We consider the issue of what should count as a resource for measurement-based quantum computation (MBQC). While a state that supports universal quantum computation clearly should be considered a resource, universality should not be necessary given the existence of interesting, but less computationally-powerful, classes of MBQCs. Here, we propose minimal criteria for a state to be considered a resource state for MBQC. Using these criteria, we explain why discord-free states cannot be resources for MBQC, contrary to recent claims [Hoban et al., arXiv:1304.2667v1]. Independently of our criteria, we also show that the arguments of Hoban et al., if correct, would imply that Shor's algorithm (for example) can be implemented by measuring discord-free states.
We consider the issue of what should count as a resource for measurement-based quantum computation (MBQC). While a state that supports universal quantum computation clearly should be considered a resource, universality should not be necessary given the existence of interesting, but less computationally-powerful, classes of MBQCs. Here, we propose minimal criteria for a state to be considered a resource state for MBQC. Using these criteria, we explain why discord-free states cannot be resources for MBQC, contrary to recent claims [Hoban et al., arXiv:1304.2667v1]. Independently of our criteria, we also show that the arguments of Hoban et al., if correct, would imply that Shor's algorithm (for example) can be implemented by measuring discord-free states.
We consider the issue of what should count as a resource for measurement-based quantum computation (MBQC). While a state that supports universal quantum computation clearly should be considered a resource, universality should not be necessary given the existence of interesting, but less computationally powerful, classes of MBQCs. Here, we propose minimal criteria for a state to be considered a resource state for MBQC. Using these criteria, we explain why discord-free states cannot be resources for MBQC, contrary to recent claims [Hoban et al., arXiv:1304.2667v1]. Independently of our criteria, we also show that the arguments of Hoban et al., if correct, would imply that Shor's algorithm (for example) can be implemented by measuring discord-free states.
In measurement-based quantum computation (MBQC), a special highly entangled state (called a resource state) allows for universal quantum computation driven by single-qubit measurements and postmeasurement corrections. The large number of qubits necessary to construct the resource state constitutes one of the main down sides to MBQC. However, in some instances it is possible to extend the resource state on the fly, meaning that not every qubit must be realized in the devices simultaneously. We consider the question of the minimal number of physical qubits that must be present in a system to directly implement a given measurement pattern. For measurement patterns which have quantum circuit representation as formalized by the notion of flow, with n inputs, n outputs, and m total qubits, we show that only a minimum of n+1 and m qubits are required, while the number of required qubits can be as high as $m\ensuremath{-}2$ for measurement patterns which implement a unitary but do not have a quantum circuit representation, as formalized by the notion of generalized flow (gflow). We discuss the implications of removing the Clifford part of a measurement pattern, using well-established transformation rules for Pauli measurements, for the presence of flow versus gflow, and hence the effect on the minimum number of physical qubits required to directly realize the measurement pattern.
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.
Abstract For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where nonlocality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.
We investigate which quantum states can serve as universal resources for approximate and stochastic measurement-based quantum computation in the sense that any quantum state can be generated from a given resource by means of single-qubit (local) operations assisted by classical communication. More precisely, we consider the approximate and stochastic generation of states, resulting, for example, from a restriction to finite measurement settings or from possible imperfections in the resources or local operations. We show that entanglement-based criteria for universality obtained in M. Van den Nest et al. [New J. Phys. 9, 204 (2007)] for the exact, deterministic case can be lifted to the much more general approximate, stochastic case. This allows us to move from the idealized situation (exact, deterministic universality) considered in previous works to the practically relevant context of nonperfect state preparation. We find that any entanglement measure fulfilling some basic requirements needs to reach its maximum value on some element of an approximate, stochastic universal family of resource states, as the resource size grows. This allows us to rule out various families of states as being approximate, stochastic universal. We prove that approximate, stochastic universality is in general a weaker requirement than deterministic, exact universality and provide resources that are efficient approximate universal, but not exact deterministic universal. We also study the robustness of universal resources for measurement-based quantum computation under realistic assumptions about the (imperfect) generation and manipulation of entangled states, giving an explicit expression for the impact that errors made in the preparation of the resource have on the possibility to use it for universal approximate and stochastic state preparation. Finally, we discuss the relation between our entanglement-based criteria and recent results regarding the uselessness of states with a high degree of geometric entanglement as universal resources [D. Gross et al., Phys. Rev. Lett. 102, 190501 (2009); M. J. Bremner et al., Phys. Rev. Lett 102, 190502 (2009)].
Fermions, as a major class of quantum particles, provide platforms for quantum information processing beyond the possibilities of spins or bosons, which have been studied more extensively. One particularly interesting model to study, in view of recent progress in manipulating ultracold fermion gases, is the fermionic version of measurement-based quantum computation (MBQC), which implements full quantum computation with only single-site measurements on a proper fermionic many-body resource state. However, it is not known which fermionic states can be used as the resource states for MBQC and how to find them. In this paper, we generalize the framework of spin MBQC to fermions. In particular, we provide a general formalism to construct many-body entangled fermion resource states for MBQC based on the fermionic projected entangled pair state representation. We give a specific fermionic state which enables universal MBQC and demonstrate that the nonlocality inherent in fermion systems can be properly taken care of with suitable measurement schemes. Such a framework opens up possibilities of finding MBQC resource states which can be more readily realized in the laboratory.