nLab qbit

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.

Related concepts

• data type

• spin resonance qbit

• one clean qbit

• Pauli gate

• bit, truth value

• bit flip channel, bit flip code

• quantum logic gate, quantum circuit

• quantum logic, quantum computing, quantum error correction

• dimer

• quantum set

References

General

The term q-bit goes back to

• Benjamin Schumacher, Quantum coding, Phys. Rev. A 51 (1995) 2738 $[$doi:10.1103/PhysRevA.51.2738$]$

and was popularized by early adoption such as in

• Peter W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R) 1995 (doi:10.1103/PhysRevA.52.R2493)

Textbook account:

• Michael A. Nielsen, Isaac L. Chuang, §1.2 in: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

See also:

• Wikipedia, Qbit

Laboratoy-realizations of qbits for use in quantum computers:

• Qubit Zoo

Spin resonance qbits

The idea of spin resonance qbits, i.e. of qbits realized on quantum mechanical spinors (e.g. electron-spin or nucleus-spin) and manipulated via spin resonance:

• Daniel Loss, David P. DiVincenzo, Quantum Computation with Quantum Dots, Phys. Rev. A 57 120 (1998) $[$arXiv:cond-mat/9701055, doi:10.1103/PhysRevA.57.120$]$

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$]$

See also:

• 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:

• Raymond Laflamme, Emanuel Knill, et al., Introduction to NMR Quantum Information Processing, Proceedings of the International School of Physics “Enrico Fermi” 148 Experimental Quantum Computation and Information [arXiv:quant-ph/0207172]

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

• Jonathan A. Jones, Controlling NMR spin systems for quantum computation, Spectroscopy 140–141 (2024) 49-85 [doi:10.1016/j.pnmrs.2024.02.002, arXiv:2402.01308]

  “Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor’s quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. […] the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.”

See also:

• Wikipedia, Spin qbit quantum computer

• Wikipedia, Nuclear magnetic resonance quantum computer

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$]$

For references on spin resonance qbits realized on a nitrogen-vacancy center in diamond, see there.

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

• CIQTEK: Diamond Quantum Computer for Education

• 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:

• Michel 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$]$

• Michel H. Devoret, R. J. Schoelkopf, Superconducting Circuits for Quantum Information: An Outlook, Science 339 6124 (2013) 1169-1174 [doi:10.1126/science.1231930]

• 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$]$

• Olivier Ezratty, Perspective on superconducting qubit quantum computing, Eur. Phys. J. A 59 94 (2023) [doi:10.1140/epja/s10050-023-01006-7]

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:

• Leonid I. Glazman, Gianluigi Catelani, Bogoliubov Quasiparticles in Superconducting Qubits, SciPost Phys. Lect. Notes 31 (2021) [arXiv:2003.04366, doi:10.21468/SciPostPhysLectNotes.31]