Germán Sierra (Instituto de Física Teórica UAM-CSIC, Madrid)
6th INFIERY Summer School Madrid, 1st September 2021 Plan of the lecture
- Part I: Historial background
- Part II: The qubit and quantum gates Quantum Computation and Quantum Information Quantum Mechanics
Quantum Computation and Quantum Information Quantum Mechanics Computer Sciences
Quantum Computation and Quantum Information Quantum Mechanics Computer Sciences
Quantum Computation and Quantum Information
Information Theory Quantum Mechanics Computer Sciences
Quantum Computation and Quantum Information
Information Theory Cryptography Quantum Mechanics Quantum Mechanics
Golden age of Physics Quantum Mechanics
Golden age of Physics
First Quantum Revolution About an improvement of Wien's spectral equation
Max Planck 1858 - 1947 On the theory of the law of energy distribution in normal spectrum
is light a wave or a particle? is light a wave or a particle?
Isaac Newton 1642-1727 is light Thomas Young 1773-1829 a wave or a particle?
Isaac Newton 1642-1727 is light Thomas Young 1773-1829 a wave or a particle?
Isaac Newton 1642-1727
James Maxwell 1831-1879 is light Thomas Young 1773-1829 a wave or a particle?
Isaac Newton 1642-1727
James Maxwell 1831-1879
Albert Einstein (1879-1955) On a heuristic point of view about the creation and generation of light
Annalen der Physik, 1905
when a ray of light propagates from a point, energy is not distributed continuously over increasing volume, but it is composed of a finite number of energy quanta, in space,that move without being divided and that they can be absorbed or emitted only as a whole. On a heuristic point of view about the creation and generation of light
Annalen der Physik, 1905
when a ray of light propagates from a point, energy is not distributed continuously over increasing volume, but it is composed of a finite number of energy quanta, in space,that move without being divided and that they can be absorbed or emitted only as a whole.
Quantum of light = photon Photoelectric effect
electrons photons
metal
(Planck 1900)
Energy of a yellow photon electrón-voltios Particles are also waves
Louis de Broglie 1892-1987 Particles are also waves
Louis de Broglie 1892-1987
Matrix Mechanics / uncertainty principle
푥, 푝 = 푖 ℏ
Werner Heisenberg 1901- 1976 Wave function / Time evolution
휕휓 ℏ2훻2 푖 = − + 푉 푥Ԧ 휓 휕푡 2푚
Erwin Schrödinger 1887 - 1961 Wave function / Time evolution
휕휓 ℏ2훻2 푖 = − + 푉 푥Ԧ 휓 휕푡 2푚
Erwin Schrödinger 1887 - 1961 Probabilistic interpretation
푑푃 = |휓 푥Ԧ |2 푑푉
Max Born 1882 - 1970 Richard Feymann : Quantum Mechanics can be understood by giving it a lot of thought 1918 - 1988 to the double slit experiment Double slit experiment with electrons Interference and superposition principle See in YouTube
Dr. Quantum – Double Slit experiment https://www.youtube.com/watch?v=rQJ4yX1l6to Paradox: the electron "passes" through both slits INTERFERING WITH ITSELF
“Quantum” Skier Classical Observer
(2010)
Protocol to create superposition of macroscopic objects including living beings. Explore the role of life and consciousness in Quantum Mechanics. Paradox: Schrödinger cat (1935)
Dead Alive Copengahen interpretation
Opening the box produces the COLLAPSE of the wave function Many world interpretation
Box closed Box opening World 1
World 2 Many world interpretation: Hugh Everett (1957)
time
Hugh Everett 1930 - 1982 Many world interpretation: Hugh Everett (1957)
time
Hugh Everett 1930 - 1982 Anticipated in literature by Jorge Luis Borges in the story "The Garden of Forking Paths" (1941)
“El jardín de senderos que se bifurcan”
“Unlike Newton and Schopenhauer, his ancestor did not believe in a uniform, absolute time. He believed in infinite series of times, in a growing and dizzying network of divergent, convergent and parallel times….” Jorge Luis Borges 1899 - 1966 Einstein: I am convinced that God does not play dice. Do you think the moon is not there when we are not looking at it?
