1087:
5130:. In 2013, a modification of similar systems (using charged rather than neutral donors) has dramatically extended this time, to 3 hours at very low temperatures and 39 minutes at room temperature. Room temperature preparation of a qubit based on electron spins instead of nuclear spin was also demonstrated by a team of scientists from Switzerland and Australia. An increased coherence of qubits is being explored by researchers who are testing the limitations of a
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3656:. In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value "0" or "1" â a most surprising circumstance that cannot be explained by classical physics.
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computable states simultaneously. Moreover, whereas a measurement of a classical bit would not disturb its state, a measurement of a qubit would destroy its coherence and irrevocably disturb the superposition state. It is possible to fully encode one bit in one qubit. However, a qubit can hold more
5811:
Seskir, Zeki C.; MigdaĆ, Piotr; Weidner, Carrie; Anupam, Aditya; Case, Nicky; Davis, Noah; Decaroli, Chiara; Ercan, Ä°lke; Foti, Caterina; Gora, PaweĆ; Jankiewicz, Klementyna; La Cour, Brian R.; Malo, Jorge Yago; Maniscalco, Sabrina; Naeemi, Azad; Nita, Laurentiu; Parvin, Nassim; Scafirimuto, Fabio;
4327:
Quantum entanglement also allows multiple states (such as the Bell state mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical
6544:
Chu, Ji; He, Xiaoyu; Zhou, Yuxuan; Yuan, Jiahao; Zhang, Libo; Guo, Qihao; Hai, Yongju; Han, Zhikun; Hu, Chang-Kang; Huang, Wenhui; Jia, Hao; Jiao, Dawei; Li, Sai; Liu, Yang; Ni, Zhongchu; Nie, Lifu; Pan, Xianchuang; Qiu, Jiawei; Wei, Weiwei; Nuerbolati, Wuerkaixi; Yang, Zusheng; Zhang, Jiajian;
4490:
can be used as a qubit. Multilevel systems can be used as well, if they possess two states that can be effectively decoupled from the rest (e.g., the ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate
286:
in which the two spin states (left-handed and the right-handed circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent
6545:
Zhang, Zhida; Zou, Wanjing; Chen, Yuanzhen; Deng, Xiaowei; Deng, Xiuhao; Hu, Ling; Li, Jian; Liu, Song; Lu, Yao; Niu, Jingjing; Tan, Dian; Xu, Yuan; Yan, Tongxing; Zhong, Youpeng; Yan, Fei; Sun, Xiaoming; Yu, Dapeng (2022-11-14). "Scalable algorithm simplification using quantum AND logic".
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There are two possible outcomes for the measurement of a qubitâusually taken to have the value "0" and "1", like a bit. However, whereas the state of a bit can only be binary (either 0 or 1), the general state of a qubit according to quantum mechanics can arbitrarily be a
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are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state
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developed a technique to dynamically control the cross-Kerr interactions between fixed-frequency qutrits, achieving high two-qutrit gate fidelities. This was followed by a demonstration of extensible control of superconducting qudits up to
2130:
6314:
Goss, Noah; Morvan, Alexis; Marinelli, Brian; Mitchell, Bradley K.; Nguyen, Long B.; Naik, Ravi K.; Chen, Larry; JĂŒnger, Christian; Kreikebaum, John Mark; Santiago, David I.; Wallman, Joel J.; Siddiqi, Irfan (2022-12-05).
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Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation".
1992:
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Kaszlikowski, Dagomir; GnaciĆski, Piotr; Ć»ukowski, Marek; Miklaszewski, Wieslaw; Zeilinger, Anton (2000). "Violations of Local
Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits".
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dephasing time are a time to characterize the physical implementation and represent their sensitivity to noise. A higher time does not necessarily mean that one or the other qubit is better suited for
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that is unique to quantum computation. A major hurdle facing quantum computing, as of 2018, in its quest to surpass classical digital computing, is noise in quantum gates that limits the size of
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In 2008 a team of scientists from the U.K. and U.S. reported the first relatively long (1.75 seconds) and coherent transfer of a superposition state in an electron spin "processing" qubit to a
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a coherent superposition, represented by a point on the surface of the Bloch sphere as described above. Coherence is essential for a qubit to be in a superposition state. With interactions,
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and the presence of current in a cable can all be used to represent bits in the same computer, an eventual quantum computer is likely to use various combinations of qubits in its design.
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will (because this state lies on the positive pole of the Y-axis) in the Y-basis always measure to the same value, while in the Z-basis results in equal probability of being measured to
3572:, i.e., she can now tell if her qubit has value "0" or "1". Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a
5812:
Sherson, Jacob F.; Surer, Elif; Wootton, James; Yeh, Lia; Zabello, Olga; Chiofalo, MarilĂč (2022). "Quantum games and interactive tools for quantum technologies outreach and education".
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two-level systems to various degrees have been successfully realized. Similarly to a classical bit where the state of a transistor in a processor, the magnetization of a surface in a
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Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtainingâwith equal probabilitiesâeither
335:, characterized as 0 or 1, is used to represent information in classical computers. When averaged over both of its states (0,1), a binary digit can represent up to one bit of
4170:
6493:
Nguyen, L.B.; Kim, Y.; Hashim, A.; Goss, N.; Marinelli, B.; Bhandari, B.; Das, D.; Naik, R.K.; Kreikebaum, J.M.; Jordan, A.; Santiago, D.I.; Siddiqi, I. (16 January 2024).
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4478:. Besides the advantage associated with the enlarged computational space, the third qutrit level can be exploited to implement efficient compilation of multi-qubit gates.
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2245:{\displaystyle {\begin{aligned}\alpha &=e^{i\delta }\cos {\frac {\theta }{2}},\\\beta &=e^{i(\delta +\varphi )}\sin {\frac {\theta }{2}}.\end{aligned}}}
7894:
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1998:
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Nguyen, Long B.; Goss, Noah; Siva, Karthik; Kim, Yosep; Younis, Ed; Qing, Bingcheng; Hashim, Akel; Santiago, David I.; Siddiqi, Irfan (2024-08-19).
4470:
is the unit of quantum information that can be realized in suitable 3-level quantum systems. This is analogous to the unit of classical information
7682:
1937:
2582:, a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles.
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5882:
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631:
8006:
7192:
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Eduardo
Berrios; Martin Gruebele; Dmytro Shyshlov; Lei Wang; Dmitri Babikov (2012). "High fidelity quantum gates with vibrational qubits".
5126:"memory" qubit. This event can be considered the first relatively consistent quantum data storage, a vital step towards the development of
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2650:
2406:{\displaystyle {\begin{aligned}\alpha &=\cos {\frac {\theta }{2}},\\\beta &=e^{i\varphi }\sin {\frac {\theta }{2}},\end{aligned}}}
1824:
2004:
7964:
753:. For example, two qubits could be represented in a four-dimensional linear vector space spanned by the following product basis states:
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2120:. This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of
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the quantum state, measuring the state in one basis hides some of the values that would have been measurable the other basis; See the
4433:
4429:'s Center for Quantum Information implemented the dual-type qubit scheme in trapped ion quantum computers using the same ion species.
3327:
6871:
Wang, Zhanning; Marcellina, Elizabeth; Hamilton, A. R.; Cullen, James H.; Rogge, Sven; Salfi, Joe; Culcer, Dimitrie (April 1, 2021).
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The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the
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7157:
7136:
5389:
3228:
1890:
472:
7490:
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Kamyar Saeedi; et al. (2013). "Room-Temperature
Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28".
6223:
Ringbauer, Martin; Meth, Michael; Postler, Lukas; Stricker, Roman; Blatt, Rainer; Schindler, Philipp; Monz, Thomas (21 July 2022).
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property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.
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4411:-level system is not an exponent of 2 cannot be mapped to arrays of qubits. It is for example possible to have 5-level qudits.
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The following is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only.
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7855:
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5758:
Shor, Peter (1997). "Polynomial-Time
Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computerâ".
2521:
7860:
7728:
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7150:
4758:
2725:, a statistical combination or "incoherent mixture" of different pure states. Mixed states can be represented by points
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of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles
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A treatment of two-level quantum systems, decades before the term "qubit" was coined, is found in the third volume of
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3513:. In other words, there is no way to tell if the first qubit has value "0" or "1" and likewise for the second qubit.
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act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the
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must be passed as fast as possible, as electrical voltage cannot change from one level to another instantaneously.
3671:(or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is
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Qubit basis states can also be combined to form product basis states. A set of qubits taken together is called a
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succeeded in developing a universal qudit quantum processor with trapped ions. In the same year, researchers at
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3276:; the qubit itself is an exhibition of quantum entanglement. In this case, quantum entanglement is a local or
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270:, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the
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An introduction to qubits for non-specialists, by the person who coined the word, is found in
Lecture 21 of
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with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint
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constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits.
3195:
may be implemented logically or physically: Logically as a measurement, followed by the application of the
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361:, and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between two
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An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit
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the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles
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Nisbet-Jones, Peter B. R.; Dilley, Jerome; Holleczek, Annemarie; Barter, Oliver; Kuhn, Axel (2013).
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is an irreversible operation in which information is gained about the state of a single qubit, and
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in classical computing, and may be mapped to (or realized by) arrays of qubits. Qudits where the
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3167:. This operation collapses the quantum state (exactly like with measurement). Initialization to
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2464:
2461:, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where
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have a direct physical representation, both easily demonstrable experimentally in a class with
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introduction to entanglement, showcasing a Bell state and the measurement of it, is found in
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J. J. L. Morton; et al. (2008). "Solid-state quantum memory using the P nuclear spin".
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750:
320:
228:
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6996:
6806:"Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres"
5898:
5855:
5850:
5232:
4654:
4514:
4341:
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2579:
1594:. Because the absolute squares of the amplitudes equate to probabilities, it follows that
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5854: This article incorporates text from this source, which is available under the
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5693:
7870:
7847:
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7098:
6873:"Qubits composed of holes could be the trick to build faster, larger quantum computers"
6848:
6805:
6804:
NĂĄfrĂĄdi, BĂĄlint; Choucair, Mohammad; Dinse, Klaus-Pete; ForrĂł, LĂĄszlĂł (July 18, 2016).
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6316:
6297:"Researchers realize two coherently convertible qubit types using a single ion species"
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qubits are represented by a superposition state vector in 2 dimensional
Hilbert space.
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are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states,
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88:
68:
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with global phase can not be represented as points on the surface of the Bloch sphere.
