Misplaced Pages

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97-687: The CNOT can be expressed in the Pauli basis as: Being both unitary and Hermitian , CNOT has the property e i θ U = ( cos ⁡ θ ) I + ( i sin ⁡ θ ) U {\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U} and U = e i π 2 ( I − U ) = e − i π 2 ( I − U ) {\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}} , and

194-399: A 2 − a 3 ) . {\displaystyle {\begin{aligned}{\vec {a}}\cdot {\vec {\sigma }}&=\sum _{k,l}a_{k}\,\sigma _{\ell }\,{\hat {x}}_{k}\cdot {\hat {x}}_{\ell }\\&=\sum _{k}a_{k}\,\sigma _{k}\\&={\begin{pmatrix}a_{3}&a_{1}-ia_{2}\\a_{1}+ia_{2}&-a_{3}\end{pmatrix}}.\end{aligned}}} More formally, this defines

291-496: A 3 → − | a → | {\displaystyle a_{3}\to -\left|{\vec {a}}\right|} . They can be rescued by letting a → = | a → | ( ϵ , 0 , − ( 1 − ϵ 2 / 2 ) ) {\displaystyle {\vec {a}}=\left|{\vec {a}}\right|(\epsilon ,0,-(1-\epsilon ^{2}/2))} and taking

388-627: A 3 + | a → | ) [ i a 2 − a 1 a 3 + | a → | ]   . {\displaystyle \psi _{+}={\frac {1}{{\sqrt {2\left|{\vec {a}}\right|\ (a_{3}+\left|{\vec {a}}\right|)\ }}\ }}{\begin{bmatrix}a_{3}+\left|{\vec {a}}\right|\\a_{1}+ia_{2}\end{bmatrix}};\qquad \psi _{-}={\frac {1}{\sqrt {2|{\vec {a}}|(a_{3}+|{\vec {a}}|)}}}{\begin{bmatrix}ia_{2}-a_{1}\\a_{3}+|{\vec {a}}|\end{bmatrix}}~.} These expressions become singular for

485-394: A k σ ℓ x ^ k ⋅ x ^ ℓ = ∑ k a k σ k = ( a 3 a 1 − i a 2 a 1 + i

582-463: A → | 2 = 0   , {\displaystyle \ ({\vec {a}}\cdot {\vec {\sigma }})^{2}-|{\vec {a}}|^{2}=0\ ,} since this can be factorised into   ( a → ⋅ σ → − | a → | ) ( a → ⋅ σ → + |

679-499: A → | ) = 0. {\displaystyle \ ({\vec {a}}\cdot {\vec {\sigma }}-|{\vec {a}}|)({\vec {a}}\cdot {\vec {\sigma }}+|{\vec {a}}|)=0.} A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies   a → ⋅ σ →   {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ }

776-495: A → | . {\displaystyle \ \pm |{\vec {a}}|.} This follows immediately from tracelessness and explicitly computing the determinant. More abstractly, without computing the determinant, which requires explicit properties of the Pauli matrices, this follows from   ( a → ⋅ σ → ) 2 − |

873-438: A → ↦ a → ⋅ σ → . {\displaystyle {\vec {a}}\mapsto {\vec {a}}\cdot {\vec {\sigma }}.} Each component of a → {\displaystyle {\vec {a}}} can be recovered from the matrix (see completeness relation below) 1 2 tr ⁡ ( (

970-425: A → ⋅ σ → ) σ → ) = a → . {\displaystyle {\frac {1}{2}}\operatorname {tr} {\Bigl (}{\bigl (}{\vec {a}}\cdot {\vec {\sigma }}{\bigr )}{\vec {\sigma }}{\Bigr )}={\vec {a}}.} This constitutes an inverse to the map a → ↦

1067-400: A → ⋅ σ → {\displaystyle {\vec {a}}\mapsto {\vec {a}}\cdot {\vec {\sigma }}} , making it manifest that the map is a bijection. The norm is given by the determinant (up to a minus sign) det ( a → ⋅ σ → ) = −

SECTION 10

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1164-470: A → ⋅ σ → ) = det ( a → ⋅ σ → ) , {\displaystyle \det(U*{\vec {a}}\cdot {\vec {\sigma }})=\det({\vec {a}}\cdot {\vec {\sigma }}),} and that U ∗ a → ⋅ σ → {\displaystyle U*{\vec {a}}\cdot {\vec {\sigma }}}

1261-534: A → ⋅ a → = − | a → | 2 . {\displaystyle \det {\bigl (}{\vec {a}}\cdot {\vec {\sigma }}{\bigr )}=-{\vec {a}}\cdot {\vec {a}}=-|{\vec {a}}|^{2}.} Then, considering the conjugation action of an SU ( 2 ) {\displaystyle {\text{SU}}(2)} matrix U {\displaystyle U} on this space of matrices, we find det ( U ∗

