SUPERCONDUCTING QUANTUM CIRCUIT

Abstract

A superconducting quantum circuit includes first to fourth qubits, and a coupler including first and second electrodes and a nonlinear element bridging the first and second electrodes, wherein each of the first to fourth qubits includes a resonator including a SQUID loop circuit and a capacitor connected in parallel to the loop circuit, the first and second qubits and the third and fourth qubits capacitively coupled to the first and second electrodes of the coupler, respectively, wherein a magnitude relationship among a capacitance value C of a capacitive coupling between each of the first to fourth qubits and the coupler, a capacitance value C.sub.J of the capacitor connected in parallel to the loop circuit for each of the first to fourth qubits, and a capacitance value C.sub.g between the first and second electrodes of the coupler, is set to C.sub.J>C.sub.g>C.

Claims

1. A superconducting quantum circuit comprising: first to fourth qubits; and a coupler that couples the first to fourth qubits with a four-body interaction, wherein the coupler includes: first and second electrodes arranged opposed to each other; and a nonlinear element including at least one Josephson junction and bridging the first and second electrodes, wherein each of the first to fourth qubits includes a resonator including: a loop circuit with a first superconducting line, a first Josephson junction, a second superconducting line and a second Josephson junction connected in a ring-shape; and a capacitor connected in parallel to the loop circuit, wherein the first and second qubits are capacitively coupled to the first electrode of the coupler, and the third and fourth qubits are capacitively coupled to the second electrode of the coupler, and wherein a magnitude relationship among a capacitance value C of a capacitive coupling between each of the first to fourth qubits and the coupler, a capacitance value C.sub.J of the capacitor connected in parallel to the loop circuit of each of the first to fourth qubits, and a capacitance value C.sub.g between the first and second electrodes of the coupler is set to C.sub.J>C.sub.g>C.

2. The superconducting quantum circuit according to claim 1, wherein a strength of the four-body interaction is expressed with a numerator that includes a fourth power of C and a denominator that includes a product of a square of C.sub.J multiplied by a cube of a sum of C.sub.g and C, and a difference between a resonance angular frequency of the coupler and a resonance angular frequency of each qubit, by making C.sub.g smaller as compared to C.sub.J, the strength of the four-body interaction being set larger.

3. The superconducting quantum circuit according to claim 1, wherein by making C smaller than C.sub.J, a two-body interaction between two qubits is weakened.

4. The superconducting quantum circuit according to claim 1, wherein as for a ratio $/{}_{-}=1,$ where is a resonance angular frequency of each of the qubits and .sub. is a resonance angular frequency of the coupler, is brought closer to .sub. within a range satisfying >C/[4{C.sub.J(C.sub.g+C)}].

5. The superconducting quantum circuit according to claim 1, wherein each of the first to fourth qubits has the resonator terminated with the loop circuit and includes a magnetic flux generator to which a pump signal is fed, the magnetic flux generator generating a magnetic flux that intersects the loop circuit and causing the resonator to perform a parametric oscillation.

6. The superconducting quantum circuit according to claim 1, wherein in each of the first to fourth qubits, the second superconducting line of the loop circuit is connected to ground, the first superconducting line side of the loop circuit is connected to a waveguide, and the capacitor connected in parallel to the loop circuit includes a stray capacitance between the waveguide and ground, wherein the waveguide of each of the first and second qubits has an end thereof capacitively coupled to the first electrode of the coupler, and the waveguide of each of the third and fourth qubits has an end thereof capacitively coupled to the second electrode of the coupler.

7. The superconducting quantum circuit according to claim 1, wherein in the coupler, the nonlinear element includes a loop circuit, in which, a first superconducting line, a first Josephson junction, a second superconducting line and a second Josephson junction are connected in a ring-shape.

8. The superconducting quantum circuit according to claim 1, wherein the first to fourth qubits configured as Josephson parametric oscillators and the coupler constitute a unit structure in a quantum computer.

9. The superconducting quantum circuit according to claim 8, comprising a plurality of the unit structures, wherein a quantum computer is configured to have at least one qubit among the first to fourth qubits constituting the unit structure shared by one or a plurality of other unit structures.

10. The superconducting quantum circuit according to claim 6, wherein the first electrode of the coupler includes first and second opposing portions extended toward the first and second qubits, respectively, the first and second opposing portions having portions opposed to ends of the waveguides of the first and second qubits, for capacitive coupling therewith, respectively, and the second electrode of the coupler includes third and fourth opposing portions extended toward third and fourth qubit, respectively, ends of the third and fourth opposing portions having portions opposed to ends of the waveguides of the third and fourth qubits, for capacitive coupling therewith, respectively.

11. The superconducting quantum circuit according to claim 10, wherein the first electrode other than the first and second opposing portions capacitively coupled with the first and second qubits and the second electrode other than the third and fourth opposing portions capacitively coupled with the third and fourth qubits each have an edge separated from an opposing edge of a ground plane surrounding the coupler with a gap width of a same order as a size of the coupler.

12. The superconducting quantum circuit according to claim 10, wherein the coupler includes a capacitor structure including a plurality of extension portions extended from the first and the second electrodes to the second and the first electrode sides, respectively, and arranged in a nested manner with a comb-teeth shape.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0032] FIG. 1A is a diagram illustrating a related art.

[0033] FIG. 1B is a diagram illustrating a related art.

[0034] FIG. 2 is a diagram illustrating a related art.

[0035] FIG. 3 is a diagram schematically illustrating an example embodiment.

[0036] FIG. 4 is a diagram schematically illustrating the example embodiment.

[0037] FIG. 5 is a diagram schematically illustrating the example embodiment.

[0038] FIG. 6 is a diagram illustrating the example embodiment.

[0039] FIG. 7 is a diagram schematically illustrating an example.

[0040] FIG. 8 is a diagram schematically illustrating a variation example.

[0041] FIG. 9 is a diagram schematically illustrating a variation example.

[0042] FIG. 10 is a diagram illustrating the example embodiment.

EXAMPLE EMBODIMENTS

[0043] The following describes example embodiments of the present disclosure. In the following description of examples and embodiments, reference is made to the accompanying drawings which form a part hereof, and in which it is shown by way of illustration specific examples that can be practiced. It is to be understood that other examples can be used and structural changes can be made without departing from the scope of the disclosed examples. It is noted that in the disclosure, the expression at least one of A and B means A, B, or (A and B). The term expressed as --(s) includes both singular and/or plural form. FIG. 3 is a diagram illustrating a circuit configuration of an example embodiment. In FIG. 3, illustration of a readout circuit connected to each JPO illustrated in FIG. 2 is omitted. In the present example embodiment, resonant frequencies of each JPO and a coupler are made close to each other to strengthen a four-body interaction. They are made close to each other in a range that satisfies a condition described below. Also, in the present example embodiment, conditions are imposed on capacitance to prevent from strengthening unnecessary interactions other than the four-body interaction.

[0044] The example embodiment provides a condition for a parameter setting to strengthen a circuit parameter dependency of the four-body interaction and the four-body interaction.

[0045] Referring to FIG. 3, four JPO1 (20A) to JPO4 (20D) and a coupler 21 are provided. JPO1 (20A) to JPO4 (20D) are connected to the coupler 21 by capacitive coupling through coupling capacitors 31A to 31D, respectively. In the following example embodiment, a lump element type is used for each JPO, but it is of course possible to use a distributed element type.

[0046] The coupler 21 includes a nonlinear element 10 that includes a Josephson junction (JJ) and a capacitor 15 connected in parallel to the nonlinear element 10.

[0047] JPO1 (20A) to JPO4 (20D) include SQUIDs (SQUID loops) 210A to 210D in which first superconducting portions 203A to 203D, first Josephson junctions 201A to 201D, second superconducting portions 204 A to 204D and second Josephson junctions 202A to 202D are connected in a ring, respectively, magnetic field generators 207A to 207D, and capacitors 206A to 206D, respectively. The magnetic field generators 207A to 207D generate magnetic flux through the SQUID loops 210A to 210D by pump signals supplied to control lines 23A to 23D from signal generation parts (not shown), respectively. The capacitors 206A to 206D are connected between the first superconducting portions 203A to 203D and the second superconducting portions 204A to 204D, respectively. The second superconducting portions 204A to 204D are connected to ground, the first superconducting portions 203A and 203B of JPO1 (20A) and JPO2 (20B) are connected to one end of the coupler 21 via coupling capacitors 31A and 31B, and the first superconducting portions 203C and 203D of JPO3 (20C) and JPO4 (20D) are connected to the other end of the coupler 21 via coupling capacitors 31C and 31D, respectively.

[0048] C.sub.J is a capacitance value of each of the capacitors 206A to 206D of JPO1 (20A) to JPO4 (20D), C.sub.g is a capacitance value of the capacitor 15 in the coupler 21, and C is a capacitance value of each of the coupling capacitor 31A through 31D.

[0049] FIG. 4 illustrates a schematic example of a configuration (wiring pattern) of JPO1 to JPO4 and the coupler 21 where JPO1 to JPO4 are made of a lumped element type configuration, as one example of FIG. 3. The coupler 21 includes first and second electrodes 16 and 18 (made of superconducting material) that are bridged by the nonlinear element 10. The first electrode 16 includes first and second opposing portions 17A and 17B (made of superconducting material) elongated toward JPO1 (20A) and JPO2 (20B), respectively. The second electrode 18 includes third and fourth opposing portions 19A and 19B (made of superconducting material) elongated toward the JPO3 (20C) and JPO4 (20D), respectively. In the coupler 21 in FIG. 4, the capacitor 15, which is shunt-connected (connected in parallel) to the Josephson junction (JJ), corresponds to a capacitance between the first and second electrodes 16 and 18 arranged opposed to each other. In FIG. 4, a ground pattern (ground plane) surrounding the first and second electrodes 16 and 18 is omitted.

[0050] JPO1 (20A) to JPO4 (20D) are configured as waveguide resonators that are terminated to ground via the SQUID 210A to 210D, respectively. The waveguide resonator modulates the magnetic flux penetrating through the SQUID loops 210A to 210D by pump signals (microwaves) from control lines 23A to 23D, at about twice the frequency of the resonance frequency. This causes parametric oscillation.

