Mark M. Wilde

Conic programs arising in physics, quantum information, machine learning, and engineering are oftentimes defined over sparse graphs. Although such problems can be solved in polynomial time using classical interior-point solvers, the computational complexity scales unfavorably with the graph size. In this context, this work proposes a variational quantum paradigm for solving conic programs, including quadratically constrained quadratic programs and semidefinite programs. We encode primal variables via the state of a parametrized quantum circuit (PQC) and dual variables via the probability mass function associated with a second PQC. The Lagrangian function can thus be expressed as scaled expectations of quantum observables. We pursue approximately stationary points of the Lagrangian by minimizing (maximizing) the Lagrangian over the parameters of the first (second) PQC. This is accomplished in a hybrid fashion: Gradients of the Lagrangian are estimated using the two PQCs, while PQC parameters are updated classically using a primal-dual method. We propose permuting the primal variables so that related observables are expressed in a banded form, enabling efficient measurement. We provide a complexity analysis that is useful to determine which problem types may enjoy quantum advantage. The proposed framework is applied to the ac optimal power flow (OPF) problem, a large-scale optimization problem central to the operation of electric power systems. Numerical tests on the IEEE 57-node power system using Pennylane's simulator corroborate that the proposed doubly variational quantum framework can find high-quality OPF solutions. While this OPF demonstration does not yield a quantum speedup, the results serve as a proof of concept and highlight challenges toward practical quantum advantage. Although showcased for the OPF, this framework has a broader scope, including conic programs with many variables and constraints, problems defined over sparse graphs, and training quantum machine-learning models to satisfy constraints.

The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum–classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semidefinite programming (SDP), by making use of variational formulas for the measured relative entropies of states and semidefinite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semidefinite programs provide the optimal values of the measured relative entropies of states and channels, but also the numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.

We study the capacity of a quantum channel for retrocausal communication, where messages are transmitted backward in time, from a sender in the future to a receiver in the past, through a noisy postselected closed timelike curve mathematically represented by the channel. We completely characterize the one-shot retrocausal quantum and classical capacities, and we show that the corresponding asymptotic capacities are equal to the average and sum, respectively, of the channel's max-information and its regularized Doeblin information. This endows these information measures with a novel operational interpretation. Furthermore, our characterization can be generalized beyond quantum channels to all completely positive maps. This imposes information-theoretic limits on transmitting messages via postselected-teleportation-like mechanisms with arbitrary initial- and final-state boundary conditions, including those considered in various black-hole final-state models.

Estimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension d with bounded Thompson metric, our theory establishes a copy complexity of O ( | Θ ( U ) | d / ϵ 2 ) for QNE with a quantum circuit parameter set Θ ( U ) , which has minimax optimal dependence on the accuracy ϵ . Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to O ( | Θ ( U ) | p o l y l o g ( d ) / ϵ 2 ) . Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.

Quantum state exclusion is an operational task with application to ontological interpretations of quantum states. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, admitting the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, when there are two hypotheses, our result provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, when all channels are classical, we show that our upper bound is achievable by a parallel strategy, thus solving the exact error exponent of classical channel exclusion.

With a view toward addressing the explosive growth in the computational demands of nuclear structure and reactions modeling, we develop a novel quantum algorithm for neutron–nucleus simulations with general potentials, which provides acceptable bound-state energies even in the presence of noise, through the noise-resilient (NR) training method. In particular, the algorithm can now solve for any band-diagonal to full Hamiltonian matrices, as needed to accommodate a general central potential. While we illustrate the approach for exponential Gaussian-like potentials and ab initio inter-cluster potentials (optical potentials), it can also accommodate the complete form of the chiral effective-field-theory nucleon–nucleon potentials used in ab initio nuclear calculations. In this study, we provide a comprehensive analysis for the efficacy of this approach for three different qubit encodings, including the one-hot, binary, and Gray encodings, in terms of the number of Pauli strings and commuting sets involved. We also discuss the advantages of the algorithm for Hamiltonians of various band-diagonal widths, especially critical for potentials of perturbative nature, leading to a drastically reduced runtime of quantum simulations. We prove that the Gray encoding allows for an efficient scaling of the model-space size N (or number of basis states used) and is more resource efficient for band-diagonal Hamiltonians having bandwidth up to N . We introduce a new commutativity scheme called distance-grouped commutativity (DGC) and compare its performance with the well-known qubit-commutativity (QC) scheme. We lay out the explicit grouping of Pauli strings and the diagonalizing unitary under the DGC scheme, and we prove that it outperforms the QC scheme, at the cost of a more complex diagonalizing unitary. Lastly, we provide first solutions of the neutron–alpha dynamics from quantum simulations suitable for noisy intermediate-scale quantum processors, using an optical potential rooted in first principles, as well as a study of the bound-state physics in neutron–Carbon systems, along with a comparison of the efficacy of the one-hot and Gray encodings.