Non-gaussian quantum state generation

Abstract

Methods are disclosed for generating, manipulating, and controlling non-Gaussian quantum states in continuous variable cluster quantum states usable for quantum computing. Some methods can be used to generate, non-Gaussian states in the CV cluster quantum state. Some of the disclosed methods generate non-Gaussian states using a sequence of photon subtraction operations and Gaussian operations controlled by signals received from homodyne measurements and photon number resolution detection. Some methods may use ancillary photon to increase the probability of photon subtraction during non-Gaussian state generation process. Some method may transform modes of a multidimensional cluster state to a non-Gaussian state using a plurality of non-Gaussian state generation processes performed on the modes of the multidimensional cluster state in time and frequency domain.

Description

NON-GAUSSIAN QUANTUM STATE GENERATIONINCORPORATION BY REFERENCE TO ANY PRIORITY APPLICATIONS

[0001] This application claims priority to U.S. Provisional Application No. 63 / 587019 filed on September 29, 2023, titled “NON-GAUSSIAN QUANTUM STATE GENERATION” that is hereby incorporated herein by reference in its entirety.BACKGROUNDTechnical Field

[0002] This disclosure relates to methods and device configurations for generation, control, and transformation of various quantum states. In particular, the methods and device configurations for generating continuous variable cluster states and non-Gaussian quantum states required for universal and fault-tolerant quantum computation (QC).Description of Related Art

[0003] Quantum computation holds the potential to solve problems intractable to classical computation by manipulating quantum information across large-scale entangled states. Implementations of quantum computing protocols and algorithms remain challenging due to the presence of errors, quantum decoherence, and difficulties in scaling the number of quantum resource (e.g., qubits). Measurement-based QC over continuous variable (CV) cluster states uses scalable CV cluster states. Certain aspects of QC using CV cluster states may benefit from non- Gaussian states generated using the modes of the CV cluster and embedding such non-Gaussian states in the CV cluster state.SUMMARY

[0004] In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state, the method including: as implemented by one or more hardware processors configured to execute computer-executable instructions: receiving first measurement data from at least one of a first photon number resolving (PNR) detector and a first homodyne detector; generating a first tuning control signal based on at least a portion of the first measurementdata; transmitting the first tuning control signal to a first tunable beam splitter that is optically connected to receive a first input mode from a first entanglement gate and transmit a first portion of the first input mode to the first PNR detector and a second portion of the first input mode to the first homodyne detector; and generating first state control signals for controlling one or more first quantum state modification modules that are optically connected to the first entanglement gate and configured to modify a second input mode received from the first entanglement gate to generate a first output mode having the non-Gaussian state.

[0005] In some aspects, the techniques described herein relate to a method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including, by a first D-PhANTM module: receiving, a pair of entangled modes of the continuous variable (CV) quantum cluster state including a first mode having a first initial state and a second mode having a second initial state; receiving first measurement results of one or both homodyne measurement or photon-number-resolving (PNR) detection performed to generate another non- Gaussian state; generating a first tuning control signal based on the first measurement results; performing photon subtraction on the first mode by splitting a first optical field into a first portion and a second portion of the first optical field based on the first tuning control signal; performing a first PNR detection on the first portion; performing a first homodyne detection on the second portion to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one first Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state.

[0006] In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state, the method including: as implemented by a hardware processor configured to execute computer-executable instructions: receiving measurement data from at least one of a photon number resolving (PNR) detector and a homodyne detector; generating a tuning control signal based on the measurement data; and transmitting the tuning control signal to a tunable beam splitter that is optically connected to an optical delay to receive a delayed mode, transmit a first portion of the delayed mode to the PNR detector and a second portion of the delayed mode to one or more quantum state modification modules are optically connected to the tunable beam splitter and configured to modify the second portion of the delayed mode and transmit the modified second portion of the delayed mode to the homodyne detector; wherein the optical delayis configured to receive an input mode including an optical field from a CV cluster state and generate the delayed mode.

[0007] In some aspects, the techniques described herein relate to a method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including: receiving a pair of entangled modes of the continuous variable (CV) quantum cluster state including a first mode having a first initial state and a second mode having a second initial state; performing photon subtraction on the first mode by splitting the first mode into a first portion and a second portion of the first mode; receiving an auxiliary photon stream; performing a photon- number-resolving (PNR) detection on the first portion and the auxiliary photon stream; performing a homodyne detection on the second portion to transform the second initial state of the second mode into a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state.

[0008] In some aspects, the techniques described herein relate to a system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system including: a bean splitter configured to receive a first mode of a pair of entangled modes of the CV quantum cluster state and generating first and second portions of the first mode; an optical source configured to generate a stream of auxiliary photons; a PNR detector configured to simultaneously receive the first portion of the first mode and the auxiliary photon stream and to generate a PNR detection result; a homodyne detector configured to receive the second portion of the first mode to generate a second detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the second detector signal.

[0009] In some aspects, the techniques described herein relate to a method of generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including: receiving a pair of entangled modes of the CV quantum cluster state including a first mode having a first initial state and a second mode having a second initial state; selecting the first mode of the pair of entangled modes in the canonical cluster state, wherein the first mode includes a first optical field; performing a first photon subtraction on the first mode by splitting the firstoptical field into a first portion and a second portion of the first optical field; receiving a first auxiliary photon stream; performing a first PNR detection on the first portion and the first auxiliary photon stream; performing a second photon subtraction on the second portion of the first optical field by splitting the second portion of the first optical field into a third portion and a fourth portion of the second portion of the first optical field; receiving a second auxiliary photon stream; performing a second PNR detection on the first portion and the first auxiliary photon stream; performing a homodyne detection on the fourth portion of the second portion of the first optical field to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the non-Gaussian state.

[0010] In some aspects, the techniques described herein relate to a system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system including: a first beam splitter configured to receive a first mode of a pair of entangled modes of the continuous variable (CV) quantum cluster state and generate a first portion and a second portion of the first mode; an optical source configured to generate a stream of auxiliary photons; a first PNR detector configured to receive the first portion of the first mode and a first portion of the stream of auxiliary photons; a second beam splitter configured to receive the second portion of the first mode and generate a transmitted portion and a reflected portion of the second portion of the first mode; a second PNR detector configured to receive the reflected portion of the second portion of the first mode and a second portion of the stream of auxiliary photons; a homodyne detector configured to receive the transmitted portion of the second portion of the first mode and generate a detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the detector signal.

[0011] In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state, the method including: as implemented by a hardware processor configured to execute computer-executable instructions: receiving first, second, and third modes of a CV macrocode cluster state; splitting optical field of the first mode to first and second portions of the optical field, providing the first portion of the optical field to a first homodyne detector; providing an auxiliary stream of photons and the second portion of the optical field to a PNRdetector; generating control signals based on first measurement data received from the PNR detector; adjusting a parameter of at least one of the first and second homodyne detectors using the control signals; and modifying the third mode of the CV macrocode cluster state using one or more quantum state modification modules based on second measurement data received from the first and second homodyne detectors.

[0012] In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state using at least one macronode of a macronode cluster state, the method including: as implemented by a hardware processor configured to execute computerexecutable instructions: receiving first measurement data from a photon number resolving (PNR) detector configured to receive a first portion of a first mode of the at least one macronode and auxiliary photons from an optical source; generating first and second HD control signals based on the first measurement data for controlling a first homodyne detector configured to receive a second portion of the first mode of the at least one macronode and a second homodyne detector configured to receive a second mode of the at least one macronode; receiving second measurement data from the first and second homodyne detectors; and generating first and second state control signals based on the second measurement data for controlling a quantum state modification module that is configured to receive a third mode of the at least one macronode and generate an output mode having the non-Gaussian state.

[0013] In some aspects, the techniques described herein relate to a method of generating a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including: receiving a plurality of entangled modes of the continuous variable (CV) quantum cluster state including modes having optical fields oscillating at different frequencies; sequentially performing one or more non-Gaussian state generation processes on individual modes of the plurality of entangled modes generates a plurality of measured output modes and at least one non-measured output mode; performing homodyne detection on the measured output modes to generate measurement results; and performing a Gaussian operation on the at least one non-measured output mode based on the measurement results.BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In the following description of the various embodiments, reference is made to the accompanying drawings which form a part hereof, and in which are shown by way of illustration various embodiments of the device. It is to be understood that other embodiments may be utilized, and structural changes may be made without departing from the scope of present invention.

[0015] Figure 1 schematically illustrates an example of a dynamic PhANTM (D- PhANTM) process that uses measurement data received from a homodyne photodetector and a photon number resolving (PNR) detector to adjust splitting ratio of a beam splitting operation and / or a Gaussian operation.

[0016] Figure 2A schematically illustrates an example process for generating nonGaussian quantum states using multiple D-PhANTM processes (shown in Figure 1) serially performed on frequency modes of a continuous variable (CV) quantum cluster state.

[0017] Figure 2B schematically illustrates an example of a two-step process for generating non-Gaussian quantum states using two D-PhANTM processes (shown in Figure 1) serially performed on frequency modes of a continuous variable (CV) quantum cluster state.

[0018] Figure 2C schematically illustrates a multi-step process and system comprising a plurality of D-PhANM processes performed in time domain.

[0019] Figures 3A-3D show calculated Wigner functions of example non-Gaussian states generated by sequentially applying two D-PhANTM process to input states of a CV cluster state.

[0020] Figure 4A schematically illustrates an example of a F-PhANTM process for generating non-Gaussian states at a high rate using an optical source of ancilla (auxiliary) photons.

[0021] Figure 4B shows calculated Wigner function of example non-Gaussian states generated by PhANTM process (top panels) and the corresponding photon subtraction probabilities (bottom panels).

[0022] Figure 4C shows calculated Wigner function of example non-Gaussian states generated by the F-PhANTM process / module 400 using auxiliary photons (top panels) and the corresponding photon subtraction probabilities (bottom panels).

[0023] Figure 4D illustrates the calculated Wigner functions of example non-Gaussian state generated by a single step of F-PhATNM process for two different squeezing levels (0.3 and0.8) of a the ancilla photons and two different reflection coefficient (3% and 11%) of the BS 408. In this calculation, three photons have been subtracted and the squeezing of the ancilla photons provided is 1.2.

[0024] Figure 4E illustrates the calculated Wigner functions of example non-Gaussian state generated by a single step of F-PhATNM process for two different squeezing levels (0.3 and 0.8) of a the ancilla photons and two different reflection coefficient (3% and 11%) of the BS 408. In this calculation, five photons have been subtracted and the squeezing of the ancilla photons provided is 1.2.

[0025] Figure 5A schematically illustrates an example embodiment of a modified F- PhANTM that includes multiple beam splitters and PNR detectors that serially subtract photons from a mode before homodyne detection.

[0026] Figures 5B-5C illustrate calculated Wigner functions for non-Gaussian states generated by the modified F-PhANTM process shown in FIG. 5A when five beam splitters (N=5) and five PNR detectors (each receiving an auxiliary photon stream) are used. For the calculations shown in FIG. 5B the angle (representing the beam splitting ratio) of the beam splitters is set to 10 degree and in FIG. 5CB the angle (representing the beam splitting ratio) of the beam splitters is set to 15 degrees. The four Wigner functions shown in each figure correspond to four different numbers of subtracted photons (n = 5, 6, 7, and 8)

[0027] Figure 6 shows an example embodiment of fast dynamic PhANTM (FD- PhANTM) model / process comprising one or more features described with respect to FIGS. 1, 2A, 2B, 2C, 4A, and 5A.

[0028] Figure 7 schematically illustrates an example fast dynamic PhANTM process / module that may generate a non-Gaussian state using a three-mode macronode received from a macronode CV cluster state.

[0029] Figure 8 schematically illustrates a multimode non-Gaussian state generation process where each non-Gaussian state generation process may comprise a PhANTM process, a D-PhANTM process, an F-PhANM process, or an FD-PhANTM process.

[0030] Figure 9 is a block diagram illustrating an example application of PhANTM on a cluster of entangled frequency and time modes comprising N time modes and M frequency modes.DETAILED DESCRIPTION

[0031] Non-Gaussian quantum states that can be produced using the disclosed methods may be used as resources for photonic quantum computation, quantum communication, and quantum sensing. In some cases, any of these applications may benefit from using high quality photonic qubits comprising cat states, squeezed cat states, GKP states, and / or squeezed GKP states, or other types of non-Gaussian states to perform at least a portion of operations (e.g., quantum operations) within a specific computational, information processing, or sensing task. It is particularly advantageous to generate and control non-Gaussian quantum states in continuous variable (CV) quantum systems that are highly scalable. However, generation and control of non- Gaussian quantum states (e.g., cat states and GKP states) in CV quantum systems remains elusive. Some of the methods and systems disclosed here may be used to generate high quality non- Gaussian states in a CV quantum platform starting from Gaussian states or low quality non- Gaussian states. In some cases, the operations (e.g., quantum operations) used for generating non- Gaussian states using the disclosed methods may be implemented using existing optical components and devices.

[0032] The methods and system disclosed here can provide a reconfigurable and highly controllable platform for generation and manipulation of non-Gaussian states, such as cat states, GKP states, grid states, and the like. Some of the methods and algorithms may use photon number resolution (PNR) detection, homodyne detection (HD), beam splitting (e.g., with controlled splitting ratio), and / or single mode continuous variable quantum optical gates to generate non- Gaussian states having desired properties (e.g., Wigner spacing, squeezing, amplitude and the like).

[0033] In some embodiments, the methods may comprise dynamic control of at least one of the beam splitting ratio in the beam splitting operation, a parameter of a quantum state rotation operation, a parameter of a squeezing operation, and a parameter of a displacement operation performed on a mode (e.g., mode of a cluster state) during a non-Gaussian state generation process. In some cases, the non-Gaussian state generation process may comprise Photon-counting-assisted Node-Teleportation Method (PhANTM).

[0034] In some cases, a system, which is configured to generate non-Gaussian states, may generate a first non-Gaussian state (e.g., a cat state) based on an initial value of a parameter and a classical controller may adjust the parameter based at least in part on a detector signalgenerated during the generation of the first non-Gaussian state before generating a second nonGaussian state. For example, a parameter of a quantum state rotation operation, a parameter of a squeezing operation, and / or a parameter of a displacement operation may be adjusted based on a PNR measurement performed to generate the second non-Gaussian state.

[0035] Additionally or alternatively, the classical controller may adjust a parameter based at least in part on a detector signal generated during the generation of the second non- Gaussian state.

