Circuit Complexity across a Topological Phase Transition

Fangli Liu,1, 2 Seth Whitsitt,1, 2 Jonathan B. Curtis,1 Rex Lundgren,1 ParajTitum,1, 2 Zhi-Cheng Yang,1, 2 James R. Garrison,1, 2 and Alexey V. Gorshkov1, 2

1Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA2Joint Center for Quantum Information and Computer Science,NIST/University of Maryland, College Park, MD 20742, USA

We use Nielsen’s geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chainacross a topological phase transition. We find that the circuit complexities of both the ground states and non-equilibrium steady states of the Kitaev model exhibit non-analytical behaviors at the critical points, and thuscan be used to detect both equilibrium and dynamical topological phase transitions. Moreover, we show that thelocality property of the real-space optimal Hamiltonian connecting two different ground states depends cruciallyon whether the two states belong to the same or different phases. This provides a concrete example of classifyingdifferent gapped phases using Nielsen’s circuit complexity. We further generalize our results to a Kitaev chainwith long-range pairing, and discuss generalizations to higher dimensions. Our result opens up a new avenuefor using circuit complexity as a novel tool to understand quantum many-body systems.

In computer science, the notion of computational complex-ity refers to the minimum number of elementary operationsfor implementing a given task. This concept readily extendsto quantum information science, where quantum circuit com-plexity denotes the minimum number of gates to implementa desired unitary transformation. The corresponding circuitcomplexity of a quantum state characterizes how difficult it isto construct a unitary transformation U which evolves a ref-erence state to the desired target state [1, 2]. Nielsen and col-laborators used a geometric approach to tackle the problem ofquantum complexity [3–5]. Suppose that the unitary transfor-mation U(t) is generated by some time-dependent Hamilto-nian H(t), with the requirement that U(tf ) = U (where tfdenotes the final time). Then, the quantum state complexityis quantified by imposing a cost functional F [H(t)] on thecontrol Hamiltonian H(t). By choosing a cost functional thatdefines a Riemannian geometry in the space of circuits, theproblem of finding the optimal control Hamiltonian synthe-sizing U then corresponds to finding minimal geodesic pathsin a Riemannian geometry [3–5].

Recently, Nielsen’s approach has been adopted in high-energy physics to quantify the complexity of quantum fieldtheory states [6–18]. This is motivated, in part, by previ-ous conjectures that relate the complexity of the boundaryfield theory to the bulk space-time geometry, i.e. the so-called“complexity equals volume” [19, 20] and “complexity equalsaction” [21, 22] proposals. Jefferson et al. used Nielsen’s ap-proach to calculate the complexity of a free scalar field [6],and found surprising similarities to the results of holographiccomplexity. A complementary study by Chapman et al., us-ing the Fubini-Study metric to quantify complexity [23], gavesimilar results. Several recent works have generalized thesestudies to other states, including coherent states [8, 24], ther-mofield double states [7, 11], and free fermion fields [12–14].However, the connection between the geometric definition ofcircuit complexity and quantum phase transitions has so farremained unexplored. This connection is important both fun-damentally, and is also intimately related to the long-standingproblem of quantum state preparations across critical points

[25–27].In this work, we consider the circuit complexity of a topo-

logical quantum system. In particular, we use Nielsen’s ap-proach to study the circuit complexity of the Kitaev chain,a prototypical model exhibiting topological phase transitionsand hosting Majorana zero modes [28–33]. Strikingly, wefind that the circuit complexity derived using this approachexhibits non-analytical behaviors at the critical points, forboth equilibrium and dynamical topological phase transitions.Moreover, the optimal Hamiltonian connecting the initial andfinal states must be non-local in real-space when evolvingacross a critical point. We further generalize our results toa Kitaev chain with long-range pairing, and discuss univer-sal features of non-analyticities at the critical points in higherdimensions. Our work establishes a connection between ge-ometrical circuit complexity and quantum phase transitions,and paves the way towards using complexity as a novel tool tostudy quantum many-body systems.

The model.—The 1D Kitaev model is described by the fol-lowing Hamiltonian [28, 29]:

H =− J

2

L∑j=1

(a†j aj+1 + H.c.

)− µ

L∑j=1

(a†j aj −

1

2

)

+∆

2

L∑j=1

(a†j a†j+1 + H.c.

),

(1)

where J is the hopping amplitude, ∆ is the superconductingpairing strength, µ is the chemical potential, L is the totalnumber of sites (assumed to be even), and a†j (aj) creates (an-nihilates) a fermion at site j. We set J = 1 and assume an-tiperiodic boundary conditions (aL+1 = −a1). Upon Fouriertransforming Eq. (1) can be written in the momentum basis

H = −∑kn

[µ+ cos kn](a†kn akn − a−kn a

†−kn

)+ i∆ sin kn

(a†kn a

†−kn − a−kn akn

),

(2)

where kn = 2πL (n + 1/2) with n = 0, 1, . . . , L/2 − 1.

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The above Hamiltonian can be diagonalized via a Bogoli-ubov transformation, which yields the excitation spectrum:

εkn =√

(µ+ cos kn)2 + ∆2 sin2 kn. The ground state ofEq. (1) can be written as

|Ψgs〉 =

L/2−1∏n=0

(cos θkn − i sin θkn a†kna†−kn) |0〉 , (3)

where tan(2θkn) = ∆ sin kn/(µ + cos kn). A topologicalphase transition occurs when the quasiparticle spectrum isgapless [28], as illustrated in Fig. 1(a). The nontrivial topo-logical phase is characterized by a nonzero winding numberand the presence of Majorana edge modes [28–33].

Complexity for a pair of fermions.—Since Hamiltonian (1)is non-interacting, the ground state wavefunction (3) couplesonly pairs of fermionic modes with momenta±kn, and differ-ent momentum pairs are decoupled. Hence, we first computethe circuit complexity of one such fermionic pair [12–14], andthen obtain the complexity of the full system by summing overall momentum contributions [6, 23].

