3.2.5 Quantum parallelism

Course subject(s) Module 3: Quantum Compiling and Quantum Dots

When performing a quantum algorithm we frequently start with a tensor product of all state at 0, i.e. |0⟩n state, followed by H⊗N Hadamard gates for all the qubits.

Right after applying the Hadamard gates the state corresponds to


This procedure allows us to prepare the initial state on a superposition of all possible combinations of 0-1 bit strings encoded in the quantum states. The quantum computer is then able to pursue 2N “different paths” of possible solutions for the algorithms that are carried out.

As we can see in the figure the final state corresponds to a superposition of tensor products between the input states |x and the solutions to a function f,|f(x). In this case, the function is encoded in the gate Uf.

This advantage provided by quantum circuits receives the name of quantum parallelism and it refers to the idea that quantum computers can look for different solutions in “parallel” and “at the same time”. But be careful, parallelism here does not precisely mean that. It is a common misconception to say that quantum parallelism allows different paths to be calculated simultaneously. This is not what happens, but rather the quantum algorithms apply gates to a superposition of quantum states. Each of these individual states can be studied via measurements, after which the wavefunction will collapse to one of the possible paths only.

In summary, we have learned a number of interesting things about quantum compiling, without needing to get too deep in the gritty details.  Continuing our musical metaphor, we learned that although there are a vast, infinite number notes, we only need a piano that can play just a few to perform any song we want.  Depending on the particular notes available, the song might be easy to play on one piano and extremely tiring on another, but at least it can be done.  We ultimately care about finding beautiful masterpieces that can be easily played, which amounts to writing the songs themselves and expressing them in terms of basic components we know how to deal with, i.e. like encoding functions in unitaries.  Then, we simply play the song and… well, the metaphor breaks down here, but that’s to be expected.  There simply is no classical analogy for quantum parallelism, and even though we can only access one possible output after any given measurement, what makes a certain algorithm or “song” a masterpiece is how cleverly it amplifies the probability of certain desired outputs over all of the undesired ones.
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