API reference

Module overview

• mpnum.mparray: Basic matrix product array (MPA) routines and compression
• mppnum.mpstruct: Underlying structure of MPAs to manage the local tensors
• mpnum.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 MPOs
• mpnum.special: Optimized versions of some routines for special cases
• mpnum.povm: Matrix product representation of Positive operator valued measures (POVM)

• mpnum.povm.localpovm: Pauli-like POVM on a single site
• mpnum.povm.mppovm: Matrix product POVM based on the Pauli-like 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 integer-for 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)

Bases: object

Efficient representation of a general N-partite 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 right-most legs (virtual indices) and the multi-index \(i_k\) corresponds to the true local legs. Open boundary conditions imply that \(A^{[1]}\) is 1-by-something and \(A^{[N]}\) is something-by-1.

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 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?

__init__(ltens)

Parameters:ltens – local tensors as instance of mpstruct.LocalTensors or simply as a list of numpy.ndarray in the format described at mpstruct.LocalTensors

__len__()

Returns the number of sites

T

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

adj()

Hermitian adjoint. Equivalent to self.T.conj().

axis_iter(axes=0)

Returns an iterator yielding Sub-MPArrays 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 site A[(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

See mpstruct.LocalTensors.canonical_form

canonicalize(left=None, right=None)

Brings the MPA to canonical form in place [Sch11, Sec. 4.4]

Note that we do not support full left- or right-canonicalization. In general, the right- (left- resp.)most local tensor cannot be in a canonical form since at least one local tensor must be non-normalized.

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 and right have the following meaning:

• self[:left] will be left-normalized
• self[right:] will be right-normalized

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 right-normalized: ValueError

compress(method='svd', **kwargs)

Compress self, modifying it in-place.

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 2-norm. 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) or None (choose depending on canonicalization). (default: None)
• canonicalize – SVD compression works best when the MPA is brought into full left-/right-cannonical 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 by scipy.sparse.linalg.svds() or a randomized SVD such as randomized_svd() might speed up computations with no or little loss of accuracy.

Parameters for 'var':

Parameters:

• rank – Maximal rank for the result. Either startmpa or rank 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:

• 'svd': Singular value truncation, [Sch11, Sec. 4.5.1]
• 'var': Variational compression, [Sch11, Sec. 4.5.2]

compression(method='svd', **kwargs)

Return a compression of self. Does not modify self.

Parameters: See compress().

Returns:(compressed_mpa, overlap) where overlap is the inner product returned by compress().

conj()

Complex conjugate

copy()

Returns a deep copy of the MPA

dtype

Returns the dtype that should be returned by to_array

dump(target)

Serializes MPArray to h5py.Group. Recover using load().

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)

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 left-canonical representation of array. [Sch11, Sec. 4.3.1]

The result is a chain of local tensors with ndims physical legs at each location and has array.ndim // ndims number of sites (assuming ndims 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 see tools.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

classmethod from_array_global(array, ndims=None, has_virtual=False)

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.

classmethod from_kron(factors)

Returns the (exact) representation of an n-fold 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)

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 site Returns:type(self) object

group_sites(sites_per_group)

Group several MPA sites into one site.

The resulting MPA has length len(self) // sites_per_group and sites_per_group * self.ndims[i] physical legs on site i. 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)

Performs the inverse operation to vleg2leg().

Parameters:pos – Number of the virtual to perform the transformation Returns:read-only MPA with transformed virtual

Todo

More appropriate naming for this functions?

classmethod load(source)

Deserializes MPArray from h5py.Group. Serialize using dump().

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)

Increase rank by padding with zeros

This function is useful to prepare initial states for variational compression. E.g. for a five-qubit 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) if None (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

ranks

Tuple of ranks

ravel()

Flatten the MPA to an MPS, shortcut for self.reshape((-1,))

reshape(newshapes)

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?

reverse()

shape

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

singularvals()

Return singular values of self for all bipartitions

Returns: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)

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)

Split MPA sites into several sites.

