The Functions in QMGsurvey III

Compiled Distributions

cqmgTSleafDistribution*

cqmgTSleafDistributionINIT

Initialization function of cqmgTSleafDistribution; cqmgTSleafDistribution calculates the target space based distribution for a given point.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgTSleafDistribution.

Output	Type	Description
cqmgTSleafDistribution	Fx	the compiled function

cqmgTSleafDistribution

Initialized with cqmgTSleafDistributionINIT; cqmgTSleafDistribution calculates the target space based distribution for a given point.

Arguments

Argument	Type	Description
x	Real(Dim)	a point in target space

cqmgTSleafDistribution is parallelizable in the variable x.

Output

The output is dist.

Output	Type	Description
out	Real(l,Dim)	compressed output. With respect to the output of cqmgTSleafDistributionEXTR out=dist

cqmgTSleafDistributionEXTR

Extraction function of cqmgTSleafDistribution; cqmgTSleafDistribution calculates the target space based distribution for a given point.

Arguments

Argument	Type	Description
out	Real(l,Dim)	the output of cqmgTSleafDistribution

Output

The output is dist.

Output	Type	Description
dist	Real(l,Dim)	dist is a list containing vectors that span the distribution of the hybrid leaf (using theta) in target space

Description

cqmgTSleafDistribution calculates the distribution of the hybrid leaf (using theta) in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
cqmgTSleafDistribution=cqmgTSleafDistributionINIT[X,l];
out=cqmgTSleafDistribution[x];
cqmgTSleafDistributionEXTR[out]

cqmgQMleafDistribution*

cqmgQMleafDistributionINIT

Initialization function of cqmgQMleafDistribution; cqmgQMleafDistribution calculates the quantum manifold based distribution for a given point.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgQMleafDistribution.

Output	Type	Description
cqmgQMleafDistribution	Fx	the compiled function

cqmgQMleafDistribution

Initialized with cqmgQMleafDistributionINIT; cqmgQMleafDistribution calculates the quantum manifold based distribution for a given point.

Arguments

Argument	Type	Description
x	Real(Dim)	a point in target space

cqmgQMleafDistribution is parallelizable in the variable x.

Output

The output is dist.

Output	Type	Description
out	Real(l,Dim)	compressed output. With respect to the output of cqmgQMleafDistributionEXTR out=dist

cqmgQMleafDistributionEXTR

Extraction function of cqmgQMleafDistribution; cqmgQMleafDistribution calculates the quantum manifold based distribution for a given point.

Arguments

Argument	Type	Description
out	Real(l,Dim)	the output of cqmgQMleafDistribution

Output

The output is dist.

Output	Type	Description
dist	Real(l,Dim)	dist is a list containing vectors that span the distribution of the hybrid leaf using omega in target space

Description

cqmgQMleafDistribution calculates the distribution of the hybrid leaf using omega in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
cqmgQMleafDistribution=cqmgQMleafDistributionINIT[X,l];
out=cqmgQMleafDistribution[x];
cqmgQMleafDistributionEXTR[out]

cqmgGQMleafDistribution*

cqmgGQMleafDistributionINIT

Initialization function of cqmgGQMleafDistribution; cqmgGQMleafDistribution calculates the quantum manifold and g based distribution for a given point.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgGQMleafDistribution.

Output	Type	Description
cqmgGQMleafDistribution	Fx	the compiled function

cqmgGQMleafDistribution

Initialized with cqmgGQMleafDistributionINIT; cqmgGQMleafDistribution calculates the quantum manifold and g based distribution for a given point.

Arguments

Argument	Type	Description
x	Real(Dim)	a point in target space

cqmgGQMleafDistribution is parallelizable in the variable x.

Output

The output is dist.

Output	Type	Description
out	Real(l,Dim)	compressed output. With respect to the output of cqmgGQMleafDistributionEXTR out=dist

cqmgGQMleafDistributionEXTR

Extraction function of cqmgGQMleafDistribution; cqmgGQMleafDistribution calculates the quantum manifold and g based distribution for a given point.

Arguments

Argument	Type	Description
out	Real(l,Dim)	the output of cqmgGQMleafDistribution

Output

The output is dist.

