RSFQ all-digital programmable multi-tone generator for quantum applications

Abstract

The technology provides a digital multi-tone generator (DMTG), and a method of generating multitone signals, comprising a clocked single flux quantum circuit configured to generate a pulse train comprising a plurality of pulses with consecutive pulse peaks separated by a period corresponding to a period of the clock; and a clocked shift register coupled with the clocked single flux quantum circuit, the clocked shift register having an encoded bit pattern, and being configured to produce an output dependent on the encoded bit pattern and the period of the clock, such that the bit pattern modulates the pulse train with a plurality of frequency components. A DMTG may be fabricated using cryogenic superconducting ICs comprising Josephson junctions, configured to generate multiple microwave signals. The DMTG may be used for control of quantum computing systems and quantum sensors, including superconducting qubits and superconducting nanowire single-photon detectors (SNSPD).

Description

SeeQC-276.1 - 1 - RSFQ ALL-DIGITAL PROGRAMMABLE MULTI-TONE GENERATOR FOR QUANTUM APPLICATIONS CROSS REFERENCE TO RELATED APPLICATIONS

[0001] The present application is a non-provisional of, and claims benefit of priority under 35 U.S.C. § 119(e) from, U.S. Provisional Patent Application No.63 / 667,802, filed July 4, 2024, the entirety of which is expressly incorporated herein by reference. FIELD OF THE INVENTION

[0002] The present invention relates to the field of digital tone generators, and more particularly to a digital superconducting circuit for generating a multitone signal. INCORPORATION BY REFERENCE AND INTERPRETATION OF LANGUAGE

[0003] Citation or identification of any reference herein, in any section of this application, shall not be construed as an admission that such reference is necessarily available as prior art to the present application. The disclosures of each reference disclosed herein, whether U.S. or foreign patent literature, or non-patent literature, are hereby incorporated by reference in their entirety in this application, and shall be treated as if the entirety thereof forms a part of this application.

[0004] All cited or identified references are provided for their disclosure of technologies to enable practice of the present invention, to provide basis for claim language, and to make clear applicant’s possession of the invention with respect to the various aggregates, combinations, and subcombinations of the respective disclosures or portions thereof (within a particular reference or across multiple references). The citation of references is intended to be part of the disclosure of the invention, and not merely supplementary background information. The incorporation by reference does not extend to teachings which are inconsistent with the invention as expressly described herein (which may be treated as counter examples), and is evidence of a proper interpretation by persons of ordinary skill in the art of the terms, phrase and concepts discussed herein, without being limiting as the sole interpretation available.

[0005] The present specification is not to be interpreted by recourse to lay dictionaries in preference to field- specific dictionaries or usage. Where a conflict of interpretation exists, the hierarchy of resolution shall be the express specification, references cited for propositions, incorporated references, the inventors’ prior publications relating to the field, academic literature in the field, commercial literature in the field, field-specific dictionaries, lay literature in the field, general purpose dictionaries, and common understanding. Where the issue of interpretation of claim amendments arises, the hierarchy is modified to include arguments made during the prosecution and accepted without retained recourse.SeeQC-276.1 - 2 - BACKGROUND OF THE INVENTION

[0007] Superconducting integrated circuits based junctions provide the fastest and lowest- power electronic circuits in any technology, but require cooling to deep cryogenic temperatures. These include both analog and digital circuits with up to thousands of devices, and similar circuits are also being applied to quantum computing and quantum sensors. Several logic families have been developed for these superconducting digital circuits, which are often based on single-flux-quantum (SFQ) voltage pulses, with an integrated voltage of ^0 = h / 2e = 2.07 mV-ps. The most established SFQ logic family is known as Rapid Single Flux Quantum (RSFQ) [1], [2] electronics and their energy efficient versions (ERSFQ) [3], [4]. These and other logic families are well developed technologies that are now being considered and implemented for the control and readout of quantum circuits such as qubits [5]-[9], and quantum sensors

[0010] -

[0012] . As a result, many of the bulky standard room-temperature electronics are being replaced by RSFQ circuits that can perform the same quantum circuit manipulation, but with potentially better scalability from decreased system overheads, better integration for cryogenic temperatures due to the much lower power dissipation, and faster operation (up to hundreds of GHz

[0013] ). Naturally, this leads to proposed architectures where a full manipulation of Qubits and single photon detectors, such as Superconducting Nanowire Single Photon Detectors (SNSPDs) is done in a fully digital and, therefore, more scalable way

[0014] .

[0008] One system of interest in this area is an arbitrary waveform generator (AWG), which at its core consists of a Digital-to-Analog Converter (DAC), that can generate programmable arbitrary analog signals. This is the quintessential device used in a myriad of control protocols, including standard microwave based single qubit gates

[0015] or time-domain measurements of RF-SNSPDs

[0016] . In particular, AWGs are able, and widely used, to generate multi-tone signals for use in frequency division multiplexing (FDM) control architectures

[0017] ,

[0018] . The RSFQ implementation of such a device usually consists of a digital pulse-based stage which encodes the signal into a binary code that is then followed by a DAC stage, using voltage multipliers or SQUID stacks

[0019] -

[0022] . The only demonstration of multi-tone signal generation using SFQ- based circuits (not limited to RSFQ logic) has been recently achieved for metrology applications using the same architecture type mentioned before, with a DAC at the output

[0021] . A Digital Multi-Tone Generator (DMTG) is an alternative way of generating programmable multi-tone signals with all-digital pulse-based signals.

[0009] Like the previously mentioned implementations of AWG-type RSFQ circuits, Circular Shift Registers (CSR)

[0023] -

[0026] are at the core of this work too, since they can be used as pulse train generators and the type of pulse train is dependent on the initial pre-loaded data on the memory cells of the CSR. This property allows the creation of evenly and unevenly spaced patterns that can be controlled to generate the desired tones on the output signal. This is also an important property when trying to encode an analog signal into a digital pulse train. The second important component is a Comb Filtering stage

[0027] like those seen in Cascaded-Integrator-SeeQC-276.1 - 3 - Comb filters, both in classical and RSFQ applications, used for decimation and interpolation filter devices

[0028] ,

[0029] ,

[0040] ,

[0041] . This component is also seen in multiplier circuits

[0019] . These comb filters are implemented to redistribute the power of the CSR output to frequencies that depend on the characteristics of the delay path. This improves the power of the desired tones and attenuates spurious tones if correctly tuned. A feedforward implementation is chosen to comply with RSFQ device restrictions, such as summation and multiplication of single SFQ pulses.

[0010] While there are several technologies for implementing quantum computing, this work focuses on Josephson junction circuits cooled to temperatures T typically much less than 1 °K, which are essentially tunable high-Q electromagnetic LC resonators, with ^ = (LC)-0.5. When cooled down to very low temperatures kT<< ħω, these become quantum oscillators, with quantized energy levels separated by E = ħω. The frequency is typically f ~ 5-10 GHz, in the microwave regime, and transitions can be induced by single microwave photons. Today, microwave technology is ubiquitous across many different quantum platforms, enabling the precise control and readout of quantum states. See, Bardin, Joseph C., Daniel H. Slichter, and David J. Reilly. "Microwaves in quantum computing." IEEE journal of microwaves 1, no.1 (2021): 403-427.

[0011] The fundamental information carriers in a quantum computer are quantum bits, or qubits, in analogy to the logical bits used in a classical (non-quantum) computer. A qubit is composed of two quantum states |0> and |1>, the ground and excited states, with kT << E = ħω01, so that these transitions will be dominated by the |1>→ |0> process. If the qubit frequency can be shifted by an environmental parameter λ (e.g. magnetic field), fluctuations in λ cause fluctuations in ω01, dephasing the qubit state on the timescale ^ф.

[0012] The |0> and |1> states are used as computational basis states, analogous to the use of “0” and “1” in classical computing. Just as the electromagnetic field can be decomposed into a linear combination of orthogonal modes (such as plane waves, spherical harmonics, or guided modes), the instantaneous state of a qubit |ψ> can be written as a linear combination of the two energy eigenstates, with complex amplitudes α0and α1: |ψ>= α0|0>+ α1|1>

[0013] The customary normalization of |ψ> is that |α0|2+ |α1|2= 1. To meet this criterion, we introduce parameterization α0 = cos(θ / 2) and α1 = ejφsin(θ / 2), where φ and θ are real and j is the imaginary unit. For a single qubit, one is free to choose a global phase convention such that α0 is purely real, but in general both α0 and α1are complex.

[0014] The quantum-mechanical nature of the state |ψ> means that the qubit can be in both states |0> and |1> simultaneously—a phenomenon known as superposition—in contrast to the behavior of classical bits, which can only be in one state at a time. When the state of the qubit is measured, however, the qubit state is said to “collapse” to just one of its eigenstates. The collapse is probabilistic, with the state being measured to be |0> with probability P|0> = |α0|2, or |1> with probability P|1> = |α1|2; for this reason, α0 and α1 are called probability amplitudes. Because of the collapse process, the post-measurement qubit state no longer containsSeeQC-276.1 - 4 - information about α0and α1; it has collapsed to either |0> or |1>. The values of P|0>and P|1>can only be determined by many rounds of preparing the same qubit state |ψ> = |α0|0 + |α1|1 and measuring it, to build statistics on the measurement collapse probabilities |α0|2and |α1|2. The complex phase φ = arg(α1) - arg(α0) can be determined using a procedure known as state tomography.

[0015] Quantum computers require many qubits to perform useful computations. For N qubits, there are 2Nbasis states of the system, from |00 … 00> to |11…11>. While N classical bits can only be in one of the 2Nbasis states at a given time, the phenomenon of superposition means that the quantum state |ψ> of N qubits can be a linear combination of any, or even all, of the 2Nbasis states at the same time, with corresponding complex probability amplitudes {α00...00, α00...01, …, α11...11}. Since these amplitudes have a physical interpretation in terms of probabilities of measurement outcomes, they have the normalization condition^ 2−1k= 0|αk| 2 = 1 , where k indexes the 2N different bitstrings corresponding to the basis states.While it is possible to access a range of superposition states by putting each qubit in its own independent superposition state, such an approach can only be used to reach a small fraction of the basis states of the qubit state vector |ψ>, since most of the possible linear combinations exhibit correlations between the qubits. For example, consider the state |ψ>= 1 / √2 |00 … 00> + 1 / √2 |11 … 11>. It is not possible to write this state as a product of separate, individual states of the constituent qubits; the state of each qubit is inextricably correlated, or entangled, with all the others. The state |ψ> above is a superposition state, where the process of measurement will cause a collapse to just one basis state. This phenomenon of entanglement is a defining feature of quantum mechanics, and an essential ingredient for quantum computing.

[0017] The 2N-dimensional state space of N qubits can hold exponentially more information than that of N classical bits, offering the hope of greatly increased computing power. In order to realize a speedup over classical algorithms, quantum algorithms generate interference between the 2Ncomplex amplitudes to increase the likelihood of measuring certain output bitstrings (ones which yield the desired outcome of the computation). In effect, a quantum computer enables 2^N parallelism with N hardware bits, which is impossible to achieve with a classical computer.

[0018] Quantum algorithms create the desired interference between probability amplitudes by controlling the states of individual qubits, and generating entanglement between qubits, in the course of the algorithm. Experimentally, these tasks are usually carried out by microwave signals, or rely critically on microwave techniques. In some qubit technologies, the measurement process is also carried out by microwave signals.

[0019] One way to think of a qubit is as a high-quality-factor electromagnetic resonator, with a resonant frequency ω01 = (E1 - E0) / ħ set by the energy difference between the qubit states, and a quality factor Q >> 1. Here Q = ω01 / γ, where γ is the decay rate of the energy in the qubit due to all sources of dissipation.

[0020] Superconducting qubits are macroscopic devices that are defined at the circuit level and implementedSeeQC-276.1 - 5 - using nominally lossless capacitors, inductors, and Josephson junctions (JJs). When operated at a low enough physical temperature—typically in the low of millikelvins—these circuits display coherent quantum mechanical behavior, as necessary for use in a quantum processor.

[0021] A transmon qubit is a nonlinear microwave LC resonator, constructed by shunting a JJ with a capacitance CQ. The nonlinearity arises from the JJ, which behaves as a current-dependent inductanceL 2 2J = L J 0 / 1 − I J / I C , where LJ0=Φ0 / 2πIC is the zero bias inductance of the JJ, Φ0=πħ / e is the magneticflux is the current through the JJ, and IC is the critical current of the JJ.The value and is typically about 40 nA for transmons, corresponding to LJ0 ≈ 8 nH

[0040] .

[0022] Some form of pulsed RF waveforms and / or baseband control signals are required to run a quantum processor. Each control signal may be generated using single-sideband mixing, with the complex envelope generated using a pair of high-speed DACs. Alternatively, the RF signals are sometimes directly generated using high speed DACs, obviating the need for analog mixing. The signals are heavily attenuated to suppress thermal noise at the qubit drive port. The digital waveforms may be generated using a field-programmable gate array (FPGA), which is configured to orchestrate quantum algorithms.

