and the second superconducting island 14, due to a physical separation of the two superconducting islands. The two Josephson elements with the capacitive coupling between the two respective capacitive islands give rise to two weakly anharmonic quantum oscillator modes ( / degrees of freedom). In some implementations the Josephson elements are fabricated such that EJ / Ec»1, to operate in the charge-insensitive regime (where EJ is the Josephson energy of the Josephson Junction in the Josephson element and Ec is the charging energy of the islands in the Josephson element). In an excited state of the quantum information element, an electric field oscillates within each Josephson element to provide a dipole moment between each superconducting island of the Josephson element. Harmonics of the dipole moment of each Josephson element are accessible in higher order excitations of the quantum information element. The large capacitive coupling between the two modes, resulting from the shared superconducting island, results in a strong hybridisation of these two modes, and in the appearance of two new normal modes of the system. In the absence of three wave mixing (e.g. when there is no symmetry breaking bias such as when the flux bias provided to the superconducting loop is zero), the excited states of the quantum information element exist as a plurality of normal modes of the system (being eigenmodes of the quantum information element Hamiltonian), in which an oscillating electric field is present with each Josephson element, across the Josephson junction. A normal mode corresponds either to an in-phase addition of the dipole moment modes of the Josephson elements, or to an out-of-phase addition of the dipole moment modes of the Josephson elements. Out-of-phase addition of dipole moments results in a dipole moment (a A-mode) across the quantum information element as a lower frequency mode, and in-phase addition of dipole moments results in a quadrupole moment (a Z-mode) across the quantum information element as a higher frequency mode. Figure 5 illustrates an energy level diagram of the quantum information element 10, where the states I1 / 7) of the system are labelled in an \nm) notation, where n indicates the number of A-mode excitations and m indicates the number of Z-mode excitations. Thus the accessible energy states are a series of modes that are separated by single mode excitations of either a normal mode of a dipole moment across the quantum information element or a normal mode of a quadrupole moment across the quantum information element. There is a splitting present between the first two normal modes (|01) and 110)) shown in Figure 5, which correspond to the addition of the dipole moments in parallel or anti-parallel to form a A-mode and a Z-mode respectively. The two states 110) and |01) represent a single-excitation manifold of the system of two normal modes. Higher excitation levels within the system also exist, and described by higher integer A-mode and Z-mode excitations (for example |02), |11) and |02)), as shown in Figure 2. A single-excitation of a mode of the quantum information element corresponds to increasing n by 1, or increasing m by 1. In this example of a quantum information element with no three-wave mixing, there is no permitted single photon excitation between the two states of the single-excitation manifold. The [nm) state energy levels are anharmonic, and addressable by photons of frequencies corresponding to the transition frequencies. The transition between |00) and 101) has a frequency of wi, the transition between 100) and 110) has a frequency of W2, the transition between |01) and |11) has a frequency of W3 and the transition between 110) and 111) has a frequency of W4, where wi # W2 and where W3 # W4. The quantum information element 10 thus provides an energy level structure including a ground state, a first excited state and a second excited state different to the first excited state. The above energy level diagram is described in the context of the quantum information element comprising Josephson junctions, but the same mode couplings arise for other forms of superconducting non-linear inductor used (i.e. the accessible energy states of the quantum information element 10 correspond to A-mode and Z-mode excitations of the quantum information element). When a three-wave mixing term is introduced (e.g. the flux bias to the superconducting loop is non-zero), the Hamiltonian is perturbed by a small amount. However, the excitations within the system still predominantly manifest as the A-mode and Z-mode excitations of the quantum information element, and the excitations of the quantum information element 10 with three-wave mixing are still in the form of two different modes, being normal modes corresponding to either to an in-phase addition of the dipole moment modes of the Josephson elements, or to an out-of-phase addition of the dipole moment modes of the Josephson elements. Thus, when a three-wave mixing element is used, there exist transitions between energy levels of the quantum information element 10 corresponding to excitations of either a first mode (e.g. a Z-mode) or a second mode (e.g. A-mode), the first mode and second mode corresponding to eigenmodes of the quantum information element when the bias flux is absent or zero. Without the three-wave mixing term, there is no single photon excitation between the energy levels of the quantum information element 10 corresponding to excitations of either the first mode or the second mode (e.g. A-mode). The presence of the three-wave mixing element enables coupling between the first and second modes of the quantum information element via the cubic phase term in the Hamiltonian. Thus, there is a coupling between energy levels of the single mode excitation manifold in the accessible energy states (e.g. between the two states |10) and 101) of Figure 5). This coupling permits the driving of a single excitation sideband transition between the states of the single-mode excitation manifold by incident electromagnetic radiation, such as a microwave pulse. A transition between states of the computational subspace corresponds to a single change in number of each of the two single modes - e.g. such that An = +1 and Am = -1, or An = -1 and Am = +1, where An + Am = 0. However, there is no single excitation transition of a single mode between the states of the computational subspace (there is no transition within the computational subspace such that either An = ±1, with Am = 0, or Am = ±1, with An = 0). As a result, relaxations between the two modes of the computational subspace has a very low probability (as compared to the single-excitation transitions of a single mode, such as transitions from the computational subspace to the ground state). As discussed above, the three-wave mixing originates from a cubic term of the potential < / >3 = ¢0(^ + 0)3. This term, when expanded, results in terms that are proportion to three operators (for example, a term proportional to a^aa). When the system is driven by an external field, the drive operator may be represented as D = n(ael"dt + a^e~ia)dt) Where Q is the strength of the drive and Wd is the drive frequency. The effect of this drive caused on the qubit mode a is a translation a -> a + 8eia>dt, where £ ~£l / (tod -Mq), such that the magnitude of the translation is proportional to the strength of the drive but inversely proportional to the detuning of the drive frequency from the qubit transition frequency a)q. Thus, the three-wave mixing term will be affected by the shift due to the drive: a^aa^ (a* + £e^l"^)(a + 8eia)dtXa + 8eia>dt) Which, when expanded, will result in terms that are linear in 8, such as the term a^a8e~i0idt. In the situation, such as in quantum information element 10, when a qubit mode a is coupled to another mode b (e.g. a A-mode coupled to a Z-mode), through an exchange interaction Vg = g^b + ab^), the operators a transform as: , 9c ,+ a -> a -I--b' ^TWM Where a>TWM is the difference in frequency between the two modes a and b (i.e. = (^b ~ "a)). and gc is a coupling value. Substituting this operator transformation into the linear term mentioned above: a^a8e~ia>dt -> (a -I———b^)(a -I———b^) 8e~ia>dt ^TWM ^TWM Which, when expanded, produces a term that is proportional to a'1 b 8e~ia>dt. Considering also that, in the Heisenberg picture, the a(a[) operators will evolve with the qubit frequency (i.e. a1' evolves as b evolves as Thus, the drive component of the Hamiltonian includes a driving term that is proportional to a^b . This term has a resonant condition when the drive pulse frequency matches the frequency of separation of the two coupled modes. As such, excitations between the two modes can be directly driven by a drive pulse at the three wave mixing frequency a>TWM to effect transfers between two normal mode states of the quantum information element. An addressable three-wave mixing transition exists between any two states that are each separated from a same third state by a single mode excitation. The addressable sideband transition is at a frequency that is either a sum of the frequencies of the two single mode excitations or a difference in the frequencies of the two single mode excitations. These three-wave mixing transitions can be considered as sideband transitions (when the two single mode excitations include an excitation of a first mode and an excitation of a second mode, such as between states 110> and 101», or can be considered as same-mode transitions (when the two single mode excitations include a first excitation of one mode and a second excitation of the same mode, such as between states |00> and |02». The quantum information element 10 thus provides a plurality of accessible states, with transitions between single mode excitation manifold states accessible by addressing the quantum information element 10 with electromagnetic radiation (e.g. a light pulse