[0036] In some cases, the parameter may comprise a squeezing level of a squeezing operator, rotation angle of a rotation operator, or a displacement (along x or z-axis) of a displacement operator of the system. In some examples, certain values of the parameters, e.g., adjusted values of a rotation and / or a squeezing operator can generate GKP states (as opposed to a cat state generated by one application of existing PhANTM protocol). In some cases, the characteristic of the resulting GKP state may be tailored by tuning one or more parameters of the system.

[0037] Some of the disclosed methods and protocols may comprise applying the PhANTM protocol on multiple modes of a cluster state. For example, the method may comprise using PhANTM to teleport entangled modes having different frequencies into one or more non- Gaussian states (e.g., cat states). Some of the method described below may comprise performing PhANTM on modes of a two-dimensional cluster state where the two dimensions may include time and frequency (e.g., discrete frequencies of entangled modes processed in time domain by discrete time steps). Advantageously, the disclosed methods, may provide a non-Gaussian state generation platform in which non-Gaussian state preparation may be spread out amongst temporal modes (e.g., modes processed and teleported in time domain), frequency modes (e.g., discrete optical modes of a resonant source that generates a plurality of photon pairs), combination of temporal and frequency modes, or any combination of the available degrees of freedom in a cluster state (e.g., a continuous variable cluster state). This flexibility allows for exploiting the tradeoffs in clock cycles and quantum resources and optimize them to provide more practical and more efficient quantum computing platforms and resources for quantum applications (e.g., quantum computation, quantum communication, and quantum sensing).

[0038] Some of the disclosed methods and protocols may comprise providing an additional photon stream, herein referred to as auxiliary photo stream (or flux) or ancilla photonsto assist the photon subtraction operation (also referred to as photon number resolution measurement or PNR) to a photon number resolution (PNR) detector in the PhANTM process preformed on modes of a cluster sate (e.g., a canonical cluster state). Advantageously, providing the ancilla photons, can increase the probability of photon detection prior to homodyne detection in a PhANTM process and thereby increase the generation rate of non-Gaussian states by the PhANTM process.

[0039] Some of the disclosed methods and protocols may comprise providing the additional photon stream (ancilla photons) to a dynamically controlled non-Gaussian quantum state generation process where at least one of the beam splitting ratio in beam splitting operation, a parameter of a quantum state rotation operation, a squeezing operation, and / or a displacement operation performed on of a mode (e.g., mode of a cluster state) is dynamically controlled. In some cases, the non-Gaussian state generation process may comprise Photon-counting-assisted Node- Teleportation Method (PhANTM).

[0040] Some of the disclosed methods and protocols may comprise generating non- Gaussian quantum states using a dynamically controlled non-Gaussian quantum state generation process performed on a macro-node cluster where a parameter of at least one homodyne detection operation (e.g., an angle of homodyne detection) is controlled based at least in part on PNR measurement performed during the non-Gaussian quantum state generation process.Photon-counting-assisted Node-Teleportation (PhANTM)

[0041] One of the methods for generating non-Gaussian quantum states (e.g., Schrodinger cat states) is Photon-counting-assisted Node-Teleportation Method (PhANTM). In some embodiments, PhANTM may comprise a teleportation process for teleporting quantum states between the nodes of a canonical cluster state where the teleportation process includes photon number resolving (PNR) detection. In some cases, PhANTM generates a Schrodinger cat state (also referred to as cat state) using one or more of a beam splitting process, a photon number resolving (PNR) detection (also referred to as PNR measurement), a squeezing process, and a homodyne detection (HD) process. In some cases, PhANTM may be used to generate and manipulate, cat states, grid states, and / or compass states within a canonical cluster state. In some cases, PhANTM may be used for teleporting macronodes, and / or generation and manipulation of cat states, grid states, and / or compass states within a macronode cluster state. In someimplementations PhANTM may be used to transform one or more nodes (or modes) of a cluster state to cat states. In some embodiments, a plurality of cat states generated by several applications of PhANTM may have random parities (even or odd) and randomly distributed squeezing levels, amplitudes.

[0042] In some cases, PhANTM process may comprise, performing photon number resolution (PNR) detection on a portion of a first input mode followed by a homodyne detection on the remaining portion of the first mode to transform the quantum state of a second input mode, which is entangled to the first input mode (e.g., by a Cz gate), to a non-Gaussian state. In some examples, the PhANTM process may comprise providing a portion of a first input mode to a first photodetector (herein referred to as PNR detector) and the remaining portion to a second photodetector (herein referred to as homodyne detector), using a beam splitter. In the existing PhANTM the beam spitter has affixed splitting ratio while, in some embodiments of the D- PhANTM the splitting ratio of the beam splitter can be tunable and may be tuned by a classical controller based on feedback received from one or both of the first and second photodetectors.

[0043] In some cases, the existing PhANTMs protocol may be performed (e.g., consecutively performed) on a one-dimensional cluster state (a canonical cluster state) comprising entangles Gaussian modes (e.g., squeezed vacuum modes) to transform the one-dimensional cluster state into a non-Gaussian (e.g., a final cat state). In some such examples, each PhANTM operation may be followed by teleporting the resulting non-Gaussian state by a rotation (e.g., Gaussian rotation or a homodyne detection) on a neighboring mode whose state is transformed to a cat state via PhANTM. In some cases, the rotation angle may be 7t / 2. The rotation may transform the state of the neighboring mode to a properly rotated cat state on which another PhANTM operation may be performed. In some cases, the homodyne measurement results may be compensated for through feed-forward displacements applied to the teleported mode or before subsequent measurements.Modified PhANTM protocols1) Dynamic PhANTM (D-PhANTM)

[0044] As described above the PHANTM, as originally provided, relies on substantially static configurations and parameters without feedback or feedforward to the input or output operations (steps of the algorithm), and beam splitting operation. The inventor have discovered that the PHANTM protocol may be modified by providing classically controlledfeedback to one or more operations of the protocol to adjust (e.g., dynamically adjust) parameters associated with the operations. Some of the disclosed methods, may comprise a modified PhANTM protocol, herein referred to as dynamic PhANTM (herein referred to as D-PhANTM) protocol and its implementation in a quantum system (e.g., a continuous variable quantum system) for generating non-Gaussian states. In some cases, D-PhANTM process may comprise a beam splitting operation controlled by a classical or classically controlled signal generated, e.g., based at least in part on a previous process (e.g., a previous D-PhANTM or a PhANTMp rocess). Additionally or alternatively, D-PhANTM process may comprise a rotation and / or a squeezing operation controlled by a classical or classically controlled signal generated, e.g., based at least in part on the PNR measurement of the same process (e.g., a previous D-PhANTM or PhANTM process).

[0045] In some cases, D-PhANTM may be used to generate cat states, GKP states, and other non-Gaussian states using the modes of a canonical continuous variable (CV) cluster state and embedding the resulting Gaussian states in the continuous cluster state.

[0046] In some embodiments, D-PhANTM may comprise controlling or adjusting (e.g., dynamically adjusting) one or more of the beam splitting operation used for PNR measurement, an amount of displacement provided to the output state, an amount of rotation provided to the output state, or amount of squeezing provided to the output state, based on outcomes of one or both of a PNR measurement and HD measurement, from the same D-PhANTM process, a previous D-PhANTM process or a PhANTM process.

[0047] In some examples, a non-Gaussian state generation may comprise consecutively performing two or more PhANTM processes (e.g., PNR measurement and homodyne detection), rotating a first output state generated by a first PhANTM process by an angle determined based on one or both a number of photons subtracted and outcome of homodyne detection performed during the first PhANTM process or a previous PhANTM process (e.g., an immediate previous PhANTM process), and performing a second PhANTM process on the rotated output state to generate a second output state.

[0048] Additionally or alternatively, D-PhANTM may comprise controlling the splitting ratio of the beam splitting operation (for performing PNR measurement) and / or the amount of displacement, rotation, and / or squeezing provided to the first output state, by the first PhANTM process, based on one or both a number of photons subtracted, and outcome ofhomodyne detection performed during the first PhANTM process. In some embodiments, D- PhANTM may comprise controlling the splitting ratio of the beam splitting operation and / or the amount of displacement, rotation, and / or squeezing provided to the second output state, by the second PhANTM process, based on one or both a number of photons subtracted, and outcome of homodyne detection performed during the first PhANTM process and / or during the second PhANTM process.

[0049] In some embodiments, a quantum state generated by the second PHANTM process and / or one of the quantum states provided as input to the second PHANTM process may be squeezed, rotated, displaced, or otherwise adjusted based on outcomes of a first PHANTM process (e.g., PNR and homodyne measurement outcomes) that provides an input state to the second PhANTM process.

[0050] Some of the disclosed methods, may comprise consecutively performing D- PhANTM without rotating the output state generated by each step of D-PhANTM.

[0051] In some embodiments, D-PhANTM may comprise consecutively performing two or more PhANTM processes, herein referred as steps of PhANTM, and providing the measurement results obtained during various PHANTM steps to a classical controller that controls one or more operations of a non-Gaussian state generation process (e.g., a subsequent or another PhANTM process) to tailor one or more input states and / or improve the quality of output states (e.g., by re-squeezing).

[0052] In FIGS 1, 2B, 2C, 4A, 5 A, 6 and 7, thin lines indicate classical information (e.g., encoded in electric signals such as photo detection signals) being fed from the photodetectors to quantum gates (quantum state modification modules) and the tunable beam splitter (TBS), and thick line indicate optical paths of flow of photons between different operation and corresponding optical modules, components or devices. Z indicates z-displacement operation, R indicates rotation operation in phase space, SQ indicates squeezing operation, and G indicates a Gaussian operation (including Z, R, and SQ).

[0053] Process illustrated in FIGS 1 , 2B, 2C, 4A, 5 A, 6, 7, and 8 may be implemented using optically and electrically interconnected electronic, optoelectronic, or photonic components, modules and devices configured to perform the corresponding operations, e.g., on modes of a CV quantum cluster state. An operation and a component performing the operation may be shown by the same symbol and labeled by the same reference numeral. Optoelectronic, or photoniccomponents, modules and devices may comprise free space or monolithically fabricated on-chip devices fabricated on one or more chips. For example, at least a portion of photonic devices may comprise silicon photonic devices fabricated on a silicon chip.

[0054] FIG. 1 is a block diagram of an example of a non-Gaussian state generation process and system that uses signals (measurement data) received from a homodyne photodetector and a photon number resolving (PNR) detector of the process (or a previous process) to adjust splitting ratio of a beam splitting operation and / or a Gaussian operation performed during the process. In some examples, the Gaussian operation may comprise rotation of a quantum state (e.g., in phase space), displacement of a quantum state (e.g., in x or z dimensions), or a squeezing of a quantum state. In some cases, the process shown in FIG. 1 may comprise a D-PhANTM process and a module 100 (e.g., hardware module) that performs the D-PhANTM process, may be referred to as D-PhANTM module 100.

[0055] In some cases, D-PhANTM may be used to teleport a quantum state of a first mode 102b entangled to a second mode 103b to transform the state of the second entangled mode 103b to a desired non-Gaussian state of an output mode 125, e.g., by performing one or more operations on the second entangled mode 103b based on PNR and HD measurements on different portions of the first entangled mode 102b.

[0056] In some cases, the two entangled modes 102b, 103 b, may be generated by an entanglement operation 104 on first and second input modes 102a, 102b. In some cases, the two entangled modes 102b, 103b, can be two modes of a cluster state entangled by an entanglement gate (e.g., a controlled Z-gate, also referred to as Cz gate). In some cases, the first and second entangled modes 102b, 103b, may be selected from a plurality of entangled modes of a cluster state.

[0057] As shown in the FIG. 1, in some cases, D-PhANTM may comprise dividing the first entanglement mode 102b received from the entanglement operator 104 into a transmitted portion and a reflected portion based at least in part on a first feedback signal 112a received from a classical controller 106, performing PNR 110 measurement on the reflected portion and preforming a homodyne measurement 114 (HD) on the transmitted portion. In some cases, the first entanglement mode 102b may be divided into the transmitted portion and the reflected portion by a tunable beam splitter 108 having a splitting ratio controlled and / or adjusted by the first feedback signal. In some cases, the PNR 110 and HD 114 measurements on the first entangled modetransforms the state of the second entangled mode 103 to an initial non-Gaussian state. In some cases, the initial non-Gaussian state and / or its characteristics may be controlled at least in part on a splitting ratio of the by the first feedback signal. In some embodiments, D-PhANTM may further comprise performing one or more of a displacement operation 118, a rotation operaitonl 19, and a squeezing operation 120 to generate the output mode having a desired non-Gaussian state. In some cases, the non-Gaussian state of the output mode may comprise a GKP state. In some embodiments, at least one of displacement operation 118, a rotation operaitonl 19, and a squeezing operation 120 may be controlled and / or adjusted by a feedback (or control) signal received from the classical controller 106. For example, a parameter of the of displacement operation 118 may be controlled or adjusted by a second feedback (or control) signal 112b, a parameter of the rotation operation may be controlled or adjusted by a third feedback (or control) signal 112c, and a parameter of the squeezing operation may be controlled or adjusted by a fourth feedback (or control) signal 112d.

[0058] In various implementations, the classical controller 106 may generate the first feedback (or control) signal 112a based on one or more of: an outcome of the HD measurement operation 114, a PNR measurement of a previously performed PhANTM or D-PhANTM, or an outcome of an HD measurement operation of a previously performed PhANTM or D-PhANTM.

[0059] In various implementations, the classical controller 106 may generate at least one the second, third, and fourth feedback (or control) signals 112b, 112c, 112d, based on one or more of: an outcome of the HD measurement operation 114, a PNR measurement of a previously performed PhANTM or D-PhANTM, or an outcome of an HD measurement operation of a previously performed PhANTM or D-PhANTM.

[0060] In some embodiment a type and / or a characteristic of the non-Gaussian state of the output mode 125 may be controlled by the first, second, third, and fourth control signals 112a, 112b, 112c, 112d, and thereby the classical controller 106.

[0061] In some embodiments, PNR measurement 110 may be performed by a photodetector (e.g., a single photon detector such as single photon avalanche photodetector, SPAD) and the HD measurement 114 may be performed by a balanced photodetector and an optical source that generates a local oscillator having an optical frequency substantially equal to that of the mode on which the HD measurement is performed.

[0062] In some cases, the D-PhANTM module 100 that may perform the operations described above, may include optically interconnected photonic components and devices configured to perform these operations, e.g., on modes of a CV quantum cluster state. An operation and a component performing the operation are shown by the same symbol and labeled by the same reference numeral.