[0051] In JPO1 (20A) and JPO2 (20B), end portions of conductive portions (waveguides made of superconducting material) 205A and 205B (corresponding to coupler connection parts 24A and 24B in FIG. 3) connecting to the first superconducting portions 203A and 203B of the SQUIDs 210A and 210B, respectively, are capacitively coupled to end portions of the first and second opposing portions 17A and 17B of the first electrode 16 of the coupler 21, respectively. Capacitance value C of the coupling capacitors 31A and 31B is a capacitance between the end portion of the conductive portions (waveguides) 205A and 205B and the first and second opposing portions 17A and 17B of the first electrode 16 of the coupler 21. In JPO3 (20C) and JPO4 (20D), end portions of conductive portions (waveguides made of superconducting material) 205C and 205D (corresponding to coupler connection parts 24C and 24D in FIG. 3) connecting to the first superconducting portions 203C and 203D of the SQUIDs 210C and 210D, respectively, are capacitively coupled to end portions of the third and fourth opposing portions 19A and 19B of the second electrode 18 of the coupler 21, respectively. Capacitance value C of the coupling capacitors 31C and 31D is a capacitance between the end portion of the conductive portions (waveguides) 205C and 205D and the third and fourth opposing portions 19A and 19B of the second electrode 18 of the coupler 21. Capacitors 206A to 206D of JPO1 (20A) to JPO4 (20D) correspond to a capacitance between the first superconducting portion 203A to 203D of the SQUID 210A to 210D and ground, respectively. In FIG. 4, ground patterns (ground planes) on both sides of the conductive portions (coplanar waveguides) 205A to 205D of JPO1 (20A) to JPO4 (20D) are omitted.

[0052] According to the example embodiment, the following relationship is held.

[00008]$\begin{array}{cc}{C}_J>C_g>C& \left(2.1\right)\end{array}$

[0053] By setting a resonance angular frequency of each JPO from JPO1 (20A) to JPO4 (20D) and a resonance angular frequency of the coupler 21 close together, a strength of the four-body interaction (magnitude of coupling constant) is able to be set large.

[0054] Derivation of a four-body interaction strength (coupling constant) g.sup.(4) for JPO1 (20A) to JPO4 (20D) is outlined below (details will be described later).

[0055] Following a standard approach in this field (e.g., NPL 2), a magnetic flux degree of freedom is set in a circuit. One magnetic flux degree of freedom is set for each JPO and two for the coupler. This is used to write down a Hamiltonian for a classical circuit. The Hamiltonian is quantized using a standard method (see NPL 2). The Hamiltonian quantized H.sub.total is given as Equation (2.1). In the present specification, the notation of {circumflex over ()}H with a hat as a quantized Hamiltonian is not used, as in the case of the PTL 2.

[00009]$\begin{array}{cc}{H}_{\mathrm{total}}=\underset{i=1}{\overset{4}{.Math.}}{H}_{\mathrm{JPO},i}+g_{+}\underset{i=1}{\overset{4}{.Math.}}\left(a_i^{+}-a_i\right)\left(a_{g_{+}}-a_{g_{+}}^{+}\right)+g_{-}\underset{i=1}{\overset{4}{.Math.}}{s}_i\left(a_i^{+}-a_i\right)\left(a_{g_{-}}-a_{g_{-}}^{+}\right)+H_{coupler}& \left(2.2\right)\end{array}$

[0056] In Equation (2.2), H.sub.JPO,i is a Hamiltonian of each JPOi (i=1, . . . , 4) and H.sub.coupler is a Hamiltonian of the coupler 21.

[0057] a.sub.i and a.sub.i.sup.+ are creation and annihilation operators of boson corresponding to JPOi (i=1, . . . , 4), respectively (in the present specification, no hat is attached to the creation and annihilation operators either).

[0058] Since the coupler 21 has two magnetic flux degrees of freedom, the coupler 21 has two modes for a boson. They are represented by creation and annihilation operators a.sub.g+ and a.sub.g+.sup.+, and a.sub.g and a.sub.g.sup.+ with regard to a boson, respectively.

[0059] H.sub.total includes an interaction of each JPO and the coupler 21.

[0060] g.sub.+ and g.sub. are strength of the interaction between each JPO and the two degrees of freedom of the coupler 21. These can be expressed in terms of circuit parameters.

[0061] s.sub.i is given as follows:

[00010]$\begin{array}{cc}s1=s2=1,s3=s4=-1& \left(2.3\right)\end{array}$

[0062] Since each JPO interacts with the coupler 21, it is conceived that the JPOs also indirectly interacts with each other through interaction with the coupler 21. By variable transforming into a form that incorporates influence of the coupler 21 on each JPO, the interaction between each JPO and the coupler 21 is transformed into the four-body interaction among the JPOs.

[0063] That is, the Hamiltonian can be transformed into a Hamiltonian in a form in which the JPOs directly interact. This variable transformation is expressed as a unitary transformation using a unitary matrix U in Equation (2.4).

[00011]$\begin{array}{cc}U=\exp \left\{-\frac{g_{+}}{-{}_{+}}\left[\underset{i=1}{\overset{4}{.Math.}}\left(a_i^{+}{a}_{g_{+}}-a_i{a}_{g_{+}}^{+}\right)\right]-\frac{g_{-}}{-{}_{-}}\left[\underset{i=1}{\overset{4}{.Math.}}\left(a_i^{+}{a}_{g_{-}}-a_i{a}_{g_{-}}^{+}\right)\right]\right\}& \left(2.4\right)\end{array}$ [0064] where , .sub.+, and .sub. are resonance angular frequencies of the coupler 21 corresponding to ai, a.sub.g+, and a.sub.g. Difference in angular frequencies between different JPOs is not taken into consideration.

[00012]$\begin{array}{cc}\frac{g_{+}}{-{}_{+}}& \left(2.5\right)\end{array}$$\mathrm{and}$$\begin{array}{cc}\frac{g_{-}}{-{}_{-}}& \left(2.6\right)\end{array}$

are assumed to be less than 1. In particular, an absolute value of Expression (2.6) is the same as g described below.

[0065] After performing the unitary transformation, the boson of each JPO is represented in a rotation coordinate system. A frequency of rotation of the coordinate system is different for each JPO. JPOi (i=1, . . . , 4) are represented by coordinate systems that rotate at half a frequency of the pump signals applied to the magnetic field generators 207A to 207D that generate magnetic fluxes to be applied to the SQUID loops of JPOi. Frequencies of the pump signals are different to each other for JPOi (i=1, . . . , 4).

[0066] With transition to the rotational coordinate system, a temporally oscillating terms appear in the Hamiltonian. These oscillating terms are averaged at a time scale of interest and can be neglected because their positive and negative values are cancelled out (rotating wave approximation). Due to a difference in pump signal frequencies, many of terms representing interactions among JPOs and between JPOs and the coupler 21 oscillate, and these terms are neglected.

[0067] However, there are some interaction terms that do not oscillate, and they remain. The transition to a rotational coordinate system and the rotating wave approximation are a standard method used to focus on a behavior in a time scale characteristic to JPO.

[0068] By focusing on the various transformations and the characteristic time scale described above, it is possible to estimate the strength of an interaction between/among JPOs that occurs through the coupler.

[0069] Hamiltonian H.sub.total obtained through the above process is given as follows (2.7).

[00013]$\begin{array}{cc}{H}_{\mathrm{total}}.fwdarw.H_{\mathrm{total}}^{}=\underset{i=1}{\overset{4}{.Math.}}{H}_{\mathrm{JPO},i}^{}-g^{\left(4\right)}\left(a_1^{+}{a}_2^{+}{a}_3{a}_4+a_1{a}_2{a}_3^{+}{a}_4^{+}+\underset{i<j}{.Math.}{a}_i^{+}{a}_i{a}_j^{+}{a}_j\right)+H_{\mathrm{coupler}}^{}& \left(2.7\right)\end{array}$

[0070] In Equation (2.7), H.sub.JPO,i and H.sub.coupler are results of transformations of H.sub.JPO,i and H.sub.coupler, respectively.

[0071] In Equation (2.7), g.sup.(4) is a strength of the four-body interaction (coupling constant), which is given by the following Equation (2.8), using parameters of the circuit.

[00014]$\begin{array}{cc}{g}^{\left(4\right)}=\frac{1}{2^8}\frac{{\left({}_{-}\right)}^2}{{\left(-{}_{-}\right)}^4}\frac{C^4}{{C_J^2\left(C_g+C\right)}^3}{e}^2& \left(2.8\right)\end{array}$

[0072] In Equation (2.8),

[0073] and .sub. are resonance angular frequencies of the JPOs and the coupler 21, respectively.

[0074] C.sub.J, C.sub.g, and C are capacitance values of capacitors (206A to 206D) of each JPO, the capacitor (15) of the coupler 21, and coupling capacitors (31A to 31D) between each of JPOs and the coupler 21, respectively, as described above.

[0075] e is the electric elementary charge (approximately 1.610.sup.(19) coulombs).

[0076] According to the present example embodiment, for example, it is possible to enlarge the coupling constant g.sup.(4) of the four-body interaction from the strength of the four-body interaction (coupling constant) expressed in Equation (2.8) by the following approaches.

[0077] <Condition 1> Bringing the resonance angular frequency of each JPO and the resonance angular frequency .sub. of the coupler 21 closer together.

[0078] However, g.sup.(4) in Equation (2.8) is derived on assumption that as for interaction between each JPO and the coupler 21, a parameter g given by the following Equation (2.9) is sufficiently smaller than 1 (g is a dimensionless quantity quantity).

[00015]$\begin{array}{cc}{g}^{}=\frac{1}{4}\frac{\sqrt{{}_{-}}}{.Math.-{}_{-}.Math.}\frac{C}{\sqrt{C_J\left(C_g+C\right)}}& \left(2.9\right)\end{array}$

[0079] In the example embodiment, while keeping a value of g small, the coupling constant g.sup.(4) for the four-body interaction is enlarged.

[0080] <Condition 2> Not bringing the resonance angular frequency of each JPO and the resonance angular frequency .sub. of the coupler 21 too close.

[0081] More specifically, letting

[00016]$\begin{array}{cc}{}_{-}/=1& \left(2.1\right)\end{array}$

is set to satisfy the following condition.

[00017]$\begin{array}{cc}>\frac{C}{4\sqrt{C_J\left(C_g+C\right)}}& \left(2.11\right)\end{array}$

[0082] In this case, the following holds.

