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Physica Scripta

Quantum operations in probabilistic representation

A. Karpati, Петер Адам, József Janszky

Published Jul 1, 2009• 2 citations
Publisher

Abstract

Representing a single qubit by using a set of totally decohered mixed-state qubits is considered. A representation is constructed where physically realizable quantum mechanical transformations correspond approximately to unitary operations on the qubit. It is shown that the representation is valid for multiple qubit systems as well, and the precision of the approximation is sufficient for the realization of quantum algorithms.

DOI
10.1088/0031-8949/2009/t135/014054

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Representing a single qubit by using a set of totally decohered mixed-state qubits is considered. A representation is constructed where physically realizable quantum mechanical transformations correspond approximately to unitary operations on the qubit. It is shown that the representation is valid for multiple qubit systems as well, and the precision of the approximation is sufficient for the realization of quantum algorithms.
We consider one-particle spin states. We show that all the information contained in the state can be written in an equivalent form using three completely decohered mixed states. The density matrix of such a system has the form of tensor product of three diagonal matrices. The unitary operators defined in the space of one-particle spin states are represented by some transformation of tensor products of diagonal matrices. We determine the explicit form of the unitary transformations acting on the representing three-qubit system and corresponding basic operators acting in the space of the initial qubit states.
Abstract Quantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct classical simulation of quantum algorithms difficult. Here we introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares these basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of n bit qubits by 2(n bit + 1) classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. We implement several well-known quantum algorithms, analyse the scaling of the needed number of realizations with the number of qubits, and highlight the role of destructive interference for the cost of the emulation.
Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary representation in usual computational basis. This decomposition may be simplified greatly with the help of the properties of the finite-dimensional multiple-quantum operator algebra spaces of a quantum system and the specific properties of a given quantum algorithm. As elementary building blocks of quantum computation, the two-qubit diagonal gates and one-qubit gates may be constructed physically with one- and two-body interactions in a two-state quantum system and hence could be conveniently realized experimentally. The present work will be helpful for implementing generally any N-qubit quantum computation in those feasible two-state quantum systems and determining conveniently the time evolution of these systems in course of quantum computation.
We present a scheme for probabilistic transformation of special quantum states assisted by auxiliary qubits. In our scheme, if quantum states can be rewritten in a particular form, it is possible to transform such states into other states using lower-dimensional unitary operations that can be more easily realized in physical experiments. Furthermore, as an important application, we propose a generalized scheme that helps construct faithful quantum channels via various probabilistic channels when considering the existence of nonmaximally-entangled states.
In the paper is discussed complete probabilistic description of quantum systems with application to multiqubit quantum computations. In simplest case it is a set of probabilities of transitions to some fixed set of states. The probabilities in the set may be represented linearly via coefficients of density matrix and it is very similar with description using mixed states, but also may give some alternative view on specific properties of quantum circuits due to possibility of direct comparison with classical statistical paradigm.
The development of classical ergodic theory has had a significant impact on the areas of mathematics, physics, and, in general, applied sciences. The quantum ergodic theory of Hamiltonian dynamics has its motivations in understanding thermodynamics and statistical mechanics. The quantum channel, a completely positive trace-preserving map, represents the most general representation of quantum dynamics and is an essential aspect of quantum information theory and quantum computation. In this work, we study the ergodic theory of quantum channels by characterizing different levels of ergodic hierarchy from integrable to mixing. The quantum channels on single systems are constructed from the unitary operators acting on bipartite states and tracing out the environment. The interaction strength of these unitary operators measured in terms of operator entanglement provides sufficient conditions for the channel to be mixing. By using block-diagonal unitary operators, we construct a set of nonergodic channels. By using the canonical form of the two-qubit unitary operator, we analytically construct channels on a single qubit ranging from integrable to mixing. Moreover, we also study interacting many-body quantum systems that include the famous Sachdev-Ye-Kitaev model and show that they display mixing within the framework of the quantum channel.