WO2021100417A1 - Electric circuit - Google Patents

Abstract

Provided is an electric circuit that operates at room temperature, can be fine-integrated, and is capable of performing a calculation that is equivalent to topological quantum calculation. An electric circuit 10 is provided with a plurality of unit circuits U1, U2 that include topological phase operation circuits TCS1, TCS2, respectively, in which at least four endpoints and impedance can be adjusted. A circuit Laplacian representing the proportional relationship between an AC current at one of the four endpoints and an AC voltage at an endpoint adjacent to the aforementioned endpoint is configured so as to correspond to the Hamiltonian of a quantum system, and the impedances of the topological phase operation circuits correspond to a parameter included in the Hamiltonian, the Hamiltonian having a topological phase in a prescribed region pertaining to the parameter.

Description

electric circuit Cross-reference of related applications

This application is based on Japanese Application Nos. 2019-209011 filed on November 19, 2019 and Japanese Application Nos. 2020-88598 filed on May 21, 2020. Invite.

The present invention relates to an electric circuit.

Quantum computing is attracting attention as a technology that breaks through the limits of Moore's Law regarding semiconductor integration. In the classical calculation, the calculation is performed using the bits of 0 and 1, whereas in the quantum calculation, the calculation is performed using the qubit by superposition of 0 and 1. By using qubits, it is expected that the calculation speed will be exponentially improved. Currently, various quantum calculation methods are in the research stage.
As hardware for performing quantum calculations, quantum dots, superconducting qubits, photon computers, nuclear magnetic resonance, ion traps, cavities QED (Quantum Electrodynamics), topological superconductors (Non-Patent Document 1), etc. have been studied. However, none of them have reached the practical stage.

In the current quantum calculation, how to deal with decoherence in which quantum information is lost in a short time is an important issue. In order to correct the error caused by decoherence, more qubits for error correction are required than the qubits required for quantum computation. Moreover, it is difficult to retain quantum information for a long period of time.

As one method for solving this problem, topological quantum calculation is known in which quantum calculation is performed by braiding the Majorana particles in a topological superconductor (Non-Patent Document 1). However, at present, a method for controlling Majorana particles has not been established, and superconductivity cannot be realized at room temperature. Therefore, a large amount of helium is required for cooling, and it is difficult to make an inexpensive and compact device. Furthermore, there is no prospect of finely integrating superconducting quantum circuits.

Therefore, the present invention provides an electric circuit that operates at room temperature, can be finely integrated, and can perform calculations equivalent to topological quantum calculations.

The electric circuit according to one aspect of the present invention includes a plurality of unit circuits including at least four endpoints and a topological phase operation circuit having adjustable impedance, and the alternating current at one of the four endpoints and the said. The circuit Laplacian, which represents the proportional relationship with the AC voltage of the end point adjacent to the end point, is configured to correspond to the Hamiltonian of the quantum system, and the impedance of the topological phase operation circuit corresponds to the parameters contained in the Hamiltonian. It has a topological phase in a given area of parameters.

According to this aspect, the circuit Laplacian is configured to correspond to the Hamiltonian of a quantum system having a topological phase in a predetermined parameter region, thereby creating an edge state between the topological phase and the topologically obvious phase. A corresponding voltage state can be created, and a calculation equivalent to a topological quantum calculation can be performed by manipulating the voltage state with a blade. Moreover, since it is a normal electric circuit, it operates at room temperature and can be finely integrated.

In the above embodiment, the plurality of unit circuits include a first circuit corresponding to an electronic band and a second circuit corresponding to a hall band, respectively, and the phase of the AC voltage of the first circuit and the phase of the AC voltage of the second circuit are set. They may be opposite to each other.

According to this aspect, a voltage state corresponding to electrons and holes can be created, and a voltage state corresponding to Majorana particles can be created between the topological phase and the topologically trivial phase.

In the above aspect, the plurality of unit circuits may include a phase operation circuit that connects the first circuit and the second circuit and changes the phase of the AC voltage.

According to this aspect, a parameter corresponding to the superconducting phase can be realized.

In the above aspect, the topological phase operation circuit may include a switch that connects the first circuit or the second circuit to the ground and switches the impedance.

