circuit described in Chapter 15 for reversible computation with classical bits. We can now conceive of it as a quantum circuit capable of performing plain addition with qubits. The corresponding building blocks are the quantum gate circuits SUM and CARRY, which are based on CNOT and CCNOT or Toffoli gates, as shown in Fig. 16.13. Given two qubit operands /x), ly) and a carry qubit |c), the outputs + y + c) represent the quantum equivalents of XOR or modulo-2 addition. Note the qubit output |c') in the CARRY gate circuit; this represents the carry result of the plain addition between 10 For the exponential-operator representation in the right-hand side, see Appendix N. Quantum bits and quantum gates CARRY x y y Figure 16.13 Quantum gate circuits SUM and CARRY, built from CNOT and CCNOT or Toffoli gates. Figure 16.14 Quantum plain-adder for two qubit registers |ao) |a1>|a2), as based on gate circuits SUM (S) and CARRY (C) shown in Fig. 16.13. The gate C* corresponds to the reverse operation of the CARRY gate. /x), ly) and c). Also note the ancilla bit |0) at the CARRY gate input. A 10-qubit plain-adder circuit, which is homologous to that described in Chapter 15 for classical computing, is shown in Fig. 16.14. This circuit performs the quantum addition of two qubit registers containing 3-qubits each, namely |a0)|a1)|az), (bo)|b1)|b2), and outputs the result as As discussed in Chapter 15, this circuit architecture can be extended to perform plain addition with registers of any size.¹1 As also discussed in the previous chapter, the circuit can be reversed (i.e., traversed from right to left) to perform subtraction, including the generation of a "negative sign" qubit. Furthermore, the con- catenation of the circuit makes it possible to perform operations such as multiplication and exponentiation in modular algebra. One must not conclude, however, that quan- As adapted from V. Vedral and M.I B. Plenio, Basics of quantum computation. Prog. Quant. Electron., 22 (1998), 1-39. 16.5 Tensor products tum algorithms simply emu