Quantum information system

Abstract

A quantum information system comprising a quantum information element and a control line, the quantum information element comprising: 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and wherein 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.

Description

[0001] QUANTUM INFORMATION SYSTEM

[0002] TECHNICAL FIELD

[0003] The present application relates to a method and system for processing of quantum information in a stable manner, and for the performance of error detection on a quantum information system.

[0004] BACKGROUND

[0005] 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.

[0006] 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.

[0007] BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Embodiments will now be described by way of example with reference to the accompanying drawings.

[0009] Figure 1A schematically illustrates a quantum information element.

[0010] Figure 1 B schematically illustrates an equivalent circuit diagram for the quantum information element of Figure 1A.

[0011] Figure 2 illustrates the energy level diagram of a quantum information element.

[0012] Figure 3A schematically illustrates a quantum information element. Figure 3B schematically illustrates an equivalent circuit diagram for the quantum information element of Figure 3A.

[0013] Figure 4A schematically illustrates a quantum information system comprising a quantum information element.

[0014] Figure 4B illustrates a readout amplitude response signal for the quantum information system of Figure 4A.

[0015] Figure 4C illustrates a readout phase response signal for the quantum information system of Figure 4A.

[0016] Figure 5 illustrates a method of quantum information processing including error detection.

[0017] Figures 6A and 6B illustrate example quantum information processing methods.

[0018] Figure 6C illustrates a phase diagram for a state measurement of the quantum information element in error detection at final readout.

[0019] Figure 6D illustrates a phase diagram for a state measurement of the quantum information element in a continuous error detection scheme.

[0020] Figures 7A and 7B illustrate example quantum information performance methods.

[0021] Figure 8 schematically illustrates an array of coupled quantum information elements.

[0022] Figure 9 schematically illustrates a quantum information system comprising the array of quantum information elements of Figure 8.

[0023] DETAILED DESCRIPTION

[0024] In a first aspect, there is described a quantum information system comprising a quantum information element and a control line, the quantum information element comprising: 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson element such that the quantum information element comprises a plurality of states comprising a first state, a second state, and a third excited state, wherein there is a single-excitation transition between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and 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.

[0025] The embodiment described above relates to a quantum information system that provides accessible quantum states that can be used for a quantum computational subspace, in which bit flip errors within the quantum computational subspace are minimised, and for which an error detection process may be performed to accurately remove errors during or at the end of a quantum computation.

[0026] The state energy levels of the quantum information element thus correspond to excitation above a ground state. Within the accessible energy levels there exist permitted singleexcitation transitions and non-permitted single-excitation transitions, with each state of the computational subspace being selected to correspond to energy levels between which there is no permitted single-excitation. A single-excitation is understood to be an single photon excitation, being the excitation between two states as the result of a single photon interaction with the quantum information element. 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 between the states of the computational subspace correspond to a creation of dipole 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. There is no permitted transverse coupling between the two states of the quantum computational subspace. 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.

[0027] In one embodiment, the coupling between the two superconducting islands within each Josephson element is substantially the same.

[0028] In a further 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 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.

[0029] In a further embodiment the control line is arranged out of plane with respect to the quantum information element.

[0030] In a further 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.

[0031] In a further embodiment, the quantum information system further comprises a processor configured to perform a quantum information processing operation on the quantum information element, the quantum information processing comprising: initialising, via the control line, a quantum state within the quantum information element; addressing, via the control line, the quantum information element with a sequence of pulses; measuring, via the readout line, the state of the quantum information element.

[0032] In a further embodiment, the processor is configured to perform the quantum information processing operation A / times, where N is a number greater than 1000, and wherein the measured state of each repetition is added to a dataset.

[0033] In a further embodiment, the quantum information system further comprises: determining, from the measured state dataset, which measurements correspond to the third state of the quantum information element; discarding, from the measured state dataset, the measurements that correspond to the third state of the quantum information element.

[0034] In a further embodiment, the step of determining which measurements correspond to the third state is repeated during the addressing of the quantum information element with the sequence of pulses.

