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].
158
questions
4
votes
2answers
68 views
How is Quantum Phase Estimation useful for simulating dynamics of a many-body system?
I am quite aware of the Quantum Fourier Transform (QFT) as well as the very closely related topic of Quantum Phase Estimation (QPE). The latter is usually motivated as follows:
Given a unitary $U$ and ...
2
votes
0answers
33 views
Could the Hamiltonian of a 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), with front, up, and right twists being labelled as $\langle F,U,R\rangle$.
Each move uses fifteen SWAP ...
-1
votes
0answers
44 views
What does $H$ being asymptotically equivalent to $(a+b) \sigma_1 \otimes \sigma_1$ mean?
For the Hamiltonian $H=a \sigma_1 \otimes \sigma_1 + b \sigma_2 \otimes \sigma_2$, where $\sigma_i$ are the well known Pauli matrices, and $a$ and $b$ are real, what does it mean that:
asymptotically ...
0
votes
0answers
18 views
In the HHL algorithm, does the controlled unitary depend on the Hermitian matrix coefficients?
In HHL algorithm, does the controlled unitary (Hamiltonian simulation part of Quantum phase estimation) depend on Hermitian matrix coefficients and how?
-1
votes
0answers
26 views
HHL algorithm explanation in layman terms [duplicate]
Can someone explain what is happening step by step in HHL algorithm in layman terms without maths starting from the first part (quantum phase estimation), than the second part (rotation gate) and ...
1
vote
0answers
46 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 ...
5
votes
1answer
53 views
Why does Hamiltonian simulation seek to find the energy minimum, if eigenvalues of unitaries are always unimodular?
I know I am wrong here and trying to find out where I am making a logical mistake. I'd appreciate it if you can help me untangle.
A. We know that the eigenvalues of Unitaries are all unimodular (...
4
votes
1answer
105 views
How to exactly implement Trotter-Suzuki formula on quantum computer
Recently, I am studying some topics related to product formula, and I am curious about how to implement such formula on real quantum devices. The $(2k)$-th order product formula can be witten as
\...
3
votes
0answers
61 views
How to find minimum time needed for Hamiltonian evolution?
Database search can be looked upon as Hamiltonian evolution, with kinetic and
potential energy operators. Let the evolution follow the Schrodinger equation:
$$i\frac{d}{dt}|\psi⟩= H|ψ⟩$$
with $H = E|s⟩...
2
votes
0answers
21 views
Nyquist–Shannon sampling theorem for Quantum Evolution
In classical digital signal processing one can try to identify the dynamics of a system by sampling its evolution from an initial time $t_0$ to a final time $t_1$. Sampling $N$ times results in a ...
1
vote
1answer
43 views
Question on Aharanov and Ta-Shma (ATS)'s Sparse Hamiltonian Simulation notation
In the equations in section 3.4.2 of Aharonov and Ta-Shma's paper (pdf, arxiv abstract), they define the operator: $$T_1:|k,0\rangle\rightarrow|b_k,m_k,M_k,\tilde{A_k},\tilde{U_k},k\rangle,$$ where $...
0
votes
0answers
26 views
How can extract reduced dynamics of a bipartite system from unitary evolution in quite
Let us assume that I have a bipartite system $A\otimes B$ and an initial product state undergoing some evolution $H^{AB} = H^A+H^B+V^{AB}$, which is time independent. I want to simulate the reduced ...
5
votes
0answers
125 views
Can we use quantum phase estimation to count how often we can walk from one Rubik's cube position to another?
Consider a state $\vert\psi\rangle$ as below, which is in a superposition of a difference between a Rubik's cube in a solved state and a Rubik's cube in the "superflip" state. Here, with $8$...
3
votes
1answer
91 views
Can I use the Lie product formula to simulate the Hamiltonian of an adjacency matrix by using the QPE to take Nth roots of permutation matrices?
I have gotten some great help recently on Hamiltonian simulation, and am interested in using Hamiltonian simulation to explore (classical) random walks on large graphs, but I'm running up against ...
1
vote
1answer
74 views
Wick rotation of the Schrödinger equation
Studying the following paper: https://www.nature.com/articles/s41534-019-0187-2.pdf
Trying to figure out how $ E_T$ shows up from (1) and (2).
Any suggestion or guidance would be appreciated.
We ...
3
votes
1answer
106 views
Is the square-root-of-SWAP for a pair of 4-dimensional qudits isomorphic to two square-root-of-SWAPS for two pairs of qubits?
This may be a very naïve question indicative of a lot of confusion, but I am trying to understand more about Hamiltonian simulation. I'm starting to intuit that the $n^{th}$-root-of-SWAP acting on a ...
8
votes
1answer
149 views
Can we simulate the Hamiltonian for the Rubik's Cube with "nth-root of SWAP" gates?