such as a microwave pulse). Quantum information element 10 can thus be understood as a multi-mode transmon, multi-mode qubit, or a dimon with accessible three-wave mixing transitions (e.g. sideband transitions) mediated by a cubic interaction term (and thus labelled as a cubic multi-mode transmon, a cubic multi-mode qubit or a cubic dimon). In some embodiments described herein, a two-mode computational subspace is created from the states of the quantum information element 10. For example, the two states of the computational subspace may correspond to the two lowest energy excited states. As shown in Figures 5 and 6, this example computational subspace may be formed as follows: |0)L= |10) |1>L = |01> This computational subspace is an example only, and other states may be used based on the |01> and |10> states of the described quantum information element 10. For example: |0>L = ^[|01> + |10)] |1)l = ^[|O1> -|10)] The chosen basis may be prepared based on appropriate logical encoding of the quantum information element 10, such as through one or more single qubit gates. The states of the computational subspace are also chosen such that transitions between the states of the computational basis may be addressed via a sideband transition, being a transition accessible via the three-wave mixing term in the Hamiltonian. The sideband transition is addressed by addressing the quantum information element 10 with a pulse of electromagnetic radiation on-resonance with (i.e. at the same frequency as) the side band frequency wTWm> as shown in Figure 6, with (oTWM = <»i - <o2, with <»! being the frequency of the single mode excitation from the third state to the first state and being the frequency of the single mode excitation from the third state to the second state. Thus, by deploying the multiple mode energy levels as a two-mode computational subspace, the multi-mode quantum information element 10 can be employed as a single logical qubit, the logical qubit having qubit states |0)L and |1)L and the addressable three-wave mixing transitions between the states being single qubit gates on the logical qubit. Using the above-described computational subspace significantly lowers the decay probability between the states of the computational subspace. The two transition frequencies 100) to 101) and 100) to 110) are separated by the large capacitive coupling strength of the original modes, which make the dipole and quadrupole modes individually addressable by control pulses of electromagnetic radiation with the appropriate frequency. An excitation from the |00) state to the |01) or the 110) state corresponds to an excitation along the longitudinal quantization axis of the computational subspace of the 101> and 110> states. The 101> or the 110> state both lie in the transverse quantization axis of the computational subspace. Due to the excited states of the computational subspace being separated from the ground state on the longitudinal quantization axis, there exists longitudinal coupling between the states of the computational subspace and the ground state, and thus a finite probability of longitudinal relaxations between the computational subspace and the ground state. For example, in lowest energy level states of Figure 2 an excited 101> state may relax to state 100>, or an excited 110> state by relax to state 100>. In both cases, the energy is lost and dissipated via lossy elements of the circuit including the dielectric, two-level systems within the fabricated device, the coupling to the coaxial control line, or via coupling to the readout resonator. The quantum information element 10 functions as a non-linear resonator with a resonance peak at the excitation frequency, comprising an internal quality factor and an external quality factor that determines the linewidth of the quantum state. Furthermore, since the states of the computational subspace lie on the transverse quantization axis, a single mode excitation between the |01> and |01> is not permitted, and a single-photon longitudinal relaxation between the |01> and |10> states is not permitted. Thus, transitions between the two states of the computational subspace can therefore only occur by a longitudinal coupling via a two-photon process (e.g. a relaxation from state |01> to state |00> and then an excitation from state |00> to state 101)), or via a transverse coupling mediated by the exchange interaction between the two states of the computational subspace. A relaxation via the two-photon longitudinal coupling or the transverse exchange interaction coupling has a significantly lower probability of occurring than a single-photon longitudinal relaxation, which is the dominant relaxation channel. By encoding quantum information in the selected computational subspace, bit flip errors between the states of the computational subspace are significantly reduced due to the forbidden longitudinal single-photon coupling between the states of the computational subspace. Furthermore, due to the permitted single photon longitudinal relaxation channel being toward a state that does not form part of the computational subspace, quantum state relaxation from the single photon decay channel does not manifest as a bit flip error. Thus, quantum information may be encoded and processed with a direct single-photon drive within the computational subspace, but also with reduced bit flip error within the computational subspace. As described above, the quantum information elements may be fabricated such that the superconducting islands of the first and second Josephson elements are symmetric. This symmetry may increase the value of the energy level splitting. In addition, the symmetry reduces the strength of transverse coupling between the first and second modes of the quantum information element (e.g. between the states |01> and 110) of the computational subspace) and significantly reduces the occurrence of errors from bit fips through the transverse coupling channel. Thus, the quantum information element 10 provides for an error detection mechanism as compared to a computational subspace of states formed from a qubit with longitudinally coupled normal modes (e.g. |0) and |1) of a transmon qubit). In such a longitudinally coupled subspace, the excited state of the computational subspace can relax to the lower state of the computational subspace. Thus, there is a significant probability of the states of the computational subspace being incorrectly populated due to single-photon longitudinal transitions between the states of the subspace. By contrast, in the quantum information elements described herein, the single photon transitions will decay each of the states of the computational subspace into a state outside the computational subspace (such as the 100) state for the subspace described above, also called the “erasure state”). These populated states can therefore be accurately flagged as errors and error detection performed more accurately. Any populated excited states |0)L and |1)L can correspondingly be accurately identified as populated states of the computational subspace. While there will exist a finite probability of a further single-photon transition from the |00) state to either the state |0)L or |1>l> the timescale of the quantum information processes may be kept short enough (e.g. 50ps) to minimise the effect of these transitions to provide a high quantum state fidelity. The choice of |01) and 110) states as the computational subspace is an example only. Alternative computational subspaces may be selected from the available energy levels formed from the coupled elements of superconducting islands. For example, the accessible energy levels of the quantum information element may include a first state, a second state and a third state, wherein there is a single-excitation transition of a first mode of the quantum information element between the first state and the second state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the second state and the third state at a second frequency. Transitions between the first state and the third state may be addressed by electromagnetic radiation at a third frequency being the sum of the first frequency and the second frequency. From these states, the first state and the third state are selected as the two states of the computational basis. For example, and referring to Figure 5, the two states of the computational subspace may be |00> and |20> (with 110> as the erasure state), or |00> and |02) (with |01> as the erasure state), or 100) and 111) (with 101) or 110) as the erasure state). Any two states may be chosen for the computational subspace where each state of the first state and second state of the computational subspace is separated by a single-excitation of a mode of the quantum information element from a third state outside the computational subspace, and a transition between the first state and second state of the computational subspace may be directly addressed by a drive field on resonance with the frequency separation between the first state and the second state of the computational subspace (being the sum or difference of the frequency of the excitation between the first state and the third state and the frequency of the excitation between the second state and the third state). The choice of 101) and 110), however, means the decay from both computational bases results in the ground state, |00), which may be easily detected. The previously described quantum information elements may be used to store quantum information and to be addressed in order to transition the quantum information element between quantum states. To prepare the quantum information element in a logical state or to transition the quantum information element between states, the quantum information element 10 is addressed via a control line. The control line is capacitively coupled to the quantum information element to provide control pulses to transition the