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].
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What does a solvable Hamiltonian model mean?
I have recently been reading about simulating the dynamics of many body Hamiltonians by means of quantum computers and I am a bit confused about some terminology. I understand that if you are able to ...
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Expectation Value of Observable from evolved statevectors using qiskit on Hardware
I want to compute this <Zi Zj> - <Zi><Zj> for an entangled n-qubit initial state under the application of a general XY Hamiltonian for a range of ...
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On the inverse of LCUs for Hamiltonian simulation
Let $H = \sum_j \alpha_j U_j$ be a linear combination of unitaries (LCU) representation for a Hamiltonian that we wish to simulate (i.e., construct a circuit approximating $e^{iHt}$). The standard ...
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Are quantum states like the W, Bell, GHZ, and Dicke state actually used in quantum computing research?
I recently started studying quantum computing and learned about several well-known quantum states such as the W state, GHZ state, and Dicke state. I noticed that there are also some questions here on ...
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Coding a hamiltonian in qiskit
I have a hamiltonian of the form:
$H=\sum_{i=1}^N Z_i Z_{i+1}-Z_NZ_1$
And another one as:
$H=-\sum_{i=1}^N X_i$
I need it to it for N terms.
I am a bit lost can anybody help. I tried looking for ...
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What are hamiltonians in the context of quantum computing
Sorry if this question is a bit too generic or basic, but my background lies only in mathematics and computer science. I am currently writing my thesis on the topic of simulating quantum computing ...
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The support of the ground state of stoquastic Hamiltonian is connected
A Hamiltonian $H$ is stoquastic in the standard basis if all the off-diagonal terms of the Hamiltonian are non-positive. If we choose $\beta$ small enough, all entries of $I-\beta H$ are non-negative. ...
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Efficient method to find square root of a Hamiltonian
I'm working with a Hamiltonian $H$ represented as a linear combination of Pauli strings:
$$H = \sum_j \alpha_j P_j,$$
where $P_j \in \{I, X, Y, Z\}^{\otimes n}$ are tensor products of Pauli matrices ...
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Reference for a proof of QMA-completeness of sparse Hamiltonian
It is well-known that the $k$-local Hamiltonian problem is QMA-complete for $k$ constants over $n$ qubits, $||{H}|| \in \text{poly}(n)$ under a reasonably large promise gap. See e.g. these notes.
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Cosine of an operator (a+a.dag()).cosm() in Qutip does not work as it should
As mentioned in the title, the .cosm() method in QuTip fails to give the correct evolution under the Hamiltonian. I am trying to define the Hamiltonian of a non-linear LCJ circuit as follows:
...
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Schrödinger equation and Hamiltonian ground state
Is it true that the Hamiltonian flow on the state space of a quantum system "pushes" quantum states towards a ground state? In mathematical therms, given a Hamiltonian $H$, an initial state $...
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Locality of Hamiltonian through basis change
Consider a Hamiltonian $H$. For efficient Trotterization, one needs a decomposition into local terms $\sum_j H_j$. However, this is a highly basis-dependent statement. Can I always find a basis ...
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What are some resources to get started in quantum computing for nuclear and high-energy physics?
One of the big hopes for quantum computing is to simulate quantum many-body systems with quantum systems and thereby solve many outstanding problems that are otherwise intractable in nuclear and high-...
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Quantum Phase Estimation for Chemical Hamiltonian
I am working on implementing QPE for calculating the ground state energy of a small molecule like H2 or HeH+. I've written some code that is getting results within about 10% accuracy, but I'm not ...
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Getting a Hermitian operator from a Quantum circuit, and taking its expectation value
I have a circuit $C$, which acting on a state $|\psi \rangle$ is equivalent to a Unitary $U$ acting on the state: $$C(|\psi \rangle) = U|\psi \rangle$$ Now, this circuit is just a Hamiltonian ...
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Hamiltonian Simulation: What's the meaning of t in $\exp(iAt)$?
My main goal is to find eigenvalues of some hamiltonian matrix $A$. When implementing Quantum Phase Estimation, I need to provide my circuit with informations about $A$. From what I have seen so far, ...
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Turning Deutsch-Josza into a continuous problem
I am wondering whether anyone has investigated if there is a notion of a continuous oracle. My starting off point is to consider the Deutsch-Josza problem, in which the oracle acts on the state in a ...
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non-stoquastic vs quantum annealer
Why non-stoquastic Hamiltonian is difficult to implement for a quantum annealer? In another way, why stoquastic vs non-stoquastic matters?
Thanks.
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What problems in chemistry or materials science could be solved with 100 fault-tolerant qubits?
Background
IBM, Infleqtion, QuEra, and other quantum hardware companies have announced roadmaps where they expect to have 100 or more fault-tolerant qubits by the end of the decade. It seems ...
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Qiskit TimeEvolutionProblem with complex operator
I have been using Qiskit to simulate some oscillators using Hamiltonian simulation. A next step I would like to make is adding dissipation of these oscillators in some way. I think it would be ...
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How to approximate the time-dependent Hamiltonian in quantum adiabatic theory by the non time-dependent Hamiltonian?
