Source code for sympy.physics.quantum.gate

"""An implementation of gates that act on qubits.

Gates are unitary operators that act on the space of qubits.

Medium Term Todo:

* Optimize Gate._apply_operators_Qubit to remove the creation of many
  intermediate Qubit objects.
* Add commutation relationships to all operators and use this in gate_sort.
* Fix gate_sort and gate_simp.
* Get multi-target UGates plotting properly.
* Get UGate to work with either sympy/numpy matrices and output either
  format. This should also use the matrix slots.
"""

from itertools import chain
import random

from sympy import Add, I, Integer, Matrix, Mul, Pow, sqrt, Tuple
from sympy.core.numbers import Number
from sympy.core.compatibility import is_sequence
from sympy.printing.pretty.stringpict import prettyForm, stringPict

from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.hilbert import ComplexSpace
from sympy.physics.quantum.operator import (UnitaryOperator, Operator,
                                            HermitianOperator)
from sympy.physics.quantum.matrixutils import matrix_tensor_product, matrix_eye
from sympy.physics.quantum.matrixcache import matrix_cache
from sympy.physics.quantum.dagger import Dagger

from sympy.matrices.matrices import MatrixBase

__all__ = [
    'Gate',
    'CGate',
    'UGate',
    'OneQubitGate',
    'TwoQubitGate',
    'IdentityGate',
    'HadamardGate',
    'XGate',
    'YGate',
    'ZGate',
    'TGate',
    'PhaseGate',
    'SwapGate',
    'CNotGate',
    # Aliased gate names
    'CNOT',
    'SWAP',
    'H',
    'X',
    'Y',
    'Z',
    'T',
    'S',
    'Phase',
    'normalized',
    'gate_sort',
    'gate_simp',
    'random_circuit',
]

#-----------------------------------------------------------------------------
# Gate Super-Classes
#-----------------------------------------------------------------------------

_normalized = True

[docs]def normalized(normalize): """Should Hadamard gates be normalized by a 1/sqrt(2). This is a global setting that can be used to simplify the look of various expressions, by leaving of the leading 1/sqrt(2) of the Hadamard gate. Parameters ---------- normalize : bool Should the Hadamard gate include the 1/sqrt(2) normalization factor? When True, the Hadamard gate will have the 1/sqrt(2). When False, the Hadamard gate will not have this factor. """ global _normalized _normalized = normalize
def _validate_targets_controls(tandc): tandc = list(tandc) # Check for integers for bit in tandc: if not bit.is_Integer and not bit.is_Symbol: raise TypeError('Integer expected, got: %r' % tandc[bit]) # Detect duplicates if len(list(set(tandc))) != len(tandc): raise QuantumError( 'Target/control qubits in a gate cannot be duplicated' )
[docs]class Gate(UnitaryOperator): """Non-controlled unitary gate operator that acts on qubits. This is a general abstract gate that needs to be subclassed to do anything useful. Parameters ---------- label : tuple, int A list of the target qubits (as ints) that the gate will apply to. Examples -------- """ _label_separator = ',' gate_name = u'G' gate_name_latex = u'G' #------------------------------------------------------------------------- # Initialization/creation #------------------------------------------------------------------------- @classmethod def _eval_args(cls, args): args = Tuple(*UnitaryOperator._eval_args(args)) _validate_targets_controls(args) return args @classmethod def _eval_hilbert_space(cls, args): """This returns the smallest possible Hilbert space.""" return ComplexSpace(2)**(max(args)+1) #------------------------------------------------------------------------- # Properties #------------------------------------------------------------------------- @property
[docs] def nqubits(self): """The total number of qubits this gate acts on. For controlled gate subclasses this includes both target and control qubits, so that, for examples the CNOT gate acts on 2 qubits. """ return len(self.targets)
