System and method for L0 regularization-based compressive sensing using coherent ising machines

Abstract

The system and method for L0 regularization-based compressive sensing (CS) can use a quantum-classical hybrid system consisting of a coherent Ising machine (CIM) and a classical digital processor (CDP). The CIM and CDP each implement an alternating minimization for L0 regularization-based compressive sensing (CS). A truncated Wigner stochastic differential equation (W-SDE) is obtained from the master equation for the density operator of a network of degenerate optical parametric oscillators.

Description

Technical Field

[0001] Cross-reference This application claims priority to U.S. Provisional Application No. 63 / 151,441, filed Feb. 19, 2021, the entire disclosure of which is incorporated herein by reference.

[0002] Appendix Appendix A (7 pages) forms part of this specification and includes supplementary materials and figures incorporated herein by reference.

[0003] Appendix B (10 pages) lists methods used for various calculations herein, and this appendix forms part of this specification and is incorporated herein by reference.

[0004] The present disclosure relates to systems and methods for compressing sensing using a coherent imaging machine.

Background Art

[0005] Currently, there are various techniques for performing simulation and optimization processes using quantum devices or machines. These quantum devices or machines are very complex and difficult / costly to construct. Examples of quantum machines include Google's quantum computer and IBM's quantum computer as well as D-WAVE's quantum annealer. All of these quantum machines are very expensive and difficult to construct and operate.

[0006] Least absolute shrinkage and selection operator (LASSO) is a very efficient approach for solving various sparse signal reconstruction problems in exploration geophysics, magnetic resonance imaging, black hole observation, and materials informatics. The LASSO operator

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[0007] L1 - regularization - based compressed sensing (CS) including LASSO can be formulated as a convex optimization problem, for which many efficient discovery algorithms are available. The above equation (1) is asymptotically equal to L1 - minimization - based CS (minimize ||x|| such that y = Ax) at the limit λ→+0. Thus, at this limit, LASSO has a non - zero element number of x to N ratio, that is, sparsity a, such that as long as a is smaller than the critical value a 1 is minimized) and, thus, at this limit, LASSO can reconstruct the sparse source signal with infinitesimal error as long as the ratio of the number of non - zero elements of x to N, i.e., the sparsity a, is smaller than the critical value a c (a c is smaller than the measurement compression ratio α (defined as α = M / N)).

[0008] On the other hand, L0 - regularization - based CS can be formulated using the following L0 - norm instead of the L1 - norm:

Equation

[0009] It has been suggested that L0 - regularization - based CS may outperform L1 - regularization - based CS. This is because L1 - regularization imposes shrinkage (soft - thresholding) on variables exceeding a certain threshold, while L0 - regularization does not impose such shrinkage (hard - thresholding). Furthermore, equation (2) is asymptotically equal to L0 - minimization - based CS (minimize ||x|| such that y = Ax) at the limit λ→+0. Thus, at this limit, L0 - regularization - based CS has a such that a is at the critical value a 0 is minimized) and, thus, at this limit, L0 - regularization - based CS has a such that a is at the critical value a cAs long as it is smaller than =α, it can be expected that x can be reconstructed with infinitesimal error. Note that the number of non-zero elements of x is the effective rank of the system of linear equations, and thus any system cannot exceed the performance limit because the system does not have a single unique solution when a > α.

[0010] The L0 regularization-based CS defined by equation (2) can be equivalently reformulated as a two-fold optimization problem: [Number] (3)

[0011] Here, the element r in r i represents the real value of the i-th element in the N-dimensional source signal. The vector σ is called a support vector that indicates the locations of the non-zero elements of the N-dimensional source signal. The element σ in σ i takes 0 or 1 to indicate whether the i-th element of the source signal is zero or non-zero. The symbol ° represents the Hadamard product. From the element-by-element representation of equation (3), the Hamiltonian (H or cost function) of the L0 regularization-based CS is [Number] (4) given as, where [Number] is an element in the M×N observation matrix A, and y u is an element in the M-dimensional observation signal.

[0012] Under the assumption that the number of 1s in the support vector σ is less than or equal to M, the set of simultaneous linear equations obtained from equation (3) with respect to r gives the solution for the non-zero elements in the source signal when the support vector σ is given. On the other hand, the minimization of H with respect to σ is identical to minimizing the Ising Hamiltonian when r is given. Interaction

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[0013] It is desirable to provide a system and method for performing L0 regularization-based compressive sensing without the difficulty of estimating support vectors, and this is the purpose for which the present disclosure is directed.

Summary of the Invention

[0014] The systems and methods herein can be used for L0 regularization-based compressive sensing using a quantum-classical hybrid system composed of a quantum machine and a classical digital processor (CDP).

[0015] In one embodiment, a hybrid system for L0 regularization-based compressive sensing of a source signal is provided. The system can include a quantum machine configured to optimize a first parameter associated with the source signal to minimize a cost function, and a classical machine configured to optimize a second parameter associated with the source signal to minimize the cost function.

