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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/... 3 First, in order to express |H'\rangle, |V'\rangle, |R\rangle and |L\rangle in terms of |H\rangle and |V\rangle, add and subtract the pair of equations (2) and add and subtract the pair (3), to get |H'\rangle + |V'\rangle = \sqrt2|H\rangle \\ |H'\rangle - |V'\rangle = \sqrt2|V\rangle \\ |R\rangle + |L\rangle = \sqrt2|H\rangle \\ |R\rangle - ...

3

One idea is to do polarimetry. By using a polarizing beam splitter, the polarization qubit can have each of its polarization components directed to a different detector for photon counting (ideally a single-photon detector, here). A polarizing beam splitter might send horizontally polarized photons in one direction and vertically polarized photons in another....

3

You mean something like $$W_{G}(\mathbf{r}) =\frac{2^{n}}{\pi^{n} \sqrt{\operatorname{Det} \sigma}} \mathrm{e}^{-(\mathbf{r}-\overline{\mathbf{r}})^{\top} \boldsymbol{\sigma}^{-1}(\mathbf{r}-\overline{\mathbf{r}})},$$ where $W_{G}(\mathbf{r})$ is the Wigner function corresponding to a Gaussian state, $\mathbf{\sigma}$ its covariance matrix, and $\overline{r}$...

2

For a general overview about hardware and the difficulties they present, I recommend section three of Quantum Computing: An Overview Across the System Stack. Another good introduction would be Quantum Computing Hardware Implementation Methods: A Survey over Categories. Let me also give you some papers specific to certain implementations and their recent ...

2

As for any platform, one has to choose a suitable $d$-dimensional "computational" subspace. Suitability depends on your application, but generally it means that one should be able to perform operations on that subspace and couple it to other qudits. In practice, these operations will couple the qudit to degrees of freedom outside of the subspace ...

2

Strictly, what you have to calculate is that for all $i$ and $j$ $$\langle 0_L|U_iU_j|1_L\rangle=0$$ and $$\langle 0_L|U_iU_j|0_L\rangle=\langle 1_L|U_iU_j|1_L\rangle.$$ (I've ignored the Hermitian conjugate because all the single-qubit errors are Hermitian.) Obviously there's a lot of work involved in calculating all $28^2$ cases of $i,j$. You can at ...

2

These are also known as SU(2)-coherent states; one original reference is https://doi.org/10.1103/PhysRevA.6.2211. In a spin system, with states labeled by the eigenvalue of the total angular momentum operator $\mathbf{J}^2$ and the z-projection of the angular momentum operator $J_z$, $$\mathbf{J}^2||J,m\rangle=J(J+1)||J,m\rangle\quad J_z ||J,m\rangle=m||J,m\... 2 Let |\psi\rangle be an eigenstate of an operator A, A|\psi\rangle=\lambda|\psi\rangle. Then$$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.$$In this particular case, A=-igt(a_2^\dagger a_1+a_1^\dagger a_2), of which |00\rangle is an eigenstate with ... 2 From the definition of the covariance matrix,$${\sigma}_{ij}=\left\langle \frac{x_i x_j+ x_j x_i}{2}\right\rangle -\langle x_i\rangle\langle x_j\rangle,$$where we define$$\boldsymbol{x}=(q_1,p_1,\cdots, q_N, p_N)^\top.$$Given that 2i\Omega_{jk}=[x_j,x_k], we learn that$${\sigma}_{ij}=\left\langle x_i x_j\right\rangle -\langle x_i\rangle\langle x_j\...

1

Modes are governed by eigenfunctions, I agree. In quantum optics, we need more than just eigenfunctions to describe a state of light: we need to know how many photons have properties corresponding to each eigenfunction. This is somewhat beyond what an eigenfunction describes, so we need a new term. For example, we can have a two-photon state of light where ...

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