Patent Description:
Superconducting qubits are one of the most promising candidates for developing commercial quantum computers. Indeed, superconducting qubits can be fabricated using standard microfabrication techniques. Moreover they operate in the few GHz bandwidth such that conventional microwave electronic technologies can be used to control qubits and readout the quantum states.

A significant challenge in quantum computation is the sensitivity of the quantum information to noise. The integrity of the quantum information is limited by the coherence time of the qubits and errors in the quantum gate operations which are both affected by the environmental noise.

One manner to address this issue is to design and use topological qubits, which are intrinsically protected against noise. Topological qubits employ quasiparticles called anyons, and more specifically non-Abelian anyons. However, non-Abelian anyons have not yet been found in nature. This has hindered the development of topological quantum computers.

Document D1 (<CIT>) relates to a qubit circuit which is constructed to operate according to a strong coupling regime, but not to a deep strong coupling regime.

In accordance with a broad aspect, there is provided a topological superconducting qubit circuit according to claim <NUM> and a corresponding method according to claim <NUM>. The circuit comprises a plurality of physical superconducting qubits and a plurality of coupling devices interleaved between pairs of the physical superconducting qubits. The coupling devices are tunable to operate the qubit circuit either in a topological regime or as a series of individual physical qubits. At least two superconducting loops, each one threadable by an external flux, are part of the qubit circuit.

In various embodiments, the circuit further comprises at least one component for generating a magnetic field for inducing the external flux in the superconducting loops.

In various embodiments, the component comprises two transmission lines, each one coupled to one of the superconducting loops through a mutual inductance.

In various embodiments, each one of the physical superconducting qubits is composed of at least one capacitor and at least one Josephson junction connected together.

In various embodiments, the Josephson junction is part of a SQUID.

In various embodiments, the capacitor and the Josephson junction are connected together at a first node, and the coupling devices are connected to the physical qubits at the first node.

In various embodiments, one of the superconducting loops comprises a second node having the same superconducting phase as the first node.

In various embodiments, the capacitor and the Josephson junction are connected together at a first node, and the coupling devices are connected to the physical qubits at a second node different from the first node.

In various embodiments, one of the superconducting loops comprises a third node having the same superconducting phase as the second node.

In various embodiments, one of the superconducting loops is a loop of superconducting material interrupted by a SQUID.

In various embodiments, another one of the superconducting loops is interrupted by a Josephson junction of the SQUID.

In accordance with another broad aspect, there is provided a method for topological protection of quantum information in a qubit circuit. The method comprises coupling a plurality of physical qubits with a plurality of interleaved coupling devices, each one of the coupling devices comprising at least one superconducting loop threadable by an external flux Φext. Parameters for the external flux Φext are selected such that <MAT>, where J is a coupling device energy and h is a physical qubit energy. The external flux Φext is applied to the superconducting loop to induce a phase shift in the coupling devices and to operate the qubit circuit in a topological regime.

In various embodiments, selecting parameters for the external flux Φext comprises selecting Φext to induce a phase shift with a value between π/<NUM> and 3π/<NUM> (mod 2π) in at least one Josephson junction of the qubit circuit.

In various embodiments, selecting parameters for the external flux Φext comprises selecting Φext to induce a phase shift of π (mod 2π) in at least one Josephson junction of the qubit circuit.

In various embodiments, the method further comprises applying an external flux ΦSQUID to the second superconducting loop of at least one of the plurality of coupling devices.

In various embodiments, applying the external flux ΦSQUID comprises applying the external flux ΦSQUID to the second superconducting loop of all of the plurality of coupling devices.

In various embodiments, the method further comprises selecting parameters for ΦSQUID = (2n+<NUM>)/<NUM> * Φo, where n is an integer and Φo is a flux quantum.

In various embodiments, the method further comprises modulating ΦSQUID for at least one of the plurality of coupling devices.

In various embodiments, modulating ΦSQUID comprises changing ΦSQUID adiabatically.

In various embodiments, modulating ΦSQUID comprises changing ΦSQUID from (2n+<NUM>)/<NUM> * Φo to another value.

Features of the systems, devices, and methods described herein may be used in various combinations, in accordance with the embodiments described herein.

Reference is now made to the accompanying Figs.

The present disclosure comprises circuits and methods for topological quantum computing using superconducting qubits. In various embodiments, a topological qubit comprises a plurality of physical superconducting qubits and a plurality of coupling devices which are interleaved between the physical qubits.

