Superconducting quantum circuit device and method for controlling superconducting quantum circuit

Abstract

To provide a superconducting quantum circuit device with high scalability in terms of scale, etc., and capable of adjusting coupling strength.SOLUTION: A superconducting quantum circuit device comprises: a SQUID (superconducting quantum interference device) in which a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction are connected in a loop; two or four Josephson parametric oscillators, each of which includes a line supplying magnetic flux interlinking with the loop of the SQUID, and parametrically oscillating according to a pump signal supplied to the line; and a combiner combining the two or four Josephson parametric oscillators, the superconducting quantum circuit device further comprises means for varying the strength of the two-body or four-body interaction by varying the relative phase between the pump signals respectively supplied to the lines of the two or four Josephson parametric oscillators for parametric oscillation.SELECTED DRAWING: Figure 2A

Description

[Technical Field]

[0001] The present invention relates to a superconducting quantum circuit device and a method for controlling a superconducting quantum circuit. [Background technology]

[0002] Qubits consisting of superconducting quantum circuits are generally coupled to waveguides or other qubits for readout and quantum state manipulation. This coupling can be broadly divided into the following two types:

[0003] (a) Bond strength fixed method The coupling strength is fixed by the geometry of the circuit (e.g., capacitance, mutual inductance, mode coupling between a three-dimensional cavity and a planar circuit, etc.).

[0004] (b) Variable coupling strength method The coupling strength can be adjusted by using a resonator or the like.

[0005] Waveguides for readout and quantum state manipulation often use the fixed coupling strength method (a) because it is easy to implement and there is little need to adjust the coupling strength.

[0006] On the other hand, for the convenience of use in quantum computing, it is desirable that the coupling between quantum bits be of the variable coupling strength type (b).

[0007] When quantum annealing is performed using a Josephson parametric oscillator (amplifier), the quantum bits are coupled to each other by a coupler that couples two-body or four-body interactions, and the coupling strength must be adjustable.

[0008] Patent Document 1 discloses a quantum gate device having a configuration for controlling the coupling strength of a resonator, the device comprising: a superconducting quantum bit coupled to a resonator; a first waveguide coupled to the resonator and into which microwave photons are incident; a second waveguide coupled to the superconducting quantum bit and into which microwave drive light is incident; and an operation unit capable of controlling at least one of the frequency of the microwave drive light, the intensity of the microwave drive light, the frequency of the resonator, the frequency of the superconducting quantum bit, and the coupling strength between the superconducting quantum bit and the resonator.

[0009] Non-Patent Document 1 discloses that when drive light with a frequency equal to the oscillation frequency is incident on a quantum bit, the effect corresponds to a local magnetic field in the Ising model and depends on the relative phase of the drive light and the pump signal.

[0010] Non-Patent Document 2 discloses a four-body interaction coupler with adjustable coupling strength that uses a Josephson ring modulator (JRM), as shown in Figure 7. Figure 7 is based on Supplementary Figure 8 Tunable four-body coupling with JRM in Non-Patent Document 2.

[0011] In Fig. 7, each Josephson parametric oscillator (JPO) has a SQUID (Superconducting Quantum Interference Device) consisting of 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 circular fashion, and two coplanar waveguides (CPW) connected to both ends of the SQUID. Note that the Josephson parametric amplifier (JPA) in Non-Patent Document 2 is the same oscillator as the Josephson parametric oscillator, and therefore will be referred to as a Josephson parametric oscillator (JPO) in this specification. A microwave drive signal (capacitor C x and C y The four-body coupling between the JPOs is activated by the unbalanced shunt-type JRM. The adjustable four-body interaction is realized by using an unbalanced shunt-type JRM. The JRM consists of two sets of two Josephson junctions (JJs) connected in series in parallel between the first node, which is the junction of JPO1 and JPO2, and the second node, which is the junction of JPO3 and JPO4. The first and second nodes are connected by a capacitor C x A microwave drive signal is applied via the junctions, and a capacitor C is connected to each of the two parallel-connected Josephson junctions (JJs). y via capacitor C x In the JRM, a microwave drive signal is applied that is in opposite phase to the microwave drive signal applied to the ex and Φ ex is the external magnetic flux applied to the large and small loops of the JRM. i (i=1~4) is the mode operator of each JPO.

[0012] 8A, 8B, and 8C are based on a, b, and c of Figure 4 in Non-Patent Document 2. As mentioned above, JPA in Figure 4 in Non-Patent Document 2 is written as JPO. r, i This system consists of four JPOs (i=1,2,3,4) interacting with each other through a single Josephson junction (JJ). To achieve time-dependent two-photon driving, the SQUID loop of each JPO is driven by a flux pump with adjustable amplitude and frequency. The frequency of the pump signal, ω, p,k (t) (k=1,2,3,4) is twice the frequency of the resonator: 2ω r,i The local four-body coupling is realized by the nonlinear inductance of the central Josephson junction (JJ).

[0013] The pump signal frequencies of the four JPOs are ω p,1 (t)+ω p,2 (t)=ω p,3 (t)+ω p,4 (t) …(1) and detune the resonator. If they are detuned from each other, the central Josephson junction JJ is -C(a1†a2†a3a4+hc) …(2) This four-body interaction is always active, and its strength C depends on the nonlinearity of the Josephson junction JJ and the detuning of the resonator consisting of the JPO and the Josephson junction. i is the operator of the resonant mode of each JPO (a i † is the creation operator, a i is the annihilation operator) (the hat symbol ^ of the operator is omitted). hc represents the Hermitian conjugate.

[0014] The group of four JPOs in Figure 8A (referred to as a plaquette in Non-Patent Document 2) is the central component of the architecture and can be expanded pyramid-like into a square lattice, which is required for implementing the LHZ (Lechner, Hauke, Zoller) scheme. Within each plaquette, the pump signals applied to the JPOs have different frequencies, but only four different pump signal frequencies are required for the entire lattice. In the LHZ scheme, the JPOs in the plaquette can be four-body interacted with each other via JRM or a single Josephson junction, thereby representing a fully coupled system in which all logical spins are coupled in two bodies. In the LHZ scheme, the four-body interactions within each plaquette are desirably configurable depending on the problem to be solved, since they provide conditions (constraints) that each JPO must satisfy to represent the fully coupled logical bits. As an example, Figure 8B shows all possible coupling combinations for the fully coupled N = 5 logical spins.

