Hamiltonian simulation

Key takeaways

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

Reference excerpt

Hamiltonian simulation (also referred to as quantum simulation) is a problem in quantum information science that attempts to find the computational complexity and quantum algorithms needed for simulating quantum systems. Hamiltonian simulation is a problem that demands algorithms which implement the evolution of a quantum state efficiently. The Hamiltonian simulation problem was proposed by Richard Feynman in 1982, where he proposed a quantum computer as a possible solution since the simulation of general Hamiltonians seem to grow exponentially with respect to the system size.

Problem statement In the Hamiltonian simulation problem, given a Hamiltonian H {\displaystyle H} ( 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} hermitian matrix acting on n {\displaystyle n} qubits), a time t {\displaystyle t} and maximum simulation error ϵ {\displaystyle \epsilon } , the goal is to find an algorithm that approximates U {\displaystyle U} such that | | U − e − i H t | | ≤ ϵ {\displaystyle ||U-e^{-iHt}||\leq \epsilon } , where e − i H t {\displaystyle e^{-iHt}} is the ideal evolution and | | ⋅ | | {\displaystyle ||\cdot ||} is the spectral norm. A special case of the Hamiltonian simulation problem is the local Hamiltonian simulation problem. This is when H {\displaystyle H} is a k-local Hamiltonian on n {\displaystyle n} qubits where H = ∑ j = ⁡ 1 m H j {\displaystyle H=\sum _{j\mathop {=} 1}^{m}H_{j}} and H j {\displaystyle H_{j}} acts non-trivially on at most k {\displaystyle k} qubits instead of n {\displaystyle n} qubits. The local Hamiltonian simulation problem is important because most Hamiltonians that occur in nature are k-local.

Techniques

Product formulas

Also known as Trotter formulas or Trotter–Suzuki decompositions, Product formulas simulate the sum-of-terms of a Hamiltonian by simulating each one separately for a small time slice. If H = A + B + C {\displaystyle H=A+B+C} , then U = e − i ( A + B + C ) t {\displaystyle U=e^{-i(A+B+C)t}} is well-approximated by ( e − i A t / r e − i B t / r e − i C t / r ) r {\displaystyle (e^{-iAt/r}e^{-iBt/r}e^{-iCt/r})^{r}} for a large r {\displaystyle r} ; where r {\displaystyle r} is the number of time steps to simulate for. The larger the r {\displaystyle r} , the more accurate the simulation. If the Hamiltonian is represented as a Sparse matrix, the distributed edge coloring algorithm can be used to decompose it into a sum of terms; which can then be simulated by a Trotter–Suzuki algorithm.

Taylor series

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hamiltonian simulation

Start with the simplest possible case. Write down what Hamiltonian simulation 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 Hamiltonian simulation 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 Hamiltonian simulation 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 Hamiltonian simulation

In research: Hamiltonian simulation 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 Hamiltonian simulation 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: Hamiltonian simulation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum information science, so understanding it makes those chapters shorter.
In everyday life: Look for Hamiltonian simulation 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.

How to study Hamiltonian simulation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hamiltonian simulation 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 Hamiltonian simulation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hamiltonian simulation in simple terms?

Hamiltonian simulation (also referred to as quantum simulation) is a problem in quantum information science that attempts to find the computational complexity and quantum algorithms needed for simulating quantum systems. Hamiltonian simulation is a problem that demands algorithms which implement th…

Why does Hamiltonian simulation 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 Hamiltonian simulation?

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 Hamiltonian simulation.