Bohr: Don't tell God what to do
Einstein and Bohr 1925 Einstein, Podolsky and Rosen paradox (1935) New York Times in May of 1935 EPR : Gedanken experiment EPR : Gedanken experiment
Superposition of two electron with opposite momentum Spooky action at a distance
We leave open the possibility that it exists, or not, end of the a complete description of reality. EPR article We believe that theory exists. Spooky action at a distance
We leave open the possibility that it exists, or not, End of the a complete description of reality. EPR article We believe that theory exists.
Hidden variable theories Schrödinger called entanglement this instantaneous action at a distance (1935)
Verschränkung (german)
Entanglement is not one, but the characteristic feature of Quantum Mechanics, which imposes its total departure from classical Physics The EPR paradox spurred the construction of hidden variable theories
Theory of “pilot waves” of de Broglie and Bohm
David Bohm 1917-1992 The EPR paradox spurred the construction of hidden variable theories
Theory of “pilot waves” of de Broglie and Bohm
David Bohm 1917-1992
Von Neumann theorem:
Hidden variable theories are incompatible with Quantum Mechanics
John von Neumann 1903-1957 Two interpretations of the physical reality
Local Realism (EPR-hidden variables)
The observable quantities have a value prior to their measurement
Quantum Mechanics
The observable quantities DO NOT have a value prior to their measurement After this discussion a DICTUM was imposed
“Shut up and calculate” After this discussion a DICTUM was imposed
“Shut up and calculate”
but one should John Bell found that von Neumann's proof was wrong: he assumed what he wanted to prove
EPR ideas were not a mere philosopical speculation about the interpretation of Quantum Mechanics
John Bell 1928-1990 John Bell found that von Neumann's proof was wrong: he assumed what he wanted to prove
EPR ideas were not a mere philosopical speculation about the interpretation of Quantum Mechanics
It would be possible to falsify "local realism" with an experiment
John Bell 1928-1990 Bell inequalities (1964) Bell experiment
EPR pair of spins + Bell experiment
EPR pair of spins +
푎Ԧ Bell experiment
EPR pair of spins +
푎Ԧ
Local realism |푃(푎Ԧ, 푏) - 푃(푎Ԧ, 푏′)| ≤ 1 + 푃(푏, 푏’) Bell experiment
EPR pair of spins +
푎Ԧ
Local realism |푃(푎Ԧ, 푏) - 푃(푎Ԧ, 푏′)| ≤ 1 + 푃(푏, 푏’)
Quantum Mechanics 푃 푎Ԧ, 푏 = −푎Ԧ ∙ 푏 can violate this inequality Experiments to verify Bell’s inequality
- electrons -> photons
- spin -> polarization
entangled photons
Monitor of coincidences Aspect’s experiments (1981)
CHSH inequality
Quantum prediction
Experimental result Alan Aspect 1947
LOCAL REALISM Bell experiment in Austria, Innsbruk (1998) Bell experiment in Canary islands (2007) QESS (Quantum Entanglement at Space Scale) 2016-2018
1200 Kms
Joint Proyect China and Austria Quantum Mechanics Computer Sciences
Quantum Computation and Quantum Information Computer Sciences Turing machine
Alan Turing (1914-1944)
Program : Alonzo Church (1903-1995) Alan Turing (1914-1944)
The Chuch-Turing thesis (1936)
Any algorithmic process can be simulated efficiently using a Turing machine Turing, computers and crytography
Enigma machine Museo scienza e tecnologia Milano,