The resulting MPA has length len(self) * sites_per_group and self.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 fewer ndims

sum(axes=None)

Element-wise 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 site i; 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()

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()

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)

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. (default None)

>>> 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)

Transforms the virtual leg between site pos and pos + 1 into local legs at those sites. The new leg will be the rightmost one at site pos and the leftmost one at site pos + 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)

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) where ax1 (ax2) is a single physical leg number or sequence of physical leg numbers referring to mpa1 (mpa2). The first (second, etc) entries of ax1 and ax2 will be contracted. Very similar to the axes argument for numpy.tensordot(). (default: (-1, 0))

Note

Note that the default value of axes is different compared to numpy.tensordot().

Parameters:astype – Return type. If None, use the type of mpa1 Returns:Dot product of the physical arrays

mpnum.mparray.inject(mpa, pos, num=None, inject_ten=None)

Interleaved chain product of an MPA and a rank 1 MPA

Return the chain product between mpa and num copies of the local tensor inject_ten, but place the copies of inject_ten before site pos inside or outside mpa. You can also supply num = 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 of mpa using pos = 0 or pos = len(mpa) is also supported, but chain() is preferred for that as it is a much simpler function.

If inject_ten is omitted, use a square identity matrix of size mpa.shape[pos][0]. If pos == 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 be None; in this case inject_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 and inject_ten must be either sequences or None, where None is interpreted as len(pos) * [None]. As above, if num[i] is None, then inject_ten[i] must be a sequence of values.

mpnum.mparray.inner(mpa1, mpa2)

Compute the inner product <mpa1|mpa2>. 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:

<mpa1|mpa2>

mpnum.mparray.local_sum(mpas, embed_tensor=None, length=None, slices=None)

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 use regular_slices(length, width, offset) with offset = 1, width = len(mpas[0]) and length = 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 of mpas[i], optional.

Returns:

An MPA.

mpnum.mparray.localouter(a, b)

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:

• a (MPArray) – MPArray
• b (MPArray) – MPArray of same length as a

Returns:

Tensor product of a and b in terms of their local tensors

mpnum.mparray.norm(mpa)

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 canonicalization

WARNING This also changes the MPA inplace by normalizing.

Parameters:mpa – MPArray Returns:l2-norm of that array

mpnum.mparray.normdist(mpa1, mpa2)

More efficient version of norm(mpa1 - mpa2)

Parameters:

• mpa1 – MPArray
• mpa2 – MPArray

Returns:

l2-norm of mpa1 - mpa2

mpnum.mparray.chain(mpas, astype=None)

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))

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 as dot().

Parameters:

• mpa2 (mpa1,) – Factors as MPArrays, length must be inequal.
• start_at – The shorter MPA will start on this site.
• axes – See axes argument to dot().

Returns:

MPA with length of the longer MPA.

mpnum.mparray.partialtrace(mpa, axes=(0, 1), mptype=None)

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 length-one MPA with zero physical legs.

For axes=(m, n) with integer m, trace over the given axes at all sites and return a length-one MPA with zero physical legs. (Use trace() to get the value directly.)

For axes=(axes1, axes2, ...) trace over axesN at site N, with axesN=(axisN_1, axisN_2) tracing the given physical legs and axesN=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)

Contract sites with zero (physical) legs.

Parameters:

• mpa (MPArray) – MPArray or iterator over local tensors
• singletons – If True, also contract sites where all physical legs have size 1 (default: False)

Returns:

An MPArray (of possibly smaller length)

mpnum.mparray.regular_slices(length, width, offset)

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 i-th slice is slice(offset * i, offset * i + width), with i 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)

Compute <mps|MPO|mps> efficiently

This 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 where D and Dp are the ranks of mps and mpo. This is more efficient than mp.inner(mps, mp.dot(mpo, mps)), whose runtime scales with D**4 * Dp**3, and also more efficient than mp.dot(mps.conj(), mp.dot(mpo, mps)).to_array(), whose runtime scales with D**6 * Dp**3.

