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Flag fault-tolerant error correction, measurement, and quantum computation for cyclic Calderbank-Shor-Steane codes

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Physical Review A 2020年第1期101卷 012342-012342页

作者： Theerapat Tansuwannont Christopher Chamberland Debbie Leung Institute for Quantum Computing and Department of Physics and Astronomy University of Waterloo Waterloo Ontario Canada N2L 3G1 IBM T. J. Watson Research Center Yorktown Heights New York 10598 USA Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario Canada N2L 3G1 Perimeter Institute for Theoretical Physics Waterloo Ontario Canada N2L 2Y5

Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to date. In this work, we prove some critical properties of Calderbank-Shor-Steane (CSS) codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla qubits and applicable to cyclic CSS codes of distance 3. The measurement protocol can be further used for logical Clifford gate implementation via quantum gate teleportation. We also provide examples of cyclic CSS codes with large encoding rates.

关键词： quantum error correction

Simulating 2D Effects in Lattice Gauge Theories on a quantum Computer

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PRX quantum 2021年第3期2卷 030334-030334页

作者： Danny Paulson Luca Dellantonio Jan F. Haase Alessio Celi Angus Kan Andrew Jena Christian Kokail Rick van Bijnen Karl Jansen Peter Zoller Christine A. Muschik Institute for Quantum Computing University of Waterloo Waterloo Ontario N2L 3G1 Canada Department of Physics and Astronomy University of Waterloo Waterloo Ontario N2L 3G1 Canada Institute of Theoretical Physics and IQST Universität Ulm Albert-Einstein-Allee 11 Ulm D-89069 Germany Departament de Física Universitat Autònoma de Barcelona Bellaterra E-08193 Spain Center of Quantum Physics University of Innsbruck Innsbruck A-6020 Austria Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences Innsbruck A-6020 Austria Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada NIC DESY Zeuthen Platanenallee 6 Zeuthen 15738 Germany Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada

quantum computing is in its greatest upswing, with so-called noisy-intermediate-scale-quantum devices heralding the computational power to be expected in the near future. While the field is progressing toward quantum advantage, quantum computers already have the potential to tackle classically intractable problems. Here, we consider gauge theories describing fundamental-particle interactions. On the way to their full-fledged quantum simulations, the challenge of limited resources on near-term quantum devices has to be overcome. We propose an experimental quantum simulation scheme to study ground-state properties in two-dimensional quantum electrodynamics (2D QED) using existing quantum technology. Our protocols can be adapted to larger lattices and offer the perspective to connect the lattice simulation to low-energy observable quantities, e.g., the hadron spectrum, in the continuum theory. By including both dynamical matter and a nonminimal gauge-field truncation, we provide the novel opportunity to observe 2D effects on present-day quantum hardware. More specifically, we present two variational-quantum-eigensolver- (VQE) based protocols for the study of magnetic field effects and for taking an important first step toward computing the running coupling of QED. For both instances, we include variational quantum circuits for qubit-based hardware. We simulate the proposed VQE experiments classically to calculate the required measurement budget under realistic conditions. While this feasibility analysis is done for trapped ions, our approach can be directly adapted to other platforms. The techniques presented here, combined with advancements in quantum hardware, pave the way for reaching beyond the capabilities of classical simulations.

关键词： quantum algorithms quantum computation quantum simulation

Pauli partitioning with respect to gate sets

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arXiv 2019年

作者： Jena, Andrew Genin, Scott Mosca, Michele Department of Combinatorics and Optimization University of Waterloo Institute for Quantum Computing University of Waterloo OTI Lumionics TorontoON Canada Department of Combinatorics and Optimization University of Waterloo WaterlooON Canada Institute for Quantum Computing University of Waterloo WaterlooON Canada Perimeter Institute for Theoretical Physics WaterlooON Canada

Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. We investigate the task of partitioning a random subset of Pauli operators into simultaneously-measurable parts. Using heuristics from coloring random graphs, we give an upper bound for the expected number of parts in our partition. We go on to conjecture that allowing arbitrary Clifford operators before measurement, rather than single-qubit operations, leads to a decrease in the number of parts which is linear with respect to the lengths of the operators. We give evidence to confirm this conjecture and comment on the importance of this result for a specific near-term application: speeding up the measurement process of the variational quantum eigensolver. Copyright © 2019, The Authors. All rights reserved.

关键词： Qubits

On the mixed-unitary rank of quantum channels

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arXiv 2020年

作者： Girard, Mark Leung, Debbie Levick, Jeremy Li, Chi-Kwong Paulsen, Vern Poon, Yiu Tung Watrous, John Institute for Quantum Computing University of Waterloo School of Computer Science University of Waterloo Department of Combinatorics & Optimization University of Waterloo Perimeter Institute for Theoretical Physics Waterloo Canada Department of Mathematics & Statistics University of Guelph Department of Mathematics College of William and Mary Department of Pure Mathematics University of Waterloo Department of Mathematics Iowa State University Center for Quantum Computing Peng Cheng Laboratory Canadian Institute for Advanced Research Toronto Canada

In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension n, are those linear maps that can be expressed as a convex combination of conjugations by n × n complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank N and the Choi rank r of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality N ≤ r2 − r + 1 is satisfied for every mixed-unitary channel (as is the equality N = 2 when r = 2), and we exhibit the first known examples of mixed-unitary channels for which N > r. Specifically, we prove that there exist mixed-unitary channels having Choi rank d + 1 and mixed-unitary rank 2d for infinitely many positive integers d, including every prime power d. We also examine the mixed-unitary ranks of the mixed-unitary Werner-Holevo channels. Copyright © 2020, The Authors. All rights reserved.

