API reference¶
Module overview¶
mpnum.mparray
: Basic matrix product array (MPA) routines and compressionmppnum.mpstruct
: Underlying structure of MPAs to manage the local tensorsmpnum.mpsmpo
: Convert matrix product state (MPS), matrix product operator (MPO) and locally purifying MPS (PMPS) representations and compute local reduced states.mpnum.factory
: Generate random, MPS, MPOs, MPDOs, MPAs, etc.mpnum.linalg
: Compute the smallest eigenvalues & vectors of MPOsmpnum.special
: Optimized versions of some routines for special casesmpnum.povm
: Matrix product representation of Positive operator valued measures (POVM)
mpnum.povm.localpovm
: Paulilike POVM on a single sitempnum.povm.mppovm
: Matrix product POVM based on the Paulilike POVM
mparray
¶
Core MPArray data structure & general purpose functions
Todo
single site MPAs – what is left?
Todo
Local tensor ownership – see MPArray class comment
Todo
Possible optimization:
 replace integerfor loops with iterator (not obviously possible everwhere)
 replace internal structure as list of arrays with lazy generator of arrays (might not be possible, since we often iterate both ways!)
 more in place operations for addition, subtraction, multiplication
Todo
Replace all occurences of self._ltens with self[…] or similar & benchmark. This will allow easier transition to lazy evaluation of local tensors

class
mpnum.mparray.
MPArray
(ltens)[source]¶ Bases:
object
Efficient representation of a general Npartite array \(A\) in matrix product form with open boundary conditions:
(1)¶\[A_{i_1, \ldots, i_N} = A^{[1]}_{i_1} \ldots A^{[N]}_{i_N}\]where the \(A^{[k]}\) are local tensors (with N legs/dimensions). The matrix products in (1) are taken with respect to the left and rightmost legs (virtual indices) and the multiindex \(i_k\) corresponds to the true local legs. Open boundary conditions imply that \(A^{[1]}\) is 1bysomething and \(A^{[N]}\) is somethingby1.
For the details on the data model used for storing the local tensors see
mpstruct.LocalTensors
.Todo
As it is now, e.g.
__imul__()
modifies items fromself._ltens
. This requires e.g.chain()
to take copies of the local tensors. The data model seems to be that an MPArray instance owns its local tensors and everyone else, including each new MPArray instance, must take copies. Is this correct?
__init__
(ltens)[source]¶ Parameters: ltens – local tensors as instance of mpstruct.LocalTensors
or simply as a list ofnumpy.ndarray
in the format described atmpstruct.LocalTensors

T
¶ Transpose (=reverse order of) physical legs on each site. See also
transpose()
for more fine grained control.

axis_iter
(axes=0)[source]¶ Returns an iterator yielding SubMPArrays of
self
by iterating over the specified physical axes.Example: If
self
represents a bipartite (i.e. length 2) array with 2 physical dimensions on each siteA[(k,l), (m,n)]
,self.axis_iter(0)
is equivalent to:(A[(k, :), (m, :)] for m in range(...) for k in range(...))
Parameters: axes – Iterable or int specifiying the physical axes to iterate over (default 0 for each site) Returns: Iterator over MPArray

canonical_form
¶

canonicalize
(left=None, right=None)[source]¶ Brings the MPA to canonical form in place [Sch11, Sec. 4.4]
Note that we do not support full left or rightcanonicalization. In general, the right (left resp.)most local tensor cannot be in a canonical form since at least one local tensor must be nonnormalized.
The following values for left and right will be needed most frequently:
Left/Right canonicalize: Do Nothing To canonicalize maximally left None
'afull'
,len(self)  1
right None
'afull'
,1
'afull'
is short for “almost full” (we do not support normalizing the outermost sites).Arbitrary integer values of
left
andright
have the following meaning:self[:left]
will be leftnormalizedself[right:]
will be rightnormalized
In accordance with the last table, the special values
None
and'afull'
will be replaced by the following integers:None
'afull'
left 0
len(self)  1
right len(self)
1
Exceptions raised:
 Integer argument too large or small:
IndexError
 Matrix would be both left and rightnormalized:
ValueError

compress
(method='svd', **kwargs)[source]¶ Compress
self
, modifying it inplace.Let \(\vert u \rangle\) the original vector and let \(\vert c \rangle\) the compressed vector. The compressions we return have the property (cf. [Sch11, Sec. 4.5.2])
\[\langle u \vert c \rangle = \langle c \vert c \rangle \in (0, \infty).\]It is a useful property because it ensures
\[\begin{split}\min_{\phi \in \mathbb R} \ u  r e^{i \phi} c \ &= \ u  r c \, \quad r > 0, \\ \min_{\mu \in \mathbb C} \ u  \mu c \ &= \ u  c \\end{split}\]for the vector 2norm. Users of this function can compute norm differences between u and a normalized c via
\[\ u  r c \^2 = \ u \^2 + r (r  2) \langle u \vert c \rangle, \quad r \ge 0.\]In the special case of \(\u\ = 1\) and \(c_0 = c/\ c \\) (pure quantum states as MPS), we obtain
\[\ u  c_0 \^2 = 2(1  \sqrt{\langle u \vert c \rangle})\]Returns: Inner product \(\langle u \vert c \rangle \in (0, \infty)\) of the original u and its compression c. Parameters: method – 'svd'
or'var'
Parameters for
'svd'
:Parameters:  rank – Maximal rank of the result. (default:
None
)  relerr – Maximal fraction of discarded singular values.
Default
0
. If both rank and relerr are given, the smaller resulting rank is used.  direction –
'right'
(sweep from left to right),'left'
(inverse) orNone
(choose depending on canonicalization). (default:None
)  canonicalize – SVD compression works best when the MPA is
brought into full left/rightcannonical form first. This variable
determines whether cannonical form is enforced before compression
(default:
True
)  svdfunc – Which SVD function to use during the compression.
It should follow the conventios of
truncated_svd()
, which is also the default choice. In some circumstances, a partial SVD as provided byscipy.sparse.linalg.svds()
or a randomized SVD such asrandomized_svd()
might speed up computations with no or little loss of accuracy.
Parameters for
'var'
:Parameters:  rank – Maximal rank for the result. Either
startmpa
orrank
is required.  num_sweeps – Number of variational sweeps (required).
 startmpa – Start vector, also fixes the rank of the result. Default: Random, with same norm as self.
 randstate –
numpy.random.RandomState
instance used for random start vector. (default:numpy.random
).  var_sites – Number of connected sites to be varied simultaneously (default 1)
Increasing
var_sites
makes it less likely to get stuck in a local minimum but is generally slower.References:
 rank – Maximal rank of the result. (default:

compression
(method='svd', **kwargs)[source]¶ Return a compression of
self
. Does not modifyself
.Parameters: See
compress()
.Returns: (compressed_mpa, overlap)
whereoverlap
is the inner product returned bycompress()
.

dtype
¶ Returns the dtype that should be returned by
to_array

dump
(target)[source]¶ Serializes MPArray to
h5py.Group
. Recover usingload()
.Parameters: target – h5py.Group
the instance should be saved to or path to h5 file (it’s then serialized to /)

classmethod
from_array
(array, ndims=None, has_virtual=False)[source]¶ Create MPA from array in local form.
See
mpnum.tools.global_to_local()
for global vs. local form.Computes the (exact up to numerical accuracy) representation of array as MPA with open boundary conditions, i.e. rank 1 at the boundary. This is done by factoring off the left and the “physical” legs from the rest of the tensor via QR decomposition and working its way through the tensor from the left. This yields a leftcanonical representation of array. [Sch11, Sec. 4.3.1]
The result is a chain of local tensors with
ndims
physical legs at each location and hasarray.ndim // ndims
number of sites (assumingndims
has the same value for each site)has_virtual
allows to treat a part of the linear chain of an MPA as MPA as well. The rank on the left and right can be different from one and different from each other in that case. This is useful to apply SVD compression only to part of an MPA.Parameters:  array (np.ndarray) – Dense array with global structure
array[(i0), ..., (iN)]
, i.e. the legs which are factorized into the same factor are already adjacent. (For me details seetools.global_to_local()
)  ndims – Number of physical legs per site (default array.ndim) or iterable over number of physical legs
 has_virtual (bool) –
True
if array already has indices for the left and right virtual legs
 array (np.ndarray) – Dense array with global structure

classmethod
from_array_global
(array, ndims=None, has_virtual=False)[source]¶ Create MPA from array in global form.
See
mpnum.tools.global_to_local()
for global vs. local form. Parameters and return value: See
from_array()
. has_virtual=True
is not supported yet.
 Parameters and return value: See

classmethod
from_kron
(factors)[source]¶ Returns the (exact) representation of an nfold Kronecker (tensor) product as MPA with ranks 1 and n sites.
Parameters: factors – A list of arrays with arbitrary number of physical legs Returns: The kronecker product of the factors as MPA

get
(indices, astype=None)[source]¶ Returns the current MPA but with the first index at each sites evaluated at the corresponding value of
indices
Parameters: indices – Length len(self)
sequence of index values for first physical leg at each siteReturns: type(self)
object

group_sites
(sites_per_group)[source]¶ Group several MPA sites into one site.
The resulting MPA has length
len(self) // sites_per_group
andsites_per_group * self.ndims[i]
physical legs on sitei
. The physical legs on each sites are in local form.Parameters: sites_per_group (int) – Number of sites to be grouped into one Returns: An MPA with sites_per_group
fewer sites and more ndims

leg2vleg
(pos)[source]¶ Performs the inverse operation to
vleg2leg()
.Parameters: pos – Number of the virtual to perform the transformation Returns: readonly MPA with transformed virtual Todo
More appropriate naming for this functions?

classmethod
load
(source)[source]¶ Deserializes MPArray from
h5py.Group
. Serialize usingdump()
.Parameters: target – h5py.Group
containing serialized MPArray or path to a single h5 File containing serialized MPArray under /

lt
¶

ndims
¶ Tuple of number of legs per site

pad_ranks
(rank=None, force_rank=False)[source]¶ Increase rank by padding with zeros
This function is useful to prepare initial states for variational compression. E.g. for a fivequbit pure state with ranks (2, 2, 4, 2) it is desirable to increase the ranks to (2, 4, 4, 2) before using it as an initial state for variational compression.
Parameters:  rank (int) – Increase rank to this value, use
max(self.rank)
ifNone
(default:None
)  force_rank – Use full rank even at the beginning and
end of the MPS. See
full_rank()
for more details. (default:False
)
Returns: MPA representation of the same array with padded rank
 rank (int) – Increase rank to this value, use

ranks
¶ Tuple of ranks

reshape
(newshapes)[source]¶ Reshape physical legs in place.
Use
shape
to obtain the shape of the physical legs.Parameters: newshapes – A single new shape or a list of new shape. Alternatively, you can pass ‘prune’ to get rid of all legs of dimension 1. Returns: Reshaped MPA Todo
Why is this here? What’s wrong with the purne function?

