Principles and symmetries of complexity in quantum field theory
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Abstract
Based on general and minimal properties of the discrete circuit complexity, we define the complexity in continuous systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU(n) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is biinvariant contrary to the rightinvariance of discrete qubit systems. We clarify why the biinvariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in klocal rightinvariant metric can also appear in our framework. Based on the biinvariance of our formalism, we propose a new interpretation for the Schrödinger’s equation in isolated systems – the quantum state evolves by the process of minimizing “computational cost”.
1 Introduction
Based on the intuition that the classical spacetime geometry encodes information theoretic properties of the dual quantum field theory (QFT) in the context of gauge/gravity duality, many quantum information concepts have been applied to investigations of gravity theories. A notable example is the holographic entanglement entropy (EE) of a subregion in a QFT [1]. Even though EE has played a crucial role in understanding the dual gravity, it turned out that EE is not enough [2], in particular, when it comes to the interior of the black hole. In the eternal AdS black hole, an Einstein–Rosen bridge (ERB) connecting two boundaries continues to grow for longer time scale even after thermalization. Because EE quickly saturates at the equilibrium, it cannot explain the growth of the ERB and another quantum information concept, complexity, was introduced as a dual to the growth of ERB [3, 4]. To ‘geometrize’ the complexity of quantum states in the dual gravity theory, two conjectures were proposed: complexityvolume (CV) conjecture [4] and complexityaction (CA) conjecture [5], which are called holographic complexity.^{1} See also Refs. [15, 16, 17].
However, note that the complexity in information theory is welldefined in discrete systems such as quantum circuits [18]. For example, the socalled circuit complexity is the minimal number of simple elementary gates required to approximate a target operator in quantum circuit. On the contrary, holographic complexity is supposed to be dual to complexity in a QFT, a continuous system. Thus, there may be a mismatch in duality if we try to compare the holographic complexity with the results purely based on the intuition from circuit complexity and it is important to develop the theory of complexity in QFT. Compared with much progress in holographic complexity, the precise meaning of the complexity in QFT is still not complete. In order to define complexity in QFT systematically, we start with the complexity of operator. The complexity between states will be obtained based on the complexity of operator. For the complexity of states we make a brief comment in Sect. 8 and refer to [19] for more detail. Our strategy to define the complexity of operator is: (i) extract minimal and essential axioms for the complexity of operator from the circuit complexity and (ii) define the complexity in continuous QFT systems based on that minimal axioms and smoothness (from continuity) (iii) consider general symmetries of QFT to give constrain on the structure of complexity. It will turn out that these steps enable us to determine the complexity of the SU(n) operators uniquely.
We want to emphasize that not all properties of circuit complexity survive in the complexity in QFT. The difference between discreteness and continuity makes some essential differences in properties of the complexity. For example, a few basic concepts in “circuit complexity” (computational complexity), such as “gates”, are not well defined in general quantum QFT so they should be modified or abandoned. Thus, we will keep only the most essential properties of the circuits complexity (which will be abstracted into the axioms G1–G3 in the following section). As another essential ingredient from QFT side, we will take advantages of basic symmetries of QFT, which may not be necessary in the case of quantum circuits or computer science. Because of the effect of this new inputs from QFT, some properties of the complexity in QFT we obtained may be incompatible with quantum circuits or qubit systems but they are more appropriate for QFT.
Our work is also inspired by a geometric approach by Nielsen et al. [20, 21, 22], where the discrete circuit complexity for a target operator is identified with the minimal geodesic distance connecting the target operator and the identity in a certain Finsler geometry [23, 24, 25, 26], which is just Riemannian geometry without the quadratic restriction. Recently, inspired by this geometric method, Refs. [13, 27, 28, 29, 30] also investigated the complexity in QFT. However, in these studies, because the Finsler metric can be chosen arbitrarily, there is a shortcoming that the complexity depends on the choice of the metric. In this paper, we show that, for SU(n) operators, the Finsler metric and complexity are uniquely determined based on four general axioms (denoted by G1–G4) and the basic symmetries of quantum QFT.
