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].
257
questions
4
votes
0
answers
62
views
How useful is it to know the ground state energy of an arbitrary $k$-local Hamiltonian, if Nature herself can never find such energy?
We know that the $2$-local Hamiltonian problem is (promise) QMA-complete, which under the reasonable assumption that BQP$\subsetneq$QMA implies that no fast quantum algorithm exists to determine the ...
- 10.7k
1
vote
0
answers
30
views
The no fast forwarding theorem and exponential advantage for many body Hamiltonians
When simulating Hamiltonians in first quantization there are $\eta$ particles occupying a grid of $N$ grid points. In the first quantization, you directly discretize the differential operators onto a ...
- 611
1
vote
1
answer
61
views
time evolution of Hamiltonian and observables
I am given the following Hamiltonian on 4 qubits.
$$
H = - J_x (X_0 X_1 + X_2 X_3) - J_z (Z_0 Z_2 + Z_1 Z_3) - h\sum_{j=0}^3 X_j + Z_j
$$
I have already implemented the time evolution of this ...
- 83
1
vote
0
answers
30
views
How is implemented the hamiltonian simulation of Hermitian operator multiplied by projection
The article "Quantum Topological Data Analysis with Linear Depth and Exponential Speedup" (Ubaru et al) discusses the implementation of the Hamiltonian $\Delta_\Gamma$, named the ...
- 671
1
vote
0
answers
24
views
How to model the transtition part of dispersive frequency shift in dispersive readout?
The phenomenon of interest is the resonance frequency shift and the larger noise in the measurements inside the intermediate-powered region as shown below (from Fig. 3.3 of Characterisation of ...
- 133
2
votes
0
answers
46
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 ...
2
votes
0
answers
39
views
Do Aharanov and Ta-Shma treat the entries of a sparse Hamiltonian as edges of a graph?
Background and history
The mid-90's to early 2000's work on Hamiltonian simulation saw some pretty rapid advances. Within two years of Shor's algorithm, Lloyd outlined how Trotterization can lead to ...
- 10.7k
0
votes
1
answer
46
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:
$$H = - \pi\delta(Z_1 - Z_2)
+ 2\pi J ~ \mathbf{I}_1 \cdot \...
2
votes
0
answers
29
views
Weird behavior when simulating vacuum Rabi oscillation on QuTip
I've been playing around with vacuum Rabi oscillation on QuTip and found an odd behavior.
My Hamiltonian is as follows:
$$ H=\omega_n n^\dagger n + \omega_c c^\dagger c -6K(n+n^\dagger)^4 - g(n^\...
- 21
1
vote
2
answers
61
views
Notation: Hamiltonian Simulation of Pauli Gates
Let $\sigma^j_x$ describe the following unitary over $n$ qubits: on the $j$-th qubit, it acts as the Pauli $x$ operator; instead, on any other qubit, it acts as the identity. A paper states now that
\...
- 53
3
votes
0
answers
31
views
Compiling pulses with time dependent $\sigma_x$ and $\sigma_y$ control
I have a Hamiltonian of the form:
$$
H = c_x(t) \sigma_x + c_y(t) \sigma_y$$
and I want to compile a pulse "P" that has both $\sigma_x$ and $\sigma_y$ control, with different time dependent ...
- 31
3
votes
1
answer
46
views
Simulating Sparse Hamiltonians: help understanding query complexity bounds
tl;dr: How can I show that $e^k/k^k$ is less than $\epsilon^2/2$ when $k=\Omega\left(\frac{\log(1/\epsilon)}{\log \log(1/\epsilon)}\right)$, where $k,\epsilon\in \mathbb{R}$ and > 0?
Context:
Berry ...
- 33
0
votes
0
answers
18
views
What is the closest experimental platform to $H=\sum_{ij} J^X_{ij}X_iX_j+J^Y_{ij}Y_iY_j+J^Z_{ij}Z_iZ_j$?
Consider the Hamiltonian
$$H=\sum_{ij} J^X_{ij}X_iX_j+J^Y_{ij}Y_iY_j+J^Z_{ij}Z_iZ_j$$
where $X,Y,Z$ are Pauli spin operators and $J_{ij}^\alpha$ are arbitrary couplings that can be positive and ...
- 360
4
votes
2
answers
213
views
When is it justified to have an oracle in a quantum algorithm?
I've always been confused as to when a quantum algorithm is allowed to have an oracle and what kind of a function the oracle can have. For instance, I know in Hamiltonian simulation algorithms, you ...
- 145
1
vote
0
answers
55
views
Problems with Exact Diagonalization Implementation in Quantum System Evolution
I've been trying to implement exact diagonalization to study the time evolution of quantum states and subsequently compute local magnetizations. I have written functions for evolving the state and ...
- 123
0
votes
0
answers
23
views
Difference in the Order of Applying Quantum Gates in Qiskit
I've been experimenting with Qiskit and came across a scenario where I'm uncertain about the effects of the order in which I apply quantum gates.
