System and method for forming continuous variable cluster states and quantum correlations with microwaves

Abstract

The present disclosure relates to a system for forming continuous variable cluster states and quantum correlations with microwaves comprising a cryogenic chamber (161) with a cryocooler, a superconducting parametric circuit (110), SPC, comprising a resonator (111), a pumping circuitry (140) arrange to pump said SPC (110) with three or more microwave signals (260) to form a pump field (280) in the resonator (111), control circuitry (180) arranged to control the pumping circuitry (140). The cryocooler is arranged to prepare a vacuum state inside the cryogenic chamber (161), and the control circuitry (180) is arranged to control the pumping circuitry (140) to pump the SPC (110) with microwave signals (260). The pump field in the resonator (111) mixes with frequency components of the quantum fluctuations (240) at the vacuum level to generate quadrature-squeezed states of microwave electromagnetic waves (290) which form a continuous variable cluster state.

Description

[0001] System and method for forming continuous variable cluster states and quantum correlations with microwaves

[0002] TECHNICAL FIELD

[0003] The present disclosure relates to continuous variable cluster state generation, quantum computing, microwave regime, quadrature measurement, programable topology of correlations between microwave frequency modes in continuous-variable quantum cluster states, and qumodes.

[0004] BACKGROUND ART

[0005] Continuous Variable (CV) Cluster States are highly entangled quantum states used in quantum information processing, particularly in measurement-based quantum computing (MBQC). Unlike discrete-variable systems that use qubits, CV systems utilize continuous variables such as the quadratures of the electromagnetic field. These states can be represented as graph states, where each node corresponds to a mode of the electromagnetic field and edges represent entanglement between modes. CV cluster states are a crucial resource for MBQC, where quantum computations are performed by making measurements on an entangled resource state. The generation of CV cluster states begins with the creation of squeezed states of light, where quantum noise in one quadrature is reduced below the standard quantum limit at the expense of increased noise in the conjugate quadrature. These squeezed states are typically produced using optical parametric oscillators (OPOs) or amplifiers (OPAs). The squeezed states are then combined using beam splitters to create entanglement between different modes. Techniques such as frequency multiplexing, where different frequency modes of light are entangled, and time multiplexing, where temporal modes of light are entangled using time-delay elements and synchronized pulses, are employed to form large-scale CV cluster states. Nonlinear optical processes like four-wave mixing in optical fibres or microresonators, driven by multiple pump lasers, can also generate entangled modes. Additionally, optomechanical systems, which use the mechanical degrees of freedom, can be utilized to generate and manipulate CV cluster states.

[0006] These CV cluster states have significant applications in quantum computing, quantum communication, and quantum metrology. In quantum computing, they serve as a resource for performing computations in MBQC. In quantum communication, they enable secure communication protocols, and in quantum metrology, they enhance precision measurements beyond classical limits. The ability to generate and control CV cluster states is essential for advancing these fields. The process involves creating squeezed states, combining them using beam splitters, and employing techniques like frequency and time multiplexing or nonlinear optical processes. These methods allow for the generation of large-scale entangled states that are crucial for the development of quantum technologies.

[0007] The challenge lies in creating and verifying the complex multipartite entanglement necessary for effective computation. As the number of modes in the system increases, establishing entanglement through traditional bipartition tests becomes impractical. The complexity of these tests scales exponentially with the size of the system.

[0008] The ability to experimentally realize such a coherent structure of quantum correlations across a large number of modes is imperative for the practical implementation of measurement-based quantum computing in CV. The challenge lies in creating and verifying the complex multipartite entanglement necessary for effective computation. As the number of modes in the system increases, establishing entanglement through traditional bipartition tests becomes impractical. The complexity of these tests scales exponentially with the size of the system.

[0009] There is a need to provide methods and systems to realize large-scale CV cluster states.

[0010] SUMMARY OF THE INVENTION

[0011] One object of the invention is to realize large-scale CVCSs in the microwave domain.

[0012] This has in accordance with the present disclosure been achieved by means of a system for forming continuous variable cluster states and quantum correlations with microwaves. The system comprises a cryogenic chamber, a cryocooler arranged to reduce the temperature inside the cryogenic chamber to allow for a vacuum state, a superconducting parametric circuit, SPC, comprising a non-linear resonator. The resonator being arranged in said cryogenic chamber. A pumping circuitry arrange to pump said SPC with three or more microwave signals to form a pump field in the resonator, control circuitry arranged to control the pumping circuitry. The cryocooler is arranged to prepare a vacuum state inside the cryogenic chamber allowing intrinsic quantum fluctuations to interact with the resonator of the SPC, and the control circuitry is arranged to control the pumping circuitry to pump the SPC with the three or more microwave signals. Whereby the pump field in the resonator mixes with frequency components of the quantum fluctuations to generate quadrature-squeezed states of microwave electromagnetic waves, and wherein said quadrate- squeezed states of microwave electromagnetic waves form the continuous variable cluster state. This has the advantage of allowing formation of continuous variable cluster state in the microwave regime. The use of microwaves may provide improved scalability, better qubit compatibility, and a reduced decoherence of cluster states.

