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Source code for qiskit.quantum_info.states.stabilizerstate

# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2021.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.

"""
Stabilizer state class.
"""

import numpy as np

from qiskit.exceptions import QiskitError
from qiskit.quantum_info.operators.op_shape import OpShape
from qiskit.quantum_info.operators.symplectic import Clifford, Pauli, PauliList
from qiskit.quantum_info.operators.symplectic.clifford_circuits import _append_x
from qiskit.quantum_info.states.quantum_state import QuantumState


[docs]class StabilizerState(QuantumState): """StabilizerState class. Stabilizer simulator using the convention from reference [1]. Based on the internal class :class:`~qiskit.quantum_info.Clifford`. .. code-block:: from qiskit import QuantumCircuit from qiskit.quantum_info import StabilizerState, Pauli # Bell state generation circuit qc = QuantumCircuit(2) qc.h(0) qc.cx(0, 1) stab = StabilizerState(qc) # Print the StabilizerState print(stab) # Calculate the StabilizerState measurement probabilities dictionary print (stab.probabilities_dict()) # Calculate expectation value of the StabilizerState print (stab.expectation_value(Pauli('ZZ'))) .. parsed-literal:: StabilizerState(StabilizerTable: ['+XX', '+ZZ']) {'00': 0.5, '11': 0.5} 1 References: 1. S. Aaronson, D. Gottesman, *Improved Simulation of Stabilizer Circuits*, Phys. Rev. A 70, 052328 (2004). `arXiv:quant-ph/0406196 <https://arxiv.org/abs/quant-ph/0406196>`_ """ def __init__(self, data, validate=True): """Initialize a StabilizerState object. Args: data (StabilizerState or Clifford or Pauli or QuantumCircuit or qiskit.circuit.Instruction): Data from which the stabilizer state can be constructed. validate (boolean): validate that the stabilizer state data is a valid Clifford. """ # Initialize from another StabilizerState if isinstance(data, StabilizerState): self._data = data._data # Initialize from a Pauli elif isinstance(data, Pauli): self._data = Clifford(data.to_instruction()) # Initialize from a Clifford, QuantumCircuit or Instruction else: self._data = Clifford(data, validate) # Initialize super().__init__(op_shape=OpShape.auto(num_qubits_r=self._data.num_qubits, num_qubits_l=0)) def __eq__(self, other): return (self._data.stab == other._data.stab).all() def __repr__(self): return f"StabilizerState({self._data.stabilizer})" @property def clifford(self): """Return StabilizerState Clifford data""" return self._data
[docs] def is_valid(self, atol=None, rtol=None): """Return True if a valid StabilizerState.""" return self._data.is_unitary()
def _add(self, other): raise NotImplementedError(f"{type(self)} does not support addition") def _multiply(self, other): raise NotImplementedError(f"{type(self)} does not support scalar multiplication")
[docs] def trace(self): """Return the trace of the stabilizer state as a density matrix, which equals to 1, since it is always a pure state. Returns: double: the trace (should equal 1). Raises: QiskitError: if input is not a StabilizerState. """ if not self.is_valid(): raise QiskitError("StabilizerState is not a valid quantum state.") return 1.0
[docs] def purity(self): """Return the purity of the quantum state, which equals to 1, since it is always a pure state. Returns: double: the purity (should equal 1). Raises: QiskitError: if input is not a StabilizerState. """ if not self.is_valid(): raise QiskitError("StabilizerState is not a valid quantum state.") return 1.0
[docs] def to_operator(self): """Convert state to matrix operator class""" return Clifford(self.clifford).to_operator()
[docs] def conjugate(self): """Return the conjugate of the operator.""" ret = self.copy() ret._data = ret._data.conjugate() return ret
[docs] def tensor(self, other): """Return the tensor product stabilzier state self ⊗ other. Args: other (StabilizerState): a stabilizer state object. Returns: StabilizerState: the tensor product operator self ⊗ other. Raises: QiskitError: if other is not a StabilizerState. """ if not isinstance(other, StabilizerState): other = StabilizerState(other) ret = self.copy() ret._data = self.clifford.tensor(other.clifford) return ret
