9

The naming started in NMR and it has become the difference between the following two experiments. Experiment one: Prepare the qubit in a superposition state (apply a H gate) and vary the wait time and then measure in the superposition basis (apply another H gate). The decay time of this experiment is $T_2^*$. We commonly call this a Ramsey experiment. ...


9

"Is Quantum Biocomputing ahead of us?" There has been some work done on biocomputing, quantum computing, spin chemistry, and magnetochemical reactions. Correlated radical pairs — pairs of transient radicals created simultaneously, such that the 2 electron spins, one on each radical, are correlated — on photoactive magnetoreceptive proteins such as ...


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 ...


5

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


5

I have worked with NVs in nanodiamonds a little bit, and you are totally right, surface characteristics have a huge influence on how far we can push them. There are definitely multiple groups working on the chemistry/material science that are working to clean up the surfaces as much as possible. I had a colleague, Carlo Bradac who worked with our chemistry ...


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\...


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/...


4

Given a quantum system in a state defined by a density matrix $\rho$, it is an accepted terminology to use the term population for the diagonal matrix elements (not necessarily in the computational basis). Since a normalized vector corresponds to a pure state, thus we can define a population of the pure state $\psi$ by: $$P_{\psi} = \langle \psi | \rho | \...


3

From Chapter 15 of NII's quantum information lecture series on "Fundamentals of Noise processes" (link here): An applied DC field $H_0$ is not completely uniform in all space points. If many spin qubits are placed in such an inhomogeneous DC field, they have different Larmor frequencies. This leads to the dephasing effect if we compare the phase ...


3

There has been a great deal of scientific debate over evidence of quantum effects in biology due to the difficulties of reproducing scientific evidence. Some have found evidence of quantum coherence while others have argued this is not the case. (Ball, 2018). The most recent research study (in Nature Chemistry, May 2018) found evidence of a specific ...


3

Much has been written about Quantum Biology. A somewhat old -and yet, solid- take is that of Phillip Ball, The dawn of Quantum Biology (Nature 2011, 474, 271-274). For now, let's not review that and instead focus on your questions. On the first question:(is it solving our problems?) A system (or process) described by Quantum Biology is non-trivially ...


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 ...


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 $A$, $A|\psi\rangle=\lambda|\psi\rangle$. Then $$e^A |\psi\rangle = \sum_{k=0}^\infty \frac{A^k}{k!}|\psi\rangle = \sum_{k=0}^\infty \frac{\lambda^k}{k!}|\psi\rangle = e^\lambda |\psi\rangle.$$ 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

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

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

Articles on technology from 10 years ago are often outdated, to some extent the same can be said of last year's information. Occasionally something will stand for decades, or fall into decline only to be revisited later. The most optimistic perspective is: someone is working on it. Here are some more recent articles on room temperature QC: "Room-...


2

I don't know the translation into physics, but the circuit you want for the most basic demonstration is the following: Here, $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$, and the gates are controlled-not gates and controlled-phase gates. The state $|\psi\rangle$ can be any input state initially. The first time the circuit is run prepares the $|\psi\rangle$ ...


1

The best I have it's this generic answer, which I put here for clarity, hoping for improvements/corrections or even to be superseded by something better: If the limiting factor for fidelity in a given architecture+algorithm are the single-qubit gates, or the two-qubit gates, or the measurement, and if this limiting factor is not optimized in a ZEFOZ point,...


1

I think your reference has the answer: nitrogen vacancy centers in diamond, where you can do one qubit gates at room temperature. In fact, even higher temperatures are possible, but you will have to play a tradeoff between fidelity and temperature at some point. That said, NV centers are not scalable, and I don't think more than 2 qubits will ever be ...


1

Let me go for a self-learner experience. After some reading, my short answer to my own question Would the calculation of the loss of entanglement be necessarily related to delocalized vibrational modes that simultaneously involve the local environment of both triplets? is: probably yes, but not necessarily/primarily. A longer answer follows. With a ...


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