Bandwidth parametric amplifier, amplifier circuit

Abstract

1. A band-pass parametric amplifier comprising a plurality of unit cells, at least one of the unit cells including a first inductor having a first node coupled to a center conductor and a second node coupled to ground, a first capacitor having a first node coupled to the center conductor and a second node coupled to ground, and a second inductor having a first node coupled to the center conductor, wherein the second capacitor has a first node coupled to a second node of the second inductor, and the second capacitor and second inductor are in series with the center conductor.

Description

[Technical field]

[0001] The present disclosure relates generally to superconducting devices, and more particularly to amplifiers used for qubit readout. [Background technology]

[0002] Superconducting quantum computing is the implementation of quantum computers in superconducting electronic circuits. Quantum computing studies the application of quantum phenomena for information processing and communication. Various models of quantum computing exist, the most common model includes the concepts of qubits and quantum gates. A qubit is a generalization of a bit that has two possible states, but can be in a quantum superposition of both states. A quantum gate is a generalization of a logical gate, but a quantum gate describes the transformation that one or more qubits will undergo after the gate is applied to the qubits, given an initial state of the qubits.

[0003] Electromagnetic energy associated with a quantum bit can be stored in so-called Josephson junctions and in the capacitive and inductive elements used to form the quantum bit. As an example, to read out the quantum bit state, a microwave signal is applied to a microwave readout cavity that couples to the quantum bit at the cavity frequency. The transmitted (or reflected) microwave signal passes through multiple thermal isolation stages and low-noise amplifiers that are used to block or reduce noise and improve the signal-to-noise ratio. The amplitude and / or phase of the return / output microwave signal carries information about the quantum bit state, such as whether the quantum bit is in a ground state, an excited state, or a superposition of two states. The microwave signal that carries the quantum information about the quantum bit state is usually weak (e.g., on the order of a few microwave photons). To measure this weak signal, a low-noise Josephson amplifier can be used as a preamplifier (i.e., first amplification stage) at the output of the quantum system to boost the quantum signal and improve the signal-to-noise ratio of the output chain. In addition to Josephson amplifiers, several Josephson microwave components that use Josephson amplifiers or Josephson mixers, such as Josephson circulators, Josephson isolators, and Josephson mixers, may be used in scalable quantum processors.

[0004] One type of Josephson amplifier is the Josephson Traveling Wave Parametric Amplifier (JTWPA), a distributed-element Josephson-based parametric amplifier. The device is formed by a nonlinear transmission line, whose central conductor comprises a large array of Josephson junctions periodically shunted to ground by lumped-element capacitors. Because the device is not a resonant structure, the amplification band in a JTWPA is usually wide (e.g., 3 to 5 GHz). Such an amplification range may be too wide for qubit readout, which may be centered around 7 GHz. Known JTWPAs may amplify signals that are passing through, unrelated to the qubit readout, such as qubit pulses, which may cause instabilities. Furthermore, known JTWPAs amplify quantum noise over a wide bandwidth. Furthermore, the wide amplification band of known JTWPAs is very sensitive to impedance mismatch, which causes multiple reflections and ripples in their gain curves. To minimize the effects of impedance mismatch at the input and output ports of the JTWPA, wideband devices that are 50 ohm matched, such as magnetic-based isolators, are often added to the input and output of the JTWPA. The addition of such wideband magnetic-based devices at the input and output of the JTWPA limits the integrability between the JTWPA and other microwave devices that are significant to the operation of a quantum processor. It is with respect to these issues and others that the present application has been written. Summary of the Invention

[0005] According to various embodiments, a bandpass parametric amplifier circuit and method are provided. The bandpass parametric amplifier circuit includes a plurality of unit cells. At least one unit cell includes a first inductor having a first node coupled to a center conductor and a second node coupled to ground. There is a first capacitor having a first node coupled to the center conductor and a second node coupled to ground. There is a second inductor having a first node coupled to the center conductor. A second capacitor has a first node coupled to a second node of the second inductor. The second capacitor and the second inductor are in series with the center conductor.

[0006] In one embodiment, the first and second inductors are non-linear.

[0007] In one embodiment, the plurality of unit cells are part of a ladder circuit structure that produces a bandpass parametric amplifier circuit. The number of unit cells in the ladder circuit structure may be an odd number. Each of the plurality of unit cells may further include a resonant structure including a third capacitor having a first node coupled to the center conductor, a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground, and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground. In one embodiment, the plurality of numbers is between 3 and 9.