[0013] In some embodiments, the SPC comprises a parametric oscillator, and / or a parametric amplifier, and / or a Josephson parametric oscillator, and / or a Josephson parametric amplifier.

[0014] In some embodiments, the system comprises a quadrature detector arranged to receive the continuous variable cluster state from the resonator of the SPC, and wherein said quadrature detector is arranged to measure the continuous variable cluster state in a determined measurement basis.

[0015] This has the advantage of allowing the state of the formed cluster state to be verified. This may further have the advantage of allowing an interaction between the formed cluster state and component to be measured.

[0016] In some embodiments, the quadrature detector comprises a multi-frequency lock-in amplifier, and / or a heterodyne detector.

[0017] This has the advantage of allowing the system to measure multiple modes simultaneously.

[0018] In some embodiments, the system is arranged to output a classical signal indicative of the measured quadratures of the continuous variable cluster state.

[0019] This has the advantage of allowing the system to provide an output indicative of the formed cluster state.

[0020] In some embodiments, the determined measurement basis and measured frequencies of the continuous variable cluster states correspond to a frequency comb.

[0021] This has the advantage of allowing the generation of cluster states and determination of measurement basis that correspond to a frequency comb, which is desirable in the context of quantum computing and quantum information processing.

[0022] In some embodiments, the control circuitry being arranged to generate quadrature-squeezed states is arranged to control a quadrature detector to measure the quadratures of the squeezed microwave electromagnetic waves, and to control the pumping circuitry based on the measured values.

[0023] This has the advantage of allowing the system to adapt the pumping to achieve a desired cluster state. This further allows the system to explore possible microwave signals for pumping and measure the resulting cluster states. The present disclosure further relates to a method for forming continuous variable cluster states and quantum correlations with microwaves, the method comprises

[0024] - providing a superconducting parametric circuit, SPC, in a cryogenic chamber with a cryocooler, wherein the SPC comprises a non-linear resonator;

[0025] - cooling said resonator to a vacuum state, whereby intrinsic quantum fluctuations interact with the resonator of the SPC; and

[0026] - pumping the resonator of said SPC with three or more microwave signals to form a pump field in the resonator, whereby, the pump field in the resonator mixes with frequency components of the quantum fluctuations to generate quadrature-squeezed states of microwave electromagnetic waves, wherein said quadrature-squeezed states of microwave electromagnetic waves form a continuous variable cluster state.

[0027] The present disclosure further relates to a computer program product comprising a non- transitory computer-readable storage medium having thereon a computer program comprising program instructions. The computer program being loadable into a processor and configured to cause the processor to perform the method for forming continuous variable cluster states.

[0028] BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Fig. 1 depicts schematically a system for forming continuous variable cluster states and quantum correlations with microwaves.

[0030] Fig. 2 depicts schematically a flowchart for a system forming continuous variable cluster states and quantum correlations with microwaves.

[0031] Fig. 3 shows a method for forming continuous variable cluster states and quantum correlations with microwaves.

[0032] Fig. 4 depicts schematically a data processing unit comprising a computer program product.

[0033] Fig. 5a-b depicts schematically the frequency and mode connections of measured continuous variable cluster states.

[0034] DETAILED DESCRIPTION Throughout the figures, same reference numerals refer to same parts, concepts, and / or elements. Consequently, what will be said regarding a reference numeral in one figure applies equally well to the same reference numeral in other figures unless explicitly stated otherwise.

[0035] Terms and expressions

[0036] The term quadrature-squeezed electromagnetic waves relates to having reduced quantum noise in one quadrature, such as amplitude or phase, at the expense of increased noise in the orthogonal quadrature.

[0037] The term continuous variable cluster states relate to multipartite entangled states defined as the continuous superposition of quadrature eigenstates, such as qumodes, characterized by a well-defined topology of correlations. Continuous variable cluster states are typically characterized by a well-defined topology of correlations, represented by a canonical graph featuring a lattice-like topology, and formally defined as a continuous superposition of qumodes.

[0038] The term measurement basis relates to how the measurement is performed. In the context of continuous variable cluster states, the measurement basis defines how the quadratures of the modes of the continuous variable cluster state are measured.

[0039] The term superconducting parametric circuit relates to a type of circuit that leverages the unique properties of superconductors to achieve parametric amplification and / or oscillation.

[0040] The term parametric oscillator relates to a type of harmonic oscillator where the oscillations are driven by varying one or more parameters of the system, such as the resonance frequency or damping, at a frequency different from the natural frequency of the oscillator.

[0041] The term parametric amplifier relates to a device used to amplify signals by modulating a parameter of the system, such as capacitance or inductance, using an external pump signal.

