Circuit Complexity and 2D Bosonisation

We consider the circuit complexity of free bosons, or equivalently free fermions, in 1+1 dimensions. Motivated by the results of [1] and [2, 3] who found different behavior in the complexity of free bosons and fermions, in any dimension, we consider the 1+1 dimensional case where, thanks to the bosonisation equivalence, we can consider the same state from both the bosonic and the fermionic perspectives. In this way the discrepancy can be attributed to a different choice of the set of gates allowed in the circuit. We study the effect in two classes of states: i) bosonic-coherent / fermionic-gaussian states; ii) states that are both bosonic- and fermionic-gaussian. We consider the complexity relative to the ground state. In the first class, the different results can be reconciled admitting a mode-dependent cost function in one of the descriptions. The differences in the second class are more important, in terms of the cutoff-dependence and the overall behavior of the complexity.

3 Fubini-Study metric method for Bosonic Coherent states 9 4 Nielsen method for gaussian states 11 4.1 Application on gaussian states in free fermionic field theory 12 4.2 Translation to gaussian states in free bosonic field theory 14 5 Application of Nielsen method on bosonic coherent states 14 5.1 Complexity between bosonic ground states 15 5.2 Bosonic coherent states with one excited mode 15 5.3 Complexity for bosonic coherent states with shifts in more modes 18 6 A class of fermionic and bosonic Gaussian states 20 6.1 A bosonisation identity 20 6.2An example with one mode 21 6. 3 An example with two modes 24 7 Conclusions 24 A Orthogonality of the rotational matrix M (n) for α ∈ R 26 A.1 (A 2 + B 2 ) ik = δ ik 27 A.2 (AB − BA) ik = 0 27

