Algebraic Quantum Codes: Linking Quantum Mechanics and Discrete Mathematics

• URL: http://arxiv.org/abs/2011.06996v1
• Date: Fri, 13 Nov 2020 16:25:31 GMT
• Title: Algebraic Quantum Codes: Linking Quantum Mechanics and Discrete Mathematics
• Authors: Markus Grassl
• Abstract summary: We present a general framework of quantum error-correcting codes (QECCs) as a subspace of a complex Hilbert space. We illustrate how QECCs can be constructed using techniques from algebraic coding theory. We discuss secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs.
• Score: 0.6091702876917279
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a general framework of quantum error-correcting codes (QECCs) as a subspace of a complex Hilbert space and the corresponding error models. Then we illustrate how QECCs can be constructed using techniques from algebraic coding theory. Additionally, we discuss secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs.

Related papers

• Separable Power of Classical and Quantum Learning Protocols Through the Lens of No-Free-Lunch Theorem [70.42372213666553]
  The No-Free-Lunch (NFL) theorem quantifies problem- and data-independent generalization errors regardless of the optimization process. We categorize a diverse array of quantum learning algorithms into three learning protocols designed for learning quantum dynamics under a specified observable. Our derived NFL theorems demonstrate quadratic reductions in sample complexity across CLC-LPs, ReQu-LPs, and Qu-LPs. We attribute this performance discrepancy to the unique capacity of quantum-related learning protocols to indirectly utilize information concerning the global phases of non-orthogonal quantum states.
  arXiv  Detail & Related papers  (2024-05-12T09:05:13Z)
• Stabilizer Formalism for Operator Algebra Quantum Error Correction [0.0]
  We introduce a stabilizer formalism for the general quantum error correction framework called operator algebra quantum error correction (OAQEC) We formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code. We show how some recent hybrid subspace code constructions are captured by the formalism.
  arXiv  Detail & Related papers  (2023-04-22T16:45:50Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• Entanglement-assisted concatenated quantum codes [77.34669920414821]
  Entanglement-assistedd quantum codes (EACQCs) constructed by concatenating two quantum codes are proposed. EACQCs show several advantages over the standard quantum codes (CQCs) EACQCs are not only competitive in quantum communication but also applicable in fault-tolerant quantum computation.
  arXiv  Detail & Related papers  (2022-02-16T14:14:02Z)
• Circuit Symmetry Verification Mitigates Quantum-Domain Impairments [69.33243249411113]
  We propose circuit-oriented symmetry verification that are capable of verifying the commutativity of quantum circuits without the knowledge of the quantum state. In particular, we propose the Fourier-temporal stabilizer (STS) technique, which generalizes the conventional quantum-domain formalism to circuit-oriented stabilizers.
  arXiv  Detail & Related papers  (2021-12-27T21:15:35Z)
• A Quantum Computing View on Unitary Coupled Cluster Theory [0.39323007669342247]
  Unitary Coupled Cluster (UCC) ansatz and related ans"atze are used to variationally solve the electronic structure problem on quantum computers. The review concludes with a unified perspective on the discussed ans"atze, attempting to bring them under a common framework.
  arXiv  Detail & Related papers  (2021-09-30T14:47:06Z)
• Depth-efficient proofs of quantumness [77.34726150561087]
  A proof of quantumness is a type of challenge-response protocol in which a classical verifier can efficiently certify quantum advantage of an untrusted prover. In this paper, we give two proof of quantumness constructions in which the prover need only perform constant-depth quantum circuits.
  arXiv  Detail & Related papers  (2021-07-05T17:45:41Z)
• Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes [3.6192409729339223]
  We develop the theory of quasi-exact fault-tolerant quantum computation, which uses qubits encoded into quasi-exact quantum error-correction codes ("quasi codes") The model of QEQ lies in between the two well-known ones: the usual noisy quantum universality without error correction and the usual fault-tolerant quantum computation, but closer to the later.
  arXiv  Detail & Related papers  (2021-05-31T08:17:30Z)
• One-shot quantum error correction of classical and quantum information [10.957528713294874]
  Quantum error correction (QEC) is one of the central concepts in quantum information science. We provide a form of capacity theorem for both classical and quantum information. We show that a demonstration of QEC by short random quantum circuits is feasible.
  arXiv  Detail & Related papers  (2020-11-02T01:24:59Z)
• Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem [77.34726150561087]
  We present a proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code with its ability to achieve a universal set of logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols.
  arXiv  Detail & Related papers  (2020-04-24T17:58:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.