Optical parametric amplification protocol for quantum nondestructive measurement

Abstract

[Solution] Methods and systems are presented that use optical parametric amplifiers in various ways to enhance the native second-order coupling strength, thereby measuring or otherwise observing photonic components of interest without destroying the components of interest at the system output. For example, such components may include a number of signal Bogoliubov excitations, pump modular quadratures, or signal quadratures squared.

Description

[Technical Field]

[0001] We claim priority over U.S. Patent No. 63 / 403,217 ("Quantum Nondemolition Measurements With Optical Parametric Amplifiers For Ultrafast Fault-Tolerant Universal Quantum Information Processing"). [Brief explanation of the drawing]

[0002] [Figure 1] Figure 1 schematically shows a system in which the first encoding unit includes one or more optical parametric amplifiers (OPAs) configured for nonlinearity enhancement of at least one photonic component incorporating one or more improvement techniques. [Figure 2] Similarly, Figure 2 shows a system suitable for nonlinearity enhancement that incorporates one or more improvement techniques. [Figure 3] Figure 3 shows a system featuring at least one phase-mismatched OPA for nonlinearity enhancement incorporating one or more improvement techniques. [Figure 4] Figure 4 plots the positive-operator-valued measure (POVM) purity as a function of the pump homodyne result for quantum nondemolition (QND) measurement protocols incorporating one or more improvement techniques. [Figure 5]Figure 5 plots the relative weights of the squeezed-Fock-state projector in POVM as another function of the homodyne result for various Na values ​​in which one or more improvement techniques can be incorporated. [Figure 6] Figure 6 shows a system featuring at least one phase mismatch OPA suitable for nonlinear enhancement, in which one or more improvement techniques may be incorporated. [Figure 7] Figure 7 shows the trajectories of signal excitation and pump displacement as a function of interaction time, in which one or more improved techniques may be incorporated. [Figure 8] Figure 8 shows the signal x-quadrature squeezing level for a given quadrature noise level, in which one or more improvement techniques may be incorporated. [Figure 9] Figure 9 shows a system featuring at least one phase-matched OPA suitable for nonlinear enhancement, in which one or more improvement techniques may be incorporated. [Figure 10] Figure 10 shows a deterministic cubic-phase state generation system featuring at least one OPA, and summarizes the results of numerical simulations in which one or more improvement techniques may be incorporated. [Figure 11] Figure 11 shows plots of nonlinear squeezing as a function of initial EPR squeezing for various values ​​of τ in which one or more improvement techniques can be incorporated. [Modes for carrying out the invention]

[0003] This invention was made possible with government support from the National Science Foundation (NCSF) grants CCF1918549, ECCS1846273, PHY2011363, and ARO grant W911NF-23-1-0048, as well as support from NASA's Jet Propulsion Laboratory. The government has certain rights to this invention. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] R. Yanagimoto, R. Nehra, R. Hamerly, E. Ng, A. Marandi, and H. Mabuchi, Quantum Nondemolition Measurements with Optical Parametric Amplifiers for Ultrafast Universal Quantum Information Processing, PRX Quantum 4, 010333 (2023) (hereinafter "PRX paper") [Non-Patent Document 2] .Yanagimoto, R. Nehra, R. Hamerly, E. Ng, A. Marandi, and H. Mabuchi, Engineering Cubic Quantum Nondemolition Hamiltonian with Mesoscopic Optical Parametric Interactions, Quantum Physics (quant-ph), arXiv:2305.03260 [quant-ph], See https: / / doi.org / 10.48550 / arXiv.2305.03260 (2023) (hereinafter “Oph paper”).

[0005] The following detailed descriptions are primarily based on the processes and symbolic representations of operations performed by conventional computer components, including processors, memory storage devices for processors, connected display devices, and input devices. Furthermore, some of these processes and operations can utilize conventional computer components in heterogeneous distributed computing environments, including remote file servers, computer servers, and memory storage devices.

[0006] The terms used in the following descriptions are intended to be interpreted in the broadest and most reasonable way, even when used in conjunction with detailed descriptions of specific exemplary embodiments. Certain terms may be emphasized below, but terms intended to be interpreted restrictively are defined so explicitly and specifically.

[0007] Expressions such as "in one embodiment," "in various embodiments," and "in several embodiments" are used repeatedly. Such expressions do not necessarily refer to the same embodiment. The terms "comprising," "having," and "including" are synonymous unless otherwise indicated by the context.

[0008] <Translator's Note (Regarding the notation used in the translated text)> Symbols such as "^" and " ~ The notation with " cannot be displayed using fonts supported by the Japan Patent Office, therefore the symbol "^" or " ~ The character "" is written shifted forward. For example, "^χ b The symbol "χ" has a "^" placed directly above it. Also, in the square root symbol "√", the bar above the root symbol is sometimes omitted. Furthermore, some symbols, such as "φ", may appear different depending on the font, but they all represent the same symbol.

[0009] "Above", "Add", "Allow", "Between", "Cat state", "Calculation", "Combination", "Destruction", "Effective", "Encoding", "Enhanced", "Established", "First", "Gaussian", "Generaldyne "dyne)", "generate", "GKP", "Hamiltonian", "homodyne", "implemented", "include", "indirect", "input", "intact", "intra-cavity", "greater", "measured", "mismatched", "further", "native", "nonlinear", "non-negative", "of", "optical", "other", "parametric", "photonic", "ponderomotive", "quadratic", "quantum", "above", "as", "squeezed", "superfast", "universal", "here", "broad", "none", or other such descriptors are used in the ordinary sense of yes or no, rather than simply as terms of degree, unless otherwise specified in the context. In light of this disclosure, a person skilled in the art will understand from the context the meaning of “remote” and other such location descriptors used herein. Similarly, a person skilled in the art will understand the meaning of “partially based” or other such descriptions of dependent computation variables / signals. As used herein, “many” means two dozen or more. As used herein, “immediate” means having a duration of less than two seconds, unless otherwise specified in the context. In this specification, "circuitry" is "invoked" unless otherwise specified in the context, when a transition of voltage states occurs and a digital signal is required to be transmitted from or through it. Software is "invoked" in this specification when it is executed / triggered, unless otherwise specified in the context. Two numbers are said to be "on the same order" as another number if, unless otherwise indicated by the context, the difference between them is less than one order of magnitude (i.e., a factor of less than 10). As used herein, "cause" is not limited to immediate causes but also includes the realization, combination, or other actual causes of an event or phenomenon. "Instances" of an item, as used herein, may be identical or dissimilar to one another.

[0010] Terms such as “processor,” “center,” “unit,” and “computer” are used herein in their usual sense to refer to inanimate structures. Such terms do not include any person, regardless of their location, employment, or other connection to what is described, unless otherwise indicated in the context. Furthermore, “for” is not used to explicitly state a mere intended purpose, such as “circuitry for” or “instruction for,” but rather to descriptively identify software or structures for a specific purpose. “Specific,” “given,” and “particular” are not intended to provide substantive descriptions with nuances related to specification, gift, or particle. Rather, these adjectives individualize items or materials to distinguish them clearly from similar or other items or materials within a given context, without using hierarchical terms such as “first.”

[0011] Next, we will refer in detail to the description of the embodiments shown in the drawings. Although embodiments are described in relation to the drawings and the associated description, there is no intention to limit the scope to the embodiments disclosed herein. Rather, the intention is to cover all alternatives, modifications, and equivalents. In alternative embodiments, additional devices, or combinations of illustrated devices, may be added or combined without limiting the scope to the embodiments disclosed herein.

[0012] Referring to Figure 1, a system 100 is schematically shown in which the first encoding unit 140 includes one or more optical parametric amplifiers (OPAs) 160. Each OPA160 includes one or more photonic components 176 (e.g., a number of signal quadrature squared 161, a pump modular quadrature 162, or signal Bogoliubov excitations 163) and exhibits a first decoherence rate (κ) 177 and a nonlinear coupling strength (g) 183. The "native" nonlinearity coupling strength (g) 183 of nonlinearity enhancement coupling can be significantly enhanced (e.g., by more than 100%) by the various configurations described herein, depending on which configuration of the (one or more) OPA160 and (one or more) photonic components 176 is used. Such significant enhancements allow many such configurations to achieve a primary photonic strength (g) It is possible to leave component 176 connected to one output port while obtaining measurement 178 or other useful results via another output port.

[0013] In some contexts, for example, the photonic component 176 characterizes the field 131 of the external pump 130 as a shift 138, state 137, mode 136, or pump field quadrature (^χ b )134 may be provided to the encoding unit 140. This allows one or more features 195 of a given input component 176 (e.g., an indicator of a pump modular quadrature 162) to be monitored as a signal output 191 (e.g., via one or more operations 197 instead of a detector 170A) so that the given input component 176 does not break down during measurement but continues to be used as a pump output 192 over long distances (e.g., over 100 kilometers).

[0014] In some contexts, the photonic component 176 may be provided to the encoding unit 140 as an element 165 or mode 166 of the input signal state 111A. This allows the number of signal Bogolyubov excitations (SBEs) 163 or other input component 176 to be monitored via the pump output 192, and the corresponding input component 176 to continue to the fiberoptic-born signal output 191 rather than being destroyed.

[0015] Referring here to Figure 2, a schematic representation of system 200 (for example, as an example of system 100) is shown, in which an external optical input 232 arrives via port 208A and a weakly nonlinear OPA 160A, and can then be matched via a dichroic mirror 209A with the input signal state 111B (for example, the number of SBEs 264) received via port 208B. The entangled input then reaches a medium 282 containing an optical parametric amplifier 160B, characterized by additional strength 253, displacement 261, field 271, and coupling 272, as further described below. As shown in the figure, a second dichroic mirror 209B then separates the pump output 192 exiting through port 208C from the signal output 291 exiting through port 208D.

[0016] In some variations of system 200, another photonic component 176 has a pump modular quadrature 162, which passes through the corresponding phase mismatch OPA 160B (example (instance)), which has a first quadratic coupling strength 183, and as a result it is enhanced with a larger additional quadratic coupling strength 253. This allows such a primary photonic component 176 to pass through output port 208C directly, while obtaining the measurement 178B or other accessible encoding via another output port 208D instead of the pump output detector 170A. For a variation in which the primary photonic component 176 of the pump input or similar external input 232 passes directly through the systems 100, 200 as a component of the pump output 192 or similar result 292 via a feedforward operator, see Figure 6 with the following attached description.

[0017] Similarly, in some variations of one or more systems 100, 200, a first particular photonic component 176 has a number 264 of signal Bogolyubov excitations 163, which passes through a corresponding phase-mismatched OPA 160B (instance), which has a first quadratic coupling strength 183, which is enhanced by a larger additional quadratic coupling strength 253. This allows such a primary photonic component 176 to pass through output port 208D directly, while obtaining state-indicative inference / measurement 178A or other accessible encoding 278 via another output port 208C instead of detector 170B. Preferably large additional quadratic coupling strengths 253 as described herein can retain the primary photonic component 176 of the fundamental harmonic passing through the signal outputs 191, 291, and 391. See Figure 3.

[0018] Referring here to Figure 3, a schematicly drawn system 300 is shown, which in several variations may include or be similar to system 100 or system 200 (or both). A phase-mismatched optical parametric amplifier 160C, as shown, receives a fundamental harmonic mode 385A (e.g., number 264 of SBEs 163) corresponding to state 311A ​​and a second harmonic mode 385B corresponding to state 311B, thereby subjecting the harmonic waves 385A-B to entanglement 322 and elective nonlinear coupling strength enhancement. See paper PRX.

[0019] Figure 1 of the paper PRX corresponds to Figure 3 herein, adapted to conform to the PCT manuscript preparation standards. Figure 3 shows our PNR QND measurement scheme using the nonlinear quantum behavior of the OPA, where the phase-space representation (i.e., Wigner function) of the system state at each step of the protocol is shown using numerical data. In numerical simulations, the cosine component axis 315A(x a 314A(p) for the sine component axis relative to the quadrature. a The initial coherent signal state |φ is shown in state 311A, where the quadrature is plotted. a Consider (0)>=|α=0.7>. The seemingly circular zone 302 (shown by the solid black line) indicates a quasi-probability value of approximately 0.2 or greater, 377A. The dashed contour line 301 indicates a smaller positive quasi-probability value of 377A compared to this state 311A. The p-squeezed vacuum state 311B with width w=1 / 4 is shown along axis 315B(xb ) with respect to axis 314B(p b ) is assumed as the initial pump state 311B to plot.

