Does The Classical Limit of Quantum Computing Exists? In any standard college book on quantum mechanics and field theory, one for sure has encountered some expressions like " the classical limit corresponds to setting hbar to zero" or quantum loops are coming in powers of hbar etc.. However, in the context of quantum computing, it is unclear how the mapping from quantum to classical computing may be quantified?
(also, the term "setting hbar to zero" is unprecise but can be used for most of cases at the level of students to understand). In geometric quantization schemes, we may multiply poisson brackets with some quantity proportional to -ihbar
What is the limit where the quantum computation theory reduces to classical?
 A: "Gate-based" quantum computers are the analog of digital classical computers. Classical computers implement logic gates such as AND, NOT, OR, XOR, NAND. Quantum computers can implement these gates but they can implement other gates. A typical example is the $\sqrt{\text{NOT}}$ gate. this is a gate which negates the quantum bit when applied twice. There is no analog for the $\sqrt{\text{NOT}}$ gate in classical digital computing. Furthermore, the existence of the $\sqrt{\text{NOT}}$ gate is a discrete fact. That is, there's not some knob we can tune that makes the $\sqrt{\text{NOT}}$ gate gradually go away until we recover classical computing with no such gate.
So the answer to your question is a flat "no". No, there is no "classical limit" of gate-based quantum computing. In fact, digital quantum computers function based on exactly the components of quantum mechanics that have no classical analog. For example spin $N$ systems with large $N$ are approximately like classical rotors, but quantum computers use spin 1/2 systems which behave maximally quantum mechanically. In quantum mechanics and classical computers there can be correlations between different bits, but quantum computers heavily leverage entanglement, a type of correlation which is impossible classically. In other words, digital quantum computing gets it's power precisely because it has no classical analog.
On the other hand, there are so-called "quantum simulators" in in which medium sized quantum systems are isolated and studied in fine detail to help us better understand other similar, larger and more applicable quantum systems. In this case we don't have control of every individual degree of freedom like in a digital quantum computer, but we have enough isolation and control that we can study something interesting. I like to call such simulators "analog quantum computers". An example of an "analog classical computer" is a wind tunnel. We can't always solve the Navier-Stokes equation, but we can build a wing and blow wind over it to see how the wing performs. We create a physical scenario, click "go", watch what happens, and use the results to inform and refine our models. For these analog quantum computers you could make a case that, in some cases, there may be classical limits for the quantum computers.
A: I think there is a very natural sense in which the "classical limit of quantum computing" does indeed exist. Of course, this depends on what exactly you mean with the question, but it's completely legitimate to ask how classical computation arises from quantum computation.
Obviously, this is not to say that quantum algorithm reduce in some limit to classical algorithms.
But you can ask in what regimes (useful) quantum algorithms are possible at all.
In the limit where decoherence and other similar factors are prominent, a quantum device cannot sustain quantum algorithms, but you might still be able to perform "classical" operations.
In other words, noise effectively constraints the space explored by the dynamics of the state. A "classical limit" can be when only "classical states" can exist in the device (that is, mixtures of states in some properly defined computational basis). Then there is no hope of performing useful quantum algorithms and the device is classical from the computational point of view.