Output	Type	Description
dist	Real(l,Dim)	dist is a list containing vectors that span the distribution of the hybrid leaf using omega and g in target space

Description

cqmgQMleafDistribution calculates the distribution of the hybrid leaf using omega and g in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
cqmgGQMleafDistribution=cqmgGQMleafDistributionINIT[X,l];
out=cqmgGQMleafDistribution[x];
cqmgGQMleafDistributionEXTR[out]

cqmgDistribution

Unification of cqmgTSleafDistribution, cqmgQMleafDistribution and cqmgGQMleafDistribution; calculates the distribution for a given point.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
x	Real(Dim)	a point in target space
l	Int	a positive even integer, the effective dimension of the quantum manifold
leaf="TSleaf"	Str	the chosen leaf, can be "TSleaf", "QMleaf" or "GQMleaf". "TSleaf" is the hybrid leaf (using theta), "QMleaf" is the hybrid leaf using omega and "GQMleaf" is the hybrid leaf using omega and g

cqmgDistribution is parallelizable in the variable x.

Output

The output is dist.

Output	Type	Description
dist	Real(l,Dim)	dist is a list containing vectors that span the distribution of the chosen leaf in target space

Description

cqmgDistribution is a unification of cqmgTSleafDistribution*, cqmgQMleafDistribution* and cqmgGQMleafDistribution*. cqmgDistribution calculates the distribution of the chosen leaf in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
leaf="QMleaf";
cqmgDistribution[X,x,l,leaf]

Compiled Curve Integration

cqmgTSleafCurveIteration*

cqmgTSleafCurveIterationINIT

Initialization function of cqmgTSleafCurveIteration; cqmgTSleafCurveIteration calculates a step in the target space based distribution.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgTSleafCurveIteration.

Output	Type	Description
cqmgTSleafCurveIteration	Fx	the compiled function

cqmgTSleafCurveIteration

Initialized with cqmgTSleafCurveIterationINIT; cqmgTSleafCurveIteration calculates a step in the target space based distribution.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length

cqmgTSleafCurveIteration is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgTSleafCurveIterationEXTR out={vPrime,xNew}

cqmgTSleafCurveIterationEXTR

Extraction function of cqmgTSleafCurveIteration; cqmgTSleafCurveIteration calculates a step in the target space based distribution.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgTSleafCurveIteration

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the distribution, normalized to length delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgTSleafCurveIteration calculates a finite step in the hybrid leaf (using theta) in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
cqmgTSleafCurveIteration=cqmgTSleafCurveIterationINIT[X,l];
out=cqmgTSleafCurveIteration[x,v,delta];
cqmgTSleafCurveIterationEXTR[out]

cqmgQMleafCurveIteration*

cqmgQMleafCurveIterationINIT

Initialization function of cqmgQMleafCurveIteration; cqmgQMleafCurveIteration calculates a step in the quantum manifold based distribution.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgQMleafCurveIteration.

Output	Type	Description
cqmgQMleafCurveIteration	Fx	the compiled function

cqmgQMleafCurveIteration

Initialized with cqmgQMleafCurveIterationINIT; cqmgQMleafCurveIteration calculates a step in the quantum manifold based distribution.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length

cqmgQMleafCurveIteration is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgQMleafCurveIterationEXTR out={vPrime,xNew}

cqmgQMleafCurveIterationEXTR

Extraction function of cqmgQMleafCurveIteration; cqmgQMleafCurveIteration calculates a step in the quantum manifold based distribution.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgQMleafCurveIteration

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the distribution, normalized to length delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgQMleafCurveIteration calculates a finite step in the hybrid leaf using omega in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
cqmgQMleafCurveIteration=cqmgQMleafCurveIterationINIT[X,l];
out=cqmgQMleafCurveIteration[x,v,delta];
cqmgQMleafCurveIterationEXTR[out]

cqmgGQMleafCurveIteration*

cqmgGQMleafCurveIterationINIT

Initialization function of cqmgGQMleafCurveIteration; cqmgGQMleafCurveIteration calculates a step in the quantum manifold and g based distribution.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgGQMleafCurveIteration.

