Quantum Stabilizer Research Articles

On the Exploration of Quantum Polar Stabilizer Codes and Quantum Stabilizer Codes with High Coding Rate.

Inspired by classical polar codes, whose coding rate can asymptotically achieve the Shannon capacity, researchers are trying to find their analogs in the quantum information field, which are called quantum polar codes. However, no one has designed a quantum polar coding scheme that applies to quantum computing yet. There are two intuitions in previous research. The first is that directly converting classical polar coding circuits to quantum ones will produce the polarization phenomenon of a pure quantum channel, which has been proved in our previous work. The second is that based on this quantum polarization phenomenon, one can design a quantum polar coding scheme that applies to quantum computing. There are several previous work following the second intuition, none of which has been verified by experiments. In this paper, we follow the second intuition and propose a more reasonable quantum polar stabilizer code construction algorithm than any previous ones by using the theory of stabilizer codes. Unfortunately, simulation experiments show that even the stabilizer codes obtained from this more reasonable construction algorithm do not work, which implies that the second intuition leads to a dead end. Based on the analysis of why the second intuition does not work, we provide a possible future direction for designing quantum stabilizer codes with a high coding rate by borrowing the idea of classical polar codes. Following this direction, we find a class of quantum stabilizer codes with a coding rate of 0.5, which can correct two of the Pauli errors.

Stabilizer Subsystem Decompositions for Single- and Multimode Gottesman-Kitaev-Preskill Codes

The Gottesman-Kitaev-Preskill (GKP) error-correcting code encodes a finite-dimensional logical space in one or more bosonic modes, and has recently been demonstrated in trapped ions and superconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the nonlogical stabilizer subsystem is equivalent to an ideal decoding of the logical state, distinguishing it from previous GKP subsystem decompositions. We describe how to decompose arbitrary states across the subsystem decomposition using a set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to consider essentially arbitrarily large photon numbers for realistic noise channels, such as loss and dephasing. Published by the American Physical Society 2024

Narain CFTs from nonbinary stabilizer codes

We generalize the construction of Narain conformal field theories (CFTs) from qudit stabilizer codes to the construction from quantum stabilizer codes over the finite field of prime power order (Fpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{F}}_{p^m} $$\\end{document} with p prime and m ≥ 1) or over the ring ℤk with k > 1. Our construction results in rational CFTs, which cover a larger set of points in the moduli space of Narain CFTs than the previous one. We also propose a correspondence between a quantum stabilizer code with non-zero logical qubits and a finite set of Narain CFTs. We illustrate the correspondence with well-known stabilizer codes.

Supersymmetric conformal field theories from quantum stabilizer codes

Supersymmetric conformal field theories from quantum stabilizer codes

Quantum symmetric conjugacy classes of non-exceptional groups

We evaluate one-dimensional representations of quantum symmetric conjugacy classes of classical matrix groups along with their quantum stabilizer subgroups.

Narain CFTs from qudit stabilizer codes

We construct a discrete subset of Narain CFTs from quantum stabilizer codes with qudit (including qubit) systems whose dimension is a prime number. Our construction exploits three important relations. The first relation is between qudit stabilizer codes and classical codes. The second is between classical codes and Lorentzian lattices. The third is between Lorentzian lattices and Narain CFTs. In particular, we study qudit Calderbank-Shor-Steane (CSS) codes as a special class of qudit stabilizer codes and the ensembles of the Narain code CFTs constructed from CSS codes. We obtain exact results for the averaged partition functions over the ensembles and discuss their implications for holographic duality.

Burst error-correcting quantum stabilizer codes designed from idempotents

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F_2-submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group C_n, for n=2^m-1 and n=2^m+1. Quantum bounds on their burst error minimum distance are subsequently determined.