[0015] The energy of the Ising model for N=5 is E=-Σ<i=1,N> h i σ i -Σ<i≠ j=1,N> J ij σ i σ j …(3) where: J ij is a two-body interaction parameter, h i is the unit parameter (local magnetic field), σ j (i=1~5) is the spin, which takes the value 1 or 0.

[0016] As an example, Figure 8C shows the case where all connections of N = 5 logical spins are realized by the LHZ scheme using plaquette. In Figure 8C, the three bits at the bottom of the LHZ triangle structure (denoted as "Fixed") are fixed in the up state. [Prior art documents] [Patent documents]

[0017] [Patent Document 1] Japanese Patent Application Publication No. 2018-180084 [Non-patent literature]

[0018] [Non-Patent Document 1] H. Goto, et. al., "Boltzmann sampling from the Ising model using quantum heating of coupled nonlinear oscillators", Nature, Sci. Rep. 8, 7154 (2018) [Non-patent document 2] Puri, et. al., "Quantum annealing with all-to-all connected nonlinear oscillators", Nature Communications 8, 15785 (2017) Summary of the Invention [Problem to be solved by the invention]

[0019] In related technologies such as Non-Patent Document 2, when realizing four-body interactions with adjustable coupling strength, an additional drive signal is required, and a large number of electronic components, circuits, etc. are required for operation. This poses a problem in that it is difficult to realize a superconducting quantum circuit with high scalability in terms of size, etc.

[0020] Therefore, an object of the present disclosure is to provide a superconducting quantum circuit device and a method for controlling a superconducting quantum circuit that solves the above problems. [Means for solving the problem]

[0021] According to one embodiment of the present disclosure, there is provided a superconducting quantum circuit device comprising: a SQUID (superconducting quantum interference device) in which a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction are connected in a loop; two or four Josephson parametric oscillators, each including a line that supplies a magnetic flux interlinked with the SQUID loop and that parametrically oscillates in accordance with a pump signal supplied to the line; a coupler that couples the two or four Josephson parametric oscillators; and means for varying the relative phase between the pump signals supplied to the lines of the two or four Josephson parametric oscillators for parametric oscillation, thereby varying the strength of a two-body or four-body interaction.

[0022] According to another embodiment of the control method of the present disclosure, there is provided a control method for a superconducting quantum circuit comprising: a SQUID (superconducting quantum interference device) in which a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction are connected in a loop; two or four Josephson parametric oscillators each including a line supplying a magnetic flux interlinking with the SQUID loop and performing parametric oscillation in accordance with a pump signal supplied to the line; and a coupler coupling the two or four Josephson parametric oscillators, The resonant frequency of each Josephson parametric oscillator is adjusted so that the resonant frequency of the central part including the coupler is not changed, and the strength of the two-body or four-body interaction is adjusted by adjusting the relative phase of the pump signal for parametric oscillation of the Josephson parametric oscillator. [Effects of the Invention]

[0023] According to the present disclosure, it is possible to realize a superconducting quantum circuit that is highly scalable in terms of size and the like and has adjustable coupling strength. [Brief explanation of the drawings]

[0024] [Figure 1] FIG. 1 is a diagram illustrating an embodiment. [Figure 2A] FIG. 1 is a diagram schematically illustrating an example of a Josephson parametric oscillator according to an embodiment. [Figure 2B] FIG. 1 is a diagram schematically illustrating an example of a Josephson parametric oscillator according to an embodiment. [Figure 2C] FIG. 10 is a diagram schematically illustrating another example of a Josephson parametric oscillator according to an embodiment. [Figure 2D] FIG. 10 is a diagram schematically illustrating another example of a Josephson parametric oscillator according to an embodiment. [Figure 2E] FIG. 10 is a diagram schematically illustrating yet another example of a Josephson parametric oscillator according to an embodiment. [Figure 2F] FIG. 10 is a diagram schematically illustrating yet another example of a Josephson parametric oscillator according to an embodiment. [Figure 3A] FIG. 1 is a diagram illustrating an example of an embodiment. [Figure 3B] FIG. 1 is a diagram illustrating an example of an embodiment. [Figure 3C] FIG. 1 is a diagram illustrating an example of an embodiment. [Figure 4A] FIG. 1 is a diagram illustrating an example of an embodiment. [Figure 4B] FIG. 10 is a diagram schematically illustrating an example of a modified example of the embodiment. [Figure 5A] FIG. 10 is a diagram schematically illustrating another example of a modified example of the embodiment. [Figure 5B] FIG. 10 is a diagram schematically illustrating another example of a modified example of the embodiment. [Figure 6] FIG. 1 is a diagram illustrating an example of a network configuration according to an embodiment. [Figure 7] FIG. 1 is a diagram illustrating an example of a related art. [Figure 8A] FIG. 1 is a diagram illustrating an example of a related art. [Figure 8B] FIG. 1 is a diagram illustrating an example of a related art. [Figure 8C]FIG. 1 is a diagram illustrating an example of a related art. DETAILED DESCRIPTION OF THE INVENTION

[0025] Several embodiments are described below, which utilize the fact that two-body and four-body interactions between Josephson parametric oscillators (JPOs) depend on the relative phases of the pump signals applied to these oscillators, and enable the effective coupling strength to be variably set (adjusted) by adjusting the relative phases of the pump signals.

[0026] FIG. 1 illustrates one embodiment. It shows a first JPO 10, a second JPO 20, and a coupler 30 that couples them. In FIG. 1, the first and second JPOs are designated JPO1 and JPO2. In this embodiment, the effective coupling strength can be variably adjusted by adjusting the relative phases θ1 and θ2 (e.g., θ1-θ2) of pump signals (whose frequency ωp is approximately twice the resonant frequency of the JPOs) applied to the two Josephson parametric oscillators (JPOs) 10 and 20.

[0027] Here, the JPO structure is typically of the following two types:

[0028] (1) One end of the SQUID (a λ / 4 type resonator in the case of a distributed constant) is connected to the superconducting part capacitively coupled to the ground plane, and the other end of the SQUID is grounded.

