Anomalous weak values and superoscillations allow a pointer to exhibit an effective shift outside the eigenvalue range of the measured observable. We reformulate this effect in a phase-space path-integral language. For each eigenvalue branch $a_j$ of the measured observable, the interaction couples $a_j$ to the pointer position $\hat{x}_p$ and thereby translates the conjugate pointer momentum by $\gamma a_j$. After postselection, the branch-conditioned amplitudes are coherently superposed with coefficients chosen to produce a superoscillatory approximation. When this conditional amplitude acts on an initial pointer momentum wavepacket for which the finite superoscillatory-window condition is satisfied, the resulting momentum wavefunction behaves as if it had been translated by $\gamma A_w$, rather than by any of the eigenvalue-conditioned shifts $\gamma a_j$. This yields an effective path-integral representation of anomalous weak pointer shifts and motivates their interpretation in terms of interference among finite coarse-grained histories. We discuss how this picture bears on conservation-law questions. Quanta 2026; 15: 31–39.
Quantum mechanics contains certain novel mathematical concepts. Among these are complex numbers, Hilbert spaces with their unitary and self-adjoint operators, states represented by complex vectors, superpositions of states, collapse of wave functions, Born's rule for probabilities and others. If we accept that quantum mechanics is probabilistic, then these concepts can be derived and they become secondary. In this work, we begin with what we call a conditional stochastic process, which is based on real numbers and probabilities. As we shall see, such processes are defined by three simple axioms. We then use conditional stochastic processes to derive quantum mechanics by employing a correspondence called a dictionary. We also show that the converse holds. That is, beginning with a quantum system, we employ the dictionary to derive a conditional stochastic process. Quanta 2026; 15: 13–18.
An interesting approach to using noisy intermediate-scale quantum (NISQ) devices is hybrid classical-quantum machine learning (QML). In these methods, classical processors handle optimisation and large-scale computation, while quantum hardware is devoted to tasks like feature mapping, nonlinear transformations, or kernel evaluation. Practical near-term demonstrations are made possible by this division of labour, which also mitigates existing hardware restrictions. In specialised fields like molecular modelling, materials discovery, and small-sample learning issues, recent advancements in variational quantum circuits, hybrid neural networks, and quantum kernel techniques have produced promising outcomes. However, scalability and wider applicability are still hampered by enduring issues including noise, barren plateaus, and the expense of repeated measurements. Long-term developments will require fault tolerance, logical qubits, and established software infrastructures, whereas near-term success depends on noise-aware algorithm design, repeatable experimental benchmarks, and enhanced error-mitigation strategies. When taken as a whole, these advancements show a viable path to achieving quantum advantage in machine learning. Quanta 2026; 15: 1–12.
It is shown that ternary qubit trees with the same number of nodes can be transformed by the naturally defined sequence of Clifford gates into each other or into standard representation as 1-dimensional chain corresponding to Jordan–Wigner transform. Quanta 2026; 15: 24–30.
What does it mean for the physically motivated computational complexity class $\mathcal{BQP}$ and the logically motivated class $\mathcal{NP}$ to be incomparable, from a physical perspective? Although it is unlikely that quantum computers can solve $\mathcal{NP}$-complete problems in polynomial time, yet it may be instructive to see how a physical theory (denoted $\Theta^{(\mathcal{NP})}$ or $\Theta^{(\mathcal{PSPACE})}$) for which $\mathcal{NP}$ or even $\mathcal{PSPACE}$, respectively, constitutes an efficient model of computation, would compare with quantum theory. The present work explores this question, after briefly reviewing some complexity results indicative of the incomparability of $\mathcal{BQP}$ and $\mathcal{NP}$. We argue that theories $\Theta^{(\mathcal{NP})}$ or $\Theta^{(\mathcal{PSPACE})}$ effectively feature degrees of nonlinearity. Thus, the linearity of quantum theory prevents complete problems in these complexity classes to be efficiently solvable on a quantum computer. On the other hand, the exponentiality of quantum theory endows quantum computers with a certain strength going beyond $\mathcal{NP}$ or even $\mathcal{PH}$. Thus the contrasting roles of the empowerment provided by quantum mechanical exponentiality and the limitation imposed by quantum mechanical linearity seem to be at the heart of the incomparability of the classes $\mathcal{BQP}$ and $\mathcal{NP}$. Quanta 2025; 14: 38–47.
