P. A. Benioff. Quantum mechanical hamiltonian models of turing machines. Journal of Statistical Physics, 29:515--546, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....described by quantum mechanics. Feynman [8, 9] pointed out first that there might be a substantial gap between computational models based on classical physics and those based on quantum mechanics. The quantum Turing machine (QTM) the first model of quantum computation, was introduced by Benioff [1, 2]. Deutsch in [5] described a universal simulator for QTMs with exponential overhead. Bernstein and Vazirani [3] were able to construct a universal QTM with only polynomial overhead. Other quantum computational models were also studied recently. Deutsch [6] has defined the model of quantum ....

P. Benioff, Quantum mechanical Hamiltonian models of Turing machines, J. Stat. Phys., 29 (1982), pp. 515--546.

....to previously identified phase transitions in search difficulty. The conditions underlying this improvement are described. Much of the algorithm is independent of particular problem instances, making it suitable for implementation as a special purpose device. 1 Introduction Quantum computers [1, 2, 7, 8, 10, 17, 9] use quantum parallelism, i.e. the ability to operate simultaneously on a superposition of many classical states, and interference among different computational paths. A measurement on a superposition gives a definite result, with probabilities determined by the amplitudes of the superposition. A ....

Benioff, P. (1982), "Quantum Mechanical Hamiltonian Models of Turing Machines", J. Stat. Phys. 29, 515--546.

....exponential precision. Hence all these suggestions for computational models do not provide counterexamples for the modern Church thesis, since they require exponential physical resources. However, note that all the suggestions mentioned above rely on classical physics. In the early 80 s Benioff[27, 28] and Feynman[94] started to discuss the question of whether computation can be done in the scale of quantum physics. In classical computers, the elementary information unit is a bit, i.e. a value which is either 0 or 1. The quantum analog of a bit would be a two state particle, called a quantum ....

Benioff P Quantum mechanical Hamiltonian models of Turing machines, J. Stat. Phys. 29 515-546 1982

....transformation [9, 18] is called the Fourier transform of the basis jx; y; bi. Thus Q scanning the first m bits of the tape t corresponds to the global transition jx; 0; 0i 7 1 2 m=2 X y jx; y; 0i: Q then simulates the deterministic computation of M on input hx; yi in a reversible manner [8, 4], using other work tapes 1 . Let b y be the one bit result of the computation of M(x; y) Q sets b = b y . The superposition is now 1 2 m=2 X y jx; y; b y i: Afterwards, Q repeats the scan it performed at the beginning, using the same local transformation rule, except that it now includes ....

P. A. Benioff. Quantum mechanical hamiltonian models of turing machines. Journal of Statistical Physics, 29:515--546, 1982.

....the result of multiplying the vector (2.4) by the matrix (2.3) The machine will thus go to the superposition of states 1 2 (j10i j11i) Gamma 1 2 (j10i Gamma j11i) j11i : 2:5) This example shows the potential effects of interference on quantum computation. Had we started with either the state j10i or the state j11i, there would have been a chance of observing the state j10i after the application of the gate (2.3) However, when we start with a superposition of these two states, the probability amplitudes for the state j10i cancel, and we have no possibility of observing j10i after the ....

....will thus go to the superposition of states 1 2 (j10i j11i) Gamma 1 2 (j10i Gamma j11i) j11i : 2:5) This example shows the potential effects of interference on quantum computation. Had we started with either the state j10i or the state j11i, there would have been a chance of observing the state j10i after the application of the gate (2.3) However, when we start with a superposition of these two states, the probability amplitudes for the state j10i cancel, and we have no possibility of observing j10i after the application of the gate. Notice that the output of the gate would have been j10i ....

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Stat. Phys. 29, 515--546. P. Benioff (1982b) "Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy," Phys. Rev. Lett. 48, 1581--1585.

....thus seems plausible that while the natural computing power of classical mechanics corresponds to Turing machines, 1 the natural computing power of quantum mechanics could be more powerful. The first person to look at the interaction between computation and quantum mechanics appears to have been Benioff [1980, 1982a, 1982b] Although he did not ask whether quantum mechanics conferred extra power to computation, he showed that reversible unitary evolution was sufficient to realize the computational power of a Turing machine, thus showing that quantum mechanics is at least as powerful computationally as ....

P. Benioff (1982a) "Quantum mechanical Hamiltonian models of Turing machines," J.

....physics is also reversible, because the reverse time evolution specified by the unitary operator U Gamma1 = U y always exists; as a consequence, several workers recognized that reversible computation could be executed within a quantum mechanical system. Quantum mechanical Turing machines [5, 6], gate arrays [7] and cellular automata [8] have been discussed, and physical realizations of Toffoli s[9, 10, 11] and Fredkin s[12, 13, 14] universal three bit gates within various quantum mechanical physical systems have been proposed. While reversible computation is contained within quantum ....

