Source code for qutip.qobj

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"""The Quantum Object (Qobj) class, for representing quantum states and
operators, and related functions.
"""

__all__ = ['Qobj', 'qobj_list_evaluate', 'ptrace', 'dag', 'isequal',
           'issuper', 'isoper', 'isoperket', 'isoperbra', 'isket', 'isbra',
           'isherm', 'shape', 'dims']

import warnings
import types

try:
    import builtins
except:
    import __builtin__ as builtins

# import math functions from numpy.math: required for td string evaluation
from numpy import (arccos, arccosh, arcsin, arcsinh, arctan, arctan2, arctanh,
                   ceil, copysign, cos, cosh, degrees, e, exp, expm1, fabs,
                   floor, fmod, frexp, hypot, isinf, isnan, ldexp, log, log10,
                   log1p, modf, pi, radians, sin, sinh, sqrt, tan, tanh, trunc)

import numpy as np
import scipy.sparse as sp
import scipy.linalg as la
import qutip.settings as settings
from qutip import __version__
from qutip.fastsparse import fast_csr_matrix, fast_identity
from qutip.cy.ptrace import _ptrace
from qutip.permute import _permute
from qutip.sparse import (sp_eigs, sp_expm, sp_fro_norm, sp_max_norm,
                          sp_one_norm, sp_L2_norm)
from qutip.dimensions import type_from_dims, enumerate_flat, collapse_dims_super
from qutip.cy.spmath import (zcsr_transpose, zcsr_adjoint, zcsr_isherm)
from qutip.cy.sparse_utils import cy_tidyup
import sys
if sys.version_info.major >= 3:
    from itertools import zip_longest
elif sys.version_info.major < 3:
    from itertools import izip_longest
    zip_longest = izip_longest


[docs]class Qobj(object): """A class for representing quantum objects, such as quantum operators and states. The Qobj class is the QuTiP representation of quantum operators and state vectors. This class also implements math operations +,-,* between Qobj instances (and / by a C-number), as well as a collection of common operator/state operations. The Qobj constructor optionally takes a dimension ``list`` and/or shape ``list`` as arguments. Parameters ---------- inpt : array_like Data for vector/matrix representation of the quantum object. dims : list Dimensions of object used for tensor products. shape : list Shape of underlying data structure (matrix shape). copy : bool Flag specifying whether Qobj should get a copy of the input data, or use the original. fast : bool Flag for fast qobj creation when running ode solvers. This parameter is used internally only. Attributes ---------- data : array_like Sparse matrix characterizing the quantum object. dims : list List of dimensions keeping track of the tensor structure. shape : list Shape of the underlying `data` array. type : str Type of quantum object: 'bra', 'ket', 'oper', 'operator-ket', 'operator-bra', or 'super'. superrep : str Representation used if `type` is 'super'. One of 'super' (Liouville form) or 'choi' (Choi matrix with tr = dimension). isherm : bool Indicates if quantum object represents Hermitian operator. iscp : bool Indicates if the quantum object represents a map, and if that map is completely positive (CP). ishp : bool Indicates if the quantum object represents a map, and if that map is hermicity preserving (HP). istp : bool Indicates if the quantum object represents a map, and if that map is trace preserving (TP). iscptp : bool Indicates if the quantum object represents a map that is completely positive and trace preserving (CPTP). isket : bool Indicates if the quantum object represents a ket. isbra : bool Indicates if the quantum object represents a bra. isoper : bool Indicates if the quantum object represents an operator. issuper : bool Indicates if the quantum object represents a superoperator. isoperket : bool Indicates if the quantum object represents an operator in column vector form. isoperbra : bool Indicates if the quantum object represents an operator in row vector form. Methods ------- copy() Create copy of Qobj conj() Conjugate of quantum object. cosm() Cosine of quantum object. dag() Adjoint (dagger) of quantum object. dnorm() Diamond norm of quantum operator. dual_chan() Dual channel of quantum object representing a CP map. eigenenergies(sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000) Returns eigenenergies (eigenvalues) of a quantum object. eigenstates(sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000) Returns eigenenergies and eigenstates of quantum object. expm() Matrix exponential of quantum object. full(order='C') Returns dense array of quantum object `data` attribute. groundstate(sparse=False, tol=0, maxiter=100000) Returns eigenvalue and eigenket for the groundstate of a quantum object. matrix_element(bra, ket) Returns the matrix element of operator between `bra` and `ket` vectors. norm(norm='tr', sparse=False, tol=0, maxiter=100000) Returns norm of a ket or an operator. permute(order) Returns composite qobj with indices reordered. ptrace(sel) Returns quantum object for selected dimensions after performing partial trace. sinm() Sine of quantum object. sqrtm() Matrix square root of quantum object. tidyup(atol=1e-12) Removes small elements from quantum object. tr() Trace of quantum object. trans() Transpose of quantum object. transform(inpt, inverse=False) Performs a basis transformation defined by `inpt` matrix. trunc_neg(method='clip') Removes negative eigenvalues and returns a new Qobj that is a valid density operator. unit(norm='tr', sparse=False, tol=0, maxiter=100000) Returns normalized quantum object. """ __array_priority__ = 100 # sets Qobj priority above numpy arrays def __init__(self, inpt=None, dims=[[], []], shape=[], type=None, isherm=None, copy=True, fast=False, superrep=None): """ Qobj constructor. """ self._isherm = isherm self._type = type self.superrep = superrep if fast == 'mc': # fast Qobj construction for use in mcsolve with ket output self._data = inpt self.dims = dims