Introduction

Quantum error correction (QEC) is critical for performing long computations involving many qubits, such as Shor’s1 or quantum chemistry algorithms2. Errors accumulating in the quantum system can be regarded as entropy or heat entering the system. In this context, the standard measurement and feedback-based QEC methods can be thought of us creating a “Maxwell Demon" keeping the system cold. These methods typically require many qubits and complex control hardware and have been demonstrated approaching the fault tolerance threshold3,4,5,6,7,8,9,10,11,12. When cooling atoms, rather than using measurement-based feedback, typically laser cooling is used. In laser cooling, the measurement and feedback are effectively encoded in the internal level structure and clever choice of laser drives. Along these lines, it is possible to perform autonomous quantum error correction (AQEC) where rather than measurements and gates, the system is “cooled” via an appropriate set of drives and couplings to engineered thermal reservoirs13. Like laser cooling, AQEC can dramatically simplify the quantum and classical hardware and control required. Both autonomous and feedback-based QEC are more challenging than simply cooling because they require that the cooling process preserves the logical manifold of the system. Here, we develop an AQEC code in a pure transmon-based14 system using scalable on-chip circuit structures and experimentally demonstrate an important step towards its full implementation.

AQEC has received growing attention in theoretical proposals15,16,17,18,19,20,21,22,23. In addition to the usual QEC conditions24, AQEC requires that the error-correction operations must commute with the system Hamiltonian at all times. This makes AQEC most appropriate for hardware efficient25,26,27,28,29,30 systems with constrained error syndromes. Thus far all demonstrations, have encoded the logical qubits into 3D superconducting cavities using an ancilla qubit as a control25,28.

In this work, we propose and implement a new AQEC protocol, called the Star code31, that only requires easy-to-realize two-photon interactions. We develop a coherence-preserving two-transmon coupler that can parametrically generate all interactions needed for the protocol. With AQEC turned on, the logical states show higher coherence times than the uncorrected case limited by stray ZZ coupling between transmons. This is a crucial step towards fully demonstrating the Star code. The structure of the paper is as follows. First, we explain the logical encoding and Hamiltonian construction of the Star code. Then we experimentally calibrate each of the parametric processes used in the code. Finally, we prepare the logical states and characterize the coherence improvement.

Results

Logical encoding

The Star code encodes a logical qubit using two orthogonal states in a nine-dimensional (two-qutrit) Hilbert space as \(\left\vert {L}_{0}\right\rangle=(\left\vert gf\right\rangle -\left\vert fg\right\rangle )/\sqrt{2}\) (logical “zero"), and \(\left\vert {L}_{1}\right\rangle=(\left\vert gg\right\rangle -\left\vert ff\right\rangle )/\sqrt{2}\) (logical “one") where \(\left\vert g\right\rangle,\,\left\vert e\right\rangle\), and \(\left\vert f\right\rangle\) represent the lowest three energy levels of a transmon. The error states after a single photon-loss (one transmon in \(\left\vert e\right\rangle\)) are orthogonal to the logical space and to each other. Further, both logical states have an equal expected photon number so that the single-photon loss (transmon decay) does not reveal information about the state it was emitted from. We engineer a parent Hamiltonian for the logical states through \(\left\vert gf\right\rangle \left\langle fg\right\vert\) and \(\left\vert gg\right\rangle \left\langle ff\right\vert\) parametric processes. These processes are all implemented by driving through \(\left\vert ee\right\rangle\) as an intermediate state, producing the star topology in Hilbert space that gives the code its name (see Fig. 1a). Using an intermediate state allows these to be achieved using only 2-photon drives (QQ sidebands). We use the convention of calling them QQ red (single-photon exchange with low-frequency drives) and blue (two-photon pumping with high-frequency drives) sidebands. Despite both sets of drives going through \(\left\vert ee\right\rangle\), the logical states can be made dark with respect to \(\left\vert ee\right\rangle\) by detuning the \(\left\vert {L}_{0}\right\rangle\) (\(\left\vert {L}_{1}\right\rangle\)) sidebands by ± νr ( ± νb) and setting equal drive strength W. When all of these processes are simultaneously applied, the two-transmon Hamiltonian in the logical-static frame (see Supplementary Note 3 for derivation) is

