Quantum speed-up for unsupervised learning

We show how the quantum paradigm can be used to speed up unsupervised learning algorithms. More precisely, we explain how it is possible to accelerate learning algorithms by quantizing some of their subroutines. Quantization refers to the process that partially or totally converts a classical algorithm to its quantum counterpart in order to improve performance. In particular, we give quantized versions of clustering via minimum spanning tree, divisive clustering and k-medians that are faster than their classical analogues. We also describe a distributed version of k-medians that allows the participants to save on the global communication cost of the protocol compared to the classical version. Finally, we design quantum algorithms for the construction of a neighbourhood graph, outlier detection as well as smart initialization of the cluster centres.     

Experimental Demonstration of Quantum Lattice Gas Computation

We report an ensemble nuclear magnetic resonance (NMR) implementation of a quantum lattice gas algorithm for the diffusion equation. The algorithm employs an array of quantum information processors sharing classical information, a novel architecture referred to as a type-II quantum computer. This concrete implementa-tion provides a test example from which to probe the strengths and limitations of this new computation paradigm. The NMR experiment consists of encoding a mass density onto an array of 16 two-qubit quantum information processors and then following the computation through 7 time steps of the algorithm. The results show good agreement with the analytic solution for diffusive dynamics. We also describe numerical simulations of the NMR implementation. The simulations aid in determining sources of experimental errors, and they help define the limits of the implementation. PACS: 03.67.Lx; 47.11.+j; 05.60.-k     

Improving Gate-Level Simulation of Quantum Circuits

Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and the vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a novel data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, a useful subset of operator matrices and state vectors can be represented in a form that grows polynomially with the number of qubits. This subset contains, but is not limited to, any equal superposition of n qubits, any computational basis state, n-qubit Pauli matrices, and n-qubit Hadamard matrices. It does not, however, contain the discrete Fourier transform (employed in Shor's algorithm) and some oracles used in Grover's algorithm. We first introduce and motivate decision diagrams and QuIDDs. We then analyze the runtime and memory complexity of QuIDD operations. Finally, we empirically validate QuIDD-based simulation by means of a general-purpose quantum computing simulator QuIDDPro implemented in C++. We simulate various instances of Grover's algorithm with QuIDDPro, and the results demonstrate that QuIDDs asymptotically outperform all other known simulation techniques. Our simulations also show that well-known worst-case instances of classical searching can be circumvented in many specific cases by data compression techniques. PACS: 03.67.Lx, 03.65.Fd, 03.65.Vd, 07.05.Bx     

Quantum Genetic Optimization

The complexity of the selection procedure of a genetic algorithm that requires reordering, if we restrict the class of the possible fitness functions to varying fitness functions, is , where is the size of the population. The quantum genetic optimization algorithm (QGOA) exploits the power of quantum computation in order to speed up genetic procedures. In QGOA, the classical fitness evaluation and selection procedures are replaced by a single quantum procedure. While the quantum and classical genetic algorithms use the same number of generations, the QGOA requires fewer operations to identify the high-fitness subpopulation at each generation. We show that the complexity of our QGOA is in terms of number of oracle calls in the selection procedure. Such theoretical results are confirmed by the simulations of the algorithm.     

Exponential Separation of Quantum and Classical Online Space Complexity

Although quantum algorithms realizing an exponential time speed-up over the best known classical algorithms exist, no quantum algorithm is known performing computation using less space resources than classical algorithms. In this paper, we study, for the first time explicitly, space-bounded quantum algorithms for computational problems where the input is given not as a whole, but bit by bit. We show that there exist such problems that a quantum computer can solve using exponentially less work space than a classical computer. More precisely, we introduce a very natural and simple model of a space-bounded quantum online machine and prove an exponential separation of classical and quantum online space complexity, in the bounded-error setting and for a total language. The language we consider is inspired by a communication problem (the disjointness function) that Buhrman, Cleve and Wigderson used to show an almost quadratic separation of quantum and classical bounded-error communication complexity. We prove that, in the framework of online space complexity, the separation becomes exponential.     

