The simplest explanation here is that the beam splitter has a 50% chance to transmit or reflect each photon. Ugo Jardonnet. Introduction to Quantum Computing.
Suppose we have an experimental set-up consisting of a photon source, a beam splitter (half silvered mirror) and a pair of photon detectors. We observe that photons hit each detector 50% of the time. The simplest explanation here is that the beam splitter has a 50% chance to transmit or reflect each photon.
Ugo Jardonnet
Introduction to Quantum Computing
4 / 20
A Simple Experiment
A Simple Experiment
We modify the experiment by adding a second beam splitter and two fully reflecting mirrors. In this modified set-up the result is non-intuitive. The photons do not hit each detector with a 50% chance. The photons arrive at the same detector, detector 2, 100% of the time.
Ugo Jardonnet
Introduction to Quantum Computing
5 / 20
A Simple Experiment
A Simple Experiment The mathematical framework of quantum physics models the experiment in a way that correctly predicts the observed outcomes. 1 i 1 The beam splitter can be model by the following operator: √2 i 1 1 0 At each step, photons can be in 2 possible paths noted and . 0 1 1
2
3
At the beginning of the experiment we know from which side photons are hitting the first beam 1 splitter. Meaning the state of the system is known to be path 1 = . 0 1 i 1 1 1 0 After the first beam splitter we get: √12 = √12 = √12 + √i 2 . i 1 0 i 0 1 2 2 We will see later that this mean a √12 = √i 2 = 12 probability for each outgoing path. 1 i √1 1 0 = , meaning a 100% probability of After the second beam splitter we get: √12 i 1 2 i 1 observing photons in path 2. Ugo Jardonnet
Introduction to Quantum Computing
6 / 20
A Simple Experiment
A Simple Experiment [1]
Photons are modeled as taking a superposition of both paths and physical ”gates” as operators modifying the probabilities of measuring photons in each of their possible states. Quantum physics can model experiment that cannot be described by classical physics. The exponential processing power needed to model some complex quantum system led to the idea of quantum computing.
Ugo Jardonnet
Introduction to Quantum Computing
7 / 20
Mathematical Framework
Outline
1
A Simple Experiment
2
Mathematical Framework Hilbert space and Dirac notation Operators in H
The state of a n-qbit system is described by a 2n -dimensional vector in a finite dimensional complex vector space, referred as a Hilbert space H. Vectors from H basis are called state vectors. They represent the possible states in which qbits can collapse when measured. Dirac Notation Vectors in H are written inside a ’ket’ as follow |v i The 2n basis vectors of H are labeled by their binary index: |00i, |01i, ...
Ugo Jardonnet
Introduction to Quantum Computing
9 / 20
Mathematical Framework
Hilbert space and Dirac notation
Dirac Notation Single-qbit System Example ~= Each possible state of a single-qbit system can be written as a vector φ
α1 , with α1 , α2 complex. α2
Following Dirac notation we get: |φi = α1 |0i + α2 |1i The pure state space of a qubit can be represented by a bloch sphere.
Ugo Jardonnet
Introduction to Quantum Computing
10 / 20
Mathematical Framework
Hilbert space and Dirac notation
Dirac Notation
4-qbit System Example - H of dimension 4 0 1 ≡ 2nd basis vector 0 0
≡ 0b01th basis vector, binary 0-indexed ≡ |01i in Dirac notation the 4 basis vectors are therefore B = {|00i, |01i, |10i, |11i}. Any vectors in H can then be written as a combination of vectors in B.
Ugo Jardonnet
Introduction to Quantum Computing
11 / 20
Mathematical Framework
Hilbert space and Dirac notation
Dirac Notation
The following Dirac notation and column vector are equivalent: r
i 2 |001i + p |111i ⇐⇒ 3 (3)
q0
2 3
0 .. . 0 √i
(1)
(3)
As you can see, this notation saves space when the dimension is high and the vectors are sparse.
Ugo Jardonnet
Introduction to Quantum Computing
12 / 20
Mathematical Framework
Operators in H
Dual vector and inner product
Dual vector hx| - x written inside a ’brac’ A dual vector is obtained by taking the corresponding row matrix of x and then the complex conjugate of every element (Hermitean conjugate). This allows writing the inner product as hx|y i: x1 y1 y ∗ ∗ hx||y i ≡ hx|y i = · = (x1 x2 ) 1 = Σ2i xi∗ yi x2 y2 y2 where c ∗ = a − bi for c = a + bi
Ugo Jardonnet
Introduction to Quantum Computing
13 / 20
Mathematical Framework
Operators in H
Tensor product Tensor product |xi ⊗ |y i also written |xi|y i or |xy i Stitch result matrices of multiplying each element xi in x by y : x0 y0 x0 y1 x y |xy i = |xi ⊗ |y i = 0 ⊗ 0 = x1 y0 x1 y1 x1 y1 Example: r
Ugo Jardonnet
2 i |01i + p |11i ⇐⇒ 3 (3)
r
2 i |0i ⊗ |1i + p |1i ⊗ |1i 3 (3)
Introduction to Quantum Computing
14 / 20
Mathematical Framework
Operators in H
Outer product and Quantum entanglement Outer product: |xihy | . Produces an operator that acts as (|xihy |)|zi = |xi(hy |zi) = (hy |zi)|xi A key point of quantum computing is that n-qbits systems can be described by the tensor product of n vector spaces V (H of dimension 1) i.e. Hn = {v1 ⊗ . . . ⊗ vn , ∀vi ∈ V, ∀i ∈ n} quantum Entanglement The quantum state of each particle cannot be described independently of the others.
Ugo Jardonnet
Introduction to Quantum Computing
15 / 20
Mathematical Framework
Operators in H
Measurement
Measurement Principle A measurement results in the system being in the eigenstate corresponding to the eigenvalue result of the measurement The measurement principle is an axiom of quantum physics, you cannot prove it is true. It just seems to match what is happening. |ψi = α|0i + β|1i. The measurement process alters the state of the qubit: the effect of the measurement is that the new state is exactly the outcome of the measurement.
Ugo Jardonnet
Introduction to Quantum Computing
16 / 20
Quantum Algorithm
Outline
1
A Simple Experiment
2
Mathematical Framework Hilbert space and Dirac notation Operators in H
Quantum algorithm for integer factorization formulated by Peter Shor in 1994.
The first part of the algorithm turns the factoring problem into the problem of finding the period of a function, and may be implemented classically. The second part finds the period using the quantum Fourier transform, and is responsible for the quantum speedup.
Ugo Jardonnet
Introduction to Quantum Computing
18 / 20
Quantum Algorithm
Groover’s algorithm
Groover’s Algorithm
Search algorithm in sub-linear time.
Ugo Jardonnet
Introduction to Quantum Computing
19 / 20
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