[0029] (2) A superconducting part (in the case of a distributed constant, a λ / 2 type resonator) capacitively coupled to the ground plane is separated by a SQUID into a first superconducting part (in the case of a distributed constant, a λ / 4 type resonator) and a second superconducting part (in the case of a distributed constant, a λ / 4 type resonator), and one end of the SQUID is connected to the first superconducting part, and the other end is connected to the second superconducting part.

[0030] Below, we will outline examples of JPO configurations. Because the 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. 2A illustrates an example of a lumped-element structure (1) for the first JPO 10 in FIG. 1. Referring to FIG. 2A, the SQUID 11 includes a first superconducting line 14, a first Josephson junction 12, a second superconducting line 15, and a second Josephson junction 13 connected in a loop. The second superconducting line 15 is connected to ground, and the first superconducting line 14 is connected to a superconducting portion (electrode) designated by node 16. FIG. 2B is a schematic plan view showing an example of the planar shape of the superconducting portion (electrode) designated by node 16 in FIG. 2A. In FIG. 2A, the inductance L between node 16 and first superconducting line 14 mainly represents the inductance component of the superconducting section (electrode). However, since it is small compared to the inductance (self-inductance) of SQUID 11, it may be omitted from the circuit diagram. Capacitance C1 between node 16 and ground represents the capacitance (capacitance component) between the superconducting section (electrode) (16 in FIG. 2B) and ground. Capacitance C1 forms a parallel LC resonant circuit with the self-inductance of SQUID 11, etc. Line (pump line) 17 in FIG. 2A is a line through which a current (pump signal) is supplied from a current control unit (not shown) to supply a magnetic flux (magnetic field) interlinked with the loop of SQUID 11. In FIG. 2A, the superconducting section (electrode) at node 16 is connected to coupling 18 with the readout circuit (not shown) via input / output capacitor Cin and to coupling 19 with the coupler (30 in FIG. 1) via capacitor C2. The same configuration may be used for the JPO 20 in Fig. 1. It should be noted that in Fig. 2B, the planar shape of the electrodes is not limited to a cross shape.

[0031] Figure 2C illustrates an example of a lumped-element structure (2) for the first JPO 10 of Figure 1. Referring to Figure 2C, the first superconducting line 14 of the SQUID 11 is connected to a junction 18 with a readout circuit (not shown) via a first superconducting section (electrode) designated by node 16A and an input / output capacitor CinA. The second superconducting line 15 of the SQUID 11 is connected to a junction 19 with a coupler (30 in Figure 1) via a second superconducting section (electrode) designated by node 16B and an input / output capacitor CinB. Figure 2D is a schematic plan view showing an example of the planar shape of the superconducting sections (electrodes) designated by nodes 16A and 16B in Figure 2C. The JPO 20 of Figure 1 may have the same configuration.

[0032] FIG. 2E illustrates an example of a distributed parameter type structure (1) for the JPO 10 of FIG. 1. In FIG. 2E, a quarter-wavelength (λ / 4) resonator 21 is shown as a distributed parameter circuit formed by cascading four-terminal circuits with inductance L and capacitance C. The first superconducting line 14 of the SQUID 11 is connected to one end 21-1 of the λ / 4 resonator 21, and the other end, node 21-2, of the λ / 4 resonator 21 is connected to a coupling 18 with a readout circuit (not shown) via an input / output capacitor Cin and to a coupling 19 with a coupler (30 in FIG. 1) via a capacitor C2. The JPO 20 of FIG. 1 may have the same configuration.

[0033] Figure 2F illustrates an example of a distributed constant type structure (2) for the JPO 10 of Figure 1. In Figure 2F, the first superconducting line 14 of the SQUID 11 is connected to one end 21A-1 of the first λ / 4 resonator 21A, and the node 21A-2 at the other end of the λ / 4 resonator 21A is connected to a coupling 18 with a readout circuit (not shown) via an input / output capacitor CinA. The second superconducting line 15 of the SQUID 11 is connected to one end 21B-1 of the second λ / 4 resonator 21B, and the node 21B-2 at the other end of the second λ / 4 resonator 21B is connected to a coupling 19 with a coupler (30 in Figure 1) via an input / output capacitor CinB. The JPO 20 of Figure 1 may have the same configuration.

[0034] The following description will be based on the example shown in Figure 2F, using the JPOs (first and second JPOs 10 and 20) of Figure 1. Figure 3A is a diagram illustrating this embodiment. In Figure 3A, the JPOs 10 and 20 of Figure 1 are denoted by reference numerals 110 and 120, and the coupler 30 is designated by a capacitor 131.

[0035] 3A, first and second Josephson junctions (JPOs) 110 and 120 are coupled by a capacitor 131. The first JPO 110 includes a SQUID 111 in which a superconducting line 111-1, a first Josephson junction (JJ), a superconducting line 111-2, and a second Josephson junction (JJ) are connected in a loop, coplanar waveguides (CPW) 112 and 113 are connected to the superconducting lines 111-1 and 111-2 of the SQUID 111, respectively, and a pump line 114 coupled to the SQUID 111 via mutual inductance. The second JPO 120 includes a second SQUID 121 in which a superconducting line 121-1, a first Josephson junction (JJ), a superconducting line 121-2, and a second Josephson junction (JJ) are connected in a loop, waveguides 122 and 123 are connected to the superconducting lines 121-1 and 121-2 of the SQUID 121, respectively, and a pump line 124 coupled to the SQUID 121 via mutual inductance. In the first JPO 110 (second JPO 120), the waveguides 112 and 113 (122 and 123) may be, for example, λ / 4 (quarter wavelength) coplanar waveguides.

[0036] A signal of frequency ω0 is input to the first and second JPOs 110 and 120, and a signal of Φ is input to the SQUIDs 111 and 121. dc The resonance frequency when a static magnetic field of ω is applied is defined as ω0. The pump lines 114 and 124 of the first and second JPOs 110 and 120 are connected to a frequency ω p By applying a pump signal (microwave) of sufficiently high intensity, the first and second JPOs 110 and 120 undergo parametric oscillation. The resonant frequencies of the first and second JPOs 110 and 120 are set to ω1 and ω2, respectively. The first and second JPOs 110 and 120 are capacitively coupled by a capacitor 131, and a frequency ω is supplied from the pump lines 114 and 124.p (ω p When driven by a pump signal (microwave current) of ω1, ω2, the Hamiltonian H (quantized Hamiltonian written using the rotating wave approximation) is given by the following equation (4). Note that the quantized Hamiltonian is generally written as ^H, but the hat ^ is removed. In the following, all Hamiltonians are quantized Hamiltonians.