[00018]$\begin{array}{cc}g<1& \left(2.12\right)\end{array}$

[0083] <Condition 3> Make the capacitance C.sub.g of the coupler 21 smaller than the capacitance C.sub.J of each JPO. The coupling constant g.sup.(4) is given as following Equation (2.13).

[00019]$\begin{array}{cc}{g}^{\left(4\right)}=g^4\frac{e^2}{C_g+C}& \left(2.13\right)\end{array}$

[0084] To enlarge the coupling constant g.sup.(4) while keeping g small, the capacitance C.sub.g of the capacitor 15 of the coupler 21 and capacitance C of each of the coupling capacitors 31A to 31D must be made small.

[0085] Since one of circuit parameters that characterizes each JPO is the capacitor C.sub.J of each JPO, the capacitance C.sub.g of the capacitor 15 of the coupler 21 is set to be smaller than the capacitor C.sub.J of each JPO. Note that C.sub.g is also included in g of Equation (2.9).

[0086] To increase the coupling constant g.sup.(4) for the four-body interaction, the capacitance C of each of the coupling capacitors 31A to 31D must also be reduced. Since this overlaps with Condition 4 below, only the capacitance C.sub.g of the capacitor 15 of the coupler 21 is considered in the following.

[0087] In order to realize a four-body interaction network which the quantum annealing of the LHZ scheme assumes, it is necessary to weaken two-body interaction in which only two of the four JPOs interact with each other.

[0088] A strength of each two-body interaction between JPO1 and JPO2, and between the JPO3 and JPO4 in FIG. 3 and FIG. 4 is given by the following Equation (2.14).

[00020]$\frac{1}{16}\frac{C}{C_J+C}\left[\frac{1}{C_J}+\frac{C}{C_{J}C+{\mathrm{CC}}_g+C_g{C}_J}\right]$

(2.14)

[0089] A strength of each two-body interaction between JPO1 and JPO3, between JPO1 and JPO4, between JPO2 and JPO3, and between JPO2 and JPO4 is given by the following Equation (2.15).

[00021]$\begin{array}{cc}\frac{{\mathrm{CC}}_g}{4C_J\left(C_{J}C+{\mathrm{CC}}_g+C_g{C}_J\right)}& \left(2.15\right)\end{array}$

[0090] In the example embodiment, the following condition is met to weaken these two-body interactions.

[0091] <Condition 4> Make the capacitance C of each of the coupling capacitors 31A to 31D between each JPO and the coupler 21 smaller than the capacitance C.sub.J of each JPO.

[0092] Qualitatively, the four-body interaction is able to be strengthened with the conditions 1-4 met.

[0093] Following settings meet the four conditions 1-4 above. [0094] Frequency of JPO /(2): 10 GHz (gigahertz) [0095] Frequency of the coupler 21 .sub./(2): 9.98 GHZ [0096] Capacitance of each JPO C.sub.J: 1000 fF (femtofarad) [0097] Capacitance of the coupler C.sub.g: 200 fF [0098] Coupling capacitance between each JPO and the coupler 21 C: 1 fF

[0099] The Josephson junction is adjusted so that the above frequencies are realized.

[0100] With this setting, a strength of the four-body interaction: g.sup.(4)/(2) is given as follows.

h1.16 MHz (megahertz)

(Dirac's constant

[00022]$=\frac{h}{2}=1{10}^{-34}{m}^2\mathrm{kg}/s$

where, h is a Planck constant.

[0101] The parameter g in Equation (2.9) (which is assumed to be sufficiently smaller than 1) is about 0.28.

[0102] The strength of the four-body interaction: g.sup.(4)/(2)

[0103] h1.16 MHz is greater than a strength of the four-body interaction which is calculated on a setting of NPL1:

h63 kHz (kilohertz)

[0104] That is, in the above Equation (1.3), when E.sub.j/(2)=600 GHz, .sub.c=0.12 .sub.0, and g.sub.k/.sub.k0.12, C/(2)=63 kHz. (reference may be made to description of Equation (23) in Supplementary Note 6 of NPL 1.)

[0105] FIG. 5 illustrates changes in coupling constants g.sup.(4) and g when only capacitance C.sub.g of the coupler 21 is changed. In FIG. 5, g.sup.(4)/(2) and g/(2) are illustrated. In FIG. 5, frequency .sub./(2) of the coupler 21 is fixed at the above setting (9.98 GHz).

[0106] It can be seen that the strength of the four-body interaction (coupling constant) g.sup.(4) increases as the capacitance C.sub.g of the coupler 21 is made smaller. The reason why the four-body interaction is strengthened can be interpreted as follows.

[0107] In the superconducting quantum circuit of the above example embodiment, the four-body interaction among JPOs is realized by the following two elements. [0108] interaction between JPOs and the coupler 21, and [0109] nonlinearity when the coupler 21 is quantized.

[0110] As capacitor C.sub.g of the coupler 21 is reduced, the nonlinearity of the coupler 21 is increased, as a result of which the four-body interaction is strengthened.

[0111] Since g in Equation (2.9) is kept small, the interaction between JPOs and the coupler 21 is not strengthened, but instead, the four-body interaction is strengthened by making the capacitance C.sub.g of the coupler 21 smaller.

[0112] As described above, in the example embodiment, by adjusting circuit parameters in consideration of the above conditions and substituting the circuit parameters into Equation (2.8) which expresses the coupling constant g.sub.(4), it is possible to realize a circuit in which the four-body interaction is strong and other interactions, i.e., the two-body interaction between two of the four JPOs are weakened.

Configuration Example 1

[0113] FIG. 6 illustrates a non-limiting example of the above example embodiment. FIG. 6 is a schematic diagram of a wiring pattern (planar circuit) of a superconducting quantum circuit fabricated on a substrate such as silicon.

[0114] FIG. 6 illustrates a planar configuration of a coupler 21, in which four qubits (quantum bits) are configured with four JPO1 (20A) to JPO4 (20D) of a lumped element type, and these four qubits (quantum bits) are coupled with the four-body interaction.

[0115] JPO1 (20A) to JPO4 (20D) of the lumped element type are configured with a linear (not nonlinear) inductance, a capacitance component, and a resonator with a nonlinear element including a Josephson junction.

[0116] In the example embodiment, the coupler 21 and JPO1 to JPO4 are realized by lines (wirings) formed by superconductors on a substrate, for example. Silicon (Si) is used as the substrate, but an electronic material such as sapphire or a compound semiconductor material (group IV, group III-V and group II-VI) may be used. The substrate is preferably a single crystal but may be a polycrystalline or amorphous. As a material of the line (wiring material), Nb (niobium) or Al (aluminum) may be used, for example, though not limited thereto. Any metal that becomes superconductive at a cryogenic temperature may be used, such as niobium nitride, indium (In), lead (Pb), tin (Sn), rhenium (Re), palladium (Pd), titanium (Ti), molybdenum (Mo), tantalum (Ta), tantalum nitride, and an alloy containing at least one of the above metals. In order to achieve superconductivity, a coupler circuit is used in a temperature environment of about 10 mK (millikelvin) achieved by a refrigerator.

[0117] Referring to FIG. 6, a planar shape of a first electrode 16 is an approximately trapezoidal and is of a shape with rotated about 45 degrees counterclockwise. First and second opposing portions 17A and 17B of the first electrode 16 are extended, respectively, from vicinities of intersections between an upper side and oblique sides (legs) of the trapezoid toward a top and left sides in the drawing, i.e., toward locations where the JPO 20A and 20B are disposed. A planar shape of a second electrode 18 is of a shape of an inverted trapezoid where a trapezoid is rotated clockwise by approximately 135 degrees. Third and fourth opposing portions 19A and 19B of the second electrode 18 are extended, respectively, from vicinities of intersections of an upper side and oblique sides (legs) of the trapezoid toward a bottom side and a right side in the drawing, i.e., toward locations where the JPO 20C and 20D are disposed. The first electrode 16 and the second electrode 18 are arranged with respective bottom sides of the trapezoids facing each other, and a planar shape thereof, when aligned with the opposing portions 17A, 17B, 19A and 19B excluded, is almost hexagonal.

[0118] The first electrode 16 has a protrusion 16C which protrudes downward in the drawing in a vicinity of an intersection of one end of a lower base and one of the oblique sides of the first electrode 16. The second electrode 18 has a cut portion 18C obtained by cutting off the second electrode 18 so as to be in parallel to the protrusion 16C of the first electrode 16 in a vicinity of an intersection of one end of the lower base and one of the oblique sides of the second electrode 18. A nonlinear element 10, such as a SQUID, is arranged in a gap between the protrusion 16C, which is located in a vicinity of an intersection of one end of the lower base and one of the oblique sides of the first electrode 16, and the cut portion 18C, which is located in a vicinity of an intersection of one end of the lower base and one of the oblique sides of the second electrode 18. A configuration illustrated in FIG. 6 makes a placement area of the coupler 21 compact with a diagonally arranged configuration of the first electrode 16 and the second electrode 18 in the coupler 21.

[0119] JPO1 (20A) to JPO4 (20D) include coplanar waveguides (Coplanar Waveguides) 25A, 25B, 25C and 25D and SQUIDs 26A, 26B, 26C, and 26D, respectively. JPO1 (20A) to JPO4 (20D) are LC resonant circuits in a microwave region which are provided with linear inductance components and capacitance components of the coplanar waveguides 25A, 25B, 25C and 25D and nonlinear inductance components of SQUIDs 26A, 26B, 26C, and 26D, respectively. By applying DC current to each of control lines 23A, 23B, 23C, and 23D which are inductively coupled with the SQUIDs 26A, 26B, 26C, and 26D, respectively, a resonance frequency thereof can be adjusted. By applying AC current to each of control lines 23A, 23B, 23C, and 23D of JPO1 (20A) to JPO4 (20D), parametric oscillation can be induced.

[0120] Connection portions (waveguides for I/O) 22A to 22D for connecting to readout circuits are capacitively coupled via capacitors 32A to 32D with JPO1 (20A) to JPO4 (20D), respectively. In FIG. 6, the connection portions (coplanar waveguides for I/O) 22A, 22B, 22C, and 22D for connecting to the readout circuits, respectively, are shown in part. Each wiring pattern of the connection portions (coplanar waveguides for I/O) 22A to 22D may have a configuration extended to a chip periphery and, for example, connected to a wiring substrate or the like (not shown) via a bump electrode or the like. Each wiring pattern of the connection portions may be connected to a measurement device or the like outside a refrigerator via a readout line (cable) or the like. In FIG. 6, the capacitor 15 illustrated in FIG. 4 is omitted.