Progress in Quantum Algorithms

We discuss the progress (or lack of it) that has been made in discovering algorithms for computation on a quantum computer. Some possible reasons are given for the paucity of quantum algorithms so far discovered, and a short survey is given of the state of the field. PACS: 03.67.Lx     

Thermodynamic Interpretation of the Quantum Error Correcting Criterion

Shannon's fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics. For example, Schumacher proved a quantum version of Shannon's 1948 classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur during the transmission, storage, and manipulation of quantum information. One can formulate a criterion of necessary and sufficient conditions for the errors to be detectable and correctable. We show that this criterion has a thermodynamical interpretation. PACS: 03.67; 05.30; 63.10     

Path Integration on a Quantum Computer

We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^-k with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an -approximation to path integrals whose integrands are at least Lipschitz. We prove: Path integration on a quantum computer is tractable. Path integration on a quantum computer can be solved roughly ^-1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.46 ^-1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. The number of qubits is polynomial in ^-1. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands. PACS: 03.67.Lx; 31.15Kb; 31.15.-p; 02.70.-c     

Approximate Quantum Error Correction

The errors that arise in a quantum channel can be corrected perfectly if and only if the channel does not decrease the coherent information of the input state. We show that, if the loss of coherent information is small, then approximate quantum error correction is possible. PACS: 03.67.H, 03.65.U     

Quantum computation with write-only memory

In classical computation, a “write-only memory” (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented with WOM can solve problems that neither a classical computer with WOM nor a quantum computer without WOM can solve, when all other resource bounds are equal. We focus on realtime quantum finite automata, and examine the increase in their power effected by the addition of WOMs with different access modes and capacities. Some problems that are unsolvable by two-way probabilistic Turing machines using sublogarithmic amounts of read/write memory are shown to be solvable by these enhanced automata.     

How Do Two Observers Pool Their Knowledge About a Quantum System?

In the theory of classical statistical inference one can derive a simple rule by which two or more observers may combine independently obtained states of knowledge together to form a new state of knowledge, which is the state which would be possessed by someone having the combined information of both observers. Moreover, this combined state of knowledge can be found without reference to the manner in which the respective observers obtained their information. However, we show that in general this is not possible for quantum states of knowledge; in order to combine two quantum states of knowledge to obtain the state resulting from the combined information of both observers, these observers must also possess information about how their respective states of knowledge were obtained. Nevertheless, we emphasize this does not preclude the possibility that a unique, well motivated rule for combining quantum states of knowledge without reference to a measurement history could be found. We examine both the direct quantum analog of the classical problem, and that of quantum state-estimation, which corresponds to a variant in which the observers share a specific kind of prior information. PACS: 03.67.-a, 02.50.-r, 03.65.Bz     

Quantum Ergodic Theory and Communication Channels

The recently developed theory of the noncommutative dynamical entropy is applied to the theory of quantum communication channels. It is argued that the speed of information transmission is bounded by the dynamical entropy of the information carrier treated as a quantum dynamical system. The proof is given for two classes of communication channels. For the first one the input and output are classical devices which produce strings of bits, while for the second one the input and output messages are quantum states.     

Quantum Algorithms for Weighing Matrices and Quadratic Residues

Abstract. In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to devise new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein and Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol χ (which indicates if an element of a finite field F _q is a quadratic residue or not). It is shown how for a shifted Legendre function f _s (i)=χ(i+s) , the unknown s ∈ F _q can be obtained exactly with only two quantum calls to f _s . This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log q + log ((1-ɛ )/2) queries to solve the same problem.     

Sharp Quantum versus Classical Query Complexity Separations

Abstract. We obtain the strongest separation between quantum and classical query complexity known to date—specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved exactly in the quantum case with a single query (and a polynomial number of auxiliary operations). The problem is simple to define and the quantum algorithm solving it is also simple when described in terms of certain quantum Fourier transforms (QFTs) that have natural properties with respect to the algebraic structures of finite fields. These QFTs may be of independent interest, and we also investigate generalizations of them to noncommutative finite rings.     

