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Quantum computers have the potential to provide exponential speedups over their classical counterparts. Quantum principles are being applied to fields such as communications, information processing, and artificial intelligence to achieve quantum advantage. However, quantum bits are extremely noisy and prone to decoherence. Thus, keeping the qubits error free is extremely important toward reliable quantum computing. Quantum error correcting codes have been studied for several decades and methods have been proposed to import classical error correcting codes to the quantum domain. Along with the exploration into novel and more efficient quantum error correction codes, it is also essential to design circuits for practical realization of these codes. This paper serves as a tutorial on designing and simulating quantum encoder and decoder circuits for stabilizer codes. We first describe Shor’s 9-qubit code which was the first quantum error correcting code. We discuss the stabilizer formalism along with the design of encoding and decoding circuits for stabilizer codes such as the five-qubit code and Steane code. We also design nearest neighbor compliant circuits for the above codes. The circuits were simulated and verified using IBM Qiskit.
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Quantifying the magic of quantum channels
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 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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- Award ID(s):
- 1714215
- PAR ID:
- 10308394
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- New Journal of Physics
- Volume:
- 21
- Issue:
- 10
- ISSN:
- 1367-2630
- Page Range / eLocation ID:
- Article No. 103002
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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