Basic operations on neutral-atoms
Warning
The digital-analog emulation framework is under construction and significant changes to the interface should be expected in the near-future. Nevertheless, the currest version serves as a prototype of the functionality, and any feedback is greatly appreciated.
Qadence includes primitives for the construction of programs implemented on a set of interacting qubits. The goal is to build digital-analog programs that better represent the reality of interacting qubit platforms, such as neutral-atoms, while maintaining a simplified interface for users coming from a digital quantum computing background that may not be as familiar with pulse-level programming.
To build the intuition for the interface in Qadence, it is important to go over some of the underlying physics. We can write a general Hamiltonian for a set of \(n\) interacting qubits as
where the driving Hamiltonian \(\mathcal{H}^\text{d}_{i}\) describes the pulses used to control single-qubit rotations, and the interaction Hamiltonian \(\mathcal{H}^\text{int}_{ij}\) describes the natural interaction between qubits.
Rydberg atoms
For the purpose of digital-analog emulation of neutral-atom systems in Qadence, we now consider a simplified time-independent global driving Hamiltonian, written as
where \(\Omega\) is the Rabi frequency, \(\delta\) is the detuning, \(\phi\) is the phase, \(X_i\) and \(Y_i\) are the standard Pauli operators, and \(N_i=\frac{1}{2}(I_i-Z_i)\) is the number operator. This Hamiltonian allows arbitrary global single-qubit rotations to be written, meaning that the values set for \((\Omega,\phi,\delta)\) are the same accross the qubit support.
For the interaction term, Rydberg atoms typically allow both an Ising and an XY mode of operation. For now, we focus on the Ising interaction, where the Hamiltonian is written as
where \(r_{ij}\) is the distance between atoms \(i\) and \(j\), and \(C_6\) is a coefficient depending on the specific Rydberg level of the excited state used in the computational logic states.
For a given register of atoms prepared in some spatial coordinates, the Hamiltonians described will generate the dynamics of some unitary operation as
where we specify the final parameter \(t\), the duration of the operation.
Qadence uses the following units for user-specified parameters:
- Rabi frequency and detuning \(\Omega\), \(\delta\): \([\text{rad}/\mu \text{s}]\)
- Phase \(\phi\): \([\text{rad}]\)
- Duration \(t\): \([\text{ns}]\)
- Atom coordinates: \([\mu \text{m}]\)
In practice
Given the Hamiltonian description in the previous section, we will now go over a few examples of the standard operations available in Qadence.
Arbitrary rotation
To start, we will exemplify the a general rotation on a set of atoms. To create an arbitrary register of atoms, we refer the user to the register creation tutorial. In this tutorial we do not use any information regarding the edges of the register graph, only the coordinates of each node that are used to compute the distance \(r_{ij}\) in the interaction term. Below, we create a line register of three qubits directly from the coordinates.
from qadence import Register
dx = 8.0 # Atom spacing in μm
reg = Register.from_coordinates([(0, 0), (dx, 0), (2*dx, 0)])
Currently, the most general rotation operation uses the AnalogRot operation, which
essentially implements \(U(t, \Omega, \delta, \phi)\) defined above.
from math import pi
from qadence import AnalogRot
rot_op = AnalogRot(
duration = 500., # [ns]
omega = pi, # [rad/μs]
delta = pi, # [rad/μs]
phase = pi, # [rad]
)
Note that in the code above a specific qubit support is not defined. By default this operation applies a global rotation on all qubits. We can define a circuit using the 3-qubit register and run it in the pyqtorch backend:
from qadence import BackendName, run
wf = run(reg, rot_op, backend = BackendName.PYQTORCH)
print(wf)
Under the hood of AnalogRot
To be fully explicit about what goes on under the hood of `AnalogRot`, we can look at the example code below.from qadence import BackendName, HamEvo, X, Y, N, add, run
from qadence.analog.utils import C6_DICT
from math import pi, cos, sin
# Following the 3-qubit register above
n_qubits = 3
dx = 8.0
# Parameters used in the AnalogRot
duration = 500.
omega = pi
delta = pi
phase = pi
# Building the terms in the driving Hamiltonian
h_x = (omega / 2) * cos(phase) * add(X(i) for i in range(n_qubits))
h_y = (-1.0 * omega / 2) * sin(phase) * add(Y(i) for i in range(n_qubits))
h_n = -1.0 * delta * add(N(i) for i in range(n_qubits))
# Building the interaction Hamiltonian
# Dictionary of coefficient values for each Rydberg level, which is 60 by default
c_6 = C6_DICT[60]
h_int = c_6 * (
1/(dx**6) * (N(0)@N(1)) +
1/(dx**6) * (N(1)@N(2)) +
1/((2*dx)**6) * (N(0)@N(2))
)
hamiltonian = h_x + h_y + h_n + h_int
# Convert duration to µs due to the units of the Hamiltonian
explicit_rot = HamEvo(hamiltonian, duration / 1000)
wf = run(n_qubits, explicit_rot, backend = BackendName.PYQTORCH)
# We get the same final wavefunction
print(wf)
When sending the AnalogRot operation to the pyqtorch backend, Qadence
automatically builds the correct Hamiltonian and the corresponding HamEvo
operation with the added qubit interactions, as shown explicitly in the
minimized section above. However, this operation is also supported in the
Pulser backend, where the correct pulses are automatically created.
RX / RY / RZ rotations
The AnalogRot provides full control over the parameters of \(\mathcal{H}^\text{d}\), but users coming from
a digital quantum computing background may be more familiar with the standard RX, RY and RZ rotations,
also available in Qadence. For the emulated analog interface, Qadence provides alternative
AnalogRX, AnalogRY and AnalogRZ operations which call AnalogRot under the hood to represent
the rotations accross the respective axis.