Visualization of the Quantum Fourier Transform Using a Quantum Computer Simulator

The quantum Fourier transform (QFT) is a key subroutine of quantum algorithms for factoring and simulation and is the heart of the hidden-subgroup problem, the solution of which is expected to lead to the development of new quantum algorithms. The QFT acts on the Hilbert space and alters the quantum mechanical phases and probability amplitudes. Unlike its classical counterpart its schematic representation and visualization are very dif.cult. The aim of this work is to develop a schematic representation and visualization of the QFT by running it on a quantum computer simulator which has been constructed in the framework of this research. Base states, superpositions of base states and entangled states are transformed and the corresponding schematic representations are presented. The visualization of the QFT presented here and the quantum computer simulator developed for this purpose may become a useful tool for introducing the QFT to students and researches without a strong background in quantum mechanics or Fourier analysis. PACS: 03.67.-a, 03.67.Lx     

Remotely Prepared Entanglement: a Quantum Web Page

Abstract. In quantum teleportation, an unknown quantum state is transmitted from one party to another using only local operations and classical communication, at the cost of shared entanglement. Is it possible similarly, using an N party entangled state, to have the state retrievable by any of the N-1 possible receivers? If the receivers cooperate, and share a suitable state, this can be done reliably. The N party GHZ is one such state; I derive a large class of such states, and show that they are in general not equivalent to the GHZ. I also briefly discuss the problem where the parties do not cooperate, and the relationship to multipartite entanglement quantification. I define a new set of entanglement monotones, the entanglements of preparation.     

The Quantum Setting with Randomized Queries for Continuous Problems

The standard setting of quantum computation for continuous problems uses deterministic queries and the only source of randomness for quantum algorithms is through measurement. Without loss of generality we may consider quantum algorithms which use only one measurement. This setting is related to the worst case setting on a classical computer in the sense that the number of qubits needed to solve a continuous problem must be at least equal to the logarithm of the worst case information complexity of this problem. Since the number of qubits must be finite, we cannot solve continuous problems on a quantum computer with infinite worst case information complexity. This can even happen for continuous problems with small randomized complexity on a classical computer. A simple example is integration of bounded continuous functions. To overcome this bad property that limits the power of quantum computation for continuous problems, we study the quantum setting in which randomized queries are allowed. This type of query is used in Shor’s algorithm. The quantum setting with randomized queries is related to the randomized classical setting in the sense that the number of qubits needed to solve a continuous problem must be at least equal to the logarithm of the randomized information complexity of this problem. Hence, there is also a limit to the power of the quantum setting with randomized queries since we cannot solve continuous problems with infinite randomized information complexity. An example is approximation of bounded continuous functions. We study the quantum setting with randomized queries for a number of problems in terms of the query and qubit complexities defined as the minimal number of queries/qubits needed to solve the problem to within ɛ by a quantum algorithm. We prove that for path integration we have an exponential improvement for the qubit complexity over the quantum setting with deterministic queries.     

Non-Markovian Quantum Error Deterrence by Dynamical Decoupling in a General Environment

A dynamical decoupling(DD) scheme for the prevention of errors in the non-Markovian (usually corresponding to low temperature, short time, and strong coupling) regimes suitable for qubits constructed out of a multilevel structure is studied. We use the effective spin-boson model (ESBM) introduced recently [K. Shiokawa and B. L. Hu, Phys. Rev. A70, 062106 (2004)] as a low temperature limit of the quantum Brownian oscillator model, where one can obtain exact solutions for a general environment with colored noises. In our decoupling scheme, a train of pairs of strong pulses are used to evolve the interaction Hamiltonian instantaneously. Using this scheme we show that the dynamical decoupling method can suppress 1/f noise with slower and hence more accessible pulses than previously studied, but it still fails to decouple super-Ohmic types of environments.      

Linking Classical and Quantum Key Agreement: Is There a Classical Analog to Bound Entanglement?

Abstract. After carrying out a protocol for quantum key agreement over a noisy quantum channel, the parties Alice and Bob must process the raw key in order to end up with identical keys about which the adversary has virtually no information. In principle, both classical and quantum protocols can be used for this processing. It is a natural question which type of protocol is more powerful. We show that the limits of tolerable noise are identical for classical and quantum protocols in many cases. More specifically, we prove that a quantum state between two parties is entangled if and only if the classical random variables resulting from optimal measurements provide some mutual classical information between the parties. In addition, we present evidence which strongly suggests that the potentials of classical and of quantum protocols are equal in every situation. An important consequence, in the purely classical regime, of such a correspondence would be the existence of a classical counterpart of so-called bound entanglement, namely ``bound information' that cannot be used for generating a secret key by any protocol. This stands in contrast to what was previously believed.