@property
[docs] def min_qubits(self): """The minimum number of qubits this gate needs to act on.""" return max(self.targets)+1
@property
[docs] def targets(self): """A tuple of target qubits.""" return self.label
@property def gate_name_plot(self): return r'$%s$' % self.gate_name_latex #------------------------------------------------------------------------- # Gate methods #-------------------------------------------------------------------------
[docs] def get_target_matrix(self, format='sympy'): """The matrix rep. of the target part of the gate. Parameters ---------- format : str The format string ('sympy','numpy', etc.) """ raise NotImplementedError('get_target_matrix is not implemented in Gate.') #------------------------------------------------------------------------- # Apply #-------------------------------------------------------------------------
def _apply_operator_IntQubit(self, qubits, **options): """Redirect an apply from IntQubit to Qubit""" return self._apply_operator_Qubit(qubits, **options) def _apply_operator_Qubit(self, qubits, **options): """Apply this gate to a Qubit.""" # Check number of qubits this gate acts on. if qubits.nqubits < self.min_qubits: raise QuantumError( 'Gate needs a minimum of %r qubits to act on, got: %r' %\ (self.min_qubits, qubits.nqubits) ) # If the controls are not met, just return if isinstance(self, CGate): if not self.eval_controls(qubits): return qubits targets = self.targets target_matrix = self.get_target_matrix(format='sympy') # Find which column of the target matrix this applies to. column_index = 0 n = 1 for target in targets: column_index += n*qubits[target] n = n<<1 column = target_matrix[:,int(column_index)] # Now apply each column element to the qubit. result = 0 for index in range(column.rows): # TODO: This can be optimized to reduce the number of Qubit # creations. We should simply manipulate the raw list of qubit # values and then build the new Qubit object once. # Make a copy of the incoming qubits. new_qubit = qubits.__class__(*qubits.args) # Flip the bits that need to be flipped. for bit in range(len(targets)): if new_qubit[targets[bit]] != (index>>bit)&1: new_qubit = new_qubit.flip(targets[bit]) # The value in that row and column times the flipped-bit qubit # is the result for that part. result += column[index]*new_qubit return result #------------------------------------------------------------------------- # Represent #------------------------------------------------------------------------- def _represent_default_basis(self, **options): return self._represent_ZGate(None, **options) def _represent_ZGate(self, basis, **options): format = options.get('format','sympy') nqubits = options.get('nqubits',0) if nqubits == 0: raise QuantumError('The number of qubits must be given as nqubits.') # Make sure we have enough qubits for the gate. if nqubits < self.min_qubits: raise QuantumError( 'The number of qubits %r is too small for the gate.' % nqubits ) target_matrix = self.get_target_matrix(format) targets = self.targets if isinstance(self, CGate): controls = self.controls else: controls = [] m = represent_zbasis( controls, targets, target_matrix, nqubits, format ) return m #------------------------------------------------------------------------- # Print methods #------------------------------------------------------------------------- def _sympystr(self, printer, *args): label = self._print_label(printer, *args) return '%s(%s)' % (self.gate_name, label) def _pretty(self, printer, *args): a = stringPict(unicode(self.gate_name)) b = self._print_label_pretty(printer, *args) return self._print_subscript_pretty(a, b) def _latex(self, printer, *args): label = self._print_label(printer, *args) return '%s_{%s}' % (self.gate_name_latex, label) def plot_gate(self, axes, gate_idx, gate_grid, wire_grid): raise NotImplementedError('plot_gate is not implemented.')