Introduction
Recently, much attention has been devoted to the study of quantum complexity in connection with the holographic correspondence.Quantum complexity is a concept that has its origin in quantum computation theory as a means of characterizing the difficulty (in the sense of the amount of resources needed) of performing a task on a quantum computer; more precisely, if the task can be described as producing a certain quantum state |ψ T from a given initial state |ψ R using a circuit made of elementary unitary operations (gates), the quantum complexity can be defined as the minimum number of gates required for such a circuit.It has been suggested by Susskind and collaborators [4][5][6][7] that the notion of complexity may be an important component in our understanding of the properties of emergent spacetime, and possibly point to a solution of the information loss paradox, by giving additional insight, beyond what can be obtained from entanglement entropy, into the information-theoretic properties associated to spacetime and in particular to the region inside black hole horizons (see the recent lectures [8]).
It was further conjectured that, similarly to entanglement entropy which is described holographically by the area of a minimal surface [9], quantum complexity is also captured, in a theory with a holographic dual, by a simple gravitational observable.However what precisely is the observable is unclear.There are two different proposals: one takes the volume of a maximal spatial slice of the geometry [5], the other takes the action evaluated on the Wheeler-DeWitt patch [10,11].Each proposal has its merits and drawbacks, and a clear-cut way to choose one over the other has not been found yet.In particular, both prescription lead to similar behavior for the late-time growth of complexity during the formation of a black-hole (i.e., linear growth in time, albeit with different coefficients [12,13]), and have similar structure of UV divergence where d is the spacetime dimension of the dual field theory, δ is a short-distance cutoff, and a is a coefficient that depends on the prescription. 1ven in cases when the two prescriptions give different results (e.g., in the case of an AdS 3 space with a defect brane, as we found in our paper with S. Chapman [16]), we do not have at present a criterion for choosing one over the other, absent any independent calculation that can serve as benchmark.By contrast, in the case of the entanglement entropy, one can compute it in a 2d CFT, for an interval of length , and obtain the famous exact result [17] which depend only on the central charge c.Because the coefficient of the log does not change under a rescaling of the cutoff, it can be considered a universal quantity with a well-defined physical meaning.By analogy, one could try to attribute a similar universal meaning to the coefficient of the log in the complexity, but the situation is less clear.
The problem is that the definition in terms of gates, which is applicable to a finite quantum system, does not translate easily to a definition that is applicable to a continuum quantum field theory.In other words, we do not know how to associate the notion of complexity to a well-defined observable in QFT.This problem was considered in [1,18], who provided a partial answer by proposing a definition of complexity for free scalar fields.The proposal of [1] used the Nielsen's approach of geometrization of quantum computation [19], while [18] used a method based on the Fubini-Study metric 2 .In Nielsen's method, described in more detail in Section 4, the circuit is replaced by a continuous version that is a path in the space of unitary operators, of the form The functions Y I (s) are determined by minimization of a certain functional F [Y I (s)] that determines the cost associated to a given path.The complexity of a unitary operator V is then defined as the minimum of the cost functional over all paths (1.3) such that U (s = 1) = V .
What is important to notice here is that this definition of complexity depends on several choices: the choice of the allowed space of operators used to build the circuit 3 , and indeed of a specific basis O I , the choice of a cost functional, and a choice of parametrization of the path.
The space of unitary operators acting on the Hilbert space of a QFT is very large, but one of the main points of [1] was to reduce it to a tractable setup by considering operators that act within the subset of Gaussian states.This allows one to consider the complexity of any state that is the ground state of any Hamiltonian quadratic in the fields.Discretizing the free scalar field to a set of N harmonic oscillators, a Gaussian state is described by a symmetric 2N × 2N matrix, and the group of unitary operators that preserves the Gaussian states is Sp(2N ).In [1] the behaviour of the complexity between Gaussian states was investigated for various choices of cost functions, mostly focusing on the one corresponding to the Cartan-Killing metric on the symplectic group.
The approach used in [18] considers a path in the space of states; the complexity is identified with the minimal length of a path connecting two states, calculated in the standard Fubini-Study metric on the space of normalized states.For a parametrized path of quantum states |ψ(σ) , the line element is (1.4) This definition may seem more canonical, but if one allows the most general path in the space of states, then the geodesic distance between normalized states is always less or equal to π/2.
In order to have a sensible measure of complexity, one must somehow restrict the possible paths.The proposal of [18] is to use paths that can be obtained using unitary operators similar to (1.3), with the basis operators being a subset of the bilinears the creation and annihilation operators; thus in both approaches one does not leave the space of Gaussian states, and the results are comparable.Nielsen's approach was extended to the case of free fermions in [2,3].In comparison to the bosonic case, the main difference is that the relevant group of operators acting on Gaussian states is SO(2N ), which is a compact group, with the consequence that the complexity (again measured using the Cartan-Killing metric) cannot grow very large.Considering a field theory in d spacetime dimension with spatial volume V and a UV cutoff Λ, it turns out that the leading divergent term in the bosonic complexity is where ω 0 is an arbitrary reference scale and κ is a parameter related to the choice of the cost function, whereas for a free fermion field theory is (1.6) The discrepancy seems at odds with the holographic interpretation, as the holographic result is relatively blind to the fermionic or bosonic nature of the dual fields.Moreover, for the case of a two-dimensional theory, we might think that the bosonization map between bosons and fermions should imply the equality of the two results.These observations do not constitute a strong objection to the results of [1][2][3]18], because states with a holographic dual are not Gaussian states, the theory is strongly coupled whereas (1.5) and (1.6) are obtained for free fields, and complexity is not a well-defined field theory observable in the usual sense, as we stressed.Nevertheless, they provide the motivation for the present paper, where we look more closely at the complexity for bosons and fermions in 1+1 dimensions, comparing states that are related by bosonisation.From this point of view, the results (1.5) and (1.6) do not constitute a discrepancy, rather they illustrate the effect of a different choice of operator spaces.
In the context of quantum computation, the question of the choice of gates is of obvious importance, and there are some known results: when the gates act on arrays of qubits, it is possible to show that there is a universal finite set of gates, such that any unitary operator can be approximated by a circuit made of universal gates with an arbitrarily small error (see e.g.[29]), and the size of the circuit is O(ln c (1/ )), for some constant c.Moreover, if a circuit has complexity m with some choice of basic gates, a different choice will give complexity O(m ln c (m/ )) [30], so the dependence on the basis choice is at most logarithmic.
It would be nice to establish similar results in the QFT context, but so far almost nothing can be said about the gate dependence.In most of the works on the complexity, the choice of gates was restricted to gates quadratic in the fields, in order not to depart from the space of Gaussian states.No fundamental reason underlies this choice except that it allows to do explicit computations, and it is not obvious how to make an alternative choice: in general it is difficult to find an algebraically closed set of gates that is larger than the set of quadratic operators but not as large as the full space [24,31].But it turns out that such a set is provided by the magic of bosonisation.
The reason for this is that the bosonisation map is very non-linear, so Gaussian states in one description generically do not correspond to Gaussian states in the other, and similarly gates that are oscillator bilinears in one description will map to more complicated gates in the other.Thus we can have a completely solvable example where it is possible to explore the effect of different choice of gate sets on the complexity.
We have started to undertaken this exploration in the present paper.We are not yet in a position to make general statements about the effect of a change of gates, since we are limited to the states whose complexity can be computed using the available technology.We have identified two such classes of states.In the first class, the states are bosonic-coherent and fermionic-gaussian.For the states in this class, we can compute the complexity analytically, and we find that it has a similar form in the two descriptions, see the results in (3.19) and (5.20), however the fermionic complexity appears to have a cost that depends on the mode number of the bosonic oscillator that is excited.Thus, the difference of complexity can become arbitrarily large, even with a single mode.
In the second class, the states are both bosonic-gaussian and fermionic-gaussian 4 .These states can be understood as ground states of a system with a free but inhomogeneous hamiltonian.They are parametrized by an arbitrary function; we considered some examples with the function having only one or only two Fourier components.We cannot find analytic results in this case but have to resort to numerics.The numerical results are given in Figs. 5, 6 and 7, and show striking differences between the bosonic and the fermionic result.The bosonic complexity for these states is cutoff-independent, and smoothly dependent on the parameters corresponding to the Fourier modes, whereas for fermions it grows like ln(Λ), and it is much less regular (it appears to be quasi-periodic in the simplest case).
The plan of the paper is the following: in Section 2 we recall the basic properties of the bosonisation equivalence and the correspondence between fermionic and bosonic states.In Section 3 we compute the complexity of a class of bosonic coherent states in terms of bosonic gates by using the Fubini-Study metric method.In Section 4 we review the Nielsen method applied on free fermonic and bosonic field theory, and in Section 5 we use it to compute the complexity of bosonic coherent states using fermionic gates.In Section 6 we describe a class of states that are of gaussian type both in the fermionic and in the bosonic description, and we compare the results for the complexity computed in either description.In Section 7 we present our conclusions.Some additional details of the computations are presented in the Appendices.