[0020] The signal state and the pump state interact via a frequency-detuned OPA160C, and its dynamics cause conditional p displacements of the pump field that depend on the number 264 of signal Bogoliubov excitations 163. At the same time, the OPA dynamics also cause a conditional rotation of the signal Bogoliubov excitations 163 depending on x b which results in a phase spread of the final unconditional signal state 331A that characterizes the signal output 391 plotted with axis 334B(p a ) with respect to axis 335B(x a ). Note that for each state 375 where Na equals from 1 to 3, some zones 303 have an elliptical or annular part showing quasiprobability values 377A of about -0.2 or less.

[0021] The complete p-homodyne measurement 370 for the final pump state 331B functions as a QND measurement of ^Na and projects the signal mode onto the squeezed photon-number state which is the eigenstate of ^Na. The final pump state 331B is represented by the p-quadrature distribution P(pb). The ensemble-averaged signal state 375 is conditioned on the result of the homodyne measurement within the interval ~ gt(Na - 1) ≤ ^pb ≤ ~ gt(Na + 1). As system parameters, Δ / g = 150,~ We use g / g=1 and total interaction time gt=1.

[0022] The realization of room-temperature ultra-fast photon-number-resolving (PNR) quantum non-destructive (QND) measurements has significant implications for photonic quantum information processing (QIP), for example, enabling deterministic quantum computation in discrete-variable architectures. However, the difficulty of performing sufficiently strong coupling has hindered the development of scalable implementations. In the paper PRX, quadratic (i.e., χ) coupling is introduced. (2) We proposed and analyzed a nonlinear optical route to PNR QND using nonlinear interactions. We show that a coherent pump field 131 driving a phase-unmatched (i.e., frequency-detuned) OPA 160C experiences a displacement 261 favorably conditioned to the number 264 of signal Bogoliubov excitations 163. Thus, the measurement of the pump displacement provides a QND measurement 178A of signal Bogoliubov excitations 163, projecting the signal mode onto a squeezed photon-number state. Next, we demonstrate how our nonlinear OPA dynamics can be used to deterministically generate Gottesman-Kitaev-Preskill states via one or more additional Gaussian resources, providing an all-optical route for fault-tolerant QIP in continuous-variable systems. Finally, by emphasizing the similarities between phase-mismatched optical parametric oscillators and multilevel atom-cavity (QED) systems, we place these QND schemes in a more classical context. Our analysis suggests that our proposal will soon χ (2) This demonstrates feasibility in nonlinear nanophotonics, highlighting the high potential of OPA160 as a universal tool for ultrafast non-Gaussian quantum state engineering and quantum computing.

[0023] Quantum information science and engineering hold immense potential to revolutionize many fields, including computing, communication, and measurement. Among the various physical systems experimented with to encode and process quantum information, photonics offers significant advantages in terms of room-temperature scalability and ultrafast operation. Optical photons cover the terahertz bandwidth and can be transmitted over long distances with almost no decoherence, making them ideal carriers for quantum information. Photonic quantum computing allows information to be encoded and processed using both discrete-variable (DV) and continuous-variable (CV) architectures. However, the lack of strong optical nonlinearity has hindered the realization of deterministic two-qubit entangled gates in DV architectures and non-Gaussian resources such as Gottesman-Kitaev-Preskill (GKP) states in CV architectures. Both of these are essential for building general-purpose fault-tolerant quantum information processors. While the limitations of weak optical nonlinearity can be circumvented by measurement-based nonlinear computations using photon number discrimination (PNR) measurements, the inherent probabilistic nature of these computations and the slowness of conventional single-photon detectors with complex cryogenic systems (e.g., superconducting nanowires and superconducting transition edge sensors) significantly limit the scalability and computational clock rate of these architectures.

[0024] In this context, the realization of ultrafast room-temperature PNR QND measurements has significant implications for both DV and CV systems. In PNR QND measurements, information about the number of photons is encoded in an auxiliary probe, and backaction is limited to (partial) projection onto the corresponding photon number eigenstate. Such ultrafast QND measurements can not only replace conventional superconducting PNR detectors, but can also directly realize deterministic two-qubit entangled gates that enable deterministic DV optical quantum computation. Furthermore, the QND characteristics of this measurement offer unique opportunities for quantum engineering, communications, and measurement. To realize PNR QND measurements with separable single-photon energy shifts, strong coupling such that g / κ > 1 (where g is the coherent coupling rate and κ is the decoherence rate) is effective. Since the pioneering research of atomic cavity quantum electrodynamics (QED), strong coupling has been demonstrated in various physical systems. However, the simultaneous realization of QND measurements on a scalable, high-bandwidth, room-temperature platform has yet to be developed.

[0025] In the PRX paper, we proposed and analyzed a nonlinear optical route to PNR QND measurements and GKP states for all-optical quantum state engineering using a quadratic optical parametric amplifier (OPA) 160. Compared to conventional PNR QND measurement proposals and GKP state generation schemes using third-order nonlinearity, our proposal using OPA160 utilizes a much stronger second-order nonlinearity, providing a more experimentally feasible route. Recently, g / κ ~ 0.01 has been demonstrated in second-order nonlinear nanophotonic resonators, but g / κ ~ 10 can also be expected with ultrafast pulses.

[0026] First, we show that the pump field 131 of the phase-mismatched OPA 160 experiences conditional displacements 261 that depend on the number of signal Bogolyubov excitations 163 (^Na) 264, while ^Na is substantially conserved under OPA dynamics. As a result, PNR QND measurements of ^Na can be performed by measuring the pump displacement. Next, we show that modulo quadrature QND measurements of the pump mode can be performed using nonlinear OPA dynamics. This demonstrates that the GKP state of the pump mode can be generated almost deterministically with only additional Gaussian resources, demonstrating a nonlinear optical route to a universal fault-tolerant CVQIP. Finally, we bridge the physics of these QND schemes to a more traditional context by establishing similarities between phase-mismatched optical parametric oscillators (OPOs) and multilevel atomic resonator QED systems. We observe the conditional localization of intracavity states into a squeezed Fock state ladder. This can be estimated experimentally from a pump homodyne record using a quantum filter without observing any signal photon loss.

[0027] Consider a phase-matched single-mode, second-order nonlinear Hamiltonian. TIFF0007840488000001.tif857(1) Here, ^a and ^b represent the annihilation operators of the signal (i.e., fundamental harmonic) and pump (i.e., second harmonic) modes, respectively, and g>0 is the nonlinear coupling strength of 183. See "Temporal trapping: a route to strong coupling and deterministic optical quantum computation" by Ryotatsu Yanagimoto et al. in Optica Vol.9, No.11 (November 2022) (hereinafter, the "Optica paper"). All papers referred to herein further provide useful context to support this disclosure, and ideally should be considered for such useful context.

[0028] Without loss of generality, we assume a non-negative phase mismatch between the signal and pump 130, δ≧0. It is noteworthy that various photonic systems, such as high-Q microring resonators, photonic crystal cavities, temporally trapped ultrashort pulses, and superconducting microwave circuits, can be described by equation (1), and our results here are consistent with any variation of these.

[0029] To parametrically handle the pump coherent amplitude (which can be large in many practical scenarios), it is converted into a unitarily given displacement frame. Here, the mean field of the pump modes is "factored out" below. TIFF0007840488000002.tif953(2) Here, |ψ(t)> and |φ(t)> are the system states in the lab frame and the displaced frame, respectively. Without loss of generality, we assume that β is a positive real number. Physically, |φ(t)> describes the quantum fluctuations around the mean field, and its dynamics are as follows: TIFF0007840488000003.tif1455(3) Here Hamiltonian TIFF0007840488000004.tif873(4) It consists of a third-order nonlinear term and a second-order nonlinear term. TIFF0007840488000005.tif11112(5) Note that r = 2gβ. From here on, unless otherwise specified, we will assume that we are in a displaced frame. OPA160 is in its initial state TIFF0007840488000006.tif899(6) This is realized for and the pump state is a coherent state of displacement β in the laboratory frame. The conventional approach to OPA analysis is to use the undepressed pump approximation, which assumes that the pump state is invariant during dynamics. This approximation is ^H D ^H NL This is equivalent to ignoring the signal, resulting in single-mode squeezing of the signal state. This is the expected behavior of OPA in the Gaussian quantum optics domain. See R. Yanagimoto, E. Ng, A. Yamamura, T. Onodera, LG Wright, M. Jankowski, MM Fejer, PL McMahon, and H. Mabuchi, Onset of Non-Gaussian Quantum Physics in Pulsed Squeezing with Mesoscopic Fields, Optica 9, 379 (2022).

[0030] Under strong nonlinearities that cause the undepleted pump approximation to break down, the contribution of nonlinear terms leads to non-Gaussian quantum features, such as signal-pump entanglement, but a qualitative physical description of this is critically lacking. Below, we present a concise description of the nonlinear quantum behavior of phase-unmatched OPA160 as an important facilitator of QND measurements of signal photons in squeezed photon bases. Our analysis employs the Hamiltonian transformation recently introduced in W. Qin, A. Miranowicz, and F. Nori, Beating the 3 dB Limit for Intracavity Squeezing and Its Application to Nondemolition Qubit Readout, Phys. Rev. Lett. 129, 123602 (2022).

[0031] Assuming a relatively large phase mismatch δ>r, ^H Q This can be rewritten as follows: TIFF0007840488000007.tif1294(7) Here, ^A = ^acoshu + ^a † sinhu corresponds to the annihilation operator 163 of Bogolyubov excitations, and Δ = √(δ 2 -r 2 ) and u=tanh -1 (r / δ) / 2. Intuitively, ^A can be interpreted as the annihilation operator for photon excitations in a squeezed photon base. The nonlinear Hamiltonian can be rewritten in terms of the Bogolyubov operator as follows: TIFF0007840488000008.tif2792(8) For the remaining tasks, ^H Q The size of " is ^H NL dominant, i.e., ge 2uAssume that ≪Δ. This can always be achieved by appropriately choosing δ and r (i.e., β). Under these conditions, ^A 2 and ^A †2 The contribution from the rapidly rotating terms, which include , is averaged out, allowing for a rotating-wave approximation. Therefore, the following occurs: TIFF0007840488000009.tif1496(9)

[0032] In the Heisenberg picture, the dynamics of the operators under the above equation can be analytically solved as follows: TIFF0007840488000010.tif14112(10) Here ^p b =(^b-^b † ) / 2i is the p-quadrature operator in pump mode. From equation (10), pump mode ^p b Note that it experiences displacement conditioned on the value of ^Na, leading to a specific signal-pump entanglement structure. Furthermore, [^HD, ^Na] ≈ 0 ensures that the value of ^Na is not disturbed during the evolution of the system. As a result, ^p b Homodyne measurements allow for the estimation of ^Na without performing signal mode destructive measurements, thus enabling QND measurements of ^Na. b Depending on the measurement results, the signal state is an eigenstate of ^Na with eigenvalue Na, i.e., a squeezed photon number state. TIFF0007840488000011.tif757(11) It is projected onto. This situation is summarized in Figure 3.

[0033] The performance of our PNR QND measurement is, ^p bIt depends on the measurement accuracy, which is limited by quadrature fluctuations in the probe pump state. Intuitively, the conditional displacement d = ~ gt is the width of p-quadrature fluctuations w = √(<φ b |^p b 2 |φ b >-<φ b |^p b |φ b > 2 This is significantly larger than , allowing for a highly reliable estimation of the value of ^Na. Figure 3 shows the results of a full quantum simulation of nonlinear OPA dynamics using the initial squeezed-vacuum pump state at w=1 / 4. The final pump state exhibits multiple Gaussian peaks in a phase space separated by distance d, each peak corresponding to a different number of signal Bogolyubov excitations 163, ^Na. The parameters used in this figure are d / w=4 (d=1, w=1 / 4), therefore ^p b When the measurement results are conditioned, the signal state can be projected onto a squeezed photon-number state with a fidelity of over 90% under the assumed system parameters.

