Patent Publication Number: US-2023163762-A1

Title: Superconducting quantum circuit apparatus and control method for a super conducting quantum circuit

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims the benefit of the priority of Japanese patent application No. 2021-189597, filed on Nov. 22, 2021, the disclosure of which is incorporated herein in its entirety by reference thereto. 
     FIELD 
     The present invention relates to a superconducting quantum circuit apparatus and a control method for a superconducting quantum circuit. 
     BAKGROUND 
     A quantum bit (qubit) formed of a superconducting quantum circuit is typically coupled to a waveguide(s) for readout or manipulation of a quantum state and to other qubit(s). This kind of coupling can be broadly classified into the following two schemes.
     (a) Fixed coupling strength (or intensity) scheme: A coupling strength is fixed according to a circuit geometry (such as capacitance and mutual inductance, mode coupling between a three-dimensional cavity and a planar circuit).   (b) Variable (Tunable) coupling strength scheme: A coupling strength can be tuned by using a resonator or the like as a coupler. In this scheme, a coupling strength can be changed without changing a circuit layout.   

     Since a waveguide used for readout or manipulation of a quantum state is rather easy to implement and there is less need to adjust a coupling strength thereof, (a) fixed coupling strength scheme is often used for coupling to the waveguide. 
     On the other hand, as for a coupling between qubits, (b) variable coupling strength scheme is desirable for convenience of use in quantum computation. 
     When Josephson parametric oscillators (amplifiers) are used as qubits to perform quantum annealing, the qubits are coupled to each other by a two-body or four-body interaction coupler, a coupling strength of which is preferably enabled to be adjustable (or tunable). 
     Patent Literature (PTL) 1 discloses a quantum gate device to control a coupling strength of a resonator. The quantum gate device includes a superconducting qubit coupled to the resonator, a first waveguide coupled to the resonator and configured to receive a microwave photon, a second waveguide coupled to the superconducting qubit and configured to receive a microwave drive light, and an operation unit capable of controlling at least one of a frequency of the microwave drive light, an intensity of the microwave drive light, a frequency of the resonator, a frequency of the superconducting qubit, and a coupling strength between the superconducting qubit and the resonator. 
     Non-Patent Literature (NPL) 1 discloses that, when a drive light with a frequency equal to an oscillation frequency is injected to a qubit, its effect corresponds to a local field in an Ising model and depends on a relative phase between the drive light and a pump signal. 
     NPL 2 discloses a four-body interaction coupler with a tunable coupling strength using a Josephson Ring Modulator (JRM), as illustrated in  FIG.  7   .  FIG.  7    is a figure cited from Supplementary  FIG.  8   : Tunable four-body coupling with JRM in the NPL 2. 
     In  FIG.  7   , each Josephson Parametric Oscillator (JPO) is provided with a Superconducting Quantum Interference Device (SQUID) and two coplanar waveguides (CPW) connected at both ends of the SQUID. The SQUID is configured to have a loop circuit in which a first superconducting line, a first Josephson Junction (JJ), a second superconducting line, and a second Josephson junction are connected in a loop. Since the Josephson Parametric Amplifier (JPA) in NPL 2 is the same oscillator as the Josephson parametric oscillator, it is referred to as a Josephson Parametric Oscillator (JPO) in the present description. Microwave drive signals with equal strength but opposite phase are applied to capacitors C x  and C y  (as shown with solid and dashed arrows) to trigger the four-body coupling between the JPOs. Tunable four-body interaction is implemented by using an imbalanced shunted Josephson Ring Modulator (JRM). JRM is provided with two pairs of Josephson Junctions (JJs). In each pair, the Josephson junctions are connected in series. The two pairs of Josephson junctions are connected in parallel between a first node and a second node, where the first node is a connection node of JPO 1  and JPO 2 , and the second node is a connection node of JPO 3  and JPO 4 . Microwave drive signals are applied to the first and second nodes via the capacitors C x , respectively, and microwave drive signals with opposite phase are applied via the capacitors C y  to connection nodes of Josephson Junctions (JJs) of each pair, respectively. In  FIG.  7   , 3Φ ex  and Φ ex  shown close to inductors of the JRM are external magnetic flux applied to big and small loops of the JRM, respectively and a i (i=1˜4) is a mode operator of each JPO. 
       FIGS.  8 A,  8 B, and  8 C  are figures based on a, b, and c of  FIG.  4    in the NPL 2. As described above, JPA in  FIG.  4    in the NPL 2 is denoted as JPO.  FIG.  8 A  illustrates a system in which four JPOs of resonance frequencies  107   r,i  (i=1,2,3,4) interact via a single Josephson Junction (JJ) disposed at a center (central JJ). To realize a time-dependent two-photon drive, the SQUID loop of each JPO is driven by a flux pump with tunable amplitude and frequency. The pump frequency ω p,k (t) (k=1,2,3,4) is able to be varied close to a frequency  2 ω r,i  that is twice the resonance frequency of JPOi. Local four-body couplings are realized through a nonlinear inductance of the Josephson junction JJ disposed at a center (central JJ). 
     The four JPOs are respectively driven by pump signals with frequencies such that: 
       ω p,1 ( t )+ω p,2 ( t )=ω p,3 ( t )+ω p,4 ( t )   (1)
 
     and the JPOs are detuned. When the JPOs are detuned from each other, the Josephson junction JJ (central JJ) at the center induces a coupling of the following form in a rotating frame (or rotating coordinate system) of the two-photon drives: 
       −C(a 1   † a 2   554 a 3 a 4 +h.c.)   (2)
 
     where 
     C is a coupling strength (coupling constant), 
     a i  is an operator of a resonance mode of each JPO (a i   †  is a creation operator, and a i  is an annihilation operator) (a hat symbol {circumflex over ( )} for the operator is omitted), and 
     h.c. inside the parentheses indicates the Hermitian conjugate of the first term inside the parentheses. 
     This four-body interaction is always active and the coupling strength C depends on nonlinearity of the Josephson junction JJ (central JJ) and/or detuning between the JPOs and the Josephson junction (central JJ). 
     A group of four JPOs (which is referred to as a plaquette in NPL 2) illustrated in  FIG.  8 A , is a main building block of an architecture. By using square lattices, it is possible to scale up to a pyramid form needed to implement an LHZ (Lechner, Hauke, Zoller) scheme. While pump signals applied to JPOs within a plaquette have different frequencies, only four distinct frequencies of pump signals are required for entire lattices. The LHZ scheme can implement a so-called full coupling type, in which all logical spins are two-body coupled to each other, by making the JPOs in the plaquette interact with each other in the four-body interaction, for example, via a JRM ( FIG.  7   ) or a single Josephson junction. In the LHZ scheme, it is desirable for a four-body interaction within each plaquette to be variable depending on a problem to be solved in order to provide a condition (constraint) that each JPO should satisfy to represent full coupling among logical bits. As an example,  FIG.  8 B  illustrates all coupling combinations in a case where N=5 logical spins are fully coupled. 
     Energy of the Ising model with M physical spins (Ising spins) is given as 
         E=Σ   i=1   M   h   i   s   i −Σ i≠j=1   M   J   i,j   s   i   s   j    (3)
 
     where
     J ij  is a parameter corresponding to a coupling coefficient for two-body interaction,   h i  is a parameter corresponding to a local field (local magnetic field), and   s i  is an i-th physical spin which takes +1 (up) or −1 (down).   