[docs] def expand(self, other): """Return the tensor product stabilzier state other ⊗ self. Args: other (StabilizerState): a stabilizer state object. Returns: StabilizerState: the tensor product operator other ⊗ self. Raises: QiskitError: if other is not a StabilizerState. """ if not isinstance(other, StabilizerState): other = StabilizerState(other) ret = self.copy() ret._data = self.clifford.expand(other.clifford) return ret
[docs] def evolve(self, other, qargs=None): """Evolve a stabilizer state by a Clifford operator. Args: other (Clifford or QuantumCircuit or qiskit.circuit.Instruction): The Clifford operator to evolve by. qargs (list): a list of stabilizer subsystem positions to apply the operator on. Returns: StabilizerState: the output stabilizer state. Raises: QiskitError: if other is not a StabilizerState. QiskitError: if the operator dimension does not match the specified StabilizerState subsystem dimensions. """ if not isinstance(other, StabilizerState): other = StabilizerState(other) ret = self.copy() ret._data = self.clifford.compose(other.clifford, qargs=qargs) return ret
[docs] def expectation_value(self, oper, qargs=None): """Compute the expectation value of a Pauli operator. Args: oper (Pauli): a Pauli operator to evaluate expval. qargs (None or list): subsystems to apply the operator on. Returns: complex: the expectation value (only 0 or 1 or -1 or i or -i). Raises: QiskitError: if oper is not a Pauli operator. """ if not isinstance(oper, Pauli): raise QiskitError("Operator for expectation value is not a Pauli operator.") num_qubits = self.clifford.num_qubits if qargs is None: qubits = range(num_qubits) else: qubits = qargs # Construct Pauli on num_qubits pauli = Pauli(num_qubits * "I") phase = 0 pauli_phase = (-1j) ** oper.phase if oper.phase else 1 for pos, qubit in enumerate(qubits): pauli.x[qubit] = oper.x[pos] pauli.z[qubit] = oper.z[pos] phase += pauli.x[qubit] & pauli.z[qubit] # Check if there is a stabilizer that anti-commutes with an odd number of qubits # If so the expectation value is 0 for p in range(num_qubits): num_anti = 0 num_anti += np.count_nonzero(pauli.z & self.clifford.stab_x[p]) num_anti += np.count_nonzero(pauli.x & self.clifford.stab_z[p]) if num_anti % 2 == 1: return 0 # Otherwise pauli is (-1)^a prod_j S_j^b_j for Clifford stabilizers # If pauli anti-commutes with D_j then b_j = 1. # Multiply pauli by stabilizers with anti-commuting destabilizers pauli_z = (pauli.z).copy() # Make a copy of pauli.z for p in range(num_qubits): # Check if destabilizer anti-commutes num_anti = 0 num_anti += np.count_nonzero(pauli.z & self.clifford.destab_x[p]) num_anti += np.count_nonzero(pauli.x & self.clifford.destab_z[p]) if num_anti % 2 == 0: continue # If anti-commutes multiply Pauli by stabilizer phase += 2 * self.clifford.stab_phase[p] phase += np.count_nonzero(self.clifford.stab_z[p] & self.clifford.stab_x[p]) phase += 2 * np.count_nonzero(pauli_z & self.clifford.stab_x[p]) pauli_z = pauli_z ^ self.clifford.stab_z[p] # For valid stabilizers, `phase` can only be 0 (= 1) or 2 (= -1) at this point. if phase % 4 != 0: return -pauli_phase return pauli_phase
[docs] def equiv(self, other): """Return True if the two generating sets generate the same stabilizer group. Args: other (StabilizerState): another StabilizerState. Returns: bool: True if other has a generating set that generates the same StabilizerState. """ if not isinstance(other, StabilizerState): try: other = StabilizerState(other) except QiskitError: return False num_qubits = self.num_qubits if other.num_qubits != num_qubits: return False pauli_orig = PauliList.from_symplectic( self._data.stab_z, self._data.stab_x, 2 * self._data.stab_phase ) pauli_other = PauliList.from_symplectic( other._data.stab_z, other._data.stab_x, 2 * other._data.stab_phase ) # Check that each stabilizer from the original set commutes with each stabilizer # from the other set if not np.all([pauli.commutes(pauli_other) for pauli in pauli_orig]): return False # Compute the expected value of each stabilizer from the original set on the stabilizer state # determined by the other set. The two stabilizer states coincide if and only if the # expected value is +1 for each stabilizer for i in range(num_qubits): exp_val = self.expectation_value(pauli_other[i]) if exp_val != 1: return False return True
