Qubit-based device

Abstract

Apparatus is disclosed comprising a device (3) which comprises a circuit (30; Fig. 19) of superconductor material disposed on substrate (31; Fig. 19). The circuit is configured to provide a probe qubit (4) and a coupling circuit (5) for coupling the probe qubit to a Josephson junction-based target circuit (7).

Description

[0001] Qubit-based device

[0002] Field

[0003] The present invention relates to a qubit-based device.

[0004] Background

[0005] Nanostructured low-dimensional materials hosting topological and interacting electronic phases can be used to explore fundamental ideas in condensed matter, and could potentially enable next generation quantum devices.

[0006] Since the 1980s, millikelvin conductance measurements have played a role in detecting interaction-dominated phases such as the fractional quantum Hall in 2D, Luttinger liquids in ID, and Kondo resonances in 0D. The search for transport signatures of topological ground states over the last decade, however, has proved challenging.

[0007] Quantised conductance signatures associated with Majorana bound states in topological superconductors are easily mimicked by disorder and compete with enhanced Coulomb and confinement interactions, and reference is made to V. Mourik et al. Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science, 336(6084), (2012). Eliminating false positive signatures may require reducing disorder and subjecting devices to elaborate measurement protocols and reference is made to S. Das Sarma and H. Pan. Disorder-induced zerobias peaks in Majorana nanowires. Phys. Rev. B, 103: 195158, (2021) and to Das Sarma and H. Pan. Inas-al hybrid devices passing the topological gap protocol. arXiv 2207.02472, (2022).

[0008] Summary

[0009] According to a first aspect of the present invention there is provided apparatus comprising a device comprising a circuit of superconductor material disposed on substrate, wherein the circuit is configured to provide a probe qubit and a coupling circuit for coupling the probe qubit to a Josephson junction-based target circuit.

[0010] This can allow projective measurements to be performed on the probe qubit so as to extract target qubit's ensemble probabilities.

[0011] The apparatus may further comprise a system configured to provide control signals to the device and to receive measurements signals from the device so as to perform charge or energy basis tomography.

[0012] The device may comprise a tuner and the system may provide a first set of control signals to the tuner for controlling coupling between the probe qubit to a Josephson junction-based target circuit.

[0013] The Josephson junction-based target circuit may be provided with at least one gate for controlling Josephson energy in the Josephson junction-based target circuit. The system may provide a second set of control signals to the at least one gate for controlling the Josephson energy in the Josephson junction-based target circuit.

[0014] The coupling may be charge-energy coupling or hybrid charge-longitudinal coupling.

[0015] The probe qubit may be couplable to the target circuit via a term of a type ~ft8zwherein n is the charge degree of freedom of the target qubit and 5Zis the longitudinal coupling of the probe qubit. The coupling circuit may comprise a capacitor. The coupling circuit may further comprise an inductor arranged in series with the capacitor to form a resonant circuit. The coupling circuit may further comprise a first capacitor arranged to couple the probe qubit to the resonant circuit, wherein the capacitor is a second capacitor, wherein the coupling circuit further comprises a third capacitor for coupling the resonant circuit to the target circuit.

[0016] The probe qubit may be a flux qubit. The flux qubit may comprise a four-junction flux qubit comprising four Josephson junctions arranged in series in a loop.

[0017] The probe qubit may be a transmon. The coupling may be a capacitive coupling. The superconductor material may be niobium.

[0018] The substrate may be a silicon substrate.

[0019] According to a second aspect of the present invention there is provided an apparatus comprising: the apparatus of the first aspect of the invention, wherein the device is a first module and the substrate is a first substrate; and a second module comprising regions of superconductor material disposed on a second substrate, the regions of superconductor material arranged to form a Josephson junction which provides the target qubit.

[0020] The superconductor material may be niobium. The second substrate may comprise a semiconductor material. The first and second modules may be connected via bump bonds such that the first and second modules are electrically connected.

[0021] According to a third aspect of the present invention there is provided a method of performing charge or energy basis tomography, the method comprising providing control signals to a device which comprises a circuit of superconductor material disposed on substrate, wherein the circuit is configured to provide a probe qubit and a coupling circuit for coupling the probe qubit to a Josephson junction-based target circuit, receiving measurements signals from the device, and processing the measurements signals so as to performing charge or energy basis tomography.

[0022] According to a fourth aspect of the present invention there is provided a method of measuring a charge state of a Josephson junction-based target circuit which is coupled to a probe qubit, the method comprising performing a measurement for a predetermined time τ.

[0023] • decoupling the probe and target qubits;

[0024] · while the probe and target qubits are decoupled and the Josephson energy of the target qubit is non-zero:

[0025] - preparing an initial state of the target qubit; and

[0026] - applying a half π-pulse to the probe qubit;

[0027] · coupling the probe and target qubits and reducing the Josephson energy of the target qubit to zero;

[0028] · while the probe and target qubits are coupled and the Josephson energy of the target qubit is at zero:

[0029] - waiting until a predetermined time τ; and

[0030] - applying a half π-pulse to the probe qubit; and - measuring the state of the probe qubit.

[0031] Thus, probabilities of the target qubit's probabilities can be extracted via projective measurements of the probe qubit.

[0032] The method may comprise repeating the measurement for different predetermined period of times τ. Preparing the initial state of the probe qubit may comprise applying a half π -pulse to the probe qubit to obtain an equal superposition of ground and excited states. Measuring the state ZPof the probe qubit may comprise applying a half n-pulse to the probe qubit.

[0033] According to a fifth aspect of the present invention there is provided a method of measuring an energy state of a target transmon which is coupled to a probe transmon, the method comprising:

[0034] · increasing a gate bias applied to a target transmon from a first value to a second value;

[0035] · after increasing the gate bias applied to the target transmon from the first value to the second value, decoupling the probe and target transmons;

[0036] · waiting a given time;

[0037] · after waiting a given time from decoupling the probe and target transmons, re- coupling probe and target transmons;

[0038] · after re-coupling the probe and target transmons, decreasing the gate bias applied to the target transmon from the second value to the first value; and

[0039] · after decreasing the gate bias applied to the target transmon from the second value to the first value, measuring the state of the probe transmon.

[0040] This sequence of pulses can help to improve mixing of on-diagonal and off-diagonal components of the density matrix describing the target transmon.

[0041] According to a sixth aspect of the present invention there is provided a computed- implemented method of reconstructing a density matrix of a target qubit-based system. The method comprises extracting a set of probabilities pnof the state of a target qubit-based system in a charge or energy basis from a set of measurements of a probe qubit configured to sense the target qubit-based system; and solving at least one mixing matrix using the set of probabilities pn.

[0042] According to a seventh aspect of the present invention there is provided a computed- implemented method of characterising a material having a predicted property exhibiting an expected tomographic pattern including a predicted entanglement entropy. The method comprises performing charge or energy basis tomography for a range of swept parameters to obtain a measured entanglement entropy of a state of a target junction-based system formed using the material for the range of swept parameters, determining whether the predicted entanglement entropy and the measured entanglement entropy match, and outputting the result of the determination.

[0043] The property may be whether the material is in a topologically non-trivial phase.

[0044] The junction may be a semiconductor junction and / or superconducting junction.

[0045] Obtaining the measured entanglement may comprise providing a density matrix for the target system, and tracing out the elements of density matrix until the entanglement entropy is reached.

[0046] Bri ef Description of the Drawings

[0047] Certain embodiments of the present invention will now be described, by way of exampie, with reference to the accompanying drawings in which:

[0048] Figure 1 is a schematic diagram of a protocol for quantum tomography reconstruction of a superconducting / hybrid circuit via projective measurements on a coupled probe qubit; the coupling term of the form containing the target's charge degree of freedom and the qubit's longitudinal one, allows the extraction of the target's ensemble probabilities via the projective measurements of the probe qubit's dynamics; combining the information obtained from the dynamics of different initial states of the target circuit, a system of equations can be built, connecting the measured diagonal elements of p to the off-diagonal ones, and the density matrix of the target system, for example, a target qubit, can be reconstructed.