According to this aspect, the lengths of the topological phase and the topologically trivial phase can be arbitrarily adjusted, and various blade operations related to the edge state can be realized.

In the above aspect, the plurality of unit circuits may be connected in a T shape including a branch connection.

According to this aspect, it is possible to realize an operation of exchanging the positions of two edge states appearing at both ends of the topological phase.

In the above aspect, the plurality of unit circuits may include a branch connection and may be connected in a cross shape.

According to this aspect, it is possible to realize an operation of exchanging the positions of two edge states appearing at both ends of different topological phases.

In the above aspect, the Hamiltonian may be a superconductor Hamiltonian described in the Bogoliubov-de Gennes format having electron-hole symmetry, or a Hamiltonian having chiral symmetry.

According to this aspect, it is possible to create a voltage state corresponding to a stable Majorana particle by electron-hole symmetry or chiral symmetry.

In the above embodiment, the AC voltage at the end point located at the boundary between the unit circuit corresponding to the topological phase of the plurality of unit circuits and the unit circuit corresponding to the topologically obvious phase of the plurality of unit circuits, or the plurality of unit circuits. The AC voltage at the end point of the unit circuit located at the end may correspond to the edge state of the Hamiltonian.

According to this aspect, a voltage state corresponding to an edge state can be created between a topological phase and a topologically trivial phase, and a calculation equivalent to a topological quantum calculation can be performed by manipulating the voltage state in a blade. Can be done.

According to the present invention, it is possible to provide an electric circuit that operates at room temperature, can be finely integrated, and can perform calculations equivalent to topological quantum calculations.

It is a figure which shows the electric circuit which concerns on embodiment of this invention. It is a figure which shows the phase operation circuit included in the electric circuit which concerns on this embodiment. It is a figure which shows the structure of the operational amplifier included in the phase operation circuit which concerns on this embodiment. It is a figure which shows the case which realizes the electron-hole interaction of φ = 0 by the phase operation circuit which concerns on this embodiment. It is a figure which shows the case which realizes the electron-hole interaction of φ = π by the phase operation circuit which concerns on this embodiment. It is a figure which shows the topological phase operation circuit which is included in the electric circuit which concerns on this embodiment, and is in an on state. It is a figure which shows the topological phase operation circuit which is included in the electric circuit which concerns on this embodiment, and is in an off state. It is a figure which shows typically the braid operation of the edge state in the electric circuit which concerns on this embodiment. It is a figure which shows typically the operation of exchanging the position of two edge states appearing at both ends of a topological phase in the electric circuit which concerns on this embodiment. It is a figure which shows typically the operation of exchanging the positions of two adjacent edge states with a topologically obvious phase in between in the electric circuit which concerns on this embodiment. It is a figure which shows the time change of the wave function in the operation which exchanges the position of two edge states appearing at both ends of the topological phase in the electric circuit which concerns on this embodiment. It is a figure which shows the time change of the energy eigenvalue and the Berry phase in the operation which exchanges the positions of two edge states appearing at both ends of a topological phase in the electric circuit which concerns on this embodiment. It is a figure which shows typically the operation of exchanging the position of the edge state which appears at both ends of the different topological phases in the electric circuit which concerns on this embodiment. It is a figure which shows an example which provided the storage circuit in the electric circuit which concerns on this embodiment. It is a figure which shows the LC resonator which concerns on this embodiment.

An embodiment of the present invention will be described with reference to the accompanying drawings. In each figure, those having the same reference numerals have the same or similar configurations.

In topological quantum computation, the blade operation of non-commutative anyon is used as the quantum gate. In a normal system in which the space is three-dimensional, a particle whose phase change of the wave function becomes 0 or π due to particle exchange and whose phase change becomes 0 is called a boson, and a particle whose phase change becomes π is called a fermion. .. On the other hand, in a system in which the space is two-dimensional, particles called anyon can be realized in which the phase change of the wave function becomes an arbitrary value by exchanging the particles. For example, in Non-Patent Document 1, non-commutative anyon is realized as an edge state of a one-dimensional superconducting wire.