[0035] In a further embodiment, the quantum information system further comprises: determining, from the measured state dataset, which measurements correspond to the first state and which measurements correspond to the second state; and after discarding the measurements that correspond to the third state of the quantum information element, renormalizing the measurements that correspond to the first state and the second state.

[0036] In a further embodiment, the quantum information system further comprises one or more additional quantum information elements, and one or more additional control lines wherein each additional 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and wherein each of the additional control lines is configured to encode, into a respective one of the additional quantum information elements, one of two states of a computational subspace, the first state of the computational subspace corresponding the first state and the second state of the computational subspace corresponding to the second state.

[0037] In another aspect there is described a method of quantum information processing in a quantum information element, the quantum information element comprising 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and 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.

[0038] In an embodiment, the encoding is performed using a control line.

[0039] In a further embodiment, the coupling between the two superconducting islands within each Josephson element is substantially the same.

[0040] In a further embodiment, the method further comprises addressing the quantum information element via a control line to control the state of the quantum information element, wherein optionally the control line is coaxially disposed with respect to the quantum information element. In a further embodiment, the method further comprises measuring the state of the quantum information element via a readout line, wherein optionally the readout line is coaxially disposed with respect to the quantum information element.

[0041] In a further embodiment, the method further comprises performing a a quantum information processing operation comprising the steps of: initialising a quantum state within the quantum information element; addressing the quantum information element with a sequence of pulses; measuring the state of the quantum information element.

[0042] In a further embodiment, the quantum information processing operation is repeated A / times, where N is a number greater than 1000, and wherein the measured state of each repetition is accumulated into a dataset.

[0043] In a further embodiment, the method further comprises classifying each of the accumulated measured states into a first measured state corresponding to the first state of the quantum information element, a second measured state corresponding to the second state of the quantum information element or a third measured state corresponding to the third state of the quantum information element, and assigning the measured states classified as into the first measured state as the first state, the measured states classified as into the second measured state as the second state of the computational subspace, and the measured states classified as into the third measured state as an error state.

[0044] In a further embodiment, the method further comprises removing the measured states assigned as error states from the dataset and renormalizing the remaining accumulated measured states on the basis of the assigned first state and assigned second state.

[0045] 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 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and providing a control output 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.

[0046] The embodiments disclosed herein provide for an improved quantum information element and system. An apparatus and method are described for quantum information processing and quantum state storage with high fidelity.

[0047] Figure 1A schematically illustrates a quantum information element 10 configured to store a quantum state. The quantum information element 10 comprises a first Josephson element and a second Josephson element. Each Josephson element comprises a Josephson junction provided between two superconducting islands, wherein one of the superconducting islands is common to both the first and second Josephson elements. Referring to Figure 1A, the first Josephson element comprises first superconducting island (also called a superconducting electrode) 12 and a third superconducting island 16, and the second Josephson element comprises a second superconducting island 14 and the third superconducting island 16.

[0048] Each pair of superconducting islands within each Josephson element are coupled by a Josephson junction and are capacitively coupled together. The capacitive coupling is due to a spatial separation between the islands of each Josephson element, and the capacitance of the Josephson junction. Referring to Figure 1 , 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 connecting the second superconducting island 14 to the third superconducting island 16, 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 and the second Josephson junction J1 has a Josephson energy Ej2. Figure 1 B illustrates a circuit diagram 18 of the element 10 of Figure 1A.

[0049] 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 1 A, 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.

[0050] 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.

[0051] 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. The excited states of the quantum information element exist as a plurality of normal modes of the system, 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 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.

[0052] Figure 2 illustrates an energy level diagram of the quantum information element 10, where the states l1^) 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. There is a splitting present between the first two normal modes (| 01> and 110» shown in Figure 2, which correspond to the addition of the dipole moments in parallel or antiparallel 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>, 111> and |02», as shown in Figure 2. A singleexcitation of the quantum information element corresponds to increasing n by 1 , or increasing m by 1 .

