I will generalize the resource-theory framework from heat exchanges to more-arbitrary thermodynamic interactions. I will introduce resource theories of non-equilibrium, a family of models that includes the Helmholtz theories. In addition to shedding new light on Helmholtz theories, the non-equilibrium framework expands the resource theories?
This research is motivated by the interplay between information and energy on small scales exemplified by single-molecule experiments, nanoscale engines, and molecular motors. This work was conducted partially arXiv Part of this work is being conducted with Joe Renes. Claudio Chamon Boston University Emergent irreversibility and entanglement spectrum statistics The well-known heuristic DMRG Density Matrix Renormalization Group has been the method of choice for the practical solution of 1D systems since its introduction two decades ago by Steve White.
However, the reasons for DMRG's success are not theoretically well understood and it is known to fail in certain cases. In this talk, I will describe the first polynomial time classical algorithm for finding ground states of 1D quantum systems described by gapped local Hamiltonians. The algorithm is based on a framework that combines recently discovered structural features of gapped 1D systems, convex programming, and a new and efficient construction of a class of operators called approximate ground state projections AGSP.
An AGSP-centric approach may help guide the search for algorithms for more general quantum systems, including for the central challenge of 2D systems, where even heuristic methods have had very limited success. Joint work with Zeph Landau and Thomas Vidick. Robin Kothari University of Waterloo Exponential improvement in precision for simulating sparse Hamiltonians We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods.
Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error.
We also significantly simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.
This is joint work with Dominic W. Berry, Andrew M. Childs, Richard Cleve, and Rolando D.
Andris Ambainis University of Latvia On physical problems that are slightly more difficult than QMA The complexity class QMA quantum counterpart of NP allows to understand the complexity of many computational problems in quantum physics but some natural problems appear to be slightly harder than QMA. We introduce new complexity classes consisting of problems that are solvable with a small number of queries to a QMA oracle and use these complexity classes to quantify the complexity of several natural computational problems for example, the complexity of estimating the spectral gap of a Hamiltonian.
However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. In this talk I will describe the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most square root of n qubits, where n is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight.
The proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. Finally, we apply the homological product to construct new small codes with low-weight stabilizers. This is a joint work with Matthew Hastings Preprint: arXiv Isaac Kim Perimeter Institute On the Informational Completeness of Local Observables For a general multipartite quantum state, we formulate a locally checkable condition, under which the expectation values of certain nonlocal observables are completely determined by the expectation values of some local observables.
The condition is satisfied by ground states of gapped quantum many-body systems in one and two spatial dimensions, assuming a widely conjectured form of area law is correct. Its implications on quantum state tomography, quantum state verification, and quantum error correcting code is discussed. Stephanie Wehner National University of Singapore The Second Laws of Quantum Thermodynamics The second law of thermodynamics tells us which state transformations are so statistically unlikely that they are effectively forbidden.
Its original formulation, due to Clausius, states that "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. Is there a second law of thermodynamics in this regime?
Here, we find that for processes which are cyclic or very close to cyclic, thesecond law for microscopic or highly correlated systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints.
In particular, we find that the quantum Renyi relative entropy distances to the equilibrium state can never increase. We further find that there are three regimes which govern which family of second laws govern state transitions, depending on how cyclic the process is.
- Eco-Efficiency, Regulation and Sustainable Business: Towards a Governance Structure for Sustainable Development (Esri Studies Series on the Environment);
- Office of Science | Department of Energy.
- An Introduction to Convex Polytopes.
In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. However, calculations are generally computationally complex and limited to weak interaction strengths. After an introduction to quantum field theory, I'll describe polynomial-time quantum algorithms for computing relativistic scattering amplitudes in both scalar and fermionic quantum field theories.
The algorithms achieve exponential speedup over known classical methods. One of the motivations for this work comes from computational complexity theory. Ultimately, we wish to know what is the computational power of our universe. Studying such quantum algorithms probes whether a universal quantum computer is powerful enough to represent quantum field theory; in other words, is quantum field theory in BQP?
Conversely, one can ask whether quantum field theory can represent a universal quantum computer; is quantum field theory BQP-hard? I'll describe our approach to addressing the question of BQP-hardness. Typically, one is interested in the unknown state generated during an experiment which can be repeated arbitrarily often in principle. However, the number of actual runs of the experiment, from which data is collected, is always finite and often small.
As pointed out recently, this may lead to unjustified or even wrong claims when employing standard statistical tools without care. In this talk, I will present a method for obtaining reliable estimates from finite tomographic data. Specifically, the method allows the derivation of confidence regions, i. For the main part of the talk, I study the following variant of the junta learning problem. We are given an oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined symmetric function h.