As a Hamiltonian that simplifies the one-dimensional superconducting wire, the Kitaev model of the following mathematical formula (1) is known.

Here, ε _k = −t cosk−μ, t is the hopping amplitude, μ is the chemical potential, φ is the superconducting phase, and Δ is the gap parameter. The Hamiltonian of the equation (1) is a topologically trivial phase when Δ is not zero and | μ |> | 2t |, and is a topological phase when | μ | << | 2t |.

The topological invariants that characterize the topological phase are typically defined by the number of wraps of the map from the momentum space to the Hilbert space, taking only discrete values such as 1, 2, ..., and topologically trivial phase phases. The invariant is 0. Inside the topological phase, the phase invariant does not change because the gap Δ is finite. On the other hand, since the topologically trivial phase invariant is 0, the gap is closed at the boundary between the topological phase and the topologically trivial phase, a gapless state occurs, and the topological invariant changes. This gapless state is called an edge state. In the Kitaev model of mathematical formula (1), Majorana particles that do not distinguish between electrons and holes (particles and antiparticles) appear as edge states.

The electric circuit 10 according to the present embodiment has a circuit Laplacian equivalent to the mathematical formula (1), and creates a voltage state corresponding to the edge state of the quantum system. Further, the electric circuit 10 according to the present embodiment can perform a calculation equivalent to the topological quantum calculation by performing a blade operation on the voltage state corresponding to the edge state.

The Hamiltonian of the equation (1) is a Hamilton of a one-dimensional superconductor described in the Bogoliubov-de Gennes form, and has electron-hole symmetry. The electron-hole symmetry is the symmetry in which the Hamiltonian ^{satisfies Ξ-1} H (k) Ξ = H (−k) for the operator Ξ that exchanges electrons and holes. By having a circuit Laplacian corresponding to such a Hamiltonian, it is possible to create a stable Majorana fermion due to electron-hole symmetry or a voltage state corresponding to a Majorana-like state having properties similar to those of Majorana fermion.

Also, when the chiral symmetry ΠH (k) + H (k) Π = 0 is satisfied instead of the electron-hole symmetry, a multiple degenerate state can be created. In this case, the same calculation as the topological quantum calculation can be performed by the blade operation of exchanging the positions of the multiple degenerate states.

FIG. 1 is a diagram showing an electric circuit 10 according to an embodiment of the present invention. The electrical circuit 10 includes a plurality of unit circuits each including at least four endpoints and a topological phase operating circuit TCS with adjustable impedance. In the figure, the adjacent unit circuit U1 and the unit circuit U2 are shown.

The circuit Laplacian, which represents the proportional relationship between the alternating current of one of the four endpoints and the alternating voltage of the endpoint adjacent to the endpoint, is configured to correspond to the Hamiltonian of the quantum system. Here, the circuit Laplacian J _ab (ω) has _{I a} (ω) = Σ _b J _ab (ω) _{with respect to the AC current I a} (ω) at the end point a and the AC voltage V _b (ω) at the end point b. It is an quantity defined by V _{b (ω).}

Specifically, the circuit Laplacian of the electric circuit 10 according to the present embodiment is represented by the following mathematical formula (2).

Here, C is the capacitance shown in FIG. 1, and L is the inductance shown in FIG. Further, C ₀ is the capacitance included in the topological phase operation circuits TCS1 and TCS2 described later, and C _X ^φ = C _X sin φ is the capacitance included in the phase operation circuit PCS described later, and R ^φ = R. _b / cosφ is a resistor included in the phase control circuit PCS described later. Here, φ is a parameter corresponding to the superconducting phase. Furthermore, σ _x , σ _y , and σ _z are 2 × 2 Pauli matrices. Note that the topological phase operation circuit TCS1, TCS2, inductance _{L 0} are also included. The phase control circuit PCS _{also includes an inductance L X} ^φ = L _X / sin φ.

