t b a b a

an arbitrary number of qubits. A quantum algorithm is typically specified as a se-
quence of unitary transformations U1, U2, U3, ..., each acting on a small number of
qubits, typically no more than three. In principle, the first step towards a physical
implementation of the algorithm is identifying the Hamiltonians which generate these
unitary transformations, i.e., U1 = e iH1t/� , U2 = e iH2t/� , U3 = e iH3t/� . The next step is
designing the physical apparatus so that the interaction Hamiltonians H1, H2, H3, ...,
are successively turned on at precise moments of time and for predetermined lengths
of time. In reality, only some types of Hamiltonians can be turned on and off as re-
quested by a quantum computation. Most of the physical implementations proposed
so far consider only two-body interactions. However, even two-qubit interactions are
not easily achievable.
• The physical system must have a qubit-specific measurement capability. The result of
a computation is read out by measuring specific qubits. If an ideal measurement is
performed on a qubit with density matrix:
ρ = p | 0〉〈0 | +(1 − p) | 1〉〈1 | +α | 0〉〈1 | +α ∗ | 1〉〈0 |,
it should give outcome “0” with probability p and “1” with probability 1 − p indepen-
dent of α, the state of neighboring qubits, or any other parameters of the system and
the state of the rest of the computer is not changed. Actually, realistic measurements
are expected to have quantum efficiencies less, even much less than 100%. The low
quantum efficiency measurement could be circumvented either by rerunning the com-
22