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In quantum computing, a quantum register is a system comprising multiple qubits.^[1] It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.^[2]

Definition

It is usually assumed that the register consists of qubits. It is also generally assumed that registers are not density matrices, but that they are pure, although the definition of "register" can be extended to density matrices.

An $n$ size quantum register is a quantum system comprising $n$ pure qubits.

The Hilbert space, ${\mathcal {H}}$ , in which the data is stored in a quantum register is given by ${\mathcal {H}}={\mathcal {H_{n-1}}}\otimes {\mathcal {H_{n-2}}}\otimes \ldots \otimes {\mathcal {H_{0}}}$ where $\otimes$ is the tensor product.^[3]

The number of dimensions of the Hilbert spaces depends on what kind of quantum systems the register is composed of. Qubits are 2-dimensional complex spaces ( $\mathbb {C} ^{2}$ ), while qutrits are 3-dimensional complex spaces ( $\mathbb {C} ^{3}$ ), et.c. For a register composed of N number of d-dimensional (or d-level) quantum systems we have the Hilbert space ${\mathcal {H}}=(\mathbb {C} ^{d})^{\otimes N}=\underbrace {\mathbb {C} ^{d}\otimes \mathbb {C} ^{d}\otimes \dots \otimes \mathbb {C} ^{d}} _{N{\text{ times}}}\cong \mathbb {C} ^{d^{N}}.$

The registers quantum state can in the bra-ket notation be written $|\psi \rangle =\sum _{k=0}^{d^{N}-1}a_{k}|k\rangle =a_{0}|0\rangle +a_{1}|1\rangle +\dots +a_{d^{N}-1}|d^{N}-1\rangle .$ The values $a_{k}$ are probability amplitudes. Because of the Born rule and the 2nd axiom of probability theory, $\sum _{k=0}^{d^{N}-1}|a_{k}|^{2}=1,$ so the possible state space of the register is the surface of the unit sphere in $\mathbb {C} ^{d^{N}}.$

Examples:

The quantum state vector of a 5-qubit register is a unit vector in $\mathbb {C} ^{2^{5}}=\mathbb {C} ^{32}.$
A register of four qutrits similarly is a unit vector in $\mathbb {C} ^{3^{4}}=\mathbb {C} ^{81}.$

Quantum vs. classical register

First, there's a conceptual difference between the quantum and classical register. An $n$ size classical register refers to an array of $n$ flip flops. An $n$ size quantum register is merely a collection of $n$ qubits.

Moreover, while an $n$ size classical register is able to store a single value of the $2^{n}$ possibilities spanned by $n$ classical pure bits, a quantum register is able to store all $2^{n}$ possibilities spanned by quantum pure qubits at the same time.

For example, consider a 2-bit-wide register. A classical register is able to store only one of the possible values represented by 2 bits - $00,01,10,11\quad (0,1,2,3)$ accordingly.

If we consider 2 pure qubits in superpositions $|a_{0}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )$ and $|a_{1}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )$ , using the quantum register definition $|a\rangle =|a_{0}\rangle \otimes |a_{1}\rangle ={\frac {1}{2}}(|00\rangle -|01\rangle +|10\rangle -|11\rangle )$ it follows that it is capable of storing all the possible values (by having non-zero probability amplitude for all outcomes) spanned by two qubits simultaneously.

References

^ Ekert, Artur; Hayden, Patrick; Inamori, Hitoshi (2008). "Basic Concepts in Quantum Computation". Coherent atomic matter waves. Les Houches - Ecole d'Ete de Physique Theorique. Vol. 72. pp. 661–701. arXiv:quant-ph/0011013. doi:10.1007/3-540-45338-5_10. ISBN 978-3-540-41047-8. S2CID 53402188.
^ Ömer, Bernhard (2000-01-20). Quantum Programming in QCL (PDF) (Thesis). p. 52. Retrieved 2021-05-24.
^ Major, Günther W., V.N. Gheorghe, F.G. (2009). Charged particle traps II : applications. Berlin: Springer. p. 220. ISBN 978-3540922605.{{cite book}}: CS1 maint: multiple names: authors list (link)