Basic results
In this section, we will review the basics of the 2D bosonisation formalisms, for free bosons and fermions, which we will use in the rest of the paper.We follow the presentation in [32].In two dimensions, the bosonisation can be proved exactly for a system on a finite size interval [−L/2, L/2].In such a system, the unbounded momentum k satisfies where δ b depends on the periodicity condition of the fermionic fields, 0 for complete periodicity and 1 for anti-periodicity.If there are M chiral fermions with periodic conditions (δ b = 0) in the system, an index η could be used to denote different types of fermions, and the mode decomposition for each fermion type is given as where η = 1, 2, . . ., M , can be spin, handedness, flavor etc.The bosonic chiral fields are given by the mode decomposition where the zero mode is omitted.Notice that only n > 0 modes are included, because of the chirality.The a appearing in the last factor is a regularisation parameter that should be sent to zero when computing physical quantities. 5 By construction, there is an operator identity at the level of annihilation and creation operators between fermions and bosons, from where we see that the bosonic operators are always an infinite sum of fermionic operators of quadratic type containing both the creation and annihilation ones.We can take (2.4) as a definition of the bosonic modes; the proof of bosonisation amounts to showing that, with this definition, the bosonic commutation relations are satisfied if the fermionic ones are: The bosonisation formula can also be stated in terms of the local fields: with the colon denoting normal ordering.The inverse formula is more complicated: where Nη = l : c † l η c l η : is the fermionic number operator with respect to the ηth spiecies.From (2.4), it is easy to see that the bosonic operators commute with the fermionic number operator, i.e.
5 Notice that the bosonisation is exact only when all the modes are included.In practice we often need to introduce a UV cutoff on the mode number; we still expect that we can match quantities that are cutoffindependent.
so only fermionic operators that don't change the fermion number can strictly speaking be bosonised.The so-called Klein factor F η in (2.7) has the role of compensating the mismatch in fermion number (this factor is often omitted in many presentations of bosonisation).However we will not need its explicit expression.The commutation relation between bosonic and fermionic modes is (2.10) Intuitively we can think that b † l creates, and b l annihilates, a particle-hole pair of total momentum l.
In the following, we will consider the simplest case with only two chiral fermions and one chiral bosons, thus the species index will be omitted.

