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Chinese Control Conference (CCC)

Cost Analysis of Manipulating Multiple Qubit Systems

Ming Zhang

Published Jul 1, 2006• 0 citations
Publisher

Abstract

For a multiple qubit system, the dimension of the system is exponentially large in the number of qubits, one may ask whether the cost of manipulating multiple qubit systems is exponentially large in the number of qubits or not? It has been found that only n(n + 3)/2 control Hamilton will guarantee open-loop controllability of n-qubit systems when only single qubit operators and two-qubit-interactive operators are permitted. Further more, the minimum cost of deterministically generating multiple-qubit GHZ entangled state will be at most linearly large in the number of qubits. To some extent, one may predict that the "cost" of manipulating multiple-qubit systems is tolerable.

DOI
10.1109/chicc.2006.4347230

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For a multiple qubit system, the dimension of the system is exponentially large in the number of qubits, one may ask whether the cost of manipulating multiple qubit systems is exponentially large in the number of qubits or not? It has been found that only n(n + 3)/2 control Hamilton will guarantee open-loop controllability of n-qubit systems when only single qubit operators and two-qubit-interactive operators are permitted. Further more, the minimum cost of deterministically generating multiple-qubit GHZ entangled state will be at most linearly large in the number of qubits. To some extent, one may predict that the "cost" of manipulating multiple-qubit systems is tolerable.
We propose how to generate genuine multipartite entanglement of electron spin qubits in a chain of quantum dots using the naturally available single-qubit rotations and two-qubit Heisenberg exchange interaction in the system. We show that the minimum number of required operations to generate entangled states of the GHZ, cluster, and W type scales linearly with the number of qubits and estimate the fidelities of the generated entangled cluster states. As the required single and two-qubit operations have recently been realized, our proposed scheme opens the way for experimental investigation of multipartite entanglement with electron spin qubits.
What is the time-optimal way of using a set of control Hamiltonians to obtain a desired interaction? Vidal, Hammerer, and Cirac [Phys. Rev. Lett. 88,88,88, 237902 (2002)] have obtained a set of powerful results characterizing the time-optimal simulation of a two-qubit quantum gate using a fixed interaction Hamiltonian and fast local control over the individual qubits. How practically useful are these results? We prove that there are two-qubit Hamiltonians such that time-optimal simulation requires infinitely many steps of evolution, each infinitesimally small, and thus is physically impractical. A procedure is given to determine which two-qubit Hamiltonians have this property, and we show that almost all Hamiltonians do. Finally, we determine some bounds on the penalty that must be paid in the simulation time if the number of steps is fixed at a finite number, and show that the cost in simulation time is not too great.
A scheme is proposed for generating Greenberger-Horne-Zeilinger (GHZ) entangled states of multiple superconducting quantum-interference device (SQUID) qubits by the use of a microwave cavity. The scheme operates essentially by creating a single photon through an auxiliary SQUID built in the cavity and performing a joint multiqubit phase shift with assistance of the cavity photon. It is shown that entanglement can be generated using this method, deterministic and independent of the number of SQUID qubits. In addition, we show that the present method can be applied to preparing many atoms in a GHZ entangled state, with tolerance to energy relaxation during the operation.
As we have seen, even a single qubit is a surprisingly interesting object. However, the real power of quantum information processing begins with systems of two or more qubits. Before studying these in detail we need to expand our notation a little. Consider a system of two qubits, labeled a and b , each of which has two basis states, |0⧽ and |1⧽. The whole system then has four basis states, which can be written as |0 a 0 b ⧽, |0 a 1 b ⧽, |1 a 0 b ⧽ and |1 a 1 b ⧽, and can be found in any general superposition of these states, so that it occupies a four-dimensional Hilbert space. In the same way, a system of three qubits inhabits an eight-dimensional Hilbert space, and so on. This exponential increase in the size of the Hilbert space with a linear increase in the number of qubits underlies the power of quantum computers. Direct products The size of the Hilbert spaces involved can also be a huge problem, however, making it difficult to describe states of systems with many qubits. A partial solution is to note that some states can be described in a simpler way, using the concept of direct products . These states, in which the individual qubits can in principle be discussed separately, make up a tiny minority of the states accessible to a multi-qubit system, but include many important states, most notably the basis states. States of this kind are said to be separable , and states which are not separable are said to be entangled . Entangled states are much more interesting than separable ones, but it is wise to begin with the simpler case.
Unlike other quantum hardware, photonic quantum architectures can produce millions of qubits from a single device. However, controlling photonic qubits remains challenging, even at small scales, due to their weak interactions, making nondeterministic gates in linear optics unavoidable. Nevertheless, a single photon can readily spread over multiple modes and create entanglement within the multiple modes deterministically. Rudolph’s concept of third quantization leverages this feature by evolving multiple single photons into multiple modes, distributing them uniformly and randomly to different parties, and creating multipartite entanglement without interactions between photons or nondeterministic gates. This method requires only classical communication and deterministic entanglement within multimode single-photon states and enables universal quantum computing. The multipartite entanglement generated within the third-quantization framework is nearly deterministic, where “deterministic” is achieved in the asymptotic limit of a large system size. In this work, we propose a near-term experiment using antimony donor in a silicon chip to realize third quantization. Utilizing the eight energy levels of antimony, one can generate two eight-mode single-photon states independently and distribute them to parties. This enables a random multipartite Bell-state experiment, achieving a Bell state with an upper-bound efficiency of 87.5% among 56 random pairs without nondeterministic entangling gates. This approach opens alternative pathways for silicon-based photonic quantum computing.