If mps2 is given, <mps2|MPO|mps> is computed instead (i.e. mp.inner(mps2, mp.dot(mpo, mps)); see also dot()).

mpnum.mparray.embed_slice(length, slice_, mpa, embed_tensor=None)

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))

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, ...) with axesN=(axisN_1, axisN_2) or axesN=None. (default: (0, 1))

Returns:

A single scalar of type mpa.dtype

mpnum.mparray.diag(mpa, axis=0)

Returns the diagonal elements mpa[i, i, ..., i]. If mpa has more than one physical dimension, the result is a numpy array with MPArray 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 an MPArray with dimension 0 is a simple number)

mpnum.mparray.sumup(mpas, weights=None)

Returns the sum of the MPArrays in mpas. Same as

functools.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)

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 to MPArray.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))

Bases: object

Core data structure to manage the local tensors of a MPArray.

The local tensors are kept in _ltens, a list of numpy.ndarrays such that _ltens[i] corresponds to the local tensor of site i.

If there are \(k\) (non-virtual) 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 \((r_{i-1}, d_1, \ldots, d_k, r_{i})\). Here, \(r_{i-1}\) 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/right-canonical form.

Returns tuple (left, right) such that

• self[:left] are left-normalized
• self[right:] are right-normalized.

copy()

Returns a deep copy of the local tensors

shape

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

update(index, tens, canonicalization=None)

Update the local tensor at site index to the new value tens. 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 case index is a slice).
• canonicalization – If tens is left-/right-normalized, pass 'left'/'right', respectively. Otherwise, pass None (default None). In case index 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)

Returns a MPA representing the identity matrix

Parameters:

• sites – Number of sites
• ldim – Int-like 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)

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'>)

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_ and np.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, -0.32222320-0.32675463j]])

mpnum.factory.random_mpdo(sites, ldim, rank, randstate=<module 'numpy.random' from '/usr/lib/python3/dist-packages/numpy/random/__init__.py'>)

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)

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)

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)

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 zero-array as MPA

mpnum.factory.diagonal_mpa(entries, sites)

Returns an MPA with entries on the diagonal and zeros otherwise.

Parameters:entries (numpy.ndarray) – one-dimensional array Returns:MPArray with rank len(entries).

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

\[\langle i_1 \ldots i_n \vert \psi \rangle = A^{(1)}_{i_1} \cdots A^{(n)}_{i_n}, \quad A^{(k)}_{i_k} \in \mathbb C^{D_{k-1} \times D_k}, \quad D_0 = 1 = D_n.\]

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

\[\langle i_1 \ldots i_n \vert M \vert j_1 \ldots j_n \rangle = A^{(1)}_{i_1 j_1} \cdots A^{(n)}_{i_n j_n}, \quad A^{(k)}_{i_k j_k} \in \mathbb C^{D_{k-1} \times D_k}, \quad D_0 = 1 = D_n.\]

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

\[\rho = \operatorname{tr}_{\mathcal H'_1, \ldots, \mathcal H'_n} (\lvert \Phi \rangle \langle \Phi \rvert)\]

and represent \(\lvert \Phi \rangle\) as

\[\langle i_1 i'_1 \ldots i_n i'_n \vert \Phi \rangle = A^{(1)}_{i_1 i'_1} \cdots A^{(n)}_{i_n i'_n}, \quad A^{(k)}_{i_k j_k} \in \mathbb C^{D_{k-1} \times D_k}, \quad D_0 = 1 = D_n.\]

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., Pérez-García, D., and Cirac, J. I. (2013). “Purifications of multipartite states: limitations and constructive methods”. New J. Phys. 15(12), p. 123021. DOI: 10.1088/1367-2630/15/12/123021. arXiv: 1308.1914.

mpnum.mpsmpo.mps_to_mpo(mps)

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)

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)

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() and to_array(). This is useful for obtaining reduced states of a PMPS on non-consecutive sites, as normalizing before using pmps_to_mpo() may not be sufficient to reduce the rank in that case.