self._isherm = False return if fast == 'mc-dm': # fast Qobj construction for use in mcsolve with dm output self._data = inpt self.dims = dims self._isherm = True return if isinstance(inpt, Qobj): # if input is already Qobj then return identical copy self._data = fast_csr_matrix((inpt.data.data, inpt.data.indices, inpt.data.indptr), shape=inpt.shape, copy=copy) if not np.any(dims): # Dimensions of quantum object used for keeping track of tensor # components self.dims = inpt.dims else: self.dims = dims self.superrep = inpt.superrep elif inpt is None: # initialize an empty Qobj with correct dimensions and shape if any(dims): N, M = np.prod(dims[0]), np.prod(dims[1]) self.dims = dims elif shape: N, M = shape self.dims = [[N], [M]] else: N, M = 1, 1 self.dims = [[N], [M]] self._data = fast_csr_matrix(shape=(N, M)) elif isinstance(inpt, list) or isinstance(inpt, tuple): # case where input is a list data = np.array(inpt) if len(data.shape) == 1: # if list has only one dimension (i.e [5,4]) data = data.transpose() _tmp = sp.csr_matrix(data, dtype=complex) self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr), shape=_tmp.shape) if not np.any(dims): self.dims = [[int(data.shape[0])], [int(data.shape[1])]] else: self.dims = dims elif isinstance(inpt, np.ndarray) or sp.issparse(inpt): # case where input is array or sparse if inpt.ndim == 1: inpt = inpt[:, np.newaxis] do_copy = copy if not isinstance(inpt, fast_csr_matrix): _tmp = sp.csr_matrix(inpt, dtype=complex, copy=do_copy) _tmp.sort_indices() #Make sure indices are sorted. do_copy = 0 else: _tmp = inpt self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr), shape=_tmp.shape, copy=do_copy) if not np.any(dims): self.dims = [[int(inpt.shape[0])], [int(inpt.shape[1])]] else: self.dims = dims elif isinstance(inpt, (int, float, complex, np.integer, np.floating, np.complexfloating)): # if input is int, float, or complex then convert to array _tmp = sp.csr_matrix([[inpt]], dtype=complex) self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr), shape=_tmp.shape) if not np.any(dims): self.dims = [[1], [1]] else: self.dims = dims else: warnings.warn("Initializing Qobj from unsupported type: %s" % builtins.type(inpt)) inpt = np.array([[0]]) _tmp = sp.csr_matrix(inpt, dtype=complex, copy=copy) self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr), shape = _tmp.shape) self.dims = [[int(inpt.shape[0])], [int(inpt.shape[1])]] if type == 'super': if self.type == 'oper': self.dims = [[[d] for d in self.dims[0]], [[d] for d in self.dims[1]]] if superrep: self.superrep = superrep else: if self.type == 'super' and self.superrep is None: self.superrep = 'super' # clear type cache self._type = None
[docs] def copy(self): """Create identical copy""" return Qobj(inpt=self)
def get_data(self): return self._data #Here we perfrom a check of the csr matrix type during setting of Q.data def set_data(self, data): if not isinstance(data, fast_csr_matrix): raise TypeError('Qobj data must be in fast_csr format.') else: self._data = data data = property(get_data, set_data) def __add__(self, other): """ ADDITION with Qobj on LEFT [ ex. Qobj+4 ] """ if isinstance(other, eseries): return other.__radd__(self) if not isinstance(other, Qobj): other = Qobj(other) if np.prod(other.shape) == 1 and np.prod(self.shape) != 1: # case for scalar quantum object dat = other.data[0, 0] if dat == 0: return self out = Qobj() if self.type in ['oper', 'super']: out.data = self.data + dat * fast_identity( self.shape[0]) else: out.data = self.data out.data.data = out.data.data + dat out.dims = self.dims if settings.auto_tidyup: out.tidyup() if isinstance(dat, (int, float)): out._isherm = self._isherm else: # We use _isherm here to prevent recalculating on self and # other, relying on that bool(None) == False. out._isherm = (True if self._isherm and other._isherm else out.isherm) out.superrep = self.superrep return out elif np.prod(self.shape) == 1 and np.prod(other.shape) != 1: # case for scalar quantum object dat = self.data[0, 0] if dat == 0: return other out = Qobj() if other.type in ['oper', 'super']: out.data = dat * fast_identity(other.shape[0]) + other.data else: out.data = other.data out.data.data = out.data.data + dat out.dims = other.dims if settings.auto_tidyup: out.tidyup() if isinstance(dat, complex): out._isherm = out.isherm else: out._isherm = self._isherm out.superrep = self.superrep return out elif self.dims != other.dims: raise TypeError('Incompatible quantum object dimensions') elif self.shape != other.shape: raise TypeError('Matrix shapes do not match') else: # case for matching quantum objects out = Qobj() out.data = self.data + other.data out.dims = self.dims if settings.auto_tidyup: out.tidyup() if self.type in ['ket', 'bra', 'operator-ket', 'operator-bra']: out._isherm = False elif self._isherm is None or other._isherm is None: out._isherm = out.isherm elif not self._isherm and not other._isherm: out._isherm = out.isherm else: out._isherm = self._isherm and other._isherm if self.superrep and other.superrep: if self.superrep != other.superrep: msg = ("Adding superoperators with different " + "representations") warnings.warn(msg) out.superrep = self.superrep return out def __radd__(self, other): """ ADDITION with Qobj on RIGHT [ ex. 4+Qobj ] """ return self + other def __sub__(self, other): """ SUBTRACTION with Qobj on LEFT [ ex. Qobj-4 ] """ return self + (-other) def __rsub__(self, other): """ SUBTRACTION with Qobj on RIGHT [ ex. 4-Qobj ] """ return (-self) + other def __mul__(self, other): """ MULTIPLICATION with Qobj on LEFT [ ex. Qobj*4 ] """ if isinstance(other, Qobj): if self.dims[1] == other.dims[0]: out = Qobj() out.data = self.data * other.data dims = [self.dims[0], other.dims[1]] out.dims = dims if settings.auto_tidyup: out.tidyup() if (not isinstance(dims[0][0], list) and not isinstance(dims[1][0], list)): # If neither left or right is a superoperator, # we should