7

The spectral gap is pretty much independent of the promise gap. (First off, my feeling is that "promise gap" is a little bit misleading, though formally correct: What it really refers to, in essence, is the accuracy you are aiming for in determining the energy of the Hamiltonian. One way is to guarantee that the problem does not have a ground state ...


6

Note that your current definitions of the projection matrices $\{P_{1},P_{2},...,P_{n}\}$ are actually not projection matrices, since $P_{i}^{2} = I \not= P_{i} \,\, \forall i$. What works 'better' is if you have something like: \begin{equation} \begin{split} P_{1}^{+1} =& |0\rangle\langle 0 | \otimes I \otimes I....\otimes I \\ P_{1}^{-1} =& |1\...


5

Without sacrificing any generality we can define an ebit as a Bell state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ shared between two parties $A$ and $B$, and since we're concerned with communicating information each party gets to use any many local operations as they want. Then $$ \text{1 qubit} \geq \text{1 ebit } $$ can be understood as "...


4

You are correct that both terms reference the central limit theorem (CLT), which states that[1] ...the average of a large number $n$ of independent measurements (each having standard deviation $\Delta \sigma$) will converge to a Gaussian distribution with standard deviation $\Delta \sigma / \sqrt{n}$, so that the error on average scales with $n^{-1/2}$. ...


4

Quirk refers to the $\frac{1}{\sqrt{2}}|0\rangle + \frac{i}{\sqrt{2}}|1\rangle$ state as $|i\rangle$ and to the $\frac{1}{\sqrt{2}}|0\rangle - \frac{i}{\sqrt{2}}|1\rangle$ state as $|-i\rangle$: When I implemented this it just seemed like a natural choice at the time. I didn't get it from a textbook or a paper.


3

In my opinion the nature of these states becomes quite clear when we look at it from an optics angle. We can identify the computational basis states with the vertical and horizontal polarization directions: $$ |0\rangle \sim |\updownarrow\,\rangle \qquad |1\rangle \sim |\leftrightarrow\,\rangle $$ The superposition states then correspond to diagonally ...


2

Note that if you are considering a projective measurement, there is no need to deal with observables at all. A projective measurement is characterised by the basis $\newcommand{\ket}[1]{\lvert #1\rangle}\{\ket{u_i}\}_i$ on which you are measuring, and therefore the associated projection probabilities $p_i\equiv \lvert\langle u_i\rvert \psi\rangle\rvert^2$ (...


2

You simply want any diagonal operator that has distinct diagonal elements (which would imply that every basis element maps to a distinct output of the measurement). One convenient way to denote this in terms of Pauli matrices is $$ \sum_{i=1}^N2^{N-i-1}(1-Z_i) $$ For a basis state $|x\rangle$ where $x$ is a binary number, the eigenvalue is the decimal ...


2

This is another reference. $|i\rangle$ and $|\mbox{-}i\rangle$ are two orthogonal y-basis states. In the above link they are called $|R\rangle$ and $|L\rangle$. $$|i\rangle = \frac{1}{\sqrt{2}}\left[ \begin{array}{c} 1 \\ i \end{array} \right] \;\; , \;\; |\mbox{-}i\rangle = \frac{1}{\sqrt{2}}\left[ \begin{array}{c} 1 \\ -i \end{array} \right]...


2

Transpiling time is the time it takes for your circuit to be translated into a circuit that can be run on the backend of your choosing. This process includes converting gates into the standard basis gates ['cx', 'u1', 'u2', 'u3', 'id'], optimizing the circuit so it is shorter, mapping virtual qubits in the circuit to physical qubits, etc. Converting gates ...


2

This is a slightly messy topic which I often find to be misinterpreted. I'm not sure you can really point at a specific bit of the system as say "the entanglement is here". If you took the bipartition of (originally clean qubit) vs (everything else), which would seem like the obvious place to look, you will find that there is no entanglement ...


2

Perfect state transfer is generally discussed in the context of continuous time evolution. For example, you might be evolving under the influence of a Hamiltonian $H$. Particularly when one is considering some sort of underlying graph structure, you probably prepare an initial state corresponding to a single vertex (perhaps $|1000\ldots 0\rangle$) and you're ...


1

Similarities ebit and qubit: An ebit is one unit of bipartite entanglement, the amount of entanglement that is contained in a maximally entangled two-qubit state (Bell state). Requirement: If a state is said to have X ebits of entanglement (quantified by some entanglement measure) it has the same amount of entanglement (in that measure) as X Bell states. If ...


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 ...


1

The term is most often used in the context of the two-slit experiment. The interference pattern that you see, which is not simply what you'd get if you added the wavefunction ($\Psi_A$) associated with going through slit A to the wavefunction ($\Psi_B$) going through slit B, but there is a "mixed" term as follows: $$ \tag{1} |\psi|^2 = |\psi_A + \...


1

Transpilation (in this context) is backend-specific. Longer transpiling time just means the software took longer to make your circuits compatible and more optimized for the backend you selected (e.g. because your circuits require a lot of remapping). In general, transpiling time is important to those who do research on that topic. Much like classical ...


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