[0042] The expression “driven by pumps” relate to providing energy with a signal that is the sum of different microwave signals, typically sine waves, with defined frequency, amplitude, and phase. The different microwave signals may also be referred to as the frequency components in the pump signal. Each microwave signal may be referred to as a pump, with the pump signal being the sum of such pumps.

[0043] The term pump field relates to providing energy to drive nonlinear interactions in a parametric circuit, such as a resonator of a parametric oscillator. The term microwave signals relate to the plurality of signals constituting the pump signal that causes the pump field in the resonator of the superconducting parametric circuit. Typically, each microwave signal is a different sine with a given amplitude, phase, and frequency.

[0044] The term frequency modes relates to distinct spectral components or frequencies at which the electromagnetic field oscillates. In continuous variable cluster states, these modes are used to create and entangle multiple quantum states.

[0045] The expression “relative phase of the pump” relates to the differences in phase between different microwave signals used to pump the resonator.

[0046] The term vacuum state relates to the lowest energy state of a quantum field, where all modes of the field are in their ground state.

[0047] The term intrinsic quantum fluctuations relates to the temporary and random changes in the amount of energy at a point in space, even in the absence of any particles or external fields.

[0048] The expression “intrinsic quantum fluctuations interact with the resonator” relates. It is to be understood that the quantum fluctuations always are present, and that the expression relates to enhancing the contribution of quantum fluctuations by cooling to the vacuum state.

[0049] The expression “mixing pumped electromagnetic waves with frequency components of quantum fluctuations” relates to utilizing the quantum noise and energy from the pumping to generate quadrature-squeezed microwave electromagnetic waves at the resonator.

[0050] The term frequency comb relates to a spectrum consisting of discrete, equally spaced spectral line. A frequency comb may be the result of the digital discretization of the measured continuous variable cluster state. It is to be understood that superconducting parametric circuit in general do not output frequency combs, instead it is by utilizing measurement devices to discretize the frequency space that the frequency comb is formed.

[0051] The expression “modes of a frequency comb” relates to the discrete, equally spaced frequency components that make up the comb.

[0052] The term square-ladder relates to a specific type of graph structure used to represent the entanglement and connectivity of modes in a continuous variable cluster state.

[0053] The term “square-ladder continuous-variable cluster states of a microwave frequency comb” relates to an advanced quantum state with a specific entanglement structure, typically used for efficient quantum computation. Fig. 1 shows an example system for forming continuous variable cluster states and quantum correlations with microwaves. The system 100 comprises

[0054] - a superconducting parametric circuit 110, SPC, comprising a non-linear resonator 111 ,

[0055] - pumping circuitry 140 arranged to pump said SPC 110 with three or more microwave signals to form a pump field in the resonator 111 ,

[0056] - a cryogenic system 160 comprising a cryogenic chamber 161 and a cryocooler (not shown), wherein the resonator 111 of the SPC 110 is arranged in the cryogenic chamber 161 , and wherein the cryogenic system 160 is arranged to reduce the temperature inside the cryogenic chamber 161 to allow for a vacuum state at the resonator 111 , and wherein the pumping circuitry 140 is arranged to, during a vacuum state at the resonator 111 , pump the SPC 110 with three or more microwave signals, whereby frequency components of intrinsic quantum fluctuations at the vacuum state mix with the pump field in the pumped resonator 111 to generate quadrature-squeezed states of microwave electromagnetic waves, wherein the quadrature-squeezed states of microwave electromagnetic waves form the continuous variable cluster state.

[0057] It is to be understood that the quadrature-squeezed states of microwave electromagnetic waves that escape the resonator carry the correlation information of the continuous variable cluster state formed in the resonator.

[0058] It is to be understood that the expression ”to reduce the temperature inside the cryogenic chamber 161 to allow for a vacuum state at the resonator 111” relates to reaching a state where the quantum fluctuations at the vacuum state constitute a significant part to the electromagnetic field, as opposed to the environments at temperatures over 1 kelvin where the quantum fluctuations are overshadowed by thermal noise. It is to be understood that, apart from the microwave signal pumping, the intrinsic quantum fluctuations typically serve as the sole input for forming said continuous variable cluster states.

[0059] Fig. 1 further schematically depicts a pump signal input 143, a resonator output 102, and a system output 103.

[0060] In some examples, the resonator 111 is arranged in the cryogenic chamber 161. In some of these examples, the SPC 110 is arranged in the cryogenic chamber 161. Typically, a low Q factor may be desired for the resonator 111 as it can allows many frequencies to fit inside the linewidth of the resonator 111.

[0061] In some examples, the cryogenic system 160 comprises a vacuum pump (not shown) arranged to evacuate the cryogenic chamber 161. In some examples, the cryogenic chamber 161 and the cryocooler are comprised in one device. Typically, the cryocooler comprises the cryogenic chamber 161.