I'm interested in, but confused about, local Hamiltonian simulation. I don't yet have enough intuition regarding even the approach set forth by Lloyd in 1997. I think Lloyd's recipe is to repeatedly ...
2
votes
1answer
90 views
Do VQE and QAOA use the same Hamiltonian?
In this paper, it talks about the 2-local Hamiltonian in the form:
$H = \sum_{(u,v)\in E} H_{uv} + \sum_{k \in v} H_k $
It also says the Ising model, Heisenberg model, XY model and QAOA are in the ...
5
votes
1answer
103 views
Simulating $e^{iX_1\otimes X_2t}$
I'm trying to figure out the quantum circuit to simulate the time-evolution of a 2-qubit Hamiltonian $e^{iX_1\otimes X_2t}$, where $X$ is a Pauli gate. From this answer, the quantum circuit performs $...
2
votes
3answers
450 views
Are these two circuits equivalent in performing controlled time-evolution?
I want to perform the controlled time-evolution of some 2 or 3-qubit Hamiltonian. Say we have this example:
$$
H= Z_0\otimes Z_1 + Z_1\otimes Z_2
$$
The circuit performing the time-evolution ...
2
votes
0answers
79 views
QAOA for Binary Optimization
I have reduced an optimization problem to a Binary Integer Linear Programming model as follows:
$$\sum_{j=1}^{2^n-1} f(C_j)x_j \rightarrow \max$$
$$\text{subject to} \sum_{j=1}^{2^n-1} S_{i,j}x_j=1\,\,...
3
votes
1answer
61 views
Relation between Jordan-Wigner transformation and Hilbert-Schmidt inner product
Given a fermionic Hamiltonian in a matrix form, we can write it as a sum over Kronecker products of Pauli matrices using the Hilbert-Schmidt inner product. However if the same Hamiltonian is given in ...
1
vote
1answer
49 views
Standard to select base hamiltonaian for Adiabatic quantum computing
I'm learning about connection between QUBO and The Ising Model.
It says
Take the base Hamiltonian of an adiabatic process as $\sum_i \big(\frac{1-\sigma_i^x}{2}\big)$
to implement Hamiltonian for ...
3
votes
0answers
35 views
What type of tasks it is possible to solve on a quantum simulator?
In this article, the author claimed that researches from Harvard and MIT created 256 qubits quantum simulator. However, we are not talking about piece of software on a classical computer but actual ...
0
votes
0answers
38 views
How does Qiskit's VQE encode binary digits into a solution from a given Hamiltonian?
So far I have been working with the VQE on different Hamiltonians that happened to have degeneracies, that is there were always at least two different global minima, because the Hamiltonians I was ...
4
votes
1answer
69 views
Hamiltonian simulation for 2 qubit gates
I have been reading this paper, and at the end they have given exact decomposition of Hamiltonian simulation step, where they have decomposed a matrix $A$ into pauli matrices and done the operation $e^...
1
vote
0answers
69 views
Implementation of the Phase Estimation algorithm
I've been working on implementing quantum phase estimation in Qiskit for a $2^n \times 2^n$ Hamiltonian as part of my bachelor project, I'm using Trotterization as my Hamiltonian simulation of choice ...
1
vote
0answers
66 views
Are inconsistent results between different VQE runs justified?
I made this post a while ago, where I learned I could use qiskit's VQE to calculate (or approximate) the Transverse Field Ising Hamiltonian and other similar Hamiltonians. After working with my code ...
1
vote
1answer
36 views
What do the numbers in the Ising sampleset mean?
I am trying to create a portfolio optimization with the DWave Quantum Computer. I wrote some code trying to somehow reconstruct the following Ising model paper:
Ai ...
2
votes
0answers
44 views
From QUBO problem to quantum circuit [duplicate]
Can you describe how to convert some QUBO problem $$f(x_1, ..., x_n) = \sum\limits_{1 \le i < j \le n} \alpha_{i,j} \, x_i \, x_j$$ into its equivalent quantum circuits? Is it preferrable to start ...
2
votes
1answer
92 views
Prove that any Hermitian Matrix is a real linear combination of Pauli operators [duplicate]
This is an important result in Quantum Computing because it means that the Hamiltonian of a Quantum System can be encoded as a sequence of real numbers and their corresponding Pauli Operator.
How do ...
4
votes
1answer
246 views
Calculating the ground states of an Ising Hamiltonian on a real quantum computer
I have followed this tutorial and based on it, I've written the following function in qiskit, which can explicitly calculate the ground states of a transverse-field Ising Hamiltonian.
...
1
vote
1answer
40 views
How to construct a Hamiltonian for an ensemble of atoms interacting with each other?
How to construct a Hamiltonian for an ensemble of atoms interacting with each other?
