Roads towards fault-tolerant universal quantum computation

Campbell, Earl T.; Terhal, Barbara M.; Vuillot, Christophe

doi:10.1038/nature23460

Cited by 481 publications

(384 citation statements)

References 76 publications

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“…, as a consequence of (121), and the channel divergence    ( ) D max 2 2 involves an optimization over all bipartite input states, one of which is  Ä F ( )( ) id 1 . + Remark 3.…”

Section: Max-thauma Of a Quantum Channelmentioning

confidence: 99%

“…One of the main obstacles to physical realizations of quantum computation is decoherence that occurs during the execution of quantum algorithms. Fault-tolerant quantum computation (FTQC) [1,2] provides a framework to overcome this difficulty by encoding quantum information into quantum error-correcting codes, and it allows reliable quantum computation when the physical error rate is below a certain threshold value.…”

Section: Introductionmentioning

confidence: 99%

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Quantifying the magic of quantum channels

2019

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Keywords: resource theory of magic quantum channels, completely positive-Wigner-preserving quantum operations, thauma of a quantum channel, distillable magic, classical simulation of noisy quantum circuits Abstract 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 dimension d, 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.states also motivates the resource theory of magic states [15][16][17][18][19][20], where the free operations are the SOs and the free states are the stabilizer states (abbreviated as 'Stab'). On the other hand, since a key step of fault-tolerant quantum computing is to implement non-SOs, a natural and fundamental problem is to quantify the nonstabilizerness or 'magic' of quantum operations. As we are at the stage of noisy intermediate-scale quantum (NISQ) technology, a resource theory of magic for noisy quantum operations is desirable both to exploit the power and to identify the limitations of NISQ devices in fault-tolerant quantum computation (FTQC). Overview of resultsIn this paper, we develop a framework for the resource theory of magic quantum channels, based on qudit systems with odd prime dimension d. Related work on this topic has appeared recently [21], but the set of free operations that we take in our resource theory is larger, given by the completely positive-Wigner-preserving (CPWP) perations as we detail below. We note here that d-level FTQC based on qudits with prime d is of considerable interest for both theoretical and practical purposes [22][23][24][25][26].Our paper is structured as follows.• In section 2, we first review the stabilizer formalism [4] and the discrete Wigner function [27][28][29]. We further review various magic m...

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Section: Max-thauma Of a Quantum Channelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantifying the magic of quantum channels

2019

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“…There are significant differences between these two code families in terms of their practical implementation. On the one hand, the surface code has a favorably high threshold error rate for fault tolerance, but only CNOT, X, and Z gates can be performed transversally [13]. On the other hand, while the color code has a smaller threshold error rate than the surface code [14,15], it allows for the transversal implementation of the full Clifford group of quantum gates (with Hadamard, π/4 phase gate, and CNOT gate as generators) [16,17].…”

Section: Introductionmentioning

confidence: 99%

Neural network decoder for topological color codes with circuit level noise

et al. 2019

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A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment-without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate ò L of the encoded logical qubit to values much below the error rate ò phys of the physical qubits-fitting the expected power law scaling   µ + ( ) d L phys 1 2 , with d the code distance. The neural network incorporates the information from 'flag qubits' to avoid reduction in the effective code distance caused by the circuit. As a test, we apply the neural network decoder to a density-matrix based simulation of a superconducting quantum computer, demonstrating that the logical qubit has a longer life-time than the constituting physical qubits with near-term experimental parameters.

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“…Magic state distillation [6][7][8] is now a leading proposal to complete a universal gate set. The key idea is to accept many low-fidelity magic states prepared by state injection or the previous round of distillation, and then to filter them into high-fidelity ones.…”

Section: Introductionmentioning

confidence: 99%

“…subsystem code in table8. In that case, the conversion between  1 and  D consists in fixing one half of the gauge qubits in encoded ñ |0 and the other half in encoded +ñ | .…”

mentioning

confidence: 99%

Universal fault-tolerant quantum computation using fault-tolerant conversion schemes

Luo

2019

New J. Phys.

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In this paper, we present the fault-tolerant conversion between quantum Reed-Muller (QRM)(2, 5) and QRM(2, 7), and also the conversion between QBCH(15, 7) and QRM(2, 7). Either of the two schemes provides a method to realize universal fault-tolerant quantum computation. In particular, the gate overhead and logical error rate of a logical T gate are provided, as well as the comparison with magic state distillation scheme. In addition, we propose two other fault-tolerant conversion schemes based on + ( | ) u u v and + + + -( | | ) a x b x a b x constructions.

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Roads towards fault-tolerant universal quantum computation

Cited by 481 publications

References 76 publications

Quantifying the magic of quantum channels

Quantifying the magic of quantum channels

Neural network decoder for topological color codes with circuit level noise

Universal fault-tolerant quantum computation using fault-tolerant conversion schemes

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