Electronic computer at the Manchester university 1950 The architect of the computers
John von Neumann 1903-1957 The first transistor (1947)
John Bardeen Walter Brattain 1908-1991 1902-1987 William Schockely 1902-1987
1982
Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's Richard Feynman a wonderful problem, because it doesn't look so easy. 1918-1988 1985
David Deutsch 1953
Proposed a quantum generalization of the Turing machines
= Quantum Computer Computational time Prime factorization
N: integer, n = log N: number of digits Peter Shor 1959 1/3 2/3 Best classical algorithm: 푂 푒1.9 푛 log 푛
Exponential speedup 2 Shor’s algorithm: 푂 푛 log 푛 log log 푛 A fast quantum mechanical algorithm for database search
arXiv:quant-ph/9605043
Lov Grover 1961 Searching an item in a list of N data takes classicaly order 푁 steps
In a quantum computer it takes order 푁
quadratic speedup 푁 → 푁 Quantum Mechanics Computer Sciences
Quantum Computation and Quantum Information
Information Theory Information theory 1948
Claude Shannon 1916-2001
Capacity of a communication channel Measure of information: Entropy
Noiseless channel coding theorem -> optimal resources to store information
Noisy channel coding theorem -> amount of information that can be reliable transmitted
Error correcting codes - > to protect information transmitted from the noise The Minivac 601, a digital computer trainer 1962 designed by Shannon. 1995
Quantum version of Shannon theory Benjamin Schumacher Shannon entropy -> von Neumann entropy
Noiseless coding theorem -> quantum noiseless coding theorem
Introduced the terminology “quantum bit = qubit” 1995
Quantum version of Shannon theory Benjamin Schumacher Shannon entropy -> von Neumann entropy
Noiseless coding theorem -> quantum noiseless coding theorem
Introduced the terminology “quantum bit = qubit” Information is physical
Physics Today 44, 5, 23 (1991)
Rolf Landauer 1927-1999 Information is physical
Physics Today 44, 5, 23 (1991)
Rolf Landauer 1927-1999
Landauer principle
퐸 = 푘퐵푇 ln 2 Cryptography
See talk “Introduction to Quantum Communication” by Vicente Martin (UPM), Thursday 2nd Sept 1984
Charles Bennet 1943 Gilles Brassard 1955 The BBVA prize Frontiers of Knowledge Award in Basic Sciences to Charles Bennett, Gilles Brassard, and Peter Shor in 2019 for their respective roles in the development of quantum computing and cryptography The 19th century was the era of steam power, the 20th century was the era of information, and the 21st century will go down in history as the quantum age, the age in which quantum technologies dominate all the changes occurring in society, in a way we cannot yet foresee.”
G. Brassard, 2019 Principles of Quantum Mechanics in a nutshell The qubit qubit = a quantum state of a two level system
Example of a qubit : spin ½ particle
ۧ↓| = 1| ۧ↑| = 0ۧ|
computational basis |0ۧ , |1ۧ
1 0 = vector notation |0ۧ = , |1ۧ 0 1
0 0 = 1 1 = 1, 0 1 = 0 Superposition principle
휓ۧ = 푎 |0ۧ + 푏 |1ۧ, 푎, 푏 ∈ ℂ|
휓 휓 = 1 → |푎|2 + |푏|2 = 1 푒푖훼|휓ۧ , 훼 휖 ℝ same state
Standard parametrization 휃 푐표푠 휃 푖 휙 휃 2 휓ۧ = cos |0ۧ + 푒 sin |1ۧ = 휃| 2 2 푒푖휙 푠푖푛 2 휃 ∈ 0, 휋 휙 ∈ [0, 2휋) Geometric representation: Bloch sphere
휃 : polar angle 휙 : azimuthal angle
푒푖휙|1ۧ + 0ۧ| 2 Pure state of a qubit Point on the Bloch sphere
How to compute the angles θ, ϕ ?