Note

The resulting array will have dimension-1 physical legs removed.

mpnum.mpsmpo.pmps_reduction(pmps, support)

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)

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)

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)

Iterate over MPO partial traces of an MPO

The support of the i-th 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)

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)

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)

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

mpnum.mpsmpo.reductions(state, mode, **kwargs)

Todo

Add docstring

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)

Iterative search for MPO eigenvalues

Note

This function can return completely inaccurate values. You are responsible for supplying a large enough startvec_rank (or startvec with large enough rank) and num_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 by var_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 “single-site DMRG” [Sch11, Sec. 6.3, p. 69]. For var_sites > 1, it is called “multi-site 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 or None
• eigs – Function which computes one eigenvector of the local eigenvalue problem on var_sites sites

Returns:

eigval, eigvec_mpa

The eigs parameter defaults to

eigs = functools.partial(scipy.sparse.linalg.eigsh, k=1, tol=1e-6)

By default, eig() computes the eigenvalue with largest magnitude. To compute e.g. the smallest eigenvalue (sign included), supply which='SA' to eigsh. For other possible values, refer to the SciPy documentation.

It is recommendable to supply a value for the tol parameter of eigsh(). 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 NP-hard (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 zero-based with one-based indices there.

Psi^A_{i-1} and Psi^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)

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 as-is. 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 in mpas, 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 in mpas to an MPO.

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.

Parameters:mpas – A sequence of MPOs or MPSs

Remaining parameters and description: See eig().

Algorithm: [Sch11, Sec. 6.3]

povm

povm.mppovm

Matrix-product representation of POVMs

This module provides the following classes:

• MPPovm: A matrix product representation of a multi-site 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 MP-POVMs.

  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 as pauli_mpp() and pauli_mpps() allow for convenient construction of MP-POVMs and MP-POVM 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

\[\begin{split}\theta_y(x) = \begin{cases} 1, & x = y, \\ 0, & \text{otherwise.} \end{cases}\end{split}\]

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 MP-POVMs 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 MP-POVMs. 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 \(\frac{m}{m-1} \operatorname{cov}_D(f, g)\) 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)

Bases: mpnum.mparray.MPArray

MPArray representation of multipartite POVM

There are two different ways to write down a POVM in matrix product form

1. As a list of matrix product operators, where each entry corresponds to

   a single POVM element

2. 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 non-zero but small). The same checks should also be implemented in localpovm.POVM.

Todo

Right now we use this class for multi-site POVMs with elements obtained from every possible combination of the elements of single-site 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)

Embed an MP-POVM 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 MP-POVMs

block_pmfs_as_array(state, mode, asarray=False, eps=1e-10, **redarg)

Todo

Add docstring

elements

Returns an iterator over all POVM elements. The result is the i-th 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)

Embed MP-POVM into larger system

Applying the resulting embedded MP-POVM to a state rho gives the same result as applying the original MP-POVM self on the reduced state of sites range(startsite, startsite + len(self)) of rho.

Parameters:

• nr_sites – Number of sites of the resulting MP-POVM
• startsite – Position of the first site of self in the resulting MP-POVM
• local_dim – Local dimension of sites to be added

Returns:

MP-POVM with self on sites range(startsite, startsite + len(self)) and MPPovm.eye() elsewhere

est_lfun(coeff, funs, samples, weights=None, eps=1e-10)

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=1e-10)

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=1e-10)

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 of x = pauli_povm() from samples for x = 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=1e-10)

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.

expectations(mpa, mode='auto')

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 n-th element is the expectation value correponding to the reduced state on sites [n,…,n + len(self) - 1]

classmethod eye(local_dims)

Construct MP-POVM with no output or measurement

Corresponds to taking the partial trace of the quantum state and a shorter MP-POVM.

