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We propose a method to enhance the performance of Large Language Models (LLMs) by integrating quantum computing and quantum-inspired techniques. Specifically, our approach involves replacing the weight matrices in the Self-Attention and Multi-layer Perceptron layers with a combination of two variational quantum circuits and a quantum-inspired tensor network, such as a Matrix Product Operator (MPO). This substitution enables the reproduction of classical LLM functionality by decomposing weight matrices through the application of tensor network disentanglers and MPOs, leveraging well-established tensor network techniques. By incorporating more complex and deeper quantum circuits, along with increasing the bond dimensions of the MPOs, our method captures additional correlations within the quantum-enhanced LLM, leading to improved accuracy beyond classical models while maintaining low memory overhead., Comment: 4 pages, 2 figures

Convolutional neural networks (CNNs) are one of the most widely used neural network architectures, showcasing state-of-the-art performance in computer vision tasks. Although larger CNNs generally exhibit higher accuracy, their size can be effectively reduced by ``tensorization'' while maintaining accuracy, namely, replacing the convolution kernels with compact decompositions such as Tucker, Canonical Polyadic decompositions, or quantum-inspired decompositions such as matrix product states, and directly training the factors in the decompositions to bias the learning towards low-rank decompositions. But why doesn't tensorization seem to impact the accuracy adversely? We explore this by assessing how \textit{truncating} the convolution kernels of \textit{dense} (untensorized) CNNs impact their accuracy. Specifically, we truncated the kernels of (i) a vanilla four-layer CNN and (ii) ResNet-50 pre-trained for image classification on CIFAR-10 and CIFAR-100 datasets. We found that kernels (especially those inside deeper layers) could often be truncated along several cuts resulting in significant loss in kernel norm but not in classification accuracy. This suggests that such ``correlation compression'' (underlying tensorization) is an intrinsic feature of how information is encoded in dense CNNs. We also found that aggressively truncated models could often recover the pre-truncation accuracy after only a few epochs of re-training, suggesting that compressing the internal correlations of convolution layers does not often transport the model to a worse minimum. Our results can be applied to tensorize and compress CNN models more effectively., Comment: 40 pages, 21 images

Large Language Models (LLMs) such as ChatGPT and LlaMA are advancing rapidly in generative Artificial Intelligence (AI), but their immense size poses significant challenges, such as huge training and inference costs, substantial energy demands, and limitations for on-site deployment. Traditional compression methods such as pruning, distillation, and low-rank approximation focus on reducing the effective number of neurons in the network, while quantization focuses on reducing the numerical precision of individual weights to reduce the model size while keeping the number of neurons fixed. While these compression methods have been relatively successful in practice, there is no compelling reason to believe that truncating the number of neurons is an optimal strategy. In this context, this paper introduces CompactifAI, an innovative LLM compression approach using quantum-inspired Tensor Networks that focuses on the model's correlation space instead, allowing for a more controlled, refined and interpretable model compression. Our method is versatile and can be implemented with - or on top of - other compression techniques. As a benchmark, we demonstrate that a combination of CompactifAI with quantization allows to reduce a 93% the memory size of LlaMA 7B, reducing also 70% the number of parameters, accelerating 50% the training and 25% the inference times of the model, and just with a small accuracy drop of 2% - 3%, going much beyond of what is achievable today by other compression techniques. Our methods also allow to perform a refined layer sensitivity profiling, showing that deeper layers tend to be more suitable for tensor network compression, which is compatible with recent observations on the ineffectiveness of those layers for LLM performance. Our results imply that standard LLMs are, in fact, heavily overparametrized, and do not need to be large at all., Comment: 5 pages, 4 figures, 2 tables, and supplementary information of 2 pages and 1 figure. Revised version with new benchmarks for LlaMA2-7B

We show how quantum-inspired 2d tensor networks can be used to efficiently and accurately simulate the largest quantum processors from IBM, namely Eagle (127 qubits), Osprey (433 qubits) and Condor (1121 qubits). We simulate the dynamics of a complex quantum many-body system -- specifically, the kicked Ising experiment considered recently by IBM in Nature 618, p. 500-505 (2023) -- using graph-based Projected Entangled Pair States (gPEPS), which was proposed by some of us in PRB 99, 195105 (2019). Our results show that simple tensor updates are already sufficient to achieve very large unprecedented accuracy with remarkably low computational resources for this model. Apart from simulating the original experiment for 127 qubits, we also extend our results to 433 and 1121 qubits, and for evolution times around 8 times longer, thus setting a benchmark for the newest IBM quantum machines. We also report accurate simulations for infinitely-many qubits. Our results show that gPEPS are a natural tool to efficiently simulate quantum computers with an underlying lattice-based qubit connectivity, such as all quantum processors based on superconducting qubits., Comment: 7 pages, 8 figures, revised version