[0008] In one embodiment, a resonant structure within each of the plurality of unit cells defines a stopband within the gain curve of the bandpass parametric amplifier.

[0009] In one embodiment, a resonant structure within each of a plurality of unit cells provides phase matching between a pump drive and a propagating in-band microwave signal amplified by a band parametric amplifier.

[0010] In one embodiment, there is a third capacitor having a first node coupled to the center conductor, a third inductor having a first node coupled to the second node of the third capacitor and a second node coupled to ground, and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

[0011] In one embodiment, the bandpass filter has a characteristic impedance greater than 50 ohms.

[0012] In one embodiment, the bandpass parametric amplifier is part of a circulator circuit.

[0013] In one embodiment, each of the first and second inductors comprises an array of Josephson junctions.

[0014] In one embodiment, each Josephson junction comprises an aluminum (Al) or niobium (Nb) superconducting electrode.

[0015] In one embodiment, each of the first and second capacitors is constructed in a coplanar waveguide geometry on a low-loss dielectric substrate.

[0016] In one embodiment, the lumped element capacitors in the bandpass parametric amplifier are plate capacitors that use low-loss dielectric materials.

[0017] In one embodiment, the bandpass parametric amplifier is a directional amplifier.

[0018] According to one embodiment, a Josephson-based directional parametric amplifier includes a plurality of nonlinear dispersive bandpass filters arranged along a transmission line and a linear element implementing periodic resonant structure loading for driving the pump.

[0019] In one embodiment, each nonlinear dispersive bandpass filter includes a first inductor having a first node coupled to the center conductor and a second node coupled to ground. There is a first capacitor having a first node coupled to the center conductor and a second node coupled to ground. There is a second capacitor having a first node coupled to the center conductor. There is a second inductor having a first node coupled to a second node of the second capacitor. The second capacitor and the second inductor are in series with the center conductor.

[0020] In one embodiment, each periodic resonant structure includes a third capacitor having a first node coupled to the center conductor, a third inductor having a first node coupled to the second node of the third capacitor and a second node coupled to ground, and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

[0021] In one embodiment, a periodic resonant structure is added every 3 to 9 nonlinear unit cells of a lumped-element bandpass filter.

[0022] These and other features will become apparent from the following detailed description of illustrative embodiments thereof, which should be read in connection with the accompanying drawings.

[0023] The drawings are of exemplary embodiments. The drawings do not show all embodiments. In addition or in the alternative, other embodiments may be used. Details that may be obvious or unnecessary may be omitted to save space or for a more effective illustration. Some embodiments may be practiced with additional components or steps and / or without all components or steps shown. When the same numbers appear in different drawings, they refer to the same or similar components or steps. [Brief description of the drawings]

[0024] [Figure 1] FIG. 1 illustrates a known JTWPA circuit. [Diagram 2] FIG. 1 is a block diagram of a band Josephson traveling wave parametric amplifier system consistent with an example embodiment. [Diagram 3] FIG. 3 illustrates an example circuit of a nonlinear dispersive filtering medium used to implement the band Josephson-based directional parametric amplifier block of FIG. 2 consistent with an example embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0025] overview In the following detailed description, numerous specific details are set forth by way of example in order to provide a thorough understanding of the relevant teachings. However, it should be apparent that the present teachings may be practiced without such details. In other instances, well-known methods, procedures, components, or circuits, or combinations thereof, have been described at a relatively high level and without detail in order to avoid unnecessarily obscuring aspects of the present teachings.

[0026] The present disclosure relates generally to superconducting devices, and more particularly to efficient multiple readout of quantum bits. Electromagnetic energy associated with a quantum bit may be stored in so-called Josephson junctions and in the capacitive and inductive elements used to form the quantum bit. As an example, to read out the quantum bit state, a microwave signal is applied to a microwave readout cavity that couples to the quantum bit at the cavity frequency. The transmitted (or reflected) microwave signal passes through multiple thermal isolation stages and low noise amplifiers that are used to block or reduce noise and improve the signal-to-noise ratio. The amplitude and / or phase of the return / output microwave signal carries information about the quantum bit state, such as whether the quantum bit is in a ground state, an excited state, or a superposition of two states. The microwave signal that carries the quantum information about the quantum bit state is usually weak (e.g., on the order of a few microwave photons).