[0053] 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 111) has a frequency of W3 and the transition between 110) and 111) has a frequency of W4, where wi W2 and where W3 t 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 quantum information element 10 can thus be labelled as a multi-mode transmon, multimode qubit, or a dimon.

[0054] In the embodiments described herein, a two-mode computational subspace is created from the two 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 Figure 2, this example computational subspace may be formed as follows:

[0055] |0)L= |10)

[0056] |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:

[0057] 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.

[0058] 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 microwave control pulses 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 |01> and 110> states. The 101) or the 110) state both lie in the transverse quantization axis of the computational subspace.

[0059] 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 |01) 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 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-photon longitudinal relaxation between the |01) and 101) states is forbidden. A transition 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 101> to state 100> and then an excitation from state 100> to state 101)), 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.

[0060] 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 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.

[0061] 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 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 the 100) state. These populated states can therefore be accurately flagged as errors and error detection performed more accurately. Any populated excited states |0)Land |1)Lcan 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 100) state to either the state |0)Lor |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 (e.g. approx. 99.5%, as described below).

[0062] 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 form the coupled josephson elements. For example, the two states may be 100) and 120> (with 110> as the erasure state), or 100) and 102) (with 101) 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 so long as there is no single-excitation transition separating them (i.e. there is no single-photon transition between the two states of the computational subspace). The choice of 101) and 110), however, means the decay from both computational bases results in the ground state, 100), which may be easily detected.

[0063] In some embodiment, the error rate of the quantum information elements described herein are improved by providing first and second Josephson elements with the same or substantially the same coupling within each Josephson element (including both the capacitive coupling and the Josephson inductive coupling). By providing the same coupling, the two Josephson elements are brought on resonance. When on resonance, the strength of the splitting between the states of the computational subspace increases, and the extent of transverse coupling between the |01> and 110) states is reduced. This results in fewer errors from both longitudinal relaxations to the ground state and transverse state switching between the excited states. In these embodiments, the superconducting islands and Josephson junctions are fabricated such that the Josephson energies EJ1 and EJ2 are substantially equal (e.g. equal to within manufacturing tolerances, which allow for a variation of 2-3% of the total Josephson energy), and the capacitances C1 and C2 are substantially equal (e.g. to within manufacturing tolerances).

[0064] In some embodiments, 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 inplane rotational symmetry.

[0065] One example of such an arrangement is the quantum information element 10 illustrated in Figure 1A. An alternative quantum information element 30 is illustrated in Figure 3A. An equivalent circuit diagram for element 30 is shown in Figure 3B, which can be seen to be identical to Figure 1A. 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 1A 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 3A, 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 1A the diameter of the outer ring is about 1 mm, and the distance between the inner islands is about 250pm. The Josephson junction inductances are each about 10nH, corresponding to a Josephson energy of about 15 - 20 GHz. The frequency unfer tile dipole mode 110> of the quantum information element is about 4.5 GHz, and the frequency coi for the quadrupole mode |01> of the quantum information element is about 6 GHz.

[0066] As described above, the fabrication of the first and second Josephson elements to be symmetric, with equal capacitances and Josephson energies, increases the value of the energy level splitting. In addition, the symmetry reduces the strength of transverse coupling between the |01> and 110> states and significantly reduces the occurrence of errors from bit fips through the transverse coupling channel. When the Josephson energies and capacitances of the Josephson elements are identical, the transverse coupling is removed entirely. Small asymmetries in the coupling may arise from manufacturing tolerances, but the resulting transverse coupling strength is significantly smaller than the longitudinal coupling between excitations. Errors from the transverse relaxations between the |01> and |10> occur on a much longer timescale than the longitudinal relaxations.

[0067] The previously described quantum information elements (e.g. 10) are used to store quantum information and to be addressed in order to transition the qubit 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.