shape
¶ List of tuples with the dimensions of each tensor leg at each site

singularvals
()[source]¶ Return singular values of
self
for all bipartitionsReturns: Iterate over bipartitions with 1, 2, … len(self)  1 sites on the left hand side. Yields a np.ndarray
containing singular values for each bipartition.Note
May decrease the rank (without changing the represented tensor).

size
¶ Returns the number of floating point numbers used to represent the MPArray
>>> from .factory import zero >>> zero(sites=3, ldim=4, rank=3).lt.shape ((1, 4, 3), (3, 4, 3), (3, 4, 1)) >>> zero(sites=3, ldim=4, rank=3).size 60

split
(pos)[source]¶ Splits the MPA into two by transforming the virtual legs into local legs according to
vleg2leg()
.Parameters: pos – Number of the virtual to perform the transformation Returns: (mpa_left, mpa_right)

split_sites
(sites_per_group)[source]¶ Split MPA sites into several sites.
The resulting MPA has length
len(self) * sites_per_group
andself.ndims[i] // sites_per_group
indices on site i.Parameters: sites_per_group (int) – Split each site in that many sites Returns: An mpa with sites_per_group
more sites and fewerndims

sum
(axes=None)[source]¶ Elementwise sum over physical legs
Parameters: axes – Physical legs to sum over axes can have the following values:
 Sequence of length zero: Sum over nothing
 Sequence of (sequences or None):
axes[i]
specifies the physical legs to sum over at sitei
;None
sums over all physical legs at a site  Sequence of integers:
axes
specifies the physical legs to sum over at each site  Single integer: Sum over physical leg
axes
at each site None
: Sum over all physical legs at each site
To not sum over any axes at a certain site, specify the empty sequence for that site.

to_array
()[source]¶ Return MPA as array in local form.
See
mpnum.tools.global_to_local()
for global vs. local form.Returns: ndarray of shape sum(self.shape, ())
Note
Full arrays can require much more memory than MPAs. (That’s why you are using MPAs, right?)

to_array_global
()[source]¶ Return MPA as array in global form.
See
mpnum.tools.global_to_local()
for global vs. local form.Returns: ndarray of shape sum(zip(*self.shape, ()))
See
to_array()
for more details.

transpose
(axes=None)[source]¶ Transpose (=reverse order of) physical legs on each site
Parameters: axes – New order of the physical axes. If None
is passed, we reverse the order of the legs on each site. (defaultNone
)>>> from .factory import random_mpa >>> mpa = random_mpa(2, (2, 3, 4), 2) >>> mpa.shape ((2, 3, 4), (2, 3, 4)) >>> mpa.transpose((2, 0, 1)).shape ((4, 2, 3), (4, 2, 3))

vleg2leg
(pos)[source]¶ Transforms the virtual leg between site
pos
andpos + 1
into local legs at those sites. The new leg will be the rightmost one at sitepos
and the leftmost one at sitepos + 1
. The new rank is 1.Also see
leg2vleg()
.Parameters: pos – Number of the virtual to perform the transformation Returns: MPA with transformed virtual Todo
More appropriate naming for this functions?


mpnum.mparray.
dot
(mpa1, mpa2, axes=(1, 0), astype=None)[source]¶  Compute the matrix product representation of the contraction of
a
 and
b
over the given axes. [Sch11, Sec. 4.2]
Parameters:  mpa2 (mpa1,) – Factors as MPArrays
 axes – Tuple
(ax1, ax2)
whereax1
(ax2
) is a single physical leg number or sequence of physical leg numbers referring tompa1
(mpa2
). The first (second, etc) entries ofax1
andax2
will be contracted. Very similar to theaxes
argument fornumpy.tensordot()
. (default:(1, 0)
)
Note
Note that the default value of
axes
is different compared tonumpy.tensordot()
.Parameters: astype – Return type. If None
, use the type ofmpa1
Returns: Dot product of the physical arrays  Compute the matrix product representation of the contraction of

mpnum.mparray.
inject
(mpa, pos, num=None, inject_ten=None)[source]¶ Interleaved chain product of an MPA and a rank 1 MPA
Return the chain product between mpa and
num
copies of the local tensorinject_ten
, but place the copies ofinject_ten
before sitepos
inside or outsidempa
. You can also supplynum = None
and a sequence of local tensors. All legs of the local tensors are interpreted as physical legs. Placing the local tensors at the beginning or end ofmpa
usingpos = 0
orpos = len(mpa)
is also supported, butchain()
is preferred for that as it is a much simpler function.If
inject_ten
is omitted, use a square identity matrix of sizempa.shape[pos][0]
. Ifpos == len(mpa)
,mpa.shape[pos  1][0]
will be used for the size of the matrix.Parameters:  mpa – An MPA.
 pos – Inject sites into the MPA before site
pos
.  num – Inject
num
copies. Can beNone
; in this caseinject_ten
must be a sequence of values.  inject_ten – Physical tensor to inject (if omitted, an identity matrix will be used; cf. above)
Returns: The chain product
pos
can also be a sequence of positions. In this case,num
andinject_ten
must be either sequences orNone
, whereNone
is interpreted aslen(pos) * [None]
. As above, ifnum[i]
isNone
, theninject_ten[i]
must be a sequence of values.

mpnum.mparray.
inner
(mpa1, mpa2)[source]¶ Compute the inner product <mpa1mpa2>. Both have to have the same physical dimensions. If these represent a MPS,
inner(...)
corresponds to the canoncial Hilbert space scalar product. If these represent a MPO,inner(...)
corresponds to the Frobenius scalar product (with Hermitian conjugation in the first argument)Parameters:  mpa1 – MPArray with same number of physical legs on each site
 mpa2 – MPArray with same physical shape as mpa1
Returns: <mpa1mpa2>

mpnum.mparray.
local_sum
(mpas, embed_tensor=None, length=None, slices=None)[source]¶ Embed local MPAs on a linear chain and sum as MPA.
We return the sum over
embed_slice(length, slices[i], mpas[i], embed_tensor)
as MPA.If
slices
is omitted, we useregular_slices(length, width, offset)
withoffset = 1
,width = len(mpas[0])
andlength = len(mpas) + width  offset
.If
slices
is omitted or if the slices just described are given, we call_local_sum_identity()
, which gives a smaller virtual dimension than naive embedding and summing.Parameters:  mpas – List of local MPAs.
 embed_tensor – Defaults to square identity matrix (see
_embed_ltens_identity()
for details)  length – Length of the resulting chain, ignored unless slices is given.
 slices –
slice[i]
specifies the position ofmpas[i]
, optional.
Returns: An MPA.

mpnum.mparray.
localouter
(a, b)[source]¶ Computes the tensor product of \(a \otimes b\) locally, that is when a and b have the same number of sites, the new local tensors are the tensorproducts of the original ones.
Parameters: Returns: Tensor product of
a
andb
in terms of their local tensors

mpnum.mparray.
norm
(mpa)[source]¶ Computes the norm (Hilbert space norm for MPS, Frobenius norm for MPO) of the matrix product operator. In contrast to
mparray.inner
, this can take advantage of the canonicalizationWARNING This also changes the MPA inplace by normalizing.
Parameters: mpa – MPArray Returns: l2norm of that array

mpnum.mparray.
normdist
(mpa1, mpa2)[source]¶ More efficient version of norm(mpa1  mpa2)
Parameters:  mpa1 – MPArray
 mpa2 – MPArray
Returns: l2norm of mpa1  mpa2

mpnum.mparray.
chain
(mpas, astype=None)[source]¶ Computes the tensor product of MPAs given in
*args
by adding more sites to the array.Parameters:  mpas – Iterable of MPAs in the order as they should appear in the chain
 astype – dtype of the returned MPA. If
None
, use the type of the first MPA.
Returns: MPA of length
len(args[0]) + ... + len(args[1])
Todo
Make this canonicalization aware
Todo
Raise warning when casting complex to real dtype

mpnum.mparray.
partialdot
(mpa1, mpa2, start_at, axes=(1, 0))[source]¶ Partial dot product of two MPAs of inequal length.
The shorter MPA will start on site
start_at
. Local dot products will be carried out on all sites of the shorter MPA. Other sites will remain unmodified.mpa1 and mpa2 can also have equal length if
start_at == 0
. In this case, we do the same asdot()
.Parameters:  mpa2 (mpa1,) – Factors as MPArrays, length must be inequal.
 start_at – The shorter MPA will start on this site.
 axes – See
axes
argument todot()
.
Returns: MPA with length of the longer MPA.

mpnum.mparray.
partialtrace
(mpa, axes=(0, 1), mptype=None)[source]¶ Computes the trace or partial trace of an MPA.
This function is most useful for computing traces of an MPO or MPA over given physical legs. For obtaining partial traces (i.e., reduced states) of an MPO,
mpnum.mpsmpo.reductions_mpo()
will be more convenient.By default
(axes=(0, 1))
compute the trace and return the value as lengthone MPA with zero physical legs.For
axes=(m, n)
with integerm
, trace over the given axes at all sites and return a lengthone MPA with zero physical legs. (Use trace() to get the value directly.)For
axes=(axes1, axes2, ...)
trace overaxesN
at siteN
, withaxesN=(axisN_1, axisN_2)
tracing the given physical legs andaxesN=None
leaving the site invariant. Afterwards,prune()
is called to remove sites with zero physical legs from the result.Parameters:  mpa – MPArray
 axes – Axes for trace, (axis1, axis2) or (axes1, axes2, …) with axesN=(axisN_1, axisN_2) or axesN=None.
 mptype – Which constructor to call with the new local tensors
(default:
type(mpa)
)
Returns: An MPArray (possibly one site with zero physical legs)

mpnum.mparray.
prune
(mpa, singletons=False)[source]¶ Contract sites with zero (physical) legs.
Parameters: Returns: An
MPArray
(of possibly smaller length)

mpnum.mparray.
regular_slices
(length, width, offset)[source]¶ Iterate over regular slices on a linear chain.
Put slices on a linear chain as follows:
>>> n = 5 >>> [tuple(range(*s.indices(n))) for s in regular_slices(n, 3, 2)] [(0, 1, 2), (2, 3, 4)] >>> n = 7 >>> [tuple(range(*s.indices(n))) for s in regular_slices(n, 3, 2)] [(0, 1, 2), (2, 3, 4), (4, 5, 6)]
The scheme is illustrated by the following figure:
###### width
#######offset
overlap
offset
####### width
######Todo
This table needs cell borders in the HTML output (> CSS) and the tabularcolumns command doesn’t work.
Note that the overlap may be larger than, equal to or smaller than zero.
We enforce that the last slice coincides with the end of the chain, i.e.
(length  width) / offset
must be integer. We produce(length  width) / offset + 1
slices and the ith slice isslice(offset * i, offset * i + width)
, withi
starting at zero.Parameters:  length (int) – The length of the chain.
 width (int) – The width of each slice.
 offset (int) – Difference between starting positions of successive slices. First slice starts at 0.
Returns: Iterator over slices.