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7692:
7510:
7436:
6908:
6672:
6395:"Empowering high-dimensional quantum computing by traversing the dual bosonic ladder"
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6224:
6101:
5366:
5134:
5012:
4693:
2849:
2714:
1282:
743:
486:
113:
72:
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6184:
6036:
5983:
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no longer experimentally accessible. If measurement is performed on a qubit that is
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7837:
7128:
7120:
7085:
7000:
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6727:
5902:
5797:
5197:
5164:
5123:
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computer. Many of the successes of quantum computation and communication, such as
3995:
A common application of the CNOT gate is to maximally entangle two qubits into the
3699:, and otherwise leaves it unchanged. With respect to the unentangled product basis
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3235:
2454:
2258:
1814:
497:
408:
395:
components, a complete description of its state in classical physics requires only
332:
210:
6633:
B. Lucatto; et al. (2019). "Charge qubit in van der Waals heterostructures".
6052:"Photonic qubits, qutrits and ququads accurately prepared and delivered on demand"
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5277:
is not measurable, because it applies to both basis states, and is on the complex
1093:
offers a straightforward way to present orthogonal states. With a typical mapping
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5278:
5146:
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4948:
4924:
4903:
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2813:
There are various kinds of physical operations that can be performed on qubits.
2718:
1750:, encode more than just the probabilities of the outcomes of a measurement; the
1022:
943:
864:
785:
742:, together called the computational basis, are said to span the two-dimensional
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A non-traditional motivation of the qubit aimed at non-physicists is found in
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5386:
instead in the X or Y Pauli basis depends on the relative phase. For example,
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78:
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6358:
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7026:
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6123:
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5131:
4887:
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347:. However, in this article, the word bit is synonymous with a binary digit.
315:. In the acknowledgments of his 1995 paper, Schumacher states that the term
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5709:
1987:{\displaystyle \beta =e^{i\varphi }\sin \left({\frac {\theta }{2}}\right)}
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denotes the unit of quantum information that can be realized in suitable
2458:
2453:
The possible quantum states for a single qubit can be visualised using a
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275:
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2862:
is lost. The result of the measurement of a single qubit with the state
2292:
has no physically observable consequences, so we can arbitrarily choose
7978:
7595:
5997:
Preskill, John (2018). "Quantum
Computing in the NISQ era and beyond".
5155:
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4385:
358:
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92:
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would lie on the equator of the sphere at the positive X-axis. In the
278:
in which the two levels can be taken as spin up and spin down; or the
30:
This article is about the quantum computing unit. For other uses, see
17:
7955:
7451:
6470:""64-dimensional quantum space" drastically boosts quantum computing"
5017:
4996:
4908:
4639:
4603:
4467:
1460:. When we measure this qubit in the standard basis, according to the
291:
of multiple states simultaneously, a property that is fundamental to
283:
155:
6317:"High-fidelity qutrit entangling gates for superconducting circuits"
5648:{\displaystyle (\mathbb {C} ^{2})^{\otimes n}\cong \mathbb {C} ^{N}}
2706:{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,}
1880:{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,}
6891:
6822:
6757:
6647:
6559:
6511:
6411:
6333:
6241:
6011:
5826:
4525:
use different implementations of qubits to suit their application.
4498:
All physical implementations are affected by noise. The so-called T
2852:
vector. The result from this multiplication is a new quantum state.
2057:{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle \,}
489:
of a qubit can be represented by a linear superposition of its two
6702:
6068:
4982:
4671:
3394:, there are equal probabilities of measuring either product state
1809:
1085:
204:
161:
38:
6388:
6386:
4400:
are frequently used to denote the dimension of a quantum system.
4376:-level quantum systems. A qubit register that can be measured to
4130:{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}}
3054:. Measurement of the state of the qubit alters the magnitudes of
2917:{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }
1434:{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }
7973:
7446:
7379:
5944:
Horodecki, Ryszard; et al. (2009). "Quantum entanglement".
5203:
5193:
4779:
3380:{\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).}
384:
209:
The general definition of a qubit as the quantum state of a two-
7146:
4360:
perform calculations by manipulating qubits within a register.
4262:{\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}
7590:
7575:
5460:{\displaystyle (|0\rangle +e^{i\pi /2}|1\rangle )/{\sqrt {2}}}
4804:
4510:
because gate times and fidelities need to be considered, too.
4336:, make use of entanglement, suggesting that entanglement is a
4068:, the inputs A (control) and B (target) to the CNOT gate are:
1927:{\displaystyle \alpha =\cos \left({\frac {\theta }{2}}\right)}
418:
340:
263:
109:
6495:"Programmable Heisenberg interactions between Floquet qubits"
3139:
Initialization or re-initialization to a known value, often
237:
5382:, where the relative phase have no effect on measurement.
266:
physically realized with a two-state device. A qubit is a
1997:
It might, at first sight, seem that there should be four
243:
7106:
The science of information: from language to black holes
6553:(1). Springer Science and Business Media LLC: 126â131.
6225:"A universal qudit quantum processor with trapped ions"
4530:
4463:
in 2024 based on programmable two-photon interactions.
3234:, by lowering the energy of the quantum system to its
1230:{\displaystyle (|0\rangle \pm |1\rangle )/{\sqrt {2}}}
5601:
5568:
5548:
5501:
5473:
5392:
5341:
5287:
5253:
5063:
5031:
4577:
4548:
4443:
4283:
4215:
4180:
4143:
4076:
4039:
4001:
3955:
3912:
3869:
3826:
3792:
3764:
3736:
3705:
3677:
3634:
3606:
3578:
3550:
3522:
3456:
3428:
3400:
3330:
3289:
3205:
3173:
3145:
3104:
3068:
3023:
2995:
2958:
2930:
2868:
2775:
2755:
2735:
2653:
2619:
2599:
2524:
2495:
2467:
2422:
2319:
2268:
2133:
2090:
2070:
2007:
1940:
1893:
1827:
1780:
1760:
1736:
1716:
1647:
1620:
1600:
1563:
1535:
1498:
1470:
1385:
1354:
1326:
1302:
1296:
of the basis states. This means that a single qubit (
1267:
1247:
1183:
1141:
1099:
998:
919:
840:
761:
700:
672:
644:
571:
506:
234:
6405:(1). Springer Science and Business Media LLC: 7117.
6327:(1). Springer Science and Business Media LLC: 7481.
3062:. For instance, if the result of the measurement is
246:
240:
7940:
7903:
7869:
7846:
7813:
7804:
7737:
7666:
7604:
7564:
7476:
7417:
7343:
7252:
7180:
2571:{\displaystyle (|0\rangle +|1\rangle )/{\sqrt {2}}}
231:
5647:
5587:
5554:
5515:
5487:
5459:
5357:
5326:
5269:
5077:
5045:
4591:
4562:
4455:
4304:
4261:
4201:
4164:
4129:
4060:
4022:
3983:
3940:
3897:
3854:
3809:
3778:
3750:
3722:
3691:
3648:
3620:
3592:
3564:
3536:
3505:
3442:
3414:
3379:
3310:
3219:
3187:
3159:
3120:
3082:
3046:
3009:
2981:
2944:
2916:
2781:
2761:
2741:
2705:
2625:
2605:
2570:
2509:
2481:
2438:
2405:
2284:
2244:
2096:
2076:
2056:
1986:
1926:
1879:
1786:
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1742:
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1699:
1626:
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1549:
1521:
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1433:
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1340:
1308:
1273:
1253:
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1169:
1127:
1063:
984:
905:
826:
734:
686:
658:
622:
557:
268:two-state (or two-level) quantum-mechanical system
6931:(5). American Physical Society (APS): 3457â3467.
2647:A pure state is fully specified by a single ket,
1056:
1015:
977:
936:
898:
857:
819:
778:
6200:"Qudits: The Real Future of Quantum Computing?"
2789:of the vector that represents the mixed state.
2310:is zero), leaving just two degrees of freedom:
354:bit is implemented by one of two levels of low
319:was created in jest during a conversation with
5723:Nielsen, Michael A.; Chuang, Isaac L. (2010).
5206:define the phase about the qubits basis states
3984:{\displaystyle |11\rangle \mapsto |10\rangle }
3941:{\displaystyle |10\rangle \mapsto |11\rangle }
3898:{\displaystyle |01\rangle \mapsto |01\rangle }
3855:{\displaystyle |00\rangle \mapsto |00\rangle }
2795:can be used to maintain the purity of qubits.
7158:
5227:. The probability to observe an outcome upon
1700:{\displaystyle |\alpha |^{2}+|\beta |^{2}=1.}
615:
588:
550:
523:
399:bits, whereas in quantum physics a system of
186:
8:
7057:Yanofsky, Noson S.; Mannucci, Mirco (2013).