[0027] Readout of the qubit state is typically facilitated by a distributed-element Josephson-based parametric amplifier, or JTWPA. The device is formed by a nonlinear transmission line whose central conductor comprises an array of Josephson junctions periodically shunted to ground by lumped-element capacitors. Directional amplification of a weak microwave signal traveling through the device is achieved by injecting a strong microwave drive, sometimes called a pump, whose frequency is near the center of the desired amplification band and whose direction of propagation coincides with that of the input weak microwave signal.

[0028] By pumping a nonlinear Kerr medium (i.e., a Josephson junction array) with a strong coherent microwave drive, a four-wave mixing process is induced, resulting in parametric amplification of the propagating weak microwave signal. In other words, a dispersive nonlinear medium allows the amplification of a weak microwave signal via coherent energy exchange with the propagating strong drive.

[0029] To ensure phase matching between the pump drive and the detuned weak microwave signal propagating in the dispersive nonlinear medium, the central conductor is capacitively loaded every few unit cells by a resonant structure that resonates near the pump frequency. This resonant loading creates a band gap in the structure, thereby facilitating phase matching.

[0030] FIG. 1 shows a known JTWPA circuit 100. The JTWPA circuit is implemented as a nonlinear lumped element transmission line 102 coupled between an input 110 and an output 112. The JTWPA unit cell 120 has a critical current I0 and an intrinsic capacitance C J , and includes a Josephson junction with a capacitive shunt to ground C. For example, every third unit cell containing a Josephson junction and a shunt capacitor to ground C has a capacitance C r and inductance L t and the coupling capacitor C cThe present invention also includes a lumped element resonator having a coupling strength set by:

[0031] The JTWPA 102 of FIG. 1 may suffer from significant insertion loss (e.g., 4 dB or more). This loss mainly comes from the dielectric loss of the plate capacitors that are part of the device. As mentioned above, the amplification bandwidth of the JTWPA 102 may be wide (e.g., 3 to 5 GHz) due to the dispersive nature of traveling wave amplifiers and the lack of resonant structures for weak signals. Therefore, due to the wide amplification bandwidth, the JTWPA 102 amplifies quantum noise in undesired bands, e.g., at the qubit frequency. Some of the amplified noise in the entire band may propagate back to the quantum system due to impedance mismatch in the output line.

[0032] Another drawback of the JTWPA 102 is that small impedance mismatches within the JTWPA 102 or at the input port 110 or output port 112 can cause multiple reflections, resulting in ripples in the gain curve of the JTWPA 102. Furthermore, for a wide bandwidth, the impedance matching between the amplifier and the 50 ohm input and output circuitry necessitates the use of wideband cryogenic isolators (e.g., 4 to 12 GHz) at the input 110 and output 112. It is noted that while providing a well-defined 50 ohm environment over a wide bandwidth, such isolators protect the quantum system from noise returning from the JTWPA or high electron mobility transistor (HEMT). However, such isolators are typically bulky, expensive, lossy, and use strong magnetic fields, which can adversely affect JTWPA operation. Furthermore, such isolators are difficult to integrate on chip, thereby hindering scalability.

[0033] Thus, in one aspect, the teachings herein provide a Josephson-based directional parametric amplifier with a medium-range bandwidth (e.g., 500 MHz to 2000 MHz) centered around a frequency close to the qubit readout frequency (e.g., 7 GHz). Instead of the nonlinear transmission lines discussed above, a nonlinear dispersive bandpass filter architecture is implemented. A linear resonator resonating at the center of the band is capacitively loaded into the nonlinear dispersive bandpass filter. Such resonators are integrated every few unit cells (e.g., 3 to 9) to ensure phase matching between the pump drive and the propagating in-band weak microwave signal.

[0034] In one embodiment, to reduce the insertion loss of the bandpass filter (which is due to the dielectric loss of the plate capacitor), the teachings herein employ two strategies. First, the bandpass filter is constructed to have a relatively high characteristic impedance of more than 50 ohms (e.g., 80 ohms). Such an increase in characteristic impedance generally reduces the capacitance in the amplifier device. To ensure a proper impedance match between the characteristic impedance of the filter and the input and output 50 ohm environment, impedance matching networks are added to the input and output of the device (e.g., adiabatically tapered transmission line).

[0035] Second, the lumped element capacitance, which is part of the linear resonant circuit incorporated into the amplifier device for phase matching, is reduced by realizing a parallel LC circuit with a high, i.e., greater than 50 ohms, characteristic impedance (e.g., using a large inductance and smaller shunt capacitance to produce a characteristic impedance higher than 50 ohms while maintaining the same resonant frequency of the linear resonant circuit). As an example, such a high inductance can be achieved using a thin, narrow superconductor layer with a large mechanical inductance.

