Showing papers on "Quantum computer published in 1996"

A fast quantum mechanical algorithm for database search

Abstract: were proposed in the early 1980’s [Benioff80] and shown to be at least as powerful as classical computers an important but not surprising result, since classical computers, at the deepest level, ultimately follow the laws of quantum mechanics. The description of quantum mechanical computers was formalized in the late 80’s and early 90’s [Deutsch85][BB92] [BV93] [Yao93] and they were shown to be more powerful than classical computers on various specialized problems. In early 1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers. This is the problem of integer factorization, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) . ----------------------------------------------

Universal Quantum Simulators

Abstract: Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.

Good quantum error-correcting codes exist

Abstract: With the realization that computers that use the interference and superposition principles of quantum mechanics might be able to solve certain problems, including prime factorization, exponentially faster than classical computers @1#, interest has been growing in the feasibility of these quantum computers, and several methods for building quantum gates and quantum computers have been proposed @2,3#. One of the most cogent arguments against the feasibility of quantum computation appears to be the difficulty of eliminating error caused by inaccuracy and decoherence @4#. Whereas the best experimental implementations of quantum gates accomplished so far have less than 90% accuracy @5#, the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions. We hope that the techniques investigated in this paper can eventually be extended so as to reduce this quantity by several orders of magnitude. In the storage and transmission of digital data, errors can be corrected by using error-correcting codes @6#. In digital computation, errors can be corrected by using redundancy; in fact, it has been shown that fairly unreliable gates could be assembled to form a reliable computer @7#. It has widely been assumed that the quantum no-cloning theorem @8# makes error correction impossible in quantum communication and computation because redundancy cannot be obtained by duplicating quantum bits. This argument was shown to be in error for quantum communication in Ref. @9#, where a code was given that mapped one qubit ~two-state quantum system! into nine qubits so that the original qubit could be recovered perfectly even after arbitrary decoherence of any one of these nine qubits. This gives a quantum code on nine qubits with a rate 1

Multiple particle interference and quantum error correction

Abstract: The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1 − 2H(2x/n), but must be less than 1 − 2H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).

Tight bounds on quantum searching

Abstract: We provide a tight analysis of Grover''s recent algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Using techniques from Shor''s quantum factoring algorithm in addition to Grover''s approach, we introduce a new technique for approximate quantum counting, which allows to estimate the number of solutions. Finally we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.

Quantum Computation and Shor's Factoring Algorithm

Abstract: Current technology is beginning to allow us to manipulate rather than just observe individual quantum phenomena. This opens up the possibility of exploiting quantum effects to perform computations beyond the scope of any classical computer. Recently Peter Shor discovered an efficient algorithm for factoring whole numbers, which uses characteristically quantum effects. The algorithm illustrates the potential power of quantum computation, as there is no known efficient classical method for solving this problem. The authors give an exposition of Shor's algorithm together with an introduction to quantum computation and complexity theory. They discuss experiments that may contribute to its practical implementation. [S0034-6861(96)00303-0]

Fault-tolerant quantum computation

Abstract: It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1/log/sup c/t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1/t). We do this by showing that operations can be performed on quantum data encoded by quantum error-correcting codes without decoding this data.

Quantum measurements and the Abelian Stabilizer Problem

Abstract: We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.

Quantum Computers and Dissipation

Abstract: We analyse dissipation in quantum computation and its destructive impact on the efficiency of quantum algorithms. Using a general model of decoherence, we study the time evolution of a quantum register of arbitrary length coupled with an environment of arbitrary coherence length. We discuss relations between decoherence and computational complexity and show that the quantum factorization algorithm must be modified in order to be regarded as efficient and realistic.

Quantum networks for elementary arithmetic operations.

Abstract: Quantum computers require quantum arithmetic We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorizing algorithm We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized \textcopyright{} 1996 The American Physical Society

Information, physics, and computation

Abstract: This paper presents several observations on the connections between information, physics, and computation. In particular, the computing power of quantum computers is examined. Quantum theory is characterized by superimposed states and nonlocal interactions. It is argued that recently studied quantum computers, which are based on local interactions, cannot simulate quantum physics.

A quantum algorithm for finding the minimum

Abstract: Let T[0..N−1] be an unsorted table of N items, eachholding a value from an ordered set. For simplicity,assume that all values are distinct. The minimumsearchingproblem is to find the index ysuch that T[y]is minimum. This clearly requires a linear number ofprobes on a classical probabilistic Turing machine.Here, we give a simple quantum algorithm whichsolves the problem using O(√N) probes. The mainsubroutine is the quantum exponential searching al-gorithm of [2], which itself is a generalization ofGrover’s recent quantum searching algorithm [3].Due to a general lower bound of [1], this is withina constant factor of the optimum.

