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High-connectivity circuits are a major roadblock for current quantum hardware. We propose a hybrid classical-quantum algorithm to simulate such circuits without swap-gate ladders. As main technical tool, we introduce quantum-classical-quantum interfaces. These replace an experimentally problematic gate (e.g. a long-range one) by single-qubit random measurements followed by state-preparations sampled according to a classical quasi-probability simulation of the noiseless gate. Each interface introduces a multiplicative statistical overhead which is remarkably independent of the on-chip qubit distance. Hence, by applying interfaces to the longest range gates in a target circuit, significant reductions in circuit depth and gate infidelity can be attained. We numerically show the efficacy of our method for a Bell-state circuit for two increasingly distant qubits and a variational ground-state solver for the transverse-field Ising model on a ring. Our findings provide a versatile toolbox for error-mitigation and circuit boosts tailored for noisy, intermediate-scale quantum computation.

As quantum technologies mature, the development of tools for benchmarking their ability to prepare and manipulate complex quantum states becomes increasingly necessary. A key concept, the state overlap between two quantum states, offers a natural tool to verify and cross-validate quantum simulators and quantum computers. Recent progress in controlling and measuring large quantum systems has motivated the development of state overlap estimators of varying efficiency and experimental complexity. Here, we demonstrate a practical approach for measuring the overlap between quantum states based on a factorable quasiprobabilistic representation of the states, and compare it with methods based on randomised measurements. Assuming realistic noisy intermediate scale quantum (NISQ) devices limitations, our quasiprobabilistic method outperforms the best circuits designed for state-overlap estimation for n-qubit states, with n > 2. For n < 7, our technique outperforms also the currently best direct estimator based on randomised local measurements, thus establishing a niche of optimality., Comment: 7 pages. Comments are welcome!

The intrinsic random nature of quantum physics offers novel tools for the generation of random numbers, a central challenge for a plethora of fields. Bell non-local correlations obtained by measurements on entangled states allow for the generation of bit strings whose randomness is guaranteed in a device-independent manner, i.e. without assumptions on the measurement and state-generation devices. Here, we generate this strong form of certified randomness on a new platform: the so-called instrumental scenario, which is central to the field of causal inference. First, we theoretically show that certified random bits, private against general quantum adversaries, can be extracted exploiting device-independent quantum instrumental-inequality violations. To that end, we adapt techniques previously developed for the Bell scenario. Then, we experimentally implement the corresponding randomness-generation protocol using entangled photons and active feed-forward of information. Moreover, we show that, for low levels of noise, our protocol offers an advantage over the simplest Bell-nonlocality protocol based on the Clauser-Horn-Shimony-Holt inequality., Comment: Modified Supplementary Information: removed description of extractor algorithm introduced by arXiv:1212.0520. Implemented security of the protocol against general adversarial attacks

We study prepare-and-measure experiments where the sender (Alice) receives trusted quantum inputs but has an untrusted state-preparation device and the receiver (Bob) has a fully-untrusted measurement device. A distributed-sampling task naturally arises in such scenario, whose goal is for Alice and Bob to reproduce the statistics of his measurements on her quantum inputs using a fixed communication channel. Their performance at such task can certify quantum communication (QC), and this is formalised by measurement-device-independent QC witnesses. Furthermore, we prove that QC can provide an advantage (over classical communication) for distributed sampling if and only if Bob's measurements are incompatible. This gives an operational interpretation to the fundamental notion of measurement incompatibility, and motivates a generalised notion of it. Our findings have both fundamental and applied implications., Comment: 4+6 pages, 3 figures. v3: improved presentation and discussion on robustness

Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory., Comment: We found an error in the proof our previous Theorem 3; therefore we are reverting to the previous version, in which all results remain correct

