Stabilizer Codes Research Articles

An explanation of the Bernstein‐Vazirani and Deustch‐Josza algorithms with the quantum stabilizer formalism

This work was supported in part by the Ministry of Economy and Competitiveness (“Ministerio de Economia y Competitividad”) from Spain/FEDER under grant MINECO‐16‐TEC2015‐67387‐C4‐3‐R, by the Ministry of Economy, Industry and Competitiveness (“Ministerio de Economia, Industria y Competitividad”) from Spain/FEDER under grant MTM‐2017‐83506‐C2‐2‐P, by Ministry of Science and Innovation (“Ministerio de Ciencia e Innovacion”) from Spain/FEDER under grant RTI2018‐098085‐B‐C44, and by he Regional Ministry of the Principality of Asturias (“Consejeria de Empleo, Industria y Turismo del Principado de Asturias”) under grants FC‐GRUPIN‐IDI/2018/000193 and FC‐GRUPIN‐IDI/2018/000226.

Flag Fault-Tolerant Error Correction for any Stabilizer Code

Conventional fault-tolerant quantum error-correction schemes require a number of extra qubits that grow linearly with the code’s maximum stabilizer generator weight. For some common distance-three codes, the recent “flag paradigm” uses just two extra qubits. Chamberland and Beverland [Quantum 2, 53 (2018)] provide a framework for flag error correction of arbitrary-distance codes. However, their construction requires conditions that only some code families are known to satisfy. We give a flag error-correction scheme that works for any stabilizer code, unconditionally. With fast qubit measurement and reset, it uses ≤d+1 extra qubits for a distance-d code.Received 24 January 2020Accepted 29 June 2020DOI:https://doi.org/10.1103/PRXQuantum.1.010302Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum error correctionQuantum memoriesQuantum Information

Universal fault-tolerant measurement-based quantum computation

Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using protocols developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource efficient proposal for photonic quantum computation.

No Quantum Ramsey Theorem for Stabilizer Codes

In this paper we study the quantum graphs of mixed-unitary channels generated by tensor products of Pauli operators, which we call Pauli channels. We show that most quantum graphs arising from Pauli channels have non-trivial quantum cliques or quantum anticliques which are stabilizer codes. However, a reformulation of Nik Weaver's quantum Ramsey theorem in terms of stabilizer codes and Pauli channels fails. Specifically, for every positive integer $n$, there exists an $n$-qubit Pauli channel for which any non-trivial quantum clique or quantum anticlique fails to be a stabilizer code.

Influence of interface pollution on the linear stability of a rotating flow

Abstract

Fault-tolerant quantum gates with defects in topological stabilizer codes

Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realised by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher dimensional defect-braiding schemes have the potential to play an important role in realising fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher dimensional codes, specifically: in self-dual surface codes; the three dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.

Combating quasiparticle poisoning with multiple Majorana fermions in a periodically-driven quantum wire

Quasiparticle poisoning has remained one of the main challenges in the implementation of Majorana-based quantum computing. It inevitably occurs when the system hosting Majorana qubits is not completely isolated from its surrounding, thus considerably limiting its computational time. We propose the use of periodic driving to generate multiple Majorana fermions at each end of a single quantum wire, which naturally provides the necessary resources to implement active quantum error corrections with minimal space overhead. In particular, we present a stabilizer code protocol that can specifically detect and correct any single quasiparticle poisoning event. Such a protocol is implementable via existing proximitized semiconducting nanowire proposals, where all of its stabilizer operators can be measured from an appropriate Majorana parity dependent four-terminal conductance.

Refined Belief Propagation Decoding of Sparse-Graph Quantum Codes

Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.

The Effects of Side-Casings Balance on Open-Deck-Ro-Ro Vessels in Term of Intact Stability

Roll-on Roll-of (Ro-Ro) is one of the most favourite types of ship, which can carry passengers, cars, and cargoes at the same time. Reports from National Committee for Safety Transportation (KNKT) indicated that there is tendency of increase of Ro-Ro vessels accidents due to stability problems. The current paper discusses an Open-Car-Deck-Ro-Ro 40 m length without side-casings which capsized on Halmahera sea. The Ro-Ro was originally designed with side-casings on the right and left sides of car deck but removed to improve car deck capacity. The side-casings reduce one truck capacity, improve 72.539 GT, and improve survival stability. The Ro-Ro stability was calculated with four loading conditions with assumption weight and center gravity (KG) of the ship with and without side-casings the same. It shows all condition with added side-casings will increase righting lever GZ at angle ≥ 10°. Also, range righting lever curve GZ positive increase, at full load arrival 0°-29° become 0°-53°, at full load departure 0°-25° become 0°-42°, at ballast load arrival 0°-44° become 0°-66°, and at ballast load departure 0°-36° become 0°-52°. Then stability curves were assessed by Intact Stability Code 2008 (ISC 2008) obtained all load case ship with side-casings pass all criteria otherwise ship without side-casings fail especially area 30°-40° ≥ 0.03 m.rad and maximum GZ ≥ 25°. The case study indicates installed side-casings reduce cargo capacity but improve survival stability.

On Optimality of CSS Codes for Transversal T

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries: (a) For any non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical transversal $T$, there exists an $[[ n,k,d ]]$ CSS code with the same property; (b) Triorthogonal codes are the most general CSS codes that realize logical transversal $T$ via physical transversal $T$; (c) Triorthogonality is necessary for physical transversal $T$ on a CSS code to realize the logical identity. The main tool we use is a recent efficient characterization of certain diagonal gates in the Clifford hierarchy (arXiv:1902.04022). We refer to these gates as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing code constructions that realize logical gates via transversal $T$. We provide several examples and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal $\pi/2^{\ell}$ $Z$-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal $\pi/2^{\ell}$ $Z$-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal $1$- or $2$-local diagonal gate.

