Elements of Quantum Computation Quantum Physics and Concepts - - PDF document

Elements of Quantum Computation

Quantum Physics and Concepts Herbert Wiklicky herbert@doc.ic.ac.uk BISS Spring School Bertinoro 2018

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Overview

Topics we will cover in this course will include:

• 1. Quantum Computation

◮ Basic Quantum Physics ◮ Mathematical Structure ◮ Quantum Circuit Model ◮ Quantum Cryptography

• 2. Quantum Algorithms

◮ Deutsch Problem ◮ Quantum Teleportation ◮ Gover’s Search Algorithm ◮ Shor’s Quantum Factorisation

• 3. Further Topics

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Text Books

◮ Noson S. Yanofsky, Mirco A. Mannucci: Quantum

Computing for Computer Scientists, Cambridge, 2008

◮ Michael A. Nielsen, Issac L. Chuang: Quantum

Computation and Quantum Information, Cambridge, 2000

◮ Phillip Kaye, Raymond Laflamme, Michael Mosca: An

Introduction to Quantum Computing, Oxford 2007

◮ N. David Mermin: Quantum Computer Science,

Cambridge University Press, 2007

◮ A. Yu. Kitaev, A. H. Shen, M. N. Vyalyi: Classical and

Quantum Computation, AMS, 2002

◮ Eleanor Rieffel, Wolfgang Polak: Quantum Computing, A

Gentle Introduction. MIT Press, 2014

◮ Richard J. Lipton, Kenneth W. Regan: Quantum Algorithms

via Linear Algebra. MIT Press, 2014

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Electronic Resources

Introductory Texts

◮ E.Rieffel, W.Polak: An introduction to quantum computing

for non-physicists. ACM Computing Surveys, 2000 doi:10.1145/367701.367709

◮ N.S.Yanofsky: An Introduction to Quantum Computing

http://arxiv.org/abs/0708.0261 Preprint Repository http://arxiv.org Physics Background

◮ Chris J. Isham: Quantum Theory – Mathematical and

Structural Foundations, Imperial College Press 1995

◮ Richard P

. Feynman, Robert B. Leighton, Matthew Sands: The Feynman Lectures on Physics, Addison-Wesley 1965

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Quantum Money (Stephen Wiesner 1960s)

Quantum Postulates: (i) It is impossible to clone a quantum states, (ii) in general, an inspection of a quantum state is irreversible and destructive. Bank of Quantum issue bank notes with a unique quantum code. Quantum Forger tries to make a copy of quantum money, however

◮ she can’t copy/clone a banknote directly, and ◮ when she inspects it, she destroys the code.

Bank of Quantum can inspect the quantum code on a banknote

◮ to confirm it is authentic, and then ◮ issue a replacement quantum banknote.

Simon Singh: Code Book, Forth Estate, 1999.

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Quantum Physics

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Quantum History

Quantum Mechanics was ‘born’ or proposed by M.Plank on 14 December 1900, 5:15pm (Berlin) 1900 Max Plank: Black Body Radiation 1905 Albert Einstein: Photoelectric Effect 1925 Werner Heisenberg: Matrix Mechanics 1926 Erwin Schrödinger: Wave Mechanics 1932 John von Neumann: Quantum Mechanics Manjit Kumar: Quantum – Einstein, Bohr and Their Great Debate about the Nature of Reality, Icon Books 2009

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Photoelectric Effect – Millikan Experiment

Experimental Setup: ν Observed: The velocity, and thus kinetic energy, of the emitted electrons depends not on the intensity of the incoming light but

• nly on its “colour”, i.e. frequency ν.

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Radiation Law

Observed relationship: Wk = hν − We Wk . . . Kinetic Energy of Electron We . . . Escape Energy of Material ν . . . Frequency of Light h . . . Plank’s Constant h = 6.62559 · 10−34Js

• =

h 2π = 1.05449 · 10−34Js

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Quantum Physical Problems

Around 1900 there were a number of experiments and

• bservations which could not be explained using classical

physics/mechanics, among them: Spectra of Elements Emission/absorption only at particular “colours”. Stern-Gerlach Experiment Interference in double slit experiment. Black Body Radiation Radiation law involves “quantised” energy levels. Photo-Electric Effect Einstein’s explanation got him the Nobel prize. These were the perhaps most exciting years in the history of theoretical physics, at the same time there were also breakthroughs in special and general relativity, etc.

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Einstein’s Explanation

Albert Einstein 1905: Not all energy levels are possible, they

• nly come in quantised portions. In Bohr’s (incomplete) “model”
• f the atom this corresponds to allowing only particular “orbits”.

In this way one can also explain the spectral emissions (and absorption) of various elements, e.g. to analyse the material composition of stars (and to make great fireworks).

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Quantum Paradoxes and Myths

There are a number of physical problems which require quantum mechanical explanations. Unfortunately, QM is not ‘really intuitive’. This leads to various Gedanken experiments which point to a contradiction with so-called common sense.

◮ Black Body Radiation ◮ Double Slit Experiment ◮ Spectral Emissions ◮ Schrödinger’s Cat ◮ Einstein-Podolsky-Rosen ◮ Quantum Teleportation

• 7. Whereof one cannot speak, thereof one must be silent.

Ludwig Wittgenstein: Tractatus Logico-Philosophicus, 1921

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From Quantum Physics to Computation

There are a number of disciplines which play an important role in trying to understand quantum mechanics and in particular quantum computation. Philosophy: What is the nature and meaning reality? Logic: How can one reason about events, objects etc.? Mathematics: How does the formal model look like? Physics: Why does it work and what does it imply? Computation: What can be computed and how? Engineering: How can it all be implemented? Each area has its own language which however often applies

• nly to classical entities – for the quantum world we often have

simply the wrong vocabulary.

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Natural Philosophy

Arguably, physics is ultimately about explaining experiments and forecasting measurement results. Observable: Entities which are (actually) measured when an experiment is conducted on a system. State: Entities which completely describe (or model) the system we are interested in. Measurement brings together/establishes a relation between states and observables of a given system. Dynamics describes how observables and/or the state changes over time. Related Questions: What is our knowledge of what? How do we obtain this information? What is a description on how the system changes?

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Harmonic Oscillator or just a “Shadow”

One can observe the same “behaviour” of the shadow of a rotating object or an object on a spring. x y φ m Observable: Shadow m State: Position (x, y) or: Phase φ Measurement: m((x, y)) = y, or: m(φ) = sin(φ) Dynamics: (x, y)(t) = (cos(t), sin(t)) or also: φ(t) = t

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Postulates for Quantum Mechanics (C*-algebra)

◮ Observables and states of a system are represented by

hermitian (i.e. self-adjoint) elements a of a C*-algebra A and by states w (i.e. normalised linear functionals) over this algebra.