$${\tilde{H}}_{QQ}= \frac{W}{2}\left(\left\vert ee\right\rangle \left\langle gf\right\vert {e}^{2\pi i{\nu }_{r}t}+\left\Vert ee\right\rangle \left\langle fg\right\vert {e}^{2\pi i{\nu }_{r}t}\right.\\ \left.+\left\vert ee\right\rangle \left\langle gg\right\vert {e}^{2\pi i{\nu }_{b}t}+\left\vert ee\right\rangle \left\langle ff\right\vert {e}^{2\pi i{\nu }_{b}t}\right)+h.c.$$
(1)
Fig. 1: Illustration of the autonomous error-correction scheme.
figure 1

The protocol requires simultaneous application of two QQ blue sidebands (\(\left\vert ee\right\rangle \leftrightarrow \left\vert gg\right\rangle\) and \(\left\vert ee\right\rangle \leftrightarrow \left\vert ff\right\rangle\)), two QQ red sidebands (\(\left\vert ee\right\rangle \leftrightarrow \left\vert fg\right\rangle\) and \(\left\vert ee\right\rangle \leftrightarrow \left\vert gf\right\rangle\)), and two QR error correcting sidebands (\(\left\vert e0\right\rangle \leftrightarrow \left\vert f1\right\rangle\)). All six drives are always-on. a Star code logical word formation. All QQ sidebands have nominally equal rates W. The two drives within a pair have opposite detunings from the on-resonance values. This opens up the energy gaps of O(W) between logical states and other states \(\{\left\vert {S}_{\pm }\right\rangle,\left\vert T\right\rangle \}\) (see Supplementary Note 3 for full expression). With only QQ sidebands on, this forms the “QQ echoed” qubit sharing the same logical states as the star code. b The AQEC cycle for \(\left\vert {L}_{0}\right\rangle\) (left) and \(\left\vert {L}_{1}\right\rangle\) (right) when a single-photon-loss event occurs. Logical state \(\left\vert {L}_{0}00\right\rangle\) \(\left\vert {L}_{1}00\right\rangle\) loses a photon from transmon Q1 at a rate 2γ1 and becomes the error state \(\left\vert eg00\right\rangle\) (\(\left\vert ef00\right\rangle\)). QR error correcting sidebands (applied on-resonance) bring the state at rate Ω1 to \(\left\vert {L}_{0}10\right\rangle\) (\(\left\vert {L}_{1}10\right\rangle\)) with one photon populating R1. R1’s photon decays quickly (at a rate κ1) and recovers the original logical state. AQEC cycle for Q2’s photon loss event is similar.

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Each transmon Qj is coupled to a lossy resonator Rj, which acts as the cold reservoir for dumping entropy. A single-photon loss, the dominant source of error in the system, populates the \(\left\vert e\right\rangle\) level, triggering autonomous correction enabled by two transmon-resonator (QR) error correcting sidebands \({\left\vert e0\right\rangle }_{j}\leftrightarrow {\left\vert f1\right\rangle }_{j},\;j=1,\, 2\) (Fig. 1(b)). These sidebands are applied resonantly at rates Ωj to the system, adding \({\tilde{H}}_{QRj}\) to the system Hamiltonian Hstatic,

$${\tilde{H}}_{QR1}=\frac{{\Omega }_{1}}{2}{a}_{r1}^{{{{\dagger}}} }\left(\left\vert fg\right\rangle \left\langle eg\right\vert+\left\vert ff\right\rangle \left\langle ef\right\vert \right)\otimes {I}_{4}+h.c.,$$
(2)
$${\tilde{H}}_{QR2}=\frac{{\Omega }_{2}}{2}{a}_{r2}^{{{{\dagger}}} }\left(\left\vert gf\right\rangle \left\langle ge\right\vert+\left\vert ff\right\rangle \left\langle fe\right\vert \right)\otimes {I}_{4}+h.c.,$$
(3)
$${\tilde{H}}_{{{{{{{{\rm{static}}}}}}}}}={\tilde{H}}_{QQ}\otimes {I}_{4}+\mathop{\sum}\limits_{j=1,2}{\tilde{H}}_{QRj}+{H}_{c}.$$
(4)

