Abstract
In this paper, we build a framework allowing for a systematic investigation of the fundamental issue: ‘Which quantum states serve as universal resources for measurement-based (one-way) quantum computation?’ We start our study by re-examining what is exactly meant by ‘universality’ in quantum computation, and what the implications are for universal one-way quantum computation. Given the framework of a measurement-based quantum computer, where quantum information is processed by local operations only, we find that the most general universal one-way quantum computer is one which is capable of accepting arbitrary classical inputs and producing arbitrary quantum outputs—we refer to this property as CQ-universality. We then show that a systematic study of CQ-universality in one-way quantum computation is possible by identifying entanglement features that are required to be present in every universal resource. In particular, we find that a large class of entanglement measures must reach its supremum on every universal resource. These insights are used to identify several families of states as being not universal, such as one-dimensional (1D) cluster states, Greenberger–Horne–Zeilinger (GHZ) states, W states, and ground states of non-critical 1D spin systems. Our criteria are strengthened by considering the efficiency of a quantum computation, and we find that entanglement measures must obey a certain scaling law with the system size for all efficient universal resources. This again leads to examples of non-universal resources, such as, e.g. ground states of critical 1D spin systems. On the other hand, we provide several examples of efficient universal resources, namely graph states corresponding to hexagonal, triangular and Kagome lattices. Finally, we consider the more general notion of encoded CQ-universality, where quantum outputs are allowed to be produced in an encoded form. Again we provide entanglement-based criteria for encoded universality. Moreover, we present a general procedure to construct encoded universal resources.
Highlights
We have investigated which states constitute universal resources for measurement-based quantum computation (MQC), and what the role of entanglement is in this issue
We argued that a distinction needs to be made between several types of universality—called CC, QC, CQand QQ-universality—depending on whether the input and output of a quantum computer are allowed to be either classical (C) or quantum (Q)
The reason that, e.g. QQ-universality is not possible, is essentially due to the fact that in one-way models, where resource states are processed with local operations only, it is not possible to have quantum states as inputs; only classical inputs are possible
Summary
The discovery of quantum algorithms, most notably Shor’s factoring algorithm [1] and Grover’s search algorithm [2], demonstrate that quantum computation can achieve a (possibly exponential) speed-up over classical devices This has put quantum computation at the focus of contemporary research, and significant progress in the theoretical understanding of quantum information processing, as well as promising steps toward an experimental realization of large-scale quantum computation, have recently been reported. In the one-way model the resource character of entanglement is highlighted, as it is clearly separated from the processing of quantum information by local, single-qubit measurements As the latter cannot increase any entanglement in the system, all entanglement required for quantum computation needs to be initially present in the system. For proposed implementations of measurement-based computation see e.g. [6], [15]–[22] (further references can be found in [23]); for recent experimental developments see [24]
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