Tracking the quantized information transfer at the edge of a chiral Floquet phase
TTracking the quantized information transfer at the edge of a chiral Floquet phase
Blake R. Duschatko, Philipp T. Dumitrescu, and Andrew C. Potter Department of Physics, University of Texas at Austin, Austin, TX 78712, USA
Two-dimensional arrays of periodically driven qubits can host inherently dynamical topologicalphases with anomalous chiral edge dynamics. These chiral Floquet phases are formally characterizedby a dynamical topological invariant, the chiral unitary index. Introducing a quantity called thechiral mutual information, we show that this invariant can be precisely interpreted in terms of aquantized chiral transfer of quantum information along the edge of the system, and devise a physicalsetup to measure it.
Time-periodic (Floquet) driving enables a host ofsymmetry-breaking and topological phases with inher-ently dynamical properties that could not occur in staticor equilibrium settings [1–12]. A striking set of examplesare Chiral Floquet (CF) phases of driven 2 d systems,which exhibit trivial bulk dynamics, but whose edges actas unidirectional “conveyer belts” for quantum states [1–6]. CF phases were first theoretically constructed ina non-interacting fermion model [1, 2, 6], and weresubsequently generalized to interacting bosonic [3, 4],fermionic [3, 13], and fractionalized (anyonic) [5] Floquetsystems.Despite superficial similarities to the more familiarquantum Hall effect, CF phases represent a distinct andintrinsically dynamical topological phenomena. Namely,CF phases have vanishing Chern number, and are insteadgoverned by a dynamical topological invariant – the chi-ral unitary index [3], ν . For non-fractionalized phases(i.e. without topological order and anyonic excitations),the index takes the form of the logarithm of a positiverational fraction, ν ∈ log Q + [3, 14], and can be heuristi-cally interpreted as the (log of the) ratio of the numberof quantum states transferred to the right divided by thenumber transferred to the left by the edge dynamics, dur-ing each period. Formally, this index has been defined interms of abstract observable algebras [3, 14] and matrixproduct operator methods [3, 15, 16], enabling a rigor-ous classification of 2 d Floquet topology in the absenceof symmetry and intrinsic topological order.Notwithstanding its theoretical utility as a formal toolfor characterizing 2 d Floquet topological phases, this al-gebraic formulation is both physically opaque and notamenable to experimental measurement. In this paper,we address both of these shortcomings by showing thatit is possible to reformulate ν in terms of a chiral im-balance in the transfer of quantum information. Ourconstruction not only yields a physically transparent in-terpretation of the chiral unitary index, but also enablesa realistic scheme to measure this dynamical topologicalinvariant using existing experimental techniques.Our strategy will be to introduce additional non-dynamical ancillary qubits that are initially locally en-tangled with the edge, and serve as “tracers” to trackthe dynamics of entanglement within the system duringits evolution. We show that this setup enables us to re-cast the chiral unitary index in terms of an appropriate chiral combination of mutual information between ancillaand system subregions. This chiral mutual information,( χ MI), can be constructed from any extensive entangle-ment measure, including Renyi entropies, which can bemeasured experimentally via “SWAP”-based many-bodyquantum interference [18, 19].In contrast to recently proposed measurement schemesbased on observing quantized magnetization [17] incharge-conserving systems, our setup does not requireany extraneous symmetries or conservation laws, whichare both inessential to the underlying topological dynam-ics and also typically absent in the qubit systems that aremost likely to realize these CF dynamics. Moreover, ourproposal avoids the use of external leads, which, whilenatural for electronic materials, are difficult to synthe-size for atomic or qubit systems.