[0068] Figure 4 shows an example quantum information system comprising the quantum information element 40 (e.g. element 10 of Figures 1A and 1 B), control line 42, readout line 44 and resonator 46. The quantum information system is coupled to a processor 50 configured to control the control line 42 and readout line 44 to perform quantum information processing. The readout line 44 and resonator 46 together form a readout element. The resonator 46 and quantum information element 40 are provided on opposite sides of a substrate 48. The control line 42 and readout element are arranged out of plane with respect to the plane of the superconducting islands of the quantum information element. The quantum information element 40, control line 42, readout line 44 and resonator 46 may be coaxially aligned, as shown in Figure 4.

[0069] 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.

[0070] The control line 42 is configured to control the state of the element by providing a control signal that couples to the modes of the element 10. For example, the control 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.

[0071] The readout element comprising the readout line 44 and resonator 46 are configured to read out a state of the quantum information element 10 in a dispersive readout regime. The resonator is coupled to the element 10, and the resonant frequency of the resonator is detuned from the frequency of excitation of the qubit state. The coupling of the qubit to the resonator thus induces a state-dependent shift in resonator frequency. The readout line 44 is configured to interrogate / address the resonator 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. 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 can be measured and the state of the element 10 can be inferred.

[0072] Figure 4B illustrates an example readout amplitude response of the readout line when addressing the resonator for a range of frequencies. Three different response lines are illustrated. Line 401 shows the resonator amplitude response, as a function of microwave probe frequency, when the element 10 is encoded with a ground state (| 00». Line 402 shows the resonator amplitude response when the element 10 is encoded with a dipole state (| 10)). Line 403 shows the resonator amplitude response when the element 10 is encoded with a quadrupole state (| 01». 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. 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 at the readout line.

[0073] In addition to an amplitude response, there will be a phase response in the pulse reflected from the resonator, having real and imaginary components. Figure 4C illustrates an example readout phase response of the readout line when addressing the resonator for a range of frequencies. Line 401 p shows the resonator phase response, as a function of microwave probe frequency, when the element 10 is encoded with a ground state (| 00». Line 402p shows the resonator phase response when the element 10 is encoded with a dipole state (| 10». Line 403p shows the resonator phase response when the element 10 is encoded with a quadrupole state (| 01». By observing the phase response of the reflected pulse it is possible to distinguish between quantum states encoded on the qubit.

[0074] In embodiments described herein, the quantum information system is configured to distinguish between all three of the states |00), |01> and 110) by addressing a single electromagnetic pulse to the resonator. The single pulse is chosen at a frequency that will drive the resonator off-resonantly to populate the resonator with photons; the readout element pulse will produce a response from the resonator 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 (410 in Figure 4B) is selected to be between the frequency response of the resonator when the quantum information element is in the 101> state, and the frequency response of the resonator when the quantum information element is in the 110> state. As shown in Figures 4B and 4C, this frequency provides a response with a maximum difference in phase shift of the pulse reflected from the resonator. The response can be measured and the in-phase and quadrature phase shifts of the response inferred to determine the measured state.

[0075] The embodiments described herein of quantum information elements including two Josephson elements having a common superconducting island (and of quantum information systems utilising such quantum information elements) provide significant improvements in coherence times, fidelity and scalability as compared to other arrangements of coupled Josephson elements (for example, where each Josephson element is a separate transmon formed of two superconducting islands, with the two transmons coupled capacitively). Due to the shared third electrode, and the fabrication to provide equal couplings between superconducting islands within a Josephson element, there is much stronger coupling between the Josephson elements of the quantum information element 10 than in a pair of coupled transmons. The strong coupling results in a much wider energy gap (e.g. 1.5GHz) between the first and second excited states in quantum information element 10 than in a pair of coupled transmons (e.g. 200MHz).