[0036] The bandpass parametric amplifiers discussed herein allow the directional amplifier to amplify weak microwave signals with very little additional noise (i.e., operate near the quantum limit). Furthermore, the directional amplifiers can be used to perform high-speed, high-fidelity quantum non-demolition (QND) measurements of qubits. Such near-quantum limited directional amplifiers with mid-range bandwidths provide various advantages. For example, such near-quantum limited directional amplifiers significantly relax the requirement of having 50 ohm matching environments on the input and output ports that match over a wide bandwidth. Such flexibility from tight matching environments allows the bandpass parametric amplifiers to be compatible with microwave devices that have similar mid-range bandwidths, such as circulators, isolators, passive filters, and the like. Furthermore, when amplifying the readout signal of a distributed coupled qubit, the mid-range bandwidth of the bandpass parametric amplifier ensures that only the qubit readout signal is amplified and not the qubit signal. Such focused readout reduces unwanted backaction on the qubits (due to small impedance mismatch) and improves device stability by rejecting out-of-band qubit pulses. Furthermore, the narrow bandwidth of the broadband parametric amplifier (compared to conventional JTWPAs) reduces ripple in the gain curve and may reduce the pump power required for device operation.

[0037] Exemplary Block Diagram FIG. 2 is a block diagram of a band Josephson traveling-wave parametric amplifier system 200, which is compatible with an exemplary embodiment. The system 200 includes a nonlinear dispersive filtering medium (NDFM) 206 coupled between external transmission lines 202 and 210. For example, the first external transmission line 202 can be an input, and the second external transmission line 210 can be an output. The external transmission lines 202 and 210 can each have a characteristic impedance Z0, which can be 50 ohms. In various embodiments, there can be at least one intermediate stage between the external transmission lines 202 and 210, represented by intermediate stages 204 and 208 in FIG. 2. In one embodiment, the intermediate stage is an impedance converter between Z1 and Z0. For example, such an intermediate impedance matching network / circuit is useful when the characteristic impedance of the nonlinear dispersive filtering medium at the center of the band Josephson traveling-wave parametric amplifier is Z1, which is different from Z0 (at the external port).

[0038] Example circuit implementation of NDFM Referring now to FIG. 3, FIG. 3 illustrates an example of a nonlinear dispersive filtering medium (NDFM) 300 used to implement the NDFM 260 block of FIG. 2 consistent with an exemplary embodiment. The NDFM 300 is a distributed element directional amplifier with a ladder of lumped element capacitors and inductors. The NDFM 300 includes a distributed nonlinear bandpass filter placed along the transmission line. There are several linear elements that implement a periodic resonant structure loading for the pump drive. An example of a periodic resonant structure is the capacitor C c 320, inductor L r 322, and capacitor C r A linear element unit cell 302 comprising 324 is shown. In one embodiment, a periodic resonant structure is added every 3 to 9 of the nonlinear filter elements to define narrow stopbands in the gain curve of the NDFM 300, sometimes referred to as the dispersion characteristic of the amplifier.

[0039] In one embodiment, a unit cell 302 of the NDFM 300 includes a first inductor 304 having a first node coupled to a center conductor 330 and a second node coupled to ground 340 (e.g., on an outer portion of the structure of the NDFM 300). There is a first capacitor 306 in parallel with the first inductor 304. In other words, the first capacitor 306 has a first node coupled to the center conductor 330 and a second node coupled to ground 340. There is a second capacitor 308 having a first node coupled to the center conductor 330. There is a second inductor 310 having a first node coupled to a second node of the second capacitor 308, the second capacitor and the second inductor being in series with the center conductor 330. A series of unit cells 302 results in a ladder circuit to produce the NDFM 300.

[0040] In one embodiment, the unit cell 302 of the NDFM 300 further includes a periodic resonant structure comprising a third capacitor 320 having a first node coupled to the center conductor 330. There is a third inductor 322 having a first node coupled to the second node of the third capacitor 320 and a second node coupled to ground 340. There is a fourth capacitor 324 having a first node coupled to the second node of the third capacitor 320 and a second node coupled to ground 340. As previously mentioned, in one embodiment, the resonant structure is repeated every 3 to 9 of the nonlinear filter elements.