1358-410: A → |   ( a 3 + | a → | )     [ a 3 + | a → | a 1 + i a 2 ] ; ψ − = 1 2 | a → | (

1455-499: A 0 when both inputs match. When searching for a specific bit pattern or PRN sequence in a very long data sequence, a series of XOR gates can be used to compare a string of bits from the data sequence against the target sequence in parallel. The number of 0 outputs can then be counted to determine how well the data sequence matches the target sequence. Correlators are used in many communications devices such as CDMA receivers and decoders for error correction and channel codes. In

1552-906: A NOT gate . If we consider the expression ( A ⋅ B ¯ ) + ( A ¯ ⋅ B ) {\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)} , we can construct an XOR gate circuit directly using AND, OR and NOT gates . However, this approach requires five gates of three different kinds. As alternative, if different gates are available we can apply Boolean algebra to transform ( A ⋅ B ¯ ) + ( A ¯ ⋅ B ) ≡ ( A + B ) ⋅ ( A ¯ + B ¯ ) {\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)\equiv (A+B)\cdot ({\overline {A}}+{\overline {B}})} as stated above, and apply de Morgan's Law to

1649-568: A 2-qubit register is etc. Viewing CNOT in this basis, the state of the second qubit remains unchanged, and the state of the first qubit is flipped, according to the state of the second bit. (For details see below.) "Thus, in this basis the sense of which bit is the control bit and which the target bit has reversed. But we have not changed the transformation at all, only the way we are thinking about it." The "computational" basis { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}}

1746-510: A C NOT interaction is of importance when considering information flow in entangled quantum systems. We now proceed to give the details of the computation. Working through each of the Hadamard basis states, the results on the right column show that the first qubit flips between | + ⟩ {\displaystyle |+\rangle } and | − ⟩ {\displaystyle |-\rangle } when

1843-410: A CDMA receiver, correlators are used to extract the polarity of a specific PRN sequence out of a combined collection of PRN sequences. A correlator looking for 11010 in the data sequence 1110100101 would compare the incoming data bits against the target sequence at every possible offset while counting the number of matches (zeros): In this example, the best match occurs when the target sequence

1940-415: A cascade of binary exclusive-or operations: the first two signals are fed into an XOR gate, then the output of that gate is fed into a second XOR gate together with the third signal, and so on for any remaining signals. The result is a circuit that outputs a 1 when the number of 1s at its inputs is odd, and a 0 when the number of incoming 1s is even. This makes it practically useful as a parity generator or

2037-660: A completeness relation. It is convenient to define a second Pauli 4-vector and allow raising and lowering using the Minkowski metric tensor. The relation can then be written x ν = 1 2 tr ⁡ ( σ ¯ ν ( x μ σ μ ) ) . {\displaystyle x_{\nu }={\tfrac {1}{2}}\operatorname {tr} {\Bigl (}{\bar {\sigma }}_{\nu }{\bigl (}x_{\mu }\sigma ^{\mu }{\bigr )}{\Bigr )}.} Similarly to

SECTION 20

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2134-843: A map R : S U ( 2 ) → S O ( 3 ) {\displaystyle R:\mathrm {SU} (2)\to \mathrm {SO} (3)} given by where R ( U ) ∈ S O ( 3 ) . {\displaystyle R(U)\in \mathrm {SO} (3).} This map is the concrete realization of the double cover of S O ( 3 ) {\displaystyle \mathrm {SO} (3)} by S U ( 2 ) , {\displaystyle \mathrm {SU} (2),} and therefore shows that SU ( 2 ) ≅ S p i n ( 3 ) . {\displaystyle {\text{SU}}(2)\cong \mathrm {Spin} (3).} The components of R ( U ) {\displaystyle R(U)} can be recovered using

2231-448: A map from R 3 {\displaystyle \mathbb {R} ^{3}} to the vector space of traceless Hermitian 2 × 2 {\displaystyle 2\times 2} matrices. This map encodes structures of R 3 {\displaystyle \mathbb {R} ^{3}} as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making

2328-498: A modulo-2 adder . For example, the 74LVC1G386 microchip is advertised as a three-input logic gate, and implements a parity generator. XOR gates and AND gates are the two most-used structures in VLSI applications. The XOR logic gate can be used as a one-bit adder that adds any two bits together to output one bit. For example, if we add 1 plus 1 in binary , we expect a two-bit answer, 10 (i.e. 2 in decimal). Since