[0016] In another embodiment, a method for L0 regularization-based compressive sensing of a source signal can be provided. The method can include optimizing, by a quantum machine, a first parameter associated with the source signal to minimize a cost function, and optimizing, by a classical machine, a second parameter associated with the source signal to minimize the cost function.

[0017] In yet another embodiment, a method for performing L0 regularization-based compression detection may be provided. The method can include injecting a plurality of pump pulses into an optical parametric oscillator formed within a fiber ring cavity having an output coupler and an input coupler, where the output coupler communicates with a homodyne detection output and a second harmonic generation (SHG) crystal.

Brief Description of the Drawings

[0018]

Fig. 1A

Fig. 1B

Figs. 2a - 2b

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Embodiments for Carrying out the Invention

[0019] The present disclosure is particularly applicable to a system and method for L0 regularization-based compressive sensing using a quantum-classical hybrid system composed of a quantum machine and a classical digital processor (CDP). A coherent ising machine (CIM) is a suitable quantum machine for this system. This is because this optimization problem can only be solved by a densely connected network. It is in this context that the present disclosure will be described. However, it will be recognized that the system and method can be implemented in other ways. Furthermore, although the exemplary data is medical data, the system and method for L0 regularization-based compressive sensing can be used for any type of data and are not limited to any specific type of data.

[0020] In a system and method for L0 regularization-based compressive sensing, the technical problem of the difficulty of estimating the support vectors described above is addressed by a technical solution that is a hybrid system having a quantum machine and a CDP. FIG. 1A shows a quantum-classical hybrid system composed of a coherent Ising machine (CIM) and a CDP that can perform L0 regularization-based compressive sensing. This system solves a two-stage optimization problem by alternately performing two minimization processes: (i) the quantum machine optimizes σ to minimize H under the condition that r is fixed, and (ii) the CDP optimizes r to minimize H under the condition that σ is fixed.

[0021] Some quantum machines such as quantum annealers, quantum approximate optimization algorithms, CIMs, etc. may perhaps be used to optimize σ. A comparison of these candidates reveals that the measurement-feedback (MFB) CIM is one of the most suitable machines for this purpose. In fact, an MFB-CIM can construct any tightly connected network composed of degenerate optical parametric oscillators (OPOs). This is because the MFB-CIM uses a time-division multiplexing scheme and MFB. In contrast, QAs and almost all other machines can only support a local graph including a chimera graph. Therefore, a tightly connected network for optimizing σ must be embedded in a fixed hardware local graph by using the Lechner-Hauke-Zoller scheme, which requires additional physical spins. Furthermore, it has been reported that the MFB CIM has experimentally outperformed the quantum annealer for two problem sets (one problem is the fully connected Sherrington-Kirkpatrick model and the other problem is the dense graph MAX-CUT). In contrast to the exponential computation time proportional to exp(O(N 2 )) for the quantum annealer, the CIM is exp( [Number] (where N is the problem size) has an exponential computation time proportional to it.

[0022] Figure 1B shows that the L0 regularization-based compressive sensing system and method can be executed in one embodiment using a quantum-classical hybrid system 100 composed of a CIM 102 and a classical digital processor (CDP) 104. This system 100 performs two minimization processes: (i) the CIM 102 optimizes σ to minimize the Hamiltonian cost function H using a given real value r of the source signal, and (ii) the CDP 104 optimizes r to minimize H using a given support vector σ, alternately. Using the system, the truncated Wigner stochastic differential equation (W-SDE) from the master equation for the density operator of the network consists of an OPO. The system and method perform a statistical mechanics method based on self-consistent signal-to-noise analysis (SCSNA), from which a macroscopic equation for the entire system can be derived. As discussed in detail below, the performance of the disclosed system and method approaches the theoretical limit of L0 minimization-based compressive sensing (CS) and perhaps exceeds the theoretical limit of the known LASSO technique.

[0023] In more detail as shown in FIG. 1B, the CIM 102 of the system 100 has a pump pulse that is injected into an optical parametric oscillator (OPO) formed within a fiber ring cavity through a second harmonic generation (SHG) crystal. A periodically poled lithium niobate (PPLN) waveguide device induces phase-sensitive degenerate optical parametric amplification of the signal pulse, and each of the OPO pulses takes a 0-phase state (corresponding to up spin) or a π-phase state (corresponding to down spin) beyond the oscillation threshold. A portion of each pulse is extracted from the main cavity by an output coupler and measured by an optical homodyne detector. A field programmable gate array (FPGA) calculates a feedback signal, which is then provided to an intensity modulator (IM) and a phase modulator (PM) to generate an injection field for each of the OPO pulses through the input coupler as described by the following equation (5). H(X i ) is the binarized value of the in-phase amplitude of the i-th OPO pulse, which is the support estimate transferred to the CDP 104, and is either 0 or 1. The CDP 104 solves a system of linear equations (Equation (7) discussed below), and the solution r i is transferred to the CIM 102 as shown in FIG. 1B.