Setup –
In interacting settings, care is required to avoiddrive-induced heating which would lead to a highly en-tangled incoherent state and destroy any Floquet topol-ogy. To this end, two strategies have emerged. First,heating can be postponed to exponentially long timesby rapid driving [20–22]. Alternatively, heating can beprevented [23] by applying random disorder to producemany-body localization (MBL) [24–27].In what follows, we will consider only rational CFphases, whose bulk Floquet evolution is trivial (i.e. lackstopological order) and MBL. In this setting, it is alwayspossible to decompose the time-evolution operator forone period, or Floquet operator, into bulk and edge com-ponents [3]: U F = T { e − i (cid:82) T H ( t ) dt } = U edge e − iH bulk T . (1)Here H bulk is a trivial, static MBL Hamiltonian, and U edge is an effective 1 d evolution that acts only on qubitswithin a few localization lengths from the boundary. Theunitary U edge is anomalous, in the sense that it cannotbe generated by any local edge Hamiltonian H d ( t ), eventhough it acts locally on the edge qubits. A precise con-struction of Eq. 1 is reviewed in Appendix A.Although our constructions generalize to generic ratio-nal CF phases, for concreteness, we will frame our discus-sion in terms of the simplest case of a 2 d qubit array with ν = log 2. The topological edge dynamics in this casewill be equivalent to a clockwise translation of the stateof each edge qubit to the right neighboring site along the a r X i v : . [ c ond - m a t . s t r- e l ] A p r s R
System Schematic –
CF system made from qubits(dots) on a square lattice. Ancilla qubits (crosses) are ini-tially entangled with corresponding edge qubits (blue bonds).During a Floquet period, the qubits in the bulk undergo atrivial MBL evolution (not shown), while qubits at the edge(gray shading) undergo a net translation – shifting the system-ancilla entanglement (yellow bonds). Dashed lines indicateentanglement cuts that divide the system edge (s) and ancilla(a) into left (L) and right (R) regions, and separate edge andbulk (B). Generically the edge region (gray shaded) can con-tain multiple layers of spins, and should be thick enough toinclude several localization lengths. edge. Initially, we will ignore the trivial MBL dynamicsof the bulk qubits and formulate a method to extract thetopological dynamics encoded in U edge . Subsequently, weexplain a method to experimentally extricate these topo-logical entanglement signatures from those arising fromtrivial bulk MBL dynamics. Chiral mutual information –
To relate the chiral uni-tary index of the edge dynamics described by U edge toa quantized chiral information transfer along the edge,we need a way to track the flow of quantum information.Here, we face a challenge: unlike charge or heat flow ata quantum Hall edge, quantum information is not as-sociated with a conserved quantity carried by a locallymeasurable current.To circumvent this difficulty, we introduce ancillaryqubits that do not participate in the system dynamics,but serve as initial location “tags” for the informationstored in the system spins. Specifically, we take an evennumber, N a , of ancillary qubits lined up with edge qubitsin the interval [ − N a / , N a / | ψ ( T ) (cid:105) = U edge | ψ (cid:105) , produces a chiral transferof states in the system. The resulting quantum informa-tion transfer from this CF dynamics can be extracted bydividing the system and ancilla spins into left (L) andright (R), and measuring the difference between mutualinformation between the left ancillas and right spins com- pared to that between the right ancillas and left spins: χ = I ( a L , s R ) − I ( a R , s L ) . (2)We will henceforth refer to this quantity as thechiral mutual information ( χ MI). Here, I ( A, B ) =[ S ( A ) + S ( B ) − S ( A ∪ B )] / A and B in the state | ψ ( T ) (cid:105) = U edge | ψ (cid:105) ,and S ( A ) = − tr ( ρ A log ρ A ) is the entanglement entropyof the reduced density matrix in region A .For the ideal case of pure chiral translation, U edge = τ ,precisely one singlet of entanglement crosses the left-right entanglement cut, so that χ = ν = log 2. Fora generic edge evolution, entanglement also spreads ina non-universal fashion in addition to this chiral shift.However, we will next show that in the limit of largesystem size and ancilla number that the χ MI remainsprecisely quantized to χ = ν . Quantization of the chiral mutual information –
We now relate χ to the chiral unitary invariant ν , beyondthe special case discussed above. As a preliminary step,we note that for a generic CF evolution with ν = log 2,we can generically parameterize the edge evolution as U edge = e − iH d t τ, (3)where τ is an operator which translates each qubit’s stateone site to the right, H d is a local 1 d Hamiltonian, and t is a parameter with units of time (see Appendix A). Wefurther note that, since we are presently considering theeffect of U edge , which does not mix bulk and edge degreesof freedom, we can use the complementarity property ofentanglement to simplify Eq. 2 to: χ edge = S ( s R ) − S ( s L ) (4)Our task is then to show that the non-topological edgeevolution produced by H d transfers equal amounts ofquantum information to s L and s R , leaving χ invariantregardless of the form of H d . The local nature of H d allows the decomposition: H d = H L + H R + V , where H L/R contain terms acting only on the left or right half ofthe entanglement cut, and V includes interactions cross-ing the cut.Moreover, unlike the anomalous chiral translation τ ,the evolution under H d can be decomposed into theproduct of many infinitesimal time-steps. This allowsus to focus on a single infinite time step evolving from t → t + ∆ t , where 0 ≤ t ≤ t is some intermediatetime, and take the limit of ∆ t →
0. Here, to O (∆ t ),we can factorize: U (∆ t ) ≈ U L U R e − iV ∆ t + O (∆ t ).Since U L/R = e − iH L/R ∆ t act only on the L/R sides ofthe entanglement cut, they do not effect the entangle-ment of either region. Hence, we must only consider U V (∆ t ) = e − iV ∆ t , which only effects spins within a finitesubset of the ancilla-entangled region.Intuitively, if the region covered by ancillas is verylarge, then the evolution up to time t cannot propa-gate unentangled degrees of freedom outside the ancilla Jt . . . . c ) C ( x, t ) x = 0 x = 1 x = 3 x = 2 x = 4 x Jt δ Jt . . . . . a ) χ/ log(2) N a = 2 N a = 4 N a = 6 N a = 8 0 1 2 3 4 Jt . . . . . b ) χ/ log(2) L = 8 L = 10 L = 12 L = 14 FIG. 2.
