Quantum::SuperpositionUser Contributed Perl DocumentQuantum::Superpositions(3)NAMEQuantum::Superpositions - QM-like superpositions in Perl
VERSION
This document describes version 1.03 of Quantum::Superpositions,
released August 11, 2000.
SYNOPSIS
use Quantum::Superpositions;
if ($x == any($a, $b, $c)) { ... }
while ($nextval < all(@thresholds)) { ... }
$max = any(@value) < all(@values);
use Quantum::Superpositions BINARY => [ CORE::index ];
print index( any("opts","tops","spot"), "o" );
print index( "stop", any("p","s") );
BACKGROUND
Under the standard interpretation of quantum mechanics, until they are
observed, particles exist only as a discontinuous probability function.
Under the Cophenhagen Interpretation, this situation is often
visualized by imagining the state of an unobserved particle to be a
ghostly overlay of all its possible observable states simultaneously.
For example, a particle that might be observed in state A, B, or C may
be considered to be in a pseudo-state where it is simultaneously in
states A, B, and C. Such a particle is said to be in a superposition
of states.
Research into applying particle superposition in construction of
computer hardware is already well advanced. The aim of such research is
to develop reliable quantum memories, in which an individual bit is
stored as some measurable property of a quantised particle (a qubit).
Because the particle can be physically coerced into a superposition of
states, it can store bits that are simultaneously 1 and 0.
Specific processes based on the interactions of one or more qubits
(such as interference, entanglement, or additional superposition) are
then be used to construct quantum logic gates. Such gates can in turn
be employed to perform logical operations on qubits, allowing logical
and mathematical operations to be executed in parallel.
Unfortunately, the math required to design and use quantum algorithms
on quantum computers is painfully hard. The Quantum::Superpositions
module offers another approach, based on the superposition of entire
scalar values (rather than individual qubits).
DESCRIPTION
The Quantum::Superpositions module adds two new operators to Perl:
"any" and "all".
Each of these operators takes a list of values (states) and
superimposes them into a single scalar value (a superposition), which
can then be stored in a standard scalar variable.
The "any" and "all" operators produce two distinct kinds of
superposition. The "any" operator produces a disjunctive superposition,
which may (notionally) be in any one of its states at any time,
according to the needs of the algorithm that uses it.
In contrast, the "all" operator creates a conjunctive superposition,
which is always in every one of its states simultaneously.
Superpositions are scalar values and hence can participate in
arithmetic and logical operations just like any other type of scalar.
However, when an operation is applied to a superposition, it is applied
(notionally) in parallel to each of the states in that superposition.
For example, if a superposition of states 1, 2, and 3 is multiplied by
2:
$result = any(1,2,3) * 2;
the result is a superposition of states 2, 4, and 6. If that result is
then compared with the value 4:
if ($result == 4) { print "fore!" }
then the comparison also returns a superposition: one that is both true
and false (since the equality is true for one of the states of $result
and false for the other two).
Of course, a value that is both true and false is of no use in an "if"
statement, so some mechanism is needed to decide which superimposed
boolean state should take precedence.
This mechanism is provided by the two types of superposition available.
A disjunctive superposition is true if any of its states is true,
whereas a conjunctive superposition is true only if all of its states
are true.
Thus the previous example does print "fore!", since the "if" condition
is equivalent to:
if (any(2,4,6) == 4)...
It suffices that any one of 2, 4, or 6 is equal to 4, so the condition
is true and the "if" block executes.
On the other hand, had the control statement been:
if (all(2,4,6) == 4)...
the condition would fail, since it is not true that all of 2, 4, and 6
are equal to 4.
Operations are also possible between two superpositions:
if (all(1,2,3)*any(5,6) < 21)
{ print "no alcohol"; }
if (all(1,2,3)*any(5,6) < 18)
{ print "no entry"; }
if (any(1,2,3)*all(5,6) < 18)
{ print "under-age" }
In this example, the string "no alcohol" is printed because the
superposition produced by the multiplication is the Cartesian product
of the respective states of the two operands: "all(5,6,10,12,15,18)".
Since all of these resultant states are less that 21, the condition is
true. In contrast, the string "no entry" is not printed, because not
all the product's states are less than 18.
Note that the type of the first operand determines the type of the
result of an operation. Hence the third string -- "underage" -- is
printed, because multiplying a disjunctive superposition by a
conjunctive superposition produces a result that is disjunctive:
"any(5,6,10,12,15,18)". The condition of the "if" statement asks
whether any of these values is less than 18, which is true.
Composite Superpositions
The states of a superposition may be any kind of scalar value -- a
number, a string, or a reference:
$wanted = any("Mr","Ms").any(@names);
if ($name eq $wanted) { print "Reward!"; }
$okay = all(\&check1,\&check2);
die unless $okay->();
my $large =
all( BigNum->new($centillion),
BigNum->new($googol),
BigNum->new($SkewesNum)
);
@huge = grep {$_ > $large} @nums;
More interestingly, since the individual states of a superposition are
scalar values and a superposition is itself a scalar value, a
superposition may have states that are themselves superpositions:
$ideal = any( all("tall", "rich", "handsome"),
all("rich", "old"),
all("smart","Australian","rich")
);
Operations involving such a composite superposition operate recursively
and in parallel on each its states individually and then recompose the
result. For example:
while (@features = get_description)
{
if (any(@features) eq $ideal)
{
print "True love";
}
}
The "any(@features) eq $ideal" equality is true if the input
characteristics collectively match any of the three superimposed
conjunctive superpositions. That is, if the characteristics
collectively equate to each of "tall" and "rich" and "handsome", or to
both "rich" and "old", or to all three of "smart" and "Australian" and
"rich".