mpnum.mparray.
sandwich
(mpo, mps, mps2=None)[source]¶ Compute
<mpsMPOmps>
efficientlyThis function computes the same value as
mp.inner(mps, mp.dot(mpo, mps))
in a more efficient way.The runtime of this method scales with
D**3 * Dp + D**2 * Dp**3
whereD
andDp
are the ranks ofmps
andmpo
. This is more efficient thanmp.inner(mps, mp.dot(mpo, mps))
, whose runtime scales withD**4 * Dp**3
, and also more efficient thanmp.dot(mps.conj(), mp.dot(mpo, mps)).to_array()
, whose runtime scales withD**6 * Dp**3
.If
mps2
is given,<mps2MPOmps>
is computed instead (i.e.mp.inner(mps2, mp.dot(mpo, mps))
; see alsodot()
).

mpnum.mparray.
embed_slice
(length, slice_, mpa, embed_tensor=None)[source]¶ Embed a local MPA on a linear chain.
Parameters:  length (int) – Length of the resulting MPA.
 slice (slice) – Specifies the position of
mpa
in the result.  mpa (MPArray) – MPA of length
slice_.stop  slice_.start
.  embed_tensor – Defaults to square identity matrix (see
_embed_ltens_identity()
for details)
Returns: MPA of length length

mpnum.mparray.
trace
(mpa, axes=(0, 1))[source]¶ Compute the trace of the given MPA.
If you specify axes (see partialtrace() for details), you must ensure that the result has no physical legs anywhere.
Parameters:  mpa – MParray
 axes – Axes for trace,
(axis1, axis2)
or(axes1, axes2, ...)
withaxesN=(axisN_1, axisN_2)
oraxesN=None
. (default:(0, 1)
)
Returns: A single scalar of type
mpa.dtype

mpnum.mparray.
diag
(mpa, axis=0)[source]¶ Returns the diagonal elements
mpa[i, i, ..., i]
. Ifmpa
has more than one physical dimension, the result is a numpy array withMPArray
entries, otherwise its a numpy array with floats.Parameters:  mpa –
MPArray
with shape >axis
 axis – The physical index to take diagonals over
Returns: Array containing the diagonal elements (each diagonal element is an
MPArray
with the physical dimension reduced by one, note that anMPArray
with dimension 0 is a simple number) mpa –

mpnum.mparray.
sumup
(mpas, weights=None)[source]¶ Returns the sum of the MPArrays in
mpas
. Same asfunctools.reduce(mp.MPArray.__add__, mpas)
but should be faster as we can get rid of intermediate allocations.
Parameters: mpas – Iterator over MPArray
Returns: Sum of mpas

mpnum.mparray.
full_rank
(ldims)[source]¶ Computes a list of maximal ranks for a tensor with given local dimesions
Parameters: ldims – Dimensions of the legs of the tensor per site. Can be either passed as one number per site ( [2, 5, 2]
) or if there are multiple legs per site as a list of tuples similar toMPArray.shape
(e.g.[(2,), (3, 4), (5,)])
)Returns: Tuple of ranks that are maximal for the local dimensions ldims
.>>> full_rank([3] * 5) [3, 9, 9, 3] >>> full_rank([2] * 8) [2, 4, 8, 16, 8, 4, 2] >>> full_rank([(2, 3)] * 4) [6, 36, 6]
mpstruct
¶
Core data structure & routines to manage local tensors

class
mpnum.mpstruct.
LocalTensors
(ltens, cform=(None, None))[source]¶ Bases:
object
Core data structure to manage the local tensors of a
MPArray
.The local tensors are kept in
_ltens
, a list ofnumpy.ndarray
s such that_ltens[i]
corresponds to the local tensor of site i.If there are \(k\) (nonvirtual) indices at site \(i\), the corresponding local tensor is a ndarray with
ndim == k + 2
. The two additional indices of the local tensor correspond to the virtual legs. We reserve the 0th index of the local tensor for the virtal leg coupling to site \(i  1\) and the last index for the virtual leg coupling to site \(i + 1\).Therefore, if the physical legs at site \(i\) have dimensions \(d_1, \ldots, d_k\), the corresponding local tensor has shape \((r_{i1}, d_1, \ldots, d_k, r_{i})\). Here, \(r_{i1}\) and \(r_i\) denote the rank between sites \((i  1, i)\) and \((i, i + 1)\), respectively.
To keep the data structure consistent, we include the left virutal leg of the leftmost local tensor as well as the right virtual leg of the rightmost local tensor as dummy indices of dimension 1.

canonical_form
¶ Tensors which are currently in left/rightcanonical form.
Returns tuple
(left, right)
such thatself[:left]
are leftnormalizedself[right:]
are rightnormalized.

shape
¶ List of tuples with the dimensions of each tensor leg at each site

update
(index, tens, canonicalization=None)[source]¶ Update the local tensor at site
index
to the new valuetens
. Checks the rank and shape of the new values to keep the MPA consistent. Therefore, some actions such as changing the rank between two sites require to update both sites at the same time, which can be done by passing in multiple values as arguments.Parameters:  index – Integer/slice. Site index/indices of the local tensor/ tensors to be updated.
 tens – New local tensor as
numpy.ndarray
. Alternatively, sequence over multiple ndarrays (in caseindex
is a slice).  canonicalization – If
tens
is left/rightnormalized, pass'left'
/'right'
, respectively. Otherwise, passNone
(defaultNone
). In caseindex
is a slice, either pass a sequence of the corresponding values or a single value, which is repeated for each site updated.

factory
¶
Module to create random test instances of matrix product arrays

mpnum.factory.
eye
(sites, ldim)[source]¶ Returns a MPA representing the identity matrix
Parameters:  sites – Number of sites
 ldim – Intlike local dimension or iterable of local dimensions
Returns: Representation of the identity matrix as MPA
>>> I = eye(4, 2) >>> I.ranks, I.shape ((1, 1, 1), ((2, 2), (2, 2), (2, 2), (2, 2))) >>> I = eye(3, (3, 4, 5)) >>> I.shape ((3, 3), (4, 4), (5, 5))

mpnum.factory.
random_local_ham
(sites, ldim=2, intlen=2, randstate=None)[source]¶ Generates a random Hamiltonian on sites sites with local dimension ldim, which is a sum of local Hamiltonians with interaction length intlen.
Parameters:  sites – Number of sites
 ldim – Local dimension
 intlen – Interaction length of the local Hamiltonians
Returns: MPA representation of the global Hamiltonian

mpnum.factory.
random_mpa
(sites, ldim, rank, randstate=None, normalized=False, force_rank=False, dtype=<class 'numpy.float64'>)[source]¶ Returns an MPA with randomly choosen local tensors (real by default)
Parameters:  sites – Number of sites
 ldim –
Physical legs, depending on the type passed:
 scalar: Single physical leg for each site with given dimension
 iterable of scalar: Same physical legs for all sites
 iterable of iterable: Generated MPA will have exactly this as ndims
 rank –
Desired rank, depending on the type passed:
 scalar: Same rank everywhere
 iterable of length
sites  1
: Generated MPA will have exactly this as ranks
 randstate – numpy.random.RandomState instance or None
 normalized – Resulting mpa has mp.norm(mpa) == 1
 force_rank – If True, the rank is exaclty rank. Otherwise, it might be reduced if we reach the maximum sensible rank.
 dtype – Type of the returned MPA. Currently only
np.float_
andnp.complex_
are implemented (default:np.float_
, i.e. real values).
Returns: Randomly choosen matrix product array
Entries of local tensors are drawn from a normal distribution of unit variance. For complex values, the real and imaginary parts are independent and have unit variance.
>>> mpa = random_mpa(4, 2, 10, force_rank=True) >>> mpa.ranks, mpa.shape ((10, 10, 10), ((2,), (2,), (2,), (2,)))
>>> mpa = random_mpa(4, (1, 2), 10, force_rank=True) >>> mpa.ranks, mpa.shape ((10, 10, 10), ((1, 2), (1, 2), (1, 2), (1, 2)))
>>> mpa = random_mpa(4, [(1, ), (2, 3), (4, 5), (1, )], 10, force_rank=True) >>> mpa.ranks, mpa.shape ((10, 10, 10), ((1,), (2, 3), (4, 5), (1,)))
The following doctest verifies that we do not change how random states are generated, ensuring reproducible results. In addition, it verifies the returned dtype:
>>> rng = np.random.RandomState(seed=3208886881) >>> random_mpa(2, 2, 3, rng).to_array() array([[0.7254321 , 3.44263486], [0.17262967, 2.4505633 ]]) >>> random_mpa(2, 2, 3, rng, dtype=np.complex_).to_array() array([[0.53552415+1.39701566j, 2.12128866+0.57913253j], [0.32652114+0.51490923j, 0.322223200.32675463j]])

mpnum.factory.
random_mpdo
(sites, ldim, rank, randstate=<module 'numpy.random' from '/usr/lib/python3/distpackages/numpy/random/__init__.py'>)[source]¶ Returns a randomly choosen matrix product density operator (i.e. positive semidefinite matrix product operator with trace 1).
Parameters:  sites – Number of sites
 ldim – Local dimension
 rank – Rank
 randstate – numpy.random.RandomState instance
Returns: randomly choosen classicaly correlated matrix product density op.
>>> rho = random_mpdo(4, 2, 4) >>> rho.ranks, rho.shape ((4, 4, 4), ((2, 2), (2, 2), (2, 2), (2, 2))) >>> rho.canonical_form (0, 4)

mpnum.factory.
random_mps
(sites, ldim, rank, randstate=None, force_rank=False)[source]¶ Returns a randomly choosen normalized matrix product state
Parameters:  sites – Number of sites
 ldim – Local dimension
 rank – Rank
 randstate – numpy.random.RandomState instance or None
 force_rank – If True, the rank is exaclty rank. Otherwise, it might be reduced if we reach the maximum sensible rank.
Returns: randomly choosen matrix product (pure) state
>>> mps = random_mps(4, 2, 10, force_rank=True) >>> mps.ranks, mps.shape ((10, 10, 10), ((2,), (2,), (2,), (2,))) >>> mps.canonical_form (0, 4) >>> round(abs(1  mp.inner(mps, mps)), 10) 0.0

mpnum.factory.
random_mpo
(sites, ldim, rank, randstate=None, hermitian=False, normalized=True, force_rank=False)[source]¶ Returns an hermitian MPO with randomly choosen local tensors
Parameters:  sites – Number of sites
 ldim – Local dimension
 rank – Rank
 randstate – numpy.random.RandomState instance or None
 hermitian – Is the operator supposed to be hermitian
 normalized – Operator should have unit norm
 force_rank – If True, the rank is exaclty rank. Otherwise, it might be reduced if we reach the maximum sensible rank.
Returns: randomly choosen matrix product operator
>>> mpo = random_mpo(4, 2, 10, force_rank=True) >>> mpo.ranks, mpo.shape ((10, 10, 10), ((2, 2), (2, 2), (2, 2), (2, 2))) >>> mpo.canonical_form (0, 4)

