Qubit circuit and method for topological protection

A qubit circuit and a method for topological protection of a qubit circuit are described. 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.

TECHNICAL FIELD

The present disclosure relates generally to quantum computation, and more particularly to superconducting topological qubits protected from noise.

BACKGROUND OF THE ART

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.

SUMMARY

In accordance with a broad aspect, there is provided a topological superconducting qubit circuit. 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 ϕextare selected such that

Jh>1,
where J is a coupling device energy and h is a physical qubit energy. The external flux ϕextis 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 ϕextcomprises selecting ϕextto induce a phase shift with a value between π/2 and 3π/2 (mod 2π) in at least one Josephson junction of the qubit circuit.

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

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

In various embodiments, applying the external flux ϕSQUIDcomprises applying the external flux ϕSQUIDto the second superconducting loop of all of the plurality of coupling devices.

In various embodiments, the method further comprises selecting parameters for ϕSQUID=(2n+1)/2*ϕo, where n is an integer and ϕois a flux quantum.

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

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

DETAILED DESCRIPTION

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 protection100is described herein and illustrated inFIG. 1. The circuit100is composed of a plurality of physical qubits102coupled together with coupling devices104. The coupling devices104are interleaved between pairs of physical qubits102, i.e a first qubit102is connected to a second qubit102by a coupling device104, the second qubit102is connected to a third qubit102by a coupling device104, the third qubit102is connected to a fourth qubit102by a coupling device104, and so on.

In some embodiments, a physical qubit102may be coupled to one or more other physical qubits102through corresponding coupling devices104, 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 qubits102in the circuit100may be of a same configuration. Alternatively, qubits102of the circuit100may have different configurations. All of the coupling devices104of the circuit100may be of a same configuration. Alternatively, coupling devices104of the circuit100may have different configurations. Although three qubits102and two coupling devices104are illustrated, these numbers are for illustrative purposes only.

The qubits102may 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 qubits102are shown inFIGS. 2A-2D.FIG. 2Aillustrates a qubit200A having a capacitor202and a Josephson junction204connected together in parallel.FIG. 2Billustrates a qubit200B having a Josephson junction206connected between a first capacitor208and a second capacitor210. This configuration is referred to as a differential architecture.FIG. 2Cillustrates a qubit200C having a capacitor212connected in series with a first Josephson junction214and a second Josephson junction216. This configuration is referred to as a two-junction architecture.FIG. 2Dillustrates a qubit200D having a Josephson junction220connected in series between a capacitor218and an inductor222. This configuration is referred to as an inductively shunted architecture. Each Josephson junction204,206,214,216,220may 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 qubits102.

The total energy of a circuit100having N qubits102may be found from its Hamiltonian. One can use the Jordan-Wgner transformation to show that circuit100, designed with proper coupling devices104, 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:

H=-∑i=1N⁢h⁢⁢σiz-∑i=1N-1⁢J⁢⁢σix⁢σi+1x,(1)
where the σiare the Pauli operators on qubit i. The first term relates to the energy of the qubits102. The second term represents the energy of the coupling between two qubits102. The coupling is said to be of ferromagnetic type for J>0 (and antiferromagnetic type for J<0). 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

Jh>1.
When this condition is met, we refer to the circuit100as having “deep strong coupling”. A circuit having deep strong coupling is said to operate in a topological regime.

FIG. 3Aillustrates an example embodiment of the qubit circuit300, where coupling devices302are SQUIDS. The coupling devices302are composed of two Josephson junctions304,306, connected in parallel. Two physical qubits308are as per the embodiment ofFIG. 2A, with a Josephson junction310of Josephson energy EJqand a capacitor312of capacitance C.

Referring toFIG. 3B, two superconducting loops314,316are illustrated for the circuit300. 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 2×N superconducting loops, although more than two loops may be provided per coupling device in the circuit.

Each loop314,316of circuit300is 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 inFIG. 3B, where a line318is coupled to superconducting loop314through a mutual inductance M1and carrying a current I1that induces a flux oft ΦSQUIDin loop314. Similarly, a line320coupled to superconducting loop316through a mutual inductance M2and carrying a current I2induces a flux Φextin loop316. Other embodiments may also apply.

