6

The $t$ in $t$-design is essentially a measure of how good a job the set of gates does in terms of randomising a state (the larger t, the more random, with properly random requiring the infinite limit). Often, you want to compute the average of some function over all possible pure input states, which is equivalent to fixing the input state and averaging over ...


6

Nielsen and Chuang in their book "Quantum Computation and Quantum Information" have section (Chapter 9) on distance measures for quantum information. Surprisingly they say in Section 9.3 " How well does a quantum channel preserve information?" that when comparing fidelity to the trace norm: Using the properties of the trace distance established in the ...


5

The numbers you are describing are very very large. To the point, it appears that they are numbers whose representation in decimal (or binary) are large enough that it there is little to no prospect of there being enough matter in the entire universe to store those numbers, in a place-value representation such as those. This being the case, no technology &...


4

I think the way that it is used in deriving bounds on quantum cloning is quite insightful, and hopefully gives you a flavour of the broader context. Where possible I'll skip some of the details in favour of description. Imagine you want to clone an unknown one-qubit quantum state $|\psi\rangle$, making two copies. We know that this is impossible to do ...


2

The Quantum Volume is a benchmark for near term, noisy quantum systems. Indeed, like other random unitary benchmarks, you need to be able to sample the ideal distribution. This distribution comes from classical simulations, so your limited to about the ~40 or so qubit limit. However, the Quantum Volume itself was designed to benchmark not only the quantum ...


2

Quantum volume is a bad metric for this purpose. For example, suppose you have a ten thousand by ten thousand grid of qubits with a gate error rate of 1 in one thousand. The quantum volume of this grid is basically 0, because if you pick two qubits at random they will on average be more than one thousand steps apart. So an error will almost certainly occur ...


2

If you look at the literature for blind quantum computation, there is the concept of a "trap state". Basically, something that isn't part of the main computation that is supposed to give specific results so that you can easily verify that the computer is behaving as expected. I believe some of these trap states are Bell pairs, and the measurements performed ...


1

An average over conjugations is known as a “twirl”. The “twirling” operation originates from invariant theory (where it is sometimes called “transfer homomorphism”). Twirling a quantum channel over $P_1^{⊗n}$, $C_1^{⊗n}$ or $C_n$ takes it to one described by a polynomial number of parameters. The twirling operation will be useful if it preserves, at least ...


1

This may not exactly answer your question (which I suspect is still very much an open question, and what you're likely to get as answers are opinions), but have you looked at blind quantum computation? See here for another perspective. One way that we can describe that premise is to imagine some company claims to have developed a fabulous universal quantum ...


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