We study the separation of geometric effects from quantum interference in quantum optimization algorithms. Constrained optimization problems such as routing, assignment, and scheduling are often encoded as product spaces of local variables, together with global feasibility penalties. The central algorithmic question we address is how a constraint-preserving mixing operator transports quantum amplitude across an exponential search space in the presence of local and global constraints. We develop a framework that separates three effects that are usually intermixed: amplitude transport, coherent interference among transported amplitudes, and problem-dependent classical postprocessing. We show that the mixing operator alone does not have a target-seeking ability. Concretely, the normalized distribution induced by its amplitude transport moves toward the distance profile of a uniformly random configuration. Thus, quantum sampling advantage may only arise when the phases of the many computational paths reaching a target configuration are sufficiently aligned for their amplitudes to reinforce. We show that, when the cost phases are engineered so that these paths add coherently, a number of circuit alternations growing only logarithmically with problem size suffices to convert the sum of their absolute contributions into a lower bound on the target amplitude, yielding a certified success probability independent of the ambient Hilbert-space dimension, the search-space size, or the feasible-set cardinality. We develop applications to problem-specific transpilation diagnostics, scalable hardware probes, constraint-induced classical maps of quantum-generated samples, the attribution of solution quality between the quantum distribution and classical post-processing in hybrid quantum-classical workflows and connections to distance-partitioned product spaces from classical coding theory.
Chinonso Onah, Stuart Hadfield, K. Michielsen· 1 citation
Qubit-efficient quantum relaxations compress classical decision variables into expectation values on substantially fewer qubits. We ask what resource tradeoffs this compression entails for quantum optimization. For the complete quadratic-Majorana encoding on $n$ qubits, pairwise correlators can represent $m=\Theta(n^2)$ binary variables. We define the universal margin as the smallest correlator magnitude that can be guaranteed with prescribed signs for every target sign assignment. We show that it is exactly $\Delta_{\rm Maj}(n)=\tan\!\left(\frac{\pi}{4n}\right)=\Theta(1/n)$, whereas uniformly random sign assignments retain $\Theta(1/\sqrt n)$ target-specific margins. The stronger $1/n$ worst-case scaling is Majorana-specific. Moreover, arbitrary density operators and fermionic Gaussian states generate the same quadratic-Majorana covariance body, so non-Gaussian state resources cannot enlarge this two-point relaxation. Beyond Majoranas, standard quantum random access code bounds provide general information-theoretic baselines. For any fixed family of $m$ designated binary observables on $n$ qubits, the universal margin is at most $\sqrt{(2\ln2\;n/m)}$, while arbitrary random access decoding from $N$ copies with constant success probability above $1/2$ requires $nN=\Omega(m)$. For a fixed Pauli correlation encoding required to work uniformly over all targets, maintaining a fixed nonzero decoded magnitude under smooth sign decoding therefore requires a rescaling parameter that grows as the available margin shrinks. Thus, while providing substantial qubit savings, compression can shift cost into restricted expectation value geometry, smaller expectation value magnitudes, or more demanding information recovery rather than eliminate it.
A noise-aware adaptive approach to quantum approximate optimization, Noise-Directed Adaptive Warm-Starting (ND-AWS), that builds on recent concepts such as Warm-Start QAOA and Noise-Directed Adaptive Remapping by leveraging bitflip gauge transformations, and exploits amplitude-damping-like noise components.
Filip B. Maciejewski, Stuart Hadfield, Oscar Wallis et al.· 4 citations