A superconducting circuit is described which can be used to artificially engineer non-Abelian anyon quasi-particle dynamics. Such a circuit may be used in developing a topological quantum processor.

For various operations of a quantum computer, such as anyon creation, braiding, and fusion, one may need to control the strength of the coupling between the physical qubits. Accordingly, a tunable qubit circuit for topological protection <NUM> is described herein and illustrated in <FIG>. The circuit <NUM> is composed of a plurality of physical qubits <NUM> coupled together with coupling devices <NUM>. The coupling devices <NUM> are interleaved between pairs of physical qubits <NUM>, i. e a first qubit <NUM> is connected to a second qubit <NUM> by a coupling device <NUM>, the second qubit <NUM> is connected to a third qubit <NUM> by a coupling device <NUM>, the third qubit <NUM> is connected to a fourth qubit <NUM> by a coupling device <NUM>, and so on.

In some embodiments, a physical qubit <NUM> may be coupled to one or more other physical qubits <NUM> through corresponding coupling devices <NUM>, thus creating a network of physical qubits which can support different configurations of topological qubits. Changing the configuration of the topological qubits is possible due to the tunability of the coupling devices interleaved between the physical qubits. All of the qubits <NUM> in the circuit <NUM> may be of a same configuration. Alternatively, qubits <NUM> of the circuit <NUM> may have different configurations. All of the coupling devices <NUM> of the circuit <NUM> may be of a same configuration. Alternatively, coupling devices <NUM> of the circuit <NUM> may have different configurations. Although three qubits <NUM> and two coupling devices <NUM> are illustrated, these numbers are for illustrative purposes only.

The qubits <NUM> may be composed of at least one capacitor and at least one Josephson junction connected together. In some embodiments, the qubits are transmon qubits, which are a specific type of superconducting qubit composed of at least one Josephson junction and at least one capacitor.

Example embodiments for the qubits <NUM> are shown in <FIG>. <FIG> illustrates a qubit 200A having a capacitor <NUM> and a Josephson junction <NUM> connected together in parallel. <FIG> illustrates a qubit 200B having a Josephson junction <NUM> connected between a first capacitor <NUM> and a second capacitor <NUM>. This configuration is referred to as a differential architecture. <FIG> illustrates a qubit 200C having a capacitor <NUM> connected in series with a first Josephson junction <NUM> and a second Josephson junction <NUM>. This configuration is referred to as a two-junction architecture. <FIG> illustrates a qubit 200D having a Josephson junction <NUM> connected in series between a capacitor <NUM> and an inductor <NUM>. This configuration is referred to as an inductively shunted architecture. Each Josephson junction <NUM>, <NUM>, <NUM>, <NUM>, <NUM> may be replaced by a pair of Josephson junctions connected in parallel, referred to herein as a SQUID (superconducting quantum interference device), for tunability of the frequency of the qubits <NUM>.

The total energy of a circuit <NUM> having N qubits <NUM> may be found from its Hamiltonian. One can use the Jordan-Wigner transformation to show that circuit <NUM>, designed with proper coupling devices <NUM>, has a similar Hamiltonian to an Ising spin chain that behaves like a topological quantum system supporting Majorana edge states, which are one type of non-Abelian quasi-particles. In the Ising spin chain model, the Hamiltonian of a chain of N coupled qubits is written as: <MAT> where the σi are the Pauli operators on qubit i. The first term relates to the energy of the qubits <NUM>. The second term represents the energy of the coupling between two qubits <NUM>. The coupling is said to be of ferromagnetic type for J > <NUM> (and antiferromagnetic type for J < <NUM>). A phase transition from a non-topological phase to a topological phase occurs when the coupling energy becomes larger than the qubit energy. In other words, the condition for achieving topological protection is <MAT>. When this condition is met, we refer to the circuit <NUM> as having "deep strong coupling". A circuit having deep strong coupling is said to operate in a topological regime.

<FIG> illustrates an example embodiment of the qubit circuit <NUM>, where coupling devices <NUM> are SQUIDS. The coupling devices <NUM> are composed of two Josephson junctions <NUM>, <NUM>, connected in parallel. Two physical qubits <NUM> are as per the embodiment of <FIG>, with a Josephson junction <NUM> of Josephson energy EJq and a capacitor <NUM> of capacitance C.

