All Questions
Tagged with hamiltonian-simulation circuit-construction
29 questions
3
votes
1
answer
277
views
How to simulate low-rank hamiltonian?
I want to implement a unitary $U\,,$
$$U=\text{exp}(-it|u\rangle\langle u|)\,,$$ where $|u\rangle$ is a known state.
Are there any methods to do this efficiently?
- 115
1
vote
1
answer
82
views
Exponentiating a tensor product of operators acting on disjoint qubit registers
Consider a problem of implementing $\operatorname{e}^{i\bigotimes_j O_j}$, where all the $O_j$ terms act on disjoint sets of qubits.
Assume that efficient circuits implementing individual $\...
- 2,221
2
votes
0
answers
85
views
Efficient gate executing the time evolution of a Hamiltonian using Runge-Kutta method
You can find a minimal working example below.
In particular, I want to replace the scipy.linalg.expm() matrix exponential by a Runge Kutta time evolution method as ...
0
votes
1
answer
110
views
How to find explicit gate decomposition of a circuit implementing a unitary using HamiltonianGate()?
I'm new to Qiskit.
I am trying to construct a gate from HamiltonianGate(), available on Qiskit. The Hamiltonian in question is:
$$H = - \pi\delta(Z_1 - Z_2)
+ 2\pi J ~ \mathbf{I}_1 \cdot \...
1
vote
1
answer
89
views
Excplicit Description of Hamiltonians?
The wikipedia article for Hamiltonian simulation lists two complexities: gate and query complexity.
These two types of complexity refer to two different things; gate complexity is the asymptotic ...
- 279
5
votes
1
answer
343
views
Is it possible to implement any random Hamiltonian using quantum circuit
Are there any restrictions on implementing the evolution of any random Hamiltonian? Suppose I want to implement Rabi oscillations using a quantum circuit, I initialize the state and the Hamiltonian ...
- 179
0
votes
1
answer
199
views
How is the P function applied in QSVT for the case of Hamiltonian simulation if it only modifies singular values?
I am watching Andras Gilyen's talk on QSVT here.
On one slide he mentions the core of QSVT:
Given $U$--- a block encoding of matrix $A$ that has singular values $\lambda$, left singular vectors, $\...
- 433
0
votes
1
answer
194
views
How to choose values of phi for Hamiltonian simulation with Quantum Singular Value Transform?
I am reading the review, Grand Unification of Quantum Algorithms, which covers the area known as "Quantum Singular Value Transform (QSVT)."
I am really trying to understand it behind the ...
- 433
4
votes
0
answers
64
views
Efficient quantum algorithms to decompose Hessian matrices into sums of unitaries
Are there efficient quantum algorithms that given a d-sparse hessian $H \in \mathbb{C}^{N \times N}$ decompose it into a sum of unitaries (e.g. Pauli matrices)?
$$H = \sum_i^q a_i U_i$$
If an ...
- 361
6
votes
1
answer
474
views
Is it essential to apply Quantum Singular Value transformation twice for Hamiltonian simulation?
I have been reading the paper A Grand Unification of Quantum Algorithms and I need clarification on the Hamiltonian simulation algorithm provided in the paper on page 23. . In procedure part point 2 ...
- 419
0
votes
1
answer
564
views
How to do rotations along arbitrary multi-qubit basis
I was trying to implement Trotterization for a $k$-local Hamiltonian simulation using qiskit. For this, say I want to apply $e^{\lambda \sigma^1_z \otimes \sigma^2_z \otimes \sigma^3_z}$ (this being ...
- 391
3
votes
1
answer
1k
views
Simulating the Ising-like model as a quantum circuit
We are interested in simulating the 1d Ising model Hamiltonian using a Quantum Circuit (QC). A similar question was posted before with no answers. Here we will assume, for simplicity, 3 lattice sites ...
- 645
6
votes
0
answers
297
views
Could the Hamiltonian of a 2x2 Rubik's Cube be simulated with a NISQ device?
Consider the four cells on each of the six faces of the 2x2x2 Rubik's cube (the pocket cube). We can construct and simulate a quarter-turn Hamiltonian as below. $^*$
Let $\langle F_1,U_1,R_1\rangle$ ...
- 14.4k
2
votes
0
answers
148
views
Eigenvalues and energy levels of 1D Heisenberg model using real Quantum Computers?
The 1D Quantum Heisenberg model is
$$H_\textrm{Heisenberg} = -~J \sum_{\langle i\ j\rangle} \hat{S}_{i} \cdot \hat{S}_{j}$$
where each spin is an operator.
