# 量子比特

where $\displaystyle{ e^{i \phi} }$ is the physically significant relative phase.

\displaystyle{ \begin{align} \alpha &= e^{i \psi} \cos\frac{\theta}{2}, \\ The possible quantum states for a single qubit can be visualised using a Bloch sphere (see diagram). Represented on such a 2-sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where \lt math\gt |0 \rangle } and $\displaystyle{ |1 \rangle }$ 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 $\displaystyle{ ((|0 \rangle +i|1 \rangle)/{\sqrt{2}}) }$ would lie on the equator of the sphere at the positive y-axis. In the classical limit, 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.

\beta &= e^{i (\psi + \phi)} \sin\frac{\theta}{2}.

\end{align}[/itex]

The surface of the Bloch sphere is a two-dimensional space, which represents the state space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles $\displaystyle{ \phi }$ and $\displaystyle{ \theta }$.

Additionally, for a single qubit the overall phase of the state ei ψ has no physically observable consequences, so we can arbitrarily choose α to be real (or β in the case that α is zero), leaving just two degrees of freedom:

\displaystyle{ \begin{align} \alpha &= \cos\frac{\theta}{2}, \\ A pure state is one fully specified by a single ket, \lt math\gt | \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\, } a coherent superposition as described above. Coherence is essential for a qubit to be in a superposition state. With interactions and decoherence, it is possible to put the qubit in a mixed state, a statistical combination or incoherent mixture of different pure states. Mixed states can be represented by points inside the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles $\displaystyle{ \phi }$ and $\displaystyle{ \theta }$, as well as the length $\displaystyle{ r }$ of the vector that represents the mixed state.

\beta &= e^{i \phi} \sin\frac{\theta}{2},

\end{align}[/itex]

where $\displaystyle{ e^{i \phi} }$ is the physically significant relative phase.

There are various kinds of physical operations that can be performed on pure qubit states.

The possible quantum states for a single qubit can be visualised using a Bloch sphere (see diagram). Represented on such a 2-sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where $\displaystyle{ |0 \rangle }$ and $\displaystyle{ |1 \rangle }$ 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 $\displaystyle{ ((|0 \rangle +i|1 \rangle)/{\sqrt{2}}) }$ would lie on the equator of the sphere at the positive y-axis. In the classical limit, 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.

The surface of the Bloch sphere is a two-dimensional space, which represents the state space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles $\displaystyle{ \phi }$ and $\displaystyle{ \theta }$.

An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.

### Mixed state混合状态

A pure state is one fully specified by a single ket, $\displaystyle{ | \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\, }$ a coherent superposition as described above. Coherence is essential for a qubit to be in a superposition state. With interactions and decoherence, it is possible to put the qubit in a mixed state, a statistical combination or incoherent mixture of different pure states. Mixed states can be represented by points inside the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles $\displaystyle{ \phi }$ and $\displaystyle{ \theta }$, as well as the length $\displaystyle{ r }$ of the vector that represents the mixed state.

The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the $\displaystyle{ |\Phi^+\rangle }$ Bell state:

### Operations on pure qubit states纯量子态的运算

$\displaystyle{ \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle). }$

(| 00 rangle + | 11 rangle) . </math >

There are various kinds of physical operations that can be performed on pure qubit states. 有各种各样的物理操作可以在纯量子态上执行。

• Quantum logic gates, building blocks for a quantum circuit in a quantum computer, operate on one, two, or three qubits: mathematically, the qubits undergo a (reversible) unitary transformation under the quantum gate. For a single qubit, unitary transformations correspond to rotations of the qubit (unit) vector on the Bloch sphere to specific superpositions. For two qubits, the Controlled NOT gate can be used to entangle or disentangle them.
• 量子逻辑门s是 量子计算机量子电路的构建块，在一个、两个或三个量子位上运行：从数学上讲，量子位在量子门下经历（可逆的）酉变换。对于单个量子位，幺正变换对应于Bloch球上的量子位（单位）矢量旋转到特定的叠加。对于两个量子比特，受控非门可以用来纠缠或解开它们。

In this state, called an equal superposition, there are equal probabilities of measuring either product state $\displaystyle{ |00\rangle }$ or $\displaystyle{ |11\rangle }$, as $\displaystyle{ |1/\sqrt{2}|^2 = 1/2 }$. In other words, there is no way to tell if the first qubit has value “0” or “1” and likewise for the second qubit.