◮ Possible results of measurements of an observable a are

given by the spectrum Sp(a) of an observable. Their probability distribution in a certain state w is given by the probability measure µ(w) induced by the state w on Sp(a). Walter Thirring: Quantum Mathematical Physics, Springer 2002 Key Notions: A quantum systems is (may be) in a certain state, but physicists have to decide which properties they want to

• bserve before a measurement is made (which instrument?).

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Postulates for Quantum Mechanics (ca. 1950)

◮ The quantum state of a (free) particle is described by a

(normalised) complex valued [wave] function:

• ψ ∈ L2 i.e.
• |

ψ(x)|2dx = 1

◮ Two quantum states can be superimposed, i.e.

ψ = α1 ψ1 + α2 ψ2 with |α1|2 + |α2|2 = 1

◮ Any observable A is represented by a linear, self-adjoint

• perator A on L2.

◮ Possible measurement results are (only) the eigen-values

λi of A corresponding to eigen-vectors/states φi ∈ L2 with A φi = λi φi

◮ Probability to measure (the possible eigenvalue) λn if the

system is in the state ψ =

i ψi

φi is Pr(A = λn | ψ) = |ψn|2

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Mathematical Framework

Quantum mechanics has a well-established and precise mathematical formulation (though its ‘common sense’ interpretation might be non-intuitive, probabilistic, etc.). The (standard) mathematical model of quantum system uses:

◮ Complex Numbers C, ◮ Vector Spaces, e.g. Cn, ◮ Hilbert Spaces, i.e. inner products .|., ◮ Unitary and Self-Adjoint Matrices/Operators, ◮ Tensor Products C2 ⊗ C2 ⊗ . . . ⊗ C2.

There are additional mathematical details in order to deal with “real” quantum physics, e.g. systems an infinite degree of freedom; for quantum computation it is however enough to study finite-dimensional Hilbert spaces.

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Quantum Postulates I – States and Observables

The standard mathematical model of (closed) quantum systems is relatively simple and just requires some basic notions in (complex) linear algebra.

◮ The information describing the state of an (isolated)

quantum mechanical system is represented mathematically by a (normalised) vector in a complex Hilbert space H.

◮ An observable is represented mathematically by a self-

adjoint matrix (operator) A acting on the Hilbert space H. Two states can be combined to form a new state α |x + β |y as long as |α|2 + |β|2 = 1, by superposition. Consequence: We can compute with many inputs in parallel.

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Quantum States and Notation

The state of a quantum mechanical system is usually denoted by |x ∈ H (rather than maybe x ∈ H). This notation is ‘inherited’ from the inner product x|y of vectors x and y in a Hilbert space – which can be seen as describing the “geometric angle” between the two vectors in H. P.A.M. Dirac “invented” the bra-ket notation (most likely inspired by the limitations of old mechanical type-writers); Simply “take the inner product apart” to denote vectors in H: inner product x|y = product x| · |y For indexed sets of vectors {xi} (maybe because typographic “typing” was problematic) different notations are used: xi = xi = xi = |i

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Quantum States and Vectors

Finite quantum states can be described by vectors in Cn, e.g.

• ψ = |ψ =

1/ √ 2 1/ √ 2

• =

1 √ 2 1 1

• r φ| =
• 1
• Observables are defined by matrices A in M(Cn) = Cn×n.

A = 1 2

• with eigenvalues λ0 = 1, λ1 = 2

Note: There are sometimes two types of indices

◮ for enumerating, for example, all eigenvectors of an

• perator like A with |0 =

1

• and |1 =

1

• ◮ to enumerate coordinates of one vector, e.g.

ψ1 = 1/ √ 2,

• r better perhaps: |01 = 0.

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Quantum Postulates II – Measurement

◮ The expected result (average) when measuring observable

A of a system in state |x ∈ H is given by: Ax = x| A |x = x| |Ax

◮ The only possible results are eigen-values λi of A. ◮ The probability of measuring λn in state |x is

Pr(A = λn|x) = x| Pn |x with Pn the orthogonal projection onto the n-th eigen- space of A generated by eigen-vector |λn Pn = |λn λn| then we have: A =

i λiPi (Spectral Theorem).

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Heisenberg’s Uncertainty Relation

Theorem

For two observables A1 and A2 we have: (∆|xA1)(∆|xA2) ≥ 1 2 |(x| [A1, A2] |x)| where the uncertainty (classically: variance) is defined by (∆|xA)2 = x| A2 |x − x| A |x2 and the commutator is defined as: [A1, A2] = A1A2 − A2A1 see e.g. Isham: Quantum Theory, ICP 1995, Section 7.3.3.

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Classical vs Quantum Mechanics

The usual interpretation of Heisenberg’s uncertainty relation is this: When one tries to measures two observables A1 and A2 then – if the commutator [A1, A2] is non-zero – a small ∆|xA1 implies a large ∆|xA2, and vice versa. A standard example of so-called incomensurable observables are position A1 = x and momentum A2 = p (on an infinite- dimensional Hilbert Space H) for which [x, p] = i and thus: ∆x∆p ≥ /2. In classical physics observables always commute, are comensurable, i.e. [A1, A2] = 0. In quantum physics for most

• bservables [A1, A2] = 0, i.e. the observable algebra is typically

non-commutative or non-abelian (cf. multiplication of (complex) numbers vs multiplication of matrices).

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Quantum Dynamics

◮ The dynamics of a (closed) system is described by the

Schrödinger Equation: id |x dt = H |x for the (self-adjoint) Hamiltonian operator H (energy).

◮ The solution is a unitary operator Ut (e.g. Isham 6.4)

Ut = exp(− i tH)

Theorem

For any self-adjoint operator A the operator exp(iA) = eiA =

∞

• n=0

(iA)n n! is a unitary operator.

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Irreversible vs Reversible

There are a number of immediate consequence of the postulates.

• 1. The state develops reversibly, i.e. |xt = Ut |x0 for some

unitary matrix (operator). Consequence: No cloning theorem, i.e. no duplication of information.

• 2. Measurement is partial (Heisenberg Uncertainty Relation).

Consequence: The full state of a quantum computer is not

• bservable.
• 3. Measurement is irreversible.

Consequence: The state of a quantum system is irrevocably destroyed if we inspect it. The mathematical structure has also consequences for any Quantum Logic, e.g. De Morgan fails, ‘Tertium non datur’ is not guaranteed, etc.

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Quantum States and Evolution

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Quantum Physics vs Quantum Computation

Quantum Physics Given a quantum system (device). What is its dynamics?