Fermionic Fock space
In the fermionic Fock space F, a unique vacuum |0 is defined in terms of the fermionic modes, In the bosonised picture, the Fock space can be reorganised as a direct sum of all the Hilbert space with fixed fermionic particle number, i.e., (2.12) Each H N with fixed fermion number can be regarded as the bosonic Hilbert space since the bosonic operators commute with the fermionic number operator as mentioned before.
The condition for a bosonic ground state is to be annihilated by all the bosonic annihilation operators b n |G B = 0, n > 0. (2.13) This condition uniquely defines a state in each module H N ; then the ground state |G B in the N -particle module is denoted as (2.14) Figure 1 gives a depiction of the Fock space as a bundle, with the base given by the ground states and the fiber by the bosonic Hilbert space.The bosonic operators act inside a single fiber, whereas fermionic operator can move between different fibers.
l q 9 3 2 l r n 4 n c u 0 r N 3 z O r f C v b 0 l D T h 6 P c 6 3 o L P e i s J W t L f e 3 N q u R z 2 D J S x j j e a 5 g S 3 8 w C 7 a 5 P 0 L l 7 j C t T f 0 n K j a w g 1 0 0 y P s K 9 3 j A o 9 f x r r 0 b 7 / Y 1 1 R s o N f P 4 s L y 7 F w i L n I s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 a J a g w e 3 9 s Z n K j a w g 1 0 0 y P s K 9 3 j A o 9 f x r r 0 b 7 / Y 1 1 R s o N f P 4 s L y 7 F w i L n I s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 a J a g w e 3 9 s Z n K j a w g 1 0 0 y P s K 9 3 j A o 9 f x r r 0 b 7 / Y 1 1 R s o N f P 4 s L y 7 F w i L n I s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 a J a g w e 3 9 s Z o p c 4 z t j l c p h T s V u 3 P / R l e G E j L i L B 5 T X R P m z r k 8 Z 9 9 5 c t e 7 P V v m 6 r + d 0 r J 2 z i t t g T 9 2 l 3 T B 4 d 3 r X A X 9 d i s M W u G n d q P T r a 5 6 A 6 + w g z d 0 n 7 v o Y A / 7 6 F H 2 N 1 z g J 3 5 5 x 9 5 3 7 4 d 3 d i 3 1 a p X n J W 4 N 7 / w v V p e d z Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o p c 4 z t j l c p h T s V u 3 P / R l e G E j L i L B 5 T X R P m z r k 8 Z 9 9 5 c t e 7 P V v m 6 r + d 0 r J 2 z i t t g T 9 2 l 3 T B 4 d 3 r X A X 9 d i s M W u G n d q P T r a 5 6 A 6 + w g z d 0 n 7 v o Y A / 7 6 F H 2 N 1 z g J 3 5 5 x 9 5 3 7 4 d 3 d i 3 1 a p X n J W 4 N 7 / w v V p e d z Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o p c 4 z t j l c p h T s V u 3 P / R l e G E j L i L B 5 T X R P m z r k 8 Z 9 9 5 c t e 7 P V v m 6 r + d 0 r J 2 z i t t g T 9 2 l 3 T B 4 d 3 r X A X 9 d i s M W u G n d q P T r a 5 6 A 6 + w g z d 0 n 7 v o Y A / 7 6 F H 2 N 1 z g J 3 5 5 x 9 5 3 7 4 d 3 d i 3 1 a p X n J W 4 N 7 / w v V p e d z Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g N b P B h 5 K q 7 H f 9 + 9 M e P 9 r 0 7 j q U m c = " l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g N b P B h 5 K q 7 H f 9 + 9 M e P 9 r 0 7 j q U m c = " l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g N b P B h 5 K q 7 H f 9 + 9 M e P 9 r 0 7 j q U m c = " r 3 j R j 2 J Z a x g j e a 5 g W 3 s o Y k W e Z / j H g 9 4 9 D r e t X f j 3 b 6 n e h W n W c K X 5 d 2 9 A c x g l X 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z 5 l P r 3 j R j 2 J Z a x g j e a 5 g W 3 s o Y k W e Z / j H g 9 4 9 D r e t X f j 3 b 6 n e h W n W c K X 5 d 2 9 A c x g l X 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z 5 l P r 3 j R j 2 J Z a x g j e a 5 g W 3 s o Y k W e Z / j H g 9 4 9 D r e t X f j 3 b 6 n e h W n W c K X 5 d 2 9 A c x g l X 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z 5 l P The base manifold G B is made of all the bosonic ground states which are denoted as the red crosses.The vertical blue lines are modules of the fermionic fock space where each represents a bosonic Hilbert space with fixed fermion number.The operators of bosonic type can only move vertically while the fermionic ones span the whole Fock space with a single fermionic operator moving horizontally.