Let us consider the reference (“R”) and target (“T ”) stateswith the same momentum but different Bogoliubov angles:|ψR,T 〉 = (cos θR,Tk − i sin θR,Tk a†ka

†−k) |0〉. Expanding the

target state in the basis of |ψR〉 and |ψR〉⊥ (i.e., the stateorthogonal to |ψR〉), we have |ψT 〉 = cos(∆θk) |ψR〉 −i sin(∆θk) |ψR〉⊥, where ∆θk = θRk − θTk . Now the goal is tofind the optimal circuit to achieve the unitary transformationconnecting |ψR〉 and |ψT 〉 :

Uk =

[cos(∆θk) −ie−iφ sin(∆θk)−i sin(∆θk) e−iφ cos(∆θk)

], (4)

where φ is an arbitrary phase. Nielsen approached this as aHamiltonian control problem, i.e. finding a time-dependentHamiltonianHk(s) that synthesizes the trajectory in the spaceof unitaries [3, 4]:

Uk(s) =←−P exp

[∫ s

0

dtH(t)

], Hk(t) =

∑I

Y Ik (t)OI (5)

with boundary conditions Uk(s = 0) = 1, and Uk(s = 1) =

Uk. Here,←−P is the path-ordering operator and OI are the

generators of U(2). The idea is then to define a cost (i.e.‘length’) functional for the various possible paths to achieveUk [3, 4, 6, 12]: D [Uk] =

∫ 1

0ds∑I |Y Ik (s)|2, and to identify

the optimal circuit or path by minimizing this functional. Thecost of the optimal path is called the circuit complexity C ofthe target state, i.e.

C [Uk] = min{Y Ik (s)}D [Uk] . (6)

Following the procedures in Refs. [12–14], one can explic-itly calculate the circuit complexity for synthesizing the uni-tary transformation (4). For quadratic Hamiltonians, it is asimple expression that depends only on the difference betweenBogoliubov angles (see Supplemental Material [34]),

C (|ψR〉 → |ψT 〉) = |∆θk|2. (7)

Note that the complexity C for two fermions is at most π2/4,since |∆θk| ∈ [0, π/2]. The maximum value is achieved whenthe target state has vanishing overlap with the reference state.

Complexity for the full wavefunction.—Given the circuitcomplexity for a pair of fermionic modes, one can readilyobtain the complexity of the full many-body wavefunction.The total unitary transformation that connects the two differ-ent ground states [Eq. (3)] is:

|ΨTgs〉 =

L/2−1∏n=0

Ukn

|ΨRgs〉 , (8)

where Ukn , given by Eq. (4), connects two fermionic stateswith momenta ±kn. By choosing the cost function to be asummation of all momentum contributions [6, 12–14], it isstraightforward to obtain the total circuit complexity

C(|ΨR

gs〉 → |ΨTgs〉)

=∑kn

|∆θkn |2, (9)

where ∆θkn is the difference of the Bogoliubov angles formomentum kn. In the infinite-system-size limit, the sum-mation can be replaced by an integral, and one can derivethat C ∝ L. This “volume law” dependence is reminiscentof the “complexity equals volume” conjecture in holography[19, 20], albeit in a different setting.

The circuit complexity given by Eq. (9) has a geometric in-terpretation, as it is the squared Euclidean distance in a highdimensional space [35]. The geodesic path (or optimal circuit)in unitary space turns out to be a straight line connecting thetwo points [i.e. Hk(s) indepedent of s) [34]. In the remain-der of this paper, we demonstrate that the circuit complexitybetween two states is able to reveal both equilibrium and dy-namical topological phase transitions.

We first choose a fixed ground state as the reference stateand calculate the circuit complexities for target ground stateswith various chemical potentials µT , crossing the phase tran-sition point. The circuit complexity increases as the differ-ence between the parameters of reference and target statesis increased [Fig. 1(b)]. More importantly, the complexitygrows rapidly around the critical points (µT = ±1), chang-ing from a convex function to a concave function at the crit-ical points. This is further illustrated in Fig. 1(c), where weplot the derivative (susceptibility) of circuit complexity withrespect to µT . The clear divergence at the critical points in-dicates that circuit complexity is nonanalytical at the criticalpoints (see Supplemental Material [34] for derivation), andthus can signal the presence of a quantum phase transition. Weemphasize that these features are robust signatures of phasetransitions, which do not change if one chooses a differentreference state in the same phase [see Figs. 1(b) and (c)].

We further plot ∆θkn versus the momentum kn, for varioustarget states (with a fixed reference state) in Fig. 1(d). Whenboth states are in the same phase, ∆θkn first increases withmomentum, and finally decreases to 0 when kn approachesπ. In contrast, when µT is beyond its critical value, ∆θkn

3

(a) (b)

-2 -10

0.1

0.2

0.3

0

µR = 0µR = 0.2µR = -0.3

-2 -10

0.2

0.4

0.6

0.8

1

0 1 2

µR = 0µR = 0.2µR = -0.3

T

T0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5µT = 0.4µT = 0.6µT = 0.9µT = 0.95µT = 1.05µT = 1.2µT = 1.9

(c) (d)0

-1

0

1

μ

Δ

W=-1 W=1

1 2

FIG. 1. (a) Phase diagram of the Kitaev chain, with W denotingthe winding number. (b) Ground state circuit complexity and (c)its derivative versus target state chemical potential (µT ) for severalreference states, each with a different chemical potential µR. (d) Bo-goliubov angle difference, ∆θkn , for different target ground states,with µR = 0. ∆R = ∆T = 1 for (b)–(d), and L = 1000 for (b) and(c).

increases monotonically with momentum, and takes the max-imal value of π/2 at kn = π. This is closely related to thetopological phase transition characterized by winding num-bers, where the Bogoliubov angles of two different states endup at the same pole (on the Bloch sphere) upon winding halfof the Brillouin zone if the states belong to the same phase[29]. Hence, the non-analytical nature of the circuit complex-ity is closely related to change of topological number (andtopological phase transition).

