We establish asymptotics of growing one dimensional self-similar fractal graphs, they are networks that allow multiple weighted edges between nodes, in terms of quantum central limit theorems for algebraic probability spaces in pure state. An additional structure is endowed with the repeating units of centro-symmetric Jacobians in the adjacency of a linear graph creating a self-similar fractal. The family of fractals induced by centro-symmetric Jacobians formulated as orthogonal polynomials that satisfy three term recurrence relations support such limits. The construction proceeds with the interacting fock spaces, T-algebras endowed with a quantum probability space, corresponding to the Jacobi coefficients of the recurrence relations and when some elements of the centro-symmetric matrix are constrained in a specific way we obtain, as the same Jacobian structure is repeated, the central limits. The generic formulation of Leonard pairs that form bases of conformal blocks and probablistic laplacians used in physics provide choice of centro-symmetric Jacobians widening the applicability of the result. We establish that the T-algebras of these 1D fractals, as they form a special class of distance-regular graphs, are thin and the induced association schemes are self-duals that lead to anyon systems with modular invariance.
We investigate whether long-run persistent chain Monte Carlo simulation of Langevin dynamics improves the quality of the representations achieved by energy-based models (EBM). We consider a scheme wherein Monte Carlo simulation of a diffusion process using a trained EBM is used to improve the adversarial robustness and the calibration score of an independent classifier network. Our results show that increasing the computational budget of Gibbs sampling in persistent contrastive divergence improves the calibration and adversarial robustness of the model, elucidating the practical merit of realizing new quantum and classical hardware and software for efficient Gibbs sampling from continuous energy potentials.
In this paper, we study Grover's search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover's algorithm on graphs that can be summarized as follows: instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph Laplacians. As a result, we search for the most appropriate analytical structure on graphs endowed with fixed topologies yielding better search outcomes. We discuss strategies to investigate the optimality of Grover's algorithm and provide an example with an easy tunable graph Laplacian to investigate our ideas.
We clarify the relations between the mathematical structures that enable fashioning quantum walks on regular graphs and their realizations in anyonic systems. Our protagonist is association schemes that may be synthesized from type-II matrices which have a canonical construction of subfactors. This way we set up quantum walks on growing distance-regular graphs induced by association schemes via interacting Fock spaces and relate them to anyon systems described by subfactors. We discuss in detail a large family of graphs that may be treated within this approach. Classification of association schemes and realizable anyon systems are complex combinatorial problems and we tackle a part of it with a quantum walk application based approach.
We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular and distance-transitive). The construction is valid for growing graphs and the interacting Fock space is well defined asymptotically for the growing graph. To realize the quantum walks defined on the graphs in terms of anyons we switch to the dual view of the association schemes and identify the corresponding modular tensor categories from the Bose-Mesner algebra. Informally, the fusion ring induced by the association scheme and a topological twist can be the basis for developing a modular tensor category and thus a system of anyons. Finally, we demonstrate the framework in the case of Grover quantum walk on distance-regular graph in terms of anyon systems for the graphs considered. In the dual perspective interacting Fock spaces gather a new meaning in terms of any
We construct relativistic quantum Markov semigroups from covariant completely positive maps. We proceed by generalizing a step in Stinespring's dilation to a general system of imprimitivity and basing it on Poincaré group. The resulting noise channels are relativistically consistent and the method is applicable to any fundamental particle, though we demonstrate it for the case of light-like particles. The Krauss decomposition of the relativistically consistent completely positive identity preserving maps (our set up is in Heisenberg picture) enables us to construct the covariant quantum Markov semigroups that are uniformly continuous. We induce representations from the little groups to ensure the quantum Markov semigroups that are ergodic due to transitive systems imprimitivity.
