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1. Quantifying the magic of quantum channels

   https://doi.org/10.1088/1367-2630/ab451d  

   Wang, Xin ; Wilde, Mark M. ; Su, Yuan ( October 2019 , New Journal of Physics)

   To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimensiond, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

    

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2. Group-theoretic error mitigation enabled by classical shadows and symmetries

   https://doi.org/10.1038/s41534-024-00854-5  

   Zhao, Andrew ; Miyake, Akimasa ( June 2024 , npj Quantum Information)

   Estimating expectation values is a key subroutine in quantum algorithms. Near-term implementations face two major challenges: a limited number of samples required to learn a large collection of observables, and the accumulation of errors in devices without quantum error correction. To address these challenges simultaneously, we develop a quantum error-mitigation strategy calledsymmetry-adjusted classical shadows, by adjusting classical-shadow tomography according to how symmetries are corrupted by device errors. As a concrete example, we highlight global U(1) symmetry, which manifests in fermions as particle number and in spins as total magnetization, and illustrate their group-theoretic unification with respective classical-shadow protocols. We establish rigorous sampling bounds under readout errors obeying minimal assumptions, and perform numerical experiments with a more comprehensive model of gate-level errors derived from existing quantum processors. Our results reveal symmetry-adjusted classical shadows as a low-cost strategy to mitigate errors from noisy quantum experiments in the ubiquitous presence of symmetry.

    

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3. Resource-efficient fault-tolerant one-way quantum repeater with code concatenation

   https://doi.org/10.1038/s41534-023-00792-8  

   Wo, Kah Jen ; Avis, Guus ; Rozpędek, Filip ; Mor-Ruiz, Maria Flors ; Pieplow, Gregor ; Schröder, Tim ; Jiang, Liang ; Sørensen, Anders S ; Borregaard, Johannes ( December 2023 , npj Quantum Information)

   One-way quantum repeaters where loss and operational errors are counteracted by quantum error-correcting codes can ensure fast and reliable qubit transmission in quantum networks. It is crucial that the resource requirements of such repeaters, for example, the number of qubits per repeater node and the complexity of the quantum error-correcting operations are kept to a minimum to allow for near-future implementations. To this end, we propose a one-way quantum repeater that targets both the loss and operational error rates in a communication channel in a resource-efficient manner using code concatenation. Specifically, we consider a tree-cluster code as an inner loss-tolerant code concatenated with an outer 5-qubit code for protection against Pauli errors. Adopting flag-based stabilizer measurements, we show that intercontinental distances of up to 10,000 km can be bridged with a minimized resource overhead by interspersing repeater nodes that each specialize in suppressing either loss or operational errors. Our work demonstrates how tailored error-correcting codes can significantly lower the experimental requirements for long-distance quantum communication. 

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4. Quantum algorithm for estimating volumes of convex bodies

   Chakrabarti, Shouvanik ; Childs, Andrew ; Hung, Shih-Han ; Li, Tongyang ; Wang, Chunhao ; Wu, Xiaodi ( January 2020 , Annual Conference on Quantum Information Processing)

   Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n-dimensional convex body within multiplicative error ϵ using Õ (n3+n2.5/ϵ) queries to a membership oracle and Õ (n5+n4.5/ϵ) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ (n4+n3/ϵ2) queries and Õ (n6+n5/ϵ2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. 

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5. Symmetric distinguishability as a quantum resource

   https://doi.org/10.1088/1367-2630/ac14aa  

   Salzmann, Robert ; Datta, Nilanjana ; Gour, Gilad ; Wang, Xin ; Wilde, Mark M ( August 2021 , New Journal of Physics)

   Abstract We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e. sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system XA , corresponding to an ensemble of two quantum states, with X being classical and A being quantum. We study the resource theory for two different classes of free operations: (i) CPTP A , which consists of quantum channels acting only on A , and (ii) conditional doubly stochastic maps acting on XA . We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability. 

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