The above form of the Hamiltonian suggests that we should study a set of coupled oscillators of the form H=ω(Ψ¯1Ψ1+Ψ¯2Ψ2)+iΩΨ¯1ρ1(Ψ1-Ψ2).(2.6)The basic idea is this: we wish to consider the simplest oscillator system which has a coupling analogous to the discretization (2.5) of the field theory. A candidate is the system with two lattice points parallel to the work for scalars considered in [4]. But if we work with d+1-dimensional Dirac fermions, the coupling term will force us to consider the d spatial directions. We wish to avoid this technical complication, and therefore in writing the above, we have restricted ourselves to a 1+1-dimensional theory. After we gain some intuition by working with this system, we will consider more general cases. In particular, we will see that the above Hamiltonian (2.6), when interpreted as a theory for Majorana fermions, is essentially a precise parallel to the bosonic two oscillator case discussed in [4]. We have adopted the notation ρa for gamma matrices in two dimensions in this subsection.

In two dimensions, gamma matrices allow a purely imaginary representation in which the spinors can be taken as Majorana. The gamma matrices take the form ρ0=(0-ii0),ρ1=(0ii0).(2.7)The general Majorana spinor in two dimensions Ψi (here i is part of the name of the spinor and not a spinor index) can be written as Ψi=(ψi-ψi+),(2.8)where the ψi± are real Grassmann variables. This means that our minimal Hamiltonian (2.6) is made of four real Majorana fermions and takes the form H=2iω(ψ1+ψ1-+ψ2+ψ2-)+iΩ(ψ1+ψ2+-ψ1-ψ2-).(2.9)Note that the Hamiltonian is real (Hermitian) because the fermions are Majorana and anticommute. Now we will rewrite this Hamiltonian in a form that is useful for future generalizations.³

Note that because the fermions are Majorana, we can use Hermitian conjugation and transposition interchangeably.

We will define complex Grassmann variables bi and bi†, bi=ψi--iψi+2bi†=ψi-+iψi+2,(2.10)and quantize by imposing the anticommutation relations {bi,bj†}=δij, see Appendix A for our conventions on fermionic oscillators. The Hamiltonian takes the form H=ω[b1†,b1]+ω[b2†,b2]-iΩ(b1†b2†+b1b2).(2.11)To bring this to the diagonalized normal mode form, we use the standard technology of Bogoliubov-Valatin (B-V) transformations (see Appendix B for a self-contained review). The result is H=λ[b˜1†,b˜1]+λ[b˜2†,b˜2],(2.12)where λ=124ω2+Ω2.(2.13)The b’s and b˜’s are related by b˜1=ηb2+iρb1†,b˜2=ηb1-iρb2†,(2.14)where ρ and η are given in Eq. (2.16).

A B-V transformation matrix that does the job of diagonalizing the Hamiltonian in the previous section is T=[0ηiρ0η00-iρ-iρ00η0iρη0],(2.15)where ρ=12-ω4ω2+Ω2,η=12+ω4ω2+Ω2.(2.16)This form suggests that we define r≡12tan-1(Ω2ω),(2.17)so that ρ=sinr,andη=cosr.(2.18)

To define complexity, we need to define a reference state from which we reach the target states via appropriate unitary transformations. A natural reference state is to choose |00⟩, defined via bi|00⟩=0,for  i=1,2,(2.19)and the target state which is the ground state of our theory is defined by bi˜|0˜0˜⟩=0,for  i=1,2.(2.20)From the B-V matrix, it follows that |0˜0˜⟩=cosr|00⟩+isinr|11⟩.(2.21)To go from the reference state to the target state by a unitary transformation, we first write the states in matrix form with the basis |ij⟩. The unitary transformation is a 4×4 unitary matrix: [cosr00isinr]=U(4×4)[1000].(2.22)Notice that the middle two components of the state vector are zero for both reference and target state. If these two components are changed along the path then we need to bring them back down to zero, when we get to the target state. So it stands to reason that (for appropriately defined Euclidean notions of distance in the space of unitaries) this would only increase the length, so we restrict ourselves to the path which only changes the first and fourth components. These transformations clearly fall inside a U(2) group: [cosrisinr]=U(2×2)[10].(2.23)Now U(2×2) is a 2×2 unitary matrix. After extracting a U(1) phase eiy and writing the remaining SU(2) as an S3, we can parametrize a general U(2) matrix as U=[eiycosρcosτ+ieiysinθsinρeiycosθsinρ+ieiycosρsinτieiycosρsinτ-eiycosθsinρeiycosρcosτ-ieiysinθsinρ].(2.24)The ranges of the various coordinates are⁴

We will work with y∈[-π,π) when we want to go across the y=0 point without changing charts.

