Beating the break-even point with a discrete-variable-encoded logical qubit

One of the main obstacles for building a quantum computer is environmentally-induced decoherence, which destroys the quantum information stored in the qubits. The errors caused by decoherence can be corrected by repetitive application of a quantum error correction (QEC) procedure, where the logical qubit is encoded in a high-dimensional Hilbert space, such that different errors project the system into different orthogonal subspaces and thus can be unambiguously identified and corrected without disturbing the stored quantum information. In conventional QEC schemes Nielsen and Chuang (2010); Terhal (2015), the codewords of a logical qubit are respectively formed by two highly symmetric entangled states of multiple physical qubits encoded with some discrete variables. The past two decades have witnessed remarkable advances in experimental demonstrations of this kind of QEC code in different systems, including nuclear spins Cory et al. (1998); Knill et al. (2001), nitrogen-vacancy centers in diamond Waldherr et al. (2014); Abobeih et al. (2022), trapped ions Chiaverini et al. (2004); Schindler et al. (2011); Egan et al. (2021); Ryan-Anderson et al. (2021); Postler et al. (2022), photonic qubits Yao et al. (2012), silicon spin qubits Takeda et al. (2022), and superconducting circuits Reed et al. (2012); Kelly et al. (2015); Córcoles et al. (2015); Chen et al. (2021); Krinner et al. (2022); Zhao et al. (2022); Acharya et al. (2022). However, in these experiments, the lifetime of the logical qubit still needs to be significantly extended to reach that of the best available physical component, which is regarded as the break-even point for judging whether or not a QEC code can benefit quantum information storage and processing.

An alternative QEC encoding scheme is to employ the large space of an oscillator, which can be used to encode either a continuous-variable or discrete-variable qubit Cai et al. (2021); Joshi et al. (2021); Flühmann et al. (2019); Campagne-Ibarcq et al. (2020); Gertler et al. (2021). Both types of codes can tolerate errors due to loss and gain of energy quanta, enabling QEC to be performed in a hardware-efficient way. Breakthrough QEC demonstrations have been reported in a circuit quantum electrodynamics (QED) system Ofek et al. (2016), where the break-even point was exceeded by distributing the quantum information over an infinite-dimensional Hilbert space, but realized with logical qubits bearing two non-orthogonal codewords. This inherent restriction can be overcome with discrete-variable encoding schemes, where the codewords of a logical qubit are encoded with mutually orthogonal Fock states of an oscillator. This feature, together with their intrinsic compatibility with error-correctable gates Ma et al. (2020); Reinhold et al. (2020), as well as with their usefulness for logically connecting modules in a quantum network Chou et al. (2018), makes such discrete-variable qubits promising in fault-tolerant quantum computation. These advantages can be turned into practical benefits in real quantum information processing only when the lifetime of the encoded logical qubits is extended beyond the break-even point, which, however, remains an elusive task, although enduring efforts have been made towards this goal Hu et al. (2019); Gertler et al. (2021).

Here, we demonstrate the exceeding of the QEC break-even point by real-time feedback correction for a discrete-variable photonic qubit in a microwave cavity, whose codewords remain mutually orthogonal and can be unambiguously discriminated. The dominate error, single photon loss, of the logical qubit is mapped to the state of a Josephson-junction based nonlinear oscillator that is dispersively coupled to the cavity and serves as an ancilla qubit, realized with a continuous pulse involving an ingeniously tailored comb of frequency components. As the driving frequencies aim at the error space where a photon loss event occurs, perturbations on the logical qubit are highly suppressed when it remains in the encoded logical space. Another intrinsic advantage of this error syndrome detection is that the continuous driving protects the system from the ancilla’s dephasing noise. We demonstrate this procedure with the lowest-order binomial code and extend the stored quantum information lifetime 16% longer than the best physical qubit, encoded in the two lowest Fock states and referred to as the Fock qubit. A more important characteristic associated with this error-detecting procedure is that neither the logical nor the error space needs to have a definite parity, which allows the implementation of QEC codes that can tolerate losses of more than one photon.

The key ingredients of a QEC procedure include encoding the quantum information to the logical qubit from the ancilla, the error syndrome measurement, the real-time error correction of the system depending on the measurement output, and the decoding process to readout the quantum information stored in the logical qubit. Our logical qubit is realized in a three-dimensional microwave cavity, and the dominant decoherence to combat is the excitation loss error. The logical qubit is encoded with a binomial code Michael et al. (2016), with the codewords

where the number in each ket denotes the photon number in the cavity. The binomial code is a typical stabilizer QEC code: When the single-photon-loss error occurs, the quantum information is projected into the error space spanned by {$\left|0_{\mathrm{E}}\right\rangle=\left|3\right\rangle,\left|1_{\mathrm{E}}\right\rangle=\left|1\right\rangle$}, with the photon number parity acting as the error syndrome to distinguish these two spaces. A general QEC protection of quantum information stored in the bosonic system is illustrated in Fig. 1. After correctly measuring the photon number parity and applying the corresponding correction operations in real time, quantum information stored in the cavity can be recovered.

The experiments are performed with a circuit QED architecture Blais et al. (2021), where a superconducting transmon qubit Koch et al. (2007) as an ancilla is dispersively coupled to a three-dimensional microwave cavity Paik et al. (2011); Axline et al. (2016); Reagor et al. (2016). The ancilla qubit has an energy relaxation time of about 98 $\mu$s and a pure dephasing time of 968 $\mu$s, while the storage cavity has a single-photon lifetime of 578 $\mu$s (corresponding to $\kappa_{s}/2\pi=0.28$ kHz), and a pure dephasing time of 4.4 ms. The universal control of the multiple photon states of the cavity can be realized by utilizing the anharmonicity of the ancilla, and thus the key ingredients of the QEC procedure, as illustrated in Fig. 1, to the logical qubit encoded in the high-dimensional Fock spaces of the bosnoic mode can be realized.

Our route towards the break-even points in the QEC is two folds: improving both the operation fidelity to the logical qubit and the error syndrome measurement fidelity. The first goal is achieved by employing a tantalum transmon qubit with high coherence Place et al. (2021); Wang et al. (2022) and an optimal quantum control technique Khaneja et al. (2005) with carefully calibrated system parameters [see Methods]. We attempt the second goal by an ingenious scheme of projection measurement of a selected collection of Fock states. The principle of the scheme is illustrated in Fig. 2a, where a classical microwave pulse containing $2M$ frequency components is applied on the ancilla to read out the Fock states. Since the frequency of the ancilla is entangled with the photon number $n$ [see Methods for more details], error syndrome detection is achieved by mapping the even (odd) parity to the ancilla ground state $\left|g\right\rangle$ (excited state $\left|e\right\rangle$) in a quantum non-demolition manner. This approach holds potential advantages of more flexible choices of error spaces and less sensitive to ancilla damping and dephasing errors since the ancilla excitation is pronounced only when loss error happened.

To characterize our syndrome measurement, the cavity is encoded to the six cardinal-point states in the Bloch spheres of both the code and error spaces based on the lowest-order binomial codewords. The measured results of the cavity photon number parities are displayed in Fig. 2b and show an average detection error of 1.1% and 2.5% for the cavity states in the code and error spaces, respectively. The encoding of the cavity, one of the most elementary processes of QEC, is further verified by the Wigner function with a high fidelity of 0.95, as shown in Fig. 2c.

Based on the above techniques, the QEC process of the binomial code can be implemented following the procedure in Fig. 1. However, practical imperfections limit the QEC performance: (i) during a waiting time of $t_{w}$, i.e. an idle process, there is a probability of about $2(\kappa_{s}t_{w})^{2}\exp{(-2\kappa_{s}t_{w})}$ for a two-photon-loss error, which is undetectable for this lowest-order binomial code. (ii) Due to the non-commutativity of the single-photon-loss error and the self-Kerr interaction of the cavity, there is a large dephasing effect of the logical qubit induced by the unpredictable photon loss event, thus destroying the stored quantum information. (iii) Quantum recovery operations are imperfect. It is worth noting that there is a logical state distortion even if no photon loss is detected Michael et al. (2016). Considering the whole system, strategies to mitigate the above imperfections are introduced: choose an optimal waiting time, employ a two-layer QEC procedure Hu et al. (2019) to avoid unnecessary operation errors introduced by the error corrections, and adopt the photon-number-resolved a.c. Stark shift (PASS) method Ma et al. (2020) during idle operations to suppress the photon-jump-error-induced decoherence in the code space (see Supplementary Information for more details). The measured Wigner functions of the cavity states after a single QEC cycle (about 90 $\mu$s of waiting) without and with performing the error correction operation are shown in Fig. 2(d, e), with state fidelities of 0.81 and 0.88 respectively.

The performance of the QEC is benchmarked by the process fidelity $F_{\chi}=\mathrm{tr}\left(\chi_{\mathrm{exp}}\chi_{\mathrm{ideal}}\right)$, which is defined by comparing the experimental measured process matrix $\chi_{\mathrm{exp}}$ for the QEC process with the ideal process matrix $\chi_{\mathrm{ideal}}$ for an identity operation. In Fig. 3a, we present the measured process matrix for the encoding and decoding process only, which indicates a reference fidelity of 0.96. In the absence of a QEC operation after a waiting time of 105 $\mu$s, the process fidelity is reduced to a value of 0.73 due to the inability to protect the quantum information stored in the cavity from the single-photon-loss error, with the corresponding measured process matrix shown in Fig. 3b. When utilizing the QEC operation, the process fidelity is indeed improved due to the protection from the single-photon-loss error, with the process matrices for the one- and two-layer QECs shown in Fig. 3(c, d), respectively.

The most important benchmark to characterize the performance of a QEC procedure is the gain in the lifetime of the protected logical qubit against that of the constituent element with the longest lifetime. For the 3D circuit QED device, the best physical qubit is encoded with the two lowest photon-number states {$\left|0\right\rangle$, $\left|1\right\rangle$}, which is more robust against decoherence effects than any other encoded photonic qubit without QEC protection. To quantitatively show the advantage of our QEC scheme, in Fig. 3e we display the measured process fidelities of the corrected binomial code as a function of the storage time with the repetitive one-layer (red triangles) and two-layer (blue circles) QECs, as well as those for the unprotected binomial code (yellow stars), the transmon qubit (green diamonds), and the Fock qubit (black squares) for comparison.

All curves are fitted according to the function $F_{\chi}=Ae^{-t/\tau}+0.25$ with $\tau$ corresponding to the lifetime of the specific encoding and $A$ being a fitting parameter. The offset in the fitting function is fixed to 0.25, implying a complete loss of information at the final time. As a result, the lifetime $\tau$ for the corrected binomial code with one-layer QECs is improved by about 8.3 times compared to the uncorrected transmon qubit, 2.8 times compared to the uncorrected binomial code. In particular, $\tau$ is about $0.1$ times longer than the uncorrected Fock qubit encoding, i.e., exceeds the break-even point of QEC in this system. Employing the two-layer QEC scheme, the corresponding lifetime $\tau$ of the logical qubit is improved to about 8.8 times that of the uncorrected transmon qubit, 2.9 times that of the uncorrected binomial code, and 1.2 times that of the break-even point. These results demonstrate that the quantum information stored in the cavity with multiphoton binomial encoding can indeed be preserved and protected from the photon loss errors via repetitive QEC operations.

Table 1 shows an overall error analysis for the one- and two-layer QEC experiments. The error sources are divided into four parts: the intrinsic errors for the lowest-order binomial code, the error detection errors, the recovery operation errors, and the ancilla thermal excitation errors during the QEC cycle. These errors can be estimated from either the numerical simulations, or the measurement results of individual calibration experiments (see Supplementary Information). The predicted lifetimes $\tau$ for the QEC experiments, calculated by $\tau=-T_{w}/\ln(1-\epsilon)$ Hu et al. (2019), with $T_{w}$ and $\epsilon$ being the total duration and the weighted total error per QEC cycle, are well consistent with those in our QEC experiments.