[docs]class CGate(Gate): """A general unitary gate with control qubits. A general control gate applies a target gate to a set of targets if all of the control qubits have a particular values (set by ``CGate.control_value``). Parameters ---------- label : tuple The label in this case has the form (controls, gate), where controls is a tuple/list of control qubits (as ints) and gate is a ``Gate`` instance that is the target operator. Examples -------- """ gate_name = u'C' gate_name_latex = u'C' # The values this class controls for. control_value = Integer(1) #------------------------------------------------------------------------- # Initialization #------------------------------------------------------------------------- @classmethod def _eval_args(cls, args): # _eval_args has the right logic for the controls argument. controls = args[0] gate = args[1] if not is_sequence(controls): controls = (controls,) controls = UnitaryOperator._eval_args(controls) _validate_targets_controls(chain(controls,gate.targets)) return (Tuple(*controls), gate) @classmethod def _eval_hilbert_space(cls, args): """This returns the smallest possible Hilbert space.""" return ComplexSpace(2)**max(max(args[0])+1,args[1].min_qubits) #------------------------------------------------------------------------- # Properties #------------------------------------------------------------------------- @property
[docs] def nqubits(self): """The total number of qubits this gate acts on. For controlled gate subclasses this includes both target and control qubits, so that, for examples the CNOT gate acts on 2 qubits. """ return len(self.targets)+len(self.controls)
@property
[docs] def min_qubits(self): """The minimum number of qubits this gate needs to act on.""" return max(max(self.controls),max(self.targets))+1
@property
[docs] def targets(self): """A tuple of target qubits.""" return self.gate.targets
@property
[docs] def controls(self): """A tuple of control qubits.""" return tuple(self.label[0])
@property
[docs] def gate(self): """The non-controlled gate that will be applied to the targets.""" return self.label[1] #------------------------------------------------------------------------- # Gate methods #-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'): return self.gate.get_target_matrix(format)
[docs] def eval_controls(self, qubit): """Return True/False to indicate if the controls are satisfied.""" return all(qubit[bit]==self.control_value for bit in self.controls)
[docs] def decompose(self, **options): """Decompose the controlled gate into CNOT and single qubits gates.""" if len(self.controls) == 1: c = self.controls[0] t = self.gate.targets[0] if isinstance(self.gate, YGate): g1 = PhaseGate(t) g2 = CNotGate(c, t) g3 = PhaseGate(t) g4 = ZGate(t) return g1*g2*g3*g4 if isinstance(self.gate, ZGate): g1 = HadamardGate(t) g2 = CNotGate(c, t) g3 = HadamardGate(t) return g1*g2*g3 else: return self #------------------------------------------------------------------------- # Print methods #-------------------------------------------------------------------------
def _print_label(self, printer, *args): controls = self._print_sequence(self.controls, ',', printer, *args) gate = printer._print(self.gate, *args) return '(%s),%s' %\ (controls, gate) def _pretty(self, printer, *args): controls = self._print_sequence_pretty(self.controls, ',', printer, *args) gate = printer._print(self.gate) gate_name = stringPict(unicode(self.gate_name)) first = self._print_subscript_pretty(gate_name, controls) gate = self._print_parens_pretty(gate) final = prettyForm(*first.right((gate))) return final def _latex(self, printer, *args): controls = self._print_sequence(self.controls, ',', printer, *args) gate = printer._print(self.gate, *args) return r'%s_{%s}{\left(%s\right)}' %\ (self.gate_name_latex, controls, gate) def plot_gate(self, circ_plot, gate_idx): min_wire = int(min(chain(self.controls, self.targets))) max_wire = int(max(chain(self.controls, self.targets))) circ_plot.control_line(gate_idx, min_wire, max_wire) for c in self.controls: circ_plot.control_point(gate_idx, int(c)) self.gate.plot_gate(circ_plot, gate_idx) #------------------------------------------------------------------------- # Miscellaneous #------------------------------------------------------------------------- def _eval_dagger(self): if isinstance(self.gate, HermitianOperator): return self else: return Gate._eval_dagger(self) def _eval_inverse(self): if isinstance(self.gate, HermitianOperator): return self else: return Gate._eval_inverse(self) def _eval_power(self, exp): if isinstance(self.gate, HermitianOperator): if exp == -1: return Gate._eval_power(self, exp) elif abs(exp) % 2 == 0: return self*(Gate._eval_inverse(self)) else: return self else: return Gate._eval_power(self, exp)