Here arj is the annihilation operator for the j-th resonator. Hc contains the diagonal terms from frame transformation (See Supplementary Note 3 for full expression). We label the full state as \(\left\vert {Q}_{1}{Q}_{2}{R}_{1}{R}_{2}\right\rangle\) and keep the lowest two levels for each resonator. Here In is the n × n identity matrix.

The Star code can correct the loss of a single photon from one of the qubits. Suppose Q1 undergoes a photon-loss (\(\left\vert f\right\rangle \to \left\vert e\right\rangle\)) at rate 2γ1, where γ1 is the \(\left\vert e\right\rangle \to \left\vert g\right\rangle\) decay rate. The logical \(\left\vert {L}_{0}00\right\rangle\), consequently, becomes the error state \(\left\vert eg00\right\rangle\) with energy \(-\frac{{\alpha }_{1}}{2}\). Here αj is the anharmonicity of j-th transmon. When \(W\, \gg \,{\Omega }_{1},\,{\tilde{H}}_{QR1}\) is a perturbation and only drives the transition \(\left\vert eg00\right\rangle \leftrightarrow \left\vert {L}_{0}10\right\rangle\) (See the left part of Fig. 1(b)). Assuming the resonator’s decay rate κ1 ≫ γ1, this oscillation quickly damps back to the original logical state \(\left\vert {L}_{0}00\right\rangle\) with no extra phase accumulated, and completes the correction cycle. The correction procedure for \(\left\vert {L}_{1}\right\rangle\) is similar through an independent path (See the right part of Fig. 1(b)). The logical superposition state preserves relative phases since the QR sidebands do not distinguish the correction path. Such a two-step logical refilling rate can be approximated with Fermi’s golden rule \({\Gamma }_{Rj}\simeq \frac{{\Omega }_{j}^{2}{\kappa }_{j}}{{\Omega }_{j}^{2}+{\kappa }_{j}^{2}}\)32. Apart from providing protection against single-photon loss, the star code also provides suppression to 1/f dephasing error19,33 through spn-locking. The continuous QQ drives create an energy gap of O(W) (right part of Fig. 1(a)) between the logical manifold and all other states, suppressing low-frequency noise. In the presence of only QQ drives (Fig. 1(a)), the “QQ echoed” qubit having the same logical manifold enjoys dephasing protection, and is a stepping stone towards the full code. Theoretical lifetime improvement of logical states is further discussed in Supplementary Note 3 and ref. 31.

We realize this protocol using the circuit shown in Supplementary Note 1. The key component is the inductive coupler based on the design in ref. 34 that enables the realization of fast parametric interactions. Two transmons Q1 and Q2 serve as the qutrits and share a common path to ground. This path is interrupted by a Superconducting Quantum Interference Device (SQUID) loop. The SQUID functions as a tunable inductor with external DC and RF magnetic fields threaded for activating the QQ sidebands. Each transmon is capacitively coupled to a lossy resonator serving both as the readout and cold reservoir. QR sidebands can be performed by sending a charge drive at the half transition frequency to the transmon35. Full circuit quantization is shown in Supplementary Note 2.

Device implementation and sideband calibration

We first characterize the individual qubits and realize the required sidebands to create and correct the logical states. We adjust the DC flux point to minimize the Cross-Kerr coupling between transmons which can dephase the logical superposition states (See Supplementary Note 4 for further discussion). The measured Cross-Kerr couplings are all lower than 320 kHz while maintaining Ramsey dephasing times \({T}_{{R}_{ge}}=15.2(9.8)\,\mu {{{{{{{\rm{s}}}}}}}}\) with relaxation time T1ge = 24.3(9.1) μs, for Q1(Q2) (See Supplementary Note 1).