Finite size dependence of chiral mutual information, χ ( t ) – normalized to the ideal value, log 2, for a fixedrandom initial state and disorder realization, for (a) fixed ancilla number N a = 6, and various system sizes L , and (b) fixed L and varying N a . (c) Behavior of the butterfly correlator, C ( x, t ) for L = 10 spins averaged over all initial states, and 100disorder realizations. Inset shows a linear fit to the butterfly velocity as described in the main text. Note, that the similaritybetween L = 12 ,
14 in (a) and for N = 6 , (cid:96) a , due toperiodic boundary conditions. region to the vicinity of the entanglement cut effected by V . Formally, this is guaranteed to accuracy e − N a /v LR t by causal bounds on local Hamiltonian evolution, where v LR is the Lieb-Robinson velocity [28]. Due to this lo-cal maximal entanglement, the the effect of U V actingon the edge is equivalent to U TV acting only on the an-cilla qubits. However, applying a unitary operation tothe ancillas cannot change the entanglement of the sys-tem spins, implying ∂S ( s L/R ) /∂t = 0. Together withthe simplification Eq. 4, this implies ∂χ edge /∂t = 0, i.e.that the χ MI is generically precisely quantized and equalto the chiral unitary index in the large system size andancilla number limit.While the above steps make use of the assumption thatthe ancilla and systems were initially maximally entan-gled, numerical simulations (Appendix B) indicate that χ remains quantized for only partial system-ancilla en-tanglement, so long as each system-ancilla pair initiallyhas the same amount of entanglement. Numerical validation and finite-size corrections –
Away from the limit of infinite system size L and ancillanumber N a , we expect finite-size induced deviations of χ from its quantized value. Intuitively, these arise whenthe non-topological parts of the edge evolution allow in-formation from outside the ancilla covered region, whichis not tracked in χ , to propagate across the entanglementcut. Such processes propagate at a maximum speed v LR and must cover a minimum distance (cid:96) a = min (cid:20) N a − , L − N a − (cid:21) , (5)were the second argument accounts for periodic bound-ary conditions. Thus for t (cid:28) (cid:96) a /v LR , χ should remainasymptotically close to its quantized value ν .To verify these simple estimates, we perform numericalsimulations for a qubit chain of length L , evolving for a single Floquet period using U edge of the form Eq. 3 with H d = L − (cid:88) i =0 (cid:16) J (cid:126)S i · (cid:126)S i +1 + h zi S zi + h xi S xi (cid:17) . (6)Here (cid:126)S i = (cid:126)σ i and the random fields are drawn fromthe uniform distribution h x/zi ∈ [ − J, J ]. For these pa-rameters, H d is non-integrable, has no symmetries, andis thermalizing (the latter feature is not essential sincethe chiral translation τ would inevitably prevent local-ization even at strong disorder [3]). We begin from aninitial state in which the ancilla-region qubits form en-tangled singlets with their corresponding ancillas, andthe remainder point along the z-axis of the Bloch sphere.Then, each qubit is rotated by a random angle θ i , drawnindependently from [0 , π ) in the z − y plane of the Blochsphere, to produce a generic initial state without specialfeatures.Figure 2 shows χ as a function of the dimensionless pa-rameter Jt , for various L and N a . In each case, χ initiallyis near its ideal quantized value log 2, before developingsystematic deviations. We observe that the near quan-tization generally persists over longer and longer timeintervals as (cid:96) a is increased, qualitatively agreeing withthe intuition outlined above. Note that in some cases,increasing N a or L does not always increase (cid:96) a .To establish a quantitative relationship between thedeviation time and (cid:96) a , we extract an estimate for theLieb-Robinson velocity by measuring the so-called “but-terfly” correlator [29]: C ( x, t ) = − (cid:104) [ e iH d t S zx e − iH d t , S z (0)] (cid:105) , (7)where, ( . . . ) indicates an average over disorder configura-tions. Heuristically, C ( x, t ) measures how perturbing aspin at site 0 effects the measurement of a spin at site x a time t later, and hence, C ( x, t ) is essentially zero untiltime t ≈ x/v LR . Thus, we can estimate v LR by identi-fying the time t δ ( x ) where C ( x, t ) reaches an arbitrarythreshold δ (cid:28)
