# This code is part of Qiskit.
#
# (C) Copyright IBM 2022.
#
# 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.
"""
ZZ Ramsey experiment
"""
from typing import List, Tuple, Union
import numpy as np
from qiskit import QuantumCircuit
from qiskit.providers.backend import Backend
from qiskit.circuit import Parameter, ParameterExpression
from qiskit_experiments.framework import BackendTiming, BaseExperiment, Options
from .analysis.zz_ramsey_analysis import ZZRamseyAnalysis
[docs]
class ZZRamsey(BaseExperiment):
r"""An experiment to characterize the static :math:`ZZ` interaction for a qubit pair.
# section: overview
This experiment assumes a two qubit Hamiltonian of the form
.. math::
H = h \left(\frac{f_0}{2} ZI + \frac{f_1}{2} IZ + \frac{f_{ZZ}}{4} ZZ\right)
and measures the strength :math:`f_{ZZ}` of the :math:`ZZ` term.
:math:`f_{ZZ}` can be described as the difference between the frequency
of qubit 0 when qubit 1 is excited and the frequency of qubit 0 when
qubit 1 is in the ground state. Because :math:`f_{ZZ}` is symmetric in
qubit index, it can also be expressed with the roles of 0 and 1
reversed. Experimentally, we measure :math:`f_{ZZ}` by performing
Ramsey sequences on qubit 0 with qubit 1 in the ground state and again
with qubit 1 in the excited state. The standard Ramsey experiment
consists of putting a qubit along the :math:`X` axis of Bloch sphere,
waiting for some time, and then measuring the qubit project along
:math:`X`. By measuring the :math:`X` projection versus time the qubit
frequency can be inferred. See
:class:`~qiskit_experiments.library.characterization.T2Ramsey` and
:class:`~qiskit_experiments.library.characterization.RamseyXY`.
Because we are interested in the difference in qubit 0 frequency
between the two qubit 1 preparations rather than the absolute
frequencies of qubit 0 for those preparations, we modify the Ramsey
sequences (the circuits for the modified sequences are shown below).
First, we add an X gate on qubit 0 to the middle of the Ramsey delay.
This would have the effect of echoing out the phase accumulation of
qubit 0 (like a Hahn echo sequence as used in
:class:`~qiskit_experiments.library.characterization.T2Hahn`), but we
add a simultaneous X gate to qubit 1 as well. Flipping qubit 1 inverts
the sign of the :math:`f_{ZZ}` term. The net result is that qubit 0
continues to accumulate phase proportional to :math:`f_{ZZ}` while the
phase due to any ZI term is canceled out. This technique allows
:math:`f_{ZZ}` to be measured using longer delay times than might
otherwise be possible with a qubit with a slow frequency drift (i.e.
the measurement is not sensitive to qubit frequency drift from shot to
shot, only to drift within a single shot).
The resulting excited state population of qubit 0 versus delay time
exhibits slow sinusoidal oscillations (assuming :math:`f_{ZZ}` is
relatively small). To help with distinguishing between qubit decay and
a slow oscillation, an extra Z rotation is applied before the final
pulse on qubit 0. The angle of this Z rotation is set proportional to
the delay time of the sequence. This angle proportional to time behaves
similarly to measuring at a fixed angle with the qubit rotating at a
constant frequency. This virtual frequency is common to the two qubit 1
preparations. By looking at the difference in frequency fitted for the
two cases, this virtual frequency (called :math:`f` in the circuits
shown below) is removed, leaving only the :math:`f_{ZZ}` value. The
value of :math:`f` in terms of the experiment options is
``num_rotations / (max(delays) - min(delays))``.
This experiment consists of the following two circuits repeated with
different ``delay`` values.