[0063] In some cases, the D-PhANTM module 100 may include: a tunable beam splitter (TBS) 108 optically connected to a entanglement device or module 104, which is configured to entangle two modes of the cluster state, to receive a first mode of a pair of entangle modes, a PNR detector 110 that is optically connected to a first port of the TBS 108 to receive photons from the first port, a homodyne detector 114 that is optically connected to a second port of the TBS 108 to receive a photons from the second port, three quantum state modification and transformation modules 118, 119, 120, optically connected (e.g., in series) to the entanglement device or module 104 to receive a second mode of the pair of entangle modes from the cluster state, and a classical controller 106 configured to receive measurement signals or data from one or both of the homodyne detector 114, the PNR detector 110, and / or receive a signal 107 comprising measurement data generated by one or more other homodyne detectors and PNR detectors (e.g., included in or associated with other D-PhANTM modules). In some embodiments, the classical controller 106 may generate one or more control signals based on signals received from one or both the homodyne detector 114 and the PNR detector 110 and provide the control signals to the TBS 118 or one or more quantum state modification modules.

[0064] In some cases, the quantum state modification modules may comprise a quantum state displacement module 118, a quantum state rotation module 119, and a quantum state squeezing module 120. In various implementations, TBS 118 may be configured to divide a first input mode 102b by dividing an optical field associated with the first input mode to first and second portions of the optical field based on a tunable splitting ratio that controls the ratio between the optical intensities of the first and second portions of the optical field. In some cases, the tunable splitting ratio may be tuned or adjusted by a tuning control signal (e.g., an electrical, acoustic, or optical signal). In some cases, the tuning control signal may comprise an electric signal generated by the classical controller 106. In some cases, the TBS can be an electro-optically tunable beam splitter configured to reroute first and second portions of an input optical field (or mode) to a first and second ports according to a splitting ratio controlled or adjusted by the electric tuning controlsignal received from the classical controller 106. In some cases, TBS 108 may comprise a monolithically fabricated on-chip photonic device. In some examples, TBS 108 may comprise a directional coupler having two on-chip waveguides mutually coupled along a coupling length where the mutual coupling is controlled by the tuning control signal via an electro-optic effect. In some cases, the photon resolution detector 110 may be configured to receive a first portion of the input mode 102b (e.g., from the first port of the TBS 108) and generate a first detector signal and transmit the first detector signal to the classical controller 106. In some cases, the homodyne detector 114 may be configured to receive a second portion of the input mode 102b (e.g., from the second port of the TBS 108) and generate a second detector signal and transmit the second detector signal to the classical controller 106.

[0065] In some cases, the classical controller 106 may generate a tuning control signal based on a signal received from one or more PNR detectors of one or more other D-PhANTM modules (e.g., a D-PhANTM module that provides its output mode to the entanglement operator 104) and provides the tuning control signal to the TBS 108 to tune a splitting ratio of the TBS 108. Additionally or alternatively, in some implementations, the classical controller 106 may generate the tuning control signal based on a signal received from one or more homodyne detectors of the one or more other D-PhANTM modules.

[0066] In some cases, the classical controller 106 may generate a tuning control signal based on a signal received from PNR detector 114 in a previous processing step of the same D- PhANTM process and provides the tuning control signal to the TBS 108 to tune a splitting ratio of the TBS 108 for a subsequent step of the same D-PhANTM process (as a feedback signal). Additionally or alternatively, in some implementations, the classical controller 106 may generate the tuning control signal based on a signal received from the homodyne detector 114 in the previous processing step of the same D-PhANTM process.

[0067] In some cases, the classical controller 106 may generate a state control signal based on a signal received from one or more PNR detectors of one or more other D-PhANTM modules (e.g., a D-PhANTM module that provides its output mode to the entanglement operator 104) and provides the state control signal to one or more quantum state modification modules 118, 119, 120 to control or tune a parameter of the respective quantum state modification module. Additionally or alternatively, in some implementations, the classical controller 106 may generate the state control signal based on a signal received from one or more homodyne detectors of theone or more other D-PhANTM modules. In some examples, a state control signal may be provided to a quantum displacement module 118 to tune or control a quantum state displacement parameter therein, to a quantum state rotation module 119 to tune or control a quantum state rotation parameter therein, and to the quantum state squeezing module to tune or control a quantum state squeezing parameter therein.

[0068] In various implementations, any of the dynamically controlled parameters may not change and have fixed value.

[0069] In some embodiments, the D-PhANTM module 100 may comprise at least one of a controlled quantum state rotation module 119, a controlled quantum state squeezing module 120, a tunable beam splitter 108 (instead of a beam splitter with fixed splitting ratio), or a controlled quantum state displacement module 118 that is controlled beast at least in part on a PNR measurement data. In some embodiments, a D-PhANTM process includes at least one operation that one of its parameters is controlled by outcomes of a PNR measurement or at least one operation other than quantum state displacement operation that one of its parameters is controlled by outcomes of a homodyne measurement.

[0070] In some implementations, the classical controller 106 may comprise a computer, a FPGA, or other devices (e.g., analog, digital electronic devices). In some cases, the classical controller 106 may include at least one non-transitory memory storing machine readable instructions and at least one electronic processor configured to execute the machine readable instructions to generate the first, second, third, and fourth control signals 112a, 112b, 112c, 112d, using signals generated by one or more of a device that performs HD measurement and device performs PNR measurement in a current or previous D-PhANTM operation.

[0071] In some cases, a classical controller, may control the dynamic feedback of the modified PHANTM by selecting and adjusting the feedback or feedforward signals. In some cases, a modified PhANTM uses active dynamic components of that can change from one step of PhANTM to the next. In some cases, the adaptive squeezing, displacement, and rotation control of disclosed method provides:

[0072] In some embodiments, squeezing an output state of one step based on the measurement outcomes of a previous step of the D-PhANTM process can sustain Wigner negativity better than an existing PhANTM process in particular in the presence of loss in the hardware. In some embodiments, the variable and controllable rotations performed in D-PhANTMprocess, instead of the constant Pi / 2 rotations performed in the existing PhANTM process, may allow generation of GKP states. In some cases, the classical controller may be configured to adjust the rotation (R) to generate a desired non-Gaussian quantum state (e.g., a GKP state, a grid state, or the like). In some such cases, the classical controller may control the properties of the resulting non-Gaussian state (e.g., amplitude, squeezing, Wingner spacing, and the like) by adjusting a displacement (X, Z), a rotation (R), and / or a squeezing (sq) of an output or input quantum state during a one or more steps of the modified PHANTM.

[0073] In some embodiments, the beam splitter used in the modified PhANTM process can be a tunable beam splitter. In these embodiments, the tunable beam splitter may be dynamically adjusted by the classical controller based on one or more signals received from the homodyne and / or ONR detector. As such, in some cases, in addition to dynamic squeezing, rotation, and displacement, the system / process shown in FIG. 1 may allow feedback control of the splitting ratio of PhANTM process based on the outcomes of previous PhANTM process that provides input to the PhANTM process. Such dynamic splitting ratio may provide additional control over the type and / or properties of the non-Gaussian states generated after two or more application the modified PhANTM process.

[0074] In some embodiments, the device shown in FIG. 1 can be reconfigured to produce cat states, GKP states and other non-Gaussian resources when applied to a single quantum node (e.g., a quantum node of one or multi-dimensional cluster state) or a chain of entangled quantum nodes.

[0075] In various implementations, D-PhANTM may not include one or more of the displacement operations (in x or z dimensions), the rotation operation (R), and the squeezing operation (SQ). In some cases, modified PhANTM may not include a tunable beam splitter. In some such cases, the beam splitter may have a fixed splitting ratio, and the classical control system may control at least one of the displacement operation (X,Z), the rotation operation (R), and the squeezing operation (SQ) based on the signals received from a photodetector performing PNR 110 and a homodyne detector performing HD 114.

[0076] FIG. 2A is a block diagram illustrating a process for generating an output mode 225 having a non-Gaussian state using a plurality of D-PhANTM processes 202-1 to 202-4 sequentially performed on modes of a CV cluster where an output mode generated by a D- PhANTM process is fed as input mode to a subsequent D-PhANTM process to generate the outputmode 225. In some embodiments, an individual D-PhANTM process of the D-PhANTM processes 202-1 to 202-4 may comprise one of more features described above with respect to the D- PhANTM process illustrated in FIG. 1. In some embodiments, at least one of the D-PhANTM processes may receive feedforward signals from one or more of prior D-PhANTM processes. In some embodiments, a feedforward signal may comprise, or can be indicative of, outcomes of one or both of a previously performed HD operation and PNR operation. In some embodiments, a feedforward signal provided to D-PhANTM process in sequence may comprise, or can be indicative of, outcomes of one or both of all previously performed HD operations and PNR operations. For example, the «thfeedforward signal may comprise a vector formed by outcomes of PNR measurements of n-1 D-PhANTM processes performed before the «thD-PhANTM process. In some cases, additionally or alternatively, the nth feedforward signal may comprise a vector formed by outcomes of HD operations of n-1 D-PhANTM processes performed before the nth D- PhANTM process.

[0077] In the example shown in FIG. 2A, the first D-PhANTM process 203-1 may receive input mode 201 and generate a first output mode 01. In some cases, first D-PhANTM 202- 1 may comprise a rotation operation (R) that may be controlled by outcomes of one or both a first PNR operation and a first HD operation of D-PhANTM 202-1. In some cases, the first D-PhANTM 202-1 may further comprise a squeezing operation that may be controlled by outcomes of one or both the first PNR operation and the first HD operation of first D-PhANTM 202- 1. In some cases, the beam splitting operation of first D-PhANTM 202-1 may not be tunable or controlled (e.g., the splitting ratio of a beam splitter used in first D-PhANTM 202-1 may have fixed predetermined value). In some cases, the first D-PhANTM 202-1 may comprise fixed and predetermined beam splitting and rotation operations.

[0078] With continued reference to FIG. 2A, the second D-PhANTM process 202-2 may receive the first output mode 01 from the first D-PhANTM 202-1 process and generate a second output mode 02. In some cases, the second D-PhANTM process 203-2 may comprise one or more features described above with respect to the first D-PhANTM operation 202-1. In some cases, the second D-PhANTM process may comprise at least beam splitting, PNR, and rotation operations. In some cases, the second D-PhANTM 202- may comprise at least one operation (e.g., a rotation operation, a beam splitting operation, a displacement operation, or a squeezing operation) controlled based at least in part on a first feedforward signal 203-1 received from thefirst D-PhANTM operation 202-1, where the first feedforward signal 203-1 comprises or is indicative of the outcomes of one or both the first PNR operation and the first HD operation of the first D-PhANTM 202-1 operation.

[0079] With continued reference to FIG. 2A, the third D-PhANTM process 203-3 may receive the first output mode 02 from the second D-PhANTM 202-2 process and generate a third output mode 03. In some cases, the third D-PhANTM 202-3 process may comprise one or more features described above with respect to the second D-PhANTM operation 202-2. In some cases, the third D-PhANTM 202-3 may comprise at least one operation (e.g., a rotation operation, a beam splitting operation, a displacement operation, or a squeezing operation) controlled based at least in part on the first feedforward signal 203-1 and a second feedforward signal 203-2 received from the first and second D-PhANTM operations 202-1, 202-2, where the second feedforward signal 203-s2comprises or is indicative of the outcomes of one or both a second PNR operation and a second HD operation of the second D-PhANTM 202-2 operation.

[0080] With continued reference to FIG. 2A, the fourth D-PhANTM process 202-4 may receive the third output mode 03 from the third D-PhANTM 202-2 process and generate the output mode 225. In some cases, the fourth D-PhANTM 202-4 process may comprise one or more features described above with respect to the second D-PhANTM 202-2 process. In some cases, the fourth D-PhANTM 202-4 process may comprise at least one operation (e.g., a rotation operation, a beam splitting operation, a displacement operation, or a squeezing operation) controlled based at least in part on the first feedforward signal 203-1, the second feedforward signal 203-2, and the third feedforward signal received from the first, second, and third D-PhANTM 202- 1 , 202-2, 202- 3, processes respectively, where the third feedforward signal 203-3 comprises or is indicative of the outcomes of one or both a third PNR operation and a third HD operation of the third D- PhANTM 202-3 process.

[0081] In some embodiments, at least one of the second, third, and fourth D-PhANTM 202-2, 202-2, 202-3 processes may comprise at least one operation (e.g., a rotation operation, a beam splitting operation, a displacement operation, or a squeezing operation) controlled based at least in part on outcomes of one or both PNR and HD processes performed in the second, third, and fourth D-PhANTM 202-2, 202-2, 202-3 processes, respectively. In some embodiments, at least one operation (e.g., a rotation operation, a beam splitting operation, a displacement operation, or a squeezing operation) of a PhANTM process of the three PhANTM process 202-2, 202-3, 202-4, may be controlled based at least in part based on feedforward signals received by the PhANTM process and the outcomes of PNR and HD operations performed in the PhANTM process.

[0082] Advantageously, the controlling different operations of an individual PhANTM process of the sequence of PhANTM processes 202-1, 202-2, 202-3, 202-4 based on outcomes of operation performed within the individual PhANTM process and / or the outcomes of operation performed within previous PhANTM processes, allows generating an output mode 225 having a non-Gaussian state that may not be generated using a sequence of static (uncontrolled) PhANTM processes, improve the quality of the resulting non-Gaussian state, and / or generate a desired nonGaussian state with higher probability or smaller number of processes compared to the sequence of static (uncontrolled) PhANTM processes. In various implementations, non-Gaussian state of the output mode 225 may comprise a cat state, a GKP state, or another a non-Gaussian state.

[0083] In some embodiments, the characteristics of the non-Gaussian state of the output mode 225 may be controlled and / or adjusted by adjusting the feedforward signals 203-1, 203-2, 203-3, and / or 203-4. For example, an effect of a feedforward signal on an operation performed in D-PhANTM process may be tailored to adjust the non-Gaussian state of the output mode 225.

[0084] In various implementations, the sequence of the PhANTM process shown in FIG. 2A may be performed (or implemented) in time or frequency domain. For example, each D- PhANTM process may receive a frequency mode of a CV cluster state, and an output mode generated by a preceding D-PhANTM process (using another frequency mode of a CV cluster) to transform the state of the mode to a non-Gaussian state. As another example, quantum state of a single frequency mode of a CV cluster may be transformed to a non-Gaussian mode by performing a plurality of D-PhANTM process performed at different time steps.

[0085] FIG. 2B schematically illustrates a two-step process and system comprising two serially D-PhANM processes performed in frequency domain (where the input modes are frequency modes of a CV cluster). In various implementations, the two-step process shown in FIG. 2B can be a portion of a larger sequential process (e.g., the process shown in FIG. 2A) or a process to generate a non-Gaussian state using two D-PhANM processes. In some cases, the two-step process shown in FIG. 2B may implemented based on two D-PhANTM modules comprising optically and electrically interconnected components and devices associated with the respective operations.