[0041] In some embodiments, at least one of the first inductor 304 and the second inductor 310 is nonlinear, as indicated by the arrows through the inductor. For example, the inductors 304 and 310 may include an array of Josephson junctions. The number of unit cells 302, sometimes referred to herein as the bandpass filter order N, is odd (e.g., 101 to 1001). The filter order N is odd to keep the device symmetric with respect to its two external ports.

[0042] The insertion loss method may be used to specify the filtering characteristics of the NDFM 300. In various embodiments, each unit cell 302 of the NDFM 300 may be designed according to maximally flat (e.g., Butterworth response) or equiripple (Chebyshev polynomials). The fractional bandwidth of the NDFM 300 is given by Equation 1 below:

[0043]

number

[0044] In one embodiment, the center frequency of the band-pass parametric amplifier 300 (eg, the geometric mean of the cutoff angular frequencies ω2 and ω1) is given by Equation 2 below:

[0045]

number

[0046] Depending on the order of the NDFM 300 and the chosen ripple magnitude (i.e., the latter relates to an equiripple filter configuration), appropriate element values ​​corresponding to the normalized source impedance and normalized cutoff frequency can be calculated using known insertion loss methods for designing filters.

[0047] Calculated or tabulated values ​​(in the literature)g i , the characteristic impedance Z of the filtering medium, the central angular frequency ω, and the desired fractional bandwidth Δ, the values ​​of the lumped element inductors and capacitors may be calculated using the following equations:

[0048] For series elements,

[0049]

number

[0050]

number

[0051] For parallel elements,

[0052]

number

[0053]

number

[0054] The linear inductance of a Josephson junction (JJ) is given by Equation 7 below.

[0055]

number

[0056]

number

[0057] In one embodiment, a lossless nonlinear inductance

number

number

number

number

[0058] In various embodiments, the JJ may include aluminum (Al) or niobium (Nb). The linear inductor L r (e.g., 322) can be a thin, highly dynamic inductor. c ) or a third capacitor 324 (C r ) or both may be superconducting plate capacitors with low loss dielectric material. r , the second capacitor 320 (C c ), and a third capacitor 324 (C r ) collectively implement a periodic resonant structural loading for driving the pump. In one embodiment, such elements are added every 3 to 9 filter elements. These elements define the stop band in the gain curve, also known as the dispersion characteristic of the amplifier. The center frequency of the NDFM 300 is given by Equation 10 below:

[0059]

number

[0060] conclusion Although the description of various embodiments of the present teachings has been presented for illustrative purposes, the description is not exhaustive and is not intended to be limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used in this specification are selected to best explain the principles of the embodiments, practical applications, or technical improvements over the technology found in the market, or to enable those skilled in the art to understand the embodiments disclosed herein.

[0061] Although what is believed to be the best mode and / or other examples have been described above, it should be understood that various modifications may be made therein, that the subject matter disclosed herein may be implemented in various forms and examples, and that the teachings have application in numerous applications, only a few of which have been described herein. It is intended by the following claims to claim any and all applications, modifications, and variations that fall within the true scope of the present teachings.

[0062] The components, steps, features, objects, benefits, and advantages discussed herein are exemplary only. Any discussion of them shall not limit the scope of protection. Although various advantages have been discussed herein, it will be understood that not all embodiments necessarily include all advantages. Unless otherwise stated, all measurements, values, ratings, positions, dimensions, sizes, and other specifications described herein, including the following claims, are approximate and not precise. They shall have a reasonable range consistent with the function to which they pertain and with what is customary in the art to which they pertain.

[0063] Numerous other embodiments are contemplated. These include embodiments having fewer, additional, or different components, steps, features, objects, benefits, and advantages, or combinations thereof. These also include embodiments in which the components and / or steps are arranged and / or ordered differently. For example, any of the signals discussed herein may be scaled, buffered, scaled and buffered, converted to another state (e.g., voltage, current, charge, time, etc.), or converted to another state (e.g., HIGH to LOW, LOW to HIGH) without significantly altering the underlying control methodology.

[0064] Although described above with exemplary embodiments, it should be understood that the term "exemplary" is meant only as an example, and not as best or optimal. Except as immediately above, nothing described or illustrated is intended to, and should not be construed as, causing the public disclosure of any component, step, feature, object, benefit, advantage, or equivalent, whether or not claimed.

