Complexity of operators generated by quantum mechanical Hamiltonians

We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS$_3$ spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal proves the complexity of the operator generated by a free Hamiltonian is zero, as expected. By studying a non-relativistic particle in compact Riemannian manifolds we find the complexity is given by the global geometric property of the space. In particular, we show that in low energy limit the critical spacetime dimension to ensure the"nonnegative"complexity is the 3+1 dimension.


I. INTRODUCTION
The recent developments of the quantum information theory and holographic duality show that some concepts from quantum information are useful in understanding foundations of gravity and quantum field theory (QFT). One of such concepts is the entanglement entropy, from which the spacetime geometry may emerge (e.g., see Refs. [1][2][3][4][5][6]). The "complexity" is another important concept since it may have a relation with physics inside black hole horizon [7][8][9][10][11][12].
Note that the deep understandings of entanglement entropy in the context of holography was possible because it is well-defined in both gravity and QFT. While there have been much progress for the complexity in gravity side, 1 the precise definition of the complexity in QFT is still not complete, which is an essential question.
The complexity is originally defined in quantum computation theory. Suppose a unitary operatorÛ is simulated by a quantum circuit, which can be constructed by combining universal elementary gates by many ways. The complexity ofÛ is defined by the minimal number of required gates to buildÛ . This definition by "counting" is based on the finite and discrete systems. However, to define the complexity in QFT or quantum mechanics (QM) we have to deal with infinite and continuous systems: we cannot "count".
This idea of complexity geometry of unitary operators is very attractive but there are two important shortcomings yet. First, the geometry is not determined by some physical principle, but given by hand. It is acceptable for quantum circuit problems since we can design the circuit as we want. However, in general QFT/QM, there must be constraints given by nature not by our hands. Second, it is only valid for SU(n) operators with finite n. It is enough for quantum circuit problems, but to deal with the operators generated by general QFT/QM systems such as H = p 2 /2m+V (x, p) we need to develop the formalism for the Hamiltonian with infinite dimensional Hilbert space.
In this paper, we propose how to remedy these shortcomings by generalizing the complexity geometry for finite qubit systems to general QFT/QM. In particular, we take into account the fact that the generating functional in QFT/QM plays a crucial role contrary to quantum circuits. We also, for the first time, note the importance of the lower boundedness of the Hamiltonian in QFT/QM. As a result, we will uncover novel interesting results which cannot be simply inferred from finite qubit systems.

II. COMPLEXITY FOR U(n) OPERATORS
Let us first review basic idea of right-invariant complexity geometry for SU(n) groups based on [26][27][28]. Consider the space of operators in SU(n) group with finite n. Suppose that a curve (c(s) ∈SU(n)) is generated by a generator H(s) as follows.

