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the auxiliary qubit. The (n+1) qubits are guided into
j the beginning of the circuit, as input state. The process
will continue until a result is read-out in the conditional
measurement device.
0
This simulation of duality computer also provides a
relativity view regarding quantum computer. In a du-
ality computer, a double-slit is located statically, and a
quantum computer is moving. However in the duality
FIG. 3: Schematic illustration of a recycling duality compu- mode,aquantumcomputerisstatic,andthedouble-slit,
tation. After the auxiliary-qubit-conditioned measurement,
theauxiliaryqubit,changestosimulatethemotionofthe
if a result is obtained, the calculation is completed and the
double-slit. Tomakeaquantumcomputermovingisvery
process is stopped. If no result is obtained, it destroys the
difficult, however it is much easier to add one additional
statein|0i auxiliaryqubit,andleavesastatewheretheaux-
qubit to an n-qubit quantum computer.
iliary is in state |1i. The state is restored to the input state
by a unitary operation V and guided to the input end. The In summary we have given a quantum computer real-
calculation processisrepeatedagainandagain untilthefinal ization of the duality computer. This realization itself
result is obtained. servesasanewmodeofquantumcomputing,theduality
mode. This provides a way to run classical algorithm in
quantumcomputersusingmuchreducedqubitresources.
Italsoprovidesamethodforerrorcorrectioninaduality
After the conditional-measurement, if a result is ob-
computer. With conditional measurement, the recycling
tained, the wave-function is collapsed, and the (U0 +
quantum computing mode is also proposed. These two
U1)ϕ result is read out. If no-result is obtained, then
modesprovidenewwaysandflexibilityinquantumalgo-
| i
the state in 0 collapses out, and the wave-function be-
rithm designs.
| i
comes
Acknowledgement
U0 U1 Helpful discussions with Prof. Dieter Suter and Dr
ψ ′ =N ′ − ϕ 1 , Qing-Yu Cai are gratefully acknowledged. This work is
| i 2 | i| i
supported by the National Fundamental Research Pro-
where N is a renormalization constant. Then perform- gramGrantNo. 2006CB921106,ChinaNationalNatural
′
ing a unitary operation V on the n qubits, the initial Science Foundation Grant Nos. 10325521,60433050.
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CLASSICAL COMPUTING, QUANTUM COMPUTING,
AND SHOR’S FACTORING ALGORITHM1
Yu. I. Manin
Max–Planck–Institut fu¨r Mathematik, Bonn, Germany
0. Why quantum computing?
9991
Information processing (computing) is the dynamical evolution of a highly orga-
nized physical system produced by technology (computer) or nature (brain). The
raM
initial state of this system is (determined by) its input; its final state is the output.
Physics describes nature in two complementary modes: classical and quantum. Up
to the nineties, the basic mathematical models of computing, Turing machines,
2
were classical objects, although the first suggestions for studying quantum models
1v8003099/hp-tnauq:viXra date back at least to 1980.
Roughly speaking, the motivation to study quantum computing comes from
several sources: physics and technology, cognitive science, and mathematics. We
will briefly discuss them in turn.
(i) Physically, the quantum mode of description is more fundamental than the
classical one. In the seventies and eighties it was remarked that, because of the
superposition principle, it is computationally unfeasible to simulate quantum pro-
cesses on classical computers ([Po], [Fe1]). Roughly speaking, quantizing a classical
system with N states we obtain a quantum system whose state space is an (N 1)–
−
dimensional complex projective space whose volume grows exponentially with N.
One can argue that the main preoccupation of quantum chemistry is the struggle
with resulting difficulties. Reversing this argument, one might expect that quan-
tum computers, if they can be built at all, will be considerably more powerful than
classical ones ([Fe1], [Ma2]).
Progress in the microfabrication techniques of modern computers has already led
us to the level where quantum noise becomes an essential hindrance to the error–
free functioning of microchips. It is only logical to start exploiting the essential