implicitly partial trace over # matching dimensions of 1. # Using izip_longest allows for the left and right dims # to have uneven length (non-square Qobjs). # We use None as padding so that it doesn't match anything, # and will never cause a partial trace on the other side. mask = [l == r == 1 for l, r in zip_longest(dims[0], dims[1], fillvalue=None)] # To ensure that there are still any dimensions left, we # use max() to add a dimensions list of [1] if all matching dims # are traced out of that side. out.dims = [max([1], [dim for dim, m in zip(dims[0], mask) if not m]), max([1], [dim for dim, m in zip(dims[1], mask) if not m])] else: out.dims = dims out._isherm = None if self.superrep and other.superrep: if self.superrep != other.superrep: msg = ("Multiplying superoperators with different " + "representations") warnings.warn(msg) out.superrep = self.superrep return out elif np.prod(self.shape) == 1: out = Qobj(other) out.data *= self.data[0, 0] out.superrep = other.superrep return out.tidyup() if settings.auto_tidyup else out elif np.prod(other.shape) == 1: out = Qobj(self) out.data *= other.data[0, 0] out.superrep = self.superrep return out.tidyup() if settings.auto_tidyup else out else: raise TypeError("Incompatible Qobj shapes") elif isinstance(other, np.ndarray): if other.dtype=='object': return np.array([self * item for item in other], dtype=object) else: return self.data * other elif isinstance(other, list): # if other is a list, do element-wise multiplication return np.array([self * item for item in other], dtype=object) elif isinstance(other, eseries): return other.__rmul__(self) elif isinstance(other, (int, float, complex, np.integer, np.floating, np.complexfloating)): out = Qobj() out.data = self.data * other out.dims = self.dims out.superrep = self.superrep if settings.auto_tidyup: out.tidyup() if isinstance(other, complex): out._isherm = out.isherm else: out._isherm = self._isherm return out else: raise TypeError("Incompatible object for multiplication") def __rmul__(self, other): """ MULTIPLICATION with Qobj on RIGHT [ ex. 4*Qobj ] """ if isinstance(other, np.ndarray): if other.dtype=='object': return np.array([item * self for item in other], dtype=object) else: return other * self.data elif isinstance(other, list): # if other is a list, do element-wise multiplication return np.array([item * self for item in other], dtype=object) elif isinstance(other, eseries): return other.__mul__(self) elif isinstance(other, (int, float, complex, np.integer, np.floating, np.complexfloating)): out = Qobj() out.data = other * self.data out.dims = self.dims out.superrep = self.superrep if settings.auto_tidyup: out.tidyup() if isinstance(other, complex): out._isherm = out.isherm else: out._isherm = self._isherm return out else: raise TypeError("Incompatible object for multiplication") def __truediv__(self, other): return self.__div__(other) def __div__(self, other): """ DIVISION (by numbers only) """ if isinstance(other, Qobj): # if both are quantum objects raise TypeError("Incompatible Qobj shapes " + "[division with Qobj not implemented]") if isinstance(other, (int, float, complex, np.integer, np.floating, np.complexfloating)): out = Qobj() out.data = self.data / other out.dims = self.dims if settings.auto_tidyup: out.tidyup() if isinstance(other, complex): out._isherm = out.isherm else: out._isherm = self._isherm out.superrep = self.superrep return out else: raise TypeError("Incompatible object for division") def __neg__(self): """ NEGATION operation. """ out = Qobj() out.data = -self.data out.dims = self.dims out.superrep = self.superrep if settings.auto_tidyup: out.tidyup() out._isherm = self._isherm return out def __getitem__(self, ind): """ GET qobj elements. """ out = self.data[ind] if sp.issparse(out): return np.asarray(out.todense()) else: return out def __eq__(self, other): """ EQUALITY operator. """ if (isinstance(other, Qobj) and self.dims == other.dims and not np.any(np.abs((self.data - other.data).data) > settings.atol)): return True else: return False def __ne__(self, other): """ INEQUALITY operator. """ return not (self == other) def __pow__(self, n, m=None): # calculates powers of Qobj """ POWER operation. """ if self.type not in ['oper', 'super']: raise Exception("Raising a qobj to some power works only for " + "operators and super-operators (square matrices).") if m is not None: raise NotImplementedError("modulo is not implemented for Qobj") try: data = self.data ** n out = Qobj(data, dims=self.dims) out.superrep = self.superrep return out.tidyup() if settings.auto_tidyup else out except: raise ValueError('Invalid choice of exponent.') def __abs__(self): return abs(self.data) def __str__(self): s = "" t = self.type shape = self.shape if self.type in ['oper', 'super']: s += ("Quantum object: " + "dims = " + str(self.dims) + ", shape = " + str(shape) + ", type = " + t + ", isherm = " + str(self.isherm) + ( ", superrep = {0.superrep}".format(self) if t == "super" and self.superrep != "super" else "" ) + "\n") else: s += ("Quantum object: " + "dims = " + str(self.dims) + ", shape = " + str(shape) + ", type = " + t + "\n") s += "Qobj data =\n" if shape[0] > 10000 or shape[1] > 10000: # if the system is huge, don't attempt to convert to a # dense matrix and then to string, because it is pointless # and is likely going to produce memory errors. Instead print the # sparse data string representation s += str(self.data) elif all(np.imag(self.data.data) == 0): s += str(np.real(self.full())) else: s += str(self.full()) return s def __repr__(self): # give complete information on Qobj without print statement in # command-line we cant realistically serialize a Qobj into a string, # so we simply return the informal __str__ representation instead.) return self.