# Tag Info

9

My quick answer: something between 4 and 4000. ^_^ The number of qubits in an electronic structure calculation depends on at least three things: Your basis set. Your qubit mapping. Your algorithm. Basis Set For perfect accuracy, you'd need an infinite number of orbitals to fully represent your system. Most quantum-computing studies today adopt the "...

7

Hint: Instead of using the BCH formula in the form usually presented, for example at the top of this Wikipedia page, use this consequence of Hadamard's Lemma: $$\tag{1} e^{iHt}\hat{a}e^{-iHt} = \hat{a} + [iHt,\hat{a}] + \frac{1}{2!}[iHt,[iHt,\hat{a}] + \cdots$$ Now substitute $H$ into the right-hand side and evaluate the commutators between $\hat{a}$ and ...

6

Since Google is one of, if not the, industry leaders in molecular simulation with quantum computers, their published work is a reasonable benchmark for what's presently within reach. As you probably know, a few years ago Google reported (commentary) successful electronic structure calculations for a Hydrogen atom, favoring the variational quantum ...

5

Use the differential form of the time evolution, $$dO/dt=i[H, O]\ .$$

5

Calculate \begin{align} \hat{U}|00\rangle &= \exp\left(-igt(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)\right)|00\rangle \\ &= \sum_{k=0}^\infty \frac{(-igt)^k}{k!}(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)^k|00\rangle \\ &= |00\rangle + \sum_{k=1}^\infty \frac{(-igt)^k}{k!}(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\... 5 Have you read Towards quantum chemistry on a quantum computer (Nature Chemistry 2010, or here in the arXiv version)? They present "a photonic implementation for the smallest problem: obtaining the energies of H_2 (the hydrogen molecule) in a minimal basis". In the figure S1 of the Supporting information there is an equivalence of the operations they ... 5 The question refers to the VQE, so let's start with this and Max_Cut; they can be built on the VQE. There used to be a vqe.ipynp but I can't find, look for an example. The VQE algorithm doesn't need much input. You can fill it with the paulis_dict. This could be a simple Z gate for finding the eigenvalues= -1. pauli_dict = { 'paulis': [{"coeff": {"imag"... 5 I'm not sure if the 286 qubit estimate has ever been fully explained, but we can backwards reason about how to get to the figure. First off, accuracy of quantum chemistry simulations via Trotterization is a function of the basis set (in both classical and quantum simulations). The basis set is kinda like a coordinatization the electron orbitals. There are a ... 5 There are several startups that have formed around QC-assisted drug discovery. The ones listed below have resources on their websites that you might find helpful. ProteinQure Qulab HQS Kuano For a general understanding of molecular simulation, Simulating Molecules using VQE from the Qiskit Textbook is a great resource if you're generally familiar with QC (... 4 If you have n bits you can combine them in 2^n different bit string (this come from combinatorics). Now take n qubits. As any qubit can in superposition of two state, i.e. 0 and 1, n qubits can be in superposition representing all 2^n possible bit strings. The notion that n qubits can hold 2^n classical bits is unfortunately misleading because ... 4 There's more than one way, and I'll suggest two of them here: Expand \hat{U} using the formula for the Taylor series of an exponential (e^\hat{A}) centered around \hat{A}=\hat{0}, and then you will have a sum of terms where each term no longer involves an exponential operator (i.e. you have just pure creation and annihilation operators and products/... 3 The oldest and most commonly known way is the Jordan-Wigner transformation. The qubit operators will be \mathcal{O}(N)-local for N occupiable orbitals. A significantly more complicated way is the Bravyi-Kitaev transformation for which the qubit operators will be \mathcal{O}(\log N)-local. There's many other ways, but the above two are by far the most ... 3 Let's start with some problems Two big problems we're interested in for drug discovery where quantum computers may do well are high accuracy prediction of receptor-ligand binding affinities and electronic structure prediction. High accuracy quantum approaches addressing these problems that yield a quantum advantage can be expected to find a home in drug ... 2 Here's a fairly thorough overview: https://arxiv.org/abs/1308.6253 For completeness I'll include the paper from the comment: https://arxiv.org/abs/quant-ph/0108146 2 Computing the exchange-correlation functional to sufficiently high accuracy is QMA-hard, where QMA is the quantum version of NP. In particular, this means that it is will all likelihood hard even for a quantum computer. 2 Based on my answer to this: Fermionic occupation operator and nearest neighbor Fermionic hopping interaction as a qubit operator, you can see that we have: \begin{align} \hat{a}_i &= \frac{1}{2} Z^{\otimes (i-1)} (X - iY),\\ \hat{a}_i^\dagger &=\frac{1}{2} Z^{\otimes (i-1)} (X + iY).\\ \end{align} If i=j we have: \begin{align} \{\hat{a}... 2 Let |\psi\rangle be an eigenstate of an operator A, A|\psi\rangle=\lambda|\psi\rangle. Thene^A |\psi\rangle = \sum_{k=0}^\infty \frac{A^k}{k!}|\psi\rangle = \sum_{k=0}^\infty \frac{\lambda^k}{k!}|\psi\rangle = e^\lambda |\psi\rangle.$$In this particular case, A=-igt(a_2^\dagger a_1+a_1^\dagger a_2), of which |00\rangle is an eigenstate with ... 2 Note that$$[(a^\dagger)^n,a] = -n(a^{\dagger})^{n-1}, \qquad [(a^\dagger)^n a^m,a] = -n (a^\dagger)^{n-1}a^m, \qquad [a^n,a]=0.$$Consider an arbitrary function of the mode operators, that we assume be written in normal formal:$$f(a,a^\dagger) = \sum_{n,m=0}^\infty c_{n,m} (a^\dagger)^n a^m.$$We know that$$e^{f(a,a^\dagger)}a e^{-f(a,a^\dagger)} = \sum_{...

