Entanglement Reduces Complexity

Transcription

1 Physics 52 Quantum Physics (Townsend) April 27, 2000 Entanglement Reduces Complexity The Physical Nature of Computing and the Advent of Quantum Computation...Nature isn t classical, dammit, and if you want to make a simulation of Nature, you d better make it quantum mechanical, and by golly it s a wonderful problem, because it doesn t look so easy. Richard P. Feynman [?] 1 Two Problems Nearly every student in recent history has had occasion to take a multiple-choice test. In answering any one question on the test, the student is faced with a finite number of choices and charged with the (relatively) simple task of selecting the correct answer from among these choices. Assuming that a student is able to recognize the correct answer when it is presented to him or her (perhaps the student attends Harvey Mudd), the problem then reduces to finding this correct answer as quickly as possible. With respect to the time it takes for the student to find the correct answer ( the running time of the student s thought process) one needs to asks the question of how many operations this process takes. This question is answered in the form of it takes x operations to find the correct answer, where an operation is some action which can be performed more or less using the same amount of resources 1 on all input. Clearly, x could be one if the first answer that the student examines is the correct answer, or x could be N where N is the number of answers to be examined. Luckily for the aforementioned student, a standard multiple-choice test typically contains at most 4 or 5 answers to be examined. However, if the maximum number of answers for each question were 100, a student would find completing the exam in a timely manner a challenging task to say the least. Thus, one observes the importance of knowing the worst-case running time, i.e. the maximum possible number of operations required for a computation to complete. The multiple-choice search outlined above is a specific case of a problem which computer science has dubed the unordered search problem. The unordered search problem supplies an 1 Usually time and space are the resources which are considered in such an analysis. 1

2 unsorted collection of data and a key value, and asks as output the unique element which corresponds to the given key value. Clearly, since the examination of an element of the collection yields no information which allows one to discard other unexamined data, finding the element corresponding to a given key can take at most N operations. Classically, there is no way around this worst case. Now consider a second problem. Given a set of m cities each with a distance defined between them, is there a path which goes through every city and ends in the city in which it began with a total combined path length of D or less (where total combined path length refers to the sum of the distances traversed to visit each city). This is referred to as the traveling salesperson problem[?]. This problem is referred to as an NP-Complete problem, which means that computer scientists have yet to find a solution that does not require an exponential number of operations in the number of cities, nor do they expect to find such a solution. In computing this problem then, if we assume that it requires 2 n operations in the worst case, a reasonable input size of 100 requires operations. For a computer capable of operations per second, this computation still requires seconds to compute, a quantity much larger than the estimated age of the universe, seconds [?]. Thus, this exponential limit is a serious barrier in computation. 2 Classical Physics, Complexity Theory, and Massive Parallelism 2.1 Computation is Physical At its core, all computation is inherently physical. Whether computation exists as marks on paper or neurons firing in the cortex of a mathematician s - physicist s brain, computation exists in some physical form. In the previous section the importance of worst-case running time was introduced with the idea of minimizing some physical resources. Without this restriction, the physical mechanism of computation would be of little importance and one could simply use some mechanism and be guaranteed an answer given some infinitude of resources. However, in the practical world we desire to minimize a limited number of resources through the mechanism of computation, and thus the physics of this mechanism becomes of paramount importance. Computation until very recently has operated on classical physics, obeying the equations of Newton and Maxwell. Reexamining the problems introduced in the opening of this paper, one 2