[docs]class UGate(Gate): """General gate specified by a set of targets and a target matrix. Parameters ---------- label : tuple A tuple of the form (targets, U), where targets is a tuple of the target qubits and U is a unitary matrix with dimension of len(targets). """ gate_name = u'U' gate_name_latex = u'U' #------------------------------------------------------------------------- # Initialization #------------------------------------------------------------------------- @classmethod def _eval_args(cls, args): targets = args[0] if not is_sequence(targets): targets = (targets,) targets = Gate._eval_args(targets) _validate_targets_controls(targets) mat = args[1] if not isinstance(mat, MatrixBase): raise TypeError('Matrix expected, got: %r' % mat) dim = 2**len(targets) if not all(dim == shape for shape in mat.shape): raise IndexError( 'Number of targets must match the matrix size: %r %r' %\ (targets, mat) ) return (targets, mat) @classmethod def _eval_hilbert_space(cls, args): """This returns the smallest possible Hilbert space.""" return ComplexSpace(2)**(max(args[0])+1) #------------------------------------------------------------------------- # Properties #------------------------------------------------------------------------- @property
[docs] def targets(self): """A tuple of target qubits.""" return tuple(self.label[0]) #------------------------------------------------------------------------- # Gate methods #-------------------------------------------------------------------------
[docs] def get_target_matrix(self, format='sympy'): """The matrix rep. of the target part of the gate. Parameters ---------- format : str The format string ('sympy','numpy', etc.) """ return self.label[1] #------------------------------------------------------------------------- # Print methods #-------------------------------------------------------------------------
def _pretty(self, printer, *args): targets = self._print_sequence_pretty(self.targets, ',', printer, *args) gate_name = stringPict(unicode(self.gate_name)) return self._print_subscript_pretty(gate_name, targets) def _latex(self, printer, *args): targets = self._print_sequence(self.targets, ',', printer, *args) return r'%s_{%s}' % (self.gate_name_latex, targets) def plot_gate(self, circ_plot, gate_idx): circ_plot.one_qubit_box( self.gate_name_plot, gate_idx, int(self.targets[0]) )
[docs]class OneQubitGate(Gate): """A single qubit unitary gate base class.""" nqubits = Integer(1) def plot_gate(self, circ_plot, gate_idx): circ_plot.one_qubit_box( self.gate_name_plot, gate_idx, int(self.targets[0]) ) def _eval_commutator(self, other, **hints): if isinstance(other, OneQubitGate): if self.targets != other.targets or self.__class__ == other.__class__: return Integer(0) return Operator._eval_commutator(self, other, **hints) def _eval_anticommutator(self, other, **hints): if isinstance(other, OneQubitGate): if self.targets != other.targets or self.__class__ == other.__class__: return Integer(2)*self*other return Operator._eval_anticommutator(self, other, **hints)
[docs]class TwoQubitGate(Gate): """A two qubit unitary gate base class.""" nqubits = Integer(2) #----------------------------------------------------------------------------- # Single Qubit Gates #-----------------------------------------------------------------------------
[docs]class IdentityGate(OneQubitGate): """The single qubit identity gate. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'1' gate_name_latex = u'1' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('eye2', format) def _eval_commutator(self, other, **hints): return Integer(0) def _eval_anticommutator(self, other, **hints): return Integer(2)*other
[docs]class HadamardGate(HermitianOperator, OneQubitGate): """The single qubit Hadamard gate. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'H' gate_name_latex = u'H' def get_target_matrix(self, format='sympy'): if _normalized: return matrix_cache.get_matrix('H', format) else: return matrix_cache.get_matrix('Hsqrt2', format) def _eval_commutator_XGate(self, other, **hints): return I*sqrt(2)*YGate(self.targets[0]) def _eval_commutator_YGate(self, other, **hints): return I*sqrt(2)*(ZGate(self.targets[0])-XGate(self.targets[0])) def _eval_commutator_ZGate(self, other, **hints): return -I*sqrt(2)*YGate(self.targets[0]) def _eval_anticommutator_XGate(self, other, **hints): return sqrt(2)*IdentityGate(self.targets[0]) def _eval_anticommutator_YGate(self, other, **hints): return Integer(0) def _eval_anticommutator_ZGate(self, other, **hints): return sqrt(2)*IdentityGate(self.targets[0])