Output	Type	Description
cqmgGQMleafCurveIteration	Fx	the compiled function

cqmgGQMleafCurveIteration

Initialized with cqmgGQMleafCurveIterationINIT; cqmgGQMleafCurveIteration calculates a step in the quantum manifold and g based distribution.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length

cqmgGQMleafCurveIteration is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgGQMleafCurveIterationEXTR out={vPrime,xNew}

cqmgGQMleafCurveIterationEXTR

Extraction function of cqmgGQMleafCurveIteration; cqmgGQMleafCurveIteration calculates a step in the quantum manifold and g based distribution.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgGQMleafCurveIteration

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the distribution, normalized to length delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgGQMleafCurveIteration calculates a finite step in the hybrid leaf using omega and g in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
cqmgGQMleafCurveIteration=cqmgGQMleafCurveIterationINIT[X,l];
out=cqmgGQMleafCurveIteration[x,v,delta];
cqmgGQMleafCurveIterationEXTR[out]

cqmgCurveIntegration

Unified function; calculates a curve in the chosen distribution.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length
n	Int	the (positive) number of steps (initial point excluded)
m	Int	the (positive) number of intermediate steps (zero means no intermediate steps). These increase the numerical precision, they do not affect the length of the curve
l	Int	a positive even integer, the effective dimension of the quantum manifold
leaf="TSleaf"	Str	the chosen leaf, can be "TSleaf", "QMleaf" or "GQMleaf". "TSleaf" is the hybrid leaf (using theta), "QMleaf" is the hybrid leaf using omega and "GQMleaf" is the hybrid leaf using omega and g
vFix=False	Bool	True: use the initial v at each step; False: use vPrime from the last step as

cqmgCurveIntegration is partially parallelizable in the variables x and v (this means lists of initial points and initial tangent vectors but no higher nesting can be handled).

Output

The output is xs.

Output	Type	Description
xs	Real(n+1,Dim)	xs is a list of all points of the discrete curve in the chosen leaf, xs(1) is the initial point x

Description

cqmgCurveIntegration is an iterated unification of cqmgTSleafCurveIteration*, cqmgQMleafCurveIteration* and cqmgGQMleafCurveIteration*. cqmgDistribution calculates a discrete curve with initial point x, initial tangent vector v and step length delta in the chosen leaf. This is done by iterating cqmgTSleafCurveIteration*, cqmgQMleafCurveIteration* or cqmgGQMleafCurveIteration*. If vFix=False the iteration uses vPrime from the last step as the new v in every step; if vFix=False v is used allways. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
n=1000;
m=3;
leaf="QMleaf";
cqmgCurveIntegration[X,x,v,delta,n,m,l,leaf]

An example for the fuzzy sphere with fixed initial tangent vector:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
n=1000;
m=3;
leaf="QMleaf";
cqmgCurveIntegration[X,x,v,delta,n,m,l,leaf,True]

An example with parallelization:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v1={0,1,0};
v2={1,0,0};
vs={v1,v2};
delta=0.01;
n=1000;
m=3;
leaf="QMleaf";
cqmgCurveIntegration[X,x,vs,delta,n,m,l,leaf,True]

Compiled Null Space Curve Integration

cqmgTSleafCurveIterationNull*

cqmgTSleafCurveIterationNullINIT

Initialization function of cqmgTSleafCurveIterationNull; cqmgTSleafCurveIterationNull calculates a step in the target space based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgTSleafCurveIterationNull.

Output	Type	Description
cqmgTSleafCurveIterationNull	Fx	the compiled function

cqmgTSleafCurveIterationNull

Initialized with cqmgTSleafCurveIterationNullINIT; cqmgTSleafCurveIterationNull calculates a step in the target space based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length

cqmgTSleafCurveIterationNull is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgTSleafCurveIterationNullEXTR out={vPrime,xNew}

cqmgTSleafCurveIterationNullEXTR

Extraction function of cqmgTSleafCurveIterationNull; cqmgTSleafCurveIterationNull calculates a step in the target space based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgTSleafCurveIterationNull

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the orthogonal distribution, normalized to length delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgTSleafCurveIterationNull calculates a finite step in the orthognal direction to the hybrid leaf (using theta) (thus the null leaf) in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
cqmgTSleafCurveIterationNull=cqmgTSleafCurveIterationNullINIT[X,l];
out=cqmgTSleafCurveIterationNull[x,v,delta];
cqmgTSleafCurveIterationNullEXTR[out]

cqmgQMleafCurveIterationNull*

cqmgQMleafCurveIterationNullINIT

Initialization function of cqmgQMleafCurveIterationNull; cqmgQMleafCurveIterationNull calculates a step in the quantum manifold based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgQMleafCurveIterationNull.

Output	Type	Description
cqmgQMleafCurveIterationNull	Fx	the compiled function

cqmgQMleafCurveIterationNull

Initialized with cqmgQMleafCurveIterationNullINIT; cqmgQMleafCurveIterationNull calculates a step in the quantum manifold based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length

cqmgQMleafCurveIterationNull is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgQMleafCurveIterationNullEXTR out={vPrime,xNew}

cqmgTSleafCurveIterationNullEXTR

Extraction function of cqmgQMleafCurveIterationNull; cqmgQMleafCurveIterationNull calculates a step in the quantum manifold based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgQMleafCurveIterationNull

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the orthogonal distribution, normalized to length delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgQMleafCurveIterationNull calculates a finite step in the orthognal direction to the hybrid leaf using omega (thus the null leaf) in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
cqmgQMleafCurveIterationNull=cqmgQMleafCurveIterationNullINIT[X,l];
out=cqmgQMleafCurveIterationNull[x,v,delta];
cqmgQMleafCurveIterationNullEXTR[out]

cqmgGQMleafCurveIterationNull*

cqmgGQMleafCurveIterationNullINIT

Initialization function of cqmgGQMleafCurveIterationNull; cqmgGQMleafCurveIterationNull calculates a step in the quantum manifold and g based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgGQMleafCurveIterationNull.

Output	Type	Description
cqmgGQMleafCurveIterationNull	Fx	the compiled function

cqmgGQMleafCurveIterationNull

Initialized with cqmgGQMleafCurveIterationNullINIT; cqmgGQMleafCurveIterationNull calculates a step in the quantum manifold and g based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length

cqmgGQMleafCurveIterationNull is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgGQMleafCurveIterationNullEXTR out={vPrime,xNew}

cqmgGQMleafCurveIterationNullEXTR

Extraction function of cqmgGQMleafCurveIterationNull; cqmgGQMleafCurveIterationNull calculates a step in the quantum manifold and g based distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgGQMleafCurveIterationNull

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the orthogonal distribution, normalized to length delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgGQMleafCurveIterationNull calculates a finite step in the orthognal direction to the hybrid leaf using omega and g (thus the null leaf) in target space. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
cqmgGQMleafCurveIterationNull=cqmgGQMleafCurveIterationNullINIT[X,l];
out=cqmgGQMleafCurveIterationNull[x,v,delta];
cqmgGQMleafCurveIterationNullEXTR[out]

cqmgCurveIntegrationNull

Unified function; calculates a curve in the chosen distribution in orthogonal (null) direction.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite step length
n	Int	the (positive) number of steps (initial point excluded)
m	Int	the (positive) number of intermediate steps (zero means no intermediate steps). These increase the numerical precision, they do not affect the length of the curve
l	Int	a positive even integer, the effective dimension of the quantum manifold
leaf="TSleaf"	Str	the chosen leaf, can be "TSleaf", "QMleaf" or "GQMleaf". "TSleaf" is the hybrid leaf (using theta), "QMleaf" is the hybrid leaf using omega and "GQMleaf" is the hybrid leaf using omega and g
vFix=False	Bool	True: use the initial v at each step; False: use vPrime from the last step as

cqmgCurveIntegrationNull is partially parallelizable in the variables x and v (this means lists of initial points and initial tangent vectors but no higher nesting can be handled).

Output

The output is xs.

Output	Type	Description
xs	Real(n+1,Dim)	xs is a list of all points of the discrete curve in the null leaf of the chosen leaf, xs(1) is the initial point x

Description

cqmgCurveIntegrationNull is an iterated unification of cqmgTSleafCurveIterationNull*, cqmgQMleafCurveIterationNull* and cqmgGQMleafCurveIterationNull*. cqmgCurveIntegrationNull calculates a discrete curve with initial point x, initial tangent vector v and step length delta in the null leaf corresponding to the chosen leaf. This is done by iterating cqmgTSleafCurveIterationNull*, cqmgQMleafCurveIterationNull* or cqmgGQMleafCurveIterationNull*. If vFix=False the iteration uses vPrime from the last step as the new v in every step; if vFix=False v is used allways. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
n=1000;
m=3;
leaf="QMleaf";
cqmgCurveIntegrationNull[X,x,v,delta,n,m,l,leaf]