Recently, I am learning how to solve the linear equation $A\left | x \right \rangle =\left | b \right \rangle $ using quantum adiabatic theory. In the solving process, people usually need to set the ...
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Calculating number of CNOT gates in Pauli evolution gate
How to calculate the number of CNOT gates for a Pauli exponentiation for given time?
I am performing Trotterization which involves performing Pauli evolution ...
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Particle number expectation value in QuTip
I am learning now to use QuTiP by going through their documentation site. I am trying to understand what does the argument - particle number expectation value in thermal density matrix do? How does it ...
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Qiskit: Evolve TrotterQRTE from Operator
I am trying to implement the method in the following paper: Exponential Quantum Speedup in Simulating Coupled Classical Oscillators using Qiskit.
All is good until I call evolve on ...
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Are commuting unitary operators related to commuting Hamiltonians?
TL/DR: Can unitary operators:
$$U_a=e^{-it(H_{a1}+H_{a2}+\cdots)}$$
and
$$U_b=e^{-it(H_{b1}+H_{b2}+\cdots)}$$
commute, even though $[H_{aj},H_{ak}]\ne 0$ and $[H_{bj},H_{bk}]\ne 0$ for all $j,k$?
...
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Modeling a chemical reaction when there is a catalyst
I'd like to construct potential energy surfaces (PES) for chemical reactions with and without the presence of a catalyst. Something like this
The closest paper that is https://arxiv.org/pdf/2007....
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Complexity of controlled-$U^j$ operations in QPE applied to Hamiltonian simulation
One method to obtain the eigenvalues of a Hamiltonian $H$ is by applying quantum phase estimation to its time-evolution operator $U(t) = e^{-iHt}$. If I want to obtain an eigenvalue to $k$ bits in ...
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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?
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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 $\...
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What is the intuition behind achieving Quantum advantage in simulating non-hermitian dynamics using Quantum computer?
There have been several works on simulating ODE for classical systems like here and here. They are quantum techniques to solve the ODE related to classical systems.
A generic methodology is:
To solve ...
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Do some Hamiltonian simulations require an irreversible process?
I just stumbled upon this research paper
https://arxiv.org/abs/2309.16596.
They claim to have found a problem which is easy to solve quantumly but hard classically: to find local minima of 2D ...
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Entanglement generation for commuting Hamiltonian
Consider an $n$ qubit Hamiltonian $H$ given by
\begin{equation}
H = \sum_{i=1}^{m} H_i,
\end{equation}
where each $H_i$ is a $k$-local term and it holds that
\begin{equation}
e^{H} = e^{H_m} \cdot e^{...
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Time-derivatives of Observables over Hamiltonian evolution
I am reading about algorithms to simulate Hamiltonian evolution by means of quantum computers, e.g. a transverse field Ising model. As far as I see one is interested in getting expectation values of ...
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What work has been done on hamiltonian simulation of operators that are integrals instead of sums?
I'm looking at a problem where I want to do hamiltonian simulation of an operator integral. That is, I want to implement the unitary
$$\mathrm U = \exp[-i \mathrm H t]$$
where $\mathrm H$ is of the ...
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Spin Hamiltonian to Quantum Circuits and are there any group theory associated with the quantum circuits?
Can we think of Quantum Circuits as another representation to describe the dynamics of a system other than its Hamiltonian? How can we go from the spin Hamiltonian version (for eg: SSH Model ...
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Deriving circuit templates for Hamiltonian simulation
Background
I've been reading the paper entitled Some improvements to product formula circuits
for Hamiltonian simulation. The authors propose three improvements motivated by phase estimation type ...
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QPE algorithm to the Hubbard Model
I'm trying to perform the Eigenvalue Estimation algorithm to the Hubbard model with two sites, one spin-up fermion and one spin-down fermion, with Qiskit. Given the Hamiltonian:
$\hat{H}=u\sum_{i=1}^N\...
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Why is "reducing Hamiltonian energy" also optimizing a Quantum Machine Learning model?
From what I observed, most hybrid qml architectures surround the ideas of Hamiltonian states, and it seems like our goal to optimize a circuit is to keep energy states as low as possible. But why is ...
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How to define an energy function, for predicting protein structure in quantum computing? [closed]
Let q denote a particular configuration of the protein in a grid, written in the form
where $x_i$ and $y_i$ are the $x$ and $y$ coordinate of the $i$th amino acid. For this configuration, how to ...
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How is geometric frustration different from (not) being frustration free?
In the context of Ising models, some Hamiltonians can be described as geometrically frustrated - such as, I think, the antiferromagnetic kagome lattice, as well as a one-dimensional anisotropic, next-...
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What properties of a local Hamiltonian are basis-(in)dependent?
Some properties of a Hamiltonian are unique to its spectrum and are basis-independent. For example, I think whether the Hamiltonian's gap remains constant as $n$ goes to infinity, or whether the ...
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If all terms of a local Hamiltonian commute, how hard is it to learn the ground state (energy)?
Suppose we have a $k$-local Hamiltonian with each of $m$ terms acting on $k$ of $n$ qudits of constant dimension $d$:
$$H=H_1+H_2+\cdots+H_m.$$
If at least some of the terms don't commute, e.g., if $[...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
