Class representing a (unary) tridiagonal matrix or direct product of such matrices. More...

#include <Tridiagonal.h>

Inheritance diagram for quantumoperator::Tridiagonal< RANK >:

Collaboration diagram for quantumoperator::Tridiagonal< RANK >:

Public Types
typedef blitzplusplus::TinyOfArrays< dcomp, RANK, LENGTH >	Diagonals
	The class is implemented in terms of a blitzplusplus::TinyOfArrays, this is the class used to store the Diagonals.

typedef Diagonals::T_numtype	Diagonal
	A unary complex blitz array.

typedef Base::Dimensions	Dimensions
	Inherited from DimensionsBookkeeper.

typedef quantumdata::Types< RANK >::StateVectorLow	StateVectorLow

Public Types inherited from DimensionsBookkeeper< RANK >
typedef ExtTiny< RANK >	Dimensions
	The dimensions as a static vector of size N_RANK.

Public Member Functions
Constructors, assignment
	Tridiagonal (const Diagonal &zero=empty, size_t k=0, const Diagonal &minus=empty, const Diagonal &plus=empty, bool toFreqs=false, IntRANK=_1_)
	Constructor for unary Tridiagonal. More...

	Tridiagonal (const Tridiagonal &tridiag)
	Copy constructor with deep-copy semantics.

template<int RANK2>
	Tridiagonal (const Tridiagonal< RANK2 > &, const Tridiagonal< RANK-RANK2 > &)
	Constructs the object as the direct product of two Tridiagonal matrices. More...

Class-specific functionality
Tridiagonal &	furnishWithFreqs (const Diagonal &mainDiagonal, IntRANK=_1_)
	Furnishes a unary tridiagonal with frequencies calculated from `mainDiagonal`. More...

void	apply (const StateVectorLow &psi, StateVectorLow &dpsidt) const
	“Applies” the tridiagonal matrix on the state vector `psiprime`, in the vein of structure::Hamiltonian::addContribution(), that is $\ket{\Psi'}+=T\ket\Psi$ . More...

Tridiagonal &	propagate (double t)
	Updates the elements of the matrix to time instant `t` using the stored frequencies. More...

const dcomp	average (const quantumdata::LazyDensityOperator< RANK > &, const typename quantumdata::LazyDensityOperator< RANK >::Idx &, IntRANK=_1_)
	Calculates the quantum average of a Tridiagonal in a quantum state described by a (unary) quantumdata::LazyDensityOperator. More...

Getters
const Diagonals &	get () const
	returns the diagonals

const Dimensions &	getDifferences () const

double	getTime () const

const Diagonals &	getFreqs () const

Algebra
Tridiagonal &	operator*= (const dcomp &dc)
	Naively implemented, could be templated if need arises – Frequencies untouched.

Tridiagonal &	operator/= (const dcomp &dc)
	”

Tridiagonal &	operator*= (double d)
	”

Tridiagonal &	operator/= (double d)
	”

const Tridiagonal	hermitianConjugate () const
	Returns a newly constructed object, which is the Hermitian conjugate of `this`. More...

const Tridiagonal	dagger () const
	Same as hermitianConjugate.

Tridiagonal &	operator+= (const Tridiagonal &)
	Naive addition. More...

const Tridiagonal	operator- () const
	Returns a (deep) copy with negated diagonals. Frequencies remain untouched.

const Tridiagonal	operator+ () const
	Returns a (deep) copy.

Tridiagonal &	operator-= (const Tridiagonal &tridiag)
	Implemented in terms of operator+=.

Public Member Functions inherited from DimensionsBookkeeper< RANK >
	DimensionsBookkeeper (mpl::bool_< IS_CONST >=mpl::false_())
	Constructor usable only in the `IS_CONST=false` case. More...

	DimensionsBookkeeper (const Dimensions &dimensions)
	Standard constructor usable also in the `IS_CONST=true` case.

const Dimensions &	getDimensions () const
	Get the Dimensions vector.

size_t	getTotalDimension () const
	Get the total dimension of a system of arbitrary arity.

size_t	getDimension (mpl::int_< RANK >=mpl::int_< 1 >()) const
	Get the (single) dimension for a unary system.

size_t	getDimension (size_t i) const

void	setDimensions (const Dimensions &dimensions)
	This will work only in the non-const case.