[0062] In some examples, the cryogenic system 160 is arranged to reduce the temperature below 1 Kelvin to generate quadrature-squeezed states of microwave electromagnetic waves. In some of these examples, the cryogenic system 160 is arranged to reduce the temperature below 500 mK, 200 mK, 100 mK, 60 mK, 40 mK, 20mK, 15 mK, 10 mK, or 5 mK.

[0063] In some examples, the pumping circuitry 140 is arranged to transmit the pump signal comprising three or more microwave signals from the pump signal generator 141 as a DC pump signal component, and an AC pump signal component. In some of these examples, each component is transmitted via its own channel. In some of these examples, each component is combined in the cryogenic chamber 161.

[0064] In some examples, the pumping circuitry 140 comprises a microwave signal generator 141 and an inductor 142, wherein the inductor 142 is arranged to generate a magnetic flux at the SPC 110 based on the three or more microwave signals generated by and transmitted from the microwave signal generator 141. For example, in embodiments wherein the pumping circuitry 140 comprises a Josephson parametric amplifier. It is to be understood that the inductor 142 shown in fig. 1 is an optional feature.

[0065] In some examples, the pumping circuitry 140 is arranged to transmit the three or more generated microwave signals into the SPC 110, and / or into the resonator 111 of the SPC 110. In some of these examples, the pumping circuitry 140 is arranged to transmit the three or more generated microwave signals into the resonator 111 of the SPC 110 as electromagnetic radiation.

[0066] In some examples, the SPC 110 comprises a superconducting quantum interference device 112, SQUID, arranged be influenced by the magnetic flux provided by the pumping circuitry 140, whereby a pump field is formed in the resonator 111. For example, Josephson parametric amplifiers and oscillators comprise SQUIDs.

[0067] In some examples, the system is arranged to generate quadrature-squeezed states with a squeezing level of at least 1 dB. In some of these examples, at least 2 dB, or at least 3 dB.

[0068] In some examples, the system is arranged to form continuous variable cluster states is in the range of 0.1 MHz to 800GHz. In some of these examples, the system is arranged to form continuous variable cluster state is in the range of 1 MHz to 600 GHz, 3 MHz to 300 GHz, 10 MHz to 100 GHz, 40 MHz to 30 GHz, or 100 MHz to 10 GHz. In some examples, the pumping circuitry 140 is arranged to generate the pump signal comprising three or more microwave signals based on a predetermined pump scheme. Typically, the pump scheme and the measurement basis are determined based on each other and the resonator 111 in order to generate measurable quadrature-squeezed microwave electromagnetic waves.

[0069] In some examples, each of the three or more microwave signals is a sinusoidal electromagnetic wave with a specific amplitude, phase, and frequency, wherein the sum of the three or more microwave signals corresponds to a complex pumping scheme.

[0070] In some examples, the pumping circuitry 140 is arranged to pump said SPC 110 with three or more microwave pump signals 260 with a specific amplitude, phase, and frequency to generate two-dimensional continuous variable cluster states.

[0071] In some examples, the pumping circuitry 140 is arranged to pump said SPC 110 with three or more microwave pump signals 260 with a specific amplitude, phase, and frequency to generate two-dimensional continuous variable cluster states with honeycomb lattice topology.

[0072] In some examples, the pumping circuitry 140 is arranged to pump said SPC 110 with four or more microwave pump signals 260 with a specific amplitude, phase, and frequency to generate two-dimensional continuous variable cluster states with square lattice topology.

[0073] In some examples, the complex pumping scheme corresponds to at least one of multifrequency pumping, and / or phase and amplitude modulation.

[0074] It is to be understood that, in a first-order approximation, pumping with a microwave signal at a given frequency a>pconnects modes / and j that follow the frequency relation: o)j = o)p- For higher-order processes with multiple microwave signals pumping these relationships become more complicated. It is to be understood that higher-order processes become relevant with multiple microwave signals, such as secondary idlers mixing with the contributions of the other microwave signals.

[0075] In some examples, the SPC 110 is operated in a non-degenerate mode.

[0076] In some examples, determining the three or more microwave signals is based on the intended measurement basis for the formed continuous variable cluster state. Correspondingly, the measurement basis is typically selected based on the three or more microwave signals and the resonator properties. In some examples, determining the three or more microwave signals is based on the centre-frequency coo for said measurement basis. In some examples, the measurement basis for the formed continuous variable cluster state corresponds to a frequency comb with a centre-frequency coo.

[0077] In some examples, the pumping circuitry 140 is arranged to parametrically pump with three coherent microwave signals centred around double the centre-frequency of a frequency comb, 2wo. Utilizing a pump signal comprising microwave signals with equidistant frequencies may result in frequency comb generation or an output suitable to be measured with a measurement frequency comb.

[0078] In some examples, the first microwave signal has the frequency of 2wo, the second microwave signal has the frequency of 2coo-4A, and the third first microwave signal has the frequency of 2coo+4A, wherein A is based on the measurement time T, such as being inversely proportional. For example, A = 0.1 MHz.

