High-Fidelity Measurement of Qubits Encoded in Multilevel Superconducting Circuits

Qubit measurements are central to quantum information processing. In the field of superconducting qubits, standard readout techniques are limited not only by the signal-to-noise ratio, but also by state relaxation during the measurement. In this work, we demonstrate that the limitation due to relaxation can be suppressed by using the many-level Hilbert space of superconducting circuits: In a multilevel encoding, the measurement is corrupted only when multiple errors occur. Employing this technique, we show that we can directly resolve transmon gate errors at the level of one part in $10^3$. Extending this idea, we apply the same principles to the measurement of a logical qubit encoded in a bosonic mode and detected with a transmon ancilla, implementing a proposal by Hann et al. [Phys. Rev. A 98, 022305 (2018)]. Qubit state assignments are made based on a sequence of repeated readouts, further reducing the overall infidelity. This approach is quite general, and several encodings are studied; the codewords are more distinguishable when the distance between them is increased with respect to photon loss. The trade-off between multiple readouts and state relaxation is explored and shown to be consistent with the photon-loss model. We report a logical assignment infidelity of $5.8\times {10}^{-5}$ for a Fock-based encoding and $4.2\times {10}^{-3}$ for a quantum error correction code (the $S=2$, $N=1$ binomial code). Our results not only improve the fidelity of quantum information applications, but also enable more precise characterization of process or gate errors.

Popular Summary

The act of observation plays a special role in quantum mechanics. In the field of quantum computing, one crucial operation is to observe, or measure, the information stored in the computer. However, because quantum states are extremely fragile, it is difficult to reliably extract the zeros and ones represented by the computer’s quantum bits (qubits). This means that sometimes zeros are incorrectly decoded as ones, and vice versa. Such measurement errors can prevent reliable quantum computation. Here, we demonstrate a method to drastically reduce measurement errors in qubits implemented with superconducting circuits.

Our approach uses redundancy by measuring the same qubit several times. In this way, the majority of outcomes will usually be correct, and measurement errors can be suppressed. An additional advantage comes from encoding information in multiphoton states: Whereas the information represented by the presence or absence of a single photon is sensitive to dissipation, multiphoton states can preserve information even if some photons are lost. We demonstrate our method for several encodings, including quantum error correction codes. Such codes can be used to protect against many types of error, not only measurement error. In a simple example of our approach, we demonstrate the use of improved qubit measurement to better characterize the error in a quantum operation.

Our results will aid in the demonstration of future experiments that depend on the ability to reliably extract information from qubits. These experiments include quantum algorithms, teleportation, and other protocols.

Figure 1

Cartoon illustrating qubit readout errors. (a) In a two-level system, qubit readout means determining the energy eigenstate of the system at time $t=0$. Dashed arrows indicate incorrect assignments $P(\uparrow ||\downarrow ⟩)$ and $P(\downarrow ||\uparrow ⟩)$, which may be due to detector noise or state transitions. (b) In a many-dimensional Hilbert space, reading out a single bit requires partitioning the Hilbert space into two sets containing the encoded 0 and 1 states. Instead of an energy eigenstate, membership in $S$ or its complement $\overline{S}=\mathcal{H}\backslash S$ is measured. The dashed arrows now indicate the assignment errors $P(S||1_L⟩)$ and $P(\overline{S}||0_L⟩)$.

Figure 2

High-fidelity measurement of a transmon. (a) Representation of the measurement performed. The first four states of the transmon are individually prepared, measured, and classified as belonging to $S$ or $\overline{S}$ based on the measurement record. (b) The probability that the ground state $|g⟩$ is incorrectly assigned as belonging to $\overline{S}$, plotted as a function of the length of the measurement signal used to make the classification. The initial shape of the curve is related to the ring-up time of the readout resonator. The misassignment probability decreases as a function of time, because collecting more signal improves the separation of readout signals. However, the improvement stops once the misassignment probability is comparable to the probability that the transmon has gained a photon. Collecting more signal causes additional misassignments. (c) The probability that an excited state $|e⟩$, $|f⟩$, or $|h⟩$ is incorrectly assigned as belonging to $S$. Again, after some initial transient behavior related to an edge case in the classification method, the misassignment probability improves as the signal is collected for a longer time. Eventually, relaxation events cause misassignments, and the curves increase for large acquisition times. The probability that all excitations are lost, causing erroneous assignment to $S$, is much smaller when a higher excited state is prepared. The exact misassignment probabilities achieved depend on many factors, especially the relaxation time $T_1^{(ge)}\approx 51\mu \mathrm{s}$ and thermal population ${\overline{n}}_t\approx 0.4\%$.

Figure 3

Improved calibration and characterization of the gate error. (a) Schematic and (b) results of the shelved measurement demonstration. After the ground state is prepared, a $g-e$ $\pi$ pulse is applied with a variable amplitude, and the transmon is measured. An $e-f$ “shelving” pulse before the measurement greatly improves the visibility of $g-e$ Rabi oscillation. The unshelved measurement is shown in blue, and the shelved measurement is shown in red.

Figure 4

Schematic of the repeated measurement procedure. (a) Block diagram showing how an encoded bit is measured. We first prepare one of several Fock states $|n⟩$ in the storage mode. We then perform several readout cycles, each time obtaining an outcome. Each cycle consists of a decode pulse which excites the ancilla conditioned on the encoded bit, followed by a readout and reset of the ancilla. (b) The initialization procedure uses number-selective pulses and ancilla readouts to verify that the correct state has been prepared. This verification is crucial in order to demonstrate a sensitive detection. (c) Each reset consists of a real-time feedforward protocol, which ensures that the ancilla is in its ground state $|g⟩$ at the end of each cycle.

Figure 5

Enhancement of measurement fidelity with the code distance. (a) In these experiments, we excite the ancilla if the storage mode has fewer than two photons in it. In doing so, we read out whether the storage state is in $S$ or $\overline{S}$ as shown. (b) Probability that, when a state in $S$ is prepared in the storage mode, it is assigned incorrectly as $\overline{S}$. The state $|0⟩$ or $|1⟩$ is prepared in the storage mode, as indicated by the labels on the plot. The data show that the fidelity improves exponentially with the number of votes, until excitation out of the $S$ subspace limits the measurement. Dash-dotted lines show the terms in the theoretical prediction corresponding to majority voting, dashed lines show terms corresponding to state transitions, and closed symbols show the experimental data. Open symbols indicate postselection on successful ancilla resets. (c) Probability that, when a state in $\overline{S}$ is prepared in the storage mode, it is assigned incorrectly as $S$. Again, we see that majority voting suppresses the misassignment probability but that for large $N$ the misassignment probability increases due to photon loss. States of higher photon number are measured with a smaller error, because the measurement is robust to more photon-loss events.

Figure 6

High-fidelity measurement of bosonic codes. Results of measuring the logical $Z$ observable in different qubit encodings. Given single-shot ancilla assignments, a Bayesian classifier outputs either “$0_L$” or “$1_L$,” according to its best estimate of the initial state. As more readouts are incorporated into the estimate, the measurement infidelity decreases monotonically. Results for the 0–2, 0–3, 0–4, and 0–5 Fock codes are shown in blue circles, green squares, red triangles, and black triangles, respectively. Results for the first- and second-order binomial codes are shown in yellow stars and cyan pentagons, respectively. As the code distance increases, the measurement fidelity improves.