The quantum walk formalism is a widely used and highly successful framework for modeling quantum systems, such as simulations of the Dirac equation, different dynamics in both the low and high energy regime, and for developing a wide range of quantum algorithms. Here we present the circuit-based implementation of a discrete-time quantum walk in position space on a five-qubit trapped-ion quantum processor. We encode the space of walker positions in particular multi-qubit states and program the system to operate with different quantum walk parameters, experimentally realizing a Dirac cellular automaton with tunable mass parameter. The quantum walk circuits and position state mapping scale favorably to a larger model and physical systems, allowing the implementation of any algorithm based on discrete-time quantum walks algorithm and the dynamics associated with the discretized version of the Dirac equation.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-qubit processor and address scalability to higher dimensions as well as larger quantum processors.
We report a scalable hybrid quantum-classical machine learning framework to build Bayesian networks (BN) that captures the conditional dependence and causal relationships of random variables. The generation of a BN consists of finding a directed acyclic graph (DAG) and the associated joint probability distribution of the nodes consistent with a given dataset. This is a combinatorial problem of structural learning of the underlying graph, starting from a single node and building one arc at a time, that fits a given ensemble using maximum likelihood estimators (MLE). It is cast as an optimization problem that consists of a scoring step performed on a classical computer, penalties for acyclicity and number of parents allowed constraints, and a search step implemented using a quantum annealer. We have assumed uniform priors in deriving the Bayesian network that can be relaxed by formulating the problem as an estimation Dirichlet parameters. We demonstrate the utility of the framework by applying to the problem of elucidating the gene regulatory network for the MAPK/Raf pathway in human T-cells using proteomics data where the concentration of proteins, nodes of the BN, are interpreted as probabilities.
We extend the spherical code based key distribution protocols to qudits with dimensions 4 and 16 by constructing equiangular frames and their companions. We provide methods for equiangular frames in arbitrary dimensions for Alice to use and the companion frames, that has one antipode to eliminate one of the possibilities, made up of qudits with $N=4,16$ as part of Bob's code. Non-orthogonal bases that form positive operator valued measures can be constructed using the tools of frames (overcomplete bases of a Hilbert space) and here we apply them to key distribution that are robust due to large size of the bases making it hard for eavesdropping. We demonstrate a method to construct a companion frame for an equiangular tight frame for $\complexes^{p-1}$ generated from the discrete Fourier transform, where $p$ is any odd prime. The security analysis is based on the assumption restricting possible attacks to intercept/resend scenario highlighting the advantages of a qudit over qubit-based protocols.
We present a scheme to describe the dynamics of accelerating discrete-time quantum walk for one- and two-particle in position space. We show the effect of acceleration in enhancing the entanglement between the particle and position space in one-particle quantum walk and in generation of entanglement between the two unentangled particle in two-particle quantum walk. By introducing the disorder in the form of phase operator we study the transition from localization to delocalization as a function of acceleration. These inter-winding connection between acceleration, entanglement generation and localization along with well established connection of quantum walks with Dirac equation can be used to probe further in the direction of understanding the connection between acceleration, mass and entanglement in relativistic quantum mechanics and quantum field theory. Expansion of operational tools for quantum simulations and for modelling quantum dynamics of accelerated particle using quantum walks is an other direction where these results can play an important role.
In our previous work [1] we described quantized computation using Horn clauses and based the semantics, dubbed as entanglement semantics as a generalization of denotational and distribution semantics, and founded it on quantum probability by exploiting the key insight classical random variables have quantum decompositions. Towards this end we built a Hilbert space of H-interpretations and a corresponding non commutative von Neu- mann algebra of bounded linear operators [2]. In this work we extend the formalism using second-quantized Horn clauses that describe processes such as Heisenberg evolutions in optical circuits, quantum walks, and quantum filters in a formally verifiable way. We base our system on a measure theoretic approach to handle infinite dimensional systems and demonstrate the expressive power of the formalism by casting an algebra used to describe interconnected quantum systems (QNET) [5] in this language. The variables of a Horn clause bounded by universal or existential quantifiers can be used to describe parameters of optical components such as beam splitter scattering paths, cavity detuning from resonance, strength of a laser beam, or input and output ports of these components. Prominent clauses in this non commutative framework are Weyl predicates, that are operators on a Boson Fock space in the language of quantum stochastic calculus, martingales and conjugate Brownian motions compactly representing statistics of quantum field fluctuations. We formulate the- orem proving as a quantum stochastic process, more precisely a martingale, in Heisenberg picture of quantum mechanics, a sequence of goals to be proved, on the Herbrand base.
