Basic Theoretical Physics A Concise Overview

36.3 Quantum Computing 28336.3 Quantum ComputingThe subject of quantum computing is based on the fact that quantum-mechanical wave functions are superposable and can interfere with each other: ψ = ψ1 + ψ2, |ψ|2 = |ψ1|2 + |ψ2|2 + 2 · Re(ψ1∗ · ψ2) .5 This is a field of research which has seen great progress in the last fewyears. Firstly we should recall that classical computing is based on binary digits;e.g., the decimal number “9” is given by 1 × 23 + 1 × 20, or the bit sequence1001; as elementary “bits” one only has zero and unity; hence N -digit binarynumbers are the vertices of a 2N -dimensional hypercube. In contrast, quan-tum computing is based on the ability to superimpose quantum mechanicalwave functions. One considers N -factor product states of the form N |Ψ = c0(ν) ψ0(ν) + c1(ν) ψ1(ν) ; ν=1i.e., as in Heisenberg spin systems with S = 1 one uses the Hilbert space 2 H“N2-level” ,a (tensor) product of N “2-level systems”, where the orthonormalized basisstates ψ0(ν) and ψ1(ν)are called “quantum bits” or qubits. Of course |Ψ is, as usual, only definedup to a complex factor. The way that large amounts of computation velocity can be gained isillustrated by the following example. Assume that (i) the unitary operator Uˆ , i.e., a complex rotation in Hilbert-space, is applied to the initial state |Ψ0 to generate the intermediate state |Ψ1 = Uˆ |Ψ0 ,before (ii) another unitary operator Vˆ is applied to this intermediate state|Ψ1 to generate the end state |Ψ2 = Vˆ |Ψ1 .Now, the unitary product operator Wˆ := Vˆ Uˆ5 Several comprehensive reviews on quantum computing and quantum cryptogra- phy can be found in the November issue 2005 of the German “Physik Journal”.

284 36 On the Interpretation of Quantum Mechanicsdirectly transforms the initial state |Ψ0 into the final state |Ψ2 . In generala classical computation of the product matrix Wˆ is very complex; it costsmany additions, and (above all!) multiplication processes, since Wi,k = Vi,j Uj,k . jIn contrast, sequential execution by some kind of “experiment”, |Ψ0 → |Ψ1 → |Ψ2 ,may under certain circumstances be a relatively easy and fast computationfor a skilled experimenter. One could well imagine that in this way in special cases (in particular forN → ∞) even an exponentially difficult task (i.e., growing exponentially withthe number of digits, N , or with the size of the system, L) can be transformedto a much less difficult problem, which does not grow exponentially but onlypolynomially with increasing N and L. In fact there are computation which are classically very difficult, e.g., thedecomposition of a very large number into prime factors. One can easily seethat 15 = 3 × 5 and 91 = 7 × 13; but even for a medium-sized number, e.g.437, factorization is not easy, and for very large numbers the task, althoughsystematically solvable, can take days, weeks, or months, even on moderncomputers. In contrast, the same problem treated by a quantum computerwith a special algorithm (the Shor algorithm), tailored for such computersand this problem, would be solved in a much shorter time. This is by no means without importance in daily life: it touches on thebasis of the encoding principles used by present-day personal computers forsecure messages on the internet, i.e., so-called PGP encoding (PGP =ˆ “prettygood privacy”). According to this encoding, every user has two “keys”, one of which, the“public key” of the receiver, is used by the sender for encoding the message.This “public key” corresponds to the afore-mentioned “large number” Z.But for fast decoding of the message the receiver also needs a private key,which corresponds to the decomposition of Z into prime factors, and this keyremains known only to the receiver6. A “spy”, knowing the computer algorithms involved and the “public key”of the receiver, e.g., his (or her) “large number”, can thus in principle calculatethe corresponding “private key”, although this may take weeks or months andwould not matter for a lot of short-term transactions by the receiver until the“spy” has finished his computation. But if the “spy” could use a “quantumcomputer”, this would be a different matter. 6 According to this so-called PGP-concept (PGP =ˆ “Pretty Good Privacy”) the private key is only “effectively private”, see below, similar to a very large integer being factorized by a finite-time computation.

36.4 2d Quantum Dots 285 Fortunately, even now (i.e., in our age of classical computers), quantummechanics offers a totally different way of encoding, called quantum crypto-graphy (see below) which, in contrast to the classical encoding schemes is ab-solutely secure. However, because of the “coherency” requirement7 of quan-tum mechanics the method suffers from the problem of range, so that atpresent it can only be applied over distances smaller than typically ∼ 10 to100 km8. After these preliminary remarks on quantum cryptography, we return toquantum computing. There are other relevant examples where quantum computing would bemuch more effective than classical computing, e.g., in the task of sorting anextremely large set of data, where on a quantum computer the Grover algo-rithm would be much faster than comparable algorithms on classical com-puters. One may ask oneself why under these circumstances have quantum com-puters not yet been realized, even though intense work has been going on inthis field for many years. The difficulties are hidden in the notion of “a skilledexperimentalist”, used above, because it is necessary to ensure that, (i), dur-ing the preparation of the initial state |ψ0 ; then, (ii), during the execution ofall operations on this state; and finally, (iii), during all measurements of theresults the coherence and superposability of all signals should be essentiallyundisturbed. This implies inter alia that errors (due to the necessary auto-matic correction) should only happen with an extremely small probability,e.g., < 10−4, and that all experiments performed in the course of a quantumcomputation should be extremely well-controlled. Additionally, the systemshould be “scalable” to N 1. Many different suggestions have indeed been made for producing a quan-tum computer, one of which will be outlined in the next section.36.4 2d Quantum DotsTwo-dimensional quantum dots can be regarded as artificial atoms with di-ameters in the region of ∼ 100˚A(= 10 nm) in a two-dimensional electron (orhole) gas [“2DEG” (or “2DHG”)]. A two-dimensional electron gas is formed,e.g., at the planar interface between a GaAs-semiconductor region (for z < 0)and an Al1−xGaxAs-region (for z > 0). At the interface there is a deflexionof the energy bands, and as a consequence an attractive potential trenchV (x, y, z) forms, which is, however, attractive only w.r.t. the z-coordinate, 7 The coherency demands do not allow, for quantum mechanical purposes, the usual amplification of the signals, which is necessary for the transmission of electromagnetic signals over hundreds or thousands of kilometers. 8 A quantum cryptographical encoding/decoding software has been commercially available since the winter of 2003/2004.