Quantum computing
While the last chapter introduced readers without a firm background in machine learning into the foundations necessary for the following parts of the thesis, this chapter gives an overview of some concepts and special methods required from the field of quantum computing. I assume the reader is familiar with quantum theory. The mathematical formalism is briefly summarised in the next section in order to introduce notation and terminology.
Box 3.1.1: Highlights in the history of quantum theory
1900 Max Planck discovers quantisation of black-body radiation
1905 Albert Einstein discovers quanti-
sation of light 1912 Max Born applies quantisation
to atomic spectrum
1923
de Broglie postulates the duality of waves and particles
1926 Erwin Schr¨odinger
formulates wave mechanics
1930 1932 Paul Dirac and John von Neumann
unite approaches
& formulate modern quantum theory
1925 Heisenberg, Jordan, Born
formulate matrix mechanics
Quantum theory branches out into physical subdisciplines
1996
Luv Grover proposes a quantum
algorithm for unordered search
1994 Peter Shor proposes a quantum
algorithm for prime factoring
Quantum Machine Learning Quantum
Information Processing
TIME
What Aaronson describes as a “complicated patchwork of ideas that physicists invented be- tween 1900 and 1926” forms the beginnings of quantum theory as it is still taught today, and which was then used to rethink the entire body of physics knowledge. The initial step is commonly attributed to the year 1900 when Max Planck introduced the idea that energy (in this case so called black-body radiation) can only be absorbed in discrete portions as if existing as ‘quanta’ or small energetic portions. With this assumption he was able to resolve a heated debate regarding the spectrum of black-body radiation [60]. A few years later, Albert Einstein made a similar discovery with light emission in the photoelectric effect, and derived the concept of a photon - a portion or energy quantum of light [61, 62]. In the following years, these early ideas of a ‘theory of energy quanta’ were applied to atomic spectroscopy (most notably by Niels Bohr), and to light (by Luois de Broglie) but still based on rather ad-hoc methods [60]. Werner Heisenberg followed by Jordan, Born and, independently, Paul Dirac formulated the first mathematically consistent version of quantum theory referred to as matrix mechanics in 1925, with which Wolfgang Pauli was able to derive the experimental results of measuring the spectrum of a hydrogen atom. Heisenberg postulated his uncertainty principle shortly after, stating that certain properties of a quantum system cannot be measured accu- rately at the same time. In 1926, following a slightly different and less abstract route, Erwin Schr¨odinger proposed his famous equation of motion for the ‘wave function’ describing the state of a quantum system. These two approaches were shown to be equivalent in the 1930s, and a more general version was finally proposed by Dirac and Jordan. In the following years, quantum theory branched out into many subdisciplines and many traditional pillars of physics were reformulated in this new framework.
The mathematical apparatus to calculate the probabilities and expectation values of measurement outcomes for a quantum system is largely based on linear algebra calculus. One associates the measurable properties of a quantum system with a vector|ψiin aHilbert spaceH=CN called
aquantum state. The scalar product defined on the Hilbert space is denoted ashψ|ψi.
Anobservableis a physical property or variable and is represented by a Hermitian operatorO on H. It follows from the spectral theoremof linear algebra that there exists a orthonormal basis of H consisting of eigenvectors of O, and the corresponding eigenvalues are real. For a discrete and finite-dimensional system associated with aN-dimensional Hilbert spaceCN, every|ψi ∈CN can hence be expressed inO’s eigenbasis{|ψii}i=1...N,
|ψi=
N
X
i=1
ai|ψii,
where theai∈Care the (quantum) amplitudes. The effect of applyingO to an element|ψi ∈CN is fully defined by theN eigenvalue equationsO|ψii=λi|ψiiwith eigenvaluesλi. Expectation valuesof the observable property are calculated by
E(O) =hψ|O|ψi=X
ij
a∗iajhψi|O|ψji=X
ij
aiajλjhψi|ψji=X
i
|ai|2λj
The dynamic evolution of a quantum state is represented by aunitary operatorU =U(t2, t1) mapping |ψ(t1)i to U(t2, t1)|ψ(t1)i = |ψ(t2)i, where the property U†U = 1 (with U† being the Hermitian conjugate) ensures that states remain normalised. U is the solution of the corresponding Schr¨odinger equationi~∂t|ψi =H|ψi with Hamiltonian H. For time-independent Hamiltonians one can write the solution asU =e−iHt.
Quantum theory can also be formulated in terms of the outer product of a vector inH, adensity operatorρ=|ψihψ|, which is a trace-1 Hermitian positive operator. A density operator allows us to elegantly incorporate classical probability theory “on top” of quantum theory by considering an ensemble ormixture of quantum states,ρ=P
kpk|ψkihψk|withP
kpk= 1 andpk being classical probabilities of the system being in state |ψki. The expectation value for a state described by a density operator is calculated by E(O) = tr{ρO}. The most general dynamics (mixing
‘coherent’ quantum and ‘incoherent’ classical probability theory) is a (completely) positive trace-preserving mapfrom a density operator to a density operator.
Ajoint quantum system is described by considering the tensor product of the Hilbert spaces of the subsystems,HAB =HA⊗ HB. If |φii, i = 1...Na is a basis of the discrete finite Hilbert spaceHA and|ϕji, j = 1...Nb a basis of HB, then a basis of the composite space is made up of theNaNb basis states|φii ⊗ |ϕji. For the tensor products of quantum states the shorthand|φiϕji is frequently used.
Thepartial traceis used to obtain the description of a subsystem from a composed system. In Dirac notation the trace over subsystem A of a joint quantum systemHB with basis{|ψbii}reads
ρA= trBρAB=
N
X
i=1
hψbi|ρAB|ψbii.
For discrete and finite-dimensional systems, quantum states and observables can be elegantly expressed by complex vectors and matrices. The ith element in the vector representation
of |ψi in the basis {|ψ1i, ...,|ψNi} can be calculated by hψi|ψi. The entry ij of the matrix representationof observable O can be calculated byhψi|O|ψji. The unitary dynamics can be expressed by a unitary matrix applied to the quantum state vector.