Imagine we have the Hamiltonian
Which one would be ...
- 123
0
votes
0
answers
189
views
Applying Trotterization to a Hamiltonian for Time Evolution in Qiskit
I'm currently working on a project where I need to simulate the time evolution of a quantum system using Qiskit. The Hamiltonian of my system is given by:
$$H = -J \sum_{j=1}^{N-1} (\sigma_j^x \...
- 123
2
votes
2
answers
422
views
What does Qubitization mean?
I was listening to an advanced lecture about quantum computing, when the professor introduced a chapter called "Qubitization and the quantum singular value transform", but never really cared ...
- 33
1
vote
1
answer
93
views
How can I get a time evolution operator for imaginary time?
I want to implement time evolution operator of a hamiltonian in qiskit. I am using circ.hamiltonian(H,time,list(range(N))) to get the circuit. This method does not ...
- 149
1
vote
0
answers
19
views
Intuitive explanation on dependence of Hamiltonian simulation on norm?
Suppose I have two Hamiltonians, $H_1$ and $H_2$, that I want to simulate for time $T$. If $\|\|H_1\|\|>\|\|H_2\|\|$, why is it more costly to simulate $H_1$ compared to $H_2$? Is there an ...
- 145
2
votes
1
answer
74
views
How many eigenstates are accessible in polynomial time?
A result of Hamiltonian complexity theory by Poulin et al. shows that only a small fraction of the volume of Hilbert space can be reached in polynomial time for any physical system or quantum computer....
- 508
1
vote
0
answers
9
views
From what distribution is QuTip's rand_herm sampled from?
I am trying to figure out how QuTip samples Hamiltonians using its rand_herm function. It seems to use SciPy's sparse.rand ...
- 199
1
vote
1
answer
70
views
How can I simulate the following 2×2 Hamiltonian $e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}}$?
How can I simulate the following 2×2 Hamiltonian $$ e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}}|\Psi\rangle$$
ie. how to rewrite that matrix exponential in terms of other, well-used ...
- 455
1
vote
0
answers
92
views
Hamiltonian of Bose-Hubbard model
In https://medium.com/qiskit/introducing-bosonic-qiskit-a-package-for-simulating-bosonic-and-hybrid-qubit-bosonic-circuits-1e1e528287bb , could anyone explain the rationale behind the use of ...
- 7
1
vote
1
answer
119
views
(When) must the ground state of a frustrated Hamiltonian be entangled?
I've only recently, and still only haphazardly and rather poorly, begun to understand Ising models with local interactions. I'm interested in particular in the simple one-dimensional Ising model with ...
- 10.7k
4
votes
1
answer
61
views
How to benchmark approximate random unitary sampling
I'm currently studying a specific sampling "quantum advantage" (sorry for the buzzword) protocol wich consist of periodically driving a random Ising chain (https://iopscience.iop.org/article/...
- 91
0
votes
1
answer
35
views
Creating a parameterized Operator in Qiskit that cannot be decomposed into Qiskit supported gates
I am trying to create a custom ansatz to use the built-in Qiskit VQE() function. My ansatz is composed of single qubit gates and a hamiltonian gate which cannot be decomposed into Qiskit supported ...
- 95
4
votes
0
answers
133
views
How does edge-coloring help for quantum walks?
Reviewing Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman's famous 2002 welded-trees problem, a quantum Theseus can find his way out of a labyrinth having an exponential number of rooms (vertices),...
- 10.7k
1
vote
1
answer
74
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 ...
- 237
2
votes
1
answer
65
views
What are the entries in the 2x2 $W$ gate used for walking along the welded-trees graph of Childs et al.?
As an extension of a famous description of non-local Hamiltonian simulation in section 4.7.3 of Nielsen and Chuang, the welded-trees paper of Childs, et al. provides the following circuit for use in ...
- 10.7k
2
votes
1
answer
53
views
The approximation of the time evolution operator U, using Trotter formula, can't hold anymore taking a time step of $\Delta t = \pi$
The problems arises from a consideration written on the book "Quantum computation and quantum information" from Michael A. Nielsen and Isaac L. Chuang on page 259.
In this chapter it's ...
2
votes
1
answer
40
views
Can a Hamiltonian of a tripartite system map an product state into a product state?
Suppose we have a finite dimensional Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$ and a Hamiltonian has the following form:
$$H = H_A \otimes I \otimes I + I \otimes H_B \...
- 2,359
2
votes
1
answer
220
views
How to perfrom a time-dependent Hamiltonian simultation using the Trotter-Suzuki formula?
I would like to know how to perform trotterization of a time-dependent operator (such as a Hamiltonian) on a gate-based quantum computer? I've seen examples for time-independent Hamiltonians, but I ...
- 155
0
votes
2
answers
112
views
circuit for quantum simulation?