Analytically, the derivatives of the circuit complexity (7)can be explicitly recast into a closed contour integral overthe complex variable z = eik (see the Supplemental Mate-rial [34] for detailed derivations). Depending on the param-eters of the target states, the poles associated with the inte-grand are located inside or outside the contour. When the tar-get state goes across a phase transition, the poles sit exactlyon the contour, resulting in the divergence of the derivativesof the circuit complexity at critical points [34]. Interestingly,the whole parameter space can be classified into four differentphase regimes depending on which poles lie inside the contour[see Fig. S1 in Supplemental Material [34]] , which agrees ex-actly with the phase diagram shown in Fig. 1(a).

Figures 2(a) and (b) show the derivative of circuit complex-ity with respect to µT and ∆T for the whole parameter regime.The derivatives show clear singular behavior at both the hor-izontal [Fig. 2(a)] and vertical [Fig. 2(b)] phase boundaries.Therefore, by using the first-order derivative of complexitywith respect to µT and ∆T , one can map out the entire equi-librium phase boundaries of the Kitaev chain.

(a)

T

ΔT ΔT

T

(b)

FIG. 2. Derivative of circuit complexity as a function of µT and∆T . Panel (a) plots the derivative with respect to µT (in units of1/∆T ), and panel (b) plots the derivative with respect to ∆T . Thereference state is chosen as the ground state of Eq. (1) with µR = 0and ∆R = −1, and L = 1000.

Real-space locality of the optimal Hamiltonian.— Since theground state [Eq. (3)] is a product of all momentum pairs, theoptimal circuit connecting two different ground states corre-sponds to the following Hamiltonian:

Hc =∑k>0

Hk =∑k>0

−i∆θ(k)ψ†kτ1ψk, (10)

where τi are the Pauli matrices, and ψk denotes the Nambu

spinor ψk =

(aka†−k

).By taking a Fourier series of the above

optimal Hamiltonian, one can show that the Hamiltonian canbe written in real space (see Supplemental Material for details[34]):

Hc =∑j

∞∑n=1

ωn (aj aj+n − H.c.) , (11)

where ∆θ(k) = 2∑∞n=1 ωn sin(nk).

One crucial observation is that when the two groundstates are in the same phase, ∆θ(0) = ∆θ(π) = 0 [seeFig. 1(d)]; hence the Fourier series of ∆θ(k) converges uni-formly. Therefore, the full series can be approximated by afinite order N∗ with arbitrarily small error. This immediatelyimplies that the real-space optimal Hamiltonian (11) is local,with a finite range N∗. In sharp contrast, if the two states be-long to different phases, ∆θ(π) = π/2 6= ∆θ(0) = 0; theFourier series of ∆θ(k) converges at most pointwise. Thusthe optimal Hamiltonian must be truly long-range (non-local)in real-space [34], given that the total evolution time is cho-sen to be a constant [Eq. (5)]. Comparing to previous workson classifying gapped phases of matter using local unitary cir-cuits [36–38], our results provide an alternative approach thathas a natural geometric interpretation.

Complexity for dynamical topological phase transition.—Dynamical phase transitions have received tremendous inter-est recently [39–51]. Studies on quench dynamics of cir-cuit complexity have mostly focused on growth rates in theshort-time regime [10, 15]. Here, we show that the long-timesteady-state value of the circuit complexity following a quan-

4

8Time t

0

0.1

0.2

0.3

0.4f = 0.4f = 0.6f = 0.8f = 0.9

f = 1.1f = 1.4f = 1.8

-2 -1f

0

0.1

0.2

0.3

0

i = 0

ii= 0.2= 0.5

(a) (b)

1 2642

FIG. 3. (a) Circuit complexity growth for various post-quench chem-ical potentials, µf . The initial state (serves as the reference state)is the ground state of Eq. (1) with µi = 0. (b) Steady-state valuesof complexity versus µf . The different lines denote different ini-tial/reference states. ∆i = ∆f = 1 and L = 1000 in both plots.

tum quench can be used to detect dynamical topological phasetransitions.

We take the initial state to be the ground state of a Hamilto-nian Hi, and consider circuit complexity growth under a sud-den quench to a different Hamiltonian, Hf . The referenceand target states are chosen as the initial state |Ψi〉 and time-evolved state |Ψ(t)〉 respectively. The time-dependent |Ψ(t)〉can be written as [52, 53]

|Ψ(t)〉 =

L2 −1∏n=0

[cos(∆θkn)−ie2iεkn t sin(∆θkn)A†knA

†−kn

]|0〉 ,

(12)where ∆θkn is the Bogoliubov angle difference betweeneigenstates of Hi and Hf , and εkn and Akn are the energylevels and normal mode operators, respectively, for the post-quench Hamiltonian. Similar to the previous derivations, onecan obtain the time-dependent circuit complexity,

C(|Ψi〉 → |Ψ(t)〉) =∑kn

φ2kn(t) (13)

where φkn(t) = arccos√

1− sin2(2∆θkn) sin2(εknt).

As shown in Fig. 3(a), the circuit complexity first increaseslinearly and then oscillates [9, 10, 15] before quickly ap-proaching a time-independent value. The steady-state valueof circuit complexity increases with µf of the post-quenchHamiltonian, until the phase transition occurs [Fig. 3(a)].Fig. 3(b) further illustrates the long-time steady-state valuesof circuit complexity versus µf for different initial states. Thesteady-state complexity clearly exhibits nonanalytical behav-ior at the critical point. This behavior arises because the time-averaged value of φkn(t) exhibits an upper bound after thephase transition (see Supplemental Material [34]), and it is arobust feature of the dynamical phase transition regardless ofthe initial state.