arXiv:1810.09405v1 [hep-th] 22 Oct 2018
We assume that the line element of this curve is given by a certain function of a generator only: dl =F (H(s))ds := g(H(s), H(s))ds , whereg(·, ·) is a metric defined in Lie algebra su(n) and we callF (H) the norm of H. In bases {e I }, H = e I Y I and the metric components can be expressed as To obtain the metric in the group manifold with the coordinate X I , the metric needs to be transformed by a coordinate transformation 2 where the transformation matrix is defined as Y I (s)ds = M I K (X)dX K . The complexity of an operatorŴ (s) := ← − P e s 0 iH(s)ds , denoted by C(Ŵ (s)), is defined by the minimal length of all curves which connectŴ (s) to identity: where H(s) satisfyŴ (s) = ← − P e s 0 iH(s)ds . Note that the geometry and complexity is rightinvariant, because H itself is invariant under the righttranslation c → cx for ∀x ∈ SU(n).
However, there is another symmetry which must be required only for the complexity of QFT/QM but not for quantum circuits. As a generator H in (1) let us consider a Hamiltonian of QFT/QM. Recall that H and U HÛ † give the same generating functional so physically equivalent. Therefore, if the complexity in QFT/QM is an observable or a physical quantity yielding observables it is natural that the complexity of the unitary operators generated by H andÛ HÛ † are also same. In other words, there is a symmetry, which we will call a unitary invarianceF For more details on this point, please see appendix A. Also, in appendix B, we show in a concrete example that Eq. (6) is a necessary condition so that the complexity should respect the fundamental symmetries in QFT/QM There are also other arguments to support Eq. (6), e.g., see Refs. [42,43]. The right-invariance and unitary invariance (6) together imply the complexity in QFT/QM should be also left-invariant, so bi-invariant. 3 This bi-invariance implies two important results. First, the theory of Lie algebra proves that the Riemannian metricg(·, ·) is uniquely determined by the Killing form up to a constant factor λ > 0 g(H, H) =F (H) 2 = λ 2 Tr(HH † ) = λ 2 Tr(H 2 ) .
Second, the geodesic in a bi-invariant metric is given by a constant generator, sayH, [52] Let us now consider an operator generated by a physical Hamiltonian, denoted by H. In general H has a nonzero trace and its Hilbert space may be infinite dimensional so Eq. (7) needs to be generalized accordingly.
First, to deal with the Hamiltonian of nonzero trace, we define a "mean value"H by such that (H −HÎ) ∈ su(n). Because U (1) is just a phase transformation, (H −HÎ) and H generate equivalent transformations and they should give the same complexity. Thus, we may use Eq. (7) for the norm of H However, if the Hilbert space of H is infinitely dimensional trace in Eq.(9) and (10) is divergent 5 so we need to renormalize it. We will show how to do it for two important cases: a quadratic Hamiltonian in one dimensional space and a non-relativistic particle in compact Riemannian manifolds.

III. ADS3 SPACETIME AS A COMPLEXITY GEOMETRY
Let us consider a general quadratic Hamiltonian in onedimension space, Here, we want to emphasize that the {Y I } should satisfy the constraints so that the Hamiltonian is bounded below. It can be seen from Eq. (14). This is another difference between QFT/QM and quantum circuits. The Hermit operators in quantum circuits have finite number of eigenvalues and are always bounded below. However, the Hermit operators in QFT/QM may not be bounded below in general, so we should be careful in defining parameter ranges.
To compute the norm of H we make a canonical transformation (x, p) → (x , p ) as follows, where c 1 , c 1 and Y ± , Y 0 satisfy One can read the eigenvalues of H −HÎ The norm of H will be divergent and we need to consider a proper renormalization of Eq. (10).
In general, we can define the ζ-function for a positive definite Hermitian operator O with eigenvalues {E n } as analytic continuation of the following sum In particular, for the operator O = H−HÎ with H in (13), we have where ζ(s, q) is the Hurwitz-ζ function, which is defined as the analytic continuation of the following sum [53] We may define the renormalized traces as and the renormalized norm square as First, by Eq. (9), Eq. (19) determinesH: which gives two solutionsH = ±ω/(2 √ 3). Thus, the Eq. (20) becomes BecauseF re (H) 2 should be nonnegative when coefficients Y I satisfy Eq. (12), we have to chooseH = −ω/(2 √ 3). Defining the parameter Indeed, our result (23) is consistent with the group theoretic consideration. The Hamiltonian (11) is written as H = Y I e I with where {ie I } consists of the su(1, 1) Lie algebra 6 7 [ As the su(1, 1) is simple, its bi-invariant metricg(·, ·) is proportional to the Killing form. In other words,g can be computed simply by the structure constants of the Lie algebra. As a result which serves as a nice consistent check for our method.
In appendix B, we provide an alternative independent method to obtain metric (26) without assuming the symmetry (6). Let us consider the case, Y + = Y 0 = 0 and Y − > 0, which corresponds to the free particle case. From (23) we find thatF which is the first explicit realization of the fact [11] (the Sec. VIII B 4) that the free Hamiltonian cannot mix information across its degrees of freedom so the complexity of the operator generated by free Hamiltonian is zero. By parameterizing the SU(1,1) group aŝ we obtain the metric in the group manifold (see Eq. (4) and appendix C for details) As the signature of this metric is (−, −, +), it is interesting to define a "spacetime interval" ds 2 = −dl 2 < 0 which yields the Riemann tensor R This means that the AdS 3 spacetime with the AdS radius AdS = λ 0 emerges as our complexity geometry! We find some interesting correspondences between the complexity geometry of SU(1,1) group and AdS 3 spacetime. For examples, the physical Hamiltonians with (12) correspond to the time-like tangent vectors in AdS 3 spacetime. In Tab. I, we make more comparison between the complexity geometry of SU(1,1) generated by quadratic Hamiltonian and spacetime geometry of AdS 3 .