Form the section perspective, this paper is organized as follows. In Sect. 2, we introduce minimal and basic concepts of the complexity and propose three axioms G1–G3 for the complexity of operators, which are inspired by the circuit complexity. In Sect. 3, we show how the Finsler metric arises from G1–G2 and the smoothness of the complexity (G4). In Sect. 4, by using fundamental symmetry properties of QFT, we investigate constraints on the Finsler metric and the complexity. In particular we show that the Finsler metric is biinvariant by several different approaches and is determined uniquely if we take the axiom G3 into account. We also compare our results with previous researches regarding biinvariance. In Sect. 5 we derive the explicit form of the Finsler metric of the SU(n) group. Thanks to the biinvariance, the geodesic in the Finsler space of SU(n) group (so the complexity) is easily computed. In Sect. 6, as one application of the geodesic in the biinvariance Finsler metric, we propose a “minimal cost principle” as a new interpretation of the Schrödinger’s equation. In Sect. 7 we make a comparison between our complexity and the complexity for Kqubit systems. In Sect. 8 we conclude.
2 Axioms for the complexity of operators
2.1 Why unitary operators?
In order to make a good definition of the “complexity of operator” we first need to clarify what kind of “operator” we intend to deal with in this paper.
If we restrict physical processes to quantum mechanical processes, Eq. (2.1) implies that realizable operators are all unitary rather than Hermitian. In other words, our target is a property of the physical process rather than a direct observable. As quantum circuits are quantum mechanical processes and Solovay–Kitaev theorem [31] says that all the unitary operators can be approximated by some quantum circuits with any nonzero tolerance, we can conclude that the realizable operators set is the set of unitary operators. As unitary operators are invertible, the realizable operators set \(\mathcal {O}\) forms a (finite dimensional or infinite dimensional) unitary group.^{2}
2.2 Definitions and axioms
 G1

[Nonnegativity]
\(\forall \hat{x}\in \mathcal {O}\), \(\mathcal {C}(\hat{x})\ge 0\) and the equality holds iff \(\hat{x}\) is the identity.
 G2

[Series decomposition rule (triangle inequality)]
\(\forall \hat{x},\hat{y}\in \mathcal {O}\), \(\mathcal {C}(\hat{x}\hat{y}) \le \mathcal {C}(\hat{x})+\mathcal {C}(\hat{y})\).
 G3

[Parallel decomposition rule]
\(\forall (\hat{x}_1 ,\hat{x}_2) \in \mathcal {N}= \mathcal {O}_1 \times \mathcal {O}_2 \subseteq \mathcal {O}\), \(\mathcal {C}\big ((\hat{x}_1,\hat{x}_2)\big )=\mathcal {C}\big ((\hat{x}_1,\hat{\mathbb {I}}_2)\big )+\mathcal {C}\big ((\hat{\mathbb {I}}_1,\hat{x}_2)\big )\).
The axiom G1 is obvious by definition. We call the axiom G2 “series decomposition rule” because the decomposition of the operator \(\hat{O}=\hat{x}\hat{y}\) to \(\hat{x}\) and \(\hat{y}\) is similar to the decomposition of a big circuit into a series of small circuits. Reversely, the ‘product’ of two operators corresponds to a serial connection of two circuits. The axiom G2 answers a basic question: what is the relationship between the complexities of two operators and the complexity of their products? Because the complexity is a kind of “minimal”, we require the inequality in G2.^{3} This \(\mathbf G2 \) will lead to the familiar “triangle inequality” in the concept of distance (see F3 in the Sect. 3) so it is also called “triangle inequality”.