Generalization to long-range Kitaev chain and higherdimensions.—We further give an example of a Kitaev chain

(b)(a)

-2 -10

0.5

1

1.5

2

2.5

= 0R

R= -0.2

R= 0.3

T

-4 -20

0.05

0.1

0.15

0.2

0.25

0

i = 0i= -0.2i= 0.3

f2 40 1 2

FIG. 4. (a) Derivative of circuit complexity with respect to µT

for three different reference ground states of the long-range Kitaevchain, with ∆R = ∆T = 1.3. (b) Steady-state value of circuitcomplexity versus µf for three different initial ground states, with∆i = ∆f = 1. L = 1000 and α = 0 in both plots.

with long-range pairing [54–57]:

HLR =− J

2

L∑j=1

(a†j aj+1 + H.c.)− µL∑j=1

(a†j aj −1

2)

+∆

2

L∑j=1

L−1∑`=1

1

dα`(a†j a

†j+` + H.c.),

(14)

where d` = min(`, L − `). In contrast to the short-rangemodel, the long-range model with α < 1 hosts topologicalphases with semi-integer winding numbers [54, 57]. As onecan see, the derivative of ground state circuit complexity onlydiverges at µT = 1 [Fig. 4(a)], in contrast with Fig. 1(c). Thisagrees perfectly with the phase diagram for the long-range in-teracting model, where a topological phase transition occursonly at µ = 1 for α = 0 [57]. Figure 4(b) shows the long-time steady-state values of the circuit complexity after a sud-den quench. Again, one observes nonanalytical behavior onlyat µT = 1.

While we have so far restricted ourselves to 1D, the re-sults we found can be readily generalized to higher dimen-sions [58], for example, to p+ ip topological superconductorsin 2D. The ground state wavefunction of a p + ip supercon-ductor essentially takes the same form as Eq. (3), with themomenta now being restricted to the 2D Brillouin zone, andtan(2θk) = |∆k|/εk, where ∆k and εk denote pairing and ki-netic terms in 2D. The circuit complexity can still be writtenas C =

∑k |∆θk|2 = L2

(2π)2

∫d2k|∆θ(k)|2. One can show

again that the derivative of the circuit complexity is given by(see Supplemental Material [34])

∂µTC =

L2

(2π)2

∫d2k

θT (k)|∆(k)|E(k)2

, (15)

where E(k)2 = ε(k)2 + |∆(k)|2 and θT (k) denotes the Bo-goliubov angle for the target state. It is thus obvious that non-analyticity happens at the critical point where E(k) = 0 [34].

Conclusions and outlook.—We use Nielsen’s approach toquantify the circuit complexity of ground states and nonequi-librium steady states of the Kitaev chain with short- and long-

5

range pairing. We find that, in both situations, circuit com-plexity can be used to detect topological phase transitions.The non-analytic behaviors can be generalized to higher-dimensional systems, such as p + ip topological supercon-ductors [59, 60].

One interesting future direction is to use the geometric ap-proach to quantify circuit complexity when the control Hamil-tonians are constrained to be local in real-space [38, 61, 62],and study its connection to quantum phase transitions [25, 63–65]. It would also be of interest to investigate the circuit com-plexity of interacting many-body systems. One particular ex-ample is the XXZ spin-half chain, whose low-energy physicscan be modeled by the Luttinger liquid [66–68]. By restrict-ing to certain classes of gates (i.e., by imposing penalties onthe cost function) [3, 6], it might be possible to find improvedmethods to efficiently prepare the ground state of the XXZmodel by calculating the geodesic path in gate space.

We thank Peizhi Du, Abhinav Deshpande, and Su-KuanChu for helpful discussions. F.L., R.L., Z.-C. Y., J.R.G.and A.V.G. acknowledge support by the DoE BES QIS pro-gram (award No. DE-SC0019449), DoE ASCR FAR-QC(award No. DE-SC0020312), NSF PFCQC program, DoEASCR Quantum Testbed Pathfinder program (award No. DE-SC0019040), AFOSR, NSF PFC at JQI, ARO MURI, andARL CDQI. Z.-C. Y. is also supported by AFOSR FA9550-16-1-0323, and ARO W911NF-15-1-0397. P.T. and S.W.were supported by NIST NRC Research Postdoctoral Asso-ciateship Awards. J.B.C. received support from the NationalScience Foundation Graduate Research Fellowship Programunder Grant No. DGE 1322106 and from the Physics Fron-tiers Center at JQI. This research was supported in part bythe Heising-Simons Foundation, the Simons Foundation, andNational Science Foundation Grant No. NSF PHY-1748958.

Note added: While finalizing this manuscript, we becameaware of Ref. [69], which used revivals in the circuit com-plexity as a qualitative probe of phase transitions in the Su-Schrieffer-Heeger model. In contrast to that work, we haveshown that the circuit complexity explicitly exhibits nonana-lyticities precisely at the critical points for the Kitaev chain.We also became aware of Ref. [58], which numerically stud-ied the complexity of a two- dimensional “ d · τ” model. Bycontrast, here we analytically study the “p + ip” model, andillustrate that the closing of the gap is essential for the nonan-alyticity of circuit complexity.

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[34] See Supplemental Material for derivations of circuit complexityfor a pair of fermions, derivations of the nonanalyticity of cir-cuit complexity at critical points, details of real-space behaviorof optimal control Hamiltonians, numerics for the nonanalyt-icity of circuit complexity after quantum quenches, and circuitcomplexity for 2D p+ ip topological superconductors.

[35] In such a space, each state is represented by one point,with its coordinates labeled by the Bogoliubov angles, i.e.(θk0 , θk1 , . . . , θkL/2−1

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[38] Y. Huang and X. Chen, “Quantum circuit complexity of one-dimensional topological phases,” Phys. Rev. B 91, 195143(2015).

[39] M. Heyl and J. C. Budich, “Dynamical topological quantumphase transitions for mixed states,” Phys. Rev. B 96, 180304(2017).

