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arXiv:2503.14506 · Physical Review A

Scalable and fault-tolerant preparation of encoded k -uniform states

Shayan Majidy, Dominik Hangleiter, Michael J. Gullans

Published Sep 17, 2025• 0 citations
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Abstract

$k$-uniform states are valuable resources in quantum information, enabling tasks such as teleportation, error correction, and accelerated quantum simulations. However, the practical realization of $k$-uniform states, at scale, faces major obstacles: Verifying $k$ uniformity is as difficult as measuring code distances, and devising fault-tolerant logical preparation protocols further adds to the complexity. To address these challenges, we present a scalable fault-tolerant method for preparing encoded $k$-uniform states and we illustrate our approach using surface and color codes. We first present a technique to determine $k$ uniformity of stabilizer states directly from their stabilizer tableau. We then identify a family of Clifford circuits that ensures both fault tolerance and scalability in preparing these states. Building on the encoded $k$-uniform states, we introduce a hybrid physical-logical strategy that retains some of the error-protection benefits of logical qubits while lowering the overhead for implementing arbitrary gates compared to fully logical algorithms. We show that this hybrid approach can outperform fully physical implementations for resource-state preparation, as demonstrated by explicit constructions of $k$-uniform states.

DOI
10.1103/cl23-c3jg

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kkk-uniform states are valuable resources in quantum information, enabling tasks such as teleportation, error correction, and accelerated quantum simulations. However, the practical realization of kkk-uniform states, at scale, faces major obstacles: Verifying kkk uniformity is as difficult as measuring code distances, and devising fault-tolerant logical preparation protocols further adds to the complexity. To address these challenges, we present a scalable fault-tolerant method for preparing encoded kkk-uniform states and we illustrate our approach using surface and color codes. We first present a technique to determine kkk uniformity of stabilizer states directly from their stabilizer tableau. We then identify a family of Clifford circuits that ensures both fault tolerance and scalability in preparing these states. Building on the encoded kkk-uniform states, we introduce a hybrid physical-logical strategy that retains some of the error-protection benefits of logical qubits while lowering the overhead for implementing arbitrary gates compared to fully logical algorithms. We show that this hybrid approach can outperform fully physical implementations for resource-state preparation, as demonstrated by explicit constructions of kkk-uniform states.
In fault-tolerant quantum computation, the preparation of logical states is a ubiquitous subroutine, yet significant challenges persist even for the simplest states required. In the present work, we present a unitary, scalable, distance-preserving encoding scheme for preparing Pauli eigenstates in surface codes. Unlike previous unitary approaches whose fault-distance remains constant with increasing code distance, our scheme ensures that the protection offered by the code is preserved during state preparation. Building on strategies discovered by reinforcement learning for the surface-17 code, we generalize the construction to arbitrary code distances and both rotated and unrotated surface codes. The proposed encoding relies only on geometrically local gates, and is therefore fully compatible with planar 2D qubit connectivity, and it achieves circuit depth scaling as O(d)\mathcal{O}(d)O(d), consistent with fundamental entanglement-generation bounds. We design explicit stabilizer-expanding circuits with and without ancilla-mediated connectivity and analyze their error-propagation behavior. Numerical simulations under depolarizing noise show that our unitary encoding without ancillas outperforms standard stabilizer-measurement-based schemes, reducing logical error rates by up to an order of magnitude. These results make the scheme particularly relevant for platforms such as trapped ions and neutral atoms, where measurements are costly relative to gates and idling noise is considerably weaker than gate noise. Our work bridges the gap between measurement-based and unitary encodings of surface-code states and opens new directions for distance-preserving state preparation in fault-tolerant quantum computation.