P. Benioff, "Quantum mechanical Hamiltonian models of Turing machines", J. Stat. Phys. 29, 515 (1982).

.... of unitary transformations U(2 n ) is continuous: To what extent is quantum computation sensitive to perturbation And is it possible to organize computation so that a moderate perturbation would not affect the result Quantum devices for doing classical computation were suggested by Benioff [1], Peres [2] and Feynmann [3] Deutsch [4, 5] was the first to give an explicit model of quantum computation. He defined both quantum Turing machines and quantum circuits. Yao [6] showed that these two models are equivalent. More specifically, quantum Turing machines can simulate, and be simulated ....

....by E(U; OE) the corresponding eigenspaces. Without risk of confusion, the corresponding observable may be denoted simply by OE. Let the operator U act on a register A. Denote by 1 an additional bit and introduce the matrix S = 1 p 2 1 1 1 Gamma1 Then the operator Xi(U ) A; 1] S[1] (U ) 1; A] S[1] 16) is a measurement operator for the observable OE. If j i 2 E(U; OE) then Xi(U)j ; 0i = j ; ji, where jji = 1 2 1 1 1 Gamma1 1 0 0 (OE) 1 1 1 Gamma1 1 0 = 1 2 (1 (OE) 1 2 (1 Gamma (OE) Hence the conditional probabilities P Xi(U ) ....

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P. Benioff, "Quantum mechanical Hamiltonian models of Turing machines", J. Stat. Phys. 29, 515 (1982).

....and integrability at least in the problem of simulating physical behavior, which will be discussed in Section 4. 2 Explicit quantum algorithms The hallmark of quantum computation is that each elementary dynamical process of the state vector of quantum computers obeys the unitary transformation[1, 2, 3]: The time evolution operator has the form of U(1t) exp [0i1tH= 1) where H is a Hamiltonian operator with Hermitian property, 1t is the time duration of each computation process and an exponential operator exp [xA] of A is defined as the sum of the convergent series exp [xA] 1 X n=0 ....

Benioff, P., "Quantum mechanical Hamiltonian models of Turing machines,"J. Stat. Phys. 29 (1982), 515-546.

....automata can be used. They are all universal in the sense that they can simulate each other with only a polynomialoverhead. However, these models are based on classical physics, whereas physicists believe that the universe is better described by quantum mechanics. Feynman [13, 14] and Benioff [4, 5] were the first who pointed out that quantum physical systems are apparently difficult to simulate on classical computers, suggesting that there may be a gap between computational models based on classical physics and models based on quantum mechanics. Deutsch [10] introduced the first formal ....

P. Benioff, Quantum mechanical Hamiltonian models of Turing machines that dissipates no energy, Physical Review Letters 48, 1581--1585, 1982.

....automata can be used. They are all universal in the sense that they can simulate each other with only a polynomialoverhead. However, these models are based on classical physics, whereas physicists believe that the universe is better described by quantum mechanics. Feynman [13, 14] and Benioff [4, 5] were the first who pointed out that quantum physical systems are apparently difficult to simulate on classical computers, suggesting that there may be a gap between computational models based on classical physics and models based on quantum mechanics. Deutsch [10] introduced the first formal ....

P. Benioff, Quantum mechanical Hamiltonian models of Turing machines, J. Stat. Phys. 29, 515--546, 1982.

....a quantum associative memory that maintains the ability to recall patterns associatively while offering a storage capacity of O(2 n ) using only n neurons. The field of quantum computation, which applies ideas from quantum mechanics to the study of computation, was introduced in the mid 1980 s [Ben82] [Deu85] Fey86] For a readable introduction to quantum computation see [Bar96] The field is still in its infancy and very 3 theoretical but offers exciting possibilities for the field of computer science perhaps the most notable to date being the discovery of quantum computational ....

Benioff, Paul, "Quantum Mechanical Hamiltonian Models of Turing Machines", Journal of Statistical Physics, vol. 29 no. 3, pp. 515-546, 1982.

....computational power and other properties of computers based on quantum mechanical principles. Its main objective is to find quantum algorithms that are significantly faster than any classical algorithm solving the same problem. The field started in the early 1980s with suggestions by Paul Benioff [Ben82] and Richard Feynman [Fey82, Fey85] and reached more rigorous ground when in 1985 David Deutsch defined the universal quantum Turing machine [Deu85] The following years saw only sparse activity, but the field accelerated explosively after Peter Shor s 1994 discovery of efficient quantum ....

P. A. Benioff. Quantum mechanical Hamiltonian models of Turing machines. Journal of Statistical Physics, 29(3):515--546, 1982.

