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quantum.h File Reference

The main header file for the Quantum library. More...

#include "linalg.h"
#include <iostream>
#include <cstdlib>
#include <stdarg.h>
#include <vector>
#include <set>
#include <algorithm>
#include <sys/types.h>
#include <sys/stat.h>
#include <unistd.h>
#include <float.h>

Include dependency graph for quantum.h:


Namespaces

namespace  ConstStates

Classes

class  TensorProductStructure
 Defines a tensor product structure on a finite dimensional Hilbert space. More...
class  TensorProductIndex
 Defines an index for iterating over a tensor product structure. More...
class  ProductState
 A tensor product state vector. More...
class  SinglyBranchingState
 A sum of product states that can be distinguished perfectly by measuring one (special) subsystem. More...
class  ProductOperator
 A tensor product of operators (complex matrices). More...
class  ProductOperatorSum
 A sum of TensorProductOperator operators. More...
class  Histogram
 Class to hold and compute histograms over the real numbers. More...
class  Accumulator
 Class to compute simple statistics of a stream of (currently real) numbers. More...
class  Timer
 The Timer class is used to precisely profile the running time of algorithms. More...

Typedefs

typedef std::set< itype, std::less<
itype > > 		ISet
 An ISet is an unordered set of distinct indices (unsigned int).
typedef std::vector< itype > IArray
 An IArray is an ordered array of indices (unsigned int).
typedef DenseVector< complex > State
 State holds a quantum state vector. It is a synonym for DenseVector<complex>.
typedef DenseMatrix< complex > Operator
 Operator holds an operator on quantum states. It is a synonym for DenseMatrix<complex>.

