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arXiv:2608.02944

The Utility of Sparse Error Detection in Quantum Simulations

Henry Froland, Dorota M. Grabowska, Sebastian Grieninger, Jeremy Hartse, Anne L. Lashbrook, Zhiyao Li, Ziyuan Li, Sarah J. M. Powell, Martin J. Savage, Xiaojun Yao, Nikita A. Zemlevskiy

quant-phhep-lathep-phnucl-thSubmitted Aug 3, 2026• Announced Aug 5, 2026• 0 citations
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Abstract

The recent success of error detecting codes points toward their potential application to fault-tolerant simulations of nature. In this work, we examine the utility of sparse error detection for simulating lattice gauge theories using quantum computers. In particular, we study the time evolution of the lattice Schwinger model embedded into the Iceberg code family, $[[N+2, N, 2]]$, as well as the Hypercube code family, $[[2^N, N, 2]]$. The lattice of electrons and positrons in the axial gauge is embedded into a single code block or into multiple code blocks, and this work finds that large codeblocks are advantageous in the absence of connectivity constraints. Noisy classical simulations with realistic near-term error rates, infrequent syndrome measurements and physics-aware postselection are found to improve observable estimation. Under realistic noise rates for near-term quantum computers, this work finds that sparse error detection in quantum simulations has the potential to improve accuracy of observable estimation. Additional rounds of error detection are found to systematically drive errors in observables to the noise floor set by the code. These findings suggest that incorporating minimal implementations of fault tolerance in the near-term will enhance the performance of quantum simulations in nuclear physics and high-energy physics.

Comments
32 pages, 37 figures, 4 tables, comments welcome

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The recent success of error detecting codes points toward their potential application to fault-tolerant simulations of nature. In this work, we examine the utility of sparse error detection for simulating lattice gauge theories using quantum computers. In particular, we study the time evolution of the lattice Schwinger model embedded into the Iceberg code family, [[N+2,N,2]][[N+2, N, 2]][[N+2,N,2]], as well as the Hypercube code family, [[2N,N,2]][[2^N, N, 2]][[2N,N,2]]. The lattice of electrons and positrons in the axial gauge is embedded into a single code block or into multiple code blocks, and this work finds that large codeblocks are advantageous in the absence of connectivity constraints. Noisy classical simulations with realistic near-term error rates, infrequent syndrome measurements and physics-aware postselection are found to improve observable estimation. Under realistic noise rates for near-term quantum computers, this work finds that sparse error detection in quantum simulations has the potential to improve accuracy of observable estimation. Additional rounds of error detection are found to systematically drive errors in observables to the noise floor set by the code. These findings suggest that incorporating minimal implementations of fault tolerance in the near-term will enhance the performance of quantum simulations in nuclear physics and high-energy physics.
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.
Many-hypercube codes, concatenated [[n,n−2,2]]{[[n,n-2,2]]}[[n,n−2,2]] quantum error-detecting codes (nnn is even), have recently been proposed as high-rate quantum codes suitable for fault-tolerant quantum computing. While the original many-hypercube codes with n=6{n=6}n=6 can achieve remarkably high encoding rates (about 30% and 20% at concatenation levels 3 and 4, respectively), they have large code block sizes at high levels (216 and 1296 physical qubits per block at levels 3 and 4, respectively), making not only experimental realization difficult but also logical error rates per code block high. Toward earlier experimental realization and lower logical error rates, here we comprehensively investigate smaller many-hypercube codes with [[6,4,2]][[6,4,2]][[6,4,2]] and/or [[4,2,2]][[4,2,2]][[4,2,2]] codes, where, e.g., D6,4,4D_{6,4,4}D6,4,4​ denotes the many-hypercube code using [[6,4,2]][[6,4,2]][[6,4,2]] at level 1 and [[4,2,2]][[4,2,2]][[4,2,2]] at levels 2 and 3. As a result, we found a counterintuitive fact that D6,4,4D_{6,4,4}D6,4,4​ (D6,6,4,4D_{6,6,4,4}D6,6,4,4​) can achieve lower logical error rates per code block than D4,4,4D_{4,4,4}D4,4,4​ (D4,4,4,4D_{4,4,4,4}D4,4,4,4​), despite its higher encoding rate and larger code block size. Focusing on level 3, we also developed efficient fault-tolerant encoders realizing about 60% overhead reduction while maintaining or even improving the performance, compared to the original design. Using them, we numerically confirmed that D6,4,4D_{6,4,4}D6,4,4​ also achieves the best performance for logical controlled-NOT gates in a circuit-level noise model. These results are important for targeting a high-rate code toward early experimental realization of efficient fault-tolerant quantum computing.
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.
Large-scale quantum computers mandate error correction and fault tolerance. Due to constraints on the number of qubits, fault tolerance is difficult to achieve in near-term quantum systems. Therefore, error correction should require minimal resources. Gates in the near-term devices are also noisy. Quantum error correction code blocks built with these noisy gates can inject further error in the circuit. The goals for error correction in near-term systems are as follows: (i) using a small number of qubits for encoding, and (ii) keeping cost of circuits for encoding and decoding low. In this paper, we propose two techniques to achieve these mutually orthogonal goals. For a binary quantum system we propose an error estimation method that can aid in reducing the number of error correcting blocks via sparse scheduling. For ternary quantum systems, we propose an approximate code that can correct errors with high probability while significantly reducing the circuit cost. These techniques are expected to be helpful for error mitigation in near-term systems in the absence of fault tolerance.
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]][[ k+2,\, k,\, 2 ]][[k+2,k,2]] iceberg quantum error detecting (QED) and [[(k2+2)(k1+2), k2k1, 4]][[ (k_2 + 2)(k_1 + 2),\, k_2k_1,\, 4 ]][[(k2​+2)(k1​+2),k2​k1​,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 484848 and 949494 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 XYXYXY 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 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]][[k+2, k, 2]][[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%44\%44% to 65%65\%65% and an increase in post-selection rate from 4%4\%4% to 33%33\%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.