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In simpler terms your question is: if noise/decoherence keeps entering the computation, how can a big computation possibly survive? The key concept you're missing is quantum error correction, which can pump noise/decoherence back out of the system. Of particular practical interest is the surface code.

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What you call a black box is simply isolating the quantum system that stores (or represents) your qubits from the environment. This can be done in several ways depending on your physical realization. For example, in an ion trap based quantum computer, one uses states of a single ion to represent a qubit, and isolates that from the environment by levitating ...

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In short: "coherence"! It's the crucial difference between quantum and classical. $\rho=|0\rangle\langle 0|+|1\rangle\langle 1|$ is just a statistical mixture, and behaves like a classical coin - any measurement that you perform on it gives a 50:50 split between the two possible outcomes. By contrast, $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ is a very ...

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To simplify things a bit, let's take a single qubit and a single qutrit for comparison. First, the amplitude damping channel (giving e.g. emission of a photon) for a qubit is $\mathcal E\left(\rho\right) = E_0\rho E_0^\dagger + E_1\rho E_1^\dagger$, where $$E_0 = \begin{pmatrix}1 && 0 \\ 0 &&\sqrt{1-\gamma}\end{pmatrix}, \quad E_1 = \begin{... 6 The statement in Wikipedia is very generic, and only cites this paper as a reference. Quoting from the abstract of the paper: We demonstrate that decoherence of many-spin systems can drastically differ from decoherence of single spin systems. The difference originates at the most basic level, being determined by parity of the central system, i.e., ... 5 There are multiple ways to mathematically express the state of a quantum system. One is to write it as a linear combination of basis states, as either a vector or a matrix, as you have here. This is useful in many cases, but it also seems to be confusing to newcomers, since things like \vert 0 \rangle + \vert 1 \rangle, \vert 0 \rangle \langle 0 \vert + \... 5 The quantum circuit model describes a quantum computer as a closed quantum system and assumes that there is a system which executes the circuit but is completely isolated from the rest of the universe. In the real world, however, there are no known mechanisms for truly isolating a quantum system from its environment. Real quantum systems are open quantum ... 5 I believe the current record is held by the Jian-Wei Pan group in China, who are able to generate entanglement via a satellite. The journal article is here, while there's plenty of media coverage that is a bit more accessible, e.g. New Scientist. This claims a distance of 1203km. 4 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 ... 4 Photons travel fast, and there's often the option to transfer their entanglement to solid state. Of course, the advantage of transferring entanglement to a solid state qubit is that one is able to operate with it (one- and two-qubit gates, for example) with ease and efficiency, whereas it is very hard to effect two-qubit quantum gates on photons themselves, ... 4 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. ... 4 Why not input one half of a maximally entangled state as the input to the black box (so that half has the same dimension as the input dimension)? Then you could test your favourite measure, such as the purity, of the full output state. If the oracle corresponds to a unitary evolution, the purity is 1. The less coherent the smaller the purity. Incidentally, ... 4 Suppose you have a state \rho, and a random process that changes this to a state \rho_j with probability p_j. If you know what the value of j is, your knowledge of the resulting state will be described by the corresponding \rho_j. If you have no information regarding j, your knowledge will be described by$$\sum_j ~ p_j ~ \rho_j$$This is a ... 3 You could also look at the following webpage: https://quantumcomputingreport.com/scorecards/qubit-quality/ where they provide recent (I'm not sure how often they update this scores) values for gate fidelities and decoherece times for IBM and Rigetti chips (unfortunately they don't give any results on ion traps and photonics, since these machines are not ... 3 I guess your best shot would be to look for experimental comparisons like this one on Arxiv. But I am not aware of a tracking. I do not think we can consider having a "state of the art" in this field. The goal being to make them always better of course with better connectivity for instance (a possible factor to take into account). 3 Josu, You might be mis-understanding something. Your equation for the Pauli channel says that when t\rightarrow \infty, all operators (X,Y,Z,I) have an equal probability (1/4) of transforming the density matrix \rho. You seem to be suggesting that t\rightarrow \infty, the probability of the I operator acting on \rho whould go to 0. Keep in mind ... 2 Let's assume that your black box processes classical inputs (i.e. a bit string) to classical outputs in a deterministic way, i.e. it defines a function f:x\mapsto y. If you can only prepare and measure separable states in that basis, all you can determine is what that function f is. Assuming that all the outputs are different, it could have been ... 2 I'm not exactly sure what you mean by quantumness of your black box. So maybe there are some more sophisticated approaches (similar to the other answer you could use an entanglement witness to show that your black box is not entanglement breaking). However, in general you could perform quantum process tomography (see e.g. arXiv:quant-ph/9611013). 2 Your question revolves implicitly around the concept of quantum decoherence and how to protect real-world implementations of qubits from it for a long time. This is an incredibly general problem, and at the same time, the details are wildly dependent on the technology used. If you have access to it, you can check chapter 5 : "Noise and decoherence" of ... 2 Imgine that there is a bit string \vec{k}\in\{0,1\}^n. We use this to specify sites (bit value 1) where an error has occurred, and sites (bit value 0) where no error has occurred. The number of 1s in the bit string is k. The probability of this particular error arising is then p^k(1-p)^{n-k} because there are k sites with an error and n-k sites ... 2 Decoherence is the very general term which, more or less, is anything resulting in a loss of purity during the evolution of a system. Sometimes, when people are being a bit non-specific, they might be thinking of a particular type of decoherence such as dephasing (or perhaps depolarising) when they use the term decoherence. Relaxation and dephasing are two ... 2 It's fundamentally similar to/the same as Baker–Campbell–Hausdorff (BCH). Generally, in quantum physics, this is most often used (or at least taught) with commuting Hamiltonians:$$e^{-i\left(H_1+H_2\right)t} = \sum_{n=1}^\infty\frac{\left(-it\right)^n}{n!}\left(H_1+H_2\right)^n = e^{-iH_1t}e^{-iH_2t}e^{\frac{1}{2}\left[H_1, H_2\right]t^2}\cdots$$where the ... 2 The state you've given, |\psi\rangle=a|0\rangle+b|1\rangle is a pure state. It has not been decohered. Decoherence is a process which turns pure states into mixed states. We usually write these in terms of density matrices. A pure state can be written in this form:$$ |\psi\rangle\langle\psi|=\left(\begin{array}{cc} |a|^2 & ab^\star \\ a^\star b & |...