Prepare N times an unknown state in the Lab
Stern-Gerlach” detector |0ۧ 푁0“ 푧 ? ? 휓ۧ| 푧 ? ? 휓ۧ| 푁1 1ۧ|
푁 = 푁0 + 푁1 푁0 = Probability of measuring |0ۧ 푝0 푁0 + 푁1
푁1 = Probability of measuring |1ۧ 푝1 푁0 + 푁1
푝0+ 푝1 = 1
Quantum Mechanical prediction
휃 휃 푝 = | 0 휓 |2 = 푐표푠2 푝 = | 1 휓 |2 = 푠푖푛2 0 2 1 2
Error ∝ 1/ 푁 푧 휃 ? ? 휓ۧ| Stern-Gerlach detector Observable 푧 1 0 휎푧 = 푍 = 0 −1
Expectation value
푧 휎 = 푝0 − 푝1
휃 휃 휓|휎푧|휓ۧ = 푐표푠2 − 푠푖푛2 = cos 휃ۦ = QM 휎푧 2 2 What about ϕ ? 1ۧ| + 0ۧ| = ۧ+| Stern-Gerlach detector 2 ? ? +휓ۧ 푁|
? ? 푁− 휓ۧ| 1ۧ| − 0ۧ| = ۧ−| 푁 = 푁 + 푁 + − 2
Stern-Gerlach detector Observable
0 1 휎푥 = 푋 = 1 0 푁+ = +Probability of measuring |+ۧ 푝 푁+ + 푁−
푁− = −Probability of measuring |−ۧ 푝 푁+ + 푁−
Quantum Mechanical prediction
1 + sin 휃 cos 휙 2 1 − sin 휃 cos 휙 푝 = | + 휓 |2 = 푝− = | − 휓 | = + 2 2
푥 휎 = 푝+ − 푝−
휓|휎푥|휓ۧ = 푠푖푛 휃 cos 휙ۦ = QM 휎푥 푧 휃 푥
푧 휃 푥
The state is still not totally fixed 푖|1ۧ + 0ۧ| = 푖ۧ+| Stern-Gerlach detector 2 ? ? 휓ۧ 푁+푖|
? ? 푁−푖 휓ۧ| 푖|1ۧ − 0ۧ| = 푁 = 푁 + 푁 i |−푖ۧ +푖 − 2
Stern-Gerlach detector Observable
0 −푖 휎푦 = 푌 = 푖 0 푁+푖 = Probability of measuring |+푖ۧ 푝+푖 푁+푖 + 푁−푖
푁−푖 = Probability of measuring |−ۧ 푝−푖 푁+푖 + 푁−푖
Quantum Mechanical prediction
1 + sin 휃 sin 휙 2 1 − sin 휃 sin 휙 푝 = | +푖 휓 |2 = 푝−푖 = | −푖 휓 | = +푖 2 2
푦 휎 = 푝+푖 − 푝−푖
휓|휎푦|휓ۧ = 푠푖푛 휃 sin 휙ۦ = QM 휎푦 푧 y 휃
휓ۧ| y 푧 휃 푥
The state is now totally fixed
푥 휎푥 = cos 휙 sin 휃 = 푥 1 + 푧 2 ۧ 푦 휎 = sin 휙 sin 휃 = 푦 |휓 = 푥 + 푖 푦 휎푧 = cos 휃 = 푧 2(1 + 푧)
0ۧ| ۧ−|
푖ۧ |+푖ۧ−|
ۧ+| 1ۧ| Density matrix Classical Mechanics
A “pure” state of a 1D particle is given by the point in the phase space (q,p)
A “mixed” state is given by a probability density 휌 푞, 푝
Expectation values of an observable 풪 푞 , 푝 is given by 풪 = න 푑푞 푑푝 휌 푞, 푝 풪 푞, 푝 Density matrix Classical Mechanics
A “pure” state of a 1D particle is given by the point in the phase space (q,p)
A “mixed” state is given by a probability density 휌 푞, 푝