Here we show that to quantize any lumped element circuit, the circuit geometry must be included in a mathematical model of either the circuit fluxes or the circuit charges. By geometry of the circuit, we refer to the so-called parasitic inductances and capacitances that arise from the details of the circuit layout, which are well known to create noise in classical circuits. In contrast, the classical lumped element model describes only the topology of the circuit, which defines how different finite element variables are connected to one another by circuit components. By geometry we also refer to the fact that the quantum variables define the circuit geometry - some are outside the wire, some are inside the wire, and some are at boundary of the wire. Just as with classical circuits, these effects create noise; this noise arises in the form of high frequency components in the Hamiltonian that are difficult to accurately simulate using a lumped element model. The presentation is appropriate for undergraduate electrical and computer engineering students learning about quantum computing and physicists learning about electrical circuits.
We review both theoretical and experimental developments in the area of quantum games since the inception of the subject circa 1999. We will also offer a narrative on the controversy that surrounded the subject in its early days, and how this controversy has affected the development of the subject.
We demonstrate a Bayesian quantum game on an ion trap quantum computer with five qubits. The players share an entangled pair of qubits and perform rotations on their qubit as the strategy choice. Two five-qubit circuits are sufficient to run all 16 possible strategy choice sets in a game with four possible strategies. The data are then parsed into player types randomly in order to combine them classically into a Bayesian framework. We exhaustively compute the possible strategies of the game so that the experimental data can be used to solve for the Nash equilibria of the game directly. Then we compare the payoff at the Nash equilibria and location of phase-change-like transitions obtained from the experimental data to the theory, and study how it changes as a function of the amount of entanglement.
In the quantum version of prisoners' dilemma, each prisoner is equipped with a single qubit that the interrogator can entangle. We enlarge the available Hilbert space by introducing a third qubit that the interrogator can entangle with the other two. We discuss an enhanced interrogation technique based on tripartite entanglement and analyze Nash equilibria. We show that for tripartite entanglement approaching a W-state, there exist Nash equilibria that coincide with the Pareto optimal choice where both prisoners cooperate. Upon continuous variation between a W-state and a pure bipartite entangled state, the game is shown to have a surprisingly rich structure. The role of bipartite and tripartite entanglement is explored to explain that structure.
We study the dynamics of discrete-time quantum walk using quantum coin operations, $\hat{C}(\theta_1)$ and $\hat{C}(\theta_2)$ in time-dependent periodic sequence. For the two-period quantum walk with the parameters $\theta_1$ and $\theta_2$ in the coin operations we show that the standard deviation [$\sigma_{\theta_1, \theta_2} (t)$] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations $\theta_1$ or $\theta_2$, $\sigma_{\theta_1, \theta_2}(t) = \min \{\sigma_{\theta_1}(t), \sigma_{\theta_2}(t) \}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using $\theta_1$ and $\theta_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\theta=0$, here we show the recovery of the massless Dirac equation with non-zero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
We discuss an efficient physical realization of topological quantum walks on a finite lattice. The $N$-point lattice is realized with $\log_2 N$ qubits, and the quantum circuit utilizes a number of quantum gates which is polynomial in the number of qubits. In a certain scaling limit, we show that a large number of steps is implemented with a number of quantum gates which is independent of the number of steps. We ran the quantum algorithm on the IBM-Q five-qubit quantum computer, thus experimentally demonstrating topological features, such as boundary bound states, on a lattice with $N=4$ points.
We discuss the connection between a class of distributed quantum games, with remotely located players, to the counter intuitive Braess' paradox of traffic flow that is an important design consideration in generic networks where the addition of a zero cost edge decreases the efficiency of the network. A quantization scheme applicable to non-atomic routing games is applied to the canonical example of the network used in Braess' Paradox. The quantum players are modeled by simulating repeated game play. The players are allowed to sample their local payoff function and update their strategies based on a selfish routing condition in order to minimize their own cost, leading to the Wardrop equilibrium flow. The equilibrium flow in the classical network has a higher cost than the optimal flow. If the players have access to quantum resources, we find that the cost at equilibrium can be reduced to the optimal cost, resolving the paradox.