.. parsed-literal::
Modified Ramsey sequence with qubit 1 initially in the ground state
┌────┐ ░ ┌─────────────────┐ ░ ┌───┐ ░ ┌─────────────────┐ ░ »
q_0: ┤ √X ├─░─┤ Delay(delay[s]) ├─░─┤ X ├─░─┤ Delay(delay[s]) ├─░─»
└────┘ ░ └─────────────────┘ ░ ├───┤ ░ └─────────────────┘ ░ »
q_1: ───────░─────────────────────░─┤ X ├─░─────────────────────░─»
░ ░ └───┘ ░ ░ »
c: 1/═════════════════════════════════════════════════════════════»
»
« ┌─────────────────────┐┌────┐ ░ ┌─┐
«q_0: ┤ Rz(4*delay*dt*f*pi) ├┤ √X ├─░─┤M├
« └────────┬───┬────────┘└────┘ ░ └╥┘
«q_1: ─────────┤ X ├────────────────░──╫─
« └───┘ ░ ║
«c: 1/═════════════════════════════════╩═
« 0
Modified Ramsey sequence with qubit 1 initially in the excited state
┌────┐ ░ ┌─────────────────┐ ░ ┌───┐ ░ ┌─────────────────┐ ░ »
q_0: ┤ √X ├─░─┤ Delay(delay[s]) ├─░─┤ X ├─░─┤ Delay(delay[s]) ├─░─»
├───┬┘ ░ └─────────────────┘ ░ ├───┤ ░ └─────────────────┘ ░ »
q_1: ┤ X ├──░─────────────────────░─┤ X ├─░─────────────────────░─»
└───┘ ░ ░ └───┘ ░ ░ »
c: 1/═════════════════════════════════════════════════════════════»
»
« ┌─────────────────────┐┌────┐ ░ ┌─┐
«q_0: ┤ Rz(4*delay*dt*f*pi) ├┤ √X ├─░─┤M├
« └─────────────────────┘└────┘ ░ └╥┘
«q_1: ──────────────────────────────░──╫─
« ░ ║
«c: 1/═════════════════════════════════╩═
« 0
# section: analysis_ref
:class:`ZZRamseyAnalysis`
"""
def __init__(
self,
physical_qubits: Tuple[int, int],
backend: Union[Backend, None] = None,
**experiment_options,
):
"""Create new experiment.
Args:
physical_qubits: The qubits on which to run the Ramsey XY experiment.
backend: Optional, the backend to run the experiment on.
experiment_options: experiment options to set
"""
super().__init__(physical_qubits, analysis=ZZRamseyAnalysis(), backend=backend)
# Override the default of get_processor() which is "1" * num_qubits. We
# only fit the probability of the target qubit.
self.analysis.set_options(outcome="1")
self.set_experiment_options(**experiment_options)
@classmethod
def _default_experiment_options(cls) -> Options:
"""Default values for the :math:`ZZ` Ramsey experiment.
Experiment Options:
delays (list[float]): The list of delays that will be scanned in
the experiment, in seconds. If not set, then ``num_delays``
evenly spaced delays between ``min_delay`` and ``max_delay``
are used. If ``delays`` is set, ``max_delay``, ``min_delay``,
and ``num_delays`` are ignored.
max_delay (float): Maximum delay time to use.
min_delay (float): Minimum delay time to use.
num_delays (int): Number of circuits to use per control state
preparation.
num_rotations (float): The extra rotation added to qubit 0 uses a
frequency that gives this many rotations in the case where
:math:`f_{ZZ}` is 0.
"""
options = super()._default_experiment_options()
options.delays = None
options.min_delay = 0e-6
options.max_delay = 10e-6
options.num_delays = 50
options.num_rotations = 5
return options
[docs]
def delays(self) -> List[float]:
"""Delay values to use in circuits
Returns:
The list of delays to use for the different circuits based on the
experiment options.
"""
# This method allows delays to be set by the user as an explicit
# sequence or as a minimum, maximum and number of values for a linearly
# spaced sequence.
options = self.experiment_options
if options.delays is not None:
return options.delays
return np.linspace(options.min_delay, options.max_delay, options.num_delays).tolist()
[docs]
def frequency(self) -> float:
"""Frequency of qubit rotation when ZZ is 0
This method calculates the simulated frequency applied to both sets of
circuits. The value is chosen to induce `num_rotations` number of
rotation within the time window that the delay is swept through.
