physics

Neural network quantum states

Neural network quantum states is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Neural network quantum states rather than just read about it. In short: Neural Network Quantum States (NQS or NNQS) is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body quantum systems.

Key takeaways

Neural network quantum states belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Neural network quantum states to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Neural network quantum states from memory before moving on to harder problems.

Reference excerpt

Neural Network Quantum States (NQS or NNQS) is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body quantum systems. Given a many-body quantum state | Ψ ⟩ {\displaystyle |\Psi \rangle } comprising N {\displaystyle N} degrees of freedom and a choice of associated quantum numbers s 1 … s N {\displaystyle s_{1}\ldots s_{N}} , then an NQS parameterizes the wave-function amplitudes

⟨ s 1 … s N | Ψ ; W ⟩ = F ( s 1 … s N ; W ) , {\displaystyle \langle s_{1}\ldots s_{N}|\Psi ;W\rangle =F(s_{1}\ldots s_{N};W),}

where F ( s 1 … s N ; W ) {\displaystyle F(s_{1}\ldots s_{N};W)} is an artificial neural network of parameters (weights) W {\displaystyle W} , N {\displaystyle N} input variables ( s 1 … s N {\displaystyle s_{1}\ldots s_{N}} ) and one complex-valued output corresponding to the wave-function amplitude. This variational form is used in conjunction with specific stochastic learning approaches to approximate quantum states of interest.

Learning the Ground-State Wave Function One common application of NQS is to find an approximate representation of the ground state wave function of a given Hamiltonian H ^ {\displaystyle {\hat {H}}} . The learning procedure in this case consists in finding the best neural-network weights that minimize the variational energy

E ( W ) = ⟨ Ψ ; W | H ^ | Ψ ; W ⟩ . {\displaystyle E(W)=\langle \Psi ;W|{\hat {H}}|\Psi ;W\rangle .}

Since, for a general artificial neural network, computing the expectation value is an exponentially costly operation in N {\displaystyle N} , stochastic techniques based, for example, on the Monte Carlo method are used to estimate E ( W ) {\displaystyle E(W)} , analogously to what is done in Variational Monte Carlo, see for example for a review. More specifically, a set of M {\displaystyle M} samples S ( 1 ) , S ( 2 ) … S ( M ) {\displaystyle S^{(1)},S^{(2)}\ldots S^{(M)}} , with S ( i ) = s 1 ( i ) … s N ( i ) {\displaystyle S^{(i)}=s_{1}^{(i)}\ldots s_{N}^{(i)}} , is generated such that they are uniformly distributed according to the Born probability density P ( S ) ∝ | F ( s 1 … s N ; W ) | 2 {\displaystyle P(S)\propto |F(s_{1}\ldots s_{N};W)|^{2}} . Then it can be shown that the sample mean of the so-called "local energy" E l o c ( S ) = ⟨ S | H ^ | Ψ ⟩ / ⟨ S | Ψ ⟩ {\displaystyle E_{\mathrm {loc} }(S)=\langle S|{\hat {H}}|\Psi \rangle /\langle S|\Psi \rangle } is a statistical estimate of the quantum expectation value E ( W ) {\displaystyle E(W)} , i.e.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Neural network quantum states

Start with the simplest possible case. Write down what Neural network quantum states claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Neural network quantum states before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Neural network quantum states ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Neural network quantum states

In research: Neural network quantum states appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Neural network quantum states in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Neural network quantum states is common in secondary-school and first-year university syllabi. It links to neighbouring topics Machine learning, Quantum Monte Carlo, Quantum states, so understanding it makes those chapters shorter.
In everyday life: Look for Neural network quantum states outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Neural network quantum states in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Neural network quantum states means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Neural network quantum states out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Neural network quantum states in simple terms?

Why does Neural network quantum states matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Neural network quantum states?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Neural network quantum states.