[0034] Referring to Figure 4, the pump homodyne result is 401(p bA plot of POVM purity for the QND measurement protocol is shown as a function of / d). See R. Nehra, M. Eaton, C. Gonzalez-Arciniegas, M. Kim, and O. Pfister, Loss tolerant quantum state tomography by number-resolving measurements without approximate displacements, arXiv:1911.00173 [quant-ph] (2019). We have a Gaussian probe pump state with various widths w |φ b Consider (0)>. Here, w less than or equal to the vacuum level w0 = 1 / 2 is |φ b (0)> indicates that it is a squeezed vacuum.

[0035] To more quantitatively link the measurement performance with the squeezing of probe-pump quadrature fluctuations, we express the Kraus operators of the QND measurement protocol. From equation (9), the Kraus operators can be expressed as follows: TIFF0007840488000012.tif1766(12) Here, C Na (p b )=e -iΔNat <p b -d(N a +1 / 2)|φ b > represents the complex probability amplitude of the measurement result, and |p b > is the eigenvalue p b saddle^p b These are eigenstates (see Appendix B of the PRX paper for the complete derivation). The Krauss operator is related to a positive operator-valued measure (POVM) with the following elements: TIFF0007840488000013.tif1594(13) Physically, the result of a complete pump homodyne measurement is pb obeys the probability distribution P(p b ) = <φ a |^F(p b )|φ a >. Given the result p b , the post-measurement signal state is as follows. TIFF0007840488000014.tif1164(14) Up to normalization.

[0036] Note that the POVM is not completely selective with respect to Na. This is because ^F(p b ) is not composed only of the single squeezed-Fock-state projector |Na><Na|. To characterize the mixture of the POVM, it is useful to consider the weights relative to the squeezed-Fock-state projector. TIFF0007840488000015.tif2074(15) Intuitively, this can be interpreted as the weight applied to |Na> based on the homodyne result (for a complete discussion, see Appendix B below). In particular, W Na (p b ) = 1 means that the postmeasurement state conditioned on the homodyne outcome p b is the pure squeezed Fock state |Na>.

[0037] In Figure 4, the purity of the POVM is plotted against the homodyne measurement result p bThis is shown as a function of . Here, we assume a squeezed vacuum state of width w as the initial pump state. As can be seen from the figure, using a pump probe state with a small w improves the purity of the POVM for a given d, and the signal can be projected onto a more fidelity squeezed photon-number state. From an experimental standpoint, squeezing the pump quadrature allows for PNR QND measurements to be performed with a shorter nonlinear interaction time, which can result in lower transmission loss.

[0038] Referring to Figure 5, the corresponding homodyne results for Na 0, 1, 2, and 3 with w / w0 = 0.5 are shown in 501(p b As a function of / d), squeezed fox state projector W in POVM Na (p b The plot of 500 with relative weights of ) is shown. In plots 400 and 500, d = ~ Assume a conditional displacement of gt = 1.0.

[0039] In contrast to conventional phase-insensitive photon-number tomography achievable with PNR QND measurements, our system 300 can perform PNR QND measurements in any squeezed photon-number basis, enabling phase-sensitive squeeze tomography and obtaining phase information about the state under tomographic reconstruction. Here, introducing a complex phase to the pump displacement β changes the rotation angle of the basis, and the ratio r / δ determines the squeezing factor. The measurement basis is further squeezed at r / δ → 1 and has a larger enhancement factor for the non-linear coupling ~ g / g. In the other limit of r / δ → 0, the measurement basis converges to the (non-squeezed) photon-number state basis, which comes at the cost of the effective non-linear coupling ~ g / g → 0. It is noted that by adding Gaussian operations 197, the measurement basis can be flexibly controlled without sacrificing the non-linear coupling. For this purpose, a pair of opposite squeezing operators ^S D and ^S a can be applied to the signal states before and after evolution under ^H † a . This transforms the measurement basis such that ^N eff =^A † eff ^A eff is measured as ^A eff =^S † a ^S a . By choosing ^S eff =^a such that ^A a , without relying on the limit of r / δ → 0, the normal photon number ^n a =^a †Implement QND measurement of $\hat{a}$. Such a pair of squeezing and anti-squeezing operations has been experimentally demonstrated in pulsed nonlinear nanophotonics and reported in R. Nehra, R. Sekine, L. Ledezma, Q. Guo, R. M. Gray, A. Roy, and A. Marandi, Few-cycle vacuum squeezing in nanophotonics, Science 377, 1333 (2022). A complete analysis of the impact of the loss of the external squeezing operation is described in Appendix E. Quantum State Engineering of Gottesman-Kitaev-Preskill States

[0040] So far, we have focused on the QND measurement of the signal excitation $\hat{a}_{163}$. Here, we show that it is also possible to perform QND measurement of the pump field quadratures using the same physics of the nonlinear OPA dynamics. For this purpose, we utilize the operator dynamics in Eq. (9) as follows. TIFF0007840488000016.tif1079(16) Here, $2$ ~ $g\tau^x$ b The information of $-\Delta t$ is encoded in the phase of $\hat{A}$ up to modulo $2\pi$. Therefore, when measuring the phase of $\hat{A}$, for example, in a general-dyne measurement, the value of $\hat{x}$ modulo $\mu = \pi / (\tilde{g}\tau')$ is indirectly estimated. This projects the pump mode onto $\hat{x}$ b $= x$ b $(\text{mod}\mu)$. Here, $x$ φ $= (\varphi+\Delta t) / (2$ φ $g\tau)(\text{mod}\mu)$. The pump quadrature $\hat{x}$ ~ itself is $[\hat{x}$ b , $\hat{H}$ b DSince the dynamics remain constant during the period of ]≈0, the QND property of the measurement is guaranteed. Such modular quadrature measurements play a central role in modern CVQIP, such as deterministic generation, stabilization, and quantum error correction using GKP states. Below, we show the generation of an approximate GKP state using the nonlinear dynamics of OPA160. Additional Gaussian resources (e.g., Gaussian initial state, measurement, and feedforward operations) are used here. Our proposal for generating GKP states applies D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001) and DJ Weigand and BM Terhal, Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit QED, Phys. Rev. A 101, 053840 (2020). However, this specification provides important technical differences for some embodiments due to the nonlinear dynamics of the phase-mismatched / frequency-detuned OPA160.

[0041] In the following discussion, the coherent excitation of the Bogolyubov signal mode is denoted as |A>. Physically, |A> is a displaced squeezed state and an eigenstate of operator ^A with eigenvalue A. As shown in Figure 6, an initial signal state |A0> with A0>0 is prepared as a "meter" state for phase shift. For the initial pump state, a p-squeezed vacuum with width w along the p-quadrature is assumed.

[0042] After transmission over time t through the nonlinear OPA 160D, the phase of ^A is measured by a complete general-dyne measurement 178A. This projects the signal mode onto the measurement basis of the displaced squeezed state. TIFF0007840488000017.tif938(17) This can occur, for example, when implementing an instance of systems 100 and 200 that does not use detector 170A so as not to damage the pump outputs 192 and 692 of systems 100, 200, and 600. Here, the measurement basis is parameterized by diameter (A0+ε)≧0 and phase φ. For details on performing the general-dyne measurement 178B, see Appendix F of the PRX paper. The performance of phase measurements can be further improved by applying measurement schemes such as HM Wiseman, Adaptive Phase Measurements of Optical Modes: Going Beyond the Marginal Q Distribution, Phys. Rev. Lett. 75, 4587 (1995) and MA Armen, JK Au, JK Stockton, AC Doherty, and H. Mabuchi, Adaptive Homodyne Measurement of Optical Phase, Phys. Rev. Lett. 89, 133602 (2002). To prepare the GKP state, a modulus μ = √(2π) modulo quadrature measurement is preferred, which involves interaction time. ~ Set gt = √(2π / 2).

[0043] If the magnitude of meter state A0 is significantly larger than the vacuum noise level, the measurement results are expected to localize exponentially around |ε|≪A0. Assuming this condition is met, the post-measurement pump state is approximately as follows: TIFF0007840488000018.tif1485(18) This approximates the GKP logical state via one or more displacement operations 197. TIFF0007840488000019.tif1787(19) It can be converted to (see Appendix C and paper PRX below for details). Here, TIFF0007840488000020.tif54 is a floor function, where |κ> is the width along x - quadrature, κ = √(<^x b 2 >-<^x b > 2 This is an x-squeezed vacuum with ) = 1 / (2√π)A0. It is worth noting that this GKP generation is nearly deterministic because of the extra displacement ^D induced by the probabilistic phase readout φ. b (^x φ This is because it is almost compensated for by the feedforward displacement operation.197 The resulting GKP state is symmetric when w=κ is true and corresponds to A0=1 / (2√π)w.

[0044] Figure 6 shows a schematic representation of system 600, as well as the results of a numerical simulation demonstrating the generation of a symmetric GKP state with a squeezing level of 15 dB (error correction threshold exceeding 10 dB). System 600 may, in some variations, include or be similar to system 100 or system 200 (or both). A phase-mismatched optical parametric amplifier 160D, as illustrated, receives a signal input 685A having state 611A (e.g., number 264 of SBEs 163) and a pump input 685B having state 611B, thereby causing inputs 685A-B to undergo entanglement 622 and selective nonlinear coupling strength enhancement.

[0045] Figure 6 shows the PNRQND measurement scheme using the nonlinear quantum behavior of OPA160D (also shown in Figure 3 of the PRX paper), where the phase space representation of the system state (i.e., the Wigner function) at each step of the protocol is shown using numerical data. In the numerical simulation, the initial coherent signal state |φ is shown in state 611A, which is obtained by plotting the sine component axis 614A (pa quadrature) against the cosine component axis 615A (xa quadrature). a Consider (0)>=|α=0.7>. The elliptical zone 602A-B (shown by the solid black line) shows a pseudoprobability value (QPV) of approximately 0.2 or greater, 677B.

[0046] The dashed contour line 601A shows the smaller positive QPV 677B of state 631A in an annular zone of weakly positive QPV surrounding an elliptical zone of negligible QPV. Similarly, the dashed contour line 601B shows the smaller positive QPV 677B of state 631B in several eccentric elliptical zones of negligible QPV. The initial pump state 611B plots axis 614B(pb) against axis 615B(xb). The signal state and pump state interact via a frequency-detuned OPA160D whose dynamics respond to the information-bearing pump modular quadrature 162 as described above.

[0047] In the final signal state 631A, the general dyne detector 670 functions as a QND measurement, providing feedforward 691 to one or more displacement operators 673, which modulates the entangle pump state 631B, and axis 655(x b ) relative to axis 654(p b This generates a pump output 692 having state 631C which plots ). As illustrated, the resulting pattern for state 631C alternates between columns and rows of black zones (each signaling a QPV of approximately 0.2 or greater) and matrices of non-black zones 603 (each signaling a QPV of approximately -0.2 or less).

[0048] Our results demonstrate that the nonlinear OPA160D is a sufficient component for realizing universal nonlinear optical QC, as fault-tolerant universal quantum computation becomes possible simply by adding Gaussian resources, by supplying GKP states as desired. Compared to existing nonlinear optical GKP state generation schemes using cross-phase modulation (XPM), our approach employs much stronger quadratic nonlinearities, which we believe opens up significant possibilities for non-Gaussian state engineering at room temperature. Nonlinear quantum fluctuations in OPO dynamics

[0049] An important application of parametric interactions is the optical parametric oscillator (OPO). An OPO is realized by exciting a quadratic nonlinear resonator with an external drive field. In the absence of signal loss, a phase-matched OPO (i.e., δ=0) has two transient states: odd and even signal cat states, which are quantum superpositions of π-phase-shifted coherent states. In the presence of finite signal loss, the parity of the cat states spontaneously switches, transforming the cat states into a non-coherent mixture of the original coherent states. This is reminiscent of the spontaneous quantum jumps observed in two-level atom-cavity QED systems. Here, we show that phase-mismatched OPOs exhibit behavior reminiscent of multilevel atom-cavity QEDs, where loss of signal photons induces quantum jumps between signal states in a squeezed Fock state ladder.