• Standard basis measurement is an irreversible operation in which information is gained about the state of a single qubit (and coherence is lost). The result of the measurement will be either $\displaystyle{ | 0 \rangle }$ (with probability $\displaystyle{ |\alpha|^2 }$) or $\displaystyle{ | 1 \rangle }$ (with probability $\displaystyle{ |\beta|^2 }$). Measurement of the state of the qubit alters the magnitudes of α and β. For instance, if the result of the measurement is $\displaystyle{ | 1 \rangle }$, α is changed to 0 and β is changed to the phase factor $\displaystyle{ e^{i \phi} }$ no longer experimentally accessible. When a qubit is measured, the superposition state collapses to a basis state (up to a phase) and the relative phase is rendered inaccessible (i.e., coherence is lost). Note that a measurement of a qubit state that is entangled with another quantum system transforms the qubit state, a pure state, into a mixed state (an incoherent mixture of pure states) as the relative phase of the qubit state is rendered inaccessible.
• 标准基测量是一种不可逆的操作，在这种操作中，获得了关于单个量子比特状态的信息（并且失去了相干性）。测量结果将是$\displaystyle{ | 0\rangle }$（概率$\displaystyle{ |\alpha | ^2 }$）或$\displaystyle{ |1\rangle }$（概率$\displaystyle{ |\beta | ^2 }$）。量子位状态的测量改变了αβ的大小。例如，如果测量结果为$\displaystyle{ | 1\rangle }$α变为0，β变为相位因子$\displaystyle{ e^{i\phi} }$。当一个量子位被测量时，叠加态坍缩成基态（直到一个相位），相对相位变得不可接近（即，相干性丢失）。注意，对与另一量子系统纠缠的量子位态的测量将量子位态（纯态）转换为混合态（纯态的非相干混合），因为量子位态的相对相位变得不可接近。

Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either $\displaystyle{ |0\rangle }$ or $\displaystyle{ |1\rangle }$, 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 $\displaystyle{ |0\rangle }$, Bob must measure the same, as $\displaystyle{ |00\rangle }$ is the only state where Alice's qubit is a $\displaystyle{ |0\rangle }$. 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” — a most surprising circumstance that can not be explained by classical physics.

## Quantum entanglement量子纠缠

An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.

The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the $\displaystyle{ |\Phi^+\rangle }$ Bell state:

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the controlled NOT gate (or CNOT or cX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is $\displaystyle{ |1\rangle }$, and otherwise leaves it unchanged. With respect to the unentangled product basis $\displaystyle{ \{|00\rangle }$, $\displaystyle{ |01\rangle }$, $\displaystyle{ |10\rangle }$, $\displaystyle{ |11\rangle\} }$, it maps the basis states as follows:

$\displaystyle{ | 0 0 \rangle \mapsto | 0 0 \rangle }$

< math > | 00地图 | 00地图 </math >

$\displaystyle{ \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle). }$

$\displaystyle{ | 0 1 \rangle \mapsto | 0 1 \rangle }$

< math > | 01地图 | 01地图 </math >

$\displaystyle{ | 1 0 \rangle \mapsto | 1 1 \rangle }$

< math > | 10个测量点 | 11个测量点

In this state, called an equal superposition, there are equal probabilities of measuring either product state $\displaystyle{ |00\rangle }$ or $\displaystyle{ |11\rangle }$, as $\displaystyle{ |1/\sqrt{2}|^2 = 1/2 }$. In other words, there is no way to tell if the first qubit has value “0” or “1” and likewise for the second qubit.

$\displaystyle{ | 1 1 \rangle \mapsto | 1 0 \rangle }$.

< math > | 11个测量点 | 10个测量点。

Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either $\displaystyle{ |0\rangle }$ or $\displaystyle{ |1\rangle }$, 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 $\displaystyle{ |0\rangle }$, Bob must measure the same, as $\displaystyle{ |00\rangle }$ is the only state where Alice's qubit is a $\displaystyle{ |0\rangle }$. 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” — a most surprising circumstance that can not be explained by classical physics.