     As an example,  FIG.  8 C  illustrates a case implementing the full coupling of N=5 logical spins with the LHZ scheme. N=5 logical spins are mapped to M=N(N−1)/2=10 physical spins. A pair of logical spins is mapped on the two levels of the physical spin. A coupling J i,j  (i≠j=1, . . . ,N) between the i-th and j-th logical pairs is encoded in a local magnetic field J k  (k=1, . . . , M) on the physical spins. In  FIG.  8 C , three bits (denoted by “Fixed”) at the bottom side of the LHZ triangular structure (or network) are fixed to up state.
     PTL 1: Japanese Patent Kokai Publication No. 2018-180084A   NPL 1: H. Goto, et. al., “Boltzmann sampling from the Ising model using quantum heating of coupled nonlinear oscillators”, Nature, Sci. Rep. 8, 7154 (2018)   NPL2: Puri, et. al., “Quantum annealing with all-to-all connected nonlinear oscillators”, Nature Communications 8, 15785 (2017)   

     SUMMARY 
     In the related art, as disclosed in NPL 2, etc., an additional drive signal(s) is(are) required to realize four-body interaction with a tunable coupling strength, and a large number of electronic components, circuits, etc., are needed for its manipulation. According to the related technology, it is difficult to realize a superconducting quantum circuit with high scalability. 
     Therefore, it is an object of the present disclosure to provide a superconducting quantum circuit apparatus and a control method thereof, each enabling to solve the above-described issues. 
     According to one aspect of the present invention, there is provided a superconducting quantum circuit apparatus, comprising 
     two or four Josephson parametric oscillators,
         each Josephson parametric oscillator including:   a SQUID (Superconducting Quantum Interference Device) including a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction connected in a loop; and   a pump line, with a pump signal supplied thereto, providing a magnetic flux penetrating through the loop of the SQUID,       

     the two or four Josephson parametric oscillators each oscillating parametrically in response to the pump signal supplied to the pump line thereof; 
     a coupler to couple the two or four Josephson parametric oscillators; and 
     a phase adjuster that varies a relative phase between or among pump signals supplied for parametric oscillation to the pump lines of the two or four Josephson parametric oscillators, respectively, to vary a strength of a two-body or four-body interaction. 
     According to another aspect of the present invention, there is provided a control method for a superconducting quantum circuit apparatus, wherein the superconducting quantum circuit comprises: 
     two or four Josephson parametric oscillators,
         each Josephson parametric oscillator including:   a SQUID (Superconducting Quantum Interference Device) including a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction connected in a loop; and   a pump line, with a pump signal supplied thereto, providing a magnetic flux penetrating through the loop of the SQUID,       

     the two or four Josephson parametric oscillators each oscillating parametrically in response to the pump signal supplied to the pump line thereof; and 
     a coupler to couple the two or four Josephson parametric oscillators. The control method comprises 
     adjusting a relative phase between or among the pump signals supplied for parametric oscillation to the two or four Josephson parametric oscillators, respectively, to tune a strength of a two-body or four-body interaction. 
     According to the present disclosure, it is possible to realize a superconducting quantum circuit with high scalability in terms of scale and with tunable coupling strength. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a diagram schematically illustrating an example embodiment. 
         FIG.  2 A  is a diagram schematically illustrating an example of a Josephson parametric oscillator according to the example embodiment. 
         FIG.  2 B  is a diagram schematically illustrating an example of a Josephson parametric oscillator according to the example embodiment. 
         FIG.  2 C  is a diagram schematically illustrating another example of the Josephson parametric oscillator according to the example embodiment. 
         FIG.  2 D  is a diagram schematically illustrating another example of the Josephson parametric oscillator according to the example embodiment. 
         FIG.  2 E  is a diagram schematically illustrating still another example of the Josephson parametric oscillator according to the example embodiment. 
         FIG.  2 F  is a diagram schematically illustrating still another example of the Josephson parametric oscillator according to the example embodiment. 
         FIG.  3 A  is a diagram schematically illustrating an example of the example embodiment. 
         FIG.  3 B  is a diagram schematically illustrating an example of the example embodiment. 
         FIG.  3 C  is a diagram schematically illustrating an example of the example embodiment. 
         FIG.  4 A  is a diagram schematically illustrating an example of the example embodiment. 
         FIG.  4 B  is a diagram schematically illustrating an example of a variation of the example embodiment. 
         FIG.  5 A  is a diagram schematically illustrating another example of a variation of the example embodiment. 
         FIG.  5 B  is a diagram schematically illustrating another example of a variation of the example embodiment. 
         FIG.  6    is a diagram schematically illustrating an example of a network configuration of the example embodiment. 
         FIG.  7    is a diagram illustrating an example of a related art. 
         FIG.  8 A  is a diagram illustrating an example of a related art. 
         FIG.  8 B  is a diagram illustrating an example of a related art. 
         FIG.  8 C  is a diagram illustrating an example of a related art. 
     
    
    
     DETAILED DESCRPTION 
     The following describes several example embodiments of the present disclosure. According to an example embodiment, it is made possible to variably set (tune) an effective coupling strength by adjusting a relative phase of pump signals, by utilizing a fact that a two-body interaction between two Josephson parametric oscillators (JPOs) and a⋅four-body interaction among four JPOs, both depend on a relative phase between or among pump signals applied, respectively, to the two or four JPOs. 
       FIG.  1    illustrates one example embodiment. In  FIG.  1   , a first JPO  10 , a second JPO  20  and a coupler coupling the first JPO  10  and the second JPO  20  are illustrated. In  FIG.  1   , the first and second JPOs are denoted as JPO 1  and JPO 2 , respectively. According to the present example embodiment, an effective coupling strength between the first and second JPOs can be variably adjusted by adjusting a relative phase (e.g., θ 1 -θ 2 ) of phases θ 1  and θ 2  of pump signals (each having a frequency ω p  close to twice a resonance frequency of the JPO) applied, respectively, to two Josephson parametric oscillators (JPOs)  10  and  20 . 
     There are typically two types of structures for the JPO.
     (1) A superconducting part (electrode) coupled capacitively to a ground plane and one end of a SQUID are connected, while the other end of the SQUID is grounded. In the case of a distributed element structure, the one end of the SQUID is connected to a λ/4 type resonator.   

