Universal Set of Scalable Dynamically Corrected Gates for Quantum Error Correction with Always-on Qubit Couplings

We construct a universal set of high fidelity quantum gates to be used on a sparse bipartite lattice with always-on Ising couplings. The gates are based on dynamical decoupling sequences using shaped pulses, they protect against low-frequency phase noise, and can be run in parallel on non-neighboring qubits. This makes them suitable for implementing quantum error correction with low-density parity check codes like the surface codes and their finite-rate generalizations. We illustrate the construction by simulating the quantum Zeno effect with the [[4, 2, 2]] toric code on a spin chain.

Figure 1

Pulse sequences used to implement two-qubit $zz$ rotations on a bipartite lattice with Ising couplings. Global sequences of $\pi$ pulses in the $x$ direction, $V_A(t)$ and $V_B(t)$, are executed on the idle qubits of the two sublattices. These decouple the interqubit Ising couplings and also the single-qubit low-frequency phase noise terms. In order to couple two neighboring qubits, the sequences $V_A$, $V_B$ are respectively replaced by $V_C$, $V_D$ on these qubits. This produces an effective Ising Hamiltonian with half of the original coupling. Shown are $Q_1(\pi )$ second-order self-refocusing pulse shapes [34, 46] used in the simulations.

Figure 2

Pulse sequence used to implement single-qubit rotations. The sequences of $\pi$ pulses in the $x$ direction, $V_1(t)$ and $V_2(t)$, are executed globally on idle qubits of the two sublattices. A single-qubit rotation is implemented as a DCG by adding three pulse-antipulse combinations and the stretched pulse in the shaded regions ($y$ axis). The sequence $V_3(t)$, where pulses in the shaded regions are $Q_1(\pi /2)$, corresponds to a $(\pi /2{)}_Y$ operation on a sublattice-A qubit. Such gates can be executed on any set of non-neighboring qubits.

Figure 3

(a) The infidelity of the gate in Fig. 2 and (b) the infidelity of the complete CNOT gate between qubits 2 and 3 of a four-qubit NN Ising chain as a function of rms chemical shift $\Delta$ (pulse sequences not shown). For small $\Delta$ the errors saturate because of the fixed $J$; when $\Delta$ is large, chemical shifts dominate the errors. The insets show the corresponding slopes; the slope of $\sim 6$ [infidelity $\propto ({\tau }_p\Delta {)}^6$] indicates that the on-site chemical shifts are decoupled to quadratic order. Analogous calculations with first-order pulses [34, 46] $S_1(\pi )$ and $S_1(\pi /2)$ give first-order [infidelity $\propto ({\tau }_p\Delta {)}^4$] decoupling and 2 orders of magnitude higher infidelities (not shown); Gaussian pulses increase infidelities by up to 5 orders of magnitude.

Figure 4

Encoding circuit for the [[4, 2, 2]] code implemented on a spin chain using four exchange gates (each implemented with three CNOTs), five CNOT gates, and a Hadamard gate $H$. Input qubits $i_1$, $i_2$ can be in an arbitrary two-qubit state, on the output the circuit returns an equivalent linear combination of the states in the code, see Eq. (6), using qubits $s_j$, $j=1,\dots ,4$, and an ancilla initialized for the stabilizer measurement circuit in Fig. 5. The decoding is done by reversing this circuit.

Figure 5

Measurement circuit for the [[4, 2, 2]] code on a spin chain with a shuttling ancilla. The ancilla is first shuttled up for the $G_z$ measurement and then shuttled down for the $G_x$ measurement. The entire circuit is repeated for every Zeno cycle.

Figure 6

Infidelities during the Zeno cycle for different noise correlation times but with the same noise amplitude $\sigma ={10}^{-3}/{\tau }_p$. The different curves correspond to cases where no pulses are applied (NP), DD pulses are applied but no measurements are made (NM), and with the syndrome measurements (WM). Closed and open symbols respectively represent the infidelities during the syndrome measurements and at the end of the final decoding. Note that the axis for the cumulative success probability (SP) is on the right.

Figure 7

Same as Fig. 6 but with fixed noise correlation time ${\tau }_c=128{\tau }_p$ with the rms noise amplitudes $\sigma$ as indicated.