g4dZddlmZddlZddlmZddlmZddl Z ddl m Z ddl mZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZmZddlmZddl m!Z!ddl m"Z"ddl#m$Z$ddl%m&Z&ddl'm(Z(m)Z)ddl*m+Z+erddl m,Z,GddeeZ-dS)z$ DensityMatrix quantum state class. ) annotationsN)Number) TYPE_CHECKING) _numpy_compat)QuantumCircuit) Instruction) QiskitError) QuantumState)TolerancesMixin)OpShape)Operator)Pauli SparsePauliOp)ScalarOp)is_hermitian_matrix)is_positive_semidefinite_matrix)QuantumChannel)SuperOp)density_expval_pauli_no_xdensity_expval_pauli_with_x) Statevector)circuitceZdZdZ dBdCfd ZdejfdZfd Zd Z e d Z dBdDdZ dZ e dZdEdZdFdZdZdZdZdGdZdGdZdZdZ dBdHd ZdId!ZdBd"ZdBdJd&Z dEdKd*ZdBdLd+ZedMd.Ze dNd2Z!edOd5Z"dBdPd7Z#dBd8Z$dBd9Z%dBd:Z&dEdQd?Z'dRdAZ(xZ)S)S DensityMatrixzDensityMatrix classNdataSnp.ndarray | list | QuantumCircuit | circuit.instruction.Instruction | QuantumStatedimsint | tuple | list | Nonec,t|ttjfr!tj|t |_nt|ttfr%t |j|_nt|dr7| }|j |_||}nRt|dr3tj|t |_nt!d|jj}|jj}|dkr|d|dkrn|dkr7tj|jtj|j|_nG|dkr2|ddkr&tj|j|d|_nt!d t-t1j|jj||  dS) aInitialize a density matrix object. Args: data: A statevector, quantum instruction or an object with a ``to_operator`` or ``to_matrix`` method from which the density matrix can be constructed. If a vector the density matrix is constructed as the projector of that vector. If a quantum instruction, the density matrix is constructed by assuming all qubits are initialized in the zero state. dims: The subsystem dimension of the state (See additional information). Raises: QiskitError: if input data is not valid. Additional Information: The ``dims`` kwarg can be None, an integer, or an iterable of integers. * ``Iterable`` -- the subsystem dimensions are the values in the list with the total number of subsystems given by the length of the list. * ``Int`` or ``None`` -- the leading dimension of the input matrix specifies the total dimension of the density matrix. If it is a power of two the state will be initialized as an N-qubit state. If it is not a power of two the state will have a single d-dimensional subsystem. dtype to_operatorN to_matrixz+Invalid input data format for DensityMatrixrz1Invalid DensityMatrix input: not a square matrix.)shapedims_ldims_r)op_shape) isinstancelistnpndarrayasarraycomplex_datarrrfrom_instructionhasattrr"r output_dimsr#r ndimr&outerconjreshapesuper__init__r auto)selfrropr4r& __class__s ^/granary/edev/quantum/lib/python3.11/site-packages/qiskit/quantum_info/states/densitymatrix.pyr9zDensityMatrix.__init__1s> dT2:. / / MD888DJJ ~{; < < M'77==CDJJ T= ) ) M!!##BDJ|~~'' T; ' ' MDNN$4$4GDDDDJJKLL Lz   199qU1X--  QYY$*bgdj.A.ABBDJJ QYY58q==DJa99DJJQRR R ',TZ5Ed[_"`"`"`aaaaacZ| |jjn|}tj|j||S)N)r!copy)rr!r,array)r;r!rAs r> __array__zDensityMatrix.__array__ts,#(= ex T::::r?ct|o+tj|j|j|j|jS)Nrtolatol)r8__eq__r,allcloser0rFrG)r;otherr=s r>rHzDensityMatrix.__eq__xsDww~~e$$  J $)$)* * *  r?cd}t|dz}|tj|jd|d|d|jdS)NzDensityMatrix( z, ) separatorprefixz, zdims=))lenr, array2stringr0 _op_shaper')r;rNpads r>__repr__zDensityMatrix.__repr__}sp!