A fully fault-tolerant implementation of the quantum error-detecting Iceberg $[[2m, 2m-2, 2]]$ code applied to a Toffoli circuit achieved beyond-break-even error detection on a leading trapped-ion quantum computer, where the effect of encoding a circuit with a quantum error-detection code enables increased fidelity compared to an unencoded circuit. This code was also applied to Bell state preparation circuits, where a lean non-fault-tolerant implementation of the Iceberg code enables a fidelity gain as well. This highlights the important point that, at least for small-scale circuits with a substantial portion of error-free runs, it can be effective simply to use error detection to filter out the runs with errors. Furthermore, experiments performed in this work highlight the necessity for judicious compilation of circuits not only for a given hardware but also within a quantum error detection code.
A fully fault-tolerant implementation of the quantum error-detecting Iceberg [[2m,2m−2,2]] code applied to a Toffoli circuit achieved beyond-break-even error detection on a leading trapped-ion quantum computer, where the effect of encoding a circuit with a quantum error-detection code enables increased fidelity compared to an unencoded circuit. This code was also applied to Bell state preparation circuits, where a lean non-fault-tolerant implementation of the Iceberg code enables a fidelity gain as well. This highlights the important point that, at least for small-scale circuits with a substantial portion of error-free runs, it can be effective simply to use error detection to filter out the runs with errors. Furthermore, experiments performed in this work highlight the necessity for judicious compilation of circuits not only for a given hardware but also within a quantum error detection code.
The rapid progress in quantum hardware is expected to make them viable tools for the study of quantum algorithms in the near term. The timeline to useful algorithmic experimentation can be accelerated by techniques that use many noisy shots to produce an accurate estimate of the observable of interest. One such technique is to encode the quantum circuit using an error detection code and discard the samples for which an error has been detected. An underexplored property of error-detecting codes is the flexibility in the circuit encoding and fault-tolerant gadgets, which enables their co-optimization with the algorthmic circuit. However, standard circuit optimization tools cannot be used to exploit this flexibility as optimization must preserve the fault-tolerance of the gadget. In this work, we focus on the [[k+2,k,2]] Iceberg quantum error detection code, which is tailored to trapped-ion quantum processors. We design new flexible fault-tolerant gadgets for the Iceberg code, which we then co-optimize with the algorithmic circuit for the quantum approximate optimization algorithm (QAOA) using tree search. By co-optimizing the QAOA circuit and the Iceberg gadgets, we achieve an improvement in QAOA success probability from 44% to 65% and an increase in post-selection rate from 4% to 33% at 22 algorithmic qubits, utilizing 330 algorithmic two-qubit gates and 744 physical two-qubit gates on the Quantinuum H2-1 quantum computer, compared to the previous state-of-the-art hardware demonstration. Furthermore, we demonstrate better-than-unencoded performance for up to 34 algorithmic qubits, employing 510 algorithmic two-qubit gates and 1140 physical two-qubit gates.
Quantum error-detecting codes offer a near-term path for improving the performance of quantum simulations on noisy hardware. Using IBM's superconducting quantum computer ibm_boston, we show that partially fault-tolerant encoded quantum simulations of the Ising model in 1+1D and 2+1D outperform their unencoded counterparts in estimating local observables. To represent 42 logical qubits on the heavy-hex quantum processor, 21 blocks of the [[4, 2, 2]] Iceberg code and up to 136 physical qubits are used. By pairing fault-tolerant syndrome extraction with non-fault-tolerant logical operations, this scheme preserves many of the benefits of error detection while avoiding the overhead typically required for a fully fault-tolerant logical gate set. The encoding's square logical connectivity, together with the freedom to place logical qubits within each block, enables simulations of a 2D spatial lattice with lower circuit depth than the unencoded implementation requires. We introduce Observable-Ranked Postselection, a selective-filtering technique based on syndrome correlations that recovers reliable results without the prohibitive shot loss of full syndrome postselection. Under the cumulative effect of device errors, this encoding improves local-observable accuracy over the unencoded baseline by 2-6% at intermediate times in 1+1D simulations, growing with circuit depth to over 200% in 2+1D at the latest times studied.