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[41] M. D. Caio, N. R. Cooper, and M. J. Bhaseen, “QuantumQuenches in Chern Insulators,” Phys. Rev. Lett. 115, 236403(2015).

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[44] C. Wang, P. Zhang, X. Chen, J. Yu, and H. Zhai, “Schemeto Measure the Topological Number of a Chern Insulator fromQuench Dynamics,” Phys. Rev. Lett. 118, 185701 (2017).

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[46] M. Heyl, F. Pollmann, and B. Dora, “Detecting Equilibriumand Dynamical Quantum Phase Transitions in Ising Chainsvia Out-of-Time-Ordered Correlators,” Phys. Rev. Lett. 121,016801 (2018).

[47] S. Roy, R. Moessner, and A. Das, “Locating topological phasetransitions using nonequilibrium signatures in local bulk ob-servables,” Phys. Rev. B 95, 041105 (2017).

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[49] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker,H. Kaplan, A. V. Gorshkov, Z. X. Gong, and C. Monroe, “Ob-servation of a many-body dynamical phase transition with a 53-qubit quantum simulator,” Nature 551, 601–604 (2017).

[50] P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges,C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos,“Direct Observation of Dynamical Quantum Phase Transitionsin an Interacting Many-Body System,” Phys. Rev. Lett. 119,080501 (2017).

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7

Supplemental Material

This Supplemental Material consists of four sections. InSec. I, we analytically derive the circuit complexity for a pairof fermions [Eq. (7) of the main text]. In Sec. II, we providedetailed derivations of the nonanalyticity of circuit complex-ity, as shown numerically in Figs. 1(b) and (c) and Figs. 2(a)and (b) of the main text. In Sec. III, we discuss the real-spacestructure of the optimal circuit. In Sec. IV, we provide numer-ical and analytical evidence of the nonanalyticity of steady-state circuit complexity after a quantum quench. Finally, inSec. V, we provide a detailed analysis of circuit complexityfor p+ ip topological superconductor.

I. DERIVATION OF CIRCUIT COMPLEXITY FOR A PAIROF FERMIONS

In this section, we present a detailed derivation of the circuitcomplexity for a pair of fermions, i.e. Eq. (7) in the main text.This expression has previously been obtained using differentapproaches in Refs. [12–14]. In order to be comprehensive,here we provide a detailed derivation following Ref. [13]. Wenote that Ref. [12] provides an alternative derivation using agroup theory approach.

By taking the derivative with respect to s in Eq. (5) of themain text, we get the following expression:∑

I

Y I(s)OI = (∂sU(s))U−1(s), (S1)

where U(s) is a unitary transformation which depends on s,and we have omitted the label k for notational clarity.

The unitary U(s) can be parametrized in matrix form:

U(s) = eiβ[e−iφ1 cosω e−iφ2 sinω−eiφ2 sinω eiφ1 cosω

], (S2)

where β, φ1, φ2, ω explicitly depend on the parameter s. Theabove matrix can be expressed in terms of the generators ofU(2), which we choose as follows:

O0 =

[i 00 i

], O1 =

[0 ii 0

], O2 =

[0 1−1 0

], O3 =

[i 00 −i

].

(S3)Using the relation

tr(OaOb) = −2δab, (S4)

one can extract the strength, Y I(s), of generator OI [cf.Eq. (5) in the main text] as follows:

Y I(s) = −1

2tr[(∂sU(s))U−1(s)OI

]. (S5)

Our cost functional can then be expressed as

D =

∫ 1

0

ds∑I

|Y I(s)|2

=

∫ 1

0

ds

[(dβ

ds

)2

+

(dω

ds

)2

+ cos2 ω

(dφ1

ds

)2

+ sin2 ω

(dφ2

ds

)2]. (S6)

Now, by exploiting the boundary condition at s = 0, i.e.U(s = 0) = I , we get

β(s = 0)φ1(s = 0)φ2(s = 0)ω(s = 0)

=

00

φ2(0)0

, (S7)

where φ2(0) is an arbitrary phase. Furthermore, we have theboundary condition at s = 1,

U(s = 1) =

[cos(∆θ) −ie−iφ sin(∆θ)−i sin(∆θ) e−iφ cos(∆θ)

], (S8)

which results in β(s = 1)φ1(s = 1)φ2(s = 1)ω(s = 1)

=

00π/2∆θ

. (S9)

The integrand in Eq. (S6) is a sum of four non-negativeterms. Setting β(s) = φ1(s) = 0 and φ2(s) = π/2 min-imizes (i.e. sets to zero) three of the four terms without im-posing any additional constraints on the minimization of theremaining (dω/ds)2 term. One can then easily check that thelinear function w(s) = s∆θ minimizes the remaining termand yields

C =

∫ 1

0

ds |∆θ|2 = |∆θ|2. (S10)

II. ANALYTICAL DERIVATION OF DIVERGENTDERIVATIVES IN GROUND STATES

In this section, we provide a detailed analytical derivation toshow that the first-order derivative indeed diverges at the crit-ical points in the thermodynamic limit. We first derive howthe derivative diverges when the reference state is in the triv-ial phase (|µR| > 1), and then we generalize our results toshow how this divergent behavior depends on the particularchoice of the reference state. Throughout this section we as-sume the reference lies within a given phase, and allow the tar-get state to approach an arbitrary point in the phase diagram.Our analytical derivations show that these divergences neces-sarily map out the phase boundary, as illustrated in Figs. 2(a)and (b) in the main text and Fig. S1 below.

8

We begin with our general expression for the complexityas a function of our reference and target states. The Bogoli-ubov angle difference ∆θk for each momentum sector k canbe expressed as

∆θk = 12 arctan sin k[∆RµT−∆TµR+(∆R−∆T ) cos k]

(µR+cos k)(µT +cos k)+∆R∆T sin2 k,

(S11)and the circuit complexity is written in terms of ∆θk:

C/L =1

2π

∫ π

0

|∆θk|2 dk. (S12)

Note that we have replaced the discrete sum in the main textwith an integral for the thermodynamic limit, and written“C(|ΨR

gs〉 → |ΨTgs〉)” as “C” for brevity.