Static Public Attributes
static const int	LENGTH =tmptools::Power<3,RANK>::value
	The number of Diagonals the class has to store.

static const int	N_RANK =RANK
	Reports the class’s rank for example towards DirectProduct.

Static Public Attributes inherited from DimensionsBookkeeper< RANK >
static const int	N_RANK
	Arity of the Hilbert space.

static const int	DIMESIONS_BOOKKEEPER_RANK
	Ditto (to break ambiguity if a class is derived from another base featuring `N_RANK`).

Related Functions
(Note that these are not member functions.)
template<int RANK>
void	apply (const typename quantumdata::Types< RANK >::StateVectorLow &psi, typename quantumdata::Types< RANK >::StateVectorLow &dpsidt, const Tridiagonal< RANK > &)
	A free-standing version of Tridiagonal::apply. More...

template<int RANK>
const Tridiagonal< RANK >	furnishWithFreqs (const Tridiagonal< RANK > &tridiag, const typename Tridiagonal< RANK >::Diagonal &mainDiagonal)
	Same as Tridiagonal::furnishWithFreqs, but returns a copy of its first argument furnished with frequencies. More...

const Tridiagonal< 1 >	zero (size_t)
	Unary zero operator as a Tridiagonal. More...

const Tridiagonal< 1 >	identity (size_t)
	Unary identity operator as a Tridiagonal. More...

template<int RANK1, int RANK2>
const Tridiagonal< RANK1+RANK2 >	operator* (const Tridiagonal< RANK1 > &t1, const Tridiagonal< RANK2 > &t2)
	Direct product. More...

template<int RANK>
const Tridiagonal< RANK >	tridiagMinusHC (const Tridiagonal< RANK > &tridiag)
	Returns the anti-Hermitian operator $T-T^\dagger$ . More...

template<int RANK>
const Tridiagonal< RANK >	tridiagPlusHC (const Tridiagonal< RANK > &tridiag)
	Returns the Hermitian operator $T+T^\dagger$ . More...

template<int RANK>
const Tridiagonal< RANK >	tridiagPlusHC_overI (const Tridiagonal< RANK > &tridiag)
	Returns the anti-Hermitian operator $(T+T^\dagger)/i$ . More...

template<int RANK>
std::ostream &	operator<< (std::ostream &, const Tridiagonal< RANK > &)

Detailed Description

template<int RANK>
class quantumoperator::Tridiagonal< RANK >

Class representing a (unary) tridiagonal matrix or direct product of such matrices.

Let us consider the following situation, which displays the full potential of the class: We have a system consisting of $M$ subsystems, each featuring a free (in general, non-Hermitian) Hamiltonian, which is diagonal in the given system’s Hilbert space. In addition, we have one more term of a special form in the Hamiltonian coupling all the subsystems (we are considering $H/i$ , because this is what appears in the framework as noted @ structure::Hamiltonian::addContribution):

$H/i=\lp H^\text{free}+H^\text{interaction}\rp/i=\sum_{m=0}^{M-1}\bigotimes_{k=0}^{m-1}\mathbf{1}_k\otimes\lp\sum_{n_m=0}^{N_m-1}\omega_{m,n}\ket{n_m}\bra{n_m}\rp\otimes\bigotimes_{k=m+1}^{M-1}\mathbf{1}_k+\bigotimes_{m=0}^{M-1}\lp\sum_{n_m=0}^{N_m-1}\alpha^{(0)}_{m,n}\ket{n_m}\bra{n_m}\right.+\left.\sum_{n_m=0}^{N_m-1-K_m}\lp\alpha^{(+)}_{m,n}\ket{n_m}\bra{n_m+K_m}+\alpha^{(-)}_{m,n}\ket{n_m+K_m}\bra{n_m}\rp\rp.$

Here, the coefficients $\omega$ and $\alpha$ are in general complex with the dimension of frequency ( $\hbar=1$ ). The $N_m$ s are the dimensions of the subsystems’ Hilbert spaces in which the vectors $\ket{n_m}$ form an orthonormal basis.