[0050] Hamiltonian term ^H drive =iλ(^b † Introducing an external pump drive for the OPO given by -^b), the outcoupling pump loss is the Lindblad operator ^L b =√κ b (^b+β) (^Lb=√κ in Lab frames) b Characterized by ^b). In the absence of signal loss, the pump operator dynamics are as follows: TIFF0007840488000021.tif1263(20) Here, ^N a λ=(κ b Selecting β) / 2 yields the following steady state. TIFF0007840488000022.tif960(21) Here, |β Na > is displacement β Na =2iκ b -1~ she(N a It is in a coherent pump state of +1 / 2). Na is N a Because it depends on the pump photon that comes out of OPO, a It propagates information about this, and this is ^N aIt serves as a weak continuous QND measurement. Therefore, by monitoring the outcoupled pump field 131, the system (pump-signal) state is conditionally steady state |N a >|β Na It is expected to collapse as one of the factors.

[0051] Here, we consider the effect of finite signal loss. The signal photon is |β Na When lost from |N, the intracavity signal state becomes |N a >→|Na> causes a quantum jump, and as a result, the signal mode becomes as follows. TIFF0007840488000023.tif11106(22) This means that the loss of signal photons corresponding to photon subtraction from the squeezed photon number state is due to Bogolyubov excitation N a →N a This means it causes discrete jumps in both positive and negative directions of ±1. Note that since coshu > sinhu, the flow is biased towards the negative direction.

[0052] Next, referring to Figure 7-8, the stochastic master equation and quantum trajectory of OPO dynamics, as shown by continuous pump-homodyne measurement, are presented. Plot 700 in Figure 7 represents signal excitation 702<^N a > and pump displacement 703<^p b >The trajectory of the plateau as a function of interaction time <^p b >=Im(β NaThe plot in Figure 8, plot 800, shows the trajectory of the x-quadrature squeezing signal compared to the quadrature noise level in vacuum (0 dB dotted line) and the squeeze limit in the OPO steady-state (negative 3 dB dotted line). System parameters, Δ / g = 100, ~ g / g = 1.5, κ a / g=0.03, and κ b Equation (9) was used with / g = 3.0.

[0053] ^N a and ^p b Because there is a quantum correlation between them, the occurrence of such quantum jumps can be estimated from the pump-homodyne measurement record without monitoring signal loss photons at all. To emulate this situation, a numerical simulation of the probabilistic master equation (SME) shown by the pump-p-homodyne measurement was performed without monitoring signal loss photons. As shown in Figure 7, <^N a > and <^p b Correlated spontaneous jumps are observed in >, and a multilevel plateau corresponding to the generation of squeezed photon number states is seen. This can only be inferred from pump homodyne recordings. Such discrete behavior emerges from a continuous-variable system where only continuous observables are monitored, and it demonstrates the essential quantum properties of photons. a=0 >|β Na=0When it is determined that the signal is in a state of >, the signal state becomes a squeezed vacuum state, and its squeeze level 802 can conditionally exceed the -3dB limit of steady-state squeeze within the OPO resonator (see Figure 8). This strong signal squeeze phenomenon is demonstrated in W. Qin, A. Miranowicz, and F. Nori, Beating the 3 dB Limit for Intracavity Squeezing and Its Application to Nondemolition Qubit Readout, Phys. Rev. Lett.129, 123602 (2022), where a squeeze exceeding 3dB is achieved in the pump mode of the OPO. Experimental Outlook

[0054] This paper discusses the experimental features necessary to realize PNR QND measurements in the single-photon region. For this purpose, to study the possibility of squeezing that enhances effective nonlinear coupling, we assume large squeezing factors for all fields involved in the dynamics (in this case, including signal Bogoliubov excitation 163 and probe pump state 137). The signal and pump losses and the squeezing factors are at the same level, i.e., κ a ~κ b and w~e -u Assuming that ≪1, the experimental characteristics of some variations of our scheme are as follows: TIFF0007840488000024.tif1015(23) (See Appendix D of the PRX paper for the full discussion) Here, the dashed symbol represents an approximate equation (inequality) that is correct down to the order of 1. It can be seen that it decreases with squeezing in probe pump mode 136. For example, if a 15 dB squeeze is applied to the initial pump, g / κ aThe constraints are approximately w -1 The factor can be reduced to ~5.6. A promising way to realize the nonlinear optics of equation (1) is by using a high-Q microring resonator, g / κ a ~0.01 has recently been achieved in indium gallium phosphide nanophotonics and thin-film lithium niobate nanophotonics. Furthermore, ultrafast pulse operations, realized through advanced dispersion engineering, can further enhance nonlinear coupling by simultaneously utilizing temporal and spatial field confinements, g / κ a ~10 may also be possible. If implemented in a single-path, such execution using ultrashort pulses could enable PNR QND measurements at terahertz slew rates. These figures suggest that this proposed scheme will soon reach χ (2) This offers a bright outlook for what can be achieved with nonlinear nanophotonics.

[0055] In relation to the above disclosure, we proposed and analyzed a scheme for PNR QND measurement and quantum state engineering utilizing the nonlinear quantum behavior of OPA 160. Phase mismatch: The pump mode driving OPA 160 is the number of Bogolyubov excitations 163. a Conditional displacement 261 is experienced according to the pump homodyne detection, and the result is nondestructively obtained by pump homodyne detection. a This demonstrates that it is possible to measure [the following]. Such PNR QND measurements enable highly efficient ultrafast PNR measurements (replacing conventional slow superconducting detectors) and deterministic execution of photon-photon entungling gates, providing all the elements necessary for deterministic room-temperature DV quantum computing at ultrafast clock rates.

[0056] Next, we demonstrate that modular quadrature QND measurements of pump modes can be realized by utilizing nonlinear OPA dynamics via signal-phase measurements, thereby naturally providing a method for deterministically generating optical GKP states with all Gaussian resources added. Our results unlock many promising possibilities for room-temperature ultrafast universal quantum computation using GKP states in CV architectures. It is also worth mentioning that our GKP state generation protocol uses Gaussian quadrature measurements, which can be purified using amplification techniques with the recently demonstrated high-gain linear OPA 160 before inefficient generaldyne measurements, thereby providing a method for generating high-purity GKP states. Finally, extending the discussion to OPO physics, we show that continuous homodyne monitoring of an outcoupled pump field 131 leads to conditional localization of signal modes on squeezed photon number states, thereby highlighting a unique opportunity to synthesize and characterize intracavity nonclassical states in real time.

[0057] The above embodiment is independent of materials with cubic nonlinearity and therefore provides a clear path to overcome the long-standing challenges of nonlinear optical PNR QND schemes based on cross-phase modulation (XPM) (where self-phase modulation, which inevitably accompanies XPM, introduces harmful phase noise into the probe field). Our research establishes a concise description of nonlinear-optical parametric interactions beyond the conventional semiclassical picture, thereby paving a practical path to large-scale, ultrafast, fault-tolerant universal photonic quantum information processors at room temperature.

[0058] Referring again to Figure 3, we see system 300, which performs a squeezed cat-state generation scheme using cubic QND measurement with optical parametric interactions. The Wigner functions of the quantum states at each stage of the protocol are shown using data from a full quantum simulation. As the initial state, ω a Prepare FH mode 385A and SH mode 385B in a p-squeezed vacuum state 311A ​​and a vacuum state 311B, respectively, with a power of =√5 / 2. After propagation through one or more external squeezers 381A-B and one or more nonlinear media 382, ​​the final unconditional FH mode 331A and SH mode (P(p b The illustrated state 331B) is obtained, which has a marginal p-quadrature distribution 335B corresponding to (d). Depending on the results of SH homodyne measurements 178, the FH mode is projected onto squeezed Schrödinger's cat states 375. Each color band in (d) yields an ensemble-averaged state with a color corresponding to state 375 with a probability of P from the SH homodyne measurement results pb This represents an interval. To generate cat states of size ∈{√4,√8,√12,√16}, the interval was set as follows. TIFF0007840488000025.tif9134(24) To accommodate a 10dB power gain, the squeezer is r a 2 =r b 2 We assumed that = 10. Appendix A: Derivation of the Rotating Frame Hamiltonian

[0059] To derive the Hamiltonian (1), we use the single-mode χ of the laboratory frame. (2) We'll start with the Hamiltonian. TIFF0007840488000026.tif989(25) The following unitary component will transition to a rotating system. TIFF0007840488000027.tif1367(26) As a result, the Hamiltonian is transformed as follows: TIFF0007840488000028.tif1962(27) Here, frequency detuning δ = ω a ―ω b / 2. Appendix B: Kraus Operators in PNR Detection

[0060] In this section, Hamiltonian^H D The Kraus operator for PNR QND measurement using p is derived. bFor the pump p-homodyne outcome, the post-measurement signal state is as follows: TIFF0007840488000029.tif1081(28) Up to standardization. Here, |p b > has an eigenvalue of p b is ^p b These are eigenstates. Target signal state TIFF0007840488000030.tif1754(29) Regarding this, we obtain the following: TIFF0007840488000031.tif3597(30) Here γ Na =id(N a (+1 / 2) and TIFF0007840488000032.tif1098(31) Equation (30) can be summarized as follows: TIFF0007840488000033.tif951(32) Using the Kraus operator, TIFF0007840488000034.tif1562(33) Pump / probe state | φ b Assuming (0)>, and a squeezed vacuum with width w along a p-quadrature, the complex probability amplitude can be written out analytically. TIFF0007840488000035.tif2089(34) This is because the center is p b =d(N a (+1 / 2) is a Gaussian function of width w.

[0061] QND measurement protocol ^F(pb The positive-operator-valued measure (POVM) of ) can be easily obtained from the Krauss operator. TIFF0007840488000036.tif2571(35) POVM is normalization condition ∫dp b ^F(p b )=1 a Note that this must be satisfied.

[0062] POVM(35) is composed of a mixture of multiple squeezed-fock-state projectors, therefore N a It should be noted that this is not entirely selective. To quantitatively characterize this mixed nature of POVM, we introduce relative weights of squeezed fock state projectors. TIFF0007840488000037.tif1661(36) W Na (p b To understand the physical interpretation of ), it is useful to consider what the squeezed photon number distribution (29) of the premeasurement state looks like. That is, TIFF0007840488000038.tif1053(37) Homodyne result p b It changes under the condition. Using equation (32), the squeezed photon number distribution of the postmeasurement state can be expressed as follows. TIFF0007840488000039.tif980(38) N is the normalization constant. Compare equations (37) and (38) and {W Na (p b )} is the squeezed photon number distribution of the input state |αNa | 2 This can be interpreted as a conditional weight that is multiplied by the given value. In particular, a certain N a W Na (p b When )=1 holds, the postmeasurement state is a pure squeezed photon number state |N a >This is the result. Appendix C: Generating all Gaussian types of GKP states

[0063] This section introduces a GKP state generation scheme using modular quadrature measurements that utilize the nonlinear quantum behavior of OPA160. Some variations incorporate protocols using ponderomotive interactions, as described in D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001) and DJWeigand and BM Terhal, Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit QED, Phys. Rev. A 101, 053840 (2020).

[0064] In the following discussion, let |A> be the coherent excitation of the signal Bogolyubov excitation. Physically, |A> is a displaced squeezed state and an eigenstate of ^A with eigenvalue A. Consider the following as the initial state of the system. TIFF0007840488000040.tif1373(39) A0 > 0, φ b (xb b This represents the x-quadrature amplitude in the initial pump state. After transmitting a phase-mismatched OPA160 at time t, the phase of the signal mode is measured by general-dyne measurement. This is the state |e iφ The measurement is projected onto a spanned measurement basis (A0+ε)> and parameterized by diameter A0+ε≧0 and phase φ. Details of the construction of the general-dyne measurement are described in Appendix F of the PRX paper.