[0037] H / hbar = ω1a1†a1+ ω2a2†a2 -(K1 / 2) a1† 2 a1 2 -(K2 / 2) a2† 2 a2 2 +(p1 / 2)[exp{-i(ω p *t-θ1)}*a1† 2 + exp{i(ω p *t-θ1)}*a1 2 ] +(p2 / 2)[(exp{-i(ω p *t-θ2)}*a2† 2 + exp{i(ω p *t-θ2)}*a2 2 )}] - g (a1†a2+ a2†a1) …(4)

[0038] In equation (4), hbar is the reduced Planck constant (=h / (2π): h is the Planck constant), ω1 and ω2 are the mode frequencies of the first JPO 110 and the second JPO 120. a i †, a i (i=1, 2) are the creation and annihilation operators of the resonance modes of the first JPO 110 and the second JPO 120, respectively, and a i † is a i is the Hermitian conjugate of a i †, a i The following exchange relationship holds between (i=1,2). [a i , a j†]=a i a j †-a j †a i =δ ij (δ ij is 1 when i=j and 0 when i≠j) [a i , a i ]=[a i †, a i †]=0 …(5) In addition, the creation and annihilation operators a i †, a i In quantum field theory, it is written as ^a with a hat ^. i †, ^a i However, this is omitted in this specification.

[0039] In equation (4), K1 and K2 are Kerr coefficients representing the amplitude of the Kerr nonlinearity in the first JPO 110 and the second JPO 120, respectively; p1 and p2 are the pump amplitudes of the parametric amplification of the first JPO 110 and the second JPO 120. ω p is the frequency of the pump signal provided for parametric amplification from pump lines 114, 124, θ1, θ2 are the phases of the pump signals provided for parametric amplification from the pump lines 114, 124; g is the coupling constant.

[0040] Here, the coupling constant g between the first JPO 110 and the second JPO 120 is a ferromagnetic coupling with a nearly constant strength.

[0041] For the above equation (5), by unitary transformation, a i →exp{-i(ω p *t-θ i ) / 2}ai (i=1,2) …(6) and describing it in a rotating coordinate system that rotates at ωp / 2, leaving only terms that do not oscillate in time, the Hamiltonian of the above equation (5) becomes the following equation (7).

[0042] H / hbar =Δ1a1†a1+ Δ2a2†a2 - (K1 / 2) a1† 2 a1 2 -(K2 / 2) a2† 2 a2 2 +(p1 / 2) (a1† 2 + a1 2 ) +(p2 / 2) (a2† 2 +a2 2 ) -g[exp{i(θ2-θ1) / 2} a1†a2+ exp{-i(θ2-θ1) / 2} a2†a1] …(7)

[0043] In equation (7), Δ1=ω1-ω p / 2 …(8a) Δ2=ω2-ω p / 2 …(8b)

[0044] In equation (7), a i †a i The coefficient of (i=1,2) is Δ i This is because the rotating coordinate system (ω p / 2), the oscillation frequency of the electromagnetic field is Δ i =ω i -ωp / 2. Also, by equation (6), ai→exp(-iω p t)ai is replaced by H0=ω i ai†ai, …(9a) H1=H - H0…(9b) This is equivalent to taking the interaction picture as follows. In this case, (ω p / 2) a i † a i is considered to be included in the unperturbed part of the Hamiltonian.

[0045] The relative phase θ of the pump signals in the first JPO 110 and the second JPO 120 p Changing (=θ2-θ1) changes the relative phase of the oscillations in the JPO by θ pThis is equivalent to rotating it 1 / 2.

[0046] On the right side of the equation (7), the terms related to the oscillation of each JPO (the first six terms) do not depend on the relative phase θp, but the last term, the two-body interaction term g[exp{i(θ2-θ1) / 2} a1†a2+ exp{-i(θ2-θ1) / 2} a2†a1] …(10) is the relative phase θ p Its real part depends on cos(θ p / 2).

[0047] Therefore, it is possible to adjust the magnitude / sign of the effective two-body interaction by adjusting the relative phase θp of the pump signals in the first and second JPOs 110 and 120. Note that when θp / 2=180 deg, this corresponds to reversing the sign of the Ising spin, and effectively reversing the ferromagnetic interaction to an antiferromagnetic interaction.

[0048] The relative phase θp of the pump signals in the first JPO 110 and the second JPO 120 can be easily adjusted. For example, a signal generated by a signal source (signal generator) 201 in FIG. 3B is split into two by a splitter 202 input to port 1, and the outputs from ports 2 and 3 are phase-shifted by phase shifters 203 and 204, respectively. The output signal from port 2 (phase: θ1) is supplied to pump line 114, and the output signal from port 3 (phase: θ1) is delayed by the relative phase θp with respect to the output signal from port 2 (phase: θ2 = θ1 + θ2). p ) may be supplied to pump line 124.

[0049] 3B, phase shifters 203 and 204 may be configured with delay lines whose delay is variable according to a control signal (not shown). In this case, phase shifters 203 and 204 may each include a plurality of delay lines with different delay times, and one of the delay lines may be selected by a switch (selector) according to a control signal and inserted into the respective signal transmission paths. Note that in FIG. 3B, only one of phase shifters 203 and 204 may be used, as long as it is possible to set the relative phase θp in the microwave signals supplied to pump lines 114 and 124.

[0050] In FIG. 3B, a Wilkinson divider (power divider) is exemplified as divider 202. The two separated signals are output to output ports 2 and 3 via parallel-connected λ / 4 (quarter wavelength) transmission lines (λ / 4 transformers). A resistor (R=2×Zo: Zo is the characteristic impedance of the transmission line on the input port 1 side) between ports 2 and 3 enables impedance matching and insulation to be maintained at the output ports (the characteristic impedance Z of each λ / 4 transmission line is √2Zo). Ports 2 and 3 contain signals with the same amplitude and phase, so no current flows through the resistor between ports 2 and 3. Note that, although a Wilkinson power divider is exemplified as divider 202, the divider is not limited to this and may be a resistive divider or the like.