For finite value of the parameter along the path, the state will be |Ψ(σ)⟩=U(σ)|R⟩,(2.26)where |R⟩ is the reference state. This means that in the explicit matrix above, we treat ρ, τ, θ, y as functions of σ. We can view this unitary as a path-ordered exponential U(σ)=P←e∫0σYI(s)OI(2.27)if we write⁵

This equation is the analogue of the time-dependent Schrodinger equation, written in a form that is usually written when one solves it via time-ordered exponentials. The left-hand side is the analogue of the Hamiltonian. So the solution, (2.27), can be directly exported here by analogy. To write (2.28), we merely note that since the right-hand side is made from a general U(2) matrix, the left-hand side must be writable in terms of the generators of U(2) whose coefficients we call YI.

Now we can use the boundary conditions to solve the system. Demanding U(s=0)=I(2.36)immediately yields (y(s=0),ρ(s=0),τ(s=0),θ(s=0))=(0,0,0,θ0).(2.37)Furthermore, we know that |ψT⟩=U(s=1)|R⟩.(2.38)Here |ψT⟩ is the target state (and our target state is the ground state). A 2×2 unitary matrix that takes the reference state to the target state is of the general form U(s=1)=[cosrisinre-iϕisinrcosre-iϕ]=e-iϕ/2[cosreiϕ/2isinre-iϕ/2isinreiϕ/2cosre-iϕ/2](2.39)for some arbitrary ϕ∈[-π,π). This translates to the end point boundary condition in terms of coordinates (y(s=1),ρ(s=1),τ(s=1),θ(s=1))=(y1,ρ1,τ1,θ1),(2.40)where y1=-ϕ/2,τ1=r,ρ1=ϕ/2,θ1=π/2-r.(2.41)Because the metric is Euclidean, one can convince oneself that the distance with the above initial and final boundary conditions is globally minimized for ϕ=0,⁶

Note that when this happens, θ is no longer determinate, including at the boundaries.

In the analogous discussion in [4], the generators (the MI in their notation) used were nonstandard and that resulted in a more complicated form of the metric and resulting geodesic equations. To solve the geodesic equations, more Killing vectors were identified. Here on the other hand, we explicitly see the S3×R form of the metric, and we only needed to use the obvious Killing vectors to find the explicit solution.

In this section, we will present an alternate approach for discussing complexity, which has a natural role in the continuum case. We will define an entangling operator K, which is also known as squeezing operator in some contexts: K=b2†b1†-b2b1=b˜1†b˜2†-b1˜b2˜.(2.44)The first equality can be viewed as the definition of the operator, the second equality is the result of a calculation, where we have used the definition of the tilde’d operators via the B-V transformation from the previous subsection.

It is also useful to define a unitary U=e-iKr,(2.45)where r∈R. The target state |0˜0˜⟩ can be reached from the reference state |00⟩ via |0˜0˜⟩=U|00⟩(2.46)for some appropriately chosen r. It is easy to see this via a similarity transformation of bi˜ using U that gives bi: U†b1˜U=eiKrb1˜e-iKr=b1˜-irb˜2†-r22b1˜+ir33!b˜2†+14!r4b˜1⋯=b1˜cosr-ib˜2†sinr.(2.47)Setting U†b1˜U=b2, we see that the transformation U does indeed take the reference state to the target state that we have worked with in the previous sections if we take the value of r to be what we found before.

Now let us consider an arbitrary path generated by the squeezing operator which takes the reference state to some more general target state: |Ψ(σ)⟩=U(σ)|00⟩=e-iKY(σ)|00⟩,(2.48)such that Y(0)=0⇒|Ψ(0)⟩=|00⟩Y(1)=r⇒|Ψ(1)⟩=|0˜0˜⟩.(2.49)We evaluate the length of this path using the Fubini-Study (FS) metric and minimizing it to get the complexity. In other words, the allowed circuits we are considering are the ones generated by K.

The Fubini-Study metric⁷

Note that the Fubini-Study metric is in the Hilbert space, while the distance we calculated in the previous subsection is in the space of unitaries.