Referring to <FIG>, two superconducting loops <NUM>, <NUM> are illustrated for the circuit <NUM>. A superconducting loop is formed by a loop of superconducting material which may be interrupted by one or more Josephson junctions. A loop of superconducting material forms a closed path in a circuit, and the path lies in the superconducting material. Magnetic flux in a loop of superconducting material is quantized, and flux quantization is maintained even if the loop of superconducting material is interrupted by one or more Josephson junctions. Generally, a circuit of N coupling devices will have <NUM> x N superconducting loops, although more than two loops may be provided per coupling device in the circuit.

Each loop <NUM>, <NUM> of circuit <NUM> is threadable by an external flux. The loop is said to be threadable by an external magnetic flux when a non-zero magnetic flux may be induced in the loop in a controlled fashion by an applied magnetic field passing through a surface defined by the loop. The magnetic field is generated by a component and/or device coupled to the loop. For example, the magnetic field can be generated by a current-carrying line such as a transmission line or a waveguide in proximity to the loop. Such current-carrying line is coupled to the loop through a mutual inductance and connected to a current source. An example is illustrated in <FIG>, where a line <NUM> is coupled to superconducting loop <NUM> through a mutual inductance M<NUM> and carrying a current I<NUM> that induces a flux ΦSQUID in loop <NUM>. Similarly, a line <NUM> coupled to superconducting loop <NUM> through a mutual inductance M<NUM> and carrying a current I<NUM> induces a flux Φext in loop <NUM>. Other embodiments may also apply.

A magnetic field is applied to the circuit <NUM> in order to induce a phase shift in the coupling devices <NUM>, so as to obtain a deep strong coupling regime. The magnetic field induces a non-zero external flux Φext threading loop <NUM>.

A superconducting node phase φi and a charge number ni are assigned to each qubit <NUM>, and the Hamiltonian of a chain of N qubits <NUM> is given by: <MAT> with <MAT> and <MAT> where Φ<NUM> is the flux quantum, e the electron charge and ΦSQUID the flux applied to the SQUID of the coupling devices <NUM>. The cosine term involving φext in the Hamiltonian can then be rewritten as: <MAT>.

Expanding the cosine and sine terms involving φi to second order Taylor series, the Hamiltonian becomes <MAT>.

The first term corresponds to the sum of the Hamiltonians of N transmon qubits having Josephson energy equal to ẼJ = EJq + 2EJccosφext while the second term represents the coupling between nearest neighbours. The last term is an additional single-qubit term stemming from the external flux. For a finite chain, the two qubits at the ends of the chain have effective Josephson energies of ẼJ = Ejq + EJccosφext. The effective qubit impedance and plasma frequency are defined as: <MAT>.

Rewriting the Hamiltonian in terms of the Pauli operators gives: <MAT> with: <MAT> <MAT> <MAT>.

In the Ising model, the condition for achieving topological protection is <MAT>. In the present case, that becomes: <MAT>.

If there is no external flux, i.e. φext = <NUM>, then the condition cannot be realised with EJc and EJq being positive. Deep strong coupling can only be satisfied if: <MAT>.

Topological order is thus attainable with such a design if an external phase having a value between π/<NUM> and 3π/<NUM> is applied to the coupler. Coupling is maximal at φext = π, in which case the condition on the design becomes EJq/<NUM> < EJc.

<FIG> shows simulation results for the energy levels of three qubits <NUM> coupled by two coupling devices <NUM> composed of SQUIDs. The external flux for the coupling devices <NUM> was set such that φext = π. <FIG> shows the energy spectrum with respect to the ground state energy when EJq=<NUM>, EJ,SQUID=<NUM> and C=<NUM> fF as a function of the SQUID magnetic frustration <MAT>. We can see that the separation between the ground state and the first excited state decreases from <NUM> to less than <NUM> when fSQUID approaches zero, where the coupling is maximal. Moreover, the derivative of the energy levels is zero at the maximal coupling point for fSQUID = <NUM>.

The calculated spectrum with four and five qubits <NUM> is shown in <FIG> and <FIG> respectively. Increasing the number of qubits <NUM> reduces the energy splitting between the ground state and the first excited state at maximal coupling. With five qubits <NUM>, the two states are almost degenerate at the fSQUID = <NUM> operating point. Degenerate ground states are indeed characteristic of a topological state in the Ising model.

<FIG> illustrates an example circuit <NUM> with a coupling device according to another embodiment. A tunable flux qubit <NUM> is used to couple two qubits <NUM>, composed of capacitor <NUM> and junction <NUM>. The coupling strength of the tunable flux qubit <NUM> used as a coupling device can be tuned by applying a flux on the SQUID formed by the two EJ,SQUID junctions <NUM>, <NUM>. In <FIG>, two superconducting loops <NUM>, <NUM> are illustrated. By using two junctions <NUM>, <NUM>, the loop <NUM> threaded by the external magnetic flux Φext is decoupled from the qubits <NUM>, which may minimize unintentional driving of the qubits <NUM>.