[0121] In an example illustrated in FIG. 6, the superconducting quantum circuit apparatus includes air bridge wirings 27A, 27B, 27C, and 27D provided for the control lines 23A, 23B, 23C, and 23D of the coplanar waveguide configuration, where the air bridge wirings 27A, 27B, 27C, and 27D are overhead wired on a wiring layer to stabilize ground plane (ground pattern) 40 surrounding JPO 20A, 20B, 20C, and 20D. The control lines 23A, 23B, 23C, and 23D have configurations that are connected to a wiring substrate or the like (not shown) via bump electrodes or the like (not shown) at a periphery of a chip or the like, for example, and are connected to a signal generation apparatus (current control unit) or the like outside the refrigerator.

[0122] In FIG. 6, a cross section along an A-A line orthogonal to the control line 23D is schematically illustrated in a balloon. A cross section for the control lines 23A to 23C is also configured in an identical manner. For example, a wiring layer is formed on a surface of a substrate 42 of silicon, on which JPOs 20 (20A to 20D), and the coupler (21) in the center are formed. The control line 23D is configured as a coplanar waveguide in the same wiring layer. Ground patterns 40-1 and 40-2 are provided on both sides of the control line 23D (wiring) in a longitudinal direction, via gaps, respectively. The air bridge wiring 27D, which is made of a superconducting material (e.g., Al or the like), is formed in such a manner that strides over the control line 23D in an arch shape and connects the ground patterns 40 on both sides of the control line 23D. An air bridge wiring structure with respect to the ground pattern 40 eliminates division of the ground pattern 40 into both sides 40-1 and 40-2 by the control line 23 (23A through 23D) which is configured as the coplanar waveguide, resulting in a stabilization of a ground potential of the ground plane 40 that surrounds JPO1 (20A) to JPO4 (20D). As a non-limiting example, in FIG. 8, the connection portions (coplanar waveguides for I/O) 22A to 22D for connecting to readout circuits are provided with air bridge wirings 28A, 28B, 28C, and 28D, which are overhead wired on the wiring layer, respectively, as well as the control lines 23A through 23D. The air bridge wirings 28A to 28D of the connection portions (coplanar waveguides for I/O) 22A to 22D also have the same cross-section configuration.

[0123] Letting C.sub.J be a capacitance value of a capacitor between each of the coplanar waveguides 25A to 25D of JPO1 (20A) to JPO4 (20D) and ground (206A to 206D in FIG. 3), C.sub.g a capacitance value of the capacitor 15 between the first and second electrodes 16 and 18 of the coupler 21, and C each of a capacitance value of a coupling capacitor (31A in FIG. 2) between the first opposing portion 17A of the coupler 21 and the coupler connection portion 24A of the coplanar waveguide 25A of JPO1 (20A), a capacitance value of a coupling capacitor (31B in FIG. 2) between the second opposing portion 17B of the coupler 21 and the coupler connection portion 24B of the coplanar waveguide 25B of JPO2 (20B), a capacitance value of a coupling capacitor (31C in FIG. 2) between the third opposing portion 19A of the coupler 21 and the coupler connection portion 24C of the coplanar waveguide 25C of JPO3 (20C), and a capacitance value of a coupling capacitor (31D in FIG. 2) between the fourth opposing portion 19B of the coupler 21 and the coupler connection portion 24D of the coplanar waveguide 25D of JPO4 (20D), the following magnitude relation is set.

C.sub.J>C.sub.g>C

[0124] By setting a resonance angular frequency of each of JPO1 (20A) to JPO4 (20D) closer to a resonance angular frequency of the coupler 21, the coupling strength of the four-body interaction can be set to be larger.

[0125] A length (width) of opposing sides of the first and second electrodes 16 and 18 of the coupler 21 is longer than width of coupler connection part 24A to 24D of the waveguides 25A to 25D of JPO1 (20A) to JPO4 (20D) which face ends of the first and second opposing portions 17A and 17B and the third and fourth opposing portions 19A and 19B of the coupler 21. A gap (cap) between the opposing sides of the first and second electrodes 16 and 18 of the coupler 21 is smaller than a gap between the ends of the coupler connection parts 24A and 24B, and the coupler connection parts 24C and 24D, and the first and second opposing parts 17A and 17B, and the third and fourth opposing parts 19A and 19B, respectively opposed thereto. Therefore, from FIG. 6, which schematically illustrates the wiring pattern, a capacitance C of the coupling capacitors 31A to 31D between the waveguides 25A and 25B of JPO1 (20A) and JPO2 (20B), the waveguides 25C and 25D of JPO3 (20C) and JPO4 (20D), and the first and second opposing portions 17A, 17B, and the third and fourth opposing portions 19A, 19B is found to be smaller than the capacitance C.sub.g between the electrodes 16 and 18 of the coupler 21 (capacitance of the capacitor 15 connected in parallel to the nonlinear element 10). In each of cross-shaped waveguides 25A to 25D (planar waveguides) of JPO1 (20A) to JPO4 (20D), a length of a side opposed to the ground plane (ground pattern) 40 via a gap, is several times longer than that of an opposing side of the first and second electrodes 16 and 18 as much as it looks like. Thus, in FIG. 6, each capacitance C.sub.J (206A to 206D in FIG. 3) of JPO1 (20A) to JP04 (20D) is found to be larger than the capacitance C.sub.g (15 in FIG. 3) between the first and second electrodes 16 and 18 of the coupler 21.

[0126] The strength g.sup.(4) of the four-body interaction depends on a detuning of resonant frequencies of the coupler (four-body interaction coupler) 21 and of JPOs (20A to 20D). Therefore, it is possible to control a coupling strength of the four-body interaction by adjusting the resonant frequencies of JPOs (20A to 20D) and the coupler 21.

[0127] The first and second electrodes 16 and 18 of the coupler 21 are coupled each other via capacitor 15 and further coupled to JPOs 20A, 20B, 20C, and 20D via the coupling capacitors 31A, 31B, 31C, and 31D, respectively, and are entirely enclosed by the ground pattern (ground plane) 40. The first and second opposing portions 17A and 17B, and the third and fourth opposing portions 19A and 19B that are capacitively coupled with JPO 20A and JPO 20B, and JPO 20C and JPO 20D, respectively, are protruded from the first electrode 16 and the second electrode 18 toward the ground plane 40, respectively, and able to be coupled with JPO 20A and JPO 20B, and JPO3 20C and JPO 20D at points spaced away from the first electrode 16 and from the second electrode 18.

[0128] By ensuring spacing between portions of the first electrode 16 and second electrode 18 other than the coupling capacitors 31A-31D and the ground pattern 40 with a large gap (almost equivalent in size as the coupler 21) where no superconductor is deposited, for example, on the order of about 100 m (micrometers) in length, a stray capacitance between each of the first 16 and the second electrode 18 and the ground pattern 40 is reduced.

[0129] The capacitor 15 between the first and second electrodes 16 and 18 makes the coupler 21 robust against a disturbance(s) caused by a magnetic field noise, etc. The reduction of the stray capacitance of the first and second electrodes 16 and 18 has an effect of strengthening the four-body interaction. The coupling strength of the four-body interaction by the coupler 21 capacitively coupled to each JPO is generally weakened due to the stray capacitance of the coupler 21, since a contribution ratio of the capacitance that contributes to coupling out of a total capacitance, which is a summation of a self-capacitance (C.sub.g) of the coupler 21 plus the capacitance C of the coupling capacitor and the stray capacitance of the coupler 21, is reduced. In the example illustrated in FIG. 6, the coupler 21 is placed separated from an edge of the ground plane (ground pattern) 40 surrounding the coupler 21 via a gap of the same extent as a size of the coupler 21, for example, thus reducing the stray capacitance of the coupler 21.

Configuration Example 2

[0130] FIG. 7 illustrates a coupler (four-body interaction coupler) 21 of another non-limiting example. In FIG. 7, a planer circuit corresponding to the coupler 21 illustrated in FIG. 4 is illustrated while qubits (quantum bits) (JPO) 20A to 20B are not shown. Referring to FIG. 7, the coupler 21 of the present variation example, includes a first electrode 16 configured with right-angle bent sides (lateral (horizontal) side 16A and longitudinal (vertical) side 16B), and a second electrode 18 configured with right-angle bent sides (lateral (horizontal) side 18A and longitudinal (vertical) side 18B). The longitudinal sides 16B and 18B of the first and second electrodes 16 and 18 are disposed opposed to each other, and the lateral sides 16A and 18A thereof are also disposed opposed to each other. The first electrode 16 includes n (seven in FIG. 7) extension portions 16D extending at a predetermined interval parallel to the lateral side 16A from the longitudinal side 16B to the longitudinal side 18B of the second electrode 18 to have a comb-teeth pattern shape. The second electrode 18 also includes n extension portions 18D extending at a predetermined interval parallel to the lateral side 18A from the longitudinal side 18B to the longitudinal side 16B of the first electrode 16 to have a comb-tooth pattern shape. The extension portions 16D and 18D each corresponding to a comb-teeth are arranged facing to each other in a nested structure. Assuming that a capacitance between neighboring extension portions 16D and 18D, a capacitance between the lateral side 16A and the extension portion 18D opposed thereto, and a capacitance between the lateral side 18A and the extension portion 16D opposed thereto are the same value C.sub.a, and a capacitance C.sub.1 between the lateral sides 16A and 18A is approximated by a configuration with parallelly connected 2n+1 capacitors of a capacitance value: C.sub.a, the capacitance C1 is given as follows.

[00023]$\begin{array}{cc}{C}_1=\left(2n+1\right)C_a& \left(2.16\right)\end{array}$

[0131] Assuming that the extension portions 16D and 18D, which are arranged opposed to each other in a nested manner and each have comb-teeth shape, are not provided, a space between the lateral sides 16A and 18A is (2n+1) times larger than the space between opposing extension portions 16D and 18D in FIG. 7. In this case, regarding a capacitance C.sub.2 between the lateral sides 16A and 18A, the following holds.

