Dominik Janzing

Thermodynamic Cost of Reliability and Low Temperatures: Tightening Landauer's Principle and the Second Law

Landauers principle states that the erasure of one bit of information requires thefree energy kT ln 2. We argue that the reliability of the bit erasure process isbounded by the accuracy inherent in the statistical state of the energy source(“the resources”) driving the process. We develop a general framework describingthe “thermodynamic worth” of the resources with respect to reliable bit erasureor good cooling. This worth turns out to be given by the distinguishability of theresources state from its equilibrium state in the sense of a statistical inferenceproblem. Accordingly, Kullback—Leibler relative information is a decisivequantity for the “worth” of the resources state. Due to the asymmetry of relativeinformation, the reliability of the erasure process is bounded rather by the relativeinformation of the equilibrium state with respect to the actual state than by therelative information of the actual state with respect to the equilibrium state (whichis the free energy up to constants).

Ergodic Quantum Computing

We propose a (theoretical) model for quantum computation where the result can be read out from the time average of the Hamiltonian dynamics of a 2-dimensional crystal on a cylinder.The Hamiltonian is a spatially local interaction among Wigner–Seitz cells containing six qubits. The quantum circuit that is simulated is specified by the initialization of program qubits. As in Margolus’ Hamiltonian cellular automaton (implementing classical circuits), a propagating wave in a clock register controls asynchronously the application of the gates. However, in our approach all required initializations are basis states. After a while the synchronizing wave is essentially spread around the whole crystal. The circuit is designed such that the result is available with probability about 1/4 despite of the completely undefined computation step. This model reduces quantum computing to preparing basis states for some qubits, waiting, and measuring in the computational basis. Even though it may be unlikely to find our specific Hamiltonian in real solids, it is possible that also more natural interactions allow ergodic quantum computing.

Simulating Hamiltonians in quantum networks: Efficient schemes and complexity bounds

We address the problem of simulating pair-interaction Hamiltonians in n-node quantum networks where the subsystems have arbitrary, possibly different, dimensions. We show that any pair interaction can be used to simulate any other by applying sequences of appropriate local control sequences. Conditions on time optimal simulation are formulated in terms of spectral majorization of matrices characterizing the coupling parameters. Efficient schemes for decoupling and time reversal can be constructed from orthogonal arrays provided that the dimensions of all nodes are equal to the same prime power. Moreover, we consider a specific system of n harmonic oscillators with bilinear interaction. In this case, decoupling can efficiently be achieved using the combinatorial concept of difference schemes. For this type of interaction we present optimal schemes for inversion.

Quasi-order of clocks and their synchronism and quantum bounds for copying timing information

The statistical state of any (classical or quantum) system with nontrivial time evolution can be interpreted as the pointer of a clock. The quality of such a clock is given by the statistical distinguishability of its states at different times. If a clock is used as a resource for producing another one the latter can at most have the quality of the resource. We show that this principle, formalized by a quasi-order, implies constraints on many physical processes. Similarly, the degree to which two (quantum or classical) clocks are synchronized can be formalized by a quasi-order of synchronism. Copying timing information is restricted by quantum no-cloning and no-broadcasting theorems since classical clocks can only exist in the limit of infinite energy. We show this quantitatively by comparing the Fisher timing information of two output systems to the inputs timing information. For classical signal processing in the quantum regime our results imply that a signal looses its localization in time if it is amplified and distributed to many devices.

NON-IDENTITY-CHECK IS QMA-COMPLETE

We describe a computational problem that is complete for the complexity class QMA, a quantum generalization of NP. It arises as a natural question in quantum computing and quantum physics. Non-identity-check is the following decision problem: Given a classical description of a quantum circuit (a sequence of elementary gates), determine whether it is almost equivalent to the identity. Explicitly, the task is to decide whether the corresponding unitary is close to a complex multiple of the identity matrix with respect to the operator norm. We show that this problem is QMA-complete. A generalization of this problem is non-equivalence check: given two descriptions of quantum circuits and a description of a common invariant subspace, decide whether the restrictions of the circuits to this subspace almost coincide. We show that non-equivalence check is also in QMA and hence QMA-complete.

