Quantum Orchestration for

With the Quantum Orchestration Platform, there’s no limit to the span of experiments you can run. Find out how to supercharge your Superconducting qubits research with these real-world use cases.

The Ramsey sequence is the most basic direct probe of quantum coherence in a two-level system. It consists of four steps: An initial \(Y/2\) pulse, followed by a free evolution, an additional \(Y/2\) pulse in the rotating frame of the qubit, and lastly, a projective measurement to determine the state of the qubit in the \(|0\rangle\), \(|1\rangle \)basis. This sequence is repeated multiple times, which allows us to build up measurement statistics. We repeat it for different free-evolution times.

The Ramsey sequence directly measures quantum coherence. During the free evolution time, the qubit will precess in the frame rotating with the drive frequency at a rate of \(\Delta f = f_{\mathrm{drive}} – f_{01}\). Due to this, it will pick up a phase that depends on both the free evolution time and on \(\Delta f \). If we repeat this many times, only a coherent and repeatable phase evolution will lead to oscillations in the measured state of the qubit. Consequently, the amplitude of these oscillations directly reflects the coherence of the phase acquired during free evolution.

We typically use this sequence to determine two vital parameters of the qubit: The \(T_2^*\) time, which is the low-frequency dephasing noise of the environment [F. Krantz et al., A quantum engineer’s guide to superconducting qubits,* Applied Physics Reviews 6, 021318 (2019)]*, as well as the qubit transition frequency, which can be accurately measured since it is the only frequency for which no oscillations are visible.

The following QUA program implements this Ramsey measurement sequence. It is crucial to emphasize that although this code is Python-embedded, it is an independent programming language. A proprietary compiler will compile the code written here into a set of machine instructions that the pulse processor inside the OPX will then execute in real-time and with nanosecond-scale latency.

The program consists of 3 nested for-loops (Fig 2). The first is an external averaging loop. The second loop loops over the frequency, which is updated to a new value when we call `update_frequency('qubit1', f_sweep)`

in line 26, and the third one loops over the wait time between the two pulses.

We play two \(Y/2\) pulses in each iteration, here generated in QUA with the statement `play('pi_op_qubit1' * amp(0.5), 'qubit1')`

in lines 27 and 29. This statement plays a \(\pi\)-pulse as defined in the program configuration, and scales its amplitude by half. The program configuration, which is not shown here for the sake of brevity, is where all the static information, including waveform samples, output frequencies, state discriminators, etc. is stored. Between the pulses, we wait for a time corresponding to the QUA variable `free_time`

. Then, in the `measure`

statement on line 31, we perform the following sequence:

Send a drive pulse to the readout resonator, wait for a fixed amount of time, acquire the output from the readout resonator, demodulate it, and store the demodulated result in the QUA variables `I`

and `Q`

. We can then either send the results back to the PC using the `save`

statements in lines 34 or 35, or use the IQ values inside the QUA program for whatever we choose.

The process is repeated for different qubit offset frequencies and wait times. As explained above, for a given frequency offset, we observe oscillations with a period that corresponds to \(1/\Delta f\), which gives the characteristic lineshapes shown in Fig. 3. The visibility of these lineshapes decays uniformly as time progresses. The run time of this program, including data acquisition, was approximately 10 minutes.

A crucial ingredient in the program is the `reset_qubit1`

method. This method is a simple yet illustrative example of an active reset sequence that demonstrates how easy it is to perform feedback operations on the OPX. All that is needed is to store the result of the measurement into a QUA variable, and then use this variable in a `while`

loop to perform a conditioned \(\pi\)-pulse operation!

```
```from qm.qua import *
def reset_qubit1():
qv = declare(fixed)
align('res', 'qubit1')
measure('meas_op_res', 'res', None, demod.full('integ_w_s', qv))
# qv is the discriminating IQ variable.
with while_(qv > -0.005):
play('pi_op_qubit1', 'qubit1')
measure('meas_op_res', 'res', None, demod.full('integ_w_s', qv))
with program() as prog:
f_sweep = declare(int)
I = declare(fixed)
Q = declare(fixed)
n = declare(int)
free_time = declare(int)
with for_(n, 0, n < 1000, n + 1):
with for_(f_sweep, 99e6, f_sweep < 101e6, f_sweep + 0.01e6):
with for_(free_time, 20, free_time < 8000, free_time + 10):
reset_qubit1()
update_frequency('qubit1', f_sweep)
play('pi_op_qubit1' * amp(0.5), 'qubit1')
wait(free_time, 'qubit1')
play('pi_op_qubit1' * amp(0.5), 'qubit1')
align('qubit1', 'res')
measure('meas_op_res', 'res', None,
demod.full('integ_w_c', I),
demod.full('integ_w_s', Q))
save(I, 'I')
save(Q, 'Q')

