Robust readout of bosonic qubits in the dispersive coupling regime

High-fidelity qubit measurements play a crucial role in quantum computation, communication, and metrology. In recent experiments, it has been shown that readout fidelity can be improved by performing repeated quantum nondemolition (QND) readouts of a qubit's state through an ancilla. For a qubit encoded in a two-level system, the fidelity of such schemes is limited by the fact that a single error can destroy the information in the qubit. On the other hand, if a bosonic system is used, this fundamental limit can be overcome by utilizing higher levels such that a single error still leaves states distinguishable. In this work, we present a robust readout scheme which leverages both repeated QND readouts and higher-level encodings to asymptotically suppress the effects of mode relaxation and individual measurement infidelity. We calculate the measurement fidelity in terms of general experimental parameters, provide an information-theoretic description of the scheme, and describe its application to several encodings, including cat and binomial codes.

Figure 1

Repeated QND readouts. Contributions to overall measurement infidelity from ancilla preparation or readout errors and noise can be exponentially suppressed by repeating measurements.

Figure 2

Robust readout scheme. (a) Quantum circuit for the readout scheme. The state of the bosonic qubit is read out through repeated QND mappings of its state onto an ancilla. (b) The bosonic mode and mapping procedure. Fock states in the bosonic mode decay with rates proportional to their excitation number. All excited states are mapped to the excited state of the two-level readout ancilla. (c) Schematic of a circuit QED system. The state of a microwave cavity mode can be read out via a dispersive coupling to a transmon qubit. The transmon is measured via its coupling to some other device, typically a resonator or cavity. Realistic parameter values for this architecture (cf. Refs. [53, 54, 55]) are shown in the table (see Sec. 2b for parameter definitions).

Figure 3

Hidden Markov Model for the robust readout scheme. (a) Markov chain and emissions. At each step of the HMM the bosonic system transitions to a new state and releases an emission. Here $B_n$ denotes the bosonic mode state after step $n$, and $A_n$ denotes the $n\mathrm{th}$ ancilla measurement outcome. (b) Transition and emission probabilities. Transitions and emissions are shown diagrammatically for the case $L=2$, where the matrix elements along the arrows are the associated probabilities. Bolded arrows indicate the intended mappings.

Figure 4

Infidelity of the robust readout scheme. The infidelity is plotted as a function of the number of measurements $N$ for $L=1$ and 2, with the parameter choices $\delta =2\%$ and ${\kappa }_{\downarrow }\tau =1\%$. The solid lines denote the approximate majority voting infidelity (14), while circles and squares respectively denote exact calculations for the majority voting and MLE schemes. Inset: the minimum attainable infidelity is plotted as a function of $L$.

Figure 5

Robust readout scheme for relaxation and heating. Decays and excitations occur between adjacent Fock states with rates proportional to the excitation number. All Fock states $|n>m\rangle$ are mapped to the excited state of the two-level ancilla.

Figure 6

Infidelity of the robust readout scheme with both relaxation and heating. The infidelity is plotted as a function of the number of measurements $N$ for $L=1,2,$ and 3, with the parameter choices $\delta =2\%$, ${\kappa }_{\downarrow }\tau =1\%$, and ${\kappa }_{\uparrow }\tau =0.5\%$. Inset: the minimum attainable infidelity is plotted as a function of $L$. For $m=0$ (red) the infidelity asymptotes to a finite value, but for optimal $m$ (black) it continues to decrease.

Figure 7

Robust readout scheme for multilevel ancilla. (a) Schematic description. Both decays and excitations occur between adjacent bosonic mode Fock states. Each Fock state is mapped to a unqiue ancilla state. In the majority voting all ancilla readouts of $|n>m\rangle$ are counted as votes for $|L\rangle$, while readouts of $|n\le m\rangle$ are counted as votes for $|0\rangle$. (b) Mapping circuit. Fourier gates on the ancilla, in combination with evolution under the dispersive coupling, implement the mapping used in the QND measurement.

Figure 8

Infidelity of the robust readout scheme with multilevel ancilla. The infidelity is plotted with the parameter choices $\delta =2\%$, ${\kappa }_{\downarrow }\tau =1\%$, and ${\kappa }_{\uparrow }\tau =0.5\%$. The dashed line is a lower bound on the fidelity determined through information-theoretic considerations (see Sec. 5).