@@ Line 1: / Line 1: @@
+{{Short description|System comprising multiple qubits}}
-In [[quantum computing]], a '''quantum register''' is a system comprising multiple [[qubit|qubits]].<ref>{{cite book |last1=Ekert |first1=Artur |last2=Hayden |first2=Patrick |last3=Inamori |first3=Hitoshi |date=2008 |title=Coherent atomic matter waves |chapter=Basic Concepts in Quantum Computation |series=Les Houches - Ecole d'Ete de Physique Theorique |volume=72 |pages=661–701 |doi=10.1007/3-540-45338-5_10 |arxiv=quant-ph/0011013|isbn=978-3-540-41047-8 |s2cid=53402188 }}</ref>  It is the quantum analogue of the classical [[processor register]].  [[Quantum computer]]s perform calculations by manipulating qubits within a quantum register.<ref>{{cite thesis |last=Ömer |first=Bernhard |date=2000-01-20 |title=Quantum Programming in QCL |url=http://tph.tuwien.ac.at/~oemer/doc/quprog.pdf |access-date=2021-05-24 |pages=52}}</ref>
+In [[quantum computing]], a '''quantum register''' is a system comprising multiple [[qubit|qubits]].<ref>{{cite book |last1=Ekert |first1=Artur |last2=Hayden |first2=Patrick |last3=Inamori |first3=Hitoshi |date=2008 |title=Coherent atomic matter waves |chapter=Basic Concepts in Quantum Computation |series=Les Houches - Ecole d'Ete de Physique Theorique |volume=72 |pages=661–701 |doi=10.1007/3-540-45338-5_10 |arxiv=quant-ph/0011013|isbn=978-3-540-41047-8 |s2cid=53402188 }}</ref>  It is the quantum analogue of the classical [[processor register]].  Quantum computers perform calculations by manipulating qubits within a quantum register.<ref>{{cite thesis |last=Ömer |first=Bernhard |date=2000-01-20 |title=Quantum Programming in QCL |url=http://tph.tuwien.ac.at/~oemer/doc/quprog.pdf |access-date=2021-05-24 |pages=52}}</ref>
 == Definition ==
 {{Further|Mathematical formulation of quantum mechanics#Description of the state of a system}}
+It is usually assumed that the register consists of qubits. It is also generally assumed that registers are not [[density matrix|density matrices]], but that they are [[pure state|pure]], although the definition of "register" can be extended to density matrices.
-An <math>n</math> size quantum register is a quantum system comprising <math>n</math> qubits.
+An <math>n</math> size quantum register is a quantum system comprising <math>n</math> [[Qubit#Qubit states|pure qubits]].
 The [[Hilbert space]], <math>\mathcal{H}</math>, in which the data is stored in a quantum register is given by <math>\mathcal{H} = \mathcal{H_{n-1}}\otimes\mathcal{H_{n-2}}\otimes\ldots\otimes\mathcal{H_0}</math> where <math>\otimes</math> is the [[tensor product]].<ref>{{cite book|last1=Major|first1=Günther W., V.N. Gheorghe, F.G.|title=Charged particle traps II : applications|date=2009|publisher=Springer|location=Berlin|isbn=978-3540922605|page=220}}</ref>
-The number of dimensions of the Hilbert spaces depend on what kind of quantum systems the register is composed of. [[Qubit]]s are 2-dimensional complex spaces, while [[qutrit]]s are 3-dimensional, et.c. For a register composed of ''N'' number of ''d''-level quantum systems we have the Hilbert space <math>\mathcal{H}=(\mathbb{C}^d)^{\otimes N} = \underbrace{\mathbb{C}^d \otimes \mathbb{C}^d \otimes \dots \otimes \mathbb{C}^d }_{N\text{ times}} \cong \mathbb{C}^{d^N}.</math>
+The number of dimensions of the Hilbert spaces depends on what kind of quantum systems the register is composed of. [[Qubit]]s are 2-dimensional [[Complex number|complex]] spaces (<math>\mathbb{C}^2</math>), while [[qutrit]]s are 3-dimensional complex spaces (<math>\mathbb{C}^3</math>), et.c. For a register composed of ''N'' number of ''d''-dimensional (or ''d''-[[Energy level|level]]) quantum systems we have the Hilbert space <math>\mathcal{H}=(\mathbb{C}^d)^{\otimes N} = \underbrace{\mathbb{C}^d \otimes \mathbb{C}^d \otimes \dots \otimes \mathbb{C}^d }_{N\text{ times}} \cong \mathbb{C}^{d^N}.</math>