[0024] Roles of the CIM and CDP

[0025] The CIM-CDP hybrid system 100 of FIG. 1B executes the L0 regularization-based CS described by the above equations (3) and (4). This system achieves optimization by alternately performing the following two minimization processes. CIM 102 optimizes σ to minimize H using a given r, and then transfers σ to CDP 104. CDP 104 optimizes r to minimize H using the given σ, and then transfers r to CIM 102. FIGS. 9A, 9B, and 10, discussed in more detail below, show the details of how the two alternating minimization processes are used. In method 900 shown in FIGS. 9A, 9B, and 10, in order to widen the gravitational sphere, a heuristically linear threshold reduction is introduced in which the threshold η linearly decreases from η init to η end as the alternating minimization proceeds.

[0026] CIM 102 estimates the support vector σ, i.e., the locations of the non-zero elements of the source signal. To optimize σ to minimize H (Hamiltonian / cost function) using a given r, CIM 102 uses a measurement feedback circuit to control an intensity modulator (IM) and a phase modulator (PM), and both modulators generate an optical injection field for the target (i-th) OPO pulse:

Equation

Equation

[0027] The local field for estimating the support vector on the CIM is [Number] (6) is set as, where X j is the in-phase amplitude (generalized coordinate) of the j-th OPO pulse measured by the homodyne detector. In the local field, H(X) is the Heaviside step function that takes 0 when X ≤ 0 or +1 when X > 0. r j is the solution given by CDP104. During the support vector estimation on CIM102, all r j are fixed. Therefore, the first term is the interaction term, and the second term corresponds to the Zeeman term.

[0028] CDP104 obtains the solution of the linear simultaneous equations from the minimization condition of H with respect to r. Without loss of generality, the expression for each element of the simultaneous equations regarding the unknown value of r i is [Number] (7) JPEG0007679043000012.jpg1693(8) can be rewritten as. To eliminate the indeterminacy, r i is set to zero when H(X i ) is zero. Here, h in Equation (8) i is the same as the local field (Equation (6)) in the case of support estimation on CIM102. X j is the solution given by CIM102. During the signal estimation on CDP104, all H(X j ) are fixed. The solution of the simultaneous equations (Equation (7)) is r = (diag[A T A] + SA T AS - diag[SA T AS]) -1 SA T y S = diag(H(X 1 ), H(X 2 ), ···, H(X N )) is as follows.

[0029] Derivation of the Wigner probability differential equation for CIM As shown in Fig. 1B, the pump pulse is injected into the main ring cavity through the second harmonic generation (SHG) crystal. The periodically poled lithium niobate (PPLN) waveguide is a very efficient nonlinear medium for an optical parametric oscillator. Assume that the amplitude of the injected pump field into the main cavity is ε, and the parametric coupling constant of the PPLN waveguide between the signal field and the pump field is κ. Then, the pumping Hamiltonian is

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[0030] The measurement feedback circuit shown in Figure 1B is connected to the main cavity by extraction and injection couplers with reflection coefficients R ex = j ex Δt and R in = j in Δt, where j ex and j in are the coarse-grained outcoupling and incoupling constants, and Δt is the cavity round-trip time. B / γ S < 1. Under the condition that vacuum fluctuations are incident on the opening ports of the extraction and injection couplers, the measurement feedback circuit can be described using a Gaussian quantum model. The master equation consists of a linear loss term, a measurement-induced state reduction term, and a coherent feedback signal injection term.

[0031] The Fokker-Planck equation is for the density operator in the master equation [Mathematics] Derived by using the Wigner W(α) representation of [Mathematics], the following truncated Wigner stochastic differential equation (W-SDE) can be achieved by applying Ito's rule: [Mathematics] where j = j ex + j in and α i is the complex Wigner amplitude, and ν i is the 〈v i (t)〉 = 0, 〈v i * (t)v j (t’)〉 = 2δ ij δ(t - t’) which satisfies the c-number noise amplitude.

[0032] After that, [Mathematics] Introduce the saturation parameter A s by [], and apply the following scale transformation [Mathematics] and [Mathematics] to obtain [Mathematics] where c i and s i are the in-phase and quadrature phase normalization amplitudes of the i-th OPO pulse. Equation (18 *) The second term on the R.H.S. of the above equation is an optical injection field having only the in-phase component. p is the normalized pump rate. p = 1 corresponds to the oscillation threshold of a single OPO without mutual coupling. When p exceeds the oscillation threshold (p > 1), each of the OPO pulses is in the 0-phase state or the π-phase state. The 0-phase of the OPO pulse is assigned to the up-state of the idling spin, while the π-phase is assigned to the down-state. The last terms in both the upper and lower equations of Equation (18 * ) represent the pump fluctuations coupled into the OPO system by the vacuum fluctuations injected from the external reservoir and the gain saturation. W i,1 and W i,2 are independent real Gaussian noise processes satisfying 〈W i,k (t)〉 = 0, 〈W i,k (t)W j,l (t’)〉 = δ ij δ kl δ(t - t’). The saturation parameter A s determines the non-linear increase (sharp rise) in the number of photons at the OPO threshold. Finally, a more general quantum model of MFB-CIM without Gaussian approximation was derived for both the discrete-time model and the continuous-time model.