For simple cases, for example, a system with ...
- 375
6
votes
2
answers
277
views
Exponentiating Pauli matrices using trapped ion native gates (single-qubit rotations + XX, YY, ZZ)
I'm wondering what are the known/good/standard ways of exponentiating Pauli terms (i.e. constructing circuits, which implement $\exp(i\alpha XIIZYI...)$) using gates supported by trapped ion quantum ...
- 2,221
5
votes
0
answers
144
views
Is there a systematic way how to generate the Hamiltonian from a given circuit?
If I have a designed circuit to solve a particular problem. Is there a systematic way how to generate the Hamiltonian from it?
6
votes
1
answer
2k
views
How can I simulate Hamiltonians composed of Pauli matrices?
Suppose I want to perform the time-evolution simulation on the following Hamiltonians:
$$
H_{1} = X_1+ Y_2 + Z_1\otimes Z_2
\\
H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2
$$
Where $X,Y,Z$ are Pauli ...
- 2,408
2
votes
1
answer
146
views
Is there a tool to get the quantum circuit corresponding to a sparse matrix?
If I know a sparse matrix, is there any tool that allows me to get the corresponding quantum circuit directly?
If not what should I do?
For example，I want to try hamilton simulation and I have the ...
- 35
5
votes
2
answers
612
views
Representation of rotation operators $e^{-i\theta(I-Z_1\otimes Z_2 \otimes Z_3)}$ about arbitrary axis for $3$ qubits
I was wondering in how to interpret and represent the operator $e^{-i\theta(I-Z_1\otimes Z_2 \otimes Z_3)}$ for a 3 qubit system in a circuit using qiskit.
I was thinking I could just perform an ...
5
votes
1
answer
272
views
Problem with building quantum circuit for Hamiltonian operation
In the book, Nielsen and Chuang, there is a section on quantum simulation of the quantum search algorithm. Hamiltonion operator is defined as follows-
$$
H = |x\rangle\langle x| + |\psi\rangle\langle\...
- 63
1
vote
2
answers
129
views
Constructing a time evolution operator $e^{it H}$ for $H^2=I$
Consider a Hamiltonian $H = \sigma_x \otimes \sigma_z$
Construct the time evolution operator $U(t) = \mathrm{e}^{-\frac{iHt}{\frac{h}{2\pi}}}$ [Hint:Write down the expansion of $\mathrm{e}^x$ and use ...
1
vote
0
answers
55
views
Primer for Quantum Circuits and Optimization
I am interested in studying physical systems and trying to build circuits to simulate them. Now, of course, all the systems I could try to work with at simple, toy, systems - and that's fine.
...
- 1,003
3
votes
0
answers
480
views
XX, YY, ZZ circuit representations?
Is there a good primer or set of lectures\examples that show entirely how to take a given matrix and developing a circuit that represents it. I am trying to implement a program to find the lowest ...
- 1,003
2
votes
3
answers
469
views
Compiling the Pauli-Z operator to the Rz operator for Hamiltonian simulation
I saw a tutorial on this long ago, but lost it. I know that the Pauli-Z operator compiles to Rz, but how? Here are the steps I remember:
First, we have to solve for $U(t)$ in the Schrodinger equation ...
- 4,228
1
vote
0
answers
20
views
Ansatz Techniques to Multi-Body Physics Problems
I have been reading this paper: https://arxiv.org/abs/1906.01563v1.
I am wondering: is it possible to use the idea behind quantum circuits to build classical Hamiltonians represented in the same way?...
- 1,003
3
votes
0
answers
184
views
Quantum Optimization via Quantum Label Classification in Quantum Circuits
I have been reading Farhi and Neven's paper on quantum neural networks on quantum circuits. I also found an example - albeit not ideal as pointed out by a couple of users - thank you - in here.
...
- 1,003
7
votes
1
answer
265
views
Simulating a 3-local Hamiltonian Term
This may be a fairly basic question, but in Nielsen & Chuang, the following circuit is given for simulating $\exp\left(-i\Delta t Z_1 \otimes Z_2 \otimes Z_3\right)$:
which uses an ancilla qubit ...
- 337
29
votes
2
answers
7k
views
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 ...
- 421
4
votes
1
answer
180
views
How are two different registers being used as "control"?
On page 2 of the paper Quantum Circuit Design for Solving Linear Systems of Equations (Cao et al.,2012) there's this circuit:
It further says:
After the inverse Fourier transform is executed on ...
- 17.8k