There Are No Easy Paths in Quantum Computing ⚡ экспресс
We investigate the computational power of non-Hermitian quantum circuits. If coherent normalized non-unitary evolution is achievable with polynomial overhead, the model enables postselection, yielding implausible complexity-theoretic power. We define the class NHBQP(U) for polynomial-size circuits with a fixed non-unitary gate U acting on O(1) qubits and renormalization. We prove: NHBQP(U) contains PostBQP; in the uniform model, NHBQP(U)=PostBQP=PP. Since PostBQP is considered intractable, scalable non-Hermitian advantage requires restrictions. We study purifications of restricted systems: unitary circuits with postselection can simulate non-Hermitian evolution and trajectories. If the purification belongs to a strongly simulable family (Clifford, matchgate, low-rank tensor networks) with event probabilities Ω(2^{-poly(n)}), classical simulation remains efficient. Adding non-Hermiticity to a universal system yields excessive power, while for strongly simulable systems it offers no benefit.
Quantum computers dance a reversible waltz: step forward — step back, and the system returns to its starting point. But in some processes, the dancer vanishes into darkness, never to return. Such irreversible steps add entropy to the system — a measure of irreversible disorder. If a quantum computer could easily perform such tricks, it would solve problems that would take ordinary machines an eternity. This would violate all known rules of computational complexity. Yet the same logic suggests: there are no easy paths. Strikingly, a similar irreversible loss of information occurs with black holes, and this puzzle forced Hawking to reconsider his own views. Nature abhors a free lunch — even in the quantum world.
🎯 Reversibility in quantum computing isn't a whim; it follows from energy conservation — a principle that has held true from steam engines to modern times.