5510:
5482:
5439:
5404:
5072:
5040:
4586:
4557:
4299:
4253:
4239:
4196:
4153:
4114:
4100:
4055:
4017:
3978:
3964:
3935:
3921:
3892:
3878:
3849:
3835:
3804:
3801:
3773:
3745:
3717:
3706:
3686:
3643:
3615:
3587:
3559:
3531:
3437:
3409:
3368:
3354:
3305:
3214:
3182:
3154:
3077:
3004:
2939:
2911:
2894:
2877:
2696:
2679:
2662:
2550:
2536:
2504:
2476:
2050:
2033:
2016:
1870:
1853:
1836:
1544:
1479:
1428:
1411:
1394:
1363:
1335:
1209:
1195:
1164:
1150:
1122:
1108:
1007:
928:
849:
770:
729:
726:
712:
701:
681:
653:
580:
515:
7005:Quantum Computation and Quantum Information
5908:Quantum Computation and Quantum Information
5726:Quantum Computation and Quantum Information
3660:Controlled gate to construct the Bell state
3628:is the only state where Alice's qubit is a
453:. Unsourced material may be challenged and
262:âthe quantum version of the classic binary
7810:
7414:
7165:
7151:
7143:
4312:Bell state forms part of the setup of the
193:
179:
43:
7059:Quantum computing for computer scientists
6936:
6890:
6847:
6821:
6756:
6701:
6646:
6558:
6528:
6510:
6444:
6410:
6366:
6332:
6240:
6150:
6067:
6010:
5957:
5825:
5771:
5639:
5635:
5634:
5621:
5611:
5607:
5606:
5600:
5579:
5567:
5547:
5542:Actually isomorphic: For a register with
5502:
5500:
5474:
5472:
5450:
5445:
5431:
5421:
5414:
5396:
5391:
5346:
5340:
5312:
5307:
5297:
5288:
5286:
5258:
5252:
5064:
5062:
5032:
5030:
4578:
4576:
4549:
4547:
4442:
4416:National Institute of Scientific Research
4293:
4284:
4282:
4245:
4231:
4216:
4214:
4190:
4181:
4179:
4156:
4144:
4142:
4121:
4106:
4092:
4077:
4075:
4049:
4040:
4038:
4011:
4002:
4000:
3970:
3956:
3954:
3927:
3913:
3911:
3884:
3870:
3868:
3841:
3827:
3825:
3793:
3791:
3765:
3763:
3737:
3735:
3709:
3704:
3678:
3676:
3635:
3633:
3607:
3605:
3579:
3577:
3551:
3549:
3523:
3521:
3495:
3483:
3478:
3470:
3465:
3457:
3455:
3429:
3427:
3401:
3399:
3360:
3346:
3331:
3329:
3299:
3290:
3288:
3206:
3204:
3174:
3172:
3146:
3144:
3109:
3103:
3069:
3067:
3038:
3033:
3024:
3022:
2996:
2994:
2973:
2968:
2959:
2957:
2931:
2929:
2903:
2886:
2869:
2867:
2832:); mathematically, the qubits undergo a (
2774:
2754:
2734:
2702:
2688:
2671:
2654:
2652:
2618:
2598:
2561:
2556:
2542:
2528:
2523:
2496:
2494:
2468:
2466:
2427:
2421:
2386:
2371:
2340:
2320:
2318:
2273:
2267:
2225:
2198:
2167:
2152:
2134:
2132:
2089:
2069:
2053:
2042:
2025:
2008:
2006:
1970:
1951:
1939:
1910:
1892:
1876:
1862:
1845:
1828:
1826:
1779:
1759:
1735:
1715:
1685:
1680:
1671:
1662:
1657:
1648:
1646:
1619:
1599:
1578:
1573:
1564:
1562:
1536:
1534:
1513:
1508:
1499:
1497:
1471:
1469:
1420:
1403:
1386:
1384:
1355:
1353:
1327:
1325:
1301:
1281:, matching the high-school definition of
1266:
1246:
1220:
1215:
1201:
1187:
1182:
1156:
1142:
1140:
1114:
1100:
1098:
1055:
1054:
1020:
1014:
1013:
999:
997:
976:
975:
941:
935:
934:
920:
918:
897:
896:
862:
856:
855:
841:
839:
818:
817:
783:
777:
776:
762:
760:
735:{\displaystyle \{|0\rangle ,|1\rangle \}}
718:
704:
699:
673:
671:
645:
643:
614:
613:
593:
587:
586:
572:
570:
549:
548:
528:
522:
521:
507:
505:
473:Learn how and when to remove this message
5378:The Pauli Z basis is usually called the
3136:the state of the other entangled qubits.
630:. They are written in the conventional
500:). These vectors are usually denoted as
378:information, e.g., up to two bits using
7683:Continuous-variable quantum information
5667:
5216:
5001:Depends on specific topological system
4352:A number of qubits taken together is a
4174:After applying CNOT, the output is the
3817:, it maps the basis states as follows:
3506:{\displaystyle |1/{\sqrt {2}}|^{2}=1/2}
3199:if the result from the measurement was
2721:, it is possible to put the qubit in a
139:
101:
58:
407:(or a single point in a 2-dimensional
350:In classical computer technologies, a
7116:, The Teaching Company (4DVDs, 2015).
5327:{\displaystyle |e^{i\delta }|^{2}=1.}
3249:operation), potentially as part of a
3227:. Physically, for example if it is a
2585:The surface of the Bloch sphere is a
2457:(see picture). Represented on such a
2255:Additionally, for a single qubit the
1634:must be constrained according to the
1170:{\displaystyle |V\rangle =|1\rangle }
1128:{\displaystyle |H\rangle =|0\rangle }
1064:{\displaystyle |11\rangle ={\biggl }}
985:{\displaystyle |10\rangle ={\biggl }}
906:{\displaystyle |01\rangle ={\biggl }}
827:{\displaystyle |00\rangle ={\biggl }}
382:. Respectively, 8 qubits is called a
7:
7101:, Cambridge University Press (2013).
6295:Fardelli, Ingrid (August 18, 2022).
451:adding citations to reliable sources
4488:two-level quantum-mechanical system
623:{\displaystyle |1\rangle ={\bigl }}
558:{\displaystyle |0\rangle ={\bigl }}
7094:Quantum Computing Since Democritus
5239:for that outcome (or basis state,
4434:University of California, Berkeley
4305:{\displaystyle |\Phi ^{+}\rangle }
4290:
4202:{\displaystyle |\Phi ^{+}\rangle }
4187:
4061:{\displaystyle |\Phi ^{+}\rangle }
4046:
4023:{\displaystyle |\Phi ^{+}\rangle }
4008:
3311:{\displaystyle |\Phi ^{+}\rangle }
3296:
3245:to a remote system or machine (an
2589:, which represents the observable
1636:second axiom of probability theory
485:In quantum mechanics, the general
25:
7036:Explorations in Quantum Computing
5871:Explorations in Quantum Computing
4432:Also in 2022, researchers at the
1021:
942:
863:
784:
594:
529:
27:Basic unit of quantum information
8012:
8011:
8002:
8001:
5849:
4857:Charged superconducting island (
3600:, Bob must measure the same, as
2828:, operate on a set of qubits (a
423:
227:
7125:Quantum entanglement for babies
7082:The Feynman Lectures on Physics
6198:Choi, Charles Q. (2017-06-28).
6122:on in the quant-ph category of
4344:that can be executed reliably.
3098:is changed to the phase factor
1817:representation of a qubit. The
1794:is for example responsible for
1529:and the probability of outcome
6468:Irving, Michael (2022-10-14).
5618:
5602:
5503:
5475:
5442:
5432:
5397:
5393:
5308:
5289:
5196:are all possible single-qubit
5065:
5033:
4579:
4550:
4285:
4256:
4246:
4232:
4228:
4182:
4165:{\displaystyle |0\rangle _{B}}
4145:
4118:
4107:
4093:
4089:
4041:
4003:
3971:
3967:
3957:
3928:
3924:
3914:
3885:
3881:
3871:
3842:
3838:
3828:
3794:
3766:
3738:
3710:
3679:
3636:
3608:
3580:
3552:
3524:
3479:
3458:
3430:
3402:
3371:
3361:
3347:
3343:
3291:
3207:
3175:
3147:
3070:
3034:
3025:
2997:
2969:
2960:
2932:
2904:
2887:
2870:
2689:
2672:
2655:
2553:
2543:
2529:
2525:
2497:
2469:
2446:is the physically significant
2214:
2202:
2043:
2026:
2009:
1863:
1846:
1829:
1681:
1672:
1658:
1649:
1574:
1565:
1537:
1509:
1500:
1472:
1421:
1404:
1387:
1356:
1328:
1212:
1202:
1188:
1184:
1157:
1143:
1115:
1101:
1000:
921:
842:
763:
719:
705:
674:
646:
573:
508:
1:
7678:Adiabatic quantum computation
6086:10.1088/1367-2630/15/5/053007
5091:van der Waals heterostructure
3810:{\displaystyle |11\rangle \}}
3723:{\displaystyle \{|00\rangle }
2982:{\displaystyle |\alpha |^{2}}
2439:{\displaystyle e^{i\varphi }}
1821:for the superposition state,
1522:{\displaystyle |\alpha |^{2}}
1464:, the probability of outcome
744:linear vector (Hilbert) space
7729:Topological quantum computer
5358:{\displaystyle e^{i\delta }}
5270:{\displaystyle e^{i\delta }}
5137:spin-orbit qubit structure.
4513:Different applications like
4421:In 2022, researchers at the
4384:-level qudit. A rarely used
3241:Sending the qubit through a
3047:{\displaystyle |\beta |^{2}}
2285:{\displaystyle e^{i\delta }}
1710:The probability amplitudes,
1587:{\displaystyle |\beta |^{2}}
8007:Quantum information science
7174:Quantum information science
6665:10.1103/PhysRevB.100.121406
6592:Journal of Chemical Physics
6169:10.1103/PhysRevLett.85.4418
5175:Physical and logical qubits
5078:{\displaystyle |10\rangle }
5046:{\displaystyle |01\rangle }
4414:In 2017, scientists at the
3779:{\displaystyle |10\rangle }
3751:{\displaystyle |01\rangle }
3621:{\displaystyle |00\rangle }
3443:{\displaystyle |11\rangle }
3415:{\displaystyle |00\rangle }
2809:Physical and logical qubits
1806:Bloch sphere representation
8074:
7402:quantum gate teleportation
7063:Cambridge University Press
7034:Colin P. Williams (2011).
7009:Cambridge University Press
6901:10.1038/s41534-021-00386-2
6569:10.1038/s41567-022-01813-7
6530:10.1038/s41567-023-02326-7
6429:10.1038/s41467-024-51434-2
6351:10.1038/s41467-022-34851-z
6259:10.1038/s41567-022-01658-0
5913:Cambridge University Press
5869:Colin P. Williams (2011).
5731:Cambridge University Press
5678:(1995). "Quantum coding".
