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.
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.
We investigate which entanglement resources allow universal measurement-based quantum computation via single-qubit operations. We find that any entanglement feature exhibited by the 2D cluster state must also be present in any other universal resource. We obtain a powerful criterion to assess the universality of graph states by introducing an entanglement measure which necessarily grows unboundedly with the system size for all universal resource states. Furthermore, we prove that graph states associated with 2D lattices such as the hexagonal and triangular lattice are universal, and obtain the first example of a universal nongraph state.
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)].
Recently, a framework was established to systematically construct novel universal resource states for measurement-based quantum computation using techniques involving finitely correlated states. With these methods, universal states were found which are in certain ways much less entangled than the original cluster-state model, and it was hence believed that with this approach, many of the extremal entanglement features of the cluster states could be relaxed. The new resources were constructed as "computationally universal" states-i.e., they allow one to efficiently reproduce the classical output of each quantum computation-whereas the cluster states are universal in a stronger sense since they are "universal state preparators." Here, we show that the new resources are universal state preparators after all, and must therefore exhibit a whole class of extremal entanglement features, similar to the cluster states.
We establish a framework which allows one to systematically construct novel schemes for measurement-based quantum computation. The technique utilizes tools from many-body physics - based on finitely correlated or projected entangled pair states - to go beyond the cluster-state based one-way computer. We also identify resource states with radically different entanglement properties than the cluster state. It is shown that there exist universal resource states which are locally arbitrarily close to a pure state. We find that non-vanishing two-point correlation functions are no obstacle to universality. An explicit example for a resource state is presented, which can partly be prepared by gates with non-maximal entangling power. Finally, we comment on the possibility of tailoring computational models to specific physical systems as, e.g. in linear optical experiments.
We introduce schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in Gross and Eisert [Phys. Rev. Lett. 98, 220503 (2007); quant-ph/0609149]. Our method makes use of tools from many-body physics---matrix product states, finitely correlated states, or projected entangled pairs states---to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem---how to realize quantum computation---was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting and present a large number of examples. We find computational schemes, which differ from the original one-way computer, for example, in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may, for example, exhibit nonvanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.