[0086] In some embodiments, a first D-PhANTM module may comprise a first classical controller 206a that receives first measurement data from a first PNR detector 110a and a first homodyne detector 114a and generates first state control signals for controlling one or more of first quantum state modification modules 118a, 119a, 120a that are optically connected to a first entanglement gate 104a of a CV cluster state. The first entanglement gate 104a may be configured to entangle two input modes 102a, 102b, to generate an entangled pair, provide a first mode of the entangled pair to a first TBS 108a and a second mode of the entangled pair to one of the one or more of the first quantum state modification modules 118a, 119a, 120a that is optically coupled to the entanglement operator 104a. In some cases, the first classical controller 206a may additionally receive the first measurement data from another PNR or homodyne detector of a previous D- PhANTM (not shown) , generate a first tuning control signal based on at least a portion of the first measurement data and transmit the first tuning control signal to the first TBS 108a that is optically connected to receive the first mode of the entangled pair from the first entanglement gate 104a, transmit a first portion of the first mode to the first PNR detector 110a and a second portion of the first mode to the first homodyne detector 114a. The first tuning control signal may control or tune the splitting ratio of the first TBS 108a. In some embodiments, the one or more first quantum state modification modules 118a, 119a, 120a may be configured to modify the second mode received from the first entanglement gate 104a to generate a first output mode having a first non-Gaussian state. In some embodiments, the one or more first quantum state modification modules 118a, 119a, 120a may modify the second mode based on the first state control signals received from the first classical controller 206a. In some embodiments, the one or more first quantum state modification modules 118a, 119a, 120a may comprise quantum state displacement module 118a, a quantum state rotation module 119a, and a quantum state squeezing module 120a.

[0087] In some embodiments, the output state received from one or more first quantum state modification modules 118a, 119a, 120a may be entangled to third mode 105a of the CV cluster state and to generate a second entangled pair. In some cases, the second entangled pair may be provided to a second D-PhANTM module comprising a second classical controller 206b that can be connected to the first classical controller 206a via a wired or wireless link 207 to receive measurement data from the first classical controller 206a. In some cases, the second classical controller 206a may receive second measurement data from the one or more of the first PNR detector 110a (via the first classical controller 206a), the first homodyne detector 114a (via thefirst classical controller 206a), a second photon number resolving (PNR) detector 110b, and a second homodyne detector 114b. In some embodiments, the second classical controller 206a may use at least a portion of the second measurement data to generate second state control signals for controlling one or more of second quantum state modification modules 118b, 119b, 120b that are optically connected to the second entanglement gate 104b of a CV cluster state. The second entanglement gate 104a may be configured to provide a first mode of the second entangled pair to a second TBS 108b and a second mode of the second entangled pair to one of the one or more second quantum state modification modules 118b, 119b, 120b that is optically coupled to the entanglement operator 104b. In some cases, may generate a second tuning control signal based on at least a portion of the second measurement data and transmit the second tuning control signal tot eh second TBS 108b that is optically connected to receive the first mode of the second entangled pair from the second entanglement gate 104b, transmit a first portion of the first mode to the second PNR detector 110b and a second portion of the first mode to the second homodyne detector 114b. In some cases, the first tuning control signal may control or tune the splitting ratio of the second t TBS 108b. In some embodiments, the one or more second quantum state modification modules 118b, 119b, 120b may be configured to modify the second mode received from the second entanglement gate 104b to generate a second output mode having a second non-Gaussian state. In some embodiments, the one or more first quantum state modification modules 118a, 119a, 120a may modify the second mode based on the second state control signals received from the second classical controller 206b. In some embodiments, the one or more second quantum state modification modules 118b, 119b, 120b may comprise quantum state displacement module 118b, a quantum state rotation module 119b, and a quantum state squeezing module 120b.

[0088] In various implementations, a single classical controller module may comprise the first and second classical controllers 206a, 206b. For example, the first and second classical controllers 206a, 206b, may be fabricated as a single integrated circuit or a portion of a single integrated circuit.

[0089] FIG. 2C schematically illustrates a multi-step process and system comprising a plurality of D-PhANM processes performed in time domain (where the input modes are frequency modes of a CV cluster) where entangled modes are sequentially received by a D- PhANM module that performs of D-PhANM processes on the sequentially received entangled modes. In some cases, the entangled modes can be modes of a time domain CV cluster state thatare received at period set by an optical delay disposed between the CV cluster (source of the entangled states) and the D-PhANM module. In some embodiments, classical controller 106 o the D-PhANTM module 200 may receive measurement data from at least one of a photon number resolving (PNR) detector 210 and a homodyne detector 214, generate a tuning control signal based on the measurement data and transmit the tuning control signal to a TBS 208 that is optically connected to an optical delay 204 to receive a delayed mode to control or tune the ratio between optical intensities of the first and second portions of the delayed mode. In some embodiments, the optical delay 204 may be configured to delay an optical mode received from the CV cluster stare by T seconds. In some embodiments, the TBS 208 may transmit a first portion of the delayed mode to the PNR detector 210 and a second portion of the delayed mode to one or more quantum state modification modules 218, 219, 220 that are optically connected to the TBS 208 and are configured to modify the second portion of the delayed mode and transmit the modified second portion of the delayed mode to the homodyne detector 214. In some cases, classical controller may generate control signals based on the measurement data and transmit the control signals to the modification modules 218, 219, 220 modules to modify the second portion of the delayed mode.

[0090] In some embodiments, at least the optical delay 204 and the beam splitter 208 can be monolithically fabricated on a single photonic chip. Additionally, in some cases, at least a portion of the one or more quantum state modification modules 218, 219, 220 can be monolithically fabricated on the single photonic chip.Example non-Gaussian state generation using D-PhANTM: tuning rotation and squeezing

[0091] Below two examples are provided to illustrate the effectiveness of controlling rotation and squeezing when performing D-PhANTM. Such feature is absent in the existing PhANTM.Example 1) Application of rotations (and squeezing) to create grid states

[0092] FIGS. 3A-3D show calculated Wigner functions of example non-Gaussian states that may be generated using two steps of D-PhANTM process. In the examples shown below the rotation is skipped (the D-PhANM process may not include a rotation operation); however in other examples, at least the first D-PhANTM may include a rotation operation that may be controlled or adjusted to generate a specific non-Gaussian quantum state (output by the second PhANTM). As shown below some cases, when the D-PhANTM processes do not include rotationoperation a grid state can be generated. In some cases, the first D-PhANTM does not include a rotation operation and the first or the second D-PhANTM include a squeezing operation (e.g., squeezing levels are changed between 0.5 dB and r 4.34 dB). In some cases, squeezing may not affect the type of the resulting non-Gaussian state. In some cases, rotation with specific values may affect the non-Gaussianity of the resulting GKP states.

[0093] If two photons are subtracted in each step (during PNR operation), the approximate square GKP state shown in FIG. 3 A may be generated.

[0094] If four photons are subtracted in step 1 and two photons are subtracted in step 2, the grid state shown in FIG. 3B may be generated.

[0095] In some cases, to provide a desired GKP Wigner negativity, Wigner spacing and squeezing, a machine learning algorithm may be trained based classical controller to adjust squeezing, rotations, and displacements at each step. The final output will be optimized algorithms that can generate the desired GKP state with a certain success probability. As such, some of the methods disclosed here may be used to generate a grid state or GKP state by determining a number of PhANTM steps, a displacement adjustment, a rotation, and a squeezing level applied to the output of each step based on the out puts of the PNR and homodyne detectors used in the PhANTM steps.Example 2) Application of squeezing to create squeezed cat states

[0096] In some cases, when the hardware used in the system has Gaussian loss in Homodyne detection and photon detection the simulations below show that squeezing after each round of photon subtraction (squeezing of 0.5) can retain the negativity of the cat Wigner function compared to squeezing once at the end of 10 PhANTM steps. The photon subtraction events are randomly chosen for this 10-step simulation (0, 1, 1, 0, 0, 2, 0, 1, 1, 1).

[0097] Resulting cat state with squeezing after each round of photon subtraction and rotation are shown in FIG. 3C.

[0098] Resulting cat state with squeezing after all 10 PhANTM rounds is shown in FIG. 3D may be generated, below (each round being photon subtraction and rotation).

[0099] These results may provide a proof-of-concept that in some applications squeezing can be beneficial after photon subtraction. Since squeezing commutes with rotation, it can be applied after the rotation.2) Fast PhANTM (F-PhANTM)

[0100] As described above, in some embodiments, the efficiency or a PhANTM or Dynamic PhANTM process may be improved by providing an auxiliary steam of photon, e.g., from a source separate from the CV cluster state. For example, in some cases, providing one or more photons having squeezed states (e.g., a momentum, p, or position, q, squeezed state) to the PNR detector, e.g., through a port of the beam splitter (e.g., a fixed or a tunable beam splitter) of a PhANTM or D-PhANTM module, which would otherwise receive non-squeezed vacuum states, may increase the rate of photon detection by the PNR detector (and thereby the rate of nonGaussian state generation). In some cases, a lightly squeezed auxiliary photon stream may increase the probability of photon subtraction by 3 times. In some cases, providing a lightly squeezed auxiliary photos stream to the PNR detector of PhANTM module (or to the PNR measurement operation of a PhANTM process), e.g., via the beam splitter, may result in generation of highly squeezed cat states with large amplitudes from a CV cluster having modes with low squeezing level. In some implementations, a lightly squeezed auxiliary photon stream may result in more photon subtraction during a PhANTM or D-PhANTM process compared to a PhANTM or D- PhANTM performed on a highly squeezed CV cluster in the absence of lightly squeezed auxiliary photon stream. In some embodiments, providing an auxiliary photon stream (e.g., lightly squeezed auxiliary photon stream) may probabilistically control the parity of the cat state by increasing the chance of even / odd photon subtractions. In some embodiments, by controllably turning the auxiliary photon stream ON or OFF the probability of the photon subtraction and thereby generation rate of cat states and / or squeezing of the generated cat states may be controlled (e.g., providing a probabilistic switch for non-Gaussian state generation). In some embodiments, by controlling a squeezing level (r) of the auxiliary photons provided to the PNR detector the probability of generating different types of non-Gaussian states (e.g., a cat state or a GKP state) may be controlled. As such, in various implementations, a PhANTM process that receives auxiliary photons (also referred to ancilla photons), for increasing the efficiency of photon subtraction operation may generate non-Gaussian state at a higher rate and may allow controlling the type and properties of the resulting non-Gaussian states (e.g., cat and GKP states). A PhANTM process that receives auxiliary or ancilla photons for increasing the efficiency of photon subtraction operation may be referred to as fast PhANTM or F-PhANTM.

[0101] FIG. 4A schematically illustrates an example of a F-PhANTM process 400 (and the corresponding module) for generating non-Gaussian states at a high rate using an optical source 407 of ancilla (auxiliary) photons. In some cases, the optical source 407 may comprise a source of squeezed light (e.g., p-squeezed of q-squeezed). In some such cases, the squeezing level of the ancilla (auxiliary) photons may controllable.

[0102] In some embodiments, the process may comprise receiving a pair of entangled modes of the continuous variable (CV) quantum cluster state by a F-PhANTM module 400. In some cases, the pair of entangled modes may comprise a first mode 403a having a first initial state and a second mode 403b having a second initial state. The first mode 403a may comprise a first optical field and the second mode may comprise a second optical field. In some examples, an entanglement operator 404 (e.g., an entanglement gate) may generate the pair of entangled modes using a first input mode 402a and a and second input mode 402b. In some cases, the state of the at least one of the first and second input modes 402a, 40b, can be p-squeezed.

[0103] In some embodiments, the process may comprise performing photon subtraction using a beam splitter (BS) 408 on the first mode 403a by splitting the first optical field into a first portion and a second portion of the first optical field. In some cases, the first portion of the optical field may be output through a first port of the BS 408 and may be transmitted to a PNR detector 410, and the second portion of the optical field may be output through a second port of the BS 408 and may be transmitted to a homodyne detector 414. In some embodiments, an auxiliary photon stream may be provided from an optical source 407 to a third port of the BS 408 and transmitted to the PNR 410 through the first port of the BS 408. The auxiliary photon stream may comprise p-squeezed or q-squeezed photons. In some cases, the squeezing level of the auxiliary photons can be larger than 1 O'4dB but less than 2 dB, less than 4 dB, less than 6 d, less than 8 dB, less than 10 dB or, in some cases, larger values. In some cases, PNR 410 may perform a photon-number-resolving (PNR) detection on the first portion of the optical field and the auxiliary photon stream that, in some examples, can be both received from the second port of BS 408. In some cases, homodyne detector 414 may perform a homodyne detection on the second portion of the optical field to transform the second initial state of the second mode 403b to a teleported non-Gaussian state of the second mode 403b. In some examples, at least one Gaussian operation (e.g., a displacement operation 421) may be performed on the second mode 403b to generate an output mode 425 having a non-Gaussian state (e.g., a cat state or GKP state). The atleast one Gaussian operation 421 may be controlled by an output (measurement results) of the homodyne detector 414.

[0104] FIG. 4B shows simulated Wigner function of example non-Gaussian states generated by a single PhANTM process and the corresponding photon subtraction probabilities (bottom panels). In the example shown in FIG. 4B the non-Gaussian state represented by its Wigner function (top panel) is generated by applying PhANTM on a cluster state having a squeezing of 11 dB. The bottom panel shows the probability of photon subtraction plotted against the number of photons subtracted (e.g., number of photons detected by the PNR). Based on this plot the probability of subtracting 1-8 photons is about 12%.

[0105]

[0088] FIG. 4C shows simulated Wigner function of example non-Gaussian states generated by a single F-PhANTM process / module 400 using auxiliary photons (top panels) and the corresponding photon subtraction probabilities (bottom panels). In the example shown in FIG. 4C the non-Gaussian state represented by its Wigner function (top panel) is generated by applying F-PhANTM on a cluster state having a squeezing of 11 dB. The bottom panel shows the probability of photon subtraction plotted against the number of photons subtracted (e.g., number of photons detected by the PNR). Based on this plot the probability of subtracting 1 - 8 photons is about 33%.

[0106] A comparison between results shown in FIGS 4B and 4C, indicate that providing the auxiliary photons in the F-PhANTM has resulted in higher probability of photon subtraction (by about 3 times) and has improved the quality (e.g., fidelity) of the resulting non- Gaussian state (e.g, cat state).

[0107] FIG. 4D illustrates the calculated Wigner functions of example non-Gaussian state generated by a single step of F-PhATNM process 400 for two different squeezing levels (0.3 and 0.8) of a the ancilla photons and two different reflection coefficient (3% and 11%) of the BS 408. In this calculation, three photons have been subtracted and the squeezing of the ancilla photons provided is 1.2.