5516:{\displaystyle |1\rangle }
5488:{\displaystyle |0\rangle }
4592:{\displaystyle |1\rangle }
4563:{\displaystyle |0\rangle }
4466:Similar to the qubit, the
4403:Qudits are similar to the
4380:states is identical to an
3692:{\displaystyle |1\rangle }
3649:{\displaystyle |0\rangle }
3593:{\displaystyle |0\rangle }
3565:{\displaystyle |1\rangle }
3537:{\displaystyle |0\rangle }
3261:
3220:{\displaystyle |1\rangle }
3188:{\displaystyle |0\rangle }
3160:{\displaystyle |0\rangle }
3121:{\displaystyle e^{i\phi }}
3083:{\displaystyle |1\rangle }
3010:{\displaystyle |1\rangle }
2945:{\displaystyle |0\rangle }
2802:
2640:
2510:{\displaystyle |1\rangle }
2482:{\displaystyle |0\rangle }
1550:{\displaystyle |1\rangle }
1485:{\displaystyle |0\rangle }
1369:{\displaystyle |1\rangle }
1341:{\displaystyle |0\rangle }
687:{\displaystyle |1\rangle }
659:{\displaystyle |0\rangle }
36:
29:
7997:
7531:Quantum Fourier transform
7427:Post-quantum cryptography
7370:Entanglement distillation
5976:10.1103/RevModPhys.81.865
5946:Reviews of Modern Physics
5790:10.1137/S0097539795293172
5760:SIAM Journal on Computing
4975:Gapped topological system
4895:Counterclockwise current
4827:
4702:
4602:
4571:
4542:
4539:
4536:
4533:
3390:In this state, called an
8017:Quantum mechanics topics
7712:Quantum machine learning
7688:One-way quantum computer
7541:Quantum phase estimation
7442:Quantum key distribution
7375:Monogamy of entanglement
6955:10.1103/physreva.52.3457
6029:10.22331/q-2018-08-06-79
5844:10.1117/1.OE.61.8.081809
5702:10.1103/PhysRevA.51.2738
5185:Two-state quantum system
5170:Electron-on-helium qubit
4482:Physical implementations
2820:, building blocks for a
2793:Quantum error correction
2769:, as well as the length
2742:{\displaystyle \varphi }
2606:{\displaystyle \varphi }
1316:) can be described by a
1289:A pure qubit state is a
307:The coining of the term
37:Not to be confused with
7624:Randomized benchmarking
7486:Amplitude amplification
6878:npj Quantum Information
6775:10.1126/science.1239584
5588:{\displaystyle N=2^{n}}
5223:This is because of the
4988:Braiding of Excitations
4423:University of Innsbruck
2762:{\displaystyle \theta }
2626:{\displaystyle \theta }
2077:{\displaystyle \alpha }
1767:{\displaystyle \alpha }
1723:{\displaystyle \alpha }
1607:{\displaystyle \alpha }
1254:{\displaystyle \alpha }
415:Standard representation
7724:Quantum Turing machine
7717:quantum neural network
7464:Quantum secret sharing
6056:New Journal of Physics
5649:
5589:
5556:
5517:
5489:
5461:
5359:
5335:Note that by removing
5328:
5271:
5111:Electron on top sheet
5079:
5047:
4943:Electron on right dot
4593:
4564:
4457:
4306:
4263:
4203:
4166:
4131:
4062:
4024:
3985:
3942:
3899:
3856:
3811:
3780:
3752:
3724:
3693:
3650:
3622:
3594:
3566:
3538:
3507:
3444:
3416:
3381:
3312:
3221:
3189:
3161:
3132:, the measurement may
3122:
3084:
3048:
3011:
2983:
2946:
2918:
2838:unitary transformation
2783:
2763:
2743:
2707:
2627:
2607:
2572:
2511:
2483:
2440:
2407:
2286:
2246:
2098:
2097:{\displaystyle \beta }
2078:
2058:
1994:
1988:
1928:
1881:
1819:probability amplitudes
1800:double-slit experiment
1788:
1787:{\displaystyle \beta }
1768:
1744:
1743:{\displaystyle \beta }
1724:
1701:
1628:
1627:{\displaystyle \beta }
1608:
1588:
1551:
1523:
1486:
1454:probability amplitudes
1435:
1370:
1342:
1310:
1286:
1275:
1274:{\displaystyle \beta }
1255:
1231:
1171:
1129:
1065:
986:
907:
828:
736:
688:
660:
624:
559:
371:coherent superposition
214:
32:Qubit (disambiguation)
8058:Australian inventions
7796:Entanglement-assisted
7757:quantum convolutional
7432:Quantum coin flipping
7397:Quantum teleportation
7358:entanglement-assisted
7188:DiVincenzo's criteria
6810:Nature Communications
6399:Nature Communications
6321:Nature Communications
5650:
5590:
5557:
5531:uncertainty principle
5518:
5490:
5462:
5360:
5329:
5272:
5237:probability amplitude
5096:Electron localization
5080:
5048:
4940:Electron on left dot
4931:Electron localization
4698:Phase-squeezed state
4617:Polarization of light
4594:
4565:
4523:Quantum communication
4458:
4330:quantum teleportation
4318:quantum teleportation
4307:
4264:
4204:
4167:
4132:
4063:
4025:
3986:
3943:
3900:
3857:
3812:
3781:
3753:
3725:
3694:
3651:
3623:
3595:
3567:
3539:
3508:
3445:
3417:
3382:
3313:
3222:
3190:
3162:
3123:
3085:
3049:
3012:
2984:
2947:
2919:
2805:DiVincenzo's criteria
2803:Further information:
2784:
2764:
2744:
2708:
2628:
2608:
2587:two-dimensional space
2573:
2512:
2484:
2441:
2408:
2287:
2247:
2099:
2079:
2059:
1989:
1929:
1882:
1813:
1789:
1769:
1745:
1725:
1702:
1629:
1609:
1589:
1552:
1524:
1487:
1436:
1371:
1343:
1311:
1309:{\displaystyle \psi }
1276:
1256:
1232:
1172:
1130:
1091:Polarization of light
1089:
1066:
987:
908:
829:
737:
689:
661:
625:
560:
343:is the basic unit of
208:
60:Information-theoretic
8053:Units of information
7607:processor benchmarks
7536:Quantum optimization
7419:Quantum cryptography
7230:physical vs. logical
7086:(2013 ebook edition)
5599:
5566:
5546:
5525:Because measurement
5499:
5471:
5390:
5339:
5285:
5251:
5189:The elements of the
5061:
5029:
4992:Depends on specific
4918:First excited state
4644:Single photon state
4575:
4546:
4540:Information support
4441:
4322:quantum cryptography
4281:
4213:
4178:
4141:
4074:
4037:
3999:
3953:
3910:
3867:
3824:
3790:
3762:
3734:
3703:
3675:
3632:
3604:
3576:
3548:
3520:
3454:
3426:
3398:
3328:
3287:
3274:quantum entanglement
3264:Quantum entanglement
3258:Quantum entanglement
3203:
3171:
3143:
3102:
3094:is changed to 0 and
3066:
3021:
2993:
2956:
2928:
2866:
2799:Operations on qubits
2773:
2753:
2733:
2651:
2617:
2597:
2522:
2493:
2465:
2420:
2317:
2266:
2131:
2088:
2068:
2005:
1938:
1891:
1825:
1796:quantum interference
1778:
1758:
1734:
1714:
1645:
1618:
1598:
1561:
1533:
1496:
1468:
1383:
1352:
1324:
1300:
1265:
1245:
1181:
1139:
1097:
996:
917:
838:
759:
698:
670:
642:
569:
504:
447:improve this section
7320:Quantum speed limit
7215:Quantum programming
7210:Quantum information
7110:Benjamin Schumacher
7088:, in chapters 9-11.
6997:Nielsen, Michael A.
6947:1995PhRvA..52.3457B
6840:10.1038/ncomms12232
6832:2016NatCo...712232N
6767:2013Sci...342..830S
6720:10.1038/nature07295
6712:2008Natur.455.1085M
6696:(7216): 1085â1088.
6657:2019PhRvB.100l1406L
6604:2012JPCA..11611347B
6598:(46): 11347â11354.
6521:2024NatPh..20..240N
6421:2024NatCo..15.7117N
6343:2022NatCo..13.7481G
6251:2022NatPh..18.1053R
6161:2000PhRvL..85.4418K
6078:2013NJPh...15e3007N
6021:2018Quant...2...79P
5968:2009RvMP...81..865H
5899:Nielsen, Michael A.
5836:2022OptEn..61h1809S
5814:Optical Engineering
5782:1995quant.ph..8027S
5694:1995PhRvA..51.2738S
5380:computational basis
4456:{\displaystyle d=4}
4427:Tsinghua University
3669:controlled NOT gate
3392:equal superposition
2856:Quantum measurement
2818:Quantum logic gates
337:Shannon information
313:Benjamin Schumacher
260:quantum information
258:is a basic unit of
141:Quantum information
7969:Forest/Rigetti QCS
7705:quantum logic gate
7491:BernsteinâVazirani
7478:Quantum algorithms
7353:Classical capacity
7237:Quantum processors
7220:Quantum simulation
5915:. pp. 13â16.
5645:
5585:
5552:
5513:
5485:
5457:
5355:
5324:
5267:
5075:
5043:
5006:Vibrational qubit
4829:Josephson junction
4589:
4560:
4453:
4364:Qudits and qutrits
4302:
4259:
4199:
4162:
4127:
4058:
4020:
3981:
3938:
3895:
3852:
3807:
3776:
3748:
3720:
3689:
3646:
3618:
3590:
3562:
3534:
3503:
3440:
3412:
3377:
3308:
3217:
3185:
3157:
3118:
3080:
3044:
3007:
2979:
2942:
2914:
2844:the quantum gates
2779:
2759:
2739:
2703:
2623:
2603:
2568:
2507:
2479:
2436:
2403:
2401:
2282:
2242:
2240:
2094:
2074:
2054:
1999:degrees of freedom
1995:
1984:
1924:
1877:
1784:
1764:
1740:
1720:
1697:
1624:
1604:
1584:
1557:with value "1" is
1547:
1519:
1492:with value "0" is
1482:
1431:
1366:
1338:
1318:linear combination
1306:
1287:
1271:
1251:
1227:
1167:
1125:
1061:
1052:
1051:
982:
973:
972:
903:
894:
893:
824:
815:
814:
732:
684:
656:
620:
611:
610:
555:
546:
545:
403:qubits requires 2
215:
8038:Quantum computing
8025:
8024:
7936:
7935:
7833:Linear optical QC
7614:Quantum supremacy
7568:complexity theory
7521:Quantum annealing
7472:
7471:
7409:Superdense coding
7198:Quantum computing
7114:The Great Courses
7072:978-0-521-87996-5
7049:978-1-84628-887-6
6925:Physical Review A
6751:(6160): 830â833.
6635:Physical Review B
6612:10.1021/jp3055729
6145:(21): 4418â4421.
5922:978-1-10700-217-3
5884:978-1-84628-887-6
5877:. pp. 9â13.