Fermi's golden rule applies to a situation in which a single quantum state $|\psi\rangle$ is coupled to a near-continuum. This "quasi-continuum coupling" structure results in a rate equation for the population of $|\psi\rangle$. Here we show that the coupling of a quantum system to the standard model of a thermal environment, a bath of harmonic oscillators, can be decomposed into a "cascade" made up of the quasi-continuum coupling structures of Fermi's golden rule. This clarifies the connection between the physics of the golden rule and that of a thermal bath, and provides a non-rigorous but physically intuitive derivation of the Markovian master equation directly from the former. The exact solution to the Hamiltonian of the golden rule, known as the Bixon-Jortner model, generalized for an asymmetric spectrum, provides a window on how the evolution induced by the bath deviates from the master equation as one moves outside the Markovian regime. Our analysis also reveals the relationship between the oscillator bath and the "random matrix model" (RMT) of a thermal bath. We show that the cascade structure is the one essential difference between the two models, and the lack of it prevents the RMT from generating transition rates that are independent of the initial state of the system. We suggest that the cascade structure is one of the generic elements of thermalizing many-body systems.
We analyze the probability distributions of the quantum walks induced from Markov chains by Szegedy (2004). The first part of this paper is devoted to the quantum walks induced from finite state Markov chains. It is shown that the probability distribution on the states of the underlying Markov chain is always convergent in the Cesaro sense. In particular, we deduce that the limiting distribution is uniform if the transition matrix is symmetric. In the cases of non-symmetric Markov chain, we exemplify that the limiting distribution of the quantum walk is not necessarily identical with the stationary distribution of the underlying irreducible Markov chain. The Szegedy scheme can be extended to infinite state Markov chains (random walks). In the second part, we formulate the quantum walk induced from a lazy random walk on the line. We then obtain the weak limit of the quantum walk. It is noted that the current quantum walk appears to spread faster than its counterpart-quantum walk on the line driven by the Grover coin discussed in literature. The paper closes with an outlook on possible future directions.
Quantum games with incomplete information can be studied within a Bayesian framework. We consider a version of prisoner's dilemma (PD) in this framework with three players and characterize the Nash equilibria. A variation of the standard PD game is set up with two types of the second prisoner and the first prisoner plays with them with probability p and 1-p respectively. The Bayesian nature of the game manifests in the uncertainty that the first prisoner faces about his opponent's type which is encoded either in a classical probability or in the amplitudes of a wave function. Here, we consider scenarios with asymmetric payoffs between the first and second prisoner for different values of the probability, p, and the entanglement. Our results indicate a class of Nash equilibria (NE) with rich structures, characterized by a phase relationship on the strategies of the players. The rich structure that can be exploited by the referee to set up rules of the game to push the players towards a specific class of NE. These results provide a deeper insight into the quantum advantages of Bayesian games over their classical counterpart.
Quantum games with incomplete information can be studied within a Bayesian framework. We analyze games quantized within the EWL framework [Eisert, Wilkens, and Lewenstein, Phys Rev. Lett. 83, 3077 (1999)]. We solve for the Nash equilibria of a variety of two-player quantum games and compare the results to the solutions of the corresponding classical games. We then analyze Bayesian games where there is uncertainty about the player types in two-player conflicting interest games. The solutions to the Bayesian games are found to have a phase diagram-like structure where different equilibria exist in different parameter regions, depending both on the amount of uncertainty and the degree of entanglement. We find that in games where a Pareto-optimal solution is not a Nash equilibrium, it is possible for the quantized game to have an advantage over the classical version. In addition, we analyze the behavior of the solutions as the strategy choices approach an unrestricted operation. We find that some games have a continuum of solutions, bounded by the solutions of a simpler restricted game. A deeper understanding of Bayesian quantum game theory could lead to novel quantum applications in a multi-agent setting.