Fault-tolerant logical entangling gates are essential for scalable quantum computing, but are limited by the error rates and overheads of physical two-qubit gates and measurements. To address this limitation, we introduce phantom codes-quantum error-correcting codes that realize entangling gates between all logical qubits in a code block purely through relabelling of physical qubits during compilation, yielding perfect fidelity with no spatial or temporal overhead. We present a systematic study of such codes. First, we identify phantom codes using complementary numerical and analytical approaches. We exhaustively enumerate all 2.71×10102.71 \times 10^{10}2.71×1010 inequivalent CSS codes up to n=14n=14n=14 and identify additional instances up to n=21n=21n=21 via SAT-based methods. We then construct higher-distance phantom-code families using quantum Reed-Muller codes and the binarization of qudit codes. Across all identified codes, we characterize other supported fault-tolerant logical Clifford and non-Clifford operations. Second, through end-to-end noisy simulations with state preparation, full QEC cycles, and realistic physical error rates, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. We observe a one-to-two order-of-magnitude reduction in logical infidelity at comparable qubit overhead for GHZ-state preparation and Trotterized many-body simulation tasks, given a modest preselection acceptance rate. Our work establishes phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure, and introduces general tools for systematically exploring the broader landscape of quantum error-correcting codes.
Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing this overhead is key for the practical development of large-scale fault-tolerant quantum computers. We propose and experimentally implement a magic-state preparation protocol to fault-tolerantly prepare a pair of logical magic states in a [[6,2,2]] quantum error-detecting code using only eight physical qubits. Implementing this protocol on H1-1, a 20 qubit trapped-ion quantum processor, we prepare magic states with experimental infidelity 7−1+3×10−57^{+3}_{-1}\times 10^{-5}7−1+3​×10−5 with a 14.8−1+1%14.8^{+1}_{-1}\%14.8−1+1​% discard rate and use these to perform a fault-tolerant non-Clifford gate, the controlled-Hadamard (CH), with logical infidelity ≤2.3−9+9×10−4\leq 2.3^{+9}_{-9}\times 10^{-4}≤2.3−9+9​×10−4. Notably, this significantly outperforms the unencoded physical CH infidelity of 10−310^{-3}10−3. Through circuit-level stabilizer simulations, we show that this protocol can be self-concatenated to produce extremely high-fidelity magic states with low space-time overhead in a [[36,4,4]] quantum error correcting code, with logical error rates of 6×10−106\times 10^{-10}6×10−10 (5×10−145\times 10^{-14}5×10−14) at two-qubit error rate of 10−310^{-3}10−3 (10−410^{-4}10−4) respectively.
Quantum computation holds the promise of solving certain complex problems exponentially faster than classical computers. However, the high prevalent noise in current quantum devices impedes the accurate execution of even basic algorithms. This can be remedied by protecting quantum information with a quantum error-correcting code, where the logical information of an algorithmic qubit is spread across multiple physical qubits. Individual quantum errors are then located and corrected by the fault-tolerant measurement of multi-qubit stabilizer operators (parity checks). Unfortunately, error correction and fault tolerance both impose large demands on the qubit overhead: hundreds to thousands of physical qubits per logical qubit. We reduce the spacetime cost of fault tolerance by redesigning key building blocks of an error-corrected quantum computer. First, we develop a combinatorial proof with flag fault tolerance that exponentially reduces the extra qubits needed to measure a stabilizer of any size, while tolerating one fault. We leverage these proofs to then design state preparation circuits for the Steane and Golay codes with 100% yield. Next, we improve error correction on a planar layout by showing that a distance-four code encoding six logical qubits protects information as well as the distance-five surface code, using one-tenth as many physical qubits. Finally, we optimize the time overhead of logical gates in surface code quantum computers by protecting measurement results with a classical code, cutting computation time by a factor of two to six. Our hardware-agnostic optimizations of fault tolerance overheads thus suggest new routes to advance the timeline of error-free quantum computing.
Quantum error correction is an essential component for practical quantum computing on noisy quantum hardware. However, logical operations on error-corrected qubits require a significant resource overhead, especially for high-precision and high-fidelity non-Clifford rotation gates. To address this issue, we propose a postselection-based algorithm to efficiently prepare resource states for gate teleportation. Our algorithm achieves fault tolerance, demonstrating the exponential suppression of logical errors with code distance, and it applies to any stabilizer codes. We provide analytical derivations and numerical simulations of the fidelity and success probability of the algorithm. We benchmark the method on surface code and show a factor of 100 to 10,000 reduction in space-time overhead compared to existing methods. Overall, our approach presents a promising path to reducing the resource requirement for quantum algorithms on error-corrected and noisy intermediate-scale quantum computers.
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.