arXiv subjects

Dominik Janzing

Publications and source records attributed to Dominik Janzing.

At least 109 records · Page 6Linked to original sources

arXiv2007

Relating the thermodynamic arrow of time to the causal arrow

Consider a Hamiltonian system that consists of a slow subsystem S and a fast subsystem F. The autonomous dynamics of S is driven by an effective Hamiltonian, but its thermodynamics is unexpected. We show that a well-defined thermodynamic arrow of time (second law) emerges for S whenever there is a well-defined causal arrow from S to F and the back-action is negligible. This is because the back-action of F on S is described by a non-globally Hamiltonian Born-Oppenheimer term that violates the Liouville theorem, and makes the second law inapplicable to S. If S and F are mixing, under the causal arrow condition they are described by microcanonic distributions P(S) and P(S|F). Their structure supports a causal inference principle proposed recently in machine learning.

cond-mat.stat-mech↗

arXiv2006

Spin-1/2 particles moving on a 2D lattice with nearest-neighbor interactions can realize an autonomous quantum computer

What is the simplest Hamiltonian which can implement quantum computation without requiring any control operations during the computation process? In a previous paper we have constructed a 10-local finite-range interaction among qubits on a 2D lattice having this property. Here we show that pair-interactions among qutrits on a 2D lattice are sufficient, too, and can also implement an ergodic computer where the result can be read out from the time average state after some post-selection with high success probability. Two of the 3 qutrit states are given by the two levels of a spin-1/2 particle located at a specific lattice site, the third state is its absence. Usual hopping terms together with an attractive force among adjacent particles induce a coupled quantum walk where the particle spins are subjected to spatially inhomogeneous interactions implementing holonomic quantum computing. The holonomic method ensures that the implemented circuit does not depend on the time needed for the walk. Even though the implementation of the required type of spin-spin interactions is currently unclear, the model shows that quite simple Hamiltonians are powerful enough to allow for universal quantum computing in a closed physical system.

quant-ph↗

arXiv2006

BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs

We describe two BQP-complete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the difference between the number of paths of length m from j to j and those which from i to j, where m is polylogarithmic in the number of vertices. The scale of the estimation accuracy is specified by some a priori known upper bound on the growth of these differences with increasing m. The problem remains BQP-hard for regular graphs with degree 4. The second problem is related to continuous-time classical random walks. The walk starts at some vertex j. The promise is that the difference of the probabilities of being at j and at i, respectively, decays with O(exp(-μt)) for some μ>0. The problem is to decide whether this difference is greater than a exp(-μT) or smaller than b exp(-μT) after some time instant T, where T is polylogarithmic and the difference a-b is inverse polylogarithmic in the number of vertices. Since the probabilities differ only by an exponentially small amount, an exponential number of trials would be necessary if one tried to answer this question by running the walk itself. A modification of this problem, asking whether there exists a pair of nodes for which the probability difference is at least a exp(-μT), is QCMA-complete.

quant-ph↗

arXiv2006

Estimating diagonal entries of powers of sparse symmetric matrices is BQP-complete

Let A be a real symmetric matrix of size N such that the number of the 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)_jj of the matrix A^m up to an error of εb^m, where b is an a priori given upper bound on the norm of A, m and εare polylogarithmic and inverse polylogarithmic in N, respectively. We show that this problem is BQP-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 quantum circuit that solves a problem in BQP can be encoded into a sparse matrix such that some basis vector |j> 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 is still in BQP when generalized to off-diagonal entries and it remains BQP-hard if A has only -1, 0, and 1 as entries.

quant-ph↗

arXiv2006

A Quantum Broadcasting Problem in Classical Low Power Signal Processing

We pose a problem called ``broadcasting Holevo-information'': given an unknown state taken from an ensemble, the task is to generate a bipartite state transfering as much Holevo-information to each copy as possible. We argue that upper bounds on the average information over both copies imply lower bounds on the quantum capacity required to send the ensemble without information loss. This is because a channel with zero quantum capacity has a unitary extension transfering at least as much information to its environment as it transfers to the output. For an ensemble being the time orbit of a pure state under a Hamiltonian evolution, we derive such a bound on the required quantum capacity in terms of properties of the input and output energy distribution. Moreover, we discuss relations between the broadcasting problem and entropy power inequalities. The broadcasting problem arises when a signal should be transmitted by a time-invariant device such that the outgoing signal has the same timing information as the incoming signal had. Based on previous results we argue that this establishes a link between quantum information theory and the theory of low power computing because the loss of timing information implies loss of free energy.