_{The circuit Laplacian of the formula (2) is J ab} (ω) = iωH _ab when ω = 1 / √ (LC) = 1 / √ (L ₀ C ₀ ) = 1 / √ (L _X C _X ). It corresponds to the Hamiltonian of the mathematical formula (1) in relation to (ω). In this case, the impedances of the topological phase operation circuits TCS1 and TCS2 correspond to the parameters t and μ included in the Hamiltonian of the equation (1). In the case of | μ |> | 2t | (in the case of a topologically obvious phase), it corresponds to the case where the topological phase operation circuits TCS1 and TCS2 are in the off state, and in the case of | μ | << | 2t | (in the case of a topological phase). ) Corresponds to the case where the topological phase operation circuits TCS1 and TCS2 are in the ON state. As described above, the Hamiltonian corresponding to the circuit Laplacian has a topological phase in a predetermined region regarding the parameter, and the electric circuit 10 is topologically self-evident with the topological phase by changing the impedance of the topological phase operation circuits TCS1 and TCS2. You can switch between different phases.

According to the electric circuit 10 according to the present embodiment, the topological phase and the topologically obvious phase can be obtained by configuring the circuit Laplacian so as to correspond to the Hamiltonian of the quantum system having the topological phase in a predetermined parameter region. A voltage state corresponding to the edge state can be created between them, and the same calculation as the topological quantum calculation can be performed by manipulating the voltage state with a blade. Further, since the electric circuit 10 according to the present embodiment is a normal electric circuit, it is relatively inexpensive, operates at room temperature, and can be finely integrated. In addition, since it simulates topological quantum computation, it is not affected by decoherence, and there is no need to increase the scale of the circuit for error correction.

The plurality of unit circuits U1 and U2 include the first circuit E1 corresponding to the electronic band and the second circuit H1 corresponding to the hall band, respectively, and have the phase of the AC voltage of the first circuit E1 and the AC voltage of the second circuit H1. The phases are opposite to each other. With such a configuration, a voltage state corresponding to electrons and holes can be created, and a Majorana particle or a Majorana-like state having properties similar to those of Majorana particles can be created between the topological phase and the topologically obvious phase. It is possible to create a corresponding voltage state.

The plurality of unit circuits U1 and U2 include a phase operation circuit PCS that connects the first circuit E1 and the second circuit H1 and changes the phase of the AC voltage. The phase manipulation circuit PCS can realize parameters corresponding to the superconducting phase included in the Hamiltonian of the quantum system.

The plurality of unit circuits U1 and U2 include a branch connection and are connected in a T shape. As will be described later, by including the T-shaped branch, it is possible to realize an operation of exchanging the positions of the two edge states appearing at both ends of the topological phase. The T-shaped branch may be a Y-shaped branch or another branch.

AC voltage or multiple units at the end point located at the boundary between the unit circuit corresponding to the topological phase of the plurality of unit circuits U1 and U2 and the unit circuit corresponding to the topologically obvious phase of the plurality of unit circuits U1 and U2. The AC voltage at the end point of the unit circuit located at the end of the circuits U1 and U2 corresponds to the edge state of the Hamiltonian. Here, the end points of the unit circuits located at the ends of the plurality of unit circuits U1 and U2 are in contact with air or an ordinary insulator, and are in contact with a topologically obvious phase. According to the electric circuit 10 according to the present embodiment, a voltage state corresponding to an edge state can be created between a topological phase and a topologically trivial phase, and the voltage state can be manipulated by a blade to perform topological quantum calculation. Equivalent calculations can be performed.

FIG. 2 is a diagram showing a phase control circuit PCS included in the electric circuit 10 according to the present embodiment. In the figure, the case where the two wires on the left side are connected to the end points of the first circuit E2 and the two wires on the right side are connected to the end points of the second circuit H2 is shown. The phase control circuit PCS includes an operational amplifier OP, a capacitance C _X ^φ and an inductance L _X ^φ .

FIG. 3 is a diagram showing a configuration of an operational amplifier OP included in the phase operation circuit PCS according to the present embodiment. The operational amplifier OP includes a resistor ^Rφ .

FIG. 4 is a diagram showing a case where the electron-hole interaction of φ = 0 is realized by the phase operation circuit PCS according to the present embodiment. The figure shows a case where the upper two wires are connected to the end points of the first circuit E1 and the lower two wires are connected to the end points of the second circuit H1.