Correspondence between states
Our main purpose in this note is to compare the circuit complexity of states in the bosonic and fermionic descriptions.We cannot consider the most general state, since the techniques developed so far only allow to deal with Gaussian states or coherent states.Our first task is to determine the correspondence between Gaussian/coherent states in both descriptions.
It is clear that, since the bosonisation relations (2.6)-(2.7)are non-linear, Gaussian states in one description will not correspond in general to Gaussian states in the other.It is shown in [2] that all Gaussian states can be obtained from the fermionic vacuum (2.11) by the action of a unitary operator of the form Equivalently, each such states is a vacuum under a set of modes related to the original ones by a Bogoliubov transformation It is easy to see then that states of the form Gaussian states; these include in particular the bosonic ground states of (2.14).
It is useful to notice the commutation relations from which it follows that where . . .denotes the vev in the state |G N B .The FS metric components can be computed: We obtain then that the FS metric is a flat Kähler metric as expected, since it is known that in quantum mechanics coherent states form a twodimensional Kähler manifold which can be parametrized by classical phase space variables [35].
For our purposes it is important to notice that the result does not depend on either the fermion number N , or the bosonic oscillator number n.We can easily understand the N -independence since all the bosonic Hilbert spaces H N are equivalent to each other.The independence on the mode number indicates that the FS metric corresponds to a cost function that assigns the same cost to all displacement operators U n .This is also natural since the FS metric is derived from the scalar product in the Hilbert space, which has no information about the energy levels apart from the fact that they are orthogonal to each other.It is trivial to compute the geodesic length in the flat metric: This agrees with the result of [34]; the result of [33] is C F S (α) = arccosh(1 + |α| 2 2 ), which agrees with (3.13) for small α, while for larger α it is smaller, which means that one can find shorter geodesics if the coherent states are embedded in the larger space mentioned at the beginning of this section.
The general case of coherent states contains displacement operators acting on several modes: with n i = n k for i = k.In order to compute the metric it is enough to consider the case of two modes: The off-diagonal metric components are ) It is easy to see that since the displacement operators of different modes commute, this applies for any number of excited modes.Therefore the metric is a direct product of the metrics for the individual modes: and the complexity of a general coherent state is 4 Nielsen method for gaussian states The Nielsen method can be understood as a way to geometrize the notion of complexity.In this section, we will review the Nielsen method and its application to fermionic and bosonic Gaussian states, along the lines developed in [1,2,23].In Nielsen's method, the circuit is replaced by a continuous path of unitary operators such that U (s = 1) gives the desired unitary operator (i.e., implements the circuit we want), and O I is a basis of operators that can be used to build the circuit (we can think of them as the generators of the infinitesimal gates e i O I ).One can think that the final state is being prepared by means of an evolution in a fictitious time s with a time-dependent Hamiltonian H = Y I (s)O I .The complexity is then determined by the choice of a suitable cost functional The minimization of this functional will determine the form of the functions Y I (s) and thus the optimal circuit.With an appropriate choice, the cost functional determines a distance on the space of unitary operators, and the problem of minimizing the complexity of a circuit is mapped to the problem of finding a geodesic path on a Riemannian manifold.The cost functional can for instance have the form Often the metric G IJ (U ) is choosen to be a right-invariant metric, but not necessarily the canonical biinvariant metric defined by A, B = tr(AB) for A, B self-adjoint.In fact, the possibility of choosing different metrics can be thought of as introducing penalty factors for moving along certain directions, corresponding to gates that may be more difficult to implement (for instance, in a system of qubits, one wants to penalize gates that act on many qubits simultaneously).A more general choice of the cost functional would lead to a Finsler geometry, where there is notion of a distance but not induced by a metric on the tangent space [1].For instance one can consider the family of cost functionals dependent on a parameter κ: Sometimes the ambiguity is reduced: the functional (4.4) with κ = 1 is parametrizationindependent.Another cost functional, which we will use in this paper, is basis-independent and makes use of the Schatten p-norm:6 ) 4.1 Application on gaussian states in free fermionic field theory In Nielsen's approach, the metric is defined not on the space of states, but on the manifold of a group of operators that act on the states.For fermions, the relevant group turns out to be the orthogonal group [2].To see this, it is convenient to use the Majorana basis ξ a = {q +∞ , . . ., q 1 , q 0 , q −1 , . . ., q −∞ , p +∞ , . . ., p 1 , p 0 , p −1 . . .p −∞ } (4.7) which is related to the annihilation and creation modes in the following way In the Majorana basis the anticommutation relations read {ξ a , ξ b } = δ ab .A general Gaussian state |ψ is completely characterized by the two-point function, or covariance matrix, which can be decomposed in the symmetric and antisymmetric part, respectively G ab and Ω ab ψ .The symmetric part is determined by the anticommutation relation, therefore it is stateindependent, while the antisymmetric part encodes the state.In terms of the covariance matrix, the fermionic vacuum state |0 defined in (2.11) would be expressed as, As discussed in sec.2.3 the Gaussian states are obtained by Bogoliubov transformations; in the Majorana basis, these are linear transformations ξ = M ξ that leave the anticommutation relation invariant, therefore they are orthogonal transformations.If we put a UV cutoff so that there are 2N modes, M ∈ SO(2N ). 7However there is a subgroup that acts trivially on the state, which only change the state by a phase.The covariance matrix transforms as The covariance matrix of the vacuum can be considered as a symplectic form, from this we see that the orthogonal matrices that leave Ω invariant should be symplectic as well, i.e., they should be unitary.The manifold of Gaussian states is then SO(2N )/U (N ).A convenient way to parametrize this coset is by using the relative covariance matrix Conversely, if we are given the reference and target states Ω 0 and Ω, we can recover the transformation matrix up to a unitary transformation.The polar decomposition of orthogonal matrices allows us to write M = u M with u ∈ U (N ) and M antisymmetric.In this manner, u and M are uniquely defined by the polar decomposition.At the level of Lie algebra, the splitting so(2N ) = u(N ) ⊕ asym(N) is an orthogonal decomposition with respect to the Killing metric of so(2N ).We expect then that the shortest path in the coset space will be obtained by moving only along the second subspace.Indeed it is shown in [2] that the geodesic connecting Ω 0 and Ω is given by a straight curve γ(s) = e sA with s ∈ [0, 1] which has a constant direction A ∈ asym(N).Finally the geodesic length is given by the norm of A = 1 2 ln ∆, which is the inner product with itself ||A|| = » A, A using the embedded metric on the Lie manifold SO(2N ).This definition of norm is basis independent and will coincide with the Schatten p = 2 norm.The Schatten p−norm for a general matrix T is defined as with s n (T ) being the singular values of the n × n matrix T , i.e., the eigenvalues of the matrix √ T † T .Due to the decreasing monotonicity of the Schatten p−norm, an interesting case is the p = 1 norm, defined as which will impose an upper bound to the p = 2 norm.The p = 1 norm has been considered before for quantum information purposes, for example it has been found that it is the only one among the p-norms to provide a consistent measure for quantum correlations, called quantum discord [36].
To summarize, the complexity of fermionic Gaussian states, defined as the geodesic length with respect to the Killing metric on the orthogonal group: In the above formula λ i are the eigenvalues of ∆ which come in pairs e ±iθ since ∆ ∈ SO(2N ).
Although the p = 1 norm loses its geometric meaning in the current case, it is interesting because in some cases discussed in the later sections, it poses an analytical bound on the Gaussian state complexity.