◮ Heisenberg Picture:

A → At = A(t) = eitHAe−itH

◮ Schrödinger Picture:

|x → |xt = |x(t) = e−itH |x Quantum Computation Given a desired computation (dynamics). What quantum device (e.g. circuit) is needed to obtain this?

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Quantum Computation

Quantum computation tries to utilise quantum systems/devices in order to perform computational tasks or to implement (secure) quantum communication protocols. 1973 C. Bennett: Reversible Computation 1980 P .A. Benioff: Quantum Turing Machine 1982 R. Feynman: Quantum Simulation 1985 D. Deutsch: Universal QTM 1994 P . Shor: Factorisations 1996 L. Grover: Database Search 2008 Harrow, Hassidim, Lloyd: Linear Equations When will (cheap) quantum computers be available? What will be a killer application for quantum computation? When will we reach quantum supremacy?

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Quantum Postulates

◮ The state of an (isolated) quantum system is represented

by a (normalised) vector in a complex Hilbert space H.

◮ An observable is represented by a self-adjoint matrix

(operator) A acting on the Hilbert space H.

◮ The expected result (average) when measuring observable

A of a system in state |x ∈ H is given by: Ax = x| A |x = x| |Ax

◮ The only possible results are eigen-values λi of A. ◮ The probability of measuring λn in state |x is given by:

Pr(A = λn|x) = x| Pn |x = x| |Pnx with Pn = |λnλn| the orthogonal projection onto the space generated by eigen-vector |λn = |n of A.

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Complex Numbers

Quantitative information, e.g. measurement results, is usually represented by real numbers R. For quantum systems we need to consider also complex numbers C. A complex number z ∈ C is a (formal) combinations of two reals x, y ∈ R: z = x + iy with i2 = −1 or i = √ −1. The complex conjugate of a complex number z = x + iy ∈ C is: z∗ = z = x + iy = x − iy = z† Hauptsatz of Algebra Complex numbers are algebraically closed: Every polynomial

• f order n over C has exactly n roots.

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Polar Coordinates

One can represent numbers z ∈ C using the complex plane. x y φ r Conversion: x = r · cos(φ) y = r · sin(φ) r =

• x2 + y2

φ = arctan(y x ) Another representation: (r, φ) = r · eiφ eiφ = cos(φ) + i sin(φ),

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Computational Quantum States

Consider a simple systems with two degrees of freedom. |0 |1

Definition

A qubit (quantum bit) is a quantum state of the form |ψ = α |0 + β |1 where α and β are complex numbers with |α|2 + |β|2 = 1. Qubits live in a two-dimensional complex vector, more precisely, Hilbert space C2 and are normalised, i.e. |ψ = ψ | ψ = 1.

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Vector Spaces

A Vector Space (over a field K, e.g. R or C) is a set V together with two operations: Scalar Product . ·. : K × V → V Vector Addition . +. : V × V → V such that ∀x, y, z ∈ V and α, β ∈ K:

• 1. x + (y + z) = (x + y) + z
• 2. x + y = y + x
• 3. ∃o : x + o = x
• 4. ∃−x : x + (−x) = o
• 5. α(x + y) = αx + αy
• 6. (α + β)x = αx + βx
• 7. (αβ)x = α(βx)
• 8. 1x = x (1 ∈ K)

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Tuple Spaces

Theorem

All finite dimensional vector spaces are isomorphic to the (finite) Cartesian product of the underlying field Kn (i.e. Rn or Cn).

• x = (x1, x2, x3, . . . , xn) represents x =

n

• i=1

xibi

• y = (y1, y2, y3, . . . , yn) represents y =

n

• i=1

yibi Finite dimensional vectors can be represented as tuples via their coordinates with respect to a base {bi}n

i=1.

α x = (αx1, αx2, αx3, . . . , αxn)

• x +

y = (x1 + y1, x2 + y2, x3 + y3, . . . , xn + yn)

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Hilbert Spaces

A complex vector space H is called an Inner Product Space or (Pre-)Hilbert Space if there is a complex valued function ., .

• n H × H that satisfies ∀x, y, z ∈ H and ∀α ∈ C:
• 1. x, x ≥ 0
• 2. x, x = 0 ⇐

⇒ x = o

• 3. αx, y = α x, y
• 4. x + y, z = x, z + y, z
• 5. x, y = y, x

The function ., . is called an inner product on H.

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Caveat: Linear in first or second argument?

Mathematical Convention: αx, y = α x, y Physical Convention: x | αy = α x | y In mathematics we have: x, αy = αy, x = ¯ αy, x = ¯ α x, y For physicists it is simply: x | αy = α x | y

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Basis Vectors

A set of vectors xi is said to be linearly independent iff

• λixi = o

implies that ∀ i : λi = 0 Two vectors in a Hilbert space are orthogonal iff x, y = 0 An orthonormal system in a Hilbert space is a set of linearly independent set of vectors with:

• bi, bj
• = δij =

1 iff i = j iff i = j

Theorem

For a Hilbert space there exists an orthonormal basis {b}. The representation of each vector is unique: x =

• i

xibi =

• i

x, bi bi

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The Finite-Dimensional Hilbert Spaces Cn

We represent vectors and their transpose using coordinates:

• x =

   x1 . . . xn    = |x ,

• y = (y1, . . . , yn) =

   y1 . . . yn   

T

= y| The adjoint of x = (x1, . . . , xn) is given by

• x† = (¯

x1, . . . , ¯ xn)T = (x∗

1, . . . , x∗ n)T

The inner product is then represented by:

• y,

x

• =
• i

¯ yixi =

• i

y∗

i xi

We can also define a norm (length) x =

• x,

x

• .

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Dual and Adjoint States

A linear functional on a vector space V is a map f : V → K such that (i) f(x + y) = f(x) + f(y) and (ii) f(αx) = αf(x) for all x, y ∈ V, α ∈ K. The space of all linear functionals on V form the dual space V∗.

Theorem (Riesz Representation Theorem)

Every linear functional f : H → C on a Hilbert space H can be represented by a vector yf in H, such that f(x) = yf, x = fy(x) Dual Hilbert spaces H∗ are isomorphic to the original Hilbert space H∗; in particular we have: (Cn)∗ = Cn. We represent vectors or ket-vectors as column vectors; and functionals, dual vector or bra-vectors as row vectors.

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Dirac Notation and Einstein Convention

We will use throughout P .A.M. Dirac’s bra-(c)-ket notation:

• xi, yj
• =
• xi,

yj

• denoted as xi|
• yj
• = i| |j

We will enumerate the (eigen-)base vectors (of an operator):

• bi = bi or

ei = ei are denoted by |i but we may need also to specify the coordinates of a vector:

◮ Ket-Vectors (column): |x = (xj)n j=1 in Cn. ◮ Bra-Vectors (row): x| = (xj)n j=1 in (Cn)∗ = Cn.

