Quantum Information Processing

classically. However, it is worth noting that the comparison is not entirely fair: A truly classical
oracle answering parity questions or implementing the black box on the states of classical
bits is useless to a quantum algorithm. To take advantage of such an algorithm, it must be
possible to use superpositions that are not implicitly collapsed. Collapse can happen if
the oracle makes a measurement or otherwise “remembers” the question that it was asked.
Resource Accounting
When trying to solve a problem using quantum information processing, an important
issue is to determine what physical resources are available and how much of each
resource is needed for the solution. As mentioned before, in classical information, the
primary resources are bits and operations. The number of bits used by an algorithm is its
space requirement; the number of operations used, its time requirement. If parallel computation
is available, one can distinguish between the total number of operations (work)
and the number of parallel steps (time).
When quantum information processing is used, the classical resources are still
relevant for running the computer that controls the quantum system and performs any
preprocessing and postprocessing tasks. The main quantum resources are analogous to
the classical ones: Quantum space is the number of qubits needed, and quantum time,
the number of quantum gates. Because it turns out that reset operations have a thermodynamic
cost, one can count irreversible quantum operations separately. This accounting
of the resource requirements of algorithms and of the minimum resources needed to
solve problems forms the foundation of quantum complexity theory.
As a simple example of resource accounting, consider the algorithm for the parity
problem. No classical computation is required to decide which quantum gates to apply
or to determine the answer from the measurement. The quantum network consists of a
total of 11 quantum gates (including add and meas operations) and one oracle call (the
application of the black box). In the case of oracle problems, one usually counts the
number of oracle calls first, as we have done in discussing the algorithm. The network is
readily parallelized to reduce the time resource to 6 steps.
Part III: Advantages of Quantum Information
The notion of quantum information as explained in this primer was established in the
1990s. It emerged from research focused on understanding how physics affects our
capabilities to communicate and process information. The recognition that usable types
of information need to be physically realizable was repeatedly emphasized by Rolf
Landauer, who proclaimed that “information is physical” (1991). Beginning in the
1960s, Landauer studied the thermodynamic cost of irreversible operations in computation
(1961). Charles Bennett showed that, by using reversible computation, this cost can
be avoided (1973). Limitations of measurement in quantum mechanics were investigated
early by researchers such as John von Neumann (1932a and 1932b) and later by
Alexander Holevo (1973b) and Carl Helstrom (1976). Holevo introduced the idea of
quantum communication channels and found bounds on their capacity for transmitting
classical information (1973a). Initially, most work focused on determining the physical
limitations placed on classical information processing. The fact that pairs of two-level
systems can have correlations not possible for classical systems was proved by John
Bell (1964). Subsequently, indications that quantum mechanics offers advantages to
information processing came from Stephen Wiesner’s studies of cryptographic applications
in the late 1960s. Wiesner’s work was not recognized, however, until the 1980s,
Quantum Information Processing
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