quantum information element between quantum states. Figure 7 shows an example quantum information system comprising a quantum information element 10 (e.g. element 10 of any one of Figures 1, 3A, 3B, 3C and 4A), control line 52, readout line 54 and resonator 56. The quantum information system is coupled to a processor 50 configured to control the control line 52 and readout line 54 to perform quantum information processing. The readout line 54 and resonator 56 together form a readout element. The resonator 46 and quantum information element 10 are provided on opposite sides of a substrate 58. The control line 52 and readout element may be arranged out of plane with respect to the plane of the superconducting islands of the quantum information element. The quantum information element 10, control line 52, readout line 54 and resonator 56 may be coaxially aligned, as shown in Figure 7. The processor 50 is included in a computer system, and is coupled to input / output interfaces, memory and other elements configured to perform data processing. The processor is configured to execute a computer program to perform steps of the computer program to perform the processes described herein, including generation of control and readout pulses and the processing of the raw data, classification steps and post processing as described below. In some examples, the computer program may be stored in a non-transitory computer-readable medium. The control line 52 is configured to control the state of the quantum information element 10 by providing a control signal that couples to the modes of the element 10. For example, the control line 52 is arranged to address the qubit with pulses of electromagnetic energy such as microwave radiation. The length, amplitude, frequency and phase of the control signal are varied to address the element 10 to prepare a quantum state based on the computational subspace as described above. The readout element comprising the readout line 54 and resonator 56 are configured to read out a state of the quantum information element 10 in a dispersive readout regime. The resonator 56 is coupled to the element 10, and the resonant frequency of the resonator is detuned from the frequency of excitation of the quantum information element 10. The coupling of the quantum information element 10 to the resonator thus induces a state-dependent shift in resonator frequency. The readout line 54 is configured to interrogate / address the resonator 56 with a control signal (e.g. an electromagnetic pulse such as a microwave pulse) at a selected frequency, and measure the response that is reflected back from the resonator 56. The reflected pulse has a shift in amplitude and phase with respect to the interrogation pulse. By measuring phase (including real and imaginary components, represented by in-phase and quadrature-phase of the voltage signal) of the response from the resonator 56 can be measured and the state of the element 10 can be inferred. In some embodiments, the quantum information system is configured to distinguish between each state of the quantum information element (e.g. |00), |01) and 110» by addressing a single electromagnetic pulse to the resonator 56. The single pulse is chosen at a frequency that will drive the resonator 56 off-resonantly to populate the resonator with photons; the readout element pulse will produce a response from the resonator 56 with a different phase response (formed of real and imaginary terms) corresponding to the shift in the resonator frequency from the |01) and 110) modes. The chosen frequency is selected to be between the frequency response of the resonator 56 when the quantum information element 10 is in the |01) state, and the frequency response of the resonator 56 when the quantum information element 10 is in the 110) state. This frequency provides a response with a maximum difference in phase shift of the pulse reflected from the resonator 56. The response can be measured and the in-phase and quadrature phase shifts of the response inferred to determine the measured state. The pulse reflected from the resonator 56 in the readout line 54 will have an amplitude response. When addressing the resonator 56 for a range of frequencies, response curves can be measured and different response curves will result when the quantum information element 10 is in each of the three states. Each response curve has a resonant peak at a different frequency, which corresponds to the excitation of the encoded state present in the quantum information element 10. By addressing the resonator at the resonant frequency of the for each state, a maximum intensity can be measured in the response received from the resonator 56 at the readout line 54. In addition to an amplitude response, there will be a phase response in the pulse reflected from the resonator 56, having real and imaginary components. For a range of frequencies, response curves can be measured, and different response curves will result when the quantum information element 10 is encoded with different states. By observing the phase response of the reflected pulse it is possible to distinguish between quantum states encoded on the quantum information element 10. Furthermore, both states of the quantum information element 10 can be addressed using a single control line 52 (control multiplexing), and a single readout element, as compared to multiple control lines and multiple readout elements required to address and read states within a coupled transmon architecture. As such, the system is more hardware efficient and is more compact, improving scalability. A coupled transmon architecture is furthermore sensitive to flux noise and requires precise flux tuning be performed between the two coupled transmons to bring the transmons into resonance (e.g. when g » △, where g is the coupling strength between the uncoupled modes due to the capacitance and △ is the detuning between the modes; when the uncoupled modes are brought into resonance, A = 0 and there is a splitting of 2g ~ 200 MHz), which is not required for the quantum information element 10 (thus making the quantum information element insensitive to flux noise, which is a large cause of decoherence in a coupled transmon architecture). While Figure 7 illustrates a quantum information system with a single quantum information element 10, it will be understood that the quantum information element may be coupled to one or more additional quantum information elements, which will be described later. The coaxial arrangement described above is a quantum information system that is compact in-plane, allowing for higher scalability of the quantum information system to include many quantum information elements without minimal control wiring on the substrate. However, it will be appreciated that the coaxial arrangement described is an example only, and alternative control line and readout line arrangements are possible, including one or more of the control line, the resonator and the readout line being in plane with the quantum information element. For example, the control line 52, readout line 54 and resonator 56 may each be formed on the same surface of substrate 58 as the quantum information element 58. The quantum information element 10 is formed on the substrate 58 via a process of photolithography, where one side of the substrate 58 is patterned with a superconducting material (e.g. aluminium or Tantalum) to form the first, second and third superconducting islands. Standard photolithography techniques may be used. For example, the process includes providing a substrate, depositing the superconducting material over the substrate, followed by covering the substrate with a photoresist. A mask is used over the photoresist layer followed by exposure of the photoresist to UV light. The exposed section of photoresist is then removed (e.g. via chemical agent) to leave a pattern on the substrate. The superconducting material is then etched away revealing a pattern on the substrate of the superconducting islands. Remaining photoresist is removed by solvents. A photolithography process is also performed on the opposing side of the substrate 58 to pattern the substrate 58 with a superconducting material (e.g. or tantalum) to form the resonator. The substrate may be sapphire or silicon. The Josephson junctions are formed via a combination of electron-beam lithography and metal deposition. The qubit side of the substrate is covered in an electron-beam resist. The pattern is defined via electron-beam lithography prior to chemical development to remove resist from the regions exposed by the electron beam. A standard double-angle deposition is performed whereby a first layer of superconducting material (e.g., aluminium) is deployed to form the superconducting junction, and may be deposited prior to oxidation to form a Josephson junction oxide barrier to form a Josephson junction. Finally, a second metal evaporation at a different angle is performed. Each Josephson junction is formed where an oxide barrier separates the superconducting material from the first and second depositions of superconducting material. As described above, quantum information element 10 can be used to store and process quantum information in a computational subspace, where transitions between states of the computational subspace can be directly driven with a single photon drive to transition states within the computational subspace (i.e. perform single qubit gates on the quantum information element 10 as a logical qubit). Due to the ground state of the quantum information element 10 not forming part of the computational subspace, error detection can be performed on the quantum information element. After error detection is performed, the qubit state can be read out with high fidelity. Example quantum information processing and readout processes for a single quantum information element 10 is described below in connection with Figures 8A - 