quant-ph↗

arXiv2006

Minimally-disturbing Heisenberg-Weyl symmetric measurements using hard-core collisions of Schrödinger particles

In a previous paper we have presented a general scheme for the implementation of symmetric generalized measurements (POVMs) on a quantum computer. This scheme is based on representation theory of groups and methods to decompose matrices that intertwine two representations. We extend this scheme in such a way that the measurement is minimally disturbing, i.e., it changes the state vector \ketΨ of the system to \sqrtΠ \ketΨ where Πis the positive operator corresponding to the measured result. Using this method, we construct quantum circuits for measurements with Heisenberg-Weyl symmetry. A continuous generalization leads to a scheme for optimal simultaneous measurements of position and momentum of a Schr"odinger particle moving in one dimension such that the outcomes satisfy Δx Δp \geq \hbar. The particle to be measured collides with two probe particles, one for the position and the other for the momentum measurement. The position and momentum resolution can be tuned by the entangled joint state of the probe particles which is also generated by a collision with hard-core potential. The parameters of the POVM can then be controlled by the initial widths of the wave functions of the probe particles. We point out some formal similarities and differences to simultaneous measurements of quadrature amplitudes in quantum optics.

quant-ph↗

arXiv2006

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

We analyse 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 non-obvious way on the parameters ratio of masses and initial widths. The asymptotic wavefunction of the two particles and its associated ellipse yield additional geometric insight into these conditions. The difference to the quantitative analysis of the amount of entanglement generated by beam splitters with squeezed light is discussed.

quant-ph↗

arXiv2005

Quantum thermodynamics with missing reference frames: Decompositions of free energy into non-increasing components

If an absolute reference frame with respect to time, position, or orientation is missing one can only implement quantum operations which are covariant with respect to the corresponding unitary symmetry group G. Extending observations of Vaccaro et al., I argue that the free energy of a quantum system with G-invariant Hamiltonian then splits up into the Holevo information of the orbit of the state under the action of G and the free energy of its orbit average. These two kinds of free energy cannot be converted into each other. The first component is subadditive and the second superadditive; in the limit of infinitely many copies only the usual free energy matters. Refined splittings of free energy into more than two independent (non-increasing) terms can be defined by averaging over probability measures on G that differ from the Haar measure. Even in the presence of a reference frame, these results provide lower bounds on the amount of free energy that is lost after applying a covariant channel. If the channel properly decreases one of these quantities, it decreases the free energy necessarily at least by the same amount, since it is unable to convert the different forms of free energies into each other.

quant-ph↗

arXiv2005

On Quantum A/D and D/A Conversion

An algorithm is proposed which transfers the quantum information of a wave function (analogue signal) into a register of qubits (digital signal) such that $n$ qubits describe the amplitudes and phases of $2^n$ points of a sufficiently smooth wave function. We assume that the continuous degree of freedom couples to one or more qubits of a quantum register via a Jaynes Cummings Hamiltonian and that we have universal quantum computation capabilities on the register as well as the possibility to perform bang-bang control on the qubits. The transfer of information is mainly based on the application of the quantum phase-estimation algorithm in both directions. Here, the running time increases exponentially with the number of qubits. We pose it as an open question which interactions would allow polynomial running time. One example would be interactions which enable exact squeezing operations.

quant-ph↗

arXiv2005

On the Computational Power of Molecular Heat Engines

A 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 such a unitary heat engine is closely connected to the ability of performing logical operations and classical computing. This is shown by several examples: (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 n non-interacting spin systems. This result suggest that there may be complexity-theoretic limitations on the efficiency of molecular heat engines.

quant-ph↗

arXiv2004

Decomposition of time-covariant operations on quantum systems with continuous and/or discrete energy spectrum

Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over (densely defined) CP-maps G_σwhere σis the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each G_σis a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. As an example, I calculate this decomposition explicitly for the rotation invariant gaussian channel on a single mode. I address the question under what conditions a covariant channel destroys superpositions between mutually orthogonal states on the same orbit. For channels which allow mutually orthogonal output states on the same orbit, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (G_σ).