     (2) A superconducting part (electrode) capacitively coupled to a ground plane is separated by a SQUID into a first superconducting part (electrode) and a second superconducting part (electrode). One end of the SQUID is connected to the first superconducting part and the other end of the SQUID is connected to the second superconducting part. In the case of a distributed element structure, a λ/2 type resonator is separated by a SQUID into a first λ/4 type resonator and a second λ/4 type resonator. The one end of the SQUID is connected to the first λ/4 type resonator and the other end of the SQUID is connected to the second λ/4 type resonator. 
     The following outlines examples of a configuration of a JPO. Since JPOs (first and second JPOs  10  and  20 ) in  FIG.  1    have the same configuration, only the configuration of the first JPO  10  will be described in the following examples.  FIG.  2 A  illustrates an example of a lumped-element structure ( 1 ) of the first JPO  10  of  FIG.  1   . Referring to  FIG.  2 A , the first JPO  10  includes a SQUID  11  in which a first superconducting line  14 , a first Josephson junction  12 , a second superconducting line  15 , and a second Josephson junction  13  are connected in a loop. The second superconducting line  15  is connected to ground and the first superconducting line  14  is connected to a superconducting part (electrode) indicated by a node  16 .  FIG.  2 B  is a schematic plan view illustrating an example of a planar shape of the superconducting part (electrode) indicated by a node  16  in  FIG.  2 A . Note that an inductance L in  FIG.  1   , which is connected between the node  16  and the first superconducting line  14 , mainly represents an inductance component of the superconducting part (electrode) and can be omitted in a circuit diagram because it is smaller than an inductance (self-inductance) of the SQUID  11 . A capacitance C 1  between the node  16  and ground designates a capacitance (capacitive component) between the superconducting part (electrode) ( 16  in  FIG.  2 B ) and ground. The capacitance C 1  configures a parallel LC resonant circuit, together with an inductance component such as a self-inductance of the SQUID  11 , etc. A line (termed as a pump line)  17  in  FIG.  2 A  is a line through which a current (pump signal) from a current control part (not shown) is supplied to provide a magnetic flux (magnetic field) penetrating through a loop of the SQUID  11 . In  FIG.  2 A , the superconducting part (electrode) (node  16 ) is connected, via an input/output capacitor C in , to a coupling portion  18  that couples with a readout circuit (not shown) and connected, via a capacitor (coupling capacitor) C 2 , to a coupling portion  19  that couples with a coupler (designated by  30  in  FIG.  1   ). The second JPO  20  in  FIG.  1    may have the same configuration as the first JPO  10 . Note that, in  FIG.  2 B , a planar shape of the electrode is, as a matter of course, not limited to a cruciform shape. 
       FIG.  2 C  illustrates an example of a lumped element structure ( 2 ) as the first JPO  10  of  FIG.  1   . Referring to  FIG.  2 C , the first superconducting line  14  of the SQUID  11  is connected, via a first superconducting portion (electrode) indicated by a node  16 A and an input/output capacitor C inA , to a coupling portion  18  that couples with the readout circuit (not shown). The second superconducting line  15  of the SQUID  11  is connected, via a second superconducting portion (electrode) indicated by a node  16 B and an input/output capacitor C inB , to a coupling portion  19  that couples with a coupler (denoted  30  in  FIG.  1   ).  FIG.  2 D  is a schematic plan view illustrating an example of a planar shape of the superconducting portions (electrodes) indicated by nodes  16 A and  16 B in  FIG.  2 C . The JPO  20  in  FIG.  1    may have the same configuration. 
       FIG.  2 E  illustrates an example of a distributed element structure ( 1 ) as the first JPO  10  of  FIG.  1   . In  FIG.  2 E , a quarter-wavelength (λ/4) resonator  21  is illustrated as a distributed element circuit with multistage connection of four-terminal circuits, each having an inductance L and a capacitance C. A first superconducting line  14  of the SQUID  11  is connected to a node  21 - 1 , which is one end of the λ/4 resonator  21 . A node  21 - 2 , which is the other end of the λ/4 resonator  21 , is connected, via an input/output capacitor C in , to a coupling portion  18  that couples with a readout circuit (not shown) and connected, via a capacitor C 2 , to a coupling portion  19  that couples with a coupler (designated by  30  in  FIG.  1   ). The JPO  20  in  FIG.  1    may have the same configuration. 
       FIG.  2 F  illustrates an example of a distributed element structure ( 2 ) as the first JPO  10  of  FIG.  1   . In  FIG.  2 F , a first superconducting line  14  of the SQUID  11  is connected to a node  21 A- 1 , which is one end of a first λ/4 resonator  21 A. A node  21 A- 2 , which is the other end of the first λ/4 resonator  21 A, is connected, via an input/output capacitor C inA , to a coupling portion  18  that couples with a readout circuit (not shown). A second superconducting line  15  of the SQUID  11  is connected to a node  21 B- 1 , which is one end of a second λ/4 resonator  21 B. A node  21 B- 2 , which is the other end of the second λ/4 resonator  21 B, is connected, via an input/output capacitor C inB , to a coupling portion  19  that couples with a coupler (designated by  30  in  FIG.  1   ). The JPO  20  in  FIG.  1    may have the same configuration. 
     Example embodiment will be described based on the example illustrated in  FIG.  2 F , as JPOs (the first and second JPOs  10  and  20 ) in  FIG.  1   .  FIG.  3 A  illustrates the example embodiment. In  FIG.  3 A , JPOs  10  and  20  in  FIG.  1    are designated by reference numerals  110  and  120 , respectively, and a coupler  30  in  FIG.  1    is designated by  131 . 
     As illustrated in  FIG.  3 A , the first and second JPOs  110  and  120  are coupled by the capacitor (Cc)  131 . The first JPO  110  includes a SQUID  111 , waveguides (Coplanar Waveguide: CPW)  112  and  113 , and a pump line  114 . The SQUID  111  includes a superconducting line  111 - 1 , a first Josephson Junction (JJ), a superconducting line  111 - 2 , and a second Josephson Junction (JJ) connected in a loop. The waveguides  112  and  113  are connected to the superconducting lines  111 - 1  and  111 - 2  of the SQUID  111 , respectively. The pump line  114  is coupled (inductively coupled) to the SQUID by mutual inductance. The second JPO  120  includes a second SQUID  121 , waveguides  122  and  123 , and a pump line  124 . The second SQUID  121  includes a superconducting line  121 - 1 , a first Josephson Junction (JJ), a superconducting line  121 - 2 , and a second Josephson Junction (JJ) connected in a loop. The waveguides  122  and  123  are connected to the superconducting lines  121 - 1  and  121 - 2  of the second SQUID  121 , respectively. The pump line  124  is coupled (inductively coupled) to the SQUID by mutual inductance. In the first JPO  110  (the second JPO  120 ), the waveguides  112  and  113  ( 122  and  123 ) may be, for example, λ/4 (quarter-wavelength) coplanar waveguides. 
     It is assumed that a resonance frequency, at a time when a signal having frequency ω 0  is supplied to the first and second JPOs  110  and  120  and a statistic magnetic field Φ dc  is applied to the SQUIDs  111  and  121 , is ω 0 . The first and second JPOs  110  and  120  are caused to oscillate parametrically when a pump signal (microwave) of sufficiently strong intensity with a frequency ω p  close to twice the resonance frequency ω 0  to each of the pump lines  114  and  124  are applied in the first and second JPOs  110  and  120 . A Hamiltonian H (quantized Hamiltonian), when resonance frequencies of the first and second JPOs  110  and  120  are ω 1  and ω 2 , respectively, the first and second JPOs  110  and  120  are capacitively coupled through a capacitor  131  and are driven with pump signals (microwave current) having frequency ω p (ω p ≈2ω 1 , ω p ≈2ω 2 ), is given by the following Equation (4). Note that the quantized Hamiltonian is generally denoted as Ĥ, but in the Equation (4), the hat{circumflex over ( )} is omitted. Hereinafter, a Hamiltonian is a quantized Hamiltonian. 
         H/h bar=ω 1   a   1   †   a   1 +ω 2   a   2   †   a   2 −( K   1 /2)  a   1   †     2     a   1   2 −( K   2 /2)  a   2   †     2     a   2   2 +(p 1 /2)[exp {− i (ω p   *t−θ   1 )}* a   1   †     2   +exp {− i (ω p   *t−θ   1 )}* a   1   2 ]+(p 2 /2)[(exp {− i (ω p   *t−θ   2 )}* a   2   †     2   +exp {− i (ω p   *t−θ   2 )}* a   2   2 )}]− g ( a   1   †   a   2   +a   2   554    a   1 )   (4)
 
     where 
     hbar is a reduced Planck constant (=h/(2π):h is the Planck constant), 
     ω 1  and ω 2  are mode frequencies of the first JPO  110  and the second JPO  120 , respectively, 
     a i   †  and a i  (i=1,2) are creation operator and annihilation operator, respectively, of resonance mode for each JPO of the first JPO  110  and the second JPO  120 , a i   †  is a Hermitian conjugate of a i . 
     The following exchange relations hold between a i   †  and a i .(i=1,2). 
       [ a   i   , a   j   †   ]=a   i   a   j   †   −a   j   †   a   i =δ ij  (δ ij  is 1 if  i=j , and 0 if  i≠j )[ a   i   , a   i   ]=[a   i   †   , a   i   † ]=0   (5)
 
     A creation operator a i   †  and an annihilation operator a i  are usually denoted as â i   † , and â i  with a hat {circumflex over ( )} in a quantum field theory, etc., but the hat{circumflex over ( )} is omitted in the present description. 
     K 1  and K 2  are Kerr coefficients representing amplitudes of Kerr-nonlinearity on the first JPO  110  and the second JPO  120 , respectively, 
     p 1  and p 2  are pump amplitudes of parametric amplifications on the first JPO  110  and the second JPO  120 , respectively, 
     ω p  is a frequency of the pump signal supplied for the parametric amplifications from pump lines  114  and  124 , 
     θ 1  and θ 2  are phases of the pump signals supplied for the parametric amplifications from pump lines  114  and  124 , respectively, and 
     g is a coupling constant of a two-body interaction between the first JPO  110  and the second JPO  120 . 
     The coupling constant g between the first JPO  110  and the second JPO  120  indicates that both are ferromagnetically coupled with a coupling strength almost constant. 
     In the Equation (4), when a unitary transformation is applied, at is replaced as follows: 
         a   i →exp {− i (ω p   *t−θ   i )/2 }a   i  ( i= 1,2)   (6)
 