[0065] Given the measurement results ε and φ, the post-measurement pump state is as follows: TIFF0007840488000041.tif25107(40) Up to the standardization stage. As a result, the Kraus operators used to represent the measurement protocol can be written as follows. TIFF0007840488000042.tif16101(41) Here, TIFF0007840488000043.tif32105(42) This is the complex amplitude.

[0066] If we employ a "meter" signal state with an amplitude far greater than the vacuum noise level, the results of the signal measurement are expected to localize exponentially around |ε|≪A0. Assuming this condition is met, equation (42) can be approximated as follows. TIFF0007840488000044.tif2197(43) Here, xn = nμ and x φ =(Δt+φ) / (2 ~ gt) (mod μ), μ = π / ( ~ gt). Equation (43) shows multiple Gaussian peaks separated by the same distance μ and with width κ = 1 / (2√π)A0.

[0067] To generate the GKP state, we consider a p-squeezed pump state. TIFF0007840488000045.tif1967(44) The width along the p-quadrature is w. Also, the interaction time is ~ Set gt = √(π / 2) and μ = √(2π). With these parameters, the post-measurement pump state is as follows: TIFF0007840488000046.tif73115(45) Here, the overall normalization constant is ignored. Here, |κ> is an x-squeezed vacuum with width κ along the x-quadrature. Assuming that |κ> is strongly squeezed, the following approximation can be made. TIFF0007840488000047.tif6996(46) Here, TIFF0007840488000048.tif54 is a floor function. Therefore, the post-measurement state can be rewritten as follows: TIFF0007840488000049.tif1280(47) Here, TIFF0007840488000050.tif1782(48) This is the approximate GKP logical state. Note that equation (47) can be converted to the approximate GKP state by feedforward displacement operations based on the general-dyne measurement result. Appendix D: Experimental Design Characteristics of PNR QND Measurement

[0068] This section studies the experimental features and conditions for realizing a PNR QND measurement scheme in the single-photon regime. When dissipation is present, the system state density matrix follows the following master equation. TIFF0007840488000051.tif20107(49) Here, {^O1, ^O2}=^O1^O2+^O2^O1 is the anti-commutator. In the text, we assumed that the dynamical time scale Δ of the phase rotation of the Bogolyubov excitation is dominant over the nonlinear coupling rate, i.e., Δ≫g. Here, we further assume that Δ is dominant over the dissipation time scale, i.e., Δ≫κa,κb. If this assumption holds, then by the rotating wave approximation, ^A in equation (49) 2 and ^A †2 It is justified to ignore the contribution from the fast rotation term, which includes . Specifically, for the term describing the signal loss, we obtain the following: TIFF0007840488000052.tif14102(50) Here, ^L + =√κ a sinh(u)^A † and ^L- =√κ a cosh(u)^A † This result indicates that, under the rotating wave approximation, the original signal Lindblad operator ^L a =√κ a The effect of ^a is two Lindblad operators ^L + and ^L - This shows that it can be broken down into its constituent parts.

[0069] In the following discussion, for the sake of concrete implementation, we consider the squeezed single-photon state |N_a=1} as the initial signal state. For the initial pump state, we assume a p-squeezed vacuum with width w along the p-quadrature. To successfully measure the QND of a PNR, the probability of a quantum jump occurring in the signal mode must be sufficiently low. In the low-loss limit, the probability of a quantum jump is given approximately as follows. TIFF0007840488000053.tif2989(51) This is a characteristic timescale of loss-induced quantum jumps. jump ~1 / (cosh 2 (u)κ a )

[0070] With "high" reliability ^N a In order to be able to measure t jumpThe conditional displacement occurring on this timescale must be greater than the characteristic width of the pump state. Here, given a finite but small pump loss, the width of the final pump state along the p-quadrature is as follows: TIFF0007840488000054.tif2854(52) Here we κ b We assumed t≪1. As a result, the experimental conditions for our scheme to succeed are: ~ gt jump TIFF0007840488000055.tif33w'(t jump ) Here, the wave symbol is used to represent an approximate equality up to a factor of order 1 (unity). Here, we assume strong squeezing for the purpose of this study (e.g., signal-Vogolubov excitation and pump state). Furthermore, we assume similar levels of loss and squeezing for both the signal and pump states, i.e., κ a ~κ b and w~e -u ≪1. Under these conditions, w'(t jump The order of magnitude of w is 1 (unity) larger than w, and w'(t jump ) can be approximated as ~w. As a result, the characteristics of the experimental design in our scheme can be expressed in a concise formula as follows. TIFF0007840488000056.tif1014(53)

[0071] Appendix E of PRX describes the loss analysis of external squeezers, modeling the loss of each squeezer by a pair of equal beam splitters placed before and after the squeezer.

[0072] Appendix F of the PRX paper describes the construction of a general-dyne measurement using two balanced homodyne detectors and one ancillary vacuum state. The overall measurement protocol projects the input state |φ>1 onto a measurement basis spanned by displaced squeezed states. The results of the general-dyne measurement are related to the results of the homodyne detectors via x+ip=secθx1+icscθp2. The squeezing level of the measurement basis ξ=tanθ in that configuration can be set by selecting the transmittance of the beam splitter (BS). Refer to the OPH paper.

[0073] We propose a scheme to realize a cubic quantum nondemolition (QND) Hamiltonian with optical parametric interactions. (2)We demonstrate that strongly squeezed fundamental and second harmonic fields, transmitted through a nonlinear medium, develop effectively under a third-order QND Hamiltonian. We highlight the versatility offered by such Hamiltonians for designing non-Gaussian quantum states, such as Schrödinger cat states and third-order phase states. We show that these generation schemes are highly resilient to various loss sources, including detector inefficiencies and outcoupling loss in off-chip measurements. Our proposal involves operating parametric interactions in the mesoscopic photon-number regime, significantly improving effective nonlinear coupling from the native single-photon coupling rate and providing a robust mechanism against photon loss. The experimental results indicate that this method is feasible in the near future, particularly in pulsed nonlinear nanophotonics.

[0074] The engineering of non-classical states of light is a central challenge in photonic quantum information processing and engineering, enabling new architectures that transcend classical limits in various fields such as measurement, sensing, communication, and computation. As the discovery of one-way optical quantum computation (QC) demonstrates, in some variations, the generation of an initial non-classical resource state can be the only non-trivial step for universal quantum operations. In continuous-variable (CV) systems, any unitary operation is only achievable by adding additional Gaussian (i.e., linear-optical) resources, allowing access to non-Gaussian resource states, such as Schrödinger's cat state, Gottessmann-Kitaev-Preskill (GKP) state, or cubic phase states.

[0075] Traditional approaches to non-Gaussian quantum state engineering rely on the nonlinearity induced by photon-number-resolving (PNR) measurements, which allows for the engineering of highly nonclassical states using complex optical circuits. However, the inherent stochastic nature of these operations and the cryogenic requirements of conventional PNR detectors (e.g., superconducting nanowires and superconducting transition-edge sensors) significantly limit the overall scalability of the architecture.

[0076] In this study, we investigated the third-order quantum nondestructive (QND) Hamiltonian ∝^x using optical parametric interactions. 2 a ^x bWe present an engineering scheme and propose a means to circumvent the limitations of conventional approaches in CV quantum information engineering. Here, the operator ^x a and ^x b These are the amplitude quadrature operators of the fundamental and second-harmonic fields, respectively. The cubic QND Hamiltonian can play a multi-purpose role in non-Gaussian quantum engineering. Firstly, it directly enables the deterministic execution of cubic QND gates, completing the universal gate set of CVQC. Secondly, it can efficiently generate non-Gaussian quantum states with only additional Gaussian operations197 and measurements178. To emphasize the latter point, we introduce a scheme for generating Schrödinger's cat state and cubic phase states and analyze its performance. Our protocol employs only homodyne conditioning without photon counting, making it compatible with pre-amplification schemes that are robust against detection-stage losses, such as detector inefficiencies and outcoupling losses in off-chip measurements. Furthermore, our method naturally incorporates mesoscopic photon counts, resulting in an order of magnitude improvement in effective nonlinear coupling and providing a means to mitigate photon loss. Experimental figures demonstrate that our approach is feasible in the near future, particularly with pulsed nonlinear nanophotonics.

[0077] The following Hamiltonian represents a resonant single-mode (resonant, single-mode) χ (2) Let's consider a nonlinear system. TIFF0007840488000057.tif1048(54) Here, g>0 is the nonlinear coupling constant, and ^a and ^b are the annihilation operators for the FH and SH modes, respectively. The Hamiltonian in equation (54) can be realized in various systems, including microresonators, time-trapped ultrashort pulses, and superconducting microwave circuits. Our results are independent of any specific physical implementation of the Hamiltonian. The initial state of the system is: TIFF0007840488000058.tif959(55) Before and after the state evolves under the Hamiltonian of equation (54), a pair of orthogonal squeezing operations ^S a ^S b and ^S † a ^S † b Apply this. As a result, the entire system will develop. TIFF0007840488000059.tif13103(56) Here, the effective Hamiltonian ^H eff is ^H's ^a→^S † a ^a^S a and ^b→^S † b ^b^S b This is obtained by substitution. In the following discussion, ^S † c ^c^S c =r c ^x c +ir c -1 ^p c ,^x c =(^c+^c † ) / 2, ^p c =(^c-^c † ) / 2, and for c∈{a, b}, the field gain r c Let ≥ 1. As a result, we obtain the following: TIFF0007840488000060.tif18109(57) geff=ra 2 r b g is the cubic QND Hamiltonian ^H eff ∝^x a ^x b To effectively achieve this. Of note is that such a cubic QND Hamiltonian enables a universal gate set for CVQC, and our scheme provides a deterministic realization. c ≫Assuming 1, ^H eff The time evolution below can be approximately solved using the Heisenberg picture, as follows: TIFF0007840488000061.tif21105(58) Here, the normalized interaction time τ = g eff t is the SH quadrature operator ^p b is ^x a 2 This means that it experiences a conditional displacement that depends on the value of [^H]. eff ,^x a 2 ] ≈ 0 is equal to ^x a 2 This ensures that it remains constant during system development, and in homodyne measurements, ^p b By measuring the second quadrature ^x, a 2 Note that this enables the performance of QND measurements.

[0078] An overview of system 900 performing the squared quadrature QND measurement protocol is shown in Figure 9, and the results of our numerical simulations, shown in Figure 1 of the Op. paper, are also summarized. System 900 may include system 100 or system 200 (or both) in some variations, or may be similar thereto. Each medium 981A-B and at least one phase-matched optical parametric amplifier 160 (e.g., including medium 982) receives a fundamental harmonic 985A having state 911A and a second harmonic 985B having state 911B. State 911A plots the sine component axis 914A (pa quadrature) against the cosine component axis 915A (xa quadrature). Within this, the elliptical zone 902 (shown as a solid black line) exhibits a quasi-probability value (QPV) of 377C greater than approximately 0.2. Similarly, in state 911B, within the elliptical zone 902 with a high QPV of 377C and small eccentricity, the axis 915B (x b ) relative to axis 914B(p b Plot the points.

[0079] Downstream of the additional media 983A-B as shown, are states 931A-B having zone 901 of high QPV377C (shown in black) and weak positive QPV377C (outside surrounded by a dashed contour line 901 and inside a solid black zone 902). Homodyne conditioning 965 is performed for various elements of the state to support the inference, as will be further explained below, p a Axis 954 corresponds to x a This is applied so that each state 975 to be plotted against axis 955A-B is shown.

[0080] The Krauss operator that characterizes the QND measurement scheme is the SHp-homodyne measurement result p b It is given as a function of . TIFF0007840488000062.tif1471(59) Here, the complex amplitude C pb (x a )=φ b (p b -τx a 2 ) is given as a function of the initial probe SH state. TIFF0007840488000063.tif968(60) Here, |p b > is the eigenvalue p b ^p b The eigenstates are (|x a (The same applies to >). Physically, homodyne result p b The probability distribution of is given by Born's rule. TIFF0007840488000064.tif9133(61) This represents the FH state after measurement, which has not been normalized. For a general discussion of optical implementations of nonlinear quantum measurement readers, see also JM Epstein, K. Birgitta Whaley, and J. Combes, Quantum limits on noise for a class of nonlinear amplifiers, Phys. Rev. A 103, 052415 (2021).