-The registers quantum state can in the [[bra-ket notation]] be written <math>|\psi\rangle = \sum_{k=0}^{d^N-1} a_k|k\rangle = a_0|0\rangle + a_1|1\rangle + \dots + a_{d^N-1}|d^N-1\rangle.</math> The values <math>a_k</math> are [[probability amplitude]]s. Because of the [[Born rule]], <math>\sum_{k=0}^{d^N-1} |a_k|^2 = 1,</math> so the possible [[State space (physics)|state space]] of the register is the surface of the [[unit sphere]] in <math>\mathbb{C}^{d^N}.</math>
+The registers [[quantum state]] can in the [[bra-ket notation]] be written <math>|\psi\rangle = \sum_{k=0}^{d^N-1} a_k|k\rangle = a_0|0\rangle + a_1|1\rangle + \dots + a_{d^N-1}|d^N-1\rangle.</math> The values <math>a_k</math> are [[probability amplitude]]s. Because of the [[Born rule]] and the [[Probability axioms#Second axiom|2nd axiom of probability theory]], <math>\sum_{k=0}^{d^N-1} |a_k|^2 = 1,</math> so the possible [[state space]] of the register is the surface of the [[unit sphere]] in <math>\mathbb{C}^{d^N}.</math>
 '''Examples:'''
-* The [[quantum state]] vector of a 5-qubit register is a [[unit vector]] in <math>\mathbb{C}^{2^5}=\mathbb{C}^{32}.</math>
+* The quantum state vector of a 5-qubit register is a [[unit vector]] in <math>\mathbb{C}^{2^5}=\mathbb{C}^{32}.</math>
-* A register of three qutrits similarly is a unit vector in <math>\mathbb{C}^{3^3}=\mathbb{C}^{27}.</math>
+* A register of four qutrits similarly is a unit vector in <math>\mathbb{C}^{3^4}=\mathbb{C}^{81}.</math>
 == Quantum vs. classical register ==
@@ Line 19: / Line 22: @@
 An <math>n</math> size classical register refers to an array of <math>n</math> [[Flip-flop_(electronics)|flip flops]]. An <math>n</math> size quantum register is merely a collection of <math>n</math> qubits.
-Moreover, while an <math>n</math> size classical register is able to store a single value of the <math>2^n</math> possibilities spanned by <math>n</math> classical pure bits, a quantum register is able to store all <math>2^n</math> possibilities spanned by quantum [[Qubit#Qubit_states |pure qubits]] at the same time.
+Moreover, while an <math>n</math> size classical register is able to store a single value of the <math>2^n</math> possibilities spanned by <math>n</math> classical pure bits, a quantum register is able to store all <math>2^n</math> possibilities spanned by quantum [[Qubit#Qubit_states|pure qubits]] at the same time.
 For example, consider a 2-bit-wide register.  A classical register is able to store only one of the possible values represented by 2 bits - <math> 00, 01, 10, 11 \quad(0, 1, 2, 3)</math> accordingly.
-If we consider 2 pure qubits in [[Quantum_superposition|superposition]]s <math>|a_0\rangle=\frac{1}{\sqrt2}(|0\rangle + |1\rangle)</math> and <math>|a_1\rangle=\frac{1}{\sqrt2}(|0\rangle - |1\rangle)</math>, using the quantum register definition <math>|a\rangle=|a_{0}\rangle\otimes|a_{1}\rangle = \frac{1}{2}(|00\rangle - |01\rangle + |10\rangle - |11\rangle)</math> it follows that it is capable of storing all the possible values spanned by two qubits simultaneously.
+If we consider 2 pure qubits in [[Quantum_superposition|superposition]]s <math>|a_0\rangle=\frac{1}{\sqrt2}(|0\rangle + |1\rangle)</math> and <math>|a_1\rangle=\frac{1}{\sqrt2}(|0\rangle - |1\rangle)</math>, using the quantum register definition <math>|a\rangle=|a_{0}\rangle\otimes|a_{1}\rangle = \frac{1}{2}(|00\rangle - |01\rangle + |10\rangle - |11\rangle)</math> it follows that it is capable of storing all the possible values (by having non-zero probability amplitude for all outcomes) spanned by two qubits simultaneously.
+== See also ==
+* [[List of proposed quantum registers]]
+* [[Quantum circuit]]
+* [[Quantum logic gate]]
-==References==
+== References ==
 {{Reflist}}

Revision as of 07:19, 13 June 2024

Definition

Quantum vs. classical register

See also

References

Further reading