8D. Figure 8A illustrates a process 810 to address, process and readout quantum states prepared in one of the physical states |10) and |01) (dipole / quadrupole states) of the quantum information element 10 described above. In process 810 the initialisation step 815 comprises the initialisation of a logical state by either providing a pulse in a single qubit gate operation to address the 110) dipole state (e.g. a microwave pulse at the resonant frequency W2 of the dipole mode to perform a n rotation from the |00) to the 110) state) to initialise the element into state |0)L = 110), or providing a pulse to address the second logical state to initialise the element into the |01) state (e.g. a microwave pulse at the resonant frequency coi of the quadrupole mode to perform a n rotation from the 100) to the 101) state) to initialise the element into state |1)L = 101). In step 825 a series of pulses may be performed to execute operations on the qubits, for example as part of a quantum algorithm. The operations include single qubit gate operations performed by providing a pulse of electromagnetic radiation at the resonant frequency w™m of the sideband transition between the physical states of the computational subspace, to drive the transition between the logical states |0)L and |1)L (in the manner described above). For example, the pulses may include different phase rotations to perform different rotations around the Bloch sphere. After completion of the algorithm, measurement process 530 is performed to read out the final quantum state of the algorithm. The quantum information element 10 may be initialised in any basis prior to the performance of the algorithm steps 825. For example, Figure 8B illustrates an example process 820 to address and readout quantum states prepared in a superposition of the physical states |10) and |01) (dipole / quadrupole states) of the quantum information element described above. In process 820 the initalisation step 815 comprises the initialisation of a logical state by generating a superposition state from the physical states by application of single qubit gates to the quantum information element 10 (for example, by a pulse configured to perform a ^-rotation in the dipole mode and a pulse configured to perform a ~ rotation in the quadrupole mode, to prepare the element into a |0)L = ^[|01) - |10)] state). In the same manner as Figure 8A, in step 825 of Figure 8B a series of pulses may be performed to execute operations on the qubits, for example as part of a quantum algorithm. After completion of the algorithm, a further series of pulses may be performed to produce a logical tt / 2 operation to rotate the state of the quantum element back into the measurement basis (the 101), 110) basis). Finally, process 530 is performed to read out the final quantum state of the algorithm. The measurement process 530 is performed by the readout element in the manner described above. The above process of Figures 8A and 8B are examples only, and any choice of orthogonal basis states formed from the physical states can be selected for the encoding of quantum information in a two-state computational basis. To encode an arbitrary basis state, a series of pulses is generated for the initialisation process 815 to address the dipole modes and quadrupole modes to rotate the physical states to the chosen basis. Similarly, any series of pulses may be generated for the pulse sequence operation prior to measurement to rotate the state generated by an algorithm back to the measurement basis. Alternatively, the method of 8A may be followed to initialise the state into the |01>, 110> basis and the quantum information element may be driven at cotwm to rotate the initialised state into a chosen basis prior to the start of the algorithm 825. Figures 8C and 8D illustrate an alternative quantum information processing process 830 where the quantum information element is initialised by driving a sideband transition between the modes of the quantum information element 10 and the resonator 56 of the quantum information system in a resonator-driven initialization sequence 815-R. As described above, the resonator 56 and the quantum information element may be coupled (e.g. capacitively coupled). This coupling results in a coupled quantum information element, having a series of accessible energy levels formed of the modes of the quantum information element coupled to the modes of the resonator. The coupled modes can be described in a \nml) notation (or equivalently as|nm)|Z)) , where n indicates the number of A-mode excitations of the quantum information element, m indicates the number of Z-mode excitations of the quantum information element and I indicates the number of resonator-mode excitations in the resonator element. Example energy states in the \nml) notation are illustrated in Figure 8C (additional accessible states are present but not illustrated in Figure 8C), which shows the states of the quantum information element 10 of Figure 3 with additional accessible resonator excitations. As shown in figure 8C, the accessible energy levels of the coupled information element can include a first state (e.g. |01>10» and a second state (e.g. 110)10)), the first state separated from a third state (e.g. the ground state |00)|0» by a first frequency and the second state separated from the ground state by a second frequency m2. The first state and second state represent a single mode excitation of the quantum information element and the resonator is not excited. Thus, for each of the first state, second state and third state, I = 0 and there is no normal mode excitation of the resonator. The first state and the second state form the computational subspace as described above. The accessible energy states of the coupled information element include a fourth state (e.g. |01>| 1» and a fifth state (e.g. 110>11», where there is a single excitation transition between the first state and the fourth state at a first coupled resonator frequency wri, and there is a single excitation transition between the second state and the fifth state at a second coupled resonator frequency wr2. The single excitations transitions between the first state and the fourth state and between the second state and the fifth state are single mode excitations of the resonator. The system is configured such that the first coupled resonator frequency wri and the second coupled resonator frequency wr2 are equal. However, in some embodiments the system is configured such that wri and wr2 are different in value. Due to the presence of the three-wave mixing element 20 in the quantum information element 10, transverse excitations between the \nml) states may be driven as sideband transitions, including between states of different resonator excitation number. For example, there exists a first sideband transition between the |00>|0> and |10>|l> states at frequency mr_twm1, and there exists a second sideband transition between the 100>10> and |01>11> states of frequency o>r-twm2- The first sideband transition may be driven by an input pulse with a drive frequency equal to o>r-twmi to perform a tt-rotation to perform the initialisation |00)|0)-> |10)|l), and the second sideband transition may be driven by an input pulse with a drive frequency equal to mr_TWM2 to perform a TT-rotation to perform the initialisation 100> 10> -> 110> 11>. Within the energy level states of the resonator-quantum information element coupled system, there exist longitudinal relaxations between states separated by △ / = ±1, including a longitudinal single-photon transition from |01>|l> to |01>|0> (or from |10>|l> to 110)10)). Thus the excited state |01)|l) quickly decays to |01)|0) after a short time period, resulting in the quantum information element initialised into state |1)L = 101) (or the excited state 110)11) quickly decays to 110)10) after a short time period, resulting in the quantum information element initialised into state |0)L = 110)). The quantum information element 10 may thus be initialized into state |0)L = 110) or |1)L = 110) by driving a sideband transition and allowing the resonator state to decay. Thus, within a system of a quantum information element 10 coupled to a resonator, an initialisation process may be performed by addressing the coupled quantum information element at an initialization frequency. In order to initialise the quantum information element in the first state, the initialisation frequency selected as the sum of the first frequency and the first coupled resonator frequency and to initialise the quantum information element in the second state, the initialisation frequency is selected as the sum of the second frequency and the second coupled resonator frequency. This initialization process 815 is shown in Figure 8C, which includes providing a pulse in a single qubit gate operation to drive a sideband transition between the ground state and an excited resonator-coupled state (e.g. a microwave pulse at the resonant frequency wr.