A common application of the CNOT gate is to maximally entangle two qubits into the $\displaystyle{ |\Phi^+\rangle }$ Bell state. To construct $\displaystyle{ |\Phi^+\rangle }$, the inputs A (control) and B (target) to the CNOT gate are:

CNOT门的一个常见应用是最大限度地将两个量子位纠缠成$\displaystyle{ |\Phi^+\rangle }$贝尔态。要构造$\displaystyle{ |\Phi^+\rangle }$，到CNOT门的输入A（控制）和B（目标）是：

### Controlled gate to construct the Bell state受控门构造钟态

$\displaystyle{ \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)_A }$ and $\displaystyle{ |0\rangle_B }$

(| 0 rangle + | 1 rangle) a </math > and < math > | 0 rangle _ b </math >

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the controlled NOT gate (or CNOT or cX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is $\displaystyle{ |1\rangle }$, and otherwise leaves it unchanged. With respect to the unentangled product basis $\displaystyle{ \{|00\rangle }$, $\displaystyle{ |01\rangle }$, $\displaystyle{ |10\rangle }$, $\displaystyle{ |11\rangle\} }$, it maps the basis states as follows:

After applying CNOT, the output is the $\displaystyle{ |\Phi^+\rangle }$ Bell State: $\displaystyle{ \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) }$.

$\displaystyle{ | 0 0 \rangle \mapsto | 0 0 \rangle }$
$\displaystyle{ | 0 1 \rangle \mapsto | 0 1 \rangle }$
$\displaystyle{ | 1 0 \rangle \mapsto | 1 1 \rangle }$
$\displaystyle{ | 1 1 \rangle \mapsto | 1 0 \rangle }$.

The $\displaystyle{ |\Phi^+\rangle }$ Bell state forms part of the setup of the superdense coding, quantum teleportation, and entangled quantum cryptography algorithms.

A common application of the CNOT gate is to maximally entangle two qubits into the $\displaystyle{ |\Phi^+\rangle }$ Bell state. To construct $\displaystyle{ |\Phi^+\rangle }$, the inputs A (control) and B (target) to the CNOT gate are:

CNOT门的一个常见应用是最大限度地将两个量子位纠缠到$\displaystyle{ |\Phi^+\rangle }$Bell state。要构造$\displaystyle{ |\Phi^+\rangle }$，到CNOT门的输入A（控制）和B（目标）是：

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 computer. Many of the successes of quantum computation and communication, such as quantum teleportation and superdense coding, make use of entanglement, suggesting that entanglement is a resource 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 quantum circuits that can be executed reliably.

$\displaystyle{ \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)_A }$ and $\displaystyle{ |0\rangle_B }$

A number of qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register. A qubyte (quantum byte) is a collection of eight qubits.

After applying CNOT, the output is the $\displaystyle{ |\Phi^+\rangle }$ Bell State: $\displaystyle{ \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) }$.

### Applications应用

Similar to the qubit, the qutrit 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 trit of ternary computers. Note, however, that not all 3-level quantum systems are qutrits. The term "qu-d-it" (quantum d-git) denotes the unit of quantum information that can be realized in suitable d-level quantum systems. In 2017, scientists at the National Institute of Scientific Research constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits.

The $\displaystyle{ |\Phi^+\rangle }$ Bell state forms part of the setup of the superdense coding, quantum teleportation, and entangled quantum cryptography algorithms.

$\displaystyle{ |\Phi^+\rangle }$Bell态密集编码量子通信和纠缠量子密码术算法设置的一部分。

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 computer. Many of the successes of quantum computation and communication, such as quantum teleportation and superdense coding, make use of entanglement, suggesting that entanglement is a resource that is unique to quantum computation.[1] 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 quantum circuits that can be executed reliably.[2]

Any two-level quantum-mechanical system 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., ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate two-level systems to various degrees were successfully realized. Similarly to a classical bit where the state of a transistor in a processor, the magnetization of a surface in a hard disk 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.

## Quantum register量子寄存器

The following is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only.

A number of qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register. A qubyte (quantum byte) is a collection of eight quits.