FIG. 5 is a diagram showing a case where the electron-hole interaction of φ = π is realized by the phase manipulation circuit according to the present embodiment. The figure shows a case where the upper two wires are connected to the end points of the first circuit E1 and the lower two wires are connected to the end points of the second circuit H1.

FIG. 6 is a diagram showing a topological phase operation circuit TCS1 included in the electric circuit 10 according to the present embodiment and in the ON state. The topological phase operation circuit TCS1 includes a capacitance C ₀ and an inductance L ₀ . The figure shows a case where one upper wire is connected to the ground and one lower wire is connected to the end point of the first circuit E1.

The topological phase operation circuit TCS1 includes a switch that connects the first circuit E1 or the second circuit H1 to the ground and switches the impedance.

Topological phase operation circuit TCS1 is in the on state, the end point of the first circuit E1, is connected to the ground via the inductor L _0.

FIG. 7 is a diagram showing a topological phase operation circuit TCS1 included in the electric circuit 10 according to the present embodiment and in an off state. The figure shows a case where one upper wire is connected to the ground and one lower wire is connected to the end point of the first circuit E1.

Topological phase operation circuit TCS1 is off state, the end point of the first circuit E1, is connected to the ground via the capacitance C _0.

The topological phase operation circuit TCS2 connected to the second circuit H2 has the same configuration as the topological phase operation circuit TCS1, but the on / off state is opposite to that of the topological phase operation circuit TCS1. That is, the topological phase operating circuit is in the ON state TCS2 is the end point of the second circuit H1, connected to ground through a capacitance C _0. Furthermore, topological phase operating circuit is off TCS2 is the end point of the second circuit H1, connected to the ground via the inductor L _0.

By switching the on / off state of a plurality of topological phase operation circuits TCS1 and TCS2 arranged in a straight line, the length of the topological phase and the topologically obvious phase can be arbitrarily adjusted. In addition, as described below, it is possible to realize various blade operations related to the edge state appearing at the boundary between the topological phase and the topologically trivial phase.

FIG. 14 is a diagram showing an example in which a storage circuit is provided in the electric circuit 10 according to the present embodiment. In the figure, by connecting the LC resonator R in parallel with the topological phase operation circuits TCS1 and TCS2 at each end point, the information of the blade operation is obtained by the amplitude and phase of the vibration current or the vibration voltage in the LC resonator R. Can be remembered.

FIG. 15 is a diagram showing an LC resonator R. As shown in the figure, in the LC resonator R, the capacitance C ₀ and the inductance L ₀ are connected in parallel, one end thereof is connected to the ground, and the other end is connected to the end point of the first circuit or the second circuit. ..

FIG. 8 is a diagram schematically showing a blade operation in an edge state in the electric circuit 10 according to the present embodiment. In the figure, the topological phase unit circuit is shown by a relatively thick solid line, and the topologically trivial unit circuit is shown by a relatively thin solid line among a plurality of linearly arranged unit circuits. The edge state (voltage state) that appears at the boundary between the topological phase and the topologically trivial phase is indicated by a hatched circle.

In the figure, the topological phase two edge states (Majorana similar state) appearing across M A ^_j-1 and M _{B ^j-1} position and operation of exchanging the topological into two adjacent members across the trivial phase The operation of exchanging the positions of the edge state (Majorana-like state) M _B ^j and M _A ^{j + 1 is shown.} Details of both operations will be described with reference to FIGS. 9 and 10. As described above, in the electric circuit 10 according to the present embodiment, a calculation equivalent to the quantum calculation is performed by performing a blade operation on the edge states arranged in a straight line.

The electric circuit 10 according to the present embodiment generates 2N topologically protected edge states at the boundary by bringing N topological phases and N topologically trivial phases into contact with each other, and exchanges them. Performs the same calculation as the topological quantum calculation by the blade operation.