Functions

std::ostream & operator<< (std::ostream &os, const TensorProductStructure &TPS)
 Prints a TensorProductStructure to a stream, as a list "{a,b,...,z}".
std::ostream & operator<< (std::ostream &os, const TensorProductIndex &TPI)
 Prints a TensorProductIndex to a stream, as a list "{a,b,...,z}".
void SimplifySubsystemDivision (TensorProductStructure &Universe, ISet &Subsystem)
 Specialized function to collapse a TensorProductStructure for partial iteration.
complex c_dot (ProductState &star, ProductState &nostar, int index=(-1))
 Computes the conjugate dot product between two ProductStates, or between two of their subsystem states if the optional index parameter is provided.
template<class T>
DenseVector< T > tensor (const DenseVector< T > &v1, const DenseVector< T > &v2)
 Creates and returns a new DenseVector containing the tensor product of v1 and v2.
template<class T>
DenseVector< T > tensor (itype N, DenseVector< T > *v_arr1[])
 Creates and returns a new DenseVector containing the tensor product of v_arr[0...N-1].
template<class T>
DenseMatrix< T > tensor (const DenseMatrix< T > &m1, const DenseMatrix< T > &m2)
 Creates and returns a new DenseMatrix containing the tensor product of m1 and m2.
template<class T>
DenseMatrix< T > tensor (itype N, DenseMatrix< T > *m_arr[])
 Creates and returns a new DenseMatrix containing the tensor product of v_arr[0...N-1].
template<class T>
DenseVector< T > & tensor_into (const DenseVector< T > &v1, const DenseVector< T > &v2, DenseVector< T > &result, T scalar=T(1))
 Overwrites result with scalar times the tensor product of v1 and v2, and returns a reference to result.
template<class T>
DenseVector< T > & tensor_into (itype N, DenseVector< T > *v_arr[], DenseVector< T > &result, T scalar=T(1))
 Overwrites result with scalar times the tensor product of v_arr[0...N-1], and returns a reference to result.
template<class T>
DenseMatrix< T > & tensor_into (const DenseMatrix< T > &m1, const DenseMatrix< T > &m2, DenseMatrix< T > &result, T scalar=T(1))
 Overwrites result with scalar times the tensor product of m1 and m2, and returns a reference to result.
template<class T>
DenseMatrix< T > & tensor_into (itype N, DenseMatrix< T > *m_arr[], DenseMatrix< T > &result, T scalar=T(1))
 Overwrites result with scalar times the tensor product of m_arr[0...N-1], and returns a reference to result.
template<class T>
DenseVector< T > & add_tensor (const DenseVector< T > &v1, const DenseVector< T > &v2, DenseVector< T > &accumulator, T scalar=T(1))
 Adds scalar times the tensor product of v1 and v2 to result, and returns a reference to result.
template<class T>
DenseVector< T > & add_tensor (itype N, DenseVector< T > *v_arr[], DenseVector< T > &accumulator, T scalar=T(1))
 Adds scalar times the tensor product of v_arr[0...N-1] to result, and returns a reference to result.
template<class T>
DenseMatrix< T > & add_tensor (const DenseMatrix< T > &m1, const DenseMatrix< T > &m2, DenseMatrix< T > &accumulator, T scalar=T(1))
 Adds scalar times the tensor product of m1 and m2 to result, and returns a reference to result.
template<class T>
DenseMatrix< T > & add_tensor (itype N, DenseMatrix< T > *m_arr[], DenseMatrix< T > &accumulator, T scalar=T(1))
 Adds scalar times the tensor product of v_arr[0...N-1] to result, and returns a reference to result.
template<class T>
DenseMatrix< T > Commutator (const DenseMatrix< T > &m1, const DenseMatrix< T > &m2)
 Creates a new DenseMatrix containing the commutator of two DenseMatrices: $[A,B] = AB-BA$ .
template<class T>
DenseMatrix< T > AntiCommutator (const DenseMatrix< T > &m1, const DenseMatrix< T > &m2)
 Creates a new DenseMatrix containing the anticommutator of two DenseMatrices: $\{A,B\} = AB+BA$ .
template<class T>
T scavenge (T scalar, typename linalg_traits< T >::abs_type minabs)
 Returns scalar if its absolute value is not less than minabs, or 0 otherwise.
complex scavenge (complex scalar, double minabs)
 Overloads scavenge<T> to scavenge the real and imaginary parts of scalar independently.
template<class T>
DenseMatrix< T > & scavenge (DenseMatrix< T > &m, typename linalg_traits< T >::abs_type minabs)
 Sets all the entries of m whose absolute value falls below minabs to zero, and returns a reference to the result.
template<class T>
DenseVector< T > & scavenge (DenseVector< T > &v, typename linalg_traits< T >::abs_type minabs)
 Sets all the entries of v whose absolute value falls below minabs to zero, and returns a reference to the result.
template<class T>
DenseMatrix< T > scavenge (const DenseMatrix< T > &m, typename linalg_traits< T >::abs_type minabs)
 Creates a new DenseMatrix which is a scavenged copy of m, and returns it.
template<class T>
DenseVector< T > scavenge (const DenseVector< T > &v, typename linalg_traits< T >::abs_type minabs)
 Creates a new DenseVector which is a scavenged copy of m, and returns it.