! scope="col" style="background: white;" | $\displaystyle{ |1 \rangle }$

!“白色背景” | < math > | 1 rangle </math >

|-

|-

### Variations of the qubit量子位的变化

| rowspan=3 |Photon

3 | Photon

Similar to the qubit, the qutrit 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 trit of ternary computers. Note, however, that not all 3-level quantum systems are qutrits.[5] The term "qu-d-it" (quantum d-git) denotes the unit of quantum information that can be realized in suitable d-level quantum systems.[6] In 2017, scientists at the National Institute of Scientific Research constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits.[7]

| Polarization encoding

| 偏振编码

| Polarization of light

| 光的偏振

## Physical implementations物理实现

| Horizontal

| 横向

| Vertical

| Vertical

Any two-level quantum-mechanical system 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., ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate two-level systems to various degrees were successfully realized. Similarly to a classical bit where the state of a transistor in a processor, the magnetization of a surface in a hard disk 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.

|-

|-

| Number of photons

| 光子数

The following is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only.

| Fock state

| 福克州

| Vacuum

| 真空

 Physical support Name Information support $\displaystyle{ |0 \rangle }$ $\displaystyle{ |1 \rangle }$ Single photon state 单光子态 Time-bin encoding 时间容器编码 Time of arrival 抵达时间 Early 早期 Late 迟到 Coherent state of light 光的相干态 Photon Squeezed light 压缩光 Polarization encoding Quadrature 正交 Polarization of light Amplitude-squeezed state 振幅压缩态 Horizontal Phase-squeezed state 相压缩态 Vertical Electrons 2 | 电子 Number of photons Electronic spin 电子自旋 Fock state Spin 旋转 Vacuum Up 起来 Single photon state Down 下来 Time-bin encoding Electron number 电子数 Time of arrival Charge 费用 Early No electron 没有电子 Late One electron 一个电子 Coherent state of light Nucleus 纽核力 Squeezed light Nuclear spin addressed through NMR 核自旋通过 NMR 处理 Quadrature 模板:Clarify Spin 旋转 Amplitude-squeezed state Up 起来 Phase-squeezed state Down 下来 Electrons Optical lattices 光学晶格 Electronic spin Atomic spin 原子自旋 Spin Spin 旋转 Up Up 起来 Down Down 下来 Electron number Josephson junction 3 | Josephson junction Charge 电荷 Superconducting charge qubit 超导电荷量子位 No electron 没有电子 Charge 费用 One electron 单电子 Uncharged superconducting island (Q=0) 无电荷超导岛(q = 0) Charged superconducting island (Q=2e, one extra Cooper pair) 带电超导岛(q = 2e，多一对库珀) Nucleus 原子核 Nuclear spin addressed through NMR 【核自旋】通过【核磁共振】 Superconducting flux qubit 超导磁通量子位 Spin Current 目前 Up Clockwise current 顺时针方向电流 Down Counterclockwise current 逆时针方向电流 Optical lattices Superconducting phase qubit 超导相量子位 Atomic spin Energy 能源 Spin Ground state 基态 Up First excited state 第一激发态 Down Singly charged quantum dot pair 单电荷量子点对 Josephson junction Electron localization 电子定位 Superconducting charge qubit Charge 费用 Charge Electron on left dot 左点上的电子 Uncharged superconducting island (Q=0) Electron on right dot 右点上的电子 Charged superconducting island (Q=2e, one extra Cooper pair) Quantum dot 量子点 Superconducting flux qubit Dot spin 点旋转 Current Spin 旋转 Clockwise current Down 下来 Counterclockwise current Up 起来 Superconducting phase qubit Gapped topological system 间隙拓扑系统 Energy 能量 Non-abelian anyons 非阿贝尔任意子 Non-abelian anyons 非阿贝尔任意子 Ground state 基态 Braiding of Excitations 编织激发 First excited state 第一激发态 Depends on specific topological system 依赖于特定的拓扑系统 Depends on specific topological system 依赖于特定的拓扑系统 Singly charged quantum dot pair 单电荷量子点对 Electron localization 电子局域化 van der Waals heterostructure 范德华异质结构 Charge Electron localization 电子定位 Electron on left dot 左点上的电子 Charge 费用 Electron on right dot 右点上的电子 Electron on bottom sheet 底片上的电子 Electron on top sheet 电子在最上面的表格 Quantum dot 量子点

|}

| Dot spin

| Spin

| Down

| Up

In a paper entitled "Solid-state quantum memory using the 31P nuclear spin", published in the October 23, 2008, issue of the journal Nature, 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 nuclear spin "memory" qubit. This event can be considered the first relatively consistent quantum data storage, a vital step towards the development of quantum computing. Recently, 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.