2425-916: A switch. XOR can also be viewed as addition modulo 2. As a result, XOR gates are used to implement binary addition in computers. A half adder consists of an XOR gate and an AND gate . The gate is also used in subtractors and comparators . The algebraic expressions A ⋅ B ¯ + A ¯ ⋅ B {\displaystyle A\cdot {\overline {B}}+{\overline {A}}\cdot B} or ( A + B ) ⋅ ( A ¯ + B ¯ ) {\displaystyle (A+B)\cdot ({\overline {A}}+{\overline {B}})} or ( A + B ) ⋅ ( A ⋅ B ) ¯ {\displaystyle (A+B)\cdot {\overline {(A\cdot B)}}} or A ⊕ B {\displaystyle A\oplus B} all represent

2522-404: A total of eight transistors, four less than in the previous design. The XOR function is implemented by passing through to the output the inverted value of A when B is high and passing the value of A when B is at a logic low. so when both inputs are low the transmission gate at the bottom is off and the one at the top is on and lets A through which is low so the output is low. When both are high only

2619-471: Is Hermitian and therefore its own inverse. The tensor product of two Hadamard transforms operating (independently) on two qubits is labelled H 2 . We can therefore write the matrices as: H 2 . C NOT . H 2 When multiplied out, this yields a matrix that swaps the | 01 ⟩ {\displaystyle |01\rangle } and | 11 ⟩ {\displaystyle |11\rangle } terms over, while leaving

2716-441: Is Hermitian , and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ 0 ), the Pauli matrices form a basis of the vector space of 2 × 2 Hermitian matrices over the real numbers , under addition. This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. The Pauli matrices satisfy

2813-598: Is involutory . The CNOT gate can be further decomposed as products of rotation operator gates and exactly one two qubit interaction gate , for example In general, any single qubit unitary gate can be expressed as U = e i H {\displaystyle U=e^{iH}} , where H is a Hermitian matrix , and then the controlled U is C U = e i 1 2 ( I 1 − Z 1 ) H 2 {\displaystyle CU=e^{i{\frac {1}{2}}(I_{1}-Z_{1})H_{2}}} . The CNOT gate

2910-479: Is Hermitian and traceless. It then makes sense to define U ∗ a → ⋅ σ → = a → ′ ⋅ σ → , {\displaystyle U*{\vec {a}}\cdot {\vec {\sigma }}={\vec {a}}'\cdot {\vec {\sigma }},} where a → ′ {\displaystyle {\vec {a}}'} has

3007-419: Is a digital logic gate that gives a true (1 or HIGH) output when the number of true inputs is odd. An XOR gate implements an exclusive or ( ↮ {\displaystyle \nleftrightarrow } ) from mathematical logic ; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents

Controlled NOT gate - Misplaced Pages Continue

3104-453: Is also used in classical reversible computing . The CNOT gate operates on a quantum register consisting of 2 qubits. The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is | 1 ⟩ {\displaystyle |1\rangle } . If { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} are

3201-478: Is an essential element of the Deutsch–Jozsa algorithm . When viewed only in the computational basis { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} , the behaviour of the C NOT appears to be like the equivalent classical gate. However, the simplicity of labelling one qubit the control and the other the target does not reflect

3298-407: Is an inverted-input AND gate . Another alternative arrangement is of five NAND gates in a topology that emphasizes the construction of the function from ( A ⋅ B ¯ ) + ( A ¯ ⋅ B ) {\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)} , noting from de Morgan's Law that a NAND gate

3395-521: Is an inverted-input OR gate . For the NAND constructions, the upper arrangement requires fewer gates. For the NOR constructions, the lower arrangement offers the advantage of a shorter propagation delay (the time delay between an input changing and the output changing). XOR chips are readily available. The most common standard chip codes are: Literal interpretation of the name "exclusive or", or observation of

3492-423: Is at a logic high using pass transistor logic to reduce the transistor count and when B is at a logic low, their output is at a high impedance state. The two in the middle are a transmission gate that drives the output to the value of A when B is at a logic low and the two rightmost transistors form an inverter needed to generate B ¯ {\displaystyle {\overline {B}}} used by

3589-515: Is diagonal with possible eigenvalues   ± | a → | . {\displaystyle \ \pm |{\vec {a}}|.} The tracelessness of   a → ⋅ σ →   {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ } means it has exactly one of each eigenvalue. Its normalized eigenvectors are ψ + = 1 2 |

3686-558: Is equivalent to a C-NOT gate except for a π / 2 {\displaystyle \pi /2} rotation of the nuclear spin around the z axis. Trapped ion quantum computers : In May, 2024, Canada implemented export restrictions on the sale of quantum computers containing more than 34 qubits and error rates below a certain CNOT error threshold , along with restrictions for quantum computers with more qubits and higher error rates. The same restrictions quickly popped up in