Noting that the junction <NUM> and junction <NUM> form an asymmetric SQUID with zero flux, we can simply replace EJq by Ejq+EJs in equation (<NUM>) to find that the same Hamiltonian as the one presented above governs the embodiment of <FIG>, and the conditions for deep strong coupling become: <MAT>.

Replacing the junctions <NUM> and <NUM> by superconducting inductors would lead to a similar result.

<FIG> illustrates an embodiment of a qubit circuit <NUM> with differential qubits <NUM> coupled with a coupling device <NUM>. The qubits <NUM> are coupled at one node <NUM> through a SQUID (junctions <NUM>, <NUM>) and at another node <NUM> by a superconducting line <NUM>. As shown in <FIG>, superconducting loops <NUM>, <NUM> are present. An external flux Φext is threaded in loop <NUM>.

The circuit <NUM> has the same Hamiltonian as the circuit <NUM> when the capacitance is replaced by C/<NUM> such that <MAT> in equation (<NUM>). The condition for reaching topological order is the same.

<FIG> illustrates an embodiment for a qubit circuit 800B using the same differential qubits <NUM> and coupling devices <NUM> as qubit circuit <NUM>. The superconducting lines <NUM> (from circuit <NUM>) between adjacent qubits are replaced by a single superconducting line <NUM> between the first qubit 802A and the last qubit 802B. Hence, superconducting loops <NUM> (from circuit <NUM>) are replaced by a single superconducting loop <NUM> that spans the entire chain of qubits. The external flux Φext threading loop <NUM> can be selected to induce a desired phase shift in the coupling devices <NUM>.

<FIG> illustrate a qubit circuit <NUM> with differential qubits <NUM> coupled with coupling device <NUM>. <FIG> illustrates three superconducting loops <NUM>, <NUM>, <NUM> formed in the circuit <NUM>.

The Hamiltonian of the circuit <NUM> is the same as the Hamiltonian of the circuit <NUM> if we set <MAT> and replace EJq and EJc in equation (<NUM>) by EJs and EJq', respectively. By inducing a phase shift of π in junction <NUM> and junction <NUM> using external fluxes, the condition for deep strong coupling becomes EJs/<NUM> < EJq'. Note that for tunability, the EJq' junction <NUM> may be implemented as a SQUID.

<FIG> illustrates an embodiment of a qubit circuit <NUM> where physical qubits <NUM> are two-junction qubits made of Josephson junctions <NUM> and <NUM> and capacitor <NUM> and are connected together with coupling device <NUM>. The coupling device <NUM> is a SQUID formed from two Josephson junctions <NUM> and <NUM>. Josephson junctions <NUM>, <NUM> have Josephson energy EJ,SQUID, while the two junctions <NUM>, <NUM> have Josephson energy EJq and EJs respectively. As shown in <FIG>, external magnetic fluxes Φext and ΦSQUID thread superconducting loops <NUM>, <NUM> formed by junctions <NUM>-<NUM>-<NUM> and <NUM>-<NUM>, respectively.

Circuit nodes <NUM>, <NUM> are associated with a node phase denoted by variables φi and ξi, respectively. The φi nodes <NUM> are associated with a charge number ni. The total Hamiltonian for such a qubit chain is: <MAT> where <MAT>and φext = 2πΦext/Φ<NUM>. The coupler <NUM> and the junctions <NUM> and <NUM> form a flux qubit with α=EJc/EJs. For α < <NUM>, the ground state of the flux qubit does not involve any persistent current such that the φi and ξi are approximated as being small. In that case, the Hamiltonian may be rewritten by expanding the cosines to second-order Taylor series: <MAT>.

Since the ξi nodes <NUM> have no capacitance, a degree of freedom may be removed from the Hamiltonian by writing ξi in terms of φi. This is done by writing the Kirchoff current law at the coupling nodes: <MAT>.

Expanding the sines to first order gives: <MAT>.

This expression shows that in general, the coupling between the qubits <NUM> is not limited to first nearest-neighbours. Indeed, the coupling term in the Hamiltonian is proportional to ξi+<NUM>ξi.