[0065] The terms and expressions used herein will be understood to have the ordinary meanings as accorded such terms and expressions with respect to the corresponding respective fields of investigation and study, unless a specific meaning is otherwise explained herein. Terms indicating relationships, such as first and second, are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any actual such relationship or order between such entities or operations. The terms "comprises", "comprising", or any other variations thereof, are intended to include a non-exclusive inclusion, such that a process, method, article, or apparatus that includes a list of elements may include other elements not expressly listed or other elements inherent to such process, method, article, or apparatus, rather than including only those elements. An element preceded by "a" or "an" does not, without further constraints, exclude the presence of additional identical elements within the process, method, article, or apparatus that includes the element.

[0066] An Abstract of the Disclosure is provided to enable the reader to quickly ascertain the nature of the technical disclosure. The Abstract is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. Moreover, in the above Detailed Description, it can be appreciated that various features are grouped together in various embodiments to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that the claimed embodiments require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed embodiment. Accordingly, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as separately claimed subject matter.

Claims

1. 1. A bandpass parametric amplifier circuit, comprising: A plurality of unit cells, at least one of the unit cells comprising: a first inductor having a first node coupled to the center conductor and a second node coupled to ground; a first capacitor having a first node coupled to the center conductor and a second node coupled to ground; a second inductor having a first node coupled to the center conductor; a second capacitor having a first node coupled to a second node of the second inductor, the second capacitor and the second inductor being in series with the center conductor; a third capacitor having a first node coupled to the center conductor; a third inductor having a first node coupled to the second node of the third capacitor and a second node coupled to ground; a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground; Equipped with a plurality of unit cells, the plurality of unit cells being part of a ladder circuit structure producing the bandpass parametric amplifier circuit, the plurality of unit cells including a resonant structure for each of a plurality of the unit cells, the resonant structure in each of the plurality of the unit cells providing phase matching between a pump drive and a propagating in-band microwave signal amplified by the bandpass parametric amplifier circuit; A bandpass parametric amplifier circuit comprising:

2. 2. The bandpass parametric amplifier circuit of claim 1, wherein said first and second inductors are non-linear.

3. 2. The bandpass parametric amplifier circuit of claim 1, wherein the number of unit cells in said ladder circuit structure is an odd number.

4. 2. The bandpass parametric amplifier circuit of claim 1, wherein said plurality of numbers is between three and nine.

5. 2. The bandpass parametric amplifier circuit of claim 1, wherein the resonant structure within each of the plurality of unit cells defines a stopband within a gain curve of the bandpass parametric amplifier circuit.

6. 2. The bandpass parametric amplifier circuit of claim 1, wherein the characteristic impedance of the resonant circuit of the bandpass parametric amplifier circuit is greater than 50 ohms.

7. 2. The bandpass parametric amplifier circuit of claim 1, wherein said bandpass parametric amplifier circuit is part of a circulator circuit.

8. 2. The bandpass parametric amplifier circuit of claim 1, wherein each of the first and second inductors comprises an array of Josephson junctions.

9. 9. The bandpass parametric amplifier circuit of claim 8, wherein each Josephson junction comprises an aluminum (Al) or niobium (Nb) superconducting electrode.

10. 2. The bandpass parametric amplifier circuit of claim 1, wherein each of the first and second capacitors is constructed in a coplanar waveguide geometry on a low-loss dielectric substrate.

11. 2. The bandpass parametric amplifier circuit of claim 1, wherein the bandpass parametric amplifier circuit is a directional amplifier.

12. 1. A method for computer-implemented bandpass parametric amplification, the method comprising: Providing a plurality of unit cells; For at least one of the unit cells, providing a first inductor having a first node coupled to a center conductor and a second node coupled to ground; providing a first capacitor having a first node coupled to the center conductor and a second node coupled to ground; providing a second inductor having a first node coupled to the center conductor; providing a second capacitor having a first node coupled to a second node of the second inductor, the second capacitor and the second inductor are in series with the central conductor; providing a third capacitor having a first node coupled to the center conductor; providing a third inductor having a first node coupled to the second node of the third capacitor and a second node coupled to ground; providing a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground; Including, the plurality of unit cells are further part of a ladder circuit structure producing a bandpass parametric amplifier, the ladder circuit structure including a resonant structure for each of a plurality of the unit cells, the resonant structure within each of the plurality of unit cells providing phase matching between a pump drive and a propagating in-band microwave signal amplified by the bandpass parametric amplifier; The method includes:

13. The method of claim 12 , wherein the first and second inductors are non-linear.

14. The method of claim 12 , wherein the number of unit cells in the ladder circuit structure is odd.

15. The method of claim 12 further comprising: defining a stopband in a gain curve with the resonant structure realized with the plurality of unit cells.

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