arXiv subjects

Thomas Decker

Publications and source records attributed to Thomas Decker.

25 records · Page 2Linked to original sources

arXiv2007

Efficient Quantum Algorithm for Hidden Quadratic and Cubic Polynomial Function Graphs

We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field F_p. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be multivariate polynomial functions of higher degree. For a fixed number of indeterminates and bounded total degree the Hidden Polynomial Function Graph Problem is hard on a classical computer as its black box query complexity is polynomial in p. In contrast, this problem can be reduced to a quantum state identification problem so that the resulting quantum query complexity does not depend on p. For univariate polynomials we construct a von Neumann measurement for distinguishing the states. We relate the success probability and the implementation of this measurement to certain classical problems involving polynomial equations. We present an efficient algorithm for hidden quadratic and cubic function graphs by establishing that the success probability of the measurement is lower bounded by a constant and that it can be implemented efficiently.

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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.

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arXiv2005

Symmetric measurements attaining the accessible information

A theorem of Davies states that for symmetric quantum states there exists a symmetric POVM maximizing the mutual information. To apply this theorem the representation of the symmetry group has to be irreducible. We obtain a similar yet weaker result for reducible representations. We apply our results to the double trines ensemble and show numerically that for this ensemble the pretty good measurement is optimal.

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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.

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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.

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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.

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arXiv2002

Performing joint measurements and transformations on several qubits by operating on a single control qubit

An n-qubit quantum register can in principle be completely controlled by operating on a single qubit that interacts with the register via an appropriate fixed interaction. We consider a hypothetical system consisting of n spin-1/2 nuclei that interact with an electron spin via a magnetic interaction. We describe algorithms that measure non-trivial joint observables on the register by acting on the control spin only. For large n this is not an efficient model for universal quantum computation but it can be modified to an efficient one if one allows n possible positions of the control particle. This toy model of measurements illustrates in which way specific interactions between the register and a probe particle support specific types of joint measurements in the sense that some joint observables can be measured by simple sequences of operations on the probe particle.

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