3 finds that the solutions to these problems are very dependent on classical physics. Specifically, in the unordered search problem classical physics dictates we must examine each element individually, and thus places an strict limit on the worst-case number of computations to be performed. Fortunately, the world as we currently know it is not strictly classical; the world behaves according to quantum mechanics. In the quantum mechanical world entities need not be unique and fixed, rather, entities can exist as a coherent superposition of states. Herein lies an important difference between quantum mechanics and the classical world:multiple states can be treated as one entity [?]. Whether this difference may be exploited to fundamentally change computation is explored in the remainder of this paper. 2.2 Classical Models of Computation Modern complexity theory, the study of resource consumption in computation, is based on both the Universal Turing Machine and the Church-Turing Thesis. A turing machine is a theoretical construct consisting of an infinitely long tape Γ, subdivided into cells on which a single symbol from the set Σ may be written. A read/write head rests at some cell on the tape and exists in some state q n out of a finite set of possible states Q. This machine is governed by a set of transition rules known as finite state controls which examine the current symbol under the read/write head, determine whether to overwrite this symbol with another in Σ, and move the head to the right or left if desired [?]. According to the quasimathematical hypothesis proposed by Alonzo Church and Alan Turing (known as the Church- Turing Thesis), the Turing Machine has the same ability to compute as any other means of computation. Formally, this means that there exists some function f that reformulates a problem computable by some computer M 1 to one computable by a Turing Machine, M T. This concept of a maximally powerful computational device is referred to as a Universal Turing Machine (UTM). According to Turing s formulation of the Church-Turing Thesis, Every function which would naturally be regarded as computable can be computed by the universal Turing machine. [?] Work has been done to show certain systems of computation equivalent to UTM s. The most important of these is the class of computations performable by Random Access Machines and High-level Programming languages, the modern method of computation [?]. As a result, if we desire to compute more efficiently we must demonstrate the existence of more powerful model of computation and perhaps alter the Church-Turing Thesis. 3

4 Figure 1: A visual representation of a Turing Machine. Note the tape Γ at the top, the finite set of states Q, and the read/write head (currently in state q 1 ). A finite state control may change the state in the read/write head to some other in Q, rewrite the symbol on the tape (currently 0), and move the head either to the left or the right. 2.3 Classical Enhancements to the Turing Machine One method popularly employed to solve computationally intensive problems is that of parallel computation, in which more than one processing device is employed to simultaneously solve different elements of the same problem. For especially difficult problems, the recent rise of massive parallel processing environments in which hundreds and thousands of processing units are employed has allowed for tenable solutions 2 [?]. However, parallelism introduces no fundamental change in the UTM. Rather, parallel computation relies on the simultaneous use of more than one UTM. Unless the number of UTMs scales with the number of computations to be performed, however, parallelism offers no more than a constant decrease in resource usage. Moreover, scaling with the size of the number of computations is not physically reasonable, even if the number of operations grows like a small polynomial with the input. Thus, parallelism offers no fundamental change in computation. The probabilistic Turing Machine (PTM), in which more than one finite state control may exist for a certain state, each governed by a probability, actually created a fundamental change in the Turing Machine model as well as a strengthening of the Church-Turing Thesis [?]. However, many classes of problems 3 still remain intractable in that they take a number of operations which is exponential in the size of the input. Moreover, in 1982 Richard Feynman demonstrated that PTMs were incapable of simulating quantum physical systems without 2 For instance, the Accelerated Strategic Computing Initiative (ASCI) project based in Sandia National Laboratory allows the US government to test nuclear detonations computationally. 3 Most notably, the NP complete problems. 4