[docs]class XGate(HermitianOperator, OneQubitGate): """The single qubit X, or NOT, gate. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'X' gate_name_latex = u'X' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('X', format) def plot_gate(self, circ_plot, gate_idx): circ_plot.not_point( gate_idx, int(self.label[0]) ) def _eval_commutator_YGate(self, other, **hints): return Integer(2)*I*ZGate(self.targets[0]) def _eval_anticommutator_XGate(self, other, **hints): return Integer(2)*IdentityGate(self.targets[0]) def _eval_anticommutator_YGate(self, other, **hints): return Integer(0) def _eval_anticommutator_ZGate(self, other, **hints): return Integer(0)
[docs]class YGate(HermitianOperator, OneQubitGate): """The single qubit Y gate. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'Y' gate_name_latex = u'Y' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('Y', format) def _eval_commutator_ZGate(self, other, **hints): return Integer(2)*I*XGate(self.targets[0]) def _eval_anticommutator_YGate(self, other, **hints): return Integer(2)*IdentityGate(self.targets[0]) def _eval_anticommutator_ZGate(self, other, **hints): return Integer(0)
[docs]class ZGate(HermitianOperator, OneQubitGate): """The single qubit Z gate. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'Z' gate_name_latex = u'Z' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('Z', format) def _eval_commutator_XGate(self, other, **hints): return Integer(2)*I*YGate(self.targets[0]) def _eval_anticommutator_YGate(self, other, **hints): return Integer(0)
[docs]class PhaseGate(OneQubitGate): """The single qubit phase, or S, gate. This gate rotates the phase of the state by pi/2 if the state is ``|1>`` and does nothing if the state is ``|0>``. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'S' gate_name_latex = u'S' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('S', format) def _eval_commutator_ZGate(self, other, **hints): return Integer(0) def _eval_commutator_TGate(self, other, **hints): return Integer(0)
[docs]class TGate(OneQubitGate): """The single qubit pi/8 gate. This gate rotates the phase of the state by pi/4 if the state is ``|1>`` and does nothing if the state is ``|0>``. Parameters ---------- target : int The target qubit this gate will apply to. Examples -------- """ gate_name = u'T' gate_name_latex = u'T' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('T', format) def _eval_commutator_ZGate(self, other, **hints): return Integer(0) def _eval_commutator_PhaseGate(self, other, **hints): return Integer(0) # Aliases for gate names.
H = HadamardGate X = XGate Y = YGate Z = ZGate T = TGate Phase = S = PhaseGate #----------------------------------------------------------------------------- # 2 Qubit Gates #-----------------------------------------------------------------------------
[docs]class CNotGate(HermitianOperator, CGate, TwoQubitGate): """Two qubit controlled-NOT. This gate performs the NOT or X gate on the target qubit if the control qubits all have the value 1. Parameters ---------- label : tuple A tuple of the form (control, target). Examples -------- """ gate_name = 'CNOT' gate_name_latex = u'CNOT' #------------------------------------------------------------------------- # Initialization #------------------------------------------------------------------------- @classmethod def _eval_args(cls, args): args = Gate._eval_args(args) return args @classmethod def _eval_hilbert_space(cls, args): """This returns the smallest possible Hilbert space.""" return ComplexSpace(2)**(max(args)+1) #------------------------------------------------------------------------- # Properties #------------------------------------------------------------------------- @property
[docs] def min_qubits(self): """The minimum number of qubits this gate needs to act on.""" return max(self.label)+1
@property
[docs] def targets(self): """A tuple of target qubits.""" return (self.label[1],)
@property
[docs] def controls(self): """A tuple of control qubits.""" return (self.label[0],)
@property
[docs] def gate(self): """The non-controlled gate that will be applied to the targets.""" return XGate(self.label[1]) #------------------------------------------------------------------------- # Properties #------------------------------------------------------------------------- # The default printing of Gate works better than those of CGate, so we # go around the overridden methods in CGate.