[0023] Any practical quantum computing system will require microwave controls signals comprising a plurality of frequencies, corresponding to controls of different qubits. This requires a multi-tone generator, as described for example in US 11,425,024, “Frequency Management for Quantum Control”. The multi-tone generator is operable to generate one or more signals at one or more desired frequencies, which will typically be dictated by the particular quantum processor (e.g., based on the resonant frequencies of quantum elements of the quantum processor). In an example implementation, two types of outputs are provided by quantum multi-tone generator: (1) one or more fixed frequency continuous wave (CW) tones; and (2) one or more synthesized frequency signals. For an implementation generating multiple tones, the tones may be multiples of a single base tone. This, together with the ability of the quantum controller to accurately and dynamically control the frequencies of pulses it generates, enables addressing qubits within a band of frequencies that covers all currently known and proposed superconducting qubit implementations. U.S.11,774,478 discloses a multitone generator that employs conventional non-cryogenic semiconductor technology.

[0024] Most current Multi-Tone Generators are room-temperature instruments connected to a superconducting quantum computer by long cables, which conduct heat down to the superconducting system. For larger systems, this becomes impractical both thermally and mechanically, and what is needed is a Multi-Tone Generator which is located in the cryogenic environment, close to the quantum computer itself. The present disclosure builds on prior-art technology of RSFQ circuits, including the following U.S. Patents and Published Patent Applications, expressly incorporated by reference in their entirety: 3705393; 3916391; 3936809; 3983419; 4186441; 5388068; 5552735; 5598105; 5818373; 5872731; 5942997; 5963351; 6127960; 6188236; 6225936; 6242939; 6310488; 6331805; 6345189; 6353330; 6388600; 6420895; 6483339; 6486694; 6507234; 6608581; 6661560; 6734699; 6750794; 6756925; 6759974; 7786748; 8260143;SeeQC-276.1 - 6 - 8301214; 8401600; 8571614; 8593141; 9461588; 9466643; 9467126; 9520180; 9646682; 9742429; 9779803; 9812192; 9876505; 10084454; 10090841; 10103736; 10147484; 10158363; 10171087; 10222416; 10243582; 10320394; 10355677; 10367483; 10374610; 10447278; 10447279; 10511276; 10554207; 10587245; 10608044; 10725361; 10726351; 10769344; 10817463; 10885974; 10892761; 11012960; 11054598; 11233516; 11283445; 11342919; 11342921; 11362656; 11385099; 11418175; 11621786; 11641194; 11791525; 11804275; 20010025012; 20020063643; 20020074544; 20020105948; 20020118903; 20020169079; 20020189533; 20020190381; 20030011398; 20030016069; 20030028338; 20030039138; 20030040440; 20030058026; 20030076251; 20030115401; 20040016883; 20040022332; 20040120299; 20040201099; 20040201400; 20050023518; 20050029512; 20050035368; 20050047245; 20050078022; 20050235027; 20050253746; 20060255987; 20060290553; 20070052441; 20070075729; 20070075752; 20070077906; 20070194958; 20070293160; 20080048902; 20080049885; 20080129368; 20080146449; 20080186064; 20080215850; 20080231353; 20090002014; 20090014714; 20090015317; 20090070402; 20090075825; 20090082209; 20090153180; 20090153381; 20090267635; 20090322374; 20100033206; 20100033252; 20100133514; 20100148841; 20110089405; 20120184445; 20120274494; 20120302446; 20120314490; 20130040818; 20130303379; 20140175380; 20150092465; 20150119253; 20150263736; 20150358022; 20160013791; 20160028402; 20160035404; 20160118570; 20160164505; 20160191060; 20160233965; 20170011305; 20170017742; 20170141769; 20170359072; 20180013052; 20180025775; 20180102470; 20180145664; 20180188107; 20180226974; 20180226975; 20180350411; 20180366634; 20190006573; 20190019938; 20190149139; 20190188596; 20190190463; 20190296743; 20190363688; 20200027502; 20200044632; 20200119251; 20200136008; 20200136626; 20200242452; 20200259483; 20200299146; 20200333263; 20210013391; 20210135084; 20210336121; 20220012623; 20220021391; 20220065954; 20220136895; 20220180038; 20220190830; 20220208726; 20220237495; 20220311401; 20220317722; 20220321072; 20220326319; 20220360342; 20220393850; 20220399145; 20230010205; 20230037252; 20230046568; 20230145418; 20230208400; 20230210022; 20230225225; 20230344432; 20230351234; 20230361776; 20240020562; 20240038298; 20240039541; 20240057484; 20240063806; 20240127100; 20240137027; 20240202560; 20240204783; 20240259002; 20240281692; 20240289672; 20240311673; 20240334844; 20240345076; 20240356555; 20240380399; 20240389477; 20250007527; 20250078920; 20250158601; 20250159983; 20250181949; 20250181955; 20250183877; and 20250183897. SUMMARY OF THE INVENTION

[0025] The present invention provides a system and method for multiplexing microwave signal tones for a quantum computer, or other quantum systems such as arrays of single photon detectors. In superconducting systems, heat leakage through interface cables represents a significant issue. Multiplexing signals allows a reduction in the number of interface cables, and therefore heat leakage. The qubits of a quantum computer require temporal coherence, which is limited by thermal noise, and therefore extremely low temperature operation, e.g., less than 0.1 °K, is used especially for transmon type qubits. These systems may be controlled and read out by superconducting electronics, e.g., SFQ circuits which employ Josephson junctions, which operate at liquid helium temperatures, e.g., about 4 °K. These circuits operate using quantized pulses, and therefore when seeking to multiplex different frequencies, amplitude modulation is infeasible. Therefore,SeeQC-276.1 - 7 - the circuit preferably produces a pulse density encoded multi-tone signal. According to the present invention, the pulse encoding is defined according to a bit The bit pattern may be defined by a circular shift register, which may have feedback and / or feed-forward logical paths. In a high-density quantum computer, the number of multiplexed signals may be high, for example 8, 16, 32 or more signals. Encoding this large number of tones requires a relatively complex encoding algorithm, especially if the bit pattern is to be kept compact. Therefore, by using feedback or feed-forward logic, the register may be made more compact. The tones may be extracted with a comb filter.

[0026] One of the most important and topical challenges of quantum circuits is their scalability. RSFQ technology is at the forefront of replacing current standard CMOS-based architectures for a number of applications, including quantum computing and quantum sensor arrays. By condensing the control and readout to SFQ-based on-chip devices that are directly connected to the quantum systems, it is possible to minimize the total system overhead, improving scalability and integration. An RSFQ device is provided that generates multi tone digital signals, based on complex pulse train sequences using a Circular Shift Register (CSR) and a comb filter stage. The frequency spectrum of the pulse trains is dependent on a preloaded pattern on the CSR, as well as on the delay line of the comb filter stage. By carefully selecting both the pattern and delay, the desired tones can be isolated and amplified as required.

[0027] Multi-tone signals can be generated using ideal pulse trains, and this approximation holds extremely well when dealing with SFQ pulses due to their sub picosecond time width a multi-tone signal, composed of fclk and its harmonics. However, the only way to control these tones is to change the frequency of the drive signal, which modifies all the tones in the signal. A. Algorithm for Spectral Analysis

[0028] A single SFQ pulse may be modelled by an infinitely narrow pulse which takes the form of a Dirac delta function: +∞ ,x^x= 0δ( ) = ^clocked at a certain frequency fclk, can also be written using Dirac functions, by infinitely summing them: +∞ Ш( x )=δ ( x−m T )equally spaced pulse trains (with period T), have a Fourier Transform (FT) given by: + ∞ −2π jfx 1 ^x^Ш x Ш fSeeQC-276.1 - 8 -

[0034] This result shows that the frequency spectrum of an ideal and equally spaced pulse train, is itself an equally spaced pulse train with ‘period’ given by shown in Fig.1A. With these types of pulse trains, it is already possible to generate a multi-tone signal, composed of fclk and its harmonics, but the only way to control these tones is to change the frequency of the drive signal, which modifies all the tones in the signal.

[0035] Using a CSR, we can redefine the Dirac Comb function to include the pattern loaded in this circuit: +∞ −ШɶN 1 N T( x ) = ^ ^ Skδ ( x − k T − m NT)m =−∞ k =0 the number of bits of the CSR, and Sk=CSR. The frequency spectrum of these unequally spaced pulse trains can be calculated in the same way and it is given by: F^N ^ШɶT ( x ) ^1 ^( f ) = Ш ( f ) ⋅ c ( f )NT1NT− c( f )=^S2 π jfke−k Tin an 8-bit CSR, with programmability focusedon tones below fclk = 10 GHz. Fig.1A shows an evenly spaced pulse train generates an evenly spaced frequency spectrum (frequency division patterns). Fig.1B shows an uneven spaced pulse train generates a richer spectrum with more tones. Pattern ‘10011001’ generates tones at 2.5 and 7.5 GHz. All spectra from the CSR are periodic with period equal to fclk. Fig.1C shows a single stage comb filter applied on same pattern as in Fig.1B, with delay τ = 0.33 ns, tunes the spectrum in a way to amplify the 2.5 GHz tone and cut off the 7.5 GHz tone. FT is normalized for the maximum power in each system, which corresponds to the pattern ‘1111 1111’. FT scale different between plots for visualization purposes.

[0042] This shows that the result of the CSR is to modulate a frequency pulse train with spacing equal to fclk / N (III 1 / NT), depending on the pattern loaded on it (Sk). Compared with Euler’s formula, the modulation function can be interpreted as a summation of sine waves, similar in concept to a Fourier Series, but limited in number and amplitudes. An example of this modulation is shown in Fig.1B, with a pattern of ‘10011001’, for a 8-bit CSR clocked at 10 GHz. With this, the tones at 2.5 and 7.5 GHz are generated at equal power, while keeping the clock frequency the same. The same tones are generated at {2.5, 7.5} + n·10 GHz, since the FT keeps its periodicity of fclk.

[0043] An important characteristic of the patterns and spectra generated by a CSR is the uniqueness of each one of them. A unique pattern for an N-bit CSR is defined as a pattern that generates a frequency spectrum with a unique combination of tones, with a unique combination of relative powers between them, that no otherSeeQC-276.1 - 9 - pattern with the same bit length can generate.

[0044] The total amount of possible patterns for CSR is given by 2N− 1. Since most of these are not unique, rules may be defined to obtain a minimal set of patterns that can generate all others. Some definitions are useful: first, for a N-bit pattern, we define ^ as the number of set bits, that is, the number of ‘ones’ in the pattern. Secondly, we define a distance variable ^ as a set of distances (cyclical) between all set bits in the pattern, where we define a distance of 1 for two consecutive set bits. This set has a cardinality equal to ^ and the sum of its values must equal the total number of bits N for a valid pattern. For instance, the pattern ‘1001 1001’ for an 8-bit CSR (see Fig.1B), has a number of set bits ^ = 4 and the distance set is ^ = {3, 1, 3, 1}.

[0045] The first rule that stands out to eliminate duplicate patterns is that any pattern with the same number of set bits ^ and the same distance set ^ generates the same pattern. For instance, patterns ‘10000000’ and ‘01000000’ both have ^ = 1 and ^ = {8}. This type of bit shift results in a phase shift of the pulse train, but has no effect on the tones generated. This rule can be extended to include patterns whose distance set is a cyclic permutation (CP) of another (e.g. {1, 3, 1, 3} which is obtained by shifting {3, 1, 3, 1}).

[0046] Heuristically, there is another rule that removes additional patterns from the unique set, despite not following completely our original definition of uniqueness. For each pattern obtained so far, there exists a pattern (defined as its dual) obtained by performing a bitwise NOT operation, that also generates the same tones, with the same power relative to each other, except the tones generated at {n·fclk} which have greater or lower power output depending on how many bits are set in the pattern. From the definition, we count every dual pattern as a unique pattern since the output power of these tones have a higher power than the original pattern, although the remaining tones powers are equal. For instance, the patterns ‘10000000’ (^ = 1) and ‘01111111’ (^ = 7) generate exactly the same tones, except the tones at {n·~ fclk} which have increased power for the latter (more sets bits correspond to higher total power at the output). By definition, every pair of dual patterns must have a sum of their set bits equal to N.

[0047] With these dual patterns, one can obtain a smaller set of unique patterns below fclk, since one needs only to look at unique patterns with set bits ^ = {1, 2, ... , N / 2}. These are shown for an 8-bit CSR in Fig.2, clocked at 10 GHz. The higher the number of set bits in the pattern, the higher the power spectral density. In a similar way, the more tones spread through the spectrum, the lower the spectral density for each one. This is easily seen with patterns ‘10000000’ and ‘11111111’, where the former has S=1 and all fclk / N tones with their amplitude reduced by two orders of magnitude, when compared to the latter, with S=8 and only one tone.

[0048] Fig.2 shows unique patterns for an 8-bit CSR clocked at fclk= 10 GHz. Uniqueness of each pattern defined by which tones it generates and the relative power between all of them. Bold patterns correspond to frequency division patterns (equally spaced in time and frequency). For each pattern shown (with exception of ‘11111111’), there exists a dual pattern that generates the same tone distribution but with the tone at fclk having increased power. Width of tones is artificially increased for visualization purposes, since we are usingSeeQC-276.1 - 10 - Dirac delta functions. FT is normalized to the maximum output power of the system, corresponding to the pattern ‘11111111’.