[0049] Figure 2 schematically illustrates a measurement protocol used for extracting the diagonal elements of the target circuit's density matrix; for t < 0, the target and qubit systems are prepared into their initial states; in particular, the qubit is prepared into an equal superposition of ground and excited states via the application of half jr-pulse; at to ~ 0 the coupling is switched on, while the Josephson energy E}parameter is fluxed down to zero; this allows the implementation of a Ramsey-like measurement of the probe qubit, whose dynamics will be characterised by the charge-basis components of the target system.

[0050] Figure 3 is a flow diagram of a method of reconstructing a density matrix.

[0051] Figure 4 is a flow diagram of a method of refining a reconstructed density matrix.

[0052] Figure 5 shows charge components of the ground state of a transmon at Ej / Ec- 50 and Ej / Ec- 10.

[0053] Figure 6 is an example of application of a tomography protocol for reconstructing the ground state's density matrix of the transmon; Figure 6a illustrates matrix elements with , used for the reconstruction of the ground state's density matrix at Figure 6b is a comparison between the ground state numerically determined from the Hamiltonian and the reconstruction using the least square regression method and ~ 20 configuration points with the lower panel shows the absolute value of the difference between each element of the two density matrices. Figure 7 is a schematic diagram of the resonator-coupled circuit; the loop on the left is the four junction flux qubit, referred to herein as the "probe qubit"; The upper two junctions are scaled relative to the size of the lower two junctions by a factor of a; flux is threaded through the circuit, denoted by fp, allowing the circuit to be biased at a half flux quantum; a gate charge is applied to the node q3 which results in a charge offset which is denoted by n5; the flux qubit is capacitively coupled to the resonator, shown as the middle loop; the resonator consists of an inductor Lrand a capacitor Cr, connected in series; the resonator is also capacitively coupled to the target qubit, the loop on the right; the target qubit consists of a capacitor and Josephson junction connected in series.

[0054] Figures 8a and 8b are plots of probe qubit gap against the charge offset applied to q3for the directly coupled circuit shown in Figure 3 for a range of probe qubit asymmetry a and the probe-resonator coupling capacitance set to zero (Figure 8a) and for a range of probe-resonator coupling capacitance with the probe qubit asymmetry a set to 0.4 (Figure 8b); the circuit is modelled using

[0055] Figure 8c is a plot of longitudinal coupling strength between the probe qubit and resonator against the charge offset applied to q3for a range of probe-resonator coupling capacitance with the probe qubit asymmetry a set to 0.4); the circuit is modelled using

[0056] Figure 8d is a plot of longitudinal coupling strength vs the probe resonator coupling capacitance with a charge offset of applied to and the probe qubit asymmetry set to 0.4); the circuit is modelled using

[0057] Figure 9 is a plot of probe qubit frequency for a range of gate charge with asymmetry introduced between the upper two junctions of the qubit.

[0058] Figure 10 is a schematic diagram of a quantum circuit describing the target qubit. Two superconductors (hatched) with Cooper pairs are coupled via a dot (cross hatched) with fermions, forming a Josephson junction. The Josephson junction is connected in parallel to another Josephson junction, with the Josephson energy, and is controlled by / two g S3ates inducing S3 offset charg S3es . An external flux controls the Josephson energy.

[0059] Figure Ila is a greyscale plot of probability density P(z?) as a function of charge difference n for a range of values of top gate charge Ng. Three points of interest for Ng, namely Ng= -2, -0.5 and 0.5, are chosen and are marked by the dashed lines.

[0060] Figures 11b to lid are plots of probability density as a function of Cooper pair number n for top gate charge Ngfor 0.5, -0.5 and -2, respectively. Figure 11b shows the probability distribution in the peak of entanglement entropy with the dot's occupation expectation value . Figure 11c corresponds to and shows a density centred around due to a non-zero offset charge gvalue. Figure lid has . Here .

[0061] Figures 12a and 12b show spectrum characteristics of a Majorana-DT. In Figure 10a fine-structure of the lowest Transmon level is presented for the exact numerical analysis (solid line) and the projected 8-dimensional Hamiltonian (dashed line) with the offset charge ng= 1 / 8; in the figure, the approximate dot's spin sector is marked alongside its corresponding energy level. Figure 12b shows entanglement entropy S as a function of the top gate charge Ng, with offset charge ng= 0 (purple, Pu) and ng- 1 / 4 (green, Gr). The blue shading B represents intermediate values of The red dotted line (R) represents the approximate formula in Equation (B14) below which is independent of the side gate ng. The peak points are marked in dashed yellow lines (Y). Here,

[0062] Figures 13a to 13c show spectrum characteristics of a Majorana-DT. Figure 13a shows Energy levels as a function of top gate charge with offset charge ; each set of lines (Black, Blue, Red) is associated with a different Transmon level, split due to the dot and Majorana degrees of freedom. Figure 13b shows the polarization of the dot in the x direction as a function of top gate charge for three side gate values, with (dot-dashed) and (solid) giving the maximal positive and negative polarization values, respectively, and (dot-dashed) a zero polarization. Figure 13c shows dot's occupation as a function of top gate charge A / ; the blue and red lines, corresponding to respectively, are numerical solutions to Equation (BIO), while the dashed lines are calculated following its projection to the Transmon's ground state. Here , but the dependence on gamounts to negligible shifts. The parameters used throughout

[0063] Figure 14 is a schematic block diagram of a system for quantum tomography reconstruction.

[0064] Figure 15 is a schematic block diagram of an arrangement for controlling coupling parameter g.

[0065] Figure 16 is a schematic block diagram of an arrangement for controlling Josephson Energy .

[0066] Figure 17 schematically illustrates a process of quantum tomography reconstruction.

[0067] Figure 18 is a plan view of a quantum entanglement spectrometer which comprises a quantum entangler module and a Josephson junction module which hosts a material under test.

[0068] Figure 19 is a schematic circuit diagram of a quantum entangler module and a Josephson junction module.

[0069] Figure 20 is a schematic perspective view of the Josephson junction module shown in Figure 19.

[0070] Figure 21 is a schematic circuit diagram of two capacitively-coupled transmons with coupling capacitance Cc.

[0071] Figure 22 schematically illustrates a measurement protocol used for extracting the diagonal elements of a target circuit's density matrix in energy basis tomography. Detailed description of Certain Embodiments

[0072] Introduction

[0073] Semiconductor-based Josephson junctions embedded in a Cooper pair box can host non-trivial, many-body states including interacting Andreev states and topological quasi-particles. A method of realising quantum state tomography based on measuring relative superconducting charge across a junction is herein disclosed. Measurements give direct access to entanglement entropy and entanglement spectrum of a partitioned system.

[0074] Semiconductor-superconductor junctions can play host to unusual excitations including Andreev states and semiconductor nanowires can, in some cases, exhibit Majorana zero modes. When the superconductors form isolated islands, the interplay between Andreev tunnelling and charging energy gives rise to collective states consisting of correlated superconducting charge states and microscopic low-lying excitations of the junction. The resulting states may have properties capable of storing quantum information. When the junction is coupled to a cavity, spectroscopic methods may be used to extract properties of the states. Spectroscopy can be a sensitive probe for the spectrum of the junction and its transition rules.