&kkC 4rtzT&QQQ 4 4 4 4..00 4 4 4 r?cD|j|jdS)zReturn settings.)rr)rrRr'r;s r>settingszDensityMatrix.settingss" 4>+@+@+B+BCCCr?output str | Nonec $ddlm}||fd|i|S)aReturn a visualization of the Statevector. **repr**: ASCII TextMatrix of the state's ``__repr__``. **text**: ASCII TextMatrix that can be printed in the console. **latex**: An IPython Latex object for displaying in Jupyter Notebooks. **latex_source**: Raw, uncompiled ASCII source to generate array using LaTeX. **qsphere**: Matplotlib figure, rendering of density matrix using `plot_state_qsphere()`. **hinton**: Matplotlib figure, rendering of density matrix using `plot_state_hinton()`. **bloch**: Matplotlib figure, rendering of density matrix using `plot_bloch_multivector()`. Args: output (str): Select the output method to use for drawing the state. Valid choices are `repr`, `text`, `latex`, `latex_source`, `qsphere`, `hinton`, or `bloch`. Default is `repr`. Default can be changed by adding the line ``state_drawer = `` to ``~/.qiskit/settings.conf`` under ``[default]``. drawer_args: Arguments to be passed directly to the relevant drawing function or constructor (`TextMatrix()`, `array_to_latex()`, `plot_state_qsphere()`, `plot_state_hinton()` or `plot_bloch_multivector()`). See the relevant function under `qiskit.visualization` for that function's documentation. Returns: :class:`matplotlib.Figure` or :class:`str` or :class:`TextMatrix` or :class:`IPython.display.Latex`: Drawing of the Statevector. Raises: ValueError: when an invalid output method is selected. r) state_drawerrX)(qiskit.visualization.state_visualizationr[)r;rX drawer_argsr[s r>drawzDensityMatrix.draws5L JIIIII|D???;???r?c|}t|trt|dSddlm}||dS)Nr)display)r^r*strprintIPython.displayr`)r;outr`s r>_ipython_display_zDensityMatrix._ipython_display_sSiikk c3    #JJJJJ / / / / / / GCLLLLLr?c|jS)z Return data.)r0rVs r>rzDensityMatrix.datas zr?c||j}||j}tj|d||sdSt |j||sdSt|j||S)z1Return True if trace 1 and positive semidefinite.Nr%rEF)rGrFr,rItracerrr)r;rGrFs r>is_validzDensityMatrix.is_validsw <9D <9D{4::<<DAAA 5"494dCCC 5.tyt$OOOOr?returnr cX|}t|j||S)zConvert to Operator input_dimsr3)rr r)r;rs r>r"zDensityMatrix.to_operators&yy{{ dEEEEr?cvttj|j|S)z+Return the conjugate of the density matrix.r)rr,r6rrrVs r> conjugatezDensityMatrix.conjugates)RWTY//diikkBBBBr?c4tj|jS)z'Return the trace of the density matrix.)r,rhrrVs r>rhzDensityMatrix.tracesx """r?cdtjtj|j|jS)z'Return the purity of the quantum state.)r,rhdotrrVs r>purityzDensityMatrix.puritys$ xty$)44555r?rJct|tst|}tj|}t j|j|j|_|j|j|_|S)u&Return the tensor product state self ⊗ other. Args: other (DensityMatrix): a quantum state object. Returns: DensityMatrix: the tensor product operator self ⊗ other. Raises: QiskitError: if other is not a quantum state. ) r*r_copyrAr,kronr0rRtensorr;rJrets r>rxzDensityMatrix.tensorsg%// )!%((EjGDJ 44 --eo>>  r?ct|tst|}tj|}t j|j|j|_|j|j|_|S)u#Return the tensor product state other ⊗ self. Args: other (DensityMatrix): a quantum state object. Returns: DensityMatrix: the tensor product state other ⊗ self. Raises: QiskitError: if other is not a quantum state. ) r*rrvrAr,rwr0rRexpandrys r>r|zDensityMatrix.expandsg%// )!%((EjGEK44 --eo>>  r?ct|tst|}|j|jt j|}|j|jz|_|S)aTReturn the linear combination self + other. Args: other (DensityMatrix): a quantum state object. Returns: DensityMatrix: the linear combination self + other. Raises: QiskitError: if other is not a quantum state, or has incompatible dimensions. )r*rrR _validate_addrvrArr0rys r>_addzDensityMatrix._add s_%// )!