Quantum Physics

The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., C n , D 2n , A 4 , S 4 , A 5 ) the recently-developed techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing (G-QHT), intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in n , the size of the channel set and group, compared to naïve repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.

We illustrate how quantum information theory and free (i.e. noncommutative) semialgebraic geometry often study similar objects from different perspectives. We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives. This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field.

Building on the quantum ensemble based classifier algorithm of Schuld and Petruccione [arXiv:1704.02146v1], we devise equivalent classical algorithms which show that this quantum ensemble method does not have advantage over classical algorithms. Essentially, we simplify their algorithm until it is intuitive to come up with an equivalent classical version. One of the classical algorithms is extremely simple and runs in constant time for each input to be classified. We further develop the idea and, as the main contribution of the paper, we propose methods inspired by combining the quantum ensemble method with adaptive boosting. The algorithms were tested and found to be comparable to the AdaBoost algorithm on publicly available data sets.

Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of 2 ?�N . We explore such nonlinearity and propose an associated quantum measurement scenario based on the nonlinear interaction of N -probe entanglement generating form. This scenario provides an enhanced precision scaling of D ?�N /(N??)! with D>2 a tunable parameter. In addition, it can be readily implemented in a variety of experimental platforms and applied to measurements of a wide range of quantities, including local gravitational acceleration g , magnetic field, and its higher-order gradients.

A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ? AB , such that both of its reduced density matrices ? A and ? B of size N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ? A and ? B , which can be bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single qubit case we provide a semi-analytic expression for the optimal transport cost between any two states, and prove that its square root satisfies the triangle inequality and yields an analogue of the Wasserstein distance of order two on the set of density matrices. Assuming that the cost matrix suffers decoherence, we study the quantum-to-classical transition of the Earth Mover's distance, propose a continuous family of interpolating distances, and demonstrate in the case of diagonal mixed states that the quantum transport is cheaper than the classical one.

A reasonable quantum information theory for fermions must respect the parity super-selection rule to comply with the special theory of relativity and the no-signaling principle. This rule restricts the possibility of any quantum state to have a superposition between even and odd parity fermionic states. It thereby characterizes the set of physically allowed fermionic quantum states. Here we introduce the physically allowed quantum operations, in congruence with the parity super-selection rule, that map the set of allowed fermionic states onto itself. We first introduce unitary and projective measurement operations of the fermionic states. We further extend the formalism to general quantum operations in the forms of Stinespring dilation, operator-sum representation, and axiomatic completely-positive-trace-preserving maps. We explicitly show the equivalence between these three representations of fermionic quantum operations. We discuss the possible implications of our results in characterization of correlations in fermionic systems.

We study learning from quantum data, in particular quantum state classification which has applications, among others, in classifying the separability of quantum states. In this learning model, there are n quantum states with classical labels as the training samples. Predictors are quantum measurements that when applied to the next unseen quantum state predict its classical label. By integrating learning theory with quantum information, we introduce a quantum counterpart of the PAC framework for learning with respect to classes of measurements. We argue that major challenges arising from the quantum nature of the problem are measurement incompatibility and the no-cloning principle -- prohibiting sample reuse. Then, after introducing a Fourier expansion through Pauli's operators, we study learning with respect to an infinite class of quantum measurements whose operator's Fourier spectrum is concentrated on low degree terms. We propose a quantum learning algorithm and show that the quantum sample complexity depends on the ``compatibility structure" of such measurement classes -- the more compatible the class is, the lower the quantum sample complexity will be. We further introduce k -junta measurements as a special class of low-depth quantum circuits whose Fourier spectrum is concentrated on low degrees.

We discuss quantum variational optimization of Ramsey interferometry with ensembles of N entangled atoms, and its application to atomic clocks based on a Bayesian approach to phase estimation. We identify best input states and generalized measurements within a variational approximation for the corresponding entangling and decoding quantum circuits. These circuits are built from basic quantum operations available for the particular sensor platform, such as one-axis twisting, or finite range interactions. Optimization is defined relative to a cost function, which in the present study is the Bayesian mean square error of the estimated phase for a given prior distribution, i.e. we optimize for a finite dynamic range of the interferometer. In analogous variational optimizations of optical atomic clocks, we use the Allan deviation for a given Ramsey interrogation time as the relevant cost function for the long-term instability. Remarkably, even low-depth quantum circuits yield excellent results that closely approach the fundamental quantum limits for optimal Ramsey interferometry and atomic clocks. The quantum metrological schemes identified here are readily applicable to atomic clocks based on optical lattices, tweezer arrays, or trapped ions.

A unitary t -design is a powerful tool in quantum information science and fundamental physics. Despite its usefulness, only approximate implementations were known for general t . In this paper, we provide for the first time quantum circuits that generate exact unitary t -designs for any t on an arbitrary number of qubits. Our construction is inductive and is of practical use in small systems. We then introduce a t -th order generalization of randomized benchmarking ( t -RB) as an application of exact 2t -designs. We particularly study the 2 -RB in detail and show that it reveals self-adjointness of quantum noise, a new metric related to the feasibility of quantum error correction (QEC). We numerically demonstrate that the 2 -RB in one- and two-qubit systems is feasible, and experimentally characterize background noise of a superconducting qubit by the 2 -RB. It is shown from the experiment that interactions with adjacent qubits induce the noise that may result in an obstacle toward the realization of QEC.

Within the last decade much progress has been made in the experimental realisation of quantum computing hardware based on a variety of physical systems. Rapid progress has been fuelled by the conviction that sufficiently powerful quantum machines will herald enormous computational advantages in many fields, including chemical research. A quantum computer capable of simulating the electronic structures of complex molecules would be a game changer for the design of new drugs and materials. Given the potential implications of this technology, there is a need within the chemistry community to keep abreast with the latest developments as well as becoming involved in experimentation with quantum prototypes. To facilitate this, here we review the types of quantum computing hardware that have been made available to the public through cloud services. We focus on three architectures, namely superconductors, trapped ions and semiconductors. For each one we summarise the basic physical operations, requirements and performance. We discuss to what extent each system has been used for molecular chemistry problems and highlight the most pressing hardware issues to be solved for a chemistry-relevant quantum advantage to eventually emerge.