What would be the circuit for operation exp(iθZ⊗Z⊗Z⊗X) by only using CNOTs and single-qubit gates. And How we can improve the circuit to implement the operation exp(iφZ⊗Z⊗X⊗Z ).exp(iθZ⊗Z⊗Z⊗X).Are ...
- 31
0
votes
3
answers
215
views
How to represent Beam-Splitter and Kerr gates as basic quantum logic gates?
I want to know how to convert these exponential forms to tensor products of well known logic gates (like the ones built into Qiskit). My goal is to program the Beam-splitter-Kerr ansatz circuit for ...
- 95
0
votes
1
answer
121
views
Do the following circuits violate the principle of "no fast-forwarding"?
The no fast-forwarding principle states roughly that, given that we can simulate a Hamiltonian for time $t$ using $r$ gates, in order to perform the Hamiltonian simulation for time $2t$, we must in ...
- 10.7k
2
votes
0
answers
41
views
Can we show that a quantum circuit with Poly(n) gates has a Hamitonian with Poly(n) terms?
It is already known that if the Hamiltonian is a sum of Poly(N) Pauli terms, it has an efficient implementation as a quantum circuit. This should mean that the circuit can be implemented with Poly(N) ...
- 21
0
votes
0
answers
16
views
Questions about ActiveSpaceTransformer code implementation
Could anyone explain how dipole moment and 2-body integrals mathematically relate to DFT embeddings ?
How is ActiveSpaceTransformer different from ECP (Effective Core Potentials) ?
If ...
- 7
2
votes
0
answers
55
views
Why do Hamiltonian simulation algorithms not depend on the size of the Hamiltonian for the gate complexity?
The wikipedia page for Hamiltonian simulation mentions the gate and query complexities for different algorithms used for the problem (Trotter-Suzuki, Taylor Series, Quantum Walks, and QSP).
They ...
- 383
1
vote
1
answer
60
views
Is Hamiltonian simulation with only real entries BQP-complete?
It is often asserted that Hamiltonian simulation (given some Hermitian matrix, $H$) is BQP-complete.
I don't see how the input to such an algorithm is done without the use of some block-encoding or ...
- 383
0
votes
1
answer
89
views
Qiskit - Approximation of Hamiltonian energy via QPE
I'm trying to study QPE with the motivation of obtaining eigenvalues of Hamiltonian, i.e. energies of a system. My problem is, that while np.linalg.eig and VQE are agreeing on the lowest energy, ...
- 245
0
votes
0
answers
299
views
Pauli decomposed Hamiltonian as Diagonal U gate
While trying to implement a quantum circuit, I had to apply Hadamard gates to all qubits to achieve equal superposition. Done.
The next operation is decomposing the Hamiltonian into a sum of tensor ...
- 231
2
votes
1
answer
50
views
time evolution of Hamiltonian to generate the Bell pair
Consider two different Hamiltonians: $H_1(t) = ZZ + \alpha(t)X_1 + \beta(t)X_2$ and $H_2(t) = XX + \alpha(t)Z_1 + \beta(t)Z_2$, where $\alpha(t)$ and $\beta(t)$ are time-dependent functions. Starting ...
- 465
1
vote
2
answers
76
views
Solving a linear optimization problem with inequality constraints in qiskit? is it possible?
This is an easy optimization problem that can be classically solved. My question is that, in qiskit, how can we solve this optimization problem using IBM real machines? Is it even possible to do that?
...
1
vote
0
answers
103
views
Block encoding for a sum of Pauli terms
Given a qubit Hamiltonian with polynomially many local Pauli terms, what is the most natural way of constructing its block encoding? I know there are multiple ways of doing so, but I am looking for ...
- 1,830
4
votes
1
answer
168
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 ...
- 169
1
vote
1
answer
40
views
Can superoperators work in Monte Carlo solver in QuTip?
The Monte Carlo solver works with kets instead of density matrices. And it doesn't allow a superoperator (which acts on density matrices or superkets) as a collapse operator. Since my master equation ...
- 173
0
votes
1
answer
46
views
How to implement the cross term (multi-qubit) in the square of the finite difference operator?
I am trying to simulate the Hamiltonian evolution of the 1+1D $\lambda\phi^4$ scalar field theory by digitising it and encoding on a quantum computer. The process of digitising is taken from this ...
- 13
3
votes
1
answer
624
views
What is the difference between Trotter, Lie-Trotter and Trotter-Suzuki approximations?
What is the difference between (1) Trotter (2) Lie-Trotter and (3) Trotter-Suzuki approximation? Are they all different? what are the formulas and errors associated in each of these approximations in ...
- 169
1
vote
1
answer
87
views
Confusing notation in Block-Encoding
I'm reading the lecture notes from Ronald de Wolf and got confused at the part where it introduces Block encoding, specifically by this notation at page 76:
More generally we can define an $a$-qubit ...
- 131