[docs] def probabilities(self, qargs=None, decimals=None): """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. """ probs_dict = self.probabilities_dict(qargs, decimals) if qargs is None: qargs = range(self.clifford.num_qubits) probs = np.zeros(2 ** len(qargs)) for key, value in probs_dict.items(): place = int(key, 2) probs[place] = value return probs
[docs] def probabilities_dict(self, qargs=None, decimals=None): """Return the subsystem measurement probability dictionary. Measurement probabilities are with respect to measurement in the computation (diagonal) basis. 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: 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: dict: The measurement probabilities in dict (ket) form. """ if qargs is None: qubits = range(self.clifford.num_qubits) else: qubits = qargs outcome = ["X"] * len(qubits) outcome_prob = 1.0 probs = {} # probabilities dictionary self._get_probablities(qubits, outcome, outcome_prob, probs) if decimals is not None: for key, value in probs.items(): probs[key] = round(value, decimals) return probs
[docs] def reset(self, qargs=None): """Reset 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: StabilizerState: the reset state. Additional Information: If all subsystems are reset this will return the ground state on all subsystems. If only some subsystems are reset this function will perform a measurement on those subsystems and evolve the subsystems so that the collapsed post-measurement states are rotated to the 0-state. The RNG seed for this sampling can be set using the :meth:`seed` method. """ # Resetting all qubits does not require sampling or RNG if qargs is None: return StabilizerState(Clifford(np.eye(2 * self.clifford.num_qubits))) randbits = self._rng.integers(2, size=len(qargs)) ret = self.copy() for bit, qubit in enumerate(qargs): # Apply measurement and get classical outcome outcome = ret._measure_and_update(qubit, randbits[bit]) # Use the outcome to apply X gate to any qubits left in the # |1> state after measure, then discard outcome. if outcome == 1: _append_x(ret.clifford, qubit) return ret
[docs] def measure(self, qargs=None): """Measure subsystems and return outcome and post-measure state. Note that this function uses the QuantumStates internal random number generator for sampling the measurement outcome. The RNG seed can be set using the :meth:`seed` method. Args: qargs (list or None): subsystems to sample measurements for, if None sample measurement of all subsystems (Default: None). Returns: tuple: the pair ``(outcome, state)`` where ``outcome`` is the measurement outcome string label, and ``state`` is the collapsed post-measurement stabilizer state for the corresponding outcome. """ if qargs is None: qargs = range(self.clifford.num_qubits) randbits = self._rng.integers(2, size=len(qargs)) ret = self.copy() outcome = "" for bit, qubit in enumerate(qargs): outcome = str(ret._measure_and_update(qubit, randbits[bit])) + outcome return outcome, ret
[docs] def sample_memory(self, shots, qargs=None): """Sample a list of qubit measurement outcomes in the computational basis. Args: shots (int): number of samples to generate. qargs (None or list): subsystems to sample measurements for, if None sample measurement of all subsystems (Default: None). Returns: np.array: list of sampled counts if the order sampled. Additional Information: This function implements the measurement :meth:`measure` method. The seed for random number generator used for sampling can be set to a fixed value by using the stats :meth:`seed` method. """ memory = [] for _ in range(shots): # copy the StabilizerState since measure updates it stab = self.copy() memory.append(stab.measure(qargs)[0]) return memory
# ----------------------------------------------------------------------- # Helper functions for calculating the measurement # ----------------------------------------------------------------------- def _measure_and_update(self, qubit, randbit): """Measure a single qubit and return outcome and post-measure state. Note that this function uses the QuantumStates internal random number generator for sampling the measurement outcome. The RNG seed can be set using the :meth:`seed` method. Note that stabilizer state measurements only have three probabilities: (p0, p1) = (0.5, 0.5), (1, 0), or (0, 1) The random case happens if there is a row anti-commuting with Z[qubit] """ num_qubits = self.clifford.num_qubits clifford = self.clifford