5744:978-1-107-00217-3
5681:Physical Review A
5555:{\displaystyle n}
5455:
5202:The circle group
5128:quantum computing
5115:
5114:
4650:Time-bin encoding
4630:Number of photons
4534:Physical support
4519:Quantum computing
4508:quantum computing
4476:ternary computers
4358:Quantum computers
4334:superdense coding
4314:superdense coding
4226:
4225:
4087:
4086:
3475:
3341:
3340:
3017:with probability
2952:with probability
2782:{\displaystyle r}
2566:
2394:
2348:
2304:in the case that
2233:
2175:
1978:
1918:
1798:, as seen in the
1239:linear polarizers
1225:
1177:, quantum states
496:states (or basis
483:
482:
475:
380:superdense coding
311:is attributed to
297:quantum computing
293:quantum mechanics
219:quantum computing
203:
202:
16:(Redirected from
8065:
8015:
8014:
8005:
8004:
7811:
7741:error correction
7670:computing models
7636:Relaxation times
7526:Quantum counting
7415:
7363:quantum capacity
7310:No-teleportation
7295:No-communication
7167:
7160:
7153:
7144:
7076:
7053:
7030:
6983:
6982:
6940:
6938:quant-ph/9503016
6919:
6913:
6912:
6894:
6868:
6862:
6861:
6851:
6825:
6801:
6795:
6794:
6760:
6738:
6732:
6731:
6705:
6683:
6677:
6676:
6650:
6630:
6624:
6623:
6587:
6581:
6580:
6562:
6541:
6535:
6534:
6532:
6514:
6490:
6484:
6483:
6481:
6480:
6465:
6459:
6458:
6448:
6414:
6390:
6381:
6380:
6370:
6336:
6311:
6305:
6304:
6292:
6286:
6285:
6283:
6281:
6244:
6235:(9): 1053â1057.
6220:
6214:
6213:
6211:
6210:
6195:
6189:
6188:
6154:
6152:quant-ph/0005028
6133:
6127:
6114:As of June 2022
6112:
6106:
6105:
6071:
6047:
6041:
6040:
6014:
5994:
5988:
5987:
5961:
5959:quant-ph/0702225
5941:
5935:
5934:
5895:
5889:
5888:
5866:
5860:
5853:
5847:
5829:
5808:
5802:
5801:
5775:
5773:quant-ph/9508027
5766:(5): 1484â1509.
5755:
5749:
5748:
5720:
5714:
5713:
5688:(4): 2738â2747.
5672:
5655:
5654:
5652:
5651:
5646:
5644:
5643:
5638:
5629:
5628:
5616:
5615:
5610:
5594:
5592:
5591:
5586:
5584:
5583:
5561:
5559:
5558:
5553:
5540:
5534:
5522:
5520:
5519:
5514:
5506:
5494:
5492:
5491:
5486:
5478:
5466:
5464:
5463:
5458:
5456:
5451:
5449:
5435:
5430:
5429:
5425:
5400:
5376:
5370:
5364:
5362:
5361:
5356:
5354:
5353:
5333:
5331:
5330:
5325:
5317:
5316:
5311:
5305:
5304:
5292:
5276:
5274:
5273:
5268:
5266:
5265:
5221:
5180:Quantum register
5108:on bottom sheet
5084:
5082:
5081:
5076:
5068:
5052:
5050:
5049:
5044:
5036:
4598:
4596:
4595:
4590:
4582:
4569:
4567:
4566:
4561:
4553:
4531:
4462:
4460:
4459:
4454:
4348:Quantum register
4342:quantum circuits
4320:, and entangled
4311:
4309:
4308:
4303:
4298:
4297:
4288:
4268:
4266:
4265:
4260:
4249:
4235:
4227:
4221:
4217:
4208:
4206:
4205:
4200:
4195:
4194:
4185:
4171:
4169:
4168:
4163:
4161:
4160:
4148:
4136:
4134:
4133:
4128:
4126:
4125:
4110:
4096:
4088:
4082:
4078:
4067:
4065:
4064:
4059:
4054:
4053:
4044:
4029:
4027:
4026:
4021:
4016:
4015:
4006:
3990:
3988:
3987:
3982:
3974:
3960:
3947:
3945:
3944:
3939:
3931:
3917:
3904:
3902:
3901:
3896:
3888:
3874:
3861:
3859:
3858:
3853:
3845:
3831:
3816:
3814:
3813:
3808:
3797:
3785:
3783:
3782:
3777:
3769:
3757:
3755:
3754:
3749:
3741:
3729:
3727:
3726:
3721:
3713:
3698:
3696:
3695:
3690:
3682:
3665:Controlled gates
3655:
3653:
3652:
3647:
3639:
3627:
3625:
3624:
3619:
3611:
3599:
3597:
3596:
3591:
3583:
3571:
3569:
3568:
3563:
3555:
3543:
3541:
3540:
3535:
3527:
3512:
3510:
3509:
3504:
3499:
3488:
3487:
3482:
3476:
3471:
3469:
3461:
3449:
3447:
3446:
3441:
3433:
3421:
3419:
3418:
3413:
3405:
3386:
3384:
3383:
3378:
3364:
3350:
3342:
3336:
3332:
3317:
3315:
3314:
3309:
3304:
3303:
3294:
3226:
3224:
3223:
3218:
3210:
3194:
3192:
3191:
3186:
3178:
3166:
3164:
3163:
3158:
3150:
3127:
3125:
3124:
3119:
3117:
3116:
3089:
3087:
3086:
3081:
3073:
3053:
3051:
3050:
3045:
3043:
3042:
3037:
3028:
3016:
3014:
3013:
3008:
3000:
2988:
2986:
2985:
2980:
2978:
2977:
2972:
2963:
2951:
2949:
2948:
2943:
2935:
2923:
2921:
2920:
2915:
2907:
2890:
2873:
2826:quantum computer
2788:
2786:
2785:
2780:
2768:
2766:
2765:
2760:
2748:
2746:
2745:
2740:
2712:
2710:
2709:
2704:
2692:
2675:
2658:
2632:
2630:
2629:
2624:
2612:
2610:
2609:
2604:
2577:
2575:
2574:
2569:
2567:
2562:
2560:
2546:
2532:
2516:
2514:
2513:
2508:
2500:
2488:
2486:
2485:
2480:
2472:
2445:
2443:
2442:
2437:
2435:
2434:
2412:
2410:
2409:
2404:
2402:
2395:
2387:
2379:
2378:
2349:
2341:
2309:
2303:
2297:
2291:
2289:
2288:
2283:
2281:
2280:
2251:
2249:
2248:
2243:
2241:
2234:
2226:
2218:
2217:
2176:
2168:
2160:
2159:
2122:Hopf coordinates
2119:
2103:
2101:
2100:
2095:
2083:
2081:
2080:
2075:
2063:
2061:
2060:
2055:
2046:
2029:
2012:
1993:
1991:
1990:
1985:
1983:
1979:
1971:
1959:
1958:
1933:
1931:
1930:
1925:
1923:
1919:
1911:
1886:
1884:
1883:
1878:
1866:
1849:
1832:
1793:
1791:
1790:
1785:
1773:
1771:
1770:
1765:
1749:
1747:
1746:
1741:
1729:
1727:
1726:
1721:
1706:
1704:
1703:
1698:
1690:
1689:
1684:
1675:
1667:
1666:
1661:
1652:
1638:by the equation
1633:
1631:
1630:
1625:
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1583:
1582:
1577:
1568:
1556:
1554:
1553:
1548:
1540:
1528:
1526:
1525:
1520:
1518:
1517:
1512:
1503:
1491:
1489:
1488:
1483:
1475:
1440:
1438:
1437:
1432:
1424:
1407:
1390:
1375:
1373:
1372:
1367:
1359:
1347:
1345:
1344:
1339:
1331:
1315:
1313:
1312:
1307:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1236:
1234:
1233:
1228:
1226:
1221:
1219:
1205:
1191:
1176:
1174:
1173:
1168:
1160:
1146:
1134:
1132:
1131:
1126:
1118:
1104:
1070:
1068:
1067:
1062:
1060:
1059:
1053:
1019:
1018:
1003:
991:
989:
988:
983:
981:
980:
974:
940:
939:
924:
912:
910:
909:
904:
902:
901:
895:
861:
860:
845:
833:
831:
830:
825:
823:
822:
816:
782:
781:
766:
751:quantum register
741:
739:
738:
733:
722:
708:
693:
691:
690:
685:
677:
665:
663:
662:
657:
649:
629:
627:
626:
621:
619:
618:
612:
592:
591:
576:
564:
562:
561:
556:
554:
553:
547:
527:
526:
511:
478:
471:
467:
464:
458:
427:
419:
391:For a system of
327:Bit versus qubit
321:William Wootters
253:
252:
249:
248:
245:
242:
239:
236:
233:
195:
188:
181:
44:
21:
8073:
8072:
8068:
8067:
8066:
8064:
8063:
8062:
8028:
8027:
8026:
8021:
7993:
7943:
7932:
7905:Superconducting
7899:
7865:
7856:Neutral atom QC
7848:Ultracold atoms
7842:
7807:implementations
7806:
7800:
7740:
7733:
7700:Quantum circuit
7668:
7662:
7656:
7646:
7606:
7600:
7567:
7560:
7516:Hidden subgroup
7468:
7457:other protocols
7413:
7390:quantum network
7385:Quantum channel
7345:
7339:
7285:No-broadcasting
7275:GottesmanâKnill
7248:
7176:
7171:
7108:, by Professor
7073:
7056:
7050:
7033:
7019:
6995:
6992:
6990:Further reading
6987:
6986:
6921:
6920:
6916:
6870:
6869:
6865:
6803:
6802:
6798:
6740:
6739:
6735:
6685:
6684:
6680:
6632:
6631:
6627:
6589:
6588:
6584:
6543:
6542:
6538:
6492:
6491:
6487:
6478:
6476:
6467:
6466:
6462:
6392:
6391:
6384:
6313:
6312:
6308:
6294:
6293:
6289:
6279:
6277:
6222:
6221:
6217:
6208:
6206:
6197:
6196:
6192:
6139:Phys. Rev. Lett
6135:
6134:
6130:
6113:
6109:
6049:
6048:
6044:
5996:
5995:
5991:
5943:
5942:
5938:
5923:
5897:
5896:
5892:
5885:
5868:
5867:
5863:
5810:
5809:
5805:
5757:
5756:
5752:
5745:
5722:
5721:
5717:
5674:
5673:
5669:
5664:
5659:
5658:
5633:
5617:
5605:
5597:
5596:
5575:
5564:
5563:
5544:
5543:
5541:
5537:
5524:
5497:
5496:
5469:
5468:
5410:
5388:
5387:
5377:
5373:
5342:
5337:
5336:
5334:
5306:
5293:
5283:
5282:
5254:
5249:
5248:
5233:modulus squared
5222:
5218:
5213:
5143:
5120:
5059:
5058:
5027:
5026:
4923:Singly charged
4901:Superconducting
4875:Superconducting
4848:superconducting
4834:Superconducting
4709:Electronic spin
4655:Time of arrival
4573:
4572:
4544:
4543:
4515:Quantum sensing
4505:
4501:
4484:
4439:
4438:
4366:
4350:
4289:
4279:
4278:
4275:
4211:
4210:
4186:
4176:
4175:
4152:
4139:
4138:
4117:
4072:
4071:
4045:
4035:
4034:
4033:. To construct
4007:
3997:
3996:
3951:
3950:
3908:
3907:
3865:
3864:
3822:
3821:
3788:
3787:
3760:
3759:
3732:
3731:
3701:
3700:
3673:
3672:
3662:
3630:
3629:
3602:
3601:
3574:
3573:
3546:
3545:
3518:
3517:
3477:
3452:
3451:
3424:
3423:
3396:
3395:
3326:
3325:
3295:
3285:
3284:
3270:
3262:Main articles:
3260:
3251:quantum network
3243:quantum channel
3229:superconducting
3201:
3200:
3169:
3168:
3141:
3140:
3105:
3100:
3099:
3097:
3093:
3064:
3063:
3061:
3057:
3032:
3019:
3018:
2991:
2990:
2967:
2954:
2953:
2926:
2925:
2924:will be either
2864:
2863:
2822:quantum circuit
2811:
2801:
2771:
2770:
2751:
2750:
2731:
2730:
2649:
2648:
2645:
2639:
2615:
2614:
2595:
2594:
2580:classical limit
2520:
2519:
2491:
2490:
2463:
2462:
2423:
2418:
2417:
2400:
2399:
2367:
2360:
2354:
2353:
2327:
2315:
2314:
2305:
2299:
2298:to be real (or
2293:
2269:
2264:
2263:
2239:
2238:
2194:
2187:
2181:
2180:
2148:
2141:
2129:
2128:
2109:
2106:complex numbers
2086:
2085:
2066:
2065:
2003:
2002:
1966:
1947:
1936:
1935:
1906:
1889:
1888:
1823:
1822:
1808:
1776:
1775:
1756:
1755:
1732:
1731:
1712:
1711:
1679:
1656:
1643:
1642:
1616:
1615:
1596:
1595:
1572:
1559:
1558:
1531:
1530:
1507:
1494:
1493:
1466:
1465:
1458:complex numbers
1456:, and are both
1451:
1447:
1381:
1380:
1350:
1349:
1322:
1321:
1298:
1297:
1263:
1262:
1243:
1242:
1179:
1178:
1137:
1136:
1095:
1094:
1084:
1050:
1049:
1043:
1042:
1036:
1035:
1029:
1028:
994:
993:
971:
970:
964:
963:
957:
956:
950:
949:
915:
914:
892:
891:
885:
884:
878:
877:
871:
870:
836:
835:
813:
812:
806:
805:
799:
798:
792:
791:
757:
756:
696:
695:
668:
667:
640:
639:
638:ânotation; the
609:
608:
602:
601:
567:
566:
544:
543:
537:
536:
502:
501:
479:
468:
462:
459:
444:
428:
417:
405:complex numbers
329:
305:
230:
226:
213:quantum system.