%((E $$U_555jI *  r?ct|tstdtj|}||jz|_|S)a Return the scalar multiplied state other * self. Args: other (complex): a complex number. Returns: DensityMatrix: the scalar multiplied state other * self. Raises: QiskitError: if other is not a valid complex number. zother is not a number)r*rr rvrArr0rys r> _multiplyzDensityMatrix._multiply sG%(( 7566 6jDI%  r?LOperator | QuantumChannel | circuit.instruction.Instruction | QuantumCircuitqargslist[int] | Nonec|t|dd}t|ttfr|||St |dr)|||St|tr|||St|ts(| |}t|||}| ||S)aEvolve a quantum state by an operator. Args: other: The operator to evolve by. qargs: a list of QuantumState subsystem positions to apply the operator on. Returns: The output density matrix. Raises: QiskitError: if the operator dimension does not match the specified QuantumState subsystem dimensions. Nrrto_quantumchannelrl) getattrr*rr_evolve_instructionr2r_evolverr r_evolve_operator)r;rJrrs r>evolvezDensityMatrix.evolve2s$ =E7D11E enk: ; ; @++E+?? ? 5- . . H**,,44T4GG G e^ , , 4==U=33 3%** G9959))DUtFFFE$$U%$888r?ctj|}tt|jjdz ddtfdDzt jt jt j|j |jj |jj |_ |j |_|S)aReturn a DensityMatrix with reversed subsystem ordering. For a tensor product state this is equivalent to reversing the order of tensor product subsystems. For a density matrix :math:`\rho = \rho_{n-1} \otimes ... \otimes \rho_0` the returned state will be :math:`\rho_0 \otimes ... \otimes \rho_{n-1}`. Returns: DensityMatrix: the state with reversed subsystem order. r%c3<K|]}t|zVdSN)rP).0iaxess r> z.DensityMatrix.reverse_qargs..gs-88aCIIM888888r?)rvrAtuplerangerR _num_qargs_lr,r7 transposer tensor_shaper&r0reverse)r;rzrs @r> reverse_qargszDensityMatrix.reverse_qargsYsjU4>6:BCCDDe88884888888J LDIt~/JKKT R R N    ..00  r?c `t|}|tj|}ntj|}tjd|z|j}tjd|z|j}|jr d|jznd}||zdkr||zStj |j d}|dkr|t||j |zS||jd} d| z} | d} |t||j ||| | zS)aCompute the expectation value of a Pauli. Args: pauli (Pauli): a Pauli operator to evaluate expval of. qargs (None or list): subsystems to apply operator on. Returns: complex: the expectation value. Nr%yrF)orderr)rPr,arangerBrsxzphaserhravelrr num_qubits_count_yr) r;paulirn_pauliqubitsx_maskz_mask pauli_phaserx_maxy_phases r>_expectation_value_pauliz&DensityMatrix._expectation_value_paulios*e** =Yw''FFXe__FV UW--V UW--.3k@su{**q F?a  - -x --- Q;;!:4RX!Y!YY Yuw#5>>+++!*8 $/667E    r?operNone | list[int]r/ct|tr|St|trDt fdt |jj|jj|j DSt|tst|}tj t |jS)aCompute the expectation value of an operator. Args: oper (Operator): an operator to evaluate expval. qargs (None or list): subsystems to apply the operator on. Returns: complex: the expectation value. c3pK|]0\}}}|t||fzV1dSr)rr)rrrcoeffrr;s r>rz2DensityMatrix.expectation_value..s\Aq%55eQFmmUKKKr?r)r*rrrsumzippaulisrrcoeffsr r,rhrsr)r;rrs` `r>expectation_valuezDensityMatrix.expectation_values dE " " >00u== = dM * * #&t{}dkmT[#Q#Q  $)) "D>>Dx**4u*==BCCCr?decimals None | int np.ndarrayc|tj|j|j|}tj|dd}|||}|S)aK Return the subsystem measurement probability vector. Measurement probabilities are with respect to measurement in the computation (diagonal) basis. Args: qargs (None or list): subsystems to return probabilities for, if None return for all subsystems (Default: None). decimals (None or int): the number of decimal places to round values. If None no rounding is done (Default: None). Returns: np.array: The Numpy vector array of probabilities. Examples: Consider a 2-qubit product state :math:`\rho=\rho_1\otimes\rho_0` with :math:`\rho_1=|+\rangle\!