To calibrate the QR sidebands for selective photon pumping, we initialize the system in \(\left\vert eg00\right\rangle\) and apply a continuous charge drive at frequency (ωr1 + ωq1 + α1)/2 to activate a 2-photon \(\left\vert e0\right\rangle \leftrightarrow \left\vert f1\right\rangle\) transition between Q1 and R1 at a rate of 0.49 MHz. The system achieves a steady state \(\left\vert fg00\right\rangle\) within 3 μs as shown by red points in Fig. 2(a). Similarly, a 0.59 MHz QR2 drive takes \(\left\vert ge00\right\rangle\) to \(\left\vert gf00\right\rangle\) in a similar time (blue points in Fig. 2(a)). The decay of transmon reduces the final average photon number slightly below 2.

Fig. 2: Different parametric oscillations.
figure 2

Photon numbers in individual transmons are measured as a function of time. a Error correcting QR sidebands \(\left\vert e0\right\rangle \leftrightarrow \left\vert f1\right\rangle\) applied separately at rates Ω1 = 0.49 MHz and Ω2 = 0.59 MHz to the transmon-resonator pairs with \(\left\vert e\right\rangle\) as initial states. Effective transitions b\(\left\vert gf\right\rangle \leftrightarrow \left\vert fg\right\rangle\) and c\(\left\vert gg\right\rangle \leftrightarrow \left\vert ff\right\rangle\) are measured when all QQ and QR sidebands are simultaneously turned on. Extracted sideband rates and detunings from simulation are Wr = 1.45 MHz, Wb = 1.25 MHz, νr = 0.8 MHz, νb = − 0.9 MHz, Ω1 = Ω2 = 0.39 MHz. Other parameters are based on experimentally measured data shown in Supplementary Note 1 and 2. Oscillation distortions qualitatively match the lab frame simulations (dashed lines). Measurement errors are smaller than the marker size.

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We achieve at least 20 MHz QQ red sidebands \(\left(\left\vert j,\,k\right\rangle \leftrightarrow \left\vert j+1,\,k-1\right\rangle \right)\) and 5 MHz QQ blue sidebands \(\left(\left\vert j,\,k\right\rangle \leftrightarrow \left\vert j+1,\,k+1\right\rangle \right)\) separately at the operating point, demonstrating a fast, coherence-preserved two-qutrit coupler with suppressed ZZ interaction. Blue sidebands have a slower rate limited by stray signals from higher flux modulation frequencies (See discussion in Supplementary Note 5). All possible sidebands realized in this coupler are shown in Supplementary Note 6.

By driving all six sidebands, the core effective 4-photon processes, \(\left\vert fg\right\rangle \leftrightarrow \left\vert gf\right\rangle\) and \(\left\vert gg\right\rangle \leftrightarrow \left\vert ff\right\rangle\) and the error-correcting QR drives can be realized simultaneously. In practice, the QQ red and blue sideband rates (Wr = 1.45 MHz and Wb = 1.25 MHz) are slightly different. When applying all sidebands, we choose a smaller W, because the coupler was found to heat and shift the readout resonator when driven at larger rates making tomographic reconstruction inaccurate. We choose almost opposite detunings (νr = 0.8 MHz, νb = − 0.9 MHz) for larger energy separation of the eigenstates and better error correction performance. Both QR sidebands are turned on at rates Ω1 = Ω2 = 0.39 MHz. Figure 2(b) shows the evolution when the initial state is \(\left\vert fg\right\rangle\). The average photon number of Q1 (in red) and Q2 (in blue) are read out simultaneously, and the oscillation between 0 and 2 forms an effective 4-photon red sideband. Note that this effective swap process is slightly different from the direct \(\left\vert fg\right\rangle \leftrightarrow \left\vert gf\right\rangle\) transition as the population in \(\left\vert ee\right\rangle\) will appear intermediately when the initial state has overlap with the eigenstates that have \(\left\vert ee\right\rangle\) component. Under this condition, \(\left\vert ee\right\rangle\) is no longer the dark state when all six sidebands are on. Oscillation damping originates from the detuning-induced slow interference and decoherence of the qutrit subspace, and this distortion is captured by the simulation as well. Similarly, by choosing the initial state as \(\left\vert gg\right\rangle\), the effective four-photon blue sideband \(\left\vert gg\right\rangle \leftrightarrow \left\vert ff\right\rangle\) can be observed in Fig. 2(c).

