Universal set of dynamically protected gates for bipartite qubit networks: Soft pulse implementation of the [[5,1,3]] quantum error-correcting code

We model repetitive quantum error correction (QEC) with the single-error-correcting five-qubit code on a network of individually controlled qubits with always-on Ising couplings. We use our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to being accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating the unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of decoherence. The combined DD and QEC protocol dramatically improves the coherence, with the QEC alone being responsible for more than an order of magnitude infidelity reduction.

Figure 1

Arrangement of qubits on a bipartite star graph for implementing the [[5,1,3]] code. This particular arrangement of the qubits was chosen to minimize the total number of gate operations during the stabilizer generator measurement cycle. (a) The qubit to be encoded, $Q_6$, is initially placed at the center. At the end of encoding, it is swapped with an ancilla qubit, initially $Q_1$. (b) During the stabilizer measurement cycle, the single ancilla qubit $Q_6$ at the center is used to sequentially measure all four stabilizer generators supported by the five qubits around it.

Figure 2

Pulse sequences used to implement the cnot gate between qubits ${\mathcal{Q}}_5$ and ${\mathcal{Q}}_6$ on a star graph. It is a combination of four single-qubit gates ($0<t\le 16{\tau }_p$, $32{\tau }_p<t\le 48{\tau }_p$, $48{\tau }_p<t\le 64{\tau }_p$, and $64{\tau }_p<t\le 80{\tau }_p$) and a single instance of the $ZZ$-coupling sequence, $16{\tau }_p<t\le 32{\tau }_p$. For better accuracy this latter sequence has to be repeated several times; we used $N_{\mathrm{rep}}=5$ repetitions (see Ref. [56] for detailed discussion of the associated errors). Second-order self-refocusing pulse shapes $Q_1\left(\pi \right)$ and $Q_1(\pi /2)$ from Refs. [38, 53] are used. The shading shows the direction of the applied pulses as indicated.

Figure 3

(a) Conceptual encoding circuit for the [[5,1,3]] code using the Hadamard, cnot, and controlled-phase gates. On the input, the first four qubits are initialized in $|0\rangle$ states and the last qubit contains the state $|\psi \rangle \equiv \alpha |0\rangle +\beta |1\rangle$ to be encoded. On the output, the corresponding logical state $\alpha |\overline{0}\rangle +\beta |\overline{1}\rangle$ of the five-qubit code is produced. The decoding is done by inverting the encoding circuit. (b) Actual encoding circuit implemented on a six-qubit star graph. Two swap operations are required since qubits on the leaves can only interact with the qubit 6 in the center. In addition, the ancilla qubit is swapped to the center at the end to prepare for the measurement cycle (see Fig. 4).

Figure 4

Circuit for a single cycle of measuring the stabilizer generators (11) for the five-qubit code on a star graph. On the input and the output, the first five qubits contain an encoded state. The same ancilla qubit 6 at the center is used for each measurement; $M$ stands for measurement and resetting to $|0\rangle$ if needed. We implemented this circuit, which uses 16 cnot gates (duration $144{\tau }_p$ each) and 8 rounds of Hadamard gates (duration $32{\tau }_p$ each) applied in parallel, with the total measurement cycle duration of $2560{\tau }_p$ ($640{\tau }_p$ per stabilizer generator). In our simulations, the entire cycle is repeated several times for repetitive QEC.

Figure 5

Infidelities during the Zeno cycle for different noise correlation times and noise amplitudes as indicated. The time axis starts after the completion of the encoding circuit, at the instance of the first measurement [see Figs. 3 and 4]. The different curves correspond to cases where no pulses are applied (NP), DD pulses are applied but no measurements are done (NM), and with the syndrome measurements (WM). Closed and open symbols respectively represent the infidelities at the end of each syndrome measurement and at the end of the final decoding. Note that the axis for the cumulative success probability (SP) is on the right.

Figure 6

Sample error correction traces for the $\left[\right[5,1,3\left]\right]$ code in the presence of the stochastic phase noise on all six qubits. The noise correlation time is ${\tau }_n=32{\tau }_p$ and noise amplitudes $\sigma$ are as indicated (in units ${\tau }_p^{-1}$). Plots show different infidelity measures $1-F$ during the stabilizer measurement cycle and at the end of the decoding. Here $F_{\mathrm{b}}$ and $F_{\mathrm{a}}$ are the regular fidelities [Eq. (17)], respectively, computed just before and right after each projective measurement; $F_{\mathrm{b}}^{\prime }$ and $F_{\mathrm{a}}^{\prime }$ are the corresponding fidelities, which include idealized recovery for single-qubit errors [see Eq. (18)]. Larger symbols indicate the infidelities at the end of the decoding circuit, where the circle corresponds to the full-system fidelity (17) and the diamond to the single-qubit fidelity after tracing out the qubits away from the center.

Figure 7

Infidelities $1-F$ for the $\left[\right[5,1,3\left]\right]$ code averaged over 25 realizations of the stochastic Gaussian noise, with the noise correlation times ${\tau }_n$ and amplitudes $\sigma$ as indicated. Large infidelities after trigger events have been excluded from the averages. Here $F$ are the regular fidelities [Eq. (17)] computed right after each projective measurement and $F^{\prime }$ are the corresponding fidelities, which include idealized recovery for single qubit errors [see Eq. (18)]. In addition, $F_D$ and $F_D^{\prime }$, respectively, are the fidelities (17) and (18) for DD-only simulations with the same pulse sequences run but error correction turned off (no projective measurements). Larger symbols indicate the respective quantities at the end of the decoding circuits, with $F^{\prime }$ and $F_D^{\prime }$ replaced by the average single-qubit decoded fidelities, with the qubits away from the center traced out.

Figure 8

Same as Fig. 6, but with longer noise correlation time ${\tau }_n=128{\tau }_p$. Different infidelities are labeled as in Fig. 7.

Figure 9

Same as Fig. 7, but comparing the effect of fast dephasing noise. (a) Infidelities averaged over 25 realizations of a Gaussian stochastic noise with a rms amplitude $\sigma =20\times {10}^{-3}/{\tau }_p$ and ${\tau }_n=128{\tau }_p$. (b) There was also a weak noise with $\sigma =2\times {10}^{-3}/{\tau }_p$ and a much shorter correlation time ${\tau }_n={\tau }_p$. Such a noise is not affected by the DD. With the addition of the fast noise, the infidelity $1-F$ increased by about an order of magnitude, while the infidelity $1-F^{\prime }$ accounting for multiqubit errors increased by two orders of magnitude, consistent with the expected absence of DD protection against the fast noise component.