arXiv subjects

Basudha Srivastava

Publications and source records attributed to Basudha Srivastava.

8 recordsLinked to original sources

arXiv2026

Duality constrains optimal thresholds in quantum error correction

Error correction thresholds are often treated as the primary figure of merit for comparing quantum error-correcting code families. We show that the optimal error correction threshold for many commonly considered codes is constrained to a single universal value at leading order in a replica limit. Through a statistical mechanical mapping, we demonstrate that duality constrains all zero-rate em-symmetric CSS codes to have the same optimal code capacity threshold. Here, em symmetry means that the X- and Z-type parity-check matrices are equivalent up to row and column permutations. Under this statistical mechanical mapping, em-symmetric CSS codes are self-dual under a generalized Kramers-Wannier duality up to a mixing of logical sectors. For zero-rate code families, this mixing contributes only subextensive corrections, so the thermodynamic bulk free energy is self-dual in the trivial logical sector. This self-duality fixes the clean critical point and constrains the disordered phase boundary. We also show that self-duality is preserved under code concatenation, and that optimal decoding of concatenated codes can be reformulated as a renormalization group flow on a hierarchical lattice. Our results provide a common framework for analyzing topological, concatenated, and more general quantum low-density parity-check code families, including both their optimal code capacity thresholds and their sub-threshold logical error suppression.

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arXiv2026

Computing with many encoded logical qubits beyond break-even

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 $[[ (k_2 + 2)(k_1 + 2),\, k_2k_1,\, 4 ]]$ 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.

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arXiv2025

Sequential decoding of the XYZ$^2$ hexagonal stabilizer code

Quantum error correction requires accurate and efficient decoding to optimally suppress errors in the encoded information. For concatenated codes, where one code is embedded within another, optimal decoding can be achieved using a message-passing algorithm that sends conditional error probabilities from the lower-level code to a higher-level decoder. In this work, we study the XYZ$^2$ topological stabilizer code, defined on a honeycomb lattice, and use the fact that it can be viewed as a concatenation of a [[2, 1, 1]] phase-flip parity check code and the surface code with $YZZY$ stabilizers, to decode the syndrome information in two steps. We use this sequential decoding scheme to correct errors on data qubits, as well as measurement errors, under various biased error models using both a maximum-likelihood decoder (MLD) and more efficient matching-based decoders. For depolarizing noise we find that the sequential matching decoder gives a threshold of 18.3%, close to optimal, as a consequence of a favorable, effectively biased, error model on the upper-level YZZY code. For phase-biased noise on data qubits, at a bias $η= \frac{p_z}{p_x+p_y} = 10$, we find that a belief-matching-based decoder reaches thresholds of 24.1%, compared to 28.6% for the MLD. With measurement errors the thresholds are reduced to 3.4% and 4.3%, for depolarizing and biased noise respectively, using the belief-matching decoder. This demonstrates that the XYZ$^2$ code has thresholds that are competitive with other codes tailored to biased noise. The results also showcase two approaches to taking advantage of concatenated codes: 1) tailoring the upper-level code to the effective noise profile of the decoded lower-level code, and 2) making use of an upper-level decoder that can utilize the local information from the lower-level code.

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arXiv2025

Data-driven decoding of quantum error correcting codes using graph neural networks

To leverage the full potential of quantum error-correcting stabilizer codes it is crucial to have an efficient and accurate decoder. Accurate, maximum likelihood, decoders are computationally very expensive whereas decoders based on more efficient algorithms give sub-optimal performance. In addition, the accuracy will depend on the quality of models and estimates of error rates for idling qubits, gates, measurements, and resets, and will typically assume symmetric error channels. In this work, instead, we explore a model-free, data-driven, approach to decoding, using a graph neural network (GNN). The decoding problem is formulated as a graph classification task in which a set of stabilizer measurements is mapped to an annotated detector graph for which the neural network predicts the most likely logical error class. We show that the GNN-based decoder can outperform a matching decoder for circuit level noise on the surface code given only simulated experimental data, even if the matching decoder is given full information of the underlying error model. Although training is computationally demanding, inference is fast and scales approximately linearly with the space-time volume of the code. We also find that we can use large, but more limited, datasets of real experimental data [Google Quantum AI, Nature {\bf 614}, 676 (2023)] for the repetition code, giving decoding accuracies that are on par with minimum weight perfect matching. The results show that a purely data-driven approach to decoding may be a viable future option for practical quantum error correction, which is competitive in terms of speed, accuracy, and versatility.