[0049] These rules allow us to create a procedure to generate the unique patterns for any N-bit CSR, as shown in Algorithm 1. This procedure grows exponentially with the number of bits, since it requires various calculations of combinations and permutations. As expected, the number of unique patterns grows in a similar way, although slower than the original 2N. Two boundaries can be defined for a quick estimation of this number: the lower boundary is defined as the sum of all s-combinations of the set ^ = {1, 2, ... , N} whose sum of elements equal N. The higher boundary is obtained from the latter, by adding the unique non-cyclic permutations (NCP) for each combination. The real number is then obtained by using Algorithm 1 and dual patterns to eliminate some of the extra NCPs from the higher boundary that generate equal spectra.

[0050] Increasing the number of bits, N, in the shift register increases the frequency resolution since the base pulse train in the frequency space, defined in Eq. (5), is equally spaced by 1 / NT = fclk / N. This means more tones can be generated within the same frequency range from 0 to fclk. Another additional benefit of increasing the number of bits, is that for a long enough CSR, it is possible to simply overlap evenly spaced pulse trains of different frequencies and obtain a pattern with these frequencies only.

[0051] Algorithm 1 Generating unique patterns on an N-bit CSR s ←0 / / Number of set bitsP t ←{ } / / Output set of unique patternswhile s≤N / 2 doEmptyset of dual patternsP t ←{} { }Get alldistanceset combinationsD← ^N^^^s^ ^

[0052] for Dj inD doif Sum(D j)=N thenfor d i in NCPs ( D j ) doif ( CP ( d ɶ i ) ∉ P t ) ∧ ( CP ( d i ) ∉P t)then Pt←d iP t←d is ← s +1

[0053] B. Comb Filtering

[0054] Using a CSR with a pre-loaded pattern increases the control of the tones generated with pulse trains. It can be further improved by using comb filtering stages. There are two types of comb filters: feedback and feedforward filters

[0027] . These can be defined as two-port circuits, where a delayed version of the output or input signal (respectively) is added onto itself. Mathematically:SeeQC-276.1 - 11 -

[0055] y(t) = x(t) + αy(t − τ), [Feedback] (7)

[0056] y(t) = x(t) + αx(t − τ), [Feedforward] (8)

[0057] where x(t) is the input signal, y(t) is theτ is the added delay and α is the scaling factor of the delayed signal. A block diagram of both types of filters is shown in Fig.4B. While the latter type has a potentially better amplitude response (specifically, it can amplify the tones more effectively), its implementation using RSFQ circuits is not as straightforward, since the scaling factor needs be less than 1 to avoid instabilities in the filter output. To have the scaling factor less than 1, the top branch of the comb filter would need to have Josephson junctions with IcRNproduct different from the rest of the circuit. Therefore, and as a first proof of concept, we look at the feedforward implementation and how it is used in the DMTG.

[0058] Taking the Fourier Transform of Eq. (8), we obtain: F[ y ( t ) ] ( f ) = F [ x ( t ) ] ( f ) + α F [ x ( t − τ ) ] ( f )−2 π jf τ(9)modulation is achievedwaves, case, can the delay of each filtering stage. There is, however, an exception to the usefulness of the comb filter, which happens when the delay matches the period spacing of the clock (1 / fclk). When such circumstances occur and because SFQ pulses cannot be added together (as in, on the same time instance, two pulses will not sum their amplitudes), the comb filter either produces no delayed pulse, so the spectrum will be exactly the same as it is seen at the end of the CSR, or it produces a delayed pulse which effectively changes the original pattern of the CSR into another pattern.

[0061] Fig.3 shows an example of the effect of using a feedforward comb filter stage on an 8-bit CSR with a preloaded pattern of ‘10011001’. As the delay is tuned from 0 to N / fclk, the tones at 2.5, 7.5 and 10 GHz are periodically tuned from 0 to a maximum amplitude. A cross section of the 2D plot can also be seen, where the periodic behavior of the tones at 2.5 and 7.5 GHz is plotted, together with a curve showing the distance between these two tones. By selecting the correct delay, we can tune both tones to be at a maximum of separation (for example to suppress the 7.5 GHz tone, with delay 0.33 ns) or a minimum of separation, which for this pattern corresponds to both tones at equal strength. Fig.1C shows the result of selecting a delay equal to 0.33 ns on the output of the CSR. As predicted, the tone at 7.5 GHz is suppressed.

[0062] According to a preferred embodiment, the DMTG may be fabricated according to an integrated process comprising superconducting films and Josephson junctions. The superconductor may comprise niobium, or aluminum, or niobium nitride, or niobium-titanium nitride, or magnesium diboride, or yttrium barium copper oxide, or any other superconducting material. The circuits may be designed to operate at a cryogenic temperature close to 4 °K, or at a temperature of 1 °K or less, or at temperature less than 100 mK. AccordingSeeQC-276.1 - 12 - to a preferred embodiment, the superconducting circuit logic may comprise RSFQ, or the low-power version ERSFQ or eSFQ. Alternatively, the may comprise another SFQ-based logic family, such as Reciprocal Quantum Logic (RQL), Adiabatic Quantum Flux Parametron (AQFP), or Pulse Conserving Logic (PCL),

[0063] It is an object to provide a digital multi-tone generator (DMTG), comprising: a clocked single flux quantum circuit configured to generate a pulse train comprising a plurality of pulses, each pulse having a pulse peak, consecutive pulse peaks being separated by a period corresponding to a period of the clock; and a clocked shift register coupled with the clocked single flux quantum circuit, the clocked shift register having an encoded bit pattern, and being configured to produce an output dependent on the encoded bit pattern and the period of the clock, such that the bit pattern modulates the pulse train with a plurality of frequency components.

[0064] It is also an object to provide a digital multi-tone generator (DMTG) comprising: a single flux quantum circuit configured to be clocked at a clock frequency and to generate a pulse train, wherein the pulse train comprises a plurality of peaks separated by a period corresponding to the clock frequency; and a shift register communicatively connected to the single flux quantum circuit, driven at the clock frequency and encoded with a bit pattern to generate a multi-tone signal, wherein the shift register is configured to apply a modulation to the generated pulse train according to the bit pattern, such that the modulated pulse train is defined by a combination of a plurality of frequency components.

[0065] It is a further object to provide a multi-tone generator, comprising: a clocked single flux quantum circuit configured to generate a pulse train comprising a plurality of pulses, each pulse having a pulse peak, consecutive pulse peaks being separated by a period corresponding to a period of the clock; and a clocked shift register coupled with the clocked single flux quantum circuit, the clocked shift register having an encoded bit pattern, and being configured to produce an output dependent on the encoded bit pattern and the period of the clock, such that the bit pattern modulates the pulse train with a plurality of frequency components.

[0066] The clocked shift register may have at least one feedback path and / or at least one feedforward path. The clocked shift register may comprise a circular shift register.

[0067] The multi-tone generator may further comprise at least two distinct paths in the clocked shift register and the method further comprising summing signals from each of the at least one feedback path and the at least one feedforward path with a summer configured to sum signals from each of at least two distinct paths.

[0068] The clocked single flux quantum circuit may comprise a rapid single flux quantum circuit.

[0069] The output may be produced by at least one Josephson junction having an overdamped output.

[0070] The multi-tone generator may further comprise a comb filter configured to filter the output and to produce a filtered output.

[0071] The multi-tone generator may further comprise a filter configured to receive the output and to produceSeeQC-276.1 - 13 - an analog microwave signal comprising the plurality of frequency components.

[0072] 10. The multi-tone generator 9, wherein the analog microwave signals selectively interact with at least one quantum device, e.g., a qubit, a transmon qubit, at least one superconducting nanowire single photon detector, or at least one quantum sensor.

[0073] The analog microwave signals may comprise an orthogonal frequency multiplexed signal.

[0074] The analog microwave signals may interact with at least one qubit, wherein the single flux quantum circuit comprises a niobium superconducting layer configured to operate at a temperature of 4 °K, and the qubit is configured to operate at a temperature of less than 0.1 °K.

[0075] The single flux quantum circuit, the shift register, and the comb filter may be formed on a common planar substrate.

[0076] The multi-tone generator may be a system that further comprises a qubit, a cryochamber surrounding the single flux quantum circuit and the qubit, and a cryocooler having a plurality of stages, wherein the single flux quantum circuit and the qubit are operated at different temperatures corresponding to different stages of the cryocooler.

[0077] It is a still further object to provide a multi-tone generator, comprising: a single flux quantum circuit clocked configured to receive a clock signal at a clock frequency of at least 10 GHz, and to generate a pulse train, wherein the pulse train comprises a plurality of peaks separated by a period corresponding to the clock frequency; and a shift register driven at the clock frequency and encoded with a bit pattern, configured to receive the pulse train, and to generate a multi-tone signal, and to produce an output having a plurality of frequency components selectively dependent on the pulse train and the bit pattern.

[0078] Another object provides a method of generating a multi-tone signal, comprising: generating a periodic pulse train comprising a plurality of pulses using a single flux quantum circuit, each pulse having a pulse peak; and producing an output dependent on an encoded bit pattern of a shift register coupled with the single flux quantum circuit, such that the bit pattern modulates the pulse train to produce a digital signal pulse density modulated output with a plurality of frequency components.

[0079] The shift register may have at least one feedback path and at least one feedforward path, further comprising a summer configured to sum signals from each of the at least one feedback path and the at least one feedforward path.

[0080] The method may further comprise: filtering the digital signal pulse density modulated output with a digital comb filter to produce a comb filtered output; generating an analog microwave signal with an analog filter from the comb filtered output; and interacting the analog microwave signal with at least one of a qubit, a superconducting nanowire single photon detector, and a quantum sensor.

[0081] It is another object to provide a method of generating a multi-tone signal, comprising: generating a pulse train comprising a plurality of pulses using a clocked single flux quantum circuit, each pulse having aSeeQC-276.1 - 14 - pulse peak, consecutive pulse peaks being separated by a period corresponding to a period of the clock; and producing an output dependent on the period of and an encoded bit pattern of a clocked shift register coupled with the clocked single flux quantum circuit, such that the bit pattern modulates the pulse train with a plurality of frequency components.

[0082] It is a still further object to provide a method of generating a multi-tone signal, comprising: generating a pulse train with by clocking a single flux quantum circuit at a clock frequency, wherein the pulse train comprises a plurality of peaks separated by a period corresponding to the clock frequency; and generating a multi-tone signal with a shift register communicatively connected to the single flux quantum circuit, driven at the clock frequency and encoded with a bit pattern, wherein the shift register is configured to apply a modulation to the generated pulse train according to the bit pattern, such that the modulated pulse train is defined by a combination of a plurality of frequency components.

[0083] The clock may have a regular or irregular period.

[0084] The clocked shift register has at least one feedback path and / or at least one feed forward path. The clocked shift register may be a circular shift register.

[0085] The shift register may have at least two distinct paths in the clocked shift register and a summer configured to sum signals from each of the at least two distinct paths.

[0086] A comb filter and / or a low pass filter configured to filter the output may be provided.

[0087] The digital multi-tone generator may comprise a feedback and / or feed forward implemented comb filter configured to filter the output.

[0088] The clocked single flux quantum circuit may comprise a rapid single flux quantum circuit (RSFQ), an SFQ / DC driver circuit, or an RSFQ / DC driver circuit.

[0089] The plurality of frequency components of the output may comprise harmonics and / or a plurality of frequency components comprising frequencies which are not harmonics.

[0090] Another object provides a multi-tone generator, comprising a single flux quantum circuit comprising a Josephson junction configured to selectively generate a single flux quantum pulses in a pulse modulated train representing a plurality of frequencies; a shift register coupled with the single flux quantum circuit, encoding a bit pattern representing the plurality of frequencies, and being configured to control the pulse density pattern in accordance with the bit pattern; a comb filter, and an analog filter configured to produce analog microwave signals at the plurality of frequencies.

[0091] The comb filter may be configured to interface with at least one qubit or a plurality of qubits.

[0092] The analog microwave signals may each selectively interact with a transmon qubit.

[0093] The analog microwave signals may interact with at least one superconducting nanowire single photon detector or other quantum sensor.

[0094] The analog microwave signals may comprise an orthogonal frequency multiplexed signal.SeeQC-276.1 - 15 -

[0095] The analog microwave signals interact may with a qubit, wherein the single flux quantum circuit comprises a niobium superconducting layer operate at a temperature of 4 °K, and the qubit is configured to operate at a temperature of less than 0.1 °K.

[0096] The plurality of frequencies may be anharmonic.

[0097] The single flux quantum circuit and the shift register may be formed on a common planar substrate.

[0098] The single flux quantum circuit, the shift register, and the comb filter may be formed on a common planar substrate.

[0099] The multi-tone generator may further comprise a qubit, a cryochamber surrounding the single flux quantum circuit and the qubit, and a cryocooler having a plurality of stages, wherein the single flux quantum circuit is operated at a temperature of a first stage of the cryocooler and the qubit is operated at a temperature of a second stage of the cryocooler. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figs.1A-1C show examples of pulse trains generated in an 8-bit CSR, with programmability focused on tones below fclk= 10GHz.

[0101] Fig.2 shows unique patterns for an 8-bit CSR clocked at fclk = 10 GHz.

[0102] Figs.3A and 3B show the comb filter effect on frequency spectrum for the same pattern and CSR shown in Fig.1B.

