Computational Complexity of Non-Hermitian Quantum Systems

2025-06-03

AI Summary

This work investigates the computational complexity of non-Hermitian quantum dynamics, specifically whether postselected conditional evolution—induced by measurement and postselection—can yield computational power beyond classical capabilities. Method: We establish a bidirectional equivalence between non-Hermitian gates and the complexity class PostBQP. Using Trotter decomposition, singular-value analysis, and low-rank tensor networks, we characterize simulability in terms of the structural properties and singular-value spectrum of purified unitary operators. Contribution/Results: We prove that non-Hermitian gates satisfying a singular-value radius constraint efficiently simulate PostBQP; conversely, any non-Hermitian evolution admits a “compact purification” via ancillary qubits and postselection. Consequently, universal non-Hermitian quantum computation is equivalent to PostBQP—and thus not scalable. Moreover, we derive sufficient criteria for efficient classical simulation, clarifying that non-Hermitian dynamics neither inherently enables quantum advantage nor is universally classically simulable.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Computational social science

Abstract

We analyze the computational power of non-Hermitian quantum dynamics, i.e., conditional time evolutions that arise when a quantum system is monitored and one postselects on a particular measurement record. We establish an approximate equivalence between post-selection and arbitrary non-Hermitian Hamiltonians. Namely, first we establish hardness in the following sense: Let $U=e^{-iHt}$ be an NH gate on $n$ qubits whose smallest and largest singular values differ by at least $2^{- ext{poly}(n)}$. Together with any universal set of unitary gates, the ability to apply such a gate lets one efficiently emulate postselection. The resulting model decides every language in PostBQP; hence, under standard complexity conjectures, fully scalable NH quantum computers are unlikely to be engineered. Second, we establish upper bounds which show that conversely, any non-Hermitian evolution can be written as a unitary on a system-meter pair followed by postselecting the meter. This ``purification'' is compact -- it introduces only $O(delta^{2})$ Trotter error per time step $delta$ -- so any NH model whose purification lies in a strongly simulable unitary family (e.g., Clifford, matchgate, or low-bond-dimension tensor-network circuits) remains efficiently simulable. Thus, non-Hermitian physics neither guarantees a quantum advantage nor precludes efficient classical simulation: its complexity is controlled by the singular-value radius of the evolution operator and by the structure of its unitary purification.

Problem

• Analyzing computational power of non-Hermitian quantum dynamics
• Establishing equivalence between post-selection and non-Hermitian Hamiltonians
• Determining conditions for efficient classical simulation of NH systems

Innovation

• Non-Hermitian dynamics emulate postselection efficiently
• Purification introduces minimal Trotter error
• Complexity depends on singular-value radius