[0051] 3C is a diagram illustrating another configuration for easily adjusting the relative phase θp of the pump signals in the first JPO 110 and the second JPO 120. For example, in the signal source that supplies the pump signal to the first JPO 110, the in-phase component (I(t)) of the intermediate frequency signal (IF) and the local oscillator signal (cos(ω LO t+θ0) (ω LO is the angular frequency of the local oscillator signal, and θ is the initial phase), a quadrature-phase component: Q(t) and a signal obtained by shifting the phase of the local oscillator signal supplied to the mixer 211 by π / 2 (90 degrees) using a π / 2 phase shifter 213: -sin(ω LOThe JPO 110 includes a mixer 212 that inputs an RF output signal (I) from mixer 211 and an RF output signal (Q) from mixer 212, and an adder 214 that adds the RF output signal (I) from mixer 211 and the RF output signal (Q) from mixer 212, and the output signal from adder 214 is supplied to pump line 114. The local oscillator 210 may variably set the relative phase θp of the pump signal from the first JPO 110 to the pump signal from the second JPO 120 by adjusting the initial phase (e.g., the above-mentioned θ0). Alternatively, in phase shift keying of the IF signal, the relative phase θp of the pump signal from the first JPO 110 to the pump signal from the second JPO 120 may variably set by adjusting the initial phase.

[0052] In FIG. 3C, the IF signal I(t) input to the mixer 211 is expressed as cos(ω IF t)(ω IF is the angular frequency of the IF signal, and the amplitude is 1), then the RF (radio frequency) output of mixer 211 is cos(ω IF t)×cos(ω LO t+θ0)=(1 / 2)[cos{(ω IF +ω LO )t+θ0} + cos{(ω IF - ω LO )t-θ0)}} …(11) In addition, the IF signal Q(t) input to the mixer 212 is expressed as sin(ω IF t), the RF output of mixer 212 is sins(ω IF t)×{-sin(ω LO t+θ0)}=-(1 / 2)[cos{(ω IF -ω LO )t-θ0} - cos{(ω IF + ω LO )t+θ0)}} …(12) Therefore, the output of the adder 214 is cos(ω IF t)×cos(ω LO t+θ0)+sins(ω IF t)×{-sin(ω LOt+θ0)} =(1 / 2)[cos{(ω IF +ω LO )t+θ0} + cos{(ω IF -ω LO )t-θ0)}} -(1 / 2)[cos{(ω IF -ω LO )t-θ0} - cos{(ω IF +ω LO )t+θ0)}} =cos{(ω IF +ω LO )t+θ0} …(13) That is, at the output of the adder 214, the lower sideband (frequency: ω IF -ω c ) are cancelled out, and the upper sideband (frequency: ω IF +ω LO =ω p ) is output to the pump line 114. In addition to the microwave, a DC (direct current) component is also simultaneously applied to the pump line 114. This DC component may be added to the microwave not on the microwave transmission line but inside the refrigerator accommodating the superconducting quantum circuit (the pump line 114 may be configured to include, for example, a DC bias line inductively coupled to the SQUID of the JPO and a pump signal (microwave signal) port inductively coupled to the SQUID). Note that the pump signal supplied to the pump line 114 may be an amplitude-modulated signal instead of the frequency-modulated signal described above.

[0053] According to this embodiment, there is an advantage that the coupling strength can be adjusted with a simpler configuration than in a configuration using a coupler with adjustable coupling strength.

[0054] 4A illustrates another embodiment of the present invention. In FIG. 4A, each of the first JPO 110, the second JPO 120, the third JPO 130, and the fourth JPO 140 includes a SQUID, first and second waveguides, and a pump line for supplying magnetic flux linking to the SQUID, as in FIG. 1 (the same applies hereinafter).

[0055] The first JPO 110 and the second JPO 120 are connected to a node 155 via capacitors 151 and 152 (AC (Alternate Current) coupling), the third JPO 130 and the fourth JPO 140 are connected to a node 156 via capacitors 153 and 154 (AC coupling), and the nodes 155 and 156 are connected via a Josephson junction 160. In the first JPO 110, the second JPO 120, the third JPO 130, and the fourth JPO 140, a frequency ω p,1 , ω p,2 , ω p,3 , ω p,4 , phase θ p,1 , θ p, 2, θ p,3 , θ p,4 A pump signal of

[0056] The Hamiltonian (quantized Hamiltonian) of the circuit in Fig. 4A is the Hamiltonian H JPO,k and the Hamiltonian Hc of the interaction,

[0057] H=Σ k=1 4 H JPO,k +Hc …(14)

[0058] The Hamiltonian (quantized Hamiltonian) for each JPO is given below, where hbar is omitted.

[0059] 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(ωp,k(t)t / 2)}a k 2 ] …(15)

[0060] In equation (15), ak † , ak is the generation and annihilation operator of the kth JPO (k=1,2,3,4) oscillation mode, ω r,k is the resonant frequency of the kth JPO, K is the Kerr coefficient, which represents the amplitude of the Kerr nonlinearity of the JPO. ε p (t) is the amplitude of the parametric pump (two-photon pump), ω p,k (t) is the angular frequency of the parametric pump of the kth JPO.

[0061] The Hamiltonian Hc (quantized Hamiltonian) of the interaction is given by the following equation (16):

[0062] H c =ω c a c †a c + g1(a c †a1+a1†a c ) + g2(a c †a2+a2†a c ) - g3(a c †a3+a3†a c ) -g4(a c †a4+a4†a c ) -Ej{cos(Φ / Φ0)+(1 / 2)(Φ / Φ0) 2} …(16)

[0063] In equation (16), a c †, a c are the generation and annihilation operators of the central Josephson junction (coupling Josephson junction) 160 modes, g i (i=1, 2, 3, 4) represents the magnitude of coupling between the ith JPO and the mode of the central Josephson junction 160. Φ0=(h / 2π)(2e) is the magnetic flux quantum, EJ is the Josephson energy at the center, which is proportional to the critical current value of the Josephson junction 160.

[0064] Φ is Φ = Φ c (a c †+a c ) …(17) is given by Φ c is the zero-point oscillation of the magnetic flux in the center of the circuit.