Expectation values of an observable 풪 푞 , 푝 is given by 풪 = න 푑푞 푑푝 휌 푞, 푝 풪 푞, 푝
Quantum Mechanics
휃 휃 휃 푐표푠2 푒−푖휙 cos sin 휓 = 2 2 2ۦ 휌 = 휓ۧ 휃 휃 휃 푒푖휙cos sin 푠푖푛2 2 2 2
푡푟 휌 = 1 푂 = 푇푟 휌 푂 휌† = 휌 휌 > 0 Mixed state
Ensemble of N pure states |휓푖ۧ with probabilities 푝푖
푁 푁
휓푖| 1 = 푝푖ۦ 휌 = 푝푖 |휓푖ۧ 푖=1 푖=1 General state
1 1 + 푧 푥 − 푖 푦 1 휌 = = 1 + 푥 휎푥 + 푦 휎푦 + 푧 휎푧 2 푥 + 푖 푦 1 − 푧 2
1 + 푟 1 − 푟 Eigenvalues , 푟 = 푥2 + 푦2 + 푧2 2 2
휌 > 0 → 0 ≤ 푟 ≤ 1 휎푥 = 푇푟 휌휎푥 = 푥, 휎푦 = 푇푟 휌휎푦 = 푦, 휎푧 = 푇푟 휌휎푧 = 푧
휓ۦ Pure states: 푟 = 1 휌 = 휓ۧ 휓ۦ Mixed states: 푟 < 1 휌 ≠ 휓ۧ Ι Maximally mixed state: 푟 = 0 휌 = 2
Bloch ball Single qubit gates
ۧ′휓ۧ |휓| U
ۧ′U |휓ۧ = |휓
휓′ 휓′ = ർ휓| 푈† 푈 |휓ۧ = 휓 휓
푈†푈 = 핀
푈 unitary matrix Pauli gates
0 1 0 −푖 1 0 푋 = 푌 = 푍 = 1 0 푖 0 0 −1
Hadamard gate S gate T gate
1 1 1 1 0 1 0 퐻 = 푆 = 푇 = 2 1 −1 0 푖 0 푒푖휋/4
1ۧ| + 0ۧ| ۧ+| = = 퐻 |0ۧ 2 1ۧ| − 0ۧ| ۧ−| = = 퐻 |1ۧ 2 Hadamard’s magic
푋 = 퐻 푍 퐻 푌 = 푆 퐻 푍 퐻 푆†
휓 퐻 푍 퐻 휓ۧ = 푍 퐻휓ۦ = 휓 푋 휓ۧۦ = 푋 휓
† 휓 푆 퐻 푍 퐻 푆 휓ۧ = 푍 퐻푆†휓ۦ = 휓 푌 휓ۧۦ = 푌 휓
푧 퐻 휓 휓
푧 푆† 퐻 휓 휓
IBM tomography General qubit gate Qubit = spin ½ representation of the rotation group SU(2)
− 푖 휃푛∙휎/2 z 푛 푅푛 휃 = 푒 휃
y 푖훼 푥 푈 = 푒 푅푛 휃
푖훼 Euler’s decomposition 푈 = 푒 푅푧 훽 푅푦 훾 푅푧 훿
− 푖훽/2 cos 훾/2 −sin 훾/2 − 푖훿/2 푈 = 푒푖훼 푒 0 푒 0 0 푒푖훽/2 sin 훾/2 cos 훾/2 0 푒푖훿/2
gimbal set Multiqubit states 1ۧ| 0ۧ| qubit |휓ۧ = 휓0 + 휓1 1
1ۧ| 0ۧ| 0ۧ| 0ۧ| qubits |휓ۧ = 휓00 + 휓01 2
1ۧ| 1ۧ| 0ۧ| 1ۧ|
+ 휓10 + 휓11
1ۧ| 0ۧ| 0ۧ| 0ۧ| 0ۧ| 0ۧ| qubits |휓ۧ = 휓000 + 휓001 3
1ۧ| 1ۧ| 0ۧ| 0ۧ| 1ۧ| 0ۧ|
+ 휓010 + 휓011
+ 4 terms State vectors
휓0 = qubit |휓ۧ 1 휓1
휓00 휓01 = qubits |휓ۧ 2 휓10 휓11
휓000 휓001 휓010 휓011 = qubits |휓ۧ 3 휓100 휓101 휓110 휓111 n qubits |휓ۧ = vector with 2푛 components Entangled states
1ۧ| 0ۧ| 0ۧ| 0ۧ| 휓ۧ = 휓00 + 휓01| 1ۧ| 1ۧ| 0ۧ| 1ۧ| + 휓10 + 휓11
1ۧ| 0ۧ| 1ۧ| 0ۧ|
휒0 + 휒1 휁0 + 휁1 EPR state : Gedanken experiment
Linear superposition of two electrons with opposite momentum How to create entanglement? How to create entanglement?