[0103] Figs.4A-4C show circuit block diagrams.

[0104] Figs.5A and 5B show frequency spectra obtained from the PSCAN2 time simulations using the Lomb-Scargle periodogram algorithm for a 5 ns pulse train.

[0105] Figs.6A-6C show the influence of pulse train characteristics on frequency spectrum for the same pattern applied to an 8-bit CSR, as seen in Figs.5A and 5B, focusing on the 2.5 GHz tone.

[0106] Figs.7A-7C show the SFQ-based architectures for qubit and SNSPDs applications utilizing the Digital Multi-Tone Generator (DMTG) device.

[0107] Fig.8A and Fig.8B show micrographs of portions of test integrated circuits of an exemplary circular shift register and the DMTG device fabricated using niobium Josephson junctions. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0108] RSFQ IMPLEMENTATION A. Circular Shift Registers

[0109] CSRs are an extension of linear shift registers, where the output of the last memory cell is connected to the input of the first, creating a circular flow of data (Fig.4A). An N-bit CSR consists of N Data Flip-Flops (DFFs) connected in series using Josephson Transmission Lines (JTLs), to control the delay between cells,SeeQC-276.1 - 16 - and a NDRO cell at the end of the loop to reset the circuit. To clock the CSR, 2-way splitters are usually used, however, in this device we use 3-way splitters to requirements of our clocking network, as will be explained below.

[0110] Shift registers are synchronous circuits, meaning they require a clock to advance to the next internal state, independent of the data inputs. The type and architecture of the clock distribution network is therefore crucial for the correct operation of this device, considering all internal and external delays on the data and clock paths. Particularly important is the clock skew, tcs, of the circuit, which for a CSR must equal to 0

[0023] ,

[0028] . This quantity is defined as the time between a clock pulse arriving at the ithmemory cell and the same clock pulse reaching the (i + 1)thcell (see Fig.4A). For a CSR: N tTotalcs=^t ics ( t t ) ( t t ) ... ( t t ) 0i = 1=1=2+2−3+ +N−1=avoid any racingthan the hold time of the cell. This must still be met, even though the total clock skew of the CSR is zero.

[0113] To satisfy this condition, a type of clocking network was designed which is based on a type of symmetrical-mixed clock design as seen in Ref.

[0021] ,

[0023] . Here, N / 2 bits of the shift register are clocked in a concurrent way and the remaining N / 2 bits are clocked in a counter flow manner, but only one input and output of data are used, as shown in Fig.4C. This ensures that the two blocks have negative and positive clock skews, respectively, but when combined in a loop the total equals to zero. Another alternative is to use a standard binary tree clock distribution, which has a higher overhead of splitters. For binary-tree clocking, the clock skew between adjacent memory cells is zero, so the sum will always also be zero. The total amount of splitters necessary for this type of clock network is N − 1, whereas for the symmetrical type it requires N / 2.

[0114] Figs.3A and 3B show the comb filter effect on frequency spectrum for the same pattern and CSR shown in Fig.1B. Fig.3A shows a 2D view of spectrum versus comb filter delay. Fig.3B shows a slice view of the two frequencies of interest (2.5 and 7.5 GHz) and a curve (green) showing the maximum distance between these two. The result is periodic with period equal to the inverse of lowest frequency tone in the spectrum, which, for this example, is 1 / (2.5 GHz) = 0.4 ns. By selecting the delay, a continuous tuning of these two tones is achieved.

[0115] B. Comb Filtering

[0116] Comb filtering in RSFQ circuits must be modified slightly compared to the standard digital signal processing filter, since SFQ pulses cannot be summed in amplitude like normal signals can. This means that the two signals generated must be merged, instead of summed. Also, the scaling factor seen in Eq. (8) is harder to achieve, specifically when α < 1, since the amplitudes of SFQ pulses are set during the design phase by the critical current density jcand IcRNproduct of the junctions. It is also possible to use an amplifyingSeeQC-276.1 - 17 - JTL to increase the voltage of the SFQ pulse and obtain α > 1, which can be useful for increasing the power of the output tones.

[0117] Figs.4A-4C show circuit block diagrams. Fig.4A shows a diagram of N-bit linear shift register being clocked in a concurrent or counter flow. tnrepresents the time when the clock signal arrives at memory cell n. The total clock skew of a circular shift register (dotted) is equal to zero. Fig.4B shows a diagram of two types of comb filters: feedback [left] and feedforward [right], both with a delay and scaling factor stage. Fig.4C shows a block diagram of the full circuit, using a N-bit CSR and a single stage comb filter step. Full symmetric clock distribution is used with N / 2 bits having concurrent flow and the remaining N / 2 having counter flow of data and clock. A non-destructive readout cell (NDRO) is used to ensure a way of writing data in the CSR (Write) and forming the loop to generate the pulse train (Read), as well as used in the comb filter to allow a zero-delay output (Set and Reset). D-JTL (delay JTL) is used as a tunable way of changing the fluxon propagation speed in the JTL.

[0118] Figs.5A and 5B show frequency spectra obtained from the PSCAN2 time simulations using the Lomb-Scargle periodogram algorithm for a 5ns pulse train. The same pattern used in Fig.1C is used for comparison, Fig.5A shows without the comb filtering stage, and Fig.5B shows with the comb filtering stage set very close to the 0.333 ns delay to obtain the best separation between the two tones, 2.5 and 7.5 GHz (as shown in Figs.3A and 3B). Dashed lines on time domain plots show the input data (pre-loading) sent to the CSR. Inset highlights one SFQ pulse, as well as the width of one of the frequency tones.

[0119] The implementation of comb filtering in this work is based on the feedforward architecture with a unity scaling factor. The circuit has three basic RSFQ cells: one splitter to generate the two data paths, one merger to combine the two pulses and one long JTL, which we call a delay JTL, which delays the propagation of SFQ pulses in one branch of the circuit. This delay (τ) is tunable by changing the bias current of the entire delay JTL, based on fluxon propagation and interaction in JTLs

[0029] . This implementation is shown in Fig.4C, alongside a full block diagram of the multi-tone generator device, comprised of a N-bit CSR and a single stage feedforward comb filtering stage, with one data input and one data output. A NDRO cell is used to break the loop of the CSR and choose whether the CSR is in write (W) or read (R) mode, which opens and closes the loop, respectively. Another NDRO switch is added to the comb filter to allow for an output with no delayed pulse (SET and RESET).

[0120] SIMULATIONS

[0121] To test the operation of the device, a simulation model was implemented using PSCAN2 superconductor circuit simulator

[0030] . A netlist of the device was designed using SeeQC’s high density fabrication process parameters, specifically devised for digital circuits and quantum applications

[0031] . A modular approach is used to optimize the design, starting with the memory cells and the shift register and further adding the comb filter stage. The optimization process was done at 10 GHz (a conventional operationSeeQC-276.1 - 18 - frequency for superconducting quantum circuits), therefore for higher frequencies the margins are expected to be narrower. The global margins XI, XJ, and XL, the total deviations of the bias currents, junction critical currents, and inductances, respectively, in the device, such that the correct operation is not compromised. These are obtained from the full circuits and tested at different clock frequencies (namely, 10, 20 and 50 GHz) and with different patterns on the CSR. For operation around 50 GHz and above, slight modifications of the CSR were made, increasing the IcRNproduct for faster practical circuit operation and removing some JTLs between data cells. A maximum frequency of 100 GHz was obtained with correct operation of the CSR without the comb filter, as a demonstration of the robustness of the design. A summary of these tests is shown in Table I.

[0122] TABLE I: Simulated Global Margins for an 8-bit Device (in percentages). Pattern ‘10001000’ Pattern ‘10011001’ 10 GHz 20 GHz 50 GHz∗ 10 GHz 20 GHz XI [-28,25] [-25,25] [-21,18] [-28,25] [-23,23] XJ [-22,25] [-22,18] [-16,23] [-20,25] [-19,21] XL [-40,40] [-40,40] [-40,40] [-40,40] [-40,40]

[0123] The output frequency spectrum was obtained from the simulations by using the Lomb-Scargle periodogram

[0032] . This algorithm provides an estimation of the discrete Fourier Transform for samples with unevenly spaced time data, which is crucial for the analysis of PSCAN2 simulations due to its dynamic time step.

[0124] The dependence of the frequency spectrum on the characteristics of the SFQ pulses comprising the pulse train was also investigated and is shown in Figs.6A-6C. Firstly, by changing the characteristic voltage Vc of the output junctions, we observe a change in the relative powers between the tones of the signal, as shown in Fig.6B. This is in accordance with the result obtained in [7], where for narrow pulses, the higher frequency components have increased power. This is a problem when working with superconducting circuits since at some critical power these high frequency tones may break Cooper pairs and poison the system with quasiparticles [8]. In practice, the advantage of having narrower pulses is to make the pulse trains and subsequently the control pulses, shorter in time. Since the area of the pulse, Φ0, remains the same, the average power output also does not change.

[0125] Increasing the pulse train length improves the bandwidth of the output tones, as seen in Fig.6C. The longer the pulse train, the narrower the tones, which in practice means our control pulse trains cannot be too short in time. The average power output is the same in the time domain (since the peak power and the duty cycle remain constant), although there is more energy in the system.

[0126] Clock jitter is another metric which exists in real experiments and can affect the frequency spectrum of the output signal. While the base simulations shown do not include a known and controllable source of jitter,SeeQC-276.1 - 19 - there is a small variation of the clock period due to the way the simulation, particularly the dynamic time step calculation, is performed, but accounts for less picoseconds. Adding more jitter into the tones tends to shift from their expected value by a few MHz (as seen on the right side of Fig.6B, where the tone at 100 GHz slightly shifts) and noise at higher frequencies increases.

[0127] When discussing powers with pulse trains and frequency spectra, one must distinguish between a couple of variables. The average power in the time domain of a pulse train is given by: σP = P ⋅ DutyCycle=pulsenavg peak P peak ⋅ ⋅

[0128] TclkN

[0129] of set bits in an N-bitpattern. power in time domain remains. For the Lomb Scargle periodogram algorithm, the result is a metric of the squared amplitudes of each Fourier component at each frequency, with units of mV2in this work. This is known as the Power Spectrum (PS). Since we are dealing with discrete signal, i.e., sampling rate and number of points are finite, one drawback is that the DFT and all estimations scale with the number of points N of our original signal, making the amplitude of the calculation arbitrary and with no real meaning. For the transform to be useful, we calculate the Power Spectral Density (PSD) which is obtained by dividing the power spectrum by the frequency resolution of the spectrum: PS PSD ≡of sample points. Using the PSD, the area under each tone represents the fraction of the signal power that is concentrated around that tone. Increasing the total length of the pulse train therefore decreases the tone width, but amplitude increases so that the total area remains constant, since the input power remains the same (higher length means more energy but averaged over longer period).

[0132] Figs.6A-6C show the influence of pulse train characteristics on frequency spectrum for the same pattern applied to an 8-bit CSR, as seen in Figs.5A and 5B, focusing on the 2.5 GHz tone. Fig.6A shows an increasing characteristic voltage, Vc=IcRN, of the output junctions in the circuit, results in a narrower pulse with increased amplitude. Difference in time shown corresponds to the small jitter in PSCAN2 simulations. Fig.6B shows increasing Vc, the output power of lower frequency tones remains approximately constant, while higher frequencies are amplified. Fig.6C shows that, to decrease the width of each tone, a longer pulse train should be considered. For a 50 ns pulse train, a bandwidth of 15 MHz is obtained, compared to 84 MHz for a 5 ns pulse train. Vcis in dimensionless PSCAN2 units, where unity equals 0.287 mV.

[0133] The DMTG device presented and discussed herein has applications in both classical RSFQ circuitry,SeeQC-276.1 - 20 - as well as a control circuit for quantum systems that also require a cryogenic environment and the higher frequencies achieved with RSFQ and ERSFQ Two examples which motivated this work are qubit and SNSPD control and readout systems.

[0134] For standard qubit control, a microwave pulse is carefully crafted to excite single qubit gates and drive qubit state rotation on the XX and YY axes. For this method, the device could be used as a way to multiplex the control and use a single line for many qubits, after it is filtered and impedance matched on a RF module, as shown in Ref.

[0035] .

[0135] Currently, SFQ-based digital control is achieved using a simple DC-SFQ converter (or a combination of such) and clocked at a subharmonic of the qubit transition frequency [8]. The idea is that each pulse rotates the qubit state slightly around the XX or YY axis of the Bloch Sphere, and by controlling the number of pulses, the desired final state can be obtained. The subharmonic is used to avoid driving the qubit directly, since the SFQ system is capacitively coupled to it. The device could be used like this in a fully digital way, or can be used to implement more complex pulse sequences as shown in Ref.

[0036] . These sequences can optimize SFQ-based qubit control by reducing leakage outside the computational space and increasing the fidelity of single qubit gates.

