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arXiv:0711.3652 · Physical Review Letters

Sequential Implementation of Global Quantum Operations

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

Published Oct 31, 2008• 32 citations
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Abstract

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.

DOI
10.1103/physrevlett.101.180506

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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.
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.
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.
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.
We draw attention to the existence of ``genuine'' fidelity gaps in an ancilla-assisted sequential decomposition of genuinely entangling isometry and unitary operations of quantum computing. The gaps arise upon a bipartite decomposition of a multiqubit operation within a one-way sequential recipe in which an ancillary system interacts locally and only once with each qubit in a row. Given the known ``no-go'' associated with such a theoretically and experimentally desirable decomposition, various figures of merit are introduced to analyze the optimal ``fidelity'' with which an arbitrary genuinely entangling operation may admit such a sequential decomposition. An efficient variational matrix-product-operator protocol is invoked in order to obtain numerically the minimal values of the fidelity gaps incurred upon the sequential decomposition of genuine entanglers. We call the values of the gaps so obtained ``genuine'' in light of possible connections to the concept of the genuine multipartite entanglement and since they turn out to be independent of the ancilla dimension and the initial states the associated isometries or unitaries act upon.
We characterize a model of universal quantum computation where the register (computational) qubits are controlled by ancillary qubits, using only a single fixed interaction between register and ancillary qubits. No additional access is required to the computational register and the dynamics of both the register and ancilla are unitary. This scheme is inspired by the measurement-based ancilla-driven quantum computation of Anders et al. [Phys. Rev. A 82, 020301(R) (2010)], but does not require measurements of the ancillas, and in this respect is similar to the original gate-based model of quantum computation. We consider what possible forms this ancilla-register interaction can take, with a proof that the interaction is necessarily locally equivalent to swap combined with an entangling controlled gate. We further show which Hamiltonians can create such interactions and discuss two examples; the two-qubit XY Hamiltonian and a particular case of the XXZ Hamiltonian. We then give an example of a simple, finite, and fault-tolerant gate set for universal quantum computation in this model.
We consider quantum circuits composed of single-qubit operations and global entangling gates generated by Ising-type Hamiltonians. It is shown that such circuits can implement a large class of unitary operators commonly used in quantum algorithms at a very low cost-using a constant or effectively constant number of global entangling gates. Specifically, we report constant-cost implementations of Clifford operations with and without ancillae, constant-cost implementation of the multiply-controlled gates with linearly many ancillae, and an O(log^{*}(n)) cost implementation of the n-controlled single-target gates using logarithmically many ancillae. This shows a significant asymptotic advantage of circuits enabled by the global entangling gates.