3783-448: Is equivalent to the XOR. An XOR gate circuit can be made from four NAND gates . In fact, both NAND and NOR gates are so-called "universal gates" and any logical function can be constructed from either NAND logic or NOR logic alone. If the four NAND gates are replaced by NOR gates , this results in an XNOR gate , which can be converted to an XOR gate by inverting the output or one of

3880-401: Is offset by 1 bit and all five bits match. When offset by 5 bits, the sequence exactly matches its inverse. By looking at the difference between the number of ones and zeros that come out of the bank of XOR gates, it is easy to see where the sequence occurs and whether or not it is inverted. Longer sequences are easier to detect than short sequences. f ( a , b ) =

3977-874: Is the Kronecker delta . I denotes the 2 × 2 identity matrix. These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for R 3 , {\displaystyle \mathbb {R} ^{3},} denoted C l 3 ( R ) . {\displaystyle \mathrm {Cl} _{3}(\mathbb {R} ).} The usual construction of generators σ j k = 1 4 [ σ j , σ k ] {\displaystyle \sigma _{jk}={\tfrac {1}{4}}[\sigma _{j},\sigma _{k}]} of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} using

Controlled NOT gate - Misplaced Pages Continue

4074-586: Is the eigenbasis for the spin in the Z-direction, whereas the Hadamard basis { | + ⟩ , | − ⟩ } {\displaystyle \{|+\rangle ,|-\rangle \}} is the eigenbasis for spin in the X-direction. Switching X and Z and qubits 1 and 2, then, recovers the original transformation." This expresses a fundamental symmetry of the CNOT gate. The observation that both qubits are (equally) affected in

4171-433: Is useful for "selecting" any one of the matrices numerically by substituting values of j = 1, 2, 3, in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations. The matrices are involutory : where I is the identity matrix . The determinants and traces of the Pauli matrices are from which we can deduce that each matrix σ j has eigenvalues +1 and −1. With

4268-475: Is written   σ μ   {\displaystyle \ \sigma ^{\mu }\ } with components This defines a map from R 1 , 3 {\displaystyle \mathbb {R} ^{1,3}} to the vector space of Hermitian matrices, which also encodes the Minkowski metric (with mostly minus convention) in its determinant: This 4-vector also has

4365-1870: The | 00 ⟩ {\displaystyle |00\rangle } and | 10 ⟩ {\displaystyle |10\rangle } terms alone. This is equivalent to a CNOT gate where qubit 2 is the control qubit and qubit 1 is the target qubit: 1 2 [ 1 1 1 1 1 − 1 1 − 1 1 1 − 1 − 1 1 − 1 − 1 1 ] . [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] . 1 2 [ 1 1 1 1 1 − 1 1 − 1 1 1 − 1 − 1 1 − 1 − 1 1 ] = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] {\displaystyle {\frac {1}{2}}{\begin{bmatrix}{\begin{array}{rrrr}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{array}}\end{bmatrix}}.{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.{\frac {1}{2}}{\begin{bmatrix}{\begin{array}{rrrr}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{array}}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0\end{bmatrix}}} A common application of

4462-999: The Greek letter sigma ( σ ), they are occasionally denoted by tau ( τ ) when used in connection with isospin symmetries. σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 1 ) . {\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\\\end{aligned}}} These matrices are named after

4559-451: The Intel 386 CPU. The XOR gate can also be implemented by the use of Transmission gates with pass transistor logic . This implementation uses two Transmission gates and two inverters not shown in the diagram to generate A ¯ {\displaystyle {\overline {A}}} and B ¯ {\displaystyle {\overline {B}}} for

4656-401: The anticommutation relations: where { σ j , σ k } {\displaystyle \{\sigma _{j},\sigma _{k}\}} is defined as σ j σ k + σ k σ j , {\displaystyle \sigma _{j}\sigma _{k}+\sigma _{k}\sigma _{j},} and δ jk

4753-427: The caret symbol ^ to denote bitwise XOR. (Note that the caret does not denote logical conjunction (AND) in these languages, despite the similarity of symbol.) The XOR gate is most commonly implemented using MOSFETs circuits. Some of those implementations include: XOR gates can be implemented using AND-OR-Invert ( AOI ) or OR-AND-Invert (OAI) logic. The metal–oxide–semiconductor ( CMOS ) implementations of

4850-458: The (unital) associative algebra generated by iσ 1 , iσ 2 , iσ 3 functions identically ( is isomorphic ) to that of quaternions ( H {\displaystyle \mathbb {H} } ). All three of the Pauli matrices can be compacted into a single expression: where the solution to i = −1 is the " imaginary unit ", and δ jk is the Kronecker delta , which equals +1 if j = k and 0 otherwise. This expression

4947-435: The C NOT gate is to maximally entangle two qubits into the | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } Bell state ; this forms part of the setup of the superdense coding , quantum teleportation , and entangled quantum cryptography algorithms. To construct | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } ,