[00024]$\begin{array}{cc}{C}_2<C_a/\left(2n+1\right)& \left(2.17\right)\end{array}$

[0132] The capacitance C.sub.1 between the first and second electrodes 16 and 18, each of which has a structure that are arranged comb-teeth opposed to each other in a nested manner, is, as a coarse estimate, greater than (2n+1).sup.2 times the capacitance C.sub.2 without a nested comb-teeth structure.

[0133] With the comb-shaped capacitor (inter-digital capacitor) between the first and second electrodes 16 and 18, a capacitance between the electrodes 16 and 18 is increased, thus further effectively reducing an impact exerted by a voltage fluctuation due to such as an electric field noise to implement a stable four-body interaction coupling.

[0134] The longitudinal side 16B of the first electrode 16 includes a first opposing portion 17A corresponding to a coupler connection part 24A of JPO 20A. and the lateral side 16A of the first electrode 16 includes a second opposing portion 17B corresponding to a coupler connection part 24B of JPO 20B. Each of the first and second opposing portions 17A and 17B has a U-shaped expanded portion and a base portion connected to the first electrode 16. In spaces of the U-shaped expanded portions of the first and second opposing portions 17A and 17B, coupler connection parts 24A and 24B, each of which is configured as a coplanar waveguide, are arranged, respectively.

[0135] The longitudinal side 18B of the second electrode 18 includes a third opposing portion 19A corresponding to a coupler connection part 24C of JPO 20C, and the lateral side 18A of the second electrode 18 includes a fourth opposing portion 19B corresponding to a coupler connection part 24D of JPO 20D. Each of the third and fourth opposing portions 19A and 19B has a U-shaped expanded portion and a base portion connecting to the electrode 18. In the U-shaped expanded portions of the third and fourth opposing portions 19A and 19B, coupler connection parts 24C and 24D of coplanar waveguide type are arranged, respectively.

[0136] A nonlinear element 10 is configured by a SQUID disposed between an end of the longitudinal side 16B of the first electrode 16 and an end of the lateral side 18A of the second electrode 18. The SQUID included in the nonlinear element 10 which bridges the first electrode 16 (an end of the longitudinal side 16B) and the second electrode 18 (an end of the lateral side 18A). In FIG. 7, the first electrode 16 (an end of the longitudinal side 16B) and the second electrode 18 (an end of the lateral side 18A), each of which are bridged by the SQUID designated by reference numeral 10, are illustrated with reference numeral 29, enclosed with a dashed circle as the SQUID bridging portion. The coupler 21 further includes a control line 51 which applies a magnetic flux to the SQUID from within the same plane. By varying a current (i.e., direct current) supplied to the control line 51, a frequency variable coupler is provided. The external magnetic flux given through the control line 51 from the current control part penetrates through the SQUID loop of the four-body interaction coupler 21, thereby varying an effective self-inductance of the SQUID loop. As a result, the resonance frequency is varied.

[0137] In the variation example illustrated in FIG. 7, a distance between the electrode and the ground (ground pattern 40) is reduced only partially in a vicinity of the SQUID bridging portion 29 of the first electrode 16 and the second electrode 18. In the vicinity of the SQUID bridging portion 29, the control line 51 is provided to allow an external magnetic flux to be applied from the immediate vicinity of the SQUID bridging portion 29, as a result of which the magnetic flux through inductive coupling from the control line 51 can be efficiently applied. As described, it is preferable that the ground pattern 40 is basically spaced apart from the first and second electrodes 16 and 18. The ground pattern 40 has a portion 30 (ground pattern protruding portion) in the vicinity of the SQUID bridging portion 29 that protrudes itself inwardly to be closer to the SQUID bridging portion 29. Thus, the length of the portion 30 is preferably or less of a circumference of an inner circumference of the ground plane 40 surrounding the electrodes 16 and 18. It may further preferably be or less, and even preferably or less.

[0138] In the coupler 21 of this configuration, letting C.sub.J be a capacitance value of a capacitor (206A to 206D in FIG. 3) of the of JPO1 (20A) to JPO4 (20D), C.sub.g be a capacitance value of coupling capacitors (31A to 31D in FIG. 3) between the coupler connection parts 24A to 24D of JP01 (20A) to JPO4 (20D) and the opposing portions 17A, 17B, 19A and 19B of the coupler 21m, the following magnitude relation holds.

C.sub.J>C.sub.g>C

Configuration Example 3

[0139] FIG. 8 illustrates a coupler (four-body interaction coupler) 21 of yet another non-limiting example. In FIG. 8, a planer circuit corresponding to the coupler 21 illustrated in FIG. 4 is illustrated while qubits (quantum bits) (JPO) 20A to 20B are not shown. Referring to FIG. 8, the coupler 21 has a structure such that the grand pattern 40, which penetrates between first second opposing portions 17A and 17B, and the third and fourth opposing portions 19A and 19B, and coupler connection parts 24A, 24B, 24C, and 24D. For example, the coupler connection parts 24A, 24B, 24C, and 24D of JPO 20A, 20B, 20C, and 20D connect to ground via capacitors 31A-2, 31B-2, 31C-2, and 31D-2, respectively, and the first and second opposing portions 17A and 17B and the third and fourth opposing portions 19A and 19B corresponding to the coupler connection parts 24A, 24B, 24C, and 24D, is connected to the ground via the capacitor 31A-1, 31B-1, 31C-1, and 31D-1, respectively. The space between the coupler connection parts 24A and 24B of JPO 20A and 20B and the first and second opposing portions 17A and 17B are shielded (theory of electrostatic shielding), and the space between the coupler connection parts 24C and 24D of JPO 20C and 20D and the third and fourth opposing portions 19A and 19B are shielded. Therefore, the capacitance between coupler connection parts 24A and 24B of JPOs 20A and 20B and the first and second opposing portions 17A and 17B, and capacitance between the coupler connection parts 24C and 24D of JPOs 20C and 20D and third and fourth opposing portions 19A and 19B operate as coupling capacitors 31A to 31D with smaller capacitance value than a configuration without ground pattern between coupler connection parts of JPOs and opposing portions (FIG. 7).

[0140] In the coupler 21 of this configuration, regarding C.sub.J which is a capacitance value of the capacitor (206A to 206D in FIG. 3) of the of JPO1 (20A) to JPO4 (20D), C.sub.g which is a capacitance value of the capacitor of the coupler 21, C which is a capacitance value of coupling capacitors (31A to 31D in FIG. 3) between the coupler connection parts 24A to 24D of JPO1 (20A) to JPO4 (20D) and the opposing portions 17A, 17B, 19A and 19B of the coupler 21, the following magnitude relation holds.

C.sub.J>C.sub.g>C

Another Example Embodiment

[0141] FIG. 10 illustrates a schematic diagram of a quantum computer 200 with JPO 20 integrated as a configuration example of another example embodiment. In the configuration illustrated in FIG. 10, each four-body interaction coupler 21 is connected to the four JPOs 20, respectively, as illustrated in FIG. 4 and/or FIG. 6. Each JPO 20 is connected to one through four four-body interaction couplers 21, and JPO 20 is arranged to be shared by a plurality of unit structures to provide a configuration in which a plurality of unit structures illustrated in FIG. 3, FIG. 4, and FIG. 6, etc., are arranged. In the quantum computer 200, at least one JPO 20 is connected to a plurality of four-body interaction couplers 21. Specially, in the example as illustrated in FIG. 10, at least one JPO 20 is connected to four four-body interaction couplers 21. The quantum computer 200 may also be described as follows. The quantum computer 200 includes a plurality of JPOs 20, each of which is connected to one through four four-body interaction couplers 21. The number of the four-body interaction couplers 21 to which each JPO 20 connects corresponds to how many unit structures share JPOs 20. Thus, in the example illustrated in FIG. 10, the quantum computer 200 includes a plurality of unit structures, and a plurality of unit structures share JPO 20. In the example illustrated in FIG. 10, 13 superconducting nonlinear JPOs 20 are integrated, but any number of JPOs 20 may be integrated in a similar manner. A current control part and a readout part are not illustrated in FIG. 10, for ease of understanding of the drawing, but as explained with reference to FIG. 2, etc., the current control part and the readout part are used to control and read out JPO 20.

<Derivation of the Above Equation for the Four-Body Interaction>

[0142] The following describes the derivation of the above equation. As described in NPL 2, FIG. 9 is a diagram corresponding to FIG. 3. In FIG. 9, Josephson junction is indicated by an x surrounded by a square, and 1 to 4 in SQUID loops 210A to 210D of JPO1 to JPO4 represent a magnetic flux penetrating through each SQUID loop. The EJs attached to the two Josephson junctions in SQUID loops 210A to 210D of JPO1 to JPO4 represent the Josephson energy.

1. CLASSICAL HAMILTONIAN

[0143] As illustrated in FIG. 9, magnetic flux 1 to 4, g1 and g2 are arranged on each node 1 to 4, g1 and g2.

[0144] In a circuit with an inductor (Josephson junction can be regarded as a nonlinear inductor) and a capacitor (non-dissipative circuit), V=d/dt with respect to a time derivative of the magnetic flux (Faraday's law). Therefore, an energy E=() CV.sup.2 of a capacitance branch (voltage V at both ends) is expressed as follows.

[00025]$\begin{array}{cc}E=\frac{1}{2}C{\stackrel{}{}}^2& \left(3.1\right)\end{array}$

[0145] That is, a capacitor energy can be treated as kinetic energy. Let be a vector (6-dimensional vector) whose elements are .sub.1 to .sub.4, .sub.g1, and .sub.g2.

[00026]$\begin{array}{cc}=\left(\begin{array}{c}{}_1\\ {}_2\\ {}_3\\ {}_4\\ {}_{g_1}\\ {}_{g_2}\end{array}\right)& \left(3.2\right)\end{array}$

[0146] In FIG. 9, the energy (kinetic energy) of a capacitor branch is given by (3.3).

[00027]$\begin{array}{cc}\frac{1}{2}{\stackrel{}{}}^t{C}_{mat}\stackrel{}{}& \left(3.3\right)\end{array}$

[0147] Here, Cmat is, as shown in the following Equation (3.4), a circuit matrix (6-rows and 6 columns matrix) of capacitance branches in FIG. 9.