A magnetic field is applied to the circuit300in order to induce a phase shift in the coupling devices302, so as to obtain a deep strong coupling regime. The magnetic field induces a non-zero external flux Φextthreading loop316.

A superconducting node phase ϕiand a charge number niare assigned to each qubit308, and the Hamiltonian of a chain of N qubits308is given by:

H=∑i=1N⁢[4⁢EC⁢ni2-EJq⁢cos⁢⁢ϕi]-∑i=1N-1⁢EJc⁢cos⁡(ϕi-ϕi+1-ϕext),(2)
with

Expanding the cosine and sine terms involving ϕito second order Taylor series, the Hamiltonian becomes

The first term corresponds to the sum of the Hamiltonians of N transmon qubits having Josephson energy equal toJ=EJq+2EJccos ϕextwhile 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 ofJ=EJq+EJccos ϕext. The effective qubit impedance and plasma frequency are defined as:

Rewriting the Hamiltonian in terms of the Pauli operators gives:

In the Ising model, the condition for achieving topological protection is

If there is no external flux, i.e. ϕext=0, then the condition cannot be realised with EJcand EJqbeing positive. Deep strong coupling can only be satisfied if:

Topological order is thus attainable with such a design if an external phase having a value between π/2 and π/2 is applied to the coupler. Coupling is maximal at ϕext=π, in which case the condition on the design becomes EJq/3<EJc.

FIG. 4shows simulation results for the energy levels of three qubits308coupled by two coupling devices302composed of SQUIDs. The external flux for the coupling devices302was set such that ϕext=π.FIG. 4shows the energy spectrum with respect to the ground state energy when EJq=20 GHz, EJ,SQUID=4.5 GHz and C=80 fF as a function of the SQUID magnetic frustration

fSQUID=ΦSQUIDΦ0.
We can see that the separation between the ground state and the first excited state decreases from 6 GHz to less than 1 GHz when fSQUIDapproaches zero, where the coupling is maximal. Moreover, the derivative of the energy levels is zero at the maximal coupling point for fSQUID=0.

The calculated spectrum with four and five qubits308is shown inFIG. 5andFIG. 6respectively. Increasing the number of qubits308reduces the energy splitting between the ground state and the first excited state at maximal coupling. With five qubits308, the two states are almost degenerate at the fSQUID=0 operating point. Degenerate ground states are indeed characteristic of a topological state in the Ising model.

FIG. 7Aillustrates an example circuit700with a coupling device according to another embodiment. A tunable flux qubit702is used to couple two qubits704, composed of capacitor706and junction708. The coupling strength of the tunable flux qubit702used as a coupling device can be tuned by applying a flux on the SQUID formed by the two EJ,SQUIDjunctions710,712. InFIG. 7B, two superconducting loops718,720are illustrated. By using two junctions714,716, the loop720threaded by the external magnetic flux Φextis decoupled from the qubits704, which may minimize unintentional driving of the qubits704.

Noting that the junction708and junction714form an asymmetric SQUID with zero flux, we can simply replace EJqby Ejq+EJsin equation (2) to find that the same Hamiltonian as the one presented above governs the embodiment ofFIGS. 7A-7B, and the conditions for deep strong coupling become:

Replacing the junctions714and716by superconducting inductors would lead to a similar result.

FIG. 8Aillustrates an embodiment of a qubit circuit800with differential qubits802coupled with a coupling device804. The qubits802are coupled at one node806through a SQUID (junctions810,812) and at another node808by a superconducting line818. As shown inFIG. 8B, superconducting loops814,816are present. An external flux ϕextis threaded in loop816.

The circuit800has the same Hamiltonian as the circuit300when the capacitance is replaced by C/2 such that

EC=e2C
in equation (2). The condition for reaching topological order is the same.

FIG. 8Cillustrates an embodiment for a qubit circuit800B using the same differential qubits802and coupling devices804as qubit circuit800. The superconducting lines818(from circuit800) between adjacent qubits are replaced by a single superconducting line820between the first qubit802A and the last qubit802B. Hence, superconducting loops816(from circuit800) are replaced by a single superconducting loop822that spans the entire chain of qubits. The external flux ϕextthreading loop822can be selected to induce a desired phase shift in the coupling devices804.