Eigenstates
It is useful to be able to determine the list of states that a given
superposition represents. In fact, it is not the states per se, but
the values to which the states may collapse -- the eigenstates that are
useful.
In programming terms this is the set of values @ev for a given
superposition $s such that "any(@ev) == $s" or "any(@ev) eq $s".
This list is provided by the "eigenstates" operator, which may be
called on any superposition:
print "The factor was: ",
eigenstates($factor);
print "Don't use any of:",
eigenstates($badpasswds);
Boolean evaluation of superpositions
The examples shown above assume the same meta-semantics for both
arithmetic and boolean operations, namely that a binary operator is
applied to the Cartesian product of the states of its two operands,
regardless of whether the operation is arithmetic or logical. Thus the
comparison of two superpositions produces a superposition of 1's and
0's, representing any (or all) possible comparisons between the
individual states of the two operands.
The drawback of applying arithmetic metasemantics to logical operations
is that it causes useful information to be lost. Specifically, which
states were responsible for the success of the comparison. For example,
it is possible to determine if any number in the array @newnums is less
than all those in the array @oldnums with:
if (any(@newnums) < @all(oldnums))
{
print "New minimum detected";
}
But this is almost certainly unsatisfactory, because it does not reveal
which element(s) of @newnum caused the condition to be true.
It is, however, possible to define a different meta-semantics for
logical operations between superpositions; one that preserves the
intuitive logic of comparisons but also gives limited access to the
states that cause those comparsions to succeed.
The key is to deviate from the arithmetic view of superpositional
comparison (namely, that a compared superposition yields a
superposition of compared state combinations). Instead, the various
comparison operators are redefined so that they form a superposition of
those eigenstates of the left operand that cause the operation to be
true. In other words, the old meta-semantics superimposed the result of
each parallel comparison, whilst the new meta-semantics superimposes
the left operands of each parallel comparison that succeeds.
For example, under the original semantics, the comparisons:
all(7,8,9) <= any(5,6,7) #A
all(5,6,7) <= any(7,8,9) #B
any(6,7,8) <= all(7,8,9) #C
would yield:
all(0,0,1,0,0,0,0,0,0) #A (false)
all(1,1,1,1,1,1,1,1,1) #B (true)
any(1,1,1,1,1,1,0,1,1) #C (true)
Under the new semantics they would yield:
all(7) #A (false)
all(5,6,7) #B (true)
any(6,7) #C (true)
The success of the comparison (the truth of the result) is no longer
determined by the values of the resulting states, but by the number of
states in the resulting superposition.
The Quantum::Superpositions module treats logical operations and
boolean conversions in exactly this way. Under these meta-semantics,
it is possible to check a comparison and also determine which
eigenstates of the left operand were responsible for its success:
$newmins = any(@newnums) < all(@oldnums);
if ($newmins)
{
print "New minima found:", eigenstates($newmins);
}
Thus, these semantics provide a mechanism to conduct parallel searches
for minima and maxima :
sub min { eigenstates( any(@_) <= all(@_) ) }
sub max { eigenstates( any(@_) >= all(@_) ) }
These definitions are also quite intuitive, almost declarative: the
minimum is any value that is less-than-or-equal-to all of the other
values; the maximum is any value that is greater-than-or-equal to all
of them.
String evaluation of superpositions
Converting a superposition to a string produces a string that encode
the simplest set of eigenstates equivalent to the original
superposition.
If there is only one eigenstate, the stringification of that state is
the string representation. This eliminates the need to explicitly
apply the "eigenstates" operator when only a single resultant state is
possible. For example:
print "lexicographically first: ",
any(@words) le all(@words);
In all other cases, superpositions are stringified in the format:
"all(eigenstates)" or "any(eigenstates)".
Numerical evaluation of superpositions
Providing an implicit conversion to numeric (for situations where
superpositions are used as operands to an arithmetic operation, or as
array indices) is more challenging than stringification, since there is
no mechanism to capture the entire state of a superposition in a single
non-superimposed number.
Again, if the superposition has a single eigenstate, the conversion is
just the standard conversion for that value. For instance, to output
the value in an array element with the smallest index in the set of
indices @i:
print "The smallest element is: ",
$array[any(@i)<=all(@i)];
If the superposition has no eigenstates, there is no numerical value to
which it could collapse, so the result is "undef".