[0076] The above-described wide energy gap between the two modes |01> and 110) improves the control and readout provided above. The separation of encoding frequencies (e.g. separation by 1 ,5GHz) improves the distinction between the driving modes when driving the modes with a single control line. The frequency separation also reduces control crosstalk in encoding (i.e. accidentally driving one mode of the element 10 when it was intended to drive the other). The greater energy gap also protects the excited states from sources of decoherence, such as charge noise, photon shot-noise dephasing, critical current noise, control electronics noise, off-resonant two-level systems and 1 / f noise.

[0077] Furthermore, both states of the quantum information element 10 can be addressed using a single control line 42 (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 » A, where g is the coupling strength between the uncoupled modes due to the capacitance and A 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).

[0078] While Figure 4 illustrates a quantum information system with a single quantum information element, it will be understood that multiple ones of the element may be combined in an array, 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 those in plane with the quantum information element.

[0079] The quantum information element 40 is formed on the substrate 48 via a process of photolithography, where one side of the substrate 48 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.

[0080] 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.

[0081] A photolithography process is also performed on the opposing side of the substrate 48 to pattern the substrate 48 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 deposited prior to oxidation to form the Josephson junctions oxide barrier. Finally, a second metal evaporation at a different angle is performed. The Josephson junction is formed where an oxide barrier separates the superconducting material from the first and second depositions of superconducting material.

[0082] As described above, quantum information element 10 can be used to store quantum information in states for which error detection can be performed. After error detection is performed, the qubit state can be read out with high fidelity. An example quantum state storage and readout process for a single quantum information element is described in connection with Figures 5 - 7.

[0083] Figure 5 illustrates a process 500 for performing quantum information processing incorporating error detection. The process begins at an initialisation step where the quantum information element is addressed to initialise the quantum information element 10 in a logical state to be stored in the quantum information element 10. For example, the quantum information element 10 is addressed by the control line in one or more single-bit quantum gate operations to excite the quantum information element into a |0)L, a |1>Lstate. The prepared logical state may then be addressed by further pulses in step 520 as part of a quantum algorithm, or may be maintained in the quantum information element 10 in the initially prepared state (i.e. step 520 is not performed when the initial state is just maintained). After a time delay, the process continues to a raw data measurement process 530.

[0084] Example embodiments of the 500 are shown in Figures 6A and 6B.

[0085] Figure 6A illustrates a process 610 to address and readout quantum states prepared in one of the physical states |10> and |01> (dipole / quadrupole states) of the quantum information element described above. In process 610 the initialisation step 510 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 of the dipole mode to perform a it rotation from the 100> to the 101> 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 of the quadrupole mode to perform a n rotation from the 100) to the 101) state). In step 520 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, measurement process 530 is performed to read out the final quantum state of the algorithm.

[0086] Figure 6B illustrates an example process 620 to address and readout quantum states prepared in a superposition of the physical states 110) and |01)(dipole / quadrupole states) of the quantum information element described above. In process 620 the initalisation step 510 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 (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 step 520 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 n / 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.

[0087] In the above processes 610 and 620, pulses are addressed to each element 10 as part of the quantum algorithm in step 520, but during the algorithm some qubits may not be addressed by pulses. In alternative embodiments no algorithm is performed and the quantum information element is used to store qubits for a finite time.

[0088] The above process of Figures 6A and 6B 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 510 to address and rotate the physical states to the chosen basis. Similarly, any series of pulses may be generated for the pulse sequence operation 522 to rotate the state generated by an algorithm back to the measurement basis.

[0089] In measurement step 530, 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 above in connection with Figure 4B.

[0090] 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 qubit due (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 510 - 530 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 A / repetitions, the dataset will comprise N entries of measured IQ values. The chosen value of A / will be selected depending on the desired number L of accurate quantum state measurements and the bit erasure fraction p, where N = Z_ / (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.

[0091] An example readout is shown in Figure 6C, which illustrates the multi-state readout for the 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. Returning to Figure 5, at step 540 the raw data signal (such as that depicted graphically at Figure 6C) is extracted from the readout line addressing the resonator. The raw data is then processed in a classification step 550. 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 100), 110) and 101» 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 classification step 550, an identification step 560 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), 110) or 100).