FIG. 9 is a diagram schematically showing an operation of exchanging the positions of two edge states appearing at both ends of the topological phase in the electric circuit 10 according to the present embodiment. In the figure, the topological phase unit circuit is shown by a relatively thick solid line, and the topologically obvious phase unit circuit is shown by a relatively thin solid line among a plurality of linearly arranged unit circuits, and appears at both ends of the topological phase. The edge state (voltage state) is indicated by a hatched circle. Further, the left-pointing arrow and the upward arrow overlaid on the solid line representing the unit circuit represent the superconducting phase φ = 0, and the right-pointing arrow and the down-pointing arrow overlaid on the solid line representing the unit circuit represent the superconducting phase φ = 0. , Represents the superconducting phase φ = π.

(A)
Of the edge states appearing at both ends of the topological phase, in order to move the rightmost edge state to the center, the switches of the topological phase operation circuits TCS1 and TCS2 of the unit circuit are switched off from the right end side. Hereinafter, the unit circuit group will be referred to as No. 1, No. 2, and No. 3 as shown in the figure. The unit circuits of No. 1, No. 2, and No. 3 are branched into a T shape.

(B)
After moving the rightmost edge state to the center, the edge state is moved to the unit circuit No. 3. This operation is performed by switching on the switches of the topological phase operation circuits TCS1 and TCS2 included in the unit circuit No. 3 from the branch side.

(C)
In order to move the edge state located at the right end to the end of the unit circuit No. 3 and then move the edge state located at the left end to the center, the topological phase operation circuit TCS1 of the unit circuit The switch of TCS2 is switched off from the left end side.

(D)
After moving the leftmost edge state to the center, the superconducting phase of the unit circuit No. 3 is changed from φ = 0 to φ = π.

(E)
After that, the edge state located in the center is moved to the right end.

(F)
The edge state moved to the end of the unit circuit of No. 3 is moved to the center.

(G)
The edge state moved to the center is moved to the left end.

(H)
After moving the edge state located in the center to the left end, the superconducting phase of the unit circuit No. 3 is changed from φ = π to φ = 0.

(I)
Finally, the edge state initially located at the left end is located at the right end, and the edge state initially located at the right end is located at the left end. In this way, the exchange operation of the two edge states can be performed.

FIG. 10 is a diagram schematically showing an operation of exchanging the positions of two adjacent edge states with a topologically obvious phase in between in the electric circuit 10 according to the present embodiment. In the figure, the topological phase unit circuit is shown by a relatively thick solid line, and the topologically obvious phase unit circuit is shown by a relatively thin solid line among a plurality of linearly arranged unit circuits, and appears at both ends of the topological phase. The edge state (voltage state) is indicated by a hatched circle. Further, the left-pointing arrow and the upward arrow overlaid on the solid line representing the unit circuit represent the superconducting phase φ = 0, and the right-pointing arrow and the down-pointing arrow overlaid on the solid line representing the unit circuit represent the superconducting phase φ = 0. , Represents the superconducting phase φ = π.

(A)
In order to move the edge state appearing at the right end of the topological phase located on the left side to the center, the switches of the topological phase operation circuits TCS1 and TCS2 of the unit circuit are switched on from the left end side of the topologically trivial phase.

(B)
The edge state, which was initially located at the left end of the topologically trivial phase, is moved to the center and then moved to the upper side of the T-shaped branch.

(C)
In order to move the edge state appearing at the left end of the topological phase located on the right side to the center, the switches of the topological phase operation circuits TCS1 and TCS2 of the unit circuit are switched on from the right end side of the topologically trivial phase.

(D)
After moving the edge state, which was initially located at the right end of the topologically trivial phase, to the center, the superconducting phase of the unit circuit located above the T-shaped branch is changed from φ = 0 to φ = π. ..

(E)
After that, the edge state located in the center is moved to the left end.

(F)
The edge state moved to the end of the unit circuit located above the T-shaped branch is moved to the center.

(G)
The edge state moved to the center is moved to the right end.

(H)
After moving the edge state located in the center to the right end, the superconducting phase of the unit circuit located above the T-shaped branch is changed from φ = π to φ = 0.

(I)
Finally, the edge state that initially appeared at the right end of the topological phase located on the left side is located at the left end of the topological phase located on the right side, and the edge state that initially appeared at the left end of the topological phase located on the right side is It will be located at the right end of the topological phase located on the left side. In this way, it is possible to perform an exchange operation of two edge states adjacent to each other with a topologically obvious phase in between.

