A Quantum-Bayesian Route to Quantum-State Space

Christopher A. Fuchs, Rüdiger Schack

In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent's personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor.

Phillip R. Dukes

Anupam Garg, Prateek Vishnoi

The Capacitated Vehicle Routing Problem (CVRP) involves finding the shortest possible routes for a fleet of vehicles, each with a fixed capacity limit, to serve a set of customers where every customer is visited exactly once. When CVRP is encoded as a Quadratic Unconstrained Binary Optimization (QUBO) model, one of the biggest practical hurdles is setting the penalty coefficients correctly — a process that is typically done by hand and has to be repeated for every new problem instance. This paper introduces a new technique called BOSA (Bayesian Optimization for Simulated Annealing), a framework that removes this manual step by using Gaussian Process-based Bayesian Optimization to find good penalty values automatically, while the underlying QUBO is solved using Simulated Annealing. Our hybrid approach integrates multi-start greedy construction and 2-opt local search to ensure solution feasibility when SA produces invalid routes. Experiments on 11 benchmark instances demonstrate solution distances of 12.04%- 33.26% relative to the best-known optima. On the two instances where comparison is possible, our proposed technique, BOSA achieves shorter routes than D-Wave (949 vs 1085 for A-n32-k5, and 2240 vs 2660 for A-n80-k10), suggesting that automated classical penalty tuning can match or exceed commercial quantum annealing solutions on these problem sizes without requiring quantum resources. Although not competitive with state-of-the-art CVRP solvers such as HGS and branch and cut algorithm, our automated tuning framework provides a foundation for future quantum-classical hybrid algorithms. It demonstrates the viability of BO-driven parameter optimization for QUBO-based combinatorial problems.

Dorje C. Brody, Bernhard K. Meister

Jon Frost, Carlos Madeira, Harald Uhlig

Joe Mashburn

In 2002 Coecke and Martin created a Bayesian order for the finite dimensional spaces of classical states in physics and used this to define a similar order, the spectral order on the finite dimensional quantum states. These orders gave the spaces a structure similar to that of a domain. This allows for measuring information content of states and for determining which partial states are approximations of which pure states. In a previous paper the author extended the Bayesian order to infinite dimensional spaces of classical states. The order on infinite dimensional spaces retains many of the characteristics important to physics, but loses the domain theoretic structure. It becomes impossible to measure information content in the same way that it is done for the finite dimensional spaces, and the sense of approximation is lost. In this paper, we will use the Bayesian order to define a spectral order on the infinite dimensional spaces of quantum states.