[0079] It is to be understood that the central pump at 2wo determines the frequency modes’ enumeration scheme, whereas the relative phase of the pump at 2wo - 4A is set to TT in order to cancel higher-order mixing processes and engineer the output correlations. In some of these examples, the first and third microwave signals have the same phase, and the second microwave signal relative phase is TT.

[0080] In some examples, the first microwave signal has the frequencies of 2wo+A, the second microwave signal has the frequency of and 2wo -A, and the third microwave signal has the frequency of 2wo + (N -1)A, wherein A is based on the measurement time T, such as being inversely proportional. For example, A = 0.1MHz. N is the width of the two-dimensional, honeycomb lattice continuous variable cluster state.

[0081] It is to be understood that coo determines the frequency modes’ enumeration scheme, whereas the relative phase of the pump at 2wo + (N -1)A is set to TT in order to cancel higher-order mixing processes and engineer the output correlations. In some of these examples, the first and third microwave signals have the same phase. In some of these examples, the second microwave signal relative phase is TT.

[0082] In some examples, the first microwave signal has the frequencies of 2wo+A, the second microwave signal has the frequency of and 2wo -A, the third microwave signal has the frequency of 2wo+NA, and the forth microwave signal has the frequency of 2wo-NA, wherein A is based on the measurement time T, such as being inversely proportional. For example, A = 0.1 MHz. N is the width of the two-dimensional, square lattice, continuous variable cluster state.

[0083] It is to be understood that coo determines the frequency modes’ enumeration scheme, whereas the relative phase of the pump at 2wo - NA is set to TT in order to cancel higher-order mixing processes and engineer the output correlations. In some of these examples, the first and third microwave signals have the same phase, and the second microwave signal relative phase is TT.

[0084] In some examples, the system 100 further comprises a quadrature detector 150 arranged to receive the continuous variable cluster state output from the resonator 111 of the SPC 110, and to measure the continuous variable cluster state according to a determined measurement basis.

[0085] In some examples, the system 100 further comprise a heterodyne detector (not shown) arranged to determining quadrature-squeezing of the continuous variable cluster state. In some of these examples, determining the quadrature-squeezing of the continuous variable cluster state comprises measuring the quadrates of the squeezed microwave electromagnetic waves of the continuous variable cluster state utilizing said heterodyne detector. In some of these examples, the system 100 is arranged to output a classical signal indicative of the measured quadratures.

[0086] In some examples, the quadrature detector comprises a lock-in amplifier 150. In some of these examples, the quadrature detector comprises a multi-frequency lock-in amplifier. In some of these examples, the multi-frequency lock-in amplifier is arranged to perform a digital heterodyne detection. Utilizing multi-frequency lock-in amplifiers provide a phase lock and a frequency reference which is stable for time scales of typical experiments.

[0087] In some examples, the quadrature detector 150 is arranged to function as a heterodyne detector.

[0088] In some examples, the quadrature detector 150 comprises a lock-in amplifier is arranged to function as a heterodyne detector.

[0089] The use of a quadrature detector 150 to measure the continuous variable cluster state for the determined measurement basis, such as with a heterodyne detector, is optional for the system 100 for forming continuous variable cluster states and quantum correlations with microwaves. For the practical applications of such continuous variable cluster states it is, however, typically desirable to be able to measure in a measurement basis in order to verify the state of the formed continuous variable cluster states.

[0090] In some examples, the quadrature detector 150 is arranged to perform coherent modulation and demodulation simultaneously across a plurality of frequencies. In some of these examples, across at least 20 frequencies, at least 50 frequencies, or at least 100 frequencies. In some examples, the measured frequencies of the continuous variable cluster states correspond to a frequency comb. In some examples, the frequency spacing is in the range of 1 Hz to 10 MHz. In some of these examples, the frequency spacing is in the range of 10 Hz to 1 MHz, 100 Hz to 100 kHz, or 1kHz to 10 kHz.

[0091] It is to be understood that for use of continuous variable cluster states in quantum computing, the formed continuous variable cluster states typically need to interact with quantum computation components and thereafter become measured by detectors.

[0092] In some examples, the system 100 further comprises the quadrature detector 150 and at least one quantum computation component (not shown), wherein the system is arranged to make at least part of the formed continuous variable cluster state interact with said at least one quantum computation component, and thereafter measure the continuous variable cluster state with the quadrature detector 150. In some of these examples, said at least one quantum computation component is arranged in the cryogenic chamber 161.

[0093] In some examples, the system 100 further comprises control circuitry 180 arranged to control the pumping circuitry 140, and / or the quadrature detector 150. In some examples, said control circuitry 180 comprises a computer.

[0094] In some examples, the control circuitry 180 is arranged to generate quadrature-squeezed states further comprises controlling a sensor to measure the quadratures of the squeezed microwave electromagnetic waves. In some of these examples, measuring the quadratures of the squeezed microwave electromagnetic waves comprises performing quadrature rotation. In some of these examples, quadrature rotation comprises adjusting the phase of a local oscillator to measure different quadrature. In some of these examples, said sensor comprises a heterodyne detector.

