Quantum complexity and the virial theorem

It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computational complexity in the thermal equilibrium.


Introduction
Given an n-qubit system, one could consider evolution operators as points living in a group manifold SU(2 n ). One can further assign a metric to this manifold, whereupon under a certain manipulation the complexity of the corresponding operator could be understood as the geodesic length that connects the operator and the identity [1,2]. This idea provides a novel geometric way to study complexity theory, and has led to deep conjectures relating the concepts of quantum complexity, holographic duality, and the nature of quantum gravity [3][4][5][6][7][8].
The geometric version of quantum complexity naturally connects quantum physics of k-local Hamiltonians with the notion of the disordered average and statistical motion of particles moving in a group manifold. In this sense, one can conjecture a relation between statistical entropy and quantum complexity. Following the second law of thermodynamics, a similar second law holding for complexity growth has also been conjectured [6]. More precisely, the conjecture states that one can decompose the entire statistical entropy of classical particles in a group manifold into kinetic and positional parts, where the former corresponds to Kolmogorov complexity, and the latter the computational complexity of the corresponding quantum system. The Kolmogorov complexity is roughly speaking the minimal cost to specify bit strings, while the computational complexity is the cost of time or the scale of depth for quantum circuits. As two different complexity measures, it is natural to ask if there exists some possible connections between them. The entropic conjectures about complexity provide us a different angle on this problem, where in the dual classical system, the physics could be understood more intuitively by addressing the property of statistical entropies.
Working in the canonical ensemble, the kinetic-positional decomposition of the statistical entropy is naively the decomposition of the whole Hamiltonian into kinetic and potential energy. In this case, from basic properties of mechanics, or more fundamentally, the equation of motion, one could naturally expect there to potentially be a relation between statistical average of potential and kinetic energy. In ordinary classical and statistical mechanics, the direct answer is celebrated virial theorem.
In this paper, we will study this problem by analyzing a modified version of the virial theorem on the group manifold SU(2 n ). By working directly in the curved geometry, one can arrive at a modified version of the usual virial theorem, where the average of potential and kinetic energies are related by the affine connection terms of the curved space. Thus, connecting with the arguments identifying complexity with entropy, we show a natural relation between two notions of complexities in quantum information theory. This paper is organized as follows. In Section 2, we discuss the extension of the virial theorem to curved space. In Section 3, we discuss the relationship between entropies, and alternatively, complexities as a consequence of this modified virial theorem. In Section 4, we conclude and discuss some possible future directions related to this research.

Traditional virial theorem
The (classical) virial theorem is a connection between the potential and kinetic energy of a statistical system. Here we will review the derivation in classical mechanics as a warmup. For a particle system with location r i and mass m i we have momenta and thus can define the function Further, we have that Here we define to be the kinetic energy. Now the force is given by Newton's law If the system is a stably bound system, we have the derivative of G vanishes after time average 1 , thus giving us This naturally relates force to potential energy. We know that typically the potential is a function depending only on distance of particles, namely, that the Lagrangian is where the potential V should only depend on positions. So the force is given by 1 For a stable, bounded system, after the time average we have resulting in the virial theorem: While this version of virial theorem works for time average of particle trajectories, one can derive a similar result from statistical ensembles, such as, for instance, the canonical ensemble. Now let us consider a system with N particles moving in d-dimensional flat space. The dimension of the phase space in this case is 2dN . We can write the indices collectively as a, b, etc. = 1, 2, · · · , dN . Now consider the quantity where here C is the normalization constant of the expectation value. One finds that specifically, we know that, taking a = b, we have that Note that the above expression has no sum. The same logic applies if one replaces x by p, obtaining As a conclusion we get This is the statistical version of the virial theorem, which is more constraining than the mechanical one (because the statistical version fixes the ensemble). We can see this by applying the Hamilton equation So we could see that, in the case of flat space, these two versions lead to the same result. The consistency between the time average and the ensemble average is a consequence of ergodicity.