     Then, the Hamiltonian is transformed into a rotating frame which rotates at ω p /2. By leaving only terms that do not oscillate in time, the Hamiltonian of the above Equation (4) is given by the following Equation (7). 
         H/h bar=Δ 1   a   1   †   a   1 +Δ 2   a   2   †   a   2 −( K   1 /2)  a   1   †     2     a   1   2 −( K   2 /2)  a   2   †     2     a   2   2 +( p   1 /2) ( a   1   †     2     +a   1   2 )+( p   2 /2) ( a   2   †     2     +a   2   2 )− g [exp { i (θ 2 −θ 1 )/2 }a   1   †   a   2 +exp {− i (θ 2 −θ 1 )/2 }a   2   †   a   1 ]  (7)
 
     where 
       Δ 1 =ω 1 −ω p /2   (8a)
 
       Δ 2 =ω 2 −ω p /2   (8b)
 
     That a coefficient of a i   † a i  (i=1,2) is Δ i  in the Equation (7), indicates that an oscillation frequency of an electromagnetic field seen from the rotating frame (rotating at ω p /2) is Δ i =ω i −ω p /2. 
     Replacing a i  by exp(−i ω p t)a i  according to the Equation (6) is equivalent to use an interaction picture (model) with 
       H 0 =ω 1   a   i   †   a   i    (9a)
 
         H   1   =H−H   0    (9b)
 
     wherein (ω p /2) a i   † a i  is regarded to have been included in a non-perturbation term of the Hamiltonian. 
     Changing a relative phase θ p (=θ 2 −θ 1 ) between the pump signals of the first JPO  110  and the second JPO  120  corresponds to rotating a relative phase of oscillation in the JPO by θ p /2. 
     On a right side of the above Equation (7), terms involving in the oscillation of each JPO (the first six terms) do not depend on the relative phase θ p , but the last term, which is a two-body interaction term: 
       g[exp {i(θ 2 −θ 1 )/2}a 1   † a 2 +exp {−i(θ 2 −θ 1 )/2}a 2   † a 1 ]  (10),
 
     depends on the relative phase θ p . That is, a real part of the term (10) depends on θ p  in the form of cos(θ p /2). 
     Therefore, a magnitude and sign of the effective strength of the two-body interaction can be adjusted by adjusting the relative phase θ p  between the pump signals of the first JPO  110  and the second JPO  120 . Note that the case where θ p /2=180 deg. corresponds to inverting a sign of an Ising spin from positive to negative, which substantially corresponds to inverting a ferromagnetic interaction to an antiferromagnetic interaction. 
     Adjustment of the relative phase θ p  between the pump signals of the first JPO  110  and the second JPO  120  can be simply implemented.  FIG.  3 B  illustrates an example of a phase adjuster to adjust the relative phase θ p  between the pump signals of the first JPO  110  and the second JPO  120 . As illustrated in  FIG.  3 B , a distributor  202  may branch a signal, which is generated by a signal source (signal generator)  201  and is received via a port  1 , into two signals for output to a port  2  and a port  3 , respectively. Phase-shifters  203  and  204  may perform phase-shifting of signals output from the ports  2  and  3 , respectively. That is, the phase-shifter  203  supplies an output signal (phase: θ 1 ) from the port  2  to the pump line  114 . The phase-shifter  204 . The phase-shifter  204  delays an output signal (phase: θ 1 ) from the port  3  by the relative phase θ p  for the signal from the port  2  and supplies an output signal (phase: θ 2 =θ 1 +θ p ) to the pump line  124 , wherein a phase of the output signal (phase: θ 1 ) from the port  3  is delayed by the relative phase θ p  to the output signal from the port  2 , to the pump line  124 . 
     In  FIG.  3 B , phase-shifters  203  and  204  may be configured with delay lines whose delay can be varied by a control signal (not shown). In this case, each of the phase-shifters  203  and  204  may be provided with two or more delay lines with delay times thereof different to each other, among which one delay line selected by a switch (selector) based on the control signal is inserted into each transmission line of the phase-shifters  203  and  204 . Note that in  FIG.  3 B , it suffices that the relative phase θ p  is set to microwave signals supplied to the pump lines  114  and  124 . Therefore, there may be provided only one of the phase-shifters  203  and  204 . 
     In  FIG.  3 B , a Wilkinson distributor (power distributor) is illustrated as the distributor  202 . Two separated signals are, via λ/4 (quarter-wavelength) transmission lines (λ/4 transformers) connected in parallel, output to the ports  2  and  3 , respectively. A characteristic impedance Z of each λ/4 transmission line is √{square root over (2)}Zo. A resistor (R=2×Zo, where Zo is a characteristic impedance of a transmission line on the input port  1  side) provided between the ports  2  and  3  enables impedance matching and maintaining isolation at the output ports  2  and  3 . Since signals with amplitudes and phases both being the same pass through the ports  2  and  3 , no current flows through the resistor between the ports  2  and  3 . The distributor  202  is, as a matter of course, not limited to Wilkinson power distributor and may be a resistive distributor or the like. 
       FIG.  3 C  illustrates another configuration to implement simplified adjusting of the relative phase θ p  between the pump signals of the first JPO  110  and the second JPO  120 . For example, a signal source supplying a pump signal to the first JPO  110 , is provided with a mixer  211 , a mixer  212  and an adder  214 . An in-phase component I(t) of an intermediate frequency signal (IF) and a local oscillation signal cos (ω LO t+θ 0 ) (where ω LO  is an angular frequency of the local oscillation signal and θ 0  is an initial phase) from the local oscillator  210  are supplied to the mixer  211 . A quadrature-phase Q(t) and a signal −sin (ω LO t+θ 0 ) output from a π/2 phase-shifter  213  that phase-shifts the local oscillation signal supplied to the mixer  211  by π/2(90 degrees) are supplied to the mixer  212 . The adder  214  adds an RF (Radio Frequency) output signal (I) from the mixer  211  and an RF output signal (Q) from the mixer  212 . An output signal from the adder  214  is supplied to the pump line  114 . By adjusting the initial phase (e.g., θ 0 ) of the local oscillation signal in the local oscillator  210 , the relative phase θ p  between the pump signals of the first JPO  110  and the second JPO  120  may be set to be variable. Alternatively, by adjusting the initial phase of the IF signal by phase modulation (phase shift keying), the relative phase θ p  between the pump signals of the first JPO  110  and the second JPO  120  may be set to be variable. 
     In  FIG.  3 C , letting the IF signal I(t) supplied to the mixer  211  cos(ω IF t) (where ω IF  is an angular frequency of the IF signal and the amplitudes thereof is 1), the RF (radio frequency) signal output from the mixer  211  is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               
                                 ω 
                                 IF 
                               
                               ⁢ 
                               t 
                             
                             ) 
                           
                           × 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               
                                 
                                   ω 
                                   LO 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 θ 
                                 0 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             ( 
                             
                               1 
                               / 
                               2 
                             
                             ) 
                           
                           [ 
                           
                             
                               cos 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           ω 
                                           IF 
                                         
                                         + 
                                         
                                           ω 
                                           LO 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 } 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           ω 
                                           IF 
                                         
                                         - 
                                         
                                           ω 
                                           LO 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     t 
                                   
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     } 
                   
                   ] 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Letting the IF signal Q(t) supplied to the mixer  212  sin(ω IF t), the RF (radio frequency) output from the mixer  212  is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             sin 
                             ⁡ 
                             ( 
                             
                               
                                 ω 
                                 IF 
                               
                               ⁢ 
                               t 
                             
                             ) 
                           
                           × 
                           
                             { 
                             
                               - 
                               
                                 sin 
                                 ⁡ 
                                 ( 
                                 
                                   
                                     
                                       ω 
                                       LO 
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                             } 
                           
                         
                         = 
                         
                           - 
                           
                             
                               ( 
                               
                                 1 
                                 / 
                                 2 
                               
                               ) 
                             
                             [ 
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             IF 
                                           
                                           - 
                                           
                                             ω 
                                             LO 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       t 
                                     