We map out the ground state phase diagram of the isotropic Kekul{\'e}-Kitaev model on the honeycomb lattice in the presence of the Heisenberg exchange couplings. Our study relies on large-scale tensor network simulations based on graph-based projected entangled pair state (gPEPS) approach in the thermodynamic limit. We find that on top of the quantum spin liquid (QSL) and conventional magnetically ordered phases which are typical of the Kitaev-Heisenberg model, the Kekul{\'e}-Heisenberg phase diagram, hosts two plaquette valance bond solid (VBS) phases with vanishing magnetic order. While the VBS phases preserve the symmetries of the original Hamiltonian, they differ markedly from the Kitaev spin liquid by having decorated plaquette ordering which is distinguished by a plaquette order parameter.

Recent advances in deep learning have enabled us to address the curse of dimensionality (COD) by solving problems in higher dimensions. A subset of such approaches of addressing the COD has led us to solving high-dimensional PDEs. This has resulted in opening doors to solving a variety of real-world problems ranging from mathematical finance to stochastic control for industrial applications. Although feasible, these deep learning methods are still constrained by training time and memory. Tackling these shortcomings, Tensor Neural Networks (TNN) demonstrate that they can provide significant parameter savings while attaining the same accuracy as compared to the classical Dense Neural Network (DNN). In addition, we also show how TNN can be trained faster than DNN for the same accuracy. Besides TNN, we also introduce Tensor Network Initializer (TNN Init), a weight initialization scheme that leads to faster convergence with smaller variance for an equivalent parameter count as compared to a DNN. We benchmark TNN and TNN Init by applying them to solve the parabolic PDE associated with the Heston model, which is widely used in financial pricing theory., Comment: 11 pages, 8 figures, minor changes. arXiv admin note: substantial text overlap with arXiv:2208.02235

Deep neural networks (NNs) encounter scalability limitations when confronted with a vast array of neurons, thereby constraining their achievable network depth. To address this challenge, we propose an integration of tensor networks (TN) into NN frameworks, combined with a variational DMRG-inspired training technique. This in turn, results in a scalable tensor neural network (TNN) architecture capable of efficient training over a large parameter space. Our variational algorithm utilizes a local gradient-descent technique, enabling manual or automatic computation of tensor gradients, facilitating design of hybrid TNN models with combined dense and tensor layers. Our training algorithm further provides insight on the entanglement structure of the tensorized trainable weights and correlation among the model parameters. We validate the accuracy and efficiency of our method by designing TNN models and providing benchmark results for linear and non-linear regressions, data classification and image recognition on MNIST handwritten digits., Comment: 21 pages, 19 figures

Partial Differential Equations (PDEs) are used to model a variety of dynamical systems in science and engineering. Recent advances in deep learning have enabled us to solve them in a higher dimension by addressing the curse of dimensionality in new ways. However, deep learning methods are constrained by training time and memory. To tackle these shortcomings, we implement Tensor Neural Networks (TNN), a quantum-inspired neural network architecture that leverages Tensor Network ideas to improve upon deep learning approaches. We demonstrate that TNN provide significant parameter savings while attaining the same accuracy as compared to the classical Dense Neural Network (DNN). In addition, we also show how TNN can be trained faster than DNN for the same accuracy. We benchmark TNN by applying them to solve parabolic PDEs, specifically the Black-Scholes-Barenblatt equation, widely used in financial pricing theory, empirically showing the advantages of TNN over DNN. Further examples, such as the Hamilton-Jacobi-Bellman equation, are also discussed., Comment: 14 pages, 11 figures, minimal changes

We combine tensor-network approaches and high-order linked-cluster expansions to investigate the quantum phase diagram of the antiferromagnetic Kitaev's honeycomb model in a magnetic field for general spin values. For the pure Kitaev model, tensor network calculations confirm the absence of fluxes and spin-spin correlations beyond nearest neighbor in the ground state, but signal a breaking of the discrete orientational symmetry for $S\in\{1,3/2,2\}$ inline with the semiclassical limit. An intermediate region between Kitaev phases and the high-field polarized phase is demonstrated for all considered spin values. In this intermediate region the tensor network results display a sequence of potential phases whose number increases with the spin value. Each of these can be characterized by distinct local magnetization patterns while the total magnetization increases smoothly as a function of the field. The analysis of the high-field zero-momentum gap and the associated spectral weight of the polarized phase for general spin $S$ obtained by linked-cluster expansions is consistent with an unconventional quantum critical breakdown of the high-field polarized phase in accordance with the presence of exotic physics at intermediate Kitaev couplings., Comment: 23 pages, 15 figures