mpnum.factory.
zero
(sites, ldim, rank, force_rank=False)[source]¶ Returns a MPA with localtensors beeing zero (but of given shape)
Parameters:  sites – Number of sites
 ldim –
Depending on the type passed (checked in the following order)
 iterable of iterable: Detailed list of physical dimensions, retured mpa will have exactly this for mpa.shape
 iterable of scalar: Same physical dimension for each site
 scalar: Single physical leg for each site with given dimension
 rank – Rank
 force_rank – If True, the rank is exaclty rank. Otherwise, it might be reduced if we reach the maximum sensible rank.
Returns: Representation of the zeroarray as MPA
mpsmpo
¶
Matrix Product State (MPS) and Operator (MPO) functions
The Introduction also covers the definitions mentioned below.
Definitions¶
We consider a linear chain of \(n\) sites with associated Hilbert spaces mathcal H_k = C^{d_k}, \(d_k\), \(k \in [1..n] := \{1, 2, \ldots, n\}\). The set of linear operators \(\mathcal H_k \to \mathcal H_k\) is denoted by \(\mathcal B_k\). We write \(\mathcal H = \mathcal H_1 \otimes \cdots \otimes \mathcal H_n\) and the same for \(\mathcal B\).
We use the following three representations:
 Matrix product state (MPS): Vector \(\lvert \psi \rangle \in \mathcal H\)
 Matrix product operator (MPO): Operator \(M \in \mathcal B\)
 Locally purified matrix product state (PMPS): Positive semidefinite operator \(\rho \in \mathcal B\)
All objects are represented by \(n\) local tensors.
Matrix product state (MPS)¶
Represent a vector \(\lvert \psi \rangle \in \mathcal H\) as
The \(k\)th local tensor is \(T_{l,i,r} = (A^{(k)}_i)_{l,r}\).
The vector \(\lvert \psi \rangle\) can be a quantum state, with the density matrix given by \(\rho = \lvert \psi \rangle \langle \psi \rvert \in \mathcal B\). Reference: E.g. [Sch11].
Matrix product operator (MPO)¶
Represent an operator \(M \in \mathcal B\) as
The \(k\)th local tensor is \(T_{l,i,j,r} = (A^{(k)}_{i j})_{l,r}\).
This representation can be used to represent a mixed quantum state \(\rho = M\), but it is not limited to positive semidefinite \(M\). Reference: E.g. [Sch11].
Locally purified matrix product state (PMPS)¶
Represent a positive semidefinite operator \(\rho \in \mathcal B\) as follows: Let \(\mathcal H_k' = \mathbb C^{d'_k}\) with suitable \(d'_k\) and \(\mathcal P = \mathcal H_1 \otimes \mathcal H'_1 \otimes \cdots \otimes \mathcal H_n \otimes \mathcal H'_n\). Find \(\vert \Phi \rangle \in \mathcal P\) such that
and represent \(\lvert \Phi \rangle\) as
The \(k\)th local tensor is \(T_{l,i,i',r} = (A^{(k)}_{i i'})_{l,r}\).
The ancillary dimensions \(d'_i\) are not determined by the \(d_i\) but depend on the state. E.g. if \(\rho\) is pure, one can set all \(d_i = 1\). Reference: E.g. [Cue13].
Todo
Are derived classes MPO/MPS/PMPS of any help?
Todo
I am not sure the current definition of PMPS is the most elegant for our purposes…
References:
[Cue13] De las Cuevas, G., Schuch, N., PérezGarcía, D., and Cirac, J. I. (2013). “Purifications of multipartite states: limitations and constructive methods”. New J. Phys. 15(12), p. 123021. DOI: 10.1088/13672630/15/12/123021. arXiv: 1308.1914.

mpnum.mpsmpo.
mps_to_mpo
(mps)[source]¶ Convert a pure MPS to a mixed state MPO.
Parameters: mps (MPArray) – An MPA with one physical leg Returns: An MPO (density matrix as MPA with two physical legs)

mpnum.mpsmpo.
mps_to_pmps
(mps)[source]¶ Convert a pure MPS into a local purification MPS mixed state.
The ancilla legs will have dimension one, not increasing the memory required for the MPS.
Parameters: mps (MPArray) – An MPA with one physical leg Returns: An MPA with two physical legs (system and ancilla)

mpnum.mpsmpo.
pmps_dm_to_array
(pmps, global_=False)[source]¶ Convert PMPS to full array representation of the density matrix
The runtime of this method scales with D**3 instead of D**6 where D is the rank and D**6 is the scaling of using
pmps_to_mpo()
andto_array()
. This is useful for obtaining reduced states of a PMPS on nonconsecutive sites, as normalizing before usingpmps_to_mpo()
may not be sufficient to reduce the rank in that case.Note
The resulting array will have dimension1 physical legs removed.

mpnum.mpsmpo.
pmps_reduction
(pmps, support)[source]¶ Convert a PMPS to a PMPS representation of a local reduced state
Parameters: support – Set of sites to keep Returns: Sites traced out at the beginning or end of the chain are removed using reductions_pmps()
and a suitable normalization. Sites traced out in the middle of the chain are converted to sites with physical dimension 1 and larger ancilla dimension.

mpnum.mpsmpo.
pmps_to_mpo
(pmps)[source]¶ Convert a local purification MPS to a mixed state MPO.
A mixed state on n sites is represented in local purification MPS form by a MPA with n sites and two physical legs per site. The first physical leg is a ‘system’ site, while the second physical leg is an ‘ancilla’ site.
Parameters: pmps (MPArray) – An MPA with two physical legs (system and ancilla) Returns: An MPO (density matrix as MPA with two physical legs)

mpnum.mpsmpo.
pmps_to_mps
(pmps)[source]¶ Convert a PMPS with unit ancilla dimensions to a simple MPS
If all ancilla dimensions of the PMPS are equal to unity, they are removed. Otherwise, an AssertionError is raised.

mpnum.mpsmpo.
reductions_mpo
(mpa, width=None, startsites=None, stopsites=None)[source]¶ Iterate over MPO partial traces of an MPO
The support of the ith result is
range(startsites[i], stopsites[i])
.Parameters:  mpa (mpnum.mparray.MPArray) – An MPO
 startsites – Defaults to
range(len(mpa)  width + 1)
.  stopsites – Defaults to
[ start + width for start in startsites ]
. If specified, we require startsites to be given and width to be None.  width – Number of sites in support of the results. Default None. Must be specified if one or both of startsites and stopsites are not given.
Returns: Iterator over partial traces as MPO

mpnum.mpsmpo.
reductions_mps_as_mpo
(mps, width=None, startsites=None, stopsites=None)[source]¶ Iterate over MPO mpdoreduced states of an MPS
width, startsites and stopsites: See
reductions_mpo()
.Parameters: mps – Pure state as MPS Returns: Iterator over reduced states as MPO

mpnum.mpsmpo.
reductions_mps_as_pmps
(mps, width=None, startsites=None, stopsites=None)[source]¶ Iterate over PMPS reduced states of an MPS
width, startsites and stopsites: See
reductions_mpo()
.Parameters: mps – Pure state as MPS Returns: Iterator over reduced states as PMPS

mpnum.mpsmpo.
reductions_pmps
(pmps, width=None, startsites=None, stopsites=None)[source]¶ Iterate over PMPS partial traces of a PMPS
width, startsites and stopsites: See
reductions_mpo()
.Parameters: pmps – Mixed state in locally purified MPS representation (PMPS, see Definitions) Returns: Iterator over reduced states as PMPS
linalg
¶
Linear algebra with matrix product arrays
Currently, we support computing extremal eigenvalues and eigenvectors of MPOs.

mpnum.linalg.
eig
(mpo, num_sweeps, var_sites=2, startvec=None, startvec_rank=None, randstate=None, eigs=None)[source]¶ Iterative search for MPO eigenvalues
Note
This function can return completely inaccurate values. You are responsible for supplying a large enough
startvec_rank
(orstartvec
with large enough rank) andnum_sweeps
.This function attempts to find eigenvalues by iteratively optimizing \(\lambda = \langle \psi \vert H \vert \psi \rangle\) where \(H\) is the operator supplied in the argument
mpo
. Specifically, we attempt to de or increase \(\lambda\) by optimizing over several neighbouring local tensors of the MPS \(\vert \psi \rangle\) simultaneously (the number given byvar_sites
).The algorithm used here is described e.g. in [Sch11, Sec. 6.3]. For
var_sites = 1
, it is called “variational MPS ground state search” or “singlesite DMRG” [Sch11, Sec. 6.3, p. 69]. Forvar_sites > 1
, it is called “multisite DMRG”.Parameters:  mpo (MPArray) – A matrix product operator (MPA with two physical legs)
 num_sweeps (int) – Number of sweeps to do (required)
 var_sites (int) – Number of neighbouring sites to be varied simultaneously
 startvec – Initial guess for eigenvector (default: random MPS with rank startvec_rank)
 startvec_rank – Rank of random start vector (required and used only if no start vector is given)
 randstate –
numpy.random.RandomState
instance orNone
 eigs – Function which computes one eigenvector of the local
eigenvalue problem on
var_sites
sites
Returns: eigval, eigvec_mpa
The
eigs
parameter defaults toeigs = functools.partial(scipy.sparse.linalg.eigsh, k=1, tol=1e6)
By default,
eig()
computes the eigenvalue with largest magnitude. To compute e.g. the smallest eigenvalue (sign included), supplywhich='SA'
toeigsh
. For other possible values, refer to the SciPy documentation.It is recommendable to supply a value for the
tol
parameter ofeigsh()
. Otherwise,eigsh()
will work at machine precision which is rarely necessary.Note
One should keep in mind that a variational method (such as the one implemented in this function) can only provide e.g. an upper bound on the lowest eigenvalue of an MPO. Deciding whether a given MPO has an eigenvalue which is smaller than a given threshold has been shown to be NPhard (in the number of parameters of the MPO representation) [KGE14].
Comments on the implementation, for
var_sites = 1
:References are to the arXiv version of [Sch11] assuming we replace zerobased with onebased indices there.
Psi^A_{i1}
andPsi^B_{i}
are identity matrices because of normalization. (See Fig. 42 on p. 67 and the text; see also Figs. 14 and 15 and pages 28 and 29.)