An example for the fuzzy sphere with fixed initial tangent vector:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=0.01;
n=1000;
m=3;
leaf="QMleaf";
cqmgCurveIntegrationNull[X,x,v,delta,n,m,l,leaf,True]

An example with parallelization:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v1={0,1,0};
v2={1,0,0};
vs={v1,v2};
delta=0.01;
n=1000;
m=3;
leaf="QMleaf";
cqmgCurveIntegrationNull[X,x,vs,delta,n,m,l,leaf,True]

Compiled Minimization of Lambda

cqmgTSleafCurveIterationAdaptiveNull*

cqmgTSleafCurveIterationAdaptiveNullINIT

Initialization function of cqmgTSleafCurveIterationAdaptiveNull; cqmgTSleafCurveIterationAdaptiveNull calculates a step in the target space based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgTSleafCurveIterationAdaptiveNull.

Output	Type	Description
cqmgTSleafCurveIterationAdaptiveNull	Fx	the compiled function

cqmgTSleafCurveIterationAdaptiveNull

Initialized with cqmgTSleafCurveIterationAdaptiveNullINIT; cqmgTSleafCurveIterationAdaptiveNull calculates a step in the target space based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite relative step length

cqmgTSleafCurveIterationAdaptiveNull is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgTSleafCurveIterationAdaptiveNullEXTR out={vPrime,xNew}

cqmgTSleafCurveIterationAdaptiveNullEXTR

Extraction function of cqmgTSleafCurveIterationAdaptiveNull; cqmgTSleafCurveIterationAdaptiveNull calculates a step in the target space based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgTSleafCurveIterationAdaptiveNull

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the orthogonal distribution, multiplied with delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgTSleafCurveIterationAdaptiveNull calculates a finite step in the orthognal direction to the hybrid leaf (using theta) (thus the null leaf) in target space. Here, vPrime is not normalized to delta but is the projection of v multiplied with delta. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=1;
cqmgTSleafCurveIterationAdaptiveNull=cqmgTSleafCurveIterationAdaptiveNullINIT[X,l];
out=cqmgTSleafCurveIterationAdaptiveNull[x,v,delta];
cqmgTSleafCurveIterationAdaptiveNullEXTR[out]

cqmgQMleafCurveIterationAdaptiveNull*

cqmgQMleafCurveIterationAdaptiveNullINIT

Initialization function of cqmgQMleafCurveIterationAdaptiveNull; cqmgQMleafCurveIterationAdaptiveNull calculates a step in the quantum manifold based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgQMleafCurveIterationAdaptiveNull.

Output	Type	Description
cqmgQMleafCurveIterationAdaptiveNull	Fx	the compiled function

cqmgQMleafCurveIterationAdaptiveNull

Initialized with cqmgQMleafCurveIterationAdaptiveNullINIT; cqmgQMleafCurveIterationAdaptiveNullcalculates a step in the quantum manifold based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite relative step length

cqmgQMleafCurveIterationAdaptiveNull is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgQMleafCurveIterationAdaptiveNullEXTR out={vPrime,xNew}

cqmgQMleafCurveIterationAdaptiveNullEXTR

Extraction function of cqmgQMleafCurveIterationAdaptiveNull; cqmgQMleafCurveIterationAdaptiveNull calculates a step in the quantum manifold based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgQMleafCurveIterationAdaptiveNull

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the orthogonal distribution, multiplied with delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgQMleafCurveIterationAdaptiveNull calculates a finite step in the orthognal direction to the hybrid leaf using omega (thus the null leaf) in target space. Here, vPrime is not normalized to delta but is the projection of v multiplied with delta. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=1;
cqmgQMleafCurveIterationAdaptiveNull=cqmgQMleafCurveIterationAdaptiveNullINIT[X,l];
out=cqmgQMleafCurveIterationAdaptiveNull[x,v,delta];
cqmgQMleafCurveIterationAdaptiveNullEXTR[out]

cqmgGQMleafCurveIterationAdaptiveNull*

cqmgGQMleafCurveIterationAdaptiveNullINIT

Initialization function of cqmgGQMleafCurveIterationAdaptiveNull; cqmgGQMleafCurveIterationAdaptiveNull calculates a step in the quantum manifold and g based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
l	Int	a positive even integer, the effective dimension of the quantum manifold

Output

The output is cqmgGQMleafCurveIterationAdaptiveNull.