def _print_label(self, printer, *args): return Gate._print_label(self, printer, *args) def _pretty(self, printer, *args): return Gate._pretty(self, printer, *args) def _latex(self, printer, *args): return Gate._latex(self, printer, *args) #------------------------------------------------------------------------- # Commutator/AntiCommutator #------------------------------------------------------------------------- def _eval_commutator_ZGate(self, other, **hints): """[CNOT(i, j), Z(i)] == 0.""" if self.controls[0] == other.targets[0]: return Integer(0) else: raise NotImplementedError('Commutator not implemented: %r' % other) def _eval_commutator_TGate(self, other, **hints): """[CNOT(i, j), T(i)] == 0.""" return self._eval_commutator_ZGate(other, **hints) def _eval_commutator_PhaseGate(self, other, **hints): """[CNOT(i, j), S(i)] == 0.""" return self._eval_commutator_ZGate(other, **hints) def _eval_commutator_XGate(self, other, **hints): """[CNOT(i, j), X(j)] == 0.""" if self.targets[0] == other.targets[0]: return Integer(0) else: raise NotImplementedError('Commutator not implemented: %r' % other) def _eval_commutator_CNotGate(self, other, **hints): """[CNOT(i, j), CNOT(i,k)] == 0.""" if self.controls[0] == other.controls[0]: return Integer(0) else: raise NotImplementedError('Commutator not implemented: %r' % other)
[docs]class SwapGate(TwoQubitGate): """Two qubit SWAP gate. This gate swap the values of the two qubits. Parameters ---------- label : tuple A tuple of the form (target1, target2). Examples -------- """ gate_name = 'SWAP' gate_name_latex = u'SWAP' def get_target_matrix(self, format='sympy'): return matrix_cache.get_matrix('SWAP', format)
[docs] def decompose(self, **options): """Decompose the SWAP gate into CNOT gates.""" i,j = self.targets[0], self.targets[1] g1 = CNotGate(i, j) g2 = CNotGate(j, i) return g1*g2*g1
def plot_gate(self, circ_plot, gate_idx): min_wire = int(min(self.targets)) max_wire = int(max(self.targets)) circ_plot.control_line(gate_idx, min_wire, max_wire) circ_plot.swap_point(gate_idx, min_wire) circ_plot.swap_point(gate_idx, max_wire) def _represent_ZGate(self, basis, **options): """Represent the SWAP gate in the computational basis. The following representation is used to compute this: SWAP = |1><1|x|1><1| + |0><0|x|0><0| + |1><0|x|0><1| + |0><1|x|1><0| """ format = options.get('format', 'sympy') targets = [int(t) for t in self.targets] min_target = min(targets) max_target = max(targets) nqubits = options.get('nqubits',self.min_qubits) op01 = matrix_cache.get_matrix('op01', format) op10 = matrix_cache.get_matrix('op10', format) op11 = matrix_cache.get_matrix('op11', format) op00 = matrix_cache.get_matrix('op00', format) eye2 = matrix_cache.get_matrix('eye2', format) result = None for i, j in ((op01,op10),(op10,op01),(op00,op00),(op11,op11)): product = nqubits*[eye2] product[nqubits-min_target-1] = i product[nqubits-max_target-1] = j new_result = matrix_tensor_product(*product) if result is None: result = new_result else: result = result + new_result return result # Aliases for gate names.
