Abstract There is good evidence that quantum computers are more powerful than classical computers, and that various simple modifications of quantum theory yield computational power that is dramatically greater still. However, these modifications also violate fundamental physical principles. This raises the question of whether there exists a physical theory, allowing computation more powerful than quantum, but which still respects those fundamental physical principles. Prior work by two of us introduced this question within a suitable framework for theories that make good operational sense, and showed that in any theory satisfying tomographic locality, the class of problems that can be solved efficiently is contained in the complexity class AWPP. Here, we show that this bound is tight, in the sense that there exists a theory, satisfying tomographic locality, as well as a basic principle of causality, which can efficiently decide everything in AWPP. Hence this theory can efficiently simulate any computation in this framework, including quantum computation.
Abstract There is good evidence that quantum computers are more powerful than classical computers, and that various simple modifications of quantum theory yield computational power that is dramatically greater still. However, these modifications also violate fundamental physical principles. This raises the question of whether there exists a physical theory, allowing computation more powerful than quantum, but which still respects those fundamental physical principles. Prior work by two of us introduced this question within a suitable framework for theories that make good operational sense, and showed that in any theory satisfying tomographic locality, the class of problems that can be solved efficiently is contained in the complexity class AWPP. Here, we show that this bound is tight, in the sense that there exists a theory, satisfying tomographic locality, as well as a basic principle of causality, which can efficiently decide everything in AWPP. Hence this theory can efficiently simulate any computation in this framework, including quantum computation.
From the general difficulty of simulating quantum systems using classical systems, and in particular \nthe existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is \nintrinsically more powerful than classical computation. At present, the best upper bound known for \nthe power of quantum computation is that BQP AWPP ⊆ , where AWPP is a classical complexity \nclass (known to be included in PP, hence PSPACE). This work investigates limits on computational \npower that are imposed by simple physical, or information theoretic, principles. To this end, we define \na circuit-based model of computation in a class of operationally-defined theories more general than \nquantum theory, and ask: what is the minimal set of physical assumptions under which the above \ninclusions still hold? We show that given only an assumption of tomographic locality (roughly, that \nmultipartite states and transformations can be characterized by local measurements), efficient \ncomputations are contained in AWPP. This inclusion still holds even without assuming a basic notion \nof causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future \nmeasurement choices). Following Aaronson, we extend the computational model by allowing postselection \non measurement outcomes. Aaronson showed that the corresponding quantum complexity \nclass, PostBQP, is equal to PP. Given only the assumption of tomographic locality, the inclusion in \nPP still holds for post-selected computation in general theories. Hence in a world with post-selection, \nquantum theory is optimal for computation in the space of all operational theories. We then consider \nwhether one can obtain relativized complexity results for general theories. It is not obvious how to \ndefine a sensible notion of a computational oracle in the general framework that reduces to the \nstandard notion in the quantum case. Nevertheless, it is possible to define computation relative to a \n‘classical oracle’. Then, we show there exists a classical oracle relative to which efficient computation in \nany theory satisfying the causality assumption does not include NP.
From the general difficulty of simulating quantum systems using classical systems, and in particular the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation is that , where is a classical complexity class (known to be included in , hence ). This work investigates limits on computational power that are imposed by simple physical, or information theoretic, principles. To this end, we define a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and ask: what is the minimal set of physical assumptions under which the above inclusions still hold? We show that given only an assumption of tomographic locality (roughly, that multipartite states and transformations can be characterized by local measurements), efficient computations are contained in . This inclusion still holds even without assuming a basic notion of causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future measurement choices). Following Aaronson, we extend the computational model by allowing post-selection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class, , is equal to . Given only the assumption of tomographic locality, the inclusion in still holds for post-selected computation in general theories. Hence in a world with post-selection, quantum theory is optimal for computation in the space of all operational theories. We then consider whether one can obtain relativized complexity results for general theories. It is not obvious how to define a sensible notion of a computational oracle in the general framework that reduces to the standard notion in the quantum case. Nevertheless, it is possible to define computation relative to a 'classical oracle'. Then, we show there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption does not include .
The advent of quantum computing has challenged classical conceptions of which problems are efficiently solvable in our physical world. This raises the general question of what broad relationships exist between physical principles and computation. The current thesis explores this question within the operationally-defined framework of generalised probabilistic theories. In particular, we investigate the limits on computational power imposed by simple physical principles. At present, the best known upper bound on the power of quantum computers is that BQP is contained in AWPP, where AWPP is a classical complexity class contained in PP. We define a circuit-based model of computation in the above mentioned operational framework and show that in theories where local measurements suffice for tomography, efficient computations are also contained in AWPP. Moreover, we explicitly construct a theory in which the class of efficiently solvable problems exactly equals AWPP, showing this containment to be tight. We also investigate how simple physical principles bound the power of computational paradigms which combine computation and communication in a non-trivial fashion, such as interactive proof systems. Additionally, we show how some of the essential components of computational algorithms arise from certain natural physical principles. We use these results to investigate the relationship between interference behaviour and computational power, demonstrating that non-trivial interference behaviour is a general resource for post-classical computation. We then investigate whether post-quantum interference is a resource for post-quantum computation. Sorkin has defined a hierarchy of possible post-quantum interference behaviours where, informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In quantum theory, at most pairs of paths can ever interact in a fundamental way. We consider how Grover's speed-up depends on the order of interference in a theory, and show that, surprisingly, the quadratic lower bound holds regardless of the order of interference.
Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that QMA is contained in PP, which does not make use of any uniquely quantum structure—such as the fact that observables correspond to self-adjoint operators—and thus may be of independent interest.
Abstract The computational abilities of theories within the generalised probabilistic theory framework has been the subject of much recent study. Such investigations aim to gain an understanding of the possible connections between physical principles and computation. Moreover, comparing and contrasting the computational properties of quantum theory with other operationally-sensible theories could shed light on the strengths and limitations of quantum computation. This paper reviews and extends some of these results, deriving new bounds on the computational ability of theories satisfying n-local tomography, and theories in which states are represented as generalised superpositions. It moreover provides a refined version of the conjecture that a quantum computer can simulate the computation in any theory within a certain sub-class of generalised probabilistic theories with at most polynomial overhead. The paper ends by describing an important relation between this conjecture and delegated computation, similar to the relation between quantum non-locality and device-independent cryptography.
Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that [Formula: see text] is contained in [Formula: see text], which does not make use of any uniquely quantum structure-such as the fact that observables correspond to self-adjoint operators-and thus may be of independent interest.