We develop connections between generalised notions of entanglement and quantum computational devices where the measurements available are restricted, either because they are noisy and/or because by design they are only along Pauli directions. By considering restricted measurements one can (by considering the dual positive operators) construct single particle state spaces that are different to the usual quantum state space. This leads to a modified notion of entanglement that can be very different to the quantum version (for example, Bell states can become separable). We use this approach to develop alternative methods of classical simulation that have strong connections to the study of non-local correlations: we construct noisy quantum computers that admit operations outside the Clifford set and can generate some forms of multiparty quantum entanglement, but are otherwise classical in that they can be efficiently simulated classically and cannot generate non-local statistics. Although the approach provides new regimes of noisy quantum evolution that can be efficiently simulated classically, it does not appear to lead to significant reductions of existing upper bounds to fault tolerance thresholds for common noise models.
We develop connections between generalised notions of entanglement and quantum computational devices where the measurements available are restricted, either because they are noisy and/or because by design they are only along Pauli directions. By considering restricted measurements one can (by considering the dual positive operators) construct single particle state spaces that are different to the usual quantum state space. This leads to a modified notion of entanglement that can be very different to the quantum version (for example, Bell states can become separable). We use this approach to develop alternative methods of classical simulation that have strong connections to the study of non-local correlations: we construct noisy quantum computers that admit operations outside the Clifford set and can generate some forms of multiparty quantum entanglement, but are otherwise classical in that they can be efficiently simulated classically and cannot generate non-local statistics. Although the approach provides new regimes of noisy quantum evolution that can be efficiently simulated classically, it does not appear to lead to significant reductions of existing upper bounds to fault tolerance thresholds for common noise models.
We develop connections between generalized notions of entanglement and quantum computational devices where the measurements available are restricted, either because they are noisy and/or because by design they are only along Pauli directions. By considering restricted measurements one can (by considering the dual positive operators) construct single-particle-state spaces that are different to the usual quantum-state space. This leads to a modified notion of entanglement that can be very different to the quantum version (for example, Bell states can become separable). We use this approach to develop alternative methods of classical simulation that have strong connections to the study of nonlocal correlations: we construct noisy quantum computers that admit operations outside the Clifford set and can generate some forms of multiparty quantum entanglement, but are otherwise classical in that they can be efficiently simulated classically and cannot generate nonlocal statistics. Although the approach provides new regimes of noisy quantum evolution that can be efficiently simulated classically, it does not appear to lead to significant reductions of existing upper bounds to fault tolerance thresholds for common noise models.
It is known that if the quantum gates in a proposed quantum computer are so noisy that they are incapable of generating entanglement, then the device can be efficiently simulated classically. If the measurements and single particle operations are restricted, then the same statement can be true for generalised non-quantum notions of entanglement. Here we show that this can improve the applicability of limited-entanglement simulation algorithms. In particular, we show that by using a non-quantum notion of separability a classical simulation algorithm of Harrow & Nielsen can efficiently simulate magic state quantum computers with noisy CNOTs (but otherwise ideal) for joint depolarising strengths of 272/489 ~ 56%, in comparison to noise levels of 2/3 ~ 66% required if the algorithm uses quantum notions of separability. This suggests that quantum entanglement may not be the most appropriate notion of entanglement to use when discussing the power of stabilizer based quantum computers.
We investigate the amount of noise required to turn a universal quantum gate set into one that can be efficiently modelled classically. This question is useful for providing upper bounds on fault tolerant thresholds, and for understanding the nature of the quantum/classical computational transition. We refine some previously known upper bounds using two different strategies. The first one involves the introduction of bi-entangling operations, a class of classically simulatable machines that can generate at most bipartite entanglement. Using this class we show that it is possible to sharpen previously obtained upper bounds in certain cases. As an example, we show that under depolarizing noise on the controlled-not gate, the previously known upper bound of 74% can be sharpened to around 67%. Another interesting consequence is that measurement based schemes cannot work using only 2-qubit non-degenerate projections. In the second strand of the work we utilize the Gottesman-Knill theorem on the classically efficient simulation of Clifford group operations. The bounds attained using this approach for the pi/8-gate can be as low as 15% for general single gate noise, and 30% for dephasing noise.
One of the core research questions in the theory of quantum computing is to find out to what precise extent the classical simulation of a noisy quantum circuits is possible and where potential quantum advantages can set in. In this work, we introduce a unified framework for the classical simulation of quantum circuits based on frame theory, encompassing and generalizing a broad class of existing simulation strategies. Within this framework, the computational cost of a simulation algorithm is determined by the one-norm of an associated quasi-probability distribution, providing a common quantitative measure across different simulation approaches. This enables a comprehensive perspective on common methods for the simulation of noisy circuits based on different quantum resources, such as entanglement or non-stabilizerness. It further provides a clear scheme for generating novel classical simulation algorithms. Indeed, by exploring different choices of frames within this formalism and resorting to tools of convex optimization, we are able not only to obtain new insights and improved bounds for existing methods -- such as stabilizer state simulation or Pauli back-propagation -- but also to discover a new approach with an improved performance based on a generalization of the Pauli frame. We, thereby, show that classical simulation techniques can directly benefit from a perspective -- that of frames -- that goes beyond the traditional classification of quantum resources.
Abstract The investigation of separable states in quantum theory has been driven by the notion that they are highly classical, in that they do not demonstrate nonlocality, and are in some contexts unable to support non-classical computation. The converse question, the extent to which entangled states do or do not support non-classical information processing, is less well understood. Motivated by this question we extend the notion of quantum separability into the entangled quantum states, by constructing separable decompositions that describe them with the ‘smallest’ possible sets of non-physical local operators. We consider a few ways to define the word ‘smallest’ and present techniques for obtaining them. The methods involve calculating certain forms of cross norm. The results generalise significantly the results obtained in our previous work on this topic (2015 New J. Phys. 17 093047), and can be be used to construct classical simulation methods and local hidden variable models for subsets of local measurements on entangled quantum states.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.