™m of the dipole mode to perform a n rotation from the |00>|0> to the |01>| 1> state). Following the single qubit gate operation pulse, the system is permitted to evolve for time t, during which time the resonator mode decays, thus initialising the element into state |1)L = |01>. The decay time from the resonator (e.g. about 1ps) is generally much shorter than the decay time of the quantum information element to the ground state (e.g. about 100ps). After initialisation, quantum algorithm 825 and readout process 835 are subsequently performed, as described above for Figures 8A or 8B. In the measurement step 835 of each of the above examples, the process measures the information stored in the quantum information element by addressing the resonator at a selected frequency as described above. When performing a final measurement of the quantum state, the measurement modes are the ground state of the quantum information element and the two excited states forming the computational subspace (i.e. |00>, |01>, and |10». Each of these modes will have a different resonator dispersive shift, with the reflected signal from the resonator having different amplitudes and complex phases. This allows a joint dispersive readout as described. The computational subspace is selected to exclude the 100) state, so any occurrence of 100) states in the final measurement is the result of an error in the logical qubit (e.g. due to a longitudinal relaxation from a state in the computational subspace). In the error detection methodology described below, the error states are to be discarded and the remaining measure states retained. Thus, in any one execution of an algorithm on a quantum information element, there is a probability that an error will or will not occur. In embodiments disclosed herein, the same process steps 815, 825 and 835 are repeated many times (for example, 1,000 repetitions) to obtain a large sample of measured qubit states, a fraction of which will be in a state of the computational subspace and a fraction in the error state. The output of the readout element for each repetition is recorded in a dataset that is an accumulated record of all measurements. For / V repetitions, the dataset will comprise N entries of measured IQ values. The chosen value of / V will be selected depending on the desired number L of accurate quantum state measurements and the bit erasure fraction p, where N = L / (1-P). For example, if there is an erasure fraction of 0.2 (20% of all bits) and it is desired to measure L = 5,000 qubits in the logical basis, an N is selected as 5000 / 0.8 = 6250. For example, the quantum information system is configured to perform a multi-state readout, with repeated measurements clustered around three locations in the IQ plane in a Gaussian pattern. The distribution pattern results from noise in the readout chain, with a width dependent on the signal to noise ratio. The raw data signal may be extracted from the readout line addressing the resonator. The raw data is then processed in a classification process. The classification is performed, for example, by a trained Gaussian Mixture Model (GMM). The GMM is trained on a data collected by performing a calibration measurement, in large number of measurements are performed (for example 10,000 or over), where the quantum information element 10 is prepared in the states to be measured (the states of the computational subspace and the erasure state, such as |00), |10) and |01)) and readout measurements are performed as described above. The preparation of the training data is performed during calibration of the system. The GMM may identify state boundaries in the IQ plane and classify each measurement result in the IQ plane as being in one of three states. Following the classification process, an identification process is performed, in which the measurement states that are classified into one of the three states in the IQ plane are assigned to a logical state - i.e. to one of state 101), |10) or |00). Following assignment of states, a post selection process is performed. In post selection, the states assigned to state |00> are identified as errors and discarded from the dataset. The remaining states are renormalized to provide a final logical state of the readout. The above-described storage and readout process incorporating error detection is configured to perform a final state measurement and to compensate for error in the final state measurement. In alternative embodiments, the error detection process may be a “continuous error detection” process, which is performed without measuring the |0)L and |1)L states of the system. In these alternative embodiments, errors may be detected by addressing the quantum information element 10 with a pulse that does not distinguish the encoded quantum states. These alternative embodiments may be useful, for example, to perform error detection during execution of a quantum algorithm. A continuous error detection process is performed in a similar manner to process described above. However, instead of providing a joint dispersive readout (where the readout line addresses the resonator at a frequency between the resonant peaks for the |0)L and |1)L), the readout element is configured to address the resonator at a frequency for which the phase response and amplitude response of the signal reflected from the resonator is the same for each of the states |0)L and |1)L. This frequency is near (e.g. within about 100kHz of) the resonant frequency of the ground state |00). The dispersive shift (i.e. the frequency separation between the frequency corresponding to the ground state 100) and each of the frequencies corresponding to the computational subspace states is larger than the frequency separation between the two frequencies and of the computational substates). Due to the pulse from the resonator having the same phase and amplitude response for the computational substates, addressing the resonator at frequency does not distinguish between the two states |0)L and |1)L. As such, the dephasing of the states of the computational subspace is minimised. The dispersive shift may be increased by adjusting the configuration of the resonator to further reduce the dephasing. Thus, the states determined to be “not-100)” maintain their quantum state and can further addressed by the control line in a subsequent series of pulses (e.g. during a quantum algorithm). The continuous error detection process therefore enables errors in a quantum state in a quantum information element to be identified during execution of a quantum information process. As discussed above, errors in a quantum state may occur during operation of the qubit via a two-photon interaction. A first single-photon interaction may cause an excited |01> or |01> state to decay into the |00> state, and a subsequent second-photon interaction may cause the 100> to be excited to a 101> or 101> state. The characteristic rate Q for single-photon relaxation is higher than the characteristic rate F-for single-photon excitation. Thus, the characteristic decay time (T; = 1 / rj for relaxation is less than the characteristic decay time for excitation (Tt = 1 / / Ft) (e.g. T; = 50 ps and T- = 100 ps). Repeated performance of the continuous error detection process can identify the errors between the two photon interactions, thus improving the accuracy of the error detection process and the reliability of the quantum information element. The continuous error detection may be performed at a predetermined interval that is optimised between identifying a higher degree of error from state relaxation and minimising the instances in which the quantum information element is addressed. For example, the predetermined interval is selected to be the average single-photon decay time for excitation Tt. As with the earlier described process 500, the continuous error detection process steps are repeated many times (for example, 1,000 repetitions), and the output of the readout element for each repetition is recorded in a dataset that is an accumulated record of all measurements. For / V repetitions, the dataset will comprise N entries of measured IQ values. The dataset for recorded measurements of the continuous error detection process are classified using a classification algorithm as described above. In this case, the classification algorithm is trained to classify the states in one of two categories ("|00)" or “not 100)”), with a boundary in the phase diagram separating the IQ regions corresponding to each category. Those measurements classified as “|00)” are identified as error states. Any results produced from using the states identified as error states in any subsequent algorithm are discounted. The above embodiments relate to the implementation of a single quantum information element with associated resonator and control lines. In further embodiments of the invention, multiple quantum information elements may be provided in a quantum information system. The multiple quantum information elements may be coupled together to allow cross-talk between quantum information elements. Multi-qubit gates can be performed on the coupled quantum information elements to perform quantum information processing. Figures 9A, 9B and 9C each illustrate the coupling of two quantum information elements together in accordance with some embodiments. Figure 9A illustrates an embodiment where a first quantum information element 10 is coupled to a second quantum information element 10A. In this embodiment, each of first quantum information element 10 and 10A comprises a three-wave mixing element, and may each be quantum information element as described above. Figure 9B illustrates an embodiment where a first quantum information element 10 is coupled to a second quantum information element 10B. The first quantum information element 10 comprises a three-wave mixing element 20, and may be as described above. The second quantum information element 10A is a multi-mode transmon element 10A which does not include a three-wave mixing element. For example, the second quantum information element comprises a first Josephson element and a second Josephson element, each Josephson element comprising a Josephson junction provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second Josephson elements. The second quantum information element 10B may have a configuration wherein the element 10B comprises a split superconducting electrode arranged coaxially with a further superconducting electrode, the split superconducting electrode forming one of the superconducting islands of each of the first Josephson element and second Josephson element, the further superconducting electrode forming the superconducting island which is common to both the first and second Josephson elements. The second quantum information element 10A may have first and second Josephson elements the same or