Let us now give a short introduction to quantum computing, which rests on a non-Kolmogorovian type of probability, namely quantum stochastics. In the following, we will present some basic facts on quantum mechanics and quantum computing. In quantum physics, the state of a quantum system is described by a vector in a (complex) Hilbert space H. It is customary to write such a state vector as a column vector, or - in the jargon of quantum physics - as ”ket vector” |ψ. The corresponding line vector is written as
ψ| and called ”bra vector”. The squared norm of the vector, or - in other words - the scalar product of the vector with itself is then written asψ|ψ, which becomes a bracket. We now come to the process of measurement in quantum mechanics. As a principle, in quantum mechanics measurements of observables are described by Hermitian operatorsAacting on the underlying Hilbert space H. If the system is in an eigenstate of A, then the measure- ment with the operatorA just reproduces the state, multiplied with a real number (since A is Hermitean). If the system is not in an eigenstate, then the outcome of the measurement will collapse to one of the observables (cor- responding to eigenstates (eigenvalues)) ofA, but what is important is that it cannot be predicted in advance to which one. Only probabilites can be indicated, which correspond to the principle of superposition. The result of any measurement of a quantum system described by the state vector |ψ is always one of the eigenvalues of the operatorA, corresponding to the observ- able being measured. If the system is in an eigenstate ofA, then the outcome of the measurement is just the corresponding eigenvalue of this eigenstate. In general, the system will be in some general stateφ. Then we may representφ
as a complex linear combination with respect to a basis{ψi}i of eigenstates
ofA:
|φ=
i
ωi|ψi, (3.1)
where theωiare called the probability amplitudes. If w.l.o.g. we assume that
i|ωi|
2= 1, then|ω
eigenstateiwith respect toA. So a quantum system can exist in a blend of all its eigenstates with respect to a certain Hermitian measurement operator simultaneously. This is called the principle of superposition and is the big difference between quantum and classical mechanics, where absolutely no such analogue exists. If the system is in a superposition of states as in (3.1), then the probability of each possible outcome of the measurementA (i.e. of each possible eigenvalue) is given by |ωi|2. An unobserved quantum system
is governed by Schr¨odinger’s equation
ih|ψ˙(t)= ˆH(t)|ψ(t),
whereh= 1.0545·10−34J sis Planck’s constant and ˆH(t) is the Hamiltonian (unitary operator) related to the total energy of the system; so the system behaves smoothly until it is measured.
Definition 3.1.A qubit (quantum bit) is a quantum 2-state system
|ψ=α|0+β|1, (3.2)
whereα, β∈CI such that|α|2+|β|2= 1.(Note that|0and|1are just names for the eigenvectors representing a classical bit and have nothing to do with the zero vector in the Hilbert space H =CI2.)
It is an easy exercise to show that for |ψas in (3.2) there exist angles θ, φ
such that
|ψ= cosθ|0+eiφsinθ|1. (3.3) So a (single) qubit can, geometrically, be interpreted as a point on the two- dimensional unit sphere, the north pole (e.g.) representing the eigenstate|1 and the south pole the eigenstate |0. It turns out that information that in classical computers use much memory can be stored with much fewer qubits in quantum computers. Let us begin with a simple example: Assume we have two classical complementary bitstrings of length 7, e.g., |0110101 and |1001010. In order to store them in a classical computer, we need two registers each of length 7 bit. However, on a quantum computer, only one register of 7 qubits suffices, since here we can just store the superposition
1
√
2(|0110101+|1001010). More generally, if we have an exponential number
of bits to store, by using the superposition principle, a polynomial number of qubits suffices. (Of course, these superpositions can be very complicated in general.) Ann-qubit memory register is realized by then-fold tensor product of the 1-qubit register. However, quantum evolvement of such a register may lead to states that are defined as a whole, but do not arise from individual qubits, i.e., the individual qubits are not defined as such. Such states are called entangled (Schr¨odinger used the german word ”verschr¨ankt” for it). A very important fact is that measurements of subsets of qubits in ann-qubit register project out the state of the whole register into a subset of eigenstates consistent with the answers (eigenvalues) obtained from the measurement.
40 3 Factorization with Quantum Computers: Shor’s Algorithm
This has to do with quantum teleportation and the Einstein-Podolsky-Rosen experiment. Another aspect is quantum parallelism. Quantum evolution is performed by unitary operators (on ”single processors”), which operate at the same time on all possible states. These phenomena will be crucial in Shor’s algorithm.