There exists a condition for which the coupling remains limited to next-nearest neighbours and the Hamiltonian is greatly simplified. Indeed, when EJc << EJq, the following can be approximated: <MAT>.

Defining <MAT>, the Hamiltonian may be rewritten as: <MAT>.

If the circuit <NUM> is operated in the regime where EJq >> EJc & EJs, and a ≈ <NUM>, then we retrieve the Hamiltonian of equation (<NUM>). Indeed, when EJq >> EJs, the inductance of the junction EJq is very small compared to the other inductances of the circuit <NUM> and can thus be considered as a short circuit. Using circuit <NUM> with a ≈ <NUM> instead of circuit <NUM> may allow the individual qubit frequency to be separately tuned in the non-topological regime, assuming junction <NUM> is implemented as a SQUID, since this junction is decoupled from the flux bias of the superconducting loop <NUM>.

In order to find the condition to obtain deep strong coupling using the architecture of <FIG>, the effective Josephson energy ẼJ for φext = π is:
<MAT>.

The condition for deep strong coupling is:
<MAT>.

If a ≈ <NUM>, this condition implies EJc / EJs > <NUM>/<NUM>, consistent with the condition previously derived for the circuit <NUM> of <FIG>.

Replacing the junctions <NUM> and <NUM> by superconducting inductors (i.e. replacing the two-junction qubits <NUM> by inductively shunted qubits) would lead to a similar result.

The energy spectrum of the coupled qubits <NUM> as a function of the flux applied to the superconducting loop <NUM>, as per the embodiment of <FIG>, was simulated. The results of the simulation are illustrated in <FIG>, where the spectrum of three coupled qubits <NUM> is shown. Here, EJ,SQUID=<NUM>, EJs=<NUM> and EJq=<NUM>, while φext = π. The spectrum is very similar to the spectrum of <FIG>, showing that adding an extra node for every qubit does not affect the physics of the coupling. This extra node may make the qubit less sensitive to external flux fluctuations.

All coupler designs presented hereinabove exhibited antiferromagnetic coupling (i.e. J<<NUM>). <FIG> shows a design allowing for ferromagnetic coupling. Qubit <NUM> is coupled to coupling device <NUM>. Junctions <NUM> and <NUM> with Josephson energies EJ1 and EJ2, respectively, connected in series are added in parallel with the junction <NUM> of Josephson energy EJq. As shown in <FIG>, the three junctions <NUM>, <NUM>, <NUM> define a superconducting loop <NUM> on which the external flux Φext is applied. Another superconducting loop <NUM> is also formed within the coupling device <NUM>.

We refer to the phase difference on junctions <NUM>, <NUM>, <NUM> as δ<NUM>i, δ<NUM>i and φi respectively. Considering the quantization of the phase around the superconducting loop threaded by the external flux, we have: <MAT> where we have defined φext = <NUM>πΦext/Φ<NUM>. For the rest of this derivation, we will assume that Φext = π.

The problem has a second constraint: that the current in junctions <NUM>, <NUM> in series must be the same, which means that: <MAT>.

We now assume that EJ1 >> EJ2. As a result, δ<NUM>i is limited to very small values around zero and δ<NUM>i approaches π due to the external flux. Combining equations (<NUM>) and (<NUM>) and assuming φext = π, we have: <MAT>.

Using a first order Taylor expansion we find: <MAT> <MAT>.

We now write the Hamiltonian: <MAT> <MAT> where in equation (<NUM>) we have shifted the argument of the cosine on the EJ2 term to make sure the argument is close to zero. We can now replace δ<NUM>i and δ<NUM>i by their equivalent in terms of φi and expand the cosine terms to second order to find: <MAT>.

We can now define an effective Josephson energy ẼJ and qubit impedance r as:
<MAT>
and <MAT>.

The Hamiltonian can be rewritten using Pauli operators as: <MAT> with h = r ẼJ/2and J = r EJc/<NUM>.

From that, we find that the condition for deep strong coupling and topological order leads to: <MAT>.

This implies that <MAT> is larger than EJq for positive EJc.

Equations (<NUM>) to (<NUM>) were derived assuming EJ1 >> EJ2. Having instead EJ1 << EJ2 leads to a swapping of EJ1 and EJ2 in the equations. Replacing either EJ1 or EJ2 by a superconducting inductor would also give a similar result.

