Download Many problems that take long time to solve on a deterministic Turing

Many problems that take long time to
solve on a deterministic Turing machine
can be often be solved very quickly on a
probabilistic Turing machine
 However there is a tradeoff between the
time it takes to return an answer to a
computation and the probability that the
returned answer is correct
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If a correct answer is required, then there is a
uncertainty in the length of time the
probabilistic algorithm must run
To a PTM a state has multiple legitimate
successors states available
The choice of which state is the one ultimately
determined by the outcome of a random choice
Models are certainly fine as mathematical
abstractions but are they consistent with known
physics?
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Quantum Turing Machines
Quantum Turing machines (QTM) are
best thought of as quantum
generalizations of probabilistic Turing
machines
 In a QTM you start the machine in some
initial configuration and allow it to evolve
for a certain time
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At time t the state is described by a
superposition of all states reachable in time t
The key difference to a PTM is that in PTM
only one particular computational trajectory is
fallowed, but in QTM all computational
trajectories are followed
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Upon interrogating the QTM for an answer you
might be told “yes your assumption is true” but
there would be no way to exhibit all computation
steps that had gone in order to arrive at the
conclusion...
Worse, if you tried to peek inside the QTM as it is
working, to get some information about the state of
the proof at that time, you would invariably disrupt
the future course of the proof...
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Model of Quantum Turing Machines by circuits
Quantum circuits/gates
Each quantum gate is reversible in the sense
that its outputs can be inferred from its input
and visa vers
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From the output of the NAND gate it is impossible to determine if the
input was (0,1), (1,0), or (0,0)
The NAND gate is irreversible: there is no logic gate capable of
inverting the NAND
AND, OR gate is irreversible :-(
NOT gate is logically reversible ;-)
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Any irreversible operation in a circuit is
necessarily accompanied by the dissipation of
heat
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information is lost, entropy grows
Can we compute without dissipating heat?
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The trick is to compute using only reversible circuit
elements!
No information loss!
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Importance to us: quantum gates are most
naturally viewed as reversible gates
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QTM algorithms are likely to be
implemented as specialized quantum
circuits rather than quantum “programs”
running on a universal quantum Turing
machine
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