mpnum.linalg.
eig_sum
(mpas, num_sweeps, var_sites=2, startvec=None, startvec_rank=None, randstate=None, eigs=None)[source]¶ Iterative search for eigenvalues of a sum of MPOs/MPSs
Try to compute the ground state of the sum of the objects in
mpas
. MPOs are taken asis. An MPS \(\vert\psi\rangle\) adds \(\vert\psi\rangle \langle\psi\vert\) to the sum.This function executes the same algorithm as
eig()
applied to an uncompressed MPO sum of the elements inmpas
, but it obtains the ingredients for the local optimization steps using less memory and execution time. In particular, this function does not have to convert an MPS inmpas
to an MPO.Todo
Add information on how the runtime of
eig()
andeig_sum()
scale with the the different ranks. For the time being, refer to the benchmark test.Parameters: mpas – A sequence of MPOs or MPSs Remaining parameters and description: See
eig()
.Algorithm: [Sch11, Sec. 6.3]
povm
¶
povm.mppovm
¶
Matrixproduct representation of POVMs
This module provides the following classes:
MPPovm
: A matrix product representation of a multisite POVM.For example, for a linear chain of n qubits this class can represent the POVM of the observable XX…X with \(2^n\) elements efficiently. It is also possible to sample from the probability distribution of this POVM efficiently.
MPPovmList
: A list of MPPOVMs.This class can be used e.g. to obtain estimated expectation values of the local observable XX1…1 on two qubits from from samples for the global observables XX…X and XXY…Y (cf. below on Linear combinations of functions of POVM outcomes).
The methods
MPPovm.embed()
,MPPovm.block()
/MPPovmList.block()
,MPPovm.repeat()
/MPPovmList.repeat()
as well aspauli_mpp()
andpauli_mpps()
allow for convenient construction of MPPOVMs and MPPOVM lists.
Linear combinations of functions of POVM outcomes¶
In order to perform the just mentioned estimation of probabilities of
one POVM from samples of another POVM with possibly larger support, we
provide a function which can estimate linear functions of functions of
POVM outcomes: Let \(M\) a finite index set with real elements
\(y \in M \subset \mathbb R\) such that \(\hat y\) are the
positive semidefinite POVM elements which sum to the identity,
\(\sum_{y \in M} \hat y = 1\). Given a state \(\rho\), the
probability mass function (PMF) of the probability distribution given
by the POVM and the state can be expressed as \(p_y =
\operatorname{tr}(\rho \hat y)\), \(y \in M\) or as \(p(x) =
\sum_{y \in M} \delta(x  y) p_y\). Let further \(D = (x_1,
\ldots, x_m)\), \(x_k \in M\) a set of samples from \(p(x)\)
and let \(f \colon M \to \mathbb R\) an arbitrary function of the
POVM outcomes. The true value \(\langle f \rangle_p = \int f(y)
p(y) \mathrm d y\) can then be estimated using the sample average
\(\langle f \rangle_D = \frac1m \sum_{k=1}^m f(x_k) p_{x_k}\). In
the same way, a linear combination \(f = \sum c_i f_i\) of
functions \(f_i \colon M \to \mathbb R\) of POVM outcomes can be
estimated by \(\langle f \rangle_D = \sum_i c_i \langle f_i
\rangle_D\). Such a linear combination of functions of POVM outcomes
can be estimated using MPPovm.est_lfun()
. More technically,
the relation \(\langle \langle f \rangle_D \rangle_{p_m} =
\langle f \rangle_p\) shows that \(\langle f \rangle_D\) is an
unbiased estimator for the true expectation value \(\langle f
\rangle_p\); the probability distribution of the dataset \(D\) is
given by the sampling distribution \(p_m(D) = p(x_1) \ldots
p(x_m)\).
Estimates of the POVM probabilities \(p_y\) can also be expressed as functions of this kind: Consider the function
The true value of this function under \(p(x)\) is \(\langle
\theta_y \rangle_p = p_y\) and the sample average \(\langle
\theta_y \rangle_D\) provides an estimator for \(p_y\). In order to
estimate probabilities of one POVM from samples for another POVM, such
a function can be used: E.g. to estimate the probability of the
\((+1, +1)\) outcome of the POVM XX1…1, we can define a
function which is equal to 1 if the outcome of the POVM XX…X on
the first two sites is equal to \((+1, +1)\) and zero
otherwise. The sample average of this function over samples for the
latter POVM XX…X will estimate the desired probability. This
approach is implemented in MPPovm.est_pmf_from()
. If samples
from more than one POVM are available for estimating a given
probability, a weighted average of estimators can be used as
implemented in MPPovm.est_pmf_from_mpps()
; the list of
MPPOVMs for which samples are available is passed as an
MPPovmList
instance. Finally, the function
MPPovmList.est_lfun_from()
allows estimation of a linear
combination of probabilities from different POVMs using samples of a
second list of MPPOVMs. This function also estimates the variance of
the estimate. In order to perform the two estimation procedures, for
each probability, we construct an estimator from a weighted average of
functions of outcomes of different POVMs, as has been explained
above. For more simple settings, MPPovmList.est_lfun()
is also
available.
True values of the functions just mentioned can be obtained from
MPPovm.lfun()
, MPPovmList.lfun()
and
MPPovmList.lfun_from()
. All functions return both the true
expectation value and the variance of the expectation value.
The variance of the (true) expectation value \(\langle f
\rangle_p\) of a function \(f\colon M \to \mathbb R\) is given by
\(\operatorname{var}_p(f) = \operatorname{cov}_p(f, f)\) with
\(\operatorname{cov}_p(f, g) = \langle fg \rangle_p  \langle f
\rangle_p \langle g \rangle_p\). The variance of the estimate
\(\langle f \rangle_D\) is given by
\(\operatorname{var}_{p_m}(\langle f \rangle_D) = \frac1m
\operatorname{var}_p(f)\) where \(p_m(D)\) is the sampling
distribution from above. An unbiased estimator for the covariance
\(\operatorname{cov}_p(f, g)\) is given by \(\frac{m}{m1}
\operatorname{cov}_D(f, g)\) where the sample covariance
\(\operatorname{cov}_D(f, g)\) is defined in terms of sample
averages in the usual way, \(\operatorname{cov}_D(f, g) = \langle
fg \rangle_D  \langle f \rangle_D \langle g \rangle_D\). This
estimator is used by MPPovm.est_lfun()
.
Todo
Explain the details of the variance estimation, in particular the
difference between the variances returned from
MPPovmList.lfun()
and MPPovmList.lfun_from()
. Check the
mean square error.
Add a good references explaining all facts mentioned above and for further reading.
Document the runtime and memory cost of the functions.
Class and function reference¶

class
mpnum.povm.mppovm.
MPPovm
(*args, **kwargs)[source]¶ Bases:
mpnum.mparray.MPArray
MPArray representation of multipartite POVM
There are two different ways to write down a POVM in matrix product form
 As a list of matrix product operators, where each entry corresponds to
a single POVM element
As a matrix proudct array with 3 physical legs:
[POVM index, column index, row index]
that is, the first physical leg of the MPArray corresponds to the index of the POVM element. This representation is especially helpful for computing expectation values with MPSs/MPDOs.
Here, we choose the second.
Todo
This class should provide a function which returns expectation values as full array. (Even though computing expectation values using the POVM struture brings advantages, we usually need the result as full array.) This function should also replace small negative probabilities by zero and canonicalize the sum of all probabilities to unity (if the deviation is nonzero but small). The same checks should also be implemented in localpovm.POVM.
Todo
Right now we use this class for multisite POVMs with elements obtained from every possible combination of the elements of singlesite POVMs: The POVM index is split across all sites. Explore whether and how this concept can also be useful in other cases.

block
(nr_sites)[source]¶ Embed an MPPOVM on local blocks
The returned
MPPovmList
will contain self embedded at every possible position on len(self) neighbouring sites in a chain of length nr_sites. The remaining sites are not measured (self.embed()
).self must a have a uniform local Hilbert space dimension.
Parameters: nr_sites – Number of sites of the resulting MPPOVMs

elements
¶ Returns an iterator over all POVM elements. The result is the ith POVM element in MPO form.
It would be nice to call this method __iter__, but this breaks mp.dot(mppovm, …). In addition, next(iter(mppovm)) would not be equal to mppovm[0].

embed
(nr_sites, startsite, local_dim)[source]¶ Embed MPPOVM into larger system
Applying the resulting embedded MPPOVM to a state rho gives the same result as applying the original MPPOVM self on the reduced state of sites range(startsite, startsite + len(self)) of rho.
Parameters:  nr_sites – Number of sites of the resulting MPPOVM
 startsite – Position of the first site of self in the resulting MPPOVM
 local_dim – Local dimension of sites to be added
Returns: MPPOVM with self on sites range(startsite, startsite + len(self)) and
MPPovm.eye()
elsewhere

est_lfun
(coeff, funs, samples, weights=None, eps=1e10)[source]¶ Estimate a linear combination of functions of POVM outcomes
This function estimates the function with exact value given by
MPPovm.lfun()
; see there for description of the parameters coeff and funs.Parameters:  samples (np.ndarray) – A shape (n_samples, len(self.nsoutdims)) with samples from self
 weights – A length n_samples array for weighted samples. You can submit counts by passing them as weights. The number of samples used in average and variance estimation is determined by weights.sum() if weights is given.
Returns: (est, var): Estimated value and estimated variance of the estimated value. For details, see Linear combinations of functions of POVM outcomes.

est_pmf
(samples, normalize=True, eps=1e10)[source]¶ Estimate probability mass function from samples
Parameters:  samples (np.ndarray) – (n_samples, len(self.nsoutdims)) array of samples
 normalize (bool) – True: Return normalized probability estimates (default). False: Return integer outcome counts.
Returns: Estimated probabilities as ndarray est_pmf with shape self.nsoutdims
n_samples * est_pmf[i1, …, ik] provides the number of occurences of outcome (i1, …, ik) in samples.

est_pmf_from
(other, samples, eps=1e10)[source]¶ Estimate PMF from samples of another MPPovm other
If other does not provide information on all elements in self, we require that the elements in self for which information is provided sum to a multiple of the identity.
Example: If we consider the MPPovm
MPPovm.from_local_povm(x, n)
for given local POVMs x, it is possible to obtain counts for the Pauli X part ofx = pauli_povm()
from samples forx = x_povm()
; this is also true if the latter is supported on a larger part of the chain.Parameters:  other (MPPovm) – Another MPPovm
 samples (np.ndarray) – (n_samples, len(other.nsoutdims)) array of samples for other
Returns: (est_pmf, n_samples_used). est_pmf: Shape self.nsoutdims ndarray of normalized probability estimates; the sum over the available probability estimates is equal to the fraction of the identity obtained by summing the corresponding POVM elements. n_samples_used: Number of samples which have contributed to the PMF estimate.

