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arXiv:2605.02051

Sub-Cubic Quantum Gate Synthesis via Stochastic Commutator Decomposition

Yevgen Kotukh

quant-phSubmitted May 3, 2026• Announced May 5, 2026• 0 citations
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Abstract

We present Stochastic Commutator Synthesis, a hybrid quantum gate compilation framework that integrates Kuperberg's sub-cubic Solovay-Kitaev exponent c near 1.44042 with the error-tailoring machinery of randomized compilation. Classical Solovay-Kitaev implementations produce known word lengths and accumulate coherent approximation errors that degrade fault-tolerant threshold estimates. Kuperberg's 2023-2025 result reduces this via doubly exponential convergence and higher-order commutator decompositions. SCS augments this geometric backbone with a Gibbs-sampled stochastic choice of commutator factors at each recursion level, converting coherent synthesis residuals into incoherent, Pauli-twirl-compatible noise -- a property exploited by RC. Combined with RL-guided pre-synthesis, SCS achieves consistent T-count reductions of 10-25 percent and demonstrates fidelity gains of up to 35 percent on multi-fold Forrelation circuits on trapped-ion hardware such as Sandia QSCOUT. We situate SCS within the complexity-theoretic landscape established by the Raz-Tal oracle separation, arguing that low-error, noise-robust compilation of Forrelation-type circuits constitutes a practical pathway toward demonstrating this separation on physical hardware.

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We present Stochastic Commutator Synthesis, a hybrid quantum gate compilation framework that integrates Kuperberg's sub-cubic Solovay-Kitaev exponent c near 1.44042 with the error-tailoring machinery of randomized compilation. Classical Solovay-Kitaev implementations produce known word lengths and accumulate coherent approximation errors that degrade fault-tolerant threshold estimates. Kuperberg's 2023-2025 result reduces this via doubly exponential convergence and higher-order commutator decompositions. SCS augments this geometric backbone with a Gibbs-sampled stochastic choice of commutator factors at each recursion level, converting coherent synthesis residuals into incoherent, Pauli-twirl-compatible noise -- a property exploited by RC. Combined with RL-guided pre-synthesis, SCS achieves consistent T-count reductions of 10-25 percent and demonstrates fidelity gains of up to 35 percent on multi-fold Forrelation circuits on trapped-ion hardware such as Sandia QSCOUT. We situate SCS within the complexity-theoretic landscape established by the Raz-Tal oracle separation, arguing that low-error, noise-robust compilation of Forrelation-type circuits constitutes a practical pathway toward demonstrating this separation on physical hardware.
We analyze the use of the Solovay Kitaev (SK) algorithm to generate an ensemble of one qubit rotations over which to perform randomized compilation. We perform simulations to compare the trace distance between the quantum state resulting from an ideal one qubit RZR_{Z}RZ​ rotation and discrete SK decompositions. We find that this simple randomized gate synthesis algorithm can reduce the approximation error of these rotations in the absence of gate errors in simulation by at least a factor of two compared to a naive gate synthesis algorithm. We test the technique under the effects of a simple coherent noise model and find that it can mitigate coherent noise. We also run our algorithm on Sandia National Laboratories' QSCOUT trapped-ion device and find that randomization is able to help in the presence of realistic noise sources.
Fault-tolerant architectures implement non-Clifford T gates through magic-state distillation, so the T-count of a synthesized circuit dominates its physical cost. The Solovay-Kitaev algorithm approximates any single-qubit unitary from a finite gate set with a sequence length that grows only polylogarithmically in the inverse target error, but it optimizes for numerical convergence rather than circuit economy, and its output carries structural redundancy that a gate-level compiler cannot see. We report a measurement of what diagrammatic post-processing recovers from that redundancy. Twelve hundred random single-qubit targets, spanning the three Pauli rotation families and the general gate U(theta, phi, lambda), are synthesized over Clifford+T at three recursion depths, translated into graph-like ZX-diagrams, simplified by automated rewriting, and extracted back to circuits. Post-processing removes 26.6-30.1% of the total gate count and 18.5-22.2% of the T-count. The absolute saving grows with recursion depth, from about 60 to about 1600 gates, while the fractional saving does not: it rises slightly from the shallowest setting and is then flat across a twenty-five-fold change in circuit length, and by the deepest setting the four target families are no longer distinguishable from one another. Because the rewrite rules preserve the implemented linear map, the approximation error is unchanged. The compile-time cost of the rewriting layer, by contrast, grows sharply with depth and comes to dominate the synthesis itself.
Current hardware for quantum computing suffers from high levels of noise, and so to achieve practical fault-tolerant quantum computing will require powerful and efficient methods to correct for errors in quantum circuits. Here, we explore the role and effectiveness of using noise tailoring techniques to improve the performance of error correcting codes. Noise tailoring methods such as randomized compiling (RC) convert complex coherent noise processes to effective stochastic noise. While it is known that this can be leveraged to design efficient diagnostic tools, we explore its impact on the performance of error correcting codes. Of particular interest is the important class of coherent errors, arising from control errors, where RC has the maximum effect---converting these into purely stochastic errors. For these errors, we show here that RC delivers an improvement in performance of the concatenated Steane code by several orders of magnitude. We also show that below a threshold rotation angle, the gains in logical fidelity can be arbitrarily magnified by increasing the size of the codes. These results suggest that using randomized compiling can lead to a significant reduction in the resource overhead required to achieve fault tolerance.
Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to performing robust computations. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, which can dramatically reduce error rates while introducing little or no experimental overhead. Moreover, we prove that our technique is robust to the inevitable variation in errors over the randomizing gates and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity---comparable to fidelities realized in current experiments---are sufficient to achieve fault-tolerant quantum computation. Furthermore, the worst-case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking protocols, enabling a rigorous certification of the performance of a quantum computer.
The rapid evolution of Noisy Intermediate-Scale Quantum (NISQ) devices has intensified the need for robust compilation strategies capable of mitigating hardware-induced errors while maximizing computational efficiency. Quantum compilers play a pivotal role in translating high-level quantum algorithms into hardware-executable instructions, yet they face significant challenges related to qubit connectivity constraints, limited coherence times, and stochastic noise. This paper examines error-resilient quantum compiler design for efficient qubit mapping, gate optimization, and noise mitigation in NISQ-era devices, proposing a framework that integrates topology-aware mapping, adaptive gate synthesis, and probabilistic error suppression techniques. By synthesizing advances in quantum circuit optimization, hardware-aware scheduling, and hybrid classical–quantum error correction strategies, the study evaluates how compiler-level innovations can substantially enhance execution fidelity and resource efficiency. The findings suggest that intelligent compiler architectures are critical enablers of practical quantum advantage during the transitional NISQ phase, bridging theoretical algorithm design with real-world hardware constraints.
Randomized compiling reduces the effects of errors on quantum computers by tailoring arbitrary Markovian errors into stochastic Pauli noise. Here we prove that randomized compiling also tailors non-Markovian errors into local stochastic Pauli noise and investigate the technique's limitations. We show through analysis and numerical results that randomized compiling alters errors in three distinct helpful ways. First, it prevents the coherent accumulation of errors (including hard to remove crosstalk effects) across gate cycles by destroying intercycle coherent correlations. Second, it converts individual gate cycle errors into Pauli noise. Finally, randomized compiling reduces the variability inherent to noisy devices. We confirm these theoretical predictions with the IBM Quantum Experience platform and describe experimental data that illustrates a drastic performance improvement across public devices. These results cement the importance of randomized compiling in near- and long-term quantum information processing.