[0108] FIG. 4E illustrates the calculated Wigner functions of example non-Gaussian state generated by a single step of F-PhATNM process 400 for two different squeezing levels (0.3 and 0.8) of a the ancilla photons and two different reflection coefficient (3% and 11%) of the BS 408. In this calculation, five photons have been subtracted and the squeezing of the ancilla photons provided is 1.2.

[0109] In some embodiments, the F-PhANTM process 400 may be modified to include additional photon subtraction steps between the entanglement operator 404 and the homodyne detector 414 to subtract more photons from the first mode 403a.

[0110] FIG. 5A schematically illustrates an example embodiment of a modified F- PhANTM that includes multiple (N) beam splitters and PNR detectors that serially subtract photons from a first mode 721a (e.g., having a squeezing level of 1) of a pair of entangled modes, before homodyne detection, to generate an output mode having a desired non-Gaussian state. In some cases, the pair of entangled modes may be received from a CV cluster state and may comprise a first mode 721a and a second mode 721b (e.g., a p-squeezed mode). In some cases, an auxiliary photon stream (p-squeezed or q-squeezed photons) may be provided to at least one of the PNR detectors, e.g., via the respective beam splitter.

[0111] In the example shown, the first mode 721a is provided to a first beam splitter (BS) 708a that transmits a first portion of the first mode along with a first auxiliary photon stream 707 (e.g., having a squeezing level of r = -1.5) to a first PNR detector 710-1. In some cases, a first beam splitter (BS) 708a may transmit a second portion of the first mode to a second beam splitter 708b. the second beam splitter (BS) 708b may transmit a portion of the second portion of the first mode along with a second auxiliary photon stream (e.g., having a squeezing level of r = -1.5) to a second PNR detector 710-2. In some cases, the second BS 708b may transmit the remaining portion of the second portion of the first mode 721a to subsequent beam splitter. In some cases, the first and the second auxiliary photon streams may be provided to BS 708a and BS 708b from the same optical source or different optical sources. In some embodiments, the process may include more assisted photon subtraction steps. In some cases, the last photon subtraction step may include a BS 708c that transmits a portion of the first mode received from a preceding BS along with an auxiliary photon stream (e.g., having a squeezing level of r = -1.5) to a last PNR detector 710-N. In some cases, BS 708c may transmit the remaining portion of the first mode 721a to a homodyne detector 714 to generate a detector signal. The homodyne detector 714 may provide the detector signal to a quantum state modification module 723 configured to perform a Gaussian operation 723 (e.g., a quantum state displacement operation) on the second mode 721b of the entangled pair received from the CV cluster, based on the detector signal. In some cases, the output mode generated by the Gaussian operation 723 may have a non-Gaussian state determined by the numbed of photon subtracted by multiple subtraction steps.

[0112] FIGS. 5B-5C illustrate calculated Wigner functions for non-Gaussian states generated by the modified F-PhANTM process shown in FIG. 5A when five beam splitters (N=5) and five PNR detectors (each receiving an auxiliary photon stream) are used. For the calculations shown in FIG. 5B the angle (representing the beam splitting ratio) of the beam splitters is set to 10 degree and in FIG. 5CB the angle (representing the beam splitting ratio) of the beam splitters is set to 15 degrees. The four Wigner functions shown in each figure correspond to four different numbers of subtracted photons (n = 5, 6, 7, and 8)Combination of D-PhANTM and F-PhANTM

[0113] In some embodiments, D-PhANTM and F-PhANTM may be combined to generate a non-Gaussian state based on a combination of features described above with respect to figures 1, 2 A, 2B, 2C, 4A, and 5 A. For example, an auxiliary photon steam (having a p-squeezed or q-squeezed state) may be provided to one or more tunable beam splitters of the D-PhANTM modules / processes shown in Figures 1, 2A, 2B, and 2C. As another example, one or more beam splitters of the F-PhANTM modules / processes shown in figures 4A, and 5 A may be replaced by a tunable beam splitter controlled by a classical controller configured to control or tune the tunable beam splitter based on one or both measurement data received from a homodyne detector or a PNR detector. Alternatively or additionally, the Gaussian module of the F-PhANTM modules / processes shown in figures 4A, and 5A may include one or more of a quantum state rotation module and a quantum state squeezing module controlled by a classical controller based on one or both measurement data received from a homodyne detector or a PNR detector. In some cases, PhANTM process / module that includes at least one of the features of a D-PhANTM and at least one PNR operation assisted by an auxiliary photon stream may be referred to as fast dynamic PhANTM or FD-PhANTM.

[0114] FIG. 6 shows an example embodiment of fast dynamic PhANTM (FD- PhANTM) 600. In some cases, the FD-PhANTM 600 module / process may comprise one or more features described above with respect to D-PhANTM 100 module / process described above with respect to FIG. 1 and the components labeled with the same reference numeral may provide functions similar to the respective component in FIG. 1 when applicable. Additionally, FD- PhANTM 600 module / process may include an auxiliary photon source 607 that is provides an auxiliary photon stream (having a p-squeezed or q-squeezed state) to the PNR 110 via the TBS108 to increase the probability of photon subtraction (as indicated in calculations shown in figures 4B and 4C).Controlled and Fast PhANTM on macronodes

[0115] In certain embodiments, some of the methods may be used to generate and control non-Gaussian states in macronode CV quantum cluster such as cluster states comprising macronodes of two or more modes. In some embodiments, a process performed on a macronode may comprise photon subtraction assisted by an auxiliary photon stream and one or more homodyne detection processes controlled based on the measurement data received from a PNR detector (e.g., by a classical controller). Additionally, in some embodiments, a process performed on a macronode may comprise controlling or tuning a tunable beam splitter used for photon subtraction base on the measurement data received from one or both a PNR detector and at least one of the homodyne detectors. In some embodiments, controlling a homodyne detector may comprise changing the angles (theta) of the homodyne detector and the corresponding homodyne detection operation to implement squeezing, or an arbitrary single mode Gaussian operation, on a state. As such, a controlled homodyne detector or homodyne detection operation may not be limited to measurement in one of the p or q basis, but in a linear combination of these bases.

[0116] FIG. 7 schematically illustrates an example process / module 600 that may generate a non-Gaussian state using a three-mode macronode 702 received from a macronode CV cluster state. I some embodiments, module 600 may receive first, second, and third modes of a macronode 702 from a macrocode CV quantum cluster state. In some cases, the macronode 702 may be generated by entangling first and second initial modes 702a, 702b, to generate an entangled pair comprising a first mode of the macronode 702 (e.g., using an entanglement operator 704) and transmitting one of the modes of the entangled pair to a beam splitter 709 that receives a third initial mode 702 to generate second and third modes of the macronode 702.

[0117] I some embodiments, the module 600 may include a beam splitter (e.g., a fixed or tunable beam splitter) that received the third mode of the macronode 702, a second homodyne detector (HD) 714b that receives the second mode of the macronode 702, and a quantum state modification module 718 that receives the first mode of the macronode 702. In some cases, the quantum state modification module 718 may comprise a quantum state displacement module and one or more of a quantum state rotation module and a quantum state squeezing module.

[0118] In some embodiments, the beamsplitter 708 (e.g., a fixed or tunable beam splitter) configured to split the optical field of the first mode of the macronode 702 to first and second portions of the optical field, provide the first portion of the optical field to a first homodyne detector (HD) 714a, and provide the second portion of the optical field and an auxiliary stream of photons, e.g., received from an optical source 707, to a PNR detector 710.

[0119] In some cases, the module 600 may include classical controller 706 configured to receive the measurement data from the PNR 710, HD 714a, and HD 714b and one or more these measurement data to generate HD control signal for controlling HD 714a, and HD 714b, a tuning control signal for controlling a beam splitting ratio of the beam splitter 708 (when the beam splitter is tunable), an state control signals for controlling the quantum state modification module 718.

[0120] In some cases, the classical controller 706 may transmit the HD control signals (e.g., generated using the measurement data from the PNR 710) to the first and second homodyne detectors 714a, 714b to control the homodyne measurement angles of the first and second homodyne detectors 714a, 714b.

[0121] In some cases, the classical controller 706 may transmit the state control signals (e.g., generated using the measurement data from one or more of PNR 710, HD 714a, and HD 714b) to the quantum state modification module 718 that modify the first mode of the macronode 702 to generate an output mode 725 having a desired non-Gaussian state.

[0122] In some cases, the classical controller 706 may additionally transmit the tuning control signal (e.g., generated previous measurement data from one or more of PNR 710, HD 714a, and HD 714b) to the beam splitter 708 to tune its splitting ratio.Multimode PhANTM

[0123] In some implementations the modified PhANTM may be used to transform modes of a multidimensional cluster states or a multimode PhANTM. The entangled modes on which multimode PhANTM is applied can be frequency modes, a combination of frequency and temporal modes (e.g., a temporal mode can be considered a frequency mode at different times), or combinations of other degrees of freedom. After several application of PhANTM on the modes (or nodes of the cluster state), the output nodes of this network of entangled modes may comprise entangled modes having a non-Gaussian Wigner function on each node. If measurements are performed on all but one of the modes, distillation of all of the non-Gaussian resources into a singlehigh-quality cat state or more a general non-Gaussian state can be accomplished. In various implementations the PhANTM operation may comprise existing PhANTM (with static parameters) or modified PhANTM (with dynamically tuned parameters via feedback).

[0124] FIG. 8 schematically illustrates an example multimode non-Gaussian state generation process where each non-Gaussian state generation process may comprise a PhANTM process, a D-PhANTM process, an F-PhANM process, or an FD-PhANTM process. In this example PhANTM or a modified PhANTM (e.g., D-PhANT, F-PhANTM, or a combination thereof) may be applied several times on the nodes of a two-dimensional cluster states, to generate a non-Gaussian state (output from the bottom row which is not measured). In some cases, the output of the detection operations at the end of each sequence of PhANTM (or a modified PhANTM) process may be used by a classical controller to displace the final state back to the origin in phase space. In some cases, two-dimensional cluster states shown in FIG. 8 may comprise a frequency and a temporal dimension. For example, each row of states in FIG.8 may represent temporal evolution of a specific frequency mode as it is processed multiple times by PhANTM or modified PhANTM and is transformed to a final non-Gaussian state that is entangles to a non- Gaussian states resulting from processing the two adjacent frequency modes. In some cases, if the final non-Gaussian states generated for all frequency modes except on of the frequency modes, the quantum state of the un-measured non-Gaussian state is transformed to a new non-Gaussian state that can be, e.g., a GKP state. In the example shown, the non-Gaussian state of the bottom row, which is not measure, can comprise a GKP state, cat state, or a gird state. The measurement (detection) of the Gaussian modes resulting from different frequency modes can be performed sequentially (e.g., tl<t2<t3<... <tn) or non-sequentially. In some cases, at least two Gaussian modes resulting from different frequency modes are measured substantially at the same time.

[0125] Advantageously, multimode PhANTM can allow a trade-off between temporal clock cycles and number of frequency modes in a quantum computer / communication network / sensor. For example, to generate a cat state having certain characteristic using PhANTM process applied on a plurality of individual and non-entangled one-dimensional clusters (single frequency mode clusters) may require more time steps compared to generating the same cat state (having the same characteristics) by applying PhANTM on a two-dimensional cluster (i.e., using multimode PhANTM). In one simulated example a cat state was generated using 20 time steps ofPhANTM operation on a ID cluster while the same cat state was generated using 4 time steps of PhANTM operation on 5 entangled frequency modes.

[0126] With reference to FIG. 8, in some embodiments, a plurality of entangled modes of the continuous variable (CV) quantum cluster state comprising modes having optical fields oscillating at different frequencies (fl, f2, f3, ... fn) may be selected for performing a multimode non-Gaussian state generation process by sequentially performing one or more non-Gaussian state generation processes on individual frequency modes(fl, f2, f3, ... fn) of the plurality of entangled modes generates to generate plurality output modes 830 (also referred to as measured output modes) that are provided to a plurality of homodyne detectors 814 and at least one output mode 831 (also referred to as non- measured output mode), which is provided to a quantum state displacement correction module 816 configured to performing a Gaussian operation (e.g., displacement in x or z dimensions) on the non-measured output mode 831 to generate an output mode 825 having a desired non-Gaussian state. In some cases, the plurality of homodyne detectors 814 may perform homodyne detection on the plurality output modes 830 to generate measurement results and provide the measurement results to the quantum state displacement correction module 816, such that the non-measured output mode 831 is displaced (or otherwise modified) based on the homodyne measurement results. In various implementations, the one non-Gaussian state generation processes (shown by squares in FIG. 8) may comprise one or more of a fast PhANTM, a dynamic PhANTM, and a Dynamic and fast PhANTM. In various implementations, the one non- Gaussian state generation processes may not be performed at the same time on different frequency modes. In the example shown in FIG. 8, a first process 802-1 may be performed on the nth frequency mode (fn) followed by a first rotation operation (R) 804-1, the output of the rotation operator may be provided to a second process 802-2 followed by a second rotation operation (R) 804-2, the output of the rotation operator may be provided to a third process 802-3 followed by a third rotation operation (R) 804-3, and the output of the rotation operator may be provided to a homodyne detector of eth plurality of homodyne detectors 814. In various implementations, at least one of the processes 802-1, 802-2, 802-3 can be different from the other processes (e.g., different type process, e.g., D-PhANTM, F-PhANTM, FD-PhANM, or having different parameters or Gaussian operations therein)

[0127] In some cases, when a non-Gaussian process that includes a rotation operation is used to transform a frequency mode, a rotation operation may not be needed before performinga subsequent process on the outcome of the first process. For example, I the process 802-1 is a D- PhANTM that includes a rotation operation the rotation 804-1 may be skipped.

[0128] In some cases, when a non-Gaussian process that does not include rotation operation is used to transform a frequency mode, a rotation operation may be performed before performing a subsequent process on the outcome of the first process.

[0129] In some aspects, the methods and systems described above may comprise: a dynamic PhANTM optical chip, fast PhANTM optical chip, or fast dynamic PhANTM optical chip, which carries out programmed non-Gaussian state preparation on input modes (e.g., modes of a CV cluster states) such that the outputs can be cat states, squeezed cat states, or GKP states and other non-Gaussian states. The dynamic PhANTM optical chip, fast PhANTM optical chip, or fast dynamic PhANTM optical chip can be used to process the modes of a general cluster or lattice state in a programmable, dynamic manner, such that the output is a configurable non-Gaussian state such as cat statePhANTM applied on a cluster state having a plurality of frequency modes: theory

[0130] An example, derivation of PhANTM applied at N given times on a cluster state composed by M frequency modes is presented below. The derivation demonstrates that polynomial gates can be applied to the final unmeasured output as a result of these quantum operations, resulting in non- Gaussian states (Cat states), that can be generated when PhANTM is applied (one time) on a single mode. The polynomial gate can be a product of the gates performed at each of the nodes, i.e., the final unmeasured mode is a distilled (strong) cat state.