substantially the same coupling within each Josephson element (including both the capacitive coupling and the Josephson inductive coupling). The same coupling within each Josephson element can be provided by providing Josephson elements with identical, physical characteristics. This can be done by manufacturing the placement and orientation of the first superconducting island with respect to the third superconducting island to be substantially the same as the placement and orientation of the first superconducting island with respect to the third superconducting island. For example, the assembly of first, second and third superconducting elements is provided in a planar arrangement with an 180 degree in-plane rotational symmetry. Figure 9C illustrates an embodiment where a first quantum information element 10 is coupled to a second quantum information element 10C. The first quantum information element 10 comprises a three-wave mixing element 20, and may be a quantum information element as described above. The second quantum information element 10C is a transmon element 10B which does not include a three-wave mixing element. The second quantum information element 10C is a transmon qubit and includes a Josephson element, the Josephson element comprising a Josephson junction coupled between a first superconducting island and a second superconducting island. The first and second superconducting islands may be arranged co-axially, with the first superconducting island being located within the first superconducting island. Figures 9A, 9B and 9C illustrate the coupling as a capacitive coupling, but other forms of coupling may also be implemented. For example a resonant coupler connecting the quantum information elements, the resonant coupler comprising an inductor and a capacitor. Alternatively, the quantum information elements may be inductively coupled together by two inductive elements (such as coupled coils). Figures 9A, 9B and 9C are examples only, and it will be understood that alternative quantum information element architectures may be deployed in the alternative to second quantum information elements 10A, 10B and 10C. In each case, the first quantum information element 10A having a three-wave mixing element is coupled to a second quantum information element. The presence of the three-wave mixing element enables multi-qubit gates in the logical two-qubit space (and it is not required that the second quantum information element comprises a three wave mixing element 20). The quantum information element 10C is a single-mode transmon (corresponding to dipole oscillations across the quantum information element), whereas both quantum information elements 10A and 10B are multi-mode transmons. Thus, when the quantum information element 10A or 10B is used, a logical four-state computational subspace of the coupled logical qubits can be selected such that there are no longitudinal transitions between the states of the logical four-state computational subspace, thus reducing the number of bit flip errors within the logical four-state computational subspace. Due to the coupling between the two quantum information elements, the normal modes on each quantum information elements are also coupled together. This results in a weak hybridisation of the two modes of the first quantum information element and the mode(s) of the second quantum information element. The excited states of the quantum information system formed of the two quantum information elements exist as a plurality of normal modes of the system, in which an oscillating electric field is present with each element across the superconducting islands of each quantum information element. As described above, for each of quantum information elements 10, 10A and 10B, within the quantum information element 10 a normal mode corresponds either to an in-phase addition of the dipole moment modes of the Josephson elements, or to an out-of-phase addition of the dipole moment modes of the Josephson element. Out-of-phase addition of dipole moments results in a dipole moment (a A-mode) across the quantum information element as a lower frequency mode, and in-phase addition of dipole moments results in a quadrupole moment (a Z-mode) across the quantum information element as a higher frequency mode. Figure 10 illustrates an energy level diagram of the first quantum information element 10 coupled to second quantum information element 10A or 10B, where the states l1^) of the system are labelled in an \nmop) notation, where n indicates the number of A-mode excitations on the first quantum information element 10, m indicates the number of Z-mode excitations on the first quantum information element 10, o indicates the number of A-mode excitations on the second quantum information element 10A or 10B, and p indicates the number of Z-mode excitations on the second quantum information element 10A or 10B. The four states |0101>,|0110>, |1001> and |1010> represent a single-excitation manifold of the system of four normal modes. Higher excitation levels within the system also exist, and described by different integer A-mode and Z-mode excitations (for example |0001>, |0011> and 11111», as shown in Figure 10. A singleexcitation of the quantum information element corresponds to increasing n by 1, increasing m by 1, increasing o by 1 or increasing p by 1. The \nmop) state energy levels are anharmonic, and addressable by photons of frequencies corresponding to the transition frequencies, with each increase in n, m ,o or p being a transition having a corresponding frequency. The quantum information elements 10 and 10B or 10C may be fabricated such that the frequency separation between each A-mode excitation within the first quantum information element is substantially the same as the corresponding A-mode excitation or the second quantum information element, and such that the frequency separation between each Z-mode excitation within the first quantum information element is substantially the same as the corresponding Z-mode excitation or the second quantum information element. The quantum information element 10 thus provides an energy level structure including a ground state, and multiple different excited states. Within the accessible energy states of the coupled quantum information elements, some energy states are separated by single mode excitations, including single mode excitations of the first quantum information element 10 (for example |0000> to 11000>, being a An = +1 transition) and single mode excitations of the second quantum information element 10B or 10C (for example |0000> to |0001), being a An = +1 transition). Sideband transitions within each of the first quantum information element 10 may be addressed by driving at a sum or difference frequency of single mode excitations of the first quantum information element (e.g. between |0100) to 11000), by driving at sum of frequencies of the |0000) -> |1000) transition and the |0000) -> |0100) transition). When quantum information element 10A is used (which contains a three-wave mixing element), sideband transitions may also be addressed in the second quantum information element 10A by driving at a sum or difference frequency of single mode excitations of the second quantum information element. When second quantum information element 10B is used (with no three-wave mixing element 20), there are no addressable sideband transitions within the second quantum information element 10B. A four-state logical subspace may be selected from the accessible energy states of the coupled quantum information elements. The four-state logical subspace is selected such that there is no single mode excitation between any of the four states of the logical subspace, and thus there is no longitudinal relaxation pathways between each state of the logical subspace. For example, as shown in Figures 10 and 11, the accessible energy levels of the coupled quantum information elements include four states that are selected as the four-state computational subspace, including a first state (e.g.|00)c = |0101», a second state (e.g. |01)c = |0110», a third state (e.g. 110)c = 11001» and a fourth state (e.g. |ll)c = 11010». The addressable energy levels include a fifth state (e.g. |0001», sixth state (e.g. |0100», seventh state (e.g. |0010», and eighth state (e.g. 11000», outside of the computational subspace. There is a single mode excitation of the first quantum information element between the fifth state and the first state and a first frequency and a single mode excitation of the first quantum information element between the fifth state and the third state at a second frequency. There is a single mode excitation of the second quantum information element between the sixth state and the first state at a third frequency and a single mode excitation of the second quantum information element between the sixth state and the second state at a fourth frequency. There is a single mode excitation of the first quantum information element between the seventh state and the second state at a fifth frequency and a single mode excitation of the first quantum information element between the seventh state and the fourth state at a sixth frequency. There is a single mode excitation of the second quantum information element between the eighth state and the third state at a seventh frequency and a single mode excitation of the second quantum information element between the eighth state and the fourth state at an eighth frequency. The fifth to eighth states do not form part of the computational subspace, and there is no longitudinal coupling between the four modes of the four-mode computational subspace. Thus, transitions between the four states of the computational subspace can therefore only occur by a longitudinal coupling via a two-photon process (e.g. a relaxation from state 11001> to state |0001> and then an excitation from state |0001> to state 10101», or via a transverse coupling mediated