....classical notion of Monte Carlo algorithms. Please note that the name EQP has been used by different authors, sometimes to mean QP [7] and sometimes to mean ZQP [4] To avoid confusion, we shall refrain from using it at all. The following results are known in quantum complexity theory: P QP [1, 2], ZPP ZQP, BPP BQP P #P [13, 4] and BQP BQP = BQP [3] It is believed that ZQP 6 BPP because Shor s quantum factorization algorithm [11] allows to recognize F = fhx; yi j x has a prime divisor smaller than yg in ZQP, whereas if F 2 BPP then factorization can be accomplished in ....

P. Benioff, "Quantum mechanical Hamiltonian models of Turing machines", Journal of Statistical Physics, Vol. 29, no. 3, 1982, pp. 515 -- 546.

....based on quantum mechanical principles to avoid this problem, thus implicitly asking the converse question: by using quantum mechanics in a computer can you compute more efficiently than on a classical computer. Other early work in the field of quantum mechanics and computing was done by Benioff [Beni]. Although he did not ask whether quantum mechanics conferred extra power to computation, he did show that a Turing machine could be simulated by the unitary evolution of a quantum process, which is a necessary prerequisite for quantum computation. Deutsch [Deu1, Deu2] was the first to give an ....

P. Benioff, Quantum mechanical Hamiltonian models of Turing machines, J. Stat. Phys., Vol. 29, pp. 515--546 (1982).

....property that has a classical cousin, entanglement is a completely quantum phenomenon for which there is no classical analog. 3. 6 Quantum Algorithms The field of quantum computation, which applies ideas from quantum mechanics to the study of computation, was introduced in the mid 1980 s [Fey86] [Ben82]. For a readable introduction to quantum computation see [Bar96] for a more rigorous treatment see for example [Deu85] The field is still in its infancy and very theoretical but offers exciting possibilities for the field of computer science the most important quantum algorithms discovered to ....

Benioff, Paul, "Quantum Mechanical Hamiltonian Models of Turing Machines", Journal of Statistical Physics, vol. 29 no. 3, pp. 515-546, 1982.

.... f (often interpreted as false ) It is customary to code the classical logical states by ptq = 1 and p f q = 0 (psq stands for the code of s) The states can, for instance, be realized by some condenser who is discharged (j cbit state 0) or charged (j cbit state 1) In quantum information theory [1, 34, 43, 6, 7, 35, 36], the most elementary unit of information is the quantum bit, henceforth called qbit. Qbits can be physically represented by a coherent superposition of the two orthonormal 8 states t and f . The qbit states x a;b = at b f (1) form a continuum, with jaj 2 jbj 2 = 1, a;b 2 C . 3.1 ....

....do not commute. The results thus have to be inferred rather than measured, and the existence of such elements of physical reality thus have to be tacitly assumed [41] 4. 3 Universal quantum computer based on the U(2) gate The brute force method of obtaining a (universal) quantum computer [6, 34, 66] by quantizing the hardware components of a Turing machine suffers from the same problem as its classical counterpart it seems technologically unreasonable to actually construct a universal quantum device with a scaled down (to nanometer size) model of a Turing machine in mind. We therefore ....

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BENIOFF, P. A. Quantum mechanical hamiltonian models of turing machines. Journal of Statistical Physics 29, 3 (1982), 515--546.

....and the exponential number required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems in fact have no solutions, unlike previously proposed quantum search algorithms. 1. Introduction Quantum computers (Benioff, 1982; Bernstein Vazirani, 1993; Deutsch, 1985, 1989; DiVincenzo, 1995; Feynman, 1986; Lloyd, 1993) offer a new approach to combinatorial search problems (Garey Johnson, 1979) with quantum parallelism, i.e. the ability to operate simultaneously on many classical search states, and interference ....

Benioff, P. (1982). Quantum mechanical hamiltonian models of Turing machines. J. Stat.

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P. A. Benioff. Quantum mechanical hamiltonian models of turing machines. Journal of Statistical Physics, 29:515--546, 1982.

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P.A. Benioff. Quantum mechanical hamiltonian models of turing machines. Journal of Statistical Physics, 29:515--546, 1982.

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P. A. Benioff. Quantum mechanical hamiltonian models of turing machines. Journal of Statistical Physics, 29:515--546, 1982.

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P. Benioff. Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy. Phys. Rev. Lett., 48 (1980), 1581--1585.

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Benioff P Quantum mechanical Hamiltonian models of Turing machines, J. Stat. Phys. 29 515-546 1982

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P. Benioff, Quantum mechanical Hamiltonian models of Turing machines. J. of Statistical Physics 29: 515-546 (1982).

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P. Benioff. Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy. Phys. Rev. Lett., 48 (1980), 1581--1585.

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