Abstract
Quantum error correction (QEC) aims to protect logical qubits from noises by using the redundancy of a large Hilbert space, which allows errors to be detected and corrected in real time^{1}. In most QEC codes^{2,3,4,5,6,7,8}, a logical qubit is encoded in some discrete variables, for example photon numbers, so that the encoded quantum information can be unambiguously extracted after processing. Over the past decade, repetitive QEC has been demonstrated with various discretevariableencoded scenarios^{9,10,11,12,13,14,15,16,17}. However, extending the lifetimes of thusencoded logical qubits beyond the best available physical qubit still remains elusive, which represents a breakeven point for judging the practical usefulness of QEC. Here we demonstrate a QEC procedure in a circuit quantum electrodynamics architecture^{18}, where the logical qubit is binomially encoded in photonnumber states of a microwave cavity^{8}, dispersively coupled to an auxiliary superconducting qubit. By applying a pulse featuring a tailored frequency comb to the auxiliary qubit, we can repetitively extract the error syndrome with high fidelity and perform error correction with feedback control accordingly, thereby exceeding the breakeven point by about 16% lifetime enhancement. Our work illustrates the potential of hardwareefficient discretevariable encodings for faulttolerant quantum computation^{19}.
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Main
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, whereby the logical qubit is encoded in a highdimensional 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^{1,9}, the code words of a logical qubit are formed by two highly symmetric entangled states of several 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^{5,6}, nitrogenvacancy centres in diamond^{10,20}, trapped ions^{7,11,21,22,23}, photonic qubits^{24}, silicon spin qubits^{25} and superconducting circuits^{12,13,14,15,16,26,27}. However, in these experiments, the lifetime of the logical qubit still needs to be greatly extended to reach that of the best available physical component, which is regarded as the breakeven point for judging whether or not a QEC code can benefit quantum information storage and processing.
An alternative QEC encoding scheme is to use the large space of an oscillator, which can be used to encode either a continuousvariable or a discretevariable qubit^{28,29,30,31,32}. Both types of code can tolerate errors due to loss and gain of energy quanta, enabling QEC to be performed in a hardwareefficient way. Circuit quantum electrodynamics (QED) systems^{18} represent an ideal platform for realizing such encoding schemes: the breakeven point has been exceeded in two breakthrough experiments^{33,34} by distributing the quantum information over an infinitedimensional Hilbert space of a continuousvariableencoded photonic qubit, but the code words of this photonic qubit are not strictly orthogonal. This inherent restriction can be overcome with discretevariable encoding schemes, whereby the code words of a logical qubit are encoded with mutually orthogonal Fock states of an oscillator. This feature, together with their intrinsic compatibility with errorcorrectable gates^{35,36} and their usefulness in logically connecting modules in a quantum network^{37}, makes such discretevariable qubits promising in faulttolerant 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 breakeven point, which, however, remains an elusive result, although enduring efforts have been made towards this goal^{17,32}.
Here, we demonstrate the exceeding of the QEC breakeven point by realtime feedback correction for a discretevariable photonic qubit in a microwave cavity, whose code words remain mutually orthogonal and can be unambiguously discriminated. The dominant error, single photon loss, of the logical qubit is mapped to the state of a Josephson junctionbased nonlinear oscillator that is dispersively coupled to the cavity and serves as an auxiliary 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 auxiliary qubit’s dephasing noise. We demonstrate this procedure with the lowestorder 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 errordetecting 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 stages of a QEC procedure are encoding the quantum information to the logical qubit from the auxiliary qubit, the error syndrome measurement, the realtime error correction of the system depending on the measurement output and the decoding process to read out the quantum information stored in the logical qubit. Our logical qubit is realized in a threedimensional microwave cavity, and the dominant decoherence to combat is the excitation loss error. The logical qubit is encoded with a binomial code^{8}, with the code words:
where the number in each ket denotes the photon number in the cavity. The binomial code is a typical stabilizer QEC code: when the singlephotonloss error occurs, the quantum information is projected into the error space spanned by \(\{\left{0}_{{\rm{E}}}\right\rangle =\left3\right\rangle ,\left{1}_{{\rm{E}}}\right\rangle =\left1\right\rangle \}\), 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^{18}, where a superconducting transmon qubit^{38} as an auxiliary qubit is dispersively coupled to a threedimensional microwave cavity^{39,40,41}. The auxiliary qubit has an energy relaxation time of about 98 μs and a pure dephasing time of 968 μs, whereas the storage cavity has a singlephoton lifetime of 578 μs (corresponding to a decay rate κ_{s}/2π = 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 using the anharmonicity of the auxiliary qubit, and thus the key stages of the QEC procedure, as illustrated in Fig. 1, can be achieved by encoding the logical qubit in the highdimensional Fock spaces of the bosonic mode.
Our route towards the breakeven points in the QEC is twofold: improving both the operation fidelity to the logical qubit and the error syndrome measurement fidelity. The first goal is achieved by using a tantalum transmon qubit with high coherence^{42,43} and an optimal quantum control technique^{44} with carefully calibrated system parameters (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 auxiliary qubit to read out the Fock states. Because the frequency of the auxiliary qubit depends on the photon number n (see Methods for more details), error syndrome detection is achieved by mapping the even parity to the auxiliary qubit ground state \(\leftg\right\rangle \) (and the odd parity to the excited state \(\lefte\right\rangle \)) in a quantum nondemolition manner. This approach holds potential advantages of more flexible choices of error spaces and less sensitivity to auxiliary qubit damping and dephasing errors because the auxiliary qubit excitation is pronounced only when loss error occurs.
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 on the basis of the lowestorder binomial code words. The measured results of the cavity photon number parities are presented 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.
On the basis of 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: (1) during a waiting time of t_{w}, that is, an idle process, there is a probability of about \(2{({\kappa }_{{\rm{s}}}{t}_{{\rm{w}}})}^{2}\exp (2{\kappa }_{{\rm{s}}}{t}_{{\rm{w}}})\) of a twophotonloss error, which is undetectable for this lowestorder binomial code. (2) Owing to the noncommutativity of the singlephotonloss error and the selfKerr 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. (3) Quantum recovery operations are imperfect. It is worth noting that there is a logical state distortion even if no photon loss is detected^{8}. Strategies to mitigate the above imperfections are introduced, taking into account the whole system: choose an optimal waiting time, use a twolayer QEC procedure^{17} to avoid unnecessary operation errors introduced by the error corrections and adopt the photonnumberresolved a.c. Stark shift (PASS) method^{35} during idle operations to suppress photonjumperrorinduced 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 μs of waiting) without and with performing the error correction operation are shown in Fig. 2d,e, with state fidelities of 0.81 and 0.88, respectively.
The performance of the QEC is benchmarked by the process fidelity \({F}_{\chi }\), which is defined as the trace of χ_{exp}χ_{ideal}, where χ_{exp} denotes the experimentally measured process matrix for the QEC process and χ_{ideal} is the ideal process matrix 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 μs, the process fidelity is reduced to a value of 0.73 because of the inability to protect the quantum information stored in the cavity from the singlephotonloss error, with the corresponding measured process matrix shown in Fig. 3b. When using the QEC operation, the process fidelity is improved because of the protection from the singlephotonloss error, with the process matrices for the onelayer and twolayer QECs shown in Fig. 3c,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 threedimensional circuit QED device, the best physical qubit is encoded with the two lowest photonnumber states \(\{\left0\right\rangle ,\left1\right\rangle \}\), 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 present the measured process fidelities of the corrected binomial code as a function of the storage time with the repetitive onelayer (red triangles) and twolayer (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_{χ} = Ae^{−t/τ} + 0.25, with τ 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 τ for the corrected binomial code with onelayer QEC is improved by about 8.3 times compared with the uncorrected transmon qubit and 2.8 times compared with the uncorrected binomial code. In particular, τ is improved to about 1.1 times that of the uncorrected Fock qubit encoding, that is, exceeding the breakeven point of QEC in this system. Using the twolayer QEC scheme, the corresponding lifetime τ 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 breakeven point. These results demonstrate that the quantum information stored in the cavity with multiphoton binomial encoding can indeed be preserved and protected from photon loss errors by means of repetitive QEC operations.