1, and performing a linear fit to extract: v LR ≈ x/t δ ( x ). Plotting the times where v LR t = (cid:96) a (redcrosses) in Fig. 2, we observe quantitative agreement withthe interpretation of the finite size errors presented at thebeginning of this section. Effect of bulk MBL dynamics –
Having discussed χ MI for the edge alone, we now consider the dynamicsof the entire system including the trivial bulk MBL mo-tion. This introduces two important changes. First, thetopological edge motion does not occur precisely on theoutermost row of qubits, but rather spreads into the bulkwith an exponentially decaying envelop with character-istic length ξ , the localization length. This can be ad-dressed by covering a fattened edge strip of width W (cid:29) ξ with ancillas, which captures the edge motion to accuracy e − W/ξ .Second, the MBL dynamics can cause information toleak between the ancilla covered region at the edge intothe bulk. Edge-bulk entanglement generated far from theentanglement cut does not effect the mutual informationterms in χ , however local cyclic motion of qubits aroundthe triple intersection of regions L , R , and B can producenon-topological contributions to χ , even for a large edgestrip W .To extract the topological edge contribution, one canevolve the system for n (cid:29) χ edge = nν , whereas the trivial bulk contributionsare bounded by ∆ χ bulk ( nT ) (cid:46) log nT due to the slowglassy MBL dynamics [30]. Hence measuring the asymp-totic slope, lim n →∞ χ ( nT ) /n enables one to extract thetopological edge contribution. Note that this procedurerequires that the length of each edge region be muchlarger than v LR nT and the width W to be much largerthan ξ log( nT ) to prevent the edge spins undergoing theCF dynamics and the bulk spins from entangling withthose beyond the ancilla-covered region during the timeevolution. Experimental proposal –
In addition to providing acomplementary, physically intuitive formulation of thechiral unitary invariant, the above construction revealsan experimental protocol to measure χ . While the vonNeumann entropies in χ are challenging to measure di-rectly, we note that Renyi versions of the χ MI, denoted χ n , can be equally well formulated using the Renyi en-tropy S n ( A ) = tr( ρ nA ) / (1 − n ), for any index n , in placeof von Neumann entropies in the mutual informationterms in Eq. 2. The proof of quantization of χ n followsthrough as above and with the same restrictions. Thesecond Renyi entropy ( n = 2) is particularly significant,as this quantity can be directly measured by making twocopies of the system, and performing an interferometricmeasurement of the SWAP operator that exchanges thecopies [18, 19].This enables a direct measurement of each term in χ , and opens the door to observing the quantized χ MIin AMO systems such as 2 d arrays of superconducting = System 1 = Ancilla 1 = System 2 = System 1 Bond= System 2 Bond = Circuit Connection = Entangled pair
SWAP | i | i U a )
Experimental setup – (a) two copies of a chiralFloquet phase along with ancilla qubits can be implementedvirtually in a 2 d planar array of qubits with only nearestneighbor two-qubit gates. (b) A unitary gate, U , betweenspins in the same system can be implemented via two inter-mediate SWAP operations using an ancilla qubit. qubits, or trapped atomic, molecular, or ion systems. Inparticular, two dimensional qubit arrays have presentlybeen fabricated by multiple groups. Furthermore, induc-tive couplings between the qubits are described by theeffective Hamiltonian [31]: H = (cid:88) (cid:104) i,j (cid:105) J i,j ( t ) (cid:0) S + i S − j + S + i S − j (cid:1) (8)where (cid:104) i, j (cid:105) denotes nearest neighbors. Applying this in-teraction for time t = π J i,j “hops” an excitation (logical | (cid:105) ) between nearest neighbor sites, and can be used toimplement idealized SWAP models for the CF phase with ν = 2 ± [3]. Note that this XY-type interaction differsfrom Heisenberg type interactions, ∼ (cid:126)σ i · (cid:126)σ j , discussedin [3] only by a σ zi σ zj term that generates an unimpor-tant conditional phase that is not required for the CFimplementation.To implement the above protocol to measure the χ MIin these systems, we nominally need four distinct 2 d lay-ers: two copies each of the CF system and ancilla spins.While such a multilayer structure is impractical to di-rectly fabricate, the layer structure can instead be “vir-tually” implemented within a single physical qubit layerwith nearest neighbor interactions as shown in Fig. 3a. Inthis setup, a four-site unit cell is used to implement thefour distinct virtual layers. To synthesize a two-qubitunitary operation U between virtual neighbors in onecopy of the system (which are next-nearest neighbors inthe physical lattice), one can use the circuit shown inFig. 3b, to first SWAP a system spin with the interven-ing ancilla, then apply U to the system-ancilla bond, andthen undo the original SWAP. We note that this proce-dure is limited by the fidelity of the SWAP operationsused to implement the virtual layering. Discussion –