[00028]$\begin{array}{cc}{C}_{\mathrm{mat}}=\left(\begin{array}{cccccc}{C}_J+C& 0& 0& 0& -C& 0\\ 0& C_J+C& 0& 0& -C& 0\\ 0& 0& C_J+C& 0& 0& -C\\ 0& 0& 0& 0C_J+C& 0& -C\\ -C& -C& 0& 0& C_g+2C& -C_g\\ 0& 0& -C& -C& -C_g& C_g+2C\end{array}\right)& \left(3.4\right)\end{array}$

[0148] An element C.sub.J+C of k-th row and k-th column (k=1 to 4) of matrix C.sub.mat is a combined capacitance of two capacitors C and C.sub.J that are connected in parallel to a node k (k=1-4). An element C.sub.J+2C in k-th row and k-th column (k=5, 6) of the matrix C.sub.mat is a combined capacitance of three capacitors C, C and C.sub.J connected in parallel to a node k (k=5, 6). An element of i-th row and j-column (ij, i,j=1-6) of the matrix C.sub.mat is a capacitance between nodes i and j (signs indicate a direction of the capacitor branch).

[0149] A Lagrangian for a circuit in FIG. 9 is given by Equation (3.5).

[00029]$\begin{array}{cc}L=\frac{1}{2}{\stackrel{}{}}^t{C}_{mat}\stackrel{}{}-V\left(\right)& \left(3.5\right)\end{array}$

[0150] In Equation (3.5), V() is potential energy and is given by Equation (3.6).

[00030]$\begin{array}{cc}V\left(\right)=-{}_{i=1}^4{E}_J\left({}_i\right)\cos \frac{{}_i}{{}_0}-E_{\mathrm{Jg}}\cos \frac{{}_{g1}-{}_{g2}}{{}_0}& \left(3.6\right)\end{array}$

where,

[00031]$\begin{array}{cc}{E}_J\left(\right)=E_J\cos \frac{}{{}_0}& \left(3.7\right)\end{array}$$\begin{array}{cc}{}_0=\frac{1}{2e}\left(\frac{h}{2}\right)& \left(3.8\right)\end{array}$

[0151] E.sub.J is Josephson energy in JPO1-JP04.

[0152] E.sub.Jg is Josephson energy in a coupler 21.

[0153] .sub.i is a magnetic flux penetrating through the SQUID loop of JPO.sub.i (i=1, . . . , 4).

[0154] .sub.g1-.sub.g2 is a magnetic flux penetrating through the coupler 21 (a difference between respective magnetic fluxes of nodes g1 and g2).

[0155] The kinetic energy of the Lagrangian

[00032]$\begin{array}{cc}\frac{1}{2}{\stackrel{}{}}^t{C}_{mat}\stackrel{}{}& \left(3.9\right)\end{array}$

is expressed using a vector q.

[00033]$$\begin{gathered} \begin{array}{cc}q={\left(q_1,q_2,q_3,q_4,q_{g1},q_{g2}\right)}^t=\frac{L}{}=C_{mat}\stackrel{}{} \\ & \left(3.1\right)\end{array} \end{gathered}$$

[0156] q.sub.1, q.sub.2, q.sub.3, q.sub.4, q.sub.g1, and q.sub.g2 are a charge of each node. t is a transpose operator.

[0157] From Equation (3.10), assuming that the matrix C.sub.mat is normal, we obtain the following.

[00034]$\begin{array}{cc}\stackrel{}{}=C_{\mathrm{mat}}^{-1}q& \left(3.11\right)\end{array}$

[0158] Therefore, the kinetic energy in Equation (3.9) can be expressed using vector q as follows.

[00035]$\begin{array}{cc}\frac{1}{2}{\stackrel{}{}}^t{C}_{mat}\stackrel{}{}=\frac{1}{2}{\left(C_{\mathrm{mat}}^{-1}q\right)}^t{C}_{mat}\left(C_{\mathrm{mat}}^{-1}q\right)=\frac{1}{2}{q^t\left(C_{\mathrm{mat}}^{-1}\right)}^t\mathrm{Iq}=\frac{1}{2}{q}^t{C}_{\mathrm{mat}}^{-1}q& \left(3.12\right)\end{array}$

[0159] In Equation (3.12), I is a unit matrix of 66. The last Expression in Equation (3.12) uses the fact that the matrix C.sub.mat is a symmetric matrix (component c.sub.ij(i,j=1, . . . , 6) is c.sub.ij=c.sub.ji with respect to subscripts i and j), which is symmetric with respect to a main diagonal, and an inverse matrix C.sub.mat.sup.1 of C.sub.mat is also a symmetric matrix. That is, C.sub.matC.sub.mat.sup.1=I. By taking the transpose of both sides of this Equation, we have (C.sub.mat.sup.1).sup.tC.sub.mat.sup.t=I. Since C.sub.mat is a symmetric matrix, by multiplying both sides by the inverse matrix C.sub.mat.sup.1 from a right side, (C.sub.mat.sup.1).sup.t=C.sub.mat.sup.1, and thus Equation (3.12) holds.

[0160] Expressing potential energy as a function of vector (nonlinear function), the Hamiltonian is given as follows.

[00036]$\begin{array}{cc}H=\frac{1}{2}{q}^t{C}_{\mathrm{mat}}^{-1}q+V\left(\right)& \left(3.13\right)\end{array}$

[0161] Obtaining the inverse matrix C.sub.mat.sup.1 of the matrix C.sub.mat of 6 rows and 6 columns concretely, the Hamiltonian H is given as the following Equation (3.14).

[00037]$\begin{array}{cc}\begin{array}{c}H=\frac{1}{2\left(C_J+C\right)}\left[1+\frac{C}{4C_J}+\frac{C^2}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\right]\underset{i=1}{\overset{4}{.Math.}}{q}_i^2+\\ \frac{C}{C_J+C}\left[\frac{C}{4C_J}+\frac{C}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\right]\left(q_1{q}_2+q_3{q}_4\right)+\\ \frac{{\mathrm{CC}}_g}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\left(q_1+q_2\right)\left(q_3+q_4\right)+\frac{1}{2C_J}\underset{i=1}{\overset{4}{.Math.}}{q}_i{q}_{g_{+}}+\\ \frac{C}{2\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\underset{i=1}{\overset{4}{.Math.}}{s}_i{q}_i{q}_{g_{-}}+\left[\frac{1}{2C}+\frac{1}{2C_J}\right]q_{g_{+}}^2+\\ \frac{C_J+C}{2\left(C_{J}C+C{C}_g+C_g{C}_J\right)}{q}_{g_{-}}^2+V\left(\right)\\ =\frac{1}{2C_J}\underset{i=1}{\overset{4}{.Math.}}{q}_i^2+\frac{1}{2C_J}\underset{i=1}{\overset{4}{.Math.}}{q}_i{q}_{g_{+}}+\frac{1}{2C_J}\frac{C^{}}{1+C^{}}\underset{i=1}{\overset{4}{.Math.}}{s}_i{q}_i{q}_{g_{-}}+\\ \frac{1}{2C_J}\left(\frac{1}{C^{}}+1\right)q_{g_{+}}^2+\frac{1}{2C_J}\left[\frac{C^{}}{\left(1+C^{}\right)}\frac{1}{C^{}}+\frac{C^2}{{\left(1+C^{}\right)}^2}\right]q_{g_{-}}^2+\\ O\left(C^{}\right)+V\left(\right)\end{array}& \left(3.14\right)\end{array}$

where,

[00038]$\begin{array}{cc}{C}^{}=C/C_J& \left(3.15\right)\end{array}$$\begin{array}{cc}{C}^{}=C/C_g& \left(3.16\right)\end{array}$$\begin{array}{cc}{q}_g=\frac{q_{g1}{q}_{g2}}{2}& \left(3.17\right)\end{array}$$\begin{array}{cc}{s}_1=s_2=1,s_3=s_4=-1& \left(3.18\right)\end{array}$

[0162] In Equation (3.14), the Expression at the bottom line, the following is assumed.

[00039]$\begin{array}{cc}{C}^{}<<1& \left(3.19\right)\end{array}$

[0163] When O(C) is neglected, a product between degrees of freedom of JPO does not appear. Details of calculations are given in section A. 1 of the appendix below.

2. QUANTIZATION

[0164] By converting variables and q in the classical Hamiltonian to operators as follows, the Hamiltonian is replaced by a function of the operator (quantization of the Hamiltonian).

[00040]$\begin{array}{cc}{}_k.fwdarw.{\stackrel{}{}}_k& \left(4.1\right)\end{array}$$q_k.fwdarw.{\stackrel{}{n}}_k$$H.fwdarw.\stackrel{}{H}$$\begin{array}{cc}\frac{{}_k}{{}_0}={}_k.fwdarw.{\stackrel{}{}}_k={}_{kZ}\left(a_k+a_k^{}\right),{}_{kZ}=\sqrt{\frac{Z_k}{2}}& \left(4.2\right)\end{array}$$\begin{array}{cc}\frac{q_k}{2e}=n_k.fwdarw.{\stackrel{}{n}}_k=-i{n}_{kZ}\left(a_k-a_k^{}\right),n_{kZ}=\sqrt{\frac{1}{2Z_k}}& \left(4.3\right)\end{array}$

where k=1, 2, 3, 4, g1, g2, and a.sup.+.sub.k and a.sub.k are creation and annihilation operators of boson. e is elementary electric charge (quantum of electricity). In this case, the following commutation relation (4.4) holds.