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   } 
                                 
                               
                               - 
                               
                                 cos 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             IF 
                                           
                                           + 
                                           
                                             ω 
                                             LO 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                     
                                       θ 
                                       0 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     } 
                   
                   ] 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The output signal from the adder  214  is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁡ 
                                         ( 
                                         
                                           
                                             ω 
                                             IF 
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                       × 
                                       
                                         cos 
                                         ⁡ 
                                         ( 
                                         
                                           
                                             
                                               ω 
                                               LO 
                                             
                                             ⁢ 
                                             t 
                                           
                                           + 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         sin 
                                         ⁡ 
                                         ( 
                                         
                                           
                                             ω 
                                             IF 
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                       × 
                                       
                                         { 
                                         
                                           - 
                                           
                                             sin 
                                             ⁡ 
                                             ( 
                                             
                                               
                                                 
                                                   ω 
                                                   LO 
                                                 
                                                 ⁢ 
                                                 t 
                                               
                                               + 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                         } 
                                       
                                     
                                   
                                   = 
                                   
                                     
                                       ( 
                                       
                                         1 
                                         / 
                                         2 
                                       
                                       ) 
                                     
                                     [ 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           { 
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     ω 
                                                     IF 
                                                   
                                                   + 
                                                   
                                                     ω 
                                                     LO 
                                                   
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               t 
                                             
                                             + 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           } 
                                         
                                       
                                       + 
                                       
                                         cos 
                                         ⁢ 
                                         
                                           { 
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     ω 
                                                     IF 
                                                   
                                                   - 
                                                   
                                                     ω 
                                                     LO 
                                                   
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               t 
                                             
                                             - 
                                             
                                               θ 
                                               0 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               } 
                             
                             ] 
                           
                           - 
                           
                             
                               ( 
                               
                                 1 
                                 / 
                                 2 
                               
                               ) 
                             
                             [ 
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             IF 
                                           
                                           - 
                                           
                                             ω 
                                             LO 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       t 
                                     
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   } 
                                 
                               
                               - 
                               
                                 cos 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             IF 
                                           
                                           + 
                                           
                                             ω 
                                             LO 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                     
                                       θ 
                                       0 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                       } 
                     
                     ] 
                   
                   = 
                   
                     cos 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               
                                 ω 
                                 IF 
                               
                               + 
                               
                                 ω 
                                 LO 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                         + 
                         
                           θ 
                           0 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In the output from the adder  214 , a lower side band (frequency: ω IF −ω LO ) is canceled out and an upper side band (frequency: ω IF +ω LO =ω p ) microwave is output to the pump line  114 . A DC (direct current) component is applied in addition to the microwave, to the pump line  114 . Addition of the DC component to the microwave may be performed inside a refrigerator in which a superconducting quantum circuit (chip) is arranged (DC biased microwave may be inductively coupled to the SQUID of the JPO). The pump signal supplied to the pump line  114  may be an amplitude modulated signal rather than a frequency modulated signal as described above. 
     According to the present example embodiment, there is an advantage that a coupling strength can be adjusted with a simpler configuration compared with a configuration that uses a coupler with a coupling strength adjustable. 
       FIG.  4 A  illustrates another example embodiment of the present invention. In  FIG.  4 A , a first JPO  110 , a second JPO  120 , a third JPO  130 , and a fourth JPO  140  are each assumed to include a SQUID, first and second waveguides, and a pump line that provides a magnetic flux penetrating through the SQUID. 
     The first JPO  110  and the second JPO  120  are connected to a node  155  via capacitors  151  and  152  (Alternate Current, AC, coupling), the third JPO  130  and the fourth JPO  140  are connected to a node  156  via capacitors  153  and  154  (AC coupling). The nodes  155  and  156  are connected via Josephson junction  160 . The pump signals with frequencies ω p,1 , ω p,2 , ω p,3 , ω p,4  and phases θ p,1 , θ p,2 , θ p,3 , θ p,4  are supplied to the pump lines (not shown) of the first JPO  110 , the second JPO  120 , the third JPO  130 , and the fourth JPO  140 , respectively. 
     A Hamiltonian (quantized Hamiltonian) of a circuit in  FIG.  4 A  can be expressed as a sum of a Hamiltonian H JPO,k  of each JPO and an interaction Hamiltonian Hc, as follows. 
         H=Σ   k=1   4   H   JPO,k   +H   c    (14)
 
     The Hamiltonian (quantized Hamiltonian) for each JPO is given as follows. Note that hbar is omitted 
       H JPO,k =ω r,k   a   k   †   a   k −( K/ 2) a   k   †     2     a   k   2 +ε p ( t )[exp {− i (ω p,k ( t ) t/ 2)} a   k   †     2   +exp { i (ω p,k ( t ) t/ 2)} a   k   2 ]  (15)
 
     where 
     a k   †  and a k  and are a creation operator and an annihilation operator for an oscillation mode across the kth JPO(k=1,2,3,4), 
     a r,k  is a resonance frequency of the kth JPO, 
     K is a Kerr coefficient representing amplitude of Kerr-nonlinearity which JPO has, p ε p (t) is an amplitude of a parametric pump (two-photon pump), and 
     ω p,k (t) is an angular frequency of a parametric pump of k-th JPO. 
     The interaction Hamiltonian Hc (quantized Hamiltonian) is given by the following Equation (16). 
         H   c =ω c   a   c   †   a   c   +g   1 ( a   c   †   a   1   +a   1   †   a   c )+ g   2 ( a   c   †   a   2   +a   2   †   a   c )− g   3 ( a   c   †   a   3   +a   3   †   a   c )− g   4 ( a   c   †   a   4   +a   4   †   a   c )− E   j {cos(Φ/Φ 0 )+(1/2)(Φ/Φ 0 ) 2 }  (16)
 
     where 
     a c   †  and a c  are a creation operator and an annihilation operator for a mode (junction mode) across the Josephson junction (coupling Josephson junction) 160 , 
     g i (i=1,2,3,4) is a magnitude of the coupling (rate at which energy is exchanged) between the i th  JPO and the mode of the Josephson junction  160 , 
     Φ 0 =(h/2π)(2e) is a flux quantum, 
     ω c  is a frequency of the junction mode, and 
     E J  is a Josephson energy of the Josephson junction  160  disposed at the center part of the circuit, which is proportional to a critical current value of the Josephson junction  160 . 
     In the Equation (16), Φ is given by 
       Φ=Φ c  ( a   c   †   +a   c )   (17).
 
     where Φ c  is a standard deviation of a zero-point magnetic flux fluctuation for the Josephson junction  160 . 
     In  FIG.  4 A , the first to fourth JPOs  110 - 140  are nonlinear resonators, each including a SQUID as a nonlinear inductor, as with the first JPO  110  and the second JPO  120  described with reference to  FIG.  1   . The Josephson junction  160  disposed in a center part is a nonlinear inductor and can be regarded as a nonlinear resonator. Therefore, the configuration illustrated in  FIG.  4 A  can be regarded as a circuit in which five nonlinear resonators are connected. A four-body coupling in the center of the circuit may be denoted as a resonator for coupling. A resonance mode of the Josephson junction  160  in the center part is far detuned from JPOs and is not driven externally, so that we have 
       &lt; a   c   &gt;=&lt;a   c   †   a   c &gt;=0. 
     The four JPOs  110 - 140  in  FIG.  4 A  are also termed as “plaquette”according to NPL 2. 
     In interaction of the Equation (17), under the condition 
       ω p,k ≠ω p,m , ω p,1 +ω p,2 =ω p,3 +ω p,4    (18),
 
     if an oscillation term such as, for example, 
       ω p,1 −ω p,2    (19),
 
     due to a frequency difference of the pump signal of JPO is negligible, the plaquette Hamiltonian is given by the following Equation (20) 
     
       
         
           
             
               
                 
                   
                     H 
                     plaquette 
                   
                   ≈ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         4 
                       
                         
                       
                         { 
                         
                           
                             H 
                             
                               JPA 
                               , 
                               k 
                             
                           
                           - 
                           
                             
                               