IV. COMPLEXITY AND QUANTUM PHASE TRANSITIONS
We will show an interesting relation between the complexity and quantum phase transitions (QPTs). For example, let us consider the Lipkin-Meshkov-Glick (LMG) model [54]. The LMG model was introduced in nuclear physics. It describes a cluster of mutually interacting spins in a transverse magnetic field and has found applications in many other areas of physics like quantum spin systems [55], ion traps [56], Bose-Einstein condensates in double wells [57], and in circuit quantum electrodynamics experiments [58]. The Hamiltonian of LMG model reads [59] where N is the number of spins and h stands for an external magnetic field along the z-direction. α and γ are two real numbers. Without loss generality, we can set α > 0 and |γ/α| ≤ 1. In the limit N → ∞, it has been shown that there is a QPT at h c = α. For h > h c the system is in paramagnetic phase with zero magnetization along the xy-plane while for h < h c it is in the symmetry-broken phase with nonzero magnetization in xy-plane [55].
In the large N limit, if h ≥ h c , the effective Hamiltonian reads [60] which may be written as By identifying Y I in (11) and using the complexity formula in (23) we obtain At the QPT point h = h c = α, the change of the complexity ∂C/∂h is not only discontinuous but also divergent. This example implies that the change of the complexity may be used as a detector of QPT.

V. COMPLEXITY IN A COMPACT RIEMANNIAN MANIFOLD
Let us consider a non-relativistic particle with mass m and a bounded below potential V (x) in a n-dimensional compact Riemannian manifold M (∂M = 0) with arbitrary positive definite metric G µν . The Hamiltonian H is given by where ∇ is the covariant derivative associated to metric G µν .
As the eigenvalues of such Hamiltonian cannot be computed analytically in general, we cannot use ζ-function method. In order to regularize the trace for this Hamiltonian, we define the regularized trace of any function of H, say f (H) as where K H (τ ) := Tr(e −τ H ) and τ → 0 + .
The operator e −τ H is usually called the "heat kernel" [61,62]. Here, we take τ > 0 so that Eq. (39) is finite since the Hamiltonian is bounded from below. First, we computeH by using Eq. (9) with the regularized trace (39) Thus, we generalize Eq. (10) as For finite dimensional cases, Eq. (42) agrees with Eq. (10) after we take limit τ → 0 + . Without loss of generality, we may set m = 1/2 in following discussion. For a general G µν , it is difficult to find even the ground state and the first eigenvalue of H. However, in the case of τ → 0 + , the trace of the heat kernel can be computed in terms of a serie of τ [62], where a 0 = M √ Gdx n , a 1 = M (R/6 + V ) √ Gdx n and Here G is the determinant of G µν , R µνσρ is the Riemannian tensor, R µν is the Ricci tensor and R is the scalar curvature. Plugging Eq. (43) into Eq. (42) and setting τ → 0 + , we obtain The divergent term does not contribute to the metric defined by Eq. (3), so it can be removed without changing the complexity geometry. Thus we obtain the renormalized norm squarẽ For a "free particle", i.e., a flat metric G µν = δ µν with a constant potential V (x) = V 0 , we obtainF re (H) 2 = 0, which is consistent with (27). Eq. (47) also gives us a new insight for the relation between the complexity and the geometry of the space. According to the definitions of coefficients a 0 , a 1 and a 2 , it turns out that the complexity (47) is always well-defined (non-negative) only if the spacetime dimension is 3+1 or less, which is compatible to the spacetime of our world in low energy limit. (see appendix D for a proof.)