5 an exponential number of operations. This suggested to Feynman that a new model of computation, one based on quantum physics and quantum probabilities, would prove to be more powerful than all classical computational models [?]. 3 Entanglement and Complexity: Quantum Mechanics Improves the Situation? 3.1 Entanglement, Superposition, and Interference Methods of encoding information quantum mechanically are readily found. Quantum mechanics demonstrates numerous particles with observable characteristics that can exist in mutually exclusive binary states. For example, the spin operator S z when applied to the state for an electron in hydrogen, Ψ e 4, yields S z Ψ e = m s ħ Ψ e (m s = ±1/2)[?] (1) The existence of these binary states in nature gives rise to the ability to code information in a sort of quantum bit, or qubit. We designate the two states of the qubit as 0 and 1, corresponding to some physical observable. These states form an orthonormal space for two-dimensional Hilbert 5 space in which this qubit exists, such that [?] 0 0 = 1 1 = = 1 0 = 0 (2) Regarded as a new means to label classical bits, qubits offer no real advance in computation. The phenomenon of entanglement, however, is the key to using qubits to a computational advantage. Imagine two electrons, one spin up and one spin down, meeting and interacting 4 Here we introduce Dirac bracket notation. ψ n (pronounced ket or ket vector ) is an alternative representation to the wavefunction of a quantum state denoted ψ n. ψ n is a complete description of a quantum state denoted by the quantum numbers referred to as a collection as n. We interpret ψ n in the same manner as a wavefunction, that is, as a probability amplitude. The complex conjugate of ψ n is ψ n (pronounced bra ). We further represent ψ na op ψ n dτ = ψ n A op ψ n and ψ nψ n dτ = ψ n ψ n [?]. 5 Hilbert space is a possibly infinitely dimensional space in which each quantum number of a state ψ is a dimension. This space is a mathematical construct that behaves much like Cartesian space but is not limited to three real, continuous dimensions [?]. 5

6 outside of our observation and then separating from one another. Due to quantum mechanics, one will emerge spin up while the other spin down. However, due to the particles being identical we cannot know how they interacted and thus cannot predict which state we will measure a given particle in. Thus, we regard an entangled electron as being a superposition of states. In general, we represent this superposition of states as [?] Ψ = cos(θ/2) 0 ie iφ sin(θ/2) 1 (3) As a superposition of states, the 0 and 1 states can be acted upon simultaneously. This is an obvious advantage over classical computation, and proves to be the basis of quantum complexity reduction. Finally, note that quantum mechanics assigns complex coefficients to the eigenstates. This allows for interference effects, and cleverly applied interference can be used to remove or reduce the modulus the coefficients of some eigenstates and thus amplify others. This phenomenon is the basis of the quantum Fourier transform, which shall be shown to be of great importance in many quantum algorithms. 3.2 Quantum Turing Machines As previously established, the classically understood limits of computation cannot be broken (assuming the Church-Turing Thesis) without a fundamental change in the universal model of computation. In 1985, motivated by quantum theory, David Deutsch proposed a strengthening of the Church-Turing Thesis which he stated as [?] Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means. 6 Deutsch notes in his paper that Turing s conception of computation as a Turing Machine actually falls short of the computational power inherent in this stronger version of the thesis [?]. Clearly, a new model of computation which fulfills the strong version of this thesis is needed to discover the computational power allowed in a quantum mechanical world. Bernstein and Vazirani in a 1993 paper entitled Quantum Complexity Theory formalized a new, quantum model of computation built on the foundation laid out by Deutsch [?]. They proposed and formalized a quantum Turing Machine (QTM). This QTM differed from 6 Deutsch justifies this strengthening in a complex argument which essentially posits the physical reasonability of this thesis. 6

