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arXiv:2106.14081 · Physical review. E

Weighted Bures length uncovers quantum state sensitivity

Paweł Kurzyński

Published Nov 18, 2021• 3 citations
arXivHTMLPDF

Abstract

The unitarity of quantum evolutions implies that an overlap between two initial states does not change in time. This property is commonly believed to explain the apparent lack of state sensitivity in quantum theory, a feature that is prevailing in classical chaotic systems. However, classical state sensitivity is based on a distance between two trajectories in phase space which is a completely different mathematical concept than an overlap between two vectors in Hilbert space. It is possible that state sensitivity in quantum theory can be detected with the help of some special metric. Here we show that the recently introduced Weighted Bures length achieves this task. We numerically investigate a unitary cellular automaton of N interacting qubits and analyze how a single-qubit perturbation affects the evolution of WBL between the unperturbed and perturbed states. We observe a linear growth of WBL if the qubits are arranged into a cyclic graph and an exponential growth if they are arranged into a random bipartite graph.

DOI
10.1103/physreve.104.l052202

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The unitarity of quantum evolutions implies that an overlap between two initial states does not change in time. This property is commonly believed to explain the apparent lack of state sensitivity in quantum theory, a feature that is prevailing in classical chaotic systems. However, classical state sensitivity is based on a distance between two trajectories in phase space which is a completely different mathematical concept than an overlap between two vectors in Hilbert space. It is possible that state sensitivity in quantum theory can be detected with the help of some special metric. Here we show that the recently introduced Weighted Bures length achieves this task. We numerically investigate a unitary cellular automaton of N interacting qubits and analyze how a single-qubit perturbation affects the evolution of WBL between the unperturbed and perturbed states. We observe a linear growth of WBL if the qubits are arranged into a cyclic graph and an exponential growth if they are arranged into a random bipartite graph.
Numerical results show that, for quantum autonomous chaotic system, the evolution of initially coherent states are sensitive to perturbation. The overlap of a perturbed state with the unperturbed one decays exponentially, which is followed by fluctuation around N-1, N being the dimension of the Hilbert space. The matrix elements of the evolution operator in interaction picture tend to be a random distribution after sufficiently long time, where the interaction is the perturbation, even when the perturbation is very weak. The difference between a regular system and the chaotic one is shown clearly. In a regular system, the overlap shows strong revival. The distribution of the evolution matrix has only a few dominant terms.
Sensitivity of the time evolution to small changes in the state is a characteristic feature of classical chaos. It has been believed that state sensitivity could not exist in quantum mechanics because of the unitary invariance of the Hilbert-space overlap of states. We argue that this Hilbert-space criterion is irrelevant and show that both quantum states and classical statistical states exhibit a similar kind of state sensitivity. This is demonstrated by the degree to which the initial state can be recovered in computational motion reversal: forward evolution for a time T, perturbation of the state, and backward time evolution. Some differences between classical and quantum state sensitivity remain, and these seem to be insensitive to decoherence. \textcopyright{} 1996 The American Physical Society.
Abstract The problem of evaluating the differences between quantum states before and after being affected by errors encoded in unitary transformations is addressed. Standard distance functions, e.g., the Bures length, are not fully adequate for such a task. Weighted distances are instead the appropriate information measures to quantify the distinguishability of multipartite states. Here, the previously introduced weighted Bures length and the newly defined weighted Hilbert‐Schmidt distance are employed to quantify how much single‐qubit Pauli errors alter cluster states. For both types of weighted distances, it is found that different errors of the same dimension change cluster states in a different way, i.e., their detectability is in general different. Indeed, they transform an ideal cluster state into a state whose weighted distances from the input depends on the specific chosen Pauli rotation, as well as the position of the affected qubit in the graph related to the cluster state. As these features are undetected by using standard distances, the study proves the usefulness of weighted distances to monitor key but elusive properties of many‐body quantum systems.
We reexamine the problem of the ``Loschmidt echo,'' that measures the sensitivity to perturbation of quantum-chaotic dynamics. The overlap squared M(t)M(t)M(t) of two wave packets evolving under slightly different Hamiltonian is shown to have the double-exponential initial decay $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\mathrm{constant}\ifmmode\times\else\texttimes\fi{}{e}^{2{\ensuremath{\lambda}}_{0}t})$ in the main part of the phase space. The coefficient ${\ensuremath{\lambda}}_{0}$ is the self-averaging Lyapunov exponent. The average decay $\overline{M}\ensuremath{\propto}{e}^{\ensuremath{-}{\ensuremath{\lambda}}_{1}t}$ is single exponential with a different coefficient ${\ensuremath{\lambda}}_{1}.$ The volume of phase space that contributes to M‾\overline{M}M vanishes in the classical limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0$ for times less than the Ehrenfest time ${\ensuremath{\tau}}_{E}=\frac{1}{2}{\ensuremath{\lambda}}_{0}^{\ensuremath{-}1}|\mathrm{ln}\ensuremath{\Elzxh}|.$ It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.
A bstract We study the quantum complexity of time evolution in large- N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e −iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between off- diagonal energy eigenstate projectors and the k -local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large- N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE "Image missing" BQP/poly and PSPACE "Image missing" BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians.
We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e -iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between offdiagonal energy eigenstate projectors and the k-local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE * BQP/poly and PSPACE * BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians.