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arXiv:1509.08675

Fermionic quantum orthogonalizations I

Gyula Lakos

Published Sep 29, 2015• 2 citations
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Abstract

We generalize classical orthogonalization procedures from real linear algebra to the setting of fermionic quantum (FQ) operations. In the case of the Gram-Schmidt orthogonalization procedure, the generalization is easy. This, however, helps to obtain general information regarding FQ operations, and to generalize the symmetric orthogonalization procedure.

DOI
10.48550/arxiv.1509.08675

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We generalize classical orthogonalization procedures from real linear algebra to the setting of fermionic quantum (FQ) operations. In the case of the Gram-Schmidt orthogonalization procedure, the generalization is easy. This, however, helps to obtain general information regarding FQ operations, and to generalize the symmetric orthogonalization procedure.
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The objective of this series of papers is to recover information regarding the behaviour of FQ operations in the case n=2n=2n=2, and FQ conform-operations in the case n=3n=3n=3. In this first part we study how the basic invariance properties of FQ operations (n=2n=2n=2) are reflected in their formal power series expansions.
A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends PT\mathcal{P}\mathcal{T}PT-symmetric quantum mechanics from bosonic systems (systems for which T2=1{\mathcal{T}}^{2}=\mathbf{1}T2=1) to fermionic systems (systems for which ${\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT\mathcal{P}\mathcal{T}PT-symmetric matrix representations for operator algebras of the form ${\ensuremath{\eta}}^{2}=0$, ${\overline{\ensuremath{\eta}}}^{2}=0$, $\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}$, where $\overline{\ensuremath{\eta}}={\ensuremath{\eta}}^{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{T}\ensuremath{\eta}{\mathcal{T}}^{\ensuremath{-}1}{\mathcal{P}}^{\ensuremath{-}1}$. It is easy to construct matrix representations for the Grassmann algebra ($\ensuremath{\alpha}=0$). However, one can only construct matrix representations for the fermionic operator algebra ($\ensuremath{\alpha}\ensuremath{\ne}0$) if $\ensuremath{\alpha}=\ensuremath{-}1$; a matrix representation does not exist for the conventional value $\ensuremath{\alpha}=1$.
We review existing classical simulation methods for performing fermionic Gaussian operations and develop new methods to address the gap by adhering to the fundamental theoretical framework established by Bravyi [Quantum Info. Comput. 5, 216 (2005)] for the most general fermionic Gaussian processes. Throughout this attempt, the focus remains on the unified approach that can be applied to generic fermionic Gaussian operations. This is beneficial since the selection of simulation methods has often been based on an ad hoc choice, heavily influenced by the specific model and circumstances, rather than on a systematic approach.
Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over Fpk\mathbb {F}_{p^k}Fpk​
We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over Fq, where q = pk, p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over Fq. We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.
The advancement of quantum computers undermines the security of classical blockchain, necessitating either a post-quantum upgrade of the existing architecture or creation of an inherently quantum blockchain. Here we propose a practically realizable model of a fully quantum blockchain based on a generalized Gram-Schmidt procedure utilizing dimensional lifting. In this model, information of transactions stored in a multi-qubit state are subsequently encoded using the generalized Gram-Schmidt process. The chain is generated as a result of the reliance of orthogonalized state on the sequence of states preceding it. Various forking scenarios and their countermeasures are considered for the proposed model. It is shown to be secure even against quantum computing attacks using the no-cloning theorem and non-democratic nature of Generalized Gram-Schmidt orthogonalization. Finally, we outline a framework for a quantum token built on the same architecture as our blockchain.