Mathematically, the construction of a bigger operator \(\hat{O}\) by \(\hat{x}_1\) and \(\hat{x}_2\) under two requirements (a) and (b) corresponds to the Cartesian product denoted by \(\hat{O}=(\hat{x}_1,\hat{x}_2)\). Note that the Cartesian product of two monoids does not correspond to the tensor product in a linear representation (i.e., a matrix representation). Instead, it corresponds to the direct sum. For example, if matrixes \(M_1\) and \(M_2\) are two representations of operators \(\hat{x}_1\) and \(\hat{x}_2\), then the representation of their Cartesian product \(\hat{O}\) is \(M_1\oplus M_2\), which is neither \(M_1\otimes M_2\) nor \(M_1M_2\).
In the language of computer science, this “totally independent” just means that one task contains two independent parallel tasks. Thus, the axiom G3 tries to answer the following question: if a task contains two parallel subtasks, what should be the relationship between the total complexity and the complexities of such subtasks? In term of mathematical language, it amounts to asking: what should be the relationship between \(\mathcal {C}\big ((\hat{x}_1,\hat{x}_2)\big )\), \(\mathcal {C}\big ((\hat{x}_1,\hat{\mathbb {I}}_2)\big )\) and \(\mathcal {C}\big ((\hat{\mathbb {I}}_1,\hat{x}_2)\big )\)?
The axioms G1–G3 are satisfied by both circuit complexity and computational complexity. We have expressed the abstract concepts extracted from circuit complexity and computational complexity in terms of mathematical language and will take them as three basic requirements to define complexity also in other systems. The axiom G1 and G2 can be satisfied by Nielson’s original works Refs. [20, 21, 22] and recent other approaches to complexity such as Refs. [13, 27, 28, 29, 30]. However, these works did not take into account the question related to G3 and broke the requirement in axiom G3 in general. From the viewpoint of quantum circuits (or computer science), series circuits (or tasks ) and parallel circuits (or tasks) are two fundamental manners to decompose a bigger circuits (or tasks) into smaller ones. Thus, the axioms G3 should be as important as G2. In this paper, we propose the concept of G3 for the first time and show that it plays a crucial role in determining the form of the complexity of SU(n) operators. We may be able to modify G3 in somewhat unnatural way, which will lead us to another form of the Finsler metric similar to (7.7). This point will be clarified in more detail in Ref. [33].
3 Emergence of the Finsler structure from the axioms for the complexity
In this section, we show that the Finsler metric arises from the minimal and general axioms for the complexity G1–G3 and the smoothness of the complexity. From here, the group element may represent either an abstract object or a faithful representation, which will be understood by context.
In Sect. 2 we have shown that the realizable operators are unitary operators, so the question now becomes how to define the complexity for unitary operators. As the unitary operators \(\hat{U}\) and \(e^{i\theta } \hat{U}\) (with \(\theta \in (0,2\pi )\)) produce equivalent quantum states, the complexity of \(\hat{U}\) and \(e^{i\theta } \hat{U}\) should be the same. Thus it is enough to study the complexity for special unitary groups, SU(n) groups. Ultimately, our aim is to investigate the complexity for operators in quantum field theory, of which Hilbert space is infinite dimensional, so we have to deal with the infinite dimensional special unitary groups. However, they involve infinite dimensional manifolds and have not been wellstudied even in mathematics so far. As an intermediate step, in this paper, we will first present our whole theory for finite dimensional cases and assume that the results can be generalized into infinite dimensional cases by some suitable limiting procedures. The subtle aspects between finite and infinite dimensional Lie groups are now under investigation [19].
The availability of two different generators can be understood also by a quantum circuit approximation to an operator, say \(\hat{O}\). As shown in Fig. 4, if a quantum circuit \(\phi _0\) is given, the operator \(\hat{O}\) can be constructed in two ways: (i) by adding a new quantum circuit \(\phi _1\) after the output of \(\phi _0\) (corresponding to Eq. (3.3)) or (ii) by adding a new quantum circuit \(\phi _2\) before the input of \(\phi _0\) (corresponding to Eq. (3.4)). The previous works such as Refs. [20, 21, 22, 27] assumed that the new operators/circuits could appear only after the output side of original operators/circuits, which corresponds to Eq. (3.3). This is one mathematically allowed choice but there is no a priori or a physical reason for that particular choice. Eq. (3.4) should be equally acceptable.