199:
51:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8071:
8069:
8061:
8060:
8055:
8050:
8045:
8043:Quantum states
8040:
8030:
8029:
8023:
8022:
8020:
8019:
8009:
7998:
7995:
7994:
7992:
7991:
7989:many others...
7986:
7981:
7976:
7971:
7962:
7948:
7946:
7938:
7937:
7934:
7933:
7931:
7930:
7925:
7920:
7915:
7909:
7907:
7901:
7900:
7898:
7897:
7892:
7887:
7882:
7876:
7874:
7867:
7866:
7864:
7863:
7861:Trapped-ion QC
7858:
7852:
7850:
7844:
7843:
7841:
7840:
7835:
7830:
7825:
7819:
7817:
7815:Quantum optics
7808:
7802:
7801:
7799:
7798:
7793:
7792:
7791:
7784:
7779:
7774:
7769:
7764:
7759:
7754:
7745:
7743:
7735:
7734:
7732:
7731:
7726:
7721:
7720:
7719:
7709:
7708:
7707:
7697:
7696:
7695:
7685:
7680:
7674:
7672:
7664:
7663:
7661:
7660:
7659:
7658:
7654:
7648:
7644:
7633:
7632:
7631:
7621:
7619:Quantum volume
7616:
7610:
7608:
7602:
7601:
7599:
7598:
7593:
7588:
7583:
7578:
7572:
7570:
7562:
7561:
7559:
7558:
7553:
7548:
7543:
7538:
7533:
7528:
7523:
7518:
7513:
7508:
7503:
7498:
7496:Boson sampling
7493:
7488:
7482:
7480:
7474:
7473:
7470:
7469:
7467:
7466:
7461:
7460:
7459:
7454:
7449:
7439:
7434:
7429:
7423:
7421:
7412:
7411:
7406:
7405:
7404:
7394:
7393:
7392:
7382:
7377:
7372:
7367:
7366:
7365:
7360:
7349:
7347:
7341:
7340:
7338:
7337:
7332:
7330:SolovayâKitaev
7327:
7322:
7317:
7312:
7307:
7302:
7297:
7292:
7287:
7282:
7277:
7272:
7267:
7262:
7256:
7254:
7250:
7249:
7247:
7246:
7245:
7244:
7234:
7233:
7232:
7222:
7217:
7212:
7207:
7206:
7205:
7195:
7190:
7184:
7182:
7178:
7177:
7172:
7170:
7169:
7162:
7155:
7147:
7141:
7140:
7117:
7102:
7099:Scott Aaronson
7089:
7077:
7071:
7054:
7048:
7031:
7017:
6991:
6988:
6985:
6984:
6914:
6863:
6796:
6733:
6678:
6641:(12): 121406.
6625:
6582:
6547:Nature Physics
6536:
6505:(1): 240â246.
6499:Nature Physics
6485:
6460:
6382:
6306:
6287:
6229:Nature Physics
6215:
6190:
6128:
6107:
6042:
5989:
5952:(2): 865â942.
5936:
5921:
5890:
5883:
5861:
5803:
5750:
5743:
5715:
5666:
5665:
5663:
5660:
5657:
5656:
5642:
5637:
5632:
5627:
5624:
5620:
5614:
5609:
5604:
5582:
5578:
5574:
5571:
5551:
5535:
5512:
5509:
5505:
5484:
5481:
5477:
5454:
5448:
5444:
5441:
5438:
5434:
5428:
5424:
5420:
5417:
5413:
5409:
5406:
5403:
5399:
5395:
5371:
5367:quantum states
5365:it means that
5352:
5349:
5345:
5323:
5320:
5315:
5310:
5303:
5300:
5296:
5291:
5264:
5261:
5257:
5215:
5214:
5212:
5209:
5208:
5207:
5200:
5187:
5182:
5177:
5172:
5167:
5162:
5149:
5142:
5139:
5119:
5116:
5113:
5112:
5109:
5103:
5098:
5093:
5087:
5086:
5085:superposition
5074:
5071:
5067:
5056:
5042:
5039:
5035:
5024:
5015:
5007:
5003:
5002:
4999:
4990:
4985:
4977:
4971:
4970:
4967:
4964:
4959:
4951:
4945:
4944:
4941:
4938:
4933:
4928:
4920:
4919:
4916:
4911:
4906:
4897:
4896:
4893:
4885:
4880:
4871:
4870:
4855:
4844:
4839:
4831:
4825:
4824:
4821:
4818:
4813:
4807:
4800:
4799:
4796:
4793:
4788:
4782:
4776:
4775:
4772:
4769:
4764:
4750:
4744:
4743:
4740:
4737:
4732:
4723:
4722:
4719:
4716:
4711:
4706:
4700:
4699:
4696:
4684:
4679:
4677:Squeezed light
4674:
4668:Coherent state
4664:
4663:
4660:
4657:
4652:
4646:
4645:
4642:
4637:
4632:
4626:
4625:
4622:
4619:
4614:
4606:
4600:
4599:
4588:
4585:
4581:
4570:
4559:
4556:
4552:
4541:
4538:
4535:
4503:
4502:lifetime and T
4499:
4483:
4480:
4452:
4449:
4446:
4365:
4362:
4354:qubit register
4349:
4346:
4301:
4296:
4292:
4287:
4274:
4271:
4258:
4255:
4252:
4248:
4244:
4241:
4238:
4234:
4230:
4224:
4220:
4198:
4193:
4189:
4184:
4159:
4155:
4151:
4147:
4124:
4120:
4116:
4113:
4109:
4105:
4102:
4099:
4095:
4091:
4085:
4081:
4057:
4052:
4048:
4043:
4019:
4014:
4010:
4005:
3993:
3992:
3980:
3977:
3973:
3969:
3966:
3963:
3959:
3948:
3937:
3934:
3930:
3926:
3923:
3920:
3916:
3905:
3894:
3891:
3887:
3883:
3880:
3877:
3873:
3862:
3851:
3848:
3844:
3840:
3837:
3834:
3830:
3806:
3803:
3800:
3796:
3775:
3772:
3768:
3747:
3744:
3740:
3719:
3716:
3712:
3708:
3688:
3685:
3681:
3661:
3658:
3645:
3642:
3638:
3617:
3614:
3610:
3589:
3586:
3582:
3561:
3558:
3554:
3533:
3530:
3526:
3502:
3498:
3494:
3491:
3486:
3481:
3474:
3468:
3464:
3460:
3439:
3436:
3432:
3411:
3408:
3404:
3388:
3387:
3376:
3373:
3370:
3367:
3363:
3359:
3356:
3353:
3349:
3345:
3339:
3335:
3307:
3302:
3298:
3293:
3259:
3256:
3255:
3254:
3239:
3216:
3213:
3209:
3184:
3181:
3177:
3156:
3153:
3149:
3137:
3115:
3112:
3108:
3095:
3091:
3079:
3076:
3072:
3059:
3055:
3041:
3036:
3031:
3027:
3006:
3003:
2999:
2976:
2971:
2966:
2962:
2941:
2938:
2934:
2913:
2910:
2906:
2902:
2899:
2896:
2893:
2889:
2885:
2882:
2879:
2876:
2872:
2853:
2846:unitary matrix
2800:
2797:
2778:
2758:
2738:
2701:
2698:
2695:
2691:
2687:
2684:
2681:
2678:
2674:
2670:
2667:
2664:
2661:
2657:
2643:Density matrix
2641:Main article:
2638:
2635:
2622:
2602:
2565:
2559:
2555:
2552:
2549:
2545:
2541:
2538:
2535:
2531:
2527:
2506:
2503:
2499:
2478:
2475:
2471:
2448:relative phase
2433:
2430:
2426:
2414:
2413:
2398:
2393:
2390:
2385:
2382:
2377:
2374:
2370:
2366:
2363:
2361:
2359:
2356:
2355:
2352:
2347:
2344:
2339:
2336:
2333:
2330:
2328:
2326:
2323:
2322:
2279:
2276:
2272:
2253:
2252:
2237:
2232:
2229:
2224:
2221:
2216:
2213:
2210:
2207:
2204:
2201:
2197:
2193:
2190:
2188:
2186:
2183:
2182:
2179:
2174:
2171:
2166:
2163:
2158:
2155:
2151:
2147:
2144:
2142:
2140:
2137:
2136:
2093:
2073:
2052:
2049:
2045:
2041:
2038:
2035:
2032:
2028:
2024:
2021:
2018:
2015:
2011:
1982:
1977:
1974:
1969:
1965:
1962:
1957:
1954:
1950:
1946:
1943:
1922:
1917:
1914:
1909:
1905:
1902:
1899:
1896:
1875:
1872:
1869:
1865:
1861:
1858:
1855:
1852:
1848:
1844:
1841:
1838:
1835:
1831:
1807:
1804:
1783:
1763:
1752:relative phase
1739:
1719:
1708:
1707:
1696:
1693:
1688:
1683:
1678:
1674:
1670:
1665:
1660:
1655:
1651:
1623:
1603:
1581:
1576:
1571:
1567:
1546:
1543:
1539:
1516:
1511:
1506:
1502:
1481:
1478:
1474:
1449:
1445:
1442:
1441:
1430:
1427:
1423:
1419:
1416:
1413:
1410:
1406:
1402:
1399:
1396:
1393:
1389:
1365:
1362:
1358:
1337:
1334:
1330:
1305:
1270:
1250:
1241:and, for real
1224:
1218:
1214:
1211:
1208:
1204:
1200:
1197:
1194:
1190:
1186:
1166:
1163:
1159:
1155:
1152:
1149:
1145:
1124:
1121:
1117:
1113:
1110:
1107:
1103:
1083:
1080:
1058:
1048:
1045:
1044:
1041:
1038:
1037:
1034:
1031:
1030:
1027:
1024:
1023:
1017:
1012:
1009:
1006:
1002:
979:
969:
966:
965:
962:
959:
958:
955:
952:
951:
948:
945:
944:
938:
933:
930:
927:
923:
900:
890:
887:
886:
883:
880:
879:
876:
873:
872:
869:
866:
865:
859:
854:
851:
848:
844:
821:
811:
808:
807:
804:
801:
800:
797:
794:
793:
790:
787:
786:
780:
775:
772:
769:
765:
746:of the qubit.