Delay master stability of inertial oscillator networks

Time lags occur in a vast range of real-world dynamical systems due to finite reaction times or propagation speeds. Here we derive an analytical approach to determine the asymptotic stability of synchronous states in networks of coupled inertial oscillators with constant delay. Building on the master stability formalism, our technique provides necessary and sufficient delay master stability conditions. We apply it to two classes of potential future power grids, where processing delays in control dynamics will likely pose a challenge as renewable energies proliferate. Distinguishing between phase and frequency delay, our method offers an insight into how bifurcation points depend on the network topology of these system designs.

Effect of pressure gradient in the connection region on the stability of the edge pedestal

We have numerically investigated the dependence of pedestal properties such as the pedestal height and the pedestal width on the normalized pressure gradient () in the connection region, . We used an ideal MHD stability code, MISHKA, and the pedestal predictive model, EPED1, to analyze the effect of on the edge stability and predict pedestal structure. Edge stability is affected by because the unstable mode extends radially into the core. Improvement of pedestal properties can be achieved by reducing . Effect of on the edge instability is largest for the intermediate mode n. Larger gradient of Shafranov shift, , also improves the pedestal width and height by stabilizing peeling-ballooning mode (PBM). Our result has been applied to JET hybrid discharges to explain the coupling between core and edge pressure profile in H-mode plasma configuration through and . The extended correlation loop between poloidal beta and the pedestal parameters is presented.

Jordan-Wigner dualities for translation-invariant Hamiltonians in any dimension: Emergent fermions in fracton topological order

Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions, we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah. We prove that given a translation-invariant fermionic system with general $q$-body interactions, where $q$ is even, a local mapping preserving global fermion parity to a dual Pauli spin model exists and is unique up to a choice of basis. Furthermore, the dual spin model is constructive, and we present various examples of these dualities. As an application, we bosonize fermionic systems where free-fermion hopping terms are absent ($q \ge 4$) and fermion parity is conserved on submanifolds such as higher-form, line, planar or fractal symmetry. For some cases in 3+1D, bosonizing such a system can give rise to fracton models where the emergent particles are immobile but yet can behave in certain ways like fermions. These models may be examples of new nonrelativistic 't Hooft anomalies. Furthermore, fermionic subsystem symmetries are also present in various Majorana stabilizer codes, such as the color code or the checkerboard model, and we give examples where their duals are cluster states or new fracton models distinct from their doubled CSS codes.

New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{5}$$

The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs). In this work, we construct some new entanglement-assisted quantum MDS (EAQMDS) codes with length $n=\frac{q^2+1}5$ from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance.

Drift Alfvén energetic particle stability with circulating particles

We develop from scratch a comprehensive linear stability eigenvalue code based on a finite element method, namely, the drift Alfvén energetic particle stability (DAEPS) code, to investigate the physics of various stable and unstable modes observed in toroidal fusion plasmas, which has the advantage of accurate calculation of the mode characteristics near marginal stability. The DAEPS code is dedicated to providing a thorough understanding of low frequency modes in collisionless plasmas, e.g., shear Alfvén wave SAW and drift Alfvén wave physics with an energetic particle (EP) effect. DAEPS can calculate the linear frequency and growth rate for these modes by keeping correct asymptotic behavior in ballooning space. In this work, we demonstrate that the DAEPS code is able to analyze linear electromagnetic modes excited by circulating particles, including the thermal particle excited beta-induced Alfvén eigenmode and EP excited toroidicity-induced Alfvén eigenmode, where the verifications are performed successfully with other codes and theories, where the finite orbit width is discovered to play an important stabilizing role, which are usually ignored by traditional theory.

Reliability analysis of GM and KG of ships from the inclining experiment

It is of paramount importance to determine the lightship characteristics of a newly built vessel correctly. Conventionally, an inclining experiment is performed after the ship is launched in order to determine the lightship GM and in turn KG values. However, there has been an ongoing debate about the validity of the results since it bears some assumptions during the test and calculation procedure. Although the inclining experiment is mandatory by the Intact Stability Code of International Maritime Organization, there emerged various alternative methods recently claiming to reveal better results. In this study, an uncertainty analysis has been carried out on GM and KG values of various sample vessels. Based on the results on the uncertainty analysis, a cumulative margin of error from various sources has been determined within a specified confidence level. Finally, the outcome for the sample vessels was detailed and evaluated in reference to that of the actual inclining experiment reports.

A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.

Delocalized information in quantum networks

We consider entanglement-based quantum networks where information is stored in a delocalized way within regions or the whole network. This offers a natural protection against failure of network nodes, loss and decoherence, and has built-in security features. Quantum information is transmitted within the network by performing local measurements on individual nodes only. Information can be localized within regions or at a specific node by collaborative actions using only entanglement within a region, or sometimes even without entanglement. We discuss several examples based on error correction stabilizer codes, Dicke states and correlation space encodings. We show how to design fully functional networks using encoded states or correlation space resources.

A unidirectional on-chip photonic interface for superconducting circuits

We propose and analyze a passive architecture for realizing on-chip, scalable cascaded quantum devices. In contrast to standard approaches, our scheme does not rely on breaking Lorentz reciprocity. Rather, we engineer the interplay between pairs of superconducting transmon qubits and a microwave transmission line, in such a way that two delocalized orthogonal excitations emit (and absorb) photons propagating in opposite directions. We show how such cascaded quantum devices can be exploited to passively probe and measure complex many-body operators on quantum registers of stationary qubits, thus enabling the heralded transfer of quantum states between distant qubits, as well as the generation and manipulation of stabilizer codes for quantum error correction.

A novel construction for quantum stabilizer codes based on binary formalism

In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.