• A. Einstein: If in an expression there are matching sub- and

super-scripts then this implicitely indicates a summation, ¯ xiyi =

• i

¯ xiyi =

• x,

y

• and xiyi∗ =
• i

xi ¯ yi =

• x |

y

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Qubit

The postulates of Quantum Mechanics simply require that a computational quantum state is represented by a normalised vector in Cn. A qubit is a two-dimensional quantum state in C2 We represent the coordinates of a qubit (state) or ket-vector as a column vector: |ψ = α β

• = α

1

• + β

1

• = α |0 + β |1

with respect to the (orthonormal) basis {|0 , |1}, i.e. the so-called standard base: |0 = 1

• and

|1 = 1

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Representing a Qubit [∗]

A qubit |ψ = α |0 + β |1 with |α|2 + |β|2 = 1 can be represented: |ψ = cos(θ/2) |0 + eiϕ sin(θ/2) |1 , where θ ∈ [0, π] and ϕ ∈ [0, 2π]. Using polar coordinates we have: |ψ = r0eiφ0 |0 + r1eiφ1 |1 , with r 2

0 + r 2 1 = 1. Take r0 = cos(ρ) and r1 = sin(ρ) for some ρ.

Set θ/2 = ρ, then |ψ = cos(θ/2)eiφ0 |0 + sin(θ/2)eiφ1 |1 , with 0 ≤ θ ≤ π, or equivalently |ψ = eiγ(cos(θ/2) |0 + eiϕ sin(θ/2) |1), with ϕ = φ1 − φ0 and γ = φ0, with 0 ≤ ϕ ≤ 2π. The global phase shift eiγ is physically irrelevant (unobservable).

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Bloch Sphere [∗]

|0 |1 cos(θ/2) |0 + eiϕ sin(θ/2) |1 θ ϕ

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Change of Basis

We can represent (the coordinates of) any vector in Cn with respect to any basis we like. For example, we can consider for qubits in C2 the (alternative)

• rthonormal basis:

|+ = 1 √ 2 (|0 + |1) |− = 1 √ 2 (|0 − |1) and thus, vice versa: |0 = 1 √ 2 (|+ + |−) |1 = 1 √ 2 (|+ − |−) A qubit is therefore represented in the two bases as: α |0 + β |1 = α √ 2 (|+ + |−) + β √ 2 (|+ − |−) = α + β √ 2 |+ + α − β √ 2 |−

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Linear Operators

Arguably, the best understood type of functions or maps between two vector spaces V and W are those preseving their basic algebraic structure.

Definition

A map T : V → W between two vector spaces V and W is called a linear map if

• 1. T(x + y) = T(x) + T(y) and
• 2. T(αx) = αT(x)

for all x, y ∈ V and all α ∈ K (e.g. K = C or R). For V = W we talk about a (linear) operator on V.

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Images of the Basis

Like vectors, we can represent a linear operator T via its “coordinates” as a matrix. Again these depend on the particular basis we use. Specifying the image of the base vectors determines – by linearity – the operator (or in general a linear map) uniquely. Suppose we know the images of the basis vectors |0 and |1 T(|0) = T00 |0 + T01 |1 T(|1) = T10 |0 + T11 |1 then this is enough to know the Tij’s to know what T is doing to all vectors (as they are representable as linear combinations of the basis vectors).

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Matrices

Using a “mathematical” indexing (starting from 1 rather ten 0), using the first index to indicate a row position and second for a column position, we can identify T with a matrix: T = T11 T12 T21 T22

• = (Tij)n

i,j=1 = (Tij)

The application of T to a general vector (qubit) then becomes a simple matrix (pre-)multiplication: T α β

• =

T11 T12 T21 T22 α β

• =

T11α + T12β T21α + T22β

• One can also express this, for |ψ = α |0 + β |1 also as:

T(|ψ) = T(α |0 + β |1) = αT(|0) + βT(|1) = T |ψ

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Matrix Multiplications

The application of a linear opertor T (represented by a matrix) to a vector x (represented via its coordinates) becomes: T(x) = Tx = (Tij)(xi) =

• i

Tijxi The standard convention is pre-multiplication so as the sequence is the same as with application. The composition of linear opertators T and S becomes also a matrix/matrix pre-multiplications: T ◦ S = TS = (Tij)(Ski) =

• i

TijSki Some authors use the more “computational” pre-multiplication. Finite-dimensional linear operators (matrices) form a vector space and with the multiplication a (linear) algebra. Adding the adjoint operation (see below) turns this into a C∗-algebra.

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Transformations

We can define a linear map B which implements the base change {|0 , |1} and {|+ , |−}: B = 1 √ 2 1 1 1 −1

• Transforming the coordinates (xi) with respect to {|0 , |1} into

coordinates (yi) using {|+ , |−} can be obtained by: B(xi)i = (yi)i and B−1(yi)i = (xi)i The matrix representation T of an operator using {|0 , |1} can be transformed into the representation S in {|+ , |−} via: S = BTB−1 Problem: It is not easy to compute inverse B−1, defined on implicitly by BB−1 = B−1B = I the identity (existence?!).

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Adjoint Operator

For a matrix T = (Tij) its transpose matrix TT is defined as TT = (T T

ij ) = (Tji)

the conjugate matrix T∗ is defined by T∗ = (T ∗

ij ) = (Tij)∗ = (Tji)

and the adjoint matrix T† is given via T† = (T †

ij ) = (T ∗ ji )

• r T† = (T∗)T = (TT)∗

Note that (TS)T = STTT and thus (TS)† = S†T†. In mathematics the adjoint operator is usually denoted by T∗ (cf. conjugate in physics) and defined implicitly via: T(x), y = x, T∗(y) or T†x | y = x | Ty

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Adjoint Vectors

Bra and ket vectors are also related using the adjoint: |x† = x|

• r using their coordinates:

(xi)† =    x1 . . . xn   

†

= ¯ x1 · · · ¯ xn

• = (¯

xi) The adjoint operator specifies the effect on dual vectors: (T |x)† = |x† T† = x| T†

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Unitary Operators

A square matrix/operator U is called unitary if U†U = I = UU† That means U’s inverse is U† = U−1. It also implies that U is invertible and the inverse is easy to compute. Quantum Mechanics requires that the dynamics or time evolution of a quantum state, e.g. qubit, is implemented via a unitary operator (as long as there is no measurement). The unitary evolution of an (isolated) quantum state/system is a mathematical consequence of being a solution of the Schrödinger equation for some Hamiltonian operator H.