Parameters:local_dims – Iterable of local dimensions

classmethod from_local_povm(lelems, width)

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=1e-10)

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=1e-10)

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

Non-singleton outcome dimensions (dimension larger one)

nsoutpos

Sites with non-singleton outcome dimension (dimension larger one)

outdims

Outcome dimensions

pack_samples(samples, dtype=None)

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')

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=1e-10, impl='auto')

Compute the POVM’s PMF for state as full array

Parameters: See MPPovm.pmf().

Parameters:impl – ‘auto’, ‘default’, ‘pmps-symm’ or ‘pmps-ltr’. ‘auto’ will use ‘pmps-symm’ 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.

pmfs_as_array(states, mode, asarray=False, eps=1e-10)

Todo

Add docstring

probability_map

Map that takes a raveled MPDO to the POVM probabilities

You can use MPPovm.expectations() or MPPovm.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)

Construct a longer MP-POVM 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 MP-POVM 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)) <= 1e-10
True

sample(rng, state, n_samples, method='cond', n_group=1, mode='auto', pack=False, eps=1e-10)

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 i-th sample is given by samples[i, :]. samples[i, j] is the outcome for the j-th non-singleton output dimension of self.

unpack_samples(samples)

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)

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)

Construct a MPPovmList

Parameters:mppseq – An iterable of MPPovm objects

All MPPovms must have the same number of sites.

block(nr_sites)

Embed MP-POVMs on local blocks

This function calls MPPovm.block(nr_sites)() for each MP-POVM in the list. Embedded MP-POVMs 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)) <= 1e-10
...  for a, b in zip(mppl.block(3).mpps, expect)]
[True, True, True, True]

block_pmfs_as_array(state, mode, asarray=False, eps=1e-10, **redarg)

Todo

Add docstring

est_lfun(coeff, funs, samples, weights=None, eps=1e-10)

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 i-th 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=1e-10)

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 using MPPovmList._lfun_estimator() and this estimator is passed to MPPovm.est_lfun() to obtain the estimate. See Linear combinations of functions of POVM outcomes for more details.

Parameters:

• other (MPPovmList) – Another MP-POVM 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=1e-10)

Estimate PMF from samples

Returns an iterator over results from MPPovm.est_pmf() (see there).

est_pmf_from(other, samples, eps=1e-10)

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=1e-10)

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 to MPPovm.pmf().

Returns:(value, var): Expectation value and variance of the expectation value

lfun_from(other, coeff, state, mode='auto', other_weights=None, eps=1e-10)

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 in MPPovmList.est_lfun_from()).

The parameter coeff is explained in MPPovmList.est_lfun_from(). state and mode are passed to MPPovm.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)

Pack samples into one integer per sample

Returns:Iterator over output from MPPovm.pack_samples()

pmf(state, mode='auto')

Compute the probability mass functions of all MP-POVMs

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=1e-10)

Compute the PMF of all MP-POVMs as full arrays

Parameters: See MPPovmList.pmf(). Sanity checks: See MPPovm.pmf_as_array().

Returns:Iterator over probabilities as ndarrays

pmfs_as_array(states, mode, asarray=False, eps=1e-10)

Todo

Add docstring

repeat(nr_sites)

Construct longer MP-POVMs by repeating each MP-POVM

This function calls MPPovm.repeat(nr_sites) for each MP-POVM 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)) <= 1e-10 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)) <= 1e-10
...  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=1e-10)

Random sample from all MP-POVMs on a quantum state

Parameters: See MPPovm.sample().