In conclusion, we experimentally demonstrate the prolonged coherence time of quantum information encoded with discrete variables in a bosonic mode by repetitive QEC. The break-even point has been reached by carefully designing the QEC procedure to balance the fidelity losses due to the undetectable errors during the idle process and the error detection and correction operations. Presently, the main infidelity is contributed by the two-photon-loss error that is beyond the ability of our current QEC code, but can be corrected by higher-order binomial codes Michael et al. (2016). Our frequency comb method could naturally be utilized to measure the generalized photon number parity of such codes, enabling detection and correction of both single- and two-photon-loss errors. Our work thus represents a key step towards scalable quantum computing and provides a practical guide for system optimization of quantum control and the design of the QEC procedure for future applications of logical qubits.

Note added–After completion of the experiment, we became aware of a QEC experiment, which also goes beyond the break-even point, but is based on a continuous-variable encoding scheme.

The fast feedback control is implemented with Zurich Instruments UHFQA and HDAWG, which are connected to each other through a DIO link cable for real time feedback control. The UHFQA generates the readout pulses, acquires the down-converted transmitted readout signals for demodulation and discrimination in hardware, and sends the digitized readout results to the HDAWG through the DIO link cable in real time. The HDAWG plays different predefined waveforms conditional on the received readout results from the DIO link cable. The feedback latency, defined as the time interval between sending out the last point of the readout pulse from UHFQA and sending out the first point of the control pulse from HDAWG, is about 511 ns in our setup, which also includes the signal travelling time through the experimental circuitry.

For the cavity in the code space, the ancilla is off-resonantly driven by the comb pulse. For the 2-photon state in the cavity, the qubit’s transition $\left|g\right\rangle\leftrightarrow\left|e\right\rangle$ is driven by $M$ pairs of frequency components with symmetric detunings, resulting in a qubit state revival at a time of $T=k\pi/\chi$ with $k$ being an integer. Similarly, for the 0- and 4-photon states in the cavity, the qubit is driven by $M-1$ pairs of symmetric components and two unpaired components, whose effects can be ignored under the condition of $2M\chi\gg\Omega$. Therefore, the ancilla also makes a cyclic evolution at $T=k\pi/\chi$ and returns to the initial ground state when the cavity is in the code space.

For the cavity in the error space with 1- and 3-photon states, the ancilla qubit’s transition $\left|g\right\rangle\leftrightarrow\left|e\right\rangle$ is driven by a resonant frequency component, $M-1$ pairs of symmetric frequency components, and an unpaired off-resonant component. Under the same condition of $2M\chi\gg\Omega$, we can neglect the off-resonant effect of the unpaired components, and the ancilla will evolve from the initial ground state to the excited state at $T=k\pi/\chi$, with $k$ being an integer when choosing the drive amplitude $\Omega=\pi/2T$. In our experiment, $\Omega=\chi/4$, while $T\approx\pi/\chi$ for an optimized parity mapping time (see Supplementary Informations).

Therefore, this frequency comb pulse achieves error syndrome detection by mapping the even (odd) parity of the cavity state to the ancilla $\left|g\right\rangle$ ($\left|e\right\rangle$) state in a QND manner. This parity mapping process can be intuitively illustrated by simultaneously applying two conditional $\pi$ rotations to the ancilla qubit to flip the qubit state to the excited state associated with the cavity’s 1- and 3-photon states, thus resulting in a minimum perturbation to the cavity states in the code space.

In order to balance the operation errors, no-parity-jump backaction errors, and photon-loss errors, we employ a two-layer QEC procedure Hu et al. (2019) to improve the QEC performance (Fig. S6 in the Supplementary Information of Section III). In our QEC experiment, there are two bottom layers in a single QEC cycle, where the first one conserves the photon number parity in the deformed code space, and the second one recovers the quantum information in the code space.

The waiting time of the idle operation in each QEC cycle is selected based on a tradeoff between the uncorrected errors occurring during this time and the operation errors occurring during the error syndrome measurements and recovery operations. On one hand, the longer the waiting time, the larger probability of the two-photon-loss event is during this time, which cannot be detected by the lowest-order binomial code. On the other hand, the more frequent the error detection, the more likely the photon-loss errors occur during the detections and corrections. We calculate the QEC lifetime as a function of the waiting time from numerical simulations and choose an optimal waiting time of about $90~{}\mu$s in our QEC experiment. (Fig. S8 in the Supplementary Information of Section IV).