# nLab qbit

Contents

### Context

#### Computation

intuitionistic mathematics

# Contents

## Idea

In quantum information theory and quantum computing, by a q-bit (or qubit) one means a quantum state in a 2-dimensional complex Hilbert space of states.

Hence the quantum data type $QBit$ is the 2-dimensional complex vector space equipped with its canonical quantum measurement-basis

$\mathbb{C}^2 \,\simeq\, \mathbb{C} \cdot \vert 0 \rangle \oplus \mathbb{C} \cdot \vert 1 \rangle \,.$

Analogous higher- but still finite- $d$-dimensional quantum data (types) are called qdits (“qtrits” for $d = 3$).

## Properties

### In terms of geometric quantization

In geometric quantization qbits are naturally understood as the states given by the geometric quantization of the 2-sphere for prequantum line bundle (plus metaplectic correction) being of unit first Chern class. See at geometric quantization of the 2-sphere – The space of quantum states.

## References

### General

The term q-bit goes back to

and was popularized by early adoption such as in

Laboratoy-realizations of qbits for use in quantum computers:

### Spin resonance qbits

The idea of qbits realized on quantum mechanical spinors (e.g. electron-spin or nucleus-spin):

The very first proof-of-principle quantum computations were made with nuclear magnetic resonance-technology:

• D. G. Cory et al, NMR Based Quantum Information Processing: Achievements and Prospects, Fortsch. Phys. 48 9-11 (2000) 875-907 $[$arXiv:quant-ph/0004104$]$

• Jonathan A. Jones, Quantum Computing and Nuclear Magnetic Resonance, PhysChemComm 11 (2001) $[$doi:10.1039/b103231n, arXiv:quant-ph/0106067$]$

• Jonathan A. Jones, Quantum Computing with NMR, Prog. NMR Spectrosc. 59 (2011) 91-120 $[$doi:10.1016/j.pnmrs.2010.11.001, arXiv:1011.1382$]$

• Dorothea Golze, Maik Icker, Stefan Berger, Implementation of two-qubit and three-qubit quantum computers using liquid-state nuclear magnetic resonance, Concepts in Magnetic Resonance 40A 1 (2012) 25-37 $[$doi:10.1002/cmr.a.21222$[$

• NMR Quantum Computing (2012) $[$slides pdf$]$

• Tao Xin et al., Nuclear magnetic resonance for quantum computing: Techniques and recent achievements (Topic Review - Solid-state quantum information processing), Chinese Physics B 27 020308 $[$doi:10.1088/1674-1056/27/2/020308$]$

• Lieven Vandersypen, Mark Eriksson: Quantum computing with semiconductor spins, Physics Today 72 8 (2019) 38 $[$doi:10.1063/PT.3.4270$]$

Exposition, review and outlook:

• Asif Equbal, Molecular spin qubits for future quantum technology, talk at CQTS (Nov 2022) $[$slides: pdf, video: rec$]$

More on implementation of quantum logic gates on qbits realized on nucleon-spin, via pulse protocols in NMR-technology:

• Price, Somaroo, Tseng, Gore, Fahmy,, Havel, Cory: Construction and Implementation of NMR Quantum Logic Gates for Two Spin Systems, Journal of Magnetic Resonance 140 2 (1999) 371-378 $[$doi;10.1006/jmre.1999.1851$]$

and analogously on electron-spin:

• M. Yu. Volkov and K. M. Salikhov, Pulse Protocols for Quantum Computing with Electron Spins as Qubits, Appl Magn Reson 41 (2011) 145–154 $[$doi:10.1007/s00723-011-0297-2$]$

There exist toy desktop quantum computers for educational purposes, operating on a couple of nuclear magnetic resonance qbits at room temperature :

• SpinQ: SpinQ Triangulum: a commercial three-qubit desktop quantum computer $[$arXiv:2202.02983$]$

### Superconducting qbits

On realizing qbits and quantum gates (hence quantum computation) via quantum states of magnetic flux through (Josephson junctions in) superconductors, manipulated via electromagnetic pulses:

• M. H. Devoret, A. Wallraff, J. M. Martinis, Superconducting Qubits: A Short Review $[$arXiv:cond-mat/0411174$]$

• John Clarke, Frank K. Wilhelm, Superconducting quantum bits, Nature 453 (2008) 1031–1042 $[$doi:10.1038/nature07128$]$

• Jerry Moy Chow, Quantum Information Processing with Superconducting Qubits (2010) $[$pdf$]$

• Jay M. Gambetta, Jerry M. Chow, Matthias Steffen, Building logical qubits in a superconducting quantum computing system, npj Quantum Information 3 2 (2017) $[$doi:10.1038/s41534-016-0004-0$]$

• Morten Kjaergaard et al. Superconducting Qubits: Current State of Play, Annual Review of Condensed Matter Physics 11 (2019) 369-395 $[$doi:10.1146/annurev-conmatphys-031119-050605$]$

• He-Liang Huang, Dachao Wu, Daojin Fan, Xiaobo Zhu, Superconducting Quantum Computing: A Review, Science China Information Sciences 63 8 (2020) 1-32 $[$arXiv:2006.10433, doi:10.1007/s11432-020-2881-9$]$

• S. Kwon et al., Gate-based superconducting quantum computing, Journal of Applied Physics 129 (2021) 041102 $[$doi:10.1063/5.0029735$]$

Fine detail of the pulse control:

• M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, S. Filipp: Leakage reduction in fast superconducting qubit gates via optimal control, npj Quantum Information 7 14 (2021) $[$doi:10.1038/s41534-020-00346-2$]$

• M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer & A. Kandala. Dynamics of superconducting qubit relaxation times, npj Quantum Information 8 132 (2022) $[$doi:10.1038/s41534-022-00643-y$]$

• Elisha Siddiqui Matekole, Yao-Lung L. Fang, Meifeng Lin, Methods and Results for Quantum Optimal Pulse Control on Superconducting Qubit Systems, 2022 IEEE International Parallel and Distributed Processing Symposium Workshops (2022) $[$arXiv:2202.03260, doi:10.1109/IPDPSW55747.2022.00102$]$

Corrections due to quasiparticle-excitations:

Last revised on March 7, 2023 at 15:22:33. See the history of this page for a list of all contributions to it.