It is often argued that entanglement is at the root of the speedup for quantum compared to classical computation, and that one needs a sufficient amount of entanglement for this speedup to be manifest. In measurement-based quantum computing, the need for a highly entangled initial state is particularly obvious. Defying this intuition, we show that quantum states can be too entangled to be useful for the purpose of computation, in that high values of the geometric measure of entanglement preclude states from offering a universal quantum computational speedup. We prove that this phenomenon occurs for a dramatic majority of all states: the fraction of useful n-qubit pure states is less than exp(-n;{2}). This work highlights a new aspect of the role entanglement plays for quantum computational speedups.
It is often argued that entanglement is at the root of the speedup for quantum compared to classical computation, and that one needs a sufficient amount of entanglement for this speedup to be manifest. In measurement-based quantum computing, the need for a highly entangled initial state is particularly obvious. Defying this intuition, we show that quantum states can be too entangled to be useful for the purpose of computation, in that high values of the geometric measure of entanglement preclude states from offering a universal quantum computational speedup. We prove that this phenomenon occurs for a dramatic majority of all states: the fraction of useful n-qubit pure states is less than exp(-n;{2}). This work highlights a new aspect of the role entanglement plays for quantum computational speedups.
Of the many and varied applications of quantum information theory, perhaps the most fascinating is the sub-field of quantum computation. In this sub-field, computational algorithms are designed which utilise the resources available in quantum systems in order to compute solutions to computational problems with, in some cases, exponentially fewer resources than any known classical algorithm. While the fact of quantum computational speedup is almost beyond doubt, the source of quantum speedup is still a matter of debate. In this paper I argue that entanglement is a necessary component for any explanation of quantum speedup and I address some purported counter-examples that some claim show that the contrary is true. In particular, I address Biham et al.'s mixed-state version of the Deutsch-Jozsa algorithm, and Knill \& Laflamme's deterministic quantum computation with one qubit (DQC1) model of quantum computation. I argue that these examples do not demonstrate that entanglement is unnecessary for the explanation of quantum speedup, but that they rather illuminate and clarify the role that entanglement does play.
We present a classical protocol to efficiently simulate any pure-state quantum computation that involves only a restricted amount of entanglement. More generally, we show how to classically simulate pure-state quantum computations on n qubits by using computational resources that grow linearly in n and exponentially in the amount of entanglement in the quantum computer. Our results imply that a necessary condition for an exponential computational speedup (with respect to classical computations) is that the amount of entanglement increases with the size n of the computation, and provide an explicit lower bound on the required growth.
The aim of this dissertation is to clarify the debate over the explanation of quantum speedup and to submit a tentative resolution to it. In particular, I argue that the physical explanation for quantum speedup is precisely the fact that the phenomenon of quantum entanglement enables a quantum computer to fully exploit the representational capacity of Hilbert space. This is impossible for classical systems, joint states of which must always be representable as product states. Chapter 2 begins with a discussion of the most popular of the candidate physical explanations for quantum speedup: the many worlds explanation. I argue that unlike the neo-Everettian interpretation of quantum mechanics it does not have the conceptual resources required to overcome the `preferred basis objection'. I further argue that the many worlds explanation, at best, can serve as a good description of the physical process which takes place in so-called network-based computation, but that it is incompatible with other models of computation such as cluster state quantum computing. I next consider, in Chapter 3, a common component of most other candidate explanations of quantum speedup: quantum entanglement. I investigate whether entanglement can be said to be a necessary component of any explanation for quantum speedup, and I consider two major purported counter-examples to this claim. I argue that neither of these, in fact, show that entanglement is unnecessary for speedup, and that, on the contrary, we should conclude that it is. In Chapters 4 and 5 I then ask whether entanglement can be said to be sufficient as well. In Chapter 4 I argue that despite a result that seems to indicate the contrary, entanglement, considered as a resource, can be seen as sufficient to enable quantum speedup. Finally, in Chapter 5 I argue that entanglement is sufficient to explain quantum speedup as well.
In a quantum computation with pure states, the generation of large amounts of entanglement is known to be necessary for a speedup with respect to classical computations. However, examples of quantum computations with mixed states are known, such as the deterministic computation with one quantum qubit (DQC1) model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which entanglement is at most marginally present, and yet a computational speedup is believed to occur. Correlations, and not entanglement, have been identified as a necessary ingredient for mixed-state quantum computation speedups. Here we show that correlations, as measured through the operator Schmidt rank, are indeed present in large amounts in the DQC1 circuit. This provides evidence for the preclusion of efficient classical simulation of DQC1 by means of a whole class of classical simulation algorithms, thereby reinforcing the conjecture that DQC1 leads to a genuine quantum computational speedup.
For any quantum algorithm operating on pure states, we prove that the presence of multi‐partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed‐up over classical computation. Furthermore, we prove that the algorithm can be efficiently simulated classically to within a prescribed tolerance η even if a suitably small amount of global entanglement is present. We explicitly identify the occurrence of increasing multi‐partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed‐up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure‐state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum‐computational power.
In this paper I argue that entanglement is a necessary component for any explanation of quantum speedup and I address some purported counter-examples that some claim show that the contrary is true. In particular, I address Biham et al.'s mixed-state version of the Deutsch-Jozsa algorithm, and Knill & Laflamme's deterministic quantum computation with one qubit (DQC1) model of quantum computation. I argue that these examples do not demonstrate that entanglement is unnecessary for the explanation of quantum speedup, but that they rather illuminate and clarify the role that entanglement does play.