Quantum error detection in qubit-resonator star architecture

TL;DR

The paper proposes a six-qubit star lattice architecture with a central resonator to enable efficient quantum error detection and correction on superconducting qubits. It demonstrates encoding two logical qubits using the [[4,2,2]] code on a single star QPU (Deneb) and analyzes the encoded states via classical shadow tomography and repeated stabilizer measurements. Key findings include logical fidelities $F_L>0.965$, logical lifetimes exceeding the best physical qubits, and logical error per cycle as low as about $0.25\%$, with a logical Bell state achieving $\tau_\Phi = 400 \pm 30$ μs and $\varepsilon_\Phi = 0.25 \pm 0.02\%$, illustrating robust entanglement preservation under error detection. These results demonstrate hardware-efficient, parallel stabilizer measurements enabled by the star topology and highlight a viable route toward tiled architectures capable of supporting higher-weight stabilizers and codes such as color codes and qLDPC codes, while acknowledging leakage and flag-qubit considerations for further improvements.

Abstract

Achieving industrial quantum advantage is unlikely without the use of quantum error correction (QEC). Other QEC codes beyond surface code are being experimentally studied, such as color codes and quantum Low-Density Parity Check (qLDPC) codes, that could benefit from new quantum processing unit (QPU) architectures. We introduce the six-qubit star lattice architecture that offers parallelism and effective local all-to-all connectivity and thus enables hardware-efficient implementation of certain QEC codes. As a first demonstration of this new architecture, we encode two logical qubits in a six-qubit superconducting QPU with a star-topology using the [[4, 2, 2]] code and characterize the logical states with the classical shadow framework. Logical life-time and logical error rate are measured over repeated quantum error detection cycles for various logical states including a logical Bell state. We measure logical state fidelities above 96 % for every cardinal logical state, find logical life-times above the best physical element, and logical error-per-cycle values ranging from 0.25(2) % to 0.91(3) %. In future, such star QPU can be tiled to enable QEC codes with high-weight and overlapping stabilizers for improved encoding rates.

Figures (14)

• Figure 1: Illustration of the six-qubit star lattice architecture with a resonator as central element. Apart from the boundary region, every central resonator connects to six qubits via respective tunable couplers and every qubit is coupled to three resonators. Through these resonators, each qubit is able to interact with its 12 nearest-neighbor qubits as shown in the highlighted sublattice.
• Figure 2: a) Schematic of the six-qubit-star QPU featuring four data qubits ($\mathrm{D}$1--$\mathrm{D}$4) and two ancilla qubits ($\mathrm{\mathrm{A}{}_\mathrm{X}{}}$ and $\mathrm{\mathrm{A}{}_\mathrm{Z}{}}$) connected via tunable couplers (green) to a central resonator (Res., orange). b) Gate sequence of the distance-2 error detection cycle consisting of state preparation sub-circuit; stabilizer measurement cycles implemented with single qubit $\sqrt{Y}$ gates, its inverse $\sqrt{Y}^\dagger$, resonator-qubit $\mathrm{CZ}{}$ and $\mathrm{MOVE}{}$ gates and ancilla qubit measurements; single qubit gates for state tomography and final data qubit readout operations, see App. \ref{['Supp_section:Detailed_QEC_circuit']}. stabilizer measurement cycles can be repeated $N$ time, with the $\mathrm{A}{}_\mathrm{X}{}$ measurement of cycle $n$ starting at the beginning of cycle $n+1$.
• Figure 3: Single stabilizer measurement expectation values $\bar{s}_{\rm exp.}^\mathrm{Z}{}$ (a) and $\bar{s}_{\rm exp.}^\mathrm{X}{}$ (b) for initial state $\psi_{\mathrm{in}}$, averaged over $10^5$ repetitions from experiment (green) and error-model simulation (blue). $\ket{\pm} = \left(\ket{0}\pm\ket{1} \right)/\sqrt{2}$.
• Figure 4: a,b) Fraction of successful run $\eta$ where no error have been detected versus number of cycle $N$ and time $t$, for the logical states in caption. c-f) Expectation values of the logical operators $Z_{\mathrm{L}{}1}$, $Z_{\mathrm{L}{}2}$, $X_{\mathrm{L}{}1}$ and $Z_{\mathrm{L}{}2}$ versus number of cycle. The error bars are one standard deviation calculated considering a binomial distribution with $\eta_N$. Solid lines are simulations.
• Figure 5: Repeated error detection for a logical Bell state. (a) Fraction of successful runs during repeated error detection cycles. (b) Logical state probability measured (markers) and simulated (solid lines) over number of cycles for the states $\ket{00}_\mathrm{L}{}$, $\ket{01}_\mathrm{L}{}$, $\ket{10}_\mathrm{L}{}$, $\ket{11}_\mathrm{L}{}$ and $\ket{\Phi{}}_{\mathrm{L}{}}$. The whiskers represent one standard deviation considering a binomial distribution of $\eta_N$. Time is calculated from the duration of the error detection cycle. (c) Logical fidelity $F_\mathrm{L}{}$, logical purity ${ p_{2, L}}$, physical purity $p_{2,\mathrm{phy}{}}$ of the Bell state given by classical shadow.