A Simple PromiseBQP-complete Matrix Problem

Let A be a real symmetric matrix of size N such that the number of non-zero entries in each row is polylogarithmic in N and the positions and the values of these entries are specified by an efficiently computable function. We consider the problem of estimating an arbitrary diagonal entry (A m ) j j of the matrix A m up to an error of e b m , where b is an a priori given upper bound on the norm of A and m and e are polylogarithmic and inverse polylogarithmic in N, respectively. We show that this problem is PromiseBQP-complete. It can be solved efficiently on a quantum computer by repeatedly applying measurements of A to the jth basis vector and raising the outcome to the mth power. Conversely, every uniform quantum circuit of polynomial length can be encoded into a sparse matrix such that some basis vector | ji corresponding to the input induces two different spectral measures depending on whether the input is accepted or not. These measures can be distinguished by estimating the mth statistical moment for some appropriately chosen m, i. e., by the jth diagonal entry of A m . The problem remains PromiseBQP-hard when restricted to matrices having only 1, 0, and 1 as entries. Estimating off-diagonal entries is also PromiseBQP-complete.

Entanglement generation via scattering of two particles with hard-core repulsion

We analyze the entanglement generation in a one-dimensional scattering process. The two colliding particles have a Gaussian wave function and interact by hard-core repulsion. In our analysis, results on the entanglement of two-mode Gaussian states are used. The produced entanglement depends in a nonobvious way on the ratio of masses and initial widths for the two particles. The asymptotic wave function of the two particles and its associated ellipse yield additional geometric insight into these conditions. We discuss the difference from the quantitative analysis of the amount of entanglement generated by beam splitters with squeezed light.

On the Computational Power of Molecular Heat Engines

AbstractA heat engine is a machine which uses the temperature difference between a hot and a cold reservoir to extract work. Here both reservoirs are quantum systems and a heat engine is described by a unitary transformation which decreases the average energy of the bipartite system. On the molecular scale, the ability of implementing a (good) unitary heat engine is closely connected to the ability of performing logical operations and classical computing. This is shown by several examples:n(1)The most elementary heat engine is a SWAP-gate acting on 1 hot and 1 cold two-level systems with different energy gaps.(2)An optimal unitary heat engine on a pair of 3-level systems can directly implement OR and NOT gates, as well as copy operations. The ability to implement this heat engine on each pair of 3-level systems taken from the hot and the cold ensemble therefore allows universal classical computation.(3)Optimal heat engines operating on one hot and one cold oscillator mode with different frequencies are able to calculate polynomials and roots approximately.(4)An optimal heat engine acting on 1 hot and n cold 2-level systems with different level spacings can even solve the NP-complete problem KNAPSACK. Whereas it is already known that the determination of ground states of interacting many-particle systems is NP-hard, the optimal heat engine is a thermodynamic problem which is NP-hard even for nnon-interacting spin systems. This result suggests that there may be complexity-theoretic limitations on the efficiency of molecular heat engines.n

Implementation of group-covariant positive operator valued measures by orthogonal measurements

We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given group-covariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.

Complexity of decoupling and time reversal for n spins with pair interactions: Arrow of time in quantum control

Well-known nuclear magnetic resonance experiments show that the time evolution according to (truncated) dipole-dipole interactions between n spins can be inverted by simple pulse sequences. Independent of n, the reversed evolution is only two times slower than the original one. Here we consider more general spin-spin couplings with long range. We prove that some are considerably more complex to invert since the number of required time steps and the slow-down of the reversed evolutions are necessarily of the order n. Furthermore, the spins have to be addressed separately. We show for which values of the coupling parameters the phase transition between simple and complex time-reversal schemes occurs.