We study the 3D Kitaev and Kitaev-Heisenberg models respectively on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph Projected Entangled-Pair States (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high-temperature leads to fractionalization of spins to itinerant Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the non-trivial Majorana band topology and the presence of Weyl nodes, which manifests itself via non-vanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory., Comment: 13 pages, 12 figures

Ultracold atoms in optical lattices are one of the most promising experimental setups to simulate strongly correlated systems. However, efficient numerical algorithms able to benchmark experiments at low-temperatures in interesting 3d lattices are lacking. To this aim, here we introduce an efficient tensor network algorithm to accurately simulate thermal states of local Hamiltonians in any infinite lattice, and in any dimension. We apply the method to simulate thermal bosons in optical lattices. In particular, we study the physics of the (soft-core and hard-core) Bose-Hubbard model on the infinite pyrochlore and cubic lattices with unprecedented accuracy. Our technique is therefore an ideal tool to benchmark realistic and interesting optical-lattice experiments., Comment: 6 pages, 4 figures + Appendix

We study the zero-temperature phase diagram of the spin-$\frac{1}{2}$ Heisenberg model with breathing anisotropy (i.e., with different coupling strength on the upward and downward triangles) on the kagome lattice. Our study relies on large scale tensor network simulations based on infinite projected entangled-pair state and infinite projected entangled-simplex state methods adapted to the kagome lattice. Our energy analysis suggests that the U(1) algebraic quantum spin-liquid (QSL) ground-state of the isotropic Heisenberg model is stable up to very large breathing anisotropy until it breaks down to a critical lattice-nematic phase that breaks rotational symmetry in real space through a first-order quantum phase transition. Our results also provide further insight into the recent experiment on vanadium oxyfluoride compounds which has been shown to be relevant platforms for realizing QSL in the presence of breathing anisotropy., Comment: 15 pages, 11 figures

We construct a short-range resonating valence-bond state (RVB) on the ruby lattice, using projected entangled-pair states (PEPS) with bond dimension $D=3$. By introducing non-local moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining non-zero topological contribution, $\gamma=-\rm{ln}\ 2$, consistent with that of a $\mathbb{Z}_2$ topological quantum spin liquid. We also calculate the ground-state energy of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum-dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function., Comment: 10 pages, 10 figures

We develop a strategy for tensor network algorithms that allows to deal very efficiently with lattices of high connectivity. The basic idea is to fine-grain the physical degrees of freedom, i.e., decompose them into more fundamental units which, after a suitable coarse-graining, provide the original ones. Thanks to this procedure, the original lattice with high connectivity is transformed by an isometry into a simpler structure, which is easier to simulate via usual tensor network methods. In particular this enables the use of standard schemes to contract infinite 2d tensor networks - such as Corner Transfer Matrix Renormalization schemes - which are more involved on complex lattice structures. We prove the validity of our approach by numerically computing the ground-state properties of the ferromagnetic spin-1 transverse-field Ising model on the 2d triangular and 3d stacked triangular lattice, as well as of the hard-core and soft-core Bose-Hubbard models on the triangular lattice. Our results are benchmarked against those obtained with other techniques, such as perturbative continuous unitary transformations and graph projected entangled pair states, showing excellent agreement and also improved performance in several regimes., Comment: 5 pages, 6 figures. Supplementary material with 8 pages, 6 figures. Revised version

We present a general graph-based Projected Entangled-Pair State (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural-matrix which codifies the details of tensor networks on any graphs in any dimension $d$, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., $D=100$), and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the $3$-state quantum Potts model in field on the kagome lattice and the spin-$1$ bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the $2d$ quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies., Comment: 12 pages, 18 figures, and Appendix with 6 pages

Using the infinite Projected Entangled Pair States (iPEPS) algorithm, we study the ground-state properties of the spin-$1/2$ quantum Heisenberg antiferromagnet on the star lattice in the thermodynamic limit. By analyzing the ground-state energy of the two inequivalent bonds of the lattice in different unit-cell structures, we identify two competing Valence-Bond-Solid (VBS) phases for different antiferromagnetic Heisenberg exchange couplings. More precisely, we observe (i) a VBS state which respects the full symmetries of the Hamiltonian, and (ii) a resonating VBS state which, in contrast to previous predictions, has a six-site unit-cell order and breaks $C_3$ symmetry. We also studied the ground-state phase diagram by measuring the ground-state fidelity and energy derivatives, and further confirmed the continuous nature of the quantum phase transition in the system. Moreover, an analysis of the isotropic point shows that its ground state is also a VBS as in (i), which is as well in contrast with previous predictions., Comment: 9 pages, 11 figures