So far we have considered just a pair of fermionic oscillators, albeit with a coupling that was motivated by our eventual interest in field theory. Now we will consider discretized versions of the field theory, and consider full lattices, with periodic and antiperiodic boundary conditions. We will work with Majorana fermions for concreteness.

The Hamiltonian of 1+1-d Majorana theory in terms of the complexified variables is (see Sec. III E for a derivation): H=∫dx(-iΨ∂1Ψ-iΨ†∂1Ψ†+m[Ψ†,Ψ]).(2.55)By placing it on a circular⁸

We use the word circular to refer to both periodic and antiperiodic boundary conditions simultaneously.

We consider the Dirac Hamiltonian in 1+1 dimensions given by H=∫dx[-iΨ¯γx∂xΨ+mΨ¯Ψ].(3.2)Here Ψ=(Ψ1,Ψ2)T is the two component complex fermion and γt=σ3 and γx=iσ2. By a Fourier transform Ψi(x)=∫dk2πΨi(k)eikxwe can write the Hamiltonian as H=∫dk[kΨ1†(k)Ψ2(k)+kΨ2†(k)Ψ1(k)+mΨ1†(k)Ψ1(k)-mΨ2†(k)Ψ2(k)].(3.3)The canonical anticommutators in momentum space become {Ψ1(k),Ψ1†(k′)}={Ψ2(k),Ψ2†(k′)}=δ(k-k′).(3.4)The reference unentangled IR state |R⟩ can be defined by Ψ1(k)|R⟩=0,Ψ2†(k)|R⟩=0∀  k∈R.(3.5)This is the ground state of the ultralocal Hamiltonian Hm=∫dx(mΨ¯Ψ)=∫dkm(Ψ1†(k)Ψ1(k)-Ψ2†(k)Ψ2(k)).(3.6)The ground state of this fermionic theory can be obtained by direct analogy with the discrete case via a Bogoliubov-Valatin transformation⁹

Note that this same ground state is precisely the one that is obtained in the standard approach to quantization of Dirac fermions, which involves the introduction of Dirac u and v modes. We demonstrate this in Appendix C.

As our target state, following the scalar case in [5], we consider the approximate ground state |m(Λ)⟩ which is defined as Ψ˜1(k)|m(Λ)⟩=0,Ψ˜2†(k)|m(Λ)⟩=0∀  k:|k|≤ΛΨ1(k)|m(Λ)⟩=0,Ψ2†(k)|m(Λ)⟩=0∀  k:|k|>Λ.(3.13)We can reach the target state from the reference state via a unitary transformation of the form |m(Λ)⟩=e-i∫|k|≤ΛdkK(k)rk|R⟩,(3.14)where the reference state is |R⟩=|Ω⟩ ∀  k and K(k) is the squeezing operator which we define as K(k)=i(Ψ1†(k)Ψ2(k)+Ψ1(k)Ψ2†(k)).(3.15)We will see that this construction shares many of the features of the scalar case discussed in [5], despite the fact that here the entanglement is happening between the two fermionic modes, and not between modes at antipodal momenta. Note also that the target ground state that we have defined is not the cMERA ground state [6,7] even though in the Λ→∞ limit they both tend to the true ground state. The cMERA can be viewed as a nongeodesic path in our language, as we will discuss in the next section.

Before we do all that, let us evaluate the complexity using our squeezing operator above. With the specific choice of the squeezing operator we have made in the previous section, the calculation goes entirely parallel to the one in [5], but we review it here largely to establish our notation. First, we calculate the distance with the Fubini-Study metric, dsFS(σ)=dσ|∂σ|Ψ(σ)⟩|2-|⟨Ψ(σ)|∂σ|Ψ(σ)⟩|2,(3.16)and then minimize the length. The circuit length of a path traced by intermediate states |Ψ(σ)⟩=U(σ)|R⟩ is l(|Ψ(σ)⟩)=∫σiσfdsFS(σ).(3.17)Using (3.1) we can write the FS metric as dsFS(σ)=dσ⟨G2(σ)⟩Ψ(σ)-⟨G(σ)⟩Ψ(σ)2.(3.18)