[0065] In Figure 4A, the first to fourth JPOs 110-140 are nonlinear resonators that include SQUIDs as nonlinear inductors, similar to the first JPO 110 and second JPO 120 in Figure 1, and the central Josephson junction 160 is a nonlinear inductor, so it can be viewed as a nonlinear resonator. For this reason, the configuration shown in Figure 4A can also be viewed as a circuit in which five nonlinear resonators are connected. The four-body coupling section in the center of the circuit is sometimes referred to as a coupling resonator. Note that the resonant mode of the central Josephson junction 160 is detuned from the JPO and is not driven externally, so <ac>=<ac†ac> = 0. The four JPO110-140 in FIG. 4A are also called "plaquette" according to Non-Patent Document 2.

[0066] In the interaction of equation (17), ω p, 1+ω p, 2=ω p, 3+ω p, 4…(18) Under the condition that, for example, ω p, 1-ω p, 2…(19) If we assume that the vibration term due to the frequency difference of the JPO pump signals such as

[0067] H plaquette ≒Σ <k=1,4> {H JPA,k -(g k 2 / Δ k )a k †a k } -E j (φ c 4 / φ0 4 ){(g1g2g3g4) / (Δ1Δ2Δ3Δ4)}(a1†a2†a3a4+hc) -E j {(φ c 4 / φ0 4 ) / (Δ k 2 Δ m 2 )}Σ <k≠m=1, 4> (g k 2 g m 2 )(a k †a k a m †a m ) …(20)

[0068] In equation (20), the second term of the first term on the right-hand side (g k 2 / Δ k )a k †a is the frequency shift of the JPO mode due to off-resonant coupling with the central Josephson junction 160.

[0069] In equation (20), the second term on the right side is the four-body coupling between the four JPOs, and from the viewpoint of the circuit parameters, C=E j (φ c 4 / φ0 4 ){(g1g2g3g4) / (Δ1Δ2Δ3Δ4)} …(21) It is also written as:

[0070] In equation (20), the last term accounts for the cross-Kerr interaction between JPOs.

[0071] In equation (20), Δ k is the mode frequency ω of the kth JPO r,k and the mode frequency (resonant frequency) ω, which is determined by the capacitance and inductance of the central Josephson junction 160. c This is the difference between Δ k =ω c -ω r,k …(twenty two)

[0072] From this, in FIG. 4A, the magnitude of the four-body interaction can be changed by changing the resonant frequencies of the first to fourth Josephson junctions 110-140 or the mode frequency of the central Josephson junction 160.

[0073] In addition, the pump lines of the first to fourth JPOs 110-140 are provided with a frequency ω p,1 , ω p,2 , ω p,3 , ω p,4 , phase θ p,1 , θ p,2 , θ p,3 , θ p,4 Since a pump signal of is supplied, the second term on the right side of equation (20) becomes -E j (φ c 4 / φ0 4 ){(g1g2g3g4) / (Δ1Δ2Δ3Δ4)}[exp{-i(θ p,3 +θ p,4 -θ p,1 -θ p,2 ) / 2}a1†a2†a3a4+ hc] …(twenty three) Therefore, by adjusting the relative phase of at least one of the four JPOs, JPO110-JPO140, the effective coupling strength can be adjusted.

[0074] Fig. 4B shows a modified example of the configuration of Fig. 4A. A capacitor 161 is shunt-connected (connected in parallel) across the Josephson junction 160, which is a four-body interaction coupler, to realize a coupler that is resistant to charge noise.

[0075] In the circuits of Figures 4A and 4B, the Josephson junction 160 acts as an inductor. The magnitude of the inductance L J is calculated by using the critical current value Ic of the Josephson junction 160. L J =Φ0 / (2πI c ) …(twenty four) The critical current value Ic is determined by the Josephson junction (the material properties, area, and thickness of the two superconductors and the insulating film between them).

[0076] Figure 5A shows a configuration in which the Josephson junction 160 in the center of Figure 4A is replaced with a SQUID 170 in which a superconducting line 171, a first Josephson junction (JJ1), a superconducting line 172, and a second Josephson junction (JJ2) are connected in a loop. By using the SQUID 170 instead of the Josephson junction (160 in Figure 4A), the resonant frequency of the coupling can be changed. Therefore, by changing the detuning between the SQUID 170 and the four JPOs, the maximum and minimum values ​​of the four-body interaction, which is variable by the phases of the four pump signals, can be adjusted.

[0077] The magnetic flux passing through the SQUID loop is Φ ext When , the critical current value of the entire SQUID is I c eff teeth, I c eff =2I c |cos(πΦ ext / Φ0) | …(25) This becomes:

[0078] The SQUID is an inductor that can be varied by the magnetic flux passing through the loop. The magnetic flux linkage of the SQUID loop can be changed relatively easily by applying a current from the outside. From the above, by replacing the JJ in the center of the circuit with a SQUID, the mode ω c This allows Δ k and E J The value of changes. As a result, the magnitude of the four-body interaction changes. Note that if the Josephson junction in the center is replaced with a SQUID170, the mode frequency in the center may change due to unintended fluctuations in magnetic flux (magnetic flux noise), etc.

[0079] In Figure 5A, the four JPO110-140 have the following resonant frequencies: ω p,1 +ω p,2 =ω p,3 +ω p,4 …(26) When the four-body interaction condition is satisfied, they bond through a four-body interaction.

[0080] In this case, the four-body interaction term of the Hamiltonian is, as shown in equation (23) above, E j (φ c 4 / φ0 4 )}(g1g2g3g4) / (Δ1Δ2Δ3Δ4)[exp{-i(θ p,3 +θ p,4 -θ p,1 -θ p,2 ) / 2}a1†a2†a3a4+hc] (hc is Hermite Conjugate) …(27) It can be written as:

[0081] Consider the expectation value of the energy for this term. When one of the phases of the parametric pump signals (pump signals for parametric oscillation) supplied to the first to fourth JPOs 110-140 is changed, the following is obtained in equation (27): exp{-i(θ p,3 +θ p,4 -θ p,1 -θ p,2 ) / 2} …(28) The value of changes.

[0082] exp{-i(θ p,3 +θ p,4 -θ p,1 -θ p,2 ) / 2}, the maximum and minimum values ​​of the real part are +1 and -1.