If a disjunctive superposition has more than one eigenstate, that
superposition could collapse to any of those values. And it is
convenient to allow it to do exactly that -- collapse (pseudo-)randomly
to one of its eigenstates. Indeed, doing so provides a useful notation
for random selection from a list:
print "And the winner is...",
$entrant[any(0..$#entrant)];
Superpositions as subroutine arguments
When a superposition is used as a subroutine argument, that subroutine
is applied in parallel to each state of the superposition and the
results re-superimposed to form the same type of superposition. For
example, given:
$n1 = any(1,4,9);
$r1 = sqrt($n1);
$n2 = all(1,4,9);
$r2 = pow($n2,3);
$r3 = pow($n1,$r1);
then $r1 contains the disjunctive superposition "any(1,2,3)", $r2
contains the conjunctive superposition "all(1,64,729)", and <$r3 >
contains the conjunctive superposition "any(1,4,9,16,64,81,729)".
Because the built-in "sqrt" and "pow" functions don't know about
superpositions, the module provides a mechanism for informing them that
their arguments may be superimposed.
If the call to "use Quantum::Superpositions" is given an argument list,
that list specifies which functions should be rewritten to handle
superpositions. Unary functions and subroutine can be "quantized" like
so:
sub incr { $_[0]+1 }
sub numeric { $_[0]+0 eq $_[0] }
use Quantum::Superpositions
UNARY => ["CORE::int", "main::incr"],
UNARY_LOGICAL => ["main::numeric"];
For binary functions and subroutines use:
sub max { $_[0] < $_[1] ? $_[1] : $_[0] }
sub same { my $failed; $IG{__WARN__}=sub{$failed=1};
return $_[0] eq $_[1] || $_[0]==$_[1] && !$failed;
}
use Quantum::Superpositions
BINARY => ['main::max', 'CORE::index'],
BINARY_LOGICAL => ['main::same'];
EXAMPLES
Primality testing
The power of programming with scalar superpositions is perhaps best
seen by returning the quantum computing's favourite adversary: prime
numbers. Here, for example is an O(1) prime-number tester, based on
naive trial division:
sub is_prime
{
my ($n) = @_;
return $n % all(2..sqrt($n)+1) != 0
}
The subroutine takes a single argument ($n) and computes (in parallel)
its modulus with respect to every integer between 2 and "sqrt($n)".
This produces a conjunctive superposition of moduli, which is then
compared with zero. That comparison will only be true if all the
moduli are not zero, which is precisely the requirement for an integer
to be prime.
Because "is_prime" takes a single scalar argument, it can also be
passed a superposition. For example, here is a constant-time filter
for detecting whether a number is part of a pair of twin primes:
sub has_twin
{
my ($n) = @_;
return is_prime($n) && is_prime($n+any(+2,-2);
}
Set membership and intersection
Set operations are particularly easy to perform using superimposable
scalars. For example, given an array of values @elems, representing
the elements of a set, the value $v is an element of that set if:
$v == any(@elems)
Note that this is equivalent to the definition of an eigenstate. That
equivalence can be used to compute set intersections. Given two
disjunctive superpositions, "$s1=any(@elems1)" and "$s2=any(@elems2)",
representing two sets, the values that constitute the intersection of
those sets must be eigenstates of both <$s1> and $s2. Hence:
@intersection = eigenstates(all($s1, $s2));
This result can be extended to extract the common elements from an
arbitrary number of arrays in parallel:
@common = eigenstates( all( any(@list1),
any(@list2),
any(@list3),
any(@list4),
)
);
Factoring
Factoring numbers is also trivial using superpositions. The factors of
an integer N are all the quotients q of N/n (for all positive integers
n < N) that are also integral. A positive number q is integral if
floor(q)==q. Hence the factors of a given number are computed by:
sub factors
{
my ($n) = @_;
my $q = $n / any(2..$n-1);
return eigenstates(floor($q)==$q);
}
Query processing
Superpositions can also be used to perform text searches. For example,
to determine whether a given string ($target) appears in a collection
of strings (@db):
use Quantum::Superpositions BINARY => ["CORE::index"];
$found = index(any(@db), $target) >= 0;
To determine which of the database strings contain the target:
sub contains_str
{
return $dbstr if (index($dbstr, $target) >= 0;
}
$found = contains_str(any(@db), $target);
@matches = eigenstates $found;
It is also possible to superimpose the target string, rather than the
database, so as to search a single string for any of a set of targets:
sub contains_targ
{
if (index($dbstr, $target) >= 0)
{
return $target;
}
}
$found = contains_targ($string, any(@targets));
@matches = eigenstates $found;
or in every target simultaneously:
$found = contains_targ($string, all(@targets));
@matches = eigenstates $found;
AUTHOR
Damian Conway (damian@conway.org)
Now maintainted by Steven Lembark (lembark@wrkhors.com)
BUGS
There are undoubtedly serious bugs lurking somewhere in code this funky
:-) Bug reports and other feedback are most welcome.
COPYRIGHT
Copyright (c) 1998-2002, Damian Conway. Copyright (c) 2002, Steven
Lembark
All Rights Reserved.
This module is free software. It may be used, redistributed and/or
modified under the stame terms as Perl-5.6.1 (or later) (see
http://www.perl.com/perl/misc/Artistic.html).
perl v5.18.1 2003-04-22 Quantum::Superpositions(3)