\langle+|`, :math:`\rho_0=|0\rangle\!\langle0|`. .. code-block:: from qiskit.quantum_info import DensityMatrix rho = DensityMatrix.from_label('+0') # Probabilities for measuring both qubits probs = rho.probabilities() print('probs: {}'.format(probs)) # Probabilities for measuring only qubit-0 probs_qubit_0 = rho.probabilities([0]) print('Qubit-0 probs: {}'.format(probs_qubit_0)) # Probabilities for measuring only qubit-1 probs_qubit_1 = rho.probabilities([1]) print('Qubit-1 probs: {}'.format(probs_qubit_1)) .. code-block:: text probs: [0.5 0. 0.5 0. ] Qubit-0 probs: [1. 0.] Qubit-1 probs: [0.5 0.5] We can also permute the order of qubits in the ``qargs`` list to change the qubit position in the probabilities output .. code-block:: from qiskit.quantum_info import DensityMatrix rho = DensityMatrix.from_label('+0') # Probabilities for measuring both qubits probs = rho.probabilities([0, 1]) print('probs: {}'.format(probs)) # Probabilities for measuring both qubits # but swapping qubits 0 and 1 in output probs_swapped = rho.probabilities([1, 0]) print('Swapped probs: {}'.format(probs_swapped)) .. code-block:: text probs: [0.5 0. 0.5 0. ] Swapped probs: [0.5 0.5 0. 0. ] rrr%)a_mina_maxN)r) _subsystem_probabilitiesr,absrdiagonalrRr'clipround)r;rrprobss r> probabilitieszDensityMatrix.probabilitiessL-- F49%%'' ( ($.*?*?*A*A.   Qa000  KKK22E r?c|Gtj|}tj|jjt }d|d<||_|S||}tt|d}tt|j |j d<|||S)axReset state or subsystems to the 0-state. Args: qargs (list or None): subsystems to reset, if None all subsystems will be reset to their 0-state (Default: None). Returns: DensityMatrix: the reset state. Additional Information: If all subsystems are reset this will return the ground state on all subsystems. If only a some subsystems are reset this function will perform evolution by the reset :class:`~qiskit.quantum_info.SuperOp` of the reset subsystems. Nr r%rrr)rr)rvrAr,zerosrRr&r/r0rrrr rrr)r;rrzstater reset_superops r>resetzDensityMatrix.resets" =*T""CHT^1AAAEE$KCIJyyQ 7 7 788 ($ 8 8 = C C E E 1{{={666r?labelracDttj|S)aReturn a tensor product of Pauli X,Y,Z eigenstates. .. list-table:: Single-qubit state labels :header-rows: 1 * - Label - Statevector * - ``"0"`` - :math:`\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}` * - ``"1"`` - :math:`\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}` * - ``"+"`` - :math:`\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}` * - ``"-"`` - :math:`\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}` * - ``"r"`` - :math:`\frac{1}{2}\begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}` * - ``"l"`` - :math:`\frac{1}{2}\begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}` Args: label (string): a eigenstate string ket label (see table for allowed values). Returns: DensityMatrix: The N-qubit basis state density matrix. Raises: QiskitError: if the label contains invalid characters, or the length of the label is larger than an explicitly specified num_qubits. )rr from_label)clsrs r>rzDensityMatrix.from_labelsD[3E::;;;r?rintint | tuple | listctj|}tj||ft}d|||f<t ||S)aReturn a computational basis state density matrix. Args: i (int): the basis state element. dims (int or tuple or list): The subsystem dimensions of the statevector (See additional information). Returns: DensityMatrix: The computational basis state :math:`|i\rangle\!