Negative Quasi-Probability as a Resource for Quantum Computation

A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speedup and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analog of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic-state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.Comment: 15 pages v4: This is a major revision. In particular, we have added a new section detailing an explicit extension of the Gottesman-Knill simulation protocol to deal with positively represented states and measurement (even when these are non-stabilizer). This paper also includes significant elaboration on the two main results of the previous versio

Cost of postselection in decision theory

© 2015 American Physical Society. Postselection is the process of discarding outcomes from statistical trials that are not the event one desires. Postselection can be useful in many applications where the cost of getting the wrong event is implicitly high. However, unless this cost is specified exactly, one might conclude that discarding all data is optimal. Here we analyze the optimal decision rules and quantum measurements in a decision theoretic setting where a prespecified cost is assigned to discarding data. Our scheme interpolates between unambiguous state discrimination (when the cost of postselection is zero) and a minimum error measurement (when the cost of postselection is maximal). We also relate our formulation to previous approaches which focus on minimizing the probability of indecision

Near-optimal quantum tomography: Estimators and bounds

© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. We give bounds on the average fidelity achievable by any quantum state estimator, which is arguably the most prominently used figure of merit in quantum state tomography. Moreover, these bounds can be computed online - that is, while the experiment is running. We show numerically that these bounds are quite tight for relevant distributions of density matrices. We also show that the Bayesian mean estimator is ideal in the sense of performing close to the bound without requiring optimization. Our results hold for all finite dimensional quantum systems

Don’t get involved: an examination of how public sector organisations in England are involving disabled people in the Disability Equality Duty

The Disability Equality Duty (DED) came into force in December 2006. It stipulated that all public sector organisations were to develop policies to promote the equality of disabled people as staff members, consumers or visitors. Its emergence comes as part of a network of social policies developed over the last 20 years to promote disability rights and citizenship in the UK. However unlike previous legislation, the DED set in place the need for organisations to be pro-active in their policies and work with disabled people to move towards change in public sector cultures and working practices. This article reports on this early stage of implementation in England. Findings show that whilst some progress has been made in securing change, practice varied greatly. Therefore if a fundamental change in the culture of work and service provision is to be secured, this key requirement will need to be given a higher priority by organisations

Minimax Quantum Tomography: Estimators and Relative Entropy Bounds

© 2016 American Physical Society. A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O(1/N) - in contrast to that of classical probability estimation, which is O(1/N) - where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states

Accelerated randomized benchmarking

© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Quantum information processing offers promising advances for a wide range of fields and applications, provided that we can efficiently assess the performance of the control applied in candidate systems. That is, we must be able to determine whether we have implemented a desired gate, and refine accordingly. Randomized benchmarking reduces the difficulty of this task by exploiting symmetries in quantum operations. Here, we bound the resources required for benchmarking and show that, with prior information, we can achieve several orders of magnitude better accuracy than in traditional approaches to benchmarking. Moreover, by building on state-of-the-art classical algorithms, we reach these accuracies with near-optimal resources. Our approach requires an order of magnitude less data to achieve the same accuracies and to provide online estimates of the errors in the reported fidelities. We also show that our approach is useful for physical devices by comparing to simulations

Practical adaptive quantum tomography

© 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. We introduce a fast and accurate heuristic for adaptive tomography that addresses many of the limitations of prior methods. Previous approaches were either too computationally intensive or tailored to handle special cases such as single qubits or pure states. By contrast, our approach combines the efficiency of online optimization with generally applicable and well-motivated data-processing techniques. We numerically demonstrate these advantages in several scenarios including mixed states, higher-dimensional systems, and restricted measurements

Experimental Demonstration of Self-Guided Quantum Tomography

© 2016 American Physical Society. Traditional methods of quantum state characterization are impractical for systems of more than a few qubits due to exponentially expensive postprocessing and data storage and lack robustness against errors and noise. Here, we experimentally demonstrate self-guided quantum tomography performed on polarization photonic qubits. The quantum state is iteratively learned by optimizing a projection measurement without any data storage or postprocessing. We experimentally demonstrate robustness against statistical noise and measurement errors on single-qubit and entangled two-qubit states

Bayesian quantum noise spectroscopy

© 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft. As commonly understood, the noise spectroscopy problem - characterizing the statistical properties of a noise process affecting a quantum system by measuring its response - is mathematically ill-posed, in the sense that no unique noise process leads to a set of responses unless extra assumptions are taken into account. Ad-hoc solutions assume an implicit structure, which is often never determined. Thus, it is unclear when the method will succeed or whether one should trust the solution obtained. Here, we propose to treat the problem from the point of view of statistical estimation theory. We develop a Bayesian solution to the problem which allows one to easily incorporate assumptions which render the problem solvable. We compare several numerical techniques for noise spectroscopy and find the Bayesian approach to be superior in many respects

Novelty, efficacy, and significance of weak measurements for quantum tomography

© 2015 American Physical Society. The use of weak measurements for performing quantum tomography is enjoying increased attention due to several recent proposals. The claimed merits of using weak measurements in this context are varied, but are generally represented by novelty, increased efficacy, and foundational significance. We critically evaluate two proposals that make such claims and find that weak measurements are not an essential ingredient for most of their claimed features