[0136] Superconducting Nanowire Single Photon Detectors (SNSPD) are another area where superconductor electronics are foreseen to improve scalability. In particular, large arrays of RF-SNSPDs (SNSPDs integrated in lumped element resonators) can be used in conjunction with RSFQ electronics to create a control and readout system that can scale more efficiently than DC-SNSPDs

[0016] . The output signal with multiple tones would be sent to an array of resonators that are coupled to the SNSPDs, and, using frequency multiplexing, all pixels could be probed simultaneously with only one input. The readout architecture would closely follow the same one used for microwave electronics, where a multi-tone input signal is sent to the signal, filtered, and down-converted using IQ mixers, to then be detected by an Analog-to-Digital Converter, which could also be implemented with RSFQ circuitry

[0037] ,

[0038] . In this case, no down-conversion would be necessary. Additionally, time domain multiplexing could be used to sweep the local oscillator signal (LO) such that the down-converted signal frequency could always fit within the bandwidth of the ADC, reducing the sample rate, but increasing the total bandwidth available for the resonators. The same architecture could be implemented to probe an array of readout resonators coupled to qubits in the standard dispersive regime

[0039] . A summary of these architectures is shown in Figs.7A-7C.

[0137] Figs.7A-7C show the SFQ-based architectures for qubit and SNSPDs applications utilizing the Digital Multi-Tone Generator (DMTG) device. Fig.7A shows qubit control using a RF module (see Ref.

[0033] ) to filter the complex pulse train into an analogue signal to drive XX and YY qubit state rotations. Fig.7B shows that by directly applying the SFQ pulses to the qubit yields a full digital control method, also working as XX and YY qubit gates (see Ref. [8],

[0034] ). Fig.7C shows a resonator readout is also possible using a heterodyne schemeSeeQC-276.1 - 21 - where both the multi tone signal and a local oscillator are generated on-chip. Down-conversion can be possibly substituted by a RSFQ based ADC to the signal from the quantum system, creating a full RSFQ based control system.

[0138] A prototype device according to this disclosure was fabricated using niobium integrated circuits. Figs. 8A and 8B show microphotographs of the prototype Circular Shift Register and the DMTG device. In Fig.8A, the test 8-bit CSR with DC / SFQ / DC conversion (D1) is shown. This device contains a total of 8 I / O ports, including 4 DC current biases and 4 I / O voltage signals. The Inset shows a detailed view of individual memory stages of the CSR, as well as the data loop with the splitter-switch-merger circuit. In Fig.8B, the test DMTG device with an 8-bit CSR and a 20 stage comb filter (D2) is shown. A total of 10 I / O ports are required for this device, with two additional DC current biases compared to the 8-bit CSR. Inset shows a detailed view of the comb filter with a 20 stage long delay Josephson Transmission Line (DJTL). Testing of these devices is presently in progress.

[0139] In conclusion, an RSFQ device is provided that uses the properties of pulse trains and their Fourier Transform to create multi-tone signals that depend on the data (pattern) loaded in the memory cells of an N-bit CSR.

[0140] All the unique patterns for an N-bit CSR may be obtained, although the search for optimal and unique patterns may be computationally complex. Together with a feedforward comb filter stage, the tones generated are further concentrated around certain frequencies, which increases the tuneability of the output, as well as increasing the amplitude of the desired tones. Additionally, stacking comb filters improves further the amplitude and spectrum.

[0141] In terms of components, the device is relatively simple to fabricate, uses well developed and studied RSFQ components, and the number of junctions in the circuit grows reasonably well with increasing bit size. The simulated circuits show a great match with the predicted frequency spectra for every pattern, and it is also shown that changing the SFQ pulse width, pulse train length, and clock periodicity has an influence on the result. This is all consistent with the results predicted theoretically. The average margins are shown to be quite good for different patterns at around 20%, which is similar to margins obtained experimentally for RSFQ devices with complexity involving few hundred junctions. It is also shown that the maximum operating frequency is dependent on the characteristic voltage of the junctions, but with these fabrication parameters, operation close to 100 GHz can be achieved.

[0142] According to a preferred embodiment, SeeQC’s #QC20uA design rules for superconducting integrated circuits were followed, based on their high density fabrication process for niobium-based integrated circuits [Yohannes, 2023]. This process uses niobium (Nb) as the fundamental superconducting metal due to its high critical temperature and overall stability. Furthermore, junctions are formed using a Nb trilayer technology, having both the top and bottom electrode of the junction consisting of Nb, with a thin layer of Al being deposited in between, allowing the controlled growth of an AlOx insulating barrier. In total, this processSeeQC-276.1 - 22 - has 5 superconducting metal layers: four niobium layers and one niobium nitride (NbN) layer. The latter consists of a high kinetic inductance metal layer, to allow the miniaturization and area reduction of the large bias inductors required to implement ERSFQ circuits. Every metal layer is separated by a dielectric bi- layer consisting of silicon nitride (SiNx) and silicon dioxide (SiO2), providing isolation between them. Additionally, a resistive layer, consisting of a non-superconducting Pd / Au mixture, defines the shunt resistors required to obtain the overdamped junctions. This process is particularly developed with quantum applications in mind, with a low critical current density of 1 kA / cm2(10 µA / µm2). This low current density allows the design of junctions with smaller critical currents, without making the junction feature sizes too small for the lithography process.

[0143] A typical fabrication process step consists of deposition of the superconducting metal layer using DC sputtering, followed by spinning of photoresist. The designed pattern is transferred onto the resist using deep-UV photolithography when exposed to light of this wavelength (193 nm). Afterwards, the photoresist is developed which dissolves either the exposed or unexposed sections, depending on whether it is positive or negative resist. Finally, the exposed metal is etched away using dry-etching tools, transferring the pattern onto the metal layer. For the dielectric layers, the deposition is done using chemical vapor deposition (CVD) techniques, while the resistive layer is deposited using physical evaporation, both followed by the same patterning and etching processes.

[0144] According to a further preferred embodiment, several prototype circuits were fabricated, tested and found to operate satisfactorily.

[0145] It will be appreciated by those of ordinary skill in the art that the diagrams, schematics, illustrations, and the like represent conceptual views or processes illustrating systems and methods embodying this invention. The functions of the various elements shown in the figures may be provided through the use of dedicated hardware as well as hardware capable of executing associated software.

[0146] Although the invention(s) have been described with reference to specific embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the true spirit and scope of the invention. In addition, modifications may be made without departing from the essential teachings of the invention. The invention is described by way of various embodiments and features. This disclosure is intended to encompass all consistent combinations, subcombinations, and permutations of the different options and features, as if expressly set forth herein individually.

[0147] The various embodiments described above can be combined to provide further embodiments. These and other changes can be made to the embodiments in light of the above-detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific embodiments disclosed in the specification and the claims, but should be construed to include all possible embodimentsSeeQC-276.1 - 23 - along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure.

[0148] The disclosure has been described with reference to various specific embodiments and techniques. However, many variations and modifications are possible while remaining within the scope of the disclosure.

[0149] As used herein in this document, the terms "coupled to" and "coupled with" are also used euphemistically to mean “communicatively coupled with” over a network, where two or more devices are able to exchange data with each other over the network, possibly via one or more intermediary device.

[0150] It should be apparent to those skilled in the art that many more modifications besides those already described are possible without departing from the inventive concepts herein. The inventive subject matter, therefore, is not to be restricted except in the spirit of the appended claims. Moreover, in interpreting both the specification and the claims, all terms should be interpreted in the broadest possible manner consistent with the context. In particular, the terms “comprises” and “comprising” should be interpreted as referring to elements, components, or steps in a non-exclusive manner, indicating that the referenced elements, components, or steps may be present, or utilized, or combined with other elements, components, or steps that are not expressly referenced. Where the specification claims refers to at least one of something selected from the group consisting of A, B, C …. and N, the text should be interpreted as requiring only one element from the group, not A plus N, or B plus N, etc.

[0151] While the foregoing describes various embodiments of the invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof. The scope of the invention is determined by the claims that follow. The invention is not limited to the described embodiments, versions or examples, which are included to enable a person having ordinary skill in the art to make and use the invention when combined with information and knowledge available to the person having ordinary skill in the art. REFERENCES

[0152] Each reference cited herein is expressly incorporated herein by reference in its entirety.

[0153] [1] O. Mukhanov, V. Semenov, and K. Likharev, “Ultimate performance of the RSFQ logic circuits,” IEEE Transactions on Magnetics, vol.23, no.2, pp.759–762.1987 doi:10.1109 / TMAG.1987.1064951

[0154] [2] K. K. Likharev and V. K. Semenov, “RSFQ logic / memory family: A new Josephson-junction technology for sub-terahertz-clock-frequency digital systems,” IEEE Transactions on Applied Superconductivity, vol.1, no.1, pp.3–28.1991 doi:10.1109 / 77.80745

[0155] [3] O. A. Mukhanov, “Energy-efficient single flux quantum technology,” IEEE Transactions on Applied Superconductivity, vol.21, no.3, pp.760–769.2011 doi:10.1109 / Tasc.2010.2096792

[0156] [4] D. E. Kirichenko, S. Sarwana, and A. F. Kirichenko, “Zero static power dissipation biasing of RSFQ circuits,” IEEE Transactions on Applied Superconductivity, vol.21, no.3, pp.776–779.2011SeeQC-276.1 - 24 - doi:10.1109 / Tasc.2010.2098432

[0157] [5] E. Leonard, M. A. Beck, J. Christensen, T. Thorbeck, C. Howington, A. Opremcak, I. V. Pechenezhskiy, K. Dodge, N. P. Dupuis, M. D. Hutchings, J. Ku, F. Schlenker, J. Suttle, C. Wilen, S. Zhu, M. G. Vavilov, B. L. T. Plourde, and R. McDermott, “Digital coherent control of a superconducting qubit,” Physical Review Applied, vol.11, no.1, pp.3–9.2019 doi:10.1103 / PhysRevApplied.11.014009

[0158] [6] M. A. Castellanos-Beltran, A. J. Sirois, L. Howe, D. Olaya, J. Biesecker, S. P. Benz, and P. F. Hopkins, “Coherence-limited digital control of a superconducting qubit using a Josephson pulse generator at 3 K,” Applied Physics Letters, vol.122, no.19.2023 doi:10.1063 / 5.0147692

[0159] [7] C. Liu, A. Ballard, D. Olaya, D. R. Schmidt, J. Biesecker, T. Lucas, J. Ullom, S. Patel, O. Rafferty, A. Opremcak, K. Dodge, V. Iaia, T. McBroom, J. L. DuBois, P. F. Hopkins, S. P. Benz, B. L. T. Plourde, and R. McDermott, “Single Flux Quantum-based digital control of supercon- ducting qubits in a multichip module,” PRX Quantum, vol.4, no.3.2023 doi:10.1103 / PRXQuantum.4.030310

[0160] [8] C. Howington, A. Opremcak, R. McDermott, A. Kirichenko, O. A. Mukhanov, and B. L. T. Plourde, “Interfacing superconducting qubits with cryogenic logic: Readout,” IEEE Transactions on Applied Superconductivity, vol.29, no.5, pp.1–5.2019 doi:10.1109 / TASC.2019.2908884

[0161] [9] L. Di Palma, A. Miano, P. Mastrovito, D. Massarotti, M. Arzeo, G. P. Pepe, F. Tafuri, and O. Mukhanov, “Discriminating the phase of a coherent tone with a flux-switchable superconducting circuit,” Physical Review Applied, vol.19, no.6.2023 doi:10.1103 / PhysRevApplied.19.064025

[0162]

[0010] D. Gupta and A. M. Kadin, “Single-photon-counting hotspot detector with integrated RSFQ readout electronics,” IEEE Transactions on Applied Superconductivity, vol.9, no.2, pp.4487–4490.1999 doi:10.1109 / 77.784022

[0163]

[0011] M. Hofherr, O. Wetzstein, S. Engert, T. Ortlepp, B. Berg, K. Ilin, D. Hen- rich, R. Stolz, H. Toepfer, H. G. Meyer, and M. Siegel, “Orthogonal sequencing multiplexer for superconducting nanowire single-photon detectors with RSFQ electronics readout circuit,” Optics Express, vol.20, no.27, pp.28683–28 697.2012 doi:10.1364 / Oe.20.028683

[0164]

[0012] A. Sahu, M. E. Çelik, D. E. Kirichenko, T. V. Filippov, and D. Gupta, “Low-power digital readout circuit for superconductor nanowire single-photon detectors,” IEEE Transactions on Applied Superconductivity, vol.29, no.5.2019 doi:10.1109 / Tasc.2019.2895051

[0165]

[0013] W. Chen, A. V. Rylyakov, V. Patel, J. E. Lukens, and K. K. Likharev, “Rapid single flux quantum T-flip flop operating up to 770 Ghz,” IEEE Transactions on Applied Superconductivity, vol.9, no.2, pp.3212–3215.1999 doi:10.1109 / 77.783712

[0166]

[0014] R. McDermott, M. G. Vavilov, B. L. T. Plourde, F. K. Wilhelm, P. J. Liebermann, O. A. Mukhanov, and T. A. Ohki, “Quantum–classical interface based on single flux quantum digital logic,” QuantumSeeQC-276.1 - 25 - Science and Technology, vol.3, no.2.2018 doi:10.1088 / 2058-9565 / aaa3a0

[0167]

[0015] P. Krantz, M. Kjaergaard, F. Yan, Orlando, S. Gustavsson, and W. D. Oliver, “A quantum engineer’s guide to superconducting qubits,” Applied Physics Reviews, vol.6, no.2.2019 doi:10.1063 / 1.5089550

[0168]