High-rate quantum error correcting (QEC) codes encode many logical qubits in a given number of physical qubits, making them promising candidates for quantum computation. Implementing high-rate codes at a scale that both frustrates classical computing and improves performance by encoding requires both high fidelity gates and long-range qubit connectivity -- both of which are offered by trapped-ion quantum computers. Here, we demonstrate computations that outperform their unencoded counterparts in the high-rate [[k+2,k,2]] iceberg quantum error detecting (QED) and [[(k2+2)(k1+2),k2k1,4]] two-level concatenated iceberg QEC codes, using the 98-qubit Quantinuum Helios trapped-ion quantum processor. Utilizing new gadgets for encoded operations, we realize this "beyond break-even" performance with reasonable postselection rates across a range of fault-tolerant (FT) and partially-fault-tolerant (pFT) component and application benchmarks with between 48 and 94 logical qubits. These benchmarks include FT state preparation and measurement, QEC cycle benchmarking, logical gate benchmarking, GHZ state preparation, and a pFT quantum simulation of the three-dimensional XY model of quantum magnetism. Additionally, we illustrate that postselection rates can be suppressed by increasing the code distance via concatenation. Our results represent state-of-the-art logical component and state fidelities and provide evidence that high-rate QED/QEC codes are viable on contemporary quantum computers for near-term beyond-classical-scale computation.
The quantum approximate optimization algorithm (QAOA) is a promising candidate for scaling up to tackle real-world applications. However, achieving better-than-classical performance with QAOA is believed to require fault tolerance. In this paper, we demonstrate a partially fault-tolerant implementation of QAOA using the [[k + 2, k, 2]] “Iceberg” error detection code. We observe that encoding the circuit with the Iceberg code improves the algorithmic performance as compared to the unencoded circuit for problems with up to 20 logical qubits on a trapped-ion quantum computer. Additionally, we propose and calibrate a model for predicting the code performance. We use this model to characterize the limits of the Iceberg code and extrapolate its performance to future hardware with improved error rates. To the best of our knowledge, our results demonstrate the largest universal quantum computing algorithm protected by partially fault-tolerant quantum error detection on practical applications to date. The authors present a partially fault-tolerant implementation of the quantum approximate optimization algorithm. By encoding circuits with the iceberg error detection code, the authors improve QAOA’s performance on problems with up to 20 logical qubits on a trapped-ion quantum computer, outlining conditions required to surpass classical algorithms.
Scaling up quantum algorithms to tackle high-impact problems in science and industry requires quantum error correction and fault tolerance. While progress has been made in experimentally realizing error-corrected primitives, the end-to-end execution of logical quantum algorithms using only fault-tolerant (FT) components has remained out of reach. We demonstrate the FT and error-corrected execution of two quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA) and the Harrow-Hassidim-Lloyd (HHL) algorithm applied to the Poisson equation, on Quantinuum H2 and Helios trapped-ion quantum processors using the [[7,1,3]] Steane code. For QAOA circuits on 5 and 6 logical qubits, we show performance improvements from increasing the number of QAOA layers and the number of T gates used to approximate logical rotations, despite increased physical circuit complexity. We further show that QAOA circuits with up to 8 logical qubits and 9 logical T gates perform similarly to unencoded circuits. For the largest QAOA circuits we run, with 12 logical (97 physical) qubits and 2132 physical two-qubit gates, we still observe better-than-random performance. Finally, we show that adding active QEC cycles and increasing the repeat-until-success limit of state preparation subroutines can improve the performance of a quantum algorithm, thereby demonstrating critical capabilities of scalable FT quantum computation. Our results are enabled by an FT logical T gate implementation with an infidelity of ∼2.6(4)×10−3 and dynamic circuits with measurement-dependent feedback. Our work demonstrates near-break-even performance of complex, error-corrected algorithmic quantum circuits using only FT components.
Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.