Returns:
The frequency at which the target qubit will rotate when ZZ is zero
based on the current experiment options.
"""
delays = self.delays()
freq = self.experiment_options.num_rotations / (max(delays) - min(delays))
return freq
def _template_circuits(
self,
frequency: Union[None, float, ParameterExpression] = None,
) -> Tuple[QuantumCircuit, QuantumCircuit]:
"""Template circuits for series 0 and 1 parameterized by delay
The generated circuits have the length of the delay instructions as the
only parameter.
Args:
dt_value: the value of the backend ``dt`` value. Used to convert
delay values into units of seconds.
delay_unit: the unit of circuit delay instructions.
Returns:
Circuits for series 0 and 1
"""
delay = Parameter("delay")
timing = BackendTiming(self.backend)
frequency = frequency if frequency is not None else self.frequency()
# frequency is always in units of Hz. delay_freq has inverse units to
# the units of `delay`.
#
# If the backend does not have a `dt`, the delays will be treated as in
# units of seconds. Otherwise they will be in units of samples. For the
# samples case, we multiply by `dt` so that `delay_freq` is in inverse
# samples per cycle.
if timing.delay_unit != "s":
delay_freq = timing.dt * frequency
else:
delay_freq = frequency
# Template circuit for series 0
# Control qubit starting in |0> state, flipping to |1> in middle
circ0 = QuantumCircuit(2, 1)
circ0.metadata["series"] = "0"
circ0.sx(0)
circ0.barrier()
circ0.delay(delay, 0, timing.delay_unit)
circ0.barrier()
circ0.x(0)
circ0.x(1)
circ0.barrier()
circ0.delay(delay, 0, timing.delay_unit)
circ0.barrier()
circ0.rz(2 * np.pi * delay_freq * (2 * delay), 0)
circ0.sx(0)
# Flip control back to 0, so control qubit is in 0 for all circuits
# when qubit 1 is measured.
circ0.x(1)
circ0.barrier()
circ0.measure(0, 0)
# Template circuit for series 1
# Control qubit starting in |1> state, flipping to |0> in middle
circ1 = QuantumCircuit(2, 1)
circ1.metadata["series"] = "1"
circ1.x(1)
circ1.sx(0)
circ1.barrier()
circ1.delay(delay, 0, timing.delay_unit)
circ1.barrier()
circ1.x(0)
circ1.x(1)
circ1.barrier()
circ1.delay(delay, 0, timing.delay_unit)
circ1.barrier()
circ1.rz(2 * np.pi * delay_freq * (2 * delay), 0)
circ1.sx(0)
circ1.barrier()
circ1.measure(0, 0)
return circ0, circ1
[docs]
def parametrized_circuits(self) -> Tuple[QuantumCircuit, QuantumCircuit]:
r"""Create circuits with parameters for numerical quantities
This method is primarily intended for generating template circuits that
visualize well. It inserts :class:`qiskit.circuit.Parameter`\ s for
:math:`π` and `dt` as well the target qubit rotation frequency `f`.
Return:
Parameterized circuits for the case of the control qubit being in 0
and in 1.
"""
f_param = Parameter("f")
dt = Parameter("dt")
pi = Parameter("pi")
freq = dt * pi * f_param / np.pi
timing = BackendTiming(self.backend)
if timing.dt is not None:
freq = freq / timing.dt
circs = self._template_circuits(frequency=freq)
return circs
[docs]
def circuits(self) -> List[QuantumCircuit]:
"""Create circuits
Returns:
A list of circuits with a variable delay.
"""
timing = BackendTiming(self.backend)
circ0, circ1 = self._template_circuits()
circs = []
for delay in self.delays():
for circ in (circ0, circ1):
assigned = circ.assign_parameters(
{circ.parameters[0]: timing.round_delay(time=delay / 2)}, inplace=False
)
assigned.metadata["xval"] = 2 * timing.delay_time(time=delay / 2)
circs.append(assigned)
return circs