For a given angle of rotation \(\theta\) provided to each of these operations, currently a set of hardcoded assumptions are made on the tunable Hamiltonian parameters:
Note that the \(\text{RZ}\) operation as defined above includes a global phase compared to the standard \(\text{RZ}\) rotation since it evolves \(\exp\left(-i\frac{\theta}{2}\frac{I-Z}{2}\right)\) instead of \(\exp\left(-i\frac{\theta}{2}Z\right)\) given the detuning operator in \(\mathcal{H}^\text{d}\).
Warning
As shown above, the values of \(\Omega\) and \(\delta\) are currently hardcoded in these operators, and the
effective angle of rotation is controlled by varying the duration of the evolution. Currently,
the best way to overcome this is to use AnalogRot directly, but more general and convenient options
will be provided soon in an improved interface.
Below we exemplify the usage of AnalogRX:
from qadence import Register, BackendName
from qadence import RX, AnalogRX, random_state, equivalent_state, kron, run
from math import pi
dx = 8.0
reg = Register.from_coordinates([(0, 0), (dx, 0), (2*dx, 0)])
n_qubits = 3
# Rotation angle
theta = pi
# Analog rotation using the Rydberg Hamiltonian
rot_analog = AnalogRX(angle = theta)
# Equivalent full-digital global rotation
rot_digital = kron(RX(i, theta) for i in range(n_qubits))
# Some random initial state
init_state = random_state(n_qubits)
# Compare the final state using the full digital and the AnalogRX
wf_analog_pyq = run(
reg,
rot_analog,
state = init_state,
backend = BackendName.PYQTORCH
)
wf_digital_pyq = run(
reg,
rot_digital,
state = init_state,
backend = BackendName.PYQTORCH
)
bool_equiv = equivalent_state(wf_analog_pyq, wf_digital_pyq, atol = 1e-03)
print("States equivalent: ", bool_equiv)
As we can see, running a global RX or the AnalogRX does not result in equivalent states at the end, given
that the digital RX operation does not include the interaction between the qubits. By setting dx very high
in the code above the interaction will be less significant and the results will match.
However, if we compare with the Pulser backend, we see that the results for AnalogRX are consistent with
the expected results from a real device:
wf_analog_pulser = run(
reg,
rot_analog,
state = init_state,
backend = BackendName.PULSER,
)
bool_equiv = equivalent_state(wf_analog_pyq, wf_analog_pulser, atol = 1e-03)
print("States equivalent: ", bool_equiv)
Evolving the interaction term
Finally, besides applying specific qubit rotations, we can also choose to evolve only the interaction term
\(\mathcal{H}^\text{int}\), equivalent to setting \(\Omega = \delta = \phi = 0\). To do so, Qadence provides the
function wait which does exactly this.
from qadence import Register, BackendName, random_state, equivalent_state, wait, run
dx = 8.0
reg = Register.from_coordinates([(0, 0), (dx, 0), (2*dx, 0)])
n_qubits = 3
duration = 1000.
op = wait(duration = duration)
init_state = random_state(n_qubits)
wf_pyq = run(reg, op, state = init_state, backend = BackendName.PYQTORCH)
wf_pulser = run(reg, op, state = init_state, backend = BackendName.PULSER)
bool_equiv = equivalent_state(wf_pyq, wf_pulser, atol = 1e-03)
print("States equivalent: ", bool_equiv)
Some technical details
Warning
The details described here are relevant in the current version but are under revision for the next version of the emulated analog interface.
In the previous section we have exemplified the main ingredients of the current user-facing functionalities of the emulated analog interface, and in the next tutorial on Quantum Circuit Learning we will exmplify its usage in a simple QML example. Here we specify some extra details of this interface.
In the block system, all the Analog rotation operators initialize a ConstantAnalogRotation
block, while the wait operation initializes a WaitBlock. As we have shown, by default,
these blocks use a global qubit support, which can be passed explicitly by setting qubit_support = "global". However, the blocks do support
local qubit supports, with some constraints. The main constraint is that using kron on operators with different durations is not allowed.
from qadence import AnalogRX, AnalogRY, Register, kron
dx = 8.0
reg = Register.from_coordinates([(0, 0), (dx, 0)])
# Does not work (the angle affects the duration, as seen above):
rot_0 = AnalogRX(angle = 1.0, qubit_support = (0,))
rot_1 = AnalogRY(angle = 2.0, qubit_support = (1,))
try:
block = kron(rot_0, rot_1)
except ValueError as error:
print("Error:", error)
# Works:
rot_0 = AnalogRX(angle = 1.0, qubit_support = (0,))
rot_1 = AnalogRY(angle = 1.0, qubit_support = (1,))
block = kron(rot_0, rot_1)
Using chain is only supported between analog blocks with global qubit support:
from qadence import chain
rot_0 = AnalogRX(angle = 1.0, qubit_support = "global")
rot_1 = AnalogRY(angle = 2.0, qubit_support = "global")
block = chain(rot_0, rot_1)
The restrictions above only apply to the analog blocks, and analog and digital blocks can currently be composed.
from qadence import RX
rot_0 = AnalogRX(angle = 1.0, qubit_support = "global")
rot_1 = AnalogRY(angle = 2.0, qubit_support = (0,))
rot_digital = RX(1, 1.0)
block_0 = chain(rot_0, rot_digital)
block_1 = kron(rot_1, rot_digital)