\langle i|`. Additional Information: The ``dims`` kwarg can be an integer or an iterable of integers. * ``Iterable`` -- the subsystem dimensions are the values in the list with the total number of subsystems given by the length of the list. * ``Int`` -- the integer specifies the total dimension of the state. If it is a power of two the state will be initialized as an N-qubit state. If it is not a power of two the state will have a single d-dimensional subsystem. r g?ro)r,prodrr/r)rrsizers r>from_intzDensityMatrix.from_int<sI.wt}}$W555ad U....r? instruction0circuit.instruction.Instruction | QuantumCircuitct|tr|}|j}t jd|zd|zft }d|d<t||dz}|||S)a1Return the output density matrix of an instruction. The statevector is initialized in the state :math:`|{0,\ldots,0}\rangle` of the same number of qubits as the input instruction or circuit, evolved by the input instruction, and the output statevector returned. Args: instruction: instruction or circuit Returns: The final density matrix. Raises: QiskitError: if the instruction contains invalid instructions for density matrix simulation. r$r r%r)r$ro) r*rto_instructionrr,rr/r_append_instruction)rrrinitvecs r>r1zDensityMatrix.from_instructionXs* k> 2 2 7%4466K + xJ: 6gFFFT DzD'8999  ,,, r?dictcl||j|j|dS)a Convert the density matrix to dictionary form. This dictionary representation uses a Ket-like notation where the dictionary keys are qudit strings for the subsystem basis vectors. If any subsystem has a dimension greater than 10 comma delimiters are inserted between integers so that subsystems can be distinguished. Args: decimals (None or int): the number of decimal places to round values. If None no rounding is done (Default: None). Returns: dict: the dictionary form of the DensityMatrix. Examples: The ket-form of a 2-qubit density matrix :math:`rho = |-\rangle\!\langle -|\otimes |0\rangle\!\langle 0|` .. code-block:: from qiskit.quantum_info import DensityMatrix rho = DensityMatrix.from_label('-0') print(rho.to_dict()) .. code-block:: text { '00|00': (0.4999999999999999+0j), '10|00': (-0.4999999999999999-0j), '00|10': (-0.4999999999999999+0j), '10|10': (0.4999999999999999+0j) } For non-qubit subsystems the integer range can go from 0 to 9. For example in a qutrit system .. code-block:: import numpy as np from qiskit.quantum_info import DensityMatrix mat = np.zeros((9, 9)) mat[0, 0] = 0.25 mat[3, 3] = 0.25 mat[6, 6] = 0.25 mat[-1, -1] = 0.25 rho = DensityMatrix(mat, dims=(3, 3)) print(rho.to_dict()) .. code-block:: text {'00|00': (0.25+0j), '10|10': (0.25+0j), '20|20': (0.25+0j), '22|22': (0.25+0j)} For large subsystem dimensions delimiters are required. The following example is for a 20-dimensional system consisting of a qubit and 10-dimensional qudit. .. code-block:: import numpy as np from qiskit.quantum_info import DensityMatrix mat = np.zeros((2 * 10, 2 * 10)) mat[0, 0] = 0.5 mat[-1, -1] = 0.5 rho = DensityMatrix(mat, dims=(2, 10)) print(rho.to_dict()) .. code-block:: text {'00|00': (0.5+0j), '91|91': (0.5+0j)} T)r string_labels)_matrix_to_dictrrRr')r;rs r>to_dictzDensityMatrix.to_dictws<X## It~,,..QU$   r?c^ |j|j|}|j|_|j|_t j|}|Y|j}tj ||j |j |_ ||_|Stj|j|jj}t!