[0075] Partitioning quantum systems can reveal unique information about topological and interacting phases. Superconductor-semiconductor junctions can be partitioned into a subsystem of bosonic Cooper pairs, spanned by the superconducting charge states, and a subsystem of low-lying fermionic excitations of the junction. There is a question as to how much information regarding the state of the junction can be extracted by observing the superconducting states alone, in particular, whether the charge projected make-up of these correlated states can reveal information about the nature of the fermionic excitations of the junction. Moreover, reconstructing the charge-basis reduced density matrix may expose non-trivial correlations between the Cooper pairs and the low-lying fermionic excitations, via properties such as the entanglement entropy and entanglement spectrum, that might be difficult to exact otherwise.

[0076] A method of tomographically reconstructing the reduced density matrix of a Cooper pair box state (herein also referred to the "target system") in the charge states basis via quantum state measurements of an additional qubit (herein also referred to as the "probe system") is herein described. In order to achieve the charge basis measurement, the probe and target systems are coupled via a strong hybrid chargelongitudinal interaction which can be obtained by combining a capacitive coupling and a voltage biased flux qubit. The direct measurement of the probe qubit frequency gives access to the diagonal element of the density matrix, while the off-diagonal terms can be extracted via a pulsing protocol similar to cavity state tomography schemes used in circuit quantum electrodynamics (QED).

[0077] These methods are applied to several types of topological and non-topological semiconductor-superconductor hybrids. When the junction comprises a dot, the fermionic state brings the junction into a superposition of odd and even charge states, that may be revealed through the charge state make-up of the collective state. When the junction also admits Majorana fermions, the system can be brought into a coherent superposition of bosons, fermions and Majorana modes, that results in superposing two sets of even and odd charge states, that are mutually shifted by a universal fractional offset charge.

[0078] Charge-basis tomography

[0079] The tomography protocol used to reconstruct the system's density matrix will now be described. In the following, it is assumed that the system is formed by an unknown circuit containing the junction j under study and characterised by one generalised coordinate only, and by a probe qubit q coupled to the target via a term of the type where n the charge degree of freedom of the target circuit and is the longitudinal coupling of the probe qubit. The total Hamiltonian of the coupled target and qubit system can be written as: where is the Hamiltonian of the target system, g is coupling and A is the frequency of the probe qubit.

[0080] Figure 1 illustrates an overall scheme for reconstructing a system's density matrix. The scheme is based on the principle that implementation of known operations on the target system (for example, a target qubit) mixes (in the sense of unitary evolution) the components of the density matrix ptJin a controlled manner leading to a state . Because of the target-qubit entangled evolution, projective measurements performed on the probe qubit can lead to the extraction of the target's ensemble probabilities , where is the state of the target circuit. The latter is a projection on the charge basis which is possible only when the coupling is implemented. The ground state of Josephson junction-based devices has a structure in the charge basis, which allows comparison with theoretical models. When this protocol is repeated using different target circuit's states, a system of equations in the density matrix components can be constructed and solved. Since the system is infinite, the reconstruction of the quantum state can be done when the basis sets are truncated. This technique was used for the reconstruction of the wavefunction of trapped atoms in harmonic potentials as described in D. Leibfried et al. : "Experimental determination of the motional quantum state of a trapped atom", Physical Review Letters, volume 77, page 4281 (1996) and has been applied for the quantum state reconstruction of both linear and nonlinear superconducting cavities via conditional measurements of the cavity photon number as described in M. Hofheinz et al. \ "Synthesizing arbitrary quantum states in a superconducting resonator", Nature, volume 459, page 546 (2009) and G. Kirchmai et al., "Observation of quantum state collapse and revival due to the single-photon Kerr effect", Nature, volume 495, page 205 (2013). In these previously-described reconstructions, the chosen basis for projection was made of eigenstates of both the coupling term and the Hamiltonian of the system leading naturally to a quantum non-demolition measurement. In the reconstruction herein described, these measurement schemes are extended to charge-based protocols for systems hosting a semiconductor-insulator-semiconductor (SIS) junction, that is transmon qubits. The approach is used for charge-based tomography of a nanowire- based junction.

[0081] The tomography protocols employed in circuit QED are usually performed for the reconstruction of the system density matrix in its energy basis. For instance, the reconstruction of the cavity states in G. Kirchmair ibid, and in M. Hofheinz et al. : "Generation of Fock states in a superconducting quantum circuit", Nature, volume 454, page 310 (2008) is done in the photon number basis, whose operator commutes with its free Hamiltonian. In the system described herein, the use of an interaction term that couples the qubit to the charge degree of freedom of the target system, together with the measurement protocol described hereinafter, enables qubit dynamics which expose the charge components of the target circuit. In this way, it allows the exploration of the system's density matrix in the charge basis rather than its energy basis, which can allow further study of the properties of hybrid junctions.

[0082] For explanatory purposes, the reconstruction of the target circuit's ground state will be described hereinafter. Measurement protocol

[0083] Referring to Figure 2, a protocol for the extraction of the target system's probabilities via projective measurements of the probe qubit will now be described.

[0084] To be able to obtain the qubit dynamics without spurious interactions, coupling g is switched on at t > 0, after preparation of the target system's state |TV(0)>| and qubit's initial state . For the same reason, the tunability of the Josephson energy E. in the target circuit for switching the system into the charging regime is desirable. This can be experimentally done using a symmetric split Josephson junction, where the effective Josephson energy can be fluxed down to zero via an external magnetic field.

[0085] The coupling g of a coupling circuit 5 (Figure 14) can be controlled using a tuner 6. For example, the coupling circuit may take the form of a tuneable resonator which contains a flux loop 21 (Figure 15) and the tuner 6 may take the form for a magnetic field generator, such as a wire or loop 22, driven by a source 12i (Figure 15) which changes the flux in the flux loop 21, thereby varying the coupling provided by the coupling circuit 5.

[0086] The Josephson energy E:of the target circuit can be controlled using a gate 10 (Figure 16).

[0087] The protocol can be summarised as follows:

[0088] (51) While g = 0, the target circuit is initialised into its ground state at a specific value of . At the same time, with the use of a half π-pulse the probe qubit is prepared in the equal superposition of ground and excited state,

[0089] (52) At to ~ 0, the Josephson energy of the target circuit is parametrically switched off, for example, using the gate 10 (Figure 16), while the coupling g is turned on using the tuner 6 (Figure 15). The speed at which the tuning is implemented is high enough not to affect the system's state

[0090] (S3) Similar to a Ramsey-like experiment, the probe qubit is left evolving for a time T according to Hamiltonian in equation (1) with in which case depends only on the charge operator n and the qubit dynamics contains the charge components of

[0091] (54) Another half jr-pulse is performed at t = T on the probe qubit allows for the projective measurement of the qubit components on the azbasis.

[0092] (55) Repetition of the previous steps ( / .e., SI to S4) at multiple values of

[0093] T leads to the reconstruction of the expectation value of the axoperator over time: whose Fourier components are proportional to the ensemble probabilities

[0094] In this scheme, the Ej- para meter tuning is used for a proper readout of the charge components. If a simple junction model is considered that can be described by only one degree of freedom, its Hamiltonian can be written in or at least approximated to the form: where are the ladder operators associated with the charge operator, are polynomial functions of n and nT, respectively and are a characteristic charging energy and Josephson energy, respectively, associated with the junction. For instance, for an SIS junction, When the coupling is switched on after preparing the combined system in the initial state: where evolves accordingly to the Hamiltonian in equation (1), above. Switching the parameter Ej to the charging regime while the coupling is turned on removes the presence of spurious contributions to the qubit dynamics originating from the presence of non-diagonal terms to the Hamiltonian in equation (2), above. This can be seen when we neglect h,in equation (2) and express the target state in the charge basis, In this way, the state evolves as: and the expectation values of the operator takes the form: where . When the evolution of is tracked, the probabilities can be read in its Fourier transform.