[0092] Following assignment of states, a post selection process is performed at step 570. 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.

[0093] The above-described storage and readout process 500 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)Land |1)Lstates 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.

[0094] A continuous error detection process is performed in a similar manner to process 500 described above. However, instead of providing a joint dispersive readout at step 524 (where the readout line addresses the resonator at a frequency between the resonant peaks for the |0)Land 11)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)Land |1)L. This frequency is near (e.g. within about 100kHz of) the resonant frequency of the ground state 100). The selected readout signal pulse frequency is shown at 440 in Figure 4B. The dispersive shift (i.e. the frequency separation between the frequency 440 corresponding to the ground state |00> and each of the frequencies 420 and 430 corresponding to the computational subspace states is larger than the frequency separation between the two frequencies 420 and 430 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 440 does not distinguish between the two states |0)Land |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-|00)” maintain their quantum state and can further addressed by the control line in a subsequent series of pulses (e.g. during a quantum algorithm).

[0095] 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 101) or 101) state to decay into the 100) state, and a subsequent second-photon interaction may cause the |00> to be excited to a |01> or |01> state. The characteristic rate Q for single-photon relaxation is higher than the characteristic rate l"Tfor singlephoton excitation. Thus, the characteristic decay time (T = 1 / 1" for relaxation is less than the characteristic decay time for excitation (TT= 1 / / Ft) (e.g. T = 50 ps and TT= 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.

[0096] As with the earlier described process 500, the continuous error detection process steps 510 - 530 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 A / repetitions, the dataset will comprise N entries of measured IQ values. An example readout for the continuous error detection process is shown in Figure 6D, which illustrates the multi-state readout for the repeated measurements clustered around three locations in the IQ plane in a Gaussian pattern. The distribution pattern results from noise or imperfections within the circuit.

[0097] 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.

[0098] In some embodiments, the quantum information system is configured to perform a quantum information element performance method. The performance method is executed to determine the performance of the quantum information element, including the logical readout fidelity of the quantum information element. The performance method may assess the rate of erroneous logical bit flips, and the extent of decoherence of a superposition state.

[0099] Executing such performance checks periodically allows for performance of a quantum information element to be tracked over time. If the fidelity falls below a performance threshold, the corresponding element can be deactivated for use in any subsequent quantum algorithms.

[0100] In one embodiment, a quantum information element performance measurement comprises performing a method as described above in connection with Figure 6A. In this method, the quantum information element is initialised into a logical state |01) or 110). The encoded quantum state is then left to decay for a given decay time, in which the quantum element is not addressed with any control pulses. After the elapse of the decay time, the state is then measured as identified above. As described above, the same process is repeated N times, and error detection process 540 is performed, and the final renormalized quantum state can be measured. From this measurement, an error fraction of measurements is determined. The error fraction characterises the bit flip rate of the system (i.e. how much of the population prepared in a first computational subspace state ends up in the second computational subspace state).

[0101] Figures 7A and 7B illustrate example processes to measure coherence of a superposition of the logical states of the quantum information element.

[0102] In one embodiment, a state is initialised in the quantum information element as a superposition state of the two states of the computational subspace. The initialised quantum state is then permitted to freely evolve for a predetermined time delay. The quantum information element is then addressed with a series of pulses to rotate the state back to the logical state subspace. At least one of the series of pulses comprises a phase-offset to induce oscillations in the measured signals. The decay constant of the oscillating signals is then determined to identify the decay time of the state. This embodiment is a logical Ramsey measurement, and is illustrated in Figure 7A.