Now we substitute Eq. (S11) into Eq. (S12), and take thederivatives with respect to µT and ∆T . We obtain

∂µTC/L =

∆T

4π

∫ π

−π

∆θk sin k

(µT + cos k)2 + ∆2T sin2 k

dk,

∂∆TC/L = − 1

4π

∫ π

−π

∆θk sin k (µT + cos k)

(µT + cos k)2 + ∆2T sin2 k

dk. (S13)

Here, we have used the fact that these functions are even ink to extend the integrals to −π. In spite of the complicatednature of these integrals, much can be learned about their an-alytic properties by recasting them as closed contour integralsin the complex plane. Defining the variable z = eik, we findthat the integrals take the form

∂µTC/L = −i∆T

∮dz

2πi

∆θ(z)(z2 − 1)

(z2 + 2µT z + 1)2 −∆2T (z2 − 1)2

,

∂∆TC/L =

i

2

∮dz

2πiz

∆θ(z)(z2 − 1

) (z2 + 2µT z + 1

)(z2 + 2µT z + 1)2 −∆2

T (z2 − 1)2,

(S14)

where the integration is taken counter-clockwise over the con-tour |z| = 1. In this form, we may use the fact that the valueof the integrals is entirely determined by the non-analyticitiesof the integrand which are located inside the contour, and thatthe value of the integration will only diverge if there is a di-vergence located on the contour.

We proceed by defining the following variables,

z1,a =−µa +

√µ2a + ∆2

a − 1

1 + ∆a,

z2,a =−µa −

√µ2a + ∆2

a − 1

1 + ∆a,

z3,a =−µa +

√µ2a + ∆2

a − 1

1−∆a,

z4,a =−µa −

√µ2a + ∆2

a − 1

1−∆a, (S15)

where a = R, T . From Eq. (S14), both derivatives containsimple poles at zi,T for i = 1, 2, 3, 4, while ∂∆T

C addition-ally has a simple pole at z = 0. Also, using the formula

FIG. S1. The phase diagram of the Kitaev chain, where in each phasewe list which of the two branch points given in Eq. (S15) lie insidethe contour integrals in Eq. (S14). The integrals can only diverge atthe phase transitions, where the branch points cross the contour,

FIG. S2. The deformation of the integration contour used to computethe gradients of the circuit complexity in the case µT > 1. There isa branch cut running between the branch points z1 and z3, where theimaginary part of the integrand is discontinuous and the integranddiverges near the branch points.

arctan(z) = (i/2) log 1−iz1+iz , we can write the Bogoliubov an-

gle as

∆θ(z) =i

4log

[(∆T + 1)(z − z1,T )(z − z2,T )

(∆T − 1)(z − z3,T )(z − z4,T )

× (∆R − 1)(z − z3,R)(z − z4,R)

(∆R + 1)(z − z1,R)(z − z2,R)

]. (S16)

The important fact we will need is that the complex logarithmcontains branch cuts running from the zeros to the infinitiesof its argument; therefore, the zia are really branch points ofthe integrand. We now note that the derivatives of the com-plexity will only diverge if the couplings are tuned to a phasetransition. This is because the zi,a can only have unit modu-lus if we are at one of the phase transitions, and at the phasetransitions the branch points cross the contour resulting in adivergent integral, see Fig. S1. In particular, we may charac-terize the phase diagram in terms of which branch points areinside or outside the contour integral.

In addition, we may actually compute the integrals exactlyin certain cases and limits, which allows us to obtain the exactanalytic dependence of the divergence on the couplings. Asa definite example, we consider the case |µT | > 1. In thiscase, there is a branch cut inside the logarithm running from

9

z1,T to z3,T , and one outside between z2,T and z4,T , and thedivergences seen at µT → 1 will be due to these branch cutsapproaching the contour. In this case we may entirely factorout the dependence on the reference state from the logarithmand focus on the terms which depend on the target state. Wedeform the contour so that it skirts the branch cut [see theparametrization into four contours in Fig. S2]. A key pointhere is that the argument of the logarithm is −π upon ap-proaching the branch cut from the bottom-half plane, whileit is +π upon approaching it from the top half. Therefore, inthe sum of the two contours running along the branch cut, thelogarithm simply contributes a phase factor and we may eval-uate the resulting simplified integrand by elementary methods,and for small ε we find∫C1 +

∫C3 = 1

16√µ2T +∆2

T−1log∣∣∣ (z3−z2)(z1−z4)ε2

(z1−z2)(z3−z4)(z1−z3)2

∣∣∣ .(S17)

We perform the integral around contour C2 by writing z =z1 + εeiθ, and integrating from −π < θ < π. At small ε, wefind∫C2 = − 1

16√µ2T +∆2

T−1log∣∣∣ (∆T +1)ε(z1−z2)

(∆T−1)(z1−z3)(z1−z4)

∣∣∣ .(S18)

The computation for contour C4 is similar, although the phasewinds around the other way:∫C4 = − 1

16√µ2T +∆2

T−1log∣∣∣ (∆T−1)ε(z3−z4)

(∆T +1)(z3−z1)(z3−z2)

∣∣∣ .(S19)

Finally, taking the sum of all four contours, we find that thelog ε divergence in each integral cancels, and we obtain thedesired result:

∂µTC/L =

1

8√µ2T + ∆2

T − 1log

∣∣∣∣ µ2T − 1

µ2T + ∆2

T − 1

∣∣∣∣+ I2(µR,∆R, µT ,∆T ), (S20)

where the function I2 depends on µR and ∆R, but is ana-lytic as the phase transition is approached. Therefore, whenapproaching from µT > 1, the quantity ∂µT

C/L diverges aslog(µT − 1)/8∆T if ∆T 6= 0, but it is analytic if one ap-proaches the multicritical point at ∆T = 0.