Table 1 shows an overall error analysis for the onelayer and twolayer QEC experiments. The error sources are divided into four parts: the intrinsic errors for the lowestorder binomial code, the error detection errors, the recovery operation errors and the auxiliary qubit 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 (Supplementary Information). The predicted lifetimes τ for the QEC experiments, calculated by \(\tau ={T}_{{\rm{w}}}/ln(1{\epsilon })\)^{17}, with T_{w} and ϵ being the total duration and the weighted total error per QEC cycle, are 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 breakeven point has been reached by carefully designing the QEC procedure to balance the fidelity losses due to undetectable errors during the idle process, and error detection and correction operations. At present, the main infidelity is contributed by the twophotonloss error that is beyond the ability of our current QEC code, but can be corrected by higherorder binomial codes^{8}. Our frequency comb method could be used to measure the generalized photon number parity of such codes, enabling detection and correction of both singlephotonloss and twophotonloss 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.
Methods
Experimental device and setup
The circuit QED device in our experiment uses a hybrid threedimensional–planar architecture^{40} and consists of a superconducting transmon qubit^{38}, a coaxial stub cavity and a Purcellfiltered stripline readout resonator (see Fig. S1 in the Supplementary Information). The highQ cavity is designed with a cylindrical reentrant quarterwave transmission line resonator^{41}, and machined from highpurity (99.9995%) aluminium. A horizontal tunnel is used to house a sapphire chip, on which the antenna pads of the transmon qubit and the striplines of the lowQ readout resonator are patterned with a thin tantalum film^{42,43}. The single AlAlO_{x}Al trilayer Josephson junction of the transmon qubit is fabricated using a doubleangle evaporation technique.
The fast feedback control is implemented with Zurich Instruments UHFQA and HDAWG, which are connected to each other through a digital input/output (DIO) link cable for realtime feedback control. The UHFQA generates the readout pulses, acquires the downconverted 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 the UHFQA and sending out the first point of the control pulse from the HDAWG, is about 511 ns in our setup, which also includes the time for the signal to travel through the experimental circuitry.
Parity mapping
The parity mapping procedure in the QEC experiment is implemented by applying a classical microwave pulse containing 2M (M = 11 in our experiment) frequency components on the auxiliary qubit, with the system dynamics governed by the Hamiltonian:
in the interaction picture. Here, \(\lefte\right\rangle \) denotes the excited state and \(\leftg\right\rangle \) denotes the ground state of the auxiliary qubit, a^{†} is the creation operator and a is the annihilation operator of the photonic field in the cavity, χ is the auxiliary qubit’s frequency shift induced per photon as a result of its dispersive coupling, δ_{n} is the frequency detuning of the nth driving component with a Rabi frequency of Ω and h.c. denotes the Hermitian conjugate. With the choice of the drive frequency detuning δ_{n} = (2M − 2n − 1)χ, the auxiliary qubit is resonantly driven when the cavity has 2m + 1 photons with m = 0, 1, …M.
For the cavity in the code space, the auxiliary qubit is offresonantly driven by the comb pulse. For the twophoton state in the cavity, the qubit’s transition \(\leftg\right\rangle \leftrightarrow \lefte\right\rangle \) is driven by M pairs of frequency components with symmetric detunings, resulting in a qubit state revival at a time of T = kπ/χ with k being an integer. Similarly, for the zerophoton and fourphoton 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 2Mχ ≫ Ω. Therefore, the auxiliary qubit also makes a cyclic evolution at T = kπ/χ and returns to the initial ground state when the cavity is in the code space.
For the cavity in the error space with onephoton and threephoton states, the auxiliary qubit’s transition \(\leftg\right\rangle \leftrightarrow \lefte\right\rangle \) is driven by a resonant frequency component, M − 1 pairs of symmetric frequency components and an unpaired offresonant component. Under the same condition of 2Mχ ≫ Ω, we can neglect the offresonant effect of the unpaired components, and the auxiliary qubit will evolve from the initial ground state to the excited state at T = kπ/χ, with k being an integer when choosing the drive amplitude Ω = π/2T. In our experiment, Ω = χ/4, and T ≈ π/χ for an optimized parity mapping time (see section II in the Supplementary Information).
