Remarks on universal quantum computer

Shi, Yu

doi:10.1016/s0375-9601(02)00015-4

Cited by 28 publications

(30 citation statements)

References 12 publications

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“…Nishimura and Ozawa [13] compared various models of quantum computation. Various papers [14,15,16,17] have discussed how these models can halt coherently (all computation paths halting simultaneously) when two programs which halt at different times are used as input in superposition. [17] showed that it is undecidable whether a quantum Turing machine halts coherently for a general input.…”

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confidence: 99%

“…States such as |ψ = 1 √ 2 (|0 + |10 ) are forbidden on the QTM but allowed on the SQTM. Second Quantized Halting Several papers [14,15,16,17] have pointed out issues in how quantum computations halt coherently (how all the computational paths halt simultaneously) on the quantum Turing machine [10,11] and Nielsen's parallel gate array [12]. The SQTM allows programs of different sizes to be input in superposition.…”

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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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The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Rogers

Vedral

2008

Mod. Phys. Lett. B

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“…Myers argued [4] that due to entanglement between a halt qubit and a system, it may be difficult to measure the halt qubit, which may spoil the computation. Subsequent discussions on the halting problem with a quantum computer have mainly focused on the superposition and entanglement of the halt qubit [5,6,7]. In this paper, we approach the halting scheme differently and use two pictures of quantum dynamics, i.e., the Schrödinger and Heisenberg pictures.…”

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Unsolvability of the Halting Problem in Quantum Dynamics

Song

2007

Int J Theor Phys

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It is shown that the halting problem cannot be solved consistently in both the Schrödinger and Heisenberg pictures of quantum dynamics. The existence of the halting machine, which is assumed from quantum theory, leads into a contradiction when we consider the case when the observer's reference frame is the system that is to be evolved in both pictures. We then show that in order to include the evolution of observer's reference frame in a physically sensible way, the Heisenberg picture with time going backwards yields a correct description. PACS numbers:With the construction of universal quantum Turing machine, Deutsch proposed [1] a quantum version of the halting problem first proved by Alan Turing in 1936 [2]. In recent years, a lot of interest has been focused on quantum computation [3], and the discussion of the halting problem using a quantum computer has also received attention. Myers argued [4] that due to entanglement between a halt qubit and a system, it may be difficult to measure the halt qubit, which may spoil the computation. Subsequent discussions on the halting problem with a quantum computer have mainly focused on the superposition and entanglement of the halt qubit [5,6,7]. In this paper, we approach the halting scheme differently and use two pictures of quantum dynamics, i.e., the Schrödinger and Heisenberg pictures. Schrödinger's wave mechanics and Heisenberg's matrix mechanics were formulated in the early twentieth century and have been considered to be equivalent, i.e., two different ways of describing the same physical phenomenon that we observe. Therefore, in order to consider a halting scheme for a quantum system, we need to examine whether the procedure is consistent in both the Schrödinger and Heisenberg pictures. We will give an example in quantum dynamics that shows this cannot be achieved. We will then argue that it is the Heisenberg picture, rather than both pictures, that yields the correct description that not only does not run into the inconsistency shown through the halting scheme but also is physically sensible.In order to discuss the halting problem, we first wish to define notations to be used. In particular we will follow a similar notation used in [8,9] such that it is convenient in both Schrödinger and Heisenberg pictures. A qubit, a basic unit of quantum information, is a two-level quantum system written as |ψ = a|0 + b|1 . Using a Bloch sphere notation, i.e., with a = exp(−iφ/2) cos(θ/2) and b = exp(iφ/2) sin(θ/2), a qubit in a density matrix form can be written as |ψ ψ| = 1 2 (1 +v · σ) where (v x , v y , v z ) = (sin θ cos φ, sin θ sin φ, cos θ) and σ = (σ x , σ y , σ z ) with σ x = |0 1| + |1 0|, σ y = −i|0 1| + i|1 0|, and σ z = |0 0| − |1 1|. Therefore a qubit, |ψ ψ|, can be represented as a unit vectorv = (v x , v y , v z ) pointing in (θ, φ) of a sphere with 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π. A unitary transformation of a qubit in the unit vector notationv can be obtained by applying U to σ i for the corresponding ith component of the vectorv, i.e., v i , where i = x, y, z (al...

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“…[7], [8], [9], [10], [11]. We will not get too deep into this discussion, but rather analyze in detail the simple definition for halting by Bernstein and Vazirani [3], which we also use in this paper.…”

Section: A Quantum Turing Machines and Their Halting Conditionsmentioning

confidence: 99%

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

Müller

2008

IEEE Trans. Inform. Theory

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Abstract-We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for an arbitrary, but preassigned number of time steps.As a corollary to this result, we give a rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant.Our proof is based on a new mathematical framework for QTMs, including a thorough analysis of their halting behaviour. We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string can always be effectively decomposed into a classical and a quantum part.

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Remarks on universal quantum computer

Cited by 28 publications

References 12 publications

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Unsolvability of the Halting Problem in Quantum Dynamics

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

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