Similar manipulations may be made for ∂∆TC/L and in

other phases. Sometimes the branch cuts take a complicatedform in the complex plane so that we cannot reduce the ex-pression into elementary integrals, but we can still deduce theform of the divergence by considering how the contour inte-grals behave as the branch points cross the contour.

Our final results are summarized as follows. The expres-sion ∂µT

C/L is always analytic unless µT → ±1. Near thesephase transitions, it diverges as

∂µTC/L ∼ sign(µT )

8√µ2T + ∆2

T − 1log

∣∣∣∣ µ2T − 1

µ2T + ∆2

T − 1

∣∣∣∣ , (S21)

so the divergence is sign(µT ) log |µT − 1|/8∆T if ∆T 6= 0,but there is not a divergence at ∆T = 0.

In contrast, the expression ∂∆TC/L is analytic whenever

∆T 6= 0. In this case, the divergence depends on whether thecouplings (µT ,∆T ) approach the phase transitions from thetopological phase or the trivial phase. If we approach the mul-ticritical points from the trivial phases, we find that ∂∆T

C/Lremains analytic. In contrast, if we approach ∆T = 0 fromthe topological phases, we find

∂∆TC/L ∼ 1

4

(1 +

|µT∆T |√|µ2T + ∆2

T − 1|

)log |∆T | . (S22)

In this case, we have a log |∆T |/4 divergence when |µT | < 1,but now we find that the divergence crosses over to log |∆T |/2as we approach the multicritical points.

III. REAL-SPACE BEHAVIOR OF THE OPTIMALCIRCUITS

In this section, we show how that the real-space optimalcircuit behaves differently depending on whether or not theinitial and target states are in the same topological phase.

As we have derived in Sec. I of the Supplemental Mate-rial, for a single momentum sector k, the circuit complexityis found to be the squared difference between the Bogoliubovangles [Eq. (7) in the main text], and the optimal circuit isgenerated by the following time-independent Hamiltonian,

Hk = −∆θk O1,k, (S23)

where O1,k is the same generator given by Eq. (S3) for mo-mentum sector k. Here, we have omitted the time label ‘s’for simplicity as the circuit is time independent (and the to-tal evolution time is fixed to be constant 1). As in the maintext and following the circuit complexity literature, we havedefinedHk to be anti-Hermitian [Eq. (5)].

Since the ground state of the Hamiltonian is a product of allmomentum sectors with k > 0, the optimal circuit which gen-erates the evolution between two ground states can be writtenas

H =∑k>0

Hk =∑k>0

−∆θk O1,k. (S24)

We are interested in the real-space behavior of the aboveHamiltonian. To discern this, we first write the above Hamil-tonian in operator form

H =∑k>0

Hk =∑k>0

−i∆θ(k)ψ†kτ1ψk, (S25)

where τi are the Pauli matrices, and ψk denotes the Nambuspinor

ψk =

(aka†−k

). (S26)

10

Utilizing the particle-hole symmetry of the Nambu spinor

ψ−k = τ1(ψ†k)T , (S27)

we can extend the sum in the evolution Hamiltonian to be overthe entire Brillouin zone

H =∑k

−iω(k)ψ†kτ1ψk, (S28)

where ω(k) satisfies

ω(k)− ω(−k) = ∆θ(k) (S29)

for k > 0. In particular, only the odd part of the function con-tributes since the even part cancels in the τ1 pairing channel.

We now proceed by performing a Fourier series expansionof the function ω(k) over the Brillouin zone. Without loss ofgenerality we may consider only the odd Fourier series sincethe even terms will cancel. Thus, we write

ω(k) =

∞∑n=1

ωn sin(nk) =∆θ(k)

2, (S30)

where the last equality is used to determine the Fourier coef-ficients.

Our crucial observation is that when the two states arewithin the same phase, the Fourier sine series for ∆θ(k) oughtto be uniformly convergent. This can be seen by consideringthe boundary conditions, which in this case read ∆θ(0) =∆θ(π) = 0, as shown in Fig. 1(d) in the main text. Thus, ifwe allow the time-evolved state |Ψ′T 〉 to be within an arbitrar-ily small error ε to the real target state |ΨT 〉, this Fourier seriescan be accurately truncated to a finite orderN∗ over the entireBrillouin zone.

This is relevant because in real-space, the Fourier har-monic sin(lk) ψ†kτ1ψk is generated by a term involving twofermionic operators separated by l sites. More specifically, asthis occurs in the τ1 channel, H must involve real-space pair-ing terms such that

H =∑j

N∗∑n=1

ωn (aj aj+n − H.c.) . (S31)

The above argument holds when the system size L is takento be infinite. In such a case, the finite-range interactingevolution Hamiltonian can be regarded as a truly short-rangeHamiltonian, and our results imply that the optimal circuit(with constant time or depth) which evolve states within thesame phase region is short-range.

On the other hand, when the two states are in differentphases, the boundary conditions ∆θ(π) = π/2 6= ∆θ(0) = 0obstruct uniform convergence, analogous to the Gibbs phe-nomenon. In this case, the Fourier sine series may still con-verge pointwise, but for fixed error the series cannot be trun-cated to finite order N∗ over the entire Brillouin zone. Insuch cases, the optimal evolution Hamiltonian H that trans-forms states between different topological phases must be

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5f = 0.3ffffff

= 0.6= 0.8= 0.95= 1.05= 1.4= 1.8

0 0.2 0.4 0.6 0.8 1kn /

0

0.05

0.1

0.15

0.2

0.25

0.3

[k n

(t)]/

(a) (b)

kn/

max

[(t)

]/k n

FIG. S3. (a) Maximum value of φkn(t) versus kn for different post-quench Hamiltonian parameters. (b) Time-averaged value of φkn(t)versus kn for different post-quench Hamiltonian parameters. In bothpanels, µi = 0, ∆i = ∆f = 1, and L = 1000. The diamond mark-ers denote the expected locations of the maxima across the phasetransition, given by solutions to 1 + µf cos kn = 0 (see text).

long-range when the evolution time is fixed to be a constant.Again, this is because the longest real-space distance requiredto generate the evolution Hamiltonian is given by the high-est order of Fourier mode appearing in the momentum spaceseries, which now cannot be accurately truncated.