 G4
 [Smoothness] The complexity of any infinitesimal operator in SU(n), \(\delta \hat{O}^{(\alpha )} = \exp (H_\alpha \delta s)\), is a smooth function of only \(H_\alpha \ne 0\) and \(\delta s \ge 0\), i.e.,where \(\tilde{F}(H_\alpha ) := \partial _{\delta s} \mathcal {C} (\delta \hat{O}^{(\alpha )})_{\delta s =0} \) and \(\mathcal {C}(\hat{\mathbb {I}}) = 0\) by G1.$$\begin{aligned} \mathcal {C} (\delta \hat{O}^{(\alpha )}) = \mathcal {C}(\hat{\mathbb {I}}) + \tilde{F}(H_\alpha ) \delta s + \mathcal {O} (\delta s^2), \end{aligned}$$(3.6)
 F1

(Nonnegativity) \(\tilde{F}(H_\alpha )\ge 0\) and \(\tilde{F}(H_\alpha )=0\) iff \(H_\alpha =0\)
 F2

(Positive homogeneity) \(\forall \lambda \in \mathbb {R}^+\), \(\tilde{F}(\lambda H_\alpha )=\lambda \tilde{F}(H_\alpha )\)
 F3

(Triangle inequality) \(\tilde{F}(H_{\alpha ,1})+\tilde{F}(H_{\alpha ,2})\ge \tilde{F}(H_1+H_2)\)
There is an invariant property in the Finsler metrics. \(F_r(c, \dot{c})\) is rightinvariant because \(H_r\) is invariant under the righttranslation \(c \rightarrow c \hat{x}\) for \(\forall \hat{x} \in \) SU(n). Similarly \(F_l(c, \dot{c})\) is leftinvariant because \(H_l\) is invariant under the lefttranslation \(c \rightarrow \hat{x} c\) for \(\forall \hat{x} \in \) SU(n). If there is no further restriction on \(F_\alpha \), there are at least two natural Finsler geometries, \(F_r\) or \(F_l\), which may give different cost or length.
4 Symmetries of the complexity inherited from QFT symmetries
In the previous section, we have shown that the complexity can be computed by the minimal length of curves in Finsler geometry. We want to emphasize again that in our work the Finsler structure is not assumed, but it has been derived based on G1, G2 and G4. This is a novel feature of our work compared to other works dealing with the Finsler geometry.
However, apart from the defining properties of the Finsler metric F1–F3, we don’t know anything on \(\tilde{F}(H_\alpha )\) so far. In this section, we will show there are constraints on \(\tilde{F}(H_\alpha )\) if we take into account some symmetries of QFT. This is another important novel feature of our work compared to others. From here, we do not rely on properties of discrete systems or circuit models, which may be incompatible with QFT so may mislead us. We will directly deal with QFT and its symmetry properties and see what kind of constraints we can impose on \(\tilde{F}(H_\alpha )\).
Note that such symmetry considerations are not necessary if we use “complexity” as a purely mathematical tool, for example, to study the “NPcompleteness” and to analyze how complex an algorithm or a quantum circuit is. However, when we use the complexity to study real physical processes and try to treat the complexity as a basic physical variable hiding in physical phenomena, symmetries relevant to physical phenomena will be a necessary requirement.