[0016] S. Doerner, A. Kuzmin, S. Wuensch, K. Ilin, and M. Siegel, “Operation of superconducting nanowire single-photon detectors embedded in lumped-element resonant circuits,” IEEE Transactions on Applied Superconductivity, vol.26, no.3.2016 doi:10.1109 / Tasc.2016.2525995

[0169]

[0017] S. Doerner, A. Kuzmin, S. Wuensch, I. Charaev, F. Boes, T. Zwick, and M. Siegel, “Frequency-multiplexed bias and readout of a 16-pixel superconducting nanowire single-photon detector array,” Applied Physics Letters, vol.111, no.3.2017 doi:10.1063 / 1.4993779

[0170]

[0018] M. Jerger, S. Poletto, P. Macha, U. Hübner, E. Il’ichev, and A. V. Ustinov, “Frequency division multiplexing readout and simultaneous manipulation of an array of flux qubits,” Applied Physics Letters, vol. 101, no.4.2012 doi:10.1063 / 1.4739454

[0171]

[0019] V. Semenov and Y. Polyakov, “Circuit improvements for a voltage multi- plier,” IEEE Transactions on Applied Superconductivity, vol.11, no.1, pp.550–553.2001 doi:10.1109 / 77.919404

[0172]

[0020] P. F. Hopkins, J. A. Brevik, M. Castellanos-Beltran, C. A. Donnelly, N. E. Flowers-Jacobs, A. E. Fox, D. Olaya, P. D. Dresselhaus, and S. P. Benz, “Rf waveform synthesizers with quantum-based voltage accuracy for communications metrology,” IEEE Transactions on Applied Superconductivity, vol.29, no.5. 2019 doi:10.1109 / Tasc.2019.2898407

[0173]

[0021] M. A. Castellanos-Beltran, D. I. Olaya, A. J. Sirois, C. A. Donnelly, P. D. Dresselhaus, S. P. Benz, and P. F. Hopkins, “Single-Flux-Quantum multiplier circuits for synthesizing gigahertz waveforms with quantum-based accuracy,” IEEE Transactions on Applied Superconductivity, vol.31, no.3.2021 doi:10.1109 / Tasc.2021.3057013

[0174]

[0022] A. Kunihiro, Y. Yamanashi, and N. Yoshikawa, “Pulse-density-modulated microwave generator using single-flux-quantum circuits for controlling qubits,” IEEE Transactions on Applied Superconductivity, vol.33, no.5.2023 doi:10.1109 / Tasc.2023.3255182

[0175]

[0023] O. A. Mukhanov, “Rapid single flux quantum (RSFQ) shift register family,” IEEE Transactions on Applied Superconductivity, vol.3, no.1, pp.2578–2581.1993 doi:10.1109 / 77.233529

[0176]

[0024] C. A. Mancini, N. Vukovic, A. M. Herr, K. Gaj, M. F. Bocko, and M. J. Feldman, “RSFQ circular shift registers,” IEEE Transactions on Applied Superconductivity, vol.7, no.2, pp.2832–2835.1997 doi:10.1109 / 77.621826

[0177]

[0025] C. A. Mancini, N. Vukovic, M. F. Bocko, M. J. Feldman, and A. M. Herr, “Experimental investigation of rsfq circular shift registers,” Applied Superconductivity, vol.6, no.10-12, pp.805–808.1998 doi:10.1016 / S0964-1807(99)00046-0SeeQC-276.1 - 26 -

[0178]

[0026] A. Rylyakov, “New design of single-bit all-digital RSFQ autocorrelator,” IEEE Transactions on Applied Superconductivity, vol.7, no.2, pp.2709– doi:10.1109 / 77.621797

[0179]

[0027] A. MacFarlane, “An analysis of a type of comb filter,” Proceedings of the IEE - Part B: Electronic and Communication Engineering, vol.107, pp.39–52(13). January 1960 doi:10.1049 / pi-b- 2.1960.0070

[0180]

[0028] E. B. Hogenauer, “An economical class of digital-filters for decimation and interpolation,” IEEE Transactions on Acoustics Speech and Signal Processing, vol.29, no.2, pp.155–162.1981 doi:10.1109 / Tassp.1981.1163535

[0181]

[0029] O. Mukhanov, A. Inamdar, T. Filippov, A. Sahu, S. Sarwana, and V. Se- menov, “Superconductor components for direct digital synthesizer,” IEEE Transactions on Applied Superconductivity, vol.17, no.2, pp.416–421.2007 doi:10.1109 / Tasc.2007.898055

[0182]

[0030] R. N. Tadros and P. A. Beerel, “A robust and tree-free hybrid clocking technique for RSFQ circuits - CSR application,” 201716th International Superconductive Electronics Conference (Isec).2017 doi:10.1109 / ISEC.2017.8314213

[0183]

[0031] V. K. Kaplunenko, “Fluxon interaction in an overdamped josephson transmission-line,” Applied Physics Letters, vol.66, no.24, pp.3365– 3367.1995 doi:10.1063 / 1.113759

[0184]

[0032] PSCAN2 Superconducting circuit simulator. pscan2sim.org /

[0185]

[0033] D. Yohannes, M. Renzullo, J. Vivalda, A. C. Jacobs, M. Yu, J. Walter, A. F. Kirichenko, I. V. Vernik, and O. A. Mukhanov, “High density fabrication process for single flux quantum circuits,” Applied Physics Letters, vol.122, no.21, p.212601.052023 doi:10.1063 / 5.0152552

[0186]

[0034] J. T. VanderPlas, “Understanding the lomb-scargle periodogram,” Astrophysical Journal Supplement Series, vol.236, no.1.2018 doi:10.3847 / 1538-4365 / aab766

[0187]

[0035] Y. X. He, H. X. Shen, S. Michibayashi, X. H. Zou, X. J. Xie, L. S. Yan, W. Pan, and N. Yoshikawa, “Compact RSFQ microwave pulse generator based on an integrated RF module for controlling superconducting qubits,” Applied Physics Letters, vol.120, no.6.2022 doi:10.1063 / 5.0083972

[0188]

[0036] K. Li, R. McDermott, and M. G. Vavilov, “Hardware-efficient qubit control with single-flux- quantum pulse sequences,” Phys. Rev. Appl., vol.12, p.014044. Jul 2019 doi:10.1103 / PhysRevApplied.12.014044

[0189]

[0037] O. A. Mukhanov, D. Gupta, A. M. Kadin, and V. K. Semenov, “Super- conductor analog-to- digital converters,” Proceedings of the IEEE, vol.92, no.10, pp.1564–1584.2004 doi:10.1109 / Jproc.2004.833660

[0190]

[0038] A. Sahu, T. Filippov, M. Radparvar, D. Kirichenko, and D. Gupta, “Digital time-division multiplexing readout circuit for sensor arrays,” IEEE Transactions on Applied Superconductivity, vol.27, no.4. 2017 doi:10.1109 / Tasc.2016.2637336SeeQC-276.1 - 27 -

[0191]

[0039] Y. Chen, D. Sank, P. O’Malley, T. White, R. Barends, B. Chiaro, J. Kelly, E. Lucero, M. Mariantoni, A. Megrant, C. Neill, A. Vainsencher, Y. Yin, A. N. Cleland, and J. M. Martinis, “Multiplexed dispersive readout of superconducting phase qubits,” Applied Physics Letters, vol.101, no.18, p. 182601.112012 doi:10.1063 / 1.4764940

[0192]

[0040] T. Filippov, O. Mukhanov, "Ultrafast Circuitry for Digital Filtering," US Patent No.7,991,814, Filed 8 / 24 / 2007, Issued 8 / 2 / 2011

[0193]

[0041] Rabiner, Lawrence R., and Bernard Gold. Theory and Application of Digital Signal Processing:(by) Lawrence R. Rabiner (and) Bernard Gold. Prentice-Hall, 1975.

[0194] J. C. Bardin, D. Sank, O. Naaman, and E. Jeffrey, “Quantum computing: An introduction for microwave engineers,” IEEE Microw. Mag., vol.21, no.8, pp.24–44, Aug.2020.

[0195] F. Arute et al., “Quantum supremacy using a programmable superconducting processor,” Nature, vol. 574, no.7779, pp.505–510, 2019.

[0196] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge, U.K.: Cambridge Univ. Press, 2000.

[0197] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, “Trapped-ion quantum computing: Progress and challenges,” Appl. Phys. Rev., vol.6, no.2, 2019, Art. no.021314.

[0198] T. P. Harty et al., “High-fidelity preparation, gates, memory, and readout of a trapped-ion quantum bit,” Phys. Rev. Lett., vol.113, no.22, 2014, Art. no.220501.

[0199] J. Levy, “Universal quantum computation with spin-1 / 2 pairs and Heisenberg exchange,” Phys. Rev. Lett., vol.89, Sep.2002, Art. no.147902.

[0200] X. Croot et al., “Device architecture for coupling spin qubits via an intermediate quantum state,” Phys. Rev. Appl., vol.10, Oct.2018, Art. no.044058.

[0201] D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K. B. Whaley, “Universal quantum computation with the exchange interaction,” Nature, vol.408, no.6810, pp.339–342, 2000.

[0202] J. Medford et al., “Quantum-dot-based resonant exchange qubit,” Phys. Rev. Lett., vol.111, Jul.2013, Art. no.050501.

[0203] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few- electron quantum dots,” Rev. Mod. Phys., vol.79, pp.1217–1265, Oct.2007.

[0204] J. M. Martinis, M. H. Devoret, and J. Clarke, “Energy-level quantization in the zero-voltage state of a current-biased Josephson junction,” Phys. Rev. Lett., vol.55, no.15, pp.1543–1546, 1985.

[0205] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature, vol.398, no.6730, pp.786–788, 1999.

[0206] J. Koch et al., “Charge-insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A, vol. 76, no.4, 2007, Art. no.042319.SeeQC-276.1 - 28 -

[0207] P. Krantz et al., “A quantum engineer’s guide to superconducting qubits,” Appl. Phys. Rev., vol.6, no. 2, 2019, Art. no.021318.

[0208] Z. Chen, “Metrology of quantum control and measurement in superconducting qubits,” Ph.D. dissertation, UC Santa Barbara, 2018.

[0209] A. P. Place et al., “New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds,” 2020, arXiv:2003.00024.

[0210] P. Jurcevic et al., “Demonstration of quantum volume 64 on a superconducting quantum computing system,” 2020, arXiv:2008.08571.

[0211] P. Klimov et al., “Fluctuations of energy-relaxation times in superconducting qubits,” Phys. Rev. Lett., vol.121, no.9, 2018, Art. no.090502.

[0212] F. H. L. Koppens et al., “Driven coherent oscillations of a single electron spin in a quantum dot,” Nature, vol.442, no.7104, pp.766–771, 2006.

[0213] J. van Dijk et al., “Impact of classical control electronics on qubit fidelity,” Phys. Rev. Appl., vol.12, no.4, 2019, Art. no.044054.

[0214] C. Ospelkaus et al., “Microwave quantum logic gates for trapped ions,”

[0215] Nature, vol.476, no.7359, pp.181–184, 2011.

[0216] M. Steffen, J. M. Martinis, and I. L. Chuang, “Accurate control of Josephson phase qubits,” Phys. Rev. B, vol.68, no.22, 2003, Art. no.224518.

[0217] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, “Simple pulses for elimination of leakage in weakly nonlinear qubits,” Phys. Rev. Lett., vol.103, no.11, 2009, Art. no.110501.

[0218] J. M. Chow et al., “Optimized driving of superconducting artificial atoms for improved single-qubit gates,” Phys. Rev. A, vol.82, no.4, 2010, Art. no.040305.

[0219] Z. Chen et al., “Measuring and suppressing quantum state leakage in a superconducting qubit,” Phys. Rev. Lett., vol.116, no.2, 2016, Art. no.020501.

[0220] D. P. L. Aude Craik et al., “High-fidelity spatial and polarization addressing of 43Ca+ qubits using near-field microwave control,” Phys. Rev. A, vol.95, no.2, 2017, Art. no.022337.

[0221] C. Piltz, T. Sriarunothai, A. F. Varón, and C. Wunderlich, “A trappedion-based quantum byte with 10−5 next-neighbour cross-talk,” Nature Commun., vol.5, no.1, 2014, Art. no.4679.

[0222] D. Leibfried, “Individual addressing and state readout of trapped ions utilizing RF micromotion,” Phys. Rev. A, vol.60, no.5, pp. R3335–R3338, 1999.

[0223] P. Staanum, and M. Drewsen, “Trapped-ion quantum logic utilizing position-dependent ac Stark shifts,” Phys. Rev. A, vol.66, no.4, 2002, Art. no.040302.

[0224] U. Warring et al., “Individual-ion addressing with microwave field gradients,” Phys. Rev. Lett., vol.110, no.17, 2013, Art. no.173002.SeeQC-276.1 - 29 -

[0225] R. Srinivas, “Laser-free trapped-ion quantum logic with a radiofrequency magnetic field gradient,” Ph.D. dissertation, Univ. Colorado, Boulder, 2020.

[0226] K. Takeda, A. Noiri, J. Yoneda, T. Nakajima, and S. Tarucha, “Resonantly driven singlet-triplet spin qubit in silicon,” Phys. Rev. Lett., vol.124, Mar.2020, Art. no.117701.

