Weiss–Weinstein Lower Bounds for Markovian Systems. Part 2: Applications to Fault-Tolerant Filtering

AbstractQuantum channels underlie the dynamics of quantum systems, but in many practical settings it is the channels themselves that require processing. We establish universal limitations on the processing of both quantum states and channels, expressed in the form of no-go theorems and quantitative bounds for the manipulation of general quantum channel resources under the most general transformation protocols. Focusing on the class of distillation tasks — which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels — we develop fundamental restrictions on the error incurred in such transformations, and comprehensive lower bounds for the overhead of any distillation protocol. In the asymptotic setting, our results yield broadly applicable bounds for rates of distillation. We demonstrate our results through applications to fault-tolerant quantum computation, where we obtain state-of-the-art lower bounds for the overhead cost of magic state distillation, as well as to quantum communication, where we recover a number of strong converse bounds for quantum channel capacity.

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WeissWeinstein Lower Bounds for Markovian Systems. Part 2: Applications to FaultTolerant Filtering

Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking ◽

10.1109/9780470544198.ch86 ◽

2009 ◽

Keyword(s):

Lower Bounds ◽

Markovian Systems

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Largest Component and Node Fault Tolerance for Grids

The Electronic Journal of Combinatorics ◽

10.37236/8376 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Jakub Przybyło ◽

Andrzej Żak

Keyword(s):

Fault Tolerance ◽

Lower Bound ◽

Lower Bounds ◽

Computer Network ◽

Fault Tolerant ◽

Subgraph Isomorphic ◽

General Connection

A graph $G$ is called $t$-node fault tolerant with respect to $H$ if $G$ still contains a subgraph isomorphic to $H$ after removing any $t$ of its vertices. The least value of $|E(G)|-|E(H)|$ among all such graphs $G$ is denoted by $\Delta(t,H)$. We study fault tolerance with respect to some natural architectures of a computer network, i.e. the $d$-dimensional toroidal grids and the hypercubes. We provide the first non-trivial lower bounds for $\Delta(1,H)$ in these cases. For this aim we establish a general connection between the notion of fault tolerance and the size of a largest component of a graph. In particular, we give for all values of $k$ (and $n$) a lower bound on the order of the largest component of any graph obtained from $C_n\Box C_n$ via removal of $k$ of its vertices, which is in general optimal.

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Quantum universality by state distillation

Quantum Information and Computation ◽

10.26421/qic9.11-12-7 ◽

2009 ◽

Vol 9 (11&12) ◽

pp. 1030-1052

Author(s):

B.W. Reichardt

Keyword(s):

Lower Bounds ◽

Upper Bound ◽

Fault Tolerant ◽

Upper Bounds ◽

Mixed States ◽

Single Qubit ◽

Noise Rate ◽

Pure States ◽

Rate Threshold ◽

Stabilizer States

Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This ``magic states distillation" question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for fault-tolerant computation. Upper bounds show the limits of threshold upper-bound arguments based on the Gottesman-Knill theorem. We extend the range of single-qubit mixed states that are known to give universality, by using a simple parity-checking operation. For applications to proving threshold lower bounds, certain practical stability characteristics are often required, and we also show a stable distillation procedure.}{No distillation upper bounds are known beyond those given by the Gottesman-Knill theorem. One might ask whether distillation upper bounds reduce to upper bounds for single-qubit ancilla states. For multi-qubit pure states and previously considered two-qubit ancilla states, the answer is yes. However, we exhibit two-qubit mixed states that are not mixtures of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. Distilling such states would require true multi-qubit state distillation methods.

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Search on a Line by Byzantine Robots

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121500209 ◽

2021 ◽

pp. 1-19

Author(s):

Jurek Czyzowicz ◽

Konstantinos Georgiou ◽

Evangelos Kranakis ◽

Danny Krizanc ◽

Lata Narayanan ◽

...

Keyword(s):

Lower Bounds ◽

Fault Tolerant ◽

Sufficient Evidence ◽

Maximum Speed ◽

Parallel Search ◽

Not Given ◽

Running Time ◽

Byzantine Faults ◽

Infinite Line ◽

Design Algorithms

We consider the problem of fault-tolerant parallel search on an infinite line by [Formula: see text] robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed [Formula: see text] and can communicate wirelessly among themselves. However, among the [Formula: see text] robots, there are [Formula: see text] robots that exhibit byzantine faults. A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient evidence that the target has been found. We aim to design algorithms that minimize the value of [Formula: see text], the time to find a target at a (unknown) distance [Formula: see text] from the origin by [Formula: see text] robots among which [Formula: see text] are faulty. We give several different algorithms whose running time depends on the ratio [Formula: see text], the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.

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New perspectives on covariant quantum error correction

Quantum ◽

10.22331/q-2021-08-09-521 ◽

2021 ◽

Vol 5 ◽

pp. 521

Author(s):

Sisi Zhou ◽

Zi-Wen Liu ◽

Liang Jiang

Keyword(s):

Error Correction ◽

Lower Bounds ◽

Fault Tolerant ◽

Symmetry Transformation ◽

Substantial Improvement ◽

Quantum Error Correction ◽

Quantum Codes ◽

Covariant Quantum ◽

Continuous Symmetries ◽

Quantum Error

Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.

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