__str__() def __call__(self, other): """ Acts this Qobj on another Qobj either by left-multiplication, or by vectorization and devectorization, as appropriate. """ if not isinstance(other, Qobj): raise TypeError("Only defined for quantum objects.") if self.type == "super": if other.type == "ket": other = qutip.states.ket2dm(other) if other.type == "oper": return qutip.superoperator.vector_to_operator( self * qutip.superoperator.operator_to_vector(other) ) else: raise TypeError("Can only act super on oper or ket.") elif self.type == "oper": if other.type == "ket": return self * other else: raise TypeError("Can only act oper on ket.") def __getstate__(self): # defines what happens when Qobj object gets pickled self.__dict__.update({'qutip_version': __version__[:5]}) return self.__dict__ def __setstate__(self, state): # defines what happens when loading a pickled Qobj if 'qutip_version' in state.keys(): del state['qutip_version'] (self.__dict__).update(state) def _repr_latex_(self): """ Generate a LaTeX representation of the Qobj instance. Can be used for formatted output in ipython notebook. """ t = self.type shape = self.shape s = r'' if self.type in ['oper', 'super']: s += ("Quantum object: " + "dims = " + str(self.dims) + ", shape = " + str(shape) + ", type = " + t + ", isherm = " + str(self.isherm) + ( ", superrep = {0.superrep}".format(self) if t == "super" and self.superrep != "super" else "" )) else: s += ("Quantum object: " + "dims = " + str(self.dims) + ", shape = " + str(shape) + ", type = " + t) M, N = self.data.shape s += r'\begin{equation*}\left(\begin{array}{*{11}c}' def _format_float(value): if value == 0.0: return "0.0" elif abs(value) > 1000.0 or abs(value) < 0.001: return ("%.3e" % value).replace("e", r"\times10^{") + "}" elif abs(value - int(value)) < 0.001: return "%.1f" % value else: return "%.3f" % value def _format_element(m, n, d): s = " & " if n > 0 else "" if type(d) == str: return s + d else: if abs(np.imag(d)) < settings.atol: return s + _format_float(np.real(d)) elif abs(np.real(d)) < settings.atol: return s + _format_float(np.imag(d)) + "j" else: s_re = _format_float(np.real(d)) s_im = _format_float(np.imag(d)) if np.imag(d) > 0.0: return (s + "(" + s_re + "+" + s_im + "j)") else: return (s + "(" + s_re + s_im + "j)") if M > 10 and N > 10: # truncated matrix output for m in range(5): for n in range(5): s += _format_element(m, n, self.data[m, n]) s += r' & \cdots' for n in range(N - 5, N): s += _format_element(m, n, self.data[m, n]) s += r'\\' for n in range(5): s += _format_element(m, n, r'\vdots') s += r' & \ddots' for n in range(N - 5, N): s += _format_element(m, n, r'\vdots') s += r'\\' for m in range(M - 5, M): for n in range(5): s += _format_element(m, n, self.data[m, n]) s += r' & \cdots' for n in range(N - 5, N): s += _format_element(m, n, self.data[m, n]) s += r'\\' elif M > 10 and N <= 10: # truncated vertically elongated matrix output for m in range(5): for n in range(N): s += _format_element(m, n, self.data[m, n]) s += r'\\' for n in range(N): s += _format_element(m, n, r'\vdots') s += r'\\' for m in range(M - 5, M): for n in range(N): s += _format_element(m, n, self.data[m, n]) s += r'\\' elif M <= 10 and N > 10: # truncated horizontally elongated matrix output for m in range(M): for n in range(5): s += _format_element(m, n, self.data[m, n]) s += r' & \cdots' for n in range(N - 5, N): s += _format_element(m, n, self.data[m, n]) s += r'\\' else: # full output for m in range(M): for n in range(N): s += _format_element(m, n, self.data[m, n]) s += r'\\' s += r'\end{array}\right)\end{equation*}' return s
[docs] def dag(self): """Adjoint operator of quantum object. """ out = Qobj() out.data = zcsr_adjoint(self.data) out.dims = [self.dims[1], self.dims[0]] out._isherm = self._isherm out.superrep = self.superrep return out
[docs] def dual_chan(self): """Dual channel of quantum object representing a completely positive map. """ # Uses the technique of Johnston and Kribs (arXiv:1102.0948), which # is only valid for completely positive maps. if not self.iscp: raise ValueError("Dual channels are only implemented for CP maps.") J = sr.to_choi(self) tensor_idxs = enumerate_flat(J.dims) J_dual = tensor.tensor_swap(J, *( list(zip(tensor_idxs[0][1], tensor_idxs[0][0])) + list(zip(tensor_idxs[1][1], tensor_idxs[1][0])) )).trans() J_dual.superrep = 'choi' return J_dual
[docs] def conj(self): """Conjugate operator of quantum object. """ out = Qobj() out.data = self.data.conj() out.dims = [self.dims[0], self.dims[1]] return out
[docs] def norm(self, norm=None, sparse=False, tol=0, maxiter=100000): """Norm of a quantum object. Default norm is L2-norm for kets and trace-norm for operators. Other ket and operator norms may be specified using the `norm` and argument. Parameters ---------- norm : str Which norm to use for ket/bra vectors: L2 'l2', max norm 'max', or for operators: trace 'tr', Frobius 'fro', one 'one', or max 'max'. sparse : bool Use sparse eigenvalue solver for trace norm. Other norms are not affected by this parameter. tol : float Tolerance for sparse solver (if used) for trace norm. The sparse solver may not converge if the tolerance is set too low. maxiter : int Maximum number of iterations performed by sparse solver (if used) for trace norm. Returns ------- norm : float The requested norm of the operator or state quantum object. Notes ----- The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it. """ if self.type in ['oper', 'super']: if norm is None or norm == 'tr': vals = sp_eigs(self.data, self.isherm, vecs=False, sparse=sparse, tol=tol, maxiter=maxiter) return np.sum(sqrt(abs(vals) ** 2)) elif norm == 'fro': return sp_fro_norm(self.data) elif norm == 'one': return sp_one_norm(self.data) elif norm == 'max': return sp_max_norm(self.data) else: raise ValueError( "For matrices, norm must be 'tr', 'fro', 'one', or 'max'.") else: if norm is None or norm == 'l2': return sp_L2_norm(self.data) elif norm == 'max': return sp_max_norm(self.data) else: raise ValueError("For vectors, norm must be 'l2', or 'max'.")