FIG. 11 is a diagram showing a time change of a wave function in an operation of exchanging the positions of two edge states appearing at both ends of the topological phase in the electric circuit 10 according to the present embodiment. In the figure, the place where the wave function is localized is shown so as to correspond to the time evolution of (a) to (i) shown in FIG. The wave function can be read as the AC voltage of the electric circuit 10 according to the present embodiment. Therefore, the probability of existence of particles can be read as the square of the amplitude of the AC voltage.

In the initial state (a), the wave function is localized at both ends of the T-shaped unit circuit (T junction), indicating that the edge state is formed at both ends of the topological phase. After that, in (b), the rightmost edge state moves to the center, and in (c), the central edge state moves to the end of the unit circuit located above the T-shaped branch.

After that, in (d), the edge state at the left end moves to the center, and the superconducting phase of the unit circuit located above the T-shaped branch is changed. The edge state located at the center in (e) moves to the right end. Further, in (f), the edge state located at the end of the unit circuit located above the T-shaped branch moves to the center, and in (g), the edge state moved to the center moves to the left end. To do. In (h), the superconducting phase of the unit circuit located above the T-shaped branch is changed. Finally, in the state (i), the edge state initially located at the left end is located at the right end, and the edge state initially located at the right end is located at the left end.

FIG. 12 is a diagram showing the time change of the energy eigenvalue and the Berry phase in the operation of exchanging the positions of the two edge states appearing at both ends of the topological phase in the electric circuit 10 according to the present embodiment. In the figure, the time variation of the energy eigenvalue and the Berry phase is shown so as to correspond to the time evolution of (a) to (i) shown in FIG. The energy eigenvalue is read as the admittance of the electric circuit 10 according to the present embodiment, and the Berry phase is read as the phase of the AC voltage.

When the operations (a) to (i) are performed, the state where the energy eigenvalue is positive remains positive, and the state where the energy eigenvalue is negative remains negative. That is, the energy of the gap mode is never zero, and the gapless edge state is not mixed with the bulk gap mode by the blade operation.

Further, the Berry phase is changed from 0 to π / 2 by the operation from (d) to (e). That is, when the positions of the edge states appearing at both ends of the topological phase are exchanged by the operations (a) to (i), the phase of e ^{iπ / 2} is generated. This means that the edge state is non-commutative anyon. It also means that a quantum gate corresponding to _{iσ x} can be realized by the operations (a) to (i). Since the Berry phase is based on the adiabatic approximation, the operations (a) to (i) need to be performed at a speed sufficiently slower than the AC frequency ω.

When the operations (a) to (i) are performed twice in succession, the positions of the two edge states are restored. However, the Berry phase becomes e ^iπ = -1, and the phase of the state is inverted. Such anyon is called Ising enion and is known as a basic component of topological quantum computing.

_{When the operations (a) to (i) are expressed as blade operations U j, j + 1} that exchange adjacent edge states j and edge states j + 1, all three blade relational expressions represented by the following mathematical formula (3) are used. Fulfill.

As described above, according to the electric circuit 10 according to the present embodiment, it is possible to perform a calculation equivalent to a quantum calculation by a blade operation in a topologically stable edge state. In addition, even if the elements that make up an actual electric circuit have some degree of messiness and thermal noise and the circuit Laplacian contains noise, the edge state is stable with respect to perturbations, so it is equivalent to topological quantum calculation. It is still possible to calculate.

FIG. 13 is a diagram schematically showing an operation of exchanging the positions of the edge states appearing at both ends of different topological phases in the electric circuit 10 according to the present embodiment. In the figure, the topological phase unit circuit is shown by a relatively thick solid line, and the topologically obvious phase unit circuit is shown by a relatively thin solid line among a plurality of unit circuits arranged in a grid pattern including a cross-shaped branch. , The edge states (voltage states) that appear at both ends of the topological phase are indicated by hatched circles. Further, the left-pointing arrow and the upward arrow overlaid on the solid line representing the unit circuit represent the superconducting phase φ = 0, and the right-pointing arrow and the down-pointing arrow overlaid on the solid line representing the unit circuit represent the superconducting phase φ = 0. , Represents the superconducting phase φ = π. In the figure, reference numerals A and B are attached to the two edge states to be exchanged. The first edge state A is an edge state located at the left end of the topological phase, and the second edge state B is an edge state located at the right end of a different topological phase.