It is known that placing a mechanical oscillator in a superposition of coherent states allows, in theory, a measurement of a linear force whose sensitivity increases with the amplitude of the mechanical oscillations, a uniquely quantum effect. Further, entangled versions of these states across a network of $n$ mechanical oscillators enables a measurement whose sensitivity increases linearly with $n$, thus improving over the classical scaling by $\sqrt{n}$. One of the key challenges in exploiting this effect is processing the signal so that it can be readily measured; linear processing is insufficient. Here we show that a Kerr oscillator will not only create the necessary states, but also perform the required processing, transforming the quantum phase imprinted by the force signal into a shift in amplitude measurable with homodyne detection. This allows us to design a relatively simple quantum electro-mechanical circuit that can demonstrate the core quantum effect at the heart of this scheme, amplitude-dependent force sensitivity. We derive analytic expressions for the performance of the circuit, including thermal mechanical noise and photon loss. We discuss the experimental challenges in implementing the scheme with near-term technology.
We derive the continuous-time limit of discrete quantum walks with topological phases. We show the existence of a continuous-time limit that preserves their topological phases. We consider both simple-step and split-step walks, and derive analytically equations of motion governing their behavior. We obtain simple analytical solutions showing the existence of bound states at the boundary of two phases, and solve the equations of motion numerically in the bulk.
Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. As shown by the examples in this article, the steady states of CTOQW can be very unusual and complicated even though the underlying graphs are simple. The examples demonstrate that the structure of a graph can affect quantum coherence in CTOQW through a long time run. Precisely, the quantum coherence persists throughout the evolution of the CTOQW when the underlying topology is certain irregular graphs (such as a path or a star as shown in the examples). In contrast, the quantum coherence will eventually vanish from the open quantum system when the underlying topology is a regular graph (such as a cycle).
We show that the ability to make direct measurements of momentum, in addition to the usual direct measurements of position, allows a simple configuration of two identical mechanical oscillators to be used for broadband back-action-free force metrology. This would eliminate the need for an optical reference oscillator in the scheme of Tsang and Caves [Phys. Rev. Lett. 105, 123601 (2010)], along with its associated disadvantages. We also show that if one is restricted to position measurements alone then two copies of the same two-oscillator configuration can be used for narrow-band back-action-free force metrology.
Associative memory models, in theoretical neuro- and computer sciences, can generally store a sublinear number of memories. We show that using quantum annealing for recall tasks endows associative memory models with exponential storage capacities. Theoretically, we obtain the radius of attractor basins, $R(N)$, and the capacity, $C(N)$, of such a scheme and their tradeoffs. Our calculations establish that for randomly chosen memories the capacity of a model using the Hebbian learning rule with recall via quantum annealing is exponential in the size of the problem, $C(N)=\mathcal{O}(e^{C_1N}),~C_1\geq0$, and succeeds on randomly chosen memory sets with a probability of $(1-e^{-C_2N}),~C_2\geq0$ with $C_1+C_2=(.5-f)^2/(1-f)$, where, $f=R(N)/N,~0\leq f\leq .5$ is the radius of attraction in terms of Hamming distance of an input probe from a stored memory as a fraction of the problem size. We demonstrate the application of this scheme on a programmable quantum annealing device - the Dwave processor.
We derive a dynamical bound on the propagation of correlations in local random quantum circuits - lattice spin systems where piecewise quantum operations - in space and time - occur with classical probabilities. Correlations are quantified by the Frobenius norm of the commutator of two positive operators acting on space-like separated local Hilbert spaces. For times $t=O(L)$ correlations spread to distances $\mathcal{D}=t$ growing, at best, diffusively for any distance within that radius with extensively suppressed distance dependent corrections whereas for $t=o(L^2)$ all parts of the system get almost equally correlated with exponentially suppressed distance dependent corrections and approach the maximum amount of correlations that may be established asymptotically.
Braneshop, and
the US National Science Foundation.
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