by the exchange interaction between the two states of the computational subspace. The two-photon longitudinal coupling and the transverse exchange interaction coupling has a significantly lower probability of occurring than a single-photon longitudinal relaxation, which is the dominant relaxation channel. By encoding quantum information in the selected four-state computational subspace, bit flip errors between the states of the computational subspace are significantly reduced due to the forbidden longitudinal single-photon coupling between the states of the computational subspace. Furthermore, due to the permitted single photon longitudinal relaxation channel being toward a state that does not form part of the computational subspace, quantum state relaxation from the single photon decay channel does not manifest as a bit flip error. Thus, quantum information may be encoded and processed with a direct single-photon drive within the computational subspace, but also with reduced bit flip error within the computational subspace. As shown in Figure 11, due to the presence of the thee-wave mixing element 20 on the first quantum information element 10, it is possible to directly address transitions between some states of the four-state computational subspace. It is possible to address a transition between the first state |00)c and third state |10)c at a frequency (<olo-i) that is the sum of the first frequency and the second frequency, and it is possible to address a transition between the second state |01)c and fourth state |11>C at a frequency (a)10-2) that is the sum of the fifth frequency and the sixth frequency. When quantum information element 10A is used (i.e. a quantum information element having a three-wave mixing element), it is also possible to directly address sideband transitions of excitations on the second quantum information element 10A. For example, it is possible to address a transition between the first state |00)c and second state 101)c at a frequency (o)1(M_i) that is the sum of the third frequency and the fourth frequency, and it is possible to address a transition between the third state |10)c and fourth state |ll)c at a frequency (^km-J that is the sum of the seventh frequency and the eighth frequency. When quantum information element 10B is used, the sideband transitions at frequencies and o)10i4_2 are not directly addressable by addressing the second quantum information element, but these transitions may be driven in a cross-resonance excitation process, as described below. In some embodiments, the quantum information elements are configured such that ^104-1 = ^104-2 and ^10-1 = ^10-2- This allows single logical qubit gates to be performed on the first quantum information element 10 (and optionally the second quantum information element 10A) independent of the state of the qubit to which the quantum information element is coupled, and improves the performance of logical two-qubit gates. As illustrated in Figure 12, the quantum information elements 10 and 10A / 10B / 10C may form part of a quantum information system, together with first control line 52A, second control line 52B, first resonator 56A, second resonator 56B, first readout line 54A and second readout line 54B. First resonator 56A and first readout line 54A together form a first readout element and second resonator 56B and second readout line 54B together form a second readout element. Thus, each of the coupled quantum information element 10 and quantum information element 10A / 10B / 10C may be separately addressed by a separate control line and separately read out by a readout line. As set out above, single excitations and sideband transitions within first quantum information element 10 and second quantum information element 10A may be addressed. The control line 52A is thus configured to perform single logical qubit operations on the first quantum information element 10 in the manner described above in connection with Figures 7 - 8D. When quantum information elements 10B and 10C are used, sideband transitions are not addressable, but single mode excitations of the quantum information element may be addressed. In each case, each of the coupled quantum information elements 10 and 10A / 10B / 10C may be individually addressed to perform single excitations within each quantum information element. As mentioned above, it is possible to perform a cross-resonance operation between the first quantum information element 10 and second quantum information element 10A / 10B / 10C. In a cross-resonance operation, one quantum information element is driven at a frequency in resonance with the other quantum information element (e.g. one quantum information element is driven at a frequency that corresponds to an addressable excitation between states in the other quantum information element, such as a single mode excitation in the other quantum information element or a sideband transition in the other quantum information element). Due to the coupling of modes between the quantum information elements, this addressing of one quantum information element has the effect of driving the transition within the other quantum information element conditional on the state of the driven quantum information element. This allows for the performance of two-qubit gates, such as a CX (CNOT) gate or a CZ (CPHASE) gate. Referring back to Figure 11, the first quantum information element 10 may be addressed at the frequency <o1O4-2 (which may be the same as corresponding to a separation of energy levels of the second quantum information element. Due to the three-wave mixing element on the first quantum information element 10, this has the effect of driving excitations within the second quantum information element 10A. However, the amplitude of this transition depends on the state of the first quantum information element 10A. For example, the transition from the third state |10)c to the fourth state 111>C is amplified, and the transition from the first state |00)c to the second state 101>c is suppressed. Given that the transition is being driven off-resonance from the sideband transitions of the first quantum information element 10, there is also no transition between the first state |00)c and third state |10)c nor between the second state |01)c and fourth state |11>C. Thus, by driving the first quantum information element 10 at a frequency separating two excitation states of the second quantum information element 10A in the computational subspace, a CNOT (CX) two-qubit gate is performed. The above CNOT operation performed by addressing the first quantum information element 10 is the same whether second quantum information element 10A is used or quantum information element 10B is used. The presence of three-wave mixing element 20 on quantum information element 10 provides the required coupling to effect the two-qubit gate. If quantum information element 10A is used, then a CNOT gate may also be performed by driving the second quantum information element 10A at a frequency corresponding to a separation of excitations of the first quantum information element 10 (e.g. driving the second quantum information element 10A at frequency ^10-1)- A universal gate set for the coupled quantum information elements may be constructed from the above-described CNOT gate. However, the quantum information system of Figure 12 is not limited to performance of a CNOT multi-qubit gate. Due to the coupling of the two quantum information elements, and the provision of separate control lines and readout lines to address each quantum information element separately, each quantum information element can be addressed at selected frequencies to perform other multi-qubit gate operations. For example, the quantum information system may further be configured to perform a CPHASE (CZ) gate on the second quantum information element 10A / 10B / 10C by addressing the first quantum information element 10 at a frequency to transition the coupled quantum information elements from a state inside the computational four-state subspace to a state outside the four-state computational subspace and back to the original state inside the four-state computational subspace. By doing so, the encoded state of the coupled quantum information elements acquires a phase conditional on the state of the first quantum information element 10. For example, the coupled quantum information element may include two states outside the computational basis (a first intermediate state and a second intermediate state), where there is a single-excitation transition of a mode of the second quantum information element between the first intermediate state and the fourth state of the computational subspace at a first intermediate frequency, and there is a single excitation transition of a mode of the first quantum information element between the first intermediate state and the second intermediate state at a second intermediate frequency. The control line is configured to address the first quantum information element 10 with two identical and sequential TT-rotation pulses, each TT-rotation pulse at a CZ gate frequency, the CZ gate frequency being the difference of the first intermediate frequency and the second intermediate frequency. For example, and referring to Figure 11, the pulses drive a transition to and from the fourth state of the computational subspace and the state |1100> as the second intermediate state (with 11000) being the first intermediate state). A universal gate set for the coupled quantum information elements may be constructed from the above-described CPHASE gate. Whilst certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions. Indeed, the novel devices, and methods described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the devices, methods and products described herein may be made without departing from the spirit of the inventions. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the inventions.