[0081] The resolution of QND measurements is critically dependent on the p-quadrature fluctuation of the probe SH state. This fluctuation is caused by the p-squeezed vacuum being subjected to the probe state |φ b By adopting (0)>, it can be naturally improved. b The squeezing of (0)> is performed by the initial SH squeezing operation ^S b Because it can be absorbed into |φ b We can assume that (0)> = |0>. Furthermore, the imbalance between the first and second SH squeezing operations can be taken into account by the trivial scaling of the final SH p-homodyne readout. Therefore, unless otherwise specified, |φ b Assume (0)> =|0>.

[0082] Vacuum probe state | φ b (0)> =|0>, C pb (x a ) = (2 / π) (1 / 4) e(-(p b -τx a 2 ) 2 ) This is p b If the coefficient of variation is much larger than the vacuum fluctuations, it can be approximated as the sum of two Gaussian distributions. TIFF0007840488000065.tif1051(62) In addition, The file is TIFF0007840488000066.tif1134. The separation and width of the Gaussian peak are given by ξ = 2√(p b / τ) and w=(2τξ) -1Therefore, intuitively, equation (62) is p b The measurement result is up to the uncertainty of w, |^x a This means estimating |=ξ / 2, which projects the FH mode onto a coherent superposition of displaced squeezed states.

[0083] Below, we analyze the squared quadrature QND measurement for generating the squeezed Schedinger cat state. The initial FH state is defined as a width w along the x-quadrature. a =√(<^x a 2 >-<^x a > 2 Consider a p-squeezed vacuum state with ). b Assuming a measurement result of >0, the post-measurement FH state is approximately as follows: TIFF0007840488000067.tif1386(63) Here we are w a 2 ≫We assume ξw (see Appendix B of the Oph paper for a complete discussion). Note that (63) is a coherent superposition of two x-squeezed states, each with width w and separated by distance ξ, which is the squeezed cat state. Figure 9 shows the results of a full quantum simulation, with the initial FH squeezed vacuum being the SH homodyne measurement result p b It is projected onto a non-Gaussian state that depends on p. b In regions where the ratio is large, the post-measurement FH state becomes a highly non-classical squeezed cat state.

[0084] The realization of a cubic QND Hamiltonian suggests more general non-Gaussian quantum state engineering. To emphasize this point, we introduce deterministic generation of cubic phase states. An overview of this system 600 is shown in Figure 6. The initial state is correlation^x a (0)-^x b (0)≈0 and ^p a (0) + ^p b Consider the EPR state (referring to Einstein, Podolski, Rosen) with (0)≈0. Using equation (59), the dynamics of the FH quadrature operator can be solved as follows. TIFF0007840488000068.tif1794(64)

[0085] Here, the first and second terms are approximately 3τ^x, respectively. a 2 (0) ≈ 3τ^x a 2 (τ) and 0. χ (2) After propagation in a nonlinear medium 682, a p-quadrature measurement is performed on the SH mode, and the third term is a real number p. b Collapse. As a result, by applying the FHp-displacement operation to compensate for this change, ^p a (τ) = 3τ^x a 2 (τ) can be deterministically forced, which indicates that the final FH state is a cubic phase state 675.

[0086] Referring to FIG. 10 here, a deterministic cubic-phase state generation system 1000 using optical parametric interaction (through a medium 1082 configured as OPA160) is shown. The phase-space portrait (Wigner function) of the state generated using 10 dB squeezing and an initial EPR pair with τ = 0.2 is shown. As a result, nonlinear quadrature squeezing Δ NL 2 = 0.255 was obtained.

[0087] Next, referring to FIG. 11, a log-log plot 1100 of the nonlinear (NL) squeezing Δ EPR 2 as a function of the initial EPR squeezing Δ NL 2 on axis 1101 is shown for various values of τ in the context of the system 1000 of FIG. 10. The black dashed line represents Δ EPR 2 = Δ NL 2 . In FIGS. 9 - 11, r a 2 = r b 2 = 10 is used.

[0088] Realistically, the EPR state can only have finite squeezing, with a finite variance Var(^x a (0) - ^x b (0)) = Var(^p a (0) + ^p b (0)) = (Δ EPR 2 ) / 4, which degrades the quality of the resultant cubic phase state. To quantify the approximate cubic phase state, Var(^p NL ) = Δ NL 2Consider non - linear squeezing characterized by / 4. This is non - linear quadriture ^p NL =^p a -3τ^x a 2 is the variance. Plot 1100 shows the trade - off between Δ NL Δ EPR and τ. Here, for a given τ, the optimal Δ NL that minimizes Δ EPR can be found

[0089] In FIG. 10, the phase - space portrait of the cubic phase state generated in the corresponding system 1000 is shown, and the results of the numerical simulation are also summarized. System 1000 can include, or be similar to, system 100 or system 200 (or both) in some variations. Each of the media 1081A - B and at least one phase - matched optical parametric amplifier 160 (for example, consisting of the medium 1082) receives the fundamental harmonic 1085A and the second harmonic 1085B. Downstream from the additional media 1083A - B, 1084 as shown, there is a detector 1070 conditioning system output 1091 corresponding to the output state 1075 represented as the p a axis 1054 with respect to the x a axis 1055 (characterized by the isolines 1001 and zones 1002 - 1003 as described above).

[0090] In general, in quantum state engineering using measurement-based post-selection, the purity of the resulting state is critically limited by the overall quantum efficiency (QE) of the measurement. In addition to detector inefficiency, photon losses in the setup (e.g., outcoupling loss in nanophotonic implementations) can reduce the overall QE. This problem is particularly serious in photon-number-resolving (PNR) measurements, where low QE directly affects the purity of the generated state. On the other hand, in quadrature measurements, such as homodyne measurements, it is possible to mitigate imperfect QE by pre-amplifying the signal using an optical parametric amplifier. Our QND measurement scheme described above involves a second-stage SH squeezing operation ^S b † The (second-stage SH squeezing operation) already includes such pre-amplification.

[0091] For a phase-space representation of heralded squeezed cat states by a homodyne detector with a finite QE instrument η, see Figure 3 in the Oph paper. As can be seen from that figure, the cat state generation scheme described herein can tolerate a fairly large detector drawback, e.g., η = 80%. By applying an additional pre-amplification with gain G, high-purity cat states can be generated even under a larger detector inefficiency, e.g., η = 20%, with G = 10. Such high robustness to low quantum efficiency is particularly attractive for counteracting large fixed losses in detector setups (which are common, e.g., out-coupling losses in off-chip detection from nanophotonic waveguides). Note that preamplifiers generally experience increased losses as their gain increases, which practically limits the maximum usable gain G.

[0092] Wigner functions for heralded squeezed cat states using third-order QND measurements (cubic QND measurements) and homodyne detectors with various QEη values. The generation of cat states with size ξ=4 is the result of SH homodyne p b =√(Gη)(τξ 2 The signal is given by ) / 4, where τ=1.0 is the normalized interaction time and G is the power gain of the preamplifier placed in front of the detector. The purity of the resulting state (abbreviated as Pur.) is shown at the bottom of each plot. The effect of loss is simulated using the Monte Carlo wavefunction (MCWF) method with a 10⁴ trajectory.

[0093] Another major cause of decoherence is propagation loss within the nonlinear medium. Nominally, the characteristic nonlinear coupling rate g is desired to be greater than the characteristic photon loss rate κ in order to observe non-Gaussian quantum features, leading to a design feature of strong coupling g / κ > 1. In our scheme, strong field squeezing leads to a mesoscopic number of photons involved in the dynamics, enhancing the effective nonlinear dynamical rate. This allows us to generate highly nonclassical states with a native nonlinear coupling rate at least an order of magnitude smaller than that of strong coupling. To see this more concretely, for FH and SH, the same squeezing gain and decoherence rate, i.e., r = r a =r b and κ=κ a =κ b Assumed, equation (57) indicates that external squeezing operations reduce the effective nonlinear coupling ratio by the field gain g. eff =r 3 The scaling factor increases by the cube of g. At the same time, the photon loss rate increases in proportion to the number of photons, and the effective decoherence rate is κ. eff =r 2 κ is obtained. As a result, the overall index of performance g is a factor proportional to the field gain of the squeezer. eff / κ eff=rg / κ is improved, resulting in increased resistance to photon loss. Good examples of enhancing nonlinear coupling using amplified quantum fluctuations include recent works such as R. Yanagimoto, T. Onodera, E. Ng, LG Wright, PLMcMahon, H. Mabuchi, Engineering a Kerr-Based Deterministic Cubic Phase Gate via Gaussian Operations, Phys. Rev. Lett. 124, 240503 (2020); C. Leroux, LCG Govia, and AA Clerk, Enhancing Cavity Quantum Electrodynamics via Antisqueezing: Synthetic Ultrastrong Coupling, Phys. 120, 093602 (2018); W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lu, JQ You, and F. Nori, Exponentially Enhanced Light-Matter Interaction, Cooperativities, and Steady-State Entanglement Using Parametric This paper is published in Amplication, Phys.120, 093601 (2018) and in Y. Michael, L. Bello, M. Rosenbluh, and A. Pe'er, Squeezing-enhanced raman spectroscopy, npj Quantum Inf.5, 1 (2019).

[0094] To verify the enhancement of nonlinearity, (Figure 4 in the Oph paper) shows the amount of negative values ​​in the Wigner function of the heralded cat state for various squeezing parameters and g / κ. As can be seen from the figure, strong squeezing operations can improve the quality of generated cat states for a given value of g / κ. The inset shows the Wigner function for states where squeezing (i.e., r=10) is achievable at g / κ ≈ 0.15 and 20 dB. It can be seen that the design feature of g / κ that produces a visible amount of negative Wigner function negativity is relaxed by an order of magnitude.

[0095] Figure 4 of the Oph paper shows the amount of Wigner function negativity in the cat states generated by cubic QND measurements with various squeezing and loss conditions. Homodyne conditioning is performed there to predict the generation of a cat state of size ξ=3.5 at τ=0.55. This nearly maximizes the nonclassical nature of the state on the parameter space studied here. The inset shows the Wigner function of the generated state with a squeezing of 20 dB and g / κ≈0.15. See Appendix C of the Oph paper for a complete discussion.

[0096] Experimentally, χ (2)Recent advances in nonlinear nanophotonics represent a significant step forward towards the strongly coupled region. Using high-Q micro-ring resonators, g / κ ~ 0.01 has been achieved in thin-film lithium niobate (TFLN) nanophotonics and indium gallium phosphide nanophotonics. Further advancements in fabrication techniques enabling material-absorption-limited loss could lead to g / κ ~ 1. Beyond conventional continuous-wave devices, utilizing three-dimensional confinement of optical fields using ultrashort pulses could potentially achieve g / κ ~ 10. These figures are related to next-generation χ². (2) In nanophotonics, our scheme shows the possibility of experimentally realizing it within reach.

[0097] We propose and analyze the engineering of a cubic QND Hamiltonian using squeezing operations and optical parametric interactions. Such a cubic QND not only directly enables deterministic CVQC but also serves as a versatile tool for efficient non-Gaussian quantum state engineering, such as cat states and cubic phase states. The generated resource states become essential components in modern quantum engineering, such as the generation of GKP states and four-component cat states. Compared to existing quantum engineering protocols using cubic nonlinear optics, our approach employs quadratic nonlinear interactions with stronger native coupling rates, potentially offering a more experimentally viable route. Our research elucidates the unique functions that nonlinear optics can realize in the mesoscopic domain. We hope our research will contribute to the rapidly developing quantum engineering toolbox of nonlinear photonics, thereby enabling us to take full advantage of the rapid advancements in experiments.

[0098] While various operation flows are described in their respective orders, it should be understood that these operations may be performed in orders other than those illustrated, and may also be performed simultaneously. Examples of such alternative orders include, unless otherwise indicated by the context, overlapping, interleaving, interrupting, reordering, incrementing, preparing, supplementing, simultaneous, reverse, or other variant orders. Furthermore, terms such as past tense adjectives like "reacting to" or "related to" are generally not intended to exclude such variants unless otherwise indicated by the context.