Error correction performance

The logical state initialization requires sequential application of multiple single-qutrit and two-qutrit rotations. For \(\left\vert {L}_{0}\right\rangle\) and \(\left\vert {L}_{1}\right\rangle\), QQ red and blue sidebands are used to generate entanglement, and for \(\left\vert {L}_{x}\right\rangle=(\left\vert {L}_{0}\right\rangle+\left\vert {L}_{1}\right\rangle )/\sqrt{2}=(\left\vert g\right\rangle+\left\vert f\right\rangle )(\left\vert g\right\rangle -\left\vert f\right\rangle )/2\), only single qutrit rotations are required. The preparation times for initial states are separately 313 ns, 142 ns, and 282 ns for \(\left\vert {L}_{0}\right\rangle,\,\left\vert {L}_{1}\right\rangle\) and \(\left\vert {L}_{x}\right\rangle\). The detailed preparation circuit is discussed in Supplementary Note 7. We perform full two-qutrit state tomography36,37 and obtain initial state fidelities of 88.1%, 89.1% and 88.7% for the three states respectively. The tomography sequences and density matrix reconstruction are shown in Supplementary Note 8.

We characterize the performance of the Star code by comparing three different cases — free decay, QQ sideband spin-locking (4 QQ echo), and full AQEC. For free decay, we do not apply any drive after the state preparation. For the 4 QQ echo case, we turn on the QQ sidebands \(\left\vert ee\right\rangle \leftrightarrow \{\left\vert gf\right\rangle,\left\vert fg\right\rangle,\left\vert gg\right\rangle,\left\vert ff\right\rangle \}\) with a similar rate-detuning configuration as shown in Fig. 1a (Wr = 1.0 MHz, Wb = 1.7 MHz, νr = 1.5 MHz, νb = 0.0 MHz). This case shows coherence improvement from spin-locking. The full AQEC (Wr = 1.45 MHz, Wb = 1.25 MHz, νr = 0.8 MHz, νb = − 0.9 MHz, Ω1 = Ω2 = 0.39 MHz) demonstrates further improvement from photon-loss correction. We plot the density matrices of the logical states after preparation and after 9 μs in Supplementary Note 8 for reference.

To demonstrate that our protocol corrects single-photon loss error, in Fig. 3, we plot the combined population of error states as a function of time for all three cases. The error populations are computed through the expectation values of \({\varepsilon }_{0}=\left\vert ge\right\rangle \left\langle ge\right\vert+\left\vert eg\right\rangle \left\langle eg\right\vert\) for \(\left\vert {L}_{0}\right\rangle,\,{\varepsilon }_{1}=\left\vert ef\right\rangle \left\langle ef\right\vert+\left\vert fe\right\rangle \left\langle fe\right\vert\) for \(\left\vert {L}_{1}\right\rangle\), and ε0 + ε1 for \(\left\vert {L}_{x}\right\rangle\) corresponding to the states after single-photon loss. We extract the error population from the density matrices reconstructed with full two-qutrit state tomography at each time point up to 27 μs using the Maximum Likelihood Estimation (MLE) from 5000 repetitions of 81 different pre-rotation measurements for each state. This is a direct demonstration of the AQEC’s effectiveness, as it measures the error state population designed to correct by the protocol. Compared to the free decay cases (black dots), turning on the AQEC clearly corrects photon loss and suppresses the error rate below the free decay cases (green dots). The error rates for all three logical states increase in the 4 QQ echo case (blue dots), as enhanced qutrit decay rates in the presence of sideband can lead to extra photon loss.