Operator & PTrace (const State &psi, Operator &rho, bool flagTraceOverMSB=false)
 Does a simple partial trace of $ |\psi\rangle\langle\psi| $ over as many dimensions as needed to fit the result in $ \rho $ .
DenseMatrix< complex > partial_trace (const DenseMatrix< complex > &m, const TensorProductStructure &TPS, ISet &indices)
 Creates and returns a new matrix which is the partial trace of m, over the indices indices in the tensor product structure TPS.
DenseMatrix< complex > partial_trace (const DenseVector< complex > &v, const TensorProductStructure &TPS, ISet &indices)
 Creates and returns a new matrix which is the partial trace of the projector onto v, over the indices indices in the tensor product structure TPS.
DenseMatrix< complex > partial_trace_asymm (const DenseVector< complex > &ket, const DenseVector< complex > &bra, const TensorProductStructure &TPS, ISet &indices)
 Creates and returns a new matrix which is the partial trace of the (non-Hermitian) outer product |ket><bra|, over the indices indices in the tensor product structure TPS.
State & Purify (const Operator &rho, State &psi)
bool PathOpen (const char *pathname, std::ofstream &os)
 Opens os for writing to the file given by pathname. Creating all necessary directories. Returns FALSE if it fails.
double Entropy (Operator &rho)
 Computes the Von Neumann entropy $ H(\rho) = -\mathrm{Tr}\rho\ln\rho $ of the positive semidefinite matrix rho.
double LinearEntropy (Operator &rho)
 Computes the linear entropy $ H(\rho) = 1-\mathrm{Tr}\rho^2 $ of the positive semidefinite matrix rho.
Operator & RandomMixedState (Operator &Rho, itype Nancillae)
 Assigns to Rho a random mixed state from the induced measure on 1+Nancillae copies.
Operator & HaarRandomize (Operator &Rho)
 Assigns to Rho a random pure state. Equivalent to RandomMixedState() with Nancillae=0.
Operator & HilbertSchmidtRandomize (Operator &Rho)
 Assigns to Rho a positive semidefinite matrix selected from Hilbert-Schmidt measure. Equivalent to RandomMixedState() with Nancillae=1.
double TraceNorm (Operator &M)
 Computes the sum of the absolute values of the eigenvalues of a Hermitian matrix M.
double InfinityNorm (Operator &M)
 Computes the square root of the largest eigenvalue of $ M^\dagger M $ . If $ M $ is Hermitian, then this is the maximum of the absolute values of $ M $ 's eigenvalues.
double Fidelity (Operator &Sigma, Operator &Rho)
 Computes the fidelity between two states as $ F = \left[\mathrm{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right]^2 $ .
double Fidelity (Operator &Sigma, State &Psi)
 Computes the fidelity between a mixed state $ \sigma $ and a pure state $ \left|\psi\right\rangle $ : $ F = \left\langle\psi\right|\sigma\left|\psi\right\rangle $ .
double Fidelity (State &Psi, Operator &Rho)
 Computes the fidelity between a pure state $ \left|\psi\right\rangle $ and a mixed state a mixed state $ \rho $ : $ F = \left\langle\psi\right|\rho\left|\psi\right\rangle $ .
double Fidelity (State &Psi, State &Phi)
 Computes the fidelity between two pure states $ \left|\phi\right\rangle $ and $ \left|\psi\right\rangle $ : $ F = \left|\left\langle\psi|\phi\right\rangle\right|^2 $ .
double NullspaceProjection (Operator &Sigma, Operator &Rho, double MinProbability=DBL_EPSILON)
 Computes the projection of $ \rho $ onto the nullspace of $ \sigma $ .
double RelativeEntropy (Operator &Sigma, Operator &Rho, double MinProbability=DBL_EPSILON)
 Computes the relative entropy, or Kullback-Leibler divergence, of $ \sigma $ relative to $ \rho $ : $ S(\rho||\sigma) = \mathrm{Tr}\rho\ln\frac{\rho}{\sigma} $.
void RelativeEntropyAndNullProjection (double &RelEnt, double &NullProj, Operator &Sigma, Operator &Rho, double MinProbability)
 Computes the relative entropy of $ \sigma $ relative to $ \rho $ and the projection of $ \rho $ onto $ \sigma $ 's null space.
Operator & Positivize (Operator &M)
 Converts M to a positive-definite matrix by replacing all its negative eigenvalues with 0.
Operator & SquareRoot (Operator &M)
 Replaces the Hermitian positive-definite matrix M with its unique positive-definite square root.
template<class Op>
double Entanglement (const State &psi, const TensorProductStructure &Universe, ISet &SubSystem, Op EntropyFunction)
 Computes the bipartite entanglement of psi between SubSystem and its complement, using the user-supplied EntropyFunction.
double Entanglement (const State &psi, const TensorProductStructure &Universe, ISet &Subsystem)
 Computes the bipartite entanglement of psi between SubSystem and its complement, using Von Neumann entropy.
void GenerateChebyshevMatrix (itype N, DenseMatrix< complex > &M)
 Computes an NxN matrix, where $ M_{m,n} $ is the coefficient of $ x^m $ in the $ n $ 'th Chebyshev polynomial.
void ResizeChebyshevMatrix (itype N, DenseMatrix< complex > &M)