|-

| Gapped topological system |间隙拓扑系统 | Non-abelian anyons |非交换anyons |Braiding of Excitations |激发编织 |Depends on specific topological system |取决于特定的拓扑系统 |Depends on specific topological system |取决于特定的拓扑系统 |-

|范德华异质结构 [11]

|Electron localization |电子局域化 |Charge |充电 Category:Quantum computing

|Electron on bottom sheet |底片上的电子 Category:Quantum information science

|Electron on top sheet

Category:Quantum states

|}

Category:Teleportation

Category:Units of information

## Qubit storage量子位存储

Category:Australian inventions

This page was moved from wikipedia:en:Qubit. Its edit history can be viewed at 量子比特/edithistory

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2. Preskill, John (2018). "Quantum Computing in the NISQ era and beyond". Quantum. 2: 79. arXiv:1801.00862. doi:10.22331/q-2018-08-06-79. Unknown parameter |s2cid= ignored (help)
3. Horodecki, Ryszard; et al. (2009). "Quantum entanglement". Reviews of Modern Physics. 81 (2): 865–942. arXiv:quant-ph/0702225. Bibcode:2009RvMP...81..865H. doi:10.1103/RevModPhys.81.865. Unknown parameter |s2cid= ignored (help)
4. R. Tanburn {; E. Okada; N. S. Dattani ! scope="col" (2015 ! scope="col"). [https://archive.org/details/arxiv-1508.04816 ! scope="col" style="background: white;" "Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 1: The "deduc-reduc" method and its application to quantum factorization of numbers ! scope="col""] Check |url= value (help). arXiv:1508.04816. Bibcode:2015arXiv150804816T. Text "- " ignored (help); Text "- " ignored (help); Text " Name " ignored (help); Unknown parameter |Physical support !Scope= ignored (help); Unknown parameter |class= ignored (help); Text " 信息支持 " ignored (help); Unknown parameter |Name !Scope= ignored (help); Text " 物理支持 " ignored (help); Text " < math > " ignored (help); Unknown parameter |$\displaystyle{ |0 \rangle }$ !“ col” style= ignored (help); Text " 0 rangle </math > " ignored (help); Unknown parameter |Information support !Scope= ignored (help); line feed character in |year= at position 5 (help); line feed character in |author= at position 11 (help); line feed character in |title= at position 189 (help); line feed character in |url= at position 45 (help); line feed character in |author3= at position 14 (help); Cite journal requires |journal= (help); Check date values in: |year= (help)
5. "Quantum systems: three-level vs qutrit". Physics Stack Exchange. Retrieved 2018-07-25.
6. Nisbet-Jones, Peter B. R.; Dilley, Jerome; Holleczek, Annemarie; Barter, Oliver; Kuhn, Axel (2013). "Photonic qubits, qutrits and ququads accurately prepared and delivered on demand". New Journal of Physics (in English). 15 (5): 053007. arXiv:1203.5614. Bibcode:2013NJPh...15e3007N. doi:10.1088/1367-2630/15/5/053007. ISSN 1367-2630. Unknown parameter |s2cid= ignored (help)
7. "Qudits: The Real Future of Quantum Computing?". IEEE Spectrum (in English). 2017-06-28. Retrieved 2017-06-29.
8. "Quantum systems: three-level vs qutrit". Physics Stack Exchange. Retrieved 2018-07-25.
9. Nisbet-Jones, Peter B. R.; Dilley, Jerome; Holleczek, Annemarie; Barter, Oliver; Kuhn, Axel (2013). "Photonic qubits, qutrits and ququads accurately prepared and delivered on demand". New Journal of Physics (in English). 15 (5): 053007. arXiv:1203.5614. Bibcode:2013NJPh...15e3007N. doi:10.1088/1367-2630/15/5/053007. ISSN 1367-2630. Unknown parameter |s2cid= ignored (help)
10. "Qudits: The Real Future of Quantum Computing?". IEEE Spectrum (in English). 2017-06-28. Retrieved 2017-06-29.
11. B. Lucatto; et al. (2019). "Charge qubit in van der Waals heterostructures". Physical Review B. 100 (12): 121406. arXiv:1904.10785. doi:10.1103/PhysRevB.100.121406. Unknown parameter |s2cid= ignored (help)