We address the problem of characterising the compatible tuples of measurements that admit a unique joint measurement. We derive a uniqueness criterion based on the method of perturbations and apply it to show that extremal points of the set of compatible tuples admit a unique joint measurement, while all tuples that admit a unique joint measurement lie in the boundary of such a set. We also provide counter-examples showing that none of these properties are both necessary and sufficient, thus completely describing the relation between joint measurement uniqueness and the structure of the compatible set. As a by-product of our investigations, we completely characterise the extremal and boundary points of the set of general tuples of measurements and of the subset of compatible tuples., Comment: 9 pages, 4 figures. v2, v3: minor corrections. v4: new proof of Thm. 3 and Cor. 4. The authors are thankful to Alessandro Toigo and Claudio Carmeli for pointing a flaw in the previous proof that restricted the theorem to pairs of POVMs

We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes used to perform the simulation of a given measurement. This general approach also allows us to identify connections between different types of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories., Comment: 25 pages, single-column, 3 figures. Closer to published version

We address the problem of characterizing the steerability of quantum states under restrictive measurement scenarios, i.e., the problem of determining whether a quantum state can demonstrate steering when subjected to $N$ measurements of $k$ outcomes. We consider the cases of either general positive operator-valued measures (POVMs) or specific kinds of measurements (e.g., projective or symmetric). We propose general methods to calculate lower and upper bounds for the white-noise robustness of a $d$-dimensional quantum state under different measurement scenarios that are also applicable to the study of the noise robustness of the incompatibility of sets of unknown qudit measurements. We show that some mutually unbiased bases, symmetric informationally complete measurements, and other symmetric choices of measurements are not optimal for steering isotropic states and provide candidates to the most incompatible sets of measurements in each case. Finally, we provide numerical evidence that nonprojective POVMs do not improve over projective ones for this task., Comment: 12 pages, 5 figures. All code is available at https://git.io/v9znv

Standard projective measurements represent a subset of all possible measurements in quantum physics, defined by positive-operator-valued measures. We study what quantum measurements are projective simulable, that is, can be simulated by using projective measurements and classical randomness. We first prove that every measurement on a given quantum system can be realised by classical processing of projective measurements on the system plus an ancilla of the same dimension. Then, given a general measurement in dimension two or three, we show that deciding whether it is projective-simulable can be solved by means of semi-definite programming. We also establish conditions for the simulation of measurements using projective ones valid for any dimension. As an application of our formalism, we improve the range of visibilities for which two-qubit Werner states do not violate any Bell inequality for all measurements. From an implementation point of view, our work provides bounds on the amount of noise a measurement tolerates before losing any advantage over projective ones., Comment: v2: 4 pages + 12.5 pages of appendices, 2 figures, updated references and narrative, computational appendix is available at https://github.com/peterwittek/ipython-notebooks/blob/master/Simulating_POVMs.ipynb, comments and suggestions are welcome

Entanglement allows for the nonlocality of quantum theory, which is the resource behind device-independent quantum information protocols. However, not all entangled quantum states display nonlocality, and a central question is to determine the precise relation between entanglement and nonlocality. Here we present the first general test to decide whether a quantum state is local, and that can be implemented by semidefinite programming. This method can be applied to any given state and for the construction of new examples of states with local hidden-variable models for both projective and general measurements. As applications we provide a lower bound estimate of the fraction of two-qubit local entangled states and present new explicit examples of such states, including those which arise from physical noise models, Bell-diagonal states, and noisy GHZ and W states., Comment: Published version with new title and abstract, improved presentation and new examples of LHV states. Codes are available at https://github.com/paulskrzypczyk/localhiddenstatemodels (please cite this paper if you use them). See also the related work by F. Hirsch et al arXiv:1512.00262

This Master's thesis has two central subjects: the simulation of correlations generated by local measurements on entangled quantum states by local hidden-variables models and the revelation of hidden nonlocality. We present and detail the Werner's local model and the hidden nonlocality of some Werner states of dimension $d\geq5$, the Gisin-Degorre's local model for a Werner state of dimension $d=2$ and the local model of Hirsch et al. for mixtures of the singlet state and noise, all of them for projective measurements. Finally, we introduce the local model for POVMs of Hirsch et al. for a state constructed upon the singlet with noise, that still violates the CHSH inequality after local filters are applied, hence presenting the so-called genuine hidden nonlocality., Comment: Master's thesis, 77 pages, 6 figures, conclusion updated in July, 2014