                                 g 
                                 k 
                                 2 
                               
                               
                                 Δ 
                                 k 
                               
                             
                             ⁢ 
                             
                               a 
                               k 
                               † 
                             
                             ⁢ 
                             
                               a 
                               k 
                             
                           
                         
                         } 
                       
                     
                     - 
                     
                       
                         E 
                         j 
                       
                       ⁢ 
                       
                         
                           Φ 
                           c 
                           4 
                         
                         
                           Φ 
                           0 
                           4 
                         
                       
                       ⁢ 
                       
                         
                           
                             g 
                             1 
                           
                           ⁢ 
                           
                             g 
                             2 
                           
                           ⁢ 
                           
                             g 
                             4 
                           
                           ⁢ 
                           
                             g 
                             4 
                           
                         
                         
                           
                             Δ 
                             1 
                           
                           ⁢ 
                           
                             Δ 
                             2 
                           
                           ⁢ 
                           
                             Δ 
                             4 
                           
                           ⁢ 
                           
                             Δ 
                             4 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               a 
                               1 
                               † 
                             
                             ⁢ 
                             
                               a 
                               2 
                               † 
                             
                             ⁢ 
                             
                               a 
                               3 
                             
                             ⁢ 
                             
                               a 
                               4 
                             
                           
                           + 
                           
                             h 
                             . 
                             c 
                             . 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         E 
                         j 
                       
                       ⁢ 
                       
                         
                           Φ 
                           c 
                           4 
                         
                         
                           Φ 
                           0 
                           4 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             
                               k 
                               ≠ 
                               m 
                             
                             = 
                             1 
                           
                           4 
                         
                           
                         
                           
                             
                               
                                 g 
                                 k 
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 m 
                                 2 
                               
                             
                             
                               
                                 Δ 
                                 k 
                                 2 
                               
                               ⁢ 
                               
                                 Δ 
                                 m 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                             a 
                             k 
                             † 
                           
                           ⁢ 
                           
                             a 
                             k 
                           
                           ⁢ 
                           
                             a 
                             m 
                             † 
                           
                           ⁢ 
                           
                             a 
                             m 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     In the Equation (20), the second part (g k   2 /Δ k )a k   † a k  of the first term of the right side results in a frequency shift of a JPO mode due to off-resonant coupling with the Josephson junction  160 . 
     In the Equation (20), the second term of the right side is a term of a four-body coupling (interaction) among the first to fourth JPOs. From the second term of the Equation (20), a coupling strength (coefficient) C of the four-body interaction can be given in terms of circuit parameters as: 
     
       
         
           
             
               
                 
                   C 
                   = 
                   
                     
                       E 
                       j 
                     
                     ⁢ 
                     
                       
                         Φ 
                         c 
                         4 
                       
                       
                         Φ 
                         0 
                         4 
                       
                     
                     ⁢ 
                     
                       
                         
                           g 
                           1 
                         
                         ⁢ 
                         
                           g 
                           2 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                       
                       
                         
                           Δ 
                           1 
                         
                         ⁢ 
                         
                           Δ 
                           2 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     In the Equation (20), the last term gives rise to a cross-Kerr interaction between the JPOs. 
     In the Equation (20), Δ k  is a difference (detuning) between a mode frequency ω r,k  of the kth JPO and a mode frequency (resonance frequency) ω c , where the mode frequency ω c  is specified by a capacitance and an inductance which the Josephson junction  160  has. 
       Δ k =ω c , −ω r,k    (22)
 
     Thus, in  FIG.  4 A , a magnitude (strength) of the four-body interaction can be varied by varying at least one of resonance frequencies of the first through fourth JPOs  110 - 140  or a mode frequency of the Josephson junction  160 . 
     In addition, since the pump signals with frequencies ω p,1 , ω p,2 , ω p,3 , ω p,4  and phases θ p,1 , θ p,2 , θ p,3 , θ p,4  are supplied to the first through the pump lines of the fourth JPO  110 - 140 , respectively, the second term of right side of the Equation (20) is given as: 
     
       
         
           
             
               
                 
                   
                     E 
                     j 
                   
                   ⁢ 
                   
                     
                       Φ 
                       c 
                       4 
                     
                     
                       Φ 
                       0 
                       4 
                     
                   
                   ⁢ 
                   
                     
                       
                         g 
                         1 
                       
                       ⁢ 
                       
                         g 
                         2 
                       
                       ⁢ 
                       
                         g 
                         4 
                       
                       ⁢ 
                       
                         g 
                         4 
                       
                     
                     
                       
                         Δ 
                         1 
                       
                       ⁢ 
                       
                         Δ 
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         4 
                       
                       ⁢ 
                       
                         Δ 
                         4 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         [ 
                         
                           exp 
                           ⁢ 
                           
                             { 
                             
                               - 
                               
                                 
                                   i 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       θ 
                                       
                                         p 
                                         , 
                                         3 
                                       
                                     
                                     + 
                                     
                                       θ 
                                       
                                         p 
                                         , 
                                         4 
                                       
                                     
                                     - 
                                     
                                       θ 
                                       
                                         p 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     
                                       θ 
                                       
                                         p 
                                         , 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             } 
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               a 
                               1 
                               † 
                             
                             ⁢ 
                             
                               a 
                               2 
                               † 
                             
                             ⁢ 
                             
                               a 
                               3 
                             
                             ⁢ 
                             
                               a 
                               4 
                             
                           
                           + 
                           
                             h 
                             . 
                             c 
                             . 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Therefore, an effective coupling strength of the four-body interaction can be adjusted by adjusting a relative phase of at least one JPO among four JPO  110  through JPO  140 . 
       FIG.  4 B  illustrates a variation of the configuration illustrated in  FIG.  4 A . Referring to  FIG.  4 B , a capacitor  161  is shunt-connected (connected in parallel) between both ends of the Josephson junction  160  to realize a coupler that is resistant to charge noise. 
     In the circuits illustrated in  FIG.  4 A  and  FIG.  4 B , the Josephson junction  160  acts as an inductor. A magnitude of inductance L J  is expressed using a critical current value I c  of the Josephson junction  160  as 
         L   J =Φ 0 /(2 πI   c )   (24)
 
     The critical current value I c  is determined by the Josephson junction (such as material properties, area (junction size), and thickness of two superconductors and an insulating film disposed therebetween). 
       FIG.  5 A  illustrates a configuration in which the Josephson junction  160  in the center part in  FIG.  4 A  is replaced by a SQUID  170 , in which a superconducting line  171 , a first Josephson Junction (JJ 1 ), a superconducting line  172 , and a second Josephson Junction (JJ 2 ) are connected in a loop. In this configuration, a resonance frequency ω c , of a coupler can be varied by using the SQUID  170  instead of the Josephson junction ( 160  in  FIG.  4 A ). Therefore, detuning between the SQUID  170  and the four JPOs can be varied to adjust the maximum and minimum values of the four-body interaction which are variable depending on phases of four pump signals. 
     When the magnetic flux passing through the loop of the SQUID is Φ ext  the critical current value I c   eff  of an entire SQUID is given by 
       I c   eff =2 I   c |cos(πΦ ext /Φ 0 )|  (25)
 
     The SQUID is an inductor with an inductance varied by a magnetic flux passing through the SQUID loop. The magnetic flux passing through the SQUID loop can be varied relatively easily by applying an external current. Therefore, a resonance frequency ω c  of the coupler can be made variable by replacing the Josephson junction  160  in the center part of the circuit with the SQUID  170 . This varies values of Δ k  and E J , resulting in a change of the magnitude of the four-body interaction. Note that when the Josephson junction  160  in the center part is replaced with the SQUID  170 , a resonance frequency of the coupler may vary due to unintended magnetic flux fluctuation (flux noise), etc. 
     In  FIG.  5 A , the four JPOs  110 - 140  are coupled by four-body interaction, when the respective resonance frequencies of the first to fourth JPOs  110 - 140  satisfy the following conditions for four-body interaction: 
       ω p,k ≠ω p,m ( k≠m= 1, . . . , 4), ω p,1 +ω p,2 =ω p,3 +ω p,4    (26)
 