The Hamiltonian of the circuit900is the same as the Hamiltonian of the circuit300if we set

EJq′=2⁢EJ,SQUID⁢cos⁡(πΦSQUIDΦ0)
and replace EJqand EJcin equation (2) by EJsand EJq′, respectively. By inducing a phase shift of π in junction906and junction910using external fluxes, the condition for deep strong coupling becomes EJs/3<EJq′. Note that for tunability, the EJq′junction906may be implemented as a SQUID.

FIG. 10Aillustrates an embodiment of a qubit circuit1000where physical qubits1002are two-junction qubits made of Josephson junctions1004and1006and capacitor1008and are connected together with coupling device1010. The coupling device1010is a SQUID formed from two Josephson junctions1012and1014. Josephson junctions1012,1014have Josephson energy EJ,SQUID, while the two junctions1004,1006have Josephson energy EJqand EJsrespectively. As shown inFIG. 10B, external magnetic fluxes ϕextand ϕSQUIDthread superconducting loops1024,1022formed by junctions1006-1014-1016and1012-1014, respectively.

Circuit nodes1018,1020are associated with a node phase denoted by variables ϕiand ξi, respectively. The ϕinodes1018are associated with a charge number ni. The total Hamiltonian for such a qubit chain is:

Since the ξinodes1020have no capacitance, a degree of freedom may be removed from the Hamiltonian by writing ξiin terms of ϕi. This is done by writing the Kirchoff current law at the coupling nodes:
EJcsin(ξi−1−ξi+ϕext)+EJqsin(ϕi−ξi)=EJssin (ξi)+EJcsin(ξi−ξi+1+ϕext).  (15)

Expanding the sines to first order gives:

This expression shows that in general, the coupling between the qubits1002is not limited to first nearest-neighbours. Indeed, the coupling term in the Hamiltonian is proportional to ξi+1ξi.

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:

Defining

a=EJqEJq+EJs+2⁢EJc,
the Hamiltonian may be rewritten as:

If the circuit1000is operated in the regime where EJq>>EJc& EJs, and a≈1, then we retrieve the Hamiltonian of equation (2). Indeed, when EJq>>EJs, the inductance of the junction EJqis very small compared to the other inductances of the circuit1000and can thus be considered as a short circuit. Using circuit1000with a≈1 instead of circuit300may allow the individual qubit frequency to be separately tuned in the non-topological regime, assuming junction1004is implemented as a SQUID, since this junction is decoupled from the flux bias of the superconducting loop1024.

In order to find the condition to obtain deep strong coupling using the architecture ofFIGS. 10A-10B, the effective Josephson energyJfor ϕext=π is:
=(a−1)2EJg+α2EJs−2a2EJc.  (19)

The condition for deep strong coupling is:
<a2EJc.  (20)

If a≈1, this condition implies EJc/EJs>⅓, consistent with the condition previously derived for the circuit300ofFIG. 3.

Replacing the junctions1016and1006by superconducting inductors (i.e. replacing the two-junction qubits1002by inductively shunted qubits) would lead to a similar result.

The energy spectrum of the coupled qubits1002as a function of the flux applied to the superconducting loop1022, as per the embodiment ofFIGS. 10A-10B, was simulated. The results of the simulation are illustrated inFIG. 11, where the spectrum of three coupled qubits1002is shown. Here, EJ,SQUID=4.5 GHz, EJs=20 GHz and EJq=80 GHz, while ϕext=π. The spectrum is very similar to the spectrum ofFIG. 4, 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<0).FIG. 12Ashows a design allowing for ferromagnetic coupling. Qubit1202is coupled to coupling device1204. Junctions1206and1208with Josephson energies EJ1and EJ2, respectively, connected in series are added in parallel with the junction1210of Josephson energy EJq. As shown inFIG. 12B, the three junctions1206,1208,1210define a superconducting loop1214on which the external flux Φextis applied. Another superconducting loop1212is also formed within the coupling device1204.

We refer to the phase difference on junctions1206,1208,1210as δ1i, δ2iand ϕirespectively. Considering the quantization of the phase around the superconducting loop threaded by the external flux, we have:
δ1i+δ2i−ϕi−ϕext=0(mod 2π),  (21)
where we have defined ϕext=2πΦext/Φ0. For the rest of this derivation, we will assume that ϕext=π.