[0033] Macroscopic equations for a quantum-classical hybrid system

[0034] To solve the macroscopic equations for a quantum-classical hybrid system, the above Equation 18 * is solved, and the system of equations (7) can be solved using statistical mechanics under the precondition of applying the statistical mechanics described below. The W-SDE (18 * ) and the system of equations (7) share the same local field (Equations (6) and (8)), and that local field can be rewritten by substituting it as

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[0035] Thus, CIM 102 and CDP 104 can be unified into a steady-state single mean eld system. Since the W-SDE for the i-th OPO only depends on the self-state and the local field h i , a formal transfer function X from h i to H(c i ) can be introduced: H(c i )=X(h i ) Substituting the formal transfer function X into equation (7), we also get: <\br >

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[0036] Accuracy of the macroscopic equation and comparison with LASSO when β = 0

[0037] To confirm the accuracy of the macroscopic equation, the above solution is compared with the solution given by Algorithm 1 and the macroscopic equation (Figs. 9A - 10 discussed in more detail below). Figs. 2a and 2b compare the solutions of the method for L0 regularization - based compressive sensing, for various values of the threshold η and the compression ratio α (red and green solid lines),

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[0038] In the case of semi-Gaussian (+), two macroscopic states with non-zero RMSE (red solid line) and nearly zero RMSE (green solid line) coexist as in the case of the CIM implementation CDMA multi-user detector. On the other hand, in the case of Gaussian (±), a single macroscopic state with nearly zero RMSE (red solid line) was found. Comparing with the simulation results in Fig. 2b, the theoretical results obtained using the macroscopic equation (13)

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[0039] Furthermore, to compare the capabilities of CIM L0 regularization-based CS and LASSO, the RMSE profiles of LASSO using the macroscopic equation (37) with the same threshold as CIM L0 regularization-based CS were calculated, and these profiles were overlaid on Fig. 2 (blue solid line). At the limit

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[0040] Phase diagrams of CIM L0 regularization-based CS and LASSO when β = 0

[0041] Phase diagrams of CIM L0 regularization-based CS for various values of the threshold η were prepared when there was no observation noise (i.e., β = 0). Fig. 3a shows the first-order phase transition lines (red lines) from the near-zero RMSE state for the semi-Gaussian (+) case and the Gaussian (±) case. The phase transition lines in Fig. 3a are the ones in the limit

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[0042] As demonstrated in Fig. 3a, in the limit

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[0043] To compare the characteristics of CIM L0 regularization-based CS with those of LASSO, Fig. 3b shows the phase diagram of LASSO: the blue line is the first-order phase transition line from the nearly zero RMSE state for various η. As η decreases, the phase transition lines from the nearly zero RMSE state in LASSO for semi-Gaussian (+) and Gaussian (±) approach the two black dotted lines, while the RMSE of the nearly zero RMSE state of LASSO decreases to zero (the blue line in Fig. 2). The black dotted lines in Fig. 3 are the critical lines indicating the boundaries of whether the L1 minimization-based CS can completely reconstruct the source signal without error for the non-negative and signed cases, respectively. Therefore, the nearly zero RMSE solution of LASSO asymptotically approaches the complete reconstruction solution of L1 minimization-based CS as η decreases.

[0044] CIM L0 regularization-based CS and LASSO have these asymptotic characteristics even for source signals from gamma (+) and bilateral gamma (±). Note that this asymptotic characteristic of CIM L0 regularization-based CS has been theoretically proven to be invariant to the difference in the probability distribution of the source signal by applying the perturbation expansion to the macroscopic equation (13) at the limit η → +0. Therefore, this theoretical result has been numerically confirmed.

[0045] On the other hand,

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[0046] The black dot-dash line in Fig. 3a is the limit

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[0047] Gravitational sphere when β = 0

[0048] To check the practicality of CIM L0 regularization-based CS, the gravitational sphere of Algorithm 1 can be verified. To widen the gravitational sphere, the method can introduce linear threshold decay heuristically, and when the minimization process is performed alternately, the threshold linearly decreases from η init to η end (see Algorithm 1 in Figs. 9A - 10). First, numerical experiments were carried out to verify the size of the gravitational sphere for various values of the initial threshold η end = 0.01 when there is no observation noise (i.e., β = 0). As shown in Fig. 4a, the gravitational sphere tended to be widened by selecting an initial threshold η init higher than η end . As the compression rate α decreased, this tendency became more prominent, especially in the case of Gaussian (±). init

[0049] Next, starting from the initial state r = 0 for various η init , it was confirmed how well Algorithm 1 converged to the almost zero RMSE state given by the macroscopic equation (13) (Fig. 4b). As demonstrated in Fig. 4b, when the sparsity a was lower than the lower limit of the first-order phase transition point (black dot-dash line in Fig. 3a), Algorithm 1 with η init = 0.6 converged to the solution (red line) of the macroscopic equation (13), while Algorithm 1 with η init ​For other values, it was not possible to converge to a solution. Compared with the RMSE profile of LASSO in Fig. 4b, Algorithm 1 exceeded the estimation accuracy of LASSO under almost all conditions where LASSO had a small error.