stab_x = self.clifford.stab_x # Check if there exists stabilizer anticommuting with Z[qubit] # in this case the measurement outcome is random z_anticommuting = np.any(stab_x[:, qubit]) if z_anticommuting == 0: # Deterministic outcome - measuring it will not change the StabilizerState aux_pauli = Pauli(num_qubits * "I") for i in range(num_qubits): if clifford.x[i][qubit]: aux_pauli = self._rowsum_deterministic(clifford, aux_pauli, i + num_qubits) outcome = aux_pauli.phase return outcome else: # Non-deterministic outcome outcome = randbit p_qubit = np.min(np.nonzero(stab_x[:, qubit])) p_qubit += num_qubits # Updating the StabilizerState for i in range(2 * num_qubits): # the last condition is not in the AG paper but we seem to need it if (clifford.x[i][qubit]) and (i != p_qubit) and (i != (p_qubit - num_qubits)): self._rowsum_nondeterministic(clifford, i, p_qubit) clifford.destab[p_qubit - num_qubits] = clifford.stab[p_qubit - num_qubits].copy() clifford.x[p_qubit] = np.zeros(num_qubits) clifford.z[p_qubit] = np.zeros(num_qubits) clifford.z[p_qubit][qubit] = True clifford.phase[p_qubit] = outcome return outcome @staticmethod def _phase_exponent(x1, z1, x2, z2): """Exponent g of i such that Pauli(x1,z1) * Pauli(x2,z2) = i^g Pauli(x1+x2,z1+z2)""" # pylint: disable=invalid-name phase = (x2 * z1 * (1 + 2 * z2 + 2 * x1) - x1 * z2 * (1 + 2 * z1 + 2 * x2)) % 4 if phase < 0: phase += 4 # now phase in {0, 1, 3} if phase == 2: raise QiskitError("Invalid rowsum phase exponent in measurement calculation.") return phase @staticmethod def _rowsum(accum_pauli, accum_phase, row_pauli, row_phase): """Aaronson-Gottesman rowsum helper function""" newr = 2 * row_phase + 2 * accum_phase for qubit in range(row_pauli.num_qubits): newr += StabilizerState._phase_exponent( row_pauli.x[qubit], row_pauli.z[qubit], accum_pauli.x[qubit], accum_pauli.z[qubit] ) newr %= 4 if (newr != 0) & (newr != 2): raise QiskitError("Invalid rowsum in measurement calculation.") accum_phase = int(newr == 2) accum_pauli.x ^= row_pauli.x accum_pauli.z ^= row_pauli.z return accum_pauli, accum_phase @staticmethod def _rowsum_nondeterministic(clifford, accum, row): """Updating StabilizerState Clifford in the non-deterministic rowsum calculation. row and accum are rows in the StabilizerState Clifford.""" row_phase = clifford.phase[row] accum_phase = clifford.phase[accum] z = clifford.z x = clifford.x row_pauli = Pauli((z[row], x[row])) accum_pauli = Pauli((z[accum], x[accum])) accum_pauli, accum_phase = StabilizerState._rowsum( accum_pauli, accum_phase, row_pauli, row_phase ) clifford.phase[accum] = accum_phase x[accum] = accum_pauli.x z[accum] = accum_pauli.z @staticmethod def _rowsum_deterministic(clifford, aux_pauli, row): """Updating an auxilary Pauli aux_pauli in the deterministic rowsum calculation. The StabilizerState itself is not updated.""" row_phase = clifford.phase[row] accum_phase = aux_pauli.phase accum_pauli = aux_pauli row_pauli = Pauli((clifford.z[row], clifford.x[row])) accum_pauli, accum_phase = StabilizerState._rowsum( accum_pauli, accum_phase, row_pauli, row_phase ) aux_pauli = accum_pauli aux_pauli.phase = accum_phase return aux_pauli # ----------------------------------------------------------------------- # Helper functions for calculating the probabilities # ----------------------------------------------------------------------- def _get_probablities(self, qubits, outcome, outcome_prob, probs): """Recursive helper function for calculating the probabilities""" qubit_for_branching = -1 ret = self.copy() for i in range(len(qubits)): qubit = qubits[len(qubits) - i - 1] if outcome[i] == "X": is_deterministic = not any(ret.clifford.stab_x[:, qubit]) if is_deterministic: single_qubit_outcome = ret._measure_and_update(qubit, 0) if single_qubit_outcome: outcome[i] = "1" else: outcome[i] = "0" else: qubit_for_branching = i if qubit_for_branching == -1: str_outcome = "".join(outcome) probs[str_outcome] = outcome_prob return for single_qubit_outcome in range(0, 2): new_outcome = outcome.copy() if single_qubit_outcome: new_outcome[qubit_for_branching] = "1" else: new_outcome[qubit_for_branching] = "0" stab_cpy = ret.copy() stab_cpy._measure_and_update( qubits[len(qubits) - qubit_for_branching - 1], single_qubit_outcome ) stab_cpy._get_probablities(qubits, new_outcome, 0.5 * outcome_prob, probs)