B [K^ @sdZddlmZmZddlZddlmZmZmZm Z m Z m Z ddl m Z ddlmZmZmZddlmZmZddlmZdd lmZdd lmZmZmZdd lmZdd lm Z dd l!m"Z"m#Z#ddl$m%Z%dddddddddddg Z&GdddeZ'Gddde'eZ(Gddde'eZ)Gddde'Z*Gd dde*e(Z+Gd!dde*e)Z,d"dZ-d#dZ.d-d%dZ/d.d'dZ0d/d(dZ1d0d)dZ2d*d+Z3d1d,dZ4dS)2zQubits for quantum computing. Todo: * Finish implementing measurement logic. This should include POVM. * Update docstrings. * Update tests. )print_functiondivisionN)IntegerlogMulAddPow conjugate)sympify) string_typesrange SYMPY_INTS)Matrixzeros) prettyForm) ComplexSpace)KetBraState) QuantumError) represent) numpy_ndarrayscipy_sparse_matrix)bitcountQubitQubitBraIntQubit IntQubitBraqubit_to_matrixmatrix_to_qubitmatrix_to_density measure_allmeasure_partialmeasure_partial_oneshotmeasure_all_oneshotc@sdeZdZdZeddZeddZeddZedd Z ed d Z d d Z ddZ ddZ dS) QubitStatez"Base class for Qubit and QubitBra.cCst|dkr$t|dtr$|djSt|dkrJt|dtrJt|d}t|}x(|D] }|dksX|dksXtd|qXW|S)Nrz$Qubit values must be 0 or 1, got: %r)len isinstancer% qubit_valuesr tupler ValueError)clsargselementr/:lib/python3.7/site-packages/sympy/physics/quantum/qubit.py _eval_args7s   zQubitState._eval_argscCstdt|S)N)rr')r,r-r/r/r0_eval_hilbert_spaceKszQubitState._eval_hilbert_spacecCs t|jS)z"The number of Qubits in the state.)r'r))selfr/r/r0 dimensionSszQubitState.dimensioncCs|jS)N)r5)r4r/r/r0nqubitsXszQubitState.nqubitscCs|jS)z,Returns the values of the qubits as a tuple.)label)r4r/r/r0r)\szQubitState.qubit_valuescCs|jS)N)r5)r4r/r/r0__len__eszQubitState.__len__cCs|jt|j|dS)Nr&)r)intr5)r4bitr/r/r0 __getitem__hszQubitState.__getitem__cGsVt|j}x<|D]4}t|j|d}||dkr`` has a 1 in the least significant qubit. Parameters ========== values : list, str The qubit values as a list of ints ([0,0,0,1,1,]) or a string ('011'). Examples ======== Create a qubit in a couple of different ways and look at their attributes: >>> from sympy.physics.quantum.qubit import Qubit >>> Qubit(0,0,0) |000> >>> q = Qubit('0101') >>> q |0101> >>> q.nqubits 4 >>> len(q) 4 >>> q.dimension 4 >>> q.qubit_values (0, 1, 0, 1) We can flip the value of an individual qubit: >>> q.flip(1) |0111> We can take the dagger of a Qubit to get a bra: >>> from sympy.physics.quantum.dagger import Dagger >>> Dagger(q) <0101| >>> type(Dagger(q)) Inner products work as expected: >>> ip = Dagger(q)*q >>> ip <0101|0101> >>> ip.doit() 1 cCstS)N)r)r4r/r/r0 dual_classszQubit.dual_classcKs |j|jkrtdStdSdS)Nr&r)r7r)r4brahintsr/r/r0_eval_innerproduct_QubitBras z!Qubit._eval_innerproduct_QubitBracKs |jd|S)N)N)_represent_ZGate)r4optionsr/r/r0_represent_default_basisszQubit._represent_default_basisc Ks|dd}d}d}x&t|jD]}|||7}|d}q Wdgd|j}d|t|<|dkrht|S|dkrddl}|j|dd S|d krdd l m } | j |dd SdS) zBRepresent this qubits in the computational basis (ZGate). formatsympyr&rr2numpyNcomplex)Zdtypez scipy.sparse)sparse) getreversedr)r5r9rrPmatrixZ transposeZscipyrRZ csr_matrix) r4ZbasisrLrNnZdefinite_stateitresultZnprRr/r/r0rKs      zQubit._represent_ZGatecKs|dg}t|}t|dkr0ttd|j}|||}x0tt|dddD]}||t||}qVWt||jkr|dSt|SdS)Nindicesrr&) rSr<r'r r6sort_reduced_densityr9r )r4rHkwargsrYZ sorted_idxZnew_matr?r/r/r0 _eval_traces  zQubit._eval_tracec Ksdd}t|f|}|j}|d}t|}xht|D]\} xVt|D]J} xDtdD]8} || | |} || | |} || | f|| | f7<qTWqFWq8W|S)zCompute