if |c1> |c2>==|1) and leaves |a) invariant when |c1>= /c2> = |0). Quantum bits and quantum gates CCR = Figure 16.8 Deutsch gate or CCR gate, with corresponding 8 X 8 matrix in reduced form, where R is a unitary rotation operator. Deutsch or CCR gate A particular type of CCU gate, called the Deutsch gate, is obtained when one chooses U = R(), as shown in Fig. 16.8. To make the Deutsch gate, a restriction applies to the rotation angle O. Indeed, in the Deutsch gate the angle O should be incommensurate with IT, which means that A/H is not a rational fraction.8 With such a property, any qubit /u) on the Bloch sphere that lies at an angle x0 from the gate's target qubit |u) can be reached with arbitrary precision by applying the CCR gate a finite number of times k, i.e., /w) = = R(+k0)|u) can be made arbitrarily close to /v) if k/x 2 1. In particular, the rotation angle ko can be made arbitrarily close to /2, which makes the Deutsch gate closely similar to a Toffoli gate (R(/2) = X). Consider, finally, that single 2 X 2 rotations defined as R = R(O) make it possible to transform any input qubit |u) into any output qubit /v) on the Bloch sphere. A controlled-R gate has the same complete transformation capability on the 2-qubit space, and a CCR or Deutsch gate on the 3-qubit space. It is beyond the scope of this chapter to establish formally that, actually, quantum circuits based only on 3-qubit CCR or Deutsch gates and CCNOT gates are capable of achieving any n-qubit transformations in the n-qubit space. Quantum circuits The matrix representation of 2- and 3-qubit gates may look somewhat impractical to handle, except in generic cases when they can be put in some reduced form. Therefore, it would seem that quantum-gate circuits with multiple gates and control qubits are not easy to model and analyze. In reality, however, gate circuits are far simpler to handle! I shall illustrate this through a few examples. Consider first the 2-qubit quantum circuit involving single-qubit gates (J, K) and controlled-U gates as shown in