The Hamiltonian is indeed in a special form because the interaction term is a direct product of such operators acting on the Hilbert spaces of the individual subsystems, whose matrix contains only three diagonals of nonzero elements. Hence the name of the class Tridiagonal, although this usually refers to the case when $K=1$ .

Now let us transform to the interaction picture defined by $H^\text{free}$ . The Hamiltonian in interaction picture reads

$H\Int(t)/i=\bigotimes_{m=0}^{M-1}\lp\sum_{n_m=0}^{N_m-1}\alpha^{(0)}_{m,n}\ket{n_m}\bra{n_m}+\sum_{n_m=0}^{N_m-1-K_m}\lp e^{\delta_{m,n}t}\alpha^{(-)}_{m,n}\ket{n_m+K_m}\bra{n_m}+e^{-\delta_{m,n}t}\alpha^{(+)}_{m,n}\ket{n_m}\bra{n_m+K_m}\rp\rp,$

where $\delta_{m,n}=\omega_{m,n}-\omega_{m,n+K_m}$ .

Quite generally, the class is designed to store and manipulate Hamiltonians of the form either $H$ or $H\Int(t)$ above for an arbitrary number of subsystems. In the latter case, it also stores and manipulates the $\delta_{m,n}$ frequencies. In particular, the Hamiltonian can be evaluated at a given time $t$ , applied on a state vector and combined with other Tridiagonal instances using algebraic operations.

Tridiagonal internally bookkeeps to which time instant its state corresponds.

Policy of storing diagonals: only the non-zeros. Policy of storing frequencies: either none or all of them. Because of these policies, Tridiagonal can represent a diagonal matrix without significant overhead.

Note: The frequencies $\omega_{m,n}$ are not necessarily real here, Tridiagonal works also for non-Hermitian operators. On non-unitary transformations in quantum mechanics cf. these notes.

Template Parameters

RANK	arity of the Hilbert space corresponding to above

Note: A serious limitation of Tridiagonal is that the composition of two such operators does not in general yield an operator of the same form. This is one of the reasons why we are planning to deprecate this class in favour of the much more general form
$H\Int(t)=\bigotimes_{m=0}^{M-1}\sum_{i_m\in\mathbb{K}_m}\sum_{n_m=0}^{N_m-1-i_m}e^{\delta^{i_m}_{m,n}t}\alpha^{i_m}_{m,n}\ket{n_m+i_m}\bra{n_m},$
with $\mathbb{K}_m=\left\{K_m^{(0)},K_m^{(1)},…\right\}$ an arbitrary set, and $\delta_{m,n}^{(i_m)}=i\lp\omega_{m,n+i_m}-\omega_{m,n}\rp$ .

Definition at line 103 of file Tridiagonal.h.

Constructor & Destructor Documentation

template<int RANK>

quantumoperator::Tridiagonal< RANK >::Tridiagonal	(	const Diagonal &	zero = `empty`,
		size_t	k = `0`,
		const Diagonal &	minus = `empty`,
		const Diagonal &	plus = `empty`,
		bool	toFreqs = `false`,
		IntRANK	= `_1_`
	)

explicit

Constructor for unary Tridiagonal.

This is the principal way to create an object of this class, which can be used in the unary case only, as ensured by the trailing dummy argument. This creates an object corresponding to the elementary operator

$H^\text{elem}/i=\sum_{n=0}^{N-1}\alpha^{(0)}_n\ket{n}\bra{n}+\sum_{n=0}^{N-1-K}\lp\alpha^{(-)}_n\ket{n+K}\bra{n}+\alpha^{(+)}_n\ket{n}\bra{n+K}\rp$

when toFreq=false and

$H\Int^\text{elem}(t=0)/i=\sum_{n=0}^{N-1-K}\lp e^{\delta_{n}t}\alpha^{(-)}_n\ket{n+K}\bra{n}+e^{-\delta_{n}t}\alpha^{(+)}_n\ket{n}\bra{n+K}\rp$

when toFreq=true. In this case, the $\delta$ frequencies are calculated out of $\alpha^{(0)}$ as $\delta_{n}=\alpha^{(0)}_{n}-\alpha^{(0)}_{n+K}$ .