Quantum codes, CFTs, and defects

We give a general construction relating Narain rational conformal field theories (RCFTs) and associated 3d Chern-Simons (CS) theories to quantum stabilizer codes. Starting from an abelian CS theory with a fusion group consisting of n even-order factors, we map a boundary RCFT to an n-qubit quantum code. When the relevant ’t Hooft anomalies vanish, we can orbifold our RCFTs and describe this gauging at the level of the code. Along the way, we give CFT interpretations of the code subspace and the Hilbert space of qubits while mapping error operations to CFT defect fields.

New quantum stabilizer codes with better parameters from the images of some RS codes and BCH codes

New quantum stabilizer codes with better parameters from the images of some RS codes and BCH codes

Universal Decoding of Quantum Stabilizer Codes via Classical Guesswork

A universal decoding scheme is conceived for quantum stabilizer codes (QSCs) by appropriately adapting the ‘guessing random additive noise decoding’ (GRAND) philosophy of classical domain codes. We demonstrate that the generalized quantum decoder conceived is eminently suitable for different QSC decoding paradigms, namely for both stabilizer-measurement-based as well as the inverse-encoder-based decoding. We then harness the resultant decoder for both quantum Bose-Chaudhuri-Hocquenghem (BCH) codes and quantum polar codes and quantify both their quantum block error rate (QBLER), and QBLER per logical qubits as well as their decoding complexity. Furthermore, we provide a parametric study of the associated design trade-offs and offer design guideline for the implementation of GRAND-based QSC decoders.

Demonstration of a Quantum Gate Using Electromagnetically Induced Transparency.

We demonstrate a native CNOT gate between two individually addressed neutral atoms based on electromagnetically induced transparency. This protocol utilizes the strong long-range interactions of Rydberg states to enable conditional state transfer on the target qubit when operated in the blockade regime. An advantage of this scheme is it enables implementation of multiqubit CNOT^{k} gates using a pulse sequence independent of qubit number, providing a simple gate for efficient implementation of digital quantum algorithms and stabilizer measurements for quantum error correction. We achieve a loss corrected gate fidelity of F_{CNOT}^{cor}=0.82(6), and prepare an entangled Bell state with F_{Bell}^{cor}=0.66(5), limited at present by laser power. We present a number of technical improvements to advance this to a level required for fault-tolerant scaling.

Exploiting degeneracy in belief propagation decoding of quantum codes

Quantum information needs to be protected by quantum error-correcting codes due to imperfect physical devices and operations. One would like to have an efficient and high-performance decoding procedure for the class of quantum stabilizer codes. A potential candidate is Gallager’s sum-product algorithm, also known as Pearl’s belief propagation (BP), but its performance suffers from the many short cycles inherent in a quantum stabilizer code, especially highly-degenerate codes. A general impression exists that BP is not effective for topological codes. In this paper, we propose a decoding algorithm for quantum codes based on quaternary BP with additional memory effects (called MBP). This MBP is like a recursive neural network with inhibitions between neurons (edges with negative weights), which enhance the perception capability of a network. Moreover, MBP exploits the degeneracy of a quantum code so that the most probable error or its degenerate errors can be found with high probability. The decoding performance is significantly improved over the conventional BP for various quantum codes, including quantum bicycle, hypergraph-product, surface and toric codes. For MBP on the surface and toric codes over depolarizing errors, we observe error thresholds of 16% and 17.5%, respectively.

Geometric graph-theoretic aspects of quantum stabilizer codes

We propose a systematic procedure for the construction of graphs associated with binary quantum stabilizer codes. The procedure is characterized by means of the following three step process. First, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code. Second, the canonical form of the CWS code is determined and third, the input vertices are attached to the graphs. In order to verify the effectiveness of the procedure, we implement the Gottesman stabilizer code characterized by multi-qubit encoding operators for the resource-efficient error correction of arbitrary single-qubit errors. Finally, the error-correcting capabilities of the Gottesman eight-qubit quantum stabilizer code is verified in graph-theoretic terms as originally advocated by Schlingemann and Werner.