est_pmf_from_mpps
(other, samples, eps=1e10)[source]¶ Estimate probability mass function from MPPovmList samples
Parameters:  other (MPPovmList) – An
MPPovmList
instance  samples – Iterable of samples (e.g. from
MPPovmList.samples()
)
Returns: (p_est, n_samples_used), both are shape self.nsoutdims ndarrays. p_est provides estimated probabilities and n_samples_used provides the effective number of samples used for each probability.
 other (MPPovmList) – An

expectations
(mpa, mode='auto')[source]¶ Computes the exp. values of the POVM elements with given state
Parameters:  mpa – State given as MPDO, MPS, or PMPS
 mode – In which form mpa is given. Possible values: ‘mpdo’, ‘pmps’, ‘mps’, or ‘auto’. If ‘auto’ is passed, we choose between ‘mps’ or ‘mpdo’ depending on the number of physical legs
Returns: Iterator over the expectation values, the nth element is the expectation value correponding to the reduced state on sites [n,…,n + len(self)  1]

classmethod
eye
(local_dims)[source]¶ Construct MPPOVM with no output or measurement
Corresponds to taking the partial trace of the quantum state and a shorter MPPOVM.
Parameters: local_dims – Iterable of local dimensions

classmethod
from_local_povm
(lelems, width)[source]¶ Generates a product POVM on width sites.
Parameters:  lelems – POVM elements as an iterator over all local elements (i.e. an iterator over numpy arrays representing the latter)
 width (int) – Number of sites the POVM lives on
Returns: MPPovm
which is a product POVM of the lelems

hdims
¶ Local Hilbert space dimensions

lfun
(coeff, funs, state, mode='auto', eps=1e10)[source]¶ Evaluate a linear combination of functions of POVM outcomes
Parameters:  coeff (np.ndarray) – A length n_funs array with the coefficients of the linear combination. If None, return the estimated values of the individual functions and the estimated covariance matrix of the estimates.
 funs (np.ndarray) – A length n_funs sequence of functions. If None, the estimated function will be a linear function of the POVM probabilities.
For further information, see also Linear combinations of functions of POVM outcomes.
The parameters state and mode are passed to
MPPovm.pmf()
.Returns: (value, var): Expectation value and variance of the expectation value

match_elems
(other, exclude_dup=(), eps=1e10)[source]¶ Find POVM elements in other which have information on self
We find all POVM sites in self which have only one possible outcome. We discard these outputs in other and afterwards check other and self for any common POVM elements.
Parameters:  other – Another MPPovm
 exclude_dup – Sequence which can include ‘self’ or ‘other’ (or both) to assert that there are no linearly dependent pairs of elements in self or other.
 eps – Threshould for values which should be treated as zero
Returns: (matches, prefactors)
matches[i_1, …, i_k, j_1, …, j_k] specifies whether outcome (i_1, …, i_k) of self has the same POVM element as the partial outcome (j_1, …, j_k) of other; outcomes are specified only on the sites mentioned in sites such that k = len(sites).
prefactors[i_1, …, i_k, j_1, …, j_k] specifies how samples from other have to be weighted to correspond to samples for self.

nsoutdims
¶ Nonsingleton outcome dimensions (dimension larger one)

nsoutpos
¶ Sites with nonsingleton outcome dimension (dimension larger one)

outdims
¶ Outcome dimensions

pack_samples
(samples, dtype=None)[source]¶ Pack samples into one integer per sample
Store one sample in a single integer instead of a list of integers with length len(self.nsoutdims). Example:
>>> p = pauli_mpp(nr_sites=2, local_dim=2) >>> p.outdims (6, 6) >>> p.pack_samples(np.array([[0, 1], [1, 0], [1, 2], [5, 5]])) array([ 1, 6, 8, 35])

pmf
(state, mode='auto')[source]¶ Compute the POVM’s probability mass function for state
If you want to compute the probabilities for reduced states of state, you can use
MPPovm.expectations()
instead of this function.Parameters:  state (mp.MPArray) – A quantum state as MPA. Must have the same length as self.
 mode – ‘mps’, ‘mpdo’ or ‘pmps’. See
MPPovm.expectations()
.
Returns: Probabilities as MPArray

pmf_as_array
(state, mode='auto', eps=1e10, impl='auto')[source]¶ Compute the POVM’s PMF for state as full array
Parameters: See
MPPovm.pmf()
.Parameters: impl – ‘auto’, ‘default’, ‘pmpssymm’ or ‘pmpsltr’. ‘auto’ will use ‘pmpssymm’ for mode ‘pmps’ and ‘default’ otherwise. Returns: PMF as shape self.nsoutdims ndarray The resulting (real or complex) probabilities pmf are passed through
project_pmf(pmf, eps, eps)
before being returned.

probability_map
¶ Map that takes a raveled MPDO to the POVM probabilities
You can use
MPPovm.expectations()
orMPPovm.pmf()
as convenient wrappers around this map.If rho is a matrix product density operator (MPDO), then
produces the POVM probabilities as MPA (similar to
mpnum.povm.localpovm.POVM.probability_map()
).

repeat
(nr_sites)[source]¶ Construct a longer MPPOVM by repetition
The resulting POVM will have length nr_sites. If nr_sites is not an integer multiple of len(self), self must factorize (have leg dimension one) at the position where it will be cut. For example, consider the tensor product MPPOVM of Pauli X and Pauli Y. Calling repeat(nr_sites=5) will construct the tensor product POVM XYXYX:
>>> import mpnum as mp >>> import mpnum.povm as mpp >>> x, y = (mpp.MPPovm.from_local_povm(lp(3), 1) for lp in ... (mpp.x_povm, mpp.y_povm)) >>> xy = mp.chain([x, y]) >>> xyxyx = mp.chain([x, y, x, y, x]) >>> mp.norm(xyxyx  xy.repeat(5)) <= 1e10 True

sample
(rng, state, n_samples, method='cond', n_group=1, mode='auto', pack=False, eps=1e10)[source]¶ Random sample from self on a quantum state
Parameters:  state (mp.MPArray) – A quantum state as MPA (see mode)
 n_samples – Number of samples to create
 method – Sampling method (‘cond’ or ‘direct’, see below)
 n_group – Number of sites to sample at a time in conditional sampling.
 mode – Passed to
MPPovm.expectations()
 eps – Threshold for small values to be treated as zero.
Two different sampling methods are available:
 Direct sampling (method=’direct’): Compute probabilities for all outcomes and sample from the full probability distribution. Usually faster than conditional sampling for measurements on a small number of sites. Requires memory linear in the number of possible outcomes.
 Conditional sampling (method=’cond’): Sample outcomes on all sites by sampling from conditional outcome probabilities on at most n_group sites at a time. Requires memory linear in the number of outcomes on n_group sites. Useful for measurements which act on large parts of a system (e.g. Pauli X on each spin).
Returns: ndarray samples with shape (n_samples, len(self.nsoutdims)) The ith sample is given by samples[i, :]. samples[i, j] is the outcome for the jth nonsingleton output dimension of self.

unpack_samples
(samples)[source]¶ Unpack samples into several integers per sample
Inverse of
MPPovm.pack_samples()
. Example:>>> p = pauli_mpp(nr_sites=2, local_dim=2) >>> p.outdims (6, 6) >>> p.unpack_samples(np.array([0, 6, 7, 12])) array([[0, 0], [1, 0], [1, 1], [2, 0]], dtype=uint8)

class
mpnum.povm.mppovm.
MPPovmList
(mppseq)[source]¶ Bases:
object
A list of
Matrix Product POVMs
This class allows you to
 Conveniently obtain samples and estimated or exact probabilities
for a list of
MPPovms
 Estimate probabilities from samples for a different MPPovmList
 Estimate linear functions of probabilities of an MPPovmList from samples for a different MPPovmList

__init__
(mppseq)[source]¶ Construct a MPPovmList
Parameters: mppseq – An iterable of MPPovm
objectsAll MPPovms must have the same number of sites.

block
(nr_sites)[source]¶ Embed MPPOVMs on local blocks
This function calls
MPPovm.block(nr_sites)()
for each MPPOVM in the list. Embedded MPPOVMs at the same position appear consecutively in the returned list:>>> import mpnum as mp >>> import mpnum.povm as mpp >>> ldim = 3 >>> x, y = (mpp.MPPovm.from_local_povm(lp(ldim), 1) for lp in ... (mpp.x_povm, mpp.y_povm)) >>> e = mpp.MPPovm.eye([ldim]) >>> xx = mp.chain([x, x]) >>> xy = mp.chain([x, y]) >>> mppl = mpp.MPPovmList((xx, xy)) >>> xxe = mp.chain([x, x, e]) >>> xye = mp.chain([x, y, e]) >>> exx = mp.chain([e, x, x]) >>> exy = mp.chain([e, x, y]) >>> expect = (xxe, xye, exx, exy) >>> [abs(mp.norm(a  b)) <= 1e10 ... for a, b in zip(mppl.block(3).mpps, expect)] [True, True, True, True]

est_lfun
(coeff, funs, samples, weights=None, eps=1e10)[source]¶ Estimate a linear combination of functions of POVM outcomes
Parameters:  coeff – Iterable of coefficient lists
 funs – Iterable of function lists
 samples – Iterable of samples
 weights – Iterable of weight lists or None
The ith item from these parameters is passed to
MPPovm.est_lfun()
on self.mpps[i].est_lfun.Returns: (est, var): Estimated value est and estimated variance var of the estimate est

est_lfun_from
(other, coeff, samples, eps=1e10)[source]¶ Estimate a linear function from samples for another MPPovmList
The function to estimate is a linear function of the probabilities of self and it is specified by coeff. Its true expectation value and variance are returned by
MPPovmList.lfun_from()
. First, an estimator is constructed usingMPPovmList._lfun_estimator()
and this estimator is passed toMPPovm.est_lfun()
to obtain the estimate. See Linear combinations of functions of POVM outcomes for more details.Parameters:  other (MPPovmList) – Another MPPOVM list
 coeff – A sequence of shape self.mpps[i].nsoutdims coefficients which specify the function to estimate
 samples – A sequence of samples for other
Returns: (est, var): Estimated value and estimated variance of the estimated value. Return (np.nan, np.nan) if other is not sufficient to estimate the function.

est_pmf
(samples, normalized=True, eps=1e10)[source]¶ Estimate PMF from samples
Returns an iterator over results from
MPPovm.est_pmf()
(see there).

est_pmf_from
(other, samples, eps=1e10)[source]¶ Estimate PMF from samples of another MPPovmList
Parameters:  other (MPPovmList) – A different MPPovmList
 samples – Samples from other
Returns: Iterator over (p_est, n_samples_used) from
MPPovm.est_pmf_from_mpps()
.

lfun
(coeff, funs, state, mode='auto', eps=1e10)[source]¶ Evaluate a linear combination of functions of POVM outcomes
coeff[i] and funs[i] are passed to
MPPovm.lfun()
on self.mpps[i]. funs = None is treated as [None] * len(self.mpps). state and mode are passed toMPPovm.pmf()
.Returns: (value, var): Expectation value and variance of the expectation value

lfun_from
(other, coeff, state, mode='auto', other_weights=None, eps=1e10)[source]¶ Evaluate a linear combination of POVM probabilities
This function computes the same expectation value as
MPPovmList.lfun()
if supplied with funs = None, but it computes the variance for a different estimation procedure: It uses weighted averages of POVM probabilities from other to obtain the necessary POVM probabilities for self (the same is done inMPPovmList.est_lfun_from()
).The parameter coeff is explained in
MPPovmList.est_lfun_from()
. state and mode are passed toMPPovm.pmf()
.You can supply the array other_weights to determine the weighted average used when a probability in a POVM in self can be estimated from probabilities in multiple different POVMs in other.
Returns: (value, var): Expectation value and variance of the expectation value. Return (np.nan, np.nan) if other is not sufficient to estimate the function.

