Quantum Information Processing

Initial State Final State
→ nand () = ,
→ nand () = ,
→ nand () = , and
→ nand () = . (3)
The nand gate can be applied to any two bits in a sequence. For example, it can be
applied to the fourth and second bits (in this order) of four bits, in which case the
initial state is transformed to , setting the fourth bit to .
Other operations on bit sequences include adding a new bit to the beginning
(prepend) or end (append) of a sequence. The new bit is always initialized to It is
also possible to discard the first or last bit regardless of its state. Versions of these
operations that are conditional on the state of another bit may also be used. An
example is the conditional append operation: “If the k th bit is in the state , then
append a bit.”
The gates just introduced suffice for implementing arbitrary state transformations
of a given bit sequence. Instructions for applying gates in a particular order are
called a circuit. An important part of investigations in information processing is to
determine the minimum resources required to perform information-processing tasks.
For a given circuit, the two primary resources are the number of gates and the total
number of bits used. The circuit complexity of a desired transformation is the minimum
number of gates needed to implement it.
The model of computation defined by the ability to apply gates in a fixed
sequence is called the circuit model. Classical computation extends the circuit
model by providing a means for repeating blocks of instructions indefinitely or until
a desired condition is achieved. In principle, it is possible to conceive of a generalpurpose
computer as a device that repeatedly applies the same circuit to the
beginnings of several bit sequences. In this article, we take for granted a traditional
programmable computer based on classical information. Thus, a quantum algorithm
is a program written for such a computer with additional instructions for applying
gates to quantum information. The computational power of this model is equivalent
to that of other general-purpose models of quantum computation, such as quantum
Turing machines (Yao 1993).
For an introduction to algorithms and their analysis, refer to Thomas Cormen et
al. (1990). Christos Papadimitriou wrote (1994) a useful textbook on computational
complexity with an introduction to classical computation and computational
machine models.
Quantum Information Processing
Number 27 2002 Los Alamos Science 5