IV. NUMERICAL EVIDENCE FOR NONANALYTICITYOF QUENCH DYNAMICS

In this section, we provide detailed numerical explana-tions for the nonanalyticity of the long-time steady-state valueof the circuit complexity at critical points, as observed inFig. 3(b) of the main text.

As derived in the main text, the time-dependent circuit com-plexity is given by

C(|Ψi〉 → |Ψ(t)〉) =∑kn

φ2kn(t), (S32)

where

φkn(t) = arccos

√1− sin2(2∆θkn) sin2(εknt). (S33)

Then the long-time steady-state complexity is just given bythe time-averaged value of the above expression,

C(|Ψi〉 → |Ψ(t)〉) =∑kn

φ2kn

(t), (S34)

where the overline denotes time averaging. Because φ2kn

(t) issuch a complex expression, it is unknown to us how to derivean analytical function for the time-averaged circuit complex-ity. Instead, we plot φkn(t) numerically, and show that thenonanalyticity indeed occurs at the phase transition.

From the expression of φkn(t), it is clear that its value oscil-lates with time, and it reaches its maximal value (envelope) foreach momentum sector kn when sin(εknt) = 1. In Fig. S3(a),we plot the maximum value of φkn(t) for different post-quench Hamiltonian parameters. As the figure clearly shows,

11

when the chemical potential µf of the post-quench Hamilto-nian is below the critical value (µf = 1), max[φkn(t)] is asmooth function of kn. However, when µf is above the criti-cal value, max[φkn(t)] exhibits a kink at a certain momentumkn, with its maximal value reaching π

2 . To understand thisbehavior, we can write down the expression for max[φkn(t)]given the choice of parameters µi = 0,∆i = ∆f = 1:

max[φkn(t)] = arccos

∣∣∣∣∣∣ 1 + µf cos kn√µ2f + 2µf cos kn + 1

∣∣∣∣∣∣. (S35)

From the above expression, it is clear that when µf < 1,max[φkn(t)] is always smaller than π/2; when µf > 1,max[φkn(t)] can obtain the maximal value of π/2 when 1 +µf cos kn = 0. Because one needs to take the absolute valuefor the arguments of arccos, the quantity max[φkn(t)] exhibitsa kink when reaching π/2, in agreement with Fig. S3(a).

We plot the time-averaged value of φkn(t) in Fig. S3(b).Again, we see an upper bound of φkn(t) when quench-ing across the critical point. Similar to Fig. S3(a), φkn(t)reaches its maximal value when 1 + µf cos kn = 0, i.e. whensin(2∆θkn) = 1. For this special momentum sector, the ex-pression for φkn(t) can be written as

φkn(t) = arcsin |sin(εknt)|. (S36)

Clearly, the time-averaged value of the above expression isjust π/4, in agreement with the numerical results shown inFig. S3(b). Therefore, after the phase transition takes place,the maximal value of φkn(t) is bounded by π/4. (This featureis independent of the parameters of the pre-quench Hamilto-nian.)

Having revealed this feature of φkn(t), the nonanalyticitycan be understood as follows: as µf increases but is still be-low the phase transition point, the integral of φ2

kn(t) increases

smoothly with µf . After reaching the phase transition, φ2kn

(t)saturates the bound, and thus the integral’s (circuit complex-ity’s) dependence on µf takes a different form. In particular,for the parameters shown in Fig. S3 [blue line in Fig. 3(b) inthe main text], the integral (i.e., the circuit complexity) be-comes a constant after the phase transition. This leads to aclear nonanalytical (kink) point at µf = 1.

V. CIRCUIT COMPLEXITY FOR TWO-DIMENSIONALp+ ip TOPOLOGICAL SUPERCONDUCTORS

In this section, we show how our results for the 1D Kitaevchain can be generalized to 2D. In particular, we consider a

p+ ip topological superconductor for which the Hamiltoniancan be written in momentum space as:

H =∑k

ψ†kHkψk, (S37)

where the summation is taken over the 2D Brillouin zone,

and ψk =

(aka†−k

)is the Nambu spinor. The single-particle

Hamiltonian takes the following form:

Hk =

(εk ∆∗k∆k −εk

), (S38)

where εk and ∆k denote the kinetic and pairing terms in 2Drespectively. The ground state wavefunction can be written as

|Ψgs〉 =∏k

(cos θk − i sin θka†ka†−k) |0〉 , (S39)

where tan(2θk) = |∆k|/εk. Similar to 1D, the circuit com-plexity of the full wavefunction is given by

C =∑k

|∆θk|2 =L2

(2π)2

∫d2k|∆θ(k)|2, (S40)

where we have replaced the summmation by an integral in theinfinite-system limit. In this continuum limit, ε(k) ≈ k2

2m − µand ∆(k) ≈ i∆(kx + iky).

We expect that the non-analyticity should not depend onthe particular choice of initial reference state, so we takeµR → −∞ [with θR(k) = 0] for simplicity. This corre-sponds to the trivial vacuum with no particle. Upon tuningµ, the system undergoes a quantum phase transition into thetopological phase at µ = 0. Taking the derivative of C withrespect to µT , we obtain

∂µTC =

L2

(2π)2

∫d2k 2θT (k)∂µT

θT (k)

=L2

(2π)2

∫d2kθT (k)∂µT

[arctan

|∆(k)|ε(k)

]=

L2

(2π)2

∫d2k

θT (k)|∆(k)|E(k)2

. (S41)