pack_samples
(samples)[source]¶ Pack samples into one integer per sample
Returns: Iterator over output from MPPovm.pack_samples()

pmf
(state, mode='auto')[source]¶ Compute the probability mass functions of all MPPOVMs
Parameters:  state – A quantum state as MPA
 mode – Passed to
MPPovm.expectations()
Returns: Iterator over probabilities as MPArrays

pmf_as_array
(state, mode='auto', eps=1e10)[source]¶ Compute the PMF of all MPPOVMs as full arrays
Parameters: See
MPPovmList.pmf()
. Sanity checks: SeeMPPovm.pmf_as_array()
.Returns: Iterator over probabilities as ndarrays

repeat
(nr_sites)[source]¶ Construct longer MPPOVMs by repeating each MPPOVM
This function calls
MPPovm.repeat(nr_sites)
for each MPPOVM in the list.For example,
pauli_mpps()
for local_dim > 3 (i.e. without Z) and two sites returns POVMs for the four tensor product observables XX, XY, YX and YY:>>> import mpnum as mp >>> import mpnum.povm as mpp >>> block_sites = 2 >>> ldim = 3 >>> x, y = (mpp.MPPovm.from_local_povm(lp(ldim), 1) for lp in ... (mpp.x_povm, mpp.y_povm)) >>> pauli = mpp.pauli_mpps(block_sites, ldim) >>> expect = ( ... mp.chain((x, x)), ... mp.chain((x, y)), ... mp.chain((y, x)), ... mp.chain((y, y)), ... ) >>> [abs(mp.norm(a  b)) <= 1e10 for a, b in zip(pauli.mpps, expect)] [True, True, True, True]
Calling repeat(5) then returns the following
MPPovmList
:>>> expect = ( ... mp.chain((x, x, x, x, x)), ... mp.chain((x, y, x, y, x)), ... mp.chain((y, x, y, x, y)), ... mp.chain((y, y, y, y, y)), ... ) >>> [abs(mp.norm(a  b)) <= 1e10 ... for a, b in zip(pauli.repeat(5).mpps, expect)] [True, True, True, True]

sample
(rng, state, n_samples, method, n_group=1, mode='auto', pack=False, eps=1e10)[source]¶ Random sample from all MPPOVMs on a quantum state
Parameters: See
MPPovm.sample()
.Return value: Iterable of return values from
MPPovm.sample()
.

unpack_samples
(samples)[source]¶ Unpack samples into several integers per sample
Returns: Iterator over output from MPPovm.unpack_samples()
 Conveniently obtain samples and estimated or exact probabilities
for a list of

mpnum.povm.mppovm.
pauli_mpp
(nr_sites, local_dim)[source]¶ Pauli POVM tensor product as MPPOVM
The resulting MPPOVM will contain all tensor products of the elements of the local Pauli POVM from
mpp.pauli_povm()
.Parameters:  nr_sites (int) – Number of sites of the returned MPPOVM
 local_dim (int) – Local dimension
Return type: For example, for two qubits the (1, 3) measurement outcome is minus X on the first and minus Y on the second qubit:
>>> nr_sites = 2 >>> local_dim = 2 >>> pauli = pauli_mpp(nr_sites, local_dim) >>> xy = np.kron([1, 1], [1, 1j]) / 2 >>> xyproj = np.outer(xy, xy.conj()) >>> proj = pauli.get([1, 3], astype=mp.MPArray) \ ... .to_array_global().reshape((4, 4)) >>> abs(proj  xyproj / 3**nr_sites).max() <= 1e10 True
The prefactor 1 / 3**nr_sites arises because X, Y and Z are in a single POVM.

mpnum.povm.mppovm.
pauli_mpps
(nr_sites, local_dim)[source]¶ Pauli POVM tensor product as MPPOVM list
The returned
MPPovmList
contains all tensor products of the singlesite X, Y (and Z if local_dim == 2) POVMs:>>> import mpnum as mp >>> import mpnum.povm as mpp >>> block_sites = 2 >>> ldim = 3 >>> x, y = (mpp.MPPovm.from_local_povm(lp(ldim), 1) for lp in ... (mpp.x_povm, mpp.y_povm)) >>> pauli = mpp.pauli_mpps(block_sites, ldim) >>> expect = ( ... mp.chain((x, x)), ... mp.chain((x, y)), ... mp.chain((y, x)), ... mp.chain((y, y)), ... ) >>> [abs(mp.norm(a  b)) <= 1e10 for a, b in zip(pauli.mpps, expect)] [True, True, True, True]
Parameters:  nr_sites (int) – Number of sites of the returned MPPOVMs
 local_dim (int) – Local dimension
Return type:
povm.localpovm
¶
An informationally complete dlevel POVM.
The POVM simplifies to measuring Paulis matrices in the case of qubits.

class
mpnum.povm.localpovm.
POVM
(elements, info_complete=False, pinv=<function pinv>)[source]¶ Bases:
object
Represent a Positive OperatorValued Measure (POVM).

classmethod
from_vectors
(vecs, info_complete=False)[source]¶ Generates a POVM consisting of rank 1 projectors based on the corresponding vectors.
Parameters:  vecs – Iterable of np.ndarray with ndim=1 representing the vectors for the POVM
 info_complete – Is the POVM informationally complete (default False)
Returns:

informationally_complete
¶

linear_inversion_map
¶ Map that reconstructs a density matrix with linear inversion.
Linear inversion is performed by taking the Moore–Penrose pseudoinverse of self.probability_map.

probability_map
¶ Map that takes a raveled density matrix to the POVM probabilities
The following two return the same:
probab = np.array([ np.trace(np.dot(elem, rho)) for elem in a_povm ]) probab = np.dot(a_povm.probability_map, rho.ravel())

classmethod

mpnum.povm.localpovm.
concat
(povms, weights, info_complete=False)[source]¶ Combines the POVMs given in povms according the weights given to a new POVM.
Parameters:  povms – Iterable of POVM
 weights – Iterable of real numbers, should sum up to one
 info_complete – Is the resulting POVM informationally complete
Returns: POVM

mpnum.povm.localpovm.
pauli_parts
(dim)[source]¶ The POVMs used by
pauli_povm()
as a listFor dim > 3,
x_povm()
andy_povm()
are returned. For dim = 2,z_povm()
is included as well.Parameters: dim – Dimension of the system Returns: Tuple of POVMs

mpnum.povm.localpovm.
pauli_povm
(dim)[source]¶ An informationally complete dlevel POVM that simplifies to measuring Pauli matrices in the case d=2.
Parameters: dim – Dimension of the system Returns: POVM
with (generalized) Pauli measurments

mpnum.povm.localpovm.
x_povm
(dim)[source]¶ The X POVM simplifies to measuring Pauli X eigenvectors for dim=2.
Parameters: dim – Dimension of the system Returns: POVM with generalized X measurments
special
¶
Optimized functions
Module contains some specialiced versions of some functions from mparray. They are tuned for speed with special applications in mind

mpnum.special.
inner_prod_mps
(mpa1, mpa2)[source]¶ Same as
mparray.inner()
, but assumes that mpa1 is a product MPSParameters:  mpa1 – MPArray with one leg per site and rank 1
 mpa2 – MPArray with same shape as mpa1 but arbitrary rank
Returns: <mpa1mpa2>

mpnum.special.
sumup
(mpas, rank, weights=None, svdfunc=<function truncated_svd>)[source]¶ Same as
mparray.sumup()
with a consequent compression, but with inplace svd compression. Also, we use a sparsematrix format for the intermediate local tensors of the sum. Therefore, the memory footprint scales only linearly in the number of summands (instead of quadratically).Right now, only the sum of product tensors is supported.
Parameters:  mpas – Iterator over MPArrays
 rank – Rank of the final result.
 weights – Iterator of same length as mpas containing weights for computing weighted sum (default: None)
 svdfunc – Function implementing the truncated svd, for required
signature see
truncated_svd()
.
Returns: Sum of mpas with max. rank rank
Possible values for
svdfunc
include:truncated_svd()
: Almost no speedup compared to the standard sumup and compression, since it computes the full SVDscipy.sparse.linalg.svds()
: Only computes the necessary singular values/vectors, but slow if rank is not small enoughmpnum.utils.extmath.randomized_svd()
: Randomized truncated SVD, fast and efficient, but only approximation.
utils
¶
utils.array_transforms
¶
Helper functions for transforming arrays

mpnum.utils.array_transforms.
global_to_local
(array, sites, left_skip=0, right_skip=0)[source]¶ Converts a general siteslocal array with fixed number p of physical legs per site from the global form
A[i_1,…, i_N, j_1,…, j_N, …](i.e. grouped by physical legs) to the local form
A[i_1, j_1, …, i_2, j_2, …](i.e. grouped by site).
Parameters:  array (np.ndarray) – Array with ndim, such that ndim % sites = 0
 sites (int) – Number of distinct sites
 left_skip (int) – Ignore that many axes on the left
 right_skip (int) – Ignore that many axes on the right
Returns: Array with same ndim as array, but reshaped
>>> global_to_local(np.zeros((1, 2, 3, 4, 5, 6)), 3).shape (1, 4, 2, 5, 3, 6) >>> global_to_local(np.zeros((1, 2, 3, 4, 5, 6)), 2).shape (1, 3, 5, 2, 4, 6)

mpnum.utils.array_transforms.
local_to_global
(array, sites, left_skip=0, right_skip=0)[source]¶ Inverse of local_to_global
Parameters:  array (np.ndarray) – Array with ndim, such that ndim % sites = 0
 sites (int) – Number of distinct sites
 left_skip (int) – Ignore that many axes on the left
 right_skip (int) – Ignore that many axes on the right
Returns: Array with same ndim as array, but reshaped
>>> ltg, gtl = local_to_global, global_to_local >>> ltg(gtl(np.zeros((1, 2, 3, 4, 5, 6)), 3), 3).shape (1, 2, 3, 4, 5, 6) >>> ltg(gtl(np.zeros((1, 2, 3, 4, 5, 6)), 2), 2).shape (1, 2, 3, 4, 5, 6)
Transform all or only the inner axes:
>>> ltg = local_to_global >>> ltg(np.zeros((1, 2, 3, 4, 5, 6)), 3).shape (1, 3, 5, 2, 4, 6) >>> ltg(np.zeros((1, 2, 3, 4, 5, 6)), 2, left_skip=1, right_skip=1).shape (1, 2, 4, 3, 5, 6)
utils.extmath
¶
Additional math functions for dealing with dense arrays

mpnum.utils.extmath.
block_diag
(summands, axes=(0, 1))[source]¶ Blockdiagonal sum for ndimensional arrays.
Perform something like a block diagonal sum (if len(axes) == 2) along the specified axes. All other axes must have identical sizes.
Parameters: axes – Along these axes, perform a blockdiagonal sum. Can be negative. >>> a = np.arange(8).reshape((2, 2, 2)) >>> b = np.arange(8, 16).reshape((2, 2, 2)) >>> a array([[[0, 1], [2, 3]], [[4, 5], [6, 7]]]) >>> b array([[[ 8, 9], [10, 11]], [[12, 13], [14, 15]]]) >>> block_diag((a, b), axes=(1, 1)) array([[[ 0, 1, 0, 0], [ 2, 3, 0, 0], [ 0, 0, 8, 9], [ 0, 0, 10, 11]], [[ 4, 5, 0, 0], [ 6, 7, 0, 0], [ 0, 0, 12, 13], [ 0, 0, 14, 15]]])

