The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Abstract

The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach -our measure of second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities it is continuous. We give examples where second quantized and quantum Kolmogorov complexity differ.PACS numbers: 03.67.LxQuantum physics, as far as we know, is the most accurate and universal description of all physical phenomena in the universe. If we therefore wish to speak about the complexity of some processes or some physical states in nature, we need to use quantum physics as our best available theory. An important question is (in very simple terms), given a physical system in some state, how difficult is it for us to reproduce it. If we wish to have a universal measure of this difficulty (which applies to all systems and states) a way to proceed is to follow the prescription of Kolmogorov. First, we define a universal computer (which is capable of simulating all other computers) and then we look for the shortest input (another physical state) to this computer that reproduces as the output the desired physical state. This way, all the complexity is defined with respect to the same universal computer, and we can thus speak about universal complexity. The universal computer (such as a universal Turing machine) needs to be fully quantum mechanical in order to capture all the possibilities available in nature. In this letter, we define a fully quantized Turing machine which we use to define a fully quantum Kolmogorov complexity and apply it to a number of different problems. Our approach is different from others in that we consider indeterminate length input programs whose expected length is our measure of complexity. Others, whose work is discussed in detail below, have perhaps avoided our approach, as allowing programs in superposition can lead to a superposition of halting times. Since traditionally computation is viewed as giving a deterministic output after a fixed amount of time, having a superposition of halting times seems to contradict the very concept of computation. We, on the other hand, view the superposition of different length inputs which can lead to a superposition of halting times as a necessity dictated by a fully quantum mechanical description of nature. Moreover allowing superpositions of different programs can lead to a program which has on average a smaller Kolmogorov complexity than when programs are restricted to having a variable but determinate length (we give examples of this in this letter). Since we want to know what is physically the shortest input which produces a given output, we need allow the computation with superpositions of different length programs. This letter is organized as follows. First we discuss the concept of a Turing machine and review previous work. Then we define the...

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

“…One example is Lemma 3.13 which essentially states that indeterminate-length QTMs are no more interesting then fixed-length QTMs, if the length ℓ(σ) of an input qubit string σ is defined as in Definition 2.1. This supports the point of view of Rogers and Vedral [17] to consider the average lengthl(σ) instead, that is, the expectation value of the length ℓ. If the halting of the underlying quantum computer is still defined as in this paper, then our result applies to their definition, too.…”

“…This corresponds to the fact that this state |ϕ needs at least 5 cells on a QTM's tape to be stored perfectly (compare Subsection II-B). An alternative approach would be to consider the expectation valuel of the length instead, which has been proposed by Rogers and Vedral [17], see also the discussion in Section IV. In contrast to classical bit strings, there are uncountably many qubit strings that cannot be perfectly distinguished by means of any quantum measurement.…”

“…This corresponds to the fact that this state |ϕ needs at least 5 cells on a QTM's tape to be stored perfectly (compare Subsection II-B). An alternative approach would be to consider the expectation value l of the length instead, which has been proposed by Rogers and Vedral [17], see also the discussion in Section IV.…”

“…Note also that Definition 2.5 depends on the definition of the length ℓ(σ) of a qubit string σ ∈ T + 1 (H {0,1} * ); there is a different approach by Rogers and Vedral [17] that uses the expected (average) length l instead and results in a different notion of quantum Kolmogorov complexity. The results of this paper are applicable to that definition, too, as long as the notion of halting of the corresponding quantum computer is defined in a deterministic way as in Equation ( 6).…”

Second Quantized Kolmogorov Complexity

“…Mora, Briegel and Kraus [16] applied algorithmic complexity to various problems in quantum communication and computation to produce new proofs of already known results. An application of a quantum Kolmogorov complexity would be to analyse a fully generalised quantum Maxwell's demon [24]. Mora, Briegel and Kraus [16] also applied the algorithmic complexity of quantum states to begin to look at the effect of entanglement on Maxwell's demon.…”