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I read in the Afterword section of Nielsen and Chuang's book Quantum Computation and Quantum Information that

A second area of progress has been in understanding of what physical resources are required to quantum compute. Perhaps the most intriguing breakthrough here has been the discovery that quantum computation can be done via measurement alone.

How is measurement and computation different? Why is the above discovery a breakthrough? I am unable to appreciate its importance.

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A good answer to this question could explain the idea of measurement in "entangled" bases, particularly parity measurement. It would be nice to see a specific example of how this kind of measurement can produce the same kind of evolution as coherent entangling interactions. – DanielSank Jun 30 at 5:45
"In theory there is no difference between theory and practice, in practice there is.". You can, by the way, compute absolutely everything with a classical potential ... in theory. In practice you can't. – CuriousOne Jun 30 at 6:45

1 Answer 1

Nielsen and Chuang are referring to a scheme known as measurement-based quantum computation, for which many good learning resources are a short google search away.

The idea is that you prepare a large, highly entangled state involving a large number of qubits at the start of the experiment. You then proceed to make measurements on the state, potentially changing which measurements you perform based on the results of previous measurements. The key strength of this scheme is that every measurement is local to a single qubit. There is then no need to perform entangling operations once the state has been prepared.

This is particularly important in photonic quantum computing, because it is very, very hard to entangle photons, and at the moment this can only be done probabilistically. For photons, you can implement the two-qubit entangling gates required for universal quantum computing, but they will only work a fraction of the time. (Nowadays they tend to be heralded, which means that you know whether they worked or not, but this does not really solve the problem.) This scales very badly, because you need all your gates to work, and even a 95% success rate per gate looks terrible if you have more than a few gates.

For MBQC, on the other hand, you can try repeatedly to get a resource state that is entangled enough, and once you have it you don't have to generate more entanglement - you just have to do local measurements on each of your photons (plus potentially some classical postprocessing and feedback).

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