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I'm trying to understand how to read results from an NMR quantum computer.

According to Nielsen and Chuang:

The principal output is the free induction decay signal $$V(t) = V_0\text{Tr}\left[e^{-iHt}\rho e^{iHt}(iX_k+Y_k)\right]$$

This signal is measured for some period of time, and then the results are Fourier-transformed into a plot like this one

enter image description here

My question is: what do these peaks mean? Suppose I want to know how much of my ensemble is in the $|110\rangle$ (or whatever) state, how do I read this off of the plot?

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With magnetic resonance based quantum computing, the amplitude(integrated area) and relative phase of the read-out spectra tells you the state of the qubits.

In this particular example, two different carbon qubits are read out simultaneously, while the third qubit, a hydrogen, would need a separate experiment to read out. The spectrum consists of two distinct doublet of doublets each one representing a qubit. The positive phase would represent being in the 1 state, and integrating the doublet of doublets would yield the probability of being in that state. A negative spectra would represent being in the 0 state.

Though you do need to be careful since the phase in these experiments is arbitrary, so your 1 could very well be a 0, but the relative phase between qubits is not arbitrary. So usually you would also gave an internal phase reference.

For further reading some of the work from J. A. Jones is a good place to start:

arXiv:quant-ph/0106067

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