Translation to gaussian states in free bosonic field theory
In the case of free bosonic theory with N degrees of freedom, the gates group corresponding to Gaussian states is Sp(2N ).Contrary to the fermionic case, the role of covariance matrices G and Ω defined in (4.9) is exchanged, in the way that G represents the state and Ω encodes the algebraic relation.The main objective is still to find the relative covariance matrix relating the reference state and the target state, given as where M ∈ Sp(2N ) encodes the basis transformation.Similar to the fermionic case, ∆ is an element in the coset space Sp(2N )/U (N ) and the geodesic connecting G and G 0 will be again γ(s) = e sA with however A ∈ sym(N ) being an element in the symmetric algebra.In the same manner, the geodesic length is given by the norm of A = 1 2 ln ∆, hence the complexity Compared to the eq.(4.15), there is an "i" difference, which is due to the fact that the eigenvalues of a real symmetric symplectic matrix are in pairs of e ±r with r ∈ R. If the covariance matrix G for the reference state is the identity matrix, the covariance matrix will simply be ∆ = M M T , which we shall use in section 6. Detailed studies of the free bosonic complexity can be found in [2,23].
5 Application of Nielsen method on bosonic coherent states In this section, we apply the method introduced in the last section 4.1 on the bosonic coherent states with instead fermionic gates.
where J r 2|α| √ n is the Bessel function of the first kind and n is the excitation mode of boson.Notice that the mixing occurs only within fermionic modes that differ by a multiple of n.In the Majorana basis the transformation reads ql and the corresponding orthogonal matrix has the form which is indeed orthogonal as we show in appendix A for the case that θ = 0. Recall that the fermionic Gaussian state |ψ is given in terms of the transformed oscillators by cq (n) |ψ = 0, q > 0; c † q (n) |ψ = 0, q ≤ 0. (5.9) We observe that the θ-dependence in (5.4),(5.5)can be eliminated by the following field redefinition which does not affect the state (5.9).It is a unitary transformation, so it does not affect the complexity, as discussed in section 4.1.This is also in agreement with the result for the bosonic complexity (3.13) which is independent of the phase of α.We find that the most convenient choice for the present calculation is to set θ = π/2.In this case the off-diagonal block B(n) of the transformation matrix M (n) vanishes identically, and the non-vanishing entries of A(n) are given by The relative covariance matrix ∆(n) is made of two identical blocks with C given in (4.10).We can observe that A T (n) and A(n) are related by an orthogonal transformation with O ij = (−1) i δ ij , which is symmetric and also commutes with C. The diagonal block can be rewritten as (A(n)OC) 2 , so the problem is reduced to the diagonalization of A(n)OC.
We have not been able to obtain the eigenvalues analytically.Instead, we notice that the matrix elements A(n) ij (and those of A(n)OC as well, they just differ by some minus signs) are of the form of Bessel functions J ν (z) with index ν = |i − j| and argument z = 2α √ n .Recalling the power series expansion we see that for small z the matrix is dominated by the diagonal elements, while off-diagonal ones are suppressed exponentially with the distance from the diagonal.This has the consequence that the eigenvalues can be computed numerically with good accuracy, and they are weakly dependent on the cutoff that we put on the length of the matrix considered.One has to keep in mind that each entry in eq. ( 5.12) contains the full contributions of all the modes.
Result with p = 1 norm In Fig. 2 we plot the complexity with p = 1 norm, as a function of α, for different values of the number n, the excited bosonic mode.The cutoff is chosen to be 10n, although the result does not depend on it.We observe that the ratio between the complexity C p=1 and |α| √ n which is the argument of the Bessel function, is constant in α.Therefore, we claim that the complexity between the bosonic ground state |G N B and the coherent state |α, n generated from it by using the Schatten p = 1 norm is where we have taken into account the two diagonal blocks and an overall factor 1/2 in (4.16).
Comparing to the result (3.13) obtained in terms of bosonic gates, we see that the fermionic complexity has an extra dependence on the excitation mode n.
commuting as a result.In general, the arguments α k are complex numbers α k = |α k |e iθ k ; one overall phase can be reabsorbed with a redefinition of c j and cj as in (5.10).The numerical results we obtained for the two-mode shifts show that the complexity is in fact independent of both phases.We conjecture that this is true in general, although we cannot give a proof.
Result with p = 1 norm Applying the method in the last section 5.2 for two-mode shifts U n 1 (α 1 ) and U n 2 (α 2 ), one could obtain the complexity C p=1 plotted as in Fig. 4a where n 1 and n 2 are taken to be n 1 = 10 and n 2 = 17.From the plot, we can see that for the |α 2 | = 0 While for the other |α 2 | = 0 curves, we see that as |α 1 | becomes dominant they asymptotes to be linear in |α 1 | again with the same slope as for the curve |α 2 | = 0. Therefore, from the shape of the curves, we could deduce that the complexity for two-mode shift is which fits nicely with the results and reduces to (5.15) when |α 2 | = 0.It is reasonable to suppose that the result will generalize to multi-mode shifts as follows: (5.20) Result with p = 2 norm We plot the complexity C p=2 with the same setup in Fig. 4b.As in Fig. 4a, the complexity increases as either α 1 or α 2 increases, and all the curves converge to a certain curve as |α 1 | √ n 1 becomes much larger than |α 2 | √ n 2 , which is expected since then the effect of the first shifting mode becomes dominant.
Finally, the upper bound for the complexity with Schatten p = 2 norm is given by and similarly for the general case.