the reduced density matrix by tracing out one qubit. The qubit argument should be of type python int, since it is used in bit operations cSs,d|d}||?d|>||@||>S)Nr2r&r/)jkqubitZbit_maskr/r/r0find_index_that_is_projecteds z.find_index_that_is_projectedr2)rZcolsrrr )r4rUrarLrbZ old_matrixZold_sizeZnew_sizeZ new_matrixr?r_r`colrowr/r/r0r\s    (zQubit._reduced_densityN) rArBrCrDrErGrJrMrKr^r\r/r/r/r0r{s5 c@seZdZdZeddZdS)raA multi-qubit bra in the computational (z) basis. We use the normal convention that the least significant qubit is on the right, so ``|00001>`` has a 1 in the least significant qubit. Parameters ========== values : list, str The qubit values as a list of ints ([0,0,0,1,1,]) or a string ('011'). See also ======== Qubit: Examples using qubits cCstS)N)r)r4r/r/r0rGszQubitBra.dual_classN)rArBrCrDrErGr/r/r/r0rsc@s<eZdZdZeddZddZddZdd ZeZ eZ d S) IntQubitStatez>A base class for qubits that work with binary representations.cstdkr$tdtr$tStdkrpddkrpttttd}fdd|D}t|Stdkrddkrttd}d|krtdddffddttdD}t|StSdS)Nr&rcsg|]}d|?d@qS)rr&r/).0r?)r-r/r0 &sz,IntQubitState._eval_args..r2z cannot represent %s with %s bitscsg|]}d|?d@qS)rr&r/)rfr?)r-r/r0rg/s) r'r(r%r1rTr rabsr+)r,r-Zrvaluesr)Zneedr/)r-r0r1s    zIntQubitState._eval_argscCs4d}d}x&t|jD]}|||7}|d>}qW|S)z(Return the numerical value of the qubit.rr&)rTr))r4ZnumberrVr?r/r/r0as_int4s   zIntQubitState.as_intcGs t|S)N)strri)r4printerr-r/r/r0 _print_label=szIntQubitState._print_labelcGs|j|f|}t|S)N)rlr)r4rkr-r7r/r/r0_print_label_pretty@sz!IntQubitState._print_label_prettyN) rArBrCrDrEr1rirlrmZ_print_label_reprZ_print_label_latexr/r/r/r0res  rec@s$eZdZdZeddZddZdS)ra'A qubit ket that store integers as binary numbers in qubit values. The differences between this class and ``Qubit`` are: * The form of the constructor. * The qubit values are printed as their corresponding integer, rather than the raw qubit values. The internal storage format of the qubit values in the same as ``Qubit``. Parameters ========== values : int, tuple If a single argument, the integer we want to represent in the qubit values. This integer will be represented using the fewest possible number of qubits. If a pair of integers, the first integer gives the integer to represent in binary form and the second integer gives the number of qubits to use. Examples ======== Create a qubit for the integer 5: >>> from sympy.physics.quantum.qubit import IntQubit >>> from sympy.physics.quantum.qubit import Qubit >>> q = IntQubit(5) >>> q |5> We can also create an ``IntQubit`` by passing a ``Qubit`` instance. >>> q = IntQubit(Qubit('101')) >>> q |5> >>> q.as_int() 5 >>> q.nqubits 3 >>> q.qubit_values (1, 0, 1) We can go back to the regular qubit form. >>> Qubit(q) |101> cCstS)N)r)r4r/r/r0rGxszIntQubit.dual_classcKs t||S)N)rrJ)r4rHrIr/r/r0_eval_innerproduct_IntQubitBra|sz'IntQubit._eval_innerproduct_IntQubitBraN)rArBrCrDrErGrnr/r/r/r0rHs/ c@seZdZdZeddZdS)rzBA qubit bra that store integers as binary numbers in qubit values.cCstS)N)r)r4r/r/r0rGszIntQubitBra.dual_classN)rArBrCrDrErGr/r/r/r0rsc s:d}t|trd}t|tr d}|jddkrL|jd}t|d}d}t}n8|jddkrx|jd}t|d}d}t}n td |t|tstd |d}x|t |D]p|r|df}n |df}|dks|dkrt |}|d