Error semitransparent universal control of a bosonic logical qubit

Abstract

Bosonic codes offer hardware-efficient approaches to logical qubit construction and hosted the first demonstration of beyond-break even logical quantum memory.However, such accomplishments were done for idling information, and realization of fault-tolerant logical operations remains a critical bottleneck for universal quantum computation in scaled systems. Error-transparent (ET) gates offer an avenue to resolve this issue, but experimental demonstrations have been limited to phase gates. Here, we introduce a framework based on dynamic encoding subspaces that enables simple linear drives to accomplish universal gates that are error semi-transparent (EsT) to oscillator photon loss. With an EsT logical gate set of {X, H, T}, we observe a five-fold reduction in infidelity conditioned on photon loss, demonstrate extended active-manipulation lifetimes with quantum error correction, and construct a composite EsT non-Clifford operation using a sequence of eight gates from the set. Our approach is compatible with methods for detectable ancilla errors, offering an approach to error-mitigated universal control of bosonic logical qubits with the standard quantum control toolkit.

Figures (5)

• Figure 1: Concept and operation of ET gate: During a logical operation $U_L(t)$ that takes the logical code space state $|\psi_L(0)\rangle$ at time $t=0$ to $|\psi_L(T)\rangle$ at time $t=T$, an error may occur at any time $t$, taking the state from code space to an orthogonal error space. (a) For an ordinary gate, an error at any point destroys the effect of the operation in the error space. Importantly, the final state depends on the exact time of the error occurring, which is a probabilistic event. (b) For an ET operation the trajectories in error and code space are identical. Hence an error happening at any time takes the state through a deterministic path in error space and ends at the same final state preserving the effect of the gate in error space. Therefore a quantum error correction (QEC) process can be used to detect and correct the error at the end of the operation, making the ET operation tolerant of errors during it. (c) An ancilla transmon qubit and an oscillator coupled to it are driven with numerically optimized pulses for realization of ET operations on an encoded logical qubit in the oscillator. (d) QEC violation for the evolved code space basis states during a logical X operation on a binomial logical qubit encoded in the oscillator. Numerically optimized EsT gate has minimal violation compared to the Ord gate. (e) Leakage out of the instantaneous error space and (f) trajectory mismatch in code and error subspaces also show smaller violation for EsT gate compared to Ord gate. Violation metrics plotted in (e) and (f) are calculated for an initial state $|0_E\rangle$ and $|0_L\rangle$, respectively.
• Figure 2: Wigner visualization of error transparency: (a) Logical Bloch sphere of code space (left) and error space (right) corresponding to single photon loss errors in the oscillator for binomial 'kitten' code. The cardinal points and their theoretical Wigner functions are marked on the Bloch sphere. Starting from the state $\left|-Y_L\right\rangle =(|0_L \rangle - i|1_L\rangle)/\sqrt{2}$ (Wigner function in solid square box), a logical X operation is performed along the X axis (marked black) to end in the state $|+Y_L\rangle =(|0_L \rangle + i|1_L\rangle)/\sqrt{2}$ (Wigner function in dashed square box). (b) Experimental pulse sequence for characterizing EsT gates. After the gate operation, we read out the qubit state and measure the parity of the oscillator state to detect errors and perform Wigner tomography. Based on this parity measurement, we post-process the Wigner functions of states in code and error space. (c) Experimentally measured Wigner functions of binomial code cardinal point $\left|-Y_L\right\rangle =(|0_L \rangle - i|1_L\rangle)/\sqrt{2}$ and the state after applying five EsT or Ord X gates are shown. Left side plots of code space Wigner function is conditioned on detection of no error ($\sim 90\%$ of shots) and shows similar performance for EsT and Ord gate. Right side plot of error space Wigner function is conditioned on detection of single photon loss error ($\sim 10\%$) and shows EsT gate significantly preserves the phase coherence and state amplitudes of the target state compared to Ord gate. Error space Wigners have a fixed global frame rotation compared to code space because of qubit being in excited state for longer during the measurement.
• Figure 3: Code and error space process tomography: (a) Experimental pulse sequence to perform code and error space process tomography. Following the gate operations, we perform qubit readout and oscillator parity measurement for error detection based on which we post-select our states into code and error space. Separate decode operations are used that respectively maps the code and error space states to the ancilla qubit state for performing process tomography. (b) Experimentally measured code (dots) and error space (triangles) process fidelities for EsT and Ord X gate with number of applied gates are shown (see App. C for details). Shaded region around the data points signify 1 standard deviation of measured fidelities in repeated experiments. Dashed lines for code space and solid lines for error space show modeled gate performance with number of gates. In codespace, both gates perform similarly while in the error space EsT gate performs significantly better than the Ord gate. The horizontal dashed line at process fidelity of 0.25 indicates a completely incoherent process. (c) We plot the modeled code and error space process fidelities without non-idealities of the error detection process. This shows a clear difference in error space process fidelity of EsT and Ord gates. (d) We summarize the infidelities for $X$, $H$ and $T$ gates extracted from our model for (left side of plot) error during the operation that takes a state from code space to error space ($1-\mathrm{F}_{\mathrm{err|jump}}$) and (right side of plot) operations in the the error space ($1-\mathrm{F}_{\mathrm{err}}$). Factor of reduction in infidelity is marked by the arrow for all the cases.
• Figure 4: EsT gates protected by AQEC: (a) Experimental sequence to check gate performance with AQEC. We perform a single AQEC step after $N$ gates and perform process tomography. (b) We show the performance of EsT and Ord X gate with number of gates applied before doing error correction. EsT gate outperforms the Ord gate and the improvement increases with number of gates and then decreases as uncorrectable qubit errors, two photon loss errors in oscillator and control errors start to dominate. The plot also shows curves for process fidelity decay for idling and for applying an AQEC pulse after idling for an equivalent duration of N gates. The blue and red dashed lines provide a visual reference for simulated idling decay curves, calculated at an average photon number equal to the average photon number during the gate operation. The horizontal dashed line at process fidelity of 0.25 indicates a completely incoherent process and shaded regions around the data points signify 1 standard deviation of fidelities for repeated measurements. The top X axis shows the total duration of the experiment that includes encode, decode, AQEC pulse and active reset of qubit. (c) Summary of improvement in gate fidelity for EsT gate compared to Ord gate by applying AQEC as a function of the number of gates for X, H and T gates.
• Figure 5: EsT protection for non-trivial sequence: (a) Logical code space Bloch sphere showing the ideal Wigner functions before (solid box) and after (dashed) the non-trivial $THXTHTHX$ operation sequence that effectively rotates the state around the marked non-trivial rotation axis (black line). The ideal error space Wigner functions are also shown. (b) We show the experimentally measured Wigner functions of the prepared $|1_L\rangle$ state, the final state post-selected in code and error subspace after applying the pulse sequence comprising of EsT and Ord gates. We measure the process fidelities as reported on the side of the measured Wigner functions. Finally, we apply the AQEC pulse and measure the process fidelity. We observe a 0.05 improvement in process fidelity for the EsT gate sequence compared to the Ord gate sequence.