Need a 2 qubit gate CNOT = Controlled NOT How to create entanglement?
Need a 2 qubit gate CNOT = Controlled NOT control target qbit qbit
00ۧ| → 00ۧ| 01ۧ| → 01ۧ| 11ۧ| → 10ۧ| 10ۧ| → 11ۧ| How to create entanglement?
Need a 2 qubit gate CNOT = Controlled NOT
control target qbit qbit
00ۧ| → 00ۧ| 01ۧ| → 01ۧ| 11ۧ| → 10ۧ| 10ۧ| → 11ۧ|
푐, 푡ۧ → |푐, 푐 ⊕ 푡ۧ|
푐 ⊕ 푡 = 푐 + 푡 푚표푑 2 How to create entanglement?
Need a 2 qubit gate CNOT = Controlled NOT
control target qbit qbit 1 0 0 0 퐶푁푂푇 = 0 1 0 0 1 0 0 0 00ۧ| → 00ۧ| 0 1 0 0 01ۧ| → 01ۧ| 11ۧ| → 10ۧ| 10ۧ| → 11ۧ|
푐, 푡ۧ → |푐, 푐 ⊕ 푡ۧ|
푐 ⊕ 푡 = 푐 + 푡 푚표푑 2 How to create entanglement?
Need a 2 qubit gate CNOT = Controlled NOT
control target qbit qbit 1 0 0 0 퐶푁푂푇 = 0 1 0 0 1 0 0 0 00ۧ| → 00ۧ| 0 1 0 0 01ۧ| → 01ۧ| 11ۧ| → 10ۧ| 10ۧ| → 11ۧ| 푐ۧ |푐ۧ| 푐, 푡ۧ → |푐, 푐 ⊕ 푡ۧ| 푡ۧ |푐 ⊕ 푡ۧ| 푐 ⊕ 푡 = 푐 + 푡 푚표푑 2 Creation of entanglement
퐻 0ۧ| 11ۧ| + 00ۧ| = ۧ+휙| 2 0ۧ|
0ۧ|0ۧ|
1ۧ| + 0ۧ| 0ۧ| 2 Creation of entanglement
퐻 0ۧ| 11ۧ| + 00ۧ| = ۧ+휙| 2 0ۧ|
0ۧ|0ۧ|
1ۧ| + 0ۧ| 0ۧ| 2
IBMQ circuit Bell basis for 2 qubit states
11ۧ| + 00ۧ| = ۧ+휙| 2
11ۧ| − 00ۧ| = ۧ−휙| 2
10ۧ| + 01ۧ| = ۧ+휓| 2
10ۧ| − 01ۧ| = ۧ−휓| 2
The qubits are maximally entangle for each of these states
Used in the teleportation protocol Creation of entanglement
퐻 0ۧ| 11ۧ| + 00ۧ| = ۧ+휙| 2 0ۧ|
1ۧ| + 0ۧ| 0ۧ| 2
퐻 |0ۧ 퐻 |1ۧ 퐻 1ۧ|
ۧ−휙−ۧ |휓+ۧ |휓| 1ۧ| 1ۧ| 0ۧ|
1ۧ| − 0ۧ| 1ۧ| + 0ۧ| 1ۧ| − 0ۧ| 0ۧ| 0ۧ| 0ۧ| 2 2 2
Computational basis -> Bell basis Toffoli gate = CCNOT
푐1ۧ |푐1ۧ|
푐2ۧ |푐2ۧ|
푡ۧ |푐1푐2 ⊕ 푡ۧ| Universal classical gates
A finite set of gates that can be used to compute any function Non reversible: AND, OR, NOT
Reversible: + Toffoli
Universal quantum gates
A finite set of quantum gates that can approximate any unitary operation to arbitrary precision
퐻, 푆, 푇, 퐶푁푂푇 퐻, 푆, 퐶푁푂푇, 푇푂퐹퐹푂퐿퐼 Deutsch’s algorithm (1985)
The problem: given a boolean function 푓 푥 determine if it is constant or balanced
푥 ∈ 0,1 , 푓 푥 ∈ 0,1
There are 4 functions of this type
푥 푓0 푓1 푓2 푓3 0 0 0 1 1 1 0 1 0 1
Constant function: 푓 0 = 푓(1) Balanced function: 푓 0 ≠ 푓(1)
푓0 , 푓3 푓1 , 푓2 Given an unknown function to determine its character we have to compute
푓 0 and 푓 1
This is refereed to as “calling TWICE an oracle”
Question: Can one call only ONCE the oracle ?