[0095] Density matrix reconstruction

[0096] To access the off-diagonal elements of the density matrix, controlled T(rf) operations (where g is a control parameter, such as Josephson Energy Ej) can be engineered and applied to the target system, for example the target qubit, before the probe measurements are performed and reference is made to M. Hofheinz ibid, and G. Kirchmair ibid. Depending on the transformation, the measured of the transformed system can be written as a combination of the elements of before the transformation:

[0097] By applying multiple transformations ?(?? / ) and measuring multiple diagonal elements n, the density matrix of the cavity can be reconstructed, up to some truncation Nt. The system of equations obtained using numerous values of can be solved if at least one of the transformations applied produces nonzero matrix elements for Given that real systems present some noise, a linear regression model can be used, as described in G. Kirchmair ibid.

[0098] The ground state of the target circuit can be reconstructed by probing the circuit at different values of the Josephson energy parameter E,. The transformation can therefore be seen as the time evolution operator obtained using an adiabatic pulse that tunes the parameter E}from the value at which it is desired to reconstruct the wavefunction, namely to the value at which we measure the ensemble probabilities, E7. If tuning is done slowly enough, the adiabatic transformation is represented by the sum of the projectors connecting each eigenfunction at the two different values of where is the dynamical phase of the eigenstate p accumulated during the adiabatic evolution, and is the eigenenergy, changing parametrically with E;during the adiabatic tuning. The matrix My' takes the form using a shortened notation

[0099] Referring to Figures 1 and 3, reconstructing the density matrix begins with a model of the target system in the form of a Hamiltonian (step S3.1) and determining the truncation needed to describe the system's ground state in charge basis (step S3.2). The Hamiltonian can include terms corresponding to, for example, charging energy, Josephson energy, and / or Majorana quasi-particles. A list of transformations is obtained based on a list of values of the control parameter (step S3.3).

[0100] The target's ensemble probabilities pnare extracted from the pulse measurements (step S3.4). The mixing matrix is calculated (step S3.5). An element of the mixing matrix can be seen as a measure of how much information the probability will contain about the matrix element of the density matrix.

[0101] Using the measured probabilities pn, the mixing matrix can be solved for each of the unknown elements using a linear regression model (step S3.6).

[0102] Thus, the density matrix can be reconstructed by evaluating the corresponding mixing matrices numerically, and solving the system of equations using an experimentally measured set of probabilities

[0103] Referring also to Figure 4, the reconstructed density matrix can be reviewed and refined. Taking the target model (step S4.1) and a list of transformations (step S4.2), the density matrix is calculated (step S4.3). The errors on the density elements are examined to check that they are sufficiently low, i.e,, do not exceed a threshold (step S4.4). If so, then the density matrix is output (step S4.5). If, however the errors exceed the threshold, then a determination is made as to whether the problem relates to the number of measured points (step S4.6). If the issue lies with the number of measured points, then the number of points is increased (step S4.7) and the process calculating the density matrix with a larger set of points is repeated (step S4.7). If the number of points is not the issue, then the model is reviewed (step S4.8).

[0104] Transmon case

[0105] As an example, the system of equations obtained when probing a simple system made of a Josephson junction-based superconducting island (charge-based qubit) in the regime of high Josephson energy (transmon regime) can be determined. In this case, the target system's Hamiltonian can be written as: where ngis the offset charge of the superconducting island written in terms of the number of Cooper pairs. Ej can be representing an effective Josephson energy of a split junction, thus it can depend on an external flux parameter, that is, The solutions of Equation (9) are superpositions of Mathieu's functions, and in the regime where they can be approximated by the solution of a quantum harmonic oscillator, with frequency where -order

[0106] Hermite poiynomial. For the ground state it just corresponds to a gaussian function with Therefore, to obtain a good resoiution of the ground state the expansion in the charge basis needs to have a truncation of . For a circuit with this corresponds to N ~ 5 as shown in Figure 5. The eigenstates coefficients in the charge basis appearing in Equation (8) can be written as

[0107] Equation (11) has the form of a Fourier transform of a function of calculated at For instance, for j = 0 (ground state) c„ is a gaussian function,

[0108] For a generic j, the coefficients can be determined by taking the Fourier transform of the completeness relation for the Hermite polynomials,

[0109] Both sides of equation (13) are integrated: The summation's terms of the right-hand side of the equation above are the expressions needed, while the left-hand side takes the form:

[0110] By equating the coefficients of the two sums and substituting the expression for is obtained: with the substitution With this expression, the matrix elements can be explicitly written:

[0111] From equation (17), it is clear that the adiabatic transformation mixes the components of the wavefunction in a nontrivial way.

[0112] Figure 6a shows, for instance, the matrix element related to the case . Because of the common scaling factors at the beginning of Equation (17), the matrix elements are nonzero only for . Note that the resolution of the probabilities is reduced because only the probabilities of the wavefunction can be measured.

[0113] Least square regression

[0114] It is desired to find a solution or at least the best approximation to the solution of the system of equations (6), the unknown variables and a certain system's configuration.

[0115] If the ground state of the system is probed, then it can be assumed therefore only a finite truncation of the wavefunction, needed. At the same time, the measured probabilities will only be visible for . Assuming to be able to solve the system of equations, the system is probed at multiple configurations with

[0116] In this way the system is overdetermined, and an approximated solution can be found using a least square regression model.

[0117] As an example, take the case of the ground state withE This state can be reconstructed with a truncation of has a dimension of 11 x 11. On the other side, when we probe a state at the resolution allows measurement of up to , giving 5 measurements of the diagonal elements of At least about 25 points are needed to be able to reconstruct the density matrix . This number can be lower depending on how many probabilities values can be measured for each configuration. A simulation of measurements, with zero-noise error, of at 21 configuration points, with 10 and perform a least square regression using Equations (6) and (8). For the latter, since an eigenstate of the system is being probed, the dynamical phases accumulation not contribute to the final density matrix, thus they can be set to zero. It can be seen from the lower panel of Figure 6b that the least-square regression model is able to reconstruct the ground state of the system within a fifth-digit error. For the reconstruction of an arbitrary superposition of eigenstates of the transmon, the time spent during the adiabatic transformation needs to be taken into account, and the dynamical phases cannot be omitted. Example of a circuit to realise tomography protocol

[0118] The measurement protocol hereinbefore described uses longitudinal coupling between a target and a probe qubit.

[0119] Referring to Figure 7, to realise this coupling, a four-junction flux qubit can be used as the probe, coupled to a tunable Cooper pair box as the target. Coupling can be provided using a resonator (as illustrated) or via direct capacitive coupling.

[0120] Reference is made to P.-M. Billangeon et al. -. "Circuit QED-based scalable architectures for quantum information processing with superconducting qubits", Physical Review B, volume 91, 094517 (2015). In P.-M. Billangeon ibid., a circuit QED architecture is described in which the coupling degree of freedom can be controlled. This can be in an arrangement providing, on the one hand, strong longitudinal coupling while, on the other, minimising the nutations on the probe qubit which are induced by transverse coupling with the target qubit.

[0121] The four-junction flux qubit can have the advantage that it can be maximised for integer charge offsets whilst at half-integer offsets the probe frequency vanishes. This is because it two Josephson junctions surround a charge there will be interference between the two paths around the charge. When the charge is equal to e ( / .e., a halfinteger number of cooper pairs), the relative phase of the two paths is equal to n, resulting in destructive interference. This destructive interference becomes complete if the two surrounding junctions are symmetric.