[0050] The characteristics shown in Fig. 4 are also satisfied when the source signals are from gamma(+) and bilateral gamma(±).

[0051]

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[0052] Furthermore, to check the practicality of CIM L0 - regularization - based CS, the accuracy and convergence of CIM L0 - regularization - based CS in the presence of observation noise (i.e.,

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[0053] Next, for the case of observation noise, the output of Algorithm 1 converged to the solution to the macroscopic equation (13) when starting from the initial state r = 0 and η init = 0.6. [Number] determined using. As shown in Figs. 5c and 6c, near or at the phase transition point, Algorithm 1 converged to the solution to the macroscopic equation (13).

[0054] The characteristics shown in Figs. 5 and 6 were similar for source signals from gamma (+) and bilateral gamma (±).

[0055] Performance of CIM L0 regularization-based CS for realistic data

[0056] The performance of CIM L0 regularization-based CS and other methods was evaluated with respect to realistic data. For the evaluation, magnetic resonance imaging (MRI) data obtained from a high-speed MRI dataset was used. The Haar-wavelet transform (HWT) was applied to the data, and 79% of the HWT coefficients were set to zero to generate a signal spanned by Haar basis functions with a sparsity of 0.21 (left panel of Figure 7a). The k-space data shown in the central panel of Figure 7a was obtained by calculating the discrete Fourier transform (DFT) from the signal in the left panel of Figure 7a, and 40% of the k-space data was undersampled at random red points within the central panel of Figure 7a to generate an observed signal with a compression ratio of 0.4. The right panel of Figure 7a shows an image with incoherent artifacts obtained by zero-filling Fourier reconstruction from the randomly undersampled k-space data.

[0057] To achieve higher reconstruction accuracy from the undersampled signal, an executable optimization problem for CIM was formulated using the L0 and L2 norms:

Equation

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[0058] Figure 7b shows the images (and RMSE) reconstructed from CIM L0 regularization-based CS (left panel of Figure 7b), LASSO (central panel of Figure 7b), and L1 minimization-based CS (right panel of Figure 7b) executed in CVX. As shown in the images surrounded by red circles in these panels, CIM L0 regularization-based CS gave the most accurate reconstruction.

[0059] The RMSEs of the three methods as a function of the threshold η were evaluated. As shown in Figure 7c, the blue line with error bars is the RMSE of CIM L0 regularization-based CS obtained from 10 trials, the red line is the RMSE of LASSO, the circles are the RMSE of L1 minimization-based CS, and the RMSE of L1 minimization-based CS is the same as that of LASSO in the limit η → 0, as demonstrated in Figure 3c. Due to the trade-off between detecting small non-zero elements and eliminating incoherent artifacts by thresholding, there exists an optimal value for minimizing the RMSEs of both CIM L0 regularization-based CS and LASSO. The RMSE of CIM L0 regularization-based CS was lower than that of the other methods within a wide range of η.

[0060] Figures 9A - 9B and 10 are flowcharts and more detailed pseudo - code of method 900 for L0 - regularization - based compression detection. In one embodiment, the CIM and CDP shown in FIG. 1B can be used to implement the method, although other devices can be used. Further, the pseudo - code of FIG. 10 is more detailed regarding the exact process implemented and shows a preferred execution form of the method, although the method may vary from the pseudo - code of FIG. 10 and can be within the scope of the present disclosure. In one embodiment, method 900 can be implemented by a computer system having a processor and a memory, and a plurality of lines of computer code / instructions stored in the memory and executed by the processor of the computer system to implement method 900, and the computer system is coupled to the CIM and CDP of the system in FIG. 1B. Alternatively, the FPGA and / or CDP of FIG. 1B can have a plurality of lines of computer code / instructions stored in the FPGA or CDP and executed by the FPGA or CDP to execute method 900.

[0061] As shown in FIG. 9A, method 900 can use an M×N observation matrix A, an M - dimensional signal y, an N - dimensional support vector σ, and an N - dimensional signal vector r. The method can initialize variables (902). In one embodiment shown in FIG. 10, the variables can be r and a threshold η = η init as shown in the pseudo - code and later in the flowchart, the method can implement a loop to decrease the threshold η, and each loop of the loop implements two minimizations and the decrease of the threshold η.