2

The answer arguably depends on the problem you wish to solve with your computation. More specifically, are you wanting to optimize near-term applications in the NISQ era, or are you wanting to build a fully scalable, fault-tolerant and universe quantum computer? For the latter, you need to think about error correction. Pretty much everything that will ...

2

Use the Jordan-Wigner transformation. For a 1D chain with NN interaction it will yield a spin Hamiltonian with NN interaction (specifically, the hopping will map to a XX term and the on-site term to a Z term). (In fact, part of this mapping is even given on the German Wikipedia site on the topic.) On the other hand, if you don't want to put this on a quantum ...

2

I think you may have misread the section - the document says: When noise mitigation is enabled, even though the result does not fall within chemical accuracy (defined as being within 0.0016 Hartree of the exact result), it is fairly close to the exact solution. So, the VQE result is not within chemical accuracy, but it is fairly close to chemical accuracy. ...

2

What I would do here is get back the raw results of your res, and there are stocked the parameters of the Ansatz you are looking for : my_res = res.raw_result # The dict of each parameter and its associated value my_param_dict = res.raw_result.optimal_parameters # The array of all the values my_param_list = res.raw_result.optimal_point print(my_param_dict) ...

2

One way to do it would be to use a transformation, such as this one: \begin{align} X_i &= \frac{1 - Z_{i,j}Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{1}\\ Y_i &= \textrm{i}\frac{Z_{i,k}-Z_{i,j}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{2}\\ Z_i &= \frac{Z_{i,j}+Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{3}\\ I_i &= \frac{1 + Z_{i,j}Z_{i,k}}{...

2

The ActiveSpaceTransformer provided by Qiskit Nature allows you to specify a list of molecular orbital indices via its active_orbitals argument. As such, your example above should just work: from qiskit_nature.transformers import ActiveSpaceTransformer transformer = ActiveSpaceTransformer(4, 4, [0, 1, 4, 5]) The first two arguments are the number of active ...

2

As you are asking specifically for the evaluation of the energy only, I will be brief. I will assume that you have a init_state (a quantum circuit) that produces the the Hartree-Fock wavefunction or any other wavefunction you like to test. I could not find a Qiskit function that provides the energy expectation value of a given wavefunction, given some ...

1

For the freeze list, as KAJ226 mentioned, it's the core orbital of LiH molecule. In this case it would be the 1$s$ orbital of Li and we only care about the 2$s$, 2$p_x$ orbital of Li, 1$s$ orbital of H, total 6 qubits considering the spin-degeneracy. It is revealed that frozen core approximation does not make any significant error in a quantitative manner. I ...

1

The question of how to order the terms when Trotterizing is an active area of research. It turns out to be beneficial to put commuting terms next to each other. It can also be beneficial to randomise the ordering of terms in each Trotter step (as this suppresses errors), or to even not include certain low value terms. A paper that gives many references to ...

1

One way to improve the accuracy of your answer is to increase the number of shots. I see that you set your shots to 1024, so you might want to increase that to 8192 shots (the maximum shots allowed by Qiskit). You are right that noise is the problem. The noise are being introduced both by the gate operations and the read-out/measurement process. To fix the ...

1

In the paper A Generic Compilation Strategy for the Unitary Coupled Cluster Ansatz they benchmark on a bunch of chemistry circuits that can be found here. I should add that these circuits contain only a single Trotterisation step of the chemistry simulation. The actual simulation circuit would repeat this structure many many times.

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