Either of the three initializing arrays might be zero-size, which signifies that the corresponding diagonal is zero, however, if they are nonzero-size, then their sizes must be compatible with each other. If both minus and plus are zero-size (purely diagonal matrix), then k might be zero as well. Violation is detected at runtime, and an exception of type TridiagonalConsistencyErrorException is thrown.

Note: It is a dilemma whether the parameter k should be considered a compile-time or a runtime parameter. In the majority of cases it is known already an compile time (e.g. ladder operators, angular momentum operators, etc.). The reason why it is treated as a runtime parameter is spatial degrees of freedom. There, operators like , , etc., are also tridiagonal in momentum space, and we wanted to have the possibility of specifying at runtime.

Parameters

zero	corresponds to $\alpha^{(0)}$
k	corresponds to above
minus	corresponds to $\alpha^{(-)}$
plus	corresponds to $\alpha^{(+)}$
toFreqs	governs whether the main diagonal ( $\alpha^{(0)}$ ) is converted to frequencies $\delta_{n}$

template<int RANK>

template<int RANK2>

quantumoperator::Tridiagonal< RANK >::Tridiagonal	(	const Tridiagonal< RANK2 > &	,
		const Tridiagonal< RANK-RANK2 > &
	)

Constructs the object as the direct product of two Tridiagonal matrices.

This is rather non-trivial, and the calculation of the resulting diagonals’ position is at the moment calculated at runtime, though it could partly be done at compile time. The eventual frequencies are also composed, direct product translates to a “direct sum” in this case. This really makes sense only if the time instants of the two tridiagonals are the same. Violation is detected at runtime, and an exception of type TridiagonalTimeMismatchException is thrown.

All tridiagonals of RANK>1 in the framework originate from direct products of unary tridiagonals.

Template Parameters

RANK2 the arity of one of the arguments (the other being RANK-RANK2)

Member Function Documentation

template<int RANK>

void quantumoperator::Tridiagonal< RANK >::apply	(	const StateVectorLow &	psi,
		StateVectorLow &	dpsidt
	)		const

“Applies” the tridiagonal matrix on the state vector psiprime, in the vein of structure::Hamiltonian::addContribution(), that is $\ket{\Psi'}+=T\ket\Psi$ .

Finding out which of the 3 to the power of arity diagonals corresponds to which state-vector slice when “applying”, is a formidable task for higher arity, and a significant portion of this calculation is done at compile time. The structure of this problem naturally maps to a recursion. There are 2 possibilities:

Explicit specializations of the function apply(). In this case, explicitly specialized code must be generated with the Python script quantumoperator/applyImpl/generate.py for each RANK (the Python script takes the rank as its first parameter in the command line). The user is encouraged to try it out to see the structure of the problem. Here, the recursion inherent in the problem is shifted to this Python script. This solution sidesteps template metaprogramming, however, it has the drawback that the amount of code such produced grows exponentially with the arity. This possibility is used if the DO_CONSIDER_EXPLICITLY_SPECIALIZED_TRIDIAGONAL_APPLIES macro is defined @ compilation.
The function is implemented using the recursive doApply, whose second version is present to break the recursivity. In this case, only this second version of the doApply function must be explicitly specialized, which results in much less code. This possibility is used if the DO_CONSIDER_EXPLICITLY_SPECIALIZED_TRIDIAGONAL_APPLIES macro is not defined @ compilation.

Note: (Perhaps somewhat surprisingly,) compile-time resource requirement is larger in the 1st case.

template<int RANK>

const dcomp quantumoperator::Tridiagonal< RANK >::average	(	const quantumdata::LazyDensityOperator< RANK > &	,
		const typename quantumdata::LazyDensityOperator< RANK >::Idx &	,
		IntRANK	= `_1_`
	)

Calculates the quantum average of a Tridiagonal in a quantum state described by a (unary) quantumdata::LazyDensityOperator.

