Einar Gabbassov

The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modeling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.

We show that, during adiabatic evolution, any changes in entanglement can be attributed to a succession of avoided energy level crossings at which eigenvalues swap their eigenvectors. These swaps mediate the generation and redistribution of entanglement in multipartite systems. The efficiency of this redistribution depends on the narrowness of the avoided level crossings and thus constrains the speed of adiabatic evolution. Moreover, we relate the amount of entanglement involved to the ruggedness of the energy landscape, which directly affects the hardness of a computational problem. This enables an analysis of computational complexity and quantum advantage from the point of view of entanglement requirements. Applied to adiabatic quantum computation, our findings directly relate the computation’s speed to its utilization of entanglement as a resource. The same principles extend to gate-based discretized adiabatic quantum algorithms, including those for Hamiltonian simulation and combinatorial optimization.

We propose an approach to solving constrained combinatorial optimization problems based on embedding the concept of Lagrangian duality into the framework of adiabatic quantum computation. Within the setting of circuit-model fault-tolerant quantum computation, we demonstrate that this approach achieves a quadratic improvement in circuit depth and maintains a constraint-independent circuit width in contrast with the prevalent approach of solving constrained problems via reformulations based on the quadratic unconstrained binary optimization (QUBO) framework. In this study, we include a detailed review of the limitations encountered when using QUBO for constrained optimization. We show that the proposed method overcomes these limitations by encoding the optimal solution at an energetically elevated level of a simpler problem Hamiltonian, which results in substantially more resource-efficient quantum circuits. We consolidate our strategy with a detailed analysis on how the concepts of Lagrangian duality such as duality gap and complementary slackness relate to the success probability of sampling the optimal solution. Our findings are illustrated by benchmarking the Lagrangian dual approach against the QUBO approach using the NP-complete binary knapsack problem.

An essential consideration in urban transit facility planning is service efficiency and accessibility. Previous research has shown that reducing the number of facilities along a route may increase efficiency but decrease accessibility. Striking a balance between these two is a critical consideration in transit planning. Transit facility consolidation is a cost-effective way to improve the quality of service by strategically determining the desirable allocation of a limited number of facilities. This paper develops an optimization framework that integrates Geographical Information systems (GIS), decision-making analysis, and quantum technologies for addressing the problem of facility consolidation. Our proposed framework includes a novel mathematical model that captures non-linear interactions between facilities and surrounding demand nodes, inter-facility competition, ridership demand and spatial coverage. The developed model can harness the power of quantum effects such as superposition and quantum tunnelling and enables transportation planners to utilize the most recent hardware solutions such as quantum and digital annealers, coherent Ising Machines and gate-based universal quantum computers. This study presents a real-world application of the framework to the public transit facility redundancy problem in the British Columbia Vancouver metropolitan area. We demonstrate the effectiveness of our framework by reducing the number of facilities by 40% while maintaining the same service accessibility. Additionally, we showcase the ability of the proposed mathematical model to take advantage of quantum annealing and classical optimization techniques.

The occurrence of multiple strains of a bacterial pathogen such as M. tuberculosis or C. difficile within a single human host, referred to as a mixed infection, has important implications for both healthcare and public health. However, methods for detecting it, and especially determining the proportion and identities of the underlying strains, from WGS (whole-genome sequencing) data, have been limited. In this paper we introduce SplitStrains, a novel method for addressing these challenges. Grounded in a rigorous statistical model, SplitStrains not only demonstrates superior performance in proportion estimation to other existing methods on both simulated as well as real M. tuberculosis data, but also successfully determines the identity of the underlying strains. We conclude that SplitStrains is a powerful addition to the existing toolkit of analytical methods for data coming from bacterial pathogens and holds the promise of enabling previously inaccessible conclusions to be drawn in the realm of public health microbiology.