[0099] Various systems, methods, articles, or other embodiments or aspects have been disclosed above, and other combinations of embodiments or aspects will also be apparent to those skilled in the art from the viewpoint of the above disclosure. The various embodiments and aspects disclosed above are for illustrative purposes only and are not intended to limit, and the true scope and spirit are set forth in the following final set of claims.

[0100] In the following numbered sections, the first combination of aspects and embodiments means that (1) in each embodiment, for each instance in which a “component” or other such identifier is introduced multiple times within a particular chain of sections (e.g., “a” or “an”), such designation can identify the same entity or different entities; and (2) what may hereafter be called “dependent” sections may, in each embodiment, incorporate or not incorporate the features of the “independent” sections they refer to, or other features described above. term

[0101] Item 1 Quantum detection methods (e.g., using one or more systems 100, 200) include: One or more optical media 282 implementing one or more optical parametric amplifiers (OPAs) 160 constitute (at least) a first quadratic coupling strength 183; Obtain a first pump state 137 or other input states 111A-B including one or more photonic components 176; A first nonlinearity enhancement coupling 272 is established such that the first secondary coupling strength 183 in one or more OPA160s is enhanced by (at least) an additional secondary coupling strength 253; Transmitting first outputs 191-192, 291 (first output) including (at least) a first photonic component 176 of a first input state (for example, via the first output port 208C or 208D); and A first extracted result 292 (e.g., a digital measurement 178) is transmitted via a first nonlinearity enhancement coupling 272, encoding the first photonic component 176 of the first input state without destroying the first photonic component 176 of the first outputs 191-192, 291 (e.g., via a second output port 208D or 208C).

[0102] Section 2 A quantum detection method described in any of the above methods, comprising: In computing systems 100 and 200, each having a continuous-variable portion, a first nonlinearity enhancement coupling 272 that realizes one or more Godesman-Kitaev-Preskill (GKP) states is used to trigger ultrafast universal quantum computing.

[0103] Section 3 A quantum detection method described in any of the above methods, comprising: In the continuous-variable portions of computing systems 100 and 200, a room-temperature universal quantum computation is triggered by one or more GKP states of the first nonlinearity enhancement coupling 272.

[0104] Item 4 A quantum detection method according to any of the items of the above method, including the following: In the continuous variable parts of the computing systems 100 and 200, perform room-temperature quantum computing with one or more GKP states via the first non-linearity enhancement coupling 272.

[0105] Item 5 A quantum detection method according to any of the items of the above method, including the following: Obtain a first Gaussian quadrature measurement 178 and After purifying the first Gaussian quadrature measurement 178 so as to generate one or more purified GKP states via the first non-linearity enhancement coupling 272 (e.g., as an output feature 195), perform a general-dyne measurement 178.

[0106] Item 6 A quantum detection method according to any of the items of the above method, including the following: Create a cat state generated in the first non-linearity enhancement coupling 272 having a cat state size of 3.5 ± 0.1 and a squeezing time of 0.55 ± 0.5 so as to achieve an appropriate non-classicality of the generated cat state.

[0107] Item 7 A quantum detection method according to any of the items of the above method, including the following: (At least temporarily) Execute a cat state generated in the first non-linearity enhancement coupling 272 having a cat state size of 3.5 ± 1.0 and a squeezing time of 0.55 ± 1.0 so as to achieve an appropriate non-classicality of the generated cat state (e.g., the magnitude of the gain or loss of the squeezer as shown in FIG. 5 of the Oph paper).

[0108] Section 8 A quantum detection method according to any of the above methods, comprising achieving sufficient nonclassicality of the generated cat state (e.g., suitable for a wide range of squeezer gains and loss parameters) by introducing the generated cat state into a first nonlinear enhancement coupling 272 having a cat state size of 3.5 ± 0.2 and a squeeze time of 0.55 ± 2.0.

[0109] Section 9 The first nonlinear enhancement coupling 272 is (At least temporarily) the first Ponderomotive (^N a ×^x b ) Configured as coupling 272 A quantum detection method as described in any of the above sections.

[0110] Section 10 One or more photonic components are configured such that a specific photonic component 176 of the photonic components is configured as a signal quadrature squared 161, and one or more OPAs 160 are configured to include a specific phase-matched OPA 160 that receives the signal quadrature square 161. A quantum detection method as described in any of the above methods, including the following.

[0111] Section 11 A quantum detection method described in any of the above methods, comprising: A specific photonic component 176 among one or more photonic components is configured as a signal quadrature squared 161, and one or more OPAs 160 are configured to include a specific phase-matched OPA 160 that receives the signal quadrature squared 161, such that the specific phase-matched OPA 160 has a first quadratic coupling strength enhanced by an additional quadratic coupling strength 253 that is 2 to 20 times greater than the first quadratic coupling strength.

[0112] Item 12 A quantum detection method described in any of the above methods, comprising: One or more photonic components are configured such that a predetermined photonic component 176 is configured as a pump modular quadrature 162, and one or more OPAs 160 are configured to include a predetermined phase mismatch OPA 160 that receives the pump modular quadrature 162.

[0113] Item 13 A quantum detection method described in any of the above methods, comprising: A predetermined photonic component 176 from one or more photonic components is configured as a pump modular quadrature 162, and one or more OPAs 160 are configured to include a predetermined phase-mismatched OPA 160 that receives the pump modular quadrature 162, so that the predetermined phase-mismatched OPA 160 has a native quadratic coupling strength 183 and is enhanced by an additional quadratic coupling strength 253 that is 50% or more greater than the native quadratic coupling strength 183 but less than 50 times greater.

[0114] Section 14 A quantum detection method described in any of the above methods, comprising: One or more photonic components are configured such that a specific photonic component 176 is the number of signal Bogolyubov excitations 163, and one or more OPAs 160 are configured to include a specific phase mismatch OPA 160 that receives a pump modular quadrature 162.

[0115] Section 15 A quantum detection method described in any of the above methods, comprising: One or more photonic components are configured such that a specific photonic component 176 is the number of signal Bogolyubov excitations 163, and one or more OPAs 160 are configured to include a particular phase-mismatched OPA 160 that receives a pump modular quadrature 162, such that the particular phase-mismatched OPA 160 has a native quadratic coupling strength 183 enhanced by an additional quadratic coupling strength 253 that is 2-20 times greater than the native quadratic coupling strength 183.

[0116] Item 16 A quantum detection method described in any of the above methods, comprising: The first OPA160 out of 1 or more OPA160s is used as the first ponderomotive (^N a ×^x b It is configured as a phase-mismatched OPA160 configured to establish coupling 272.

[0117] Section 17 A quantum detection method described in any of the above methods, comprising: One or more OPA160s are configured (at least temporarily) as a phase-matched OPA160 configured to establish a squeezed cat state.

[0118] Section 18 A quantum detection method described in any of the above methods, comprising: The first quadratic coupling strength (183) in OPA160 of 1 or more is the first ponderomotive (^N a ×^x b The first ponderomotive (^N) is enhanced by the (at least) additional quadratic coupling strength 253 resulting from coupling 272. a ×^x b ) Establish coupling 272.

[0119] Section 19 A quantum detection method described in any of the above methods, comprising: The steps include configuring a quadratic nonlinear resonator as the first nonlinearity enhancement coupling 272 and An external drive field 131 with a finite decoherence rate 177(κ) excites a first nonlinearity enhancement coupling 272 in which the quantum superposition of transient signal cat states in a squeezed Fock state ladder is developed. The first nonlinearity enhancement coupling 272 becomes an optical parametric oscillator (OPO), thereby inducing quantum jumps between signal states in the transient signal cat states due to signal photon loss.

[0120] Section 20 A quantum detection method described in any of the above methods, comprising: The steps include transmitting a first output via a first output port 208D, which includes at least a primary feature 195 of the first input state, and The first ponderomotive (^N) is at least one element of the first nonlinear enhancement coupling 272. a ×^x bA first extraction result 292 (e.g., digital measurement 178 or other encoding 278) is transmitted via coupling 272 through the second output port 208C without destroying the first output, encoding one or more elements of the first photonic component 176 (e.g., state-indicative modes 166) including the number 264 of the signal Bogolyubov excitations 163 of the first input state.

[0121] Section 21 The phase noise induced by self-phase modulation is sufficiently mitigated, and the first extraction result 292 is obtained without destroying the first photonic component 176. A quantum detection method as described in any of the above sections.

[0122] Section 22 One or more non-Gaussian quantum states are generated within the Hamiltonian medium 282 and used to obtain the first extraction result 292 without destroying the first photonic component 176. A quantum detection method as described in any of the above sections.

[0123] Section 23 One or more non-Gaussian quantum states are generated within the Hamiltonian medium 282 (optical resonator) and used to obtain the first extraction result 292 without destroying the first photonic component 176. A quantum detection method as described in any of the above sections.

[0124] Section 24 A quantum detection method described in any of the above methods, comprising: A Hamiltonian medium 282 is implemented as a component of an optical parametric oscillator (OPO), and the extraction result 292 within it has an outcoupled pump field 131 monitored by a homodyne detector 170, allowing the intra-cavity squeezed photon-number state to be estimated without destroying the first photonic component 176.

[0125] Section 25 The number of signal Bogoliubov excitations (^N) in the first input state. a A displacement of 1 or more is induced in the pump mode 136 conditioned to ), thereby indirectly obtaining a quantum non-destructive measurement 178 of the signal Bogolyubov excitation 163 via the homodyne detector 170. A quantum detection method as described in any of the above methods, including the following.

[0126] Section 26 The mesoscopic number of photons inside the Hamiltonian medium 282 (of the optical resonator) enables an effective native nonlinear coupling rate of 0.1 < g / κ < 1, thereby enabling the first extraction result 292 to be obtained without destroying the first photonic component 176. The quantum detection method according to any one of the items of the above method.

[0127] Item 27 The phase noise induced by self-phase modulation is sufficiently relaxed, and the first extraction result 292 is obtained without destroying the first photonic component 176. The quantum detection method according to any one of the items of the above method.

[0128] Item 28 The photon-number-resolving (PNR) quantum nondemolition (QND) measurement 178 is obtained using a homodyne detector 170 between 15 °C and 30 °C (e.g., room temperature), and the quantum detection method according to any one of the items of the above method.

[0129] Item 29 The photon-number-resolving (PNR) quantum nondemolition (QND) measurement 178 is obtained in less than 10 microseconds through the Hamiltonian medium 282 between 15 ° and 30 °C (e.g., room temperature), and the quantum detection method according to any one of the items of the above method.

[0130] Item 30 The photon-number-resolving (PNR) quantum nondemolition (QND) measurement 178 is obtained in less than 100 nanoseconds (e.g., as an "ultrafast" measurement 178) through a homodyne detector 170 between 0 °C and 55 °C, and the quantum detection method according to any one of the items of the above method.

[0131] Section 31 A quantum detection method as described in any of the above sections, wherein a photon-number-resolved (PNR) quantum non-destructive (QND) measurement 178 is obtained in less than 100 nanoseconds (e.g., as an "ultrafast" measurement) 178 via a homodyne detector 170 between 15° and 30°C (e.g., room temperature).

[0132] Section 32 The first nonlinearity enhancement coupling (^N) a ×^x b ) Coupling 272 is connected to the first input state and one or more pump field quadrature (^x b ) A quantum detection method according to any of the above methods, which is established at least temporarily with 134.

[0133] Item 33 First Ponderomotive as First Nonlinear Enhancement Coupling 272 (^N a ×^x b ) Coupling 272 is the number of signal Bogolyubov excitations 163 of the first input state (^N a )264 and 1 or more pump field quadrature (^x b ) Established between 134, A quantum detection method as described in any of the above sections.

[0134] Section 34 Number of signal-bogolyubov excitations 163 (^N a A first preparatory operation to configure the encoding unit 140 so that one or more OPA160s that receive )264 include at least one phase-mismatched OPA160B; and A second preparatory operation that constitutes an encoding unit 140 in universal photonic quantum information processing (QIP) systems 100, 200 The quantum detection method according to any one of the items of the above method.