SECTION 50

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5044-470: The CNOT gate can be represented by the matrix ( permutation matrix form): The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state and into the vibrational state of the ion within the trap. At the time of the experiment, the reliability of the CNOT-operation was measured to be on

5141-1132: The Clifford algebra recovers the commutation relations above, up to unimportant numerical factors. A few explicit commutators and anti-commutators are given below as examples: Each of the ( Hermitian ) Pauli matrices has two eigenvalues : +1 and −1 . The corresponding normalized eigenvectors are The Pauli vector is defined by σ → = σ 1 x ^ 1 + σ 2 x ^ 2 + σ 3 x ^ 3 , {\displaystyle {\vec {\sigma }}=\sigma _{1}{\hat {x}}_{1}+\sigma _{2}{\hat {x}}_{2}+\sigma _{3}{\hat {x}}_{3},} where x ^ 1 {\displaystyle {\hat {x}}_{1}} , x ^ 2 {\displaystyle {\hat {x}}_{2}} , and x ^ 3 {\displaystyle {\hat {x}}_{3}} are an equivalent notation for

5238-2239: The Hadamard view, the control and target qubits have conceptually swapped and qubit A is inverted when qubit B is | − ⟩ B {\displaystyle |-\rangle _{B}} . The output state after applying the C NOT gate is 1 2 ( | + + ⟩ + | − − ⟩ ) {\textstyle {\frac {1}{\sqrt {2}}}(|++\rangle +|--\rangle )} which can be shown as follows: = 1 2 ( | + ⟩ A | + ⟩ B + | − ⟩ A | − ⟩ B ) {\displaystyle ={\frac {1}{\sqrt {2}}}(|+\rangle _{A}|+\rangle _{B}+|-\rangle _{A}|-\rangle _{B})} = 1 2 2 ( ( | 0 ⟩ A + | 1 ⟩ A ) ( | 0 ⟩ B + | 1 ⟩ B ) + ( | 0 ⟩ A − | 1 ⟩ A ) ( | 0 ⟩ B − | 1 ⟩ B ) ) {\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|0\rangle _{A}+|1\rangle _{A})(|0\rangle _{B}+|1\rangle _{B})+(|0\rangle _{A}-|1\rangle _{A})(|0\rangle _{B}-|1\rangle _{B}))} = 1 2 2 ( ( | 00 ⟩ + | 01 ⟩ + | 10 ⟩ + | 11 ⟩ ) + ( | 00 ⟩ − | 01 ⟩ − | 10 ⟩ + | 11 ⟩ ) ) {\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|00\rangle +|01\rangle +|10\rangle +|11\rangle )+(|00\rangle -|01\rangle -|10\rangle +|11\rangle ))} = 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} . The C-ROT gate (controlled Rabi rotation )

5335-492: The IEC rectangular symbol, raises the question of correct behaviour with additional inputs. If a logic gate were to accept three or more inputs and produce a true output if exactly one of those inputs were true, then it would in effect be a one-hot detector (and indeed this is the case for only two inputs). However, it is rarely implemented this way in practice. It is most common to regard subsequent inputs as being applied through

5432-848: The Pauli 3-vector case, we can find a matrix group that acts as isometries on   R 1 , 3   ; {\displaystyle \ \mathbb {R} ^{1,3}\ ;} in this case the matrix group is   S L ( 2 , C )   , {\displaystyle \ \mathrm {SL} (2,\mathbb {C} )\ ,} and this shows   S L ( 2 , C ) ≅ S p i n ( 1 , 3 ) . {\displaystyle \ \mathrm {SL} (2,\mathbb {C} )\cong \mathrm {Spin} (1,3).} Similarly to above, this can be explicitly realized for   S ∈ S L ( 2 , C )   {\displaystyle \ S\in \mathrm {SL} (2,\mathbb {C} )\ } with components In fact,

5529-456: The Pauli matrices span the space of observables of the complex two-dimensional Hilbert space . In the context of Pauli's work, σ k represents the observable corresponding to spin along the k th coordinate axis in three-dimensional Euclidean space R 3 . {\displaystyle \mathbb {R} ^{3}.} The Pauli matrices (after multiplication by i to make them anti-Hermitian ) also generate transformations in

5626-623: The UK, France, Spain and the Netherlands. They offered few explanations for this action, but all of them are Wassenaar Arrangement states, and the restrictions seem related to national security concerns potentially including quantum cryptography or protection from competition . Pauli matrices In mathematical physics and mathematics , the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless , Hermitian , involutory and unitary . Usually indicated by

5723-452: The XOR gate corresponding to the AOI logic above are shown below. On the left, the nMOS and pMOS transistors are arranged so that the input pairs A ⋅ B ¯ {\displaystyle A\cdot {\overline {B}}} and A ¯ ⋅ B {\displaystyle {\overline {A}}\cdot B} activate the 2 pMOS transistors of