CNOT = CNotGate SWAP = SwapGate #----------------------------------------------------------------------------- # Represent #----------------------------------------------------------------------------- def represent_zbasis(controls, targets, target_matrix, nqubits, format='sympy'): """Represent a gate with controls, targets and target_matrix. This function does the low-level work of representing gates as matrices in the standard computational basis (ZGate). Currently, we support two main cases: 1. One target qubit and no control qubits. 2. One target qubits and multiple control qubits. For the base of multiple controls, we use the following expression [1]: 1_{2**n} + (|1><1|)^{(n-1)} x (target-matrix - 1_{2}) Parameters ---------- controls : list, tuple A sequence of control qubits. targets : list, tuple A sequence of target qubits. target_matrix : sympy.Matrix, numpy.matrix, scipy.sparse The matrix form of the transformation to be performed on the target qubits. The format of this matrix must match that passed into the `format` argument. nqubits : int The total number of qubits used for the representation. format : str The format of the final matrix ('sympy', 'numpy', 'scipy.sparse'). Examples -------- References ---------- [1] http://www.johnlapeyre.com/qinf/qinf_html/node6.html. """ controls = [int(x) for x in controls] targets = [int(x) for x in targets] nqubits = int(nqubits) # This checks for the format as well. op11 = matrix_cache.get_matrix('op11', format) eye2 = matrix_cache.get_matrix('eye2', format) # Plain single qubit case if len(controls) == 0 and len(targets) == 1: product = [] bit = targets[0] # Fill product with [I1,Gate,I2] such that the unitaries, # I, cause the gate to be applied to the correct Qubit if bit != nqubits-1: product.append(matrix_eye(2**(nqubits-bit-1), format=format)) product.append(target_matrix) if bit != 0: product.append(matrix_eye(2**bit, format=format)) return matrix_tensor_product(*product) # Single target, multiple controls. elif len(targets) == 1 and len(controls) >= 1: target = targets[0] # Build the non-trivial part. product2 = [] for i in range(nqubits): product2.append(matrix_eye(2, format=format)) for control in controls: product2[nqubits-1-control] = op11 product2[nqubits-1-target] = target_matrix - eye2 return matrix_eye(2**nqubits, format=format) +\ matrix_tensor_product(*product2) # Multi-target, multi-control is not yet implemented. else: raise NotImplementedError( 'The representation of multi-target, multi-control gates ' 'is not implemented.' ) #----------------------------------------------------------------------------- # Gate manipulation functions. #-----------------------------------------------------------------------------
[docs]def gate_simp(circuit): """Simplifies gates symbolically It first sorts gates using gate_sort. It then applies basic simplification rules to the circuit, e.g., XGate**2 = Identity """ # Bubble sort out gates that commute. circuit = gate_sort(circuit) # Do simplifications by subing a simplification into the first element # which can be simplified. We recursively call gate_simp with new circuit # as input more simplifications exist. if isinstance(circuit, Add): return sum(gate_simp(t) for t in circuit.args) elif isinstance(circuit, Mul): circuit_args = circuit.args elif isinstance(circuit, Pow): b, e = circuit.as_base_exp() circuit_args = (gate_simp(b)**e,) else: return circuit # Iterate through each element in circuit, simplify if possible. for i in xrange(len(circuit_args)): # H,X,Y or Z squared is 1. # T**2 = S, S**2 = Z if isinstance(circuit_args[i], Pow): if isinstance(circuit_args[i].base, (HadamardGate, XGate, YGate, ZGate))\ and isinstance(circuit_args[i].exp, Number): # Build a new circuit taking replacing the # H,X,Y,Z squared with one. newargs = (circuit_args[:i] +\ (circuit_args[i].base**(circuit_args[i].exp % 2),) +\ circuit_args[i+1:]) # Recursively simplify the new circuit. circuit = gate_simp(Mul(*newargs)) break