[0227] D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, “Efficient Z gates for quantum computing,” Phys. Rev. A, vol.96, no.2, 2017, Art. no.022330.

[0228] R. Srinivas et al., “Trapped-ion spin-motion coupling with microwaves and a near-motional oscillating magnetic field gradient,” Phys. Rev. Lett., vol.122, no.16, 2019, Art. no.163201.

[0229] F. Yan et al., “Tunable coupling scheme for implementing highfidelity two-qubit gates,” Phys. Rev. App., vol.10, no.5, 2018, Art. no.054062.

[0230] C. Rigetti and M. Devoret, “Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies,” Phys. Rev. B, vol.81, no.13, 2010, Art. no.134507.

[0231] M. D. Shulman et al., “Demonstration of entanglement of electrostatically coupled singlet-triplet qubits,” Science, vol.336, no.6078, pp.202–205, 2012.

[0232] A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, “Hybrid quantum systems with circuit quantum electrodynamics,” Nature Phys., vol.16, no.3, pp.257–267, 2020.

[0233] F. Mintert and C. Wunderlich, “Ion-trap quantum logic using longwavelength radiation,” Phys. Rev. Lett., vol.87, no.25, 2001, Art. no.257904.

[0234] M. Johanning et al., “Individual addressing of trapped ions and coupling of motional and spin states using RF radiation,” Phys. Rev. Lett., vol.102, no.7, 2009, Art. no.073004.

[0235] R. T. Sutherland et al., “Versatile laser-free trapped-ion entangling gates,” New J. Phys., vol.21, no. 3, 2019, Art. no.033033.

[0236] C. Ospelkaus et al., “Trapped-ion quantum logic gates based on oscillating magnetic fields,” Phys. Rev. Lett., vol.101, no.9, 2008, Art. no.090502.

[0237] U. Warring et al., “Techniques for microwave near-field quantum control of trapped ions,” Phys. Rev. A, vol.87, no.1, 2013, Art. no.013437.

[0238] J. Welzel, F. Stopp, and F. Schmidt-Kaler, “Spin and motion dynamics with zigzag ion crystals in transverse magnetic gradients,” J. Phys. B, Atomic Mol. Opt. Phys., vol.52, no.2, 2019, Art. no.025301.

[0239] T. P. Harty et al., “High-fidelity trapped-ion quantum logic using near-field microwaves,” Phys. Rev. Lett., vol.117, no.14, 2016, Art. no.140501.

[0240] S. Weidt et al., “Trapped-ion quantum logic with global radiation fields,” Phys. Rev. Lett., vol.117, no. 22, 2016, Art. no.220501.

[0241] H. Hahn et al., “Integrated 9Be+ multi-qubit gate device for the ion-trap quantum computer,” npj Quantum Inf., vol.5, no.1, 2019, Art. no.70.SeeQC-276.1 - 30 -

[0242] G. Zarantonello et al., “Robust and resource-efficient microwave nearfield entangling 9Be+ gate,” Phys. Rev. Lett., vol.123, no.26, 2019, Art.

[0243] V. M. Schäfer et al., “Fast quantum logic gates with trapped-ion qubits,” Nature, vol.555, no.7694, pp.75–78, 2018.

[0244] J. Raftery et al., “Direct digital synthesis of microwave waveforms for quantum computing,” 2017, arXiv:1703.00942.

[0245] W. D. Kalfus et al., “High-fidelity control of superconducting qubits using direct microwave synthesis in higher Nyquist zones,” 2020, arXiv:2008.02873.

[0246] A. Wallraff et al., “Approaching unit visibility for control of a super-conducting qubit with dispersive readout,” Phys. Rev. Lett., vol.95, no.6, 2005, Art. no.060501.

[0247] D. H. Slichter et al., “Measurement-induced qubit state mixing in circuit QED from up-converted dephasing noise,” Phys. Rev. Lett., vol.109, no.15, 2012, Art. no.153601.

[0248] J. Aumentado, “Superconducting parametric amplifiers: The state of the art in Josephson parametric amplifiers,” IEEE Microw. Mag., vol.21, no.8, pp.45–59, Aug.2020.

[0249] R. Vijay, D. H. Slichter, and I. Siddiqi, “Observation of quantum jumps in a superconducting artificial atom,” Phys. Rev. Lett., vol.106, no.11, 2011, Art. no.110502.

[0250] E. Jeffrey et al., “Fast accurate state measurement with superconducting qubits,” Phys. Rev. Lett., vol.112, no.19, 2014, Art. no.190504.

[0251] M. D. Reed et al., “Fast reset and suppressing spontaneous emission of a superconducting qubit,” Appl. Phys. Lett., vol.96, no.20, 2010, Art. no.203110.

[0252] A. Opremcak et al., “Measurement of a superconducting qubit with a microwave photon counter,” Science, vol.361, no.6408, pp.1239–1242, 2018.

[0253] A. Opremcak et al., “High-fidelity measurement of a superconducting qubit using an on-chip microwave photon counter,” 2020, arXiv:2008.02346.

[0254] W. Macalpine and R. Schildknecht, “Coaxial resonators with helical inner conductor,” Proc. IRE, vol. 47, no.12, pp.2099–2105, 1959.

[0255] D. Gandolfi, M. Niedermayr, M. Kumph, M. Brownnutt, and R. Blatt, “Compact radio-frequency resonator for cryogenic ion traps,” Rev. Sci. Instrum., vol.83, no.8, 2012, Art. no.084705.

[0256] M. F. Brandl, P. Schindler, T. Monz, and R. Blatt, “Cryogenic resonator design for trapped ion experiments in Paul traps,” Appl. Phys. B, vol.122, no.6, 2016, Art. no.157.

[0257] M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: Progress and challenges,” J. Phys. B, Atomic Mol. Opt. Phys., vol.49, no.20, 2016, Art. no.202001.

[0258] M. A. Castellanos-Beltran and K. W. Lehnert, “Widely tunable parametric amplifier based on a superconducting quantum interference device array resonator,” Appl. Phys. Lett., vol.91, no.8, 2007, Art. no.SeeQC-276.1 - 31 - 083509.

[0259] T. Yamamoto et al., “Flux-driven amplifier,” Appl. Phys. Lett., vol.93, no.4, 2008, Art. no.042510.

[0260] R. Vijay, “Josephson bifurcation amplifier: Amplifying quantum signals using a dynamical bifurcation,” Ph.D. dissertation, Yale Univ., 2008.

[0261] N. Bergeal et al., “Phase-preserving amplification near the quantum limit with a Josephson ring modulator,” Nature, vol.465, no.7294, pp.64–68, 2010.

[0262] M. Hatridge et al., “Quantum back-action of an individual variable-strength measurement,” Science, vol.339, no.6116, pp.178–181, 2013.

[0263] J. Y. Mutus et al., “Strong environmental coupling in a Josephson parametric amplifier,” Appl. Phys. Lett., vol.104, no.26, 2014, Art. no.263513.

[0264] T. Roy et al., “Broadband parametric amplification with impedance engineering: Beyond the gain- bandwidth product,” Appl. Phys. Lett., vol.107, no.26, 2015, Art. no.262601.

[0265] O. Naaman et al., “High saturation power Josephson parametric amplifier with GHz bandwidth,” in Proc. IEEE MTT-S Int. Microw. Symp., vol.2019, Jun.2019, pp.259–262.

[0266] A. Macklin et al., “A near-quantum-limited Josephson traveling-wave parametric amplifier,” Science, vol.350, no.6258, pp.307–310, 2015.

[0267] S. Pauka et al., “A cryogenic interface for controlling many qubits,” 2019, arXiv:1912.01299.

[0268] B. Patra, et al., 2020. A scalable cryo-CMOS 2-to-20GHz digitally intensive controller for 4× 32 frequency multiplexed spin qubits / transmons in 22nm FinFET technology for quantum computers. In 2020 IEEE International Solid-State Circuits Conference, ISSCC 2020 (pp.304-306). Institute of Electrical and Electronics Engineers (IEEE).

[0269] J. C. Bardin et al., “A 28 nm bulk-CMOS 4-to-8 GHz 2 mW cryogenic pulse modulator for scalable quantum computing,” in Proc. IEEE Int. Solid-State Circuits Conf., 2019, pp.456–458.

[0270] J. C. Bardin et al., “Design and characterization of a 28-nm bulk-cmos cryogenic quantum controller dissipating less than 2 mW at 3 K,” IEEE J. Solid-State Circuits, vol.54, no.11, pp.3043–3060, 2019.

[0271] I. Bashir, et al., 2019, "A mixed-signal control core for a fully integrated semiconductor quantum computer system-on-chip." In ESSCIRC 2019-IEEE 45th European Solid State Circuits Conference (ESSCIRC), pp.125-128. IEEE, 2019.

[0272] M. Rol et al., “Fast, high-fidelity conditional-phase gate exploiting leakage interference in weakly anharmonic superconducting qubits,” Phys. Rev. Lett., vol.123, no.12, 2019, Art. no.120502.

[0273] N. Takeuchi et al., "Microwave-multiplexed qubit controller using adiabatic superconductor logic", npj Quantum Information volume 10, Article number: 53 (2024) www.nature.com / articles / s41534-024-00849-2

[0274] A.F. Kirichenko, et al., “ERSFQ 8-bit parallel adders as a process benchmark,” IEEE Trans. Appl. Supercond.,SeeQC-276.1 - 32 - (Volume: 25, Issue: 3, June 2015), article 1300505.

[0275] D. Yohannes, et al., “High-density fabrication for single-flux-quantum circuits”, Appl. Phys. Lett.122, 212601 (2023).

[0276] Lin, Wen-Bin, and Bin-Da Liu. "The high-resolution multi-tone signal generators." In Asia-Pacific Conference on Circuits and Systems, vol.1, pp.245-250. IEEE, 2002.

[0277] Prathapan, Mridula, Peter Mueller, Christian Menolfi, Matthias Brändli, Marcel Kossel, Pier Andrea Francese, David Heim et al. "A cryogenic SRAM based arbitrary waveform generator in 14 nm for spin qubit control." In ESSCIRC 2022-IEEE 48th European Solid State Circuits Conference (ESSCIRC), pp.57-60. IEEE, 2022.

[0278] Van Dijk, Jeroen Petrus Gerardus, Bishnu Patra, Sushil Subramanian, Xiao Xue, Nodar Samkharadze, Andrea Corna, Charles Jeon et al. "A scalable cryo-CMOS controller for the wideband frequency-multiplexed control of spin qubits and transmons." IEEE Journal of Solid-State Circuits 55, no.11 (2020): 2930-2946.

[0279] Toubeix, Mathieu, Eric Guthmuller, Adrian Evans, Antoine Faurie, and Tristan Meunier. "FASQuiC: Flexible Architecture for Scalable spin Qubit Control." IEEE Transactions on Quantum Engineering (2024).

[0280] Takeuchi, Naoki, Taiki Yamae, Taro Yamashita, Tsuyoshi Yamamoto, and Nobuyuki Yoshikawa. "Microwave-multiplexed qubit controller using adiabatic superconductor logic." npj Quantum Information 10, no.1 (2024): 53.

[0281] Yu, He, Cong Wang, David A. Humphreys, Maoliu Lin, Luqman Ali, and Fanyi Meng. "Analysis of Time and Frequency Response for Single / Multi-Tone Stimulus Harmonic Phase-Reference Based on Prime Number Algorithm." Electronics 9, no.11 (2020): 1836.

[0282] Perelshtein, Mikhail. "Harnessing Quantum Resources in Superconducting Devices for Computing and Sensing." (2024).

[0283] Sypkens, Sasha, Lorenzo Minutolo, Sahil Patel, Emanuel Knehr, Alexander B. Walter, Henry G. Leduc, Lautaro Narváez et al. "Frequency-domain multiplexing of SNSPDs with tunable superconducting resonators." arXiv preprint arXiv:2401.17454 (2024).

[0284] Simons, Rainee N., and Edwin G. Wintucky. Multi-Band (K-Q-and E-Band) Multi-Tone Millimeter- Wave Frequency Synthesizer for Radio Wave Propagation Studies. No. NASA / TM-2014-218438.2014.

[0285] Fu, Yongqing, Shengnan Guo, and Zhixin Yu. "The modulation technology of chaotic multi-tone and its application in covert communication system." IEEE Access 7 (2019): 122289-122301.

[0286] Bulzacchelli, John Francis. "A superconducting bandpass delta-sigma modulator for direct analog-to- digital conversion of microwave radio." PhD diss., Massachusetts Institute of Technology, 2003.

[0287] Gebauer, Richard. "A Flexible FPGA-based Control Platform for Superconducting Multi-Qubit Experiments." PhD Dissertation, Karlsruhe, Karlsruher Institut für Technologie (KIT), 2021, 2022.SeeQC-276.1 - 33 -

[0288] Jerger, Markus, Stefano Poletto, Pascal Macha, Uwe Hübner, Evgeni Il’ichev, and Alexey V. Ustinov. "Frequency division multiplexing readout and manipulation of an array of flux qubits." Applied Physics Letters 101, no.4 (2012).

[0289] Chen, Yu, D. Sank, P. O'Malley, T. White, R. Barends, B. Chiaro, J. Kelly et al. "Multiplexed dispersive readout of superconducting phase qubits." Applied Physics Letters 101, no.18 (2012).