5820-649: The XOR gate with inputs A and B . The behavior of XOR is summarized in the truth table shown on the right. There are three schematic symbols for XOR gates: the traditional ANSI and DIN symbols and the IEC symbol. In some cases, the DIN symbol is used with ⊕ instead of ≢. For more information see Logic Gate Symbols . The "=1" on the IEC symbol indicates that the output is activated by only one active input. The logic symbols ⊕, J pq , and ⊻ can be used to denote an XOR operation in algebraic expressions. C-like languages use

5917-399: The XOR gate. The trade-off with the previous implementation is that since transmission gates are not ideal switches, there is resistance associated with them, so depending on the signal strength of the input, cascading them may degrade the output levels. The previous transmission gate implementation can be further optimized from eight to six transistors by implementing the functionality of

SECTION 60

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6014-519: The bottom to Vss for a logic low. If inverted inputs (for example from a flip-flop ) are available, this gate can be used directly. Otherwise, two additional inverters with two transistors each are needed to generate A ¯ {\displaystyle {\overline {A}}} and B ¯ {\displaystyle {\overline {B}}} , bringing the total number of transistors to twelve. The AOI implementation without inverted input has been used, for example, in

6111-436: The commutator to the anticommutator gives so that,     σ j σ k = δ j k I + i ε j k ℓ σ ℓ   .   {\displaystyle ~~\sigma _{j}\sigma _{k}=\delta _{jk}I+i\varepsilon _{jk\ell }\,\sigma _{\ell }~.~} Contracting each side of

6208-407: The complexity of what happens for most input values of both qubits. Insight can be won by expressing the CNOT gate with respect to a Hadamard transformed basis { | + ⟩ , | − ⟩ } {\displaystyle \{|+\rangle ,|-\rangle \}} . The Hadamard transformed basis of a one-qubit register is given by and the corresponding basis of

6305-540: The computational basis, it appears that qubit A is affecting qubit B. Changing our viewpoint to the Hadamard basis demonstrates that, in a symmetrical way, qubit B is affecting qubit A. The input state can alternately be viewed as: | + ⟩ A {\displaystyle |+\rangle _{A}} and 1 2 ( | + ⟩ + | − ⟩ ) B {\displaystyle {\frac {1}{\sqrt {2}}}(|+\rangle +|-\rangle )_{B}} In

6402-470: The cross-terms vanish. It then follows, now showing summation explicitly, det ( ∑ μ x μ σ μ ) = ∑ μ det ( x μ σ μ ) . {\textstyle \det \left(\sum _{\mu }x_{\mu }\sigma ^{\mu }\right)=\sum _{\mu }\det \left(x_{\mu }\sigma ^{\mu }\right).} Since

6499-597: The determinant property follows abstractly from trace properties of the   σ μ . {\displaystyle \ \sigma ^{\mu }.} For   2 × 2   {\displaystyle \ 2\times 2\ } matrices, the following identity holds: That is, the 'cross-terms' can be written as traces. When   A , B   {\displaystyle \ A,B\ } are chosen to be different   σ μ   , {\displaystyle \ \sigma ^{\mu }\ ,}

6596-726: The eigenvectors ψ + = ( cos ⁡ ( ϑ / 2 ) , sin ⁡ ( ϑ / 2 ) exp ⁡ ( i φ ) ) {\displaystyle \psi _{+}=(\cos(\vartheta /2),\sin(\vartheta /2)\exp(i\varphi ))} and ψ − = ( − sin ⁡ ( ϑ / 2 ) exp ⁡ ( − i φ ) , cos ⁡ ( ϑ / 2 ) ) {\displaystyle \psi _{-}=(-\sin(\vartheta /2)\exp(-i\varphi ),\cos(\vartheta /2))} . The Pauli 4-vector, used in spinor theory,

6693-479: The equation with components of two 3 -vectors a p and b q (which commute with the Pauli matrices, i.e., a p σ q = σ q a p ) for each matrix σ q and vector component a p (and likewise with b q ) yields Finally, translating the index notation for the dot product and cross product results in XOR gate XOR gate (sometimes EOR , or EXOR and pronounced as Exclusive OR )

6790-472: The exclusive-or operation. Hence, a suitable setup of XOR gates can model a linear-feedback shift register, in order to generate random numbers. XOR gates may be used in simplest phase detectors . An XOR gate may be used to easily change between buffering or inverting a signal. For example, XOR gates can be added to the output of a seven-segment display decoder circuit to allow a user to choose between active-low or active-high output. XOR gates produce