[docs] def tr(self): """Trace of a quantum object. Returns ------- trace : float Returns ``real`` if operator is Hermitian, returns ``complex`` otherwise. """ if self.isherm: return float(np.real(np.sum(self.data.diagonal()))) else: return complex(np.sum(self.data.diagonal()))
[docs] def full(self, order='C', squeeze=False): """Dense array from quantum object. Parameters ---------- order : str {'C', 'F'} Return array in C (default) or Fortran ordering. squeeze : bool {False, True} Squeeze output array. Returns ------- data : array Array of complex data from quantum objects `data` attribute. """ if squeeze: return self.data.toarray(order=order).squeeze() else: return self.data.toarray(order=order)
[docs] def diag(self): """Diagonal elements of quantum object. Returns ------- diags : array Returns array of ``real`` values if operators is Hermitian, otherwise ``complex`` values are returned. """ out = self.data.diagonal() if np.any(np.imag(out) > settings.atol) or not self.isherm: return out else: return np.real(out)
[docs] def expm(self, method='dense'): """Matrix exponential of quantum operator. Input operator must be square. Parameters ---------- method : str {'dense', 'sparse'} Use set method to use to calculate the matrix exponentiation. The available choices includes 'dense' and 'sparse'. Since the exponential of a matrix is nearly always dense, method='dense' is set as default.s Returns ------- oper : qobj Exponentiated quantum operator. Raises ------ TypeError Quantum operator is not square. """ if self.dims[0][0] != self.dims[1][0]: raise TypeError('Invalid operand for matrix exponential') if method == 'dense': F = sp_expm(self.data, sparse=False) elif method == 'sparse': F = sp_expm(self.data, sparse=True) else: raise ValueError("method must be 'dense' or 'sparse'.") out = Qobj(F, dims=self.dims) return out.tidyup() if settings.auto_tidyup else out
[docs] def check_herm(self): """Check if the quantum object is hermitian. Returns ------- isherm : bool Returns the new value of isherm property. """ self._isherm = None return self.isherm
[docs] def sqrtm(self, sparse=False, tol=0, maxiter=100000): """Sqrt of a quantum operator. Operator must be square. Parameters ---------- sparse : bool Use sparse eigenvalue/vector solver. tol : float Tolerance used by sparse solver (0 = machine precision). maxiter : int Maximum number of iterations used by sparse solver. Returns ------- oper : qobj Matrix square root of operator. Raises ------ TypeError Quantum object is not square. Notes ----- The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it. """ if self.dims[0][0] == self.dims[1][0]: evals, evecs = sp_eigs(self.data, self.isherm, sparse=sparse, tol=tol, maxiter=maxiter) numevals = len(evals) dV = sp.spdiags(np.sqrt(evals, dtype=complex), 0, numevals, numevals, format='csr') if self.isherm: spDv = dV.dot(evecs.T.conj().T) else: spDv = dV.dot(np.linalg.inv(evecs.T)) out = Qobj(evecs.T.dot(spDv), dims=self.dims) return out.tidyup() if settings.auto_tidyup else out else: raise TypeError('Invalid operand for matrix square root')
[docs] def cosm(self): """Cosine of a quantum operator. Operator must be square. Returns ------- oper : qobj Matrix cosine of operator. Raises ------ TypeError Quantum object is not square. Notes ----- Uses the Q.expm() method. """ if self.dims[0][0] == self.dims[1][0]: return 0.5 * ((1j * self).expm() + (-1j * self).expm()) else: raise TypeError('Invalid operand for matrix square root')
[docs] def sinm(self): """Sine of a quantum operator. Operator must be square. Returns ------- oper : qobj Matrix sine of operator. Raises ------ TypeError Quantum object is not square. Notes ----- Uses the Q.expm() method. """ if self.dims[0][0] == self.dims[1][0]: return -0.5j * ((1j * self).expm() - (-1j * self).expm()) else: raise TypeError('Invalid operand for matrix square root')
[docs] def unit(self, norm=None, sparse=False, tol=0, maxiter=100000): """Operator or state normalized to unity. Uses norm from Qobj.norm(). Parameters ---------- norm : str Requested norm for states / operators. sparse : bool Use sparse eigensolver for trace norm. Does not affect other norms. tol : float Tolerance used by sparse eigensolver. maxiter : int Number of maximum iterations performed by sparse eigensolver. Returns ------- oper : qobj Normalized quantum object. """ out = self / self.norm(norm=norm, sparse=sparse, tol=tol, maxiter=maxiter) if settings.auto_tidyup: return out.tidyup() else: return out
[docs] def ptrace(self, sel): """Partial trace of the quantum object. Parameters ---------- sel : int/list An ``int`` or ``list`` of components to keep after partial trace. Returns ------- oper : qobj Quantum object representing partial trace with selected components remaining. Notes ----- This function is identical to the :func:`qutip.qobj.ptrace` function that has been deprecated. """ q = Qobj() q.data, q.dims, _ = _ptrace(self, sel) return q.tidyup() if settings.auto_tidyup else q
[docs] def permute(self, order): """Permutes a composite quantum object. Parameters ---------- order : list/array List specifying new tensor order. Returns ------- P : qobj Permuted quantum object. """ q = Qobj() q.data, q.dims = _permute(self, order) return q.tidyup() if settings.auto_tidyup else q
[docs] def tidyup(self, atol=None): """Removes small elements from the quantum object. Parameters ---------- atol : float Absolute tolerance used by tidyup. Default is set via qutip global settings parameters. Returns ------- oper : qobj Quantum object with small elements removed. """ if atol is None: atol = settings.auto_tidyup_atol if self.data.nnz: cy_tidyup(self.data.data,atol,self.data.nnz) self.data.eliminate_zeros() return self else: return self
[docs] def transform(self, inpt, inverse=False, sparse=True): """Basis transform defined by input array. Input array can be a ``matrix`` defining the transformation, or a ``list`` of kets that defines the new basis. Parameters ---------- inpt : array_like A ``matrix`` or ``list`` of kets defining the transformation. inverse : bool Whether to return inverse transformation. sparse : bool Use sparse matrices when possible. Can be slower. Returns ------- oper : qobj Operator in new basis. Notes ----- This function is still in development. """ if isinstance(inpt, list) or (isinstance(inpt, np.ndarray) and len(inpt.shape) == 1): if len(inpt) != max(self.shape): raise TypeError( 'Invalid size of ket list for basis transformation') if sparse: S = sp.hstack([psi.data for psi in inpt], format='csr', dtype=complex).conj().T else: S = np.hstack([psi.full() for psi in inpt], dtype=complex).conj().T elif isinstance(inpt, Qobj) and inpt.isoper: S = inpt.data elif isinstance(inpt, np.ndarray): S = inpt.conj() sparse = False else: raise TypeError('Invalid operand for basis transformation') # transform data if inverse: if self.isket: data = (S.conj().T) * self.data elif self.isbra: data = self.data.dot(S) else: if sparse: data = (S.conj().T) * self.data * S else: data = (S.conj().T).dot(self.data.dot(S)) else: if self.isket: data = S * self.data elif self.isbra: data = self.data.dot(S.conj().T) else: if sparse: data = S * self.data * (S.conj().T) else: data = S.dot(self.data.dot(S.conj().T)) out = Qobj(data, dims=self.dims) out._isherm = self._isherm out.superrep = self.superrep if settings.auto_tidyup: return out.tidyup() else: return out