[0062] During the loop shown in FIG. 9A, the method can minimize a cost function (H shown in the pseudo - code in the preferred embodiment) with respect to the support vector (904). The method can use CIM102 to perform this minimization. As shown in the pseudo - code, in the preferred embodiment, this process 904 is

Equation

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[0063] During the loop shown in Figure 9A, the method can minimize (906) a cost function (H shown in pseudocode in the preferred embodiment) with respect to the real value (r) of the signal source. The method can use CDP104 to perform this minimization. As shown in the pseudocode, in the preferred embodiment, this process 906 can be: S = diag(σ) and r=(diag[A T A]+SA T AS - diag[SA T AS]) -1 SA T Set to y.

[0064] As shown in Figure 9B, the method can decrement (908) a threshold value (η in the preferred embodiment). As shown in the pseudocode, in the preferred embodiment, this process 909 can decrement η:

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[0065] Effectiveness of CIM in support estimation In Algorithm 1 shown in FIGS. 9A, 9B, and 10, the c amplitude is always initialized to c = 0 at the initial stage of support estimation, even when r is initialized to the true signal value, i.e., x°ξ. In this situation, the solution of Algorithm 1 fits very well into the almost zero RMSE state of the macroscopic equation, as demonstrated in FIGS. 2 and Supplementary FIG. 1B. Therefore, the simulated CIM can reconstruct the support vector to the theoretical limit. Limit

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[0066] Correctness of the assumption

[0067] To derive the macroscopic equation (12), by substituting the state variables within the second coefficient of the power of the quantum noise with the average values of the state variables, an approximation for 〈H(c i )〉 for each OPO pulse was derived (see Equation (19)). As shown in FIGS. 2b, 5c, and 6c, the macroscopic equation derived under this approximation has good accuracy at the values of CIM used in the actual device. However, as shown in FIG. 2a, some solutions of the macroscopic equation did not match the numerical solutions of Algorithm 1 for smaller values of

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[0068] Gravitational sphere and its dependence on the threshold value

[0069] To broaden the gravitational sphere of Algorithm 1, a linearly decreasing threshold decay was discovered and introduced, where the threshold decreases linearly as the alternating minimization progresses. A higher initial threshold η init to a lower final threshold η end results in a broader gravitational sphere (see Figure 4).

[0070] According to the definition of the injection field for each OPO pulse in Equation (5), the threshold η acts as an external field that gives a negative bias for the OPO pulse to take the downward state. By first applying a large negative external field, almost all of the OPO pulses take the π-phase state, and thus almost all of {H(X j )} j=1、...、N take zero at the initial stage of the alternating minimization process. At the initial stage, the system can easily reach the ground state under a strong negative bias because the phase space consisting of a small number of upward-state OPO pulses is simple. Then, throughout the alternating minimization process, the system tracks the gradual change of the ground state due to the incremental increase in the number of upward-state OPO pulses by gradually expelling the negative external field. Finally, the system reaches the ground state at the final threshold η end . This is a qualitative explanation of the mechanism for broadening the gravitational sphere of Algorithm 1 by linearly decreasing the threshold.

[0071] However, as demonstrated in Figure 4b, when there is no observational noise, the system could not converge to a nearly zero RMSE solution beyond the lower bound line of the first-order phase transition point. Similar to the spin glass phase, there may be many quasi-steady states under the condition of exceeding the lower bound line, and thus the system may be trapped in one of the quasi-steady states.

[0072] On the one hand, in the presence of observation noise, as demonstrated in FIGS. 5c and 6c, the system converged to a nearly zero RMSE solution even near the phase transition line when starting from the actual initial condition r = 0. It was suggested that the symmetry of the system enables the generation of a pseudo-steady state. Observation noise may break the symmetry with respect to the pseudo-steady state.

[0073] Conclusion

[0074] The above description has referred to specific embodiments for purposes of illustration. However, the above exemplary discussion is not intended to be exhaustive or to limit the disclosure to the exact forms disclosed. Many modifications and variations are possible in light of the above teachings. Embodiments have been selected and described in order to best explain the principles of the disclosure and its practical applications, thereby enabling one of ordinary skill in the art to best utilize the disclosure and various embodiments with various modifications as are suited to the particular use contemplated.

[0075] The systems and methods disclosed herein can be implemented or distributed among one or more components, systems, servers, applications, and other sub-components. When implemented as a system, such a system can include and / or require components such as software modules, general-purpose CPUs, RAM, etc., which are found in general-purpose computers. In embodiments where the innovation resides on a server, such a server can include and / or require components such as CPUs, RAM, etc., as are found in general-purpose computers.

[0076] Furthermore, the systems and methods of this specification can be achieved by implementation forms using heterogeneous or completely different software, hardware, and / or firmware components beyond those described above. For example, with respect to such other components (e.g., software, processing components, etc.) and / or computer-readable media related to or embodying the present invention, the innovative aspects of this specification can be implemented without conflict in a number of general-purpose or dedicated computing systems or configurations. Various exemplary computing systems, environments, and / or configurations that can be considered suitable for use with the innovations of this specification include software or other components embodied in or on a personal computer, server, or server computing device, such as routing / connectivity components, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, set-top boxes, consumer electronic devices, network PCs, other existing computer platforms, distributed computing environments including one or more of the above systems or devices, etc., but are not limited thereto.