[0095] In some examples, the quadrature detector 150, and / or a sensor for measuring the formed continuous variable cluster state is connected to the pumping circuitry 140, and the pumping circuitry 140 is arranged to pump based on the measurement values obtained from said components. In some of these examples, the control circuitry 180 is arranged to generate quadrature-squeezed states is arranged to control a quadrature detector 150 to measure the quadratures of the squeezed microwave electromagnetic waves, and to dynamically control the pumping circuitry 140 based on the measured values in a feedback loop to optimize the formation of the continuous variable cluster state. The use of a feedback loop may facilitate the prototyping of systems 100.

[0096] In some examples, the system 100 comprises at least one amplifier (not shown) arranged between the microwave signal generator 141 and part of the pumping circuitry 140 generating magnetic flux at the SPC 110, such as the inductor 142. In some of these examples, said at least one amplifier is arranged between the microwave signal generator 141 and pump signal input 143.

[0097] In some examples, the system 100 comprises at least one amplifier (not shown) arranged between the SPC 110 and the output 103 of the system 100. In some of these examples, said at least one amplifier arranged between the SPC 110 and the output 103 of the system 100. In some of these examples, the at least one amplifier is arranged between the output 102 of the resonator 111 and the output 103 of the system 100.

[0098] Typically, the part of the pumping circuitry that generates the three or more microwave signals, the output 103 of the system 100, and the quadrature detector 150 are arranged outside of the cryogenic chamber 161 , such as in a room temperature environment.

[0099] Fig. 2 depicts schematically an example flowchart for a system forming continuous variable cluster states and quantum correlations with microwaves. The flowchart 200 comprises a branch relating to quantum fluctuations in the resonator 210,220,230,240 and branch relating to pumping the resonator 250,260,270,280, wherein the final contributions of each branch are mixed in the resonator to form continuous variable cluster states.

[0100] The branch relating to quantum fluctuations starts with the cryocooler receiving instructions 210 to decrease the temperature at the resonator of a superconducting parametric circuit. Heat 220 is removed from the resonator resulting in a vacuum state 230 at the resonator at which the quantum fluctuations 240 provide a significant contribution to the electromagnetic field.

[0101] The branch relating to pumping starts with the pump generator receiving pump instructions 250 to generate microwave signals forming a pump signal 260. The pump signal 260 is transmitted from the pump generator to the resonator. At the resonator and / or at the superconducting parametric circuit a magnetic flux 270 is generated based on the pump signal 260. The pump signal 260 and / or magnetic flux 270 causes a pump field 280 to form in the resonator.

[0102] Mixing 201 between the frequency components of the quantum fluctuations 240 and the pump field 280 generates quadrature-squeezed microwave electromagnetic waves 290 in the resonator, wherein the pump signal have been selected such that the quadrature-squeezed microwave electromagnetic waves 290 form a continuous variable cluster state. It is to be understood that said mixing 201 occurs by itself under these conditions in a correctly configured superconducting parametric circuit.

[0103] In the example the superconducting parametric circuit comprises a superconducting quantum interference device, SQUID, that is arranged to interact with the generated magnetic flux 270 to in turn generate the pump field in the resonator, such as for a Josephson parametric oscillator. It is to be understood that in other types of systems the pump signal 260 may be directly transmitted into the resonator, thereby omitting the step of generating a magnetic flux 270.

[0104] Fig. 3 shows a method for forming continuous variable cluster states and quantum correlations with microwaves. The method 300 comprises

[0105] - providing 310 a superconducting parametric circuit, SPC, in a cryogenic chamber (161) with a cryocooler, wherein the SPC comprises a non-linear resonator;

[0106] - cooling 320 said resonator to a vacuum state, whereby intrinsic quantum fluctuations interact with the resonator of the SPC;

[0107] - pumping 330 the resonator of said SPC with three or more microwave signals to form a pump field in the resonator, whereby, the pump field in the resonator mixes with frequency components of the quantum fluctuations to generate quadrature-squeezed states of microwave electromagnetic waves, wherein said quadrature-squeezed states of microwave electromagnetic waves form a continuous variable cluster state.

[0108] In some examples, the method 300 comprises providing 370 an output comprising the formed continuous variable cluster state.

[0109] In some examples, pumping 330 the resonator comprises

[0110] - determining 331 the parameters for said three or more microwave signals,

[0111] - generating 332 said three or more microwave signals based on the determined parameters, and / or

[0112] - generating 333 a magnetic flux at the SPC arranged to form a pump field in the resonator of the SPC based on said three or more microwave signals.

[0113] In some examples, determining the parameters for said three or more microwave signals comprises for each microwave signal determining the phase, amplitude, and / or frequency. Typically, the term “phase” here relates to a relative phase between signals.