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Properties of Unitary Operators

Unitary operators generalise in some sense permutations (in fact every permutation of base vectors gives rise to a simple unitary map). They can also be seen as generalised rotations. Unitary operators also preserve the “geometry” of a Hilbert space, i.e. they preserve the inner prduct: x| U†U |y = x | y . Any single qubit operation, i.e. unitary 2 × 2 matrix U can be expressed as via 4 (real) parameters: U =

• ei(α−β/2−δ/2) cos γ/2

ei(α+β/2−δ/2) sin γ/2 −ei(α−β/2+δ/2) sin γ/2 ei(α+β/2+δ/2) cos γ/2

• where α, β, δ and γ are real numbers.

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Basic 1-Qubit Operators

Pauli X-Gate X = 1 1

• X

Pauli Y-Gate Y = −i i

• Y

Pauli Z-Gate Z = 1 −1

• Z

Hadamard Gate H =

1 √ 2

1 1 1 −1

• H

Phase Gate Φ = 1 eiφ

• Φ

Φ The Pauli-X gate is often referred to as NOT gate. Note that the notation for Hamiltonian and Hadamard gate are both H.

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Graphical “Notation”

The product (combination) of unitary operators results in a unitary operator, i.e. with U1, . . . , Un unitary, the product U = Un . . . U1 is also unitary (Note: (TS)† = S†T†). |x U |x H

π 2

H X Z A simple example: |y = HH |x or (|x ; H; H = |y): |x |y H H ≡ |x |y = |x I because H2 = I, i.e. 1 √ 2 1 1 1 −1 1 √ 2 1 1 1 −1

• =

1 1

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Quantum Measurement

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Quantum Postulates

◮ The state of an (isolated) quantum system is represented

by a (normalised) vector in a complex Hilbert space H.

◮ An observable is represented by a self-adjoint matrix

(operator) A acting on the Hilbert space H.

◮ The expected result (average) when measuring observable

A of a system in state |x ∈ H is given by: Ax = x| A |x = x| |Ax

◮ The only possible results are eigen-values λi of A. ◮ The probability of measuring λn in state |x is given by:

Pr(A = λn|x) = x| Pn |x = x| |Pnx with Pn = |λnλn| the orthogonal projection onto the space generated by eigen-vector |λn = |n of A.

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Basic Measurement Principle

The values α and β describing a qubit are often called probability amplitudes. If we measure a qubit |φ = α |0 + β |1 = α β

• in the computational basis {|0 , |1} then we observe state

|0 with probability |α|2 and |1 with probability |β|2. Furthermore, the state |φ changes: it collapses into state |0 with probability |α|2 or |1 with probability |β|2, respectively.

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Self Adjoint Operators

An operator A is called self-adjoint or hermitian iff A = A† The postulates of Quantum Mechanics require that a quantum

• bservable A is represented by a self-adjoint operator A.

Possible measurement results are eigenvalues λi of A (always real for self-adjoint operators) defined as A |i = λi |i or A ai = λi ai or Aai = λiai Probability to observe λk in state |x =

i αi |i is

Pr(A = λk, |x) = |αk|2 Physicist refer to αk as probability amplitude.

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Spectrum

The set of eigen-values {λ1, λ2, . . .} of an operator T is called its spectrum σ(T). σ(T) = {λ | λI − T is not invertible} It is possible that for an eigen-value λi in the equation T |i = λi |i we may have more than one eigen-vector |i for an eigen-value λi, i.e. the dimension of the eigen-space d(i) > 1. We will not consider these degenerate cases here. Terminology: “eigen” means “self” or “own” in German (cf also Italian “auto-valore”), it characterises a matrix/operator.

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Projections

Projections An operator P on Cn is called projection (or idempotent) iff P2 = PP = P Orthogonal Projection An operator P on Cn is called (orthogonal) projection iff P2 = P = P† We say that an (orthogonal) projection P projects onto its image space P(Cn), which is always a linear sub-spaces of Cn. Birkhoff-von Neumann: Projections on Hilbert space form an (ortho-)lattice which gives rise to non-classical “quantum logic”.

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Outer Product

The outer product |xy| for vectors |x = (x1, . . . , xn)T and y| = (y1, . . . , yn) is an operator/matrix (actually: |x ⊗ y|): (|xy|)ij = xiyj e.g. |01| = 1 1

• =

1

• It could be treated just as a formal combination, e.g. we can

express the identity as I = |00| + |11| because (|00| + |11|) |ψ = (|00| + |11|)(α |0 + β |1) = α |00||0 + α |11||0 + β |00||1 + β |11||1 = α |0 + β |1

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Spectral Theorem

In the bra-ket notation we can represent a projection onto the sub-space generated by |x by the outer product Px = |xx|.

Theorem

A self-adjoint operator A (on a finite dimensional Hilbert space, e.g. Cn) can be represented uniquely as a linear combination A =

• i

λiPi with λi ∈ R and Pi the (orthogonal) projection onto the eigen-space generated by the eigen-vector |i, i.e. Pi = |ii|

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Measurement Process

If we perform a measurement of the observable represented by: A =

• i

λi |ii| with eigen-values λi and eigen-vectors |i in a state |x we have to decompose the state according to the observable, i.e. |x =

• i

Pi |x =

• i

|ii|x =

• i

i|x |i =

• i

αi |i With probability |αi|2 = | i|x |2 two things happen

◮ The measurement instrument will the display λi. ◮ The state |x collapses to |i.

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Do-It-Yourself Observable

We can take any (orthonormal) basis {|i}n

0 of Cn+1 to act as

computational basis. We are free to choose (different) measurement results λi to indicate different states in {|i}. |x =

i i|x|i

A =

i λi |ii|

|n|x|2 λn |n . . . |0|x|2 λ0 |0 The “display” values λi are essential for physicists, in a quantum computing context they are just side-effects.

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Reversibility

Quantum Dynamics For unitary transformations describing qubit dynamics: U† = U−1 The quantum dynamics is invertible or reversible Quantum Measurement For projection operators in quantum measurement (typically): P† = P−1 i.e. the quantum measurement is not reversible. However P2 = P i.e. the quantum measurement is idempotent.

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Composite Quantum Systems

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Beyond Qubits – Quantum Registers

Operations on a single Qubit are nice and interesting but don’t give us much computational power. We need to consider “larger” computational states which contain more information. There could be two options:

◮ Quantum Systems with a larger number of freedoms. ◮ Quantum Registers as a combination of several Qubits.

Though it might one day be physically more realistic/cheaper to build quantum devices based on not just binary basic states, even then it will be necessary to combine these larger “Qubits”.

