288 Quantum computing
(a) (b)EnergyFig. 13.4(a) The energy levels of two spin-1/2 particles which do not interact with each other. The frequencies that drive
the transition between the levels of Qubit 1 and Qubit 2 areω 1 andω 2 , respectively. In NMR these levels correspond to the up
and down orientations of two protons in a strong magnetic field, i.e. the states|mI=± 1 / 2 〉for each proton. The difference in
the resonance frequenciesω 1
=ω 2 arises from interaction with their surroundings (other nearby atoms in the molecule). (b)
The energy levels of two interacting particles, or qubits. The interaction between the qubits makes the frequency required to
change the orientation of Qubit 2 depend on the state of the other qubit (and similarly for Qubit 1). To denote this, the two
new resonance frequencies close toω 2 are labelled with the superscripts| 0 x〉and| 1 x〉. (Note this rather cumbersome notation
is not generally used but it is useful in this introductory example.) The absorption spectrum is drawn below the corresponding
transitions. Aπ-pulse of radio-frequency radiation at angular frequencyω| 21 x〉switches Qubit 2 (| 0 〉 2 ↔| 1 〉 2 )if and only if
Qubit 1 is in| 1 〉 1. This gives the CNOT gate of Table 13.1. For details of the NMR technique see Atkins (1994). In NMR the
interaction of the nuclear magnetic moment with a strong external magnetic fieldgIμBI·Bdominates. For a field of 10 T the
protons at the centre of the hydrogen atoms in the sample have resonance frequencies of 400 MHz. This frequency corresponds
to approximatelyω 1 / 2 π ω 2 / 2 πin the figure. Thechemical shiftcausesω 1 andω 2 to differ by only a few parts per million,
but this is resolved by standard NMR equipment. Atkins (1994) describes how the fine structure in NMR spectra, shown in
(b), arises from spin–spin coupling between the two nuclear spins.
(^5) The amplitudes and phases of the to know that spontaneous emission is negligible. (^5) To make our discussion
states within the superposition must
not change spontaneously, i.e. the de-
coherence must be very small over the
time-scale of the experiment.
less abstract, it is also useful to know that in NMR experiments the
energy differences between the levels shown in Fig. 13.4 arise from the
orientation of the magnetic moment of the nuclei, proportional to their
spin, in a strong magnetic field and that radio-frequency radiation drives
transitions between the states. There is a splitting ofω 1 between the
up and down states of the first qubit (| 0 〉and| 1 〉)andω 2 for the
second qubit. Therefore a pulse of radio-frequency radiation at angular
frequencyω 1 changes the orientation of the first spin (Qubit 1); for
example, aπ-pulse (as defined in Section 7.3.1) swaps the states of this