[docs] def trunc_neg(self, method="clip"): """Truncates negative eigenvalues and renormalizes. Returns a new Qobj by removing the negative eigenvalues of this instance, then renormalizing to obtain a valid density operator. Parameters ---------- method : str Algorithm to use to remove negative eigenvalues. "clip" simply discards negative eigenvalues, then renormalizes. "sgs" uses the SGS algorithm (doi:10/bb76) to find the positive operator that is nearest in the Shatten 2-norm. Returns ------- oper : qobj A valid density operator. """ if not self.isherm: raise ValueError("Must be a Hermitian operator to remove negative " "eigenvalues.") if method not in ('clip', 'sgs'): raise ValueError("Method {} not recognized.".format(method)) eigvals, eigstates = self.eigenstates() if all([eigval >= 0 for eigval in eigvals]): # All positive, so just renormalize. return self.unit() idx_nonzero = eigvals != 0 eigvals = eigvals[idx_nonzero] eigstates = eigstates[idx_nonzero] if method == 'clip': eigvals[eigvals < 0] = 0 elif method == 'sgs': eigvals = eigvals[::-1] eigstates = eigstates[::-1] acc = 0.0 dim = self.shape[0] n_eigs = len(eigvals) for idx in reversed(range(n_eigs)): if eigvals[idx] + acc / (idx + 1) >= 0: break else: acc += eigvals[idx] eigvals[idx] = 0.0 eigvals[:idx+1] += acc / (idx + 1) return sum([ val * qutip.states.ket2dm(state) for val, state in zip(eigvals, eigstates) ], Qobj(np.zeros(self.shape), dims=self.dims) ).unit()
[docs] def matrix_element(self, bra, ket): """Calculates a matrix element. Gives the matrix element for the quantum object sandwiched between a `bra` and `ket` vector. Parameters ----------- bra : qobj Quantum object of type 'bra'. ket : qobj Quantum object of type 'ket'. Returns ------- elem : complex Complex valued matrix element. Raises ------ TypeError Can only calculate matrix elements between a bra and ket quantum object. """ if isinstance(bra, Qobj) and isinstance(ket, Qobj): if self.isoper: if bra.isbra and ket.isket: return (bra.data * self.data * ket.data)[0, 0] if bra.isket and ket.isket: return (bra.data.T * self.data * ket.data)[0, 0] raise TypeError("Can only calculate matrix elements for operators " + "and between ket and bra Qobj")
[docs] def overlap(self, state): """Overlap between two state vectors. Gives the overlap (scalar product) for the quantum object and `state` state vector. Parameters ----------- state : qobj Quantum object for a state vector of type 'ket' or 'bra'. Returns ------- overlap : complex Complex valued overlap. Raises ------ TypeError Can only calculate overlap between a bra and ket quantum objects. """ if isinstance(state, Qobj): if self.isbra: if state.isket: return (self.data * state.data)[0, 0] elif state.isbra: return (self.data * state.data.H)[0, 0] elif self.isket: if state.isbra: return (self.data.H * state.data.H)[0, 0] elif state.isket: return (self.data.H * state.data)[0, 0] raise TypeError("Can only calculate overlap for state vector Qobjs")
[docs] def eigenstates(self, sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000): """Eigenstates and eigenenergies. Eigenstates and eigenenergies are defined for operators and superoperators only. Parameters ---------- sparse : bool Use sparse Eigensolver sort : str Sort eigenvalues (and vectors) 'low' to high, or 'high' to low. eigvals : int Number of requested eigenvalues. Default is all eigenvalues. tol : float Tolerance used by sparse Eigensolver (0 = machine precision). The sparse solver may not converge if the tolerance is set too low. maxiter : int Maximum number of iterations performed by sparse solver (if used). Returns ------- eigvals : array Array of eigenvalues for operator. eigvecs : array Array of quantum operators representing the oprator eigenkets. Order of eigenkets is determined by order of eigenvalues. Notes ----- The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it. """ evals, evecs = sp_eigs(self.data, self.isherm, sparse=sparse, sort=sort, eigvals=eigvals, tol=tol, maxiter=maxiter) new_dims = [self.dims[0], [1] * len(self.dims[0])] ekets = np.array([Qobj(vec, dims=new_dims) for vec in evecs], dtype=object) norms = np.array([ket.norm() for ket in ekets]) return evals, ekets / norms
[docs] def eigenenergies(self, sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000): """Eigenenergies of a quantum object. Eigenenergies (eigenvalues) are defined for operators or superoperators only. Parameters ---------- sparse : bool Use sparse Eigensolver sort : str Sort eigenvalues 'low' to high, or 'high' to low. eigvals : int Number of requested eigenvalues. Default is all eigenvalues. tol : float Tolerance used by sparse Eigensolver (0=machine precision). The sparse solver may not converge if the tolerance is set too low. maxiter : int Maximum number of iterations performed by sparse solver (if used). Returns ------- eigvals : array Array of eigenvalues for operator. Notes ----- The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it. """ return sp_eigs(self.data, self.isherm, vecs=False, sparse=sparse, sort=sort, eigvals=eigvals, tol=tol, maxiter=maxiter)
[docs] def groundstate(self, sparse=False, tol=0, maxiter=100000, safe=True): """Ground state Eigenvalue and Eigenvector. Defined for quantum operators or superoperators only. Parameters ---------- sparse : bool Use sparse Eigensolver tol : float Tolerance used by sparse Eigensolver (0 = machine precision). The sparse solver may not converge if the tolerance is set too low. maxiter : int Maximum number of iterations performed by sparse solver (if used). safe : bool (default=True) Check for degenerate ground state Returns ------- eigval : float Eigenvalue for the ground state of quantum operator. eigvec : qobj Eigenket for the ground state of quantum operator. Notes ----- The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it. """ if safe: evals = 2 else: evals = 1 grndval, grndvec = sp_eigs(self.data, self.isherm, sparse=sparse, eigvals=evals, tol=tol, maxiter=maxiter) if safe: if tol == 0: tol = 1e-15 if (grndval[1]-grndval[0]) <= 10*tol: print("WARNING: Ground state may be degenerate. " "Use Q.eigenstates()") new_dims = [self.dims[0], [1] * len(self.dims[0])] grndvec = Qobj(grndvec[0], dims=new_dims) grndvec = grndvec / grndvec.norm() return grndval[0], grndvec