[0290] João N. F. Barbosa, “Single Flux Quantum Control Chip for Integrated Quantum Circuits”, Ph.D. dissertation, University of Glasgow (2024).

[0291] US 7,969,178, "Method and apparatus for controlling qubits with SFQ logic".

[0292] US 8,099,179, "Systems, methods, and apparatus for digital-to-analog conversion of superconducting magnetic flux signals".

[0293] US 8,138,784, "Method and apparatus for controlling qubits with SFQ logic" .

[0294] US 10,726,351, "System and method for controlling superconducting qubits with SFQ logic”.

[0295] US 10,797,684, "Superconducting waveform synthesizer".

[0296] US 9,853,645, “Low-power biasing networks for superconducting integrated circuits”.

[0297] WO 2017 / 078735, “Individual Qubit Excitation Control”.

[0298] What is claimed is:

Claims

SeeQC-276.1 - 34 - CLAIMS 1. A multi-tone generator, a clocked single flux quantum circuit configured to generate a pulse train comprising a plurality of pulses, each pulse having a pulse peak, consecutive pulse peaks being separated by a period corresponding to a period of the clock; and a clocked shift register coupled with the clocked single flux quantum circuit, the clocked shift register having an encoded bit pattern, and being configured to produce an output dependent on the encoded bit pattern and the period of the clock, such that the bit pattern modulates the pulse train with a plurality of frequency components.

2. The multi-tone generator according to claim 1, wherein the clocked shift register has at least one feedback path.

3. The multi-tone generator according to claims 1 or 2, wherein the clocked shift register has at least one feedforward path.

4. The multi-tone generator according to any of claims 1 to 3, wherein the clocked shift register comprises a circular shift register.

5. The multi-tone generator according to any of claims 1 to 4, further comprising at least two distinct paths in the clocked shift register and a summer configured to sum signals from each of the at least two distinct paths.

6. The multi-tone generator according to any of claims 1 to 5, wherein the clocked single flux quantum circuit comprises a rapid single flux quantum circuit.

7. The multi-tone generator according to any of claims 1 to 6, wherein the output is produced by at least one Josephson junction having an overdamped output.

8. The multi-tone generator according to any of claims 1 to 7, further comprising a comb filter configured to filter the output and to produce a filtered output.

9. The multi-tone generator according to any of claims 1 to 8, further comprising a filter configured to receive the output and to produce an analog microwave signal comprising the plurality of frequency components which interact with at least one quantum device.

10. The multi-tone generator according to claim 9, wherein the analog microwave signals selectively interact with at least one transmon qubit.

11. The multi-tone generator according to claim 9 or 10, wherein the analog microwave signals interact with at least one superconducting nanowire single photon detector.

12. The multi-tone generator according to any of claims 9 to 11, wherein the analog microwave signals interact with at least one quantum sensor.

13. The multi-tone generator according to any of claims 9 to 12, wherein the analog microwaveSeeQC-276.1 - 35 - signals comprise an orthogonal frequency multiplexed signal.

14. The multi-tone generator of claims 9 to 13, wherein the analog microwave signals interact with at least one qubit, wherein the single flux quantum circuit comprises a niobium superconducting layer configured to operate at a temperature of 4 °K, and the qubit is configured to operate at a temperature of less than 0.1 °K.

15. The multi-tone generator according to any of claims 1 to 14, wherein the single flux quantum circuit, the shift register, and the comb filter are formed on a common planar substrate.

16. The multi-tone generator according to any of claims 1 to 15, further comprising a qubit, a cryochamber surrounding the single flux quantum circuit and the qubit, and a cryocooler having a plurality of stages, wherein the single flux quantum circuit and the qubit are operated at different temperatures corresponding to different stages of the cryocooler.

17. A multi-tone generator, comprising: a single flux quantum circuit clocked configured to receive a clock signal at a clock frequency of at least 10 GHz, and to generate a pulse train, wherein the pulse train comprises a plurality of peaks separated by a period corresponding to the clock frequency; and a shift register driven at the clock frequency and encoded with a bit pattern, configured to receive the pulse train, and to generate a multi-tone signal, and to produce an output having a plurality of frequency components selectively dependent on the pulse train and the bit pattern. 18 A method of generating a multi-tone signal, comprising: generating a periodic pulse train comprising a plurality of pulses using a single flux quantum circuit, each pulse having a pulse peak; and producing an output dependent on an encoded bit pattern of a shift register coupled with the single flux quantum circuit, such that the bit pattern modulates the pulse train to produce a digital signal pulse density modulated output with a plurality of frequency components.

19. The method according to claim 18, wherein the shift register has at least one feedback path and at least one feedforward path, further comprising summing signals from each of the at least one feedback path and the at least one feedforward path with a summer.

20. The method according to claim 18 or 19, further comprising: filtering the digital signal pulse density modulated output with a digital comb filter to produce a comb filtered output; generating an analog microwave signal with an analog filter from the comb filtered output; and interacting the analog microwave signal with at least one of a qubit, a superconducting nanowire single photon detector, and a quantum sensor.

21. A digital multi-tone generator (DMTG), comprising:SeeQC-276.1 - 36 - a clocked single flux quantum circuit configured to generate a pulse train comprising a plurality of pulses, each pulse having a pulse peak, peaks being separated by a period corresponding to a period of the clock; and a clocked shift register coupled with the clocked single flux quantum circuit, the clocked shift register having an encoded a bit pattern, and being configured to produce an output dependent on the encoded bit pattern and the period of the clock, such that the bit pattern modulates the pulse train with a plurality of frequency components.

22. The digital multi-tone generator according to claim 21, wherein the clock has a regular period.

23. The digital multi-tone generator according to claims 21 or 22, wherein the clocked shift register has at least one feedback path.

24. The digital multi-tone generator according to any of claims 21 to 23, wherein the clocked shift register has at least one feedforward path.

25. The digital multi-tone generator according to any of claims 21 to 24, wherein the clocked shift register is a circular shift register.

26. The digital multi-tone generator according to any of claims 21 to 25, further comprising at least two distinct paths in the clocked shift register and a summer configured to sum signals from each of the at least two distinct paths.

27. The digital multi-tone generator according to any of claims 21 to 26, further comprising a comb filter configured to filter the output.

28. The digital multi-tone generator according to any of claims 21 to 27, further comprising a feedback implemented comb filter configured to filter the output.

29. The digital multi-tone generator according to any of claims 21 to 28, further comprising a feed forward implemented comb filter configured to filter the output.

30. The digital multi-tone generator according to any of claims 21 to 29, further comprising a low pass filter configured to filter the output.

31. The digital multi-tone generator according to any of claims 21 to 30, wherein the clocked single flux quantum circuit comprises a rapid single flux quantum circuit (RSFQ).

32. The digital multi-tone generator according to any of claims 21 to 31, wherein the clocked single flux quantum circuit comprises a rapid single flux quantum circuit (RSFQ) / DC driver circuit.

33. The digital multi-tone generator according to any of claims 21 to 32 wherein the plurality of frequency components of the output comprise harmonics.

34. The digital multi-tone generator according to any of claims 21 to 33, wherein the plurality of frequency components of the output comprise frequencies which are not harmonics.

35. A digital multi-tone generator (DMTG) comprising:SeeQC-276.1 - 37 - a single flux quantum circuit configured to be clocked at a clock frequency and to generate a pulse train, wherein the pulse train comprises a plurality separated by a period corresponding to the clock frequency; and a shift register communicatively connected to the single flux quantum circuit, driven at the clock frequency and encoded with a bit pattern to generate a multi-tone signal, wherein the shift register is configured to apply a modulation to the generated pulse train according to the bit pattern, such that the modulated pulse train is defined by a combination of a plurality of frequency components. 36 A method of generating a multi-tone signal, comprising: generating a pulse train comprising a plurality of pulses using a clocked single flux quantum circuit, each pulse having a pulse peak, consecutive pulse peaks being separated by a period corresponding to a period of the clock; and producing an output dependent on the period of the clock and an encoded bit pattern of a clocked shift register coupled with the clocked single flux quantum circuit, such that the bit pattern modulates the pulse train with a plurality of frequency components.

37. The method according to claim 36, wherein the clock has a regular period.

38. The method according to claim 36 or 37, wherein the clocked shift register has at least one feedback path.

39. The method according to any of claims 36 to 38, wherein the clocked shift register has at least one feedforward path.

40. The method according to any of claims 36 to 39, wherein the clocked shift register is a circular shift register.

41. The method according to any of claims 36 to 40, wherein the clocked shift register further comprises at least two distinct paths and a summer configured to sum signals from each of the at least two distinct paths.

42. The method according to any of claims 36 to 41, further comprising filtering the output with a comb filter.

43. The method according to any of claims 36 to 42, further comprising filtering output of the clocked shift register with a feedback implemented comb filter.

44. The method according to any of claims 36 to 43, further comprising filtering output of the clocked shift register with a feed forward implemented comb filter.

45. The method according to any of claims 36 to 44, further comprising filtering the output of the clocked shift register with a low pass filter.

46. The method according to any of claims 36 to 45, wherein the clocked single flux quantumSeeQC-276.1 - 38 - circuit comprises a rapid single flux quantum circuit (RSFQ).

47. The method according to any of 36 to 46, wherein the clocked single flux quantum circuit comprises a rapid single flux quantum circuit (RSFQ) / DC driver circuit.

48. The method according to any of claims 36 to 47, wherein the plurality of frequency components of the output comprise harmonics.

49. The method according to any of claims 36 to 48, wherein the plurality of frequency components of the output comprise frequencies which are not harmonics.

50. A method of generating a multi-tone signal, comprising: generating a pulse train with by clocking a single flux quantum circuit at a clock frequency, wherein the pulse train comprises a plurality of peaks separated by a period corresponding to the clock frequency; and generating a multi-tone signal with a shift register communicatively connected to the single flux quantum circuit, driven at the clock frequency and encoded with a bit pattern, wherein the shift register is configured to apply a modulation to the generated pulse train according to the bit pattern, such that the modulated pulse train is defined by a combination of a plurality of frequency components.

51. A multi-tone generator, comprising: a single flux quantum circuit comprising a Josephson junction configured to selectively generate single flux quantum pulses in a pulse modulated train representing a plurality of frequencies; and a shift register coupled with the single flux quantum circuit, encoding a bit pattern representing the plurality of frequencies, and being configured to control the pulse density pattern in accordance with the bit pattern.

52. The multi-tone generator according to claim 51, further comprising a comb filter circuit configured to receive the encoded bit pattern.

53. The multi-tone generator according to claim 52, further comprising an analog filter configured to receive an output of the comb filter and to produce an analog microwave signal comprising the plurality of frequencies.

54. The multi-tone generator according to any of claims 51 to 53, wherein the comb filter is configured to produce a digital representation of the plurality of the plurality of frequencies.

55. The multi-tone generator according to claim 54, wherein the digital representation of the plurality of the plurality of frequencies are filtered with an analog filter to produce a multi-tone microwave signal which interfaces with a plurality of qubits.

56. The multi-tone generator according to claim 54 or 55, wherein the digital representation of the plurality of the plurality of frequencies is produced by at least one Josephson junction having an overdamped output.SeeQC-276.1 - 39 - 57. The multi-tone generator according to any of claims 51 to 56, wherein the analog microwave signals each selectively interact with a transmon 58. The multi-tone generator according to any of claims 51 to 56, wherein the analog microwave signals interact with at least one superconducting nanowire single photon detector.

59. The multi-tone generator according to any of claims 51 to 58, wherein the analog microwave signals interact with at least one quantum sensor.

60. The multi-tone generator according to any of claims 51 to 59, wherein the analog microwave signals comprise an orthogonal frequency multiplexed signal.

61. The multi-tone generator according to any of claims 51 to 60, wherein the analog microwave signals interact with a qubit, wherein the single flux quantum circuit comprises a niobium superconducting layer configured to operate at a temperature of 4K, and the qubit is configured to operate at a temperature of less than 0.1K.

62. The multi-tone generator according to any of claims 51 to 61, wherein the plurality of frequencies are anharmonic.

63. The multi-tone generator according to any of claims 51 to 62, wherein the single flux quantum circuit and the shift register are formed on a common planar substrate.

64. The multi-tone generator according to any of claims 51 to 63, wherein the single flux quantum circuit, the shift register, and the comb filter are formed on a common planar substrate.

65. The multi-tone generator according to any of claims 51 to 64, further comprising a qubit, a cryochamber surrounding the single flux quantum circuit and the qubit, and a cryocooler having a plurality of stages, wherein the single flux quantum circuit is operated at a temperature of a first stage of the cryocooler and the qubit is operated at a temperature of a second stage of the cryocooler.

66. The multi-tone generator according to any of claims 51 to 65, wherein the circuit is designed according to at least one single flux quantum logic family, such families including rapid-single-flux-quantum, energy-efficient rapid-single-flux-quantum, reciprocal quantum logic, adiabatic quantum flux parametron, or pulse-conserving logic.

67. The multi-tone generator according to any of claims 51 to 66, further located within a cryogenic enclosure, further configured to transmit tones to a cryogenic quantum computing system or a cryogenic quantum sensor system located within the same cryogenic enclosure.

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