| fd|D}tj|j|jj}t%j|||}|} tj| j| jj} t%j|| | d}tj||j|_ ||_|S)z$Evolve density matrix by an operatorrNc g|] }dz |z  S)r%)rqubit num_indicess r> z2DensityMatrix._evolve_operator..s">>>u;?U*>>>r?T)rRcompose_dims_l_dims_rr _num_qargs_rrvrArr,rsTr6r0r7rrPrr _einsum_matmuladjointr&) r;rJr new_shaperzop_matrxindicesmatadjmat_adjrs @r>rzDensityMatrix._evolve_operators^N**5?%*HH %- !*!7 j =ZFvty1155fhmmooFFCI%CMJDIt~'BCC$))++&& >>>>>>>jU_%ABB(g>>mmoo*SXs}'ABB(';PTUUJvy77 !  r?c ddlm}ddlm}t j|}|0|t |j|_dSt||r!| j|_dSt||rdStj |}|#| |j|_dS|jtd|jt|jt"s,t|jdt%|jdd t'|jjD |jD]k}|jrtd |jj fd |jD}n fd |jD}||j|ldS) z:Update the current Statevector by applying an instruction.r)Reset)BarrierNrzCannot apply Instruction: z instruction definition is z; expected QuantumCircuitci|]\}}|| Srr)ridxbits r> z5DensityMatrix._append_instruction.. sUUUhc3cUUUr?z.Cannot apply instruction with classical bits: c g|] }| Srr)rtup qubit_indicess r>rz5DensityMatrix._append_instruction..sNNNC]3/NNNr?c,g|]}|Srr)rrrrs r>rz5DensityMatrix._append_instruction..s#UUU3U=#56UUUr?)qiskit.circuit.resetrqiskit.circuit.barrierrr _instruction_to_matrixrrr0r*rr_instruction_to_superopr definitionr namertype enumeraterclbits operationr) r;rJrrrrchanr new_qargsrs ` @r>rz!DensityMatrix._append_instructions/......222222-e44 ?..x}}E.JJODJ F eU # # E**0DJ F eW % %  F.u55  d%88=DJ F   #G5:GGHH H%*N;; :,,$u?O:P:P,,, VU)Erz!DensityMatrix._evolve_instructionsR c> * * '$$&&Cj 5111 r?rG float | NonerFrc||j}||j}t|j||st dt j|j\}}|t||k}t|dkst j |dd||st d|ddt j |f}t|S)aReturn a statevector from a pure density matrix. Args: atol (float): Absolute tolerance for checking operation validity. rtol (float): Relative tolerance for checking operation validity. Returns: Statevector: The pure density matrix's corresponding statevector. Corresponds to the eigenvector of the only non-zero eigenvalue. Raises: QiskitError: if the state is not pure. N)rGrFz+Not a valid density matrix (non-hermitian).r%rz"Density matrix is not a pure state) rGrFrr0r r,linalgeigrrPiscloseargmaxr)r;rGrFevalsevecs nonzero_evalspsis r>to_statevectorzDensityMatrix.to_statevectors <9D <9D"4:DtDDD MKLL Ly}}TZ00 uc%jj4/0 }   " ""*]15EqtZ^*_*_*_ "BCC CAAAry'''(3r? list[int]c8|j|jj}t |jdz t j|z }t |}ttd|z}|D]}|||z||c||<|||z< t j ||}t j||jj }t||S)zReturn partially transposed density matrix. Args: qargs (list): The subsystems to be transposed. Returns: DensityMatrix: The partially transposed density matrix. r%r$ro)r0r7rRrrPr'r,rBrr+rrr&r)r;rarrnlstrrhos r>partial_transposezDensityMatrix.partial_transpose>sj  !<==DN))++,,q028E??B   5Q<<   4 4A!$QUSV CFCAJJl3$$jdn233Styy{{3333r?r)rrrr)rXrY)NN)rjr )rJrrjr)rJrrrrjr)rjr)rr rrrjr/)rrrrrjr)rrrjr)rrarjr)rrrrrjr)rrrjr)rrrjr)rGrrFrrjr)rrrjr)*__name__ __module__ __qualname____doc__r9rCOPY_ONLY_IF_NEEDEDrCrHrTpropertyrWr^rerrir"rprhrtrxr|rrrrrrrr classmethodr staticmethodrr1rrrrrr __classcell__)r=s@r>rr.sD +/AbAbAbAbAbAbAbF#)J;;;;        DDXD(@(@(@(@(@TX P P P PFFFF CCC###666&&(*#'%9%9%9%9%9N,    BDDDDD0FJPPPPPd77777>!<!<!<[!r<s#"""""  888888222222))))))AAAAAAKKKKKK::::::;;;;;;IIIIIIII<<<<<<HHHHHHTTTTTTPPPPPPAAAAAAbbbbbbbb>>>>>>a 4a 4a 4a 4a 4L/a 4a 4a 4a 4a 4r?