[0122] Figure 8(a) shows how the longitudinal coupling strength depends on the gate charge. The relationship can be understood by noting that, from the Feynman-Hellmann theorem, it can be expected that This is seen in Figure 8(c), where the longitudinal coupling strength is seen to follow the gradient of Figure 8(a). Given this relationship, it can be seen that the probe frequency and the longitudinal coupling can be simultaneously tuned using the three degrees of freedom:

[0123] Referring to Figure 9, the perpendicular coupling strength is shown as a function of offset charge for different variations in Josephson energy of the upper two junctions of the four-junction flux-based probe qubit shown in Figure 7. A 1% variation in these alpha junctions can lead to undesirable perpendicular coupling. This should be compared, however, to the relative strength of the longitudinal coupling that is required for charge-energy coupling. Circuit Analysis

[0124] A method of nodes approach is applied to the circuit shown in Figure 7, and reference is made to S. Rasmussen et al. "Superconducting circuit companion—an introduction with worked examples", PRX Quantum 2, 040204 (2021).

[0125] In the method of nodes, the circuit is divided into a graph of nodes, such that every component is separated by a node. These nodes are categorised into ground, passive and active nodes. The ground node is connected to the ground, whilst the passive and active nodes are distinguished by whether they passively connect components of the same type (either only capacitors or only inductors, including Josephson Junctions (JJs)) or if they actively connect different component types.

[0126] The graph of nodes can be divided into capacitive and inductive subgraphs. Since the capacitive subgraph only contains linear capacitance, we can express the capacitive energy in terms of the flux derivative. If the flux co-ordinate is considered as analogous to position, capacitive and kinetic energy can be seen as equivalent, likewise, inductive and potential energy are equivalent.

[0127] With the graph of nodes representation, the concept of a spanning tree - a tree connecting every node in the circuit by only one path - can be introduced. Although the spanning tree can be defined in many ways, the exact choice does not change the underlying calculation.

[0128] Given an arbitrary choice of spanning tree, the circuit is stepped through, starting with the tree's ground node. For each node, an energy term is added to the Lagrangian, if the node closes a loop in the circuit, an external flux is added into the energy term. These energies derive from the flux difference between neighbouring nodes seen in Table Al. Table Al

[0129] The calculated Lagrangian will be a sum of capacitive and inductive energies. This can be simplified by introducing the capacitance and inductance matrices, resulting in a Lagrangian of the form: where C and L-1denote the capacitance and inductance matrices respectively.

[0130] From the Lagrangian we find the conjugate momenta associated with the node flux as:

[0131] From equation Al it can be seen that this yields the charge-flux matrix equation q = Given this relationship, a Legendre transformation can be performed on the Lagrangian in equation Al, resulting in a Hamiltonian of the form: where the last term sums over the energy contributions of each Josephson junctions in the circuit. Applying this method to the circuit shown in Figure 7, the following Lagrangian is found: (A4)

[0132] From this Lagrangian and equation A2 we can then calculate the node charges as:

[0133] For convenience, these equations can be grouped into the capacitance matrix, defined by the relation q = C<t>, resulting in the capacitance matrix:

[0134] The Hamiltonian of a superconducting circuit is given by:

[0135] Using the matrix form of the Hamiltonian, given in equation A3, the Hamiltonian can then be expressed in terms of charge and flux operators.

[0136] In the charge basis the Hamiltonian is cumbersome, therefore it is convenient to consider only the two lowest energy states of the probe qubit, resulting in a Hamiltonian of the form: denotes the bare Hamiltonian of the target qubit, the gap between the ground and first excited state of the probe qubit, the longitudinal (transverse) coupling strength between the probe qubit and the resonator, are annihilation (creation) operators on the resonator in the Fock basis, is the frequency of the resonator, is the transverse coupling between the resonator and the target qubit and the number operator on the target qubit (equal to in Figure 7),

[0137] There is no analytical expression to relate the charge basis Hamiltonian and equation A8, therefore to determine the parameters in equation A8 one needs to diagonalise the circuit and evaluate the following inner products: using the notation of | probe, cavity, target).

[0138] Applications

[0139] The reduced density matrix of relative charge states can provide insight into the properties of weak links in various scenarios.

[0140] In a first scenario, a Transmon is coupled to a normal system acting as a weak link, such as a short wire or an interacting dot, or an array of interacting dots. The superconducting islands comprising the Transmon give rise to quantized plasma oscillations of Cooper pairs, which interact with the fermionic states upon their injection into and out of the dot. These interactions endow the bosonic plasma oscillations with additional structure coming from the low-lying excitations of the junction; we shall call such resonances "interacting Andreev states". In the case of topological superconductivity, Majorana states further dress the dot states. The resulting interacting Andreev and Majorana states can be challenging to disentangle, making a full tomographic reconstruction attractive. Additionally, when multiple interacting dots are present, non-trivial phases may arise, such as a Mott insulator at half-filling. We are interested in how the presence of these phases is reflected in the properties of the reduced density matrix. Specifically, the behaviour of the entanglement entropy associated with the relative charge subspace and what information it may reveal regarding the excitations present in the weak link is described. Tomography of a single, interacting Andreev state

[0141] A model for the "Dot-Transmon" (DT) is considered which is a prototypical effective model consisting of the charge and phase dynamics of a modified Transmon and its interactions and charge exchange with a normal dot (the latter can also represent the low energy dynamics of a short wire). For convenience, the full Hamiltonian H is divided into two parts,

[0142] The first Hamiltonian is a modified Transmon Hamiltonian, which accounts for the Transmon's dynamics as well as for the dot's charging energy and interaction with a nearby gate:

[0143] The second Hamiltonian HDis responsible for the interaction of the dot's electrons with the magnetic field and with the nearby superconductors via the proximity effect:

[0144] Here is the relative number of Cooper pairs between the islands (where and nRbeing the number of Cooper pairs on the left-hand side or right-hand side, respectively) and is conjugate to the phase difference (where and is the phase on the left and right island, respectively), counts the total number of Cooper pairs exceeding neutrality in the superconducting islands and is conjugate to the average phase is controlled by a side gate, Nqis controlled by a top gate.

[0145] Referring also to Figure 10, Ecand are the charging and Josephson energy scales associated with the transmon. It is assumed that Ej is predominantly set by a parallel Josephson junction, neglecting the dot's contribution. controls the energy of a given charge occupation of the dot, with the phase serving as the generator for charge transfer into the dot, a process that is absent in the classical transmon. The operator creates a fermion on the dot with spin B represents its Zeeman splitting and its induced pairing. To solve the model, it is advantageous to span the space by the four possible dot occupation states . Since the system conserves the total number of particles (with each Cooper pair contributing 1 and each fermion contributing 1 / 2), the dot's occupation ndis locked to the number of Cooper pairs in the islands N via . The model can be further simplified by focusing on an even dot occupation nd, being one of the allowed sectors, with the resulting Hamiltonian: operating on the wavefunction associated with the occupation a. The notation represents a projection of on a specific value of . While in this case, the quantum numbers N and n are constrained by the fermionic parity in the system and by the value of Nt, as summarized in Table BI below:

[0146] Table BI

[0147] Table BI lists charge state composition and periodicity of the two states, and , associated with an empty and filled dot, namely, in the first line, the dot occupation, in the second line, the total number of Cooper pairs in the superconducting islands, in the third line, the quantization of the relative charge n, and, in the fourth line, the periodicity of the wavefunctions under

[0148] The last two conditions, namely the quantization of n and periodicity (or antiperiodicity) of the wavefunctions in cp, are equivalent to each other, as reflected in the charge basis decomposition of the wavefunctions . From now on, in order to simplify the representation of the results, only the case is considered, i.e., N is constrained to the two values, namely N - 0,-1.