Answer: YES using quantum parallelism and interference Quantum oracle
푥ۧ |푥ۧ| 푈푓 ۧ(푦ۧ |푦 ⊕ 푓(푥| Quantum oracle
푥ۧ |푥ۧ| 푈푓 ۧ(푦ۧ |푦 ⊕ 푓(푥|
Step 1: call to the oracle
푥ۧ |푥ۧ| 푈푓 푓(푥) |0ۧ − |1ۧ(−1) 1ۧ| − 0ۧ| Quantum oracle
푥ۧ |푥ۧ| 푈푓 ۧ(푦ۧ |푦 ⊕ 푓(푥|
Step 1: call to the oracle
푥ۧ |푥ۧ| 푈푓 푓(푥) |0ۧ − |1ۧ(−1) 1ۧ| − 0ۧ|
Step 2: interference of the answers
푓(0)|0ۧ + (−1)푓(1)|1ۧ(−1) 1ۧ| + 0ۧ| 푈푓 1ۧ| − 0ۧ| 1ۧ| − 0ۧ| Step 3: quantum data mining 푓(0)+(−1)푓(1))|0ۧ(−1)) + 푓(0)|0ۧ + (−1)푓(1)|1ۧ 퐻(−1) 푓(0)−(−1)푓(1))|1ۧ(−1)) Step 3: quantum data mining 푓(0)+(−1)푓(1))|0ۧ(−1)) + 푓(0)|0ۧ + (−1)푓(1)|1ۧ 퐻(−1) 푓(0)−(−1)푓(1))|1ۧ(−1)) Step 4: getting the answer
If 푓 0 = 푓(1) |0ۧ
If 푓 0 ≠ 푓(1) |1ۧ Step 3: quantum data mining 푓(0)+(−1)푓(1))|0ۧ(−1)) + 푓(0)|0ۧ + (−1)푓(1)|1ۧ 퐻(−1) 푓(0)−(−1)푓(1))|1ۧ(−1)) Step 4: getting the answer
If 푓 0 = 푓(1) |0ۧ
If 푓 0 ≠ 푓(1) |1ۧ
Deustch circuit
퐻 퐻 0ۧ| 푈푓 퐻 1ۧ| Step 3: quantum data mining 푓(0)+(−1)푓(1))|0ۧ(−1)) + 푓(0)|0ۧ + (−1)푓(1)|1ۧ 퐻(−1) 푓(0)−(−1)푓(1))|1ۧ(−1)) Step 4: getting the answer
If 푓 0 = 푓(1) |0ۧ
If 푓 0 ≠ 푓(1) |1ۧ
Deustch circuit
퐻 퐻 0ۧ| 푈푓 퐻 1ۧ|
One does not know which constant or balanced is Step 3: quantum data mining 푓(0)+(−1)푓(1))|0ۧ(−1)) + 푓(0)|0ۧ + (−1)푓(1)|1ۧ 퐻(−1) 푓(0)−(−1)푓(1))|1ۧ(−1)) Step 4: getting the answer
If 푓 0 = 푓(1) |0ۧ
If 푓 0 ≠ 푓(1) |1ۧ
Deustch circuit
퐻 퐻 0ۧ| 푈푓 퐻 1ۧ|
One does not know which You can't always get what you want constant or balanced is But if you try sometime you find You get what you need Rolling Stones References Thanks so much for your attention
Muchas gracias por su atención