Return value: Iterable of return values from MPPovm.sample().

unpack_samples(samples)

Unpack samples into several integers per sample

Returns:Iterator over output from MPPovm.unpack_samples()

mpnum.povm.mppovm.pauli_mpp(nr_sites, local_dim)

Pauli POVM tensor product as MP-POVM

The resulting MP-POVM 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 MP-POVM
• local_dim (int) – Local dimension

Return type:

MPPovm

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() <= 1e-10
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)

Pauli POVM tensor product as MP-POVM list

The returned MPPovmList contains all tensor products of the single-site 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)) <= 1e-10 for a, b in zip(pauli.mpps, expect)]
[True, True, True, True]

Parameters:

• nr_sites (int) – Number of sites of the returned MP-POVMs
• local_dim (int) – Local dimension

Return type:

MPPovmList

povm.localpovm

An informationally complete d-level 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>)

Bases: object

Represent a Positive Operator-Valued Measure (POVM).

classmethod from_vectors(vecs, info_complete=False)

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())

mpnum.povm.localpovm.concat(povms, weights, info_complete=False)

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)

The POVMs used by pauli_povm() as a list

For dim > 3, x_povm() and y_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)

An informationally complete d-level 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)

The X POVM simplifies to measuring Pauli X eigenvectors for dim=2.

Parameters:dim – Dimension of the system Returns:POVM with generalized X measurments

mpnum.povm.localpovm.y_povm(dim)

The Y POVM simplifies to measuring Pauli Y eigenvectors for dim=2.

Parameters:dim – Dimension of the system Returns:POVM with generalized Y measurments

mpnum.povm.localpovm.z_povm(dim)

The Z POVM simplifies to measuring Pauli Z eigenvectors for dim=2.

Parameters:dim – Dimension of the system Returns:POVM with generalized Z 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)

Same as mparray.inner(), but assumes that mpa1 is a product MPS

Parameters:

• mpa1 – MPArray with one leg per site and rank 1
• mpa2 – MPArray with same shape as mpa1 but arbitrary rank

Returns:

<mpa1|mpa2>

mpnum.special.sumup(mpas, rank, weights=None, svdfunc=<function truncated_svd>)

Same as mparray.sumup() with a consequent compression, but with in-place svd compression. Also, we use a sparse-matrix 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 SVD
• scipy.sparse.linalg.svds(): Only computes the necessary singular values/vectors, but slow if rank is not small enough
• mpnum.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)

Converts a general sites-local 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)

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))

Block-diagonal sum for n-dimensional 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 block-diagonal 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, )))

np.tensordot with sane defaults for matrix multiplication

mpnum.utils.extmath.mkron(*args)

np.kron() with an arbitrary number of n >= 1 arguments

mpnum.utils.extmath.partial_trace(array, traceout)

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

mpnum.utils.extmath.truncated_svd(A, k)

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: left-singular vectors s: singular values in descending order v: right-singular 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/dist-packages/numpy/random/__init__.py'>)

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, unless n_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 step-by-step 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 if n_iter <= 2 and switches to LU otherwise.
• transpose – True, False or 'auto' Whether the algorithm should be applied to M.T instead of M. The result should approximately be the same. The 'auto' mode will trigger the transposition if M.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 is np.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 Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
• An implementation of a randomized algorithm for principal component analysis A. Szlam et al. 2014

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)

Ground state energy of the cyclic XY model

Parameters:

• nr_sites – Number of spin one-half 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. (A-12), (2.20) of [LSM61])

\[E_0 = -\frac12 \sum_{l=0}^{N-1} \Lambda_{k(l)}\]

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)

Local terms of the cyclic XY model (MPOs)

Parameters:

• nr_sites – Number of spin one-half sites
• gamma – Asymmetry parameter

Returns:

List terms of length nr_sites (MPOs)

The term terms[i] acts on spins (i, i + 1) and spin nr_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)

Construct an MPO cyclic nearest-neighbour Hamiltonian

Parameters:terms – List of nearst-neighbour 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 Frobenius-normalized 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)

Construct a sparse cyclic nearest-neighbour Hamiltonian

Parameters:

• terms – List of nearst-neighbour terms (square array or MPO, see return value of cXY_local_terms())
• ldim – Local dimension

Returns:

The Hamiltonian as sparse matrix

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 integer-for 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 non-zero 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 multi-site POVMs with elements obtained from every possible combination of the elements of single-site 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 sphinx-autobuild) does not rebuild this list.