[0077] In some cases, aspects of the systems and methods can be achieved or implemented, for example, by logic and / or logic instructions including program modules executed in relation to such components or circuitry. Generally, program modules can include routines, programs, objects, components, data structures that perform particular tasks or execute particular instructions herein. The present invention can also be implemented in a distributed software, computer, or circuit configuration where components are connected by a communication bus, circuitry, or link. In a distributed configuration, control / instructions can originate from both local and remote computer storage media including memory storage media.

[0078] The software, circuitry, and components of this specification can also include and / or utilize one or more types of computer-readable media. A computer-readable media can be any available media that can be present on, associated with, or accessed by such circuitry and / or computing components. By way of example and not limitation, computer-readable media can comprise computer storage media and communication media. Computer storage media can include volatile and non-volatile media, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules, or other data. Computer storage media includes RAM, ROM, EEPROM, flash memory, or other memory technology, CD-ROM, digital versatile disks (DVDs), or other optical storage, magnetic tape, magnetic disk storage, or other magnetic storage devices, or any other media that can be used to store the desired information and can be accessed by a computing component, including but not limited to any other media. Communication media can include computer-readable instructions, data structures, program modules, and / or other components. Additionally, communication media can include wired media such as a wired network or a direct-wired connection. However, any such type of media in this specification does not include transient media. Any combination of the foregoing is also included within the scope of computer-readable media.

[0079] In this description, terms, components, modules, devices, etc. can refer to any type of logical or functional software elements, circuits, blocks, and / or processes that can be executed in various ways. For example, the functions of various circuits and / or blocks can be combined with each other to form any other number of modules. Each module can be further executed as a software program stored on a tangible memory (e.g., random access memory, read-only memory, CD-ROM memory, hard disk drive, etc.) that is read by a central processing unit to perform the innovative functions of this specification. Or, a module can include programming instructions that are transmitted to a general-purpose computer or processing / graphics hardware by a communication carrier wave. Similarly, a module can be executed as a hardware logic circuit section that performs the functions encompassed by the innovation of this specification. Finally, a module can be executed using dedicated instructions (SIMD instructions), a field programmable logic array, or any mixture thereof that provides the desired level of performance and cost.

[0080] As discussed herein, features that are not inconsistent with the present disclosure may be implemented by computer hardware, software, and / or firmware. For example, the systems and methods disclosed herein may be embodied in a variety of forms or combinations thereof, including, for example, a data processor such as a computer that also includes a database, digital electronic circuitry, firmware, software. Further, while some of the disclosed embodiments describe specific hardware components, systems and methods that are not inconsistent with the innovations herein may be implemented using any combination of hardware, software, and / or firmware. Additionally, the features described above and other aspects and principles of the innovations herein may be implemented in a variety of environments. Such environments and associated applications may be specifically constructed to perform various routines, processes, and / or operations in accordance with the present invention, or may include a general-purpose computer or computing platform selectively activated or reconfigured by code to provide the required functionality. The processes disclosed herein are not inherently related to any particular computer, network, architecture, environment, or other apparatus, and may be implemented by a suitable combination of hardware, software, and / or firmware. For example, various general-purpose machines may be used with programs written in accordance with the teachings of the present invention, or it may be more convenient in some cases to construct a dedicated device or system to perform the required methods and techniques.

[0081] The methods and system aspects described herein, such as logic, can be similarly executed as functions programmed within any of a variety of circuit components, including programmable logic devices (PLDs) such as field programmable gate arrays (FPGAs), programmable array logic (PAL) devices, electronically programmable logic and memory devices, and standard cell-based devices, as well as application-specific integrated circuits. Some other possibilities for executing the aspects include memory devices, microcontrollers with memory (such as EEPROM), embedded microprocessors, firmware, software, etc. Further, the aspects can be embodied in a microprocessor having software-based circuit emulation, discrete logic (sequential and combinational), custom devices, fuzzy (neural) logic, quantum devices, and hybrids of any of the above device types. The underlying device technology can be provided by a variety of component types, such as metal-oxide semiconductor field-effect transistor (MOSFET) technology like complementary metal-oxide semiconductor (CMOS), bipolar technology like emitter-coupled logic (ECL), polymer technology (e.g., silicon conjugated polymers and metal conjugated polymer-metal structures), analog-digital hybrids, etc.