[00041]$\begin{array}{cc}\left[{\hat{}}_k,{\hat{n}}_k\right]=i& \left(4.4\right)\end{array}$$\begin{array}{cc}{Z}_1,Z_2,Z_3,Z_4=\sqrt{\frac{8E_C}{E_J}}& \left(4.5\right)\end{array}$$\begin{array}{cc}{Z}_{g1}=Z_{g2}=\sqrt{\frac{2E_{\mathrm{Cg}}^{}}{E_{\mathrm{Jg}}}}& \left(4.6\right)\end{array}$

[0165] In Equations (4.5) and (4.6),

[00042]$\begin{array}{cc}{E}_C=\frac{e^2}{2C_J}& \left(4.7\right)\end{array}$$E_{\mathrm{Cg}}^{}=E_{\mathrm{Cg}}\left[\frac{1}{1+C^{}}+\frac{C^{}{C}^{}}{{\left(1+C^{}\right)}^2}\right]$

where

[00043]$\begin{array}{cc}{E}_{\mathrm{Cg}}=\frac{e^2}{2C_g}& \left(4.8\right)\end{array}$

[0166] Since Z.sub.1 and Z.sub.2 are equal,

[00044]$\begin{array}{cc}{}_Z:={}_{1Z}={}_{2Z}& \left(4.9\right)\end{array}$

[0167] We have

[00045]$\begin{array}{cc}{n}_Z:=n_{1Z}=n_{2Z}& \left(4.1\right)\end{array}$$\begin{array}{cc}{}_g={}_{\mathrm{g1}}{}_{\mathrm{g2}}.fwdarw.{\stackrel{}{}}_g={\stackrel{}{}}_{g1}{\stackrel{}{}}_{g2}={}_{\mathrm{gZ}}\left(a_q+a_g^{}\right)& \left(4.11\right)\end{array}$$\begin{array}{cc}{}_{\mathrm{gZ}}=\sqrt{\frac{Z_g}{2}}& \left(4.12\right)\end{array}$$\begin{array}{cc}{n}_g=\frac{n_{g1}{n}_{g2}}{2}.fwdarw.{\stackrel{}{n}}_g=\frac{{\stackrel{}{n}}_{g1}{\stackrel{}{n}}_{g2}}{2}=-i{n}_{\mathrm{gZ}}\left(a_g-a_g^{}\right)& \left(4.13\right)\end{array}$$\begin{array}{cc}{n}_{\mathrm{gZ}}=\sqrt{\frac{1}{2Z_g}}& \left(4.14\right)\end{array}$

[0168] In Equations (4.11) and (4.13),

[00046]$\begin{array}{cc}{a}_g=\frac{a_{g1}{a}_{g2}}{\sqrt{2}}& \left(4.15\right)\end{array}$

[0169] In Equations (4.12) and (4.14),

[00047]$\begin{array}{cc}{Z}_{g}2Z_{g1}=2Z_{g2}=2z_{g2}=\sqrt{\frac{8E^{}{C}_g}{E_{Jg}}}& \left(4.16\right)\end{array}$

[0170] Following holds as a commutator (commutation relation) of an operator.

[00048]$\begin{array}{cc}\left[a_g,a_g^{}\right]=1& \left(4.17\right)\end{array}$$\begin{array}{cc}\left[{\stackrel{}{}}_g,{\stackrel{}{q}}_g\right]=i& \left(4.18\right)\end{array}$

[0171] Therefore, by quantizing the Hamiltonian of a classical system as above, the following is derived. A quantized Hamiltonian is written H.sub.Q instead of using a hat.

[00049]$\begin{array}{cc}H.fwdarw.H_Q=-4E_C{n}_Z^2\underset{i=1}{\overset{4}{.Math.}}{\left(a_i-a_i^{+}\right)}^2-4E_C{n}_Z{n}_{\mathrm{gZ}}\underset{i=1}{\overset{4}{.Math.}}\left(a_i-a_i^{+}\right)\left(a_{g+}-a_{g+}^{+}\right)-4E_C\frac{C^{}}{1+C^{}}{n}_Z{n}_{\mathrm{gZ}}\underset{i=1}{\overset{4}{.Math.}}{s}_i\left(a_i-a_i^{+}\right)\left(a_{g_{-}}-a_{g_{-}}^{+}\right)-4E_C\left(\frac{1}{C^{}}+1\right){n_{\mathrm{gZ}}^2\left(a_{g_{+}}-a_{g_{+}}^{+}\right)}^2-4E_{\mathrm{Cg}}{n_{\mathrm{gZ}}^2\left(a_{g_{-}}-a_{g_{-}}^{+}\right)}^2-\underset{i=1}{\overset{4}{.Math.}}{E}_J\left({}_i\right)\left[1-\frac{{}_Z^2}{2}{\left(a_i+a_i^{+}\right)}^2+\frac{{}_Z^4}{24}{\left(a_i+a_i^{+}\right)}^4\right]-E_{J_g}\left[1-\frac{{}_{\mathrm{gZ}}^2}{2}{\left(a_{g_{-}}+a_{g_{-}}^{+}\right)}^2+\frac{{}_{\mathrm{gZ}}^4}{24}{\left(a_{g_{-}}+a_{g_{-}}^{+}\right)}^4\right]& \left(4.19\right)\end{array}$

[0172] Here, O(C) is neglected and higher order terms in an expansion of cos(.sub.i/.sub.0) are neglected.

[0173] Next, let i be time-varying so that it can be approximated as follows (reference may be made to NPL 3).

[00050]$\begin{array}{cc}{E}_J\left({}_i\right)=E_J+E_J\cos \left({}_{p,i}t\right)& \left(4.2\right)\end{array}$

[0174] As a quantized Hamiltonian H.sub.Q, Equation (4.21) and following equations are derived.

[00051]$\begin{array}{cc}{H}_Q=\underset{i=1}{\overset{4}{.Math.}}\left[a_i^{+}{a}_i-\frac{E_c}{12}{\left(a_i+a_i^{+}\right)}^4+\frac{E_J}{4E_J}{\left(a_i+a_i^{+}\right)}^2\cos \left({}_{p,i}t\right)\right]+g_{+}\underset{i=1}{\overset{4}{.Math.}}\left(a_i^{+}-a_i\right)\left(a_{g_{+}}-a_{g_{+}}^{+}\right)+g_{-}\underset{i=1}{\overset{4}{.Math.}}{s}_i\left(a_i^{+}-a_i\right)\left(a_{g_{-}}-a_{g_{-}}^{+}\right)+{}_{+}{a}_{g_{+}}^{+}{a}_{g_{+}}-\frac{{}_{+}}{2}\left(a_{g_{+}}^2+a_{g_{+}}^{+2}\right)+{}_{-}{a}_{g_{-}}^{+}{a}_{g_{-}}-\frac{E_{C_g}^{}}{12}{\left(a_{g_{-}}+a_{g_{-}}^{+}\right)}^4& \left(4.21\right)\end{array}$

where,

[00052]$\begin{array}{cc}=\sqrt{8E_C{E}_J}& \left(4.22\right)\end{array}$$\begin{array}{cc}{}_{+}=4E_c\left(\frac{1}{C^{}}+1\right)\sqrt{\frac{E_{Jg}}{8E_{\mathrm{Cg}}^{}}}\frac{C_g+C}{2C}{}_{-}& \left(4.23\right)\end{array}$$\begin{array}{cc}{}_{-}=\sqrt{8E_{\mathrm{Cg}}^{}{E}_{JG}}& \left(4.24\right)\end{array}$$\begin{array}{cc}{g}_{+}=\frac{E_C}{\sqrt{2}}{\left(\frac{E_J{E}_{Jg}}{E_C{E}_{\mathrm{Cg}}^{}}\right)}^{1/4}=\frac{\sqrt{{}_{-}}}{4}\sqrt{\frac{E_c}{E_{\mathrm{Cg}}^{}}}& \left(4.25\right)\end{array}$$\begin{array}{cc}{g}_{-}=\frac{Ec}{\sqrt{2}}{\left(\frac{E_J{E}_{Jg}}{E_C{E}_{\mathrm{Cg}}^{}}\right)}^{1/4}\frac{C^{}}{1+C^{}}=\frac{\sqrt{{}_{-}}}{4}\sqrt{\frac{E_C}{E_{\mathrm{Cg}}^{}}}\frac{C}{C_g+C}& \left(4.26\right)\end{array}$

[0175] In Equation (4.21), a constant term(s) are excluded.

[0176] In addition, E.sub.J is treated as small and, therefore, E.sub.J.sup.4.sub.Z is neglected.

[0177] Furthermore, at the rightmost side of Equation (4.23), quantities as large as C (=C/C.sub.J) is neglected.

[0178] Next, a unitary transformation using a unitary matrix Ug of Equation (4.27) is applied on the Hamiltonian of Equation (4.21).

[00053]$\begin{array}{cc}{U}_g=\exp \left\{-g_{+}^{}\left[\underset{i=1}{\overset{4}{.Math.}}\left(a_i^{+}{a}_{g_{+}}-a_i{a}_{g_{+}}^{+}\right)\right]-g_{-}^{}\left[\underset{i=1}{\overset{4}{.Math.}}{s}_i\left(a_i^{+}{a}_{g_{-}}-a_i{a}_{g_{-}}^{+}\right)\right]\right\}& \left(4.27\right)\end{array}$

[0179] In Equation (4.27),

[00054]$\begin{array}{cc}{g}_{+}^{}=\frac{g_{+}}{-{}_{+}}& \left(4.28\right)\end{array}$$\begin{array}{cc}{g}_{-}^{}=\frac{g_{-}}{-{}_{-}}& \left(4.29\right)\end{array}$

s.sub.i is given by Equation (3.18).

[0180] This unitary transformation is a transformation that incorporates an interaction (g.sub.+(a.sub.i.sup.a.sub.i)(a.sub.g+.sup.a.sub.g+) etc.) between JPOs 1 to 4 and the coupler 21 in the HQ into the creation/annihilation operator a.sup.+.sub.i/a.sub.i, etc.

[0181] In other words, it is a transformation that perturbatively incorporates an influence of the coupler 21 on JPO1 to JPO4.

[0182] After this unitary transformation, the creation/annihilation operator a.sup.+.sub.i/a.sub.i is transferred to the rotational coordinate system with oscillation frequency .sub.p,i/2. (<Note 1>: Since this is after the above unitary transformation, this a.sub.i is the degree of freedom of each JPO perturbatively incorporating influence of the coupler 21).

[00055]$\begin{array}{cc}{H^{}}_{Q:=}{U}_{{}_p/2}^{}{U}_g^{}{H}_Q{U}_g{U}_{{}_p/2}-{\mathrm{iU}}_{{}_p/2}^{}{\stackrel{.}{U}}_{{}_p/2}& \left(4.3\right)\end{array}$$\begin{array}{cc}{U}_{{}_p/2}=\exp \left(-i{.Math.}_{k=1}^4\frac{{}_{p,k}t}{2}{a}_k^{}{a}_k\right)& \left(4.31\right)\end{array}$

will be examined. Here, angular frequencies of pump signals of JPO1 to JPO4 are set to satisfy the following Equation (4.32).