7 its classical counterpart the PTM through the use of quantum finite state controls. These quantum finite state controls consisted of δ such that δ(p, σ, τ, q, d) gives the probability amplitude of a machine in state in p reading the current symbol σ writing a symbol τ, transitioning to state q and moving a direction d. Thus, δ is a function such that δ : Q Σ Σ Q {L, R} C 7 (4) Note that the only difference between a QTM and a PTM is the mapping into C as opposed to [0, 1]. The effect of this change is to allow interference effects in quantum computation. To see such interference, assume a QTM in configurations c 1 and c 2 both may transition to state c with amplitude p. Clearly, starting the machine in one of these configurations and observing after a time step yields c with probability p. However, a QTM may be started in a superposition of states, α 1 c 1 + α 2 c 2. If we observe after a time step, we will observe c with a probability ( pα 1 ) + ( pα 2 ) 2 = p α 1 + α 2 2. Thus, interference results due to observation of this state can occur, even to the extent of making such a transition impossible (α 1 = α 2 ) [?]. A final result from Bernstein and Vazirani s paper is the construction of a universal QTM, thus providing for a new model of computation. This established a new, quantum limit on computational power. Knowing this, one is free to establish useful computational mechanism which are equivalent to the universal QTM [?]. 3.3 Elements of a Quantum Circuit Classical boolean logic tells us that any boolean operation can be composed of only two operations, AND and NOT. A result of the Schrödinger equation requires that any time evolution operator acting on Ψ must be unitary, that is, it must be reversible 8 [?]. This result requires that all quantum boolean operations be time reversible. The requirement of reversibility disallows the choice use of AND in a quantum logic circuit (AND is fed two bits of information and returns only one.). Happily, with the use of a two qubit controlled-not operator and a unitary rotation operator, a quantum computer retains this ability to impose any boolean operation on a qubit or set of qubits [?] 9. 7 Recall Q is the finite set of states, Σ is the finite alphabet of symbols used in writing to the tape Γ, and L and R represent left and right head moves, respectively. 8 In fact, a unitary time evolution operator is an equivalent form of the well-formedness condition on a QTM [?] 9 This is not a statement that quantum algorithms only use these two operations. Rather, this result states that all possible boolean operations we desire to use are available to us in that we can build them out of a 7

8 The controlled-not operation is typical of all quantum operators. For the reversibility requirement it returns the same number of arguments as it accepts 10. Its function is to flip the value of the target bit if and only if the control bit is 1. It is implemented as [?] ( denotes addition mod 2) CNOT c,t : a c b t a c a b, a, b = 0, 1 (5) Since quantum computing derives its power from the ability to represent states as a superposition of eigenstates, another important (unitary) quantum operator is [?] V (θ, φ) : 0 cos(θ/2) 0 ie iφ sin(θ/2) 1 1 cos(θ/2) 1 ie iφ sin(θ/2) 0 (6) To generalize, a quantum circuit takes as input an array of qubits a (called a register) and an output register with all bits in the 0 state 11. The quantum computation F is then computed such that the process yields [?] a 0 a F (a). (7) Observe that if the register a is composed of qubits in a superposition of 0 and 1, and a is a register of size L qubits, then this process yields every 2 L output in a superposition. This is the power of quantum computation that separates it from its classical subset. 4 Quantum Algorithms 4.1 Two Problems Redux The beginning of this paper introduced two computational problems, the first of which was the unordered search problem. This problem was examined in a paper by L. Grover in 1996 in which he proposed a quantum algorithm that required less than the classically required N steps in the worst case. This algorithm works by letting some state ψ 0 be a superposition of all states to be searched. Then, a function which negates the amplitude of the desired state (the state associated to a given key) can be applied to this state ψ 0. However, before a measurement can be made another operation is required to amplify the amplitude of the composition of operations that we know how to build already. 10 Feynman demonstrated that the number of extra arguments that needed to be carried through a quantum boolean operation is in general never more than the input itself [?] 11 Hughes demonstrates that such a system is readily created. 8