The infinite Projected Entangled-Pair State (iPEPS) algorithm is one of the most efficient techniques for studying the ground-state properties of two-dimensional quantum lattice Hamiltonians in the thermodynamic limit. Here, we show how the algorithm can be adapted to explore nearest-neighbor local Hamiltonians on the ruby and triangle-honeycomb lattices, using the Corner Transfer Matrix (CTM) renormalization group for 2D tensor network contraction. Additionally, we show how the CTM method can be used to calculate the ground state fidelity per lattice site and the boundary density operator and entanglement entropy (EE) on an infinite cylinder. As a benchmark, we apply the iPEPS method to the ruby model with anisotropic interactions and explore the ground-state properties of the system. We further extract the phase diagram of the model in different regimes of the couplings by measuring two-point correlators, ground state fidelity and EE on an infinite cylinder. Our phase diagram is in agreement with previous studies of the model by exact diagonalization., Comment: 12 pages, 18 figure

The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge symmetry. We mapped out the phase digram of the model and found gapped and gapless spin-liquid phases. While the low energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in a small magnetic field. We argue that the latter phases could be possibly described by exotic excitations, whose their spectrum is richer than the Ising phase of the Kitaev model.

We use the topological entanglement entropy (TEE) as an efficient tool to fully characterize the Abelian phase of a $\mathbb{Z}_2 \times \mathbb{Z}_2$ spin liquid emerging as the ground state of topological color code (TCC), which is a class of stabilizer states on the honeycomb lattice. We provide the fusion rules of the quasiparticle (QP) excitations of the model by introducing single- or two-body operators on physical spins for each fusion process which justify the corresponding fusion outcome. Beside, we extract the TEE from Renyi entanglement entropy (EE) of the TCC, analytically and numerically by finite size exact diagonalization on the disk shape regions with contractible boundaries. We obtain that the EE has a local contribution, which scales linearly with the boundary length in addition to a topological term, i.e. the TEE, arising from the condensation of closed strings in the ground state. We further investigate the ground state dependence of the TEE on regions with non-contractible boundaries, i.e. by cutting the torus to half cylinders, from which we further identify multiple independent minimum entropy states (MES) of the TCC and then extract the U and S modular matrices of the system, which contain the self and mutual statistics of the anyonic QPs and fully characterize the topological phase of the TCC. Eventually, we show that, in spite of the lack of a local order parameter, TEE and other physical quantities obtained from ground state wave function such as entanglement spectrum (ES) and ground state fidelity are sensitive probes to study the robustness of a topological phase. We find that the topological order in the presence of a magnetic field persists until the vicinity of the transition point, where the TEE and fidelity drops to zero and the ES splits severely, signaling breakdown of the topological phase of the TCC.

The stability of the topological order phase induced by the $Z_3$ Kitaev model, which is a candidate for fault-tolerant quantum computation, against the local order phase induced by the 3-State Potts model is studied. We show that the low energy sector of the Kitaev-Potts model is mapped to the Potts model in the presence of transverse magnetic field. Our study relies on two high-order series expansion based on continuous unitary transformations in the limits of small- and large-Potts couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the $Z_3$ Kitaev model breaks down to the Potts model through a first order phase transition. We capture the phase transition by analysis of the ground state energy, one-quasiparticle gap and geometric measure of entanglement., Comment: 12 pages, 10 figures, Accepted for publication in Physical Review B

We investigate the low-energy properties as well as quantum and thermal phase transitions of the Baxter-Wu model in a transverse magnetic field. Our study relies on stochastic series expansion quantum Monte Carlo and on series expansions about the low- and high-field limits at zero temperature using the quantum finite-lattice method on the triangular lattice. The phase boundary consists of a second-order critical line in the 4-state Potts model universality class starting from the pure Baxter-Wu limit meeting a first-order line connected to the zero-temperature transition point ($h\approx2.4$, $T=0$). Both lines merge at a tricritical point approximatively located at ($h\approx 2.3J$, $T\approx J$)., Comment: 13 pages, 17 figures

We investigate the low-energy spectral properties and robustness of the topological phase of color code, which is a quantum spin model for the aim of fault-tolerant quantum computation, in the presence of a uniform magnetic field or Ising interactions, using high-order series expansion and exact diagonalization. In a uniform magnetic field, we find 1st-order phase transitions in all field directions. In contrast, our results for the Ising interactions unveil that for strong enough Ising couplings, the Z2 x Z2 topological phase of color code breaks down to symmetry broken phases by 1st- or 2nd-order phase transitions., Comment: 10 pages, 11 figures