 Efficiently resizes an existing Chebyshev polynomial matrix to NxN, using the coefficients already computed.
void GenerateChebyshevEvolutionVector (double time, itype N, DenseVector< complex > &V)
 Computes an N-element vector, where $ v_n $ is the coefficient of the $ n $ 'th Chebyshev polynomial in the expansion of $ e^{-ix} $ .
int ChebyshevPrep (double timestep, double prec, DenseVector< complex > &PowerCoeffs, DenseVector< double > &CoeffsBoundSeq)
 Precomputes the coefficients needed to evolve a state using ChebyshevStep(), for time timestep, with accuracy prec.
template<typename T>
int ChebyshevStep (const State &initial, State &final, DenseVector< complex > &PowerCoeffs, DenseVector< double > &CoeffsBoundSeq, const T &Hamiltonian, double E_bound, double prec)
 Transforms the quantum state initial by $ e^{-i\mathbf{H}} $ , placing the result in final. Requires precomputation by ChebyshevPrep().
template<typename T>
int ChebyshevEvolve (const State &initial, State &final, double time, const T &Hamiltonian, double E_bound, double prec=1e-10)
 Evolves the quantum state initial for a single step of duration time according to Hamiltonian, placing the result in final.
template<typename T>
int ChebyshevStepEvolve (const State &initial, State &final, double time, double maxtimestep, const T &Hamiltonian, double E_bound, double prec=1e-10)
 Evolves the quantum state initial ==> final, for duration time, in steps no longer than maxtimestep, according to Hamiltonian.
Operator ConstStates::Identity (itype D)
 Returns the identity operator in D dimensions.
Operator ConstStates::Zero (itype D)
 Returns the zero operator in D dimensions.
Operator ConstStates::Hadamard ()
 Returns the Hadamard matrix for qubits.
Operator ConstStates::HalfPhase ()
 Returns the phase-flip matrix (square root of SigmaZ) for qubits.
Operator ConstStates::SigmaX ()
 Returns the J=1/2 Pauli spin operator along the X axis.
Operator ConstStates::SigmaY ()
 Returns the J=1/2 Pauli spin operator along the Y axis.
Operator ConstStates::SigmaZ ()
 Returns the J=1/2 Pauli spin operator along the Z axis.
Operator ConstStates::SigmaI ()
 Returns the J=1/2 Identity operator.
Operator ConstStates::Sigma (itype i)
 Returns the ith matrix from the Pauli group, {X,Y,Z,I}.
Operator ConstStates::QubitSpinOperator (double x, double y, double z)
 Returns an arbitrary J=1/2 spin operator $ x\sigma_x+y\sigma_y+z\sigma_z $.
State ConstStates::QubitSpinState (double x, double y, double z)
 Returns a J=1/2 pure state so that <X>=x, <Y>=y, and <Z>=z, up to normalization.
State ConstStates::XPlus ()
 Returns the +1 eigenstate of SigmaX.
State ConstStates::XMinus ()
 Returns the -1 eigenstate of SigmaX.
State ConstStates::YPlus ()
 Returns the +1 eigenstate of SigmaY.
State ConstStates::YMinus ()
 Returns the -1 eigenstate of SigmaY.
State ConstStates::ZPlus ()
 Returns the +1 eigenstate of SigmaZ.
State ConstStates::ZMinus ()
 Returns the -1 eigenstate of SigmaZ.
State ConstStates::QubitMUBState (itype i)
 Returns the ith state from the 6-element qubit MUB set: {|0>, |1>, |+>, |->, |+i>, |-i>}.
State ConstStates::QubitSICPState (itype i)
 Returns the ith state from a 4-element qubit SICPOVM chosen so that the 1st is |0>, and the 2nd is in the X-Z plane.
Operator ConstStates::Jz (itype d)
 Returns the Jz (angular momentum along the z-axis) operator in d dimensions.
Operator ConstStates::Jx (itype d)
 Returns the Jz (angular momentum along the z-axis) operator in d dimensions.
Operator ConstStates::Jy (itype d)
 Returns the Jz (angular momentum along the z-axis) operator in d dimensions.
Operator ConstStates::Jn (itype d, double x, double y, double z)
 Returns the d-dimensional angular-momentum-along-the-(x,y,z)-axis operator.
template<class T>
DenseVector< T > & internal_tensor (const DenseVector< T > &v1, const DenseVector< T > &v2, DenseVector< T > &result, T scalar, bool overwrite)
 Internal method used to place scalar times the tensor product of v1 and v2 into result.
template<class T>
DenseVector< T > & internal_tensor (itype N, DenseVector< T > *v_arr[], DenseVector< T > &result, T scalar, bool overwrite)
 Internal method used to place scalar times the tensor product of the first N elements of v_arr into result.
template<class T>
DenseMatrix< T > & internal_tensor (const DenseMatrix< T > &m1, const DenseMatrix< T > &m2, DenseMatrix< T > &result, T scalar, bool overwrite)
 Internal method used to place scalar times the tensor product of m1 and m2 into result.
template<class T>
DenseMatrix< T > & internal_tensor (itype N, DenseMatrix< T > *m_arr[], DenseMatrix< T > &result, T scalar, bool overwrite)
 Internal method used to place scalar times the tensor product of the first N elements of v_arr into result.

Detailed Description

The main header file for the Quantum library.

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