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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

On the impossibility to perform a global unitary operation sequentially

Lucas Lamata, Juan León, David Pérez-Garcı́a, D. Salgado, E. Solano

Published Jun 1, 2009• 0 citations
Publisher

Abstract

In quantum information, sequential protocols are of utmost importance which involve a very demanding task, especially in the implementation of a global unitary acting on N qubits. In principle, this can be simplified by decomposing the unitary in sequential two-body interactions, between each qubit and an ancillary system, in a manner where each qubit is acted upon only once by the ancilla. This study shows that the sequential decomposition of a global unitary that entangles two or more qubits is impossible in general, even in the case of an infinite-dimensional ancilla.

DOI
10.1109/cleoe-eqec.2009.5192130

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In quantum information, sequential protocols are of utmost importance which involve a very demanding task, especially in the implementation of a global unitary acting on N qubits. In principle, this can be simplified by decomposing the unitary in sequential two-body interactions, between each qubit and an ancillary system, in a manner where each qubit is acted upon only once by the ancilla. This study shows that the sequential decomposition of a global unitary that entangles two or more qubits is impossible in general, even in the case of an infinite-dimensional ancilla.
We study the possibility for a global unitary applied on an arbitrary number of qubits to be decomposed in a sequential unitary procedure, where an ancillary system is allowed to interact only once with each qubit. We prove that sequential unitary decompositions are in general impossible for genuine entangling operations, even with an infinite-dimensional ancilla, being the controlled-NOT gate a paradigmatic example. Nevertheless, we find particular nontrivial operations in quantum information that can be performed in a sequential unitary manner, as is the case of quantum error correction and quantum cloning.
We consider the recently proposed ``no-go'' theorem of Lamata et al. [Phys. Rev. Lett. 101, 180506 (2008)] on the impossibility of sequential implementation of global unitary operations with the aid of an itinerant ancillary system and view the claim within the language of Kraus representation. By virtue of an extremely useful tool for analyzing entanglement properties of quantum operations, namely, operator-Schmidt decomposition, we provide alternative proof to the no-go theorem and also study the role of initial correlations between the qubits and ancilla in sequential preparation of unitary entanglers. Despite the negative response from the no-go theorem, we demonstrate explicitly how the matrix-product operator (MPO) formalism provides a flexible structure to develop protocols for sequential implementation of such entanglers with an optimal fidelity. The proposed numerical technique, which we call variational matrix-product operator (VMPO), offers a computationally efficient tool for characterizing the ``globalness'' and entangling capabilities of nonlocal unitary operations.
Not all unitary operations upon a set of qubits can be implemented by sequential interactions between each qubit and an ancillary system. We analyze the specific case of sequential quantum cloning, 1-->M, and prove that the minimal dimension D of the ancilla grows linearly with the number of clones M. In particular, we obtain D=2M for symmetric universal quantum cloning and D=M+1 for symmetric phase-covariant cloning. Furthermore, we provide a recipe for the required ancilla-qubit interactions in each step of the sequential procedure for both cases.
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.
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Schemes of universal quantum computation in which the interactions between the computational elements, in a computational register, are mediated by some ancillary system are of interest due to their relevance to the physical implementation of a quantum computer. Furthermore, reducing the level of control required over both the ancillary and register systems has the potential to simplify any experimental implementation. In this paper we consider how to minimise the control needed to implement universal quantum computation in an ancilla-mediated fashion. Considering computational schemes which require no measurements and hence evolve by unitary dynamics for the global system, we show that when employing an ancilla qubit there are certain fixed-time ancilla-register interactions which, along with ancilla initialisation in the computational basis, are universal for quantum computation with no additional control of either the ancilla or the register. We develop two distinct models based on locally inequivalent interactions and we then discuss the relationship between these unitary models and the measurement-based ancilla-mediated models known as ancilla-driven quantum computation.