A class of fermionic and bosonic Gaussian states
In the previous sections, we have shown that the bosonic coherent states of the type in (3.14) are fermionic gaussian states and we have also seen how to obtain the corresponding complexity by using Fubini-Study method and Nielsen method in terms of bosonic gates and fermionic gates respectively.However the bosonic coherent states are only a subset of all the fermionic gaussian state, whereas a state that is fermionic gaussian is not generically also bosonic gaussian.
In this section we show that there is a large class of states that are simultaneously bosonic and fermionic gaussian.This allows us to study the effect that different choices of gates have on the complexity of a given state, since in the fermionic description we allow gates that are fermion bilinears, whereas in the bosonic description we use gates that are bosonic bilinears.Notice that neither set of gates is a subset of the other, so we cannot establish a priori a bound between the two complexities.

A bosonisation identity
Starting with the fundamental bosonisation formula (2.7), it is possible to obtain the following relation [32]: Taking the integral over space of this relation, we obtain on the two sides the fermionic and bosonic hamiltonian respectively, which is the basic property of the bosonisation formalism.Formula (6.1) shows that the equality holds at the level of the hamiltonian density, up to total derivative terms which, crucially for our purposes, are linear in the boson field, hence fermion bilinears, if we work on a subspace of fixed particle number.For definiteness we can consider the case N = 0, i.e., the module with fermion number zero.Therefore if we consider the inhomogeneous hamiltonian H f = dxf (x) :iψ † (x)∂ x ψ(x):, dependent on an arbitrary function f (x), the corresponding ground state will be fermionic gaussian, and also bosonic gaussian.
It is unlikely that further identities of this type exist, therefore we conjecture that the states obtained this way are the only ones that are gaussian in both descriptions.If we write down the same relation in Fourier components, taking Once again we see that identifying the coefficient of f 0 , we obtain the usual identity for fermionic and bosonic Hamiltonians