Output	Type	Description
cqmgGQMleafCurveIterationAdaptiveNull	Fx	the compiled function

cqmgGQMleafCurveIterationAdaptiveNull

Initialized with cqmgGQMleafCurveIterationAdaptiveNullINIT; cqmgGQMleafCurveIterationAdaptiveNull calculates a step in the quantum manifold and g based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
x	Real(Dim)	an initial point in target space
v	Real(Dim)	an initial tangent vector in target space
delta	Real	a positive finite relative step length

cqmgGQMleafCurveIterationAdaptiveNull is parallelizable in the variables x, v and delta.

Output

The output is out.

Output	Type	Description
out	Real(2,Dim)	compressed output. With respect to the output of cqmgGQMleafCurveIterationAdaptiveNullEXTR out={vPrime,xNew}

cqmgGQMleafCurveIterationAdaptiveNullEXTR

Extraction function of cqmgGQMleafCurveIterationAdaptiveNull; cqmgGQMleafCurveIterationAdaptiveNull calculates a step in the quantum manifold and g based distribution in orthogonal (null) direction and with variable length.

Arguments

Argument	Type	Description
out	Real(2,Dim)	the output of cqmgGQMleafCurveIterationAdaptiveNull

Output

The output is {vPrime,xNew}.

Output	Type	Description
vPrime	Real(Dim)	vPrime is the projection of the initial tangent vector v into the orthogonal distribution, multiplied with delta
xNew	Real(Dim)	xNew=x+vPrime is the new point after a finite step with vPrime

Description

cqmgGQMleafCurveIterationAdaptiveNull calculates a finite step in the orthognal direction to the hybrid leaf using omega and g (thus the null leaf) in target space. Here, vPrime is not normalized to delta but is the projection of v multiplied with delta. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
v={0,1,0};
delta=1;
cqmgGQMleafCurveIterationAdaptiveNull=cqmgGQMleafCurveIterationAdaptiveNullINIT[X,l];
out=cqmgGQMleafCurveIterationAdaptiveNull[x,v,delta];
cqmgGQMleafCurveIterationAdaptiveNullEXTR[out]

cqmgMinimizeLambda

Unified function; finds the minimum of lambda in the orthogonal direction to the chosen distribution.

Arguments

Argument	Type	Description
X	MatConf(Dim,Nim)	a matrix configuration
x	Real(Dim)	an initial point in target space
l	Int	a positive even integer, the effective dimension of the quantum manifold
leaf="TSleaf"	Str	the chosen leaf, can be "TSleaf", "QMleaf" or "GQMleaf". "TSleaf" is the hybrid leaf (using theta), "QMleaf" is the hybrid leaf using omega and "GQMleaf" is the hybrid leaf using omega and g
delta=1	Real	a positive finite relative step length
nMax=100	Int	the (positive) maximal number of steps (initial point excluded)
epsilon=10^-8	Real	the (positive) numerical tolerance for the minimum

cqmgMinimizeLambda is not parallelizable.

Output

The output is {xNew,Successful,i}.

Output	Type	Description
xNew	Real(Dim)	the point that (locally) minimizes the lowest eigenvalue of the Hamiltonian in the null leaf of the chosen leaf
Successful	Bool	True: a local minimum has been found up to tolerance epsilon; Flase: no local minimum has been found within nMax steps
i	Int	the number of steps the were needed for success

Description

cqmgCurveIntegrationNull finds a local minimum of lambda in the null leaf of the chosen leaf. This is done via a gradient descent method, using that $\partial_a\lambda=-(\mathbf{x}^a-x^a)$. delta is multiplied to the projection of the current gradient of lambda in every step. The method stops when $\vert \partial_a\lambda\vert<\epsilon$. Compare to [1] section 3.3.

Example(s)

An example for the fuzzy sphere:

X=qmgXsu2[4];
l=2;
x={0,0,1};
leaf="QMleaf";
cqmgMinimizeLambda[X,x,l,leaf]