In the context of our discussion in the previous subsection, we can consider a general path generated by the squeezing operator as a unitary transformation that takes the state through U(σ)=e-i∫|k|≤ΛdkK(k)Yk(σ)|R⟩.(3.19)Here Yk(σ)=∫siσyk(s′)ds′andYk(sf)=rk,(3.20)where the last condition arises from our specific choice of target state. The order of k and σ integrals can be exchanged because K(k) at different k are all commuting. By direct calculation one sees that ⟨G(σ)⟩=0,and⟨G2(σ)⟩=Vol∫dkyk2(σ).(3.21)Here we write δ(0)=Vol. The complexity is the length of the minimal path: C=min{G(s)}l(dsFS)=min{Yk(σ)}∫sisfdσVol∫|k|≤Λdk(∂σYk(σ))2.(3.22)We recognize a flat Euclidean geometry associated with coordinate Yk(σ), implying that the geodesic is a straight line path. Under a simple affine parametrization σ the geodesic is Yk(σ)=σ-sisf-siYk(sf).(3.23)The complexity for the target state is the length of this straight line path, C=Vol∫k≤Λdk(rk2),(3.24)and rk is given by (3.11). As we see, the essential difference between complexity in the scalar case and the fermionic case is in the form of the rk, both in the discrete case and the continuum case. We find that the parallel between the form of the complexity in the discrete [4] and continuum [5] cases for the scalar, holds for the fermionic case as well.

These integrals are divergent in the cutoff Λ. In the massive case, the above expression can be explicitly evaluated to C2=-m2Vol log(Λ2m2+1)tan-1(Λm)-14mVol(-log(16)tan-1(Λm)-iLi2(-e-2itan-1(Λm))+iLi2(-e2itan-1(Λm)))+12ΛVol(tan-1(Λm))2.(3.25)Its behavior at large Λ takes the form C2=π28Vol(Λ+4mπlog(2mΛ)-4mπ+O(1/Λ)).(3.26)When m=0 the integral is simple and the complexity simplifies: C=π2ΛVol8.(3.27)

Our discussion so far used a single generator K(k) to reach the target state. This can be viewed as a specific choice of gate in the circuit complexity language. Since we were able to construct a parallel between the entangling operator in the scalar case in [5] and the Dirac fermion case above, it is interesting to ask whether a construction analogous to the more general SU(1,1) generators discussed in [5] is possible here. It turns out that the answer is yes, except that the generators satisfy an SU(2) algebra now. Our calculations in this section are direct adaptations of those in the Letter of [5], but we include some tricks here that simplify the calculation.

The basic idea is to note that the target state can also be reached by using a more general set of generators: K+(k)=iΨ1†(k)Ψ2(k)K-(k)=iΨ1(k)Ψ2†(k)K0(k)=Ψ1†(k)Ψ1(k)-Ψ2†(k)Ψ2(k)2.(3.28)These commute with the number preserving operators (n1-n2) where n1=Ψ1†Ψ1 and n2=Ψ2Ψ2†. These generators satisfy the following commutations relations: [K+(k),K-(k′)]=2K0(k)δ(k-k′)[K0(k),K+(k′)]=K+(k)δ(k-k′)[K0(k),K-(k′)]=-K-(k)δ(k-k′).(3.29)This is easily seen to be a set of decoupled SU(2) algebras at each k, once one rescales the generators appropriately with δ(0)=Vol. Now as in [5] let us consider a general path of the form: |Ψ(σ)⟩=e∫|k|<Λdkg(k,σ)|R⟩(3.30)generated by g(k,σ), given as g(k,σ)=α+(k,σ)K+(k)+α-(k,σ)K-(k)+ω(k,σ)K0(k).(3.31)The unitarity condition for the above transformation implies α+*(σ)=-α-(σ) and ω*(σ)=-ω(σ). We can decompose the unitary transformation above using [8] U(σ)=e∫|k|<Λdkβ+(k,σ)K+(k)e∫|k|<Λdklogβ0(k,σ)K0(k)⁢e∫|k|<Λdkβ-(k,σ)K-(k),(3.32)where the coefficients are β±=2α±sinhΞ2ΞcoshΞ-ωsinhΞ,β0=(coshΞ-ω2ΞsinhΞ)-2,with Ξ defined via Ξ2=ω24+α+α-.(3.33)Analogous to [5] now we can observe that K- annihilates the reference state while the exponential of K0 only changes the reference state up to a phase K-|R⟩=0;K0|R⟩=-δ(0)2|R⟩.(3.34)So |Ψ(σ)⟩ can be written as |Ψ(σ)⟩=Ne∫Λdkβ+(k,σ)K+(k)|R⟩(3.35)with N=e-δ(0)2∫|k|≤Λdklogβ0(k,σ).(3.36)