VI. CONCLUSIONS
For the complexity to be a useful tool for gravity and QFT/QM, we first have to fill up the conceptual gap between the quantum circuits and QFT/QM in defining complexity. By noting that the generating functional plays a central role in QFT/QM contrary to quantum circuits we propose an additional symmetry (6) for the complexity in QFT/QM. It leads a simple and unique formula for the complexity of SU(n) operators. Even though the formula is unique, its result is still rich. In particular, when we study complexity of the operators generated by Hamiltonians in the infinite dimensional Hilbert space, it shows novel results which can not be obtained from finite qubit systems. Interestingly enough, the complexity geometry corresponding to a general quadratic Hamiltonian in one-dimension is equivalent to AdS 3 space. Here, we noted the lower boundedness of the Hamiltonian gives an constraint for the non-negativity of the complexity, which has not been appreciated before.
Our formula proves that the complexity of the operator generated by free Hamiltonians vanishes, which was intuitively plausible. We also showed that the complexity can be used as an indicator of a QPT. It will be interesting to analyze the critical behaviors of the complexity in detail for more QPT systems. We uncovered the connection between the complexity and the background geometry. In particular, the fact that the critical dimension to insure a nonnegative complexity in low energy limit is just 3+1 dimension is worthy of more investigations.
This paper focused on a few quantum mechanical systems in order to simplify technical details. However, we expect that our proposal may result in more novel properties in QFT, especially at strong interactions or in the curved spacetime. The symmetry of the complexity of operators is implied by the symmetry ofF as shown in Eq. (5). If complexity is observable or a physical quantity yielding any observable, so is normF , and vice verse. Thus, we discuss the symmetry of the normF .
Let us consider two systems of which dynamics are given by time independent Hamiltonians H and U HU † respectively. Recall that the generating functional in QFT is the same for both H and U HU † so two systems are indistinguishable as far as the observables derived from the generating function. Therefore, if the complexity is an observable, it is natural to haveF (H) = F (Û HÛ † ). Note that a generating functional is not essential for the quantum circuit system soF (H) = F (Û HÛ † ) may not be necessary for this perspective.
However, it is possible that the complexity itself is not an observable but a physical quantity yielding observables. For example, a wavefunction of the shrodinger equation and the gauge potential in electromagnetism are such quantities. We will show also in this case the con-ditionF (H) =F (Û HÛ † ) is required.
Let us first consider the case that the complexity itself leads to an observable. This means that the normF can lead to an observable, i.e., an observable O(F ) as a function ofF . Indeed, this one variable case is trivial. In order for O to be an observableF must be an observable sinceF is a scalar. Notwithstanding, we describe it for completeness. We assume that O(x) is a non-constant and smooth function, which means Because O is an observable, for arbitrary H andÛ (s), where s is a continuous parameter forÛ (s). In more general cases, one complexity itself is not an observable but we can obtain an observable by several complexities. For example, one may think that the complexity is not a direct observable but the difference between two complexities is observable, which implies that neitherF (H 1 ) norF (H 2 ) is observable but In general an observable O may have more than two arguments.
If O(x 1 , x 2 , · · · , x n ) is a non-constant smooth function, we can find that ∀x 1 , x 2 , · · · , x n , ∃k ∈ N + and ∃l such that ∂ k O/∂x k l = 0. Without loss generality, we take l = 1 and obtain ∃k ∈ N + , ∂ k O/∂x k 1 = 0, ∀x 1 , x 2 , · · · , x n . (A5) Next, as O(F (H 1 ),F (H 2 ), · · · ,F (H n )) is an observable, we require: for arbitrary H j andÛ . i.e. H j andÛ H jÛ † gives the same observables. The basic idea of the proof is to reduce Eq. (A6) to the trivial one variable case. for all φ( x). For the Hamiltonian (B3), we can take a canonical transformation ( x, p) → (γ x, γ −1 p), which leads to Because this transformation is canonical, H 0 (γ) and H 0 describe the same physical system sõ for all γ = 0. Let us first consider what we can obtain from the symmetry (B4). For φ( x) = am x 2 with an arbitrary real number a we obtain Comparing with the bases defined in Eq. (24), we find that this Hamiltonian contains the triple copies of su(1, 1) Lie algebra. The symmetry (B4) implies for arbitrary a, k and m. In general,F (H) 2 =g IJ Y I Y J , whereg IJ is a general metric for su(1, 1) Lie algebra. In our case, F (H(a, k, m)) 2 = 3 g −− 1 m 2 +g 00 16a 2 +g ++ (4a 2 m + k) 2 The overall coefficient 3 comes from the fact that H(a, k, m) contains triple copies of su(1, 1) Lie algebra. If k = 0 Eq. (B8) implies for all a. This means thatg ++ =g +0 =g −0 = 0 and Thus, Eq. (B9) boils down tõ We see that the gauge symmetry gives us very strong restriction on the metric for quadratic Hamiltonians.
for all γ = 0. This impliesg −− = 0. Thus, respecting the symmetries (B4) and (B6) we find that the nonzero components of metricg IJ are only {g +− ,g 00 } and they satisfy the Eq. (B11). As a result, for a quadratic Hamiltonian H = Y I e I with {e I } defined in Eq. (24), we havẽ which is the same as Eq. (23). It shows that the complexity metric of quadratic Hamiltonians must be bi-invariant! By this alternative approach we may conclude that the bi-invariance (and so the symmetry (6)) is a necessary condition for the complexity to respect the fundamental symmetries of QFT/QM.