9 keyed state. For this operation, Grover applied an inversion around the mean defined to take the superposition Σ k α k k to the state Σ k β k k where β k = α k + α and α represents the mean of α 12 k. Grover showed that π N/4 repetitions of these two steps were required to find the arbitrary element [?]. This quantum algorithm illustrates the clear computational advantage of a quantum computer over its classical analog, demonstrating an asymptotically faster 13 implementation of the same algorithm. The second problem introduced in the beginning of the paper was that of the Traveling Salesperson. While the quantum algorithm for this problem is outside the scope of this paper, the result of this algorithm is pertinent to the discussion of quantum computational power. In a 1998 paper Daniel S. Abrams and Seth Lloyd constructed a set of two qubit unitary quantum operators which demonstrated that the class of problems equivalent to the Traveling Salesperson problem 14 was solvable in a polynomial, and in fact a linear number of operations in the input (i.e the number of cities in the Salesperson problem) [?]. While this result is produces an asymptotically faster algorithm in much the same manner as the unordered search problem, this result is of greater importance to computer science because the reduction in the number of steps is exponential and thus opens the possibility of the computation of useful but previously intractable problems. Thus, for this important class of problems the ability to compute a superposition of answers proves to be an incredible increase to the power of computation. 4.2 Quantum Cryptography Keeping data secret in a world of information requires cryptography, or message encoding. Classical methods of two-way, public-key cryptography rely on the intractability (computational difficulty) of factoring two large prime numbers. While given someone s public key (a number available freely which anyone may use to encrypt a message to this person) it is theoretically feasible to calculate that person s private key (a number held privately by this person which allows him or her to decrypt messages encrypted with his or her public key), the difficulty of factoring large primes makes this task almost impossible given a large enough key value. Specifically, factoring requires N operations, where N is the number to be factored. 12 As an exercise to the reader, consider the algorithm acting on N = 4, i.e. using the set x = {0, 1, 2, 3}, where the chosen element x 0 = 3 and ψ 0 = ( )/2. One application of the flip and inversion operations will yield exactly the state ψ 1 = Saying that f(n) is asymptotically faster than g(n) implies that lim N f(n)/g(n) = 0, where f(n) and g(n) represent the number of steps required to perform a computation. 14 Referred to as NP-Complete problems. 9

10 For a 100 digit number (N ), this is not a tractable problem. Number theory offers an alternate and efficient method of factoring large numbers. Given r where r is the period of F (x) N = a x (mod N) (that is, the smallest r for which a = a r (mod N)), one can use the Euclidean algorithm to find the greatest common divisor of a r/2 ± 1 and N [?]. While these mathematical details are of only ancillary interest, one should appreciate that given r this method very efficiently (in a polynomial number of steps in l where 10 l 1 N 10 l ) factors large numbers. Why this is not the classically employed solution rests on the fact that finding r is classically as difficult as brute force factoring[?]. Quantum computing provides a efficient method for finding this r, known as Shor s algorithm. Shor s algorithm operates by letting ψ 0 be a superposition of the sequence who s period is in question. Through a fairly technical series of operations known as the Quantum Fourier Transform (QTF), ψ 0 evolves into ψ 1 such that when ψ 1 is observed after this time evolution interference effects allow only those states which are multiples of 1/r to be observed. This allows the ready computation of r, and thus the factorization of N. As a result, the implementation of large scale quantum computers will quickly destroy modern cryptography. One does not, however, need to give up neither hope nor security. Quantum physics offers a solution to this cryptographic dilemma by allowing for cryptographic protocols which cannot be violated without first violating the laws of quantum mechanics. The mechanism of such protocols is the transmission of entangled pairs between two parties such that the pairs are correlated. Due to the nature of quantum mechanics, if some observer measures a transmitted member of this pair and retransmits it in an attempt to eavesdrop this observation will destroy the pair s correlation. Thus, the key to quantum cryptographic protocols is to publicly share the result of a random sample of the transmitted data. If any pair of this set of shared pairs fails to show a correlation, then the communication line can be declared insecure and a new method found. Hence, the inability to make measurements and preserve the measured system in quantum mechanics allows for the discovery of secure lines of communication which cannot be eavesdropped on without the knowledge of the communicating parties. 5 Modeling Reality: Quantum Computers May Pass the Turing Test In computer science there exists a famous test, dubbed the Turing Test, which stands as the accepted criteria for making the claim that a computer system simulates some physical 10