So far we have focused on the case of thesimplest rational CF phase with ν = log 2, where we in-troduced the chiral mutual information, χ MI as a meansto characterize and possibly experimentally measure thechiral unitary invariant, ν . This setup works equallywell for more general rational CF phases of bosons, with ν = log r for arbitrary rational fraction r .In the presence of strong interactions, the periodicdrive can also induce dynamical fractionalization, inwhich the 2 d bulk develops dynamical Abelian topo-logical order [5]. In such fractionalized settings, thechiral unitary invariant can take “radical” values of theform ν = log √ r where r ∈ Q + is a rational fraction.In these settings, the Floquet evolution for the systemdoes not simply decompose into commuting edge andbulk pieces [5], and it is presently unclear whether χ MIcan be used to directly detect the radical CF invariant.However, in this case, one could instead consider thesame setup and evaluate χ MI for two periods, forwhich the evolution U (2 T ) becomes rational, and theconstruction above applies. In this case, one could inferthe chiral unitary invariant from: ν = χ [ U (2 T )]. Acknowledgements –
We thank K. Hazzard for insightfulconversations. This work was supported by NSF DMR-1653007 (ACP) and was performed in part at Aspen Cen-ter for Physics, which is supported by National ScienceFoundation grant PHY-1607611.
Appendix A: Review of formal aspects of ChiralFloquet phases1. Effective edge evolution
In this Appendix, we review the results of [3], whichformulated a precise notion of the effective boundary dy-namics for a Floquet MBL system. Specifically, sup-pose the dynamics are produced by a time-periodic localHamiltonian, H ( t ) = H ( t + T ) = (cid:80) r h a ( t ), where a in-dexes nearby groups of qubits. The long-time dynamicsat time t = nT + δ is captured by n = (cid:98) t/T (cid:99) appli-cations of the Floquet unitary, U F = T (cid:110) e − i (cid:82) T H ( t ) dt (cid:111) ,followed by micro-motion for time δ = t − nT . Sincewe have considered a system in which H ( t ) is MBL inthe bulk, then in an infinite 2d plane without bound-aries, we may write the Floquet operator, U F = e − iH F T ,as time-evolution with respect to a static MBL Hamil-tonian: H F = E ( { τ zi } ). Here, τ zi , are local integrals ofmotion (LIOM), consisting of qubit operators near site i dressed by a cloud of virtual fluctuations that decay ex-ponentially in distance from i , and E is some quasi-localfunction.One can truncate the evolution onto a finite region, A , with a boundary, ∂A , in one of two distinct ways. First, one can truncate the time-dependent Hamiltonianto omit terms residing outside A , H ( A ) ( t ) = (cid:80) a ⊂ A h a ( t ),which produces a truncated Floquet unitary U ( A ) = T (cid:110) e − i (cid:82) T H ( A ) ( t ) dt (cid:111) . Second, one can truncate the ef-fective MBL Floquet Hamiltonian by dropping terms inthe Taylor expansion of E , involving LIOM centered out-side of A . Denote the resulting truncated MBL Hamil-tonian as H ( A ) F . The bulk dynamics produced by U ( A ) and e − iH ( A ) F T at a distance R from ∂A are identical toaccuracy e − R/ξ , where ξ is the localization length. How-ever, only U ( A ) captures any anomalous topological edgedynamics near ∂A .Comparing these two truncation schemes produces aneffective 1 d edge evolution: U edge = e + iH ( A ) F T U ( A ) (A1)which is exponentially well localized to ∂A , and whichcaptures any non-trivial topological edge dynamics. Thisquantity plays an analogous role to that of the effectiveboundary field theory for a zero temperature equilibriumtopological phase. Such equilibrium boundary field the-ories are anomalous, in the sense that it cannot emergeas the low energy description of microscopic quantumdegrees of freedom living only at the sample boundary.Similarly, for chiral Floquet phases, the effective edge dy-namics described by U edge can be considered anomalousif it cannot arise from time-evolution under a local 1 d time-dependent Hamiltonian, H d ( t ), acting only on thesample edge.