4.1 Independence of left/right generators from unitary invariance
In this subsection we consider the effect of the unitary invariance of the quantum field theory on the Finsler metric, cost, and complexity. Let us consider an arbitrary quantum field \(\Phi \) with a Hilbert space \(\mathcal {H}\) and a vacuum \(\Omega \rangle \), which are collectively denoted by \(\{\Phi ,\mathcal {H}, \Omega \rangle \}\). Its unitary partner is \(\tilde{\Phi }(\vec {x},t):=\hat{U}\Phi (\vec {x},t)\hat{U}^{\dagger }\), \(\tilde{\mathcal {H}}:=\{\hat{U}\psi \, \rangle \, \forall \psi \rangle \in \mathcal {H}\}\) and \(\tilde{\Omega }\rangle :=\hat{U}\Omega \rangle \), which are denoted by \(\{\tilde{\Phi },\tilde{\mathcal {H}},\tilde{\Omega }\rangle \}\).
4.2 Comparison between SU(n) groups and qubit systems
In order to clarify why the adjoint invariance of the complexity is natural, which may not be the case in discrete systems, we make a comparison between SU(n) groups and qubit systems in Fig. 6. For qubit systems, the operators set forms a countable monoid \(\mathcal {O}\) and can be obtained from a countable fundamental gates set g. The complexity of any operator in \(\mathcal {O}\) is given by the minimal gates number when we use the gates in g to form the target operator. For SU(n) groups, the operators set forms an SU(n) Lie group and the fundamental gates are replaced by infinitesimal operators, which form the Lie algebra \(\mathfrak {su}(n)\).
It is the difference between Eqs. (4.12) and (4.15) that leads the difference between qubit systems and SU(n) regarding the invariance under adjoint transformations. Because Eq. (4.16) is valid also for any infinitesimal operator, it implies Eq. (4.6). This is another derivation of Eq. (4.6). We have presented two arguments to support the idea that the complexity of SU(n) group should be invariant under adjoint transformations. In Appendix D, we will give the third and the fourth arguments to support this conclusion.
4.3 Reversibility of Finsler metric from the CPT symmetry
5 Complexity of SU(n) operators
5.1 Finsler metric of SU(n) operators
5.2 Geodesics and complexity of SU(n) operators
6 Minimal cost principle
 Minimal cost principle: For isolated systems, the timeevolution operator \(\hat{U}(t)\) will be along the curve to reach the target operator so that the cost during this process is locally minimal, i.e. the evolution curve will make the following integral to be locally minimal:where we used Eq. (5.1), where \( H \rightarrow \dot{\hat{U}}(t)[\hat{U}(t)]^\dagger \).$$\begin{aligned} L[\hat{U}(t)]=\int _0^{t_0}\text {Tr}\sqrt{\dot{\hat{U}}(t)[\dot{\hat{U}}(t)]^\dagger }\text {d}t,~~\hat{U}(0)=\hat{\mathbb {I}},~~~\hat{U}(t_0)=\hat{O},\nonumber \\ \end{aligned}$$(6.2)
7 Comparison with the complexity for Klocal qubit systems
For a better understanding of the novel aspects of our work compared to previous research it is useful to compare our complexity and the complexity for Kqubit systems [27]. In particular, our complexity is biinvariant but the complexity geometry in Ref. [27] is only rightinvariant. At first glance, two theories may look different, but the difference in complexity turns out to be little and most of physical results in Ref. [27] can also appear in our theory.
 (1)
In our paper, the only basic assumptions are G1–G4. All conclusions such as Finsler geometry and the Finsler metric Eq. (5.1) are the results of these four assumptions and fundamental symmetries of QFTs. In Ref. [27], the Riemannian geometry and metric (7.6) in Kqubit system were proposed directly as the basic assumptions.
