Quantum Computing Simulation using the Auxiliary Field Decomposition

Abstract

Instead of using classical bits which are 0 or 1, quantum computers make use of “quantum bits” which are similar to XY-Spins. The total information described by N quantum bits is vector in the Kronecker product space

$$\left( {{a_1}{b_1}} \right)\,\otimes\,\left( {{a_2}{b_2}} \right)\,\otimes\,\left( {{a_3}{b_3}}\right)\, \otimes \, \ldots \,\left( {{a_N}{b_N}} \right)\, with\,a_i^2\, + \,b_i^2\, = \,1.$$

(11.1)

The dimension is the same as the space spanned by the same number of classical bits

$${a_1}{a_2}{a_3} \ldots {a_n},{a_1}\, \in \,\left\{ {0,1} \right\}\,$$

(11.2)

but the intention is to sample the problem via a quantum mechanical wave-function “quantum parallel”. To allow the representation of the gates of a quantum circuit with quantum mechanical states, the minimum requirement is that the circuit is reversible, so that the number of input states must be the same as the number of output states. Examples for forbidden and allowed states are given in Fig. 11.1. All the final output of the quantum computation must be must be represent able in the sense of quantum mechanics. The difference between quantum computing and mere “reversible computing” is that in the quantum circuit a quantum mechanical wave function is propagated. The aim is to realize the propagation of a quantum mechanical wave function in such a way that the “all the solutions” are obtained “at once”, an idea which usually referred to under as “quantum parallelism”. Most of the quantum parallel algorithms proposed so far seem only to work for algorithms which select from discrete alternatives, like Shor’s prime factoring [8] or Grover’s database search [9], with the exception of a proposal for the computation of densities of states [2].