2. Chiral unitary invariant
In this section, we review the operator-algebraic def-inition of the chiral unitary invariant [3, 14]. Considercutting the 1 d edge of a 2 d MBL-Floquet system intotwo halves, and defining a large region, L to the left, andanother, R , to the right of one of the cuts. Here, by“large”, we mean that each region is much larger thanthe Lieb-Robinson velocity at the edge times the driv-ing period. Consider the set of all operators acting onthe region L . Since any product or linear combinationof such operators gives another operator in the set, thesesets each form an algebra, A L/R .To simplify notation, let us temporarily focus on theleft. Given an orthonormal basis {| i (cid:105)} for the states inthis region, we can define an orthonormal basis for theoperator algebra: e ij = | i (cid:105)(cid:104) j | . This operator basis formsan orthonormal set under the inner product: (cid:104) O, O (cid:48) (cid:105) =tr O † O (cid:48) where the trace is taken over all states in theregion (e.g. over the basis {| i (cid:105)} ). From this inner-producton operators, we can define the overlap of two algebras, A , B , by: (cid:104)A , B(cid:105) ≡ √ D A D B D A ∪ B (cid:118)(cid:117)(cid:117)(cid:116) D A (cid:88) i,j =1 D B (cid:88) k,l =1 (cid:12)(cid:12) (cid:104) e ( A ) ij , e ( B ) kl (cid:105) (cid:12)(cid:12) (A2)where D A is the dimension of the Hilbert space of region A . As defined, this overlap is 1 for independent (com-muting) algebras, and the auto-overlap is (cid:104)A , A(cid:105) = D A .The chiral unitary invariant is then constructed bytaking the algebra, A L , for the region left of the cut,time-evolving it by one period, and measuring its over-lap with the algebra, A R to the right of the cut: N R = (cid:104) U A L U † , A R (cid:105) . N R quantifies, how many states worth ofinformation can we deduce about L at t by measuringoperators in R at t = t + T , or roughly: “how manystates are transferred from L to R during the drivingperiod”. Then, one should also compute the number ofstates transferred from right to left: N L = (cid:104) U A R U † , A L (cid:105) .Neither N R/L are separately quantized. However, theirratio r = N R N L is, and its logarithm defines the chiral uni-tary invariant: ν = log N R N L = log (cid:104) U A L U † , A R (cid:105)(cid:104) U A R U † , A L (cid:105) (A3)Directly measuring this quantity as formulated wouldrequire measuring a large number of operator overlaps– for a complete set of operators in A L/R , and eachoperator overlap requires averaging over a complete setof states for that region. The number of such requiredmeasurements clearly grows exponentially in the sizeof the
L/R regions, with each involving a multi-spinmeasurement that is effectively as challenging as theSWAP-operator based entanglement measurementsketched in the main text. Hence, the ancilla degrees offreedom yield a potentially large reduction in measure-ment complexity, which is exponential in the parameter v LR T .
3. Generic form of edge evolution
The chiral unitary invariant, ν , of the edge evolution U edge is invariant under modifying U edge by a finite depthlocal unitary (FDLU) transformation. As a corollary, wecan always write the Floquet evolution operator for theedge of a (rational) CF phase of qubits (or spins-1/2)with ν = log 2 as pure translation τ modified by an FDLUtransformation: U edge = U † FDLU τ U
FDLU = (cid:16) U † FDLU τ U
FDLU τ † (cid:17) τ (A4)Then, note that τ U FDLU τ † is also a 1 d FDLU (in fact,simply U FDLU translated one site to the left). In abosonic system, any 1 d FDLU has trivial chiral unitaryindex, and in the absence of any extraneous symmetries, can be generated by an effective time independent Hamil-tonian, which we denote as H d .We note, in passing, that for rational fermion CFphases, there are intrinsically dynamical topologicalphases for which we cannot always reduce to time-independent evolution, but we can still write: U edge = Jt . . . . . . χ / χ ( ) α = π/ α = π/ α = 3 π/ α = π/ FIG. 4.