Fig. 3: Error population under different conditions.
figure 3

Black, blue, and green points represent tomographic measurement results under free decay, 4 QQ echo, and full AQEC. The y-axes represent the combined population of error states for initial states a\(\left\vert {L}_{0}\right\rangle\), b\(\left\vert {L}_{1}\right\rangle\), and c\(\left\vert {L}_{x}\right\rangle\). Population accumulates at the error states in the free decay case, enhanced in the 4 QQ echo case, and corrected with AQEC drive on. The experimental data is explained with master equation simulations shown in solid lines. Error bars (one standard deviation) are smaller than the marker size38. Detailed fitting parameters used for the solid lines are shown in Supplementary Note 10.

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In addition to correcting photon loss, it is also important to characterize how well the AQEC protocol preserves the coherence of the logical states. To quantify the coherence, we plot the fidelity decay for each logical state. Fidelities are calculated as \(F={(tr\sqrt{\sqrt{{\rho }_{m}}{\rho }_{th}\sqrt{{\rho }_{m}}})}^{2}\), where \({\rho }_{th}=\left\vert {L}_{i}\right\rangle \left\langle {L}_{i}\right\vert,\,i=0,\,1,\,x\) depending on the compared logical states, and ρm is the experimentally measured density matrix. All expectation values’ error bars are calculated using the Tomographer package38. Fitting the data to the exponential decays for \(\left\vert {L}_{0}\right\rangle\) and \(\left\vert {L}_{1}\right\rangle\)39, the logical states’ coherence are improved from 17.9 μs (\(\left\vert {L}_{0}\right\rangle\)) and 3.4 μs (\(\left\vert {L}_{1}\right\rangle\)) in the free decay cases, to 20.3 μs and 16.8 μs in the four QQ echo cases, and up to 28.1 μs and 16.3 μs in the error correction cases (see Fig. 4(a, b)). This demonstrates a factor of 1.6 and 4.8 improvement in logical state coherence against the free decay case. We believe the coherence limit of \(\left\vert {L}_{0}\right\rangle\) and \(\left\vert {L}_{1}\right\rangle\) is limited by QR sidebands not being sufficiently small compared to the QQ sidebands, which causes leakage into the \(\{\left\vert T\right\rangle,\left\vert {S}_{\pm }\right\rangle \}\) states. \(\left\vert {L}_{x}\right\rangle\)’s lifetime after error correction is on par with the free decay case. The uncompensated ZZ coupling between transmons dephases the \(\left\vert {L}_{x}\right\rangle\) after error correction, but not for \(\left\vert {L}_{0}\right\rangle\) and \(\left\vert {L}_{1}\right\rangle\). Therefore, we expect \(\left\vert {L}_{x}\right\rangle\)’s coherence time should be worse than that for \(\left\vert {L}_{0}\right\rangle\) and \(\left\vert {L}_{1}\right\rangle\). The fact that turning on QR sidebands does not significantly worsen the \(\left\vert {L}_{x}\right\rangle\) coherence shows that without photon loss error, QR sidebands do not introduce significant dephasing error.

Fig. 4: Coherence improvement.
figure 4

Black, blue, and green circles are experimentally measured state fidelities at a given time. The fidelities are extracted from the tomographic reconstruction of states with 5000 repeated measurements. Error bars (one standard deviation) are smaller than the marker size38. The improvement with AQEC turned on is explained by the master equation simulation. a\(\left\vert {L}_{0}\right\rangle\) and b\(\left\vert {L}_{1}\right\rangle\) traces are fitted to the exponential decay curve \(A\exp (-t/\tau )+C\), and c\(\left\vert {L}_{x}\right\rangle\) traces are fitted to \(A\exp (-t/\tau )\). The offset C is necessary since fidelity will achieve steady-state values. The error (one standard deviation) for τ are obtained from the fitting. The large uncertainty comes from treating C as a free variable in the fitting. The fast transition period (first \(1.5\,\mu {{{{{{{\rm{s}}}}}}}} \sim {\Omega }_{j}^{-1}\) in the AQEC case is not included in the fitting* for a better representation of logical coherence. *First 10.5 μs data are used in \(\left\vert {L}_{1}\right\rangle\)’s free decay case for better fitting.