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Is it possible for quantum computers to exhibit behavior similar to flip errors in classical computers where a state |0⟩ becomes |1⟩ due to this error? Yes. It would be a very abrupt error if you're talking about errors on physical qubits. Usually, we'd think of an error as being a little bit of an X rotation (for example). However, the effect of performing ...

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If the measurement is an irreversible process then the probability of the resultant state is towards the initial quantum state it was in, that is, if the resultant state is $1$ with a probability of, say, $63\%$, that means that is has a $63\%$ probability that the initial state was $|\psi \rangle= |1\rangle$. This is a fundamental misinterpretation....

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In the Copenhagen interpretation, there are only two kinds of things that one can do, one is evolution and other is the measurement. Measuring but not looking is equivalent to measuring the system and hence projecting it to one of the possible eigenstates. (Or maybe you can clarify more what you meant by not looking?) And after the system is probed in the ...

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This question is (in my opinion) the most important question to ask when trying to understand the mathematics of "quantum superposition." Quantum superposition is the essence of how quantum computations are made. If I have a coin, and I flip it 50% of the times I'll get heads and 50% of the time I can get tails: P(Heads) = 50% P(Tails) = 50% But if I ...

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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 ...

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There are two possible answers. Let's say the universe evolves from $t=0$ to $t_f$ then the unitary evolution $U$ from $0$ to $t_f$ induces a CP evolution on the subsystem. To see this, note that the composition of CP maps is CP. Now, the reduced (system) evolution is $Tr_E U\rho_s\otimes\rho_E U^\dagger$ which is a composition of the map \$\rho_s\...

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