krfd d t |D}| ||||}qWt|t t tfr6|}|S)aConvert from the matrix repr. to a sum of Qubit objects. Parameters ---------- matrix : Matrix, numpy.matrix, scipy.sparse The matrix to build the Qubit representation of. This works with sympy matrices, numpy matrices and scipy.sparse sparse matrices. Examples ======== Represent a state and then go back to its qubit form: >>> from sympy.physics.quantum.qubit import matrix_to_qubit, Qubit >>> from sympy.physics.quantum.gate import Z >>> from sympy.physics.quantum.represent import represent >>> q = Qubit('01') >>> matrix_to_qubit(represent(q)) |01> rOrPz scipy.sparserr&r2FTz*Matrix must be a row/column vector, got %rz>Matrix must be a row/column vector of size 2**nqubits, got: %rgcs g|]}td|>@dkqS)r&r)r9)rfx)r?r/r0rgsz#matrix_to_qubit..)r(rrshaperrrrrr rQreverserrrexpand) rUrNZmlistlenr6Zketr,rXr.Z qubit_arrayr/)r?r0rsD         cCs>ddlm}|}dd|D}t|dkr2dS||SdS)z Works by finding the eigenvectors and eigenvalues of the matrix. We know we can decompose rho by doing: sum(EigenVal*|Eigenvect>.N)Zsympy.physics.quantum.densityrsZ eigenvectsr')ZmatrsZeigenr-r/r/r0r s  rOcCs t||dS)zConverts an Add/Mul of Qubit objects into it's matrix representation This function is the inverse of ``matrix_to_qubit`` and is a shorthand for ``represent(qubit)``. )rN)r)rarNr/r/r0rsTcCst||}|dkrg}|r"|}t|j}tt|td}xDt|D]8}||dkrN|t t ||||t ||fqNW|St ddS)aPerform an ensemble measurement of all qubits. Parameters ========== qubit : Qubit, Add The qubit to measure. This can be any Qubit or a linear combination of them. format : str The format of the intermediate matrices to use. Possible values are ('sympy','numpy','scipy.sparse'). Currently only 'sympy' is implemented. Returns ======= result : list A list that consists of primitive states and their probabilities. Examples ======== >>> from sympy.physics.quantum.qubit import Qubit, measure_all >>> from sympy.physics.quantum.gate import H, X, Y, Z >>> from sympy.physics.quantum.qapply import qapply >>> c = H(0)*H(1)*Qubit('00') >>> c H(0)*H(1)*|00> >>> q = qapply(c) >>> measure_all(q) [(|00>, 1/4), (|01>, 1/4), (|10>, 1/4), (|11>, 1/4)] rOr2gz7This function can't handle non-sympy matrix formats yetN) r normalizedmaxrpr9mathrr appendrrr NotImplementedError)rarN normalizemZresultssizer6r?r/r/r0r!s"   (c Cst||}t|ttfr"t|f}|dkr|r6|}t||}g}xR|D]J}d}||j|d7}|dkrJ|r~t|} nt|} | | |fqJW|St ddS)aPerform a partial ensemble measure on the specified qubits. Parameters ========== qubits : Qubit The qubit to measure. This can be any Qubit or a linear combination of them. bits : tuple The qubits to measure. format : str The format of the intermediate matrices to use. Possible values are ('sympy','numpy','scipy.sparse'). Currently only 'sympy' is implemented. Returns ======= result : list A list that consists of primitive states and their probabilities. Examples ======== >>> from sympy.physics.quantum.qubit import Qubit, measure_partial >>> from sympy.physics.quantum.gate import H, X, Y, Z >>> from sympy.physics.quantum.qapply import qapply >>> c = H(0)*H(1)*Qubit('00') >>> c H(0)*H(1)*|00> >>> q = qapply(c) >>> measure_partial(q, (0,)) [(sqrt(2)*|00>/2 + sqrt(2)*|10>/2, 