Abstract:Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements. As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial. We empirically study the scaling of restricted Boltzmann machines (RBMs) applied to reconstruct ground-state wavefunctions of the one-dimensional transverse-field Isi… Show more
“…The question of efficiency of classification in this setting thus simplifies to a question of learnability of a target state via the NNS ansatz. Recent studies into the learnability scaling of positive, real valued, pure quantum states via NNS (in the context of tomographic reconstruction of ground states) provide a systematic evaluation of the scaling of computational resources in the RBM setting [15]. Whilst our learning procedure is contextually simpler (as the NNS are trained on complete phase/amplitude knowledge, not projective measurements), these scaling principles are still appropriate, and affirm NNS state reconstruction techniques as highly efficient and powerful models.…”
Section: Efficiency Of Entanglement Classification Via Nnsmentioning
confidence: 99%
“…Hence, complex-valued target states do not pose a great threat to classification efficiency. Classifications of up to nine-qubit systems can be performed on a standard laptop, and much larger systems are readily investigated with greater computational resources, as seen in [10,15].…”
Section: Efficiency Of Entanglement Classification Via Nnsmentioning
The task of classifying the entanglement properties of a multipartite quantum state poses a remarkable challenge due to the exponentially increasing number of ways in which quantum systems can share quantum correlations. Tackling such challenge requires a combination of sophisticated theoretical and computational techniques. In this paper we combine machinelearning tools and the theory of quantum entanglement to perform entanglement classification for multipartite qubit systems in pure states. We use a parameterisation of quantum systems using artificial neural networks in a restricted Boltzmann machine architecture, known as Neural Network Quantum States, whose entanglement properties can be deduced via a constrained, reinforcement learning procedure. In this way, Separable Neural Network States can be used to build entanglement witnesses for any target state.
OPEN ACCESS RECEIVED
“…The question of efficiency of classification in this setting thus simplifies to a question of learnability of a target state via the NNS ansatz. Recent studies into the learnability scaling of positive, real valued, pure quantum states via NNS (in the context of tomographic reconstruction of ground states) provide a systematic evaluation of the scaling of computational resources in the RBM setting [15]. Whilst our learning procedure is contextually simpler (as the NNS are trained on complete phase/amplitude knowledge, not projective measurements), these scaling principles are still appropriate, and affirm NNS state reconstruction techniques as highly efficient and powerful models.…”
Section: Efficiency Of Entanglement Classification Via Nnsmentioning
confidence: 99%
“…Hence, complex-valued target states do not pose a great threat to classification efficiency. Classifications of up to nine-qubit systems can be performed on a standard laptop, and much larger systems are readily investigated with greater computational resources, as seen in [10,15].…”
Section: Efficiency Of Entanglement Classification Via Nnsmentioning
The task of classifying the entanglement properties of a multipartite quantum state poses a remarkable challenge due to the exponentially increasing number of ways in which quantum systems can share quantum correlations. Tackling such challenge requires a combination of sophisticated theoretical and computational techniques. In this paper we combine machinelearning tools and the theory of quantum entanglement to perform entanglement classification for multipartite qubit systems in pure states. We use a parameterisation of quantum systems using artificial neural networks in a restricted Boltzmann machine architecture, known as Neural Network Quantum States, whose entanglement properties can be deduced via a constrained, reinforcement learning procedure. In this way, Separable Neural Network States can be used to build entanglement witnesses for any target state.
OPEN ACCESS RECEIVED
“…An optimal set of weights and hidden layer biases are found through optimization of a Boltzmann distribution of energies given a set of input data applied to the visible layer of the RBM. Training such networks, via optimization and sampling, requires large computational resources as the size of the network grows 2 . Figure 1 Illustration of a Restricted Boltzmann Machine (RBM) bipartite graph where are visible nodes, are hidden nodes and are the weights connecting the hidden and visible nodes.…”
Restricted Boltzmann Machines (RBMs) have been proposed for developing neural networks for a variety of unsupervised machine learning applications such as image recognition, drug discovery, and materials design. The Boltzmann probability distribution is used as a model to identify network parameters by optimizing the likelihood of predicting an output given hidden states trained on available data. Training such networks often requires sampling over a large probability space that must be approximated during gradient based optimization. Quantum annealing has been proposed as a means to search this space more efficiently which has been experimentally investigated on D-Wave hardware. D-Wave implementation requires selection of an effective inverse temperature or hyperparameter ($$\beta $$
β
) within the Boltzmann distribution which can strongly influence optimization. Here, we show how this parameter can be estimated as a hyperparameter applied to D-Wave hardware during neural network training by maximizing the likelihood or minimizing the Shannon entropy. We find both methods improve training RBMs based upon D-Wave hardware experimental validation on an image recognition problem. Neural network image reconstruction errors are evaluated using Bayesian uncertainty analysis which illustrate more than an order magnitude lower image reconstruction error using the maximum likelihood over manually optimizing the hyperparameter. The maximum likelihood method is also shown to out-perform minimizing the Shannon entropy for image reconstruction.
“…ollowing fascinating success in image and speech recognition tasks, machine-learning (ML) methods have recently been shown to be useful in physical sciences. For example, ML has been used to classify phases of matter 1 , to enhance quantum state tomography 2,3 , to bypass expensive dynamic ab initio calculations 4 , and more 5 . Currently, artificial neural networks (NNs) are being explored as variational approximations for many-body quantum systems in the context of variational Monte Carlo (vMC) approach.…”
mentioning
confidence: 99%
“…While it can be proven mathematically that NNs can in principle approximate any smooth function to arbitrary accuracy 31 , it might require an impractically large number of parameters. Thus an important feature of any ansatz is its expressibility-a potential capacity to represent a many-body wave function with high accuracy using a moderate number of parameters [32][33][34] , and so far, significant effort has been put into the search for NQS architectures that possess this property 3,35 . At the same time, there is another issue that is not widely discussed in this context -the generalization properties of an ansatz.…”
Neural quantum states (NQS) attract a lot of attention due to their potential to serve as a very expressive variational ansatz for quantum many-body systems. Here we study the main factors governing the applicability of NQS to frustrated magnets by training neural networks to approximate ground states of several moderately-sized Hamiltonians using the corresponding wave function structure on a small subset of the Hilbert space basis as training dataset. We notice that generalization quality, i.e. the ability to learn from a limited number of samples and correctly approximate the target state on the rest of the space, drops abruptly when frustration is increased. We also show that learning the sign structure is considerably more difficult than learning amplitudes. Finally, we conclude that the main issue to be addressed at this stage, in order to use the method of NQS for simulating realistic models, is that of generalization rather than expressibility.
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