 (2)The complexity given by Eq. (7.7) satisfies our axioms G1 and G2 but breaks G3. It can be shown by considering a simple case, \(\mathcal {I}^{ab}=\delta ^{ab}\), which corresponds to biinvariant case without any penalty (isotropy). In this case, the complexity of the operator \(\hat{O} = {\text {exp}}(H)\) is given by F(H) because the geodesic is generated by a constant generator (due to biinvariance) i.e.By using Eqs. (7.8) and (7.7) we have$$\begin{aligned} C(\hat{O}) = F(H). \end{aligned}$$(7.8)and in more general cases,$$\begin{aligned}&\mathcal {C}(\hat{O}_1\oplus \hat{O}_2)=\sqrt{\mathcal {C}^2(\hat{O}_1\oplus \hat{\mathbb {I}}_2)+\mathcal {C}^2(\hat{\mathbb {I}}_1\oplus \hat{O}_2)}\nonumber \\&\quad =\sqrt{\mathcal {C}^2(\hat{O}_1)+\mathcal {C}^2(\hat{O}_2)}, \end{aligned}$$(7.9)where \(p_i\) is a weighting factor if \(\mathcal {I}^{ab}\ne \delta ^{ab}\). This means that the total complexity of parallel operations is not the sum of the complexity of the individual operations, so breaks G3. We want to stress again that G3 is a very natural requirement that has not been considered in previous research.$$\begin{aligned} \mathcal {C}\left( \bigoplus _i\hat{O}_i\right) =\sqrt{\sum _ip_i[\mathcal {C}(\hat{O}_i)]^2}\ne \sum _i\mathcal {C}(\hat{O}_i), \end{aligned}$$(7.10)
 (3)
For the same curve in SU(n) group, the tangent vector at a point is unique but the generator is not. It can be a left or right generator. We admit of two ways (left generator or right generator) but Ref. [27] considers only one way (right generator). As there is no reason to assume that physics favors “left world” or “right world”, a simple and natural possibility is that two generators yield the same complexity. It is the case that is realized in our framework unlike the complexities in [27] and Neilsen’s original works [20, 21, 22] where the left and right generator will give different complexities.
One may argue that Eq. (7.7) could be valid only for the right generator and, for the left generator, there might be another leftinvariant metric which has different penalty \(\mathcal {I}'^{ab}\) and could give out the same curve length. In our upcoming work Ref. [33], we will show this is possible only if the geometry is biinvariant. This gives us another argument that the complexity for SU(n) group should be biinvariant.
 (4)
When Ref. [27] discusses some particular physical situations such as “particle on complexity geometry”, “complexity equals to action” and “the complexity growth”, the authors restricted the generators in a “klocal” subspace \(\mathfrak {g}_k\) and assumed \(\mathcal {I}_{ab}_{\mathfrak {g}_k}=\delta _{ab}\) (see the Sect. IV.C and V in Ref. [27] for detailed explanations). As a result, the geodesics in the submanifold generated by \(\mathfrak {g}_k\) are also given by constant generators, which are the same as our biinvariant Finsler geometry. The lengths of such geodesics in Ref. [27] and in our paper are only different by multiplicative factors, which implies that all the results given by “klocal” subspace can also appear similarly in our biinvariant Finsler geometry.
Moreover, in order to obtain the complexity geometry as was proposed in Ref. [27], we can choose some twodimensional submanifold in SU(n) geometry. As described in our upcoming paper [33], by using the Gauss–Codazzi equation, we can show that the submanifold can have negative induced sectional curvature somewhere despite the SU(n) geometry is positively curved. So it satisfies the same properties as shown in Ref. [27], where the sectional curvature is made negative near the identity by choosing an appropriate penalty factor.
8 Discussion and outlook
In this paper we proposed four basic axioms for the complexity of operators: nonnegativity (G1), series decomposition rule (triangle inequality) (G2), parallel decomposition rule (G3) and smoothness (G4). Combining these four axioms and basic symmetries in QFT, we have obtained the complexity of the SU(n) operator without ambiguity: Eq. (5.11). In our derivation the biinvariance of the Finsler structure plays an important role and this biinvariance is a natural implication of the symmetry in QFTs rather than an artificial assumption. Our logical flows are shown in the Fig. 1.