[0149] Due to the induced pairing interaction in the dot, each Transmon level is split into a doublet separated by . Assuming that where the plasma frequency, the Hamiltonian can be projected onto the Transmon's ground state, whose energy is with dispersion , in order to find an explicit form for the pairing term in equation (B3) ;the ground state wave function in the harmonic approximation, namely

[0150] , is taken. Here, . In this approximation, the periodicity of the wave function is lost, so the limits of integration are extended to , which leads to:

[0151] Here ), and and are the Pauli matrices operating in the dot's occupation space \<p), |U>. The eigenvalues of Equation (12) are: and the eigenstates where The density matrix for the ground state is or explicitly:

[0152] From here, the entanglement entropy can be extracted via , where is the reduced density matrix of the ground state obtained after tracing out the dot and the total-charge degrees of freedom. The full calculation is presented in Appendix Bl and gives the result:

[0153] The maximally entangled state, i.e., the equal amplitude superposition of and , occurs for \ where the entanglement entropy attains its maximal value, while concurrently the dot gets occupied (on average) by a single fermion. Tomography of coupled, interacting Andreev and Majorana states

[0154] Next, Majorana fermions are introduced to the Dot-Transmon analyzed in the previous section. This is achieved by including a third Hamiltonian: accounting for the hybridization of the Majorana fermions localized near the weak link, y2and y3, with the dot. The combined model is herein referred to as the Majorana- dot-transmon (MDT). Together with the Majorana fermions y-, and y4at a nanowire's remote ends, they form the non-local fermions . The coupling between the dot and Majorana fermions is given by Reference is made to K. Yavilberg et al.: "Differentiating Majorana from Andreev bound states in a superconducting circuit", Physical Review B, volume 100, 241408 (2019).

[0155] To solve the model, the basis of the previous section is extended to , where consists of the fermionic occupation in the dot, while describes the occupation of the non-local fermions in the superconducting regions of the wire. Since the total parity of the system is conserved, the zero-parity case is considered which consists of only eight possible fermion occupation subspaces. These are summarized in Table BII below, where we use the notation fK,a{<p) = {<p\K,a) for the wavefunctions in each subspace.

[0156] Table BII

[0157] Table BII lists charge state composition and periodicity of states of the interacting Majorana-Andreev system, namely in the first line, the dot occupation, in the second line, the total number of Cooper pairs in the superconducting islands, in the third line, the quantization of the relative charge n, and, in the fourth line, the periodicity of the wavefunctions under

[0158] The corresponding Hamiltonian is: and has eight subspaces which are ordered as in Table BIT. The blocks on the diagonal deal with the interaction between the Transmon and the dot, similar to the form in the previous section but with some added complexity due to the single particle occupation in the dot which also introduces the effect of the Zeeman field. The off- diagonal term W is responsible for the single fermion charge transfer via the Majorana-dot coupling (see Equation (B9)). A complete solution need not be determined since only a solution for the ground state is of interest. Thus, only the lowest energy subspace is considered by projecting on the Transmon's ground state. This reduces the problem to an 8-dimensional Hamiltonian. The diagonal elements of can be diagonalized by modifying the Transmon wavefunctions to comply with the boundary conditions of Table II, with corresponding eigenenergies:

[0159] Here we introduced the notation for the occupation of the left and right non local fermions, respectively. The off-diagonal terms of Equation (18) can be found using the Harmonic approximation. This approach is valid if are lower than the Transmon plasma frequency. Figure 12(a) demonstrates that the approximation works well in this regime.

[0160] The effect of the Majorana fermions addition can be clearly seen in the entanglement entropy. Following a projection on the Transmon's ground state subspace, the maximal value of the entropy, compared to the result of the previous section, increases to . Figure 12(a) shows the entanglement entropy as a function of Ne, where the points corresponding to are given by

[0161] These values indicate the points of avoided crossings and can be easily found by setting r = t0= 0 in the projected Hamiltonian. Near each of these points a slightly more general solution can be obtained by focusing on the regime, where the high magnetic field allows the dot's -states to be neglected and the Hamiltonian is reduced to one which is easily diagonalizable. Thus, near the points , the ground state takes the form: where and

[0162] It can be seen that near the Ndffpoints, the ground state approaches an equal superposition of four states and thus gives us the value . To get the functional form of the entanglement entropy near the points, all degrees of freedom can be traced out, except the implicit Transmon state, from the ground state density matrix . This gives:

[0163] Figure 12(a) shows a fairly good agreement of the entanglement entropy between the full numerical calculation and the analytical results which are valid only near the regions of the peaks.

[0164] Tracing out degrees of freedom

[0165] This section explains how degrees of freedom are traced out in the model and how the entanglement entropy is calculated. In cases of a numerical calculation, the wavefunction is represented in the charge basis. The DT model has three quantum numbers: N , and n which count the charge, and o- for dot states. The Majorana-DT introduces additional parity variations due the Majorana states, denoted by with their corresponding occupations . As in the description hereinbefore described, the total amount of cooper pairs N is constrained by particle conservation and set

[0166] DT case

[0167] As a single occupation of the dot will not mix with the Cooper pairs in this model, we are interested only in subspaces. The charge difference n has a selection rule that determines the possible values in each subspace:

[0168] A pure state that obeys Nt= 0 can be written as: with N . Its corresponding density matrix is given by and by tracing out the degrees of freedom ct and it assumes a biock-diagonai form: with each block represented in the n-basis:

[0169] Hereinbefore, there was a focus on two variations of Transmon's ground state, which differ by their boundary conditions and thus their dependence oniwhich is denoted here >: where corresponds to the two subspaces:

[0170] A pure state of the DT in the Transmon ground state sector is given by

[0171] Tracing out the degrees of freedom of its density matrix gives where the blocks are:

[0172] The entanglement entropy for this state is:

[0173] MDT case

[0174] The tracing-out procedure of the DT case can be expanded to include the Majorana fermions. For simplicity, the notation is used throughout the section. The charge difference n has the selection rule:

[0175] A pure state that obeys N can be written as: Tracing out the dot degrees of freedom a and the total charge N yields the reduced density matrix:

[0176] Since , the result is a block diagonal matrix. Each is spanned with n and K states as follows:

[0177] If we continue and trace over the reduced density has a block diagonal form, and each block has different sets of :

[0178] As in the previous section, we focus on projected Transmon ground state under the constraints given above the four projected states, where: a pure state in this subspace likewise, we trace over and get: and the entanglement entropy in this case .

[0179] System for performing quantum tomography reconstruction

[0180] When an extended quantum system is partitioned, the entropy of each subregion X is a measure of the lack of information, or uncertainty. At a sufficiently low temperature, the entropy reflects non-local correlations, in other words, the entanglement, between each subregion, and is thus called the "entanglement entropy". The entanglement entropy and the related entanglement spectrum are sensitive to interactions and topology, making it a robust, unique fingerprint of different ground states. Theoretical dependencies of entanglement entropy and entanglement spectrum on temperature, electron density, and external fields can be calculated and compared with experiment in order to identify these phases and reference is made to C. Castelnovo and C. Chamon, "Entanglement and topological entropy of the toric code at finite temperature" Physical Review B, 76(18), 184442 (2007), A. Kitaev and J. Preskill: "Topological entanglement entropy", Physical Review Letters, 96(11), 110404 (2006), M. Levin and X.-G. Wen: "Detecting topological order in a ground state wave function" Physicai Review Letters, 96(11), 110405, (2006), F. Pollmann et a / . : "Entanglement spectrum of a topological phase in one dimension", Physical Review B, 81(6), 064439 (2010) and O. Zozulya et al. : "Bipartite entanglement entropy in fractional quantum hall states", Physical Review B, 76(12), 125310 (2007).