Partial entanglement – χ ( t ) for a state in whicheach system/ancilla spin pair has variable entanglement char-acterized by the parameter α ∈ [0 , π/ L = 12 and N a = 6. Each curve is normalized by the initial value of χ ( t = 0). These results give numerical evidence that χ ( t ) isquantized for any (uniform) amount of initial system/ancillaentanglement. T { e − (cid:82) t H ( t (cid:48) ) dt (cid:48) } τ , and the arguments for the quantiza-tion in χ applies equally well. Appendix B: Partial entanglement
In the arguments presented in the main text, we con-sidered CF evolution for states in which the ancillaryspins are initially maximally entangled with their systemcounterparts. Here, we were able to analytically estab-lish that χ is a quantized invariant that reproduces thechiral unitary invariant, ν . We can also consider startingwith an arbitrary amount of entanglement where eachsystem/ancilla pair starts in a state: | ψ (cid:105) = cos( α ) | ↑ s (cid:105)| ↓ a (cid:105) − sin( α ) | ↓ s (cid:105)| ↑ a (cid:105) (B1)where α ∈ (cid:2) , π (cid:3) allows one to continuously adjust theentanglement per ancilla: s ( α ) = − cos α log cos α − sin α log sin α , from 0 ( α = 0), to log 2 ( α = π ). Fromnumerical simulation, we observe that χs ( α ) appears tobe quantized to 1 for the chiral Floquet phase (up tofinite size corrections), for any value of α >
0. We notethat the (asymptotic) quantization requires taking α tobe spatially uniform, for example an uniform gradient of ∇ α (cid:54) = 0 would relax via a non-topological chiral flow ofentanglement, spoiling the quantization of χ . [1] Takuya Kitagawa, Erez Berg, Mark Rudner, and Eu-gene Demler, “Topological characterization of periodi-cally driven quantum systems,” Phys. Rev. B , 235114(2010).[2] Mark S Rudner, Netanel H Lindner, Erez Berg,and Michael Levin, “Anomalous edge states and thebulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X , 031005 (2013).[3] Hoi Chun Po, Lukasz Fidkowski, Takahiro Morimoto,Andrew C. Potter, and Ashvin Vishwanath, “Chiral flo-quet phases of many-body localized bosons,” Phys. Rev.X , 041070 (2016).[4] Fenner Harper and Rahul Roy, “Floquet topological or-der in interacting systems of bosons and fermions,” Phys.Rev. Lett. , 115301 (2017).[5] Hoi Chun Po, Lukasz Fidkowski, Ashvin Vishwanath,and Andrew C Potter, “Radical chiral floquet phases in aperiodically driven kitaev model and beyond,” PhysicalReview B , 245116 (2017).[6] Paraj Titum, Erez Berg, Mark S. Rudner, Gil Refael,and Netanel H. Lindner, “Anomalous floquet-andersoninsulator as a nonadiabatic quantized charge pump,”Phys. Rev. X , 021013 (2016).[7] Liang Jiang, Takuya Kitagawa, Jason Alicea, A. R.Akhmerov, David Pekker, Gil Refael, J. Ignacio Cirac,Eugene Demler, Mikhail D. Lukin, and Peter Zoller,“Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. , 220402(2011).[8] CW von Keyserlingk and SL Sondhi, “Phase structureof one-dimensional interacting floquet systems. i. abeliansymmetry-protected topological phases,” Phys. Rev. B , 245145 (2016).[9] Dominic V Else and Chetan Nayak, “Classification oftopological phases in periodically driven interacting sys-tems,” Phys. Rev. B , 201103 (2016).[10] Andrew C. Potter, Takahiro Morimoto, and AshvinVishwanath, “Classification of interacting topological flo-quet phases in one dimension,” Phys. Rev. X , 041001(2016).[11] Rahul Roy and Fenner Harper, “Abelian floquetsymmetry-protected topological phases in one dimen-sion,” Phys. Rev. B , 125105 (2016).[12] Rahul Roy and Fenner Harper, “Periodic table for floquettopological insulators,” Phys. Rev. B , 155118 (2017).[13] Lukasz Fidkowski, Hoi Chun Po, Andrew C Potter, andAshvin Vishwanath, “Interacting invariants for floquetphases of fermions in two dimensions,” arXiv preprintarXiv:1703.07360 (2017).[14] D Gross, V Nesme, H Vogts, and RF Werner, “Indextheory of one dimensional quantum walks and cellular au-tomata,” Communications in Mathematical Physics ,419–454 (2012).