     In this case, a term of the four-body interaction in the Hamiltonian, is given by 
     
       
         
           
             
               
                 
                   
                     E 
                     j 
                   
                   ⁢ 
                   
                     
                       Φ 
                       c 
                       4 
                     
                     
                       Φ 
                       0 
                       4 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           g 
                           1 
                         
                         ⁢ 
                         
                           g 
                           2 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                       
                       
                         
                           Δ 
                           1 
                         
                         ⁢ 
                         
                           Δ 
                           2 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                       
                     
                     [ 
                     
                       exp 
                       ⁢ 
                       
                         { 
                         
                           
                             - 
                             
                               i 
                               ⁡ 
                               ( 
                               
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     3 
                                   
                                 
                                 + 
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     4 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     1 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           / 
                           2 
                         
                         } 
                       
                     
                     ] 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           a 
                           1 
                           † 
                         
                         ⁢ 
                         
                           a 
                           2 
                           † 
                         
                         ⁢ 
                         
                           a 
                           3 
                         
                         ⁢ 
                         
                           a 
                           4 
                         
                       
                       + 
                       
                         h 
                         . 
                         c 
                         . 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Here, the expression (23) is cited again for convenience of explanation. 
     The following considers an expected value of an energy of the Expression (27). When varying one of the phases of pump signals for parametric oscillation supplied to the first to fourth JPOs  110 - 140 , respectively, a value of 
       exp{−i(θ p,3 +θ p,4 −θ p,1 −θ p,2 )/2}  (28)
 
     varies. 
     Maximum and minimum value of a real part of exp{−i(θ p,3 +θ p,4 −θ p,1 −θ p,2 )/2} in the Expression (28) are +1 and −1, respectively. 
     Therefore, a range that is able to be varied only by varying the phase of the pump signals supplied for parametric oscillation to the first to fourth JPOs  110 - 140 , respectively, is given by 
     
       
         
           
             
               
                 
                   
                     ± 
                     
                       E 
                       j 
                     
                   
                   ⁢ 
                   
                     
                       Φ 
                       c 
                       4 
                     
                     
                       Φ 
                       0 
                       4 
                     
                   
                   ⁢ 
                   
                     
                       
                         g 
                         1 
                       
                       ⁢ 
                       
                         g 
                         2 
                       
                       ⁢ 
                       
                         g 
                         4 
                       
                       ⁢ 
                       
                         g 
                         4 
                       
                     
                     
                       
                         Δ 
                         1 
                       
                       ⁢ 
                       
                         Δ 
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         4 
                       
                       ⁢ 
                       
                         Δ 
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     When varying the magnetic flux Φ ext  that passes through the loop of the SQUID  170 , values the detuning Δ k  and the Josephson energy E J  in the Equation (29) are varied, respectively, as a result of which a value of 
     
       
         
           
             C 
             = 
             
               
                 E 
                 j 
               
               ⁢ 
               
                 
                   Φ 
                   c 
                   4 
                 
                 
                   Φ 
                   0 
                   4 
                 
               
               ⁢ 
               
                 
                   
                     g 
                     1 
                   
                   ⁢ 
                   
                     g 
                     2 
                   
                   ⁢ 
                   
                     g 
                     4 
                   
                   ⁢ 
                   
                     g 
                     4 
                   
                 
                 
                   
                     Δ 
                     1 
                   
                   ⁢ 
                   
                     Δ 
                     2 
                   
                   ⁢ 
                   
                     Δ 
                     4 
                   
                   ⁢ 
                   
                     Δ 
                     4 
                   
                 
               
             
           
         
       
     
     in the Equation (21) varies. 
     Thus, it is possible to adjust maximum and the minimum values of the four-body interaction, which can be varied by the phases of the pump signals supplied for parametric oscillation to the first to fourth JPOs  110 - 140 , respectively. 
     Note that when the resonance frequency ω c  of the SQUID  170  is varied, not only detuning Δ k  but also the Josephson energy E J  is varies. 
     As described above, in example illustrated in  FIG.  5 A , by replacing the Josephson junction  160  in  FIG.  4 A  with the SQUID  170 , the maximum and minimum values of the four-body interaction 
     
       
         
           
             
               
                 
                   
                     
                       ± 
                       
                         E 
                         j 
                       
                     
                     ⁢ 
                     
                       
                         Φ 
                         c 
                         4 
                       
                       
                         Φ 
                         0 
                         4 
                       
                     
                     ⁢ 
                     
                       
                         
                           g 
                           1 
                         
                         ⁢ 
                         
                           g 
                           2 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                       
                       
                         
                           Δ 
                           1 
                         
                         ⁢ 
                         
                           Δ 
                           2 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                       
                     
                   
                   ± 
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     can be varied. 
     Dependency of the four-body interaction to the resonance frequency ω c  of the SQUID  170  is complicated. Therefore, in actual experiments, basically without changing resonance frequency ω c  of the SQUID  170  in  FIG.  5 A , the respective resonance frequencies ω r,k  (k=1˜4) of the first to fourth JPOs  110 - 140  may be adjusted so that 
     
       
         
           
             C 
             = 
             
               
                 E 
                 j 
               
               ⁢ 
               
                 
                   Φ 
                   c 
                   4 
                 
                 
                   Φ 
                   0 
                   4 
                 
               
               ⁢ 
               
                 
                   
                     g 
                     1 
                   
                   ⁢ 
                   
                     g 
                     2 
                   
                   ⁢ 
                   
                     g 
                     4 
                   
                   ⁢ 
                   
                     g 
                     4 
                   
                 
                 
                   
                     Δ 
                     1 
                   
                   ⁢ 
                   
                     Δ 
                     2 
                   
                   ⁢ 
                   
                     Δ 
                     4 
                   
                   ⁢ 
                   
                     Δ 
                     4 
                   
                 
               
             
           
         
       
     
     becomes large as compared to a required magnitude (strength) of the four-body interaction, and then, the value of the four-body interaction may be fine-tuned by adjusting the phases of the pump signals supplied for parametric oscillation to the first to fourth JPOs  110 - 140 , respectively. 
     As described above, according to the present example embodiment, an effective coupling strength can be adjusted by adjusting a relative phase of pump signals supplied to the first to fourth JPOs  110 - 140  for parametric oscillation. When the resonance frequencies ω p,1 , ω p,2 , ω p,3 , ω p,4  of the first to fourth JPOs  110 - 140  satisfy 
       ω p,1 +ω p,2 =ω p,3 +ω p,4    (31),
 
     a value of 
       θ p,1 +θ p,2 +θ p,3 , −θ p,4    (32)
 
     is adjusted for the phases of pump signals. 
     Therefore, the effective coupling strength can be adjusted by adjusting a relative phase at least in one JPO among four JPOs  110 - 140  in  FIG.  5 A . For example, four-body interaction term can be adjusted by adjusting a relative phase of phase θ p,3  of the pump signal in JPO  130  to phase θ p,1  of the pump signal in JPO  110 , while fixing the values of θ p,1 , θ p,2 , θ p,3 , and θ p,4 . 
       FIG.  5 B  illustrates a variation of the configuration illustrated in  FIG.  5 A . Referring to  FIG.  5 B , a capacitor  173  is shunt-connected between both ends of the SQUID  170 , which is a coupler of the four-body interaction to realize a coupler that is resistant to charge noise. 
     In still another example embodiment, a JPO network is configured using two-body interaction which is described with reference to  FIG.  3 A . In the JPO network configured such that two-body interaction coupling portions do not form a loop, signs and magnitudes of all two-body interaction can be adjusted by adjusting a phase of pump signals (relative phase). When two-body interaction coupling portions in the JPO network form a loop (JPO 2 -JPO 4 -JPO 5 ) as illustrated in  FIG.  6   , an adjusting range of signs and magnitudes of all the two-body interaction is limited. The reason is that phases of some pump signals cannot be freely determined. 
     In the LHZ (Lechner, Hauke, Zoller) scheme, which planarly couples four-body interaction couplers described with reference to  FIG.  4 A  and  FIG.  5 A , there is no loop in a JPO network that limits a degree of freedom, because they are coupled loosely, as illustrated in  FIG.  8 C . Any combination of coupling strengths can be realized without restriction. 
     In two-body interaction coupling and/or four-body interaction coupling between JPOs, a polarity (positive and negative) and a magnitude of a coupling strength can be adjusted by adjusting (varying) a relative phase of the pump signals supplied to JPOs for parametric oscillation. 
     Note that same effect as above can be obtained by using lumped constant type JPO described with reference to  FIG.  2 A  to  FIG.  2 D , instead of the distributed element JPO. 
     In the above-described example embodiments, a two-body and/or four-body coupling portion (capacitors, Josephson junctions) that does not have an ability to adjust a coupling strength is described, however, the present disclosure is also applicable to a coupling portion whose coupling strength is able to be variably adjusted. For example, the technique (adjusting a strength of the four-body interaction by the phase of pump signals for parametric oscillation of JPOs) of the present disclosure can be applied to a variable four-body coupling portion (JRM) described with reference to  FIG.  7   . 
     In this case, in order to realize the four-body interaction, when the combination (or relation) of the resonance frequencies of the JPOs 1, 2, 3, and 4, and a frequency ω d  of the drive signals inputted from capacitors Cx, Cy is, for example, given by 
       ω d =ω p,1 +ω p,2 +ω p,3 −ω p,4    (33)
 
