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Quantum logic gate constructions with one-bit "teleportation"

Xinlan Zhou, Debbie Leung, Isaac L. Chuang

Published Feb 14, 2000• 2 citations
OpenAlex

Abstract

We present a general method to construct quantum logic gates using one-bit``teleportation'' as a basic primitive, extending previous results based ontraditional two-bit teleportation (D. Gottesman and I. L. Chuang, Nature {\\bf402}, 390, 1999). This allows the realization of a variety of useful quantumoperations that cannot be directly performed due to some physical constraints.In particular, this technique leads to straightforward and systematicconstruction of many fault-tolerant encoded operations, including the $\\pi/8$and Toffoli gates, and remote quantum operations.

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We present a general method to construct quantum logic gates using one-bit``teleportation'' as a basic primitive, extending previous results based ontraditional two-bit teleportation (D. Gottesman and I. L. Chuang, Nature {\bf402}, 390, 1999). This allows the realization of a variety of useful quantumoperations that cannot be directly performed due to some physical constraints.In particular, this technique leads to straightforward and systematicconstruction of many fault-tolerant encoded operations, including the pi/8\\pi/8pi/8and Toffoli gates, and remote quantum operations.
We present a general method to construct fault-tolerant quantum logic gates with a simple primitive, which is an analog of quantum teleportation. The technique extends previous results based on traditional quantum teleportation [Gottesman and Chuang, Nature (London) 402, 390 (1999)] and leads to straightforward and systematic construction of many fault-tolerant encoded operations, including the $\ensuremath{\pi}/8$ and Toffoli gates. The technique can also be applied to the construction of remote quantum operations that cannot be directly performed.
We present a general method to construct a universal set of quantum gates using probabilistic teleportation as a basic primitive. The technique generalizes the teleportation method of gate construction to partially entangled quantum channels. Without recourse to local filtering or entanglement concentration, using local rotation and CNOT operations followed by measurements in the computational basis, one can construct many encoded quantum operations with unit fidelity but less than unit probability. The technique can also be applied to the construction of remote quantum gates that cannot be directly performed.
To build a general-purpose quantum computer, it is crucial for the quantum devices to implement classical boolean logic. A straightforward realization of quantum boolean logic is to use auxiliary qubits as intermediate storage. This inefficient implementation causes a large number of auxiliary qubits to be used. In this paper, we have derived a systematic way of realizing any general m-to-n bit combinational boolean logic using elementary quantum gates. Our approach transforms the m-to-n bit classical mapping into a t-bit unitary quantum operation with minimum number of auxiliary qubits, then a variation of Toffoli gate is used as the basic building block to construct the unitary operation. Finally, each of these building blocks can be decomposed into one-bit rotation and two-bit control-U gates. The efficiency of the network is taken into consideration by formulating it as a constrained set partitioning problem.
We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values (x,y) to (x,x\ensuremath{\bigoplus}y)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(2n{2}^{\mathit{n}}2n)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.
Quantum error correction is an essential tool for reliably performing tasks for processing quantum information on a large scale. However, integration into quantum circuits to achieve these tasks is problematic when one realizes that nontransverse operations, which are essential for universal quantum computation, lead to the spread of errors. Quantum gate teleportation has been proposed as an elegant solution for this. Here, one replaces these fragile, nontransverse inline gates with the generation of specific, highly entangled offline resource states that can be teleported into the circuit to implement the nontransverse gate. As the first important step, we create a maximally entangled state between a physical and an error-correctable logical qubit and use it as a teleportation resource. We then demonstrate the teleportation of quantum information encoded on the physical qubit into the error-corrected logical qubit with fidelities up to 0.786. Our scheme can be designed to be fully fault tolerant so that it can be used in future large-scale quantum technologies.
Deterministic and exact teleportation can be achieved via two partially entangled pairs of particles [Gu Y J 2005 Opt. Comm. ]. The key point of the protocol is a generalized measurement described by a positive operator-valued measure, which can be realized by performing a unitary operation in the extended space and a conventional Von Neumann orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates and thus provide a physical means to realize the teleportation. Our method for constructing quantum circuits differs from the usual methods based on decomposition of unitary matrices and is convenient for a large class of quantum processes involving generalized measurements.