Note: Not implemented.

Todo:: Implement.

template<int RANK>

Tridiagonal& quantumoperator::Tridiagonal< RANK >::furnishWithFreqs	(	const Diagonal &	mainDiagonal,
		IntRANK	= `_1_`
	)

Furnishes a unary tridiagonal with frequencies calculated from mainDiagonal.

Note that the matrix may have its own main diagonal in this case, which remains untouched. (This is actually exploited if we want to transform out only part of a main diagonal with interaction picture.)

Returns: *this

Note: Works in the unary case only

Todo:: Extend to arbitrary arity

template<int RANK>

const Tridiagonal quantumoperator::Tridiagonal< RANK >::hermitianConjugate ( ) const

Returns a newly constructed object, which is the Hermitian conjugate of this.

Transposition involves a non-trivial permutation of diagonals, which could be done @ compile-time, but at the moment it’s runtime.

Frequencies need not be transposed, because they are identical to their transpose.

Note: Eventually, an in-place version of Hermitian conjugation could be also implemented, if need for this arises.

Todo:: Implement an in-place version.

template<int RANK>

Tridiagonal& quantumoperator::Tridiagonal< RANK >::operator+= ( const Tridiagonal< RANK > & )

Naive addition.

The structures of the two objects must match, and there is a rather complicated set of rules as to what is meant by “matching”:

Tridiagonal freely mixes with a purely diagonal matrix throught addition.
If both have off-diagonals as well, the ks must match.
If both store frequencies, they must be equal.

Any violation of these rules is detected at runtime, and an exception of type TridiagonalStructureMismatchException is thrown.

template<int RANK>

Tridiagonal& quantumoperator::Tridiagonal< RANK >::propagate ( double t )

Updates the elements of the matrix to time instant t using the stored frequencies.

Returns: *this

Friends And Related Function Documentation

template<int RANK>

void apply	(	const typename quantumdata::Types< RANK >::StateVectorLow &	psi,
		typename quantumdata::Types< RANK >::StateVectorLow &	dpsidt,
		const Tridiagonal< RANK > &
	)

template<int RANK>

const Tridiagonal< RANK > furnishWithFreqs	(	const Tridiagonal< RANK > &	tridiag,
		const typename Tridiagonal< RANK >::Diagonal &	mainDiagonal
	)

Parameters

tridiag	Tridiagonal whose copy is to be furnished with frequencies
mainDiagonal	main diagonal determining the frequencies to be used for furnishing

template<int RANK>

const Tridiagonal< 1 > identity ( size_t )

template<int RANK1, int RANK2>

const Tridiagonal< RANK1+RANK2 > operator*	(	const Tridiagonal< RANK1 > &	t1,
		const Tridiagonal< RANK2 > &	t2
	)

Definition at line 338 of file Tridiagonal.h.

template<int RANK>

std::ostream & operator<<	(	std::ostream &	,
		const Tridiagonal< RANK > &
	)

const Tridiagonal< RANK > tridiagMinusHC ( const Tridiagonal< RANK > & tridiag )

Definition at line 348 of file Tridiagonal.h.

template<int RANK>

const Tridiagonal< RANK > tridiagPlusHC ( const Tridiagonal< RANK > & tridiag )

Definition at line 355 of file Tridiagonal.h.

template<int RANK>

const Tridiagonal< RANK > tridiagPlusHC_overI ( const Tridiagonal< RANK > & tridiag )

Definition at line 362 of file Tridiagonal.h.

template<int RANK>

const Tridiagonal< 1 > zero ( size_t )

The documentation for this class was generated from the following file:

quantumoperator/Tridiagonal.h

Public Types

Public Member Functions

Static Public Attributes

Related Functions

Detailed Description

template<int RANK> class quantumoperator::Tridiagonal< RANK >

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

template<int RANK>
class quantumoperator::Tridiagonal< RANK >