The task is to identify the variables the function depends on. This is a generalization of the Bernstein-Vazirani problem when h is the XOR function and the combinatorial group testing problem when h is the OR function. Additionally, I describe an application of these techniques for the problem of testing juntas, that is a joint work with Andris Ambainis, Oded Regev, and Ronald de Wolf.
Sean Hallgren Penn State visiting MIT A quantum algorithm for finding the group of units in arbitrary degree number fields Computing the unit group of a number field is one of the main problems in computational algebraic number theory. Polynomial time algorithms for problems for arbitrary degree number fields should be polynomial time in both the degree and log of the discriminant.
The best classical algorithms for computing the unit group takes time exponential in both parameters. There is a quantum algorithm that runs in time polynomial in log the discriminant but exponential in the degree. We give a quantum algorithm that is polynomial in both parameters of the number field. The proof works via a reduction to a continuous Hidden Subgroup Problem. Juerg Froehlich ETH Zurich Quantum probability theory After a brief general introduction to the subject of quantum probability theory, quantum dynamical systems are introduced and some of their probabilistic features are described.
Volume 64, Number 4
On the basis of a few general principles - "duality between observables and indeterminates", "loss of information" and "entanglement generation" - a quantum theory of experiments and measurements is developed, and the "theory of von Neumann measurements" is outlined. Finally, a theory of non-demolition measurements is sketched, and, as an application of the Martingale Convergence Theorem, it is shown how facts emerge in non-demolition measurements.
Matthias Troyer ETH Zurich Validating quantum devices About a century after the development of quantum mechanics we have now reached an exciting time where non-trivial devices that make use of quantum effects can be built. While a universal quantum computer of non-trivial size is still out of reach there are a number commercial and experimental devices: quantum random number generators, quantum encryption systems, and analog quantum simulators.
In this colloquium I will present some of these devices and validation tests we performed on them.
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Quantum random number generators use the inherent randomness in quantum measurements to produce true random numbers, unlike classical pseudorandom number generators which are inherently deterministic. Optical lattice emulators use ultracold atomic gases in optical lattices to mimic typical models of condensed matter physics. Finally, I will discuss the devices built by Canadian company D-Wave systems, which are special purpose quantum simulators for solving hard classical optimization problems.
Joseph Traub Columbia University on sabbatical at Harvard Algorithms and complexity for quantum computing We introduce the notion of strong quantum speedup. To compute this speedup one must know the classical computational complexity. What is it about the problems of quantum physics and quantum chemistry that enable us to get lower bounds on the classical complexity?
We then turn to a particular problem, the ground state of the time-independent Schroedinger equation for a system of p particles. The classical deterministic complexity of this problem is exponential in p. We provide an algorithm for solving this problem on a quantum computer with cost linear in p. Thus this problem can be easily solved on a quantum computer.
Some researchers in discrete complexity theory believe that quantum computation is not effective for eigenvalue problems. One of our goals is to explain this dissonance. We do not claim separation of the complexity hierarchy since our complexity estimates are obtained using specific kinds of oracle calls. We end with a selection of research directions and where to learn more. Thomas Vidick MIT Fully device-independent quantum key distribution The laws of quantum mechanics allow unconditionally secure key distribution protocols. Nevertheless, security proofs of traditional quantum key distribution QKD protocols rely on a crucial assumption, the trustworthiness of the quantum devices used in the protocol.
In device-independent QKD, even this last assumption is relaxed: the devices used in the protocol may have been adversarially prepared, and there is no a priori guarantee that they perform according to specification. Proving security in this setting had been a central open problem in quantum cryptography. We give the first device-independent proof of security of a protocol for quantum key distribution that guarantees the extraction of a linear amount of key even when the devices are subject to a constant rate of noise. The only assumptions required are that the laboratories in which each party holds his or her own device are spatially isolated, and that both devices as well as the eavesdropper, are bound by the laws of quantum mechanics.
At the heart of the security proof lie novel tools for the manipulation of a fundamental limitation of quantum entanglement, its monogamy. I will present the intuition relating monogamy and security and give an overview of our security proof. Based on joint work with U. Toby Cubitt University of Cambridge Undecidability of the spectral gap question The spectral gap of a quantum many-body Hamiltonian -- the difference between the ground state energy lowest eigenvalue and lowest excited state next-lowest eigenvalue, ignoring degeneracies in the thermodynamic limit limit of arbitrarily large system size -- plays an important role in determining the physical properties of a many-body system.
In particular, it determines the phase diagram of the system, with quantum phase transitions occurring at critical points where the spectral gap vanishes. A number of famous open problems in mathematical physics concern whether or not particular many-body models are gapped. For example, the "Haldane conjecture" states that Heisenberg spin chains are gapped for integer spin, and gapless for half-integer spin.