Fault-tolerant quantum computation

Abstract: Recently, it was realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties of realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, thus making long computations impossible. A futher difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering the output of long computations unreliable. It was previously known that a quantum circuit with t gates could tolerate O(1/t) amounts of inaccuracy and decoherence per gate. We show, for any quantum computation with t gates, how to build a polynomial size quantum circuit that can tolerate O(1/(log t)^c) amounts of inaccuracy and decoherence per gate, for some constant c. We do this by showing how to compute using quantum error correcting codes. These codes were previously known to provide resistance to errors while storing and transmitting quantum data.

Semiclassical Fourier transform for quantum computation.

Abstract: It is shown that the Fourier transform preceding the final measurement in Shor's algorithm for factorization on a quantum computer can be carried out in a semiclassical way by using the ``classical'' (macroscopic) signal resulting from measuring one bit to determine the type of measurement carried out on the next bit, and so forth. In this way all the two-bit gates in the Fourier transform can be replaced by a smaller number of one-bit gates controlled by classical signals. This suggests that it may be worthwhile looking for other uses of semiclassical methods in quantum computing.

Conscious events as orchestrated space-time selections

Abstract: What is consciousness? Some philosophers have contended that "qualia," or an experiential medium from which consciousness is derived, exists as a fundamental component of reality. Whitehead, for example, described the universe as being comprised of "occasions of experience." To examine this possibility scientifically, the very nature of physical reality must be re-examined. We must come to terms with the physics of space-time--as is described by Einstein's general theory of relativity--and its relation to the fundamental theory of matter--as described by quantum theory. This leads us to employ a new physics of objective reduction: " OR" which appeals to a form of quantum gravity to provide a useful description of fundamental processes at the quantum/classical borderline (Penrose, 1994; 1996). Within the OR scheme, we consider that consciousness occurs if an appropriately organized system is able to develop and maintain quantum coherent superposition until a specific "objective" criterion (a threshold related to quantum gravity) is reached; the coherent system then self-reduces (objective reduction: OR). We contend that this type of objective self-collapse introduces non-computability, an essential feature of consciousness. OR is taken as an instantaneous event--the climax of a self-organizing process in fundamental space-time--and a candidate for a conscious Whitehead "occasion" of experience. How could an OR process occur in the brain, be coupled to neural activities, and account for other features of consciousness? We nominate an OR process with the requisite characteristics to be occurring in cytoskeletal microtubules within the brain's neurons (Penrose and Hameroff, 1995; Hameroff and Penrose, 1995; 1996). In this model, quantum-superposed states develop in microtubule subunit proteins ("tubulins"), remain coherent and recruit more superposed tubulins until a mass-time-energy threshold (related to quantum gravity) is reached. At that point, self-collapse, or objective reduction (OR) abruptly occurs. We equate the pre-reduction, coherent superposition ("quantum computing") phase with pre-conscious processes, and each instantaneous (and non-computable) OR, or self-collapse, with a discrete conscious event. Sequences of OR events give rise to a "stream" of consciousness. Microtubule-associated-proteins can "tune" the quantum oscillations of the coherent superposed states; the OR is thus self-organized, or "orchestrated" ("Orch OR"). Each Orch OR event selects (non-computably) microtubule subunit states which regulate synaptic/neural functions using classical signaling. The quantum gravity threshold for self-collapse is relevant to consciousness, according to our arguments, because macroscopic superposed quantum states each have their own space-time geometries (Penrose, 1994; 1996). These geometries are also superposed, and in some way "separated," but when sufficiently separated, the superposition of space-time geometries becomes significantly unstable, and reduce to a single universe state. Quantum gravity determines the limits of the instability; we contend that the actual choice of state made by Nature is non-computable. Thus each Orch OR event is a self-selection of space-time geometry, coupled to the brain through microtubules and other biomolecules. If conscious experience is intimately connected with the very physics underlying space-time structure, then Orch OR in microtubules indeed provides us with a completely new and uniquely promising perspective on the hard problem of consciousness.

Efficient networks for quantum factoring.

Abstract: We consider how to optimize memory use and computation time in operating a quantum computer. In particular, we estimate the number of memory quantum bits (qubits) and the number of operations required to perform factorization, using the algorithm suggested by Shor [in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 124]. A K-bit number can be factored in time of order K3 using a machine capable of storing 5K+1 qubits. Evaluation of the modular exponential function (the bottleneck of Shor’s algorithm) could be achieved with about 72K3 elementary quantum gates; implementation using a linear ion trap would require about 396K3 laser pulses. A proof-of-principle demonstration of quantum factoring (factorization of 15) could be performed with only 6 trapped ions and 38 laser pulses. Though the ion trap may never be a useful computer, it will be a powerful device for exploring experimentally the properties of entangled quantum states.