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The large difference in free-decay coherence times between \(\left\vert {L}_{0}\right\rangle\) and \(\left\vert {L}_{1}\right\rangle\) originates from the low-frequency dephasing noise on ΦDC through the flux line. It causes a shift in both transmons’ frequencies in the same direction, which \(\left\vert {L}_{1}\right\rangle\) is sensitive to but \(\left\vert {L}_{0}\right\rangle\) is not. The passive echo protection from the Star code drives suppresses this; consequently, in the 4 QQ echo case both logical states have similar coherence time.

The AQEC performance is primarily limited by three factors in our experiment. The most important fact is that the QQ sideband rates Wr and Wb are well below their ideal values. Stronger drives would further suppress phase noise (lifetimes in the 4 QQ echo experiment are well below 2T1, indicating room for improvement), and the increased energy separation would also allow us to use stronger QR drives, correcting photon loss more quickly. Although the coupler supports 9 MHz QQ sidebands for short periods, when Wb goes beyond 5 MHz the readout resonator frequency starts to shift, introducing systematic measurement distortion (See Supplementary Note 5 for details). This problem worsens with all six tones applied and we stay well below this limit to ensure reliable tomography results. The second limit is the ZZ coupling between the transmons, an extra dephasing channel for superposition states (see Supplementary Note 4 for details). Our coupler is operated at the minimum ZZ flux bias of the coupler to minimize the effect. It could be further mitigated by stronger QR sidebands enabling faster error correction, or through additional off-resonant QQ drive terms to dynamically cancel it. The third limit comes from heating and physical coherence drop when sidebands are turned on. The average photon number in the readout increases from < 0.01 (free decay and 4 QQ echo cases) to 0.03 (AQEC case), which will actively convert a logical state to the error state under the QR interactions and significantly reduces logical lifetime. Photon-excitation events in the transmon are also non-correctable errors, but they should make smaller contributions to logical coherence (see Supplementary Note 10). Further improvement can thus come from two paths—improving isolation between control signals or improving physical qubit coherence so that weaker drives can be more effective. Other limits are in the order of milliseconds (see Supplementary Note 10) and do not affect our results considerably.

Discussion

In summary, we have experimentally demonstrated all interactions required for a hardware-efficient AQEC code, the Star code, utilizing only two transmon-resonator pairs and a linear coupler to perform the second-order transitions. Three levels per transmon are used to store information, with the middle level capturing photon loss error, and entropy is dumped to the resonator autonomously through the always-on cooling sidebands. Inter-transmon parametric drives are applied to the coherence-preserving coupler for separating the Star code logical space from other eigenstates. We demonstrated a clear low-frequency dephasing suppression by turning on all QQ sidebands and a minor improvement after turning on the additional error correction drives because of the presence of residual ZZ coupling. The static ZZ is suppressed with the inductive coupler while engineering the cancellation of the dynamical ZZ arising in the presence of all sidebands remains a topic of future research (see Supplementary Note 6 for details). Our system is entirely constructed from scalable components and fundamentally avoids the need for fast and accurate error detection and feedback error correction pulses. The Star code can be a self-corrected building block for the surface code3,40 to further correct higher-order errors when scaled up, and can be a fault-tolerant qubit for the bosonic system.

In the future, engineering a ZZ-free coupler would remove the primary source of decoherence in this work. Error-transparent single-qubit and two-qubit gates have been proposed theoretically to extend the Star code beyond single qubits17. The Star code can also be implemented in other platforms with an anharmonic three-level structure.

Methods

Error bars (one standard deviation) for all expectation values calculated from the Maximum Likelihood Estimation(MLE) reconstructed density matrix (See Supplementary Note 8) use the Tomographer package38.