The problem has a second constraint: that the current in junctions1206,1208in series must be the same, which means that:
EJ1sin δ1i=EJ2sin δ2i.  (22)

We now assume that EJ1>>EJ2. As a result, δ1iis limited to very small values around zero and δ2iapproaches π due to the external flux. Combining equations (21) and (22) and assuming ϕext=π, we have:
EJ1sin δ1i=−EJ2sin(−δ1i+ϕi).  (23)

Using a first order Taylor expansion we find:

We now write the Hamiltonian:

We can now define an effective Josephson energyand qubit impedance r as:

=EJq-EJ⁢⁢1⁢EJ⁢⁢2EJ⁢⁢1-EJ⁢⁢2+2⁢EJc(29)
and

The Hamiltonian can be rewritten using Pauli operators as:

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

This implies that

EJ⁢⁢1⁢EJ⁢⁢2EJ⁢⁢1-EJ⁢⁢2
is larger than EJqfor positive EJc.

Equations (23) to (32) were derived assuming EJ1>>EJ2. Having instead EJ1<<EJ2leads to a swapping of EJ1and EJ2in the equations. Replacing either EJ1or EJ2by a superconducting inductor would also give a similar result.

As will be understood, the circuits100,300,700,800,900,1000,1200may be operated as topologically protected qubit circuits.FIG. 13illustrates a method1300for topological protection of quantum information in a qubit circuit, such as circuits100,300,700,800,900,1000,1200. At step1302, a plurality of physical qubits, such as qubits102,308,704,802,902,1002,1202, are coupled with a plurality of interleaved coupling devices, such as coupling devices104,302,702,804,904,1010,1204. In some embodiments, the qubits each comprise at least one capacitor and at least one Josephson junction connected together, as illustrated in the embodiments ofFIGS. 2A-2D. The coupling devices each comprise at least one superconducting loop threadable by an external flux ϕext.

At step1304, parameters are selected for the external flux ϕextsuch that

Jh>1,
where J is the coupling devices and h is the energy of the physical qubits. In some embodiments, selecting parameters as per step1304comprises selecting ϕextto induce a phase shift with a value between π/2 and 3π/2 (mod 2π) in at least one Josephson junction of the qubit circuit. In some embodiments, selecting parameters as per step1304comprises selecting ϕextto induce a phase shift of π (mod 2π) in at least one Josephson junction of the qubit circuit.

At step1306, the external flux ϕextis 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, ϕextis 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 method1300may thus, in some embodiments, also comprise a step1308of selecting parameters for the external flux ϕSQUID, and/or a step1310of applying the external flux ϕSQUIDto the second superconducting loop. The parameters for ϕampmay be selected such that ϕSQUID=(2n+1)/2*ϕo, where n is an integer and ϕois 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=+/−0.5 ϕoprovides such capability. A flux ϕSQUID=(2n+1)/2*ϕocan also be applied only to selected couplers. For example, if a flux ϕSQUID=(2n+1)/2*ϕois 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/2 physical qubits.

The flux in the SQUID of one coupler may be changed from a value of (2n+1)/2*ϕoto a different value. For example, the flux in a coupler between two chains of N/2 coupled physical qubits can be modified to a value different from (2n+1)/2*ϕo, such as a value of nϕo, in order to convert the two topological qubits made of N/2 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, ϕSQUIDis changed adiabatically to ensure that the symmetry of the wave function is preserved during the procedure.

Although illustrated as sequential, the steps1304-1310of the method1300may be performed in any desired order, and in some cases concurrently. For example, parameters for both fluxes may be selected concurrently, as per steps1304and1308, but applied sequentially as a function of a desired implementation. Steps1306and1310will necessarily be performed sequentially, but not necessarily in the order illustrated.

Various aspects of the circuits and methods described herein may be used alone, in combination, or in a variety of arrangements not specifically discussed in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments. In addition, all of the embodiments described above with regards to circuit100may be used conjointly with the method1300.

Although particular embodiments have been shown and described, it will be apparent to those skilled in the art that changes and modifications may be made without departing from this invention in its broader aspects. The scope of the following claims should not be limited by the embodiments set forth in the examples, but should be given the broadest reasonable interpretation consistent with the description as a whole.