Therefore, this frequency comb pulse achieves error syndrome detection by mapping the even parity of the cavity state to the auxiliary qubit \(\leftg\right\rangle \) state (and the odd parity to the \(\lefte\right\rangle \) state) in a quantum nondemolition manner. This parity mapping process can be intuitively illustrated by simultaneously applying two conditional π rotations to the auxiliary qubit to flip the qubit state to the excited state associated with the cavity’s onephoton and threephoton states, thus resulting in a minimum perturbation to the cavity states in the code space.
Strategies for system optimization
The PASS method^{35} is adopted to mitigate the photonlossinduced dephasing effect of the logical code words, due to the noncommutativity of the annihilation operation and the selfKerr term. In our experiment, we apply an offresonant drive pulse with a frequency detuning of about −3.5χ on the auxiliary qubit during the idle operation, resulting in different phase accumulation rates f_{n} for Fock state \(\leftn\right\rangle \) with n = 1, 2, 3, 4 relative to the vacuum state. By choosing an optimal amplitude of the detuned drive, we could achieve the errortransparent condition^{35} of (f_{4} − f_{2}) − (f_{3} − f_{1}) = 0 to mitigate the dephasing effect of the logical qubit (Supplementary Fig. 4).
To balance the operation errors, the noparityjump backaction errors and the photonloss errors, we use a twolayer QEC procedure^{17} to improve the QEC performance (see Fig. S6 in the Supplementary Information). In our QEC experiment, there are two bottom layers in a single QEC cycle: the first layer conserves the photon number parity in the deformed code space and the second layer recovers the quantum information in the code space.
The waiting time of the idle operation in each QEC cycle is selected on the basis of a tradeoff between the uncorrected errors occurring during this time and the operation errors occurring during the error syndrome measurements and recovery operations. On the one hand, the longer the waiting time, the larger the probability of the twophotonloss event occurring during this time, which cannot be detected by the lowestorder binomial code. On the other hand, the more frequent the error detection, the more likely it is that the photonloss 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 μs in our QEC experiment (Supplementary Fig. 8).
Data availability
Source data for Figs. 2 and 3 are available with the paper. All other data relevant to this study are available from the corresponding author upon reasonable request.
Code availability
The code used for simulations is available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the KeyArea Research and Development Program of Guangdong Province (Grants No. 2018B030326001 and No. 2020B0303030001); the Shenzhen Science and Technology Program (Grant No. RCYX20210706092103021); the National Natural Science Foundation of China (Grants No. 12274198, No. 11904158, No. U1801661, No. 12274080, No. 12061131011, No. 92265210, No. 92165209, No. 11925404, No. 11890704 and No. 11875108); the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515010324); the Guangdong Provincial Key Laboratory (Grant No. 2019B121203002); the Program (Grant No. 2016ZT06D348); the Science, Technology and Innovation Commission of Shenzhen Municipality (Grant No. KYTDPT20181011104202253); the Shenzhen–Hong Kong cooperation zone for technology and innovation (Contract No. HZQBKCZYB2020050); the National Key Research and Development Program of China (Grant No. 2017YFA0304303); the China Postdoctoral Science Foundation (BX2021167); the Innovation Program for Quantum Science and Technology (Grants No. ZD0301703 and No. ZD0102040201); and the Natural Science Foundation of Beijing (Grant No. Z190012).
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Y.X. and D.Y. supervised the project. Y.X. conceived and designed the experiment. Z.N. performed the experiment. Z.N. and Y.X. analysed the data and carried out the numerical simulations. Z.N. and S. Li developed the feedback control technique under the supervision of Y.X., X.D., Y.C., W.W., Z.B.Y. and F.Y. contributed to the experimental and theoretical optimization. L.Z., S. Liu and H.Y. provided supports in device fabrication. S.B.Z. proposed the theoretical scheme of the frequency comb method. S.B.Z., C.L.Z. and L.S. provided theoretical and experimental supports. C.L.Z., S.B.Z., L.S. and Y.X. wrote the manuscript, and all authors provided feedback.
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This file contains the following four sections and additional references: I. Experimental method; II. Frequency comb control method; III. Details of the QEC procedure; and IV. Error analysis.
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Ni, Z., Li, S., Deng, X. et al. Beating the breakeven point with a discretevariableencoded logical qubit. Nature 616, 56–60 (2023). https://doi.org/10.1038/s41586023057844
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DOI: https://doi.org/10.1038/s41586023057844
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