[0082] It should also be noted that the various logics and / or functions disclosed herein can be enabled using any number of combinations of hardware, firmware, and / or as data and / or instructions embodied in various machine-readable or computer-readable media, with respect to their behavior, register transfers, logic components, and / or other characteristics. Computer-readable media in which such formatted data and / or instructions can be embodied include, but are not limited to, various forms of non-volatile memory media (e.g., optical, magnetic, or semiconductor memory media), but do not include transient media. Unless the context clearly requires otherwise, throughout the description, the words "comprise," "comprising," and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is, in the sense of "including, but not limited to." Words using the singular or plural number also include the plural or singular number, respectively. Further, the words "herein," "hereunder," "above," "below," and words of similar import refer to this application as a whole and not to any particular part of this application. When the word "or" is used in reference to a list of two or more items, that word covers all of the following interpretations of the word: any of the items in the list, all of the items in the list, and any combination of the items in the list.

[0083] While particular presently preferred embodiments of the invention have been specifically described herein, it will become apparent to those skilled in the art to which the invention pertains that various modifications and variations of the various embodiments shown and described herein can be made without departing from the spirit and scope of the invention. Accordingly, it is intended that the invention be limited only to the extent required by the applicable statutes.

[0084] Although the above has been with respect to particular embodiments of the present disclosure, it will be recognized by those skilled in the art that changes to these embodiments may be made without departing from the principles and spirit of the present disclosure, and that the scope of the present disclosure is defined by the appended claims.

Claims

1. A hybrid system for L0 regularization based compressive detection of a source signal, comprising: a quantum machine configured to optimize a first parameter of a source signal, the first parameter including a support vector indicating the location of non-zero elements in the source signal, to minimize a cost function, the support vector being modeled by the quantum machine as a plurality of amplified pump pulses, each of the amplified pump pulses assuming a 0-phase state or a π-phase state; a classical machine configured to optimize a second parameter of a source signal, the second parameter including real values ​​in the source signal, to minimize the cost function; and A hybrid system equipped with

2. The hybrid system of claim 1 , wherein the source signal comprises an N-dimensional source signal.

3. the quantum machine and the classical machine are configured to alternatively perform corresponding optimizations of the quantum machine and the classical machine; the classical machine is configured to maintain the second parameter constant as the quantum machine optimizes the first parameter; 2. The hybrid system of claim 1, wherein the quantum machine is configured to maintain the first parameter constant as the classical machine optimizes the second parameter.

4. The hybrid system of claim 1 , wherein the cost function comprises a Hamiltonian cost function.

5. The hybrid system of claim 1 , wherein the quantum machine is a coherent Ising machine.

6. The hybrid system of claim 1 , wherein the classical machine comprises a digital processor or a field programmable gate array.

7. The hybrid system of claim 1 , wherein the source signal is a magnetic resonance imaging signal.

8. A method for L0 regularization based compressive detection of a source signal, comprising: optimizing, by a quantum machine, first parameters of a source signal including support vectors indicating locations of non-zero elements in the source signal, the support vectors being modeled by the quantum machine as a plurality of amplified pump pulses, each of which assumes a 0-phase state or a π-phase state, to minimize a cost function; optimizing, by a classical machine, second parameters of the source signals, including real values ​​in the source signals, to minimize the cost function; A method comprising:

9. The method of claim 8 , wherein the source signal comprises an N-dimensional source signal.

10. the quantum machine and the classical machine alternatively perform corresponding optimizations of the quantum machine and the classical machine; as the quantum machine optimizes the first parameter, the classical machine holds the second parameter constant; 9. The method of claim 8, wherein the quantum machine holds the first parameter constant as the classical machine optimizes the second parameter.

11. The method of claim 8 , wherein the cost function comprises a Hamiltonian cost function.

12. The method of claim 8 , wherein the quantum machine is a coherent Ising machine.

13. The method of claim 8 , wherein the classical machine comprises a digital processor or a field programmable gate array.

14. The method of claim 8 , wherein the source signal is a magnetic resonance imaging signal.

15. 1. A method for performing L0 regularization based compressive sensing, comprising: injecting a plurality of pump pulses into a coherent Ising machine optical parametric oscillator having an optical parametric oscillator formed in a fiber ring cavity having an output coupler and an input coupler in communication with a homodyne detection output and a second harmonic generation (SHG) crystal; amplifying the plurality of pump pulses such that each of the plurality of pump pulses assumes a 0-phase state or a π-phase state to model a support vector indicating locations of non-zero elements in a source signal; optimizing the support vectors to minimize a cost function.

16. extracting a portion of each pump pulse of the plurality of pump pulses from the fiber ring cavity by the output coupler on the fiber ring cavity; measuring the extracted pulse using an optical homodyne detector; The method of claim 15 further comprising:

17. 17. The method of claim 16, further comprising: computing a feedback signal by a classical digital processor or an optical delay line system, and providing the computation result to an intensity modulator (IM) and a phase modulator (PM) to generate an injection field for each of a plurality of optical parametric oscillator (OPO) pulses through the input coupler on the fiber ring cavity.

18. 20. The method of claim 17, further comprising generating a solution to linear simultaneous equations with a classical machine and transferring the solution to the coherent Ising machine with a buffer.

19. 20. The method of claim 18, further comprising using the solution from the classical digital processor to provide a feedback pulse to the input coupler of the fiber ring cavity.

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