[0181] Superconducting circuits comprising inductors and capacitors provide a versatile platform for probing and generating entanglement via circuit quantum electrodynamics (cQED). These circuits, in particular transmon qubits with hybrid superconductorsemiconductor Josephson junctions, known as "gatemons", can be used to measure entanglement entropy and entanglement spectrum in a variety of electronic materials. Gatemon circuits can be realised with materials hosting putative topological phases such as graphene, WSez, III-V semiconductors, and V-VI topological insulators. The interface between the fermionic Andreev Bound states and bosonic condensate of Cooper pairs is a natural sub-region partition.

[0182] Referring to Figure 14, a system 1 for quantum tomography reconstruction, including performing projective measurements on a coupled probe qubit, is shown.

[0183] The system 1 includes a quantum entanglement spectrometer device 2 which includes a quantum entangler module 3, which comprises a probe qubit 4, a coupling circuit 5 and a tuner 6 for controlling coupling g, and a Josephson junction module 7, which comprises a target qubit 8 in a material under test 9 and a gate 10 for controlling Josephson Energy Ej. A computer system 11 is used to control signal sources 12, for example in the form of microwave sources, and signal detectors 14 to obtain measurements 16. The computer system 11 or another computer system 17 is used to carry out state reconstruction using readouts 16 and a model 18.

[0184] Referring also to Figure 15, the tuner 6 for controlling coupling g can take the form of a loop or coil 22 driven by a signal source 12i which varies the flux in a flux loop 21.

[0185] Referring also to Figure 16, the Josephson Energy Ej can be controlled by gating one of the Josephson junctions in a flux loop in the target system with an electrode 10 driven by a voltage source 122. Additionally or alternatively, the Josephson energy Ej can be varied using a loop or coil and source similar to that shown in Figure 15.

[0186] Referring also to Figure 17, the state ZPof the probe qubit 4 is conditioned by the number Nrof Cooper pairs in the target qubit and can be used to extract a probability distribution P(NT). Pulsing a voltage on a junction gate electrode 10 modulates the P(N) enables the Wigner function to be extracted using charge-phase basis tomography, and the entanglement entropy spectrum to be inspected for signatures of interactions and topology.

[0187] Referring to Figures 18 and 19, the quantum entangler module 3 takes the form of a superconducting circuit 30 disposed on a substrate 31. The superconducting circuit 30 comprises input terminals 32, 33, 34, traces 35 forming waveguides, inductors 36, capacitors 37, qubits 38, for example in the form of transmon qubits, and output terminals 39. The superconducting circuit 40 may be formed from niobium, aluminium, or other suitable superconductor material. The substrate 31 may take the form of a low-loss silicon substrate.

[0188] The Josephson junction module 4 is placed on top of a region 41 of the superconducting circuit 30.

[0189] Referring to Figure 20, the Josephson junction module 4 comprises superconducting regions 50 (for example, traces and / or pads of superconducting material) disposed on a substrate (or "chip") 51 separated by a suitable gap. The substrate 51 comprises or supports the material under test 7, such as a semiconductor material. The quantum entangler module 3 and the Josephson junction module 4 can be ohmically coupled using bump bonds 42.

[0190] The Josephson junction module 4 can mediate transport of Cooper pairs 53 via Andreev bound states: an electron (or hole) can move from the right-hand side (or, in the case of a hole, left-hand side) to the left-hand side (right-hand side) and can be absorbed along with an electron with opposite spin and momentum, forming a bound state in the junction and a Cooper pair on the transmon island. The probability P(W) of observing N Cooper pairs in the ground state depends on the collective wavefunction of the transmon and Josephson junction. Properties of the Josephson junction that influence ground state entanglement also modify the probability P(N).

[0191] As described herein before, circuit quantum entanglement spectroscopy can be used to perform charge state tomography, extract the entanglement entropy and entanglement spectrum of a condensate sCPand investigate the ground state phases of the Josephson junction. Whereas circuit quantum electrodynamics can be used to perform tomography of qubits and cavities, the relevant conjugate variables for accessing the condensate SCPin the ground state are charge and phase. Herein, measurements in the charge basis are used.

[0192] Several aspects can help achieve this. For example, the spectrometer employs a superconducting coupling circuit and a control sequence used for charge-basis tomography. The spectrometer has two separate modules that can be assembled, namely one module comprising superconducting and metal-oxide elements that comprise the quantum entangler module (QEM), and another that providing the Josephson junction that hosts the electronic material capable of forming an interface with a superconducting contact. This allows sophisticated circuits to be fabricated by lithography, and can help avoid lossy Josephson junction substrates that usually degrade qubit lifetime. Moreover, the circuit quantum entanglement spectroscopy operates on the ground state, relaxing onerous requirements for long lifetimes.

[0193] Energy basis tomography

[0194] Charge basis tomography hereinbefore described employs a circuit comprising probe and target qubits and flux as a probe. The probe and target qubits can be coupled by direct capacitive coupling or by indirect coupling, in other words, by a resonator. Charge basis tomography can be used in entanglement spectroscopy and in topological qubits.

[0195] Tomography may, however, be performed in an energy basis.

[0196] Referring to Figure 21, energy basis tomography uses a circuit 60 comprising a probe transmon 64 and a target transmon 68 with purely capacitive coupling. Energy basis tomography can be used in qudit processing and neuromorphic computing.

[0197] Gate charge ngcan be varied by varying a voltage Vgapplied by a voltage source 123 applied to a gate capacitor Cgcoupled to a node 70 between the a coupling capacitor Ccand a target transmon 68. Coupling between probe and target transmons 64, 68 can be varied by varying tuning between the probe and target transmons 64, 68 by varying the Josephson energy Ej.

[0198] Josephson energy &can be varied using a gate 10 and voltage source 12a. If a target transmon includes a flux loop 21 (Figure 15), then the Josephson energy £5 can be varied additionally or alternatively using a magnetic field source 22 (Figure 15) and source 112 (Figure 15).

[0199] A tomographic measurement can be used to experimentally recover the state of a quantum mechanical system, encoded in the density matrix p = \ip) bjj\. To see how this works for the transmon, consider the time evolution of the system, described by the unitary operator :

[0200] Inserting identities, it is found that the on diagonal elements of the density matrix are given by:

[0201] It is found that the initial off diagonal elements can be related to the final on diagonal elements by a set of matrices defining the set of linear equations:

[0202] The on diagonal elements correspond physically to the probability of finding the transmon in a given state . In principle, this could be an eigenstate of any physical observable. However, as there are already established techniques for measuring the probabilities for energy states, consider is considered to be the probability of the system being in the energy eigenstate . This gives an energy basis representation of the density matrix and it is referred to as energy basis tomography.

[0203] Using the measured it is possible to solve the system defined by equation (C4.3), requiring at least one equation for each of the unknown elements . As the density matrix is Hermitian and the on diagonal elements can be measured, solving for the elements above the diagonal is sufficient to uniquely determine the entire density matrix. Therefore, to reconstruct an N dimension density matrix, there are unknowns.

[0204] To generate the full set of equations, different mixing matrices for each j are needed. In total, N times this many equations are obtained, one from each measured probability. This is achieved by performing an ensemble measurement, preparing the transmon in the same initial state times, and applying a different sequence of pulses each time.

[0205] By evaluating the corresponding mixing matrices numerically, and solving the system of equations using an experimentally measured set of , the density matrix can be reconstructed.

[0206] Pulse sequence

[0207] As the potential has a finite depth of , it is only able to confine the lowest few eigenstates, states for which . Above this threshold, the eigenstates will act as free particle states. Free from the influence of the potential, these states have a very different functional form to the bound states within the potential. Specifically, as the bound states are confined in they will have a broad spread in the n . Conversely the free states having a broad spread in will be confined in n.