6887-503: The existence of a norm follows from the fact that R 3 {\displaystyle \mathbb {R} ^{3}} is a Lie algebra (see Killing form ). This cross-product can be used to prove the orientation-preserving property of the map above. The eigenvalues of   a → ⋅ σ →   {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ } are   ± |

6984-593: The following commutation relations: where the Levi-Civita symbol ε jkl is used. These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra ( R 3 , × ) ≅ s u ( 2 ) ≅ s o ( 3 ) . {\displaystyle (\mathbb {R} ^{3},\times )\cong {\mathfrak {su}}(2)\cong {\mathfrak {so}}(3).} They also satisfy

7081-743: The inclusion of the identity matrix I (sometimes denoted σ 0 ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt ) of the Hilbert space H 2 {\displaystyle {\mathcal {H}}_{2}} of 2 × 2 Hermitian matrices over R {\displaystyle \mathbb {R} } , and the Hilbert space M 2 , 2 ( C ) {\displaystyle {\mathcal {M}}_{2,2}(\mathbb {C} )} of all complex 2 × 2 matrices over C {\displaystyle \mathbb {C} } . The Pauli matrices obey

7178-426: The inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "must have one or the other but not both". An XOR gate may serve as a "programmable inverter" in which one input determines whether to invert the other input, or to simply pass it along with no change. Hence it functions as a inverter (a NOT gate) which may be activated or deactivated by

7275-399: The inputs (e.g. with a fifth NOR gate ). An alternative arrangement is of five NOR gates in a topology that emphasizes the construction of the function from ( A + B ) ⋅ ( A ¯ + B ¯ ) {\displaystyle (A+B)\cdot ({\overline {A}}+{\overline {B}})} , noting from de Morgan's Law that a NOR gate

7372-589: The inputs A (control) and B (target) to the C NOT gate are: 1 2 ( | 0 ⟩ + | 1 ⟩ ) A {\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}} and | 0 ⟩ B {\displaystyle |0\rangle _{B}} After applying C NOT , the resulting Bell State 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\textstyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} has

7469-534: The inverter that generates A ¯ {\displaystyle {\overline {A}}} and the bottom pass-gate with just two transistors arranged like an inverter but with the source of the pMOS connected to B {\displaystyle B} instead of Vdd and the source of the nMOS connected to B ¯ {\displaystyle {\overline {B}}} instead of GND. The two leftmost transistors mentioned above, perform an optimized conditional inversion of A when B

7566-519: The last term to get ( A + B ) ⋅ ( A ⋅ B ) ¯ {\displaystyle (A+B)\cdot {\overline {(A\cdot B)}}} which can be implemented using only four gates as shown on the right. intuitively, XOR is equivalent to OR except for when both A and B are high. So the AND of the OR with then NAND that gives a low only when both A and B are high

7663-643: The limit ϵ → 0 {\displaystyle \epsilon \to 0} , which yields the correct eigenvectors (0,1) and (1,0) of σ z {\displaystyle \sigma _{z}} . Alternatively, one may use spherical coordinates a → = a ( sin ⁡ ϑ cos ⁡ φ , sin ⁡ ϑ sin ⁡ φ , cos ⁡ ϑ ) {\displaystyle {\vec {a}}=a(\sin \vartheta \cos \varphi ,\sin \vartheta \sin \varphi ,\cos \vartheta )} to obtain

7760-565: The map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory. Another way to view the Pauli vector is as a 2 × 2 {\displaystyle 2\times 2} Hermitian traceless matrix-valued dual vector, that is, an element of Mat 2 × 2 ( C ) ⊗ ( R 3 ) ∗ {\displaystyle {\text{Mat}}_{2\times 2}(\mathbb {C} )\otimes (\mathbb {R} ^{3})^{*}} that maps

7857-515: The matrices are   2 × 2   , {\displaystyle \ 2\times 2\ ,} this is equal to ∑ μ x μ 2 det ( σ μ ) = η ( x , x ) . {\textstyle \sum _{\mu }x_{\mu }^{2}\det(\sigma ^{\mu })=\eta (x,x).} Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding

7954-510: The more familiar x ^ {\displaystyle {\hat {x}}} , y ^ {\displaystyle {\hat {y}}} , and z ^ {\displaystyle {\hat {z}}} . The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows: a → ⋅ σ → = ∑ k , l

8051-402: The one at the bottom is active and lets the inverted value of A through and since A is high the output will again be low. Similarly if B stays high but A is low the output would be A ¯ {\displaystyle {\overline {A}}} which is high as expected and if B is low but A is high the value of A passes through and the output is high completing the truth table for

8148-601: The only allowed input values for both qubits, then the TARGET output of the CNOT gate corresponds to the result of a classical XOR gate . Fixing CONTROL as | 1 ⟩ {\displaystyle |1\rangle } , the TARGET output of the CNOT gate yields the result of a classical NOT gate . More generally, the inputs are allowed to be a linear superposition of { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} . The CNOT gate transforms

