Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [hamiltonian-simulation]

Hamiltonian simulation is a class of algorithms that, given a Hermitian matrix A, output a quantum circuit implementing an approximation to the unitary exp[iAt].

309 questions
29
votes
2 answers

Circuit construction for Hamiltonian simulation

I would like to know how to design a quantum circuit that given a Hermitian matrix $\hat{H}$ and time $t$, maps $|\psi\rangle$ to $e^{\frac{i\hat{H}t}{\hbar}} |\psi\rangle$ with $\hbar =1$. Thank you for your answer.
Gradius
  • 421
  • 4
  • 5
27
votes
1 answer

What are examples of Hamiltonian simulation problems that are BQP-complete?

Many papers assert that Hamiltonian simulation is BQP-complete (e.g., Hamiltonian simulation with nearly optimal dependence on all parameters and Hamiltonian Simulation by Qubitization). It is easy to see that Hamiltonian simulation is BQP-hard…
groupsgroupsgroups
  • 389
  • 2
  • 6
18
votes
1 answer

Obtaining gate $e^{-i\Delta t Z}$ from elementary gates

I am currently reading "Quantum Computation and Quantum Information" by Nielsen and Chuang. In the section about Quantum Simulation, they give an illustrative example (section 4.7.3), which I don't quite understand: Suppose we have the Hamiltonian…
brzepkowski
  • 1,069
  • 7
  • 19
15
votes
3 answers

Simulate hamiltonian evolution

I'm trying to figure out how to simulate the evolution of qubits under the interaction of Hamiltonians with terms written as a tensor product of Pauli matrices in a quantum computer. I have found the following trick in Nielsen and Chuang's book…
Apo
  • 575
  • 2
  • 10
13
votes
1 answer

Where does precisely the difficulty in exponentiating a Hamiltonian $H$ in the quantum simulation problem lay?

I've read in the Nielsen's, Chuang's "Quantum Computation and Quantum Information": Classical simulation begins with the realization that in solving a simple differential equation such as $dy/dt = f(y)$, to first order, it is known that $y(t +…
brzepkowski
  • 1,069
  • 7
  • 19
13
votes
2 answers

Number of Qubits Required for Simulation of Caffeine and Penicillin Molecules

I recently read this report from BCG, which stated: For scientists trying to design a compound that will attach itself to, and modify, a target disease pathway, the critical first step is to determine the electronic structure of the molecule.…
Greenstick
  • 1,086
  • 8
  • 23
13
votes
1 answer

Advantage of simulating sparse Hamiltonians

In @DaftWullie's answer to this question he showed how to represent in terms of quantum gates the matrix used as example in this article. However, I believe it to be unlikely to have such well structured matrices in real life examples, therefore I…
FSic
  • 889
  • 6
  • 18
13
votes
2 answers

Hamiltonian simulation with complex coefficients

As part of a variational algorithm, I would like to construct a quantum circuit (ideally with pyQuil) that simulates a Hamiltonian of the form: $H = 0.3 \cdot Z_3Z_4 + 0.12\cdot Z_1Z_3 + [...] + - 11.03 \cdot Z_3 - 10.92 \cdot Z_4 + \mathbf{0.12i…
Mark Fingerhuth
  • 1,125
  • 7
  • 13
13
votes
1 answer

How to implement a matrix exponential in a quantum circuit?

Maybe it is a naive question, but I cannot figure out how to actually exponentiate a matrix in a quantum circuit. Assuming to have a generic square matrix A, if I want to obtain its exponential, $e^{A}$, i can use the series $$e^{A} \simeq I+…
FSic
  • 889
  • 6
  • 18
13
votes
1 answer

How can I decompose a matrix in terms of Pauli matrices?

I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices. I would prefer an option to do this in larger than 2 dimensions, if that is possible.
yishairasowsky
  • 241
  • 2
  • 4
12
votes
2 answers

How are elementary quantum gates realised?

When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions. In some sense, these are rather mysterious objects, in that they perform "magic" discrete operations on the states. They are…
glS
  • 27,510
  • 7
  • 37
  • 125
11
votes
1 answer

Why was Feynman hesitant about simulating fermions with a quantum computer?

Richard Feynman has a number of foundational publications from the early-mid 80's on quantum computing that I continue to read with awe and inspiration. As earlier discussed, the 1985 article "Quantum Mechanical Computers" is concerned with…
Mark Spinelli
  • 15,378
  • 3
  • 26
  • 83
11
votes
3 answers

Why are diagonal Hamiltonians considered classical?

I've been following UT QML course (http://localhost:8888/tree/UNI/PHD/UT-QML) and during their lecture on the Ising hamiltonian, they point out that the hamiltonian of an Ising model without a transverse field commutes $$ H=-\sum_{} J_{ij}…
César Leonardo Clemente López
  • 801
  • 4
  • 12
10
votes
1 answer

Ground state energy estimation - VQE vs. Ising vs. Trotter–Suzuki

Disclaimer: I am a software engineer who is curious about quantum computing. Although I understand some basic concepts, theory and math behind it, I am by no means experienced in this domain. I am doing some preliminary research on the state of…
Anurag Bhandari
  • 203
  • 1
  • 6
10
votes
2 answers

Is there a Hamiltonian simulation technique implemented somewhere?

I was wondering if there was some code available for Hamiltonian simulation for sparse matrix. And also if they exist, they correspond to a divide and conquer approach or a Quantum walk approach?
cnada
  • 4,802
  • 1
  • 9
  • 22
1
2 3
…
20 21