[0083] Therefore, the range that can be changed only by changing the phase of the parametric pump signal supplied to each of the first to fourth JPOs 110-140 is as follows: ±E j (φ c 4 / φ0 4 )(g1g2g3g4) / (Δ1Δ2Δ3Δ4) …(29) This becomes:

[0084] On the other hand, the magnetic flux φ that passes through the loop of SQUID170 in the center of the circuit ext When changing the detuning Δ k and Josephson energy E J Since the value of changes, the above equation (21) C=E j (φ c 4 / φ0 4 ){(g1g2g3g4) / (Δ1Δ2Δ3Δ4)} The value of can be changed.

[0085] Therefore, the maximum and minimum values ​​of the variable four-body interaction can be adjusted by adjusting the phases of the parametric pump signals supplied to the first to fourth JPOs 110-140, respectively.

[0086] When the resonant frequency ωc is changed, not only the detuning Δk but also the Josephson energy E J also changes.

[0087] As described above, in the example of FIG. 5A, by replacing the Josephson junction 160 in FIG. 4A with the SQUID 170, the maximum and minimum values ​​of the four-body interaction ±E j (φ c 4 / φ0 4 )(g1g2g3g4) / (Δ1Δ2Δ3Δ4) …(30) can be varied.

[0088] The central resonance frequency ω of the four-body interaction c The dependence on the resonant frequency of the SQUID 170 is complex. For this reason, in actual experiments, for example, the resonant frequency ωc of the center portion in FIG. 3A is not changed, and C=E j (φ c 4 / φ0 4 The resonant frequencies ω of the four JPO110-140 are set so that )(g1g2g3g4) / (Δ1Δ2Δ3Δ4) is large compared to the magnitude of the required interaction. r,k After adjusting (k=1 to 4), fine adjustment may be made by adjusting the phase of the pump signal for parametric oscillation.

[0089] As described above, in this embodiment, the effective coupling strength can be adjusted by adjusting the relative phases of the pump signals for parametric oscillation of the four JPOs 110-140. p,1 , ω p,2 , ω p,3 , ω p,4 but ω p,1 +ω p,2 =ω p,3 +ω p,4 …(31) and the phase of the pump signal is θ p,3 +θ p,4 -θ p,1 -θ p,2 …(32) Adjust the value of

[0090] Therefore, in FIG. 5A, the effective coupling strength can be adjusted by adjusting the relative phase of one of the four JPOs 110-140. For example, θ p,1 , θ p,2 , θ p,4 is fixed, and the phase of the pump signal of JPO130 is p,3 and the pump signal θ of JPO110 p,1 By adjusting the relative phase of the four-body interaction terms,

[0091] Fig. 5B is a diagram showing a modified example of the configuration of Fig. 5A. A capacitor 173 is shunt-connected across both ends of SQUID 170, which is a four-body interaction coupler, to realize a coupler that is resistant to charge noise.

[0092] In yet another embodiment, when a JPO network is constructed using the two-body interactions described with reference to Fig. 3A, for example, the JPO network is connected by two-body interaction coupling sections to eliminate loops, as shown in Fig. 6. The positive and negative signs and magnitudes of all two-body interactions can be adjusted by adjusting the phase of the pump signals. However, if a loop (JPO2-JPO4-JPO5) exists in the network, the phase of some of the pump signals cannot be freely determined, limiting the adjustment range.

[0093] On the other hand, in the case of the LHZ (Lechner, Hauke, Zoller) method, which couples four-body interaction couplers in a planar manner as described with reference to Figures 4A, 5A, etc., as shown in Figure 8C, a loosely coupled graph does not have any loops that restrict the degrees of freedom, and any combination of coupling strengths can be realized without any restrictions.

[0094] In the two-body and four-body coupling between JPOs, the polarity (positive / negative) and magnitude of the coupling strength can be adjusted by adjusting (varying) the relative phase of the pump signals supplied to the JPOs for parametric oscillation.

[0095] The same effect as above can be obtained by using the lumped constant type JPO described with reference to FIGS. 2A to 2D instead of the distributed constant type JPO.

[0096] In the above embodiments, two-body / four-body coupling units (capacitors, Josephson junctions) that do not have the function of adjusting the coupling strength have been described as examples, but the present invention can also be applied to coupling units whose coupling strength is variably adjustable. For example, the technique of the present invention (adjusting the strength of the four-body interaction by the phase of the parametric pump) can be applied to the variable four-body coupling unit (JRM) described with reference to FIG.

[0097] In this case, to realize the four-body interaction, the oscillation frequency of JPO1, 2, 3, and 4 and the frequency ω of the drive signal input from the capacitors Cx and Cy are d For example, the combination of ω d =ω p,1 +ω p,2 +ω p,3 -ω p,4 …(33) If , the drive signal 2Φ Z (√n)cos (ω d t) …(34) For this, the Hamiltonian shown in the following equation (35) is derived.

[0098] H plaquette ≒Σ<k=1,4> (H JPA,k -{(g x k ) 2 / Δ x k }a k †a k )-C jrm (a1†a2†a3†a4+hc) …(35)

[0099] however, C jrm =E J (√n){φx 4 φz / (4φ0 5 )}(g1g2g3g4) / (Δ1Δ2Δ3Δ4) …(36)

[0100] The second term on the right side of equation (35) is the four-body interaction term. The effect of the phase of the pump signals supplied to JPO1-JPO4 on this second term is explicitly expressed as follows:

[0101] E J (√n){φx 4 φz / (4φ0 5 )}(g1g2g3g4) / (Δ1Δ2Δ3Δ4)[exp{-i(θ p,1 +θ p,2 +θ p,3 -θ p,4 ) / 2}](a1†a2†a3†a4+hc) …(37)

[0102] Therefore, in the circuit of Fig. 7, the effective coupling strength of the four JPOs is determined by the phase θ p,1 , θ p,2 , θ p,3 , θ p,4 In contrast, exp{-i(θ p,1 +θ p,2 +θ p,3 -θ p,4 ) / 2} …(38) That is, even in the configuration with the shunt-type JRM shown in Fig. 7 as the four-body interaction coupling part, the polarity (positive / negative) and magnitude of the four-body coupling strength can be adjusted by adjusting (varying) the relative phases of the pump signals supplied for the parametric oscillation of the four JPOs.

[0103] In a network of JPOs coupled in a planar fashion by four-body interaction couplings, the sign and magnitude of all four-body interactions can be adjusted by adjusting the phase of the pump signals supplied to the JPOs. For example, as described with reference to Figure 8C, a quantum annealer can be constructed using a network of JPOs.