[0114] It is to be understood that the step of “generating 333 a magnetic flux at the SPC arranged to form a pump field in the resonator” typically relates to passing the generated microwave pump signals through an inductor arranged to impact a SQUID component, such as in a Josephson parametric oscillator. In some examples, said step may be replaced with introducing the generated three or more microwave signals into the resonator, thereby forming the pump field by a direct interaction between the generated three or more microwave signals and the resonator. In some examples, the method 300 comprises pumping 330 the resonator of said SPC with three or more microwave signals to form a pump field in the resonator, whereby, the pump field in the resonator mixes with frequency components of the quantum fluctuations of the prepared vacuum state in the absence of an injected coherent signal to generate quadrature-squeezed states of microwave electromagnetic waves, wherein said quadrature-squeezed states of microwave electromagnetic waves form a continuous variable cluster state.

[0115] In some examples, the method 300 comprises pumping 330 the resonator of said SPC with three or more microwave signals to form a pump field in the resonator, whereby, the pump field in the resonator mixes with frequency components of the quantum fluctuations to generate quadrature-squeezed states of microwave electromagnetic waves, wherein said quadrature-squeezed states of microwave electromagnetic waves form a continuous variable cluster state as a usable quantum resource output.

[0116] It is to be understood that the method 300, typically, serves to provide a usable quantum resource output from the system in which it was produced, and not merely measuring a continuous variable cluster state.

[0117] In some examples, the method comprises determining 350 the measurement basis for the continuous variable cluster state, and measuring 360 the continuous variable cluster state with a quadrature detector utilizing said determined measurement basis.

[0118] It is to be understood that the step of determining 350 the measurement basis and utilizing said determined measurement basis may in practice correspond to leaving a detector setting unchanged.

[0119] In some examples, said quadrature detector comprises a heterodyne detector, a lock-in amplifier, and / or a multi-frequency lock-in amplifier.

[0120] In some examples, measuring 360 the continuous variable cluster state comprises measuring 360 quadratures of the generated quadrature-squeezed states of microwave electromagnetic waves. In some of these examples, the measuring 360 comprises providing an output indicative of the measured quadrature.

[0121] In some examples, the formed continuous variable cluster state comprises a square-ladder cluster state.

[0122] In some examples, pumping the resonator comprises measuring the generated quadrature- squeezed states of microwave electromagnetic waves and adapting the generation of quadrature-squeezed states by adapting said three or more microwave signals based the measurement.

[0123] In some examples, the method further comprises pathing (not shown) the formed continuous variable cluster state to interact with one of more quantum computation components, and thereafter measuring 360 the continuous variable cluster state.

[0124] It is to be understood that to fully utilize the formed continuous variable cluster state a complex interaction of some kind, such as with the components of a quantum computer, preferably occurs before detection of the continuous variable cluster state. In a simple embodiment of the method 300 the last step of the method 300 is to provide a continuous variable cluster state output that can be used as an input for a quantum computer, or similar system. In a more complex embodiment, the method 300 may further comprise the step of interactions with quantum computation components, and the subsequent measurement of the continuous variable cluster state.

[0125] Fig. 4 depicts schematically a data processing unit comprising a computer program product for forming continuous variable cluster states and quantum correlations with microwaves. Fig. 4 depicts a data processing unit 410 comprising a computer program product comprising a non- transitory computer-readable storage medium 412. The non-transitory computer-readable storage medium 412 having thereon a computer program comprising program instructions.

[0126] The computer program is loadable into a data processing unit 410 and is configured to cause a processor 411 to carry out the method for forming continuous variable cluster states and quantum correlations with microwaves accordance with the description of fig. 3.

[0127] The data processing unit 410 may be comprised in a device 400.

[0128] The data processing unit 410 may be comprised in the system 100 described in fig. 1.

[0129] Experimental example of forming continuous variable cluster states.

[0130] An example experiment to create a square-ladder topology cluster state, see fig. 5a-b. Fig. 5a depicts the frequency of 95 modes centred around >0(corresponding to the measurement basis as a frequency comb) and at approximately twice the frequency the three microwave signals comprising the pump signal. Fig. 5b represents three cluster states 501,502,503 among the 95 modes.

[0131] This was done utilizing a Josephson parametric amplifier, JPA, featuring a resonance frequency around >0= 4.2 GHz, by suitably tuning the DC bias voltage, and set a measuring time window to T = 10 zs . This results in a frequency resolution A = 0.1 MHz. The multifrequency amplifier was configured to set the measurement window around >0between 4.1953 GHz and 4.2047 GHz. This results in 95 modes symmetrically placed aroundospaced by A, and constitutes our measurement frequency comb. The modes were labelled as shown in fig. 5a-b with negative label numbers for c < a)0and positive for co > coo. Now, by driving the JPA with three pumps of equal strength at frequencies = 2coo- 4A= 8.3996 GHz, c2= 2&)0= 8.4 GHz and c2= 2coo+ 4A= 8.4004 GHz, and setting the relative phase of the pump to TT, thereby recovering the three independent square ladder structures of the mode correlations shown in fig. 5b.