quant-ph↗

arXiv2004

Implementation of group-covariant POVMs 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 an implementation of a given group-covariant POVM by a quantum circuit using its symmetry. 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.

quant-ph↗

arXiv2004

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 6 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.

quant-ph↗

arXiv2003

Reliable and Efficient Inference of Bayesian Networks from Sparse Data by Statistical Learning Theory

To learn (statistical) dependencies among random variables requires exponentially large sample size in the number of observed random variables if any arbitrary joint probability distribution can occur. We consider the case that sparse data strongly suggest that the probabilities can be described by a simple Bayesian network, i.e., by a graph with small in-degree Δ. Then this simple law will also explain further data with high confidence. This is shown by calculating bounds on the VC dimension of the set of those probability measures that correspond to simple graphs. This allows to select networks by structural risk minimization and gives reliability bounds on the error of the estimated joint measure without (in contrast to a previous paper) any prior assumptions on the set of possible joint measures. The complexity for searching the optimal Bayesian networks of in-degree Δincreases only polynomially in the number of random varibales for constant Δand the optimal joint measure associated with a given graph can be found by convex optimization.

cs.LG↗

arXiv2003

Quantum circuits for single-qubit measurements corresponding to platonic solids

Each platonic solid defines a single-qubit positive operator valued measure (POVM) by interpreting its vertices as points on the Bloch sphere. We construct simple circuits for implementing this kind of measurements and other simple types of symmetric POVMs on one qubit. Each implementation consists of a discrete Fourier transform and some elementary quantum operations followed by an orthogonal measurement in the computational basis.

quant-ph↗

arXiv2003

Measuring 4-local n-qubit observables could probabilistically solve PSPACE

We consider a hypothetical apparatus that implements measurements for arbitrary 4-local quantum observables A on n qubits. The apparatus implements the ``measurement algorithm'' after receiving a classical description of A. We show that a few precise measurements, applied to a basis state would provide a probabilistic solution of PSPACE problems. The error probability decreases exponentially with the number of runs if the measurement accuracy is of the order of the spectral gaps of A. Moreover, every decision problem which can be solved on a quantum computer in T time steps can be encoded into a 4-local observable such that the solution requires only measurements of accuracy O(1/T). Provided that BQP<>PSPACE, our result shows that efficient algorithms for precise measurements of general 4-local observables cannot exist. We conjecture that the class of physically existing interactions is large enough to allow the conclusion that precise energy measurements for general many-particle systems require control algorithms with high complexity.

quant-ph↗

arXiv2003

Synchronizing quantum clocks with classical one-way communication: Bounds on the generated entropy

We describe separable joint states on bipartite quantum systems that cannot be prepared by any thermodynamically reversible classical one-way communication protocol. We argue that the joint state of two synchronized microscopic clocks is always of this type when it is considered from the point of view of an ``ignorant'' observer who is not synchronized with the other two parties. We show that the entropy generation of a classical one-way synchronization protocol is at least ΔS = \hbar^2/(4ΔE Δt)^2 if Δt is the time accuracy of the synchronism and ΔE is the energy bandwidth of the clocks. This dissipation can only be avoided if the common time of the microscopic clocks is stored by an additional classical clock. Furthermore, we give a similar bound on the entropy cost for resetting synchronized clocks by a classical one-way protocol. The proof relies on observations of Zurek on the thermodynamic relevance of quantum discord. We leave it as an open question whether classical multi-step protocols may perform better. We discuss to what extent our results imply problems for classical concepts of reversible computation when the energy of timing signals is close to the Heisenberg limit.

quant-ph↗

arXiv2003

Two QCMA-complete problems

QMA and QCMA are possible quantum analogues of the complexity class NP. In QCMA the verifier is a quantum program and the proof is classical. In contrast, in QMA the proof is also a quantum state. We show that two known QMA-complete problems can be modified to QCMA-complete problems in a natural way: (1) Deciding whether a 3-local Hamiltonian has low energy states (with energy smaller than a given value) that can be prepared with at most k elementary gates is QCMA-complete, whereas it is QMA-complete when the restriction on the complexity of preparation is dropped. (2) Deciding whether a (classically described) quantum circuit acts almost as the identity on all basis states is QCMA-complete. It is QMA-complete to decide whether it acts on all states almost as the identity.

quant-ph↗