[docs] def trans(self): """Transposed operator. Returns ------- oper : qobj Transpose of input operator. """ out = Qobj() out.data = zcsr_transpose(self.data) out.dims = [self.dims[1], self.dims[0]] return out
[docs] def extract_states(self, states_inds, normalize=False): """Qobj with states in state_inds only. Parameters ---------- states_inds : list of integer The states that should be kept. normalize : True / False Weather or not the new Qobj instance should be normalized (default is False). For Qobjs that represents density matrices or state vectors normalized should probably be set to True, but for Qobjs that represents operators in for example an Hamiltonian, normalize should be False. Returns ------- q : Qobj A new instance of :class:`qutip.Qobj` that contains only the states corresponding to the indices in `state_inds`. Notes ----- Experimental. """ if self.isoper: q = Qobj(self.data[states_inds, :][:, states_inds]) elif self.isket: q = Qobj(self.data[states_inds, :]) elif self.isbra: q = Qobj(self.data[:, states_inds]) else: raise TypeError("Can only eliminate states from operators or " + "state vectors") return q.unit() if normalize else q
[docs] def eliminate_states(self, states_inds, normalize=False): """Creates a new quantum object with states in state_inds eliminated. Parameters ---------- states_inds : list of integer The states that should be removed. normalize : True / False Weather or not the new Qobj instance should be normalized (default is False). For Qobjs that represents density matrices or state vectors normalized should probably be set to True, but for Qobjs that represents operators in for example an Hamiltonian, normalize should be False. Returns ------- q : Qobj A new instance of :class:`qutip.Qobj` that contains only the states corresponding to indices that are **not** in `state_inds`. Notes ----- Experimental. """ keep_indices = np.array([s not in states_inds for s in range(self.shape[0])]).nonzero()[0] return self.extract_states(keep_indices, normalize=normalize)
[docs] def dnorm(self, B=None): """Calculates the diamond norm, or the diamond distance to another operator. Parameters ---------- B : Qobj or None If B is not None, the diamond distance d(A, B) = dnorm(A - B) between this operator and B is returned instead of the diamond norm. Returns ------- d : float Either the diamond norm of this operator, or the diamond distance from this operator to B. """ return mts.dnorm(self, B)
@property def ishp(self): # FIXME: this needs to be cached in the same ways as isherm. if self.type in ["super", "oper"]: try: J = sr.to_choi(self) return J.isherm except: return False else: return False @property def iscp(self): # FIXME: this needs to be cached in the same ways as isherm. if self.type in ["super", "oper"]: try: J = ( self # We can test with either Choi or chi, since the basis # transformation between them is unitary and hence # preserves the CP and TP conditions. if self.superrep in ('choi', 'chi') else sr.to_choi(self) ) # If J isn't hermitian, then that could indicate either # that J is not normal, or is normal, but has complex eigenvalues. # In either case, it makes no sense to then demand that the # eigenvalues be non-negative. if not J.isherm: return False eigs = J.eigenenergies() return all(eigs >= -settings.atol) except: return False else: return False @property def istp(self): import qutip.superop_reps as sr if self.type in ["super", "oper"]: try: # Normalize to a super of type choi or chi. # We can test with either Choi or chi, since the basis # transformation between them is unitary and hence # preserves the CP and TP conditions. if self.type == "super" and self.superrep in ('choi', 'chi'): qobj = self else: qobj = sr.to_choi(self) # Possibly collapse dims. if any([len(index) > 1 for super_index in qobj.dims for index in super_index]): qobj = Qobj(qobj, dims=collapse_dims_super(qobj.dims)) else: qobj = qobj # We use the condition from John Watrous' lecture notes, # Tr_1(J(Phi)) = identity_2. tr_oper = qobj.ptrace([0]) ident = ops.identity(tr_oper.shape[0]) return isequal(tr_oper, ident) except: return False else: return False @property def iscptp(self): from qutip.superop_reps import to_choi if self.type == "super" or self.type == "oper": reps = ('choi', 'chi') q_oper = to_choi(self) if self.superrep not in reps else self return q_oper.iscp and q_oper.istp else: return False @property def isherm(self): if self._isherm is not None: # used previously computed value return self._isherm self._isherm = bool(zcsr_isherm(self.data)) return self._isherm @isherm.setter def isherm(self, isherm): self._isherm = isherm @property def type(self): if not self._type: self._type = type_from_dims(self.dims) return self._type @property def shape(self): if self.data.shape == (1, 1): return tuple([np.prod(self.dims[0]), np.prod(self.dims[1])]) else: return tuple(self.data.shape) @property def isbra(self): return self.type == 'bra' @property def isket(self): return self.type == 'ket' @property def isoperbra(self): return self.type == 'operator-bra' @property def isoperket(self): return self.type == 'operator-ket' @property def isoper(self): return self.type == 'oper' @property def issuper(self): return self.type == 'super' @staticmethod
[docs] def evaluate(qobj_list, t, args): """Evaluate a time-dependent quantum object in list format. For example, qobj_list = [H0, [H1, func_t]] is evaluated to Qobj(t) = H0 + H1 * func_t(t, args) and qobj_list = [H0, [H1, 'sin(w * t)']] is evaluated to Qobj(t) = H0 + H1 * sin(args['w'] * t) Parameters ---------- qobj_list : list A nested list of Qobj instances and corresponding time-dependent coefficients. t : float The time for which to evaluate the time-dependent Qobj instance. args : dictionary A dictionary with parameter values required to evaluate the time-dependent Qobj intance. Returns ------- output : Qobj A Qobj instance that represents the value of qobj_list at time t. """ q_sum = 0 if isinstance(qobj_list, Qobj): q_sum = qobj_list elif isinstance(qobj_list, list): for q in qobj_list: if isinstance(q, Qobj): q_sum += q elif (isinstance(q, list) and len(q) == 2 and isinstance(q[0], Qobj)): if isinstance(q[1], types.FunctionType): q_sum += q[0] * q[1](t, args) elif isinstance(q[1], str): args['t'] = t q_sum += q[0] * float(eval(q[1], globals(), args)) else: raise TypeError('Unrecognized format for ' + 'specification of time-dependent Qobj') else: raise TypeError('Unrecognized format for specification ' + 'of time-dependent Qobj') else: raise TypeError( 'Unrecongized format for specification of time-dependent Qobj') return q_sum
