Superconducting Qubits

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 which 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!

Ramsey with optimal \(\pi/2\) pulses

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 which are not centered around \(\langle S_z\rangle = 0\) since the qubit will not precess 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 0.251 MHz 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.

Fig. 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 Fig. 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 Fig. 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

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.