As will be understood, the circuits <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM> may be operated as topologically protected qubit circuits. <FIG> illustrates a method <NUM> for topological protection of quantum information in a qubit circuit, such as circuits <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>. At step <NUM>, a plurality of physical qubits, such as qubits <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, are coupled with a plurality of interleaved coupling devices, such as coupling devices <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>. In some embodiments, the qubits each comprise at least one capacitor and at least one Josephson junction connected together, as illustrated in the embodiments of <FIG>. The coupling devices each comprise at least one superconducting loop threadable by an external flux Φext.

At step <NUM>, parameters are selected for the external flux Φext such that <MAT> <NUM>, where J is the energy of the coupling devices and h is the energy of the physical qubits. In some embodiments, selecting parameters as per step <NUM> comprises selecting Φext, to induce a phase shift with a value between π/<NUM> and 3π/<NUM> (mod 2π) in at least one Josephson junction of the qubit circuit. In some embodiments, selecting parameters as per step <NUM> comprises selecting Φext to induce a phase shift of π (mod 2π) in at least one Josephson junction of the qubit circuit.

At step <NUM>, the external flux Φext is applied to the at least one superconducting loop to induce a phase shift in the coupler and operate the circuit in a topological regime. In some embodiments, Φext is selected to induce a phase shift of π in the coupler.

In some embodiments, the qubit circuit comprises at least a first superconducting loop threadable by the external flux Φext, and at least a second superconducting loop threadable by an external flux ΦSQUID. The method <NUM> may thus, in some embodiments, also comprise a step <NUM> of selecting parameters for the external flux ΦSQUID, and/or a step <NUM> of applying the external flux ΦSQUID to the second superconducting loop. The parameters for ΦSQUID may be selected such that ΦSQUID = (2n+<NUM>)/<NUM> * Φo, where n is an integer and Φo is the flux quantum.

The qubits may be decoupled and operated as individual physical qubits with the appropriate choice of external flux ΦSQUID. In some embodiments, ΦSQUID= +/- <NUM>Φo provides such capability. A flux ΦSQUID= (2n+<NUM>)/<NUM> * Φo can also be applied only to selected couplers. For example, if a flux ΦSQUID= (2n+<NUM>)/<NUM> * Φo is applied to a coupler in the middle of a chain of N coupled qubits (N even), then the topological qubit can be broken into two topological qubits each made of N/<NUM> physical qubits.

The flux in the SQUID of one coupler may be changed from a value of (2n+<NUM>)/<NUM> * Φo to a different value. For example, the flux in a coupler between two chains of N/<NUM> coupled physical qubits can be modified to a value different from (2n+<NUM>)/<NUM> * Φo, such as a value of nΦo, in order to convert the two topological qubits made of N/<NUM> physical qubits into a single one made of N qubits.

In general, the strength of the coupling can be modulated by modulating ΦSQUID. In some embodiments, ΦSQUID is changed adiabatically to ensure that the symmetry of the wave function is preserved during the procedure.

Although illustrated as sequential, the steps <NUM>-<NUM> of the method <NUM> may be performed in any desired order, and in some cases concurrently. For example, parameters for both fluxes may be selected concurrently, as per steps <NUM> and <NUM>, but applied sequentially as a function of a desired implementation. Steps <NUM> and <NUM> will necessarily be performed sequentially, but not necessarily in the order illustrated.

Claim 1:
A topological superconducting qubit circuit (<NUM>; <NUM>; <NUM>; <NUM>; <NUM>; <NUM>; <NUM>) comprising:
a plurality of physical superconducting qubits (<NUM>; <NUM>; <NUM>; <NUM>; <NUM>; <NUM>; <NUM>); and
a plurality of coupling devices (<NUM>; <NUM>; <NUM>; <NUM>; <NUM>; <NUM>; <NUM>) interleaved between pairs of the physical superconducting qubits,
wherein the coupling devices are tunable to operate the qubit circuit in a topological regime and as a series of individual physical qubits; and
wherein the topological superconducting qubit circuit further comprises at least two superconducting loops (<NUM>, <NUM>; <NUM>, <NUM>; <NUM>, <NUM>; <NUM>, <NUM>, <NUM>; <NUM>, <NUM>; <NUM>, <NUM>) per coupling device, each one of the at least two loops being threadable by an external flux,
wherein the coupling devices are tunable by selecting parameters for the external flux and applying the external flux with the selected parameters to one of the at least two superconducting loops,
wherein the parameters are selected for the external flux such that |J/h| > <NUM>, where J is a coupling device energy and h is a physical qubit energy; and
wherein the qubit circuit operates in the topological regime when the external flux is applied to the at least one of the at least two loops to induce a phase shift in the coupling devices.