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gates are used to add the result bitwise to the bits in y. Now the computation
reads (suppressing the initial x-copying),
(x,¯0,¯0,y) (x,f(x),g(x),y) (x,f(x),g(x),y f(x)). (2.21)
→ → ⊕
All steps performed in this computation up to the last ones involving y are
reversible and none of them affects the fourth register, so reversing this part of
the computation, yields
(x,f(x),g(x),y f(x)) (x,¯0,¯0,y f(x)). (2.22)
⊕ → ⊕
The complete computation now reads
(x,¯0,¯0,y) (x,¯0,¯0,y f(x)). (2.23)
→ ⊕
Note that this is still a reversible computation, as the part could also be
⊕
reversed,giving the bit stringy back. But, of course,we don’twantto do that.
So, suppressing the ancilla bit strings, we have simply
(x,y) (x,y f(x)). (2.24)
→ ⊕
2.4.4 Reversible computation and physics
Uptothispointwehaveonlydiscussed”logical”reversibility. Thecomputation
is reversible in the sense that the input can be recovered from the output by
carefullykeepingtrackofeverybitduringthecomputation. Thisisverycloseto
”physical” reversibility by which is meant that the time evolution of a physical
system can be reversed so that an initial state of the system can be recovered
from a final state. When a computational process is carried out by a physical
system16 there is precisely such a time evolution involved, so the two concepts
of reversibility must be closely connected.
16Theremustalwaysbesomeunderlyingphysicalsystemperformingthecomputation,even
whencomputationisviewedabstractlyasmeresymbolshuffling. Someoneorsomethingmust
shufflethesymbols.
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The microscopical laws of dynamics are all reversible, whether classical or
quantum. Reversible dynamics does not dissipate any energy. In order for this
to make sense one must really talk about closed physical systems, i.e. systems
that do not in any way interact with the environment. One must also have full
controloveralldegreesoffreedom,i.e. the dynamicsofeverydegreeoffreedom
must be governed by fundamental (time-reversible) equations of motion. Such
systems are conservative, meaning that the time evolution is reversible and no
energy is dissipated into the environment.
Comparingthistoacomputationalprocesswecanguessthatitisthelossof
control of some of the individual bits, inadvertently or deliberately, that leads
to energy dissipation in a computational process.
Indeed, as has been studied by Landauer [5], erasure of one bit of informa-
tion leads to an energy dissipation given by k T ln2. Here k is Boltzmann’s
B B
constant, a fundamental constant of physics relating mechanical quantities to
thermodynamical quantities like energy and entropy. T is the temperature of
the environment into which the energy is dissipated.
Physically, this possibility of performing computations reversibly is of more
theoretical interest than practical. The solid state hardware of today dissi-
pate energy far above the k T ln2 limit. Even if solid state circuits can be
b
manufactured that performs reversible logical operations like the Toffoli gate,
these devices must be poweredby some voltage source like any other electronic
gate. Tiny electric current will flow and there will be heat dissipation from
electric resistance. Even if this effect can be minimized, perhaps by exploiting
superconductivity, the inevitable weak interaction with the environment will
generate noise that will have to be corrected. The error correction registers
employed must eventually be erased, since memory is always finite, leading to
energy dissipation.
For now, classicalreversiblecomputationservesjust as a backdropto quan-
tumcomputation,whichisinherentlyreversible. Wewillreturntothesubjectof
reversiblecomputationinchapter8onphysicsofcomputation,andinparticular
to the question of the thermodynamics of computation.
2.5 Comparison to real computers
Neither the Turing machine model nor the circuit model is very close to the
actual workings of a modern digital computer. In what sense then are they
models of real world computers? It is generally agreed that all present day
general purpose digital computers are von Neumann machines, machines that
store both data and program in a memory and which works in a cyclic way of
(fetching instructions and data, executing instructions, storing data). This is a
rathervaguedescriptionofthebasicworkingsofacomputerandcannotbyitself
serveasa modelofcomputation. Thereis howevera computationalmodelthat
very closely captures the workingsof a modern computer - the Random Access
Machine-model. It has a CPU with temporary storage registers, a program
counterandanALU-muchasinarealworldprocessor. TheCPUisconnected
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to a (random access) memory which stores both data and program. In order
to read and write in arbitrary memory locations, every memory cell have an
address. The model can be programmed in an assembly-like language. It is
clear from the close analogy to real computers that the model has expressive
strength enough to support compilers and higher level languages.
The maindifference betweenthe Turingmachine andrealcomputersis that
its memory is not accessible immediately. In order to read a square away from
the present position of the read/write head, all intermediate squares must be
traversedand read.
The RandomAccess Machine(RAM) canreachanarbitrarymemorycellin
a single step. It can be considered a simplified model of real world computers.
The same functions are computable on Turing machines and on the RAM.
2.6 Non-deterministic Turing Machines
Themodelsofcomputationconsideredsofararedeterministic,i.e. ateachstep
inthecomputation,thenextstepisexactlydeterminedbytheprogramandthe
data. However,non-deterministic models of computation are theoretically very
important and often lead to simplification of analysis, though they cannot in
generalbe efficiently implemented. Referring back to the definition of a Turing
machine, we see that in the set of instructions defining the program,there is at
most one instruction with a certain combination of scanned tape symbol and
machine state. This makes the computation deterministic, i.e. the action of
the machine is uniquely determined. Removing this restriction leads to non-
determinism.