# ----------------------------------------------------------------------------- # This functions evaluates a time-dependent quantum object on the list-string # and list-function formats that are used by the time-dependent solvers. # Although not used directly in by those solvers, it can for test purposes be # conventient to be able to evaluate the expressions passed to the solver for # arbitrary value of time. This function provides this functionality. # def qobj_list_evaluate(qobj_list, t, args): """ Depracated: See Qobj.evaluate """ warnings.warn("Deprecated: Use Qobj.evaluate", DeprecationWarning) return Qobj.evaluate(qobj_list, t, args) # ----------------------------------------------------------------------------- # # A collection of tests used to determine the type of quantum objects, and some # functions for increased compatibility with quantum optics toolbox. # def dag(A): """Adjont operator (dagger) of a quantum object. Parameters ---------- A : qobj Input quantum object. Returns ------- oper : qobj Adjoint of input operator Notes ----- This function is for legacy compatibility only. It is recommended to use the ``dag()`` Qobj method. """ if not isinstance(A, Qobj): raise TypeError("Input is not a quantum object") return A.dag() def ptrace(Q, sel): """Partial trace of the Qobj with selected components remaining. Parameters ---------- Q : qobj Composite quantum object. sel : int/list An ``int`` or ``list`` of components to keep after partial trace. Returns ------- oper : qobj Quantum object representing partial trace with selected components remaining. Notes ----- This function is for legacy compatibility only. It is recommended to use the ``ptrace()`` Qobj method. """ if not isinstance(Q, Qobj): raise TypeError("Input is not a quantum object") return Q.ptrace(sel) def dims(inpt): """Returns the dims attribute of a quantum object. Parameters ---------- inpt : qobj Input quantum object. Returns ------- dims : list A ``list`` of the quantum objects dimensions. Notes ----- This function is for legacy compatibility only. Using the `Qobj.dims` attribute is recommended. """ if isinstance(inpt, Qobj): return inpt.dims else: raise TypeError("Input is not a quantum object") def shape(inpt): """Returns the shape attribute of a quantum object. Parameters ---------- inpt : qobj Input quantum object. Returns ------- shape : list A ``list`` of the quantum objects shape. Notes ----- This function is for legacy compatibility only. Using the `Qobj.shape` attribute is recommended. """ if isinstance(inpt, Qobj): return Qobj.shape else: return np.shape(inpt) def isket(Q): """ Determines if given quantum object is a ket-vector. Parameters ---------- Q : qobj Quantum object Returns ------- isket : bool True if qobj is ket-vector, False otherwise. Examples -------- >>> psi = basis(5,2) >>> isket(psi) True Notes ----- This function is for legacy compatibility only. Using the `Qobj.isket` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.isket else False def isbra(Q): """Determines if given quantum object is a bra-vector. Parameters ---------- Q : qobj Quantum object Returns ------- isbra : bool True if Qobj is bra-vector, False otherwise. Examples -------- >>> psi = basis(5,2) >>> isket(psi) False Notes ----- This function is for legacy compatibility only. Using the `Qobj.isbra` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.isbra else False def isoperket(Q): """Determines if given quantum object is an operator in column vector form (operator-ket). Parameters ---------- Q : qobj Quantum object Returns ------- isoperket : bool True if Qobj is operator-ket, False otherwise. Notes ----- This function is for legacy compatibility only. Using the `Qobj.isoperket` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.isoperket else False def isoperbra(Q): """Determines if given quantum object is an operator in row vector form (operator-bra). Parameters ---------- Q : qobj Quantum object Returns ------- isoperbra : bool True if Qobj is operator-bra, False otherwise. Notes ----- This function is for legacy compatibility only. Using the `Qobj.isoperbra` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.isoperbra else False def isoper(Q): """Determines if given quantum object is a operator. Parameters ---------- Q : qobj Quantum object Returns ------- isoper : bool True if Qobj is operator, False otherwise. Examples -------- >>> a = destroy(5) >>> isoper(a) True Notes ----- This function is for legacy compatibility only. Using the `Qobj.isoper` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.isoper else False def issuper(Q): """Determines if given quantum object is a super-operator. Parameters ---------- Q : qobj Quantum object Returns ------- issuper : bool True if Qobj is superoperator, False otherwise. Notes ----- This function is for legacy compatibility only. Using the `Qobj.issuper` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.issuper else False def isequal(A, B, tol=None): """Determines if two qobj objects are equal to within given tolerance. Parameters ---------- A : qobj Qobj one B : qobj Qobj two tol : float Tolerence for equality to be valid Returns ------- isequal : bool True if qobjs are equal, False otherwise. Notes ----- This function is for legacy compatibility only. Instead, it is recommended to use the equality operator of Qobj instances instead: A == B. """ if tol is None: tol = settings.atol if not isinstance(A, Qobj) or not isinstance(B, Qobj): return False if A.dims != B.dims: return False Adat = A.data Bdat = B.data elems = (Adat - Bdat).data if np.any(np.abs(elems) > tol): return False return True def isherm(Q): """Determines if given operator is Hermitian. Parameters ---------- Q : qobj Quantum object Returns ------- isherm : bool True if operator is Hermitian, False otherwise. Examples -------- >>> a = destroy(4) >>> isherm(a) False Notes ----- This function is for legacy compatibility only. Using the `Qobj.isherm` attribute is recommended. """ return True if isinstance(Q, Qobj) and Q.isherm else False # TRAILING IMPORTS # We do a few imports here to avoid circular dependencies. from qutip.eseries import eseries import qutip.superop_reps as sr import qutip.tensor as tensor import qutip.operators as ops import qutip.metrics as mts import qutip.states import qutip.superoperator