A Construction of Optimal Nonbinary Pure Quantum Stabilizer Codes

A Construction of Optimal Nonbinary Pure Quantum Stabilizer Codes

Hat problem: a new strategy based on quantum stabilizer codes

Hat problem: a new strategy based on quantum stabilizer codes

Dynamical signatures of rank-2 U(1) spin liquids

Emergent $U(1)$ gauge theories and artificial photons in frustrated magnets are outstanding examples of many-body collective phenomena. The classical and quantum regimes of these systems provide platforms for classical and quantum spin liquids, and are the subject of current active theoretical and experimental investigations. Recently, realizations of rank-2 $U(1)$ (R2-U1) gauge theories in three-dimensional frustrated magnets have been proposed. Such systems in the quantum regime may lead to the so-called fracton ordered phases---a new class of topological order that has been associated with quantum stabilizer codes and holography. However, there exist few distinguishing characteristics of these states for their detection in real materials. Here we focus on the classical limit and present the dynamical spin structure factor for a R2-U1 spin liquid state on a breathing pyrochlore lattice. Remarkably, we find unique signatures of the R2-U1 state, and we contrast them with the results obtained from a more conventional $U(1)$ spin liquid. These results provide a path of investigation for future inelastic neutron scattering experiments on candidate materials.

Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applications

<p style='text-indent:20px;'>Hermitian self-orthogonal codes form an essential ingredient in the construction of quantum stabilizer codes. In this paper, we present two new classes of Hermitian self-orthogonal MDS codes from twisted generalized Reed-Solomon codes. The constructed codes are monomially inequivalent to generalized Reed-Solomon codes. Two new classes of Hermitian LCD MDS codes can then be derived. Finally, based on the constructed Hermitian self-orthogonal MDS codes, we present two classes of quantum MDS codes with new parameters.</p>

Low-overhead fault-tolerant error correction scheme based on quantum stabilizer codes

Fault-tolerant error-correction (FTEC) circuit is the foundation for achieving reliable quantum computation and remote communication. However, designing a fault-tolerant error correction scheme with a solid error-correction ability and low overhead remains a significant challenge. In this paper, a low-overhead fault-tolerant error correction scheme is proposed for quantum communication systems. Firstly, syndrome ancillas are prepared into Bell states to detect errors caused by channel noise. We propose a detection approach that reduces the propagation path of quantum gate fault and reduces the circuit depth by splitting the stabilizer generator into X-type and Z-type. Additionally, a syndrome extraction circuit is equipped with two flag qubits to detect quantum gate faults, which may also introduce errors into the code block during the error detection process. Finally, analytical results are provided to demonstrate the fault-tolerant performance of the proposed FTEC scheme with the lower overhead of the ancillary qubits and circuit depth.

Non-rational Narain CFTs from codes over F4

We construct a map between a class of codes over F4 and a family of non-rational Narain CFTs. This construction is complementary to a recently introduced relation between quantum stabilizer codes and a class of rational Narain theories. From the modular bootstrap point of view we formulate a polynomial ansatz for the partition function which reduces modular invariance to a handful of algebraic easy-to-solve constraints. For certain small values of central charge our construction yields optimal theories, i.e. those with the largest value of the spectral gap.

Comments on the holographic description of Narain theories

We discuss the holographic description of Narain U(1)c× U(1)c conformal field theories, and their potential similarity to conventional weakly coupled gravitational theories in the bulk, in the sense that the effective IR bulk description includes “U(1) gravity” amended with additional light degrees of freedom. Starting from this picture, we formulate the hypothesis that in the large central charge limit the density of states of any Narain theory is bounded by below by the density of states of U(1) gravity. This immediately implies that the maximal value of the spectral gap for primary fields is ∆1 = c/(2πe). To test the self-consistency of this proposal, we study its implications using chiral lattice CFTs and CFTs based on quantum stabilizer codes. First we notice that the conjecture yields a new bound on quantum stabilizer codes, which is compatible with previously known bounds in the literature. We proceed to discuss the variance of the density of states, which for consistency must be vanishingly small in the large-c limit. We consider ensembles of code and chiral theories and show that in both cases the density variance is exponentially small in the central charge.