An example with one mode
We consider now one particular example of states in this family, corresponding to a function that has only Fourier modes n = ±2.The operator identity for f 2 which can be read from eq. (6.7) is and for f −2 we have the hermitian conjugate of the above.The two identities will help build the following unitary operator W f (β) = W b (β), given in terms of fermionic modes and bosonic modes, respectively, as .11)We will assume β ∈ R and focus on the sector N = 0 in the following part with the reference state being the fermionic vacuum |0 .The adjoint action of these operators on the modes cannot be obtained in an analytically closed form as in (5.4), (5.5).We must resort to a fully numerical computation.We put a cutoff on the number of modes, find the matrices representing 1  β ln W f and 1 β ln W b , respectively R f and R b , and compute the exponential of those two matrices, which give the change of basis ξf i = (e βR f ) ij ξ f j and ξb i = (e βR b ) ij ξ b j .The relative covariance matrix is then computed as in (4.12),(4.17We notice that if one of the two blocks is positive definite, the other would be negative definite, due to the fact that the eigenvalues of ∆ b always come in pairs e ±r (this is because ∆ b is symplectic and orthogonal).In this case one can show that the antisymmetric part does not contribute to the complexity; then it is easy to see that, setting A b = 0 in (6.13), the complexity C p=1 can be computed exactly and is linear in the trace of the symmetric part S b .However we do not have a criterion to determine when the blocks will be postive definite.
Result with p = 1 norm The results are shown in Fig. 5.The behavior appears to be very different for bosons and fermions.The bosonic complexity (see Fig. 5a) grows linearly without bound; this linear growth can be explained, as discussed above, by the positivedefiniteness of the blocks of the covariance matrix.On comparison, the fermionic complexity is oscillating (see Fig. 5b) in a way that suggests a quasi-periodic function.The maximum complexity of the first peak seems to grow linearly w.r.t. the logarithm of the cutoff ln(N ) as in Fig. 6a, this is consistent with the fact that the orthogonal group SO(2N ) becomes non-compact as N → ∞.For bosons, on the other hand, this particular class of states has a complexity that is independent of the cutoff, even though the transformation at finite β mixes all the modes among themselves and not just a finite number of them.
Result with p = 2 norm The complexities for gaussian states complexity with p = 2 norm are plotted in Fig. 5c for the bosonic case and in Fig. 5d for fermionic one.The trends appear to be similar as the p = 1 case, except that the values of the complexity are smaller.
-22 -    The cutoff is chosen to be N = 24 and the ratio varies from 0 to 1 with an interval of 0.25.
Fubini-Study metric method on the bosonic side, with the analytical results given in (3.13) for a single mode shift and in (3.19) a generalization to multi-mode shift.The general multi-mode result shows that different modes are orthogonal in the FS metric which is a flat Euclidean metric.On the fermionic side, we applied instead the Nielsen method developed in [2], with numerical result plotted in Fig. 2, 4a for Schatten p = 1 norm, and Fig. 3, 4b for p = 2 norm.Based on those numerics, an analytical result for multimode shift with p = 1 norm is obtained in eq.(5.20), which differ from the bosonic result (3.19) by an extra dependence on the mode number.This means that if we assign a mode-number dependent cost in the bosonic case, the two results would be equivalent up to a total factor.This result is a bit surprising, since it is for the p = 2 norm that we can interpret the complexity in terms of Riemannian geometry, so we should expect that the FS metric would compare more directly to that case.The p = 1 norm gives the space the structure of a Finsler manifold.It would be interesting to understand better the structure of this geometry; this could also shed some light on the relation between bosonic states and fermionic states holographically, since bosonic coherent states in the bulk constitute a sector of states that can be explcitly described both in the bulk and in the boundary (see the recent work [37]).
The second class of states is identified through the relation (6.1) which is a local (in space) relation between bosonic bilinears and fermionic ones, thus relating the gaussian states on both sides.These states can be understood as the ground states for an inhomogeneous Hamiltonian,

2 B↵
H M 9 S O o p c 4 z t j l c p h T s V u 3 P / R l e G E j L i L B 5 T X R P m z r k 8 Z 9 9 5 c t e 7 P V v m 6 r + d 0 r J 2 z i t t g T 9 2 l 3 T B 4 d 3 r X A X 9 d i s M W u G n d q P T r a 5 6 A 6 + w g z d 0 n 7 v o Y A / 7 6 F H 2 N 1 z g J 3 5 5 x 9 5 3 7 4 d 3 d i 3 1 a p X n J W 4 N 7 / w v V p e d z Q = = < / l a t e x i t > G < l a t e x i t s h a 1 _ b a s e 6 4 = " g N b P B h 5 K q 7 H f 9 + 9 M e P 9 r 0 7 j q

Figure 2 :
Figure 2: Complexity C p=1 over α √ n is constant in |α|, for n ∈ [20, 100] with an interval of 20, where n labels the bosonic excitation b † n in (3.3).The cutoff on the size of the matrices A(n) and B(n) is set equal to 10n.

Figure 4 :
Figure 4: Complexity for two-mode shifts with n 1 = 10, n 2 = 17.The cutoff is chosen to be N = 200 for the two-mode shift covariance matrice, which is similar to one of the diagonal blocks in (5.12).curve, the complexity is a linear function of |α 1 | with a proportionality constant being 4 √ n 1 .

Figure 5 :
Figure 5: The complexity for bosonic and fermionic gaussian states with p = 1 and p = 2 norms.(a) and (c) represent the bosonic case while (b) and (d) represent the fermionic case.The cutoff is chosen to be N = 40.

Figure 6 :
Figure 6: The maximum complexity for the fermionic case grows w.r.t. the logarithm of the cutoff, linearly for the p = 1 norm and quasi-linearly for the p = 2 norm.

Figure 7 :
Figure 7: The complexity for bosonic and fermionic gaussian states with p = 1 and p = 2 norms.(a) and (c) represent the bosonic case while (b) and (d) represent the fermionic case.The cutoff is chosen to be N = 24 and the ratio varies from 0 to 1 with an interval of 0.25.