[0103] The process 710 of Figure 7A, which begins with a step to prepare the element in a superposition of the physical states |01> and |10>. For example, by a pulse at the dipole mode frequency is provided to perform a it rotation of the dipole mode, and a pulse at the quadrupole mode frequency is provided to perform a rotation of the quadrupole mode, to prepare the element into a \ )L= - = [|0)L- |1)L] state). The superposition state is then stored in the element for later readout and left to freely evolve for a time delay. After a time delay further pulses are performed to rotate the quantum state stored in the quantum information element back to the measurement basis of the control line (i.e. back to the physical state basis |01), |10». An additional stepping phase (p is provided to one of the physical states to create oscillations in the system. The measurement process 530 is then performed, in which the physical states are addressed directly as described above. The logical population fraction of each of the logical states 101) and 110), which oscillate in time, are measured. The decay time of the population fractions can thus be measured, which corresponds to the decay time of each of the logical basis states.

[0104] Figure 7B illustrates a further example process 720 to store and readout quantum states prepared in a superposition of the physical states |01> and |10> (dipole / quadrupole states) of the quantum information element described above, with an additional stabilisation of the stored quantum state. In process 720 the initialization step 510 is performed in the same manner as described above for Figure 6B. After a first time delay further pulses are performed to refocus the element (for example, by a pulse configured to perform a it rotation in the dipole mode, a pulse configured to perform a n rotation in the quadrupole mode and then a further pulse configured to perform a n rotation in the dipole mode). After a second delay further pulses are performed to rotate the state |0)Lback to the measurement basis of the control line (i.e. back to the physical state basis |01>, 110)). An additional stepping phase (p is provided to one of the physical states to create oscillations in the system. The measurement process 530 is then performed, in which the two physical states are addressed directly as described above. In this example, step 520 serves to rotate the stored quantum information into the measurement basis of the readout line and also to stabilise the system. Process 720 forms a spin echo measurement.

[0105] Table 1 below shows a physical state assignment matrix, showing the overall fraction of each assigned state for a given prepared state.

[0106] Table 1 - Physical states assignment matrix

[0107] Following the post selection process, the “00” rows and columns are removed and the remaining measured “01” and “10” states are renormalized, producing Table 2 below.

[0108] Following the error detection process 500, a high logical readout fidelity can be achieved at the expense of an erasure fraction of a number of measurement states. In the example results above, a fidelity of about 99.5% is achieved at the expense of an average erasure fraction of 20%. There is still a small amount of error present in the system (approx. 0.5%), which includes contributions from low-probability two-photon processes, and also logical assignment error (where the element was prepared in a state different to the one which was indented.

[0109] The above described examples are discussed with reference to example computational subspace states |0)L= |01> and |1)L= 110), and the state of |00> being classified as an erasure state. However, these specific states are referenced by way of example only. As explained above, the computational subspace and the erasure state may be selected from other energy level states of the quantum information element, such that there is no single-excitation transition between the |0)Land |1)Lstates, and there is a single excitation between the erasure state and each of the |0)Land |1)Lstates. The control line 42, readout element 44 and processor 50 are each configured to perform the same methodology as described above for the example states. Each of the selected states can be encoded in the quantum information element by a pulse or series of pulses, and each state will couple to the resonator to produce a corresponding resonator shift that can be interrogated from the readout line. The classifier can be trained on corresponding training data, and the classification and identification algorithms can select the chosen states to perform the error detection and post selection process.

[0110] 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 together to form an array. Figure 8 illustrates a planar schematic of an example array of quantum information elements 810. It will be understood, however, that a grid-like arrangement is only one example and other arrangements of qubits are possible including hexagonal arrays, circular arrays or other planar arrangements. The quantum information elements 810 are coupled together by a coupling 820. Figure 8 illustrates the coupling as a capacitive coupling, but other forms of coupling may also be implemented (for example a resonant coupling comprising an inductor and a capacitor). The coupled array of quantum information elements in the array can be individually addressed as part of a quantum algorithm, where the coupling between the elements enables two- and multi-qubit gate operations to be performed. Figure 9 illustrates an example quantum information system comprising the example array of quantum information elements of Figure 8 above, where each quantum information element is aligned coaxially with a separate control line, readout line and resonator in the same manner as shown in Figure 4. Only three are illustrated in Figure 9, but it will be understood that every quantum information element is provided with said coaxial components.

