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Operator Complexity for Continuous Variable Systems

S. Shajidul Haque, Chandan Jana, Bret Underwood

Published Oct 15, 2021• 5 citations
OpenAlex

Abstract

We extend the method of computing operator complexity as a minimal length geodesic on a group manifold to infinite-dimensional Hilbert spaces by computing the complexity for the displacement, squeeze and rotation operators of a quantum harmonic oscillator. The resulting complexity of these operators is state-independent since this approach is independent of the particular reference and target states chosen. We show how the group manifold geometry associated with the Heisenberg generators of the displacement operator is non-compact 3-dimensional hyperbolic space, with a time-independent complexity equal to the magnitude of the coherent state parameter. A generic quadratic Hamiltonian for a quantum harmonic oscillator can be decomposed as a product of squeeze and rotation operators, parameterized by a squeezing parameter, squeezing angle, and rotation angle. The squeeze and rotation operators are elements of the group SU(1,1), and we again find an associated geometry of the group manifold that is negatively curved. The corresponding complexity of the squeezing operator is proportional to the squeezing parameter and is independent of the squeezing angle, in contrast the complexity of squeezed states found in some state-based approaches. These results are a step towards characterizing the properties of quantum information for continuous variable systems in a state-independent way.

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Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.

Journal of Mathematical Physics•Jun 1, 2019•11 citations
We extend the method of computing operator complexity as a minimal length geodesic on a group manifold to infinite-dimensional Hilbert spaces by computing the complexity for the displacement, squeeze and rotation operators of a quantum harmonic oscillator. The resulting complexity of these operators is state-independent since this approach is independent of the particular reference and target states chosen. We show how the group manifold geometry associated with the Heisenberg generators of the displacement operator is non-compact 3-dimensional hyperbolic space, with a time-independent complexity equal to the magnitude of the coherent state parameter. A generic quadratic Hamiltonian for a quantum harmonic oscillator can be decomposed as a product of squeeze and rotation operators, parameterized by a squeezing parameter, squeezing angle, and rotation angle. The squeeze and rotation operators are elements of the group SU(1,1), and we again find an associated geometry of the group manifold that is negatively curved. The corresponding complexity of the squeezing operator is proportional to the squeezing parameter and is independent of the squeezing angle, in contrast the complexity of squeezed states found in some state-based approaches. These results are a step towards characterizing the properties of quantum information for continuous variable systems in a state-independent way.
A bstract According to the pioneering work of Nielsen and collaborators, the length of the minimal geodesic in a geometric realization of a suitable operator space provides a measure of the quantum complexity of an operation. Compared with the original concept of complexity based on the minimal number of gates required to construct the desired operation as a product, this geometrical approach amounts to a more concrete and computable definition, but its evaluation is nontrivial in systems with a high-dimensional Hilbert space. The geometrical formulation can more easily be evaluated by considering the geometry associated with a suitable finite-dimensional group generated by a small number of relevant operators of the system. In this way, the method has been applied in particular to the harmonic oscillator, which is also of interest in the present paper. However, subtle and previously unrecognized issues of group theory can lead to unforeseen complications, motivating a new formulation that remains on the level of the underlying Lie algebras for most of the required steps. Novel insights about complexity can thereby be found in a low-dimensional setting, with the potential of systematic extensions to higher dimensions as well as interactions. Specific examples include the quantum complexity of various target unitary operators associated with a harmonic oscillator, inverted harmonic oscillator, and coupled harmonic oscillators. The generality of this approach is demonstrated by an application to an anharmonic oscillator with a cubic term.
We evaluate the complexity of the free scalar field by the operator approach in which the transformation matrix between the second quantization operators of the reference state and target state is regarded as the quantum gate. We first examine the system in which the reference state is two noninteracting oscillators with the same frequency ${\ensuremath{\omega}}_{0}$ while the target state is two interacting oscillators with frequency ${\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{2}$. We calculate the geodesic length on the associated group manifold of gate matrix and reproduce the known value of ground-state complexity. Next, we study the complexity in the excited states. Although the gate matrix is very large we can transform it to a diagonal matrix and obtain the associated complexity. We explicitly calculate the complexity in several excited states and prove that the square of the geodesic length in the general state ∣n,m⟩|\mathrm{n},\mathrm{m}⟩∣n,m⟩ is ${D}_{(\mathrm{n},\mathrm{m})}^{2}=(n+1){(\mathrm{ln}\sqrt{\frac{{\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{1}}{{\ensuremath{\omega}}_{0}}})}^{2}+(m+1){(\mathrm{ln}\sqrt{\frac{{\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{2}}{{\ensuremath{\omega}}_{0}}})}^{2}$. The results are extended to the NNN couple harmonic oscillators which correspond to the lattice version of the free scalar field.
We generalized the squeeze and displacement operators of the one-dimensional harmonic oscillator to the three-dimensional case and based on these operators we construct the corresponding coherent and squeezed states. We have also calculated the Wigner function for the three-dimensional harmonic oscillator and from the analysis of time evolution of this function, the quantum Liouville equation is also presented. Further properties of the quantum states including Mandel's Q and quadrature squeezing parameters are discussed as well.
We introduce a quantifier of phase-space complexity for discrete-variable (DV) quantum systems. Motivated by a recent framework developed for continuous-variable systems, we construct a complexity measure of quantum states based on the Husimi Q-function defined over spin coherent states. The quantifier combines into a single scalar quantity two complementary information-theoretic quantities, the Wehrl entropy, which captures phase-space spread, and the Fisher information, which captures localization. We derive fundamental properties of this measure, including its invariance under SU(2) displacements. The complexity is normalized such that coherent states have unit complexity, while the completely mixed state has zero complexity, a feature distinct from the continuous-variable case. We provide analytic expressions for several relevant families of states, including Gibbs and Dicke states, and perform a numerical analysis of spin-squeezed states, NOON states, and randomly generated states. Numerical results reveal a monotonic, but not deterministic, relationship between complexity and purity, leading us to conjecture that maximal complexity is attained by pure states, thereby connecting the problem to the optimization of Wehrl entropy via Majorana constellations. Finally, we extend the framework to quantum channels, defining measures for both the generation and breaking of complexity. We analyze the performance of common unitary gates and the amplitude damping channel, showing that while low-dimensional systems can achieve maximal complexity via spin squeezing or NOON states, this becomes impossible in higher dimensions. These results highlight dimension-dependent limitations in the generation of phase-space complexity and establish a unified phase-space approach to complexity across both continuous and discrete variable regimes.
We present a systematic method to expand the quantum complexity of interacting theory in a series of coupling constants. The complexity is evaluated by the operator approach in which the transformation matrix between the second quantization operators of the reference state and the target state defines the quantum gate. We start with two coupled oscillators and perturbatively evaluate the geodesic length of the associated group manifold of the gate matrix. Next, we generalize the analysis to NNN coupled oscillators, which describes the lattice $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ theory. Especially, we introduce simple diagrams to represent the perturbative series and construct simple rules to efficiently calculate the complexity. General formulas are obtained for the higher-order complexity of excited states. We present several diagrams to illuminate the properties of complexity and show that the interaction correction to complexity may be positive or negative depending on the magnitude of the reference-state frequency.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.