[0131] FIG. 9 is a block diagram illustrating an example application of PhANTM on a cluster of entangled frequency and time modes comprising N time modes and M frequency modes. The input state is represented on the right and the output state, resulting from all the homodyne detection, on the bottom left. The frequency modes are labeled) and the time steps by i, where) is from 0 to M and i is from 0 to N. In some cases, it can be assumed:

[0132] In some cases, it can be assumed that there is no loss (e.g., optical loss or photon loss) during an individual PhANTM operation and that the squeezing of the vacuum statesinvolved in the process is infinite. In some embodiments, a PhANTM process performed on a mode j at time i may be derived as a Kraus operator. When the measurement result of an individual homodyne detection (my) is zero, the Kraus operator can be expressed as:

[0133] Where Rj is the rotation operator applied on the mode j and Tfnj.i is the Hermite polynomial and Q the quadrature operator for mode j. Since, the rotation operation can be implemented in free space or on-chip, the rotation operator may be corrected after the application of the Kraus operator. In some cases, the PhANTM process can be reduced to the application of the Hermite polynomial of degree WQJ), where ?2(j,i) is the number of photon subtracted on mode j at the given time i. After the N PhANTM steps we perform homodyne detection on the p quadrature on modes 2 to M. Using the communication between Cz gates and polynomials, the teleported output state can be written as:

[0134] Replacing the input state by its integral expression results in:

[0135] Applying the product of Cz gates and the product of polynomial in Q to the state inside the integral provides:

[0136] Next, using the relation

[0137] Replacing t by Q in the coefficient of the state and taking out the new operator from the integral.

[0138] From (7) the form of the state which is a infinite squeezed vacuum state in addition of the Z displacement on the left side of the equation:Example embodiments

[0139] Various additional example embodiments of the disclosure can be described by the following clauses:

[0140] Clause 1. A method for generating a non-Gaussian state, the method comprising: as implemented by one or more hardware processors configured to execute computerexecutable instructions: receiving first measurement data from at least one of a first photon number resolving (PNR) detector and a first homodyne detector; generating a first tuning control signal based on at least a portion of the first measurement data; transmitting the first tuning control signal to a first tunable beam splitter that is optically connected to receive a first input mode from a first entanglement gate and transmit a first portion of the first input mode to the first PNR detector and a second portion of the first input mode to the first homodyne detector; and generating first state control signals for controlling one or more first quantum state modification modules that are optically connected to the first entanglement gate and configured to modify a second input mode received from the first entanglement gate to generate a first output mode having the non-Gaussian state.

[0141] Clause 2. The method of clause 1, wherein generating the first state control signals comprises generating the first state control signals based on the first measurement data.

[0142] Clause 3. The method of clause 1, wherein transmitting the first tuning control signal to the first tunable beam splitter comprises tuning splitting ratio of the first tunable beam splitter based on the first tuning control signal.

[0143] Clause 4. The method of clause 1, wherein the one or more first quantum state modification modules comprise a displacement operation module, a rotation operation module, and a squeezing operation module.

[0144] Clause 5. The method of clause 4, wherein a rotation angle of the rotation operation module, an X-displacement parameter of the displacement operation module, or a Z- displacement parameter of the displacement operation module is adjusted based on the first state control signals.

[0145] Clause 6. The method of clause 4, wherein the one or more first quantum state modification modules comprise at least one displacement operation module.

[0146] Clause 7. The method of clause 4, wherein an X-displacement parameter or a Z-displacement parameter of the displacement operation module is adjusted based on at least a portion of the first state control signals.

[0147] Clause 8. The method of clause 7, wherein the at least a portion of the first state control signals is proportional to a signal generated by the first homodyne detector.

[0148] Clause 9. The method of clause 1, further comprising providing first auxiliary photons to the first PNR detector.

[0149] Clause 10. The method of clause 9, wherein the first measurement data comprises data generated by the first PNR detector in response to receiving the first portion of the first input mode and the first auxiliary photons.

[0150] Clause 11. The method of clause 10, wherein the first auxiliary photons comprise p-squeezed or q-squeezed photons.

[0151] Clause 12. The method of clause 1, wherein the first entanglement gate is a controlled-z (Cz) gate.

[0152] Clause 13. The method of clause 1, wherein the first output mode comprises a GKP state.

[0153] Clause 14. The method of clause 1, further comprising: receiving second measurement data from at least one of the first PNR detector, the first homodyne detector, a second PNR detector, and a second homodyne detector; generating a second tuning control signal based on at least a portion of the second measurement data; transmitting the second tuning control signal to a second tunable beam splitter that is optically connected to receive a third input mode from a second entanglement gate and transmit a first portion of the third input mode to the second PNRdetector and a second portion of the third input mode to the second homodyne detector; and generating second state control signals for controlling one or more second quantum state modification modules that are optically connected to the second entanglement gate and configured to modify a fourth input mode received from the second entanglement gate; wherein the second entanglement gate is optically connected to the one or more first quantum state modification modules to receive the first output mode and generate the third input mode and the fourth input mode using at least the first output mode.

[0154] Clause 15. The method of clause 14, wherein generating the second state control signals comprises generating the second state control signals based on the second measurement data.

[0155] Clause 16. The method of clause 14, wherein transmitting the second tuning control signal to the second tunable beam splitter comprises tuning splitting ratio of the second tunable beam splitter based on the second tuning control signal.

[0156] Clause 17. The method of clause 14, wherein the one or more second quantum state modification modules comprise a displacement operation module, a rotation operation module, and a squeezing operation module.

[0157] Clause 18. The method of clause 14, further comprising providing second auxiliary photons to the second PNR detector.

[0158] Clause 19. The method of clause 18, wherein the second measurement data comprises data generated by the second PNR detector in response to receiving the second portion of the third input mode and the second auxiliary photons.

[0159] Clause 20. The method of clause 19, wherein the second auxiliary photons comprise p-squeezed or q-squeezed photons.

[0160] Clause 21. The method of clause 14, wherein the second tunable beam splitter is monolithically fabricated on a photonic chip.

[0161] Clause 22. The method of clause 21, wherein at least a portion of the one or more second quantum state modification modules is monolithically fabricated on the photonic chip.

[0162] Clause 23. A method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising, by a first D-PhANTM module:receiving, a pair of entangled modes of the continuous variable (CV) quantum cluster state comprising a first mode having a first initial state and a second mode having a second initial state; receiving first measurement results of one or both homodyne measurement or photon-number- resolving (PNR) detection performed to generate another non-Gaussian state; generating a first tuning control signal based on the first measurement results; performing photon subtraction on the first mode by splitting a first optical field into a first portion and a second portion of the first optical field based on the first tuning control signal; performing a first PNR detection on the first portion; performing a first homodyne detection on the second portion to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one first Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state.

[0163] Clause 24. The method of clause 23, wherein a ratio between optical intensities of the first portion and the second portion of the first optical field is tuned based on the first tuning control signal.

[0164] Clause 25. The method of clause 23, wherein performing the at least one first Gaussian operation generates the first output mode by transforming the teleported non-Gaussian state of the second mode to the first non-Gaussian state.

[0165] Clause 26. The method of clause 23, wherein performing the first PNR detection transforms the first initial state of the first mode to a non-Gaussian state, and wherein performing the first homodyne detection on the second portion teleports the non-Gaussian state of the first mode to the second mode.

[0166] Clause 27. The method of clause 23, further comprising: generating a first state control signal based on at least one of the first measurement results or the first tuning control signal; and adjusting at least one parameter of the at least one first Gaussian operation based on the first state control signal.

[0167] Clause 28. The method of clauses 23, wherein the pair of entangled modes are generated using a CZ gate.

[0168] Clause 29. The method of clause 23, wherein the at least one first Gaussian operation comprises a displacement operation.

[0169] Clause 30. The method of clause 29, wherein the at least one first Gaussian operation further comprises one or both rotation and squeezing.

[0170] Clause 31. The method of clauses 23, wherein at least one of the first initial state or the second initial state comprises a squeezed state.

[0171] Clause 32. The method of clauses 23, wherein the first non-Gaussian state comprises a GKP state.

[0172] Clause 33. The method of clause 23, wherein performing the first PNR detection on the first portion comprises receiving a first auxiliary photon stream and performing the first PNR detection on both the first portion and the first auxiliary photon stream.

[0173] Clause 34. The method of clause 33, wherein the first auxiliary photon stream comprises p-squeezed or q-squeezed photons.

[0174] Clause 35. The method of clause 23, further comprising processing the CV quantum cluster state by a second D-PHANTM module, wherein processing the CV quantum cluster state comprises: receiving a third mode comprising a third optical field from the CV quantum cluster state, wherein the third mode has a third initial state and is entangled to the first output mode; receiving second measurement results of one or both of the first homodyne detection and the first PNR detection; generating a second tuning control signal based on the second measurement results; performing photon subtraction on the first output mode by splitting an output optical field of the first output mode into a third portion and a fourth portion of the output optical field based on the second tuning control signal; performing a second PNR detection on the third portion; performing a second homodyne detection on the fourth portion to transform the third initial state of the third mode to a second teleported non-Gaussian state; and performing at least one second Gaussian operation on the third mode to generate a second output mode having the second teleported non-Gaussian state.

[0175] Clause 36. The method of clause 35, wherein a ratio between optical intensities of the third portion and the fourth portion of the output optical field is tuned based on the second tuning control signal.

[0176] Clause 37. The method of clause 35, further comprising: generating a second state control signal based on at least one of the second measurement results or the second tuning control signal; and adjusting at least one parameter of the at least one second Gaussian operation based on the second state control signal.

[0177] Clause 38. The method of clause 35, wherein the at least one second Gaussian operation comprises a displacement operation.

[0178] Clause 39. The method of clause 35, wherein the at least one second Gaussian operation further comprises one or both rotation and squeezing.

[0179] Clause 40. The method of clause 35, wherein the second teleported nonGaussian state comprises a GKP state.

[0180] Clause 41. The method of clause 35, wherein the second teleported nonGaussian state has a larger fidelity compared to the first non-Gaussian state.

[0181] Clause 42. The method of clause 35, wherein performing the second PNR detection on the third portion, comprises receiving second auxiliary photons and performing the second PNR detection on both the third portion and the second auxiliary photons.

[0182] Clause 43. The method of clause 42, wherein the second auxiliary photons comprise p-squeezed or q-squeezed photons.

[0183] Clause 44. A method for generating a non-Gaussian state, the method comprising: as implemented by a hardware processor configured to execute computer-executable instructions: receiving measurement data from at least one of a photon number resolving (PNR) detector and a homodyne detector; generating a tuning control signal based on the measurement data; and transmitting the tuning control signal to a tunable beam splitter that is optically connected to an optical delay to receive a delayed mode, transmit a first portion of the delayed mode to the PNR detector and a second portion of the delayed mode to one or more quantum state modification modules are optically connected to the tunable beam splitter and configured to modify the second portion of the delayed mode and transmit the modified second portion of the delayed mode to the homodyne detector; wherein the optical delay is configured to receive an input mode comprising an optical field from a CV cluster state and generate the delayed mode.

[0184] Clause 45. The method of clause 44, further comprising generating control signals based on the measurement data and transmitting the control signals to the one or more quantum state modification modules to modify the second portion of the delayed mode.

[0185] Clause 46. The method of clause 44, wherein the one or more quantum state modification modules comprise a displacement operation module, a rotation operation module, or a squeezing operation module.

[0186] Clause 47. The method of clause 44, wherein the one or more quantum state modification modules comprise at least one displacement operator.

[0187] Clause 48. The method of clause 47, wherein the one or more quantum state modification modules further comprise one or both a rotation operator and a squeezing operator.

[0188] Clause 49. The method of clause 44, wherein a ratio between optical intensities of the first portion and the second portion of the delayed mode is tuned based on the tuning control signal.

[0189] Clause 50. The method of clause 44, wherein at least the optical delay and the tunable beam splitter are monolithically fabricated on a single photonic chip.

[0190] Clause 51. The method of clause 50, wherein at least a portion of the one or more quantum state modification modules is monolithically fabricated on the single photonic chip.

[0191] Clause 52. A method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising: receiving a pair of entangled modes of the continuous variable (CV) quantum cluster state comprising a first mode having a first initial state and a second mode having a second initial state; performing photon subtraction on the first mode by splitting the first mode into a first portion and a second portion of the first mode; receiving an auxiliary photon stream; performing a photon-number-resolving (PNR) detection on the first portion and the auxiliary photon stream; performing a homodyne detection on the second portion to transform the second initial state of the second mode into a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state.

[0192] Clause 53. The method of clause 52, wherein the auxiliary photon stream comprises p-squeezed or q-squeezed photons.

[0193] Clause 54. A system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system comprising: a bean splitter configured to receive a first mode of a pair of entangled modes of the CV quantum cluster state and generating first and second portions of the first mode; an optical source configured to generate a stream of auxiliary photons; a PNR detector configured to simultaneously receive the first portion of the first mode and the auxiliary photon stream and to generate a PNR detection result; a homodyne detector configured to receive the second portion of the first mode to generate a second detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair ofentangled modes and generate an output mode having the non-Gaussian state based on the second detector signal.

[0194] Clause 55. The system of clause 54, wherein the auxiliary photon stream comprises p-squeezed or q-squeezed photons.

[0195] Clause 56. The system of clause 54, wherein the one or more state modification modules comprise at least one displacement operator.

[0196] Clause 57. The system of clause 54, wherein the one or more state modification modules further comprise one or both a rotation operator and a squeezing operator.

[0197] Clause 58. A method of generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising: receiving a pair of entangled modes of the CV quantum cluster state comprising a first mode having a first initial state and a second mode having a second initial state; selecting the first mode of the pair of entangled modes in the canonical cluster state, wherein the first mode comprises a first optical field; performing a first photon subtraction on the first mode by splitting the first optical field into a first portion and a second portion of the first optical field; receiving a first auxiliary photon stream; performing a first PNR detection on the first portion and the first auxiliary photon stream; performing a second photon subtraction on the second portion of the first optical field by splitting the second portion of the first optical field into a third portion and a fourth portion of the second portion of the first optical field; receiving a second auxiliary photon stream; performing a second PNR detection on the first portion and the first auxiliary photon stream; performing a homodyne detection on the fourth portion of the second portion of the first optical field to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the non-Gaussian state.

[0198] Clause 59. The method of clause 58, further comprising generating a tuning control signal based on the first PNR detection, wherein performing the second photon subtraction comprises turning a ratio between optical intensities of the third portion and the fourth portion of the second portion of the first optical field based on the tuning control signal.

[0199] Clause 60. The method of clause 58, wherein the first and second auxiliary photons streams comprise p-squeezed or q-squeezed photons.

[0200] Clause 61. A system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system comprising: a first beam splitter configured to receive a first mode of a pair of entangled modes of the continuous variable (CV) quantum cluster state and generate a first portion and a second portion of the first mode; an optical source configured to generate a stream of auxiliary photons; a first PNR detector configured to receive the first portion of the first mode and a first portion of the stream of auxiliary photons; a second beam splitter configured to receive the second portion of the first mode and generate a transmitted portion and a reflected portion of the second portion of the first mode; a second PNR detector configured to receive the reflected portion of the second portion of the first mode and a second portion of the stream of auxiliary photons; a homodyne detector configured to receive the transmitted portion of the second portion of the first mode and generate a detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the detector signal.

[0201] Clause 62. The system of clause 61, wherein the one or more quantum state modification modules comprise at least one displacement operator.

[0202] Clause 63. The system of clause 62, wherein the one or more quantum state modification modules further comprise one or both of a rotation operator and a squeezing operator.

[0203] Clause 64. The system of clause 61, wherein the first and second auxiliary photon streams comprise p-squeezed or q-squeezed photons.

[0204] Clause 65. A method for generating a non-Gaussian state, the method comprising: as implemented by a hardware processor configured to execute computer-executable instructions: receiving first, second, and third modes of a CV macrocode cluster state; splitting optical field of the first mode to first and second portions of the optical field, providing the first portion of the optical field to a first homodyne detector; providing an auxiliary stream of photons and the second portion of the optical field to a PNR detector; generating control signals based on first measurement data received from the PNR detector; adjusting a parameter of at least one of the first and second homodyne detectors using the control signals; and modifying the third mode of the CV macrocode cluster state using one or more quantum state modification modules based on second measurement data received from the first and second homodyne detectors.

[0205] Clause 66. A method for generating a non-Gaussian state using at least one macronode of a macronode cluster state, the method comprising: as implemented by a hardware processor configured to execute computer-executable instructions: receiving first measurement data from a photon number resolving (PNR) detector configured to receive a first portion of a first mode of the at least one macronode and auxiliary photons from an optical source; generating first and second HD control signals based on the first measurement data for controlling a first homodyne detector configured to receive a second portion of the first mode of the at least one macronode and a second homodyne detector configured to receive a second mode of the at least one macronode; receiving second measurement data from the first and second homodyne detectors; and generating first and second state control signals based on the second measurement data for controlling a quantum state modification module that is configured to receive a third mode of the at least one macronode and generate an output mode having the non-Gaussian state.

[0206] Clause 67. The method of clause 66, further comprising generating a tuning control signal based on the second measurement data and use the second measurement data to tune a splitting ratio of a beam splitter configured to receive the first mode of the at least one macronode and generate the first and second portions of the at least one macronode.

[0207] Clause 68. A method of generating a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising: receiving a plurality of entangled modes of the continuous variable (CV) quantum cluster state comprising modes having optical fields oscillating at different frequencies; sequentially performing one or more non-Gaussian state generation processes on individual modes of the plurality of entangled modes generates a plurality of measured output modes and at least one non-measured output mode; performing homodyne detection on the measured output modes to generate measurement results; and performing a Gaussian operation on the at least one non-measured output mode based on the measurement results.

[0208] Clause 69. The method of clause 68, wherein at least one non-Gaussian state generation process of the one or more non-Gaussian state generation processes comprises a fast PhANTM process, a dynamic PhANTM process, or a dynamic and fast PhANTM process. SUMM In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state, the method including: as implemented by one or more hardware processors configured to execute computer-executable instructions: receiving first measurement data from at least one of afirst photon number resolving (PNR) detector and a first homodyne detector; generating a first tuning control signal based on at least a portion of the first measurement data; transmitting the first tuning control signal to a first tunable beam splitter that is optically connected to receive a first input mode from a first entanglement gate and transmit a first portion of the first input mode to the first PNR detector and a second portion of the first input mode to the first homodyne detector; and generating first state control signals for controlling one or more first quantum state modification modules that are optically connected to the first entanglement gate and configured to modify a second input mode received from the first entanglement gate to generate a first output mode having the non-Gaussian state. In some aspects, the techniques described herein relate to a method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including, by a first D-PhANTM module: receiving, a pair of entangled modes of the continuous variable (CV) quantum cluster state including a first mode having a first initial state and a second mode having a second initial state; receiving first measurement results of one or both homodyne measurement or photon-number-resolving (PNR) detection performed to generate another non- Gaussian state; generating a first tuning control signal based on the first measurement results; performing photon subtraction on the first mode by splitting a first optical field into a first portion and a second portion of the first optical field based on the first tuning control signal; performing a first PNR detection on the first portion; performing a first homodyne detection on the second portion to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one first Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state. In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state, the method including: as implemented by a hardware processor configured to execute computer-executable instructions: receiving measurement data from at least one of a photon number resolving (PNR) detector and a homodyne detector; generating a tuning control signal based on the measurement data; and transmitting the tuning control signal to a tunable beam splitter that is optically connected to an optical delay to receive a delayed mode, transmit a first portion of the delayed mode to the PNR detector and a second portion of the delayed mode to one or more quantum state modification modules are optically connected to the tunable beam splitter and configured to modify the second portion of the delayed mode and transmit the modified second portion of the delayed mode to the homodynedetector; wherein the optical delay is configured to receive an input mode including an optical field from a CV cluster state and generate the delayed mode. In some aspects, the techniques described herein relate to a method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including: receiving a pair of entangled modes of the continuous variable (CV) quantum cluster state including a first mode having a first initial state and a second mode having a second initial state; performing photon subtraction on the first mode by splitting the first mode into a first portion and a second portion of the first mode; receiving an auxiliary photon stream; performing a photon-number-resolving (PNR) detection on the first portion and the auxiliary photon stream; performing a homodyne detection on the second portion to transform the second initial state of the second mode into a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state. In some aspects, the techniques described herein relate to a system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system including: a bean splitter configured to receive a first mode of a pair of entangled modes of the CV quantum cluster state and generating first and second portions of the first mode; an optical source configured to generate a stream of auxiliary photons; a PNR detector configured to simultaneously receive the first portion of the first mode and the auxiliary photon stream and to generate a PNR detection result; a homodyne detector configured to receive the second portion of the first mode to generate a second detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the second detector signal. In some aspects, the techniques described herein relate to a method of generating an output mode having a non- Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including: receiving a pair of entangled modes of the CV quantum cluster state including a first mode having a first initial state and a second mode having a second initial state; selecting the first mode of the pair of entangled modes in the canonical cluster state, wherein the first mode includes a first optical field; performing a first photon subtraction on the first mode by splitting the first optical field into a first portion and a second portion of the first optical field; receiving a first auxiliary photon stream; performing afirst PNR detection on the first portion and the first auxiliary photon stream; performing a second photon subtraction on the second portion of the first optical field by splitting the second portion of the first optical field into a third portion and a fourth portion of the second portion of the first optical field; receiving a second auxiliary photon stream; performing a second PNR detection on the first portion and the first auxiliary photon stream; performing a homodyne detection on the fourth portion of the second portion of the first optical field to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the non-Gaussian state. In some aspects, the techniques described herein relate to a system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system including: a first beam splitter configured to receive a first mode of a pair of entangled modes of the continuous variable (CV) quantum cluster state and generate a first portion and a second portion of the first mode; an optical source configured to generate a stream of auxiliary photons; a first PNR detector configured to receive the first portion of the first mode and a first portion of the stream of auxiliary photons; a second beam splitter configured to receive the second portion of the first mode and generate a transmitted portion and a reflected portion of the second portion of the first mode; a second PNR detector configured to receive the reflected portion of the second portion of the first mode and a second portion of the stream of auxiliary photons; a homodyne detector configured to receive the transmitted portion of the second portion of the first mode and generate a detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the detector signal. In some aspects, the techniques described herein relate to a method for generating a non- Gaussian state, the method including: as implemented by a hardware processor configured to execute computer-executable instructions: receiving first, second, and third modes of a CV macrocode cluster state; splitting optical field of the first mode to first and second portions of the optical field, providing the first portion of the optical field to a first homodyne detector; providing an auxiliary stream of photons and the second portion of the optical field to a PNR detector; generating control signals based on first measurement data received from the PNR detector; adjusting a parameter of at least one of the first and second homodyne detectors using the control signals; and modifying the third mode of the CV macrocode cluster state using one or morequantum state modification modules based on second measurement data received from the first and second homodyne detectors. In some aspects, the techniques described herein relate to a method for generating a non-Gaussian state using at least one macronode of a macronode cluster state, the method including: as implemented by a hardware processor configured to execute computer-executable instructions: receiving first measurement data from a photon number resolving (PNR) detector configured to receive a first portion of a first mode of the at least one macronode and auxiliary photons from an optical source; generating first and second HD control signals based on the first measurement data for controlling a first homodyne detector configured to receive a second portion of the first mode of the at least one macronode and a second homodyne detector configured to receive a second mode of the at least one macronode; receiving second measurement data from the first and second homodyne detectors; and generating first and second state control signals based on the second measurement data for controlling a quantum state modification module that is configured to receive a third mode of the at least one macronode and generate an output mode having the non-Gaussian state. In some aspects, the techniques described herein relate to a method of generating a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method including: receiving a plurality of entangled modes of the continuous variable (CV) quantum cluster state including modes having optical fields oscillating at different frequencies; sequentially performing one or more non-Gaussian state generation processes on individual modes of the plurality of entangled modes generates a plurality of measured output modes and at least one nonmeasured output mode; performing homodyne detection on the measured output modes to generate measurement results; and performing a Gaussian operation on the at least one non-measured output mode based on the measurement results.Terminology

[0209] It is to be understood that not necessarily all objects or advantages may be achieved in accordance with any particular embodiment described herein. Thus, for example, those skilled in the art will recognize that certain embodiments may be configured to operate in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other objects or advantages as may be taught or suggested herein.

[0210] All of the processes described herein may be embodied in, and fully automated via, software code modules executed by a computing system that includes one or more computers or processors. The code modules may be stored in any type of non-transitory computer-readable medium or other computer storage device. Some or all the methods may be embodied in specialized computer hardware. Further, the computing system may include, be implemented as part of, or communicate with, a computation network or a cloud computing system.

[0211] Many other variations than those described herein will be apparent from this disclosure. For example, depending on the embodiment, certain acts, events, or functions of any of the algorithms described herein can be performed in a different sequence, can be added, merged, or left out altogether (for example, not all described acts or events are necessary for the practice of the algorithms). Moreover, in certain embodiments, acts or events can be performed concurrently, for example, through multi-threaded processing, interrupt processing, or multiple processors or processor cores or on other parallel architectures, rather than sequentially. In addition, different tasks or processes can be performed by different machines and / or computing systems that can function together. The terms “in one / some case(s)” or “in one / some implementation(s)” represent various embodiments.

[0212] The various illustrative logical blocks and modules described in connection with the embodiments disclosed herein can be implemented or performed by a machine, such as a processing unit or processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A processor can be a microprocessor, but in the alternative, the processor can be a controller, microcontroller, or state machine, combinations of the same, or the like. A processor can include electrical circuitry configured to process computer-executable instructions. In another embodiment, a processor includes an FPGA or other programmable device that performs logic operations without processing computer-executable instructions. A processor can also be implemented as a combination of computing devices, for example, a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. Although described herein primarily with respect to digital technology, a processor may also include primarily analog components. A computing environment can include any type of computer system,including, but not limited to, a computer system based on a microprocessor, a mainframe computer, a digital signal processor, a portable computing device, a device controller, or a computational engine within an appliance, to name a few.

[0213] Conditional language such as, among others, “can,” “could,” “might” or “may,” unless specifically stated otherwise, are otherwise understood within the context as used in general to convey that certain embodiments include, while other embodiments do not include, certain features, elements and / or steps. Thus, such conditional language is not generally intended to imply that features, elements and / or steps are in any way required for one or more embodiments or that one or more embodiments necessarily include logic for deciding, with or without user input or prompting, whether these features, elements and / or steps are included or are to be performed in any particular embodiment.

[0214] Disjunctive language such as the phrase “at least one of X, Y, or Z,” unless specifically stated otherwise, is otherwise understood with the context as used in general to present that an item, term, etc. , may be either X, Y, or Z, or any combination thereof (for example, X, Y, and / or Z). Thus, such disjunctive language is not generally intended to, and should not, imply that certain embodiments require at least one of X, at least one of Y, or at least one of Z to each be present.

[0215] Any process descriptions, elements or blocks in the flow diagrams described herein and / or depicted in the attached figures should be understood as potentially representing modules, segments, or portions of code which include one or more executable instructions for implementing specific logical functions or elements in the process. Alternate implementations are included within the scope of the embodiments described herein in which elements or functions may be deleted, executed out of order from that shown, or discussed, including substantially concurrently or in reverse order, depending on the functionality involved as would be understood by those skilled in the art.

[0216] Unless otherwise explicitly stated, articles such as “a” or “an” should generally be interpreted to include one or more described items. Accordingly, phrases such as “a device configured to” are intended to include one or more recited devices. Such one or more recited devices can also be collectively configured to carry out the stated recitations. For example, “a processor configured to carry out recitations A, B and C” can include a first processor configuredto carry out recitation A working in conjunction with a second processor configured to carry out recitations B and C.

[0217] Many variations and modifications may be made to the above-described embodiments, the elements of which are to be understood as being among other acceptable examples. All such modifications and variations are intended to be included herein within the scope of this disclosure.

Claims

WHAT IS CLAIMED IS:

1. A method for generating a non-Gaussian state, the method comprising: as implemented by one or more hardware processors configured to execute computer-executable instructions: receiving first measurement data from at least one of a first photon number resolving (PNR) detector and a first homodyne detector; generating a first tuning control signal based on at least a portion of the first measurement data; transmitting the first tuning control signal to a first tunable beam splitter that is optically connected to receive a first input mode from a first entanglement gate and transmit a first portion of the first input mode to the first PNR detector and a second portion of the first input mode to the first homodyne detector; and generating first state control signals for controlling one or more first quantum state modification modules that are optically connected to the first entanglement gate and configured to modify a second input mode received from the first entanglement gate to generate a first output mode having the non-Gaussian state.

