High-threshold and low-overhead fault-tolerant quantum memory.

The accumulation of physical errors1-3 prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction4 promises a solution by encoding k logical qubits onto a larger number n of physical qubits, such that the physical errors are suppressed enough to allow running a desired computation with tolerable fidelity. Quantum error correction becomes practically realizable once the physical error rate is below a threshold value that depends on the choice of quantum code, syndrome measurement circuit and decoding algorithm5. We present an end-to-end quantum error correction protocol that implements fault-tolerant memory on the basis of a family of low-density parity-check codes6. Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code7-10 that for 20 years was the leading code in terms of error threshold. The syndrome measurement cycle for a length-n code in our family requires n ancillary qubits and a depth-8 circuit with CNOT gates, qubit initializations and measurements. The required qubit connectivity is a degree-6 graph composed of two edge-disjoint planar subgraphs. In particular, we show that 12 logical qubits can be preserved for nearly 1 million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of 0.1%, whereas the surface code would require nearly 3,000 physical qubits to achieve said performance. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors.

Encoding a magic state with beyond break-even fidelity.

To run large-scale algorithms on a quantum computer, error-correcting codes must be able to perform a fundamental set of operations, called logic gates, while isolating the encoded information from noise1-8. We can complete a universal set of logic gates by producing special resources called magic states9-11. It is therefore important to produce high-fidelity magic states to conduct algorithms while introducing a minimal amount of noise to the computation. Here we propose and implement a scheme to prepare a magic state on a superconducting qubit array using error correction. We find that our scheme produces better magic states than those that can be prepared using the individual qubits of the device. This demonstrates a fundamental principle of fault-tolerant quantum computing12, namely, that we can use error correction to improve the quality of logic gates with noisy qubits. Moreover, we show that the yield of magic states can be increased using adaptive circuits, in which the circuit elements are changed depending on the outcome of mid-circuit measurements. This demonstrates an essential capability needed for many error-correction subroutines. We believe that our prototype will be invaluable in the future as it can reduce the number of physical qubits needed to produce high-fidelity magic states in large-scale quantum-computing architectures.

Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders.

Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.

Calibrated Decoders for Experimental Quantum Error Correction.

Arbitrarily long quantum computations require quantum memories that can be repeatedly measured without being corrupted. Here, we preserve the state of a quantum memory, notably with the additional use of flagged error events. All error events were extracted using fast, midcircuit measurements and resets of the physical qubits. Among the error decoders we considered, we introduce a perfect matching decoder that was calibrated from measurements containing up to size-four correlated events. To compare the decoders, we used a partial postselection scheme shown to retain ten times more data than full postselection. We observed logical errors per round of 2.2±0.1×10^{-2} (decoded without postselection) and 5.1±0.7×10^{-4} (full postselection), which was less than the physical measurement error of 7×10^{-3} and therefore surpasses a pseudothreshold for repeated logical measurements.

Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits.

Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting code words through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.

Demonstration of a quantum error detection code using a square lattice of four superconducting qubits.

The ability to detect and deal with errors when manipulating quantum systems is a fundamental requirement for fault-tolerant quantum computing. Unlike classical bits that are subject to only digital bit-flip errors, quantum bits are susceptible to a much larger spectrum of errors, for which any complete quantum error-correcting code must account. Whilst classical bit-flip detection can be realized via a linear array of qubits, a general fault-tolerant quantum error-correcting code requires extending into a higher-dimensional lattice. Here we present a quantum error detection protocol on a two-by-two planar lattice of superconducting qubits. The protocol detects an arbitrary quantum error on an encoded two-qubit entangled state via quantum non-demolition parity measurements on another pair of error syndrome qubits. This result represents a building block towards larger lattices amenable to fault-tolerant quantum error correction architectures such as the surface code.

Implementing a strand of a scalable fault-tolerant quantum computing fabric.

With favourable error thresholds and requiring only nearest-neighbour interactions on a lattice, the surface code is an error-correcting code that has garnered considerable attention. At the heart of this code is the ability to perform a low-weight parity measurement of local code qubits. Here we demonstrate high-fidelity parity detection of two code qubits via measurement of a third syndrome qubit. With high-fidelity gates, we generate entanglement distributed across three superconducting qubits in a lattice where each code qubit is coupled to two bus resonators. Via high-fidelity measurement of the syndrome qubit, we deterministically entangle the code qubits in either an even or odd parity Bell state, conditioned on the syndrome qubit state. Finally, to fully characterize this parity readout, we develop a measurement tomography protocol. The lattice presented naturally extends to larger networks of qubits, outlining a path towards fault-tolerant quantum computing.

Flexor tendon injuries following locked volar plating of distal radius fractures.

We present 2 cases showing that flexor pollicis longus and flexor digitorum profundus index injury can occur after placement of 2 commonly used locked volar plates. In contrast with the literature, the radii healed in an anatomic position without plate lift-off. The patients presented 6 and 8 months after surgery with new onset of radial wrist pain and tenderness at the site of the plate and absence or weakness of the flexor pollicis longus. In both cases, the plate was positioned anterior to the distal radial rim on the lateral radiograph. We suggest close follow-up of all fractures in which the distal end of the plate is anterior to the radial rim and removal of hardware if symptoms suggest tendon irritation.

Subsystem fault tolerance with the Bacon-Shor code.

We discuss how the presence of gauge subsystems in the Bacon-Shor code [D. Bacon, Phys. Rev. A 73, 012340 (2006)10.1103/PhysRevA.73.012340 (2006)] leads to remarkably simple and efficient methods for fault-tolerant error correction (FTEC). Most notably, FTEC does not require entangled ancillary states, and it can be implemented with nearest-neighbor two-qubit measurements. By using these methods, we prove a lower bound on the quantum accuracy threshold, 1.94 x 10(-4) for adversarial stochastic noise, that improves previous lower bounds by nearly an order of magnitude.