Quantum Error Detection For Early Term Fault-Tolerant Quantum Algorithms

TL;DR

This work investigates quantum error detection (QED) using the $[[n,n-2,2]]$ iceberg code as a low-overhead route toward early fault-tolerant quantum computing. It develops an end-to-end fault-tolerant compilation and simulation framework, applies it to Grover's algorithm, and derives a statistical model to optimize mid-circuit syndrome scheduling under circuit-level noise. The key finding is that, with carefully chosen syndrome schedules, QED can yield up to about a 2x improvement in success probability for shallow circuits in moderate noise, though benefits diminish with increased circuit size, depth, or shot-budget constraints due to post-selection overhead. The results offer actionable guidelines for deploying QED in near-term experiments and position QED as a practical, constant-overhead error-mitigation layer rather than a scalable fault-tolerance solution, with open data and a compiler released for broader validation.

Abstract

Quantum error detection (QED) offers a promising pathway to fault tolerance in near-term quantum devices by balancing error suppression with minimal resource overhead. However, its practical utility hinges on optimizing design parameters-such as syndrome measurement frequency-to avoid diminishing returns from detection overhead. In this work, we present a comprehensive framework for fault-tolerant compilation and simulation of quantum algorithms using [[n, n-2, 2]] codes, which enable low-qubit-overhead error detection and a simple nearly fault-tolerant universal set of operations. We demonstrate and analyze our pipeline with a purely statistical interpretation and through the implementation of Grover's search algorithm. Our results are used to answer the question is quantum error detection a worthwhile avenue for early-term fault tolerance, and if so how can we get the most out of it? Simulations under the circuit-level noise model reveal that finding optimal syndrome schedules improves algorithm success probabilities by an average of 6.7x but eventual statistical limits from post-selection in noisy/resource-limited regimes constrain scalability. Furthermore, we propose a simple data-driven approach to predict fault tolerant compilation parameters, such as optimal syndrome schedules, and expected fault tolerant performance gains based on circuit and noise features. These results provide actionable guidelines for implementing QED in early-term quantum experiments and underscore its role as a pragmatic, constant-overhead error mitigation layer for shallow algorithms. To aid in further research, we release all simulation data computed for this work and provide an experimental QED compiler at https://codeqraft.xyz/qed.

Figures (18)

• Figure 1: Heatmaps showing experimental results for Grover's algorithm with a single marked state as functions of circuit locations ($L$) and noise model parameter ($p$). (Left: Optimal Syndrome Rate) The rate of syndrome measurements per logical operation that maximizes the probability of measuring the marked state. A white boundary indicates configurations where the optimal syndrome count is one, signifying no utility from adding mid-circuit syndrome measurements beyond the final readout. (Right: Normalized Success rate improvement), The ratio of success probabilities between the encoded implementation $p_{\text{enc}}$ using quantum error detection (with the optimal syndrome count) and a bare implementation $p_{\text{bare}}$ without error detection under identical noise conditions normalized up to $p_{\text{ideal}}$ the noiseless success probability: $p_{\text{enc}} - p_{\text{bare}}/p_{\text{ideal}} -p_{\text{bare}}$. The white boundary corresponds to a ratio where adding fault tolerance does not improve performance.
• Figure 2: Logical Gates: Logical gate implementations on the $\llbracket n,n-2,2 \rrbracket$ code. The figure illustrates the synthesis of various logical gates, including single-qubit gates ($X$, $Y$, $Z$, $S$, $H$), encoded multi-qubit gates (CNOT, CZ). Each subfigure shows the physical circuit for the corresponding logical operation on $n$ physical qubits.
• Figure 3: Optimal syndrome scheduling in error-detected circuits. When a user has the ability to increase the sensitivity of their fault-tolerant scheme for detecting errors through the implementation of syndrome measurements, they will simultaneously increase the false detection of errors via the syndrome circuits themselves. Such a trade-off presents non-obvious optimal syndrome scheduling strategies that will often necessitate experimental validation.
• Figure 4: Probability of successful state identification in 4 qubit Grover's search encoded with a $\llbracket 6,4,2 \rrbracket$ code under various noise conditions $p \in \{0.4\%, 0.6\%, 0.8\% \}$. The simulations delineate the success probabilities across $k = 1, 2, 3$ Grover iterations with different syndrome measurement schedules. The dashed lines represent the performance of an unencoded circuit under identical conditions. The uppermost black dashed line indicates the theoretical maximum success probability achievable in an ideal, noiseless Grover's search. The shaded areas around each line indicate the 95% confidence intervals, computed via percentile bootstrap over 50 independent trials.
• Figure 5: Shot count on success probability. We show $k=2$ iteration Grover search on the $\llbracket 6,4,2 \rrbracket$ code at noise levels averaged over runs with $p \in \{0.4\%, 0.6\%, 0.8\%\}$ (see the setup in \ref{['fig:grover_4q']}). The success probability is shown with both $10^3$ shots and $50\times 10^3$ shots per round to emphasize that in shot-limited environments adding syndrome measurements yields diminishing returns compared to the "shot-rich environment"---an effect explained in this work by the phenomenon of catastrophic failures. Survival probability (the fraction of shots that make it past post-selection) for the $50\times 10^3$ shot run is shown on the complementary axis. The dashed grey line shows the average success rate of the unencoded simulation under the same noise conditions.