Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, non-oscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures.

1 aBrady, Lucas, T.1 aKocia, Lucas1 aBienias, Przemyslaw1 aBapat, Aniruddha1 aKharkov, Yaroslav1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2107.0121802049nas a2200145 4500008004100000245005200041210005100093260001500144300001100159490000700170520165600177100001701833700001601850856003701866 2018 eng d00aMeasurement Contextuality and Planck's Constant0 aMeasurement Contextuality and Plancks Constant c2018/07/12 a0730200 v203 aContextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck's constant, ℏ, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of ℏ---all orders higher than ℏ0 can be interpreted as quantum corrections to the order ℏ0 term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order ℏ0 terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is equivalent to orders of ℏ as a resource within the WWM formalism. This explains why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres-Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, we find that qubit Pauli observables lack an order ℏ0 contribution in their Weyl symbol and so exhibit contextuality regardless of the state being measured. On the other hand, odd-dimensional qudit observables generally possess non-zero order ℏ0 terms, and higher, in their WWM formulation, and so exhibit contextuality depending on the state measured: odd-dimensional qudit states that exhibit measurement contextuality have an order ℏ1 contribution that allows for the violation of classical bounds while states that do not exhibit measurement contextuality have insufficiently large order ℏ1 contributions.

1 aKocia, Lucas1 aLove, Peter uhttps://arxiv.org/abs/1711.0806602361nas a2200109 4500008004100000245014200041210006900183520192900252100001702181700001602198856003702214 2018 eng d00aThe Non-Disjoint Ontic States of the Grassmann Ontological Model, Transformation Contextuality, and the Single Qubit Stabilizer Subtheory0 aNonDisjoint Ontic States of the Grassmann Ontological Model Tran3 aWe show that it is possible to construct a preparation non-contextual ontological model that does not exhibit "transformation contextuality" for single qubits in the stabilizer subtheory. In particular, we consider the "blowtorch" map and show that it does not exhibit transformation contextuality under the Grassmann Wigner-Weyl-Moyal (WWM) qubit formalism. Furthermore, the transformation in this formalism can be fully expressed at order ℏ0 and so does not qualify as a candidate quantum phenomenon. In particular, we find that the Grassmann WWM formalism at order ℏ0 corresponds to an ontological model governed by an additional set of constraints arising from the relations defining the Grassmann algebra. Due to this additional set of constraints, the allowed probability distributions in this model do not form a single convex set when expressed in terms of disjoint ontic states and so cannot be mapped to models whose states form a single convex set over disjoint ontic states. However, expressing the Grassmann WWM ontological model in terms of non-disjoint ontic states corresponding to the monomials of the Grassmann algebra results in a single convex set. We further show that a recent result by Lillystone et al. that proves a broad class of preparation and measurement non-contextual ontological models must exhibit transformation contextuality lacks the generality to include the ontological model considered here; Lillystone et al.'s result is appropriately limited to ontological models whose states produce a single convex set when expressed in terms of disjoint ontic states. Therefore, we prove that for the qubit stabilizer subtheory to be captured by a preparation, transformation and measurement non-contextual ontological theory, it must be expressed in terms of non-disjoint ontic states, unlike the case for the odd-dimensional single-qudit stabilizer subtheory.

1 aKocia, Lucas1 aLove, Peter uhttps://arxiv.org/abs/1805.0951401897nas a2200109 4500008004100000245010200041210006900143520150500212100001701717700001601734856003701750 2018 eng d00aStationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits0 aStationary Phase Method in Discrete Wigner Functions and Classic3 aWe apply the periodized stationary phase method to discrete Wigner functions of systems with odd prime dimension using results from p-adic number theory. We derive the Wigner-Weyl-Moyal (WWM) formalism with higher order ℏ corrections representing contextual corrections to non-contextual Clifford operations. We apply this formalism to a subset of unitaries that include diagonal gates such as the π8 gates. We characterize the stationary phase critical points as a quantum resource injecting contextuality and show that this resource allows for the replacement of the p2t points that represent t magic state Wigner functions on p-dimensional qudits by ≤pt points. We find that the π8 gate introduces the smallest higher order ℏ correction possible, requiring the lowest number of additional critical points compared to the Clifford gates. We then establish a relationship between the stabilizer rank of states and the number of critical points necessary to treat them in the WWM formalism. This allows us to exploit the stabilizer rank decomposition of two qutrit π8 gates to develop a classical strong simulation of a single qutrit marginal on t qutrit π8 gates that are followed by Clifford evolution, and show that this only requires calculating 3t2+1 critical points corresponding to Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼30.8t for 10−2 precision) for any number of π8 gates to full precision.

1 aKocia, Lucas1 aLove, Peter uhttps://arxiv.org/abs/1810.03622