Logical-qubit operations in an error-detecting surface code
Jorge Marques, Boris Varbanov, M. S. Moreira, Hany A. Ali, Nandini Muthusubramanian, Christos Zachariadis, Francesco Battistel, Marc Beekman, Nadia Haider, Wouter Vlothuizen, Alessandro Bruno, Barbara M. Terhal, L. DiCarlo
Abstract Future fault-tolerant quantum computers will require storing and processing quantum data in logical qubits. We realize a suite of logical operations on a distance-two logical qubit stabilized using repeated error detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference through detailed characterization. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.
Abstract Future fault-tolerant quantum computers will require storing and processing quantum data in logical qubits. We realize a suite of logical operations on a distance-two logical qubit stabilized using repeated error detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference through detailed characterization. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.
Fault-tolerant logical operations are fundamental for scalable quantum computation. Here, we report the experimental realization of lattice-surgery operations between a pair of distance-three surface-code logical qubits on a planar superconducting processor. During repeated syndrome extraction cycles, the logical qubits exhibit per-cycle error rates of 0.0365(2) and 0.0282(1), respectively, after leakage events are rejected. By leveraging joint initialization and lattice splitting, we deterministically prepare a logical Bell state, confirming genuine bipartite entanglement via the error-corrected logical state fidelity. We further execute a two-qubit Deutsch-Jozsa algorithm at the logical level to demonstrate algorithmic utility in a fault-tolerant framework. Finally, to achieve universal control, we implement magic-state injection and gate teleportation to realize continuous non-Clifford rotations about the logical X axis. For the logical RX(π/4) gate, we achieve a logical gate fidelity of 0.943−9+10 conditioned on the absence of detected errors. These results establish lattice surgery as a practical and versatile paradigm for logical computation in near-term surface-code architectures, representing a critical milestone toward scalable fault-tolerant quantum advantage in superconducting circuits.
Fault-tolerant quantum computing (FTQC) is essential for achieving large-scale practical quantum computation. Implementing arbitrary FTQC requires the execution of a universal gate set on logical qubits, which is highly challenging. Particularly, in the superconducting system, two-qubit gates on surface code logical qubits have not been realized. Here, we experimentally implement a logical CNOT gate along with arbitrary single-qubit rotation gates on distance-2 surface codes using the superconducting quantum processor Wukong , thereby demonstrating a universal logical gate set. In the experiment, we demonstrate the transversal CNOT gate on a two-dimensional topological processor based on a tailored encoding circuit, at the cost of removing the ancilla qubits required for stabilizer measurements. Furthermore, we fault-tolerantly prepare logical Bell states and observe a violation of CHSH inequality, confirming the entanglement between logical qubits. Using the logical CNOT gate and an ancilla logical state, arbitrary single-qubit rotation gates are realized through gate teleportation. All logical gates are characterized on a complete state set and their fidelities are evaluated by logical Pauli transfer matrices. The demonstration of a universal logical gate set and the entangled logical states highlights significant aspects of FTQC on superconducting quantum processors.
Stabilizer quantum error correction (QEC) codes, in particular topological surface codes, are prime candidates to enable practical quantum computing. While it is widely believed that strictly fault-tolerant protocols can only be implemented using single- and two-qubit gates, several quantum computing platforms, including trapped ions, neutral atoms, and superconducting qubits, support native multi-qubit operations. In this Letter, we show that stabilizer measurement circuits for unrotated surface codes can be fault tolerant using single auxiliary qubits and three-qubit gates. These gates enable lower-depth circuits with fewer fault locations and potentially shorter QEC cycle times. We find that in an optimistic parameter regime where fidelities of three-qubit gates are the same as those of two-qubit gates, the logical error rate can be up to one order of magnitude lower and the threshold significantly higher, increasing from ≈0.63% to ≈0.83%. Our results, applicable to a wide range of platforms, motivate further investigation into multi-qubit gates for fault-tolerant QEC as they can offer substantial time and physical qubit resource advantages to reach a given target logical error rate.
Fault-tolerant quantum computation requires logical operations that manipulate encoded information while preserving quantum error-correction protection. In planar surface-code architectures, code deformation and lattice surgery provide a local, measurement-based route to such operations. Here we experimentally realize key elements of patch-based surface-code logical processing on a 107-qubit superconducting quantum processor. We first implement a reusable primitive layer comprising merge and split, patch expansion and shrinkage, and deformations mediated by domain walls and twist defects. We then compose these primitives to realize logical state routing, the logical controlled-NOT gate, and the single-qubit Hadamard and phase gates, which together form a Clifford-generating set. All operations are implemented on distance-three rotated surface-code patches with multi-round syndrome extraction and neural-network decoding, without post-selection. Our results advance superconducting surface-code experiments from protected logical memory to active, patch-based fault-tolerant logical operations.
Quantum error correction (QEC) protects quantum systems against inevitable noises and control inaccuracies, providing a pathway toward fault-tolerant (FT) quantum computation. Stabilizer codes, including surface code and color code, have long been the focus of research and have seen significant experimental progress in recent years. Recently proposed time-dynamical QEC, including Floquet codes and generalized time-dynamical code implementations, opens up new opportunities for FT quantum computation. By employing a periodic schedule of low-weight parity checks, Floquet codes can generate additional dynamical logical qubits, offering enhanced error correction capabilities and, potentially, higher code performance. Here, we experimentally implement the Floquet-Bacon-Shor code on a superconducting quantum processor. We encode a dynamical logical qubit within a 3×3 lattice of data qubits alongside a conventional static logical qubit. We demonstrate FT encoding and measurement of the two-qubit logical states and stabilize these states using repeated error detection. We showcase universal single-qubit logical gates on the dynamical qubit. Furthermore, by implementing a logical controlled-not (cnot) gate, we entangle the dynamical and static logical qubits, generating an error-detected logical Bell state with a fidelity of 75.9%. Our results highlight the potential of Floquet codes for resource-efficient FT quantum computation.
Fault-tolerant logical entangling gates are essential for scalable quantum computing, but are limited by the error rates and overheads of physical two-qubit gates and measurements. To address this limitation, we introduce phantom codes-quantum error-correcting codes that realize entangling gates between all logical qubits in a code block purely through relabelling of physical qubits during compilation, yielding perfect fidelity with no spatial or temporal overhead. We present a systematic study of such codes. First, we identify phantom codes using complementary numerical and analytical approaches. We exhaustively enumerate all 2.71×1010 inequivalent CSS codes up to n=14 and identify additional instances up to n=21 via SAT-based methods. We then construct higher-distance phantom-code families using quantum Reed-Muller codes and the binarization of qudit codes. Across all identified codes, we characterize other supported fault-tolerant logical Clifford and non-Clifford operations. Second, through end-to-end noisy simulations with state preparation, full QEC cycles, and realistic physical error rates, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. We observe a one-to-two order-of-magnitude reduction in logical infidelity at comparable qubit overhead for GHZ-state preparation and Trotterized many-body simulation tasks, given a modest preselection acceptance rate. Our work establishes phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure, and introduces general tools for systematically exploring the broader landscape of quantum error-correcting codes.