The plurality of unit circuits in this example include a branch connection and are connected in a cross shape. As described below, such a configuration can realize an operation of exchanging the positions of two edge states appearing at both ends of different topological phases.

(A)
First, the first edge state A and its paired edge state are moved to the upper side of the cross-shaped branch, and the second edge state B and its paired edge state are moved to the lower side of the cross-shaped branch. The state of (b) is assumed.

(B)
Either one of the first edge state A and the second edge state B is retracted to the left side or the right side of the cross-shaped branch, and the positions of the first edge state A and the second edge state B are exchanged with the state of (c). To do.

(C)
The first edge state A located below the cross-shaped branch and its paired edge state are moved to the location where the second edge state B was initially located, and the second edge state located above the cross-shaped branch is located. The edge state B and its paired edge state are moved to the position where the first edge state A was initially located to be in the state (d).

(D)
Finally, the first edge state A, which initially appeared at the left end of the topological phase located at the upper side, is located at the right end of the topological phase located at the lower side and appears at the right end of the topological phase located at the lower side at the beginning. The second edge state B is located at the left end of the topological phase located on the upper side. In this way, it is possible to perform an edge state exchange operation that appears at both ends of different topological phases.

The embodiments described above are for facilitating the understanding of the present invention, and are not for limiting and interpreting the present invention. Each element included in the embodiment and its arrangement, material, condition, shape, size, and the like are not limited to those exemplified, and can be changed as appropriate. In addition, the configurations shown in different embodiments can be partially replaced or combined.

10 ... Electric circuit, E1, E2 ... 1st circuit, H1, H2 ... 2nd circuit, OP ... Operator, PCS ... Phase operation circuit, TCS ... Topological phase operation circuit, U1, U2 ... Unit circuit, R ... LC resonator (Memory circuit)

Claims (8)

1. It has multiple unit circuits, each containing at least four endpoints and a topological phase manipulation circuit with adjustable impedance.
   The circuit Laplacian, which represents the proportional relationship between the AC current at one of the four endpoints and the AC voltage at the endpoint adjacent to the endpoint, is configured to correspond to the Hamiltonian of the quantum system.
   The impedance of the topological phase operation circuit corresponds to the parameters contained in the Hamiltonian.
   The Hamiltonian has a topological phase in a predetermined region with respect to the parameter.
   electric circuit.
2. The plurality of unit circuits include a first circuit corresponding to an electronic band and a second circuit corresponding to a hole band, respectively.
   The phase of the AC voltage of the first circuit and the phase of the AC voltage of the second circuit are opposite to each other.
   The electric circuit according to claim 1.
3. The plurality of unit circuits include a phase operation circuit that connects the first circuit and the second circuit and changes the phase of the AC voltage.
   The electric circuit according to claim 2.
4. The topological phase operation circuit includes a switch that connects the first circuit or the second circuit to the ground and switches the impedance.
   The electric circuit according to any one of claims 1 to 3.
5. The plurality of unit circuits include a branch connection and are connected in a T shape.
   The electric circuit according to any one of claims 1 to 4.
6. The plurality of unit circuits include a branch connection and are connected in a cross shape.
   The electric circuit according to any one of claims 1 to 4.
7. The Hamiltonian is a superconductor Hamiltonian described in the Bogoliubov-de Gennes form having electron-hole symmetry, or a Hamiltonian having chiral symmetry.
   The electric circuit according to any one of claims 1 to 6.
8. Of the plurality of unit circuits, the AC voltage at the end point located at the boundary between the unit circuit corresponding to the topological phase and the unit circuit corresponding to the topologically obvious phase of the plurality of unit circuits, or among the plurality of unit circuits. The AC voltage at the end of the unit circuit located at the end corresponds to the Hamiltonian edge state.
   The electric circuit according to claim 7.

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