We argue the importance of the biinvariance in four ways based on: (i) the unitaryinvariance of QFTs (Sect. 4.1); (ii) the nature of continuous operators rather than discrete ones (Sect. 4.2); (iii) inverseinvariance of the relative complexity (Appendix D.1); and (iv) the “ketworld”  “braworld” equivalence (Appendix D.2). The biinvariance here is different from the only rightinvariance for qubit systems [20, 21, 22, 27]. We clarify the differences and similarities of our proposal (biinvariance) from previous researches (only rightinvariance) in Sects. 4.2 and 7. It can be shown that most of results in only rightinvariant complexity geometry proposed by Ref. [27] can also appear in our framework. We want to emphasis that the complexity cannot be a welldefined physical observable in general finite dimensional systems if the complexity geometry is not biinvariant.
Thanks to the biinvariance of the Finsler metric the process of minimal cost (complexity) is generated by a constant generator. This observation leads us to make a novel interpretation for the Schrödinger’s equation: the quantum state evolves by the process of minimizing “computational cost,” which we call “ minimal cost principle.”
In a more general context, geometrizing the complexity in continuous operators sets amounts to giving positive homogeneous norms in some Lie algebras. Our paper deals with only SU(n) group so we gives the norm for Lie algebra \(\mathfrak {su}(n)\). For more general Lie algebra \(\mathfrak {g}\), though we cannot determine the norm uniquely, it is natural that such a norm is determined only by the properties of \(\mathfrak {g}\), for example the structure constants, without any other extra information. As in general relativity where the spacetime metric is determined by matter distribution through Einstein’s equations, can we find any physical equation to determine this norm?
Footnotes
 1.
 2.
Hermit operators, which correspond to observable quantities and are not unitary in general, cannot be approximated by quantum circuits if the tolerance is small enough.
 3.
 4.
In some cases, the parallel decomposition may be impossible. However, in local computations/gates, the parallel decomposition is permitted, for example see Ref. [32]. Our axiom G3 does not talk about the possibility of the parallel decomposition, but says what will happen to the complexity if the parallel decomposition is permitted.
 5.
This condition will be relaxed in our upcoming work [33], where we will allow different contributions to the total complexity when \(\delta \hat{O}\) is added to the leftside or rightside.
 6.
Strictly speaking, \(\tilde{F}\) and \(F_\alpha \) are the Minkowski norm and the Finsler metric respectively if we make a further requirement that, in F3, the equality holds only when \(H_1\) and \(H_2\) are linearly dependent. However, this subtle mismatch is not important physically.
 7.
The CPT symmetry is a theorem for local relativistic quantum field theories in Minkowski spacetime. Here, C means ‘charge conjugation’, P ‘parity transformation’ (‘space inversion’), and T ‘time reversal’. This theorem states that the local Lorentz quantum field theories are invariant under the combined transformations of C, P, and T.
 8.
It seems that the complexity is invariant under the unitary transformation is weaker than the requirement that the cost function is invariant under the unitary transformation. However, it can be shown that they are equivalent as shown in Appendix E.
 9.
The equality means the equality between two sets, i.e, \(\widetilde{\mathfrak {su}}(n)\) and \(\mathfrak {su}(n)\) contain the same elements.
 10.
If the Hamiltonian is not static, the complexity curve and the curve generated by Hamiltonian will be different. In this case, we may extend the system to include backreaction of the system on its background, rendering the total Hamiltonian static.
 11.
In some textbook, there is an overall factor F in the definition so that the Cartan tensor is scale invariant under the transformation \(v\rightarrow \lambda v\).
 12.
This condition is necessary to have a positive definite metric tensor.
 13.
This can be obtained, for example, by setting \(H=\text {diag}(\gamma ,\gamma ,0,0,\ldots ,0)\) in Eq. (C.9).
Notes
Acknowledgements
The work of K.Y. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF2017R1A2B4004810) and GIST Research Institute (GRI) Grant funded by the GIST in 2018 and 2019. C. Y. Zhang is supported by National Postdoctoral Program for Innovative Talents 938 BX201600005 and China Postdoctoral Science Foundation. C. Niu is supported by the Natural Science Foundation of China under Grant No. 11805083.
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