Setup
Now we will discuss the virial theorem in the curved space. Existing literature has extended the virial to the curved spacetime in the language of relativity with the application of astrophysics, for instance, in the context of studying dark matter (see [9][10][11][12][13]). However, currently we are interested in only the non-relativistic case, where the goal is to study trajectories of particles moving in a general curved space instead of spacetime, as this is the problem relevant for studying the Nielsen complexity geometry.
We start by considering the Lagrangian in the curved space. Let the space M be a Euclidean manifold. The coordinate of particle i is denoted by x µ i , where µ are the indices for vectors on the manifold. The metric on M is given by g µν . We know that the Lagrangian for many free particles labeled by i is given by its kinetic energy The whole Hamiltonian is where the momenta are defined by Now we can also define curved phase space. The positional part of the phase space is given by the manifold coordinates, while the element volume is given by the invariant volume For given x, p is located in the tangent space of x. Therefore, for any x, p lives in flat space. If we set dim(M) = d, then the space for momentum to be integrated over is R d . So we define the phase space to be For a canonical ensemble we have the inverse temperature β, giving a phase factor of In particular, one can decompose it as P (x, p) = exp(−βH) = exp(−βV ) exp(−βK) (2.25)

Statistical version
Now we start to derive a virial theorem for a canonical ensemble. We begin by considering how to interpret the expression One could write it as Using the formula One can alternatively write it in terms of connections Note that the momentum part is the same, but with no geometric contribution Combining terms, we obtain a new version of the virial theorem One can also take a summation over these quantities; however, such a sum will produce a new term if we expand the total energy in terms of the kinetic and potential energies, giving

Mechanical version
One can alternatively study the mechanical version of the virial theorem by taking a time average. One can define the quantity So similar with the previous derivation, we take the derivative over t to get So take the average will give a modified version of the virial theorem, where we use the equation of motion Using general relativity identity Note here that the mechanical and statistical virial theorems are, in fact, different. The reason is that now in a curved space, the argument of ergodicity is broken. The different points in space are not equaly likely to be accessed at late time; the true probability depends sensatively on the shape of the manifold and on the initial positions and momenta of the particles. This reflects the argument that we made in the previous section, where it was noted that the statistical mechanical version of the virial theorem is more constrained than the mechanical one due to ensemble fixing. For our usage, we will use the claim to identify the quantum complexity and statistical entropy, so we could use the statistical virial theorem to derive a relation between two parts of entropy.

A relation between complexities
We can decompose the entire entropy in the following way: To make this relation work with tractible computations, we make the assumption that all coordinates x µ i are very close to the origin x µ i = 0 2 . In this case, we could write x as ∆x. Then, defining V (x µ i = 0) = 0, then we could have In this limit, we obtain where for our current purpose we set the mass to one according to [6]. Based on the conjectures in [6], the ensemble average of computational (Kolmogorov) complexity of a disordered k-local quantum system, should be proportional to positional (kinetic) part of statistical entropy of a dual statistical gas living in the group manifold. From the entropy relation derived above, we arrive at a direct relation between computational and Kolmogorov complexity. The relation is pedagogically Kolmogorov complexity = Computational complexity + geometric corrections (3.5) where the geometric corrections can be computed directly from Nielsen's geometric construction of quantum computing [1,2]. In this metric definition, the metric as a bilinear for Hamiltonian representation of vector fields H and J near point U is H, J = Tr (HP(J)) + qTr (HQ(J)) 2 n (3.6) where P and Q are super operators that takes the Hamitonian to the one and two body term components and three or more body components respectively. Defining G = P + qQ, we have where the indices σ, τ etc denote the possibilities of Paulis. One can also derive the connection with one upper index where F = G −1 . Using this framework, we can in principle compute the geometric correction concretely in any circumstance. A trivial example for this computation is that for twoqubit system, where n = 2, the correction term vanishes so we arrive at the conclusion that computational complexity is simply proportional to Kolmogorov complexity. For n ≥ 3, one can obtain the correction mechanically when specifying the location and speed of classical trajectories, though it should be noted that even for this small number of qubits the calculation becomes quite intensive.

Conclusion and discussion
In this paper, we study a possible consequence of conjectures in [6] for identifying complexities in the k-local disordered Hamiltonian and classical systems of particulate gas living in the group manifold of unimodular matrices. After a discussion of the virial theorem in a general curved space, we arrive at a nontrivial relation between Kolmogorov complexity and computational complexity by identifying complexities with entropies.
In future work it would be interesting to study such an argument from the complexity theoretic and quantum resource theoretic points of view (where, the Kolmogorov complexitycomputational complexity decomposition, or moreover, the complexity-uncomplexity decomposition, claimed in [6], may have a meaningful interpretation as quantum resource). It will also be interesting to work out some specific examples for k-local disordered Hamiltonians (like the SYK model [14][15][16][17][18][19] 3 ) to verify validity of the statement in practice, as perhaps a nontrivial check to the conjectures of [6].