In the previous example, we glossed over an important detail: when we played the pulses, the frequency detuning was also on, and thus our pulses were slightly detuned and contained a (small) \(Z\) component. This can lead to asymmetric oscillations that are not centered around \(\langle S_z\rangle = 0\) since the qubit will not process on the equatorial plane exactly (see Fig. 4). If we wanted to get rid of this \(Z\) component, we would have to detune the frequency only during the free evolution time. We can do this by artificially adding a phase during the free evolution stage, proportional to the wait time.

In QUA, we can easily do this using the` frame_rotation_2pi`

statement, as seen in the example program. This statement is accumulative, which means that calling it with a value \(a\) will add \(2\pi a\) to the phase of subsequent pulses. This allows it to function as an actual frame rotation or virtual Z-rotation operation [D. C. McKay et al., Efficient Z gates for quantum computing, Phys. Rev. A 96, 022330 (2017)].

In fig. 4 we compare two scenarios: in the first, we detune the drive frequency by 0.251MHz throughout the experiment. In the second, we update the frame by \(0.251\mathrm{MHz} \times t_{\mathrm{wait}}\) using `frame_rotation_2pi`

. We can see that both lead to the same oscillation frequency, however, in the second case the oscillations are more symmetric and centered around \(\langle S_z \rangle =0\), thus enabling us to perform a better fitting procedure for \(T_2^*\) and the frequency detuning.

```
```from qm.qua import *
free_time_step = 20
offset_freq = 0.251e6 # the artificial offset frequency added during free evolution time only
# factor of 4 is due to free time being in units of 4nsec
rot_angle = free_time_step * 4 * offset_freq * 1e-9
with program() as ramsey_1d:
I = declare(fixed)
Q = declare(fixed)
n = declare(int)
free_time = declare(int)
total_angle = declare(fixed, value=0)
with for_(n, 0, n < 1000, n + 1):
assign(total_angle, 0)
with for_(free_time, free_time_step, free_time < 4000, free_time + free_time_step):
reset_qubit1()
assign(total_angle, total_angle + rot_angle)
play('pi_op_qubit1' * amp(0.5), 'qubit1')
wait(free_time, 'qubit1')
frame_rotation_2pi(total_angle, "qubit1")
play('pi_op_qubit1' * amp(0.5), 'qubit1')
align('qubit1', 'res')
measure('meas_op_res', 'res', None, demod.full('integ_w_c', I), demod.full('integ_w_s', Q))
save(I, 'I')
save(Q, 'Q')

Quantum Error-Correction is a great example demonstrating the integration of many of the benefits of the Quantum Orchestration Platform (QOP) and the OPX. Here we show an example of how you can easily implement a 3-qubit bit-flip code on a Superconducting qubit device with the QOP.

Figure 1 shows the experimental setup. Five transmon qubits are controlled using microwave signals that are IQ modulated by ten analog output channels of the two OPXs, for XY control. Flux bias signals are generated directly by five additional analog output channels of the OPXs, for Z control.

Each qubit is coupled to a readout resonator and all five resonators are coupled to the same transmission line. The transmission line is probed using another microwave signal that is IQ modulated by two analog output channels of an OPX and measured after downconversion by the OPX analog input channel.

Figure 2 shows the 3-qubit bit-flip code circuit. A logical qubit state \(|\psi\rangle=\alpha |0\rangle +\beta |1\rangle\) is encoded using three physical qubits in the state \(\alpha|000\rangle + \beta|111\rangle\). If the first qubit was prepared in the original state \(|\psi\rangle\), then this can be done by performing two CNOT gates as shown in the Encoding stage of the circuit in the figure.

The idea of the 3-qubit bit-flip code is that a single bit flip in the encoded state can be detected by measuring and tracking the parity of two pairs of qubits (Repeat Error Tracking stage of the circuit in Figure 2). This can be repeated to track the bit-flips during some time, after which a correction is applied to the three qubits according to the error tracking results (Correction stage in Figure 2). Finally, the state is decoded back to the state of a single physical qubit.

The parity measurements can be done by employing two more ancilla qubits initialized in the \(|0\rangle\) state before every measurement sequence, as shown in Figure 2. To measure the parity of a pair of qubits, say qubit 1 and 2, one CNOT gate is applied to qubit 1 and the ancilla qubit, and one CNOT gate is applied to qubit 2 and the same ancilla. This entangles the parity of the two qubits with the state of the ancilla, which can then be measured to determine the parity.

From the control perspective, running such a protocol is very demanding since it requires:

**Fast real-time processing:**to manage the error tracking in every iteration of the Repeated Error Tracking stage. Faster than the desired duration of the iteration, which is typically 100s of ns in superconducting circuits [M. Reed*et al.,*Realization of three-qubit quantum error correction with superconducting circuits,*Nature*482, 382–385 (2012)]

**Control flow:**to enable Error Tracking repetition for as many times as one requires**Low-latency feedback:**to correct the error based on the error tracking results

```
```from qm.qua import *
drive_elements = ['q0_xy', 'q1_xy', 'q2_xy', 'a0_xy', 'a1_xy']
readout_elements = ['q0_resonator', 'q1_resonator', 'q2_resonator', 'a0_resonator', 'a1_resonator']
all_elements = drive_elements + readout_elements
repetitions = 10
with program() as bit_flip_code:
qb_states = declare(bool, size=3)
an_states = declare(bool, size=2)
flips = declare(bool, size=3, value=[False, False, False])
i = declare(int)
# Preparation
reset_qubits(drive_elements, readout_elements)
play('pi/2', 'q0_xy')
# Encoding
CNOT('q0_xy', 'q1_xy')
CNOT('q0_xy', 'q2_xy')
# Error tracking
with for_(i, 0, i < repetitions, i+1):
CNOT('q0_xy', 'a0_xy')
CNOT('q1_xy', 'a0_xy')
CNOT('q1_xy', 'a1_xy')
CNOT('q2_xy', 'a1_xy')
measure_states(['a0_resonator', 'a1_resonator'], an_states, [0,0])
save_vector(an_states, 'states')
with if_(an_states[0]==0 & an_states[1]==1):
play('pi', 'a1_xy')
assign(flips[2], ~flips[2])
with if_(an_states[0]==1 & an_states[1]==0):
play('pi', 'a0_xy')
assign(flips[0], ~flips[0])
with if_(an_states[0]==1 & an_states[1]==1):
play('pi', 'a0_xy')
play('pi', 'a1_xy')
assign(flips[1], ~flips[1])
save_vector(flips, 'all_ground')
# Error correction
with if_(flips[0]):
play('pi', 'q0_xy')
with if_(flips[1]):
play('pi', 'q1_xy')
with if_(flips[2]):
play('pi', 'q2_xy')
# Decoding
CNOT('q0_xy', 'q1_xy')
CNOT('q0_xy', 'q2_xy')

To implement the 3-qubit bit-flip code in QUA, first, classical variables are initialized. We store measurements of the three data qubits in a boolean vector `qb_states`

, measurements of the two ancilla states in the boolean vector `an_states`

*,* and the parity of the number of flips on each data qubit in the boolean vector `flips`

.

Then, the preparation stage of the circuit is defined. Here we first reset the five qubits using the function `reset_qubits(elements)`

. For specific implementation of an active reset function of a single qubit, which uses real-time feedback and resets the qubits quickly and efficiently with QUA, please refer to the Ramsey example above. Here we focus on the error correction code. [S. J. Devitt et al., Quantum error correction for beginners, *Reports on Progress in Physics*, *76*(7) (2013)].

After we initialize the qubits we apply a \(\pi/2\) pulse to the first data qubit in order to prepare the qubit in an initial state. Then, we entangle the data qubit with the two ancillary qubits.

The error tracking phase inside a for-loop repeats blocks of CNOT gates followed by a measurement of the output resonators, from which an error syndrome can be extracted. If an error is detected, one of the three conditional statements can apply to the appropriate recovery gate. The actual waveforms required to perform the CNOT gate vary between quantum computer implementations, and the specific CNOT method can be implemented straightforwardly using the same constructs we’ve seen so far.

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