Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits

Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting code words through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.

Figure 1

(a) False-colored micrograph of a five-qubit lattice. (b) Cartoon representation of a five-qubit lattice and the logical operators on the data qubits as given in Eq. (1). Arrows represent the directions of the two-qubit cross resonance gate and point from the control to the target qubit.

Figure 2

cnot pulse sequences and state preparation circuit. (a) Decomposition of the four-pulse echoed cnot gate (FPCX). Pulses are applied to the physical channels representing control, cross resonance (CR), target, and spectator qubits (SQ). The pulses comprise a frame change (FC) with an angle parameter, Gaussian derivative (GD) with an amplitude and angle, and a Gaussian flattop (GF) with an amplitude and angle. FC is a virtual $Z$ gate applied in software, where $Z(\theta )=\mathrm{FC}(-\theta )$ [32]. (b) Logical state preparation circuit. $|{\overline{0}}_p{\overline{0}}_g⟩$ is prepared without any postrotations (PR). Other states in the logical $Z$ basis are prepared by applying ${\overline{X}}_{L1}$ and/or ${\overline{X}}_{L2}$. $|{\overline{+}}_g{\overline{+}}_p⟩$ is prepared by applying the Hadamard gates on all four data qubits at PR. Other states in logical $X$ basis are prepared by applying ${\overline{Z}}_{L1}$ and/or ${\overline{Z}}_{L2}$ following the Hadamard gates at PR. Note that the first (left) logical qubit is the protected qubit in the $Z$ basis but becomes the gauge qubit in the $X$ basis due to the application of Hadamard gates at PR.

Figure 3

Magnitude of the reconstructed $|{\overline{1}}_p{\overline{1}}_g⟩$ state. We show the absolute differences between the actual and ideal matrix elements in the basis consisting of 16 states $\{|L_1{L}_2,s_z{s}_x⟩\}$. Labels $L_1$ and $L_2$ run over the four states of the logical qubits in the $Z$ basis. Syndrome bits $s_z$ and $s_x$ run over the four possible syndromes and represent the presence of phase-flip and bit-flip errors, respectively. The Supplemental Material [23] describes the physical to logical change of basis.

Figure 4

(a) Acceptance and (b) error probability of logical states with phase error $Z(\theta )$ inserted on the syndrome qubit at various sites. We fit the data to Eqs. (2) and (3) with an additional systematic offset parameter $\delta$ that is added to $\theta$ [23]. (c) Acceptance and (d) error probability of logical states with error $Y(\theta )$ inserted on the control and syndrome qubit after each cnot gate.

Figure 5

Encoded $|{\overline{1}}_p{\overline{1}}_g⟩$ lifetime and acceptance probability, $P(\tilde{0}\tilde{0})$. The ideal curve corresponds to Eq. (4), with $T_1=76.75\mu \mathrm{s}$. Data for the encoded state is plotted with model fits described in the text, with standard errors (statistical errors) added in the inset. Relaxation times of the four data qubits obtained from the model are $T_{1(i)}=\{57,84,85,81\}\mu \mathrm{s}$ with $i\in \{1,2,3,4\}$, which are within one standard deviation of the mean $T_1$ measured for each qubit [23]. The shaded region contains the curves for each qubit from the values of $T_{1(i)}$ obtained from the model fit; $p_0=0.05$ and $p_1=0.015$ are measurement errors from the fit. The Supplemental Material contains $|\overline{+}\overline{+}⟩$ data [23].