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Free Vector Spaces

In the theory of formal languages we have the construction of words out of some (finite) set of letters, i.e. alphabet Σ or S. For vector spaces there is similar construction: Take any (finite) set of objects B and “declare” it a base. The free vector space is the set of all linear combinations of elements in B = {b1, b2, . . .}, i.e. V(B) =

• i

λibi | λi ∈ C and bi ∈ B

• r

V(B) =

• i

λi |i | λi ∈ C and |i ∈ B

• with the obvious algebraic operations (incl. inner product).

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Multi Qubit State

We encountered already the state space of a single qubit with B = {0, 1} but also with B = {+, −}. The state space of a two qubit system is given by V({0, 1} × {0, 1}) or V({+, −} × {+, −}) i.e. the base vectors are (in the standard base): B2 = {(0, 0), (1, 0), (0, 1), (1, 1)}

• r we use a “short-hand” notation B2 = {00, 01, 10, 11}

Issue: What about V(B × B × B)? What is its dimension, or how many base vectores are there in B3?

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Tensor Product

Given a n × m matrix A and a k × l matrix B: A =    a11 . . . a1m . . . ... . . . an1 . . . anm    B =    b11 . . . b1l . . . ... . . . bk1 . . . bkl    The tensor or Kronecker product A ⊗ B is a nk × ml matrix: A ⊗ B =    a11B . . . a1mB . . . ... . . . an1B . . . anmB    Special cases are square matrices (n = m and k = l) and vectors (row n = k = 1, column m = l = 1).

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Tensor Product of Vectors

The tensor product of (ket) vectors fulfils a number of nice algebraic properties, such as

• 1. The bilinearity property:

(αv + α′v′) ⊗ (βw + β′w′) = = αβ(v ⊗ w) + αβ′(v ⊗ w′) + α′β(v′ ⊗ w) + α′β′(v′ ⊗ w′) with α, α′, β, β′ ∈ C, and v, v′ ∈ Ck, w, w′ ∈ Cl.

• 2. For v, v′ ∈ Ck and w, w′ ∈ Cl we have:
• v ⊗ w, v′ ⊗ w′

=

• v, v′

w, w′

• 3. We denote by bm

i ∈ Bm ⊆ Cm the i’th basis vector in Cm

then bk

i ⊗ bl j = bkl (i−1)l+j

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Tensor Product of Matrices

For the tensor product of square matrices we also have:

• 1. The bilinearity property:

(αM + α′M′) ⊗ (βN + β′N′) = = αβ(M ⊗ N) + αβ′(M ⊗ N′) + α′β(M′ ⊗ N) + α′β′(M′ ⊗ N′) α, α′, β, β′ ∈ C, M, M′ m × m matrices N, N′ n × n matrices.

• 2. We have, with v ∈ Cm and w ∈ Cn:

(M ⊗ N)(v ⊗ w) = (Mv) ⊗ (Nw) (M ⊗ N)(M′ ⊗ N′) = (MM′) ⊗ (NN′)

• 3. If M and N are unitary (or invertible) so is M ⊗ N, and:

(M ⊗ N)T = MT ⊗ NT and (M ⊗ N)† = M† ⊗ N†

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The Two Qubit States

Given two Hilbert spaces H1 with basis B1 and H2 with basis B2 we can define the tensor product of spaces as H1 ⊗ H2 = V({bi ⊗ bj | bi ∈ B1, bj ∈ B2}) Using the notation |i ⊗ |j = |i |j = |ij the standard base of the state space of a two qubit system C4 = C2 ⊗ C2 are: |00 =     1     , |01 =     1     , |10 =     1     , |11 =     1     Often one also represents them using a “decimal” notation, i.e. |00 ≡ |0, |01 ≡ |1, |10 ≡ |2, and |11 ≡ |3.

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Entanglement

The important relation between V(B), e.g. V({0, 1}), and V(Bn), e.g. V({0, 1}n) is given by V(Bn) = (V(B))⊗n, i.e.: V(B × B × . . . × B) = V(B) ⊗ V(B) ⊗ . . . ⊗ V(B) Every n qubit state in C2n can be represented as a linear combination of the base vectors |0 . . . 00 , |0 . . . 10 , . . . , |1 . . . 11 or decimal |0 , |1 , |2 , . . . , . . . , |2n − 1. A two-qubit quantum state |ψ ∈ C22 is said to be separable iff there exist two single-qubit states |ψ1 and |ψ2 in C2 such that |ψ = |ψ1 ⊗ |ψ2 If |ψ is not separable then we say that |ψ is entangled.

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Entanglement and Classical Probabilities

In quantum physics the state is given by a vector in a complex Hilbert space. Instead of probability amplitudes in Cn let us consider probability distributions in a real vector space, i.e. Rd. All the normalised (using the 1-norm, i.e. (pi)i1 =

i |pi|)

elements ρ in Rd represent probability distributions on a d element probability space Ωd = {ω1, ω2, . . . , ωd} i.e. ρ = (ρi) ∈ D(Ωd) with ρi = P(ωi) ∈ [0, 1]. The normalised elements in Rd1 ⊗ Rd2 correspond to the joint probability distributions on Ωd1 × Ωd1, with ρij = P(ωi ∧ ωj), i.e. D(Ωd1 × Ωd1) = D(Ωd1) ⊗ D(Ωd1)

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Classical Correlations

If the events in Ωd1 and Ωd2 are independent (“uncorrelated”) then their joint distribution is given as a product of distributions

• n Ωd1 and Ωd1, i.e. ρ = ρ1 ⊗ ρ2 or P(ωi ∧ ωj) = P(ωi) · P(ωj).

If there is a “correlation” or “dependency” then it is impossible to express a certain joint distribution as a simple (tensor product) but only as a sum of (tensor) products. Consider, for example, two coins which “miraculously” always fall on the same side, i.e. a joint distribution: ρij H T H

1 2

T

1 2

ρ = 1 2(1, 0) ⊗ (1, 0)T + 1 2(0, 1) ⊗ (0, 1)T = ρ1 ⊗ ρ2

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Classical Gates

At heart of classical (electronic) circuits we have to consider gates like for example: AND ≡ ∧ 1 1 1 1 1 XOR ≡ ⊕ 1 1 1 1 1 1 NAND 1 1 1 1 1 1 1 The idea is to define similar quantum gates, taking two (or n) qubits at input and producing some output. Contrary to classical gates we have to use unitary, i.e. reversible, gates in quantum circuits.