Fault Tolerant Quantum Computation with Constant Error

Abstract: This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, $\eta$, is smaller than a constant threshold, $\eta_c$. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.

Multiple Particle Interference and Quantum Error Correction

Abstract: The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than 1 - 2 H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).

Quantum cryptographic network based on quantum memories.

Abstract: Quantum correlations between two particles show nonclassical properties that can be used for providing secure transmission of information. We present a quantum cryptographic system in which users store particles in a transmission center, where their quantum states are preserved using quantum memories. Correlations between the particles stored by two users are created upon request by projecting their product state onto a fully entangled state. Our system allows for secure communication between any pair of users who have particles in the same center. Unlike other quantum cryptographic systems, it can work without quantum channels and it is suitable for building a quantum cryptographic network. We also present a modified system with many centers. \textcopyright{} 1996 The American Physical Society.

Efficient Simulation of Quantum Systems by Quantum Computers

Abstract: We show that the time evolution of the wave function of a quantum mechanical many particle system can be implemented very efficiently on a quantum computer. The computational cost of such a simulation is comparable to the cost of a conventional simulation of the corresponding classical system. We then sketch how results of interest, like the energy spectrum of a system, can be obtained. We also indicate that ultimately the simulation of quantum field theory might be possible on large quantum computers. We want to demonstrate that in principle various interesting things can be done. Actual applications will have to be worked out in detail also depending on what kind of quantum computer may be available one day...

Conventions for quantum pseudocode

Abstract: A few conventions for thinking about and writing quantum pseudocode are proposed. The conventions can be used for presenting any quantum algorithm down to the lowest level and are consistent with a quantum random access machine (QRAM) model for quantum computing. In principle a formal version of quantum pseudocode could be used in a future extension of a conventional language.

Simulations of Many-Body Quantum Systems by a Quantum Computer

Abstract: We suggest that quantum computers can solve quantum many-body problems that are impracticable to solve on a classical computer.

Concatenated Quantum Codes

Abstract: One of the main problems for the future of practical quantum computing is to stabilize the computation against unwanted interactions with the environment and imperfections in the applied operations. Existing proposals for quantum memories and quantum channels require gates with asymptotically zero error to store or transmit an input quantum state for arbitrarily long times or distances with fixed error. In this report a method is given which has the property that to store or transmit a qubit with maximum error $\epsilon$ requires gates with error at most $c\epsilon$ and storage or channel elements with error at most $\epsilon$, independent of how long we wish to store the state or how far we wish to transmit it. The method relies on using concatenated quantum codes with hierarchically implemented recovery operations. The overhead of the method is polynomial in the time of storage or the distance of the transmission. Rigorous and heuristic lower bounds for the constant $c$ are given.

Quantum Physics and Computers

Abstract: Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers. This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This ‘quantum parallelism’ outstrips by far any parallelism that can be thought of in classical computation and is responsible for the ‘exponential’ speed-up of computation. Experimentally, however, it will be extremely difficult to ‘decouple’ a quantum computer from its environment. Noise fluctuations due to the outside world, no matte...

Quantum Computing: Dream or Nightmare?

Abstract: The principles of quantum computing were laid out about 15 years ago by computer scientists applying the superposition principle of quantum mechanics to computer operation. Quantum computing has recently become a hot topic in physics, with the recognition that a two‐level system can be presented as a quantum bit, or “qubit,” and that an interaction between such systems could lead to the building of quantum gates obeying nonclassical logic. (See PHYSICS TODAY, October 1995, page 24 and March 1996, page 21.)

Teleportation as a quantum computation

Abstract: An explicit quantum circuit is given to implement quantum teleportation. This circuit makes teleportation straightforward to anyone who believes that quantum computation is a reasonable proposition. It could also be genuinely used inside a quantum computer if teleportation is needed to move quantum information around. An unusual feature of this circuit is that there are points in the computation at which the quantum information can be completely disrupted by a measurement (or some types of interaction with the environment) without ill effects: the same final result is obtained whether or not these measurements takes place.

Realistic lower bounds for the factorization time of large numbers on a quantum computer

Abstract: We investigate the time T a quantum computer requires to factorize a given number dependent on the number of bits L required to represent this number We stress the fact that in most cases one has to take into account that the execution time of a single quantum gate is related to the decoherence time of the quantum bits (qubits) that are involved in the computation Although exhibited here only for special systems, this interdependence of decoherence and computation time seems to be a restriction in many current models for quantum computers and leads to the result that the computation time T scales much stronger with L than previously expected \textcopyright{} 1996 The American Physical Society

Low temperature phase diagrams for quantum perturbations of classical spin systems

Abstract: We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH (0) is diagonal in a basis ∣s〉=⊗ x ∣s x 〉 which may be labeled by the configurationss={sx} of a suitable classical spin system on ℤ d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH (0)(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH (0), provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.