731:
728:
725:
721:
717:
714:
711:
707:
703:
683:
680:
676:
655:
652:
648:
617:
607:
604:
603:
600:
597:
596:
590:
585:
582:
579:
575:
552:
542:
539:
538:
535:
532:
531:
525:
520:
517:
514:
510:
481:
480:
431:
429:
422:
416:
413:
328:
325:
304:
301:
201:
200:
198:
197:
190:
183:
175:
172:
171:
170:
169:
159:
153:
144:
143:
137:
136:
135:
134:
124:
117:
104:
103:
99:
98:
97:
96:
86:
76:
63:
62:
56:
55:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8070:
8059:
8056:
8054:
8051:
8049:
8048:Teleportation
8046:
8044:
8041:
8039:
8036:
8035:
8033:
8018:
8010:
8008:
8000:
7999:
7996:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7970:
7966:
7963:
7961:
7957:
7953:
7950:
7949:
7947:
7945:
7939:
7929:
7926:
7924:
7921:
7919:
7916:
7914:
7911:
7910:
7908:
7906:
7902:
7896:
7893:
7891:
7888:
7886:
7885:Spin qubit QC
7883:
7881:
7878:
7877:
7875:
7872:
7868:
7862:
7859:
7857:
7854:
7853:
7851:
7849:
7845:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7820:
7818:
7816:
7812:
7809:
7803:
7797:
7794:
7790:
7789:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7768:
7765:
7763:
7760:
7758:
7755:
7753:
7750:
7749:
7747:
7746:
7744:
7742:
7736:
7730:
7727:
7725:
7722:
7718:
7715:
7714:
7713:
7710:
7706:
7703:
7702:
7701:
7698:
7694:
7693:cluster state
7691:
7690:
7689:
7686:
7684:
7681:
7679:
7676:
7675:
7673:
7671:
7665:
7657:
7653:
7649:
7647:
7643:
7639:
7638:
7637:
7634:
7630:
7627:
7626:
7625:
7622:
7620:
7617:
7615:
7612:
7611:
7609:
7603:
7597:
7594:
7592:
7589:
7587:
7584:
7582:
7579:
7577:
7574:
7573:
7571:
7569:
7563:
7557:
7554:
7552:
7549:
7547:
7544:
7542:
7539:
7537:
7534:
7532:
7529:
7527:
7524:
7522:
7519:
7517:
7514:
7512:
7509:
7507:
7504:
7502:
7501:DeutschâJozsa
7499:
7497:
7494:
7492:
7489:
7487:
7484:
7483:
7481:
7479:
7475:
7465:
7462:
7458:
7455:
7453:
7450:
7448:
7445:
7444:
7443:
7440:
7438:
7437:Quantum money
7435:
7433:
7430:
7428:
7425:
7424:
7422:
7420:
7416:
7410:
7407:
7403:
7400:
7399:
7398:
7395:
7391:
7388:
7387:
7386:
7383:
7381:
7378:
7376:
7373:
7371:
7368:
7364:
7361:
7359:
7356:
7355:
7354:
7351:
7350:
7348:
7346:communication
7342:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7258:
7257:
7255:
7251:
7243:
7240:
7239:
7238:
7235:
7231:
7228:
7227:
7226:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7204:
7201:
7200:
7199:
7196:
7194:
7191:
7189:
7186:
7185:
7183:
7179:
7175:
7168:
7163:
7161:
7156:
7154:
7149:
7148:
7145:
7138:
7137:9781492670261
7134:
7130:
7126:
7122:
7118:
7115:
7111:
7107:
7103:
7100:
7096:
7095:
7090:
7087:
7084:
7083:
7078:
7074:
7068:
7064:
7060:
7055:
7051:
7045:
7041:
7037:
7032:
7028:
7024:
7020:
7014:
7010:
7007:. Cambridge:
7006:
7002:
7001:Chuang, Isaac
6998:
6994:
6993:
6989:
6980:
6976:
6972:
6968:
6964:
6960:
6956:
6952:
6948:
6944:
6939:
6934:
6930:
6926:
6918:
6915:
6910:
6906:
6902:
6898:
6893:
6888:
6884:
6880:
6879:
6874:
6867:
6864:
6859:
6855:
6850:
6845:
6841:
6837:
6833:
6829:
6824:
6819:
6815:
6811:
6807:
6800:
6797:
6792:
6788:
6784:
6780:
6776:
6772:
6768:
6764:
6759:
6754:
6750:
6746:
6745:
6737:
6734:
6729:
6725:
6721:
6717:
6713:
6709:
6704:
6699:
6695:
6691:
6690:
6682:
6679:
6674:
6670:
6666:
6662:
6658:
6654:
6649:
6644:
6640:
6636:
6629:
6626:
6621:
6617:
6613:
6609:
6605:
6601:
6597:
6593:
6586:
6583:
6578:
6574:
6570:
6566:
6561:
6556:
6552:
6548:
6540:
6537:
6531:
6526:
6522:
6518:
6513:
6508:
6504:
6500:
6496:
6489:
6486:
6475:
6471:
6464:
6461:
6456:
6452:
6447:
6442:
6438:
6434:
6430:
6426:
6422:
6418:
6413:
6408:
6404:
6400:
6396:
6389:
6387:
6383:
6378:
6374:
6369:
6364:
6360:
6356:
6352:
6348:
6344:
6340:
6335:
6330:
6326:
6322:
6318:
6310:
6307:
6302:
6298:
6291:
6288:
6276:
6272:
6268:
6264:
6260:
6256:
6252:
6248:
6243:
6238:
6234:
6230:
6226:
6219:
6216:
6205:
6204:IEEE Spectrum
6201:
6194:
6191:
6186:
6182:
6178:
6174:
6170:
6166:
6162:
6158:
6153:
6148:
6144:
6140:
6132:
6129:
6125:
6121:
6117:
6111:
6108:
6103:
6099:
6095:
6091:
6087:
6083:
6079:
6075:
6070:
6065:
6062:(5): 053007.
6061:
6057:
6053:
6046:
6043:
6038:
6034:
6030:
6026:
6022:
6018:
6013:
6008:
6004:
6000:
5993:
5990:
5985:
5981:
5977:
5973:
5969:
5965:
5960:
5955:
5951:
5947:
5940:
5937:
5932:
5928:
5924:
5918:
5914:
5911:. Cambridge:
5910:
5909:
5904:
5903:Chuang, Isaac
5900:
5894:
5891:
5886:
5880:
5876:
5872:
5865:
5862:
5859:
5857:
5852:
5845:
5841:
5837:
5833:
5828:
5823:
5820:(8): 081809.