The robustness of the topological color code, which is a class of error correcting quantum codes, is investigated under the influence of an uniform magnetic field on the honeycomb lattice. Our study relies on two high-order series expansions using perturbative continuous unitary transformations in the limit of low and high fields, exact diagonalization and a classical approximation. We show that the topological color code in a single parallel field is isospectral to the Baxter-Wu model in a transverse field on the triangular lattice. It is found that the topological phase is stable up to a critical field beyond which it breaks down to the polarized phase by a first-order phase transition. The results also suggest that the topological color code is more robust than the toric code, in the parallel magnetic field., Comment: 11 pages, 8 figures

The growth of a rough and porous thin surface by deposition of randomly shaped clusters with different sizes over an initially flat linear substrate is simulated, using Monte Carlo technique. Unlike the ordinary Random Deposition, our approach results in aggregation of clusters which produces a porous bulk with correlation along the surface and the surface saturation occurs in long enough deposition times. The scaling exponents; the growth, roughness, and dynamic exponents are calculated based on the time scale. Moreover, the porosity and its dependency to the time and clusters size are also calculated. We also study the influence of clusters size on the scaling exponent, as well as on the global porosity.

In the past decade, several neutron reflectometry methods for determining the modulus and phase of the complex reflection coefficient of an unknown multilayer thin film have been worked out among which the method of variation of surroundings and reference layers are of highest interest. These methods were later modified for measurement of the polarization of the reflected beam instead of the measurement of the intensities. In their new architecture, these methods not only suffered from the necessity of change of experimental setup, but also another difficulty was added to their experimental implementations. This deficiency was related to the limitations of the technology of the neutron reflectometers that could only measure the polarization of the reflected neutrons in the same direction as the polarization of the incident beam. As the instruments are limited, the theory has to be optimized so that the experiment could be performed. In a recent work, we developed the method of variation of surroundings for one directional polarization analysis. In this new work, the method of reference layer with polarization analysis has been optimized to determine the phase and modulus of the unknown film with measurement of the polarization of the reflected neutrons in the same direction as the polarization of the incident beam.

Polarized neutron reflectometry (PNR) have developed theoretically and experimentally in the past decades. In order to resolve the phase problem in neutron reflectometry, several simulation methods have been proposed such as reference layer and the variation of surroundings. By considering some factors such as smoothness or roughness in simulations, it is tried to gain more compatible results with experiment. In this paper, by using four different functions "Linear, Airy, Eckart and Error function", we investigate the effects of the smoothness of the interfacial potential on reflectivity, polarization of reflected neutrons and determination of the SLD of the sample, using reference layer method with polarized neutrons. We have also proposed a solution in order to gain better results with less noise in output reflectivity data, at the presence of smoothness.

By knowing the phase and modules of the reflection coefficient in neutron reflectometry problems, a unique result for the scattering length density (SLD) of a thin film can be determined which will lead to the exact determination of type and thickness of the film. In the past decade, several methods have been worked out to resolve the phase problem such as dwell time method, reference layer method and variation of surroundings, among which the reference method and variation of surroundings by using a magnetic substrate and polarized neutrons is of the most applicability. All of these methods are based on the solution of Schrodinger equation for a discontinuous and step-like potential at each interface. As in real sample there are some smearing and roughness at boundaries, consideration of smoothness and roughness of interfaces would affect the final output result. In this paper, we have investigated the effects of smoothness of interfaces on determination of the phase of reflection as well as the retrieval process of the SLD, by using a smooth varying function (Eckart potential). The effects of roughness of interfaces on the same parameters, have also been investigated by random variation of the interface around it mean position.

Recently it has been shown that the modules and phase of complex reflection coefficient can be determined by using a magnetic substrate and polarized neutrons. Several other methods have also been worked out based on measurement of polarizations of reflected neutrons from magnetic reference layers and magnetic substrate. However, due to the fact that available reflectometers are limited in the choice of polarization of reflected beam in the same direction as the polarization of the incident beam, neither of the methods which are based on polarization analysis, has been proven to be experimentally practical. In this paper, we have proposed a new method for determining the phase of reflection coefficient which is based on two measurements of polarization which correspond to two magnetic fields with the same magnitudes and different orientations. The polarization analysis is performed in the same direction as the polarization of the incident beam and is well suited for available reflectometers. The problems envisaged in implementation of the method are also discussed., Comment: 5 pages, 6 figures