2. The method of claim 1, wherein generating the first state control signals comprises generating the first state control signals based on the first measurement data.

3. The method of claim 1, wherein transmitting the first tuning control signal to the first tunable beam splitter comprises tuning splitting ratio of the first tunable beam splitter based on the first tuning control signal.

4. The method of claim 1, wherein the one or more first quantum state modification modules comprise a displacement operation module, a rotation operation module, and a squeezing operation module.

5. The method of claim 4, wherein a rotation angle of the rotation operation module, an X-displacement parameter of the displacement operation module, or a Z-displacement parameter of the displacement operation module is adjusted based on the first state control signals.

6. The method of claim 4, wherein the one or more first quantum state modification modules comprise at least one displacement operation module.

7. The method of claim 4, wherein an X-displacement parameter or a Z-displacement parameter of the displacement operation module is adjusted based on at least a portion of the first state control signals.

8. The method of claim 7, wherein the at least a portion of the first state control signals is proportional to a signal generated by the first homodyne detector.

9. The method of claim 1, further comprising providing first auxiliary photons to the first PNR detector.

10. The method of claim 9, wherein the first measurement data comprises data generated by the first PNR detector in response to receiving the first portion of the first input mode and the first auxiliary photons.

11. The method of claim 10, wherein the first auxiliary photons comprise p-squeezed or q- squeezed photons.

12. The method of claim 1, wherein the first entanglement gate is a controlled-z (Cz) gate.

13. The method of claim 1, wherein the first output mode comprises a GKP state.

14. The method of claim 1, further comprising: receiving second measurement data from at least one of the first PNR detector, the first homodyne detector, a second PNR detector, and a second homodyne detector; generating a second tuning control signal based on at least a portion of the second measurement data; transmitting the second tuning control signal to a second tunable beam splitter that is optically connected to receive a third input mode from a second entanglement gate and transmit a first portion of the third input mode to the second PNR detector and a second portion of the third input mode to the second homodyne detector; and generating second state control signals for controlling one or more second quantum state modification modules that are optically connected to the second entanglement gate and configured to modify a fourth input mode received from the second entanglement gate; wherein the second entanglement gate is optically connected to the one or more first quantum state modification modules to receive the first output mode and generate the third input mode and the fourth input mode using at least the first output mode.

15. The method of claim 14, wherein generating the second state control signals comprises generating the second state control signals based on the second measurement data.

16. The method of claim 14, wherein transmitting the second tuning control signal to the second tunable beam splitter comprises tuning splitting ratio of the second tunable beam splitter based on the second tuning control signal.

17. The method of claim 14, wherein the one or more second quantum state modification modules comprise a displacement operation module, a rotation operation module, and a squeezing operation module.

18. The method of claim 14, further comprising providing second auxiliary photons to the second PNR detector.

19. The method of claim 18, wherein the second measurement data comprises data generated by the second PNR detector in response to receiving the second portion of the third input mode and the second auxiliary photons.

20. The method of claim 19, wherein the second auxiliary photons comprise p-squeezed or q-squeezed photons.

21. The method of claim 14, wherein the second tunable beam splitter is monolithically fabricated on a photonic chip.

22. The method of claim 21, wherein at least a portion of the one or more second quantum state modification modules is monolithically fabricated on the photonic chip.

23. A method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising, by a first D-PhANTM module: receiving, a pair of entangled modes of the continuous variable (CV) quantum cluster state comprising a first mode having a first initial state and a second mode having a second initial state; receiving first measurement results of one or both homodyne measurement or photon-number-resolving (PNR) detection performed to generate another non-Gaussian state; generating a first tuning control signal based on the first measurement results; performing photon subtraction on the first mode by splitting a first optical field into a first portion and a second portion of the first optical field based on the first tuning control signal; performing a first PNR detection on the first portion;performing a first homodyne detection on the second portion to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one first Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state.

24. The method of claim 23, wherein a ratio between optical intensities of the first portion and the second portion of the first optical field is tuned based on the first tuning control signal.

25. The method of claim 23, wherein performing the at least one first Gaussian operation generates the first output mode by transforming the teleported non-Gaussian state of the second mode to the first non-Gaussian state.

26. The method of claim 23, wherein performing the first PNR detection transforms the first initial state of the first mode to a non-Gaussian state, and wherein performing the first homodyne detection on the second portion teleports the non-Gaussian state of the first mode to the second mode.

27. The method of claim 23, further comprising: generating a first state control signal based on at least one of the first measurement results or the first tuning control signal; and adjusting at least one parameter of the at least one first Gaussian operation based on the first state control signal.

28. The method of claims 23, wherein the pair of entangled modes are generated using a CZ gate.

29. The method of claim 23, wherein the at least one first Gaussian operation comprises a displacement operation.

30. The method of claim 29, wherein the at least one first Gaussian operation further comprises one or both rotation and squeezing.

31. The method of claims 23, wherein at least one of the first initial state or the second initial state comprises a squeezed state.

32. The method of claims 23, wherein the first non-Gaussian state comprises a GKP state.

33. The method of claim 23, wherein performing the first PNR detection on the first portion comprises receiving a first auxiliary photon stream and performing the first PNR detection on both the first portion and the first auxiliary photon stream.

34. The method of claim 33, wherein the first auxiliary photon stream comprises p- squeezed or q-squeezed photons.

35. The method of claim 23, further comprising processing the CV quantum cluster state by a second D-PHANTM module, wherein processing the CV quantum cluster state comprises: receiving a third mode comprising a third optical field from the CV quantum cluster state, wherein the third mode has a third initial state and is entangled to the first output mode; receiving second measurement results of one or both of the first homodyne detection and the first PNR detection; generating a second tuning control signal based on the second measurement results; performing photon subtraction on the first output mode by splitting an output optical field of the first output mode into a third portion and a fourth portion of the output optical field based on the second tuning control signal; performing a second PNR detection on the third portion; performing a second homodyne detection on the fourth portion to transform the third initial state of the third mode to a second teleported non-Gaussian state; and performing at least one second Gaussian operation on the third mode to generate a second output mode having the second teleported non-Gaussian state.

36. The method of claim 35, wherein a ratio between optical intensities of the third portion and the fourth portion of the output optical field is tuned based on the second tuning control signal.

37. The method of claim 35, further comprising: generating a second state control signal based on at least one of the second measurement results or the second tuning control signal; and adjusting at least one parameter of the at least one second Gaussian operation based on the second state control signal.

38. The method of claim 35, wherein the at least one second Gaussian operation comprises a displacement operation.

39. The method of claim 35, wherein the at least one second Gaussian operation further comprises one or both rotation and squeezing.

40. The method of claim 35, wherein the second teleported non-Gaussian state comprises a GKP state.

41. The method of claim 35, wherein the second teleported non-Gaussian state has a larger fidelity compared to the first non-Gaussian state.

42. The method of claim 35, wherein performing the second PNR detection on the third portion, comprises receiving second auxiliary photons and performing the second PNR detection on both the third portion and the second auxiliary photons.

43. The method of claim 42, wherein the second auxiliary photons comprise p-squeezed or q-squeezed photons.

44. A method for generating a non-Gaussian state, the method comprising: as implemented by a hardware processor configured to execute computerexecutable instructions: receiving measurement data from at least one of a photon number resolving (PNR) detector and a homodyne detector; generating a tuning control signal based on the measurement data; and transmitting the tuning control signal to a tunable beam splitter that is optically connected to an optical delay to receive a delayed mode, transmit a first portion of the delayed mode to the PNR detector and a second portion of the delayed mode to one or more quantum state modification modules are optically connected to the tunable beam splitter and configured to modify the second portion of the delayed mode and transmit the modified second portion of the delayed mode to the homodyne detector; wherein the optical delay is configured to receive an input mode comprising an optical field from a CV cluster state and generate the delayed mode.

45. The method of claim 44, further comprising generating control signals based on the measurement data and transmitting the control signals to the one or more quantum state modification modules to modify the second portion of the delayed mode.

46. The method of claim 44, wherein the one or more quantum state modification modules comprise a displacement operation module, a rotation operation module, or a squeezing operation module.

47. The method of claim 44, wherein the one or more quantum state modification modules comprise at least one displacement operator.

48. The method of claim 47, wherein the one or more quantum state modification modules further comprise one or both a rotation operator and a squeezing operator.

49. The method of claim 44, wherein a ratio between optical intensities of the first portion and the second portion of the delayed mode is tuned based on the tuning control signal.

50. The method of claim 44, wherein at least the optical delay and the tunable beam splitter are monolithically fabricated on a single photonic chip.

51. The method of claim 0, wherein at least a portion of the one or more quantum state modification modules is monolithically fabricated on the single photonic chip.

52. A method of generating an output mode having a first non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising: receiving a pair of entangled modes of the continuous variable (CV) quantum cluster state comprising a first mode having a first initial state and a second mode having a second initial state; performing photon subtraction on the first mode by splitting the first mode into a first portion and a second portion of the first mode; receiving an auxiliary photon stream; performing a photon-number-resolving (PNR) detection on the first portion and the auxiliary photon stream; performing a homodyne detection on the second portion to transform the second initial state of the second mode into a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the first non-Gaussian state.

53. The method of claim 0, wherein the auxiliary photon stream comprises p-squeezed or q-squeezed photons.

54. A system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system comprising: a bean splitter configured to receive a first mode of a pair of entangled modes of the CV quantum cluster state and generating first and second portions of the first mode; an optical source configured to generate a stream of auxiliary photons;a PNR detector configured to simultaneously receive the first portion of the first mode and the auxiliary photon stream and to generate a PNR detection result; a homodyne detector configured to receive the second portion of the first mode to generate a second detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the second detector signal.

55. The system of claim 54, wherein the auxiliary photon stream comprises p-squeezed or q-squeezed photons.

56. The system of claim 54, wherein the one or more state modification modules comprise at least one displacement operator.

57. The system of claim 54, wherein the one or more state modification modules further comprise one or both a rotation operator and a squeezing operator.

58. A method of generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising: receiving a pair of entangled modes of the CV quantum cluster state comprising a first mode having a first initial state and a second mode having a second initial state; selecting the first mode of the pair of entangled modes in the canonical cluster state, wherein the first mode comprises a first optical field; performing a first photon subtraction on the first mode by splitting the first optical field into a first portion and a second portion of the first optical field; receiving a first auxiliary photon stream; performing a first PNR detection on the first portion and the first auxiliary photon stream; performing a second photon subtraction on the second portion of the first optical field by splitting the second portion of the first optical field into a third portion and a fourth portion of the second portion of the first optical field; receiving a second auxiliary photon stream; performing a second PNR detection on the first portion and the first auxiliary photon stream;performing a homodyne detection on the fourth portion of the second portion of the first optical field to transform the second initial state of the second mode to a teleported non-Gaussian state; and performing at least one Gaussian operation on the second mode to generate a first output mode having the non-Gaussian state.

59. The method of claim 58, further comprising generating a tuning control signal based on the first PNR detection, wherein performing the second photon subtraction comprises turning a ratio between optical intensities of the third portion and the fourth portion of the second portion of the first optical field based on the tuning control signal.

60. The method of claim 58, wherein the first and second auxiliary photons streams comprise p-squeezed or q-squeezed photons.

61. A system for generating an output mode having a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the system comprising: a first beam splitter configured to receive a first mode of a pair of entangled modes of the continuous variable (CV) quantum cluster state and generate a first portion and a second portion of the first mode; an optical source configured to generate a stream of auxiliary photons; a first PNR detector configured to receive the first portion of the first mode and a first portion of the stream of auxiliary photons; a second beam splitter configured to receive the second portion of the first mode and generate a transmitted portion and a reflected portion of the second portion of the first mode; a second PNR detector configured to receive the reflected portion of the second portion of the first mode and a second portion of the stream of auxiliary photons; a homodyne detector configured to receive the transmitted portion of the second portion of the first mode and generate a detector signal; and one or more quantum state modification modules configured to receive a second mode of the pair of entangled modes and generate an output mode having the non-Gaussian state based on the detector signal.

62. The system of claim 61, wherein the one or more quantum state modification modules comprise at least one displacement operator.

63. The system of claim 62, wherein the one or more quantum state modification modules further comprise one or both of a rotation operator and a squeezing operator.

64. The system of claim 61, wherein the first and second auxiliary photon streams comprise p-squeezed or q-squeezed photons.

65. A method for generating a non-Gaussian state, the method comprising: as implemented by a hardware processor configured to execute computerexecutable instructions: receiving first, second, and third modes of a CV macrocode cluster state; splitting optical field of the first mode to first and second portions of the optical field, providing the first portion of the optical field to a first homodyne detector; providing an auxiliary stream of photons and the second portion of the optical field to a PNR detector; generating control signals based on first measurement data received from the PNR detector; adjusting a parameter of at least one of the first and second homodyne detectors using the control signals; and modifying the third mode of the CV macrocode cluster state using one or more quantum state modification modules based on second measurement data received from the first and second homodyne detectors.

66. A method for generating a non-Gaussian state using at least one macronode of a macronode cluster state, the method comprising: as implemented by a hardware processor configured to execute computerexecutable instructions: receiving first measurement data from a photon number resolving (PNR) detector configured to receive a first portion of a first mode of the at least one macronode and auxiliary photons from an optical source; generating first and second HD control signals based on the first measurement data for controlling a first homodyne detector configured to receive asecond portion of the first mode of the at least one macronode and a second homodyne detector configured to receive a second mode of the at least one macronode; receiving second measurement data from the first and second homodyne detectors; and generating first and second state control signals based on the second measurement data for controlling a quantum state modification module that is configured to receive a third mode of the at least one macronode and generate an output mode having the non-Gaussian state.

67. The method of claim 66, further comprising generating a tuning control signal based on the second measurement data and use the second measurement data to tune a splitting ratio of a beam splitter configured to receive the first mode of the at least one macronode and generate the first and second portions of the at least one macronode.

68. A method of generating a non-Gaussian state embedded in a canonical cluster state, wherein the canonical cluster state is a continuous variable (CV) quantum cluster state, the method comprising: receiving a plurality of entangled modes of the continuous variable (CV) quantum cluster state comprising modes having optical fields oscillating at different frequencies; sequentially performing one or more non-Gaussian state generation processes on individual modes of the plurality of entangled modes generates a plurality of measured output modes and at least one non-measured output mode; performing homodyne detection on the measured output modes to generate measurement results; and performing a Gaussian operation on the at least one non-measured output mode based on the measurement results.

69. The method of claim 68, wherein at least one non-Gaussian state generation process of the one or more non-Gaussian state generation processes comprises a fast PhANTM process, a dynamic PhANTM process, or a dynamic and fast PhANTM process.

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