1/2), (sqrt(2)*|01>/2 + sqrt(2)*|11>/2, 1/2)] rOrz7This function can't handle non-sympy matrix formats yetN) rr(r rr9rt_get_possible_outcomesHrrwrx) rar>rNryrzpossible_outcomesoutputoutcomeZprob_of_outcomeZ next_matrixr/r/r0r")s*$     c Cszddl}t||}|dkrn|}t||}|}d}x<|D]*}||j|d7}||kr>t|Sq>WntddS)aPerform a partial oneshot measurement on the specified qubits. A oneshot measurement is equivalent to performing a measurement on a quantum system. This type of measurement does not return the probabilities like an ensemble measurement does, but rather returns *one* of the possible resulting states. The exact state that is returned is determined by picking a state randomly according to the ensemble probabilities. Parameters ---------- qubits : Qubit The qubit to measure. This can be any Qubit or a linear combination of them. bits : tuple The qubits to measure. format : str The format of the intermediate matrices to use. Possible values are ('sympy','numpy','scipy.sparse'). Currently only 'sympy' is implemented. Returns ------- result : Qubit The qubit that the system collapsed to upon measurement. rNrOz7This function can't handle non-sympy matrix formats yet)randomrrtr|r}rrx) rar>rNrrzr~ random_numberZ total_probrr/r/r0r#ts   c Cst|j}tt|dd}g}x,tdt|>D]}|td|dq4Wg}x|D]}|d|>qZWxVtd|D]F}d}x,tt|D]} ||| @r|| d7}qW|||||<q~W|S)aGet the possible states that can be produced in a measurement. Parameters ---------- m : Matrix The matrix representing the state of the system. bits : tuple, list Which bits will be measured. Returns ------- result : list The list of possible states which can occur given this measurement. These are un-normalized so we can derive the probability of finding this state by taking the inner product with itself r2g?r&r) rurpr9rvrr r'rwr) rzr>r{r6Zoutput_matricesr?Z bit_masksr:Ztruenessr_r/r/r0r|s   r|cCsddl}t|}|dkr|}|}d}d}x.|D]&}|||7}||krTP|d7}q6Wtt|ttt |j ddSt ddS)adPerform a oneshot ensemble measurement on all qubits. A oneshot measurement is equivalent to performing a measurement on a quantum system. This type of measurement does not return the probabilities like an ensemble measurement does, but rather returns *one* of the possible resulting states. The exact state that is returned is determined by picking a state randomly according to the ensemble probabilities. Parameters ---------- qubits : Qubit The qubit to measure. This can be any Qubit or a linear combination of them. format : str The format of the intermediate matrices to use. Possible values are ('sympy','numpy','scipy.sparse'). Currently only 'sympy' is implemented. Returns ------- result : Qubit The qubit that the system collapsed to upon measurement. rNrOr&r2g?z7This function can't handle non-sympy matrix formats yet) rrrtr rrr9rvrrurprx)rarNrrzrZtotalrXr?r/r/r0r$s  $)rO)rOT)rOT)rO)rO)5rDZ __future__rrrvrOrrrrrr Zsympy.core.basicr Zsympy.core.compatibilityr r r Zsympy.matricesrrZ sympy.printing.pretty.stringpictrZsympy.physics.quantum.hilbertrZsympy.physics.quantum.staterrrZsympy.physics.quantum.qexprrZsympy.physics.quantum.representrZ!sympy.physics.quantum.matrixutilsrrZmpmath.libmp.libintmathr__all__r%rrrerrrr rr!r"r#r|r$r/r/r/r0sN       K/7 G  8 K 03