[0135] Item 35 Establish a first ponderomotive (^N a ×^x b ) coupling 272 as a first nonlinearity enhancement coupling 272 such that 0.1 < g / κ < 10000 The quantum detection method according to any one of the items of the above method.

[0136] Item 36 Establish the first ponderomotive (^N a ×^x b ) coupling 272 as (at least temporarily) a first nonlinearity enhancement coupling 272 such that 0.3 < g / κ < 3000 The quantum detection method according to any one of the items of the above method. Here, g is a nonlinear coupling constant 176, and κ is a decoherence rate 177(κ) of the first ponderomotive coupling 272.

[0137] Item 37 So that 0.1 < g / κ < 1, The first ponderomotive (^N a ×^x b) Establish coupling 272 as the first nonlinearity enhancement coupling 272 (at least temporarily) A quantum detection method according to any of the above methods, including the above. Here, g is its nonlinear coupling constant of 176, and κ is its decoherence rate of 177(κ).

[0138] Section 38 0.2 <g / κとなるように、 The first Ponderomotive (^N a ×^x b The coupling 272 is established as the first nonlinearity enhancement coupling 272. A quantum detection method according to any of the above methods, including the above.

[0139] Item 39 0.5 <g / κとなるように、 The first Ponderomotive (^N a ×^x b The coupling 272 is established as the first nonlinearity enhancement coupling 272. A quantum detection method according to any of the above methods, including the above. Here, g is its nonlinear coupling constant of 176, and κ is its decoherence rate of 177(κ).

[0140] Section 40 The measurement 178 or other encoding 278 of the first photonic component in the first input state is transmitted without losing the first photonic component 176. A quantum detection method according to any of the above methods, including the above.

[0141] Section 41 The additional quadratic coupling strength of 253 is more than 50% greater than the first quadratic coupling strength of 183, but less than 50 times the first quadratic coupling strength of 183. A quantum detection method as described in any of the above sections.

[0142] Section 42 The additional quadratic coupling strength of 253 is 2-20 times that of the first quadratic coupling strength of 183. A quantum detection method as described in any of the above sections.

[0143] Section 43 The additional quadratic coupling strength of 253 is 4-40 times that of the first quadratic coupling strength of 183. A quantum detection method as described in any of the above sections.

[0144] Section 44 Without destroying the mode 166 or other photonic component 176 of the first output 191, the number of signal Bogoliubov excitations 163 of the first input state (^N a ) Send the first pump output 192 which encodes 264. A quantum detection method according to any of the above methods, including the above.

[0145] Section 45 One or more photonic components 176 include a number of signal quadrature squared 161, pump modular quadrature 162, or signal Bogoliubov excitations 163. A quantum detection method as described in any of the above sections.

[0146] Section 46 One of the one or more photonic components 176 is configured to include a signal quadrature squared 161 or a pump modular quadrature 162 (or both). A quantum detection method according to any of the above methods, including the above.

[0147] Section 47 One of the one or more photonic components 176 is configured to include a number of signal quadrature squareds 161 or signal Bogoliubov excitations 163. A quantum detection method according to any of the above methods, including the above.

[0148] Section 48 One of one or more photonic components 176 is composed of a number of pump modular quadratures 162 or signal Bogoliubov excitations 163 (or both). A quantum detection method according to any of the above methods, including the above.

[0149] Section 49 A specific photonic component 176 among one or more photonic components 176 is configured to include a signal quadrature squared 161. A quantum detection method according to any of the above methods, including the above.

[0150] Item 50 One or more photonic components 176 are configured to include a pump modular quadrature 162. A quantum detection method according to any of the above methods, including the above.

[0151] Section 51 A particular photonic component 176 out of 1 or more photonic components 176 is determined by the number of signal Bogoliubov excitations 163 (^N a Configure to include 264 A quantum detection method according to any of the above methods, including the above.

[0152] Section 52 Without destroying the first output, the number of signal Bogoliubov excitations (^N) is (at least) as the first element 165 of the first input state. a The pump output 192 is transmitted as a component of the first extraction result 292 which encodes 264. A quantum detection method according to any of the above methods, including the above.

[0153] Section 53 Without destroying the first output, the number of signal Bogoliubov excitations (^N) is (at least) as the first element 165 of the first input state. a )264 encodes pump output 192 or transmits other first result 292 A quantum detection method according to any of the above methods, including the above.

[0154] Section 54 Without reducing the first output by more than 1%, a digital measurement 178 of the first photonic component 176 of the first input state in the pump output 192 or other extraction result 292 is acquired and transmitted. A quantum detection method according to any of the above methods, including the above.

[0155] Section 55 (At least) the first nonlinearity enhancement coupling 272 as the first ponderomotive (^N a ×^x b Coupling 272 is established in system 200 as shown in Figure 2. A quantum detection method as described in any of the above sections.

[0156] Section 56 The first nonlinearity enhancement coupling is the first ponderomotive (^N a ×^x b Coupling 272 is established in system 100 as shown in Figure 1. A quantum detection method as described in any of the above sections.

[0157] Section 57 Systems 100, 200 configured to perform the method described in any of the above sections.

[0158] Section 58 Systems 100, 200 manufactured by the method described in any one of the above methods.

[0159] With respect to the numbered claims set forth below, a person skilled in the art will understand that the operations described herein may generally be performed in any order. Furthermore, while various operation flows are shown in sequence, it should be understood that the various operations may be performed in any order other than those illustrated, or simultaneously. Examples of such alternative orders include, unless otherwise indicated by the context, overlapping orders, interleaved orders, interrupted orders, reordering, incremental orders, preparation orders, supplementary orders, simultaneous orders, reverse orders, or other variant orders. Terms such as “responding to,” “related to,” or other such transitive, relational, or other conjunctions generally do not exclude such variants unless otherwise indicated by the context. Moreover, each of the following claims is intended to give the least restrictive interpretation reasonable to a person skilled in the art.

[0160] The following claims are fully supported by the above description, independently of any documents referenced herein. However, they may be made more concise and clearer by using a medium that includes color, shading, searchable text, and hyperlinks to relevant content. Therefore, to more quickly learn the techniques supporting the content of this specification, it is recommended, where possible, to refer to the color-enhanced online versions of the publications cited herein.

Claims

1. A first secondary coupling intensity is constructed in one or more optical media on which one or more optical parametric amplifiers (OPAs) are implemented. Obtain a first input state that includes one or more photonic components, A first nonlinear enhancement coupling is established such that the first secondary coupling strength in the one or more OPAs is enhanced by an additional secondary coupling strength greater than the first secondary coupling strength. A first output including the first photonic component of the one or more photonic components is transmitted via the first output port. Without destroying the first photonic component of the first output, a first extraction result encoding the first photonic component of the first input state is transmitted via the second output port through the first nonlinear enhancement coupling. A quantum detection method that includes the following.

2. The quantum detection method according to claim 1, comprising acquiring and transmitting, via a first nonlinear enhancement coupling, a first extraction result that encodes the first photonic component of the first input state via the second output port without reducing the first photonic component by 1% or more.

3. The quantum detection method according to claim 1, comprising triggering a room-temperature ultrafast universal quantum computation using the first nonlinear enhancement coupling, and realizing one or more Gottesman-Kitaev-Preskill (GKP) states in the computation system.

4. The first photonic component is configured as a signal quadrature square, and the one or more OPAs are configured to include a specific phase-matched OPA that receives the signal quadrature square, the specific phase-matched OPA having a first secondary coupling intensity, and the first secondary coupling intensity is enhanced by an additional secondary coupling intensity that is less than 20 times greater than the first secondary coupling intensity. A quantum detection method according to claim 1, which includes the following:

5. The first photonic component is configured as a pump modular quadrature, and the one or more OPAs are configured to include a predetermined phase mismatch OPA that receives the pump modular quadrature, and the predetermined phase mismatch OPA has the native secondary coupling strength enhanced by an additional secondary coupling strength less than 50 times greater than the native secondary coupling strength. A quantum detection method according to claim 1, which includes the following:

6. The first photonic component is configured as the number of signal-Vogolubov excitations, and the one or more OPAs are configured to include a specific phase mismatch OPA that receives a pump modular quadrature, and the specific phase mismatch OPA has an additional quadratic coupling strength that is less than 20 times greater than the native secondary coupling strength, thereby enhancing the native secondary coupling strength. A quantum detection method according to claim 1, which includes the following:

7. The additional secondary coupling strength is 50% or more greater than the first secondary coupling strength, and less than 50 times the first secondary coupling strength. The quantum detection method according to claim 1, characterized in that it is a feature of the present invention.

8. The first OPA of the one or more OPAs is to be fitted with the first Ponderomotive (^N) ａ ×^x ｂ ) Configured as a phase mismatch OPA configured to establish coupling A quantum detection method according to claim 1, which includes the following:

9. A specific OPA among the one or more OPAs is configured (at least temporarily) as a phase-matched OPA configured to establish a squeezed cat state. A quantum detection method according to claim 1, which includes the following:

10. The first Ponderomotive (^N) ａ ×^x ｂ The coupling is established as the first nonlinear enhancement coupling, The first secondary coupling strength in the one or more OPAs is the first pondelomotive (^N) ａ ×^x ｂ ) Enhanced by the additional secondary coupling strength resulting from the coupling. A quantum detection method according to claim 1, which includes the following:

11. A second-order nonlinear resonator is configured as the first nonlinear enhancement coupling. By exciting the first nonlinear enhancement coupling with an externally driven field having a finite decoherence rate (κ) that evolves the quantum superposition of transient signal cat states in a squeezed fock state ladder, the first nonlinear enhancement coupling becomes an optical parametric oscillator (OPO), and the loss of signal photons induces quantum jumps between signal states in the transient signal cat state. A quantum detection method according to claim 1, which includes the following:

12. Phase noise induced by self-phase modulation is mitigated, and the first extraction result is obtained in the first output without destroying the first photonic component. The quantum detection method according to claim 1, characterized in that it is a feature of the present invention.

13. The Hamiltonian medium is configured as an optical parametric oscillator (OPO), and has an out-coupled pump field whose first extraction result is monitored by a homodyne detector, enabling estimation of the in-resonator squeezed photon number state without destroying the first photonic component. A quantum detection method according to claim 1, which includes the following:

14. Photon number discrimination (PNR) quantum non-destructive (QND) measurement, Obtained in less than 10 microseconds via a Hamiltonian medium between 0°C and 55°C The quantum detection method according to claim 1, characterized in that it is a feature of the present invention.

15. The number of non-negative signal-bogolyubov excitations (^N) ａ The encoding unit is configured such that at least one phase mismatch OPA is included among the one or more OPAs that receive ) The steps include: configuring the encoding unit into a universal quantum information processing (QIP) system; A quantum detection method according to claim 1, which includes the following:

16. The quantum detection method according to claim 1, including establishing a first ponderomotive ( ａ ａ × ^x ｂ coupling as the first non-linearity enhancement coupling so that 0.1 < g / κ < 10000). Here, g is the nonlinear coupling constant, and κ is the decoherence rate (κ) of the first coupling.

17. Without destroying the first output, the non-negative number (^N) of signal Bogolyubov excitations is used as the first element of the first input state. ａ ) transmits the pump output or other first result which encodes the result. A quantum detection method according to claim 1, which includes the following:

18. The primary of the one or more photonic components is configured as a quadrature squared or pump modular quadrature. A quantum detection method according to claim 1, which includes the following:

19. A means for acquiring a first input state including one or more photonic components, Means for establishing a first nonlinear enhancement coupling such that the first secondary coupling strength in the one or more OPAs is enhanced by an additional secondary coupling strength greater than the first secondary coupling strength, Means for transmitting a first output, including the first photonic component of the one or more photonic components, via a first output port. Means for transmitting a first extraction result encoding the first photonic component of the first input state via a second output port through a first nonlinear enhancement coupling without destroying the first photonic component of the first output. A quantum detection system that includes this.

Images