mpnum.utils.extmath.
matdot
(A, B, axes=((1, ), (0, )))[source]¶ np.tensordot with sane defaults for matrix multiplication

mpnum.utils.extmath.
partial_trace
(array, traceout)[source]¶ Return the partial trace of an array over the sites given in traceout.
Parameters:  array (np.ndarray) – Array in global form (see
global_to_local()
above) with exactly 2 legs per site  traceout – List of sites to trace out, must be in _ascending_ order
Returns: Partial trace over input array
 array (np.ndarray) – Array in global form (see

mpnum.utils.extmath.
truncated_svd
(A, k)[source]¶ Compute the truncated SVD of the matrix A i.e. the k largest singular values as well as the corresponding singular vectors. It might return less singular values/vectors, if one dimension of A is smaller than k.
In the background it performs a full SVD. Therefore, it might be inefficient when k is much smaller than the dimensions of A.
Parameters:  A – A real or complex matrix
 k – Number of singular values/vectors to compute
Returns: u, s, v, where u: leftsingular vectors s: singular values in descending order v: rightsingular vectors

mpnum.utils.extmath.
randomized_svd
(M, n_components, n_oversamples=10, n_iter='auto', piter_normalizer='auto', transpose='auto', randstate=<module 'numpy.random' from '/usr/lib/python3/distpackages/numpy/random/__init__.py'>)[source]¶ Computes a truncated randomized SVD. Uses the same convention as
scipy.sparse.linalg.svds()
. However, we guarantee to return the singular values in descending order.Parameters:  M – The input data matrix, can be any type that can be converted
into a
scipy.linalg.LinarOperator
, e.g.numpy.ndarray
, or a sparse matrix.  n_components (int) – Number of singular values and vectors to extract.
 n_oversamples (int) – Additional number of random vectors to sample the
range of M so as to ensure proper conditioning. The total number of
random vectors used to find the range of M is
n_components + n_oversamples
. Smaller number can improve speed but can negatively impact the quality of approximation of singular vectors and singular values. (default 10)  n_iter – Number of power iterations. It can be used to deal with very
noisy problems. When
'auto'
, it is set to 4, unlessn_components
is small (< .1 * min(X.shape)
). Then,n_iter
is set to 7. This improves precision with few components. (default'auto'
)  piter_normalizer (str) –
'auto'
(default),'QR'
,'LU'
,'none'
. Whether the power iterations are normalized with stepbystep QR factorization (the slowest but most accurate),'none'
(the fastest but numerically unstable when n_iter is large, e.g. typically 5 or larger), or'LU'
factorization (numerically stable but can lose slightly in accuracy). The ‘auto’ mode applies no normalization ifn_iter <= 2
and switches to LU otherwise.  transpose –
True
,False
or'auto'
Whether the algorithm should be applied toM.T
instead ofM
. The result should approximately be the same. The'auto'
mode will trigger the transposition ifM.shape[1] > M.shape[0]
since then the computational overhead in the randomized SVD is generally smaller. (default'auto'
).  randstate – An instance of
numpy.random.RandomState
(default isnp.random
))
Notes
This algorithm finds a (usually very good) approximate truncated singular value decomposition using randomization to speed up the computations. It is particularly fast on large matrices on which you wish to extract only a small number of components. In order to obtain further speed up,
n_iter
can be set <=2 (at the cost of loss of precision).References
 Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions Halko, et al., 2009 http://arxiv.org/abs/arXiv:0909.4061
 A randomized algorithm for the decomposition of matrices PerGunnar Martinsson, Vladimir Rokhlin and Mark Tygert
 An implementation of a randomized algorithm for principal component analysis A. Szlam et al. 2014
 M – The input data matrix, can be any type that can be converted
into a
utils.physics
¶
Code related to physical models
Contents:
 Hamiltonian and analytic ground state energy of the cyclic XY model
References:
[LSM61]  Lieb, Schultz and Mattis (1961). Two soluble models of an antiferromagnetic chain. 

mpnum.utils.physics.
cXY_E0
(nr_sites, gamma)[source]¶ Ground state energy of the cyclic XY model
Parameters:  nr_sites – Number of spin onehalf sites
 gamma – Asymmetry parameter
Returns: Exact energy of the ground state
This function is implemented for
nr_sites
which is an odd multiple of two. In this case, the ground state energy of the XY model is given by (Eqs. (A12), (2.20) of [LSM61])\[E_0 = \frac12 \sum_{l=0}^{N1} \Lambda_{k(l)}\]with (Eqs. (2.18b), (2.18c))
\[\Lambda_k^2 = 1  (1  \gamma^2) [\sin(k)]^2, \quad k(l) = \frac{2\pi}{N} \left( l  \frac N2 \right)\]and \(\Lambda_k \ge 0\).

mpnum.utils.physics.
cXY_local_terms
(nr_sites, gamma)[source]¶ Local terms of the cyclic XY model (MPOs)
Parameters:  nr_sites – Number of spin onehalf sites
 gamma – Asymmetry parameter
Returns: List
terms
of lengthnr_sites
(MPOs)The term
terms[i]
acts on spins(i, i + 1)
and spinnr_sites
is the same as the first spin.The Hamiltonian of the cyclic XY model is given by [LSM61, Eq. 2.1]:
\[H_\gamma = \sum_{i=1}^{N} (1+\gamma) S^x_i S^x_{i+1} + (1\gamma) S^y_i S^y_{i+1}\]with \(S^j_{N+1} = S^j_{1}\). The function
cXY_E0()
returns the exact ground state energy of this Hamiltonian.

mpnum.utils.physics.
mpo_cH
(terms)[source]¶ Construct an MPO cyclic nearestneighbour Hamiltonian
Parameters: terms – List of nearstneighbour terms (MPOs, see return value of cXY_local_terms()
)Returns: The Hamiltonian as MPO Note
It may not be advisable to call
mp.MPArray.canonicalize()
on a Hamiltonian, e.g.:>>> mpoH = mpo_cH(cXY_local_terms(nr_sites=100, gamma=0)) >>> abs1 = max(abs(lt).max() for lt in mpoH.lt) >>> mpoH.canonicalize() >>> abs2 = np.round(max(abs(lt).max() for lt in mpoH.lt), 3) >>> print('{:.3f} {:.2e}'.format(abs1, abs2)) 1.000 2.79e+15
The Hamiltonian generally has a large Frobenius norm because local terms are embedded with identity matrices. This causes large tensor entries of canonicalization which will eventually overflow the numerical maximum (the overflow happens somewhere between 2000 and 3000 sites in this example). One could embed local terms with Frobeniusnormalized identity matrices instead, but this would make the eigenvalues of H exponentially (in
nr_sites
) small. This would eventually cause numerical underflows.

mpnum.utils.physics.
sparse_cH
(terms, ldim=2)[source]¶ Construct a sparse cyclic nearestneighbour Hamiltonian
Parameters:  terms – List of nearstneighbour terms (square array or MPO,
see return value of
cXY_local_terms()
)  ldim – Local dimension
Returns: The Hamiltonian as sparse matrix
 terms – List of nearstneighbour terms (square array or MPO,
see return value of
Todo list (autogenerated)¶
Todo
single site MPAs – what is left?
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray, line 3.)
Todo
Local tensor ownership – see MPArray class comment
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray, line 4.)
Todo
Possible optimization:
 replace integerfor loops with iterator (not obviously possible everwhere)
 replace internal structure as list of arrays with lazy generator of arrays (might not be possible, since we often iterate both ways!)
 more in place operations for addition, subtraction, multiplication
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray, line 5.)
Todo
Replace all occurences of self._ltens with self[…] or similar & benchmark. This will allow easier transition to lazy evaluation of local tensors
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray, line 12.)
Todo
As it is now, e.g. __imul__()
modifies
items from self._ltens
. This requires
e.g. chain()
to take copies of the local
tensors. The data model seems to be that an MPArray
instance owns its local tensors and everyone else,
including each new MPArray instance, must take
copies. Is this correct?
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.MPArray, line 18.)
Todo
More appropriate naming for this functions?
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.MPArray.leg2vleg, line 6.)
Todo
Why is this here? What’s wrong with the purne function?
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.MPArray.reshape, line 10.)
Todo
More appropriate naming for this functions?
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.MPArray.vleg2leg, line 11.)
Todo
Make this canonicalization aware
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.chain, line 10.)
Todo
Raise warning when casting complex to real dtype
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.chain, line 11.)
Todo
This table needs cell borders in the HTML output (> CSS) and the tabularcolumns command doesn’t work.
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mparray.py:docstring of mpnum.mparray.regular_slices, line 24.)
Todo
Are derived classes MPO/MPS/PMPS of any help?
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mpsmpo.py:docstring of mpnum.mpsmpo, line 99.)
Todo
I am not sure the current definition of PMPS is the most elegant for our purposes…
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mpsmpo.py:docstring of mpnum.mpsmpo, line 101.)
Todo
Add docstring
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/mpsmpo.py:docstring of mpnum.mpsmpo.reductions, line 1.)
Todo
Add information on how the runtime of eig()
and
eig_sum()
scale with the the different ranks. For
the time being, refer to the benchmark test.
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/linalg.py:docstring of mpnum.linalg.eig_sum, line 13.)
Todo
Explain the details of the variance estimation, in particular the
difference between the variances returned from
MPPovmList.lfun()
and MPPovmList.lfun_from()
. Check the
mean square error.
Add a good references explaining all facts mentioned above and for further reading.
Document the runtime and memory cost of the functions.
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm, line 116.)
Todo
This class should provide a function which returns expectation values as full array. (Even though computing expectation values using the POVM struture brings advantages, we usually need the result as full array.) This function should also replace small negative probabilities by zero and canonicalize the sum of all probabilities to unity (if the deviation is nonzero but small). The same checks should also be implemented in localpovm.POVM.
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm.MPPovm, line 19.)
Todo
Right now we use this class for multisite POVMs with elements obtained from every possible combination of the elements of singlesite POVMs: The POVM index is split across all sites. Explore whether and how this concept can also be useful in other cases.
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm.MPPovm, line 28.)
Todo
Add docstring
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm.MPPovm.block_pmfs_as_array, line 1.)
Todo
Add docstring
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm.MPPovm.pmfs_as_array, line 1.)
Todo
Add docstring
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm.MPPovmList.block_pmfs_as_array, line 1.)
Todo
Add docstring
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/mpnum/povm/mppovm.py:docstring of mpnum.povm.mppovm.MPPovmList.pmfs_as_array, line 1.)
Todo
Reference to Schollwoeck not working anymore.
(The original entry is located in /home/docs/checkouts/readthedocs.org/user_builds/mpnum/checkouts/latest/docs/mpnum.rst, line 118.)
Todo
Reference to Schollwoeck not working anymore.
Note
make livehtml
(based on sphinxautobuild) does not rebuild
this list.