Operation

Unsolved problem in physics:

Is a universal quantum computer sufficient toefficiently simulate an arbitrary physical system?
(more unsolved problems in physics)

While a classical 3-bit state and a quantum 3-qubit state are each eight-dimensionalvectors, they are manipulated quite differently for classical or quantum computation. For computing in either case, the system must be initialized, for example into the all-zeros string,

{\displaystyle |000\rangle }

, corresponding to the vector (1,0,0,0,0,0,0,0). In classical randomized computation, the system evolves according to the application of stochastic matrices, which preserve that the probabilities add up to one (i.e., preserve the L1 norm). In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively rotations (they preserve that the sum of the squares add up to one, the Euclidean or L2 norm). (Exactly what unitaries can be applied depend on the physics of the quantum device.) Consequently, since rotations can be undone by rotating backward, quantum computations are reversible. (Technically, quantum operations can be probabilistic combinations of unitaries, so quantum computation really does generalize classical computation. Seequantum circuit for a more precise formulation.)

Finally, upon termination of the algorithm, the result needs to be read off. In the case of a classical computer, we sample from theprobability distribution on the three-bit register to obtain one definite three-bit string, say 000. Quantum mechanically, onemeasures the three-qubit state, which is equivalent to collapsing the quantum state down to a classical distribution (with the coefficients in the classical state being the squared magnitudes of the coefficients for the quantum state, as described above), followed by sampling from that distribution. This destroys the original quantum state. Many algorithms will only give the correct answer with a certain probability. However, by repeatedly initializing, running and measuring the quantum computer's results, the probability of getting the correct answer can be increased. In contrast, counterfactual quantum computation allows the correct answer to be inferred when the quantum computer is not actually running in a technical sense, though earlier initialization and frequent measurements are part of the counterfactual computation protocol.

For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer,Shor's algorithm, Grover's algorithm,Deutsch–Jozsa algorithm, amplitude amplification, quantum Fourier transform,quantum gate, quantum adiabatic algorithmand quantum error correction.