For example if the one atom hamiltonian can be written as:
$$\hat{H}=\left(\begin{matrix}0&\Omega_p(t)&0\\\...
3
votes
2answers
211 views
Why is the time ordering omitted in the trotterised version of the time-dependent evolution operator?
The unitary evolution of a time-dependent hamiltonian is given by the time-ordered matrix exponential
$$\begin{aligned}
U(t)&=\mathcal T\exp\left[-i\int_0^tH(\tau)d\tau\right]\\
&=I-i\int_0^td\...
5
votes
1answer
153 views
Finding the norm of a Hamiltonian
I am experimenting with https://journals.aps.org/prx/pdf/10.1103/PhysRevX.8.041015 and in equation 36 I find that they use the norm of the Hamiltonian.
Is there a clean way to compute it, or an upper ...
3
votes
1answer
63 views
How to get eigenvectors of Hamiltonian in OpenFermion
In OpenFermion you can create a Hamiltonian in terms of creation and annihilation pretty easily:
...
4
votes
1answer
166 views
How do we determine the direction of a time evolution $e^{-itH}$ on the Bloch sphere?
I have a question about the direction of time evolution on a Bloch sphere: suppose I'm performing a unitary time-evolution $\exp(-iHt/\hbar)$ for a single qubit, then on the Bloch sphere it ...
1
vote
0answers
42 views
Has any analogue quantum simulator showed quantum advantage yet?
Quantum advantage/supremacy was achieved by Google using a quantum computer and more recently by Pan Jianwei's group using photons. So I was wondering, has any analog quantum simulator showed quantum ...
7
votes
3answers
275 views
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 ...
3
votes
1answer
220 views
Quantum Circuit for $e^{iAt}$ Hamiltonian Simulation in HHL algorithm
In HHL algorithm, there is a step in Quantum Phase Estimation where we have to apply powers of $e^{iAt}$ to the register (see pic). I am not able to understand how to find the quantum circuit ...
3
votes
1answer
251 views
VQE - How to get from expectation value to eigenvalue?
In VQE, for a single-qubit Hamiltonian, I can use a standard ansatz to make a state $\psi$ and use two products to compute the expectation value $\langle\psi|{\cal H}|\psi\rangle$. As I vary the ...
5
votes
2answers
196 views
How to construct the two qubit gate generated by the Hamiltonian $H= X\otimes X + Y \otimes Y + Z \otimes Z $?
I know that the two qubit gate generated by $H=X\otimes X$ is $\exp\{-\text{i}\theta X\otimes X\}=\cos{\theta} \mathbb1 \otimes \mathbb1 - \text{i} \sin{\theta} X \otimes X$, where $X$ is the $\...
2
votes
1answer
113 views
Fermionic commutation relation using Jordan-Wigner transformation
How to show in detailed steps that Fermionic annihilation and creation operators under Jordan-Wigner transformation satisfy the Fermionic commutation relation
$\{\hat{a}_i,\hat{a}_j\}= \{\hat{a}_i^\...
4
votes
3answers
101 views
Why does joint ground state not change under action of beam splitting unitary operator?
How can one show that $\hat{U}|00\rangle=|00\rangle$ where $\hat{U}=e^{-igt(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)}$ and $|00\rangle$ is the unique joint zero eigenstate of the ...
7
votes
3answers
316 views
If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?
How can one show that $\hat{U}^\dagger\hat{a}\hat{U}$ (with $\hat{U} =e^{-i\hat{H}t}$) involves only linear orders of the ladder operator, when $H$ is the general quadratic Hamiltonian $(\hat{H} = \...
6
votes
2answers
114 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 ...
4
votes
2answers
130 views
Fermionic occupation operator and nearest neighbor Fermionic hopping interaction as a qubit operator
How to express Fermionic occupation operator $(\hat{a}_j^\dagger\hat{a}_j)$ and nearest neighbor Fermionic hopping interaction ($H_h= J\sum_{i=1}\hat{a}_i^\dagger \hat{a}_{i+1}+\hat{a}_{i+1}^\dagger \...
3
votes
1answer
31 views
Distant quantum gates between uncoupled qubits
Is there any formalism to perform quantum gates between two qubits (let's say in a superconducting quantum network) to perform a quantum gate between two qubits which are not directly coupled? I want ...
5
votes
1answer
140 views
What's the state-of-the-art to calculate $|Ab\rangle$, given a matrix $|A\rangle$ and a vector $|b\rangle$ in QRAM encoding
Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e.
$$|A\rangle= \frac{1}{\|A\|_F}\sum_{i,j=0}^{n-1}{a_{ij}}|i,j\rangle$$
and a vector $b\in \mathbb{R}^...
2
votes
0answers
74 views
Matrix multiplication through Block Encodings
For a project, I want to simulate a matrix multiplication on a simulated quantum circuit. Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e.
$$|A\...