5819:
5815:
5807:
5804:
5799:
5795:
5791:
5787:
5783:
5779:
5774:
5769:
5765:
5761:
5754:
5751:
5746:
5740:
5736:
5732:
5728:
5727:
5719:
5716:
5711:
5707:
5703:
5699:
5695:
5691:
5687:
5683:
5682:
5677:
5676:B. Schumacher
5671:
5668:
5661:
5640:
5630:
5625:
5622:
5612:
5580:
5576:
5572:
5569:
5549:
5539:
5536:
5532:
5528:
5507:
5479:
5452:
5446:
5436:
5426:
5422:
5418:
5415:
5411:
5407:
5401:
5385:
5381:
5375:
5372:
5368:
5350:
5347:
5343:
5321:
5318:
5313:
5301:
5298:
5294:
5280:
5262:
5259:
5255:
5246:
5242:
5238:
5234:
5230:
5226:
5220:
5217:
5210:
5205:
5201:
5199:
5198:quantum gates
5195:
5192:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5157:
5153:
5150:
5148:
5145:
5144:
5140:
5138:
5136:
5133:
5129:
5125:
5118:Qubit storage
5117:
5110:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5088:
5069:
5057:
5055:
5054:superposition
5037:
5025:
5023:
5019:
5016:
5014:
5011:
5008:
5005:
5004:
5000:
4998:
4995:
4991:
4989:
4986:
4984:
4981:
4978:
4976:
4973:
4972:
4968:
4965:
4963:
4960:
4958:
4955:
4952:
4950:
4947:
4946:
4942:
4939:
4937:
4934:
4932:
4929:
4926:
4922:
4921:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4899:
4898:
4894:
4892:
4889:
4886:
4884:
4881:
4879:
4876:
4873:
4872:
4868:
4864:
4860:
4856:
4853:
4849:
4845:
4843:
4840:
4838:
4835:
4832:
4830:
4826:
4822:
4819:
4817:
4814:
4812:
4808:
4806:
4802:
4801:
4797:
4794:
4792:
4789:
4787:
4783:
4781:
4778:
4777:
4773:
4770:
4768:
4765:
4763:
4760:
4757:
4754:
4751:
4749:
4746:
4745:
4742:Two electron
4741:
4738:
4736:
4733:
4731:
4728:
4725:
4724:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4701:
4697:
4695:
4692:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4669:
4666:
4665:
4661:
4658:
4656:
4653:
4651:
4648:
4647:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4623:
4620:
4618:
4615:
4613:
4610:
4607:
4605:
4601:
4583:
4554:
4532:
4529:
4526:
4524:
4520:
4516:
4511:
4509:
4496:
4494:
4489:
4481:
4479:
4477:
4473:
4469:
4464:
4450:
4447:
4444:
4435:
4430:
4428:
4424:
4419:
4417:
4412:
4410:
4406:
4405:integer types
4401:
4399:
4395:
4392:, since both
4391:
4388:for qudit is
4387:
4383:
4379:
4375:
4371:
4363:
4361:
4359:
4355:
4347:
4345:
4343:
4339:
4335:
4331:
4325:
4323:
4319:
4315:
4294:
4272:
4270:
4250:
4242:
4236:
4222:
4218:
4191:
4172:
4157:
4149:
4122:
4111:
4103:
4097:
4083:
4079:
4069:
4050:
4032:
4012:
3975:
3961:
3949:
3932:
3918:
3906:
3889:
3875:
3863:
3846:
3832:
3820:
3819:
3818:
3798:
3770:
3742:
3714:
3683:
3670:
3666:
3659:
3657:
3640:
3612:
3584:
3556:
3528:
3514:
3500:
3496:
3492:
3489:
3484:
3472:
3466:
3462:
3434:
3406:
3393:
3374:
3365:
3357:
3351:
3337:
3333:
3324:
3323:
3322:
3320:
3300:
3281:
3279:
3275:
3269:
3265:
3257:
3252:
3248:
3244:
3240:
3237:
3233:
3230:
3211:
3198:
3179:
3151:
3138:
3135:
3131:
3113:
3110:
3106:
3074:
3039:
3029:
3001:
2974:
2964:
2936:
2908:
2900:
2897:
2891:
2883:
2880:
2874:
2861:
2857:
2854:
2851:
2850:quantum state
2847:
2843:
2840:described by
2839:
2835:
2831:
2827:
2823:
2819:
2816:
2815:
2814:
2810:
2806:
2798:
2796:
2794:
2790:
2776:
2756:
2736:
2728:
2724:
2720:
2716:
2715:quantum noise
2699:
2693:
2685:
2682:
2676:
2668:
2665:
2659:
2644:
2636:
2634:
2620:
2600:
2592:
2588:
2583:
2581:
2563:
2557:
2547:
2539:
2533:
2501:
2473:
2460:
2456:
2451:
2449:
2431:
2428:
2424:
2396:
2391:
2388:
2383:
2380:
2375:
2372:
2368:
2364:
2362:
2357:
2350:
2345:
2342:
2337:
2334:
2331:
2329:
2324:
2313:
2312:
2311:
2308:
2302:
2296:
2277:
2274:
2270:
2262:of the state
2261:
2260:
2235:
2230:
2227:
2222:
2219:
2211:
2208:
2205:
2199:
2195:
2191:
2189:
2184:
2177:
2172:
2169:
2164:
2161:
2156:
2153:
2149:
2145:
2143:
2138:
2127:
2126:
2125:
2123:
2117:
2113:
2107:
2091:
2071:
2047:
2039:
2036:
2030:
2022:
2019:
2013:
2000:
1980:
1975:
1972:
1967:
1963:
1960:
1955:
1952:
1948:
1944:
1941:
1920:
1915:
1912:
1907:
1903:
1900:
1897:
1894:
1887:are given by
1873:
1867:
1859:
1856:
1850:
1842:
1839:
1833:
1820:
1816:
1812:
1805:
1803:
1801:
1797:
1781:
1761:
1753:
1737:
1717:
1694:
1691:
1686:
1676:
1668:
1663:
1653:
1641:
1640:
1639:
1637:
1621:
1601:
1579:
1569:
1541:
1514:
1504:
1476:
1463:
1459:
1455:
1425:
1417:
1414:
1408:
1400:
1397:
1391:
1379:
1378:
1377:
1360:
1332:
1319:
1303:
1295:
1294:superposition
1292:
1284:
1283:orthogonality
1268:
1248:
1240:
1222:
1216:
1206:
1198:
1192:
1161:
1153:
1147:
1119:
1111:
1105:
1092:
1088:
1081:
1079:
1077:
1072:
1046:
1039:
1032:
1025:
1010:
1004:
967:
960:
953:
946:
931:
925:
888:
881:
874:
867:
852:
846:
809:
802:
795:
788:
773:
767:
754:
752:
747:
745:
723:
715:
709:
678:
650:
637:
633:
605:
598:
583:
577:
540:
533:
518:
512:
499:
495:
492:
488:
487:quantum state
477:
474:
466:
456:
452:
448:
442:
441:
437:
432:This section
430:
426:
421:
420:
414:
412:
410:
406:
402:
398:
394:
389:
387:
386:
381:
376:
372:
366:
364:
360:
357:
353:
348:
346:
342:
338:
334:
326:
324:
322:
318:
314:
310:
302:
300:
298:
294:
290:
289:superposition
285:
281:
277:
273:
269:
265:
261:
257:
251:
224:
220:
212:
207:
196:
191:
189:
184:
182:
177:
176:
174:
173:
168:-dimensional)
167:
163:
160:
157:
154:
151:
148:
147:
146:
145:
142:
138:
132:
128:
125:
122:
118:
115:
111:
108:
107:
106:
105:
100:
94:
90:
87:
84:
80:
77:
74:
70:
67:
66:
65:
64:
61:
57:
54:
49:
46:
45:
40:
33:
19:
7913:Charge qubit
7838:KLM protocol
7787:
7651:
7641:
7335:Purification
7265:EastinâKnill
7224:
7129:Chris Ferrie
7124:
7121:picture book
7105:
7092:
7080:
7058:
7035:
7004:
6928:
6924:
6917:
6882:
6876:
6866:
6813:
6809:
6799:
6748:
6742:
6736:
6693:
6687:
6681:
6638:
6634:
6628:
6595:
6591:
6585:
6550:
6546:
6539:
6502:
6498:
6488:
6477:. Retrieved
6473:
6463:
6402:
6398:
6324:
6320:
6309:
6290:
6278:. Retrieved
6232:
6228:
6218:
6207:. Retrieved
6203:
6193:
6142:
6138:
6131:
6110:
6059:
6055:
6045:
6002:
5998:
5992:
5949:
5945:
5939:
5907:
5893:
5870:
5864:
5848:
5817:
5813:
5806:
5763:
5759:
5753:
5725:
5718:
5685:
5679:
5670:
5538:
5379:
5374:
5245:global phase
5244:
5219:
5165:Bloch sphere
5124:nuclear spin
5121:
4914:Ground state
4865:, one extra
4862:
4858:
4851:
4837:charge qubit
4811:energy level
4786:energy level
4780:Neutral atom
4753:Nuclear spin
4739:No electron
4609:Polarization
4527:
4512:
4497:
4485:
4465:
4431:
4420:
4413:
4408:
4402:
4397:
4393:
4389:
4381:
4377:
4373:
4369:
4367:
4351:
4326:
4324:algorithms.
4276:
4273:Applications
4209:Bell State:
4173:
4070:
3994:
3663:
3515:
3391:
3389:
3282:
3271:
3236:ground state
3197:Pauli-X gate
2812:
2791:
2726:
2646:
2584:
2455:Bloch sphere
2452:
2447:
2415:
2306:
2300:
2294:
2256:
2254:
2115:
2111:
1996:
1815:Bloch sphere
1751:
1709:
1443:
1288:
1082:Qubit states
1075:
1074:In general,
1073:
755:
748:
484:
469:
460:
445:Please help
433:
409:vector space
400:
396:
392:
390:
383:
374:
367:
363:logic levels
351:
349:
333:binary digit
330:
316:
308:
306:
282:of a single
280:polarization
255:
222:
216:
165:
149:
102:Data storage
7944:programming
7923:Phase qubit
7828:Circuit QED
7300:No-deleting
7242:cloud-based
5279:unit circle
5229:measurement
5147:Ancilla bit
5010:Vibrational
4994:topological
4980:Non-abelian
4949:Quantum dot
4925:quantum dot
4904:phase qubit
4867:Cooper pair
4621:Horizontal
3232:phase qubit
2842:multiplying
2723:mixed state
2719:decoherence
2637:Mixed state
2591:state space
491:orthonormal
345:information
256:quantum bit
53:information
8032:Categories
7984:libquantum
7918:Flux qubit
7823:Cavity QED
7772:BaconâShor
7762:stabilizer
7290:No-cloning
7018:0521632358
6892:1911.11143
6823:1611.07690
6758:2303.17734
6648:1904.10785
6560:2112.14922
6512:2211.10383
6479:2022-10-14
6412:2312.17741
6334:2206.07216
6242:2109.06903
6209:2017-06-29
6012:1801.00862
5827:2202.07756
5733:. p.
5662:References
5241:eigenstate
5152:Bell state
4878:flux qubit
4846:Uncharged
4682:Quadrature
4635:Fock state
4031:Bell state
3319:Bell state
3268:Bell state
2834:reversible
339:, where a
7890:NV center
7325:Threshold
7305:No-hiding
7270:Gleason's
6963:1050-2947
6909:232486360
6816:: 12232.
6703:0803.2021
6673:129945636
6577:1745-2473
6474:New Atlas
6437:2041-1723
6359:2041-1723
6275:237513730
6267:1745-2481
6124:arxiv.org
6116:1150 uses
6102:110606655
6094:1367-2630
6069:1203.5614
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636:"braâket"
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463:July 2023
434:does not
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303:Etymology
158:(ternary)
7952:OpenQASM
7928:Transmon
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7506:Grover's
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7131:(2017).
7040:Springer
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6791:42906250
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274:of the
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