Classical vs. Quantum Computers

This section introduces the most basic differences between classical and quantum computers in a phenomenologic manner. For a more rigid and formal explanation, please refer to section 2.

Quantum Bits

In a classical computer, the logical state is determined by the expectation value of its register contents (e.g. tension of a capacitor). The interpretation as (classical) bits is performed by comparing the measured value to a defined threshold, while the great number of particles guarantees that the uncertainty of the measurement is small enough to make errors practically impossible.

In a quantum computer, information is represented as the common quantum state of many subsystems. Each subsystem is described by a combination of two ``pure'' states interpreted as ${\vert \rangle}$ and ${\vert 1 \rangle}$ (quantum bit, qubit). This can e.g. be realised by the spin of a particle, the polarisation of a photon or by the ground state and an excited state of an ion.

For a single qubit, this state can be described by the complex amplitudes $a$ and $b$ of each of the two states ( $a {\vert \rangle} + b {\vert 1 \rangle}$) with the condition $a a^*+b b^*=1$.

It is obvious, that this interpretation stands in contradiction to classic boolean logic, where intermediate states between 0 and 1 are not possible.

Entanglement of States

The logical state of a classical register is determined by the states of all bits this register contains. Those bits can be changed locally i.e. independently form one another. The state of an $n$ bit register, can therefore be described by $n$ binary values.

A quantum register containing more than one qubit can not be described by simply listing the states of each qubit, moreover it is not even possible to define the state of an isolated qubit:

Given an isolated system of two qubits, its state can be described by four complex amplitudes $a {\vert,0 \rangle} + b {\vert 1,0 \rangle} + c {\vert,1 \rangle} + d {\vert 1,1 \rangle}$. You can define the expectation value for the first qubit, which is $\sqrt{b b^*+d d^*}$ but there is no isolated state for the first qubit anymore like e.g. $(a+c) {\vert \rangle} + (b+d) {\vert 1 \rangle}$ since $\vert a\vert^2+\vert b\vert^2+\vert c\vert^2+\vert d\vert^2=1$ does not implicate that $\vert a+c\vert^2+\vert b+d\vert^2=1$.

Therefore, manipulations on a single qubit effect the complex amplitudes of the overall state and have a global character. To describe the combined state of $n$ entangled qubits, $2^n$ complex numbers are necessary.

Measurement

In a classical computer, the formal description of the inner state and the measurement of this state (i.e. the output of the program) is the same and given by the binary values of the concerned bits. Moreover, the inner state is not effected by the process of measurement (non destructive measurement).

According to the Kopenhagner interpretation of quantum physics, the outcome of measurements on quantum systems (qubits) must be formulated in classical terms (binary bits). The quantum state of the system is thereby reduced: If the first bit in the above mentioned 2 qubit state is measured, and a value of 1 is observed, then the state will be reduced to $b' {\vert 1,0 \rangle}+ d' {\vert 1,1 \rangle}$ with $\vert b'\vert^2+\vert d'\vert^2=1$, thus all basevectors which 0 in the first bit ( ${\vert,0 \rangle}$ and ${\vert,1 \rangle}$) will be set to an amplitude of zero.

Therefore it is principally not possible to measure the state of a quantum register itself; it is however possible, to estimate the expectation value of a qubit by repeated measurements after the same calculation.

Reversibility of Computation

Heat dissipation is one of the major problems with the miniaturisation of classical computers and constant cooling of all components is required. This is achieved by the thermic coupling of the circuits to a heat reservoir like e.g. the surrounding air.

For a quantum computer, cooling by heat coupling is no option since its logical state is directly represented by the common quantum state of its registers. Any heat coupling would necessarily result in the entanglement of this state with the outside world and destroy the coherence of the computation.

The second law of thermodynamics postulates that any non-reversible state change of a system must dissipate heat. Many common logical operations like AND, OR or resetting a bit to 0 or 1 are non-reversible is the sense that the input cannot be calculated from the output. Therefore, these operations cannot directly be implemented in a quantum computer.