[0208] Referring to Figure 22, starting with a large , many of the states in the initial superposition will be bound states. Pulsing down to zero, the final eigenstates will all be free states. Reducing down to zero has the effect of decoupling the probe and target transmons. Each of the bound states in the initial superposition will correspond to a broad spread of eigenstates post pulse so that the full breadth of its charge space distribution is covered. Using this insight, a very large initial E, can be used and pulse it down to as close to zero as possible. This can help ensure that the initial superposition will be composed almost entirely of bound states, and that good mixing will be achieved in accommodating the resulting spread in momentum.

[0209] Due to the even symmetry of the potential, the eigenstates have odd or even parity. Evaluating overlaps before and after an pulse, it is found that that even(odd) states pre-pulse will be composed entirely of even(odd) states post pulse. This gives rise to the chequerboard effect seen mixing matrices (that is, a rectangular mesh of zerovalue elements in the mixing matrix). To address this, the gate parameter is also pulsed. This changes the boundary condition imposed on the energy eigenstates. Therefore, a state with definite parity pre-pulse, will be forced into a mixture of even and odd states to fit the boundary condition. This overcomes the limitations of pulsing alone and helps give a much broader spread in the mixing matrices. This is most effective when pulsing between integer and half integer values of , as this applies a full n phase shift to the interval in which the boundary condition is enforced, flipping the parity of the states. As the effect of the pulsing is felt due to the influence of the periodic potential, it is best to apply this procedure while the potential is still high. For this reason, the pulse sequence shown in Figure 22 is used to maximise mixing.

[0210] Increasing back to a sufficiently large non-zero value has the effect of re-coupling the probe and target transmons.

[0211] By varying , the target system can be perturbed. As explained earlier, on- diagonal elements of the density matrix correspond physically to the probability of finding the target system in a given state and can be measured. Although off-diagonal elements cannot be directly measured, by applying pulses (and varying the durations of the pulses) on-diagonal and off-diagonal elements can be mixed (and the mixing varied) and, through, projective measurements, the state of the target system to be determined.

[0212] Modifications

[0213] It will be appreciated that various modifications may be made to the embodiments hereinbefore described. Such modifications may involve equivalent and other features which are already known in the fields of quantum tomography reconstruction, and which may be used instead of or in addition to features already described herein. Features of one embodiment may be replaced or supplemented by features of another embodiment.

[0214] Although measurements of a state can be based on a (time-based) Ramsey sequence (involving applying a first n / 2 pulse, allowing the system to evolve for a given time and applying a second n / 2 pulse), measurements of a state can follow a (frequencybased) Kirchmayr approach.

[0215] Although claims have been formulated in this application to particular combinations of features, it should be understood that the scope of the disclosure of the present invention also includes any novel features or any novel combination of features disclosed herein either explicitly or implicitly or any generalization thereof, whether or not it relates to the same invention as presently claimed in any claim and whether or not it mitigates any or all of the same technical problems as does the present invention. The applicants hereby give notice that new claims may be formulated to such features and / or combinations of such features during the prosecution of the present application or of any further application derived therefrom.

Claims

Claims1. Apparatus comprising: a device (3) comprising a circuit (30) of superconductor material disposed on substrate (31), wherein the circuit is configured to provide a probe qubit (4) and a coupling circuit (5) for coupling the probe qubit to a Josephson junction-based target circuit (7).

2. The apparatus of claim 1, further comprising: a system (11, 17) configured to provide control signals to the device (3), to receive measurement signals from the device and to process the measurement signals so as to perform charge or energy basis tomography.

3. The apparatus of claim 1 or 2, wherein the probe qubit (4) is couplable to the target circuit (7) via a term of a typewherein n is the charge degree of freedom of the target qubit and $zis the longitudinal coupling of the probe qubit.

4. The apparatus of claim 1, 2 or 3, wherein the coupling circuit (5) comprises a capacitor.

5. The apparatus of claim 4, wherein the coupling circuit (5) further comprises an inductor (26) arranged in series with the capacitor (272) to form a resonant circuit.

6. The apparatus of claim 4, wherein the coupling circuit (5) further comprises a first capacitor (27i) arranged to couple the probe qubit (3) to the resonant circuit, wherein the capacitor is a second capacitor, wherein the coupling circuit further comprises a third capacitor (27B) for coupling the resonant circuit to the target circuit (7).

7. The apparatus of any one of claims 1 to 6, wherein the probe qubit (4) is a flux qubit.

8. The apparatus of claim 7, wherein the flux qubit (4) comprises a four-junction flux qubit comprising four Josephson junctions arranged in series in a loop.

9. The apparatus of any one of claims 1 to 8, wherein the probe qubit is a transmon.

10. The apparatus of claim 9, wherein the coupling is a capacitive coupling.

11. The apparatus of any one of claims 1 to 10, wherein the superconductor material is niobium.

12. The apparatus of any one of claims 1 to 11, wherein the substrate is a silicon substrate.

13. Apparatus comprising: the apparatus of any one of claims 1 to 12, wherein the device (3) is a first module and the substrate (21) is a first substrate; and a second module (6) comprising regions (30) of superconductor material disposed on a second substrate (31), the regions of superconductor material arranged to form a Josephson junction which provides the target qubit (7).

14. The apparatus of claim 13, wherein the superconductor material is niobium.

15. The apparatus of claim 13 or 14, wherein the second substrate comprises a semiconductor material.

16. The apparatus of any one of claims 13 to 15, wherein the first and second modules (3, 6) are connected via bump bonds (32) such that the first and second modules are electrically connected.

17. A method of performing charge or energy basis tomography, the method comprising providing control signals to a device (3) which comprises a circuit (30) of superconductor material disposed on substrate (31), wherein the circuit is configured to provide a probe qubit (4) and a coupling circuit (5) for coupling the probe qubit to a Josephson junction-based target circuit (7); receiving measurement signals from the device; and processing the measurement signals so as to perform charge or energy basis tomography.

18. A method of measuring a charge state of a Josephson junction-based target circuit (7) which is coupled to a probe qubit (4), the method comprising performing a measurement for a predetermined time r:· decoupling the probe and target qubits;· while the probe and target qubits are decoupled and the Josephson energy of the target qubit is non-zero:- preparing an initial state of the target qubit; and- applying a half K-pulse to the probe qubit;· coupling the probe and target qubits and reducing the Josephson energy of the target qubit to zero;· while the probe and target qubits are coupled and the Josephson energy of the target qubit is at zero:- waiting until a predetermined time v, and- applying half K-pulse to the probe qubit; and- measuring the state of the probe qubit.

19. The method of claim 17 or 18, comprising repeating the measurement for different predetermined period of times r.

20. A method of measuring an energy state of a target transmon which is coupled to a probe transmon, the method comprising:· increasing a gate bias applied to a target transmon from a first value to a second value;· after increasing the gate bias applied to the target transmon from the first value to the second value, decoupling the probe and target transmons;· waiting a given time;· after waiting a given time from decoupling the probe and target transmons, recoupling probe and target transmons;· after re-coupling the probe and target transmons, decreasing the gate bias applied to the target transmon from the second value to the first value; and· after decreasing the gate bias applied to the target transmon from the second value to the first value, measuring the state of the probe transmon.

21. A computed-implemented method of reconstructing a density matrix of a target qubit-based system, the method comprising: extracting a set of probabilities pnof the state of the target qubit-based system in a charge or energy basis from a set of measurements of a probe qubit configured to sense the target qubit-based system; andsolving at least one mixing matrix using the set of probabilities pn.

22. A computed-impiemented method of characterising a material having a predicted property (for example, whether it is in a topologically non-trivial phase) having an expected tomography pattern including a predicted (unique) entanglement entropy, the method comprising: performing charge or energy basis tomography for a range of swept parameters to obtain a measured entanglement entropy of a state of a target junction-based system formed using the material for the range of swept parameters; determining whether the predicted entanglement entropy and the measured entanglement entropy match; and outputting the result of the determination.

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