Quantum Kolmogorov Complexity

Berthiaume, André; Dam, Wim van; Laplante, Sophie

doi:10.1006/jcss.2001.1765

Cited by 101 publications

(121 citation statements)

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“…Svozil [2] originally defined quantum Kolmogorov complexity on a circuit based model and then Berthiaume, van Dam and Laplante [3] defined quantum Kolmogorov complexity [4] on a polynomial-time bounded quantum Turing machine. [2] and [3] both restricted inputs to be of variable but determined length (i.e. quantum but not second quantized inputs).…”

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confidence: 99%

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Rogers

Vedral

2008

Mod. Phys. Lett. B

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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...

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confidence: 99%

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Rogers

Vedral

2008

Mod. Phys. Lett. B

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“…The first attempt to define a quantum Kolmogorov complexity was by Svozil [29] in 1996. The attempts to define a quantum Kolmogorov complexity [3,7,14,27,29,30,31,32] assume that the descriptions have determinate lengths. Second quantised Kolmogorov complexity differs from the others in that it uses indeterminate length quantum strings.…”

Section: Previous Workmentioning

confidence: 99%

“…Berthiaume, van Dam and Laplante [3] defined a quantum Kolmogorov complexity based on Bernstein and Vazirani's model of a quantum Turing machine [2]. This quantum Kolmogorov complexity is invariant up to an additive constant with respect to a universal quantum Turing machine U which can simulate any other quantum Turing machine from its classical description.…”

Section: B Quantum Kolmogorov Complexitymentioning

confidence: 99%

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Second Quantized Kolmogorov Complexity

Rogers

Vedral

Nagarajan

2008

Int. J. Quantum Inform.

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The Kolmogorov complexity of a string is the length of its shortest description. We define a second quantised Kolmogorov complexity where the length of a description is defined to be the average length of its superposition. We discuss this complexity's basic properties. We define the corresponding prefix complexity and show that the inequalities obeyed by this prefix complexity are also obeyed by von Neumann entropy.

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“…Among the different measures of state complexity, quantum Kolmogorov complexity is defined by length of the shortest possible program that would generate the state [15][16][17][18]. This very common definition suffers from the setback that it is not computable.…”

Section: Introductionmentioning

confidence: 99%

State complexity and quantum computation

Cai

Scarani

2015

Annalen der Physik

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The complexity of a quantum state may be closely related to the usefulness of the state for quantum computation. We discuss this link using the tree size of a multiqubit state, a complexity measure that has two noticeable (and, so far, unique) features: it is in principle computable, and non-trivial lower bounds can be obtained, hence identifying truly complex states. In this paper, we first review the definition of tree size, together with known results on the most complex three and four qubits states. Moving to the multiqubit case, we revisit a mathematical theorem for proving a lower bound on tree size that scales superpolynomially in the number of qubits. Next, states with superpolynomial tree size, the Immanant states, the Deutsch-Jozsa states, the Shor's states and the subgroup states, are described. We show that the universal resource state for measurement based quantum computation, the 2D-cluster state, has superpolynomial tree size. Moreover, we show how the complexity of subgroup states and the 2D cluster state can be verified efficiently. The question of how tree size is related to the speed up achieved in quantum computation is also addressed. We show that superpolynomial tree size of the resource state is essential for measurement based quantum computation. The necessary role of large tree size in the circuit model of quantum computation is still a conjecture; and we prove a weaker version of the conjecture.

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Quantum Kolmogorov Complexity

Cited by 101 publications

References 28 publications

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Second Quantized Kolmogorov Complexity

State complexity and quantum computation

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