11 phenomenon. The conditions of the test require that an impartial observer interacts, through computer communication or some other means of communications which prevents this user from knowing with whom he or she is interacting, with both a human and another computer. The Turing Test states that if the impartial observer cannot differentiate between the human and the computer if allowed to ask some set of questions, then the computer is said to simulate the phenomena in question 15. While this paper looks at the question of how one can use physics to do computation, one might imagine asking the converse, that is, whether computation can be used to do physics. Through the Turing Test that was just introduced, we have a criterion by which to measure the answer to this question. Feynman, in a 1982 speech, showed that any attempt to model an N particle quantum system on a classical computer resulted in a computation requiring approximately N N computations [?]. While this computation can be made, the intractability of such a computation on even a modest input size would cause the impartial observer in the Turing Test to quickly tell apart the computation from physical measurement. Clearly, classical computation cannot simulate the physical world (at least, it cannot simulate the quantum mechanical world). Feynman also noted that individual quantum phenomena cannot be simulated at all since individual particles are subject to probabilities and thus their path cannot be predetermined. Thus, the question of the ability to do physics with computation becomes the question of whether one may use quantum computers to simulate macroscopic quantum phenomena. The answer to this problem, however, has yet to be resolved. 6 The Promise and the Reality Quantum computing is a fascinating, interdisciplinary exploration that seeks to answer the question of what human beings have in their power to know. However, like any exploration in physics, quantum computation only finds importance if it proves to be physically realizable. The principles which enable the theoretical concept of quantum computation have been shown in experiment, i.e. one can do experiments on the effects of entanglement. Moreover, small quantum algorithms have begun to be implemented. For example, the unordered search problem mentioned in this paper was solved for a n = 7 system in 35 milliseconds using techniques of nuclear magnetic resonance spectroscopy [?]. However, systems of the size needed to implement Shor s algorithm have yet to appear and it is not at all clear whether 15 This criterion is the standard by which computer scientists judge artificial intelligence. So far, no general artificial intelligence has passed this test. 11

12 they will. Finally, questions of our ability to simulate physics on a quantum computer have yet to be resolved. All of these questions rest on our understanding of the new form of the Church-Turing Thesis. Whether this thesis correctly describes the power of computation one may obtain, however, will remain an open question. References [1] Abrams, D.S. and S. Lloyd. Computational Complexity and Physical Law. Lecture Notes in Computer Science 1509: Quantum Computing and Quantum Communications: First NASA International Conference, QCQC 98 Palm Springs, CA USA, Feb 17-20, 1998 Selected Papers. Berlin: Springer, [2] Accelerated Strategic Computing Initiative (ASCI). < [3] Atallah, Mikhail J. ed. Algorithms and Theory of Computation Handbook. London: CRC Press LLC, [4] Baggot, Jim. The Meaning of Quantum Theory. Oxford: Oxford University Press, [5] Bennett, Charle H. Quantum Information Theory. Feynman and Computation: Exploring the Limits of Computers. Anthony J.G. Hey. ed. Reading, Mass.: Perseus Books, [6] Bernstien, Ethan and Umesh Vazirani. Quantum Complexity Theory. Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing, May 16-18, 1993, San Diego, CA USA. < bernstein/> [7] Chuang, Isaac L. et al. Experimental Implementation of Fast Quantum Searching. Physical Review Letters, Volume 80, Issue 15. April 13, pp [8] Deutsch, David. Quantum theory, the Church-Turing principle, and the universal quantum computer. Proceedings of the Royal Society of London., Vol A400, 1985, pp [9] Eisberg, Robert and Robert Resnick. Quantum Physics: Of Atoms, Molecules, Solids, Nuclei, and Particles. 2 nd Ed.. New York: John Wiley & Sons,

13 [10] Feynman, Richard P. Simulating Physics with Computers. Feynman and Computation: Exploring the Limits of Computers. Anthony J.G. Hey. ed. Reading, Mass.: Perseus Books, [11] Hughes, Richard J. Quantum Computation. Feynman and Computation: Exploring the Limits of Computers. Anthony J.G. Hey. ed. Reading, Mass.: Perseus Books, [12] Landauer, Rolf. Information is Inevitably Physical. Feynman and Computation: Exploring the Limits of Computers. Anthony J.G. Hey. ed. Reading, Mass.: Perseus Books,