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arXiv2024

Exact results on finite size corrections for surface codes tailored to biased noise

The code-capacity threshold of a scalable quantum error correcting stabilizer code can be expressed as a thermodynamic phase transition of a corresponding random-bond Ising model. Here we study the XY and XZZX surface codes under phase-biased noise, $p_x=p_y=p_z/(2η)$, with $η\geq 1/2$, and total error rate $p=p_x+p_y+p_z$. By appropriately formulating the boundary conditions, in the rotated code geometry, we find exact solutions at a special disordered point, $p=\frac{1+η^{-1}}{2+η^{-1}}\gtrsim 0.5$, for arbitrary odd code distance $d$, where the codes reduce to one-dimensional Ising models. The total logical failure rate is given by $P_{f}=\frac{3}{4}-\frac{1}{4}e^{-2d_Z\,\text{artanh}(1/2η)}$, where $d_{Z}=d^2$ and $d$ for the two codes respectively, is the effective code distance for pure phase-flip noise. As a consequence, for code distances $d\ll η$, and error rates near the threshold, the XZZX code is effectively equivalent to the phase-flip correcting repetition code over $d$ qubits. The large finite size corrections for $d_Z<η$ also make threshold extractions, from numerical calculations at moderate code distances, unreliable. We show that calculating thresholds based not only on the total logical failure rate, but also independently on the phase- and bit-flip logical failure rates, gives a more confident estimate. Using this method for the XZZX code with a tensor-network based decoder and code distances up to $d\approx 100$, we find that the thresholds converge to a single value at moderate bias ($η=30, 100$), at an error rate above the hashing bound.

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arXiv2022

Error-rate-agnostic decoding of topological stabilizer codes

Efficient high-performance decoding of topological stabilizer codes has the potential to crucially improve the balance between logical failure rates and the number and individual error rates of the constituent qubits. High-threshold maximum-likelihood decoders require an explicit error model for Pauli errors to decode a specific syndrome, whereas lower-threshold heuristic approaches such as minimum weight matching are "error agnostic". Here we consider an intermediate approach, formulating a decoder that depends on the bias, i.e., the relative probability of phase-flip to bit-flip errors, but is agnostic to error rate. Our decoder is based on counting the number and effective weight of the most likely error chains in each equivalence class of a given syndrome. We use Metropolis-based Monte Carlo sampling to explore the space of error chains and find unique chains, that are efficiently identified using a hash table. Using the error-rate invariance the decoder can sample chains effectively at an error rate which is higher than the physical error rate and without the need for "thermalization" between chains in different equivalence classes. Applied to the surface code and the XZZX code, the decoder matches maximum-likelihood decoders for moderate code sizes or low error rates. We anticipate that, because of the compressed information content per syndrome, it can be taken full advantage of in combination with machine-learning methods to extrapolate Monte Carlo-generated data.

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arXiv2022

The XYZ$^2$ hexagonal stabilizer code

We consider a topological stabilizer code on a honeycomb grid, the "XYZ$^2$" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ($XYZXYZ$) plaquette stabilizers and weight-two ($XX$) link stabilizers on a planar hexagonal grid composed of $2d^2$ qubits for code distance $d$, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure $X$, $Y$, or $Z$ noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance $d^2$ only for pure $Y$ noise, here the code distance is $2d^2$ for both pure $Z$ and pure $Y$ noise. Thresholds for noise with finite $Z$ bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.

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arXiv2022

Dimension towers of SICs. II. Some constructions

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension $d$, there is good evidence that there always exists an aligned SIC in dimension $d(d-2)$, having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension $d(d-2)$ that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the $d$ dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.

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