关键词： quantum entanglement

Rigidity of superdense coding

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arXiv 2020年

作者： Nayak, Ashwin Yuen, Henry Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo 200 University Ave. W. WaterlooONN2L 3G1 Canada Department of Computer Science Columbia University New York United States

The famous superdense coding protocol of Bennett and Wiesner demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared EPR pair. Our first result is that an arbitrary protocol for achieving this task (where there are no assumptions on the sender’s encoding operations or the dimension of the shared entangled state) is locally equivalent to the canonical Bennett-Wiesner protocol. In other words, the superdense coding task is rigid. In particular, we show that the sender and receiver only use additional entanglement (beyond the EPR pair) as a source of classical randomness. We also investigate several questions about higher-dimensional superdense coding, where the goal is to communicate one of d2 possible messages by sending a d-dimensional quantum state, for general dimensions d. Unlike the d = 2 case (i.e. sending a single qubit), there can be inequivalent superdense coding protocols for higher d. We present concrete constructions of inequivalent protocols, based on constructions of inequivalent orthogonal unitary bases for all d > 2. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. Our analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest. © 2020, CC BY.

关键词： quantum entanglement

Fault-tolerant quantum error correction using error weight parities

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arXiv 2020年

作者： Tansuwannont, Theerapat Leung, Debbie Institute for Quantum Computing Department of Physics and Astronomy University of Waterloo WaterlooONN2L 3G1 Canada Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo WaterlooONN2L 3G1 Canada Perimeter Institute for Theoretical Physics WaterlooONN2L 2Y5 Canada

In quantum error correction using imperfect primitives, errors of high weight arising from a few faults are major concerns since they might not be correctable by the quantum error correcting code. Fortunately, some errors of different weights are logically equivalent and the same correction procedure is applicable to all equivalent errors, thus correcting high-weight errors is sometimes possible. In this work, we introduce a technique called weight parity error correction (WPEC) which can correct Pauli error of any weight in some stabilizer codes provided that the parity of the weight of the error is known. We show that the technique is applicable to concatenated codes constructed from the [[7,1,3]] Steane code or the [[23,1,7]] Golay code. We also provide a fault-tolerant error correction protocol using WPEC for the [[49,1,9]] concatenated Steane code which can correct up to 3 faults and requires only 2 ancillas. Copyright © 2020, The Authors. All rights reserved.

关键词： Error correction

Robust Interior Point Method for quantum Key Distribution Rate Computation

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arXiv 2021年

作者： Hu, Hao Im, Jiyoung Lin, Jie Lütkenhaus, Norbert Wolkowicz, Henry Department of Combinatorics and Optimization Faculty of Mathematics University of Waterloo WaterlooONN2L 3G1 Canada Department of Mathematical Sciences Clemson University ClemsonSC29634 United States Institute for Quantum Computing Department of Physics and Astronomy University of Waterloo WaterlooONN2L 3G1 Canada Department of Electrical & Computer Engineering University of Toronto TorontoONM5S 3G4 Canada

Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems. © 2021, CC BY.

关键词： Convex optimization

Additive entanglemement measures cannot be more than asymptotically continuous

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arXiv 2019年

作者： Coladangelo, Andrea Leung, Debbie Computing and Mathematical Sciences Caltech PasadenaCA United States Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo Perimeter Institute WaterlooON Canada

In this short note, we show that any non-constant quantity defined on density matrices that is additive on tensor products and invariant under permutations cannot be "more than asymptotically continuous". The proof can be adapted to show that any additive entanglement measure (on any number of parties) that is invariant under local unitary operations also cannot be more than asymptotically continuous. The proof is a direct consequence of generalizing a protocol in [1] for embezzling entanglement. Copyright © 2019, The Authors. All rights reserved.

关键词： Additives

One-shot quantum state redistribution and quantum Markov chains

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arXiv 2021年

作者： Anshu, Anurag Hadiashar, Shima Bab Jain, Rahul Nayak, Ashwin Touchette, Dave School of Engineering and Applied Sciences Harvard University CambridgeMA United States Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo 200 University Ave. W. WaterlooONN2L 3G1 Canada Department of Computer Science Center for Quantum Technologies National University of Singapore 21 Lower Kent Ridge Rd Singapore119077 Singapore Department of Computer Science Institut Quantique Université de Sherbrooke 2500 Boulevard de l’Université SherbrookeQCJ1K 2R1 Canada

We revisit the task of quantum state redistribution in the one-shot setting, and design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to operationally connect quantum state redistribution and quantum Markov chains, and can be interpreted as an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. Thus, our work takes a step towards an optimal characterization of the resources required for one-shot quantum state redistribution, an important open problem in quantum Shannon theory. © 2021, CC BY.

关键词： Entropy