[15] M. Burak S¸ahino˘glu, S. K. Shukla, F. Bi, and X. Chen,“Matrix Product Representation of Locality Preserv-ing Unitaries,” ArXiv e-prints (2017), arXiv:1704.01943[quant-ph].[16] J Ignacio Cirac, David Perez-Garcia, Norbert Schuch,and Frank Verstraete, “Matrix product unitaries: struc-ture, symmetries, and topological invariants,” J. Stat.Mech. , 083105 (2017). [17] Frederik Nathan, Mark S Rudner, Netanel H Lindner,Erez Berg, and Gil Refael, “Quantized magnetizationdensity in periodically driven systems,” Physical reviewletters , 186801 (2017).[18] Pawe(cid:32)l Horodecki and Artur Ekert, “Method for directdetection of quantum entanglement,” Phys. Rev. Lett. , 127902 (2002).[19] Rajibul Islam, Ruichao Ma, Philipp M. Preiss,M. Eric Tai, Alexander Lukin, Matthew Rispoli, andMarkus Greiner, “Measuring entanglement entropy in aquantum many-body system,” Nature , 77–83 (2015).[20] Dmitry A Abanin, Wojciech De Roeck, Wen Wei Ho, andFran¸cois Huveneers, “Effective hamiltonians, prethermal-ization, and slow energy absorption in periodically drivenmany-body systems,” Physical Review B , 014112(2017).[21] Tomotaka Kuwahara, Takashi Mori, and Keiji Saito,“Floquet–magnus theory and generic transient dynam-ics in periodically driven many-body quantum systems,”Annals of Physics , 96–124 (2016).[22] Dominic V Else, Bela Bauer, and Chetan Nayak,“Prethermal phases of matter protected by time-translation symmetry,” Physical Review X , 011026(2017).[23] While MBL has been firmly established in 1 d sys-tems [32], its stability to rare-region effects in higher di-mensional systems has been questioned [33]. While an im-portant point of principle these effects are practically ir-relevant, as they occur on time-scales that are doubly ex-ponentially long in the disorder strength, which can eas-ily be made to exceed any practical experimental lifetime(or even the age of the universe!) for moderate disorderstrength. Moreover, these rare-region worries can likelybe side-stepped by implementing quasi-periodic “disor-der.[24] Rahul Nandkishore and David A. Huse, “Many-body lo-calization and thermalization in quantum statistical me-chanics,” Ann. Rev. Cond. Matt. Phys. , 15–38 (2015).[25] Ehud Altman and Ronen Vosk, “Universal dynamics andrenormalization in many-body-localized systems,” Annu.Rev. Condens. Matter Phys. , 383–409 (2015).[26] Achilleas Lazarides, Arnab Das, and Roderich Moessner,“Fate of many-body localization under periodic driving,”Physical review letters , 030402 (2015).[27] Pedro Ponte, Z Papi´c, Fran¸cois Huveneers, andDmitry A Abanin, “Many-body localization in peri-odically driven systems,” Physical review letters ,140401 (2015).[28] Elliott H Lieb and Derek W Robinson, “The fi-nite group velocity of quantum spin systems,” inStatistical Mechanics (Springer, 1972) pp. 425–431.[29] Stephen H Shenker and Douglas Stanford, “Black holesand the butterfly effect,” Journal of High Energy Physics , 67 (2014).[30] Jens H Bardarson, Frank Pollmann, and Joel E Moore,“Unbounded growth of entanglement in models of many-body localization,” Physical review letters , 017202(2012).[31] P. Roushan, C. Neill, J. Tangpanitanon, V. M. Basti-das, A. Megrant, R. Barends, Y. Chen, Z. Chen,B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus,M. Neeley, C. Quintana, D. Sank, A. Vainsencher,J. Wenner, T. White, H. Neven, D. G. Ange-lakis, and J. Martinis, “Spectroscopic signaturesof localization with interacting photons in super-conducting qubits,” Science , 1175–1179 (2017),http://science.sciencemag.org/content/358/6367/1175.full.pdf.[32] John Z Imbrie, “On many-body localization for quantumspin chains,” Journal of Statistical Physics , 998–1048(2016).[33] Wojciech De Roeck and Fran¸cois Huveneers, “Stabil-ity and instability towards delocalization in many-bodylocalization systems,” Physical Review B95