8245-422: The only combination for which the OR and XOR gate outputs differ, an OR gate may be replaced by an XOR gate (or vice versa) without altering the resulting logic. This is convenient if the circuit is being implemented using simple integrated circuit chips which contain only one gate type per chip. Pseudo-random number (PRN) generators , specifically linear-feedback shift registers (LFSR), are defined in terms of

8342-408: The order of 90%. In addition to a regular controlled NOT gate, one could construct a function-controlled NOT gate, which accepts an arbitrary number n +1 of qubits as input, where n +1 is greater than or equal to 2 (a quantum register ). This gate flips the last qubit of the register if and only if a built-in function, with the first n qubits as input, returns a 1. The function-controlled NOT gate

8439-413: The physicist Wolfgang Pauli . In quantum mechanics , they occur in the Pauli equation , which takes into account the interaction of the spin of a particle with an external electromagnetic field . They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix

8536-432: The property that the individual qubits can be measured using any basis and will always present a 50/50 chance of resolving to each state. In effect, the individual qubits are in an undefined state. The correlation between the two qubits is the complete description of the state of the two qubits; if we both choose the same basis to measure both qubits and compare notes, the measurements will perfectly correlate. When viewed in

8633-522: The quantum state: a | 00 ⟩ + b | 01 ⟩ + c | 10 ⟩ + d | 11 ⟩ {\displaystyle a|00\rangle +b|01\rangle +c|10\rangle +d|11\rangle } into: a | 00 ⟩ + b | 01 ⟩ + c | 11 ⟩ + d | 10 ⟩ {\displaystyle a|00\rangle +b|01\rangle +c|11\rangle +d|10\rangle } The action of

8730-399: The same norm as a → , {\displaystyle {\vec {a}},} and therefore interpret U {\displaystyle U} as a rotation of three-dimensional space. In fact, it turns out that the special restriction on U {\displaystyle U} implies that the rotation is orientation preserving. This allows the definition of

8827-409: The second qubit is | − ⟩ {\displaystyle |-\rangle } : A quantum circuit that performs a Hadamard transform followed by C NOT then another Hadamard transform, can be described as performing the CNOT gate in the Hadamard basis (i.e. a change of basis ): (H 1 ⊗ H 1 ) . C NOT . (H 1 ⊗ H 1 ) The single-qubit Hadamard transform, H 1 ,

8924-480: The sense of Lie algebras : the matrices iσ 1 , iσ 2 , iσ 3 form a basis for the real Lie algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , which exponentiates to the special unitary group SU(2) . The algebra generated by the three matrices σ 1 , σ 2 , σ 3 is isomorphic to the Clifford algebra of R 3 , {\displaystyle \mathbb {R} ^{3},} and

9021-401: The top left or the 2 pMOS transistors of the top right respectively, connecting Vdd to the output for a logic high. The remaining input pairs A ⋅ B {\displaystyle A\cdot B} and A ¯ ⋅ B ¯ {\displaystyle {\overline {A}}\cdot {\overline {B}}} activate each one of the two nMOS paths in

9118-657: The tracing process above: The cross-product is given by the matrix commutator (up to a factor of 2 i {\displaystyle 2i} ) [ a → ⋅ σ → , b → ⋅ σ → ] = 2 i ( a → × b → ) ⋅ σ → . {\displaystyle [{\vec {a}}\cdot {\vec {\sigma }},{\vec {b}}\cdot {\vec {\sigma }}]=2i\,({\vec {a}}\times {\vec {b}})\cdot {\vec {\sigma }}.} In fact,

9215-458: The trailing sum bit in this output is achieved with XOR, the preceding carry bit is calculated with an AND gate . This is the main principle in Half Adders . A slightly larger Full Adder circuit may be chained together in order to add longer binary numbers. In certain situations, the inputs to an OR gate (for example, in a full-adder) or to an XOR gate can never be both 1's. As this is

9312-527: The transmission gate and the pass transistor logic circuit. As with the previous implementation, the direct connection of the inputs to the outputs through the pass gate transistors or through the two leftmost transistors, should be taken into account, especially when cascading them. If a specific type of gate is not available, a circuit that implements the same function can be constructed from other available gates. A circuit implementing an XOR function can be trivially constructed from an XNOR gate followed by

9409-415: The useful product relation: σ i σ j = δ i j + i ϵ i j k σ k . {\displaystyle {\begin{aligned}\sigma _{i}\sigma _{j}=\delta _{ij}+i\epsilon _{ijk}\sigma _{k}.\end{aligned}}} Hermitian operators represent observables in quantum mechanics, so

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