Appendix C: Metric for SU(1,1) group
We will explain how to obtain Eq. (C8). Let us start with Eq. (1), where U (X I ) is an element of the group G parameterized by the coordinate X I and the Lie algebra g is spanned by the bases {ie I }.
the term in the parenthesis in Eq. (D1) is nonnegative so we focus on the last term, 2ca 0 .
For a flat space, c = 0 so Eq. (D1) is always nonnegative in arbitrary dimension. It can be zero only for free particles, i.e., V (x) is constant. It is consistent with Eq. (27).
For a curved space, as a 0 is positive, we only need to consider the sign of c. For n = 1, R µνσρ = 0 so c = 0. For n = 2, R µνσρ has only one independent term. Under a local orthonormalized frame {e i } the metric components read G ij = δ ij and the nonzero curvature component is K := R 1212 . Thus, R µνσρ R µνσρ − R µν R µν = 2K 2 ≥ 0 and c ≥ 0. For n ≥ 3, we rewrite c as where the Gauss-Bonnet term is introduced: (D4) For n = 3, c GB = 0. In the local orthonormalized frame {e i }, we can diagonalize the Ricci tensor and obtain three eigenvalues {k 1 , k 2 , k 3 }. Thus, 3R µν R µν − R 2 = 3(k 2 1 + k 2 2 + k 2 3 ) − (k 1 + k 2 + k 3 ) 2 ≥ 0 and c ≥ 0. However, for n ≥ 4, 3R µν R µν − R 2 and c GB can be negative for some geometries, so c and Eq. (D1) can be negative.
The Hamiltonian given by Eq. (38) can only be used in low energy limit of quantum field theory. If the spacetime dimension in low energy limit is 3+1 or less, then Eq. (47) is always nonnegative.