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The Controlled-NOT or CNOT Gate

The quantum analog of a classical XOR-gate is the CNOT-gate. The behaviour of the CNOT-gate (on two qubits, i.e. C2 ⊗ C2), is for base vectors |x , |y ∈ {|0 , |1}: |x, y → |x, y ⊕ x with y ⊕ x = (y + x) mod 2 i.e. |00 → |00 , |01 → |01 , |10 → |11 , |11 → |10. We represent the CNOT-gate graphically and as a matrix (with respect to the standard basis {|00 , |01 , |10 , |11}) as: |x |x |y |x ⊕ y CNOT =     1 1 1 1    

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Swapping Gate

We can exploit the CNOT-Gate to SWAP two qubits: |x |y |y |x is depicted by (shorthand): |x |y |y |x Exercise: Check that this really maps |x ⊗ |y into |y ⊗ |x (for all |x and |y not just base vectors?).

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Controlled Phase Gate

The controlled phase-gate is depicted as follows (for base vectors |x , |y ∈ {|0 , |1}): |x |x |y eixyφ |y Φ Its matrix/operator representation is given by: Φ =     1 1 1 eiφ    

• n any two qubits, i.e. vectors in C2 ⊗ C2.

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General Controlled Gate

In general, we can control any single qubit transformation U : C2 → C2 by another qubit, i.e. such that for all |y ∈ C2: |0 ⊗ |y → |0 ⊗ |y |1 ⊗ |y → |1 ⊗ U |y The diagrammatic representation is: |x |x |y U |y U

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Toffoli Gate

The Toffoli-gate is a 3-qubit quantum gate on C2 ⊗ C2 ⊗ C2 = = C8 with the following behaviour T : |x, y, z → |x′, y′, z′ and matrix representation (standard base enumeration): input

• utput

x y z x′ y′ z′ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 T =             1 1 1 1 1 1 1 1            

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Toffoli Gate Usage

The Toffoli gate can be used can be used to implement a reversible version of NAND and a FANOUT gate. |x |x |y |y |z |z ⊕ xy Toffoli |x |x |y |y |1 |¬xy NAND |1 |1 |y |y |0 |y FANOUT This works only with x, y ∈ {0, 1}.

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Linear Maps from Functions

In general, we can take any (binary) function f : {0, 1}n → {0, 1}m and define a corresponding linear map Tf Tf : (V({0, 1}))⊗n → (V({0, 1}))⊗m or Tf : (C2)⊗n → (C2)⊗m We just have to read the map f as an instruction on how base vectors should be transformed under Tf (into base vectors). Once we know or specify the image of all base vectors we know the (matrix representation) of Tf via Tf |x = |f(x) E.g. with f(011) = 10101 we have Tf : |011 → |10101. Problem: Tf is, in general, not unitary, i.e. reversible.

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Reversible Operators from General Functions

Reversibility makes it impossible to have a quantum device Uf which just computes a general function f, i.e. Uf : |x → |f(x). However, we can always “pack” up a function f as a unitary

• perator Uf using an ancilla qubit to remember the initial state,

e.g. |x ⊗ |0 → |x ⊗ |f(x) . The standard implementation of f : {0, 1}n → {0, 1}m as unitary operator Uf on C2n ⊗ C2m is: Uf : |x ⊗ |y → |x ⊗ |y ⊕ f(x) Graphically represented by the diagram/quantum circuit: |x |x |y |y ⊕ f(x) Uf

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Quantum Computation

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Quantum Circuit Model

We can specify a quantum algorithm on qubit registers – i.e. a unitary operator U : (C2)⊗n → (C2)⊗n – using a combination of (standardised) quantum gates – like Hadamard, Pauli, etc. – and maybe “oracles” like Uf as well as measurements. For example, the quantum circuit for teleportation (without correction) as an operator on (C2)⊗3 is given as follows: |ψ1 |φ1 |ψ2 |φ2 |ψ3 |φ3 H H

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Calculations for Small Quantum Circuits

Circuits with few qubits can “implemented”, e.g. in octave, etc. q0 = [1,0]’ q1 = [0,1]’ H = (1/sqrt(2))*[1, 1;1,-1] CX = [1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 0, 1; 0, 0, 1, 0] S1 = kron(eye(2),H,eye(2)) S2 = kron(eye(2),CX) S3 = kron(CX,eye(2)) S4 = kron(H,eye(2),eye(2)) T = S1*S2*S3*S4

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Computational Expressivness

The question arises: What we can compute with a given set of basic quantum gates? What can we compute with a quantum circuit? For permutations it is well known that all permutations can be decomposed into elementary so-called transpositions which

• nly exchange two elements. Similar results also exist for

rotations. For general unitary operators U on Cn – in particular on m qubits, i.e. C2m = (C2)⊗m – an analogue results gurantees that 2 × 2 unitary matrices make up all unitary operators. See e.g.: A. Yu. Kitaev, A. H. Shen, M. N. Vyalyi: Classical and Quantum Computation, AMS, 2002, p70.

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Unitary Operators on Cn

Theorem

An arbritary unitary operator U on the space Cn can be represented as a product of n(n−1)

2

matrices of the form:                  1 . . . . . . . . . . . . . . . . . . ... . . . . . . . . . 1 . . . . . . a b . . . . . . c d . . . . . . 1 . . . . . . . . . ... . . . . . . . . . . . . . . . . . . 1                  with a b c d

• a 2 × 2 unitary matrix (on C2).

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Approximation of Unitary Operators

If we are only interested in “about the right result” we have: Given two unitary transformations U and V. The error of approximation is defined by e(U, V) = sup

|φ

(U − V) |φ

Definition

A set of gates G = {G1, . . .} is said to be approximatly universal if any n-qubit operator U (with n ≥ 1) can be approximated to arbitrary accuracy, i.e. for all ε > 0 there exists a circuit V which is constructed of gates in G and their controlled versions such that we have e(U, V) < ε.

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(Approximatly) Universal Gates

A possible set of approximatly universal gates (e.g. Kaye, Laflamme, Mosca: Introduction to Quantum Computing, p71): H = 1 √ 2 1 1 1 −1

• Φ

π 4

• =

1 ei π

4

• CNOT =

    1 1 1 1    

Theorem

The set G = {H, Φ( π

4)} is universal for 1-qubits.

Theorem

The set G = {CNOT, H, Φ( π

4)} is a universal set of gates.

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Cloning of Qubits?

Is it possible to create a second copy of a general qubit |ψ using a unitary operation U. |ψ |ψ |0 |ψ U

Theorem (No Cloning Theorem)

The exists no unitary transformation U such that U |ψ |0 = |ψ |ψ for all qubits |ψ ∈ C2.

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Argument

Consider two qubits |ψ and |φ.Then by linearity: U(α |ψ + β |φ) |0 = αU(|ψ) |0 + βU(|φ) |0 = α |ψ |ψ + β |φ |φ but also if U is a cloning operator: U(α |ψ + β |φ) |0 = (α |ψ + β |φ)(α |ψ + β |φ) = α2 |ψ |ψ + β2 |φ |φ +αβ |ψ |φ + αβ |φ |ψ Only for α = 0 or β = 0 we have α |ψ |ψ + β |φ |φ = α2 |ψ |ψ + β2 |φ |φ +αβ |ψ |φ + αβ |φ |ψ

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Approximate Cloning?

Is it not even possible to approximately clone a qubit. Consider two qubits |ψ and |φ with 0 < ψ|φ < 1 such that U(|ψ ⊗ |0) ≈ |ψ ⊗ |ψ and U(|φ ⊗ |0) ≈ |φ ⊗ |φ By unitarity – U preserving inner products – we get (|ψ |0)†(|φ |0) = ψ|φ 0|0 = ψ|φ ≈ ψ|φ2 Thus ψ|φ ≈ 0 or ψ|φ ≈ 1.

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Quantum Cryptography

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Communication on Insecure Channels

Alice Bob Eve ENC DEC ENC(T, KA) = M DEC(M, KB) = T DEC(ENC(T, KA), KB) = T

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One-Time-Pad or Vernam Cipher

Gilbert Sandford Vernam, 1917 Step 0. Alice and Bob share a common, random key K. Step 1. Alice calculates M = T ⊕ K. Step 2. Message M is sent along the insecure channel. Step 3. Bob retrieves plain text T = M ⊕ K. K = KA = KB ENC(T, K) = DEC(T, K) = T ⊕ K. Caveat: Never ever reuse random key K!

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Example

T 1 1 1 1 K ⊕ 1 1 1 1 M 1 1 ↓ ↓ ↓ ↓ ↓ ↓ M 1 1 K ⊕ 1 1 1 1 T 1 1 1 1

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Quantum Key Distribution

The problem with One-Time-Pads is Key Distribution. Quantum Key Distribution aims to exploit quantum features in

• rder to protect the keys, utilising:

No-Cloning. The message cannot be duplicated.

• Measurement. Observing the message changes it.

These quantum techniques aim in addressing two security aims:

• Authentication. Is sender really Alice?

Intrusion Detection. Is Eve eavesdropping?

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BB84

Charles Bennett and Gilles Brassard 1984 The aim is to exchange a key K (e.g. a One-Time-Pad). Alice and Bob communicate over two insecure channels: a quantum channel and a classical one. The protocol is based on the use of two (computational) bases: = {

• , |

} = {(1, 0)T, (0, 1)T} = {| , | } = { 1 √ 2 (−1, 1)T, 1 √ 2 (1, 1)T} Interpretation of messages in both basis M |

• |
• 1
• |
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Measuring in Wrong Base

As long as Alice and Bob send and receive qubits in the same basis, Bob will always measure the same qubit Alice has sent. However, if they don’t agree on the measurement base, Bob will make the wrong assumption of what Alice has sent. Assume that Alice sends 0 encoded as | in the basis but Bob uses to measure it: In this case he will measure

• r

| with 50% chance, i.e. concludes with a 50:50 chance that Alice intended to send 0 or 1 respectively. This is due to the following obvious facts that: |

• =

1 √ 2(

• − |

) |

• =

1 √ 2(

• + |

)

• =

1 √ 2(|

+ | ) |

• =

1 √ 2(|

− | )

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BB84 Protocol

Step 1.a Alice chooses n random bits to send (e.g. to be used as One-Time-Pad). Step 1.b Alice randomly chooses n times whether to use

• r

to encode each bit. Step 2.a Alice encodes the bits accordingly in the bases and sends the qubits to Bob. Step 2.b Bob randomly chooses n times whether to use

• r

to measure the qubits he got and measures them. Step 3. Over the classical channel Alice and Bob compare which basis they used for each bit. If they agree they keep it otherwise they drop it. Step 4.a Bob choose a part (e.g. half) of the transmitted bits (drops them) and compares them openly with Alice. Step 4.b If these test bits do not agree (subject to transmission errors) Alice and Bob conclude that Eve was eavesdropping and abandon transmission.

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Example

KA 1 1 1 1 1 1 1 BA ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ BB

• bs

KB 1 1 1 1 1 1 1 √ √ √ √ √ √ √ √ K 1 1 1 1 1

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B92

Charles Bennett 1992 The idea is to use a non-orthogonal basis to encode 0 and 1, e.g. B = {| , | } = {(1, 0)T, 1 √ 2 (1, 1)T} Step 1. Alice chooses n random bits and encodes them, e.g. 0 ≡ | and 1 ≡ | and send these qubits to Bob. Step 2. Bob measures these qubits in randomly chosen base

• r

. Step 3. Bob tells Alice over an open classical which qubits he considers ambiguous in order to drop them. Again – as in BB84 – some bits can be sacrificed to see if an extensive number of “transmission errors” indicates that Eve was eavesdropping and abandon transmission.

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Ambiguous Bits

When Bob measures the qubits received from Alice he will conclude that certain observations are inconclusive. Using . If Bob observes

• Bob knows that Alice sent 1 ≡ |

. | Bob drops this bit. Using . If Bob observes | Bob knows that Alice sent 0 ≡ | . | Bob drops this bit. In the average three quarters of the qubits have to be discarded.

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Example

KA 1 1 1 1 1 1 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ BB

• bs

KB ? ? 1 ? ? ? 1 ? ? √ √ √ √ √ K 1 1

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EPR

Artur Ekert 1991 The idea is to distribute a key K via pairs of entangled states, for example the Bell states: 1 √ 2 (|00 + |11) The key K is effectively generated only after the distribution of these states to Alice and Bob. They do this independently but entanglement guarantees they obtain the same key. This protocol is inspired by the Einstein-Podolsky-Rosen (EPR, 1935) Gedanken-Experiment.

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EPR Protocol

Step 1. A random sequence of entangled 2-qubit states – e.g.

1 √ 2(|00 + |11) – is created.

For each such state one of the qubits is given to Alice and Bob, respectively. Step 2. Bob and Alice measure each of their qubits in a randomly chosen base

• r

. Step 3. Over the classical channel Alice and Bob compare which basis they used for each bit. If they agree they keep it otherwise they drop it. As in BB84 too many “transmission errors” indicate that Eve was eavesdropping and the transmission is abandoned. Ekert proposed a more sophisticated eavesdropping detection (Bell’s theorem).

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Example

BA

• bs

KA 1 1 1 BB

• bs

KB 1 1 √ √ √ √ √ √ √ K

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