Claims

1. A quantum information element comprising:a first element and a second element;the first element comprising a first superconducting non-linear inductor provided between two superconducting islands; andthe second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements.

2. The quantum information element of claim 1, wherein the three-wave mixing element comprises a second superconducting non-linear inductor and a superconducting loop, the superconducting loop comprising the second superconducting non-linear inductor, the superconducting loop being configured to receive a bias flux through the superconducting loop.

3. The quantum information element of claim 2, wherein the first superconducting non-linear inductor is a first Josephson junction and the second superconducting nonlinear inductor is a second Josephson junction.

4. The quantum information element of claim 3, wherein the superconducting loop further comprises at least two third Josephson junctions, wherein the at least two third Josephson junctions are connected in series between the superconducting islands of the second element, and in parallel with the second Josephson junction.

5. The quantum information element of claim 3, wherein the superconducting loop further comprises an inductor connected between the superconducting islands of the second element, and in parallel with the second Josephson junction.

6. The quantum information element of claim 1, wherein the three-wave mixing element comprises a second superconducting non-linear inductor and wherein the superconducting junction is configured to be current-biased by a current source.

7. The quantum information element of claim 6, wherein the three-wave mixing element comprises a conductive loop, the conductive loop comprising the second superconducting non-linear inductor and the current source, wherein the current sourceis connected in parallel to the second superconducting non-linear inductor and wherein the conductive loop is configured to receive a bias flux through the conductive loop, wherein, optionally, the conductive loop is a superconducting loop.

8. The quantum information element of any preceding claim, wherein the quantum information element comprises a split superconducting electrode arranged coaxially with a further superconducting electrode, the split superconducting electrode forming one of the superconducting islands of each of the first element and second element, the further superconducting electrode forming the superconducting island which is common to both the first and second elements.

9. A quantum information system comprising a quantum information element according to any preceding claim and a control line,wherein the superconducting islands of the first element are coupled to the superconducting islands of the second element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency,wherein the control line is configured to encode into the quantum information element one of two states of a computational subspace, the first state of the computational subspace corresponding to the first state and the second state of the computational subspace corresponding to the second state;and wherein the control line is configured to address the quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency.wherein, optionally, the control line is arranged out of plane with respect to the quantum information element.

10. The quantum information system of claim 9, further comprising a readout element configured to measure the quantum state of the quantum information element wherein, optionally, the readout element is arranged out of plane with respect to the quantum information element.

11. A quantum information system comprising a quantum information element according to any one of claims 1- 8 and a readout element, the readout element comprising a resonator and a readout line,wherein the quantum information element is coupled to the resonator such that the coupled resonator element and quantum information element together comprise a plurality of states comprising a first state, a second state, a third state and a fourth state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency, a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency, and a singleexcitation transition of a mode of the resonator between the second state and the fourth state at a third frequency, wherein the first state corresponds to a first state of a computational subspace of the quantum information element, and the second state corresponds to a second state of the computational subspace; andwherein the readout line is configured to perform an initialization operation to encode into the quantum information element one of the first state and second state of the computational subspace, the initialization operation comprising addressing the quantum information element at an initialization frequency, the initialization frequency being the sum of the second frequency and the third frequency.

12. A quantum information system comprising a first quantum information element and a second quantum information element coupled to the first quantum information element, wherein the first quantum information element is a quantum information element according to any one of claims 1-8, and wherein the second quantum information element comprises a first element, the first element comprising a first superconducting non-linear inductor provided between two superconducting islands.

13. The quantum information system of claim 12, wherein the second quantum information element further comprises a second element, the second element comprising a second superconducting non-linear inductor provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements.

14. The quantum information system of claim 12, wherein the second quantum information element further comprises a second element, the second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements.

15. The quantum information system of any one of claims 12 - 14, further comprising a first control line and a second control line, wherein the first control line is configured to address the first quantum information element and the second control line is configured to address the second quantum information element,wherein, optionally, the quantum information system further comprises a first readout element configured to measure the quantum state of the first quantum information element and a second readout element configured to measure the quantum state of the second quantum information element.

16. The quantum information system of claim 15, wherein the first control line is configured to address the first quantum information element at a cross resonance frequency on resonance with the second quantum information element.

17. The quantum information system of any one of claims 15 or 16, wherein the first quantum information element and the second quantum information element are coupled together such that the first quantum information element and the second quantum information element together comprise plurality of states comprising a first state, a second state, and a third state,wherein there is a single-excitation transition of a first mode of the first quantum information element between the third state and the first state at a first frequency, and a single-excitation transition of a second mode of the first quantum information element between the third state and the second state at a second frequency, andwherein the first control line is configured to address the first quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency.

18. The quantum information system of claim 15 or 16, when dependent from claim 13 or claim 14, wherein the first quantum information element and the second quantum information element are coupled together such that the first quantum informationelement and the second quantum information element together comprise plurality of states comprising a first state, a second state, a third state, a fourth state, a fifth state, a sixth state, a seventh state and an eighth state, wherein there is:a single-excitation transition of a first mode of the first quantum information element between the fifth state and the first state and a first frequency;a single-excitation transition of a second mode of the first quantum information element between the fifth state and the third state at a second frequency; asingle-excitation transition of a first mode of the second quantum information element between the sixth state and the first state at a third frequency;a single-excitation transition of a second mode of the second quantum information element between the sixth state and the second state at a fourth frequency;a single-excitation transition of the first mode of the first quantum information element between the seventh state and the second state at a fifth frequency;a single-excitation transition of the second mode of the first quantum information element between the seventh state and the fourth state at a sixth frequency;a single-excitation transition of the first mode of the second quantum information element between the eighth state and the third state at a seventh frequency;a single-excitation transition of the second mode of the second quantum information element between the eighth state and the fourth state at an eighth frequency;wherein the first control line and second control line are configured to encode into the quantum information element one of four states of a computational subspace, the first state of the computational subspace corresponding to the first state, the second state of the computational subspace corresponding to the second state, the third state of the computational subspace corresponding to the third state, and the fourth state of the computational subspace corresponding to the fourth state;wherein the control line is configured to address the first quantum information element at a first cross-resonance frequency, the cross-resonance frequency being the difference between the third frequency and the fourth frequency or being the difference between the seventh frequency and the eighth frequency.

19. A method of quantum information processing in a quantum information element, the quantum information element comprising a first element and a second element;the first element comprising a first superconducting non-linear inductor provided between two superconducting islands; andthe second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements;wherein the superconducting islands of the first element are coupled to the superconducting islands of the second element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the quantum information element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency,the method comprising:encoding into the quantum information element one of two states of a computational subspace, the first state of the computational subspace corresponding to the first state and the second state of the computational subspace corresponding to the second state; andaddressing the quantum information element at a three-wave mixing frequency, the three-wave mixing frequency being the difference between the first frequency and the second frequency.

20. A computer program, the computer program comprising instructions that, when executed by a processor, cause the processor perform a method of quantum information processing in a quantum information element, the quantum information element comprising a first element and a second element;the first element comprising a first superconducting non-linear inductor provided between two superconducting islands; andthe second element comprising a three-wave mixing element provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second elements;wherein the superconducting islands of the first element are coupled to the superconducting islands of the second element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third state, wherein there is a single-excitation transition of a first mode of the quantuminformation element between the third state and the first state at a first frequency and wherein there is a single-excitation transition of a second mode of the quantum information element between the third state and the second state at a second frequency,5 the method comprising:encoding into the quantum information element one of two states of a computational subspace, the first state of the computational subspace corresponding to the first state and the second state of the computational subspace corresponding to the second state; and10 addressing the quantum information element at a three-wave mixing frequency,the three-wave mixing frequency being the difference between the first frequency and the second frequency.15

Images