[0132] The JPA is operated at a temperature of TE= lOmK where kBTE« hco0. This ensures the dominance of quantum over thermal fluctuations in the JPA. The three coherent pumps squeeze the vacuum fluctuations and physically realize the continuous-variable quantum correlations between the modes of the frequency comb with the topology described by the graph in fig. 5b.

[0133] To prove the presence of quantum correlations with the specified topologies, statistics of IQ quadrature measurements of each mode through the multifrequency lock-in amplifier was collected. This is done by repeating the experiment on the order of 106times and computing the first and second statistical moments. These measurements allow estimation of the expectation value of suitable quantum operators signalling the presence of quantum correlations between modes.

Claims

1. CLAIMS1 . A system for forming continuous variable cluster states and quantum correlations with microwaves, the system (100) comprises- a cryogenic chamber (161),- a cryocooler arranged to reduce the temperature inside the cryogenic chamber (161) to allow for a vacuum state,- a superconducting parametric circuit (110), SPC, comprising a non-linear resonator (111), wherein said resonator (111) is arranged in said cryogenic chamber (161),- a pumping circuitry (140) arrange to pump said SPC (110) with three or more microwave signals (260) to form a pump field (280) in the resonator (111),- control circuitry (180) arranged to control the pumping circuitry (140), wherein the cryocooler is arranged to prepare a vacuum state inside the cryogenic chamber (161) allowing intrinsic quantum fluctuations to interact with the resonator (111) of the SPC (110), and wherein the control circuitry (180) is arranged to control the pumping circuitry (140) to pump the SPC (110) with the three or more microwave signals (260), whereby the pump field in the resonator (111) mixes with frequency components of the quantum fluctuations (240) to generate quadrature-squeezed states of microwave electromagnetic waves (290), and wherein said quadrate- squeezed states of microwave electromagnetic waves form the continuous variable cluster state.

2. The system according to claim 1 , wherein the SPC (110) comprises a parametric oscillator, a parametric amplifier, a Josephson parametric oscillator, and / or a Josephson parametric amplifier.

3. The system according to claim 1 or 2, wherein the system (100) comprises a quadrature detector (150) arranged to receive the continuous variable cluster state from the resonator (111) of the SPC (110), and wherein said quadrature detector (150) is arranged to measure the continuous variable cluster state in a determined measurement basis.

4. The system according to any preceding claim, wherein the quadrature detector (150) comprises a multi-frequency lock-in amplifier, and / or a heterodyne detector.

5. The system according to claim 3 or 4, wherein the system (100) is arranged to output a classical signal indicative of the measured quadratures of the continuous variable cluster state.

6. The system according to claim 3 to 5, wherein the determined measurement basis and measured frequencies of the continuous variable cluster states correspond to a frequency comb.

7. The system according to claim 3 to 6, wherein the control circuitry (180) being arranged to generate quadrature-squeezed states is arranged to control a quadrature detector (150) to measure the quadratures of the squeezed microwave electromagnetic waves, and to control the pumping circuitry (140) based on the measured values.

8. The system according to any preceding claim, wherein the formed continuous variable cluster state comprises a square-ladder variable cluster state.

9. A method for forming continuous variable cluster states and quantum correlations with microwaves, the method (300) comprises - providing (310) a superconducting parametric circuit (110), SPC, in a cryogenic chamber (161) with a cryocooler, wherein the SPC comprises a non-linear resonator (111);- cooling (320) said resonator (111) to a vacuum state, whereby intrinsic quantum fluctuations interact with the resonator (111) of the SPC (110); and- pumping (330) the resonator (111) of said SPC (110) with three or more microwave signals to form a pump field in the resonator (111), whereby, the pump field in the resonator (111) mixes with frequency components of the quantum fluctuations to generate quadrature-squeezed states of microwave electromagnetic waves, wherein said quadrature-squeezed states of microwave electromagnetic waves form a continuous variable cluster state.

10. The method according to claim 9, wherein the formed continuous variable cluster state comprises a square-ladder cluster state.

11. The method according to claim 9 or 10, where the method comprises determining (350) a measurement basis, and measuring (360) the continuous variable cluster state in the determined measurement basis by utilizing one or more quadrature detectors (150).

12. The method according to claim 11 , wherein measuring (360) the continuous variable cluster state comprises measuring the generated quadrature-squeezed states of microwave electromagnetic waves utilizing a multi-frequency lock-in amplifier, and / or a heterodyne detector.

13. The method according to any of claims 12, wherein measuring (360) the continuous variable cluster state comprises providing an output indicative of the measured quadratures.

14. A computer program product comprising a non-transitory computer-readable storage medium (412) having thereon a computer program comprising program instructions, the computer program being loadable into a processor (411) and configured to cause the processor (411) to perform the method (300) for forming continuous variable cluster states and quantum correlations with microwaves according to any one of claims 9-13.

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