[00056]$\begin{array}{cc}{}_{p,1}+{}_{p,2}={}_{p,3}+{}_{p,4}& \left(4.32\right)\end{array}$

[0183] Then, assuming that .sub.p,k is sufficiently large for a time scale of interest, a rotating wave approximation that neglects the oscillating term is performed.

[0184] At this time,

[00057]$\begin{array}{cc}{U}_{{}_p/2}^{}{a}_k{U}_{{}_p/2}=\exp \left(-\frac{i{}_{p,k}t}{2}{a}_k\right)& \left(4.33\right)\end{array}$

is used.

[0185] O(g.sub.+.sup.2), etc., are neglected as small quantity. However, it is assumed that g.sub..sup.4/C.sub.g is not negligible as compared to other terms (e.g., .sub.i below).

[0186] We have the following Hamiltonian H.sub.Q. Details of calculations are given in section A.2.

[00058]$\begin{array}{cc}\begin{array}{c}{H}_Q^{}=\underset{i=1}{\overset{4}{.Math.}}\left[{}_i{a}_i^{+}{a}_i-\frac{K^{}}{2}{a_i^{+}}^2{a}_i^2+\frac{p}{2}\left({a_i^{+}}^2+a_i^2\right)\right]\\ -g^{\left(4\right)}\left(\underset{k=1}{\overset{3}{.Math.}}\underset{l=k+1}{\overset{4}{.Math.}}{a}_k^{+}{a}_k{a}_l^{+}{a}_l+a_1{a}_2{a}_3^{+}{a}_4^{+}+a_1^{+}{a}_2^{+}{a}_3{a}_4\right)\\ -\frac{g^{\left(4\right)}}{2}\underset{k=1}{\overset{4}{.Math.}}{a}_k^{+}{a_k\left(a_{g-}+a_{g-}^{+}\right)}^2+{}_{-}^{}{a}_g^{+}{\_a}_{g-}+\frac{{}_g^{}}{2}\left(a_{g-}^{+2}+a_{g-}^2\right)\\ -g_{g-}^{\left(4\right)}{\left(a_{g-}+a_{g-}^{+}\right)}^4\end{array}& \left(4.34\right)\end{array}$

[0187] In Equation (4.34),

[00059]$\begin{array}{cc}{}_i=+g_{+}{g}_{+}^{}+g_{-}{g}_{-}^{}-\frac{{}_{p,i}}{2}+E_c-2E_{Cq}^{}{g}_{-}^4& \left(4.35\right)\end{array}$$\begin{array}{cc}{K}^{}=E_C+g_{-}^4{E}_{\mathrm{Cg}}^{}& \left(4.36\right)\end{array}$$\begin{array}{cc}p=\frac{E_J}{4E_J}& \left(4.37\right)\end{array}$$\begin{array}{cc}{g}^{\left(4\right)}=2g_{-}^4{E}_{\mathrm{Cg}}^{}& \left(4.38\right)\end{array}$$\begin{array}{cc}{}_{-}^{}={}_{-}-4g_{-}{g}_{-}^{}-4g_{-}^2{E}_{\mathrm{Cg}}^{}& \left(4.39\right)\end{array}$$\begin{array}{cc}{}_g^{}=8g_{-}{g}_{-}^{}-4g_{-}^2{E}_{\mathrm{Cg}}^{}& \left(4.40\right)\end{array}$$\begin{array}{cc}{g}_{g-}^{\left(4\right)}=\frac{E_{\mathrm{Cg}}^{}}{12}& \left(4.41\right)\end{array}$

g.sub.+, g.sub., and g.sup.(4) are expressed using circuit parameters and resonance angular frequencies and .sub. as follows.

[00060]$\begin{array}{cc}{g}_{+}^{}=\frac{\sqrt{{}_{-}}}{4\left(-{}_{+}\right)}\sqrt{\frac{C_g+C}{C_J}}& \left(4.42\right)\end{array}$$\begin{array}{cc}{g}_{-}^{}=\frac{\sqrt{{}_{-}}}{4\left(-{}_{-}\right)}\frac{C}{\sqrt{C_J\left(C_g+C\right)}}& \left(4.43\right)\end{array}$$\begin{array}{cc}{g}^{\left(4\right)}=\frac{{\left({}_{-}\right)}^2}{{\left[4\left(-{}_{-}\right)\right]}^4}\frac{C^4}{{C_J^2\left(C_g+C\right)}^3}{e}^2& \left(4.44\right)\end{array}$

[0188] Note that, since in a result of NPL 1 corresponding to Equation (4.44), g.sup.(4) diverges infinity at capacitance C.fwdarw.0, it is physically wrong. According to NPL 1, the strength of the four-body interaction is proportional to g1g2g3g4 (Equation (1.3) above), each gk (k=1 to 4) is expressed as (1/C) with respect to a coupling capacitor C between a coupler and a qubit (Equation (1.5) above), and strength of the four-body interaction in Equation (1.3) above diverges to infinity when coupling capacitor C.fwdarw.0.

3. CONCLUSION

[0189] In the circuit illustrated in FIG. 9, each JPO interacts with other JPOs through the coupler 21. To represent this as a direct interaction between JPOs, in the example embodiment, influence of the coupler 21 is perturbatively incorporated into JPOs. The JPO, which incorporates the influence of the coupler 21, realizes the four-body interaction. The following conditions are assigned in a calculation process.

[0190] (1)

[00061]$C/\mathrm{CJ}1$

[0191] (2) g+ and g given by equations (4.43) and (4.44), respectively, satisfy the following relationship.

|g.sub.+|,|g.sub.|<<1(See Note 2 below)

[0192] (3) angular frequencies .sub.p,k (k=1, 2, 3, 4) of pump signals respectively differ greatly, and the rotating wave approximation is valid.

[0193] These conditions have physically following meaning.

[0194] Condition (1) weakens direct interactions between/among JPOs performed without via the coupler 21.

[0195] Condition (2) causes the coupler 21 not to have a too strong influence and the JPO to have a behavior not deviating from that of the original JPO.

[0196] Condition (3) weakens an interaction (two-body interaction between JPOs and an interaction between the JPO and the coupler 21) other than the four-body interaction among JPOs into which the influence of coupler 21 is incorporated.

[0197] When the above conditions are met, the effective Hamiltonian of the circuit is given by Equation (4.34) and the coupling coefficient g.sup.(4) of the four-body interaction among JPOs is given by Equation (4.44).

[0198] <Note 2> Note that, when

[00062]${}_{-}\stackrel{}{=}\mathrm{and},C/\mathrm{Cg}1,$

[00063]$\begin{array}{cc}.Math.g_{+}^{}.Math.=.Math.\frac{-{}_{-}}{-\frac{C_g+C}{2C}{}_{-}}\frac{C_g+C}{C}{g}_{-}^{}.Math..Math.\frac{2\left(-{}_{-}\right)}{{}_{-}}{g}_{-}^{}.Math.<.Math.g_{-}^{}.Math.& \left(4.45\right)\end{array}$

holds. Thus,

[00064]$\begin{array}{cc}.Math.g_{-}^{}.Math.1& \left(4.46\right)\end{array}$

is a sufficient condition.

APPENDIX

[0199] In the following, details of the above derivation process are explained supplementally.

A. Calculation

<A.1 Capacitance>

[0200] C.sub.mat.sup.1 in Equation (3.4) is an inverse matrix of the matrix C.sub.mat, and a value C.sub.ij.sup.1 of element (i,j) of the inverse matrix is as follows. Since C.sub.mat.sup.1 is a symmetric matrix, only an upper triangular part of the matrix (including diagonal components) C.sub.ij.sup.1 (j>=i) is shown.

[00065]$\begin{array}{cc}{C}_{11}^{-1}=C_{22}^{-1}=C_{33}^{-1}=C_{44}^{-1}=\frac{1}{C_J+C}+\frac{1}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\frac{C}{C_J+C}\left(2C_{J}C+C{C}_g+C_g{C}_J\right)=\frac{1}{C_J+C}\left[1+\frac{C}{4C_J}+\frac{C^2}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\right]& \left(A\text{.1}\right)\end{array}$$\begin{array}{cc}{C}_{12}^{-1}=C_{34}^{-1}=\frac{1}{4C_J\left(C_{J}C+{\mathrm{CC}}_g{C}_g{C}_J\right)}\frac{C}{C_J+C}\left(2C_{J}C+C{C}_g+C_g{C}_J\right)=\frac{C}{C_J+C}\left[\frac{1}{4C_J}+\frac{C}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\right]& \left(A\text{.2}\right)\end{array}$$\begin{array}{cc}{C}_{13}^{-1}=C_{14}^{-1}=C_{23}^{-1}=C_{24}^{-1}=\frac{C{C}_g}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}& \left(A\text{.3}\right)\end{array}$$$\begin{gathered} \begin{array}{cc}{C}_{15}^{-1}=C_{25}^{-1}=C_{36}^{-1}=C_{46}^{-1}=\frac{1}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\left(2C_{J}C+C{C}_g+C_g{C}_J\right)= \\ \frac{1}{4C_J}+\frac{C}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}& \left(A\text{.4}\right)\end{array} \end{gathered}$$$$\begin{gathered} \begin{array}{cc}{C}_{16}^{-1}=C_{26}^{-1}=C_{35}^{-1}=C_{45}^{-1}=\frac{1}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\left({\mathrm{CC}}_g+C_g{C}_J\right)= \\ \frac{1}{4C_J}-\frac{C}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}& \left(A\text{.5}\right)\end{array} \end{gathered}$$$$\begin{gathered} \begin{array}{cc}{C}_{55}^{-1}=C_{66}^{-1}=\frac{1}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\frac{C_J+C}{C}\left(2C_{J}C+C{C}_g+C_g{C}_J\right)= \\ \frac{1}{4C}+\frac{1}{4C_J}+\frac{C_J+C}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}& \left(A\text{.6}\right)\end{array} \end{gathered}$$$\begin{array}{cc}{C}_{56}^{-1}=\frac{1}{4C_J\left(C_{J}C+C{C}_g+C_g{C}_J\right)}\frac{C_J+C}{C}\left(C{C}_g+C_g{C}_J\right)=\frac{1}{4C}+\frac{1}{4C_J}-\frac{C_J+C}{4\left(C_{J}C+C{C}_g+C_g{C}_J\right)}& \left(A\text{.7}\right)\end{array}$