[0104] The superconducting quantum circuits according to the above embodiments may be realized, for example, by lines (wiring) formed on a substrate using a superconductor. In this case, for example, silicon is used as the substrate, but other electronic materials such as sapphire or compound semiconductor materials (Group IV, Group III-V, Group II-VI) may also be used. Furthermore, the substrate is preferably single-crystal, but may also be polycrystalline or amorphous. The material of the superconducting lines may be, for example, niobium (Nb) or aluminum (Al), but is not limited thereto. Any metal that becomes superconducting when cooled to a cryogenic temperature may also be used, such as niobium nitride, indium (In), lead (Pb), tin (Sn), rhenium (Re), palladium (Pd), titanium (Ti), molybdenum (Mo), tantalum (Ta), tantalum nitride, or an alloy containing at least one of these. Furthermore, to achieve a superconducting state, the superconducting quantum circuit is used in a temperature environment of, for example, about 10 mK (millikelvin) achieved by a refrigerator.

[0105] The disclosures of Patent Document 1, Non-Patent Documents 1, and 2 are incorporated herein by reference. Modifications and adjustments of the embodiments and examples are possible within the scope of the entire disclosure of the present invention (including the scope of the claims), and further based on the basic technical ideas thereof. Furthermore, various combinations and selections of the various disclosed elements (including each element of each claim, each element of each example, each element of each drawing, etc.) are possible within the scope of the claims of the present invention. In other words, the present invention naturally includes various modifications and alterations that would be possible for a person skilled in the art to make in accordance with the entire disclosure, including the scope of the claims, and the technical ideas thereof.

[0106] <Addendum> For reference, the correspondence between the formula numbers in the specification and those in Supplementary Notes 6 and 8 (Notes 6 and 8) of Non-Patent Document 2 is shown below. TIFF0007767866000001.tif21144 [Explanation of symbols]

[0107] 10 JPO1 11 SQUID 12 The first Josephson junction 13 Second Josephson junction 14 The first superconducting line 15 Second superconducting line 16, 16A, 16B Nodes (electrodes) 17 Pump Line 18 Connection with readout circuit 19 Coupler connection 20 JPO2 21, 21A, 21B λ / 4 resonator 21-1, 21A-1, 21B-1 One end of λ / 4 resonator 21-2, 21A-2, 21B-2: other end nodes of λ / 4 resonators 30 Combiner 110 JPO1 111 SQUID 111-1, 111-2 Superconducting lines 112, 113 CPW 114 Pump Line 115 Input Capacitor 116 Circulator 120 JPO2 121 SQUID 121-1, 121-2 Superconducting lines 122, 123 CPW 124 Pump Line 125 Input Capacitor 126 Circulator 130 JPO3 131 Capacitor 140 JPO4 151-154 Capacitor 155 1st Node 156 Second Node 160 Josephson junction 161 Capacitor 170 SQUID 171, 172 Superconducting lines 173 Capacitor 201 Signal source 202 Distributor 203, 204 Phase shifter 210 Local Oscillator 211, 212 Mixer 213 π / 2 phase shifter 214 Adder< / ac>

Claims

1. two or four Josephson parametric oscillators, each including a SQUID (superconducting quantum interference device) in which a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction are connected in a loop, and a line that supplies a magnetic flux that interlinks with the SQUID loop, and that parametrically oscillates in accordance with a pump signal supplied to the line; a combiner for combining the two or four Josephson parametric oscillators; a means for varying the relative phase between pump signals supplied to the lines of the two or four Josephson parametric oscillators, respectively, for parametric oscillation, thereby varying the strength of two-body or four-body interaction; A superconducting quantum circuit device equipped with the above.

2. 2. The superconducting quantum circuit device according to claim 1, wherein the coupler that couples the two Josephson parametric oscillators comprises a capacitor.

3. 2. The superconducting quantum circuit device according to claim 1, wherein the coupler that couples the four Josephson parametric oscillators comprises a Josephson junction.

4. 2. The superconducting quantum circuit device according to claim 1, wherein the coupler that couples the four Josephson parametric oscillators comprises a SQUID.

5. 5. The superconducting quantum circuit device according to claim 3, wherein one end and the other end of the coupler are connected to a connection point between first and second Josephson parametric oscillators and a connection point between third and fourth Josephson parametric oscillators among the four Josephson parametric oscillators.

6. 6. The superconducting quantum circuit device according to claim 3, wherein a capacitor is shunt-connected to the coupler.

7. 7. The superconducting quantum circuit device according to claim 1, wherein the relative phase is varied by varying the phase of one of the pump signals of the two Josephson parametric oscillators, thereby varying the strength of the two-body interaction.

8. 7. The superconducting quantum circuit device according to claim 1, wherein the relative phase is varied by varying the phase of at least one pump signal among the four pump signals of the four Josephson parametric oscillators, thereby varying the strength of the four-body interaction.

9. the coupler includes a shunt-type ring modulator; The ring modulator comprises:

2. The superconducting quantum circuit device according to claim 1, wherein two sets of two Josephson junctions connected in series are connected in parallel between a first node which is a connection point of first and second Josephson parametric oscillators among the four Josephson parametric oscillators, and a second node which is a connection point of third and fourth Josephson parametric oscillators, and a first drive signal is applied to the first and second nodes via first and second capacitors, and a second drive signal having the same signal intensity but an opposite phase to the first drive signal is applied to third and fourth nodes which are connection points of the two sets of Josephson junctions connected in parallel, via third and fourth capacitors.

10. A method for controlling a superconducting quantum circuit comprising: a SQUID (superconducting quantum interference device) in which a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction are connected in a loop; two or four Josephson parametric oscillators each including a line for supplying a magnetic flux interlinked with the SQUID loop, the two or four Josephson parametric oscillators performing parametric oscillation in response to a pump signal supplied to the line; and a coupler for coupling the two or four Josephson parametric oscillators, a control method for a superconducting quantum circuit, characterized in that the resonant frequency of each of the Josephson parametric oscillators is adjusted so that the resonant frequency of the central part including the coupler is not changed, and the strength of the two-body or four-body interaction is adjusted by adjusting the relative phases of pump signals for parametric oscillation of the two or four Josephson parametric oscillators.

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