elif isinstance(circuit_args[i].base, PhaseGate): # Build a new circuit taking old circuit but splicing # in simplification. newargs = circuit_args[:i] # Replace PhaseGate**2 with ZGate. newargs = newargs + (ZGate(circuit_args[i].base.args[0])**\ (Integer(circuit_args[i].exp/2)), circuit_args[i].base**\ (circuit_args[i].exp % 2)) # Append the last elements. newargs = newargs + circuit_args[i+1:] # Recursively simplify the new circuit. circuit = gate_simp(Mul(*newargs)) break elif isinstance(circuit_args[i].base, TGate): # Build a new circuit taking all the old elements. newargs = circuit_args[:i] # Put an Phasegate in place of any TGate**2. newargs = newargs + (PhaseGate(circuit_args[i].base.args[0])**\ Integer(circuit_args[i].exp/2), circuit_args[i].base**\ (circuit_args[i].exp % 2)) # Append the last elements. newargs = newargs + circuit_args[i+1:] # Recursively simplify the new circuit. circuit = gate_simp(Mul(*newargs)) break return circuit
[docs]def gate_sort(circuit): """Sorts the gates while keeping track of commutation relations This function uses a bubble sort to rearrange the order of gate application. Keeps track of Quantum computations special commutation relations (e.g. things that apply to the same Qubit do not commute with each other) circuit is the Mul of gates that are to be sorted. """ # Make sure we have an Add or Mul. if isinstance(circuit, Add): return sum(gate_sort(t) for t in circuit.args) if isinstance(circuit, Pow): return gate_sort(circuit.base)**circuit.exp elif isinstance(circuit, Gate): return circuit if not isinstance(circuit, Mul): return circuit changes = True while changes: changes = False circ_array = circuit.args for i in xrange(len(circ_array)-1): # Go through each element and switch ones that are in wrong order if isinstance(circ_array[i], (Gate, Pow)) and\ isinstance(circ_array[i+1], (Gate, Pow)): # If we have a Pow object, look at only the base first_base, first_exp = circ_array[i].as_base_exp() second_base, second_exp = circ_array[i+1].as_base_exp() # Use sympy's hash based sorting. This is not mathematical # sorting, but is rather based on comparing hashes of objects. # See Basic.compare for details. if first_base.compare(second_base) > 0: if Commutator(first_base, second_base).doit() == 0: new_args = (circuit.args[:i] + (circuit.args[i+1],) +\ (circuit.args[i],) + circuit.args[i+2:]) circuit = Mul(*new_args) circ_array = circuit.args changes = True break if AntiCommutator(first_base, second_base).doit() == 0: new_args = (circuit.args[:i] + (circuit.args[i+1],) +\ (circuit.args[i],) + circuit.args[i+2:]) sign = Integer(-1)**(first_exp*second_exp) circuit = sign*Mul(*new_args) circ_array = circuit.args changes = True break return circuit #----------------------------------------------------------------------------- # Utility functions #-----------------------------------------------------------------------------
[docs]def random_circuit(ngates, nqubits, gate_space=(X, Y, Z, S, T, H, CNOT, SWAP)): """Return a random circuit of ngates and nqubits. This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP) gates. Parameters ---------- ngates : int The number of gates in the circuit. nqubits : int The number of qubits in the circuit. gate_space : tuple A tuple of the gate classes that will be used in the circuit. Repeating gate classes multiple times in this tuple will increase the frequency they appear in the random circuit. """ qubit_space = range(nqubits) result = [] for i in xrange(ngates): g = random.choice(gate_space) if g == CNotGate or g == SwapGate: qubits = random.sample(qubit_space,2) g = g(*qubits) else: qubit = random.choice(qubit_space) g = g(qubit) result.append(g) return Mul(*result)
def zx_basis_transform(self, format='sympy'): """Transformation matrix from Z to X basis.""" return matrix_cache.get_matrix('ZX', format) def zy_basis_transform(self, format='sympy'): """Transformation matrix from Z to Y basis.""" return matrix_cache.get_matrix('ZY', format)