     Hamiltonian with respect to the drive signal of 
       2ω Z √{square root over (n)} cos(ω d t)   (34)
 
     is given by a following Equation (35). 
     
       
         
           
             
               
                 
                   
                     H 
                     plaquette 
                   
                   ≈ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         4 
                       
                         
                       
                         { 
                         
                           
                             H 
                             
                               JPA 
                               , 
                               k 
                             
                           
                           - 
                           
                             
                               
                                 
                                   ( 
                                   
                                     g 
                                     k 
                                     x 
                                   
                                   ) 
                                 
                                 2 
                               
                               
                                 Δ 
                                 k 
                                 x 
                               
                             
                             ⁢ 
                             
                               a 
                               k 
                               † 
                             
                             ⁢ 
                             
                               a 
                               k 
                             
                           
                         
                         } 
                       
                     
                     - 
                     
                       
                         C 
                         jrn 
                       
                       ( 
                       
                         
                           
                             a 
                             1 
                             † 
                           
                           ⁢ 
                           
                             a 
                             2 
                             † 
                           
                           ⁢ 
                           
                             a 
                             3 
                             † 
                           
                           ⁢ 
                           
                             a 
                             4 
                           
                         
                         + 
                         
                           h 
                           . 
                           c 
                           . 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 
                   
                     C 
                     jrn 
                   
                   = 
                   
                     
                       E 
                       j 
                     
                     ⁢ 
                     
                       n 
                     
                     ⁢ 
                     
                       
                         
                           Φ 
                           x 
                           4 
                         
                         ⁢ 
                         
                           Φ 
                           z 
                         
                       
                       
                         4 
                         ⁢ 
                         
                           Φ 
                           0 
                           5 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           g 
                           1 
                         
                         ⁢ 
                         
                           g 
                           2 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                       
                       
                         
                           Δ 
                           1 
                         
                         ⁢ 
                         
                           Δ 
                           2 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The second term on the right side of Equation (35) is the four-body interaction term. In this second term, an effect caused by phases of the pump signals supplied to the first to fourth JPO 1 -JPO 4 , respectively, is given explicitly by the following Expression (37). 
     
       
         
           
             
               
                 
                   
                     E 
                     j 
                   
                   ⁢ 
                   
                     n 
                   
                   ⁢ 
                   
                     
                       
                         Φ 
                         x 
                         4 
                       
                       ⁢ 
                       
                         Φ 
                         z 
                       
                     
                     
                       4 
                       ⁢ 
                       
                         Φ 
                         0 
                         5 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           g 
                           1 
                         
                         ⁢ 
                         
                           g 
                           2 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                         ⁢ 
                         
                           g 
                           4 
                         
                       
                       
                         
                           Δ 
                           1 
                         
                         ⁢ 
                         
                           Δ 
                           2 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                         ⁢ 
                         
                           Δ 
                           4 
                         
                       
                     
                     [ 
                     
                       exp 
                       ⁢ 
                       
                         { 
                         
                           
                             - 
                             
                               i 
                               ⁡ 
                               ( 
                               
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     3 
                                   
                                 
                                 + 
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     4 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     1 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   
                                     p 
                                     , 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           / 
                           2 
                         
                         } 
                       
                     
                     ] 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           a 
                           1 
                           † 
                         
                         ⁢ 
                         
                           a 
                           2 
                           † 
                         
                         ⁢ 
                         
                           a 
                           3 
                           † 
                         
                         ⁢ 
                         
                           a 
                           4 
                         
                       
                       + 
                       
                         h 
                         . 
                         c 
                         . 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     Therefore, even in the circuit illustrated in  FIG.  7   , the effective coupling strength of the four JPOs has dependency in the form of 
       exp{−i(θ p,1 +θ p,2 +θ p,3 −θ p,4 )/2}  (38)
 
     with respect to the phases θ p,1 , θ p,2 , θ p,3 , and θ p,4  of the pump signals supplied to the first to fourth JPOs  1 - 4 , respectively. Even in the configuration with a shunted-type JRM as illustrated in  FIG.  7   , a sign (positive and negative) and magnitude of the coupling strength of the four-body interaction is enabled to be adjusted by adjusting (varying) a relative phase of the pump signals supplied to JPOs  1 - 4  for parametric oscillation. 
     Even in a JPO network in which a plurality of JPOs are planarly coupled by a four-body interaction coupling portions, signs and magnitudes of each four-body interaction can be adjusted by adjusting the phase of the pump signals supplied to the JPOs. For example, a JPO network can be used to configure a quantum annealer, as illustrated in  FIG.  8 C . 
     Note that a superconducting quantum circuit according to each of the above-mentioned example embodiments may be implemented by, for example, lines (wirings) of a superconducting material formed on a substrate. In this case, while silicon may be used as a material for the substrate, any other electric materials such as sapphire or compound semiconductor materials (Group IV, III-V, II-VI) may be used. The substrate is preferably monocrystalline, but may also be polycrystalline or amorphous. While Nb (niobium) or Al (aluminum) may be used as a material of the superconducting line, the material is not limited to them and any other metal which is in a superconducting state when it is cooled to an extremely low temperature (cryogenic temperature), such as niobium nitride (NbN), indium (In), lead (Pb), tin (Sn), rhenium (Re), palladium (Pd), titanium (Ti), molybdenum (Mo), tantalum (Ta), tantalum nitride and alloys containing at least any one of those, may be used. In order to achieve a superconducting state, a superconducting quantum circuit is used in a temperature environment such as at 10 mK (milli-Kelvin) achieved by a cryogenic refrigerating machine. 
     Each disclosure of PTL 1 and NPLs 1 and 2 cited above is incorporated herein in its entirety by reference thereto. It is to be noted that it is possible to modify or adjust the example embodiments or examples within the whole disclosure of the present invention (including the Claims) and based on the basic technical concept thereof. Further, it is possible to variously combine or select a wide variety of the disclosed elements (including the individual elements of the individual claims, the individual elements of the individual examples and the individual elements of the individual figures) within the scope of the Claims of the present invention. That is, it is self-explanatory that the present invention includes any types of variations and modifications to be done by a skilled person according to the whole disclosure including the Claims, and the technical concept of the present invention. 
     &lt;Appendix&gt; 
     For reference, the correspondence between the equation numbers in the present disclosure and those in Supplementary Note 6,8 (Note 6,8) of NPL 2 is provided. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
               
               
                 Specification 
                 (14) 
                 (15) 
                 (16) 
                 (20) 
                 (35) 
                 (36) 
               
               
                   
               
             
            
               
                 NPL 2 
                 Note 6 
                 Note 6 
                 Note 6 
                 Note 6 
                 Note 8 
                 Note 8 
               
               
                   
                 (18) 
                 (17) 
                 (19) 
                 (23) 
                 (30) 
                 (31)