[0111] The array of M quantum information elements provide M independently addressable quantum memory units, each configured to store a quantum state. . Each of the quantum information elements can be separately addressed and measured in the manner described above for one quantum information element, and the error detection and compensation methods provided above can be applied to each element independently. Thus the array is configured to operate as an M-bit quantum processor with high fidelity of stored and processed quantum information.

[0112] 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

CLAIMS:1 . A quantum information system comprising a quantum information element and a control line, the quantum information element comprising: 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and 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.

2. The quantum information system of claim 1 , wherein the coupling between the two superconducting islands within each Josephson element is substantially the same.

3. The quantum information system 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 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.

4. The quantum information system of any preceding claim, wherein the control line is arranged out of plane with respect to the quantum information element.

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

6. The quantum information system of claim 5, further comprising a processor configured to perform a quantum information processing operation on the quantum information element, the quantum information processing comprising: initialising, via the control line, a quantum state within the quantum information element; addressing, via the control line, the quantum information element with a sequence of pulses; measuring, via the readout line, the state of the quantum information element.

7. The quantum information system of claim 6, wherein the processor is configured to perform the quantum information processing operation A / times, where N is a number greater than 1000, and wherein the measured state of each repetition is added to a dataset.

8. The quantum information system of claim 7, further comprising: determining, from the measured state dataset, which measurements correspond to the third state of the quantum information element; discarding, from the measured state dataset, the measurements that correspond to the third state of the quantum information element.

9. The quantum information system of claim 8, wherein the step of determining which measurements correspond to the third state is repeated during the addressing of the quantum information element with the sequence of pulses.

10. The quantum information system of claim 8 or claim 9, further comprising: determining, from the measured state dataset, which measurements correspond to the first state and which measurements correspond to the second state; and after discarding the measurements that correspond to the third state of the quantum information element, renormalizing the measurements that correspond to the first state and the second state.11 . The quantum information system of any preceding claim, further comprising one or more additional quantum information elements, and one or more additional control lines wherein each additional 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and wherein each of the additional control lines is configured to encode, into a respective one of the additional quantum information elements, one of two states of a computational subspace, the first state of the computational subspace corresponding the first state and the second state of the computational subspace corresponding to the second state.

12. A method of quantum information processing in a quantum information element, the quantum information element comprising 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and 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.

13. The method of claim 11 , wherein the coupling between the two superconducting islands within each Josephson element is substantially the same.

14. The method of any one of claims 11 - 13, further comprising addressing the quantum information element via a control line to control the state of the quantum information element, wherein optionally the control line is coaxially disposed with respect to the quantum information element.

15. The method of any one of claims 11 - 14, further comprising measuring the state of the quantum information element via a readout line, wherein optionally the readout line is coaxially disposed with respect to the quantum information element.

16. The method of any one of claims 11 - 15, further comprising performing a a quantum information processing operation comprising the steps of: initialising a quantum state within the quantum information element; addressing the quantum information element with a sequence of pulses; measuring the state of the quantum information element.

17. The method of any one of claims 11 - 16, wherein the quantum information processing operation is repeated A / times, where N is a number greater than 1000, and wherein the measured state of each repetition is accumulated into a dataset.

18. The method of claim 17, further comprising classifying each of the accumulated measured states into a first measured state corresponding to the first state of the quantum information element, a second measured state corresponding to the second state of the quantum information element or a third measured state corresponding to the third state of the quantum information element, and assigning the measured states classified as into the first measured state as the first state, the measured states classified as into the second measured state as the second state of the computational subspace, and the measured states classified as into the third measured state as an error state.

19. The method of claim 18, further comprising removing the measured states assigned as error states from the dataset and renormalizing the remaining accumulated measured states on the basis of the assigned first state and assigned second state.

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 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; wherein the superconducting islands of the first Josephson element are coupled to the superconducting islands of the second Josephson 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 between the third state and the first state and wherein there is a single-excitation transition between the third state and the second state; and providing a control output 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.

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